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25aa754c2c8351ad03fc7cf7ce87510646624933 | 38bf3fd2bb651ab70511408fcf70e2029e2ba310 | /src/order/filter/basic.lean | 094934f31de7628bc3380e59f998ae4b11eb3882 | [
"Apache-2.0"
] | permissive | JaredCorduan/mathlib | 130392594844f15dad65a9308c242551bae6cd2e | d5de80376088954d592a59326c14404f538050a1 | refs/heads/master | 1,595,862,206,333 | 1,570,816,457,000 | 1,570,816,457,000 | 209,134,499 | 0 | 0 | Apache-2.0 | 1,568,746,811,000 | 1,568,746,811,000 | null | UTF-8 | Lean | false | false | 82,068 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
Theory of filters on sets.
-/
import order.galois_connection order.zorn
import data.set.finite
open lattice set
universes u v w x y
open_locale classical
namespace lattice
variables {α : Type u} {ι : Sort v}
def complete_lattice.copy (c : complete_lattice α)
(le : α → α → Prop) (eq_le : le = @complete_lattice.le α c)
(top : α) (eq_top : top = @complete_lattice.top α c)
(bot : α) (eq_bot : bot = @complete_lattice.bot α c)
(sup : α → α → α) (eq_sup : sup = @complete_lattice.sup α c)
(inf : α → α → α) (eq_inf : inf = @complete_lattice.inf α c)
(Sup : set α → α) (eq_Sup : Sup = @complete_lattice.Sup α c)
(Inf : set α → α) (eq_Inf : Inf = @complete_lattice.Inf α c) :
complete_lattice α :=
begin
refine { le := le, top := top, bot := bot, sup := sup, inf := inf, Sup := Sup, Inf := Inf, ..};
subst_vars,
exact @complete_lattice.le_refl α c,
exact @complete_lattice.le_trans α c,
exact @complete_lattice.le_antisymm α c,
exact @complete_lattice.le_sup_left α c,
exact @complete_lattice.le_sup_right α c,
exact @complete_lattice.sup_le α c,
exact @complete_lattice.inf_le_left α c,
exact @complete_lattice.inf_le_right α c,
exact @complete_lattice.le_inf α c,
exact @complete_lattice.le_top α c,
exact @complete_lattice.bot_le α c,
exact @complete_lattice.le_Sup α c,
exact @complete_lattice.Sup_le α c,
exact @complete_lattice.Inf_le α c,
exact @complete_lattice.le_Inf α c
end
end lattice
open set lattice
section order
variables {α : Type u} (r : α → α → Prop)
local infix ` ≼ ` : 50 := r
lemma directed_on_Union {r} {ι : Sort v} {f : ι → set α} (hd : directed (⊆) f)
(h : ∀x, directed_on r (f x)) : directed_on r (⋃x, f x) :=
by simp only [directed_on, exists_prop, mem_Union, exists_imp_distrib]; exact
assume a₁ b₁ fb₁ a₂ b₂ fb₂,
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂,
⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂) in
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
end order
theorem directed_of_chain {α β r} [is_refl β r] {f : α → β} {c : set α}
(h : zorn.chain (f ⁻¹'o r) c) :
directed r (λx:{a:α // a ∈ c}, f (x.val)) :=
assume ⟨a, ha⟩ ⟨b, hb⟩, classical.by_cases
(assume : a = b, by simp only [this, exists_prop, and_self, subtype.exists];
exact ⟨b, hb, refl _⟩)
(assume : a ≠ b, (h a ha b hb this).elim
(λ h : r (f a) (f b), ⟨⟨b, hb⟩, h, refl _⟩)
(λ h : r (f b) (f a), ⟨⟨a, ha⟩, refl _, h⟩))
structure filter (α : Type*) :=
(sets : set (set α))
(univ_sets : set.univ ∈ sets)
(sets_of_superset {x y} : x ∈ sets → x ⊆ y → y ∈ sets)
(inter_sets {x y} : x ∈ sets → y ∈ sets → x ∩ y ∈ sets)
/-- If `F` is a filter on `α`, and `U` a subset of `α` then we can write `U ∈ F` as on paper. -/
@[reducible]
instance {α : Type*}: has_mem (set α) (filter α) := ⟨λ U F, U ∈ F.sets⟩
namespace filter
variables {α : Type u} {f g : filter α} {s t : set α}
lemma filter_eq : ∀{f g : filter α}, f.sets = g.sets → f = g
| ⟨a, _, _, _⟩ ⟨._, _, _, _⟩ rfl := rfl
lemma filter_eq_iff : f = g ↔ f.sets = g.sets :=
⟨congr_arg _, filter_eq⟩
protected lemma ext_iff : f = g ↔ ∀ s, s ∈ f ↔ s ∈ g :=
by rw [filter_eq_iff, ext_iff]
@[extensionality]
protected lemma ext : (∀ s, s ∈ f ↔ s ∈ g) → f = g :=
filter.ext_iff.2
lemma univ_mem_sets : univ ∈ f :=
f.univ_sets
lemma mem_sets_of_superset : ∀{x y : set α}, x ∈ f → x ⊆ y → y ∈ f :=
f.sets_of_superset
lemma inter_mem_sets : ∀{s t}, s ∈ f → t ∈ f → s ∩ t ∈ f :=
f.inter_sets
lemma univ_mem_sets' (h : ∀ a, a ∈ s) : s ∈ f :=
mem_sets_of_superset univ_mem_sets (assume x _, h x)
lemma mp_sets (hs : s ∈ f) (h : {x | x ∈ s → x ∈ t} ∈ f) : t ∈ f :=
mem_sets_of_superset (inter_mem_sets hs h) $ assume x ⟨h₁, h₂⟩, h₂ h₁
lemma congr_sets (h : {x | x ∈ s ↔ x ∈ t} ∈ f) : s ∈ f ↔ t ∈ f :=
⟨λ hs, mp_sets hs (mem_sets_of_superset h (λ x, iff.mp)),
λ hs, mp_sets hs (mem_sets_of_superset h (λ x, iff.mpr))⟩
lemma Inter_mem_sets {β : Type v} {s : β → set α} {is : set β} (hf : finite is) :
(∀i∈is, s i ∈ f) → (⋂i∈is, s i) ∈ f :=
finite.induction_on hf
(assume hs, by simp only [univ_mem_sets, mem_empty_eq, Inter_neg, Inter_univ, not_false_iff])
(assume i is _ hf hi hs,
have h₁ : s i ∈ f, from hs i (by simp),
have h₂ : (⋂x∈is, s x) ∈ f, from hi $ assume a ha, hs _ $ by simp only [ha, mem_insert_iff, or_true],
by simp [inter_mem_sets h₁ h₂])
lemma exists_sets_subset_iff : (∃t ∈ f, t ⊆ s) ↔ s ∈ f :=
⟨assume ⟨t, ht, ts⟩, mem_sets_of_superset ht ts, assume hs, ⟨s, hs, subset.refl _⟩⟩
lemma monotone_mem_sets {f : filter α} : monotone (λs, s ∈ f) :=
assume s t hst h, mem_sets_of_superset h hst
end filter
namespace tactic.interactive
open tactic interactive
/-- `filter_upwards [h1, ⋯, hn]` replaces a goal of the form `s ∈ f`
and terms `h1 : t1 ∈ f, ⋯, hn : tn ∈ f` with `∀x, x ∈ t1 → ⋯ → x ∈ tn → x ∈ s`.
`filter_upwards [h1, ⋯, hn] e` is a short form for `{ filter_upwards [h1, ⋯, hn], exact e }`.
-/
meta def filter_upwards
(s : parse types.pexpr_list)
(e' : parse $ optional types.texpr) : tactic unit :=
do
s.reverse.mmap (λ e, eapplyc `filter.mp_sets >> eapply e),
eapplyc `filter.univ_mem_sets',
match e' with
| some e := interactive.exact e
| none := skip
end
end tactic.interactive
namespace filter
variables {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
section principal
/-- The principal filter of `s` is the collection of all supersets of `s`. -/
def principal (s : set α) : filter α :=
{ sets := {t | s ⊆ t},
univ_sets := subset_univ s,
sets_of_superset := assume x y hx hy, subset.trans hx hy,
inter_sets := assume x y, subset_inter }
instance : inhabited (filter α) :=
⟨principal ∅⟩
@[simp] lemma mem_principal_sets {s t : set α} : s ∈ principal t ↔ t ⊆ s := iff.rfl
lemma mem_principal_self (s : set α) : s ∈ principal s := subset.refl _
end principal
section join
/-- The join of a filter of filters is defined by the relation `s ∈ join f ↔ {t | s ∈ t} ∈ f`. -/
def join (f : filter (filter α)) : filter α :=
{ sets := {s | {t : filter α | s ∈ t} ∈ f},
univ_sets := by simp only [univ_mem_sets, mem_set_of_eq]; exact univ_mem_sets,
sets_of_superset := assume x y hx xy,
mem_sets_of_superset hx $ assume f h, mem_sets_of_superset h xy,
inter_sets := assume x y hx hy,
mem_sets_of_superset (inter_mem_sets hx hy) $ assume f ⟨h₁, h₂⟩, inter_mem_sets h₁ h₂ }
@[simp] lemma mem_join_sets {s : set α} {f : filter (filter α)} :
s ∈ join f ↔ {t | s ∈ filter.sets t} ∈ f := iff.rfl
end join
section lattice
instance : partial_order (filter α) :=
{ le := λf g, ∀ ⦃U : set α⦄, U ∈ g → U ∈ f,
le_antisymm := assume a b h₁ h₂, filter_eq $ subset.antisymm h₂ h₁,
le_refl := assume a, subset.refl _,
le_trans := assume a b c h₁ h₂, subset.trans h₂ h₁ }
theorem le_def {f g : filter α} : f ≤ g ↔ ∀ x ∈ g, x ∈ f := iff.rfl
/-- `generate_sets g s`: `s` is in the filter closure of `g`. -/
inductive generate_sets (g : set (set α)) : set α → Prop
| basic {s : set α} : s ∈ g → generate_sets s
| univ {} : generate_sets univ
| superset {s t : set α} : generate_sets s → s ⊆ t → generate_sets t
| inter {s t : set α} : generate_sets s → generate_sets t → generate_sets (s ∩ t)
/-- `generate g` is the smallest filter containing the sets `g`. -/
def generate (g : set (set α)) : filter α :=
{ sets := generate_sets g,
univ_sets := generate_sets.univ,
sets_of_superset := assume x y, generate_sets.superset,
inter_sets := assume s t, generate_sets.inter }
lemma sets_iff_generate {s : set (set α)} {f : filter α} : f ≤ filter.generate s ↔ s ⊆ f.sets :=
iff.intro
(assume h u hu, h $ generate_sets.basic $ hu)
(assume h u hu, hu.rec_on h univ_mem_sets
(assume x y _ hxy hx, mem_sets_of_superset hx hxy)
(assume x y _ _ hx hy, inter_mem_sets hx hy))
protected def mk_of_closure (s : set (set α)) (hs : (generate s).sets = s) : filter α :=
{ sets := s,
univ_sets := hs ▸ (univ_mem_sets : univ ∈ generate s),
sets_of_superset := assume x y, hs ▸ (mem_sets_of_superset : x ∈ generate s → x ⊆ y → y ∈ generate s),
inter_sets := assume x y, hs ▸ (inter_mem_sets : x ∈ generate s → y ∈ generate s → x ∩ y ∈ generate s) }
lemma mk_of_closure_sets {s : set (set α)} {hs : (generate s).sets = s} :
filter.mk_of_closure s hs = generate s :=
filter.ext $ assume u,
show u ∈ (filter.mk_of_closure s hs).sets ↔ u ∈ (generate s).sets, from hs.symm ▸ iff.refl _
/- Galois insertion from sets of sets into a filters. -/
def gi_generate (α : Type*) :
@galois_insertion (set (set α)) (order_dual (filter α)) _ _ filter.generate filter.sets :=
{ gc := assume s f, sets_iff_generate,
le_l_u := assume f u h, generate_sets.basic h,
choice := λs hs, filter.mk_of_closure s (le_antisymm hs $ sets_iff_generate.1 $ le_refl _),
choice_eq := assume s hs, mk_of_closure_sets }
/-- The infimum of filters is the filter generated by intersections
of elements of the two filters. -/
instance : has_inf (filter α) := ⟨λf g : filter α,
{ sets := {s | ∃ (a ∈ f) (b ∈ g), a ∩ b ⊆ s },
univ_sets := ⟨_, univ_mem_sets, _, univ_mem_sets, inter_subset_left _ _⟩,
sets_of_superset := assume x y ⟨a, ha, b, hb, h⟩ xy, ⟨a, ha, b, hb, subset.trans h xy⟩,
inter_sets := assume x y ⟨a, ha, b, hb, hx⟩ ⟨c, hc, d, hd, hy⟩,
⟨_, inter_mem_sets ha hc, _, inter_mem_sets hb hd,
calc a ∩ c ∩ (b ∩ d) = (a ∩ b) ∩ (c ∩ d) : by ac_refl
... ⊆ x ∩ y : inter_subset_inter hx hy⟩ }⟩
@[simp] lemma mem_inf_sets {f g : filter α} {s : set α} :
s ∈ f ⊓ g ↔ ∃t₁∈f.sets, ∃t₂∈g.sets, t₁ ∩ t₂ ⊆ s := iff.rfl
lemma mem_inf_sets_of_left {f g : filter α} {s : set α} (h : s ∈ f) : s ∈ f ⊓ g :=
⟨s, h, univ, univ_mem_sets, inter_subset_left _ _⟩
lemma mem_inf_sets_of_right {f g : filter α} {s : set α} (h : s ∈ g) : s ∈ f ⊓ g :=
⟨univ, univ_mem_sets, s, h, inter_subset_right _ _⟩
lemma inter_mem_inf_sets {α : Type u} {f g : filter α} {s t : set α}
(hs : s ∈ f) (ht : t ∈ g) : s ∩ t ∈ f ⊓ g :=
inter_mem_sets (mem_inf_sets_of_left hs) (mem_inf_sets_of_right ht)
instance : has_top (filter α) :=
⟨{ sets := {s | ∀x, x ∈ s},
univ_sets := assume x, mem_univ x,
sets_of_superset := assume x y hx hxy a, hxy (hx a),
inter_sets := assume x y hx hy a, mem_inter (hx _) (hy _) }⟩
lemma mem_top_sets_iff_forall {s : set α} : s ∈ (⊤ : filter α) ↔ (∀x, x ∈ s) :=
iff.refl _
@[simp] lemma mem_top_sets {s : set α} : s ∈ (⊤ : filter α) ↔ s = univ :=
by rw [mem_top_sets_iff_forall, eq_univ_iff_forall]
section complete_lattice
/- We lift the complete lattice along the Galois connection `generate` / `sets`. Unfortunately,
we want to have different definitional equalities for the lattice operations. So we define them
upfront and change the lattice operations for the complete lattice instance. -/
private def original_complete_lattice : complete_lattice (filter α) :=
@order_dual.lattice.complete_lattice _ (gi_generate α).lift_complete_lattice
local attribute [instance] original_complete_lattice
instance : complete_lattice (filter α) := original_complete_lattice.copy
/- le -/ filter.partial_order.le rfl
/- top -/ (filter.lattice.has_top).1
(top_unique $ assume s hs, by have := univ_mem_sets ; finish)
/- bot -/ _ rfl
/- sup -/ _ rfl
/- inf -/ (filter.lattice.has_inf).1
begin
ext f g : 2,
exact le_antisymm
(le_inf (assume s, mem_inf_sets_of_left) (assume s, mem_inf_sets_of_right))
(assume s ⟨a, ha, b, hb, hs⟩, show s ∈ complete_lattice.inf f g, from
mem_sets_of_superset (inter_mem_sets
(@inf_le_left (filter α) _ _ _ _ ha)
(@inf_le_right (filter α) _ _ _ _ hb)) hs)
end
/- Sup -/ (join ∘ principal) (by ext s x; exact (@mem_bInter_iff _ _ s filter.sets x).symm)
/- Inf -/ _ rfl
end complete_lattice
lemma bot_sets_eq : (⊥ : filter α).sets = univ := rfl
lemma sup_sets_eq {f g : filter α} : (f ⊔ g).sets = f.sets ∩ g.sets :=
(gi_generate α).gc.u_inf
lemma Sup_sets_eq {s : set (filter α)} : (Sup s).sets = (⋂f∈s, (f:filter α).sets) :=
(gi_generate α).gc.u_Inf
lemma supr_sets_eq {f : ι → filter α} : (supr f).sets = (⋂i, (f i).sets) :=
(gi_generate α).gc.u_infi
lemma generate_empty : filter.generate ∅ = (⊤ : filter α) :=
(gi_generate α).gc.l_bot
lemma generate_univ : filter.generate univ = (⊥ : filter α) :=
mk_of_closure_sets.symm
lemma generate_union {s t : set (set α)} :
filter.generate (s ∪ t) = filter.generate s ⊓ filter.generate t :=
(gi_generate α).gc.l_sup
lemma generate_Union {s : ι → set (set α)} :
filter.generate (⋃ i, s i) = (⨅ i, filter.generate (s i)) :=
(gi_generate α).gc.l_supr
@[simp] lemma mem_bot_sets {s : set α} : s ∈ (⊥ : filter α) :=
trivial
@[simp] lemma mem_sup_sets {f g : filter α} {s : set α} :
s ∈ f ⊔ g ↔ s ∈ f ∧ s ∈ g :=
iff.rfl
@[simp] lemma mem_Sup_sets {x : set α} {s : set (filter α)} :
x ∈ Sup s ↔ (∀f∈s, x ∈ (f:filter α)) :=
iff.rfl
@[simp] lemma mem_supr_sets {x : set α} {f : ι → filter α} :
x ∈ supr f ↔ (∀i, x ∈ f i) :=
by simp only [supr_sets_eq, iff_self, mem_Inter]
@[simp] lemma le_principal_iff {s : set α} {f : filter α} : f ≤ principal s ↔ s ∈ f :=
show (∀{t}, s ⊆ t → t ∈ f) ↔ s ∈ f,
from ⟨assume h, h (subset.refl s), assume hs t ht, mem_sets_of_superset hs ht⟩
lemma principal_mono {s t : set α} : principal s ≤ principal t ↔ s ⊆ t :=
by simp only [le_principal_iff, iff_self, mem_principal_sets]
lemma monotone_principal : monotone (principal : set α → filter α) :=
by simp only [monotone, principal_mono]; exact assume a b h, h
@[simp] lemma principal_eq_iff_eq {s t : set α} : principal s = principal t ↔ s = t :=
by simp only [le_antisymm_iff, le_principal_iff, mem_principal_sets]; refl
@[simp] lemma join_principal_eq_Sup {s : set (filter α)} : join (principal s) = Sup s := rfl
/- lattice equations -/
lemma empty_in_sets_eq_bot {f : filter α} : ∅ ∈ f ↔ f = ⊥ :=
⟨assume h, bot_unique $ assume s _, mem_sets_of_superset h (empty_subset s),
assume : f = ⊥, this.symm ▸ mem_bot_sets⟩
lemma inhabited_of_mem_sets {f : filter α} {s : set α} (hf : f ≠ ⊥) (hs : s ∈ f) :
∃x, x ∈ s :=
have ∅ ∉ f.sets, from assume h, hf $ empty_in_sets_eq_bot.mp h,
have s ≠ ∅, from assume h, this (h ▸ hs),
exists_mem_of_ne_empty this
lemma filter_eq_bot_of_not_nonempty {f : filter α} (ne : ¬ nonempty α) : f = ⊥ :=
empty_in_sets_eq_bot.mp $ univ_mem_sets' $ assume x, false.elim (ne ⟨x⟩)
lemma forall_sets_neq_empty_iff_neq_bot {f : filter α} :
(∀ (s : set α), s ∈ f → s ≠ ∅) ↔ f ≠ ⊥ :=
by
simp only [(@empty_in_sets_eq_bot α f).symm, ne.def];
exact ⟨assume h hs, h _ hs rfl, assume h s hs eq, h $ eq ▸ hs⟩
lemma mem_sets_of_neq_bot {f : filter α} {s : set α} (h : f ⊓ principal (-s) = ⊥) : s ∈ f :=
have ∅ ∈ f ⊓ principal (- s), from h.symm ▸ mem_bot_sets,
let ⟨s₁, hs₁, s₂, (hs₂ : -s ⊆ s₂), (hs : s₁ ∩ s₂ ⊆ ∅)⟩ := this in
by filter_upwards [hs₁] assume a ha, classical.by_contradiction $ assume ha', hs ⟨ha, hs₂ ha'⟩
lemma infi_sets_eq {f : ι → filter α} (h : directed (≥) f) (ne : nonempty ι) :
(infi f).sets = (⋃ i, (f i).sets) :=
let ⟨i⟩ := ne, u := { filter .
sets := (⋃ i, (f i).sets),
univ_sets := by simp only [mem_Union]; exact ⟨i, univ_mem_sets⟩,
sets_of_superset := by simp only [mem_Union, exists_imp_distrib];
intros x y i hx hxy; exact ⟨i, mem_sets_of_superset hx hxy⟩,
inter_sets :=
begin
simp only [mem_Union, exists_imp_distrib],
assume x y a hx b hy,
rcases h a b with ⟨c, ha, hb⟩,
exact ⟨c, inter_mem_sets (ha hx) (hb hy)⟩
end } in
subset.antisymm
(show u ≤ infi f, from le_infi $ assume i, le_supr (λi, (f i).sets) i)
(Union_subset $ assume i, infi_le f i)
lemma mem_infi {f : ι → filter α} (h : directed (≥) f) (ne : nonempty ι) (s) :
s ∈ infi f ↔ s ∈ ⋃ i, (f i).sets :=
show s ∈ (infi f).sets ↔ s ∈ ⋃ i, (f i).sets, by rw infi_sets_eq h ne
lemma infi_sets_eq' {f : β → filter α} {s : set β}
(h : directed_on (f ⁻¹'o (≥)) s) (ne : ∃i, i ∈ s) :
(⨅ i∈s, f i).sets = (⋃ i ∈ s, (f i).sets) :=
let ⟨i, hi⟩ := ne in
calc (⨅ i ∈ s, f i).sets = (⨅ t : {t // t ∈ s}, (f t.val)).sets : by rw [infi_subtype]; refl
... = (⨆ t : {t // t ∈ s}, (f t.val).sets) : infi_sets_eq
(assume ⟨x, hx⟩ ⟨y, hy⟩, match h x hx y hy with ⟨z, h₁, h₂, h₃⟩ := ⟨⟨z, h₁⟩, h₂, h₃⟩ end)
⟨⟨i, hi⟩⟩
... = (⨆ t ∈ {t | t ∈ s}, (f t).sets) : by rw [supr_subtype]; refl
lemma infi_sets_eq_finite (f : ι → filter α) :
(⨅i, f i).sets = (⋃t:finset (plift ι), (⨅i∈t, f (plift.down i)).sets) :=
begin
rw [infi_eq_infi_finset, infi_sets_eq],
exact (directed_of_sup $ λs₁ s₂ hs, infi_le_infi $ λi, infi_le_infi_const $ λh, hs h),
apply_instance
end
lemma mem_infi_finite {f : ι → filter α} (s) :
s ∈ infi f ↔ s ∈ ⋃t:finset (plift ι), (⨅i∈t, f (plift.down i)).sets :=
show s ∈ (infi f).sets ↔ s ∈ ⋃t:finset (plift ι), (⨅i∈t, f (plift.down i)).sets,
by rw infi_sets_eq_finite
@[simp] lemma sup_join {f₁ f₂ : filter (filter α)} : (join f₁ ⊔ join f₂) = join (f₁ ⊔ f₂) :=
filter_eq $ set.ext $ assume x,
by simp only [supr_sets_eq, join, mem_sup_sets, iff_self, mem_set_of_eq]
@[simp] lemma supr_join {ι : Sort w} {f : ι → filter (filter α)} :
(⨆x, join (f x)) = join (⨆x, f x) :=
filter_eq $ set.ext $ assume x,
by simp only [supr_sets_eq, join, iff_self, mem_Inter, mem_set_of_eq]
instance : bounded_distrib_lattice (filter α) :=
{ le_sup_inf :=
begin
assume x y z s,
simp only [and_assoc, mem_inf_sets, mem_sup_sets, exists_prop, exists_imp_distrib, and_imp],
intros hs t₁ ht₁ t₂ ht₂ hts,
exact ⟨s ∪ t₁,
x.sets_of_superset hs $ subset_union_left _ _,
y.sets_of_superset ht₁ $ subset_union_right _ _,
s ∪ t₂,
x.sets_of_superset hs $ subset_union_left _ _,
z.sets_of_superset ht₂ $ subset_union_right _ _,
subset.trans (@le_sup_inf (set α) _ _ _ _) (union_subset (subset.refl _) hts)⟩
end,
..filter.lattice.complete_lattice }
/- the complementary version with ⨆i, f ⊓ g i does not hold! -/
lemma infi_sup_eq {f : filter α} {g : ι → filter α} : (⨅ x, f ⊔ g x) = f ⊔ infi g :=
begin
refine le_antisymm _ (le_infi $ assume i, sup_le_sup (le_refl f) $ infi_le _ _),
rintros t ⟨h₁, h₂⟩,
rw [infi_sets_eq_finite] at h₂,
simp only [mem_Union, (finset.inf_eq_infi _ _).symm] at h₂,
rcases h₂ with ⟨s, hs⟩,
suffices : (⨅i, f ⊔ g i) ≤ f ⊔ s.inf (λi, g i.down), { exact this ⟨h₁, hs⟩ },
refine finset.induction_on s _ _,
{ exact le_sup_right_of_le le_top },
{ rintros ⟨i⟩ s his ih,
rw [finset.inf_insert, sup_inf_left],
exact le_inf (infi_le _ _) ih }
end
lemma mem_infi_sets_finset {s : finset α} {f : α → filter β} :
∀t, t ∈ (⨅a∈s, f a) ↔ (∃p:α → set β, (∀a∈s, p a ∈ f a) ∧ (⋂a∈s, p a) ⊆ t) :=
show ∀t, t ∈ (⨅a∈s, f a) ↔ (∃p:α → set β, (∀a∈s, p a ∈ f a) ∧ (⨅a∈s, p a) ≤ t),
begin
simp only [(finset.inf_eq_infi _ _).symm],
refine finset.induction_on s _ _,
{ simp only [finset.not_mem_empty, false_implies_iff, finset.inf_empty, top_le_iff,
imp_true_iff, mem_top_sets, true_and, exists_const],
intros; refl },
{ intros a s has ih t,
simp only [ih, finset.forall_mem_insert, finset.inf_insert, mem_inf_sets,
exists_prop, iff_iff_implies_and_implies, exists_imp_distrib, and_imp, and_assoc] {contextual := tt},
split,
{ intros t₁ ht₁ t₂ p hp ht₂ ht,
existsi function.update p a t₁,
have : ∀a'∈s, function.update p a t₁ a' = p a',
from assume a' ha',
have a' ≠ a, from assume h, has $ h ▸ ha',
function.update_noteq this,
have eq : s.inf (λj, function.update p a t₁ j) = s.inf (λj, p j) :=
finset.inf_congr rfl this,
simp only [this, ht₁, hp, function.update_same, true_and, imp_true_iff, eq] {contextual := tt},
exact subset.trans (inter_subset_inter (subset.refl _) ht₂) ht },
assume p hpa hp ht,
exact ⟨p a, hpa, (s.inf p), ⟨⟨p, hp, le_refl _⟩, ht⟩⟩ }
end
/- principal equations -/
@[simp] lemma inf_principal {s t : set α} : principal s ⊓ principal t = principal (s ∩ t) :=
le_antisymm
(by simp; exact ⟨s, subset.refl s, t, subset.refl t, by simp⟩)
(by simp [le_inf_iff, inter_subset_left, inter_subset_right])
@[simp] lemma sup_principal {s t : set α} : principal s ⊔ principal t = principal (s ∪ t) :=
filter_eq $ set.ext $
by simp only [union_subset_iff, union_subset_iff, mem_sup_sets, forall_const, iff_self, mem_principal_sets]
@[simp] lemma supr_principal {ι : Sort w} {s : ι → set α} : (⨆x, principal (s x)) = principal (⋃i, s i) :=
filter_eq $ set.ext $ assume x, by simp only [supr_sets_eq, mem_principal_sets, mem_Inter];
exact (@supr_le_iff (set α) _ _ _ _).symm
lemma principal_univ : principal (univ : set α) = ⊤ :=
top_unique $ by simp only [le_principal_iff, mem_top_sets, eq_self_iff_true]
lemma principal_empty : principal (∅ : set α) = ⊥ :=
bot_unique $ assume s _, empty_subset _
@[simp] lemma principal_eq_bot_iff {s : set α} : principal s = ⊥ ↔ s = ∅ :=
⟨assume h, principal_eq_iff_eq.mp $ by simp only [principal_empty, h, eq_self_iff_true],
assume h, by simp only [h, principal_empty, eq_self_iff_true]⟩
lemma inf_principal_eq_bot {f : filter α} {s : set α} (hs : -s ∈ f) : f ⊓ principal s = ⊥ :=
empty_in_sets_eq_bot.mp ⟨_, hs, s, mem_principal_self s, assume x ⟨h₁, h₂⟩, h₁ h₂⟩
theorem mem_inf_principal (f : filter α) (s t : set α) :
s ∈ f ⊓ principal t ↔ { x | x ∈ t → x ∈ s } ∈ f :=
begin
simp only [mem_inf_sets, mem_principal_sets, exists_prop], split,
{ rintros ⟨u, ul, v, tsubv, uvinter⟩,
apply filter.mem_sets_of_superset ul,
intros x xu xt, exact uvinter ⟨xu, tsubv xt⟩ },
intro h, refine ⟨_, h, t, set.subset.refl t, _⟩,
rintros x ⟨hx, xt⟩,
exact hx xt
end
end lattice
section map
/-- The forward map of a filter -/
def map (m : α → β) (f : filter α) : filter β :=
{ sets := preimage m ⁻¹' f.sets,
univ_sets := univ_mem_sets,
sets_of_superset := assume s t hs st, mem_sets_of_superset hs $ preimage_mono st,
inter_sets := assume s t hs ht, inter_mem_sets hs ht }
@[simp] lemma map_principal {s : set α} {f : α → β} :
map f (principal s) = principal (set.image f s) :=
filter_eq $ set.ext $ assume a, image_subset_iff.symm
variables {f : filter α} {m : α → β} {m' : β → γ} {s : set α} {t : set β}
@[simp] lemma mem_map : t ∈ map m f ↔ {x | m x ∈ t} ∈ f := iff.rfl
lemma image_mem_map (hs : s ∈ f) : m '' s ∈ map m f :=
f.sets_of_superset hs $ subset_preimage_image m s
lemma range_mem_map : range m ∈ map m f :=
by rw ←image_univ; exact image_mem_map univ_mem_sets
lemma mem_map_sets_iff : t ∈ map m f ↔ (∃s∈f, m '' s ⊆ t) :=
iff.intro
(assume ht, ⟨set.preimage m t, ht, image_preimage_subset _ _⟩)
(assume ⟨s, hs, ht⟩, mem_sets_of_superset (image_mem_map hs) ht)
@[simp] lemma map_id : filter.map id f = f :=
filter_eq $ rfl
@[simp] lemma map_compose : filter.map m' ∘ filter.map m = filter.map (m' ∘ m) :=
funext $ assume _, filter_eq $ rfl
@[simp] lemma map_map : filter.map m' (filter.map m f) = filter.map (m' ∘ m) f :=
congr_fun (@@filter.map_compose m m') f
end map
section comap
/-- The inverse map of a filter -/
def comap (m : α → β) (f : filter β) : filter α :=
{ sets := { s | ∃t∈ f, m ⁻¹' t ⊆ s },
univ_sets := ⟨univ, univ_mem_sets, by simp only [subset_univ, preimage_univ]⟩,
sets_of_superset := assume a b ⟨a', ha', ma'a⟩ ab,
⟨a', ha', subset.trans ma'a ab⟩,
inter_sets := assume a b ⟨a', ha₁, ha₂⟩ ⟨b', hb₁, hb₂⟩,
⟨a' ∩ b', inter_mem_sets ha₁ hb₁, inter_subset_inter ha₂ hb₂⟩ }
end comap
/-- The cofinite filter is the filter of subsets whose complements are finite. -/
def cofinite : filter α :=
{ sets := {s | finite (- s)},
univ_sets := by simp only [compl_univ, finite_empty, mem_set_of_eq],
sets_of_superset := assume s t (hs : finite (-s)) (st: s ⊆ t),
finite_subset hs $ @lattice.neg_le_neg (set α) _ _ _ st,
inter_sets := assume s t (hs : finite (-s)) (ht : finite (-t)),
by simp only [compl_inter, finite_union, ht, hs, mem_set_of_eq] }
lemma cofinite_ne_bot (hi : set.infinite (@set.univ α)) : @cofinite α ≠ ⊥ :=
forall_sets_neq_empty_iff_neq_bot.mp
$ λ s hs hn, by change set.finite _ at hs;
rw [hn, set.compl_empty] at hs; exact hi hs
/-- The monadic bind operation on filter is defined the usual way in terms of `map` and `join`.
Unfortunately, this `bind` does not result in the expected applicative. See `filter.seq` for the
applicative instance. -/
def bind (f : filter α) (m : α → filter β) : filter β := join (map m f)
/-- The applicative sequentiation operation. This is not induced by the bind operation. -/
def seq (f : filter (α → β)) (g : filter α) : filter β :=
⟨{ s | ∃u∈ f, ∃t∈ g, (∀m∈u, ∀x∈t, (m : α → β) x ∈ s) },
⟨univ, univ_mem_sets, univ, univ_mem_sets, by simp only [forall_prop_of_true, mem_univ, forall_true_iff]⟩,
assume s₀ s₁ ⟨t₀, t₁, h₀, h₁, h⟩ hst, ⟨t₀, t₁, h₀, h₁, assume x hx y hy, hst $ h _ hx _ hy⟩,
assume s₀ s₁ ⟨t₀, ht₀, t₁, ht₁, ht⟩ ⟨u₀, hu₀, u₁, hu₁, hu⟩,
⟨t₀ ∩ u₀, inter_mem_sets ht₀ hu₀, t₁ ∩ u₁, inter_mem_sets ht₁ hu₁,
assume x ⟨hx₀, hx₁⟩ x ⟨hy₀, hy₁⟩, ⟨ht _ hx₀ _ hy₀, hu _ hx₁ _ hy₁⟩⟩⟩
instance : has_pure filter := ⟨λ(α : Type u) x, principal {x}⟩
instance : has_bind filter := ⟨@filter.bind⟩
instance : has_seq filter := ⟨@filter.seq⟩
instance : functor filter := { map := @filter.map }
section
-- this section needs to be before applicative, otherwise the wrong instance will be chosen
protected def monad : monad filter := { map := @filter.map }
local attribute [instance] filter.monad
protected lemma is_lawful_monad : is_lawful_monad filter :=
{ id_map := assume α f, filter_eq rfl,
pure_bind := assume α β a f, by simp only [has_bind.bind, pure, bind, Sup_image, image_singleton,
join_principal_eq_Sup, lattice.Sup_singleton, map_principal, eq_self_iff_true],
bind_assoc := assume α β γ f m₁ m₂, filter_eq rfl,
bind_pure_comp_eq_map := assume α β f x, filter_eq $
by simp only [has_bind.bind, pure, functor.map, bind, join, map, preimage, principal,
set.subset_univ, eq_self_iff_true, function.comp_app, mem_set_of_eq, singleton_subset_iff] }
end
instance : applicative filter := { map := @filter.map, seq := @filter.seq }
instance : alternative filter :=
{ failure := λα, ⊥,
orelse := λα x y, x ⊔ y }
@[simp] lemma pure_def (x : α) : pure x = principal {x} := rfl
@[simp] lemma mem_pure {a : α} {s : set α} : a ∈ s → s ∈ (pure a : filter α) :=
by simp only [imp_self, pure_def, mem_principal_sets, singleton_subset_iff]; exact id
@[simp] lemma mem_pure_iff {a : α} {s : set α} : s ∈ (pure a : filter α) ↔ a ∈ s :=
by rw [pure_def, mem_principal_sets, set.singleton_subset_iff]
@[simp] lemma map_def {α β} (m : α → β) (f : filter α) : m <$> f = map m f := rfl
@[simp] lemma bind_def {α β} (f : filter α) (m : α → filter β) : f >>= m = bind f m := rfl
/- map and comap equations -/
section map
variables {f f₁ f₂ : filter α} {g g₁ g₂ : filter β} {m : α → β} {m' : β → γ} {s : set α} {t : set β}
@[simp] theorem mem_comap_sets : s ∈ comap m g ↔ ∃t∈ g, m ⁻¹' t ⊆ s := iff.rfl
theorem preimage_mem_comap (ht : t ∈ g) : m ⁻¹' t ∈ comap m g :=
⟨t, ht, subset.refl _⟩
lemma comap_id : comap id f = f :=
le_antisymm (assume s, preimage_mem_comap) (assume s ⟨t, ht, hst⟩, mem_sets_of_superset ht hst)
lemma comap_comap_comp {m : γ → β} {n : β → α} : comap m (comap n f) = comap (n ∘ m) f :=
le_antisymm
(assume c ⟨b, hb, (h : preimage (n ∘ m) b ⊆ c)⟩, ⟨preimage n b, preimage_mem_comap hb, h⟩)
(assume c ⟨b, ⟨a, ha, (h₁ : preimage n a ⊆ b)⟩, (h₂ : preimage m b ⊆ c)⟩,
⟨a, ha, show preimage m (preimage n a) ⊆ c, from subset.trans (preimage_mono h₁) h₂⟩)
@[simp] theorem comap_principal {t : set β} : comap m (principal t) = principal (m ⁻¹' t) :=
filter_eq $ set.ext $ assume s,
⟨assume ⟨u, (hu : t ⊆ u), (b : preimage m u ⊆ s)⟩, subset.trans (preimage_mono hu) b,
assume : preimage m t ⊆ s, ⟨t, subset.refl t, this⟩⟩
lemma map_le_iff_le_comap : map m f ≤ g ↔ f ≤ comap m g :=
⟨assume h s ⟨t, ht, hts⟩, mem_sets_of_superset (h ht) hts, assume h s ht, h ⟨_, ht, subset.refl _⟩⟩
lemma gc_map_comap (m : α → β) : galois_connection (map m) (comap m) :=
assume f g, map_le_iff_le_comap
lemma map_mono (h : f₁ ≤ f₂) : map m f₁ ≤ map m f₂ := (gc_map_comap m).monotone_l h
lemma monotone_map : monotone (map m) | a b := map_mono
lemma comap_mono (h : g₁ ≤ g₂) : comap m g₁ ≤ comap m g₂ := (gc_map_comap m).monotone_u h
lemma monotone_comap : monotone (comap m) | a b := comap_mono
@[simp] lemma map_bot : map m ⊥ = ⊥ := (gc_map_comap m).l_bot
@[simp] lemma map_sup : map m (f₁ ⊔ f₂) = map m f₁ ⊔ map m f₂ := (gc_map_comap m).l_sup
@[simp] lemma map_supr {f : ι → filter α} : map m (⨆i, f i) = (⨆i, map m (f i)) :=
(gc_map_comap m).l_supr
@[simp] lemma comap_top : comap m ⊤ = ⊤ := (gc_map_comap m).u_top
@[simp] lemma comap_inf : comap m (g₁ ⊓ g₂) = comap m g₁ ⊓ comap m g₂ := (gc_map_comap m).u_inf
@[simp] lemma comap_infi {f : ι → filter β} : comap m (⨅i, f i) = (⨅i, comap m (f i)) :=
(gc_map_comap m).u_infi
lemma le_comap_top (f : α → β) (l : filter α) : l ≤ comap f ⊤ :=
by rw [comap_top]; exact le_top
lemma map_comap_le : map m (comap m g) ≤ g := (gc_map_comap m).l_u_le _
lemma le_comap_map : f ≤ comap m (map m f) := (gc_map_comap m).le_u_l _
@[simp] lemma comap_bot : comap m ⊥ = ⊥ :=
bot_unique $ assume s _, ⟨∅, by simp only [mem_bot_sets], by simp only [empty_subset, preimage_empty]⟩
lemma comap_supr {ι} {f : ι → filter β} {m : α → β} :
comap m (supr f) = (⨆i, comap m (f i)) :=
le_antisymm
(assume s hs,
have ∀i, ∃t, t ∈ f i ∧ m ⁻¹' t ⊆ s, by simpa only [mem_comap_sets, exists_prop, mem_supr_sets] using mem_supr_sets.1 hs,
let ⟨t, ht⟩ := classical.axiom_of_choice this in
⟨⋃i, t i, mem_supr_sets.2 $ assume i, (f i).sets_of_superset (ht i).1 (subset_Union _ _),
begin
rw [preimage_Union, Union_subset_iff],
assume i,
exact (ht i).2
end⟩)
(supr_le $ assume i, monotone_comap $ le_supr _ _)
lemma comap_Sup {s : set (filter β)} {m : α → β} : comap m (Sup s) = (⨆f∈s, comap m f) :=
by simp only [Sup_eq_supr, comap_supr, eq_self_iff_true]
lemma comap_sup : comap m (g₁ ⊔ g₂) = comap m g₁ ⊔ comap m g₂ :=
le_antisymm
(assume s ⟨⟨t₁, ht₁, hs₁⟩, ⟨t₂, ht₂, hs₂⟩⟩,
⟨t₁ ∪ t₂,
⟨g₁.sets_of_superset ht₁ (subset_union_left _ _), g₂.sets_of_superset ht₂ (subset_union_right _ _)⟩,
union_subset hs₁ hs₂⟩)
(sup_le (comap_mono le_sup_left) (comap_mono le_sup_right))
lemma map_comap {f : filter β} {m : α → β} (hf : range m ∈ f) : (f.comap m).map m = f :=
le_antisymm
map_comap_le
(assume t' ⟨t, ht, sub⟩, by filter_upwards [ht, hf]; rintros x hxt ⟨y, rfl⟩; exact sub hxt)
lemma comap_map {f : filter α} {m : α → β} (h : ∀ x y, m x = m y → x = y) :
comap m (map m f) = f :=
have ∀s, preimage m (image m s) = s,
from assume s, preimage_image_eq s h,
le_antisymm
(assume s hs, ⟨
image m s,
f.sets_of_superset hs $ by simp only [this, subset.refl],
by simp only [this, subset.refl]⟩)
le_comap_map
lemma le_of_map_le_map_inj' {f g : filter α} {m : α → β} {s : set α}
(hsf : s ∈ f) (hsg : s ∈ g) (hm : ∀x∈s, ∀y∈s, m x = m y → x = y)
(h : map m f ≤ map m g) : f ≤ g :=
assume t ht, by filter_upwards [hsf, h $ image_mem_map (inter_mem_sets hsg ht)]
assume a has ⟨b, ⟨hbs, hb⟩, h⟩,
have b = a, from hm _ hbs _ has h,
this ▸ hb
lemma le_of_map_le_map_inj_iff {f g : filter α} {m : α → β} {s : set α}
(hsf : s ∈ f) (hsg : s ∈ g) (hm : ∀x∈s, ∀y∈s, m x = m y → x = y) :
map m f ≤ map m g ↔ f ≤ g :=
iff.intro (le_of_map_le_map_inj' hsf hsg hm) map_mono
lemma eq_of_map_eq_map_inj' {f g : filter α} {m : α → β} {s : set α}
(hsf : s ∈ f) (hsg : s ∈ g) (hm : ∀x∈s, ∀y∈s, m x = m y → x = y)
(h : map m f = map m g) : f = g :=
le_antisymm
(le_of_map_le_map_inj' hsf hsg hm $ le_of_eq h)
(le_of_map_le_map_inj' hsg hsf hm $ le_of_eq h.symm)
lemma map_inj {f g : filter α} {m : α → β} (hm : ∀ x y, m x = m y → x = y) (h : map m f = map m g) :
f = g :=
have comap m (map m f) = comap m (map m g), by rw h,
by rwa [comap_map hm, comap_map hm] at this
theorem le_map_comap_of_surjective' {f : α → β} {l : filter β} {u : set β} (ul : u ∈ l)
(hf : ∀ y ∈ u, ∃ x, f x = y) :
l ≤ map f (comap f l) :=
assume s ⟨t, tl, ht⟩,
have t ∩ u ⊆ s, from
assume x ⟨xt, xu⟩,
exists.elim (hf x xu) $ λ a faeq,
by { rw ←faeq, apply ht, change f a ∈ t, rw faeq, exact xt },
mem_sets_of_superset (inter_mem_sets tl ul) this
theorem map_comap_of_surjective' {f : α → β} {l : filter β} {u : set β} (ul : u ∈ l)
(hf : ∀ y ∈ u, ∃ x, f x = y) :
map f (comap f l) = l :=
le_antisymm map_comap_le (le_map_comap_of_surjective' ul hf)
theorem le_map_comap_of_surjective {f : α → β} (hf : function.surjective f) (l : filter β) :
l ≤ map f (comap f l) :=
le_map_comap_of_surjective' univ_mem_sets (λ y _, hf y)
theorem map_comap_of_surjective {f : α → β} (hf : function.surjective f) (l : filter β) :
map f (comap f l) = l :=
le_antisymm map_comap_le (le_map_comap_of_surjective hf l)
lemma comap_neq_bot {f : filter β} {m : α → β}
(hm : ∀t∈ f, ∃a, m a ∈ t) : comap m f ≠ ⊥ :=
forall_sets_neq_empty_iff_neq_bot.mp $ assume s ⟨t, ht, t_s⟩,
let ⟨a, (ha : a ∈ preimage m t)⟩ := hm t ht in
neq_bot_of_le_neq_bot (ne_empty_of_mem ha) t_s
lemma comap_neq_bot_of_surj {f : filter β} {m : α → β}
(hf : f ≠ ⊥) (hm : ∀b, ∃a, m a = b) : comap m f ≠ ⊥ :=
comap_neq_bot $ assume t ht,
let
⟨b, (hx : b ∈ t)⟩ := inhabited_of_mem_sets hf ht,
⟨a, (ha : m a = b)⟩ := hm b
in ⟨a, ha.symm ▸ hx⟩
@[simp] lemma map_eq_bot_iff : map m f = ⊥ ↔ f = ⊥ :=
⟨by rw [←empty_in_sets_eq_bot, ←empty_in_sets_eq_bot]; exact id,
assume h, by simp only [h, eq_self_iff_true, map_bot]⟩
lemma map_ne_bot (hf : f ≠ ⊥) : map m f ≠ ⊥ :=
assume h, hf $ by rwa [map_eq_bot_iff] at h
lemma sInter_comap_sets (f : α → β) (F : filter β) :
⋂₀(comap f F).sets = ⋂ U ∈ F, f ⁻¹' U :=
begin
ext x,
suffices : (∀ (A : set α) (B : set β), B ∈ F → f ⁻¹' B ⊆ A → x ∈ A) ↔
∀ (B : set β), B ∈ F → f x ∈ B,
by simp only [mem_sInter, mem_Inter, mem_comap_sets, this, and_imp, mem_comap_sets, exists_prop, mem_sInter,
iff_self, mem_Inter, mem_preimage, exists_imp_distrib],
split,
{ intros h U U_in,
simpa only [set.subset.refl, forall_prop_of_true, mem_preimage] using h (f ⁻¹' U) U U_in },
{ intros h V U U_in f_U_V,
exact f_U_V (h U U_in) },
end
end map
lemma map_cong {m₁ m₂ : α → β} {f : filter α} (h : {x | m₁ x = m₂ x} ∈ f) :
map m₁ f = map m₂ f :=
have ∀(m₁ m₂ : α → β) (h : {x | m₁ x = m₂ x} ∈ f), map m₁ f ≤ map m₂ f,
begin
intros m₁ m₂ h s hs,
show {x | m₁ x ∈ s} ∈ f,
filter_upwards [h, hs],
simp only [subset_def, mem_preimage, mem_set_of_eq, forall_true_iff] {contextual := tt}
end,
le_antisymm (this m₁ m₂ h) (this m₂ m₁ $ mem_sets_of_superset h $ assume x, eq.symm)
-- this is a generic rule for monotone functions:
lemma map_infi_le {f : ι → filter α} {m : α → β} :
map m (infi f) ≤ (⨅ i, map m (f i)) :=
le_infi $ assume i, map_mono $ infi_le _ _
lemma map_infi_eq {f : ι → filter α} {m : α → β} (hf : directed (≥) f) (hι : nonempty ι) :
map m (infi f) = (⨅ i, map m (f i)) :=
le_antisymm
map_infi_le
(assume s (hs : preimage m s ∈ infi f),
have ∃i, preimage m s ∈ f i,
by simp only [infi_sets_eq hf hι, mem_Union] at hs; assumption,
let ⟨i, hi⟩ := this in
have (⨅ i, map m (f i)) ≤ principal s, from
infi_le_of_le i $ by simp only [le_principal_iff, mem_map]; assumption,
by simp only [filter.le_principal_iff] at this; assumption)
lemma map_binfi_eq {ι : Type w} {f : ι → filter α} {m : α → β} {p : ι → Prop}
(h : directed_on (f ⁻¹'o (≥)) {x | p x}) (ne : ∃i, p i) :
map m (⨅i (h : p i), f i) = (⨅i (h: p i), map m (f i)) :=
let ⟨i, hi⟩ := ne in
calc map m (⨅i (h : p i), f i) = map m (⨅i:subtype p, f i.val) : by simp only [infi_subtype, eq_self_iff_true]
... = (⨅i:subtype p, map m (f i.val)) : map_infi_eq
(assume ⟨x, hx⟩ ⟨y, hy⟩, match h x hx y hy with ⟨z, h₁, h₂, h₃⟩ := ⟨⟨z, h₁⟩, h₂, h₃⟩ end)
⟨⟨i, hi⟩⟩
... = (⨅i (h : p i), map m (f i)) : by simp only [infi_subtype, eq_self_iff_true]
lemma map_inf' {f g : filter α} {m : α → β} {t : set α} (htf : t ∈ f) (htg : t ∈ g)
(h : ∀x∈t, ∀y∈t, m x = m y → x = y) : map m (f ⊓ g) = map m f ⊓ map m g :=
begin
refine le_antisymm
(le_inf (map_mono inf_le_left) (map_mono inf_le_right))
(assume s hs, _),
simp only [map, mem_inf_sets, exists_prop, mem_map, mem_preimage, mem_inf_sets] at hs ⊢,
rcases hs with ⟨t₁, h₁, t₂, h₂, hs⟩,
refine ⟨m '' (t₁ ∩ t), _, m '' (t₂ ∩ t), _, _⟩,
{ filter_upwards [h₁, htf] assume a h₁ h₂, mem_image_of_mem _ ⟨h₁, h₂⟩ },
{ filter_upwards [h₂, htg] assume a h₁ h₂, mem_image_of_mem _ ⟨h₁, h₂⟩ },
{ rw [image_inter_on],
{ refine image_subset_iff.2 _,
exact λ x ⟨⟨h₁, _⟩, h₂, _⟩, hs ⟨h₁, h₂⟩ },
{ exact λ x ⟨_, hx⟩ y ⟨_, hy⟩, h x hx y hy } }
end
lemma map_inf {f g : filter α} {m : α → β} (h : ∀ x y, m x = m y → x = y) :
map m (f ⊓ g) = map m f ⊓ map m g :=
map_inf' univ_mem_sets univ_mem_sets (assume x _ y _, h x y)
lemma map_eq_comap_of_inverse {f : filter α} {m : α → β} {n : β → α}
(h₁ : m ∘ n = id) (h₂ : n ∘ m = id) : map m f = comap n f :=
le_antisymm
(assume b ⟨a, ha, (h : preimage n a ⊆ b)⟩, f.sets_of_superset ha $
calc a = preimage (n ∘ m) a : by simp only [h₂, preimage_id, eq_self_iff_true]
... ⊆ preimage m b : preimage_mono h)
(assume b (hb : preimage m b ∈ f),
⟨preimage m b, hb, show preimage (m ∘ n) b ⊆ b, by simp only [h₁]; apply subset.refl⟩)
lemma map_swap_eq_comap_swap {f : filter (α × β)} : prod.swap <$> f = comap prod.swap f :=
map_eq_comap_of_inverse prod.swap_swap_eq prod.swap_swap_eq
lemma le_map {f : filter α} {m : α → β} {g : filter β} (h : ∀s∈ f, m '' s ∈ g) :
g ≤ f.map m :=
assume s hs, mem_sets_of_superset (h _ hs) $ image_preimage_subset _ _
section applicative
@[simp] lemma mem_pure_sets {a : α} {s : set α} :
s ∈ (pure a : filter α) ↔ a ∈ s :=
by simp only [iff_self, pure_def, mem_principal_sets, singleton_subset_iff]
lemma singleton_mem_pure_sets {a : α} : {a} ∈ (pure a : filter α) :=
by simp only [mem_singleton, pure_def, mem_principal_sets, singleton_subset_iff]
@[simp] lemma pure_neq_bot {α : Type u} {a : α} : pure a ≠ (⊥ : filter α) :=
by simp only [pure, has_pure.pure, ne.def, not_false_iff, singleton_ne_empty, principal_eq_bot_iff]
lemma mem_seq_sets_def {f : filter (α → β)} {g : filter α} {s : set β} :
s ∈ f.seq g ↔ (∃u ∈ f, ∃t ∈ g, ∀x∈u, ∀y∈t, (x : α → β) y ∈ s) :=
iff.refl _
lemma mem_seq_sets_iff {f : filter (α → β)} {g : filter α} {s : set β} :
s ∈ f.seq g ↔ (∃u ∈ f, ∃t ∈ g, set.seq u t ⊆ s) :=
by simp only [mem_seq_sets_def, seq_subset, exists_prop, iff_self]
lemma mem_map_seq_iff {f : filter α} {g : filter β} {m : α → β → γ} {s : set γ} :
s ∈ (f.map m).seq g ↔ (∃t u, t ∈ g ∧ u ∈ f ∧ ∀x∈u, ∀y∈t, m x y ∈ s) :=
iff.intro
(assume ⟨t, ht, s, hs, hts⟩, ⟨s, m ⁻¹' t, hs, ht, assume a, hts _⟩)
(assume ⟨t, s, ht, hs, hts⟩, ⟨m '' s, image_mem_map hs, t, ht, assume f ⟨a, has, eq⟩, eq ▸ hts _ has⟩)
lemma seq_mem_seq_sets {f : filter (α → β)} {g : filter α} {s : set (α → β)} {t : set α}
(hs : s ∈ f) (ht : t ∈ g) : s.seq t ∈ f.seq g :=
⟨s, hs, t, ht, assume f hf a ha, ⟨f, hf, a, ha, rfl⟩⟩
lemma le_seq {f : filter (α → β)} {g : filter α} {h : filter β}
(hh : ∀t ∈ f, ∀u ∈ g, set.seq t u ∈ h) : h ≤ seq f g :=
assume s ⟨t, ht, u, hu, hs⟩, mem_sets_of_superset (hh _ ht _ hu) $
assume b ⟨m, hm, a, ha, eq⟩, eq ▸ hs _ hm _ ha
lemma seq_mono {f₁ f₂ : filter (α → β)} {g₁ g₂ : filter α}
(hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : f₁.seq g₁ ≤ f₂.seq g₂ :=
le_seq $ assume s hs t ht, seq_mem_seq_sets (hf hs) (hg ht)
@[simp] lemma pure_seq_eq_map (g : α → β) (f : filter α) : seq (pure g) f = f.map g :=
begin
refine le_antisymm (le_map $ assume s hs, _) (le_seq $ assume s hs t ht, _),
{ rw ← singleton_seq, apply seq_mem_seq_sets _ hs,
simp only [mem_singleton, pure_def, mem_principal_sets, singleton_subset_iff] },
{ rw mem_pure_sets at hs,
refine sets_of_superset (map g f) (image_mem_map ht) _,
rintros b ⟨a, ha, rfl⟩, exact ⟨g, hs, a, ha, rfl⟩ }
end
@[simp] lemma map_pure (f : α → β) (a : α) : map f (pure a) = pure (f a) :=
le_antisymm
(le_principal_iff.2 $ sets_of_superset (map f (pure a)) (image_mem_map singleton_mem_pure_sets) $
by simp only [image_singleton, mem_singleton, singleton_subset_iff])
(le_map $ assume s, begin
simp only [mem_image, pure_def, mem_principal_sets, singleton_subset_iff],
exact assume has, ⟨a, has, rfl⟩
end)
@[simp] lemma seq_pure (f : filter (α → β)) (a : α) : seq f (pure a) = map (λg:α → β, g a) f :=
begin
refine le_antisymm (le_map $ assume s hs, _) (le_seq $ assume s hs t ht, _),
{ rw ← seq_singleton, exact seq_mem_seq_sets hs
(by simp only [mem_singleton, pure_def, mem_principal_sets, singleton_subset_iff]) },
{ rw mem_pure_sets at ht,
refine sets_of_superset (map (λg:α→β, g a) f) (image_mem_map hs) _,
rintros b ⟨g, hg, rfl⟩, exact ⟨g, hg, a, ht, rfl⟩ }
end
@[simp] lemma seq_assoc (x : filter α) (g : filter (α → β)) (h : filter (β → γ)) :
seq h (seq g x) = seq (seq (map (∘) h) g) x :=
begin
refine le_antisymm (le_seq $ assume s hs t ht, _) (le_seq $ assume s hs t ht, _),
{ rcases mem_seq_sets_iff.1 hs with ⟨u, hu, v, hv, hs⟩,
rcases mem_map_sets_iff.1 hu with ⟨w, hw, hu⟩,
refine mem_sets_of_superset _
(set.seq_mono (subset.trans (set.seq_mono hu (subset.refl _)) hs) (subset.refl _)),
rw ← set.seq_seq,
exact seq_mem_seq_sets hw (seq_mem_seq_sets hv ht) },
{ rcases mem_seq_sets_iff.1 ht with ⟨u, hu, v, hv, ht⟩,
refine mem_sets_of_superset _ (set.seq_mono (subset.refl _) ht),
rw set.seq_seq,
exact seq_mem_seq_sets (seq_mem_seq_sets (image_mem_map hs) hu) hv }
end
lemma prod_map_seq_comm (f : filter α) (g : filter β) :
(map prod.mk f).seq g = seq (map (λb a, (a, b)) g) f :=
begin
refine le_antisymm (le_seq $ assume s hs t ht, _) (le_seq $ assume s hs t ht, _),
{ rcases mem_map_sets_iff.1 hs with ⟨u, hu, hs⟩,
refine mem_sets_of_superset _ (set.seq_mono hs (subset.refl _)),
rw ← set.prod_image_seq_comm,
exact seq_mem_seq_sets (image_mem_map ht) hu },
{ rcases mem_map_sets_iff.1 hs with ⟨u, hu, hs⟩,
refine mem_sets_of_superset _ (set.seq_mono hs (subset.refl _)),
rw set.prod_image_seq_comm,
exact seq_mem_seq_sets (image_mem_map ht) hu }
end
instance : is_lawful_functor (filter : Type u → Type u) :=
{ id_map := assume α f, map_id,
comp_map := assume α β γ f g a, map_map.symm }
instance : is_lawful_applicative (filter : Type u → Type u) :=
{ pure_seq_eq_map := assume α β, pure_seq_eq_map,
map_pure := assume α β, map_pure,
seq_pure := assume α β, seq_pure,
seq_assoc := assume α β γ, seq_assoc }
instance : is_comm_applicative (filter : Type u → Type u) :=
⟨assume α β f g, prod_map_seq_comm f g⟩
lemma {l} seq_eq_filter_seq {α β : Type l} (f : filter (α → β)) (g : filter α) :
f <*> g = seq f g := rfl
end applicative
/- bind equations -/
section bind
@[simp] lemma mem_bind_sets {s : set β} {f : filter α} {m : α → filter β} :
s ∈ bind f m ↔ ∃t ∈ f, ∀x ∈ t, s ∈ m x :=
calc s ∈ bind f m ↔ {a | s ∈ m a} ∈ f : by simp only [bind, mem_map, iff_self, mem_join_sets, mem_set_of_eq]
... ↔ (∃t ∈ f, t ⊆ {a | s ∈ m a}) : exists_sets_subset_iff.symm
... ↔ (∃t ∈ f, ∀x ∈ t, s ∈ m x) : iff.refl _
lemma bind_mono {f : filter α} {g h : α → filter β} (h₁ : {a | g a ≤ h a} ∈ f) :
bind f g ≤ bind f h :=
assume x h₂, show (_ ∈ f), by filter_upwards [h₁, h₂] assume s gh' h', gh' h'
lemma bind_sup {f g : filter α} {h : α → filter β} :
bind (f ⊔ g) h = bind f h ⊔ bind g h :=
by simp only [bind, sup_join, map_sup, eq_self_iff_true]
lemma bind_mono2 {f g : filter α} {h : α → filter β} (h₁ : f ≤ g) :
bind f h ≤ bind g h :=
assume s h', h₁ h'
lemma principal_bind {s : set α} {f : α → filter β} :
(bind (principal s) f) = (⨆x ∈ s, f x) :=
show join (map f (principal s)) = (⨆x ∈ s, f x),
by simp only [Sup_image, join_principal_eq_Sup, map_principal, eq_self_iff_true]
end bind
lemma infi_neq_bot_of_directed {f : ι → filter α}
(hn : nonempty α) (hd : directed (≥) f) (hb : ∀i, f i ≠ ⊥) : (infi f) ≠ ⊥ :=
let ⟨x⟩ := hn in
assume h, have he: ∅ ∈ (infi f), from h.symm ▸ (mem_bot_sets : ∅ ∈ (⊥ : filter α)),
classical.by_cases
(assume : nonempty ι,
have ∃i, ∅ ∈ f i,
by rw [mem_infi hd this] at he; simp only [mem_Union] at he; assumption,
let ⟨i, hi⟩ := this in
hb i $ bot_unique $
assume s _, (f i).sets_of_superset hi $ empty_subset _)
(assume : ¬ nonempty ι,
have univ ⊆ (∅ : set α),
begin
rw [←principal_mono, principal_univ, principal_empty, ←h],
exact (le_infi $ assume i, false.elim $ this ⟨i⟩)
end,
this $ mem_univ x)
lemma infi_neq_bot_iff_of_directed {f : ι → filter α}
(hn : nonempty α) (hd : directed (≥) f) : (infi f) ≠ ⊥ ↔ (∀i, f i ≠ ⊥) :=
⟨assume neq_bot i eq_bot, neq_bot $ bot_unique $ infi_le_of_le i $ eq_bot ▸ le_refl _,
infi_neq_bot_of_directed hn hd⟩
lemma mem_infi_sets {f : ι → filter α} (i : ι) : ∀{s}, s ∈ f i → s ∈ ⨅i, f i :=
show (⨅i, f i) ≤ f i, from infi_le _ _
@[elab_as_eliminator]
lemma infi_sets_induct {f : ι → filter α} {s : set α} (hs : s ∈ infi f) {p : set α → Prop}
(uni : p univ)
(ins : ∀{i s₁ s₂}, s₁ ∈ f i → p s₂ → p (s₁ ∩ s₂))
(upw : ∀{s₁ s₂}, s₁ ⊆ s₂ → p s₁ → p s₂) : p s :=
begin
rw [mem_infi_finite] at hs,
simp only [mem_Union, (finset.inf_eq_infi _ _).symm] at hs,
rcases hs with ⟨is, his⟩,
revert s,
refine finset.induction_on is _ _,
{ assume s hs, rwa [mem_top_sets.1 hs] },
{ rintros ⟨i⟩ js his ih s hs,
rw [finset.inf_insert, mem_inf_sets] at hs,
rcases hs with ⟨s₁, hs₁, s₂, hs₂, hs⟩,
exact upw hs (ins hs₁ (ih hs₂)) }
end
/- tendsto -/
/-- `tendsto` is the generic "limit of a function" predicate.
`tendsto f l₁ l₂` asserts that for every `l₂` neighborhood `a`,
the `f`-preimage of `a` is an `l₁` neighborhood. -/
def tendsto (f : α → β) (l₁ : filter α) (l₂ : filter β) := l₁.map f ≤ l₂
lemma tendsto_def {f : α → β} {l₁ : filter α} {l₂ : filter β} :
tendsto f l₁ l₂ ↔ ∀ s ∈ l₂, f ⁻¹' s ∈ l₁ := iff.rfl
lemma tendsto_iff_comap {f : α → β} {l₁ : filter α} {l₂ : filter β} :
tendsto f l₁ l₂ ↔ l₁ ≤ l₂.comap f :=
map_le_iff_le_comap
lemma tendsto.congr' {f₁ f₂ : α → β} {l₁ : filter α} {l₂ : filter β}
(hl : {x | f₁ x = f₂ x} ∈ l₁) (h : tendsto f₁ l₁ l₂) : tendsto f₂ l₁ l₂ :=
by rwa [tendsto, ←map_cong hl]
theorem tendsto.congr'r {f₁ f₂ : α → β} {l₁ : filter α} {l₂ : filter β}
(h : ∀ x, f₁ x = f₂ x) : tendsto f₁ l₁ l₂ ↔ tendsto f₂ l₁ l₂ :=
iff_of_eq (by congr'; exact funext h)
theorem tendsto.congr {f₁ f₂ : α → β} {l₁ : filter α} {l₂ : filter β}
(h : ∀ x, f₁ x = f₂ x) : tendsto f₁ l₁ l₂ → tendsto f₂ l₁ l₂ :=
(tendsto.congr'r h).1
lemma tendsto_id' {x y : filter α} : x ≤ y → tendsto id x y :=
by simp only [tendsto, map_id, forall_true_iff] {contextual := tt}
lemma tendsto_id {x : filter α} : tendsto id x x := tendsto_id' $ le_refl x
lemma tendsto.comp {f : α → β} {g : β → γ} {x : filter α} {y : filter β} {z : filter γ}
(hg : tendsto g y z) (hf : tendsto f x y) : tendsto (g ∘ f) x z :=
calc map (g ∘ f) x = map g (map f x) : by rw [map_map]
... ≤ map g y : map_mono hf
... ≤ z : hg
lemma tendsto_le_left {f : α → β} {x y : filter α} {z : filter β}
(h : y ≤ x) : tendsto f x z → tendsto f y z :=
le_trans (map_mono h)
lemma tendsto_le_right {f : α → β} {x : filter α} {y z : filter β}
(h₁ : y ≤ z) (h₂ : tendsto f x y) : tendsto f x z :=
le_trans h₂ h₁
lemma tendsto_map {f : α → β} {x : filter α} : tendsto f x (map f x) := le_refl (map f x)
lemma tendsto_map' {f : β → γ} {g : α → β} {x : filter α} {y : filter γ}
(h : tendsto (f ∘ g) x y) : tendsto f (map g x) y :=
by rwa [tendsto, map_map]
lemma tendsto_map'_iff {f : β → γ} {g : α → β} {x : filter α} {y : filter γ} :
tendsto f (map g x) y ↔ tendsto (f ∘ g) x y :=
by rw [tendsto, map_map]; refl
lemma tendsto_comap {f : α → β} {x : filter β} : tendsto f (comap f x) x :=
map_comap_le
lemma tendsto_comap_iff {f : α → β} {g : β → γ} {a : filter α} {c : filter γ} :
tendsto f a (c.comap g) ↔ tendsto (g ∘ f) a c :=
⟨assume h, tendsto_comap.comp h, assume h, map_le_iff_le_comap.mp $ by rwa [map_map]⟩
lemma tendsto_comap'_iff {m : α → β} {f : filter α} {g : filter β} {i : γ → α}
(h : range i ∈ f) : tendsto (m ∘ i) (comap i f) g ↔ tendsto m f g :=
by rw [tendsto, ← map_compose]; simp only [(∘), map_comap h, tendsto]
lemma comap_eq_of_inverse {f : filter α} {g : filter β} {φ : α → β} (ψ : β → α)
(eq : ψ ∘ φ = id) (hφ : tendsto φ f g) (hψ : tendsto ψ g f) : comap φ g = f :=
begin
refine le_antisymm (le_trans (comap_mono $ map_le_iff_le_comap.1 hψ) _) (map_le_iff_le_comap.1 hφ),
rw [comap_comap_comp, eq, comap_id],
exact le_refl _
end
lemma map_eq_of_inverse {f : filter α} {g : filter β} {φ : α → β} (ψ : β → α)
(eq : φ ∘ ψ = id) (hφ : tendsto φ f g) (hψ : tendsto ψ g f) : map φ f = g :=
begin
refine le_antisymm hφ (le_trans _ (map_mono hψ)),
rw [map_map, eq, map_id],
exact le_refl _
end
lemma tendsto_inf {f : α → β} {x : filter α} {y₁ y₂ : filter β} :
tendsto f x (y₁ ⊓ y₂) ↔ tendsto f x y₁ ∧ tendsto f x y₂ :=
by simp only [tendsto, lattice.le_inf_iff, iff_self]
lemma tendsto_inf_left {f : α → β} {x₁ x₂ : filter α} {y : filter β}
(h : tendsto f x₁ y) : tendsto f (x₁ ⊓ x₂) y :=
le_trans (map_mono inf_le_left) h
lemma tendsto_inf_right {f : α → β} {x₁ x₂ : filter α} {y : filter β}
(h : tendsto f x₂ y) : tendsto f (x₁ ⊓ x₂) y :=
le_trans (map_mono inf_le_right) h
lemma tendsto_infi {f : α → β} {x : filter α} {y : ι → filter β} :
tendsto f x (⨅i, y i) ↔ ∀i, tendsto f x (y i) :=
by simp only [tendsto, iff_self, lattice.le_infi_iff]
lemma tendsto_infi' {f : α → β} {x : ι → filter α} {y : filter β} (i : ι) :
tendsto f (x i) y → tendsto f (⨅i, x i) y :=
tendsto_le_left (infi_le _ _)
lemma tendsto_principal {f : α → β} {a : filter α} {s : set β} :
tendsto f a (principal s) ↔ {a | f a ∈ s} ∈ a :=
by simp only [tendsto, le_principal_iff, mem_map, iff_self]
lemma tendsto_principal_principal {f : α → β} {s : set α} {t : set β} :
tendsto f (principal s) (principal t) ↔ ∀a∈s, f a ∈ t :=
by simp only [tendsto, image_subset_iff, le_principal_iff, map_principal, mem_principal_sets]; refl
lemma tendsto_pure_pure (f : α → β) (a : α) :
tendsto f (pure a) (pure (f a)) :=
show filter.map f (pure a) ≤ pure (f a),
by rw [filter.map_pure]; exact le_refl _
lemma tendsto_const_pure {a : filter α} {b : β} : tendsto (λa, b) a (pure b) :=
by simp [tendsto]; exact univ_mem_sets
lemma tendsto_if {l₁ : filter α} {l₂ : filter β}
{f g : α → β} {p : α → Prop} [decidable_pred p]
(h₀ : tendsto f (l₁ ⊓ principal p) l₂)
(h₁ : tendsto g (l₁ ⊓ principal { x | ¬ p x }) l₂) :
tendsto (λ x, if p x then f x else g x) l₁ l₂ :=
begin
revert h₀ h₁, simp only [tendsto_def, mem_inf_principal],
intros h₀ h₁ s hs,
apply mem_sets_of_superset (inter_mem_sets (h₀ s hs) (h₁ s hs)),
rintros x ⟨hp₀, hp₁⟩, simp only [mem_preimage],
by_cases h : p x,
{ rw if_pos h, exact hp₀ h },
rw if_neg h, exact hp₁ h
end
section prod
variables {s : set α} {t : set β} {f : filter α} {g : filter β}
/- The product filter cannot be defined using the monad structure on filters. For example:
F := do {x ← seq, y ← top, return (x, y)}
hence:
s ∈ F ↔ ∃n, [n..∞] × univ ⊆ s
G := do {y ← top, x ← seq, return (x, y)}
hence:
s ∈ G ↔ ∀i:ℕ, ∃n, [n..∞] × {i} ⊆ s
Now ⋃i, [i..∞] × {i} is in G but not in F.
As product filter we want to have F as result.
-/
/-- Product of filters. This is the filter generated by cartesian products
of elements of the component filters. -/
protected def prod (f : filter α) (g : filter β) : filter (α × β) :=
f.comap prod.fst ⊓ g.comap prod.snd
lemma prod_mem_prod {s : set α} {t : set β} {f : filter α} {g : filter β}
(hs : s ∈ f) (ht : t ∈ g) : set.prod s t ∈ filter.prod f g :=
inter_mem_inf_sets (preimage_mem_comap hs) (preimage_mem_comap ht)
lemma mem_prod_iff {s : set (α×β)} {f : filter α} {g : filter β} :
s ∈ filter.prod f g ↔ (∃ t₁ ∈ f, ∃ t₂ ∈ g, set.prod t₁ t₂ ⊆ s) :=
begin
simp only [filter.prod],
split,
exact assume ⟨t₁, ⟨s₁, hs₁, hts₁⟩, t₂, ⟨s₂, hs₂, hts₂⟩, h⟩,
⟨s₁, hs₁, s₂, hs₂, subset.trans (inter_subset_inter hts₁ hts₂) h⟩,
exact assume ⟨t₁, ht₁, t₂, ht₂, h⟩,
⟨prod.fst ⁻¹' t₁, ⟨t₁, ht₁, subset.refl _⟩, prod.snd ⁻¹' t₂, ⟨t₂, ht₂, subset.refl _⟩, h⟩
end
lemma tendsto_fst {f : filter α} {g : filter β} : tendsto prod.fst (filter.prod f g) f :=
tendsto_inf_left tendsto_comap
lemma tendsto_snd {f : filter α} {g : filter β} : tendsto prod.snd (filter.prod f g) g :=
tendsto_inf_right tendsto_comap
lemma tendsto.prod_mk {f : filter α} {g : filter β} {h : filter γ} {m₁ : α → β} {m₂ : α → γ}
(h₁ : tendsto m₁ f g) (h₂ : tendsto m₂ f h) : tendsto (λx, (m₁ x, m₂ x)) f (filter.prod g h) :=
tendsto_inf.2 ⟨tendsto_comap_iff.2 h₁, tendsto_comap_iff.2 h₂⟩
lemma prod_infi_left {f : ι → filter α} {g : filter β} (i : ι) :
filter.prod (⨅i, f i) g = (⨅i, filter.prod (f i) g) :=
by rw [filter.prod, comap_infi, infi_inf i]; simp only [filter.prod, eq_self_iff_true]
lemma prod_infi_right {f : filter α} {g : ι → filter β} (i : ι) :
filter.prod f (⨅i, g i) = (⨅i, filter.prod f (g i)) :=
by rw [filter.prod, comap_infi, inf_infi i]; simp only [filter.prod, eq_self_iff_true]
lemma prod_mono {f₁ f₂ : filter α} {g₁ g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) :
filter.prod f₁ g₁ ≤ filter.prod f₂ g₂ :=
inf_le_inf (comap_mono hf) (comap_mono hg)
lemma prod_comap_comap_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
{f₁ : filter α₁} {f₂ : filter α₂} {m₁ : β₁ → α₁} {m₂ : β₂ → α₂} :
filter.prod (comap m₁ f₁) (comap m₂ f₂) = comap (λp:β₁×β₂, (m₁ p.1, m₂ p.2)) (filter.prod f₁ f₂) :=
by simp only [filter.prod, comap_comap_comp, eq_self_iff_true, comap_inf]
lemma prod_comm' : filter.prod f g = comap (prod.swap) (filter.prod g f) :=
by simp only [filter.prod, comap_comap_comp, (∘), inf_comm, prod.fst_swap,
eq_self_iff_true, prod.snd_swap, comap_inf]
lemma prod_comm : filter.prod f g = map (λp:β×α, (p.2, p.1)) (filter.prod g f) :=
by rw [prod_comm', ← map_swap_eq_comap_swap]; refl
lemma prod_map_map_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
{f₁ : filter α₁} {f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} :
filter.prod (map m₁ f₁) (map m₂ f₂) = map (λp:α₁×α₂, (m₁ p.1, m₂ p.2)) (filter.prod f₁ f₂) :=
le_antisymm
(assume s hs,
let ⟨s₁, hs₁, s₂, hs₂, h⟩ := mem_prod_iff.mp hs in
filter.sets_of_superset _ (prod_mem_prod (image_mem_map hs₁) (image_mem_map hs₂)) $
calc set.prod (m₁ '' s₁) (m₂ '' s₂) = (λp:α₁×α₂, (m₁ p.1, m₂ p.2)) '' set.prod s₁ s₂ :
set.prod_image_image_eq
... ⊆ _ : by rwa [image_subset_iff])
((tendsto.comp (le_refl _) tendsto_fst).prod_mk (tendsto.comp (le_refl _) tendsto_snd))
lemma map_prod (m : α × β → γ) (f : filter α) (g : filter β) :
map m (f.prod g) = (f.map (λa b, m (a, b))).seq g :=
begin
simp [filter.ext_iff, mem_prod_iff, mem_map_seq_iff],
assume s,
split,
exact assume ⟨t, ht, s, hs, h⟩, ⟨s, hs, t, ht, assume x hx y hy, @h ⟨x, y⟩ ⟨hx, hy⟩⟩,
exact assume ⟨s, hs, t, ht, h⟩, ⟨t, ht, s, hs, assume ⟨x, y⟩ ⟨hx, hy⟩, h x hx y hy⟩
end
lemma prod_eq {f : filter α} {g : filter β} : f.prod g = (f.map prod.mk).seq g :=
have h : _ := map_prod id f g, by rwa [map_id] at h
lemma prod_inf_prod {f₁ f₂ : filter α} {g₁ g₂ : filter β} :
filter.prod f₁ g₁ ⊓ filter.prod f₂ g₂ = filter.prod (f₁ ⊓ f₂) (g₁ ⊓ g₂) :=
by simp only [filter.prod, comap_inf, inf_comm, inf_assoc, lattice.inf_left_comm]
@[simp] lemma prod_bot {f : filter α} : filter.prod f (⊥ : filter β) = ⊥ := by simp [filter.prod]
@[simp] lemma bot_prod {g : filter β} : filter.prod (⊥ : filter α) g = ⊥ := by simp [filter.prod]
@[simp] lemma prod_principal_principal {s : set α} {t : set β} :
filter.prod (principal s) (principal t) = principal (set.prod s t) :=
by simp only [filter.prod, comap_principal, principal_eq_iff_eq, comap_principal, inf_principal]; refl
@[simp] lemma prod_pure_pure {a : α} {b : β} : filter.prod (pure a) (pure b) = pure (a, b) :=
by simp
lemma prod_eq_bot {f : filter α} {g : filter β} : filter.prod f g = ⊥ ↔ (f = ⊥ ∨ g = ⊥) :=
begin
split,
{ assume h,
rcases mem_prod_iff.1 (empty_in_sets_eq_bot.2 h) with ⟨s, hs, t, ht, hst⟩,
rw [subset_empty_iff, set.prod_eq_empty_iff] at hst,
cases hst with s_eq t_eq,
{ left, exact empty_in_sets_eq_bot.1 (s_eq ▸ hs) },
{ right, exact empty_in_sets_eq_bot.1 (t_eq ▸ ht) } },
{ rintros (rfl | rfl),
exact bot_prod,
exact prod_bot }
end
lemma prod_neq_bot {f : filter α} {g : filter β} : filter.prod f g ≠ ⊥ ↔ (f ≠ ⊥ ∧ g ≠ ⊥) :=
by rw [(≠), prod_eq_bot, not_or_distrib]
lemma tendsto_prod_iff {f : α × β → γ} {x : filter α} {y : filter β} {z : filter γ} :
filter.tendsto f (filter.prod x y) z ↔
∀ W ∈ z, ∃ U ∈ x, ∃ V ∈ y, ∀ x y, x ∈ U → y ∈ V → f (x, y) ∈ W :=
by simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff, exists_prop, iff_self]
end prod
/- at_top and at_bot -/
/-- `at_top` is the filter representing the limit `→ ∞` on an ordered set.
It is generated by the collection of up-sets `{b | a ≤ b}`.
(The preorder need not have a top element for this to be well defined,
and indeed is trivial when a top element exists.) -/
def at_top [preorder α] : filter α := ⨅ a, principal {b | a ≤ b}
/-- `at_bot` is the filter representing the limit `→ -∞` on an ordered set.
It is generated by the collection of down-sets `{b | b ≤ a}`.
(The preorder need not have a bottom element for this to be well defined,
and indeed is trivial when a bottom element exists.) -/
def at_bot [preorder α] : filter α := ⨅ a, principal {b | b ≤ a}
lemma mem_at_top [preorder α] (a : α) : {b : α | a ≤ b} ∈ @at_top α _ :=
mem_infi_sets a $ subset.refl _
@[simp] lemma at_top_ne_bot [nonempty α] [semilattice_sup α] : (at_top : filter α) ≠ ⊥ :=
infi_neq_bot_of_directed (by apply_instance)
(assume a b, ⟨a ⊔ b, by simp only [ge, le_principal_iff, forall_const, set_of_subset_set_of,
mem_principal_sets, and_self, sup_le_iff, forall_true_iff] {contextual := tt}⟩)
(assume a, by simp only [principal_eq_bot_iff, ne.def, principal_eq_bot_iff]; exact ne_empty_of_mem (le_refl a))
@[simp] lemma mem_at_top_sets [nonempty α] [semilattice_sup α] {s : set α} :
s ∈ (at_top : filter α) ↔ ∃a:α, ∀b≥a, b ∈ s :=
let ⟨a⟩ := ‹nonempty α› in
iff.intro
(assume h, infi_sets_induct h ⟨a, by simp only [forall_const, mem_univ, forall_true_iff]⟩
(assume a s₁ s₂ ha ⟨b, hb⟩, ⟨a ⊔ b,
assume c hc, ⟨ha $ le_trans le_sup_left hc, hb _ $ le_trans le_sup_right hc⟩⟩)
(assume s₁ s₂ h ⟨a, ha⟩, ⟨a, assume b hb, h $ ha _ hb⟩))
(assume ⟨a, h⟩, mem_infi_sets a $ assume x, h x)
lemma map_at_top_eq [nonempty α] [semilattice_sup α] {f : α → β} :
at_top.map f = (⨅a, principal $ f '' {a' | a ≤ a'}) :=
calc map f (⨅a, principal {a' | a ≤ a'}) = (⨅a, map f $ principal {a' | a ≤ a'}) :
map_infi_eq (assume a b, ⟨a ⊔ b, by simp only [ge, le_principal_iff, forall_const, set_of_subset_set_of,
mem_principal_sets, and_self, sup_le_iff, forall_true_iff] {contextual := tt}⟩)
(by apply_instance)
... = (⨅a, principal $ f '' {a' | a ≤ a'}) : by simp only [map_principal, eq_self_iff_true]
lemma tendsto_at_top [preorder β] (m : α → β) (f : filter α) :
tendsto m f at_top ↔ (∀b, {a | b ≤ m a} ∈ f) :=
by simp only [at_top, tendsto_infi, tendsto_principal]; refl
lemma tendsto_at_top' [nonempty α] [semilattice_sup α] (f : α → β) (l : filter β) :
tendsto f at_top l ↔ (∀s ∈ l, ∃a, ∀b≥a, f b ∈ s) :=
by simp only [tendsto_def, mem_at_top_sets]; refl
theorem tendsto_at_top_principal [nonempty β] [semilattice_sup β] {f : β → α} {s : set α} :
tendsto f at_top (principal s) ↔ ∃N, ∀n≥N, f n ∈ s :=
by rw [tendsto_iff_comap, comap_principal, le_principal_iff, mem_at_top_sets]; refl
/-- A function `f` grows to infinity independent of an order-preserving embedding `e`. -/
lemma tendsto_at_top_embedding {α β γ : Type*} [preorder β] [preorder γ]
{f : α → β} {e : β → γ} {l : filter α}
(hm : ∀b₁ b₂, e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀c, ∃b, c ≤ e b) :
tendsto (e ∘ f) l at_top ↔ tendsto f l at_top :=
begin
rw [tendsto_at_top, tendsto_at_top],
split,
{ assume hc b,
filter_upwards [hc (e b)] assume a, (hm b (f a)).1 },
{ assume hb c,
rcases hu c with ⟨b, hc⟩,
filter_upwards [hb b] assume a ha, le_trans hc ((hm b (f a)).2 ha) }
end
lemma tendsto_at_top_at_top [nonempty α] [semilattice_sup α] [preorder β] (f : α → β) :
tendsto f at_top at_top ↔ ∀ b : β, ∃ i : α, ∀ a : α, i ≤ a → b ≤ f a :=
iff.trans tendsto_infi $ forall_congr $ assume b, tendsto_at_top_principal
lemma tendsto_at_top_at_bot [nonempty α] [decidable_linear_order α] [preorder β] (f : α → β) :
tendsto f at_top at_bot ↔ ∀ (b : β), ∃ (i : α), ∀ (a : α), i ≤ a → b ≥ f a :=
@tendsto_at_top_at_top α (order_dual β) _ _ _ f
lemma tendsto_finset_image_at_top_at_top {i : β → γ} {j : γ → β} (h : ∀x, j (i x) = x) :
tendsto (λs:finset γ, s.image j) at_top at_top :=
tendsto_infi.2 $ assume s, tendsto_infi' (s.image i) $ tendsto_principal_principal.2 $
assume t (ht : s.image i ⊆ t),
calc s = (s.image i).image j :
by simp only [finset.image_image, (∘), h]; exact finset.image_id.symm
... ⊆ t.image j : finset.image_subset_image ht
lemma prod_at_top_at_top_eq {β₁ β₂ : Type*} [inhabited β₁] [inhabited β₂] [semilattice_sup β₁]
[semilattice_sup β₂] : filter.prod (@at_top β₁ _) (@at_top β₂ _) = @at_top (β₁ × β₂) _ :=
by simp [at_top, prod_infi_left (default β₁), prod_infi_right (default β₂), infi_prod];
exact infi_comm
lemma prod_map_at_top_eq {α₁ α₂ β₁ β₂ : Type*} [inhabited β₁] [inhabited β₂]
[semilattice_sup β₁] [semilattice_sup β₂] (u₁ : β₁ → α₁) (u₂ : β₂ → α₂) :
filter.prod (map u₁ at_top) (map u₂ at_top) = map (prod.map u₁ u₂) at_top :=
by rw [prod_map_map_eq, prod_at_top_at_top_eq, prod.map_def]
/-- A function `f` maps upwards closed sets (at_top sets) to upwards closed sets when it is a
Galois insertion. The Galois "insertion" and "connection" is weakened to only require it to be an
insertion and a connetion above `b'`. -/
lemma map_at_top_eq_of_gc [semilattice_sup α] [semilattice_sup β] {f : α → β} (g : β → α) (b' : β)(hf : monotone f) (gc : ∀a, ∀b≥b', f a ≤ b ↔ a ≤ g b) (hgi : ∀b≥b', b ≤ f (g b)) :
map f at_top = at_top :=
begin
rw [@map_at_top_eq α _ ⟨g b'⟩],
refine le_antisymm
(le_infi $ assume b, infi_le_of_le (g (b ⊔ b')) $ principal_mono.2 $ image_subset_iff.2 _)
(le_infi $ assume a, infi_le_of_le (f a ⊔ b') $ principal_mono.2 _),
{ assume a ha, exact (le_trans le_sup_left $ le_trans (hgi _ le_sup_right) $ hf ha) },
{ assume b hb,
have hb' : b' ≤ b := le_trans le_sup_right hb,
exact ⟨g b, (gc _ _ hb').1 (le_trans le_sup_left hb),
le_antisymm ((gc _ _ hb').2 (le_refl _)) (hgi _ hb')⟩ }
end
lemma map_add_at_top_eq_nat (k : ℕ) : map (λa, a + k) at_top = at_top :=
map_at_top_eq_of_gc (λa, a - k) k
(assume a b h, add_le_add_right h k)
(assume a b h, (nat.le_sub_right_iff_add_le h).symm)
(assume a h, by rw [nat.sub_add_cancel h])
lemma map_sub_at_top_eq_nat (k : ℕ) : map (λa, a - k) at_top = at_top :=
map_at_top_eq_of_gc (λa, a + k) 0
(assume a b h, nat.sub_le_sub_right h _)
(assume a b _, nat.sub_le_right_iff_le_add)
(assume b _, by rw [nat.add_sub_cancel])
lemma tendso_add_at_top_nat (k : ℕ) : tendsto (λa, a + k) at_top at_top :=
le_of_eq (map_add_at_top_eq_nat k)
lemma tendso_sub_at_top_nat (k : ℕ) : tendsto (λa, a - k) at_top at_top :=
le_of_eq (map_sub_at_top_eq_nat k)
lemma tendsto_add_at_top_iff_nat {f : ℕ → α} {l : filter α} (k : ℕ) :
tendsto (λn, f (n + k)) at_top l ↔ tendsto f at_top l :=
show tendsto (f ∘ (λn, n + k)) at_top l ↔ tendsto f at_top l,
by rw [← tendsto_map'_iff, map_add_at_top_eq_nat]
lemma map_div_at_top_eq_nat (k : ℕ) (hk : k > 0) : map (λa, a / k) at_top = at_top :=
map_at_top_eq_of_gc (λb, b * k + (k - 1)) 1
(assume a b h, nat.div_le_div_right h)
(assume a b _,
calc a / k ≤ b ↔ a / k < b + 1 : by rw [← nat.succ_eq_add_one, nat.lt_succ_iff]
... ↔ a < (b + 1) * k : nat.div_lt_iff_lt_mul _ _ hk
... ↔ _ :
begin
cases k,
exact (lt_irrefl _ hk).elim,
simp [mul_add, add_mul, nat.succ_add, nat.lt_succ_iff]
end)
(assume b _,
calc b = (b * k) / k : by rw [nat.mul_div_cancel b hk]
... ≤ (b * k + (k - 1)) / k : nat.div_le_div_right $ nat.le_add_right _ _)
/- ultrafilter -/
section ultrafilter
open zorn
variables {f g : filter α}
/-- An ultrafilter is a minimal (maximal in the set order) proper filter. -/
def is_ultrafilter (f : filter α) := f ≠ ⊥ ∧ ∀g, g ≠ ⊥ → g ≤ f → f ≤ g
lemma ultrafilter_unique (hg : is_ultrafilter g) (hf : f ≠ ⊥) (h : f ≤ g) : f = g :=
le_antisymm h (hg.right _ hf h)
lemma le_of_ultrafilter {g : filter α} (hf : is_ultrafilter f) (h : f ⊓ g ≠ ⊥) :
f ≤ g :=
le_of_inf_eq $ ultrafilter_unique hf h inf_le_left
/-- Equivalent characterization of ultrafilters:
A filter f is an ultrafilter if and only if for each set s,
-s belongs to f if and only if s does not belong to f. -/
lemma ultrafilter_iff_compl_mem_iff_not_mem :
is_ultrafilter f ↔ (∀ s, -s ∈ f ↔ s ∉ f) :=
⟨assume hf s,
⟨assume hns hs,
hf.1 $ empty_in_sets_eq_bot.mp $ by convert f.inter_sets hs hns; rw [inter_compl_self],
assume hs,
have f ≤ principal (-s), from
le_of_ultrafilter hf $ assume h, hs $ mem_sets_of_neq_bot $
by simp only [h, eq_self_iff_true, lattice.neg_neg],
by simp only [le_principal_iff] at this; assumption⟩,
assume hf,
⟨mt empty_in_sets_eq_bot.mpr ((hf ∅).mp (by convert f.univ_sets; rw [compl_empty])),
assume g hg g_le s hs, classical.by_contradiction $ mt (hf s).mpr $
assume : - s ∈ f,
have s ∩ -s ∈ g, from inter_mem_sets hs (g_le this),
by simp only [empty_in_sets_eq_bot, hg, inter_compl_self] at this; contradiction⟩⟩
lemma mem_or_compl_mem_of_ultrafilter (hf : is_ultrafilter f) (s : set α) :
s ∈ f ∨ - s ∈ f :=
classical.or_iff_not_imp_left.2 (ultrafilter_iff_compl_mem_iff_not_mem.mp hf s).mpr
lemma mem_or_mem_of_ultrafilter {s t : set α} (hf : is_ultrafilter f) (h : s ∪ t ∈ f) :
s ∈ f ∨ t ∈ f :=
(mem_or_compl_mem_of_ultrafilter hf s).imp_right
(assume : -s ∈ f, by filter_upwards [this, h] assume x hnx hx, hx.resolve_left hnx)
lemma mem_of_finite_sUnion_ultrafilter {s : set (set α)} (hf : is_ultrafilter f) (hs : finite s)
: ⋃₀ s ∈ f → ∃t∈s, t ∈ f :=
finite.induction_on hs (by simp only [empty_in_sets_eq_bot, hf.left, mem_empty_eq, sUnion_empty,
forall_prop_of_false, exists_false, not_false_iff, exists_prop_of_false]) $
λ t s' ht' hs' ih, by simp only [exists_prop, mem_insert_iff, set.sUnion_insert]; exact
assume h, (mem_or_mem_of_ultrafilter hf h).elim
(assume : t ∈ f, ⟨t, or.inl rfl, this⟩)
(assume h, let ⟨t, hts', ht⟩ := ih h in ⟨t, or.inr hts', ht⟩)
lemma mem_of_finite_Union_ultrafilter {is : set β} {s : β → set α}
(hf : is_ultrafilter f) (his : finite is) (h : (⋃i∈is, s i) ∈ f) : ∃i∈is, s i ∈ f :=
have his : finite (image s is), from finite_image s his,
have h : (⋃₀ image s is) ∈ f, from by simp only [sUnion_image, set.sUnion_image]; assumption,
let ⟨t, ⟨i, hi, h_eq⟩, (ht : t ∈ f)⟩ := mem_of_finite_sUnion_ultrafilter hf his h in
⟨i, hi, h_eq.symm ▸ ht⟩
lemma ultrafilter_map {f : filter α} {m : α → β} (h : is_ultrafilter f) : is_ultrafilter (map m f) :=
by rw ultrafilter_iff_compl_mem_iff_not_mem at ⊢ h; exact assume s, h (m ⁻¹' s)
lemma ultrafilter_pure {a : α} : is_ultrafilter (pure a) :=
begin
rw ultrafilter_iff_compl_mem_iff_not_mem, intro s,
rw [mem_pure_sets, mem_pure_sets], exact iff.rfl
end
lemma ultrafilter_bind {f : filter α} (hf : is_ultrafilter f) {m : α → filter β}
(hm : ∀ a, is_ultrafilter (m a)) : is_ultrafilter (f.bind m) :=
begin
simp only [ultrafilter_iff_compl_mem_iff_not_mem] at ⊢ hf hm, intro s,
dsimp [bind, join, map, preimage],
simp only [hm], apply hf
end
/-- The ultrafilter lemma: Any proper filter is contained in an ultrafilter. -/
lemma exists_ultrafilter (h : f ≠ ⊥) : ∃u, u ≤ f ∧ is_ultrafilter u :=
let
τ := {f' // f' ≠ ⊥ ∧ f' ≤ f},
r : τ → τ → Prop := λt₁ t₂, t₂.val ≤ t₁.val,
⟨a, ha⟩ := inhabited_of_mem_sets h univ_mem_sets,
top : τ := ⟨f, h, le_refl f⟩,
sup : Π(c:set τ), chain r c → τ :=
λc hc, ⟨⨅a:{a:τ // a ∈ insert top c}, a.val.val,
infi_neq_bot_of_directed ⟨a⟩
(directed_of_chain $ chain_insert hc $ assume ⟨b, _, hb⟩ _ _, or.inl hb)
(assume ⟨⟨a, ha, _⟩, _⟩, ha),
infi_le_of_le ⟨top, mem_insert _ _⟩ (le_refl _)⟩
in
have ∀c (hc: chain r c) a (ha : a ∈ c), r a (sup c hc),
from assume c hc a ha, infi_le_of_le ⟨a, mem_insert_of_mem _ ha⟩ (le_refl _),
have (∃ (u : τ), ∀ (a : τ), r u a → r a u),
from zorn (assume c hc, ⟨sup c hc, this c hc⟩) (assume f₁ f₂ f₃ h₁ h₂, le_trans h₂ h₁),
let ⟨uτ, hmin⟩ := this in
⟨uτ.val, uτ.property.right, uτ.property.left, assume g hg₁ hg₂,
hmin ⟨g, hg₁, le_trans hg₂ uτ.property.right⟩ hg₂⟩
/-- Construct an ultrafilter extending a given filter.
The ultrafilter lemma is the assertion that such a filter exists;
we use the axiom of choice to pick one. -/
noncomputable def ultrafilter_of (f : filter α) : filter α :=
if h : f = ⊥ then ⊥ else classical.epsilon (λu, u ≤ f ∧ is_ultrafilter u)
lemma ultrafilter_of_spec (h : f ≠ ⊥) : ultrafilter_of f ≤ f ∧ is_ultrafilter (ultrafilter_of f) :=
begin
have h' := classical.epsilon_spec (exists_ultrafilter h),
simp only [ultrafilter_of, dif_neg, h, dif_neg, not_false_iff],
simp only at h',
assumption
end
lemma ultrafilter_of_le : ultrafilter_of f ≤ f :=
if h : f = ⊥ then by simp only [ultrafilter_of, dif_pos, h, dif_pos, eq_self_iff_true, le_bot_iff]; exact le_refl _
else (ultrafilter_of_spec h).left
lemma ultrafilter_ultrafilter_of (h : f ≠ ⊥) : is_ultrafilter (ultrafilter_of f) :=
(ultrafilter_of_spec h).right
lemma ultrafilter_of_ultrafilter (h : is_ultrafilter f) : ultrafilter_of f = f :=
ultrafilter_unique h (ultrafilter_ultrafilter_of h.left).left ultrafilter_of_le
/-- A filter equals the intersection of all the ultrafilters which contain it. -/
lemma sup_of_ultrafilters (f : filter α) : f = ⨆ (g) (u : is_ultrafilter g) (H : g ≤ f), g :=
begin
refine le_antisymm _ (supr_le $ λ g, supr_le $ λ u, supr_le $ λ H, H),
intros s hs,
-- If s ∉ f.sets, we'll apply the ultrafilter lemma to the restriction of f to -s.
by_contradiction hs',
let j : (-s) → α := subtype.val,
have j_inv_s : j ⁻¹' s = ∅, by
erw [←preimage_inter_range, subtype.val_range, inter_compl_self, preimage_empty],
let f' := comap j f,
have : f' ≠ ⊥,
{ apply mt empty_in_sets_eq_bot.mpr,
rintro ⟨t, htf, ht⟩,
suffices : t ⊆ s, from absurd (f.sets_of_superset htf this) hs',
rw [subset_empty_iff] at ht,
have : j '' (j ⁻¹' t) = ∅, by rw [ht, image_empty],
erw [image_preimage_eq_inter_range, subtype.val_range, ←subset_compl_iff_disjoint,
set.compl_compl] at this,
exact this },
rcases exists_ultrafilter this with ⟨g', g'f', u'⟩,
simp only [supr_sets_eq, mem_Inter] at hs,
have := hs (g'.map subtype.val) (ultrafilter_map u') (map_le_iff_le_comap.mpr g'f'),
rw [←le_principal_iff, map_le_iff_le_comap, comap_principal, j_inv_s, principal_empty,
le_bot_iff] at this,
exact absurd this u'.1
end
/-- The `tendsto` relation can be checked on ultrafilters. -/
lemma tendsto_iff_ultrafilter (f : α → β) (l₁ : filter α) (l₂ : filter β) :
tendsto f l₁ l₂ ↔ ∀ g, is_ultrafilter g → g ≤ l₁ → g.map f ≤ l₂ :=
⟨assume h g u gx, le_trans (map_mono gx) h,
assume h, by rw [sup_of_ultrafilters l₁]; simpa only [tendsto, map_supr, supr_le_iff]⟩
/- The ultrafilter monad. The monad structure on ultrafilters is the
restriction of the one on filters. -/
def ultrafilter (α : Type u) : Type u := {f : filter α // is_ultrafilter f}
def ultrafilter.map (m : α → β) (u : ultrafilter α) : ultrafilter β :=
⟨u.val.map m, ultrafilter_map u.property⟩
def ultrafilter.pure (x : α) : ultrafilter α := ⟨pure x, ultrafilter_pure⟩
def ultrafilter.bind (u : ultrafilter α) (m : α → ultrafilter β) : ultrafilter β :=
⟨u.val.bind (λ a, (m a).val), ultrafilter_bind u.property (λ a, (m a).property)⟩
instance ultrafilter.has_pure : has_pure ultrafilter := ⟨@ultrafilter.pure⟩
instance ultrafilter.has_bind : has_bind ultrafilter := ⟨@ultrafilter.bind⟩
instance ultrafilter.functor : functor ultrafilter := { map := @ultrafilter.map }
instance ultrafilter.monad : monad ultrafilter := { map := @ultrafilter.map }
noncomputable def hyperfilter : filter α := ultrafilter_of cofinite
lemma hyperfilter_le_cofinite (hi : set.infinite (@set.univ α)) : @hyperfilter α ≤ cofinite :=
(ultrafilter_of_spec (cofinite_ne_bot hi)).1
lemma is_ultrafilter_hyperfilter (hi : set.infinite (@set.univ α)) : is_ultrafilter (@hyperfilter α) :=
(ultrafilter_of_spec (cofinite_ne_bot hi)).2
theorem nmem_hyperfilter_of_finite (hi : set.infinite (@set.univ α)) {s : set α} (hf : set.finite s) :
s ∉ @hyperfilter α :=
λ hy,
have hx : -s ∉ hyperfilter :=
λ hs, (ultrafilter_iff_compl_mem_iff_not_mem.mp (is_ultrafilter_hyperfilter hi) s).mp hs hy,
have ht : -s ∈ cofinite.sets := by show -s ∈ {s | _}; rwa [set.mem_set_of_eq, lattice.neg_neg],
hx $ hyperfilter_le_cofinite hi ht
theorem compl_mem_hyperfilter_of_finite (hi : set.infinite (@set.univ α)) {s : set α} (hf : set.finite s) :
-s ∈ @hyperfilter α :=
(ultrafilter_iff_compl_mem_iff_not_mem.mp (is_ultrafilter_hyperfilter hi) s).mpr $
nmem_hyperfilter_of_finite hi hf
theorem mem_hyperfilter_of_finite_compl (hi : set.infinite (@set.univ α)) {s : set α} (hf : set.finite (-s)) :
s ∈ @hyperfilter α :=
have h : _ := compl_mem_hyperfilter_of_finite hi hf,
by rwa [lattice.neg_neg] at h
section
local attribute [instance] filter.monad filter.is_lawful_monad
instance ultrafilter.is_lawful_monad : is_lawful_monad ultrafilter :=
{ id_map := assume α f, subtype.eq (id_map f.val),
pure_bind := assume α β a f, subtype.eq (pure_bind a (subtype.val ∘ f)),
bind_assoc := assume α β γ f m₁ m₂, subtype.eq (filter_eq rfl),
bind_pure_comp_eq_map := assume α β f x, subtype.eq (bind_pure_comp_eq_map _ f x.val) }
end
lemma ultrafilter.eq_iff_val_le_val {u v : ultrafilter α} : u = v ↔ u.val ≤ v.val :=
⟨assume h, by rw h; exact le_refl _,
assume h, by rw subtype.ext; apply ultrafilter_unique v.property u.property.1 h⟩
lemma exists_ultrafilter_iff (f : filter α) : (∃ (u : ultrafilter α), u.val ≤ f) ↔ f ≠ ⊥ :=
⟨assume ⟨u, uf⟩, lattice.neq_bot_of_le_neq_bot u.property.1 uf,
assume h, let ⟨u, uf, hu⟩ := exists_ultrafilter h in ⟨⟨u, hu⟩, uf⟩⟩
end ultrafilter
end filter
namespace filter
variables {α β γ : Type u} {f : β → filter α} {s : γ → set α}
open list
lemma mem_traverse_sets :
∀(fs : list β) (us : list γ),
forall₂ (λb c, s c ∈ f b) fs us → traverse s us ∈ traverse f fs
| [] [] forall₂.nil := mem_pure_sets.2 $ mem_singleton _
| (f::fs) (u::us) (forall₂.cons h hs) := seq_mem_seq_sets (image_mem_map h) (mem_traverse_sets fs us hs)
lemma mem_traverse_sets_iff (fs : list β) (t : set (list α)) :
t ∈ traverse f fs ↔
(∃us:list (set α), forall₂ (λb (s : set α), s ∈ f b) fs us ∧ sequence us ⊆ t) :=
begin
split,
{ induction fs generalizing t,
case nil { simp only [sequence, pure_def, imp_self, forall₂_nil_left_iff, pure_def,
exists_eq_left, mem_principal_sets, set.pure_def, singleton_subset_iff, traverse_nil] },
case cons : b fs ih t {
assume ht,
rcases mem_seq_sets_iff.1 ht with ⟨u, hu, v, hv, ht⟩,
rcases mem_map_sets_iff.1 hu with ⟨w, hw, hwu⟩,
rcases ih v hv with ⟨us, hus, hu⟩,
exact ⟨w :: us, forall₂.cons hw hus, subset.trans (set.seq_mono hwu hu) ht⟩ } },
{ rintros ⟨us, hus, hs⟩,
exact mem_sets_of_superset (mem_traverse_sets _ _ hus) hs }
end
lemma sequence_mono :
∀(as bs : list (filter α)), forall₂ (≤) as bs → sequence as ≤ sequence bs
| [] [] forall₂.nil := le_refl _
| (a::as) (b::bs) (forall₂.cons h hs) := seq_mono (map_mono h) (sequence_mono as bs hs)
end filter
|
57c3b7c85eb2deac807afe9ee7f4b862c54119fa | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/topology/algebra/open_subgroup.lean | bb53777ebb19e5c1f3769a08c686294be37e035f | [
"Apache-2.0"
] | permissive | dexmagic/mathlib | ff48eefc56e2412429b31d4fddd41a976eb287ce | 7a5d15a955a92a90e1d398b2281916b9c41270b2 | refs/heads/master | 1,693,481,322,046 | 1,633,360,193,000 | 1,633,360,193,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 8,437 | lean | /-
Copyright (c) 2019 Johan Commelin All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import order.filter.lift
import topology.opens
import topology.algebra.ring
open topological_space
open_locale topological_space
set_option old_structure_cmd true
/-- The type of open subgroups of a topological additive group. -/
@[ancestor add_subgroup]
structure open_add_subgroup (G : Type*) [add_group G] [topological_space G]
extends add_subgroup G :=
(is_open' : is_open carrier)
/-- The type of open subgroups of a topological group. -/
@[ancestor subgroup, to_additive]
structure open_subgroup (G : Type*) [group G] [topological_space G] extends subgroup G :=
(is_open' : is_open carrier)
/-- Reinterpret an `open_subgroup` as a `subgroup`. -/
add_decl_doc open_subgroup.to_subgroup
/-- Reinterpret an `open_add_subgroup` as an `add_subgroup`. -/
add_decl_doc open_add_subgroup.to_add_subgroup
-- Tell Lean that `open_add_subgroup` is a namespace
namespace open_add_subgroup
end open_add_subgroup
namespace open_subgroup
open function topological_space
variables {G : Type*} [group G] [topological_space G]
variables {U V : open_subgroup G} {g : G}
@[to_additive]
instance has_coe_set : has_coe_t (open_subgroup G) (set G) := ⟨λ U, U.1⟩
@[to_additive]
instance : has_mem G (open_subgroup G) := ⟨λ g U, g ∈ (U : set G)⟩
@[to_additive]
instance has_coe_subgroup : has_coe_t (open_subgroup G) (subgroup G) := ⟨to_subgroup⟩
@[to_additive]
instance has_coe_opens : has_coe_t (open_subgroup G) (opens G) := ⟨λ U, ⟨U, U.is_open'⟩⟩
@[simp, to_additive] lemma mem_coe : g ∈ (U : set G) ↔ g ∈ U := iff.rfl
@[simp, to_additive] lemma mem_coe_opens : g ∈ (U : opens G) ↔ g ∈ U := iff.rfl
@[simp, to_additive]
lemma mem_coe_subgroup : g ∈ (U : subgroup G) ↔ g ∈ U := iff.rfl
attribute [norm_cast] mem_coe mem_coe_opens mem_coe_subgroup open_add_subgroup.mem_coe
open_add_subgroup.mem_coe_opens open_add_subgroup.mem_coe_add_subgroup
@[to_additive] lemma coe_injective : injective (coe : open_subgroup G → set G) :=
λ U V h, by cases U; cases V; congr; assumption
@[ext, to_additive]
lemma ext (h : ∀ x, x ∈ U ↔ x ∈ V) : (U = V) := coe_injective $ set.ext h
@[to_additive]
lemma ext_iff : (U = V) ↔ (∀ x, x ∈ U ↔ x ∈ V) := ⟨λ h x, h ▸ iff.rfl, ext⟩
variable (U)
@[to_additive]
protected lemma is_open : is_open (U : set G) := U.is_open'
@[to_additive]
protected lemma one_mem : (1 : G) ∈ U := U.one_mem'
@[to_additive]
protected lemma inv_mem {g : G} (h : g ∈ U) : g⁻¹ ∈ U := U.inv_mem' h
@[to_additive]
protected lemma mul_mem {g₁ g₂ : G} (h₁ : g₁ ∈ U) (h₂ : g₂ ∈ U) : g₁ * g₂ ∈ U := U.mul_mem' h₁ h₂
@[to_additive]
lemma mem_nhds_one : (U : set G) ∈ 𝓝 (1 : G) :=
is_open.mem_nhds U.is_open U.one_mem
variable {U}
@[to_additive]
instance : has_top (open_subgroup G) := ⟨{ is_open' := is_open_univ, .. (⊤ : subgroup G) }⟩
@[to_additive]
instance : inhabited (open_subgroup G) := ⟨⊤⟩
@[to_additive]
lemma is_closed [has_continuous_mul G] (U : open_subgroup G) : is_closed (U : set G) :=
begin
apply is_open_compl_iff.1,
refine is_open_iff_forall_mem_open.2 (λ x hx, ⟨(λ y, y * x⁻¹) ⁻¹' U, _, _, _⟩),
{ intros u hux,
simp only [set.mem_preimage, set.mem_compl_iff, mem_coe] at hux hx ⊢,
refine mt (λ hu, _) hx,
convert U.mul_mem (U.inv_mem hux) hu,
simp },
{ exact U.is_open.preimage (continuous_mul_right _) },
{ simp [U.one_mem] }
end
section
variables {H : Type*} [group H] [topological_space H]
/-- The product of two open subgroups as an open subgroup of the product group. -/
@[to_additive "The product of two open subgroups as an open subgroup of the product group."]
def prod (U : open_subgroup G) (V : open_subgroup H) : open_subgroup (G × H) :=
{ carrier := (U : set G).prod (V : set H),
is_open' := U.is_open.prod V.is_open,
.. (U : subgroup G).prod (V : subgroup H) }
end
@[to_additive]
instance : partial_order (open_subgroup G) :=
{ le := λ U V, ∀ ⦃x⦄, x ∈ U → x ∈ V,
.. partial_order.lift (coe : open_subgroup G → set G) coe_injective }
@[to_additive]
instance : semilattice_inf_top (open_subgroup G) :=
{ inf := λ U V, { is_open' := is_open.inter U.is_open V.is_open, .. (U : subgroup G) ⊓ V },
inf_le_left := λ U V, set.inter_subset_left _ _,
inf_le_right := λ U V, set.inter_subset_right _ _,
le_inf := λ U V W hV hW, set.subset_inter hV hW,
top := ⊤,
le_top := λ U, set.subset_univ _,
..open_subgroup.partial_order }
@[simp, to_additive] lemma coe_inf : (↑(U ⊓ V) : set G) = (U : set G) ∩ V := rfl
@[simp, to_additive] lemma coe_subset : (U : set G) ⊆ V ↔ U ≤ V := iff.rfl
@[simp, to_additive] lemma coe_subgroup_le : (U : subgroup G) ≤ (V : subgroup G) ↔ U ≤ V := iff.rfl
attribute [norm_cast] coe_inf coe_subset coe_subgroup_le open_add_subgroup.coe_inf
open_add_subgroup.coe_subset open_add_subgroup.coe_add_subgroup_le
variables {N : Type*} [group N] [topological_space N]
/-- The preimage of an `open_subgroup` along a continuous `monoid` homomorphism
is an `open_subgroup`. -/
@[to_additive "The preimage of an `open_add_subgroup` along a continuous `add_monoid` homomorphism
is an `open_add_subgroup`."]
def comap (f : G →* N)
(hf : continuous f) (H : open_subgroup N) : open_subgroup G :=
{ is_open' := H.is_open.preimage hf,
.. (H : subgroup N).comap f }
@[simp, to_additive]
lemma coe_comap (H : open_subgroup N) (f : G →* N) (hf : continuous f) :
(H.comap f hf : set G) = f ⁻¹' H := rfl
@[simp, to_additive]
lemma mem_comap {H : open_subgroup N} {f : G →* N} {hf : continuous f} {x : G} :
x ∈ H.comap f hf ↔ f x ∈ H := iff.rfl
@[to_additive]
lemma comap_comap {P : Type*} [group P] [topological_space P]
(K : open_subgroup P) (f₂ : N →* P) (hf₂ : continuous f₂) (f₁ : G →* N) (hf₁ : continuous f₁) :
(K.comap f₂ hf₂).comap f₁ hf₁ = K.comap (f₂.comp f₁) (hf₂.comp hf₁) :=
rfl
end open_subgroup
namespace subgroup
variables {G : Type*} [group G] [topological_space G] [has_continuous_mul G] (H : subgroup G)
@[to_additive]
lemma is_open_of_mem_nhds {g : G} (hg : (H : set G) ∈ 𝓝 g) :
is_open (H : set G) :=
begin
simp only [is_open_iff_mem_nhds, set_like.mem_coe] at hg ⊢,
intros x hx,
have : filter.tendsto (λ y, y * (x⁻¹ * g)) (𝓝 x) (𝓝 $ x * (x⁻¹ * g)) :=
(continuous_id.mul continuous_const).tendsto _,
rw [mul_inv_cancel_left] at this,
have := filter.mem_map'.1 (this hg),
replace hg : g ∈ H := set_like.mem_coe.1 (mem_of_mem_nhds hg),
simp only [set_like.mem_coe, H.mul_mem_cancel_right (H.mul_mem (H.inv_mem hx) hg)] at this,
exact this
end
@[to_additive]
lemma is_open_of_open_subgroup {U : open_subgroup G} (h : U.1 ≤ H) :
is_open (H : set G) :=
H.is_open_of_mem_nhds (filter.mem_of_superset U.mem_nhds_one h)
@[to_additive]
lemma is_open_mono {H₁ H₂ : subgroup G} (h : H₁ ≤ H₂) (h₁ : is_open (H₁ :set G)) :
is_open (H₂ : set G) :=
@is_open_of_open_subgroup _ _ _ _ H₂ { is_open' := h₁, .. H₁ } h
end subgroup
namespace open_subgroup
variables {G : Type*} [group G] [topological_space G] [has_continuous_mul G]
@[to_additive]
instance : semilattice_sup_top (open_subgroup G) :=
{ sup := λ U V,
{ is_open' := show is_open (((U : subgroup G) ⊔ V : subgroup G) : set G),
from subgroup.is_open_mono le_sup_left U.is_open,
.. ((U : subgroup G) ⊔ V) },
le_sup_left := λ U V, coe_subgroup_le.1 le_sup_left,
le_sup_right := λ U V, coe_subgroup_le.1 le_sup_right,
sup_le := λ U V W hU hV, coe_subgroup_le.1 (sup_le hU hV),
..open_subgroup.semilattice_inf_top }
end open_subgroup
namespace submodule
open open_add_subgroup
variables {R : Type*} {M : Type*} [comm_ring R]
variables [add_comm_group M] [topological_space M] [topological_add_group M] [module R M]
lemma is_open_mono {U P : submodule R M} (h : U ≤ P) (hU : is_open (U : set M)) :
is_open (P : set M) :=
@add_subgroup.is_open_mono M _ _ _ U.to_add_subgroup P.to_add_subgroup h hU
end submodule
namespace ideal
variables {R : Type*} [comm_ring R]
variables [topological_space R] [topological_ring R]
lemma is_open_of_open_subideal {U I : ideal R} (h : U ≤ I) (hU : is_open (U : set R)) :
is_open (I : set R) :=
submodule.is_open_mono h hU
end ideal
|
5fdbcdfee7661930ab107cb843c83e3f05645b85 | 07c6143268cfb72beccd1cc35735d424ebcb187b | /src/combinatorics/composition.lean | 95724f168d66b9a7584fe7386d3504a6b5fb0d68 | [
"Apache-2.0"
] | permissive | khoek/mathlib | bc49a842910af13a3c372748310e86467d1dc766 | aa55f8b50354b3e11ba64792dcb06cccb2d8ee28 | refs/heads/master | 1,588,232,063,837 | 1,587,304,803,000 | 1,587,304,803,000 | 176,688,517 | 0 | 0 | Apache-2.0 | 1,553,070,585,000 | 1,553,070,585,000 | null | UTF-8 | Lean | false | false | 26,808 | lean | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import data.fintype.card tactic.omega
/-!
# Compositions
A composition of an integer `n` is a decomposition `n = i₀ + ... + i_{k-1}` of `n` into a sum of
positive integers. Combinatorially, it corresponds to a decomposition of `{0, ..., n-1}` into
non-empty blocks of consecutive integers, where the `iⱼ` are the lengths of the blocks.
This notion is closely related to that of a partition of `n`, but in a composition of `n` the
order of the `iⱼ`s matters.
We implement two different structures covering these two viewpoints on compositions. The first
one, made of a list of positive integers summing to `n`, is the main one and is called
`composition n`. The second one is useful for combinatorial arguments (for instance to show that
the number of compositions of `n` is `2^(n-1)`). It is given by a subset of `{0, ..., n}`
containing `0` and `n`, where the elements of the subset (other than `n`) correspond to the leftmost
points of each block. The main API is built on `composition n`, and we provide an equivalence
between the two types.
## Main functions
* `c : composition n` is a structure, made of a list of positive integers summing to `n`.
* `composition_card` states that the cardinality of `composition n` is exactly
`2^(n-1)`, which is proved by constructing an equiv with `composition_as_set n` (see below), which
is itself in bijection with the subsets of `fin (n-1)` (this holds even for `n = 0`, where `-` is
nat subtraction).
Let `c : composition n` be a composition of `n`. Then
* `c.blocks_pnat` is the list of blocks in `c`, as positive integers
* `c.blocks` is the list of blocks in `c`, as elements of `ℕ`
* `c.length` is the number of blocks in the composition;
* `c.blocks_fun : fin c.length → ℕ` is the realization of `c.blocks` as a function on
`fin c.length`. This is the main object when using compositions to understand the composition of
analytic functions.
* `c.size_up_to : ℕ → ℕ` is the sum of the size of the blocks up to `i`.;
* `c.embedding i : fin (c.blocks_fun i) → fin n` is the increasing embedding of the `i`-th block in
`fin n`;
* `c.index j`, for `j : fin n`, is the index of the block containing `j`.
We turn to the second viewpoint on compositions, that we realize as a finset of `fin (n+1)`.
`c : composition_as_set n` is a structure made of a finset of `fin (n+1)` called `c.boundaries`
and proofs that it contains `0` and `n`. (Taking a finset of `fin n` containing `0` would not
make sense in the edge case `n = 0`, while the previous description works in all cases).
The elements of this set (other than `n`) correspond to leftmost points of blocks.
Thus, there is an equiv between `composition n` and `composition_as_set n`. We
only construct basic API on `composition_as_set` (notably `c.length` and `c.blocks`) to be able
to construct this equiv, called `composition_equiv n`. Since there is a straightforward equiv
between `composition_as_set n` and finsets of `{1, ..., n-1}` (obtained by removing `0` and `n`
from a `composition_as_set` and called `composition_as_set_equiv n`), we deduce that
`composition_as_set n` and `composition n` are both fintypes of cardinality `2^(n - 1)`
(see `composition_as_set_card` and `composition_card`).
## Implementation details
The main motivation for this structure and its API is in the construction of the composition of
formal multilinear series, and the proof that the composition of analytic functions is analytic.
The representation of a composition as a list is very handy as lists are very flexible and already
have a well-developed API.
## Tags
Composition, partition
## References
<https://en.wikipedia.org/wiki/Composition_(combinatorics)>
-/
open list
open_locale classical
variable {n : ℕ}
/-- A composition of `n` is a list of positive integers summing to `n`. -/
@[ext] structure composition (n : ℕ) :=
(blocks_pnat : list ℕ+)
(blocks_pnat_sum : (blocks_pnat.map (λ n : ℕ+, (n : ℕ))).sum = n)
instance {n : ℕ} : inhabited (composition n) :=
⟨⟨repeat (1 : ℕ+) n, (by simp)⟩⟩
/-- Combinatorial viewpoint on a composition of `n`, by seeing it as non-empty blocks of
consecutive integers in `{0, ..., n-1}`. We register every block by its left end-point, yielding
a finset containing `0`. As this does not make sense for `n = 0`, we add `n` to this finset, and
get a finset of `{0, ..., n}` containing `0` and `n`. This is the data in the structure
`composition_as_set n`. -/
@[ext] structure composition_as_set (n : ℕ) :=
(boundaries : finset (fin n.succ))
(zero_mem : (0 : fin n.succ) ∈ boundaries)
(last_mem : (fin.last n ∈ boundaries))
instance {n : ℕ} : inhabited (composition_as_set n) :=
⟨⟨finset.univ, finset.mem_univ _, finset.mem_univ _⟩⟩
/-!
### Compositions
A composition of an integer `n` is a decomposition `n = i₀ + ... + i_{k-1}` of `n` into a sum of
positive integers.
-/
namespace composition
variables (c : composition n)
/-- The list of blocks in a composition, as natural numbers. -/
def blocks : list ℕ := c.blocks_pnat.map (λ n : ℕ+, (n : ℕ))
lemma blocks_sum : c.blocks.sum = n := c.blocks_pnat_sum
/-- The length of a composition, i.e., the number of blocks in the composition. -/
def length : ℕ := c.blocks.length
@[simp] lemma blocks_length : c.blocks.length = c.length := rfl
@[simp] lemma blocks_pnat_length : c.blocks_pnat.length = c.length :=
by rw [← c.blocks_length, blocks, length_map]
/-- The blocks of a composition, seen as a function on `fin c.length`. When composing analytic
functions using compositions, this is the main player. -/
def blocks_fun : fin c.length → ℕ := λ i, nth_le c.blocks i.1 i.2
lemma sum_blocks_fun : finset.univ.sum c.blocks_fun = n :=
begin
conv_rhs { rw [← c.blocks_sum, ← of_fn_nth_le c.blocks, of_fn_sum] },
simp [blocks_fun, length],
refl
end
@[simp] lemma one_le_blocks {i : ℕ} (h : i ∈ c.blocks) : 1 ≤ i :=
begin
simp only [mem_map, blocks] at h,
rcases h with ⟨h, ha, rfl⟩,
exact h.2
end
@[simp] lemma blocks_pos {i : ℕ} (h : i ∈ c.blocks) : 0 < i :=
c.one_le_blocks h
@[simp] lemma one_le_blocks' {i : ℕ} (h : i < c.length) : 1 ≤ nth_le c.blocks i h:=
c.one_le_blocks (nth_le_mem (blocks c) i h)
@[simp] lemma blocks_pos' (i : ℕ) (h : i < c.length) : 0 < nth_le c.blocks i h:=
c.one_le_blocks' h
lemma length_le : c.length ≤ n :=
begin
conv_rhs { rw ← c.blocks_sum },
exact length_le_sum_of_one_le _ (λ i hi, c.one_le_blocks hi)
end
lemma length_pos_of_pos (h : 0 < n) : 0 < c.length :=
begin
apply length_pos_of_sum_pos,
convert h,
exact c.blocks_sum
end
/-- The sum of the sizes of the blocks in a composition up to `i`. -/
def size_up_to (i : ℕ) : ℕ := (c.blocks.take i).sum
@[simp] lemma size_up_to_zero : c.size_up_to 0 = 0 := by simp [size_up_to]
lemma size_up_to_of_length_le (i : ℕ) (h : c.length ≤ i) : c.size_up_to i = n :=
begin
dsimp [size_up_to],
convert c.blocks_sum,
exact take_all_of_le h
end
@[simp] lemma size_up_to_length : c.size_up_to c.length = n :=
c.size_up_to_of_length_le c.length (le_refl _)
lemma size_up_to_le (i : ℕ) : c.size_up_to i ≤ n :=
begin
conv_rhs { rw [← c.blocks_sum, ← sum_take_add_sum_drop _ i] },
exact nat.le_add_right _ _
end
lemma size_up_to_succ {i : ℕ} (h : i < c.length) :
c.size_up_to (i+1) = c.size_up_to i + c.blocks.nth_le i h :=
by { simp only [size_up_to], rw sum_take_succ _ _ h }
lemma size_up_to_succ' (i : fin c.length) :
c.size_up_to ((i : ℕ) + 1) = c.size_up_to i + c.blocks_fun i :=
c.size_up_to_succ i.2
lemma size_up_to_strict_mono {i : ℕ} (h : i < c.length) : c.size_up_to i < c.size_up_to (i+1) :=
by { rw c.size_up_to_succ h, simp }
/-- The `i`-th boundary of a composition, i.e., the leftmost point of the `i`-th block. We include
a virtual point at the right of the last block, to make for a nice equiv with
`composition_as_set n`. -/
def boundary : fin (c.length + 1) → fin (n+1) :=
λ i, ⟨c.size_up_to i, nat.lt_succ_of_le (c.size_up_to_le i)⟩
@[simp] lemma boundary_zero : c.boundary 0 = 0 :=
by simp [boundary, fin.ext_iff]
@[simp] lemma boundary_last : c.boundary (fin.last c.length) = fin.last n :=
by simp [boundary, fin.ext_iff]
lemma strict_mono_boundary : strict_mono c.boundary :=
begin
apply fin.strict_mono_iff_lt_succ.2 (λ i hi, _),
exact c.size_up_to_strict_mono ((add_lt_add_iff_right 1).mp hi)
end
/-- The boundaries of a composition, i.e., the leftmost point of all the blocks. We include
a virtual point at the right of the last block, to make for a nice equiv with
`composition_as_set n`. -/
def boundaries : finset (fin (n+1)) :=
finset.univ.image c.boundary
lemma card_boundaries_eq_succ_length : c.boundaries.card = c.length + 1 :=
begin
dsimp [boundaries],
rw finset.card_image_of_injective finset.univ c.strict_mono_boundary.injective,
simp
end
/-- To `c : composition n`, one can associate a `composition_as_set n` by registering the leftmost
point of each block, and adding a virtual point at the right of the last block. -/
def to_composition_as_set : composition_as_set n :=
{ boundaries := c.boundaries,
zero_mem := begin
simp only [boundaries, finset.mem_univ, exists_prop_of_true, finset.mem_image],
exact ⟨0, rfl⟩,
end,
last_mem := begin
simp only [boundaries, finset.mem_univ, exists_prop_of_true, finset.mem_image],
exact ⟨fin.last c.length, c.boundary_last⟩,
end }
/-- The canonical increasing bijection between `fin (c.length + 1)` and `c.boundaries` is
exactly `c.boundary`. -/
lemma mono_of_fin_boundaries :
c.boundary = finset.mono_of_fin c.boundaries c.card_boundaries_eq_succ_length :=
begin
apply finset.mono_of_fin_unique' _ _ c.strict_mono_boundary,
assume i hi,
simp [boundaries, - set.mem_range, set.mem_range_self]
end
/-- Embedding the `i`-th block of a composition (identified with `fin (c.blocks_fun i)`) into
`fin n` at the relevant position. -/
def embedding (i : fin c.length) : fin (c.blocks_fun i) → fin n :=
λ j, ⟨c.size_up_to i.1 + j.val,
calc c.size_up_to i.1 + j.val
< c.size_up_to i.1 + c.blocks.nth_le i.1 i.2 : add_lt_add_left j.2 _
... = c.size_up_to (i.1 + 1) : (c.size_up_to_succ _).symm
... ≤ n :
by { conv_rhs { rw ← c.size_up_to_length }, exact monotone_sum_take _ i.2 } ⟩
lemma embedding_inj (i : fin c.length) : function.injective (c.embedding i) :=
λ a b hab, by simpa [embedding, fin.ext_iff] using hab
/--
`index_exists` asserts there is some `i` so `j < c.size_up_to (i+1)`.
In the next definition we use `nat.find` to produce the minimal such index.
-/
lemma index_exists {j : ℕ} (h : j < n) :
∃ i : ℕ, j < c.size_up_to i.succ ∧ i < c.length :=
begin
have n_pos : 0 < n := lt_of_le_of_lt (zero_le j) h,
have : 0 < c.blocks.sum, by rwa [← c.blocks_sum] at n_pos,
have length_pos : 0 < c.blocks.length := length_pos_of_sum_pos (blocks c) this,
refine ⟨c.length.pred, _, nat.pred_lt (ne_of_gt length_pos)⟩,
have : c.length.pred.succ = c.length := nat.succ_pred_eq_of_pos length_pos,
simp [this, h]
end
/-- `c.index j` is the index of the block in the composition `c` containing `j`. -/
def index (j : fin n) : fin c.length :=
⟨nat.find (c.index_exists j.2), (nat.find_spec (c.index_exists j.2)).2⟩
lemma lt_size_up_to_index_succ (j : fin n) : j.val < c.size_up_to (c.index j).succ :=
(nat.find_spec (c.index_exists j.2)).1
lemma size_up_to_index_le (j : fin n) : c.size_up_to (c.index j) ≤ j :=
begin
by_contradiction H,
set i := c.index j with hi,
push_neg at H,
have i_pos : (0 : ℕ) < i,
{ by_contradiction i_pos,
push_neg at i_pos,
simp [le_zero_iff_eq.mp i_pos, c.size_up_to_zero] at H,
exact nat.not_succ_le_zero j H },
let i₁ := (i : ℕ).pred,
have i₁_lt_i : i₁ < i := nat.pred_lt (ne_of_gt i_pos),
have i₁_succ : i₁.succ = i := nat.succ_pred_eq_of_pos i_pos,
have := nat.find_min (c.index_exists j.2) i₁_lt_i,
simp [lt_trans i₁_lt_i (c.index j).2, i₁_succ] at this,
exact nat.lt_le_antisymm H this
end
/-- Mapping an element `j` of `fin n` to the element in the block containing it, identified with
`fin (c.blocks_fun (c.index j))` through the canonical increasing bijection. -/
def inv_embedding (j : fin n) : fin (c.blocks_fun (c.index j)) :=
⟨j - c.size_up_to (c.index j),
begin
rw [nat.sub_lt_right_iff_lt_add, add_comm, ← size_up_to_succ'],
{ exact lt_size_up_to_index_succ _ _ },
{ exact size_up_to_index_le _ _ }
end⟩
lemma embedding_comp_inv (j : fin n) :
j = c.embedding (c.index j) (c.inv_embedding j) :=
begin
rw fin.ext_iff,
apply (nat.add_sub_cancel' (c.size_up_to_index_le j)).symm,
end
lemma mem_range_embedding_iff {j : fin n} {i : fin c.length} :
j ∈ set.range (c.embedding i) ↔
c.size_up_to i ≤ j ∧ (j : ℕ) < c.size_up_to (i : ℕ).succ :=
begin
split,
{ assume h,
rcases set.mem_range.2 h with ⟨k, hk⟩,
rw fin.ext_iff at hk,
change c.size_up_to i + k.val = (j : ℕ) at hk,
rw ← hk,
simp [size_up_to_succ', k.2] },
{ assume h,
apply set.mem_range.2,
refine ⟨⟨j.val - c.size_up_to i, _⟩, _⟩,
{ rw [nat.sub_lt_left_iff_lt_add, ← size_up_to_succ'],
{ exact h.2 },
{ exact h.1 } },
{ rw fin.ext_iff,
exact nat.add_sub_cancel' h.1 } }
end
/-- The embeddings of different blocks of a composition are disjoint. -/
lemma disjoint_range {i₁ i₂ : fin c.length} (h : i₁ ≠ i₂) :
disjoint (set.range (c.embedding i₁)) (set.range (c.embedding i₂)) :=
begin
classical,
wlog h' : i₁ ≤ i₂ using i₁ i₂,
{ by_contradiction d,
obtain ⟨x, hx₁, hx₂⟩ :
∃ x : fin n, (x ∈ set.range (c.embedding i₁) ∧ x ∈ set.range (c.embedding i₂)) :=
set.not_disjoint_iff.1 d,
have : i₁ < i₂ := lt_of_le_of_ne h' h,
have A : (i₁ : ℕ).succ ≤ i₂ := nat.succ_le_of_lt this,
apply lt_irrefl (x : ℕ),
calc (x : ℕ) < c.size_up_to (i₁ : ℕ).succ : (c.mem_range_embedding_iff.1 hx₁).2
... ≤ c.size_up_to (i₂ : ℕ) : monotone_sum_take _ A
... ≤ x : (c.mem_range_embedding_iff.1 hx₂).1 },
{ rw disjoint.comm,
apply this (ne.symm h) }
end
lemma mem_range_embedding (j : fin n) :
j ∈ set.range (c.embedding (c.index j)) :=
begin
have : c.embedding (c.index j) (c.inv_embedding j)
∈ set.range (c.embedding (c.index j)) := set.mem_range_self _,
rwa ← c.embedding_comp_inv j at this
end
lemma mem_range_embedding_iff' {j : fin n} {i : fin c.length} :
j ∈ set.range (c.embedding i) ↔ i = c.index j :=
begin
split,
{ rw ← not_imp_not,
assume h,
exact set.disjoint_right.1 (c.disjoint_range h) (c.mem_range_embedding j) },
{ assume h,
rw h,
exact c.mem_range_embedding j }
end
/-- Two compositions (possibly of different integers) coincide if and only if they have the
same sequence of blocks of positive integers. -/
lemma sigma_eq_iff_blocks_pnat_eq {c : Σ n, composition n} {c' : Σ n, composition n} :
c = c' ↔ c.2.blocks_pnat = c'.2.blocks_pnat :=
begin
refine ⟨λ H, by rw H, λ H, _⟩,
rcases c with ⟨n, c⟩,
rcases c' with ⟨n', c'⟩,
have : n = n', by { rw [← c.blocks_pnat_sum, ← c'.blocks_pnat_sum, H] },
induction this,
simp only [true_and, eq_self_iff_true, heq_iff_eq],
ext1,
exact H
end
/-- Two compositions (possibly of different integers) coincide if and only if they have the
same sequence of blocks. -/
lemma sigma_eq_iff_blocks_eq {c : Σ n, composition n} {c' : Σ n, composition n} :
c = c' ↔ c.2.blocks = c'.2.blocks :=
begin
refine ⟨λ H, by rw H, λ H, _⟩,
rwa [sigma_eq_iff_blocks_pnat_eq, ← (injective_map_iff.2 subtype.val_injective).eq_iff]
end
end composition
/-!
### Compositions as sets
Combinatorial viewpoints on compositions, seen as finite subsets of `fin (n+1)` containing `0` and
`n`, where the points of the set (other than `n`) correspond to the leftmost points of each block.
-/
/-- Bijection between compositions of `n` and subsets of `{0, ..., n-2}`, defined by
considering the restriction of the subset to `{1, ..., n-1}` and shifting to the left by one. -/
def composition_as_set_equiv (n : ℕ) : composition_as_set n ≃ finset (fin (n - 1)) :=
{ to_fun := λ c, {i : fin (n-1) |
(⟨1 + i.val, by { have := i.2, omega }⟩ : fin n.succ) ∈ c.boundaries}.to_finset,
inv_fun := λ s,
{ boundaries := {i : fin n.succ | (i = 0) ∨ (i = fin.last n)
∨ (∃ (j : fin (n-1)) (hj : j ∈ s), i.val = j.val + 1)}.to_finset,
zero_mem := by simp,
last_mem := by simp },
left_inv := begin
assume c,
ext i,
simp only [exists_prop, add_comm, set.mem_to_finset, true_or, or_true, set.mem_set_of_eq,
fin.last_val],
split,
{ rintro (rfl | rfl | ⟨j, hj1, hj2⟩),
{ exact c.zero_mem },
{ exact c.last_mem },
{ convert hj1, rwa fin.ext_iff } },
{ simp only [classical.or_iff_not_imp_left],
assume i_mem i_ne_zero i_ne_last,
simp [fin.ext_iff] at i_ne_zero i_ne_last,
refine ⟨⟨i.val - 1, _⟩, _, _⟩,
{ have : i.val < n + 1 := i.2, omega },
{ convert i_mem, rw fin.ext_iff, simp, omega },
{ simp, omega } },
end,
right_inv := begin
assume s,
ext i,
have : 1 + i.val ≠ n,
by { apply ne_of_lt, have := i.2, omega },
simp only [fin.ext_iff, this, exists_prop, fin.val_zero, false_or, add_left_inj, add_comm,
set.mem_to_finset, true_or, add_eq_zero_iff, or_true, one_ne_zero, set.mem_set_of_eq,
fin.last_val, false_and],
split,
{ rintros ⟨j, js, hj⟩, convert js, exact (fin.ext_iff _ _).2 hj },
{ assume h, exact ⟨i, h, rfl⟩ }
end }
instance composition_as_set_fintype (n : ℕ) : fintype (composition_as_set n) :=
fintype.of_equiv _ (composition_as_set_equiv n).symm
lemma composition_as_set_card (n : ℕ) : fintype.card (composition_as_set n) = 2 ^ (n - 1) :=
begin
have : fintype.card (finset (fin (n-1))) = 2 ^ (n - 1), by simp,
rw ← this,
exact fintype.card_congr (composition_as_set_equiv n)
end
namespace composition_as_set
variables (c : composition_as_set n)
lemma boundaries_nonempty : c.boundaries.nonempty :=
⟨0, c.zero_mem⟩
lemma card_boundaries_pos : 0 < finset.card c.boundaries :=
finset.card_pos.mpr c.boundaries_nonempty
/-- Number of blocks in a `composition_as_set`. -/
def length : ℕ := finset.card c.boundaries - 1
lemma card_boundaries_eq_succ_length : c.boundaries.card = c.length + 1 :=
(nat.sub_eq_iff_eq_add c.card_boundaries_pos).mp rfl
lemma length_lt_card_boundaries : c.length < c.boundaries.card :=
by { rw c.card_boundaries_eq_succ_length, exact lt_add_one _ }
lemma lt_length (i : fin c.length) : i.val + 1 < c.boundaries.card :=
nat.add_lt_of_lt_sub_right i.2
lemma lt_length' (i : fin c.length) : i.val < c.boundaries.card :=
lt_of_le_of_lt (nat.le_succ i.val) (c.lt_length i)
/-- Canonical increasing bijection from `fin c.boundaries.card` to `c.boundaries`. -/
def boundary : fin c.boundaries.card → fin (n+1) :=
finset.mono_of_fin c.boundaries rfl
@[simp] lemma boundary_zero : c.boundary ⟨0, c.card_boundaries_pos⟩ = 0 :=
begin
rw [boundary, finset.mono_of_fin_zero rfl c.boundaries_nonempty c.card_boundaries_pos],
exact le_antisymm (finset.min'_le _ _ _ c.zero_mem) (fin.zero_le _),
end
@[simp] lemma boundary_length : c.boundary ⟨c.length, c.length_lt_card_boundaries⟩ = fin.last n :=
begin
convert finset.mono_of_fin_last rfl c.boundaries_nonempty c.card_boundaries_pos,
exact le_antisymm (finset.le_max' _ _ _ c.last_mem) (fin.le_last _)
end
/-- Size of the `i`-th block in a `composition_as_set`, seen as a function on `fin c.length`. -/
def blocks_fun (i : fin c.length) : ℕ :=
(c.boundary ⟨i.val + 1, c.lt_length i⟩).val - (c.boundary ⟨i.val, c.lt_length' i⟩).val
lemma blocks_fun_pos (i : fin c.length) : 0 < c.blocks_fun i :=
begin
have : (⟨i.val, c.lt_length' i⟩ : fin c.boundaries.card) < ⟨i.val + 1, c.lt_length i⟩ :=
nat.lt_succ_self _,
exact nat.lt_sub_left_of_add_lt (finset.mono_of_fin_strict_mono c.boundaries rfl this)
end
/-- List of the sizes of the blocks in a `composition_as_set`. -/
def blocks (c : composition_as_set n) : list ℕ :=
of_fn c.blocks_fun
@[simp] lemma blocks_length : c.blocks.length = c.length :=
length_of_fn _
lemma blocks_partial_sum {i : ℕ} (h : i < c.boundaries.card) :
(c.blocks.take i).sum = c.boundary ⟨i, h⟩ :=
begin
induction i with i IH, { simp },
have A : i < c.blocks.length,
{ rw c.card_boundaries_eq_succ_length at h,
simp [blocks, nat.lt_of_succ_lt_succ h] },
have B : i < c.boundaries.card := lt_of_lt_of_le A (by simp [blocks, length, nat.sub_le]),
rw [sum_take_succ _ _ A, IH B],
simp only [blocks, blocks_fun, fin.coe_eq_val, nth_le_of_fn'],
rw nat.add_sub_cancel',
refine le_of_lt (@finset.mono_of_fin_strict_mono _ _ c.boundaries _ rfl
(⟨i, B⟩ : fin c.boundaries.card) _ _),
exact nat.lt_succ_self _
end
lemma mem_boundaries_iff_exists_blocks_sum_take_eq {j : fin (n+1)} :
j ∈ c.boundaries ↔ ∃ i < c.boundaries.card, (c.blocks.take i).sum = j.val :=
begin
split,
{ assume hj,
rcases (c.boundaries.mono_of_fin_bij_on rfl).surj_on hj with ⟨i, _, hi⟩,
refine ⟨i.1, i.2, _⟩,
rw [← hi, c.blocks_partial_sum i.2],
refl },
{ rintros ⟨i, hi, H⟩,
convert (c.boundaries.mono_of_fin_bij_on rfl).maps_to (set.mem_univ ⟨i, hi⟩),
have : c.boundary ⟨i, hi⟩ = j, by rwa [fin.ext_iff, ← fin.coe_eq_val, ← c.blocks_partial_sum hi],
exact this.symm }
end
lemma blocks_sum : c.blocks.sum = n :=
begin
have : c.blocks.take c.length = c.blocks := take_all_of_le (by simp [blocks]),
rw [← this, c.blocks_partial_sum c.length_lt_card_boundaries, c.boundary_length],
refl
end
/-- List of the sizes of the blocks in a `composition_as_set`, seen as positive integers. -/
def blocks_pnat : list (ℕ+ ) :=
of_fn (λ i, ⟨c.blocks_fun i, c.blocks_fun_pos i⟩)
lemma coe_blocks_pnat_eq_blocks : c.blocks_pnat.map coe = c.blocks :=
by { simp [blocks_pnat, blocks], refl }
@[simp] lemma blocks_pnat_length : c.blocks_pnat.length = c.length :=
length_of_fn _
/-- Associating a `composition n` to a `composition_as_set n`, by registering the sizes of the
blocks as a list of positive integers. -/
def to_composition : composition n :=
{ blocks_pnat := c.blocks_pnat,
blocks_pnat_sum := begin
rw [blocks_pnat, map_of_fn],
conv_rhs { rw ← c.blocks_sum },
congr
end }
end composition_as_set
/-!
### Equivalence between compositions and compositions as sets
In this section, we explain how to go back and forth between a `composition` and a
`composition_as_set`, by showing that their `blocks` and `length` and `boundaries` correspond to
each other, and construct an equivalence between them called `composition_equiv`.
-/
@[simp] lemma composition.to_composition_as_set_length (c : composition n) :
c.to_composition_as_set.length = c.length :=
by simp [composition.to_composition_as_set, composition_as_set.length,
c.card_boundaries_eq_succ_length]
@[simp] lemma composition_as_set.to_composition_length (c : composition_as_set n) :
c.to_composition.length = c.length :=
by simp [composition_as_set.to_composition, composition.length, composition.blocks]
@[simp] lemma composition.to_composition_as_set_blocks (c : composition n) :
c.to_composition_as_set.blocks = c.blocks :=
begin
let d := c.to_composition_as_set,
change d.blocks = c.blocks,
have length_eq : d.blocks.length = c.blocks.length,
{ convert c.to_composition_as_set_length,
simp [composition_as_set.blocks] },
suffices H : ∀ (i ≤ d.blocks.length), (d.blocks.take i).sum = (c.blocks.take i).sum,
from eq_of_sum_take_eq length_eq H,
assume i hi,
have i_lt : i < d.boundaries.card,
{ convert nat.lt_succ_iff.2 hi,
convert d.card_boundaries_eq_succ_length,
exact length_of_fn _ },
have i_lt' : i < c.boundaries.card := i_lt,
have i_lt'' : i < c.length + 1, by rwa c.card_boundaries_eq_succ_length at i_lt',
have A : finset.mono_of_fin d.boundaries rfl ⟨i, i_lt⟩
= finset.mono_of_fin c.boundaries rfl ⟨i, i_lt'⟩ := rfl,
have B : c.size_up_to i = c.boundary ⟨i, i_lt''⟩ := rfl,
rw [d.blocks_partial_sum i_lt, composition_as_set.boundary, A, ← composition.size_up_to, B,
fin.coe_eq_val, fin.coe_eq_val, ← fin.ext_iff,
c.mono_of_fin_boundaries, finset.mono_of_fin_eq_mono_of_fin_iff]
end
@[simp] lemma composition.to_composition_as_set_blocks_pnat (c : composition n) :
c.to_composition_as_set.blocks_pnat = c.blocks_pnat :=
begin
rw ← (injective_map_iff.2 subtype.val_injective).eq_iff,
convert c.to_composition_as_set_blocks,
exact composition_as_set.coe_blocks_pnat_eq_blocks _
end
@[simp] lemma composition_as_set.to_composition_blocks_pnat (c : composition_as_set n) :
c.to_composition.blocks_pnat = c.blocks_pnat := rfl
@[simp] lemma composition_as_set.to_composition_blocks (c : composition_as_set n) :
c.to_composition.blocks = c.blocks :=
by simp [composition.blocks, composition_as_set.coe_blocks_pnat_eq_blocks]
@[simp] lemma composition_as_set.to_composition_boundaries (c : composition_as_set n) :
c.to_composition.boundaries = c.boundaries :=
begin
ext j,
simp [c.mem_boundaries_iff_exists_blocks_sum_take_eq, c.card_boundaries_eq_succ_length,
composition.boundary, fin.ext_iff, composition.size_up_to, exists_prop, finset.mem_univ,
take, exists_prop_of_true, finset.mem_image, composition_as_set.to_composition_blocks,
composition.boundaries],
split,
{ rintros ⟨i, hi⟩,
refine ⟨i.1, _, hi⟩,
convert i.2,
simp },
{ rintros ⟨i, i_lt, hi⟩,
have : i < c.to_composition.length + 1, by simpa using i_lt,
exact ⟨⟨i, this⟩, hi⟩ }
end
@[simp] lemma composition.to_composition_as_set_boundaries (c : composition n) :
c.to_composition_as_set.boundaries = c.boundaries := rfl
/-- Equivalence between `composition n` and `composition_as_set n`. -/
def composition_equiv (n : ℕ) : composition n ≃ composition_as_set n :=
{ to_fun := λ c, c.to_composition_as_set,
inv_fun := λ c, c.to_composition,
left_inv := λ c, by { ext1, exact c.to_composition_as_set_blocks_pnat },
right_inv := λ c, by { ext1, exact c.to_composition_boundaries } }
instance composition_fintype (n : ℕ) : fintype (composition n) :=
fintype.of_equiv _ (composition_equiv n).symm
lemma composition_card (n : ℕ) : fintype.card (composition n) = 2 ^ (n - 1) :=
begin
rw ← composition_as_set_card n,
exact fintype.card_congr (composition_equiv n)
end
|
4cbd352c68572ff4c714a38b8c65fce3643a4196 | 54c9ed381c63410c9b6af3b0a1722c41152f037f | /Binport/Run.lean | e89396cc545a6090e0d579beb292adc5826c1e35 | [
"Apache-2.0"
] | permissive | dselsam/binport | 0233f1aa961a77c4fc96f0dccc780d958c5efc6c | aef374df0e169e2c3f1dc911de240c076315805c | refs/heads/master | 1,687,453,448,108 | 1,627,483,296,000 | 1,627,483,296,000 | 333,825,622 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,979 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Selsam
-/
import Binport.Basic
import Binport.Rules
import Binport.ParseExport
import Binport.ProcessActionItem
import Binport.Path
import Lean
import Std.Data.HashMap
import Std.Data.HashSet
open Lean Lean.Meta
open Std (HashMap HashSet)
namespace Binport
namespace Run -- TODO: better name. runAll? schedule?
abbrev Job := Task (Except IO.Error Unit)
instance : Inhabited Job := ⟨⟨pure ()⟩⟩
structure Context where
proofs : Bool
structure State where
path2task : HashMap String Job := {}
abbrev RunM := ReaderT Context (StateRefT State IO)
-- TODO: weave
def rulesFilename := "rules.txt"
def parseTLeanImports (target : Path34) : IO (List Path34) := do
let line ← IO.FS.withFile target.toTLean IO.FS.Mode.read fun h => h.getLine
let tokens := line.trim.splitOn " " |>.toArray
let nImports := tokens[1].toNat!
let mut paths := #[]
for i in [:nImports] do
if tokens[2+2*i+1] ≠ "-1" then throw $ IO.userError "found relative import!"
let dotPath : DotPath := ⟨tokens[2+2*i]⟩
paths := paths.push $ ← resolveDotPath dotPath
return paths.toList
def bindTasks (tasks : List Job) (k : Unit → IO Job) : IO Job :=
match tasks with
| [] => k ()
| task::tasks => IO.bindTask task λ
| Except.ok _ => bindTasks tasks k
| Except.error err => throw err
@[implementedBy withImportModules]
constant withImportModulesConst {α : Type} (imports : List Import) (opts : Options) (trustLevel : UInt32 := 0) (x : Environment → IO α) : IO α :=
throw $ IO.userError "NYI"
def genOLeanFor (proofs : Bool) (target : Path34) : IO Unit := do
println! s!"\n[genOLeanFor] START {target.mrpath.path}\n"
createDirectoriesIfNotExists target.toLean4olean.toString
let imports : List Import := [{ module := `Init : Import }, { module := `PrePort : Import }]
++ (← parseTLeanImports target).map fun path => { module := parseName path.toLean4dot }
withImportModulesConst imports (opts := {}) (trustLevel := 0) $ λ env₀ => do
let env₀ := env₀.setMainModule target.mrpath.toDotPath.path
let _ ← PortM.toIO (ctx := { proofs := proofs, path := target }) (env := env₀) do
parseRules rulesFilename
IO.FS.Handle.mk target.toTLean IO.FS.Mode.read >>= fun h => do
let _ ← h.getLine -- discard imports
let mut actionItems := #[]
while (not (← h.isEof)) do
let line := (← h.getLine).dropRightWhile λ c => c == '\n'
if line == "" then continue
actionItems := actionItems.append (← processLine line).toArray
let mut isIrreducible : Bool := false
for actionItem in actionItems do
processActionItem actionItem
let env ← getEnv
writeModule env target.toLean4olean
println! s!"\n[genOLeanFor] END {target.mrpath.path}\n"
pure ()
partial def visit (target : Path34) : RunM Job := do
match (← get).path2task.find? target.toTLean.toString with
| some task => pure task
| none => do
if (← target.toLean4olean.pathExists) then
IO.asTask (pure ())
else
let paths ← parseTLeanImports target
let cjobs ← paths.mapM visit
let ctx ← read
let job : Job ← IO.asTask $ do
for cjob in cjobs do
match ← IO.wait cjob with
| Except.ok _ => pure ()
| Except.error err => throw err
genOLeanFor ctx.proofs target
modify λ s => { s with path2task := s.path2task.insert target.toTLean.toString job }
pure job
end Run
open Run
unsafe def run (proofs : Bool) (target : Path34) : IO Unit := do
initSearchPath s!"{Lean4LibPath}:{Lib4Path}"
let job ← (visit target) { proofs := proofs } |>.run' (s := {})
let result ← IO.wait job
match result with
| Except.ok _ => pure ()
| Except.error err => throw err
end Binport
|
80c8e20414e3f7dba7b2045cc0619894963b1827 | de91c42b87530c3bdcc2b138ef1a3c3d9bee0d41 | /old/eval/axisOrientationEval.lean | b397f739bcefdc7a2839fd0601aae2d7e25a6820 | [] | no_license | kevinsullivan/lang | d3e526ba363dc1ddf5ff1c2f36607d7f891806a7 | e9d869bff94fb13ad9262222a6f3c4aafba82d5e | refs/heads/master | 1,687,840,064,795 | 1,628,047,969,000 | 1,628,047,969,000 | 282,210,749 | 0 | 1 | null | 1,608,153,830,000 | 1,595,592,637,000 | Lean | UTF-8 | Lean | false | false | 298 | lean | import ..imperative_DSL.environment
open lang.axisOrientation
def axisOrientationEval : lang.axisOrientation.orientationExpr → environment.env → orientation.AxisOrientation 3
| (lang.axisOrientation.orientationExpr.lit V) i := V
| (lang.axisOrientation.orientationExpr.var v) i := i.ax.or v
|
476872a1e329b10a97857a2bc137d306caa8e805 | 7cef822f3b952965621309e88eadf618da0c8ae9 | /src/algebra/geom_sum.lean | 8a427b8f3342d929dcdb47600c7f36da79b45a89 | [
"Apache-2.0"
] | permissive | rmitta/mathlib | 8d90aee30b4db2b013e01f62c33f297d7e64a43d | 883d974b608845bad30ae19e27e33c285200bf84 | refs/heads/master | 1,585,776,832,544 | 1,576,874,096,000 | 1,576,874,096,000 | 153,663,165 | 0 | 2 | Apache-2.0 | 1,544,806,490,000 | 1,539,884,365,000 | Lean | UTF-8 | Lean | false | false | 6,240 | lean | /-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
Sums of finite geometric series
-/
import algebra.big_operators algebra.commute
universe u
variable {α : Type u}
open finset
/-- Sum of the finite geometric series $∑_{i=0}^{n-1} x^i$. -/
def geom_series [semiring α] (x : α) (n : ℕ) :=
(range n).sum (λ i, x ^ i)
theorem geom_series_def [semiring α] (x : α) (n : ℕ) :
geom_series x n = (range n).sum (λ i, x ^ i) := rfl
@[simp] theorem geom_series_zero [semiring α] (x : α) :
geom_series x 0 = 0 := rfl
@[simp] theorem geom_series_one [semiring α] (x : α) :
geom_series x 1 = 1 :=
by { rw [geom_series_def, sum_range_one, pow_zero] }
/-- Sum of the finite geometric series $∑_{i=0}^{n-1} x^i y^{n-1-i}$. -/
def geom_series₂ [semiring α] (x y : α) (n : ℕ) :=
(range n).sum (λ i, x ^ i * (y ^ (n - 1 - i)))
theorem geom_series₂_def [semiring α] (x y : α) (n : ℕ) :
geom_series₂ x y n = (range n).sum (λ i, x ^ i * y ^ (n - 1 - i)) := rfl
@[simp] theorem geom_series₂_zero [semiring α] (x y : α) :
geom_series₂ x y 0 = 0 := rfl
@[simp] theorem geom_series₂_one [semiring α] (x y : α) :
geom_series₂ x y 1 = 1 :=
by { have : 1 - 1 - 0 = 0 := rfl,
rw [geom_series₂_def, sum_range_one, this, pow_zero, pow_zero, mul_one] }
@[simp] theorem geom_series₂_with_one [semiring α] (x : α) (n : ℕ) :
geom_series₂ x 1 n = geom_series x n :=
sum_congr rfl (λ i _, by { rw [one_pow, mul_one] })
/-- $x^n-y^n=(x-y)∑x^ky^{n-1-k}$ reformulated without `-` signs. -/
protected theorem commute.geom_sum₂_mul_add [semiring α] {x y : α} (h : commute x y) (n : ℕ) :
(geom_series₂ (x + y) y n) * x + y ^ n = (x + y) ^ n :=
begin
let f := λ (m i : ℕ), (x + y) ^ i * y ^ (m - 1 - i),
change ((range n).sum (f n)) * x + y ^ n = (x + y) ^ n,
induction n with n ih,
{ rw [range_zero, sum_empty, zero_mul, zero_add, pow_zero, pow_zero] },
{ have f_last : f n.succ n = (x + y) ^ n :=
by { dsimp [f],
rw [nat.sub_sub, nat.add_comm, nat.sub_self, pow_zero, mul_one] },
have f_succ : ∀ i, i ∈ range n → f n.succ i = y * f n i :=
λ i hi, by {
dsimp [f],
have : commute y ((x + y) ^ i) :=
(h.symm.add_right (commute.refl y)).pow_right i,
rw [← mul_assoc, this.eq, mul_assoc, ← pow_succ y (n - 1 - i)],
congr' 2,
rw [nat.succ_eq_add_one, nat.add_sub_cancel, nat.sub_sub, add_comm 1 i],
have := nat.add_sub_of_le (mem_range.mp hi),
rw [add_comm, nat.succ_eq_add_one] at this,
rw [← this, nat.add_sub_cancel, add_comm i 1, ← add_assoc,
nat.add_sub_cancel] },
rw [pow_succ (x + y), add_mul, sum_range_succ, f_last, add_mul, add_assoc],
rw [(((commute.refl x).add_right h).pow_right n).eq],
congr' 1,
rw[sum_congr rfl f_succ, ← mul_sum, pow_succ y],
rw[mul_assoc, ← mul_add y, ih] }
end
/-- $x^n-y^n=(x-y)∑x^ky^{n-1-k}$ reformulated without `-` signs. -/
theorem geom_sum₂_mul_add [comm_semiring α] (x y : α) (n : ℕ) :
(geom_series₂ (x + y) y n) * x + y ^ n = (x + y) ^ n :=
(commute.all x y).geom_sum₂_mul_add n
theorem geom_sum_mul_add [semiring α] (x : α) (n : ℕ) :
(geom_series (x + 1) n) * x + 1 = (x + 1) ^ n :=
begin
have := (commute.one_right x).geom_sum₂_mul_add n,
rw [one_pow, geom_series₂_with_one] at this,
exact this
end
theorem geom_sum₂_mul_comm [ring α] {x y : α} (h : commute x y) (n : ℕ) :
(geom_series₂ x y n) * (x - y) = x ^ n - y ^ n :=
begin
have := (h.sub_left (commute.refl y)).geom_sum₂_mul_add n,
rw [sub_add_cancel] at this,
rw [← this, add_sub_cancel]
end
theorem geom_sum₂_mul [comm_ring α] (x y : α) (n : ℕ) :
(geom_series₂ x y n) * (x - y) = x ^ n - y ^ n :=
geom_sum₂_mul_comm (commute.all x y) n
theorem geom_sum_mul [ring α] (x : α) (n : ℕ) :
(geom_series x n) * (x - 1) = x ^ n - 1 :=
begin
have := geom_sum₂_mul_comm (commute.one_right x) n,
rw [one_pow, geom_series₂_with_one] at this,
exact this
end
theorem geom_sum_mul_neg [ring α] (x : α) (n : ℕ) :
(geom_series x n) * (1 - x) = 1 - x ^ n :=
begin
have := congr_arg has_neg.neg (geom_sum_mul x n),
rw [neg_sub, ← mul_neg_eq_neg_mul_symm, neg_sub] at this,
exact this
end
theorem geom_sum [division_ring α] {x : α} (h : x ≠ 1) (n : ℕ) :
(geom_series x n) = (x ^ n - 1) / (x - 1) :=
have x - 1 ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← geom_sum_mul, mul_div_cancel _ this]
theorem geom_sum_Ico_mul [ring α] (x : α) {m n : ℕ} (hmn : m ≤ n) :
((finset.Ico m n).sum (pow x)) * (x - 1) = x^n - x^m :=
by rw [sum_Ico_eq_sub _ hmn, ← geom_series_def, ← geom_series_def, sub_mul,
geom_sum_mul, geom_sum_mul, sub_sub_sub_cancel_right]
theorem geom_sum_Ico_mul_neg [ring α] (x : α) {m n : ℕ} (hmn : m ≤ n) :
((finset.Ico m n).sum (pow x)) * (1 - x) = x^m - x^n :=
by rw [sum_Ico_eq_sub _ hmn, ← geom_series_def, ← geom_series_def, sub_mul,
geom_sum_mul_neg, geom_sum_mul_neg, sub_sub_sub_cancel_left]
theorem geom_sum_Ico [division_ring α] {x : α} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :
(finset.Ico m n).sum (λ i, x ^ i) = (x ^ n - x ^ m) / (x - 1) :=
by simp only [sum_Ico_eq_sub _ hmn, (geom_series_def _ _).symm, geom_sum hx, div_sub_div_same,
sub_sub_sub_cancel_right]
lemma geom_sum_inv [division_ring α] {x : α} (hx1 : x ≠ 1) (hx0 : x ≠ 0) (n : ℕ) :
(geom_series x⁻¹ n) = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x) :=
have h₁ : x⁻¹ ≠ 1, by rwa [inv_eq_one_div, ne.def, div_eq_iff_mul_eq hx0, one_mul],
have h₂ : x⁻¹ - 1 ≠ 0, from mt sub_eq_zero.1 h₁,
have h₃ : x - 1 ≠ 0, from mt sub_eq_zero.1 hx1,
have h₄ : x * x⁻¹ ^ n = x⁻¹ ^ n * x,
from nat.cases_on n (by simp)
(λ _, by conv { to_rhs, rw [pow_succ', mul_assoc, inv_mul_cancel hx0, mul_one] };
rw [pow_succ, ← mul_assoc, mul_inv_cancel hx0, one_mul]),
by rw [geom_sum h₁, div_eq_iff_mul_eq h₂, ← domain.mul_left_inj h₃,
← mul_assoc, ← mul_assoc, mul_inv_cancel h₃];
simp [mul_add, add_mul, mul_inv_cancel hx0, mul_assoc, h₄]
|
4b2c7ba8e8e591ea79def8e0f3c8c1dcc2db9e26 | c777c32c8e484e195053731103c5e52af26a25d1 | /archive/imo/imo1962_q1.lean | 0afc5e6be6726f0b2deb045058db464532ba6a26 | [
"Apache-2.0"
] | permissive | kbuzzard/mathlib | 2ff9e85dfe2a46f4b291927f983afec17e946eb8 | 58537299e922f9c77df76cb613910914a479c1f7 | refs/heads/master | 1,685,313,702,744 | 1,683,974,212,000 | 1,683,974,212,000 | 128,185,277 | 1 | 0 | null | 1,522,920,600,000 | 1,522,920,600,000 | null | UTF-8 | Lean | false | false | 5,480 | lean | /-
Copyright (c) 2020 Kevin Lacker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Lacker
-/
import data.nat.digits
/-!
# IMO 1962 Q1
Find the smallest natural number $n$ which has the following properties:
(a) Its decimal representation has 6 as the last digit.
(b) If the last digit 6 is erased and placed in front of the remaining digits,
the resulting number is four times as large as the original number $n$.
Since Lean does not explicitly express problems of the form "find the smallest number satisfying X",
we define the problem as a predicate, and then prove a particular number is the smallest member
of a set satisfying it.
-/
namespace imo1962_q1
open nat
def problem_predicate (n : ℕ) : Prop :=
(digits 10 n).head = 6 ∧ of_digits 10 ((digits 10 n).tail.concat 6) = 4 * n
/-!
First, it's inconvenient to work with digits, so let's simplify them out of the problem.
-/
abbreviation problem_predicate' (c n : ℕ) : Prop :=
n = 10 * c + 6 ∧ 6 * 10 ^ (digits 10 c).length + c = 4 * n
lemma without_digits {n : ℕ} (h1 : problem_predicate n) :
∃ c : ℕ, problem_predicate' c n :=
begin
use n / 10,
cases n,
{ have h2 : ¬ problem_predicate 0, by norm_num [problem_predicate],
contradiction },
{ rw [problem_predicate, digits_def' (dec_trivial : 2 ≤ 10) n.succ_pos,
list.head, list.tail_cons, list.concat_eq_append] at h1,
split,
{ rw [← h1.left, div_add_mod (n+1) 10], },
{ rw [← h1.right, of_digits_append, of_digits_digits,
of_digits_singleton, add_comm, mul_comm], }, },
end
/-!
Now we can eliminate possibilities for `(digits 10 c).length` until we get to the one that works.
-/
lemma case_0_digit {c n : ℕ} (h1 : (digits 10 c).length = 0) : ¬ problem_predicate' c n :=
begin
intro h2,
have h3 : 6 * 10 ^ 0 + c = 6 * 10 ^ (digits 10 c).length + c, by rw h1,
have h4 : 6 * 10 ^ 0 + c = 4 * (10 * c + 6), by rw [h3, h2.right, h2.left],
linarith,
end
lemma case_1_digit {c n : ℕ} (h1 : (digits 10 c).length = 1) : ¬ problem_predicate' c n :=
begin
intro h2,
have h3 : 6 * 10 ^ 1 + c = 6 * 10 ^ (digits 10 c).length + c, by rw h1,
have h4 : 6 * 10 ^ 1 + c = 4 * (10 * c + 6), by rw [h3, h2.right, h2.left],
have h6 : c > 0, by linarith,
linarith,
end
lemma case_2_digit {c n : ℕ} (h1 : (digits 10 c).length = 2) : ¬ problem_predicate' c n :=
begin
intro h2,
have h3 : 6 * 10 ^ 2 + c = 6 * 10 ^ (digits 10 c).length + c, by rw h1,
have h4 : 6 * 10 ^ 2 + c = 4 * (10 * c + 6), by rw [h3, h2.right, h2.left],
have h5 : c > 14, by linarith,
linarith
end
lemma case_3_digit {c n : ℕ} (h1 : (digits 10 c).length = 3) : ¬ problem_predicate' c n :=
begin
intro h2,
have h3 : 6 * 10 ^ 3 + c = 6 * 10 ^ (digits 10 c).length + c, by rw h1,
have h4 : 6 * 10 ^ 3 + c = 4 * (10 * c + 6), by rw [h3, h2.right, h2.left],
have h5 : c > 153, by linarith,
linarith
end
lemma case_4_digit {c n : ℕ} (h1 : (digits 10 c).length = 4) : ¬ problem_predicate' c n :=
begin
intro h2,
have h3 : 6 * 10 ^ 4 + c = 6 * 10 ^ (digits 10 c).length + c, by rw h1,
have h4 : 6 * 10 ^ 4 + c = 4 * (10 * c + 6), by rw [h3, h2.right, h2.left],
have h5 : c > 1537, by linarith,
linarith
end
/-- Putting this inline causes a deep recursion error, so we separate it out. -/
lemma helper_5_digit {c : ℤ} (h : 6 * 10 ^ 5 + c = 4 * (10 * c + 6)) : c = 15384 := by linarith
lemma case_5_digit {c n : ℕ} (h1 : (digits 10 c).length = 5) (h2 : problem_predicate' c n) :
c = 15384 :=
begin
have h3 : 6 * 10 ^ 5 + c = 6 * 10 ^ (digits 10 c).length + c, by rw h1,
have h4 : 6 * 10 ^ 5 + c = 4 * (10 * c + 6), by rw [h3, h2.right, h2.left],
zify at *,
exact helper_5_digit h4
end
/--
`linarith` fails on numbers this large, so this lemma spells out some of the arithmetic
that normally would be automated.
-/
lemma case_more_digits {c n : ℕ} (h1 : (digits 10 c).length ≥ 6) (h2 : problem_predicate' c n) :
n ≥ 153846 :=
begin
have h3 : c ≠ 0,
{ intro h4,
have h5 : (digits 10 c).length = 0, by simp [h4],
exact case_0_digit h5 h2 },
have h6 : 2 ≤ 10, from dec_trivial,
calc
n ≥ 10 * c : le.intro h2.left.symm
... ≥ 10 ^ (digits 10 c).length : base_pow_length_digits_le 10 c h6 h3
... ≥ 10 ^ 6 : (pow_le_iff_le_right h6).mpr h1
... ≥ 153846 : by norm_num,
end
/-!
Now we combine these cases to show that 153846 is the smallest solution.
-/
lemma satisfied_by_153846 : problem_predicate 153846 :=
by norm_num [problem_predicate, digits_def', of_digits]
lemma no_smaller_solutions (n : ℕ) (h1 : problem_predicate n) : n ≥ 153846 :=
begin
cases without_digits h1 with c h2,
have h3 : (digits 10 c).length < 6 ∨ (digits 10 c).length ≥ 6, by apply lt_or_ge,
cases h3,
case or.inr
{ exact case_more_digits h3 h2, },
case or.inl
{ interval_cases (digits 10 c).length with h,
{ exfalso, exact case_0_digit h h2 },
{ exfalso, exact case_1_digit h h2 },
{ exfalso, exact case_2_digit h h2 },
{ exfalso, exact case_3_digit h h2 },
{ exfalso, exact case_4_digit h h2 },
{ have h4 : c = 15384, from case_5_digit h h2,
have h5 : n = 10 * 15384 + 6, from h4 ▸ h2.left,
norm_num at h5,
exact h5.ge, }, },
end
end imo1962_q1
open imo1962_q1
theorem imo1962_q1 : is_least {n | problem_predicate n} 153846 :=
⟨satisfied_by_153846, no_smaller_solutions⟩
|
76dcbb984bdad79a551a94bc15f6a9aae03c986f | 4376c25f060c13471bb89cdb12aeac1d53e53876 | /Lean Game/world3_le.lean | b201b1d95e9ccbf5cb01d825e7eaaa9a54a010eb | [
"MIT"
] | permissive | RaitoBezarius/projet-maths-lean | 8fa7df563d64c256561ab71893c523fc1424b85c | 42356e980e021a20c3468f5ca1639fec01bb934f | refs/heads/master | 1,613,002,128,339 | 1,589,289,282,000 | 1,589,289,282,000 | 244,431,534 | 0 | 1 | MIT | 1,584,312,574,000 | 1,583,169,883,000 | TeX | UTF-8 | Lean | false | false | 3,573 | lean | import solutions.world2_multiplication
import mynat.le
/- Here's what you get from the import:
1) The following data:
* a binary relation called mynat.le, and notation a ≤ b for this relation.
The definition is: a ≤ b ↔ ∃ c : mynat, b = a + c
2) The following axiom:
* `le_def (a b : mynat) : a ≤ b ↔ ∃ (c : mynat), b = a + c`
You can rewrite `le_def`.
If a goal is of the form `∃ c, ...` then to make progress you can use the `use` tactic.
For example `use 7` will replace all c's in the goal with 7's.
-/
namespace mynat
-- example
theorem le_refl (a : mynat) : a ≤ a :=
begin
rw le_def,
use 0,
rw add_zero,
end
example : one ≤ one := le_refl one
-- ignore this; it's making the "refl" tactic work with goals of the form a ≤ a
attribute [_refl_lemma] le_refl
theorem le_succ {a b : mynat} (h : a ≤ b) : a ≤ (succ b) :=
begin
sorry
end
lemma zero_le (a : mynat) : 0 ≤ a :=
begin
sorry
end
lemma le_zero {a : mynat} : a ≤ 0 → a = 0 :=
begin
sorry
end
theorem le_trans ⦃a b c : mynat⦄ (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c :=
begin
sorry
end
instance : preorder mynat := by structure_helper
-- ignore this, it's the definition.
theorem lt_iff_le_not_le {a b : mynat} : a < b ↔ a ≤ b ∧ ¬ b ≤ a := iff.rfl
theorem le_antisymm : ∀ {{a b : mynat}}, a ≤ b → b ≤ a → a = b :=
begin
sorry
end
instance : partial_order mynat := by structure_helper
theorem lt_iff_le_and_ne ⦃a b : mynat⦄ : a < b ↔ a ≤ b ∧ a ≠ b :=
begin
sorry
end
lemma succ_le_succ {a b : mynat} (h : a ≤ b) : succ a ≤ succ b :=
begin
sorry
end
theorem le_total (a b : mynat) : a ≤ b ∨ b ≤ a :=
begin
sorry
end
instance : linear_order mynat := by structure_helper
theorem add_le_add_right (a b : mynat) : a ≤ b → ∀ t, (a + t) ≤ (b + t) :=
begin
sorry
end
theorem le_succ_self (a : mynat) : a ≤ succ a :=
begin
sorry
end
theorem le_of_succ_le_succ {a b : mynat} : succ a ≤ succ b → a ≤ b :=
begin
sorry
end
theorem not_succ_le_self {{d : mynat}} (h : succ d ≤ d) : false :=
begin
sorry
end
theorem add_le_add_left : ∀ (a b : mynat), a ≤ b → ∀ (c : mynat), c + a ≤ c + b :=
begin
sorry
end
def succ_le_succ_iff (a b : mynat) : succ a ≤ succ b ↔ a ≤ b :=
begin
sorry
end
def succ_lt_succ_iff (a b : mynat) : succ a < succ b ↔ a < b :=
begin
sorry
end
theorem lt_of_add_lt_add_left : ∀ {{a b c : mynat}}, a + b < a + c → b < c :=
begin
sorry
end
theorem le_iff_exists_add : ∀ (a b : mynat), a ≤ b ↔ ∃ (c : mynat), b = a + c :=
begin
sorry
end
theorem zero_ne_one : (0 : mynat) ≠ 1 :=
begin
sorry
end
instance : ordered_comm_monoid mynat := by structure_helper
theorem le_of_add_le_add_left ⦃ a b c : mynat⦄ : a + b ≤ a + c → b ≤ c :=
begin
sorry
end
instance : ordered_cancel_comm_monoid mynat := by structure_helper
theorem mul_le_mul_of_nonneg_left ⦃a b c : mynat⦄ : a ≤ b → 0 ≤ c → c * a ≤ c * b :=
begin
sorry
end
theorem mul_le_mul_of_nonneg_right ⦃a b c : mynat⦄ : a ≤ b → 0 ≤ c → a * c ≤ b * c :=
begin
sorry
end
theorem ne_zero_of_pos ⦃a : mynat⦄ : 0 < a → a ≠ 0 :=
begin
sorry
end
theorem mul_lt_mul_of_pos_left ⦃a b c : mynat⦄ : a < b → 0 < c → c * a < c * b :=
begin
sorry
end
theorem mul_lt_mul_of_pos_right ⦃a b c : mynat⦄ : a < b → 0 < c → a * c < b * c :=
begin
sorry
end
instance : ordered_semiring mynat := by structure_helper
lemma lt_irrefl (a : mynat) : ¬ (a < a) :=
begin
sorry
end
end mynat
|
e76140ee6f14db30e36b4db802caee632ac06c15 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/topology/category/Profinite/default.lean | 615e55709b93db5bdd624c1e087afc9b10338173 | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 11,365 | lean | /-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Calle Sönne
-/
import topology.category.CompHaus
import topology.connected
import topology.subset_properties
import topology.locally_constant.basic
import category_theory.adjunction.reflective
import category_theory.monad.limits
import category_theory.limits.constructions.epi_mono
import category_theory.Fintype
/-!
# The category of Profinite Types
We construct the category of profinite topological spaces,
often called profinite sets -- perhaps they could be called
profinite types in Lean.
The type of profinite topological spaces is called `Profinite`. It has a category
instance and is a fully faithful subcategory of `Top`. The fully faithful functor
is called `Profinite_to_Top`.
## Implementation notes
A profinite type is defined to be a topological space which is
compact, Hausdorff and totally disconnected.
## TODO
0. Link to category of projective limits of finite discrete sets.
1. finite coproducts
2. Clausen/Scholze topology on the category `Profinite`.
## Tags
profinite
-/
universe u
open category_theory
/-- The type of profinite topological spaces. -/
structure Profinite :=
(to_CompHaus : CompHaus)
[is_totally_disconnected : totally_disconnected_space to_CompHaus]
namespace Profinite
/--
Construct a term of `Profinite` from a type endowed with the structure of a
compact, Hausdorff and totally disconnected topological space.
-/
def of (X : Type*) [topological_space X] [compact_space X] [t2_space X]
[totally_disconnected_space X] : Profinite := ⟨⟨⟨X⟩⟩⟩
instance : inhabited Profinite := ⟨Profinite.of pempty⟩
instance category : category Profinite := induced_category.category to_CompHaus
instance concrete_category : concrete_category Profinite := induced_category.concrete_category _
instance has_forget₂ : has_forget₂ Profinite Top := induced_category.has_forget₂ _
instance : has_coe_to_sort Profinite Type* := ⟨λ X, X.to_CompHaus⟩
instance {X : Profinite} : totally_disconnected_space X := X.is_totally_disconnected
-- We check that we automatically infer that Profinite sets are compact and Hausdorff.
example {X : Profinite} : compact_space X := infer_instance
example {X : Profinite} : t2_space X := infer_instance
@[simp]
lemma coe_to_CompHaus {X : Profinite} : (X.to_CompHaus : Type*) = X :=
rfl
@[simp] lemma coe_id (X : Profinite) : (𝟙 X : X → X) = id := rfl
@[simp] lemma coe_comp {X Y Z : Profinite} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g : X → Z) = g ∘ f := rfl
end Profinite
/-- The fully faithful embedding of `Profinite` in `CompHaus`. -/
@[simps, derive [full, faithful]]
def Profinite_to_CompHaus : Profinite ⥤ CompHaus := induced_functor _
/-- The fully faithful embedding of `Profinite` in `Top`. This is definitionally the same as the
obvious composite. -/
@[simps, derive [full, faithful]]
def Profinite.to_Top : Profinite ⥤ Top := forget₂ _ _
@[simp] lemma Profinite.to_CompHaus_to_Top :
Profinite_to_CompHaus ⋙ CompHaus_to_Top = Profinite.to_Top :=
rfl
section Profinite
/--
(Implementation) The object part of the connected_components functor from compact Hausdorff spaces
to Profinite spaces, given by quotienting a space by its connected components.
See: https://stacks.math.columbia.edu/tag/0900
-/
-- Without explicit universe annotations here, Lean introduces two universe variables and
-- unhelpfully defines a function `CompHaus.{max u₁ u₂} → Profinite.{max u₁ u₂}`.
def CompHaus.to_Profinite_obj (X : CompHaus.{u}) : Profinite.{u} :=
{ to_CompHaus :=
{ to_Top := Top.of (connected_components X),
is_compact := quotient.compact_space,
is_hausdorff := connected_components.t2 },
is_totally_disconnected := connected_components.totally_disconnected_space }
/--
(Implementation) The bijection of homsets to establish the reflective adjunction of Profinite
spaces in compact Hausdorff spaces.
-/
def Profinite.to_CompHaus_equivalence (X : CompHaus.{u}) (Y : Profinite.{u}) :
(CompHaus.to_Profinite_obj X ⟶ Y) ≃ (X ⟶ Profinite_to_CompHaus.obj Y) :=
{ to_fun := λ f, f.comp ⟨quotient.mk', continuous_quotient_mk⟩,
inv_fun := λ g,
{ to_fun := continuous.connected_components_lift g.2,
continuous_to_fun := continuous.connected_components_lift_continuous g.2},
left_inv := λ f, continuous_map.ext $ connected_components.surjective_coe.forall.2 $ λ a, rfl,
right_inv := λ f, continuous_map.ext $ λ x, rfl }
/--
The connected_components functor from compact Hausdorff spaces to profinite spaces,
left adjoint to the inclusion functor.
-/
def CompHaus.to_Profinite : CompHaus ⥤ Profinite :=
adjunction.left_adjoint_of_equiv Profinite.to_CompHaus_equivalence (λ _ _ _ _ _, rfl)
lemma CompHaus.to_Profinite_obj' (X : CompHaus) :
↥(CompHaus.to_Profinite.obj X) = connected_components X := rfl
/-- Finite types are given the discrete topology. -/
def Fintype.discrete_topology (A : Fintype) : topological_space A := ⊥
section discrete_topology
local attribute [instance] Fintype.discrete_topology
/-- The natural functor from `Fintype` to `Profinite`, endowing a finite type with the
discrete topology. -/
@[simps] def Fintype.to_Profinite : Fintype ⥤ Profinite :=
{ obj := λ A, Profinite.of A,
map := λ _ _ f, ⟨f⟩ }
end discrete_topology
end Profinite
namespace Profinite
-- TODO the following construction of limits could be generalised
-- to allow diagrams in lower universes.
/-- An explicit limit cone for a functor `F : J ⥤ Profinite`, defined in terms of
`Top.limit_cone`. -/
def limit_cone {J : Type u} [small_category J] (F : J ⥤ Profinite.{u}) :
limits.cone F :=
{ X :=
{ to_CompHaus := (CompHaus.limit_cone.{u u} (F ⋙ Profinite_to_CompHaus)).X,
is_totally_disconnected :=
begin
change totally_disconnected_space ↥{u : Π (j : J), (F.obj j) | _},
exact subtype.totally_disconnected_space,
end },
π := { app := (CompHaus.limit_cone.{u u} (F ⋙ Profinite_to_CompHaus)).π.app } }
/-- The limit cone `Profinite.limit_cone F` is indeed a limit cone. -/
def limit_cone_is_limit {J : Type u} [small_category J] (F : J ⥤ Profinite.{u}) :
limits.is_limit (limit_cone F) :=
{ lift := λ S, (CompHaus.limit_cone_is_limit.{u u} (F ⋙ Profinite_to_CompHaus)).lift
(Profinite_to_CompHaus.map_cone S),
uniq' := λ S m h,
(CompHaus.limit_cone_is_limit.{u u} _).uniq (Profinite_to_CompHaus.map_cone S) _ h }
/-- The adjunction between CompHaus.to_Profinite and Profinite.to_CompHaus -/
def to_Profinite_adj_to_CompHaus : CompHaus.to_Profinite ⊣ Profinite_to_CompHaus :=
adjunction.adjunction_of_equiv_left _ _
/-- The category of profinite sets is reflective in the category of compact hausdroff spaces -/
instance to_CompHaus.reflective : reflective Profinite_to_CompHaus :=
{ to_is_right_adjoint := ⟨CompHaus.to_Profinite, Profinite.to_Profinite_adj_to_CompHaus⟩ }
noncomputable
instance to_CompHaus.creates_limits : creates_limits Profinite_to_CompHaus :=
monadic_creates_limits _
noncomputable
instance to_Top.reflective : reflective Profinite.to_Top :=
reflective.comp Profinite_to_CompHaus CompHaus_to_Top
noncomputable
instance to_Top.creates_limits : creates_limits Profinite.to_Top :=
monadic_creates_limits _
instance has_limits : limits.has_limits Profinite :=
has_limits_of_has_limits_creates_limits Profinite.to_Top
instance has_colimits : limits.has_colimits Profinite :=
has_colimits_of_reflective Profinite_to_CompHaus
noncomputable
instance forget_preserves_limits : limits.preserves_limits (forget Profinite) :=
by apply limits.comp_preserves_limits Profinite.to_Top (forget Top)
variables {X Y : Profinite.{u}} (f : X ⟶ Y)
/-- Any morphism of profinite spaces is a closed map. -/
lemma is_closed_map : is_closed_map f :=
CompHaus.is_closed_map _
/-- Any continuous bijection of profinite spaces induces an isomorphism. -/
lemma is_iso_of_bijective (bij : function.bijective f) : is_iso f :=
begin
haveI := CompHaus.is_iso_of_bijective (Profinite_to_CompHaus.map f) bij,
exact is_iso_of_fully_faithful Profinite_to_CompHaus _
end
/-- Any continuous bijection of profinite spaces induces an isomorphism. -/
noncomputable def iso_of_bijective (bij : function.bijective f) : X ≅ Y :=
by letI := Profinite.is_iso_of_bijective f bij; exact as_iso f
instance forget_reflects_isomorphisms : reflects_isomorphisms (forget Profinite) :=
⟨by introsI A B f hf; exact Profinite.is_iso_of_bijective _ ((is_iso_iff_bijective f).mp hf)⟩
/-- Construct an isomorphism from a homeomorphism. -/
@[simps hom inv] def iso_of_homeo (f : X ≃ₜ Y) : X ≅ Y :=
{ hom := ⟨f, f.continuous⟩,
inv := ⟨f.symm, f.symm.continuous⟩,
hom_inv_id' := by { ext x, exact f.symm_apply_apply x },
inv_hom_id' := by { ext x, exact f.apply_symm_apply x } }
/-- Construct a homeomorphism from an isomorphism. -/
@[simps] def homeo_of_iso (f : X ≅ Y) : X ≃ₜ Y :=
{ to_fun := f.hom,
inv_fun := f.inv,
left_inv := λ x, by { change (f.hom ≫ f.inv) x = x, rw [iso.hom_inv_id, coe_id, id.def] },
right_inv := λ x, by { change (f.inv ≫ f.hom) x = x, rw [iso.inv_hom_id, coe_id, id.def] },
continuous_to_fun := f.hom.continuous,
continuous_inv_fun := f.inv.continuous }
/-- The equivalence between isomorphisms in `Profinite` and homeomorphisms
of topological spaces. -/
@[simps] def iso_equiv_homeo : (X ≅ Y) ≃ (X ≃ₜ Y) :=
{ to_fun := homeo_of_iso,
inv_fun := iso_of_homeo,
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, refl } }
lemma epi_iff_surjective {X Y : Profinite.{u}} (f : X ⟶ Y) : epi f ↔ function.surjective f :=
begin
split,
{ contrapose!,
rintros ⟨y, hy⟩ hf,
let C := set.range f,
have hC : is_closed C := (is_compact_range f.continuous).is_closed,
let U := Cᶜ,
have hU : is_open U := is_open_compl_iff.mpr hC,
have hyU : y ∈ U,
{ refine set.mem_compl _, rintro ⟨y', hy'⟩, exact hy y' hy' },
have hUy : U ∈ nhds y := hU.mem_nhds hyU,
obtain ⟨V, hV, hyV, hVU⟩ := is_topological_basis_clopen.mem_nhds_iff.mp hUy,
classical,
letI : topological_space (ulift.{u} $ fin 2) := ⊥,
let Z := of (ulift.{u} $ fin 2),
let g : Y ⟶ Z := ⟨(locally_constant.of_clopen hV).map ulift.up, locally_constant.continuous _⟩,
let h : Y ⟶ Z := ⟨λ _, ⟨1⟩, continuous_const⟩,
have H : h = g,
{ rw ← cancel_epi f,
ext x, dsimp [locally_constant.of_clopen],
rw if_neg, { refl },
refine mt (λ α, hVU α) _,
simp only [set.mem_range_self, not_true, not_false_iff, set.mem_compl_eq], },
apply_fun (λ e, (e y).down) at H,
dsimp [locally_constant.of_clopen] at H,
rw if_pos hyV at H,
exact top_ne_bot H },
{ rw ← category_theory.epi_iff_surjective,
apply faithful_reflects_epi (forget Profinite) },
end
lemma mono_iff_injective {X Y : Profinite.{u}} (f : X ⟶ Y) : mono f ↔ function.injective f :=
begin
split,
{ intro h,
haveI : limits.preserves_limits Profinite_to_CompHaus := infer_instance,
haveI : mono (Profinite_to_CompHaus.map f) := infer_instance,
rwa ← CompHaus.mono_iff_injective },
{ rw ← category_theory.mono_iff_injective,
apply faithful_reflects_mono (forget Profinite) }
end
end Profinite
|
20c76e1a7ffa7de5a38292a9d9bfa4f987b079b9 | b2fe74b11b57d362c13326bc5651244f111fa6f4 | /src/topology/algebra/ordered.lean | 4cb5b44632cbc0f2aa77ff59edce1584a3627dce | [
"Apache-2.0"
] | permissive | midfield/mathlib | c4db5fa898b5ac8f2f80ae0d00c95eb6f745f4c7 | 775edc615ecec631d65b6180dbcc7bc26c3abc26 | refs/heads/master | 1,675,330,551,921 | 1,608,304,514,000 | 1,608,304,514,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 150,346 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import tactic.linarith
import tactic.tfae
import algebra.archimedean
import algebra.group.pi
import algebra.ordered_ring
import order.liminf_limsup
import data.set.intervals.image_preimage
import data.set.intervals.ord_connected
import data.set.intervals.surj_on
import data.set.intervals.pi
import topology.algebra.group
import topology.extend_from_subset
import order.filter.interval
/-!
# Theory of topology on ordered spaces
## Main definitions
The order topology on an ordered space is the topology generated by all open intervals (or
equivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `preorder.topology α`.
However, we do *not* register it as an instance (as many existing ordered types already have
topologies, which would be equal but not definitionally equal to `preorder.topology α`). Instead,
we introduce a class `order_topology α`(which is a `Prop`, also known as a mixin) saying that on
the type `α` having already a topological space structure and a preorder structure, the topological
structure is equal to the order topology.
We also introduce another (mixin) class `order_closed_topology α` saying that the set of points
`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear
order with the order topology.
We prove many basic properties of such topologies.
## Main statements
This file contains the proofs of the following facts. For exact requirements (`order_closed_topology`
vs `order_topology`, `preorder` vs `partial_order` vs `linear_order` etc) see their statements.
### Open / closed sets
* `is_open_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;
* `is_open_Iio`, `is_open_Ioi`, `is_open_Ioo` : open intervals are open;
* `is_closed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;
* `is_closed_Iic`, `is_closed_Ici`, `is_closed_Icc` : closed intervals are closed;
* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`
and `{x | f x < g x}` are included by `{x | f x = g x}`;
* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any
neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood
of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.
### Convergence and inequalities
* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually
`f x ≤ g x`, then `a ≤ b`
* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`
(resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a); we also provide primed versions
that assume the inequalities to hold for all `x`.
### Min, max, `Sup` and `Inf`
* `continuous.min`, `continuous.max`: pointwise `min`/`max` of two continuous functions is
continuous.
* `tendsto.min`, `tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise
`min`/`max` tend to `min a b` and `max a b`, respectively.
* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,
sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`
both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.
### Connected sets and Intermediate Value Theorem
* `is_preconnected_I??` : all intervals `I??` are preconnected,
* `is_preconnected.intermediate_value`, `intermediate_value_univ` : Intermediate Value Theorem for
connected sets and connected spaces, respectively;
* `intermediate_value_Icc`, `intermediate_value_Icc'`: Intermediate Value Theorem for functions
on closed intervals.
### Miscellaneous facts
* `is_compact.exists_forall_le`, `is_compact.exists_forall_ge` : extreme value theorem, a continuous
function on a compact set takes its minimum and maximum values.
* `is_closed.Icc_subset_of_forall_mem_nhds_within` : “Continuous induction” principle;
if `s ∩ [a, b]` is closed, `a ∈ s`, and for each `x ∈ [a, b) ∩ s` some of its right neighborhoods
is included `s`, then `[a, b] ⊆ s`.
* `is_closed.Icc_subset_of_forall_exists_gt`, `is_closed.mem_of_ge_of_forall_exists_gt` : two
other versions of the “continuous induction” principle.
## Implementation
We do _not_ register the order topology as an instance on a preorder (or even on a linear order).
Indeed, on many such spaces, a topology has already been constructed in a different way (think
of the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),
and is in general not defeq to the one generated by the intervals. We make it available as a
definition `preorder.topology α` though, that can be registered as an instance when necessary, or
for specific types.
-/
open classical set filter topological_space
open function (curry uncurry)
open_locale topological_space classical filter
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w}
/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the
set of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.
This property is satisfied for the order topology on a linear order, but it can be satisfied more
generally, and suffices to derive many interesting properties relating order and topology. -/
class order_closed_topology (α : Type*) [topological_space α] [preorder α] : Prop :=
(is_closed_le' : is_closed {p:α×α | p.1 ≤ p.2})
instance : Π [topological_space α], topological_space (order_dual α) := id
section order_closed_topology
section preorder
variables [topological_space α] [preorder α] [t : order_closed_topology α]
include t
lemma is_closed_le_prod : is_closed {p : α × α | p.1 ≤ p.2} :=
t.is_closed_le'
lemma is_closed_le [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :
is_closed {b | f b ≤ g b} :=
continuous_iff_is_closed.mp (hf.prod_mk hg) _ is_closed_le_prod
lemma is_closed_le' (a : α) : is_closed {b | b ≤ a} :=
is_closed_le continuous_id continuous_const
lemma is_closed_Iic {a : α} : is_closed (Iic a) :=
is_closed_le' a
lemma is_closed_ge' (a : α) : is_closed {b | a ≤ b} :=
is_closed_le continuous_const continuous_id
lemma is_closed_Ici {a : α} : is_closed (Ici a) :=
is_closed_ge' a
instance : order_closed_topology (order_dual α) :=
⟨(@order_closed_topology.is_closed_le' α _ _ _).preimage continuous_swap⟩
lemma is_closed_Icc {a b : α} : is_closed (Icc a b) :=
is_closed_inter is_closed_Ici is_closed_Iic
@[simp] lemma closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=
is_closed_Icc.closure_eq
@[simp] lemma closure_Iic (a : α) : closure (Iic a) = Iic a :=
is_closed_Iic.closure_eq
@[simp] lemma closure_Ici (a : α) : closure (Ici a) = Ici a :=
is_closed_Ici.closure_eq
lemma le_of_tendsto_of_tendsto {f g : β → α} {b : filter β} {a₁ a₂ : α} [ne_bot b]
(hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) :
a₁ ≤ a₂ :=
have tendsto (λb, (f b, g b)) b (𝓝 (a₁, a₂)),
by rw [nhds_prod_eq]; exact hf.prod_mk hg,
show (a₁, a₂) ∈ {p:α×α | p.1 ≤ p.2},
from t.is_closed_le'.mem_of_tendsto this h
lemma le_of_tendsto_of_tendsto' {f g : β → α} {b : filter β} {a₁ a₂ : α} [ne_bot b]
(hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) :
a₁ ≤ a₂ :=
le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)
lemma le_of_tendsto {f : β → α} {a b : α} {x : filter β}
[ne_bot x] (lim : tendsto f x (𝓝 a)) (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=
le_of_tendsto_of_tendsto lim tendsto_const_nhds h
lemma le_of_tendsto' {f : β → α} {a b : α} {x : filter β}
[ne_bot x] (lim : tendsto f x (𝓝 a)) (h : ∀ c, f c ≤ b) : a ≤ b :=
le_of_tendsto lim (eventually_of_forall h)
lemma ge_of_tendsto {f : β → α} {a b : α} {x : filter β} [ne_bot x]
(lim : tendsto f x (𝓝 a)) (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=
le_of_tendsto_of_tendsto tendsto_const_nhds lim h
lemma ge_of_tendsto' {f : β → α} {a b : α} {x : filter β} [ne_bot x]
(lim : tendsto f x (𝓝 a)) (h : ∀ c, b ≤ f c) : b ≤ a :=
ge_of_tendsto lim (eventually_of_forall h)
@[simp]
lemma closure_le_eq [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :
closure {b | f b ≤ g b} = {b | f b ≤ g b} :=
(is_closed_le hf hg).closure_eq
lemma closure_lt_subset_le [topological_space β] {f g : β → α} (hf : continuous f)
(hg : continuous g) :
closure {b | f b < g b} ⊆ {b | f b ≤ g b} :=
by { rw [←closure_le_eq hf hg], exact closure_mono (λ b, le_of_lt) }
lemma continuous_within_at.closure_le [topological_space β]
{f g : β → α} {s : set β} {x : β} (hx : x ∈ closure s)
(hf : continuous_within_at f s x)
(hg : continuous_within_at g s x)
(h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=
show (f x, g x) ∈ {p : α × α | p.1 ≤ p.2},
from order_closed_topology.is_closed_le'.closure_subset ((hf.prod hg).mem_closure hx h)
/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,
then the set `{x ∈ s | f x ≤ g x}` is a closed set. -/
lemma is_closed.is_closed_le [topological_space β] {f g : β → α} {s : set β} (hs : is_closed s)
(hf : continuous_on f s) (hg : continuous_on g s) :
is_closed {x ∈ s | f x ≤ g x} :=
(hf.prod hg).preimage_closed_of_closed hs order_closed_topology.is_closed_le'
omit t
lemma nhds_within_Ici_ne_bot {a b : α} (H₂ : a ≤ b) :
ne_bot (𝓝[Ici a] b) :=
nhds_within_ne_bot_of_mem H₂
@[instance] lemma nhds_within_Ici_self_ne_bot (a : α) :
ne_bot (𝓝[Ici a] a) :=
nhds_within_Ici_ne_bot (le_refl a)
lemma nhds_within_Iic_ne_bot {a b : α} (H : a ≤ b) :
ne_bot (𝓝[Iic b] a) :=
nhds_within_ne_bot_of_mem H
@[instance] lemma nhds_within_Iic_self_ne_bot (a : α) :
ne_bot (𝓝[Iic a] a) :=
nhds_within_Iic_ne_bot (le_refl a)
end preorder
section partial_order
variables [topological_space α] [partial_order α] [t : order_closed_topology α]
include t
private lemma is_closed_eq : is_closed {p : α × α | p.1 = p.2} :=
by simp only [le_antisymm_iff];
exact is_closed_inter t.is_closed_le' (is_closed_le continuous_snd continuous_fst)
@[priority 90] -- see Note [lower instance priority]
instance order_closed_topology.to_t2_space : t2_space α :=
{ t2 :=
have is_open {p : α × α | p.1 ≠ p.2}, from is_closed_eq,
assume a b h,
let ⟨u, v, hu, hv, ha, hb, h⟩ := is_open_prod_iff.mp this a b h in
⟨u, v, hu, hv, ha, hb,
set.eq_empty_iff_forall_not_mem.2 $ assume a ⟨h₁, h₂⟩,
have a ≠ a, from @h (a, a) ⟨h₁, h₂⟩,
this rfl⟩ }
end partial_order
section linear_order
variables [topological_space α] [linear_order α] [order_closed_topology α]
lemma is_open_lt_prod : is_open {p : α × α | p.1 < p.2} :=
by { simp_rw [← is_closed_compl_iff, compl_set_of, not_lt],
exact is_closed_le continuous_snd continuous_fst }
lemma is_open_lt [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :
is_open {b | f b < g b} :=
by simp [lt_iff_not_ge, -not_le]; exact is_closed_le hg hf
variables {a b : α}
lemma is_open_Iio : is_open (Iio a) :=
is_open_lt continuous_id continuous_const
lemma is_open_Ioi : is_open (Ioi a) :=
is_open_lt continuous_const continuous_id
lemma is_open_Ioo : is_open (Ioo a b) :=
is_open_inter is_open_Ioi is_open_Iio
@[simp] lemma interior_Ioi : interior (Ioi a) = Ioi a :=
is_open_Ioi.interior_eq
@[simp] lemma interior_Iio : interior (Iio a) = Iio a :=
is_open_Iio.interior_eq
@[simp] lemma interior_Ioo : interior (Ioo a b) = Ioo a b :=
is_open_Ioo.interior_eq
variables [topological_space γ]
/-- Intermediate value theorem for two functions: if `f` and `g` are two continuous functions
on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`. -/
lemma intermediate_value_univ₂ [preconnected_space γ] {a b : γ} {f g : γ → α} (hf : continuous f)
(hg : continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b) :
∃ x, f x = g x :=
begin
obtain ⟨x, h, hfg, hgf⟩ : (univ ∩ {x | f x ≤ g x ∧ g x ≤ f x}).nonempty,
from is_preconnected_closed_iff.1 preconnected_space.is_preconnected_univ _ _
(is_closed_le hf hg) (is_closed_le hg hf) (λ x hx, le_total _ _) ⟨a, trivial, ha⟩
⟨b, trivial, hb⟩,
exact ⟨x, le_antisymm hfg hgf⟩
end
/-- Intermediate value theorem for two functions: if `f` and `g` are two functions continuous
on a preconnected set `s` and for some `a b ∈ s` we have `f a ≤ g a` and `g b ≤ f b`,
then for some `x ∈ s` we have `f x = g x`. -/
lemma is_preconnected.intermediate_value₂ {s : set γ} (hs : is_preconnected s)
{a b : γ} (ha : a ∈ s) (hb : b ∈ s) {f g : γ → α}
(hf : continuous_on f s) (hg : continuous_on g s) (ha' : f a ≤ g a) (hb' : g b ≤ f b) :
∃ x ∈ s, f x = g x :=
let ⟨x, hx⟩ := @intermediate_value_univ₂ α s _ _ _ _ (subtype.preconnected_space hs) ⟨a, ha⟩ ⟨b, hb⟩
_ _ (continuous_on_iff_continuous_restrict.1 hf) (continuous_on_iff_continuous_restrict.1 hg)
ha' hb'
in ⟨x, x.2, hx⟩
/-- Intermediate Value Theorem for continuous functions on connected sets. -/
lemma is_preconnected.intermediate_value {s : set γ} (hs : is_preconnected s)
{a b : γ} (ha : a ∈ s) (hb : b ∈ s) {f : γ → α} (hf : continuous_on f s) :
Icc (f a) (f b) ⊆ f '' s :=
λ x hx, mem_image_iff_bex.2 $ hs.intermediate_value₂ ha hb hf continuous_on_const hx.1 hx.2
/-- Intermediate Value Theorem for continuous functions on connected spaces. -/
lemma intermediate_value_univ [preconnected_space γ] (a b : γ) {f : γ → α} (hf : continuous f) :
Icc (f a) (f b) ⊆ range f :=
λ x hx, intermediate_value_univ₂ hf continuous_const hx.1 hx.2
/-- Intermediate Value Theorem for continuous functions on connected spaces. -/
lemma mem_range_of_exists_le_of_exists_ge [preconnected_space γ] {c : α} {f : γ → α}
(hf : continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b) :
c ∈ range f :=
let ⟨a, ha⟩ := h₁, ⟨b, hb⟩ := h₂ in intermediate_value_univ a b hf ⟨ha, hb⟩
/-- If a preconnected set contains endpoints of an interval, then it includes the whole interval. -/
lemma is_preconnected.Icc_subset {s : set α} (hs : is_preconnected s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) :
Icc a b ⊆ s :=
by simpa only [image_id] using hs.intermediate_value ha hb continuous_on_id
/-- If a preconnected set contains endpoints of an interval, then it includes the whole interval. -/
lemma is_connected.Icc_subset {s : set α} (hs : is_connected s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) :
Icc a b ⊆ s :=
hs.2.Icc_subset ha hb
/-- If preconnected set in a linear order space is unbounded below and above, then it is the whole
space. -/
lemma is_preconnected.eq_univ_of_unbounded {s : set α} (hs : is_preconnected s) (hb : ¬bdd_below s)
(ha : ¬bdd_above s) :
s = univ :=
begin
refine eq_univ_of_forall (λ x, _),
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := not_bdd_below_iff.1 hb x,
obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bdd_above_iff.1 ha x,
exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩
end
/-!
### Neighborhoods to the left and to the right on an `order_closed_topology`
Limits to the left and to the right of real functions are defined in terms of neighborhoods to
the left and to the right, either open or closed, i.e., members of `𝓝[Ioi a] a` and
`𝓝[Ici a] a` on the right, and similarly on the left. Here we simply prove that all
right-neighborhoods of a point are equal, and we'll prove later other useful characterizations which
require the stronger hypothesis `order_topology α` -/
/-!
#### Right neighborhoods, point excluded
-/
lemma Ioo_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Ioo a c ∈ 𝓝[Ioi b] b :=
mem_nhds_within.2 ⟨Iio c, is_open_Iio, H.2,
by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩
lemma Ioc_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Ioc a c ∈ 𝓝[Ioi b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Ioi H) Ioo_subset_Ioc_self
lemma Ico_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Ico a c ∈ 𝓝[Ioi b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Ioi H) Ioo_subset_Ico_self
lemma Icc_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Icc a c ∈ 𝓝[Ioi b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Ioi H) Ioo_subset_Icc_self
@[simp] lemma nhds_within_Ioc_eq_nhds_within_Ioi {a b : α} (h : a < b) :
𝓝[Ioc a b] a = 𝓝[Ioi a] a :=
le_antisymm (nhds_within_mono _ Ioc_subset_Ioi_self) $
nhds_within_le_of_mem $ Ioc_mem_nhds_within_Ioi $ left_mem_Ico.2 h
@[simp] lemma nhds_within_Ioo_eq_nhds_within_Ioi {a b : α} (h : a < b) :
𝓝[Ioo a b] a = 𝓝[Ioi a] a :=
le_antisymm (nhds_within_mono _ Ioo_subset_Ioi_self) $
nhds_within_le_of_mem $ Ioo_mem_nhds_within_Ioi $ left_mem_Ico.2 h
@[simp]
lemma continuous_within_at_Ioc_iff_Ioi [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioc a b) a ↔ continuous_within_at f (Ioi a) a :=
by simp only [continuous_within_at, nhds_within_Ioc_eq_nhds_within_Ioi h]
@[simp]
lemma continuous_within_at_Ioo_iff_Ioi [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioo a b) a ↔ continuous_within_at f (Ioi a) a :=
by simp only [continuous_within_at, nhds_within_Ioo_eq_nhds_within_Ioi h]
/-!
#### Left neighborhoods, point excluded
-/
lemma Ioo_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Ioo a c ∈ 𝓝[Iio b] b :=
by simpa only [dual_Ioo] using @Ioo_mem_nhds_within_Ioi (order_dual α) _ _ _ _ _ _ ⟨H.2, H.1⟩
lemma Ico_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Ico a c ∈ 𝓝[Iio b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Iio H) Ioo_subset_Ico_self
lemma Ioc_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Ioc a c ∈ 𝓝[Iio b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Iio H) Ioo_subset_Ioc_self
lemma Icc_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Icc a c ∈ 𝓝[Iio b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Iio H) Ioo_subset_Icc_self
@[simp] lemma nhds_within_Ico_eq_nhds_within_Iio {a b : α} (h : a < b) :
𝓝[Ico a b] b = 𝓝[Iio b] b :=
by simpa only [dual_Ioc] using @nhds_within_Ioc_eq_nhds_within_Ioi (order_dual α) _ _ _ _ _ h
@[simp] lemma nhds_within_Ioo_eq_nhds_within_Iio {a b : α} (h : a < b) :
𝓝[Ioo a b] b = 𝓝[Iio b] b :=
by simpa only [dual_Ioo] using @nhds_within_Ioo_eq_nhds_within_Ioi (order_dual α) _ _ _ _ _ h
@[simp] lemma continuous_within_at_Ico_iff_Iio [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ico a b) b ↔ continuous_within_at f (Iio b) b :=
by simp only [continuous_within_at, nhds_within_Ico_eq_nhds_within_Iio h]
@[simp] lemma continuous_within_at_Ioo_iff_Iio [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioo a b) b ↔ continuous_within_at f (Iio b) b :=
by simp only [continuous_within_at, nhds_within_Ioo_eq_nhds_within_Iio h]
/-!
#### Right neighborhoods, point included
-/
lemma Ioo_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ioo a c) :
Ioo a c ∈ 𝓝[Ici b] b :=
mem_nhds_within_of_mem_nhds $ mem_nhds_sets is_open_Ioo H
lemma Ioc_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ioo a c) :
Ioc a c ∈ 𝓝[Ici b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Ici H) Ioo_subset_Ioc_self
lemma Ico_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ico a c) :
Ico a c ∈ 𝓝[Ici b] b :=
mem_nhds_within.2 ⟨Iio c, is_open_Iio, H.2,
by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩
lemma Icc_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ico a c) :
Icc a c ∈ 𝓝[Ici b] b :=
mem_sets_of_superset (Ico_mem_nhds_within_Ici H) Ico_subset_Icc_self
@[simp] lemma nhds_within_Icc_eq_nhds_within_Ici {a b : α} (h : a < b) :
𝓝[Icc a b] a = 𝓝[Ici a] a :=
le_antisymm (nhds_within_mono _ Icc_subset_Ici_self) $
nhds_within_le_of_mem $ Icc_mem_nhds_within_Ici $ left_mem_Ico.2 h
@[simp] lemma nhds_within_Ico_eq_nhds_within_Ici {a b : α} (h : a < b) :
𝓝[Ico a b] a = 𝓝[Ici a] a :=
le_antisymm (nhds_within_mono _ (λ x, and.left)) $
nhds_within_le_of_mem $ Ico_mem_nhds_within_Ici $ left_mem_Ico.2 h
@[simp]
lemma continuous_within_at_Icc_iff_Ici [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Icc a b) a ↔ continuous_within_at f (Ici a) a :=
by simp only [continuous_within_at, nhds_within_Icc_eq_nhds_within_Ici h]
@[simp]
lemma continuous_within_at_Ico_iff_Ici [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ico a b) a ↔ continuous_within_at f (Ici a) a :=
by simp only [continuous_within_at, nhds_within_Ico_eq_nhds_within_Ici h]
/-!
#### Left neighborhoods, point included
-/
lemma Ioo_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioo a c) :
Ioo a c ∈ 𝓝[Iic b] b :=
mem_nhds_within_of_mem_nhds $ mem_nhds_sets is_open_Ioo H
lemma Ico_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioo a c) :
Ico a c ∈ 𝓝[Iic b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Iic H) Ioo_subset_Ico_self
lemma Ioc_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioc a c) :
Ioc a c ∈ 𝓝[Iic b] b :=
by simpa only [dual_Ico] using @Ico_mem_nhds_within_Ici (order_dual α) _ _ _ _ _ _ ⟨H.2, H.1⟩
lemma Icc_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioc a c) :
Icc a c ∈ 𝓝[Iic b] b :=
mem_sets_of_superset (Ioc_mem_nhds_within_Iic H) Ioc_subset_Icc_self
@[simp] lemma nhds_within_Icc_eq_nhds_within_Iic {a b : α} (h : a < b) :
𝓝[Icc a b] b = 𝓝[Iic b] b :=
by simpa only [dual_Icc] using @nhds_within_Icc_eq_nhds_within_Ici (order_dual α) _ _ _ _ _ h
@[simp] lemma nhds_within_Ioc_eq_nhds_within_Iic {a b : α} (h : a < b) :
𝓝[Ioc a b] b = 𝓝[Iic b] b :=
by simpa only [dual_Ico] using @nhds_within_Ico_eq_nhds_within_Ici (order_dual α) _ _ _ _ _ h
@[simp]
lemma continuous_within_at_Icc_iff_Iic [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Icc a b) b ↔ continuous_within_at f (Iic b) b :=
by simp only [continuous_within_at, nhds_within_Icc_eq_nhds_within_Iic h]
@[simp]
lemma continuous_within_at_Ioc_iff_Iic [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioc a b) b ↔ continuous_within_at f (Iic b) b :=
by simp only [continuous_within_at, nhds_within_Ioc_eq_nhds_within_Iic h]
end linear_order
section linear_order
variables [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α}
section
variables [topological_space β]
lemma frontier_le_subset_eq (hf : continuous f) (hg : continuous g) :
frontier {b | f b ≤ g b} ⊆ {b | f b = g b} :=
begin
rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg],
rintros b ⟨hb₁, hb₂⟩,
refine le_antisymm hb₁ (closure_lt_subset_le hg hf _),
convert hb₂ using 2, simp only [not_le.symm], refl
end
lemma frontier_lt_subset_eq (hf : continuous f) (hg : continuous g) :
frontier {b | f b < g b} ⊆ {b | f b = g b} :=
by rw ← frontier_compl;
convert frontier_le_subset_eq hg hf; simp [ext_iff, eq_comm]
@[continuity] lemma continuous.min (hf : continuous f) (hg : continuous g) :
continuous (λb, min (f b) (g b)) :=
have ∀b∈frontier {b | f b ≤ g b}, f b = g b, from assume b hb, frontier_le_subset_eq hf hg hb,
continuous_if this hf hg
@[continuity] lemma continuous.max (hf : continuous f) (hg : continuous g) :
continuous (λb, max (f b) (g b)) :=
@continuous.min (order_dual α) _ _ _ _ _ _ _ hf hg
end
lemma continuous_min : continuous (λ p : α × α, min p.1 p.2) := continuous_fst.min continuous_snd
lemma continuous_max : continuous (λ p : α × α, max p.1 p.2) := continuous_fst.max continuous_snd
lemma tendsto.max {b : filter β} {a₁ a₂ : α} (hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) :
tendsto (λb, max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=
(continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)
lemma tendsto.min {b : filter β} {a₁ a₂ : α} (hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) :
tendsto (λb, min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=
(continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)
end linear_order
end order_closed_topology
/-- The order topology on an ordered type is the topology generated by open intervals. We register
it on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.
We define it as a mixin. If you want to introduce the order topology on a preorder, use
`preorder.topology`. -/
class order_topology (α : Type*) [t : topological_space α] [preorder α] : Prop :=
(topology_eq_generate_intervals : t = generate_from {s | ∃a, s = Ioi a ∨ s = Iio a})
/-- (Order) topology on a partial order `α` generated by the subbase of open intervals
`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an
instance as many ordered sets are already endowed with the same topology, most often in a non-defeq
way though. Register as a local instance when necessary. -/
def preorder.topology (α : Type*) [preorder α] : topological_space α :=
generate_from {s : set α | ∃ (a : α), s = {b : α | a < b} ∨ s = {b : α | b < a}}
section order_topology
instance {α : Type*} [topological_space α] [partial_order α] [order_topology α] :
order_topology (order_dual α) :=
⟨by convert @order_topology.topology_eq_generate_intervals α _ _ _;
conv in (_ ∨ _) { rw or.comm }; refl⟩
section partial_order
variables [topological_space α] [partial_order α] [t : order_topology α]
include t
lemma is_open_iff_generate_intervals {s : set α} :
is_open s ↔ generate_open {s | ∃a, s = Ioi a ∨ s = Iio a} s :=
by rw [t.topology_eq_generate_intervals]; refl
lemma is_open_lt' (a : α) : is_open {b:α | a < b} :=
by rw [@is_open_iff_generate_intervals α _ _ t]; exact generate_open.basic _ ⟨a, or.inl rfl⟩
lemma is_open_gt' (a : α) : is_open {b:α | b < a} :=
by rw [@is_open_iff_generate_intervals α _ _ t]; exact generate_open.basic _ ⟨a, or.inr rfl⟩
lemma lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=
mem_nhds_sets (is_open_lt' _) h
lemma le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=
(𝓝 b).sets_of_superset (lt_mem_nhds h) $ assume b hb, le_of_lt hb
lemma gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=
mem_nhds_sets (is_open_gt' _) h
lemma ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=
(𝓝 a).sets_of_superset (gt_mem_nhds h) $ assume b hb, le_of_lt hb
lemma nhds_eq_order (a : α) :
𝓝 a = (⨅b ∈ Iio a, 𝓟 (Ioi b)) ⊓ (⨅b ∈ Ioi a, 𝓟 (Iio b)) :=
by rw [t.topology_eq_generate_intervals, nhds_generate_from];
from le_antisymm
(le_inf
(le_binfi $ assume b hb, infi_le_of_le {c : α | b < c} $ infi_le _ ⟨hb, b, or.inl rfl⟩)
(le_binfi $ assume b hb, infi_le_of_le {c : α | c < b} $ infi_le _ ⟨hb, b, or.inr rfl⟩))
(le_infi $ assume s, le_infi $ assume ⟨ha, b, hs⟩,
match s, ha, hs with
| _, h, (or.inl rfl) := inf_le_left_of_le $ infi_le_of_le b $ infi_le _ h
| _, h, (or.inr rfl) := inf_le_right_of_le $ infi_le_of_le b $ infi_le _ h
end)
lemma tendsto_order {f : β → α} {a : α} {x : filter β} :
tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ (∀ a' > a, ∀ᶠ b in x, f b < a') :=
by simp [nhds_eq_order a, tendsto_inf, tendsto_infi, tendsto_principal]
instance tendsto_Icc_class_nhds (a : α) : tendsto_Ixx_class Icc (𝓝 a) (𝓝 a) :=
begin
simp only [nhds_eq_order, infi_subtype'],
refine ((has_basis_infi_principal_finite _).inf
(has_basis_infi_principal_finite _)).tendsto_Ixx_class (λ s hs, _),
refine (ord_connected_bInter _).inter (ord_connected_bInter _); intros _ _,
exacts [ord_connected_Ioi, ord_connected_Iio]
end
instance tendsto_Ico_class_nhds (a : α) : tendsto_Ixx_class Ico (𝓝 a) (𝓝 a) :=
tendsto_Ixx_class_of_subset (λ _ _, Ico_subset_Icc_self)
instance tendsto_Ioc_class_nhds (a : α) : tendsto_Ixx_class Ioc (𝓝 a) (𝓝 a) :=
tendsto_Ixx_class_of_subset (λ _ _, Ioc_subset_Icc_self)
instance tendsto_Ioo_class_nhds (a : α) : tendsto_Ixx_class Ioo (𝓝 a) (𝓝 a) :=
tendsto_Ixx_class_of_subset (λ _ _, Ioo_subset_Icc_self)
instance tendsto_Ixx_nhds_within (a : α) {s t : set α} {Ixx}
[tendsto_Ixx_class Ixx (𝓝 a) (𝓝 a)] [tendsto_Ixx_class Ixx (𝓟 s) (𝓟 t)]:
tendsto_Ixx_class Ixx (𝓝[s] a) (𝓝[t] a) :=
filter.tendsto_Ixx_class_inf
/-- Also known as squeeze or sandwich theorem. This version assumes that inequalities hold
eventually for the filter. -/
lemma tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : filter β} {a : α}
(hg : tendsto g b (𝓝 a)) (hh : tendsto h b (𝓝 a))
(hgf : ∀ᶠ b in b, g b ≤ f b) (hfh : ∀ᶠ b in b, f b ≤ h b) :
tendsto f b (𝓝 a) :=
tendsto_order.2
⟨assume a' h',
have ∀ᶠ b in b, a' < g b, from (tendsto_order.1 hg).left a' h',
by filter_upwards [this, hgf] assume a, lt_of_lt_of_le,
assume a' h',
have ∀ᶠ b in b, h b < a', from (tendsto_order.1 hh).right a' h',
by filter_upwards [this, hfh] assume a h₁ h₂, lt_of_le_of_lt h₂ h₁⟩
/-- Also known as squeeze or sandwich theorem. This version assumes that inequalities hold
everywhere. -/
lemma tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : filter β} {a : α}
(hg : tendsto g b (𝓝 a)) (hh : tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :
tendsto f b (𝓝 a) :=
tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh
(eventually_of_forall hgf) (eventually_of_forall hfh)
lemma nhds_order_unbounded {a : α} (hu : ∃u, a < u) (hl : ∃l, l < a) :
𝓝 a = (⨅l (h₂ : l < a) u (h₂ : a < u), 𝓟 (Ioo l u)) :=
have ∃ u, u ∈ Ioi a, from hu, have ∃ l, l ∈ Iio a, from hl,
by { simp only [nhds_eq_order, inf_binfi, binfi_inf, *, inf_principal, Ioi_inter_Iio], refl }
lemma tendsto_order_unbounded {f : β → α} {a : α} {x : filter β}
(hu : ∃u, a < u) (hl : ∃l, l < a) (h : ∀l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :
tendsto f x (𝓝 a) :=
by rw [nhds_order_unbounded hu hl];
from (tendsto_infi.2 $ assume l, tendsto_infi.2 $ assume hl,
tendsto_infi.2 $ assume u, tendsto_infi.2 $ assume hu, tendsto_principal.2 $ h l u hl hu)
end partial_order
theorem induced_order_topology' {α : Type u} {β : Type v}
[partial_order α] [ta : topological_space β] [partial_order β] [order_topology β]
(f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
(H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b)
(H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :
@order_topology _ (induced f ta) _ :=
begin
letI := induced f ta,
refine ⟨eq_of_nhds_eq_nhds (λ a, _)⟩,
rw [nhds_induced, nhds_generate_from, nhds_eq_order (f a)],
apply le_antisymm,
{ refine le_infi (λ s, le_infi $ λ hs, le_principal_iff.2 _),
rcases hs with ⟨ab, b, rfl|rfl⟩,
{ exact mem_comap_sets.2 ⟨{x | f b < x},
mem_inf_sets_of_left $ mem_infi_sets _ $ mem_infi_sets (hf.2 ab) $ mem_principal_self _,
λ x, hf.1⟩ },
{ exact mem_comap_sets.2 ⟨{x | x < f b},
mem_inf_sets_of_right $ mem_infi_sets _ $ mem_infi_sets (hf.2 ab) $ mem_principal_self _,
λ x, hf.1⟩ } },
{ rw [← map_le_iff_le_comap],
refine le_inf _ _; refine le_infi (λ x, le_infi $ λ h, le_principal_iff.2 _); simp,
{ rcases H₁ h with ⟨b, ab, xb⟩,
refine mem_infi_sets _ (mem_infi_sets ⟨ab, b, or.inl rfl⟩ (mem_principal_sets.2 _)),
exact λ c hc, lt_of_le_of_lt xb (hf.2 hc) },
{ rcases H₂ h with ⟨b, ab, xb⟩,
refine mem_infi_sets _ (mem_infi_sets ⟨ab, b, or.inr rfl⟩ (mem_principal_sets.2 _)),
exact λ c hc, lt_of_lt_of_le (hf.2 hc) xb } },
end
theorem induced_order_topology {α : Type u} {β : Type v}
[partial_order α] [ta : topological_space β] [partial_order β] [order_topology β]
(f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
(H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) :
@order_topology _ (induced f ta) _ :=
induced_order_topology' f @hf
(λ a x xa, let ⟨b, xb, ba⟩ := H xa in ⟨b, hf.1 ba, le_of_lt xb⟩)
(λ a x ax, let ⟨b, ab, bx⟩ := H ax in ⟨b, hf.1 ab, le_of_lt bx⟩)
/-- On an `ord_connected` subset of a linear order, the order topology for the restriction of the
order is the same as the restriction to the subset of the order topology. -/
instance order_topology_of_ord_connected {α : Type u}
[ta : topological_space α] [linear_order α] [order_topology α]
{t : set α} [ht : ord_connected t] :
order_topology t :=
begin
letI := induced (coe : t → α) ta,
refine ⟨eq_of_nhds_eq_nhds (λ a, _)⟩,
rw [nhds_induced, nhds_generate_from, nhds_eq_order (a : α)],
apply le_antisymm,
{ refine le_infi (λ s, le_infi $ λ hs, le_principal_iff.2 _),
rcases hs with ⟨ab, b, rfl|rfl⟩,
{ refine ⟨Ioi b, _, λ _, id⟩,
refine mem_inf_sets_of_left (mem_infi_sets b _),
exact mem_infi_sets ab (mem_principal_self (Ioi ↑b)) },
{ refine ⟨Iio b, _, λ _, id⟩,
refine mem_inf_sets_of_right (mem_infi_sets b _),
exact mem_infi_sets ab (mem_principal_self (Iio b)) } },
{ rw [← map_le_iff_le_comap],
refine le_inf _ _,
{ refine le_infi (λ x, le_infi $ λ h, le_principal_iff.2 _),
by_cases hx : x ∈ t,
{ refine mem_infi_sets (Ioi ⟨x, hx⟩) (mem_infi_sets ⟨h, ⟨⟨x, hx⟩, or.inl rfl⟩⟩ _),
exact λ _, id },
simp only [set_coe.exists, mem_set_of_eq, mem_map],
convert univ_sets _,
suffices hx' : ∀ (y : t), ↑y ∈ Ioi x,
{ simp [hx'] },
intros y,
revert hx,
contrapose!,
-- here we use the `ord_connected` hypothesis
exact λ hx, ht y.2 a.2 ⟨le_of_not_gt hx, le_of_lt h⟩ },
{ refine le_infi (λ x, le_infi $ λ h, le_principal_iff.2 _),
by_cases hx : x ∈ t,
{ refine mem_infi_sets (Iio ⟨x, hx⟩) (mem_infi_sets ⟨h, ⟨⟨x, hx⟩, or.inr rfl⟩⟩ _),
exact λ _, id },
simp only [set_coe.exists, mem_set_of_eq, mem_map],
convert univ_sets _,
suffices hx' : ∀ (y : t), ↑y ∈ Iio x,
{ simp [hx'] },
intros y,
revert hx,
contrapose!,
-- here we use the `ord_connected` hypothesis
exact λ hx, ht a.2 y.2 ⟨le_of_lt h, le_of_not_gt hx⟩ } }
end
lemma nhds_top_order [topological_space α] [order_top α] [order_topology α] :
𝓝 (⊤:α) = (⨅l (h₂ : l < ⊤), 𝓟 (Ioi l)) :=
by simp [nhds_eq_order (⊤:α)]
lemma nhds_bot_order [topological_space α] [order_bot α] [order_topology α] :
𝓝 (⊥:α) = (⨅l (h₂ : ⊥ < l), 𝓟 (Iio l)) :=
by simp [nhds_eq_order (⊥:α)]
lemma tendsto_nhds_top_mono [topological_space β] [order_top β] [order_topology β] {l : filter α}
{f g : α → β} (hf : tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) :
tendsto g l (𝓝 ⊤) :=
begin
simp only [nhds_top_order, tendsto_infi, tendsto_principal] at hf ⊢,
intros x hx,
filter_upwards [hf x hx, hg],
exact λ x, lt_of_lt_of_le
end
lemma tendsto_nhds_bot_mono [topological_space β] [order_bot β] [order_topology β] {l : filter α}
{f g : α → β} (hf : tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) :
tendsto g l (𝓝 ⊥) :=
@tendsto_nhds_top_mono α (order_dual β) _ _ _ _ _ _ hf hg
lemma tendsto_nhds_top_mono' [topological_space β] [order_top β] [order_topology β] {l : filter α}
{f g : α → β} (hf : tendsto f l (𝓝 ⊤)) (hg : f ≤ g) :
tendsto g l (𝓝 ⊤) :=
tendsto_nhds_top_mono hf (eventually_of_forall hg)
lemma tendsto_nhds_bot_mono' [topological_space β] [order_bot β] [order_topology β] {l : filter α}
{f g : α → β} (hf : tendsto f l (𝓝 ⊥)) (hg : g ≤ f) :
tendsto g l (𝓝 ⊥) :=
tendsto_nhds_bot_mono hf (eventually_of_forall hg)
section linear_order
variables [topological_space α] [linear_order α] [order_topology α]
lemma exists_Ioc_subset_of_mem_nhds' {a : α} {s : set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :
∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=
begin
rw [nhds_eq_order a] at hs,
rcases hs with ⟨t₁, ht₁, t₂, ht₂, hts⟩,
-- First we show that `t₂` includes `(-∞, a]`, so it suffices to show `(l', ∞) ⊆ t₁`
suffices : ∃ l' ∈ Ico l a, Ioi l' ⊆ t₁,
{ have A : 𝓟 (Iic a) ≤ ⨅ b ∈ Ioi a, 𝓟 (Iio b),
from (le_infi $ λ b, le_infi $ λ hb, principal_mono.2 $ Iic_subset_Iio.2 hb),
have B : t₁ ∩ Iic a ⊆ s,
from subset.trans (inter_subset_inter_right _ (A ht₂)) hts,
from this.imp (λ l', Exists.imp $ λ hl' hl x hx, B ⟨hl hx.1, hx.2⟩) },
clear hts ht₂ t₂,
-- Now we find `l` such that `(l', ∞) ⊆ t₁`
rw [mem_binfi] at ht₁,
{ rcases ht₁ with ⟨b, hb, hb'⟩,
exact ⟨max b l, ⟨le_max_right _ _, max_lt hb hl⟩,
λ x hx, hb' $ Ioi_subset_Ioi (le_max_left _ _) hx⟩ },
{ intros b hb b' hb', simp only [mem_Iio] at hb hb',
use [max b b', max_lt hb hb'],
simp [le_refl] },
exact ⟨l, hl⟩
end
lemma exists_Ico_subset_of_mem_nhds' {a : α} {s : set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :
∃ u' ∈ Ioc a u, Ico a u' ⊆ s :=
begin
convert @exists_Ioc_subset_of_mem_nhds' (order_dual α) _ _ _ _ _ hs _ hu,
ext, rw [dual_Ico, dual_Ioc]
end
lemma exists_Ioc_subset_of_mem_nhds {a : α} {s : set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :
∃ l < a, Ioc l a ⊆ s :=
let ⟨l', hl'⟩ := h in let ⟨l, hl⟩ := exists_Ioc_subset_of_mem_nhds' hs hl' in ⟨l, hl.fst.2, hl.snd⟩
lemma exists_Ico_subset_of_mem_nhds {a : α} {s : set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :
∃ u (_ : a < u), Ico a u ⊆ s :=
let ⟨l', hl'⟩ := h in let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl' in ⟨l, hl.fst.1, hl.snd⟩
lemma order_separated {a₁ a₂ : α} (h : a₁ < a₂) :
∃u v : set α, is_open u ∧ is_open v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ (∀b₁∈u, ∀b₂∈v, b₁ < b₂) :=
match dense_or_discrete a₁ a₂ with
| or.inl ⟨a, ha₁, ha₂⟩ := ⟨{a' | a' < a}, {a' | a < a'}, is_open_gt' a, is_open_lt' a, ha₁, ha₂,
assume b₁ h₁ b₂ h₂, lt_trans h₁ h₂⟩
| or.inr ⟨h₁, h₂⟩ := ⟨{a | a < a₂}, {a | a₁ < a}, is_open_gt' a₂, is_open_lt' a₁, h, h,
assume b₁ hb₁ b₂ hb₂,
calc b₁ ≤ a₁ : h₂ _ hb₁
... < a₂ : h
... ≤ b₂ : h₁ _ hb₂⟩
end
@[priority 100] -- see Note [lower instance priority]
instance order_topology.to_order_closed_topology : order_closed_topology α :=
{ is_closed_le' :=
is_open_prod_iff.mpr $ assume a₁ a₂ (h : ¬ a₁ ≤ a₂),
have h : a₂ < a₁, from lt_of_not_ge h,
let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h in
⟨v, u, hv, hu, ha₂, ha₁, assume ⟨b₁, b₂⟩ ⟨h₁, h₂⟩, not_le_of_gt $ h b₂ h₂ b₁ h₁⟩ }
lemma order_topology.t2_space : t2_space α := by apply_instance
@[priority 100] -- see Note [lower instance priority]
instance order_topology.regular_space : regular_space α :=
{ regular := assume s a hs ha,
have hs' : sᶜ ∈ 𝓝 a, from mem_nhds_sets hs ha,
have ∃t:set α, is_open t ∧ (∀l∈ s, l < a → l ∈ t) ∧ 𝓝[t] a = ⊥,
from by_cases
(assume h : ∃l, l < a,
let ⟨l, hl, h⟩ := exists_Ioc_subset_of_mem_nhds hs' h in
match dense_or_discrete l a with
| or.inl ⟨b, hb₁, hb₂⟩ := ⟨{a | a < b}, is_open_gt' _,
assume c hcs hca, show c < b,
from lt_of_not_ge $ assume hbc, h ⟨lt_of_lt_of_le hb₁ hbc, le_of_lt hca⟩ hcs,
inf_principal_eq_bot $ (𝓝 a).sets_of_superset (mem_nhds_sets (is_open_lt' _) hb₂) $
assume x (hx : b < x), show ¬ x < b, from not_lt.2 $ le_of_lt hx⟩
| or.inr ⟨h₁, h₂⟩ := ⟨{a' | a' < a}, is_open_gt' _, assume b hbs hba, hba,
inf_principal_eq_bot $ (𝓝 a).sets_of_superset (mem_nhds_sets (is_open_lt' _) hl) $
assume x (hx : l < x), show ¬ x < a, from not_lt.2 $ h₁ _ hx⟩
end)
(assume : ¬ ∃l, l < a, ⟨∅, is_open_empty, assume l _ hl, (this ⟨l, hl⟩).elim,
nhds_within_empty _⟩),
let ⟨t₁, ht₁o, ht₁s, ht₁a⟩ := this in
have ∃t:set α, is_open t ∧ (∀u∈ s, u>a → u ∈ t) ∧ 𝓝[t] a = ⊥,
from by_cases
(assume h : ∃u, u > a,
let ⟨u, hu, h⟩ := exists_Ico_subset_of_mem_nhds hs' h in
match dense_or_discrete a u with
| or.inl ⟨b, hb₁, hb₂⟩ := ⟨{a | b < a}, is_open_lt' _,
assume c hcs hca, show c > b,
from lt_of_not_ge $ assume hbc, h ⟨le_of_lt hca, lt_of_le_of_lt hbc hb₂⟩ hcs,
inf_principal_eq_bot $ (𝓝 a).sets_of_superset (mem_nhds_sets (is_open_gt' _) hb₁) $
assume x (hx : b > x), show ¬ x > b, from not_lt.2 $ le_of_lt hx⟩
| or.inr ⟨h₁, h₂⟩ := ⟨{a' | a' > a}, is_open_lt' _, assume b hbs hba, hba,
inf_principal_eq_bot $ (𝓝 a).sets_of_superset (mem_nhds_sets (is_open_gt' _) hu) $
assume x (hx : u > x), show ¬ x > a, from not_lt.2 $ h₂ _ hx⟩
end)
(assume : ¬ ∃u, u > a, ⟨∅, is_open_empty, assume l _ hl, (this ⟨l, hl⟩).elim,
nhds_within_empty _⟩),
let ⟨t₂, ht₂o, ht₂s, ht₂a⟩ := this in
⟨t₁ ∪ t₂, is_open_union ht₁o ht₂o,
assume x hx,
have x ≠ a, from assume eq, ha $ eq ▸ hx,
(ne_iff_lt_or_gt.mp this).imp (ht₁s _ hx) (ht₂s _ hx),
by rw [nhds_within_union, ht₁a, ht₂a, bot_sup_eq]⟩,
..order_topology.t2_space }
/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,
provided `a` is neither a bottom element nor a top element. -/
lemma mem_nhds_iff_exists_Ioo_subset' {a : α} {s : set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :
s ∈ 𝓝 a ↔ ∃l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=
begin
split,
{ assume h,
rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩,
rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩,
refine ⟨l, u, ⟨la, au⟩, λx hx, _⟩,
cases le_total a x with hax hax,
{ exact hu ⟨hax, hx.2⟩ },
{ exact hl ⟨hx.1, hax⟩ } },
{ rintros ⟨l, u, ha, h⟩,
apply mem_sets_of_superset (mem_nhds_sets is_open_Ioo ha) h }
end
/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.
-/
lemma mem_nhds_iff_exists_Ioo_subset [no_top_order α] [no_bot_order α] {a : α} {s : set α} :
s ∈ 𝓝 a ↔ ∃l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=
mem_nhds_iff_exists_Ioo_subset' (no_bot a) (no_top a)
lemma nhds_basis_Ioo' {a : α} (hl : ∃l, l < a) (hu : ∃u, a < u) :
(𝓝 a).has_basis (λ b : α × α, b.1 < a ∧ a < b.2) (λ b, Ioo b.1 b.2) :=
⟨λ s, (mem_nhds_iff_exists_Ioo_subset' hl hu).trans $ by simp⟩
lemma nhds_basis_Ioo [no_top_order α] [no_bot_order α] {a : α} :
(𝓝 a).has_basis (λ b : α × α, b.1 < a ∧ a < b.2) (λ b, Ioo b.1 b.2) :=
nhds_basis_Ioo' (no_bot a) (no_top a)
lemma filter.eventually.exists_Ioo_subset [no_top_order α] [no_bot_order α] {a : α} {p : α → Prop}
(hp : ∀ᶠ x in 𝓝 a, p x) :
∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ {x | p x} :=
mem_nhds_iff_exists_Ioo_subset.1 hp
lemma Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=
mem_nhds_sets is_open_Iio h
lemma Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=
mem_nhds_sets is_open_Ioi h
lemma Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=
mem_sets_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self
lemma Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=
mem_sets_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self
lemma Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=
mem_nhds_sets is_open_Ioo ⟨ha, hb⟩
lemma Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=
mem_sets_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self
lemma Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=
mem_sets_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self
lemma Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=
mem_sets_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self
section pi
/-!
### Intervals in `Π i, π i` belong to `𝓝 x`
For each leamma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because
sometimes Lean fails to unify different instances while trying to apply the dependent version to,
e.g., `ι → ℝ`.
-/
variables {ι : Type*} {π : ι → Type*} [fintype ι] [Π i, linear_order (π i)]
[Π i, topological_space (π i)] [∀ i, order_topology (π i)] {a b x : Π i, π i} {a' b' x' : ι → α}
lemma pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=
pi_univ_Iic a ▸ set_pi_mem_nhds (finite.of_fintype _) (λ i _, Iic_mem_nhds (ha _))
lemma pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=
pi_Iic_mem_nhds ha
lemma pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=
pi_univ_Ici a ▸ set_pi_mem_nhds (finite.of_fintype _) (λ i _, Ici_mem_nhds (ha _))
lemma pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=
pi_Ici_mem_nhds ha
lemma pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=
pi_univ_Icc a b ▸ set_pi_mem_nhds (finite.of_fintype _) (λ i _, Icc_mem_nhds (ha _) (hb _))
lemma pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=
pi_Icc_mem_nhds ha hb
variables [nonempty ι]
lemma pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x :=
begin
refine mem_sets_of_superset (set_pi_mem_nhds (finite.of_fintype _) (λ i _, _))
(pi_univ_Iio_subset a),
exact Iio_mem_nhds (ha i)
end
lemma pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=
pi_Iio_mem_nhds ha
lemma pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=
@pi_Iio_mem_nhds ι (λ i, order_dual (π i)) _ _ _ _ _ _ _ ha
lemma pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=
pi_Ioi_mem_nhds ha
lemma pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x :=
begin
refine mem_sets_of_superset (set_pi_mem_nhds (finite.of_fintype _) (λ i _, _))
(pi_univ_Ioc_subset a b),
exact Ioc_mem_nhds (ha i) (hb i)
end
lemma pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=
pi_Ioc_mem_nhds ha hb
lemma pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x :=
begin
refine mem_sets_of_superset (set_pi_mem_nhds (finite.of_fintype _) (λ i _, _))
(pi_univ_Ico_subset a b),
exact Ico_mem_nhds (ha i) (hb i)
end
lemma pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=
pi_Ico_mem_nhds ha hb
lemma pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x :=
begin
refine mem_sets_of_superset (set_pi_mem_nhds (finite.of_fintype _) (λ i _, _))
(pi_univ_Ioo_subset a b),
exact Ioo_mem_nhds (ha i) (hb i)
end
lemma pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=
pi_Ioo_mem_nhds ha hb
end pi
lemma disjoint_nhds_at_top [no_top_order α] (x : α) :
disjoint (𝓝 x) at_top :=
begin
rw filter.disjoint_iff,
cases no_top x with a ha,
use [Iio a, Iio_mem_nhds ha, Ici a, mem_at_top a],
rw [inter_comm, Ici_inter_Iio, Ico_self]
end
@[simp] lemma inf_nhds_at_top [no_top_order α] (x : α) :
𝓝 x ⊓ at_top = ⊥ :=
disjoint_iff.1 (disjoint_nhds_at_top x)
lemma disjoint_nhds_at_bot [no_bot_order α] (x : α) :
disjoint (𝓝 x) at_bot :=
@disjoint_nhds_at_top (order_dual α) _ _ _ _ x
@[simp] lemma inf_nhds_at_bot [no_bot_order α] (x : α) :
𝓝 x ⊓ at_bot = ⊥ :=
@inf_nhds_at_top (order_dual α) _ _ _ _ x
lemma not_tendsto_nhds_of_tendsto_at_top [no_top_order α]
{F : filter β} [ne_bot F] {f : β → α} (hf : tendsto f F at_top) (x : α) :
¬ tendsto f F (𝓝 x) :=
hf.not_tendsto (disjoint_nhds_at_top x).symm
lemma not_tendsto_at_top_of_tendsto_nhds [no_top_order α]
{F : filter β} [ne_bot F] {f : β → α} {x : α} (hf : tendsto f F (𝓝 x)) :
¬ tendsto f F at_top :=
hf.not_tendsto (disjoint_nhds_at_top x)
lemma not_tendsto_nhds_of_tendsto_at_bot [no_bot_order α]
{F : filter β} [ne_bot F] {f : β → α} (hf : tendsto f F at_bot) (x : α) :
¬ tendsto f F (𝓝 x) :=
hf.not_tendsto (disjoint_nhds_at_bot x).symm
lemma not_tendsto_at_bot_of_tendsto_nhds [no_bot_order α]
{F : filter β} [ne_bot F] {f : β → α} {x : α} (hf : tendsto f F (𝓝 x)) :
¬ tendsto f F at_bot :=
hf.not_tendsto (disjoint_nhds_at_bot x)
/-!
### Neighborhoods to the left and to the right on an `order_topology`
We've seen some properties of left and right neighborhood of a point in an `order_closed_topology`.
In an `order_topology`, such neighborhoods can be characterized as the sets containing suitable
intervals to the right or to the left of `a`. We give now these characterizations. -/
-- NB: If you extend the list, append to the end please to avoid breaking the API
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`
1. `s` is a neighborhood of `a` within `(a, b]`
2. `s` is a neighborhood of `a` within `(a, b)`
3. `s` includes `(a, u)` for some `u ∈ (a, b]`
4. `s` includes `(a, u)` for some `u > a` -/
lemma tfae_mem_nhds_within_Ioi {a b : α} (hab : a < b) (s : set α) :
tfae [s ∈ 𝓝[Ioi a] a, -- 0 : `s` is a neighborhood of `a` within `(a, +∞)`
s ∈ 𝓝[Ioc a b] a, -- 1 : `s` is a neighborhood of `a` within `(a, b]`
s ∈ 𝓝[Ioo a b] a, -- 2 : `s` is a neighborhood of `a` within `(a, b)`
∃ u ∈ Ioc a b, Ioo a u ⊆ s, -- 3 : `s` includes `(a, u)` for some `u ∈ (a, b]`
∃ u ∈ Ioi a, Ioo a u ⊆ s] := -- 4 : `s` includes `(a, u)` for some `u > a`
begin
tfae_have : 1 ↔ 2, by rw [nhds_within_Ioc_eq_nhds_within_Ioi hab],
tfae_have : 1 ↔ 3, by rw [nhds_within_Ioo_eq_nhds_within_Ioi hab],
tfae_have : 4 → 5, from λ ⟨u, umem, hu⟩, ⟨u, umem.1, hu⟩,
tfae_have : 5 → 1,
{ rintros ⟨u, hau, hu⟩,
exact mem_sets_of_superset (Ioo_mem_nhds_within_Ioi ⟨le_refl a, hau⟩) hu },
tfae_have : 1 → 4,
{ assume h,
rcases mem_nhds_within_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩,
rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩,
refine ⟨u, au, λx hx, _⟩,
refine hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩,
exact hx.1 },
tfae_finish
end
lemma mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[Ioi a] a ↔ ∃u ∈ Ioc a u', Ioo a u ⊆ s :=
(tfae_mem_nhds_within_Ioi hu' s).out 0 3
/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u < u'`, provided `a` is not a top element. -/
lemma mem_nhds_within_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[Ioi a] a ↔ ∃u ∈ Ioi a, Ioo a u ⊆ s :=
(tfae_mem_nhds_within_Ioi hu' s).out 0 4
/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u`. -/
lemma mem_nhds_within_Ioi_iff_exists_Ioo_subset [no_top_order α] {a : α} {s : set α} :
s ∈ 𝓝[Ioi a] a ↔ ∃u ∈ Ioi a, Ioo a u ⊆ s :=
let ⟨u', hu'⟩ := no_top a in mem_nhds_within_Ioi_iff_exists_Ioo_subset' hu'
/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
lemma mem_nhds_within_Ioi_iff_exists_Ioc_subset [no_top_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Ioi a] a ↔ ∃u ∈ Ioi a, Ioc a u ⊆ s :=
begin
rw mem_nhds_within_Ioi_iff_exists_Ioo_subset,
split,
{ rintros ⟨u, au, as⟩,
rcases exists_between au with ⟨v, hv⟩,
exact ⟨v, hv.1, λx hx, as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ },
{ rintros ⟨u, au, as⟩,
exact ⟨u, au, subset.trans Ioo_subset_Ioc_self as⟩ }
end
/-- The following statements are equivalent:
0. `s` is a neighborhood of `b` within `(-∞, b)`
1. `s` is a neighborhood of `b` within `[a, b)`
2. `s` is a neighborhood of `b` within `(a, b)`
3. `s` includes `(l, b)` for some `l ∈ [a, b)`
4. `s` includes `(l, b)` for some `l < b` -/
lemma tfae_mem_nhds_within_Iio {a b : α} (h : a < b) (s : set α) :
tfae [s ∈ 𝓝[Iio b] b, -- 0 : `s` is a neighborhood of `b` within `(-∞, b)`
s ∈ 𝓝[Ico a b] b, -- 1 : `s` is a neighborhood of `b` within `[a, b)`
s ∈ 𝓝[Ioo a b] b, -- 2 : `s` is a neighborhood of `b` within `(a, b)`
∃ l ∈ Ico a b, Ioo l b ⊆ s, -- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`
∃ l ∈ Iio b, Ioo l b ⊆ s] := -- 4 : `s` includes `(l, b)` for some `l < b`
begin
have := @tfae_mem_nhds_within_Ioi (order_dual α) _ _ _ _ _ h s,
-- If we call `convert` here, it generates wrong equations, so we need to simplify first
simp only [exists_prop] at this ⊢,
rw [dual_Ioi, dual_Ioc, dual_Ioo] at this,
convert this; ext l; rw [dual_Ioo]
end
lemma mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[Iio a] a ↔ ∃l ∈ Ico l' a, Ioo l a ⊆ s :=
(tfae_mem_nhds_within_Iio hl' s).out 0 3
/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`, provided `a` is not a bottom element. -/
lemma mem_nhds_within_Iio_iff_exists_Ioo_subset' {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[Iio a] a ↔ ∃l ∈ Iio a, Ioo l a ⊆ s :=
(tfae_mem_nhds_within_Iio hl' s).out 0 4
/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`. -/
lemma mem_nhds_within_Iio_iff_exists_Ioo_subset [no_bot_order α] {a : α} {s : set α} :
s ∈ 𝓝[Iio a] a ↔ ∃l ∈ Iio a, Ioo l a ⊆ s :=
let ⟨l', hl'⟩ := no_bot a in mem_nhds_within_Iio_iff_exists_Ioo_subset' hl'
/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`
with `l < a`. -/
lemma mem_nhds_within_Iio_iff_exists_Ico_subset [no_bot_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Iio a] a ↔ ∃l ∈ Iio a, Ico l a ⊆ s :=
begin
convert @mem_nhds_within_Ioi_iff_exists_Ioc_subset (order_dual α) _ _ _ _ _ _ _,
simp only [dual_Ioc], refl
end
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `[a, +∞)`
1. `s` is a neighborhood of `a` within `[a, b]`
2. `s` is a neighborhood of `a` within `[a, b)`
3. `s` includes `[a, u)` for some `u ∈ (a, b]`
4. `s` includes `[a, u)` for some `u > a` -/
lemma tfae_mem_nhds_within_Ici {a b : α} (hab : a < b) (s : set α) :
tfae [s ∈ 𝓝[Ici a] a, -- 0 : `s` is a neighborhood of `a` within `[a, +∞)`
s ∈ 𝓝[Icc a b] a, -- 1 : `s` is a neighborhood of `a` within `[a, b]`
s ∈ 𝓝[Ico a b] a, -- 2 : `s` is a neighborhood of `a` within `[a, b)`
∃ u ∈ Ioc a b, Ico a u ⊆ s, -- 3 : `s` includes `[a, u)` for some `u ∈ (a, b]`
∃ u ∈ Ioi a, Ico a u ⊆ s] := -- 4 : `s` includes `[a, u)` for some `u > a`
begin
tfae_have : 1 ↔ 2, by rw [nhds_within_Icc_eq_nhds_within_Ici hab],
tfae_have : 1 ↔ 3, by rw [nhds_within_Ico_eq_nhds_within_Ici hab],
tfae_have : 4 → 5, from λ ⟨u, umem, hu⟩, ⟨u, umem.1, hu⟩,
tfae_have : 5 → 1,
{ rintros ⟨u, hau, hu⟩,
exact mem_sets_of_superset (Ico_mem_nhds_within_Ici ⟨le_refl a, hau⟩) hu },
tfae_have : 1 → 4,
{ assume h,
rcases mem_nhds_within_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩,
rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩,
refine ⟨u, au, λx hx, _⟩,
refine hv ⟨hu ⟨hx.1, hx.2⟩, _⟩,
exact hx.1 },
tfae_finish
end
lemma mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[Ici a] a ↔ ∃u ∈ Ioc a u', Ico a u ⊆ s :=
(tfae_mem_nhds_within_Ici hu' s).out 0 3 (by norm_num) (by norm_num)
/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u < u'`, provided `a` is not a top element. -/
lemma mem_nhds_within_Ici_iff_exists_Ico_subset' {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[Ici a] a ↔ ∃u ∈ Ioi a, Ico a u ⊆ s :=
(tfae_mem_nhds_within_Ici hu' s).out 0 4 (by norm_num) (by norm_num)
/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u`. -/
lemma mem_nhds_within_Ici_iff_exists_Ico_subset [no_top_order α] {a : α} {s : set α} :
s ∈ 𝓝[Ici a] a ↔ ∃u ∈ Ioi a, Ico a u ⊆ s :=
let ⟨u', hu'⟩ := no_top a in mem_nhds_within_Ici_iff_exists_Ico_subset' hu'
/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
with `a < u`. -/
lemma mem_nhds_within_Ici_iff_exists_Icc_subset' [no_top_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Ici a] a ↔ ∃u ∈ Ioi a, Icc a u ⊆ s :=
begin
rw mem_nhds_within_Ici_iff_exists_Ico_subset,
split,
{ rintros ⟨u, au, as⟩,
rcases exists_between au with ⟨v, hv⟩,
exact ⟨v, hv.1, λx hx, as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ },
{ rintros ⟨u, au, as⟩,
exact ⟨u, au, subset.trans Ico_subset_Icc_self as⟩ }
end
/-- The following statements are equivalent:
0. `s` is a neighborhood of `b` within `(-∞, b]`
1. `s` is a neighborhood of `b` within `[a, b]`
2. `s` is a neighborhood of `b` within `(a, b]`
3. `s` includes `(l, b]` for some `l ∈ [a, b)`
4. `s` includes `(l, b]` for some `l < b` -/
lemma tfae_mem_nhds_within_Iic {a b : α} (h : a < b) (s : set α) :
tfae [s ∈ 𝓝[Iic b] b, -- 0 : `s` is a neighborhood of `b` within `(-∞, b]`
s ∈ 𝓝[Icc a b] b, -- 1 : `s` is a neighborhood of `b` within `[a, b]`
s ∈ 𝓝[Ioc a b] b, -- 2 : `s` is a neighborhood of `b` within `(a, b]`
∃ l ∈ Ico a b, Ioc l b ⊆ s, -- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`
∃ l ∈ Iio b, Ioc l b ⊆ s] := -- 4 : `s` includes `(l, b]` for some `l < b`
begin
have := @tfae_mem_nhds_within_Ici (order_dual α) _ _ _ _ _ h s,
-- If we call `convert` here, it generates wrong equations, so we need to simplify first
simp only [exists_prop] at this ⊢,
rw [dual_Icc, dual_Ioc, dual_Ioi] at this,
convert this; ext l; rw [dual_Ico]
end
lemma mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[Iic a] a ↔ ∃l ∈ Ico l' a, Ioc l a ⊆ s :=
(tfae_mem_nhds_within_Iic hl' s).out 0 3 (by norm_num) (by norm_num)
/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`, provided `a` is not a bottom element. -/
lemma mem_nhds_within_Iic_iff_exists_Ioc_subset' {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[Iic a] a ↔ ∃l ∈ Iio a, Ioc l a ⊆ s :=
(tfae_mem_nhds_within_Iic hl' s).out 0 4 (by norm_num) (by norm_num)
/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`. -/
lemma mem_nhds_within_Iic_iff_exists_Ioc_subset [no_bot_order α] {a : α} {s : set α} :
s ∈ 𝓝[Iic a] a ↔ ∃l ∈ Iio a, Ioc l a ⊆ s :=
let ⟨l', hl'⟩ := no_bot a in mem_nhds_within_Iic_iff_exists_Ioc_subset' hl'
/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
with `l < a`. -/
lemma mem_nhds_within_Iic_iff_exists_Icc_subset' [no_bot_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Iic a] a ↔ ∃l ∈ Iio a, Icc l a ⊆ s :=
begin
convert @mem_nhds_within_Ici_iff_exists_Icc_subset' (order_dual α) _ _ _ _ _ _ _,
simp_rw (show ∀ u : order_dual α, @Icc (order_dual α) _ a u = @Icc α _ u a, from λ u, dual_Icc),
refl,
end
/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
with `a < u`. -/
lemma mem_nhds_within_Ici_iff_exists_Icc_subset [no_top_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Ici a] a ↔ ∃u, a < u ∧ Icc a u ⊆ s :=
begin
rw mem_nhds_within_Ici_iff_exists_Ico_subset,
split,
{ rintros ⟨u, au, as⟩,
rcases exists_between au with ⟨v, hv⟩,
exact ⟨v, hv.1, λx hx, as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ },
{ rintros ⟨u, au, as⟩,
exact ⟨u, au, subset.trans Ico_subset_Icc_self as⟩ }
end
/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
with `l < a`. -/
lemma mem_nhds_within_Iic_iff_exists_Icc_subset [no_bot_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Iic a] a ↔ ∃l, l < a ∧ Icc l a ⊆ s :=
begin
rw mem_nhds_within_Iic_iff_exists_Ioc_subset,
split,
{ rintros ⟨l, la, as⟩,
rcases exists_between la with ⟨v, hv⟩,
refine ⟨v, hv.2, λx hx, as ⟨lt_of_lt_of_le hv.1 hx.1, hx.2⟩⟩, },
{ rintros ⟨l, la, as⟩,
exact ⟨l, la, subset.trans Ioc_subset_Icc_self as⟩ }
end
end linear_order
section linear_ordered_add_comm_group
variables [topological_space α] [linear_ordered_add_comm_group α] [order_topology α]
variables {l : filter β} {f g : β → α}
/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`
and `g` tends to `at_top` then `f + g` tends to `at_top`. -/
lemma filter.tendsto.add_at_top {C : α} (hf : tendsto f l (𝓝 C)) (hg : tendsto g l at_top) :
tendsto (λ x, f x + g x) l at_top :=
begin
nontriviality α,
obtain ⟨C', hC'⟩ : ∃ C', C' < C := no_bot C,
refine tendsto_at_top_add_left_of_le' _ C' _ hg,
exact (hf.eventually (lt_mem_nhds hC')).mono (λ x, le_of_lt)
end
/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`
and `g` tends to `at_bot` then `f + g` tends to `at_bot`. -/
lemma filter.tendsto.add_at_bot {C : α} (hf : tendsto f l (𝓝 C)) (hg : tendsto g l at_bot) :
tendsto (λ x, f x + g x) l at_bot :=
@filter.tendsto.add_at_top (order_dual α) _ _ _ _ _ _ _ _ hf hg
/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to
`at_top` and `g` tends to `C` then `f + g` tends to `at_top`. -/
lemma filter.tendsto.at_top_add {C : α} (hf : tendsto f l at_top) (hg : tendsto g l (𝓝 C)) :
tendsto (λ x, f x + g x) l at_top :=
by { conv in (_ + _) { rw add_comm }, exact hg.add_at_top hf }
/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to
`at_bot` and `g` tends to `C` then `f + g` tends to `at_bot`. -/
lemma filter.tendsto.at_bot_add {C : α} (hf : tendsto f l at_bot) (hg : tendsto g l (𝓝 C)) :
tendsto (λ x, f x + g x) l at_bot :=
by { conv in (_ + _) { rw add_comm }, exact hg.add_at_bot hf }
end linear_ordered_add_comm_group
section linear_ordered_field
variables [linear_ordered_field α]
variables {l : filter β} {f g : β → α}
variables [topological_space α] [order_topology α]
/-- In a linearly ordered field with the order topology, if `f` tends to `at_top` and `g` tends to
a positive constant `C` then `f * g` tends to `at_top`. -/
lemma filter.tendsto.at_top_mul {C : α} (hC : 0 < C) (hf : tendsto f l at_top)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_top :=
begin
refine tendsto_at_top_mono' _ _ (hf.at_top_mul_const (half_pos hC)),
filter_upwards [hg.eventually (lt_mem_nhds (half_lt_self hC)),
hf.eventually (eventually_ge_at_top 0)],
exact λ x hg hf, mul_le_mul_of_nonneg_left hg.le hf
end
/-- In a linearly ordered field with the order topology, if `f` tends to a positive constant `C` and
`g` tends to `at_top` then `f * g` tends to `at_top`. -/
lemma filter.tendsto.mul_at_top {C : α} (hC : 0 < C) (hf : tendsto f l (𝓝 C))
(hg : tendsto g l at_top) :
tendsto (λ x, (f x * g x)) l at_top :=
by simpa only [mul_comm] using hg.at_top_mul hC hf
/-- In a linearly ordered field with the order topology, if `f` tends to `at_top` and `g` tends to
a negative constant `C` then `f * g` tends to `at_bot`. -/
lemma filter.tendsto.at_top_mul_neg {C : α} (hC : C < 0) (hf : tendsto f l at_top)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_bot :=
begin
rcases exists_between hC with ⟨C', hCC', hC'0⟩,
refine tendsto_at_bot_mono' _ _ (hf.at_top_mul_neg_const hC'0),
filter_upwards [hg.eventually (gt_mem_nhds hCC'), hf.eventually (eventually_ge_at_top 0)],
exact λ x hg hf, mul_le_mul_of_nonneg_left hg.le hf
end
end linear_ordered_field
section linear_ordered_field
variables [linear_ordered_field α] [topological_space α] [order_topology α]
/-- The function `x ↦ x⁻¹` tends to `+∞` on the right of `0`. -/
lemma tendsto_inv_zero_at_top : tendsto (λx:α, x⁻¹) (𝓝[set.Ioi (0:α)] 0) at_top :=
begin
refine (at_top_basis' 1).tendsto_right_iff.2 (λ b hb, _),
have hb' : 0 < b := zero_lt_one.trans_le hb,
filter_upwards [Ioc_mem_nhds_within_Ioi ⟨le_rfl, inv_pos.2 hb'⟩],
exact λ x hx, (le_inv hx.1 hb').1 hx.2
end
/-- The function `r ↦ r⁻¹` tends to `0` on the right as `r → +∞`. -/
lemma tendsto_inv_at_top_zero' : tendsto (λr:α, r⁻¹) at_top (𝓝[set.Ioi (0:α)] 0) :=
begin
refine (has_basis.tendsto_iff at_top_basis ⟨λ s, mem_nhds_within_Ioi_iff_exists_Ioc_subset⟩).2 _,
refine λ b hb, ⟨b⁻¹, trivial, λ x hx, _⟩,
have : 0 < x := lt_of_lt_of_le (inv_pos.2 hb) hx,
exact ⟨inv_pos.2 this, (inv_le this hb).2 hx⟩
end
lemma tendsto_inv_at_top_zero : tendsto (λr:α, r⁻¹) at_top (𝓝 0) :=
tendsto_inv_at_top_zero'.mono_right inf_le_left
variables {l : filter β} {f : β → α}
lemma tendsto.inv_tendsto_at_top (h : tendsto f l at_top) : tendsto (f⁻¹) l (𝓝 0) :=
tendsto_inv_at_top_zero.comp h
lemma tendsto.inv_tendsto_zero (h : tendsto f l (𝓝[set.Ioi 0] 0)) : tendsto (f⁻¹) l at_top :=
tendsto_inv_zero_at_top.comp h
/-- The function `x^(-n)` tends to `0` at `+∞` for any positive natural `n`.
A version for positive real powers exists as `tendsto_rpow_neg_at_top`. -/
lemma tendsto_pow_neg_at_top {n : ℕ} (hn : 1 ≤ n) : tendsto (λ x : α, x ^ (-(n:ℤ))) at_top (𝓝 0) :=
tendsto.congr' (eventually_eq_of_mem (Ioi_mem_at_top 0) (λ x hx, (fpow_neg x n).symm))
(tendsto.inv_tendsto_at_top (tendsto_pow_at_top hn))
end linear_ordered_field
lemma preimage_neg [add_group α] : preimage (has_neg.neg : α → α) = image (has_neg.neg : α → α) :=
(image_eq_preimage_of_inverse neg_neg neg_neg).symm
lemma filter.map_neg [add_group α] : map (has_neg.neg : α → α) = comap (has_neg.neg : α → α) :=
funext $ assume f, map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)
section topological_add_group
variables [topological_space α] [ordered_add_comm_group α] [topological_add_group α]
lemma neg_preimage_closure {s : set α} : (λr:α, -r) ⁻¹' closure s = closure ((λr:α, -r) '' s) :=
have (λr:α, -r) ∘ (λr:α, -r) = id, from funext neg_neg,
by rw [preimage_neg]; exact
(subset.antisymm (image_closure_subset_closure_image continuous_neg) $
calc closure ((λ (r : α), -r) '' s) = (λr, -r) '' ((λr, -r) '' closure ((λ (r : α), -r) '' s)) :
by rw [←image_comp, this, image_id]
... ⊆ (λr, -r) '' closure ((λr, -r) '' ((λ (r : α), -r) '' s)) :
monotone_image $ image_closure_subset_closure_image continuous_neg
... = _ : by rw [←image_comp, this, image_id])
end topological_add_group
section order_topology
variables [topological_space α] [topological_space β]
[linear_order α] [linear_order β] [order_topology α] [order_topology β]
lemma is_lub.nhds_within_ne_bot {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :
ne_bot (𝓝[s] a) :=
let ⟨a', ha'⟩ := hs in
forall_sets_nonempty_iff_ne_bot.mp $ assume t ht,
let ⟨t₁, ht₁, t₂, ht₂, ht⟩ := mem_inf_sets.mp ht in
by_cases
(assume h : a = a',
have a ∈ t₁, from mem_of_nhds ht₁,
have a ∈ t₂, from ht₂ $ by rwa [h],
⟨a, ht ⟨‹a ∈ t₁›, ‹a ∈ t₂›⟩⟩)
(assume : a ≠ a',
have a' < a, from lt_of_le_of_ne (ha.left ‹a' ∈ s›) this.symm,
let ⟨l, hl, hlt₁⟩ := exists_Ioc_subset_of_mem_nhds ht₁ ⟨a', this⟩ in
have ∃a'∈s, l < a',
from classical.by_contradiction $ assume : ¬ ∃a'∈s, l < a',
have ∀a'∈s, a' ≤ l, from assume a ha, not_lt.1 $ assume ha', this ⟨a, ha, ha'⟩,
have ¬ l < a, from not_lt.2 $ ha.right this,
this ‹l < a›,
let ⟨a', ha', ha'l⟩ := this in
have a' ∈ t₁, from hlt₁ ⟨‹l < a'›, ha.left ha'⟩,
⟨a', ht ⟨‹a' ∈ t₁›, ht₂ ‹a' ∈ s›⟩⟩)
lemma is_glb.nhds_within_ne_bot : ∀ {a : α} {s : set α}, is_glb s a → s.nonempty →
ne_bot (𝓝[s] a) :=
@is_lub.nhds_within_ne_bot (order_dual α) _ _ _
lemma is_lub_of_mem_nhds {s : set α} {a : α} {f : filter α}
(hsa : a ∈ upper_bounds s) (hsf : s ∈ f) [ne_bot (f ⊓ 𝓝 a)] : is_lub s a :=
⟨hsa, assume b hb,
not_lt.1 $ assume hba,
have s ∩ {a | b < a} ∈ f ⊓ 𝓝 a,
from inter_mem_inf_sets hsf (mem_nhds_sets (is_open_lt' _) hba),
let ⟨x, ⟨hxs, hxb⟩⟩ := nonempty_of_mem_sets this in
have b < b, from lt_of_lt_of_le hxb $ hb hxs,
lt_irrefl b this⟩
lemma is_glb_of_mem_nhds : ∀ {s : set α} {a : α} {f : filter α},
a ∈ lower_bounds s → s ∈ f → ne_bot (f ⊓ 𝓝 a) → is_glb s a :=
@is_lub_of_mem_nhds (order_dual α) _ _ _
lemma is_lub_of_is_lub_of_tendsto {f : α → β} {s : set α} {a : α} {b : β}
(hf : ∀x∈s, ∀y∈s, x ≤ y → f x ≤ f y) (ha : is_lub s a) (hs : s.nonempty)
(hb : tendsto f (𝓝[s] a) (𝓝 b)) : is_lub (f '' s) b :=
have hnbot : ne_bot (𝓝[s] a), from ha.nhds_within_ne_bot hs,
have ∀a'∈s, ¬ b < f a',
from assume a' ha' h,
have ∀ᶠ x in 𝓝 b, x < f a', from mem_nhds_sets (is_open_gt' _) h,
let ⟨t₁, ht₁, t₂, ht₂, hs⟩ := mem_inf_sets.mp (hb this) in
by_cases
(assume h : a = a',
have a ∈ t₁ ∩ t₂, from ⟨mem_of_nhds ht₁, ht₂ $ by rwa [h]⟩,
have f a < f a', from hs this,
lt_irrefl (f a') $ by rwa [h] at this)
(assume h : a ≠ a',
have a' < a, from lt_of_le_of_ne (ha.left ha') h.symm,
have {x | a' < x} ∈ 𝓝 a, from mem_nhds_sets (is_open_lt' _) this,
have {x | a' < x} ∩ t₁ ∈ 𝓝 a, from inter_mem_sets this ht₁,
have ({x | a' < x} ∩ t₁) ∩ s ∈ 𝓝[s] a,
from inter_mem_inf_sets this (subset.refl s),
let ⟨x, ⟨hx₁, hx₂⟩, hx₃⟩ := hnbot.nonempty_of_mem this in
have hxa' : f x < f a', from hs ⟨hx₂, ht₂ hx₃⟩,
have ha'x : f a' ≤ f x, from hf _ ha' _ hx₃ $ le_of_lt hx₁,
lt_irrefl _ (lt_of_le_of_lt ha'x hxa')),
and.intro
(assume b' ⟨a', ha', h_eq⟩, h_eq ▸ not_lt.1 $ this _ ha')
(assume b' hb', by exactI (le_of_tendsto hb $
mem_inf_sets_of_right $ assume x hx, hb' $ mem_image_of_mem _ hx))
lemma is_glb_of_is_glb_of_tendsto {f : α → β} {s : set α} {a : α} {b : β}
(hf : ∀x∈s, ∀y∈s, x ≤ y → f x ≤ f y) : is_glb s a → s.nonempty →
tendsto f (𝓝[s] a) (𝓝 b) → is_glb (f '' s) b :=
@is_lub_of_is_lub_of_tendsto (order_dual α) (order_dual β) _ _ _ _ _ _ f s a b
(λ x hx y hy, hf y hy x hx)
lemma is_glb_of_is_lub_of_tendsto : ∀ {f : α → β} {s : set α} {a : α} {b : β},
(∀x∈s, ∀y∈s, x ≤ y → f y ≤ f x) → is_lub s a → s.nonempty →
tendsto f (𝓝[s] a) (𝓝 b) → is_glb (f '' s) b :=
@is_lub_of_is_lub_of_tendsto α (order_dual β) _ _ _ _ _ _
lemma is_lub_of_is_glb_of_tendsto : ∀ {f : α → β} {s : set α} {a : α} {b : β},
(∀x∈s, ∀y∈s, x ≤ y → f y ≤ f x) → is_glb s a → s.nonempty →
tendsto f (𝓝[s] a) (𝓝 b) → is_lub (f '' s) b :=
@is_glb_of_is_glb_of_tendsto α (order_dual β) _ _ _ _ _ _
lemma mem_closure_of_is_lub {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :
a ∈ closure s :=
by rw closure_eq_cluster_pts; exact ha.nhds_within_ne_bot hs
lemma mem_of_is_lub_of_is_closed {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty)
(sc : is_closed s) : a ∈ s :=
by rw ←sc.closure_eq; exact mem_closure_of_is_lub ha hs
lemma mem_closure_of_is_glb {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) :
a ∈ closure s :=
by rw closure_eq_cluster_pts; exact ha.nhds_within_ne_bot hs
lemma mem_of_is_glb_of_is_closed {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty)
(sc : is_closed s) : a ∈ s :=
by rw ←sc.closure_eq; exact mem_closure_of_is_glb ha hs
/-- A compact set is bounded below -/
lemma is_compact.bdd_below {α : Type u} [topological_space α] [linear_order α]
[order_closed_topology α] [nonempty α] {s : set α} (hs : is_compact s) : bdd_below s :=
begin
by_contra H,
rcases hs.elim_finite_subcover_image (λ x (_ : x ∈ s), @is_open_Ioi _ _ _ _ x) _
with ⟨t, st, ft, ht⟩,
{ refine H (ft.bdd_below.imp $ λ C hC y hy, _),
rcases mem_bUnion_iff.1 (ht hy) with ⟨x, hx, xy⟩,
exact le_trans (hC hx) (le_of_lt xy) },
{ refine λ x hx, mem_bUnion_iff.2 (not_imp_comm.1 _ H),
exact λ h, ⟨x, λ y hy, le_of_not_lt (h.imp $ λ ys, ⟨_, hy, ys⟩)⟩ }
end
/-- A compact set is bounded above -/
lemma is_compact.bdd_above {α : Type u} [topological_space α] [linear_order α]
[order_topology α] : Π [nonempty α] {s : set α}, is_compact s → bdd_above s :=
@is_compact.bdd_below (order_dual α) _ _ _
end order_topology
section linear_order
variables [topological_space α] [linear_order α] [order_topology α] [densely_ordered α]
/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
lemma closure_Ioi' {a b : α} (hab : a < b) :
closure (Ioi a) = Ici a :=
begin
apply subset.antisymm,
{ exact closure_minimal Ioi_subset_Ici_self is_closed_Ici },
{ assume x hx,
by_cases h : x = a,
{ rw h, exact mem_closure_of_is_glb is_glb_Ioi ⟨_, hab⟩ },
{ exact subset_closure (lt_of_le_of_ne hx (ne.symm h)) } }
end
/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/
@[simp] lemma closure_Ioi (a : α) [no_top_order α] :
closure (Ioi a) = Ici a :=
let ⟨b, hb⟩ := no_top a in closure_Ioi' hb
/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom
element. -/
lemma closure_Iio' {a b : α} (hab : b < a) :
closure (Iio a) = Iic a :=
begin
apply subset.antisymm,
{ exact closure_minimal Iio_subset_Iic_self is_closed_Iic },
{ assume x hx,
by_cases h : x = a,
{ rw h, exact mem_closure_of_is_lub is_lub_Iio ⟨_, hab⟩ },
{ apply subset_closure, by simpa [h] using lt_or_eq_of_le hx } }
end
/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/
@[simp] lemma closure_Iio (a : α) [no_bot_order α] :
closure (Iio a) = Iic a :=
let ⟨b, hb⟩ := no_bot a in closure_Iio' hb
/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp] lemma closure_Ioo {a b : α} (hab : a < b) :
closure (Ioo a b) = Icc a b :=
begin
apply subset.antisymm,
{ exact closure_minimal Ioo_subset_Icc_self is_closed_Icc },
{ have hab' : (Ioo a b).nonempty, from nonempty_Ioo.2 hab,
assume x hx,
by_cases h : x = a,
{ rw h, exact mem_closure_of_is_glb (is_glb_Ioo hab) hab' },
by_cases h' : x = b,
{ rw h', refine mem_closure_of_is_lub (is_lub_Ioo hab) hab' },
exact subset_closure ⟨lt_of_le_of_ne hx.1 (ne.symm h), by simpa [h'] using lt_or_eq_of_le hx.2⟩ }
end
/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp] lemma closure_Ioc {a b : α} (hab : a < b) :
closure (Ioc a b) = Icc a b :=
begin
apply subset.antisymm,
{ exact closure_minimal Ioc_subset_Icc_self is_closed_Icc },
{ apply subset.trans _ (closure_mono Ioo_subset_Ioc_self),
rw closure_Ioo hab }
end
/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp] lemma closure_Ico {a b : α} (hab : a < b) :
closure (Ico a b) = Icc a b :=
begin
apply subset.antisymm,
{ exact closure_minimal Ico_subset_Icc_self is_closed_Icc },
{ apply subset.trans _ (closure_mono Ioo_subset_Ico_self),
rw closure_Ioo hab }
end
@[simp] lemma interior_Ici [no_bot_order α] {a : α} : interior (Ici a) = Ioi a :=
by rw [← compl_Iio, interior_compl, closure_Iio, compl_Iic]
@[simp] lemma interior_Iic [no_top_order α] {a : α} : interior (Iic a) = Iio a :=
by rw [← compl_Ioi, interior_compl, closure_Ioi, compl_Ici]
@[simp] lemma interior_Icc [no_bot_order α] [no_top_order α] {a b : α}:
interior (Icc a b) = Ioo a b :=
by rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]
@[simp] lemma interior_Ico [no_bot_order α] {a b : α} : interior (Ico a b) = Ioo a b :=
by rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]
@[simp] lemma interior_Ioc [no_top_order α] {a b : α} : interior (Ioc a b) = Ioo a b :=
by rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]
@[simp] lemma frontier_Ici [no_bot_order α] {a : α} : frontier (Ici a) = {a} :=
by simp [frontier]
@[simp] lemma frontier_Iic [no_top_order α] {a : α} : frontier (Iic a) = {a} :=
by simp [frontier]
@[simp] lemma frontier_Ioi [no_top_order α] {a : α} : frontier (Ioi a) = {a} :=
by simp [frontier]
@[simp] lemma frontier_Iio [no_bot_order α] {a : α} : frontier (Iio a) = {a} :=
by simp [frontier]
@[simp] lemma frontier_Icc [no_bot_order α] [no_top_order α] {a b : α} (h : a < b) :
frontier (Icc a b) = {a, b} :=
by simp [frontier, le_of_lt h, Icc_diff_Ioo_same]
@[simp] lemma frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} :=
by simp [frontier, h, le_of_lt h, Icc_diff_Ioo_same]
@[simp] lemma frontier_Ico [no_bot_order α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} :=
by simp [frontier, h, le_of_lt h, Icc_diff_Ioo_same]
@[simp] lemma frontier_Ioc [no_top_order α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} :=
by simp [frontier, h, le_of_lt h, Icc_diff_Ioo_same]
lemma nhds_within_Ioi_ne_bot' {a b c : α} (H₁ : a < c) (H₂ : a ≤ b) :
ne_bot (𝓝[Ioi a] b) :=
mem_closure_iff_nhds_within_ne_bot.1 $ by { rw [closure_Ioi' H₁], exact H₂ }
lemma nhds_within_Ioi_ne_bot [no_top_order α] {a b : α} (H : a ≤ b) :
ne_bot (𝓝[Ioi a] b) :=
let ⟨c, hc⟩ := no_top a in nhds_within_Ioi_ne_bot' hc H
lemma nhds_within_Ioi_self_ne_bot' {a b : α} (H : a < b) :
ne_bot (𝓝[Ioi a] a) :=
nhds_within_Ioi_ne_bot' H (le_refl a)
@[instance]
lemma nhds_within_Ioi_self_ne_bot [no_top_order α] (a : α) :
ne_bot (𝓝[Ioi a] a) :=
nhds_within_Ioi_ne_bot (le_refl a)
lemma nhds_within_Iio_ne_bot' {a b c : α} (H₁ : a < c) (H₂ : b ≤ c) :
ne_bot (𝓝[Iio c] b) :=
mem_closure_iff_nhds_within_ne_bot.1 $ by { rw [closure_Iio' H₁], exact H₂ }
lemma nhds_within_Iio_ne_bot [no_bot_order α] {a b : α} (H : a ≤ b) :
ne_bot (𝓝[Iio b] a) :=
let ⟨c, hc⟩ := no_bot b in nhds_within_Iio_ne_bot' hc H
lemma nhds_within_Iio_self_ne_bot' {a b : α} (H : a < b) :
ne_bot (𝓝[Iio b] b) :=
nhds_within_Iio_ne_bot' H (le_refl b)
@[instance]
lemma nhds_within_Iio_self_ne_bot [no_bot_order α] (a : α) :
ne_bot (𝓝[Iio a] a) :=
nhds_within_Iio_ne_bot (le_refl a)
end linear_order
section linear_order
variables [topological_space α] [linear_order α] [order_topology α] [densely_ordered α]
{a b : α} {s : set α}
lemma comap_coe_nhds_within_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.nonempty → ∃ a < b, Ioo a b ⊆ s) :
comap (coe : s → α) (𝓝[Iio b] b) = at_top :=
begin
nontriviality,
haveI : nonempty s := nontrivial_iff_nonempty.1 ‹_›,
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩,
ext u, split,
{ rintros ⟨t, ht, hts⟩,
obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=
(mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht,
obtain ⟨y, hxy, hyb⟩ := exists_between hxb,
refine mem_sets_of_superset (mem_at_top ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _,
rintros ⟨z, hzs⟩ (hyz : y ≤ z),
refine hts (hxt ⟨hxy.trans_le _, hb _⟩); assumption },
{ intros hu,
obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_at_top_sets.1 hu,
exact ⟨Ioo x b, Ioo_mem_nhds_within_Iio (right_mem_Ioc.2 $ hb x.2), λ z hz, hx _ hz.1.le⟩ }
end
lemma comap_coe_nhds_within_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)
(hs : s.nonempty → ∃ b > a, Ioo a b ⊆ s) :
comap (coe : s → α) (𝓝[Ioi a] a) = at_bot :=
begin
refine @comap_coe_nhds_within_Iio_of_Ioo_subset (order_dual α) _ _ _ _ _ _ ha (λ h, _),
rcases hs h with ⟨b, hab, h⟩,
use [b, hab],
rwa dual_Ioo
end
lemma map_coe_at_top_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map (coe : s → α) at_top = 𝓝[Iio b] b :=
begin
rcases eq_empty_or_nonempty (Iio b) with (hb'|⟨a, ha⟩),
{ rw [filter_eq_bot_of_not_nonempty at_top, map_bot, hb', nhds_within_empty],
exact λ ⟨⟨x, hx⟩⟩, not_nonempty_iff_eq_empty.2 hb' ⟨x, hb hx⟩ },
{ rw [← comap_coe_nhds_within_Iio_of_Ioo_subset hb (λ _, hs a ha), map_comap],
rw subtype.range_coe,
exact (mem_nhds_within_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha) },
end
lemma map_coe_at_bot_of_Ioo_subset (ha : s ⊆ Ioi a)
(hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :
map (coe : s → α) at_bot = (𝓝[Ioi a] a) :=
begin
refine @map_coe_at_top_of_Ioo_subset (order_dual α) _ _ _ _ a s ha (λ b' hb', _),
rcases hs b' hb' with ⟨b, hab, hbs⟩,
use [b, hab],
rwa dual_Ioo
end
/-- The `at_top` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at
the right endpoint in the ambient order. -/
lemma comap_coe_Ioo_nhds_within_Iio (a b : α) :
comap (coe : Ioo a b → α) (𝓝[Iio b] b) = at_top :=
comap_coe_nhds_within_Iio_of_Ioo_subset Ioo_subset_Iio_self $
λ h, ⟨a, nonempty_Ioo.1 h, subset.refl _⟩
/-- The `at_bot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at
the left endpoint in the ambient order. -/
lemma comap_coe_Ioo_nhds_within_Ioi (a b : α) :
comap (coe : Ioo a b → α) (𝓝[Ioi a] a) = at_bot :=
comap_coe_nhds_within_Ioi_of_Ioo_subset Ioo_subset_Ioi_self $
λ h, ⟨b, nonempty_Ioo.1 h, subset.refl _⟩
lemma comap_coe_Ioi_nhds_within_Ioi (a : α) : comap (coe : Ioi a → α) (𝓝[Ioi a] a) = at_bot :=
comap_coe_nhds_within_Ioi_of_Ioo_subset (subset.refl _) $
λ ⟨x, hx⟩, ⟨x, hx, Ioo_subset_Ioi_self⟩
lemma comap_coe_Iio_nhds_within_Iio (a : α) :
comap (coe : Iio a → α) (𝓝[Iio a] a) = at_top :=
@comap_coe_Ioi_nhds_within_Ioi (order_dual α) _ _ _ _ a
@[simp] lemma map_coe_Ioo_at_top {a b : α} (h : a < b) :
map (coe : Ioo a b → α) at_top = 𝓝[Iio b] b :=
map_coe_at_top_of_Ioo_subset Ioo_subset_Iio_self $ λ _ _, ⟨_, h, subset.refl _⟩
@[simp] lemma map_coe_Ioo_at_bot {a b : α} (h : a < b) :
map (coe : Ioo a b → α) at_bot = 𝓝[Ioi a] a :=
map_coe_at_bot_of_Ioo_subset Ioo_subset_Ioi_self $ λ _ _, ⟨_, h, subset.refl _⟩
@[simp] lemma map_coe_Ioi_at_bot (a : α) :
map (coe : Ioi a → α) at_bot = 𝓝[Ioi a] a :=
map_coe_at_bot_of_Ioo_subset (subset.refl _) $ λ b hb, ⟨b, hb, Ioo_subset_Ioi_self⟩
@[simp] lemma map_coe_Iio_at_top (a : α) :
map (coe : Iio a → α) at_top = 𝓝[Iio a] a :=
@map_coe_Ioi_at_bot (order_dual α) _ _ _ _ _
variables {l : filter β} {f : α → β}
@[simp] lemma tendsto_comp_coe_Ioo_at_top (h : a < b) :
tendsto (λ x : Ioo a b, f x) at_top l ↔ tendsto f (𝓝[Iio b] b) l :=
by rw [← map_coe_Ioo_at_top h, tendsto_map'_iff]
@[simp] lemma tendsto_comp_coe_Ioo_at_bot (h : a < b) :
tendsto (λ x : Ioo a b, f x) at_bot l ↔ tendsto f (𝓝[Ioi a] a) l :=
by rw [← map_coe_Ioo_at_bot h, tendsto_map'_iff]
@[simp] lemma tendsto_comp_coe_Ioi_at_bot :
tendsto (λ x : Ioi a, f x) at_bot l ↔ tendsto f (𝓝[Ioi a] a) l :=
by rw [← map_coe_Ioi_at_bot, tendsto_map'_iff]
@[simp] lemma tendsto_comp_coe_Iio_at_top :
tendsto (λ x : Iio a, f x) at_top l ↔ tendsto f (𝓝[Iio a] a) l :=
by rw [← map_coe_Iio_at_top, tendsto_map'_iff]
@[simp] lemma tendsto_Ioo_at_top {f : β → Ioo a b} :
tendsto f l at_top ↔ tendsto (λ x, (f x : α)) l (𝓝[Iio b] b) :=
by rw [← comap_coe_Ioo_nhds_within_Iio, tendsto_comap_iff]
@[simp] lemma tendsto_Ioo_at_bot {f : β → Ioo a b} :
tendsto f l at_bot ↔ tendsto (λ x, (f x : α)) l (𝓝[Ioi a] a) :=
by rw [← comap_coe_Ioo_nhds_within_Ioi, tendsto_comap_iff]
@[simp] lemma tendsto_Ioi_at_bot {f : β → Ioi a} :
tendsto f l at_bot ↔ tendsto (λ x, (f x : α)) l (𝓝[Ioi a] a) :=
by rw [← comap_coe_Ioi_nhds_within_Ioi, tendsto_comap_iff]
@[simp] lemma tendsto_Iio_at_top {f : β → Iio a} :
tendsto f l at_top ↔ tendsto (λ x, (f x : α)) l (𝓝[Iio a] a) :=
by rw [← comap_coe_Iio_nhds_within_Iio, tendsto_comap_iff]
end linear_order
section complete_linear_order
variables [complete_linear_order α] [topological_space α] [order_topology α]
[complete_linear_order β] [topological_space β] [order_topology β] [nonempty γ]
lemma Sup_mem_closure {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) :
Sup s ∈ closure s :=
mem_closure_of_is_lub (is_lub_Sup _) hs
lemma Inf_mem_closure {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) :
Inf s ∈ closure s :=
mem_closure_of_is_glb (is_glb_Inf _) hs
lemma is_closed.Sup_mem {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) (hc : is_closed s) :
Sup s ∈ s :=
mem_of_is_lub_of_is_closed (is_lub_Sup _) hs hc
lemma is_closed.Inf_mem {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) (hc : is_closed s) :
Inf s ∈ s :=
mem_of_is_glb_of_is_closed (is_glb_Inf _) hs hc
/-- A monotone function continuous at the supremum of a nonempty set sends this supremum to
the supremum of the image of this set. -/
lemma map_Sup_of_continuous_at_of_monotone' {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Mf : monotone f) (hs : s.nonempty) :
f (Sup s) = Sup (f '' s) :=
--This is a particular case of the more general is_lub_of_is_lub_of_tendsto
(is_lub_of_is_lub_of_tendsto (λ x hx y hy xy, Mf xy) (is_lub_Sup _) hs $
Cf.mono_left inf_le_left).Sup_eq.symm
/-- A monotone function `s` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. -/
lemma map_Sup_of_continuous_at_of_monotone {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Mf : monotone f) (fbot : f ⊥ = ⊥) :
f (Sup s) = Sup (f '' s) :=
begin
cases s.eq_empty_or_nonempty with h h,
{ simp [h, fbot] },
{ exact map_Sup_of_continuous_at_of_monotone' Cf Mf h }
end
/-- A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. -/
lemma map_supr_of_continuous_at_of_monotone' {ι : Sort*} [nonempty ι] {f : α → β} {g : ι → α}
(Cf : continuous_at f (supr g)) (Mf : monotone f) :
f (⨆ i, g i) = ⨆ i, f (g i) :=
by rw [supr, map_Sup_of_continuous_at_of_monotone' Cf Mf (range_nonempty g), ← range_comp, supr]
/-- If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/
lemma map_supr_of_continuous_at_of_monotone {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : continuous_at f (supr g)) (Mf : monotone f) (fbot : f ⊥ = ⊥) :
f (⨆ i, g i) = ⨆ i, f (g i) :=
by rw [supr, map_Sup_of_continuous_at_of_monotone Cf Mf fbot, ← range_comp, supr]
/-- A monotone function continuous at the infimum of a nonempty set sends this infimum to
the infimum of the image of this set. -/
lemma map_Inf_of_continuous_at_of_monotone' {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Mf : monotone f) (hs : s.nonempty) :
f (Inf s) = Inf (f '' s) :=
@map_Sup_of_continuous_at_of_monotone' (order_dual α) (order_dual β) _ _ _ _ _ _ f s Cf
Mf.order_dual hs
/-- A monotone function `s` sending `top` to `top` and continuous at the infimum of a set sends
this infimum to the infimum of the image of this set. -/
lemma map_Inf_of_continuous_at_of_monotone {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Mf : monotone f) (ftop : f ⊤ = ⊤) :
f (Inf s) = Inf (f '' s) :=
@map_Sup_of_continuous_at_of_monotone (order_dual α) (order_dual β) _ _ _ _ _ _ f s Cf
Mf.order_dual ftop
/-- A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. -/
lemma map_infi_of_continuous_at_of_monotone' {ι : Sort*} [nonempty ι] {f : α → β} {g : ι → α}
(Cf : continuous_at f (infi g)) (Mf : monotone f) :
f (⨅ i, g i) = ⨅ i, f (g i) :=
@map_supr_of_continuous_at_of_monotone' (order_dual α) (order_dual β) _ _ _ _ _ _ ι _ f g Cf
Mf.order_dual
/-- If a monotone function sending `top` to `top` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed infimum of the composition. -/
lemma map_infi_of_continuous_at_of_monotone {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : continuous_at f (infi g)) (Mf : monotone f) (ftop : f ⊤ = ⊤) :
f (infi g) = infi (f ∘ g) :=
@map_supr_of_continuous_at_of_monotone (order_dual α) (order_dual β) _ _ _ _ _ _ ι f g Cf
Mf.order_dual ftop
end complete_linear_order
section conditionally_complete_linear_order
variables [conditionally_complete_linear_order α] [topological_space α] [order_topology α]
[conditionally_complete_linear_order β] [topological_space β] [order_topology β] [nonempty γ]
lemma cSup_mem_closure {s : set α} (hs : s.nonempty) (B : bdd_above s) : Sup s ∈ closure s :=
mem_closure_of_is_lub (is_lub_cSup hs B) hs
lemma cInf_mem_closure {s : set α} (hs : s.nonempty) (B : bdd_below s) : Inf s ∈ closure s :=
mem_closure_of_is_glb (is_glb_cInf hs B) hs
lemma is_closed.cSup_mem {s : set α} (hc : is_closed s) (hs : s.nonempty) (B : bdd_above s) :
Sup s ∈ s :=
mem_of_is_lub_of_is_closed (is_lub_cSup hs B) hs hc
lemma is_closed.cInf_mem {s : set α} (hc : is_closed s) (hs : s.nonempty) (B : bdd_below s) :
Inf s ∈ s :=
mem_of_is_glb_of_is_closed (is_glb_cInf hs B) hs hc
/-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. -/
lemma map_cSup_of_continuous_at_of_monotone {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Mf : monotone f) (ne : s.nonempty) (H : bdd_above s) :
f (Sup s) = Sup (f '' s) :=
begin
refine ((is_lub_cSup (ne.image f) (Mf.map_bdd_above H)).unique _).symm,
refine is_lub_of_is_lub_of_tendsto (λx hx y hy xy, Mf xy) (is_lub_cSup ne H) ne _,
exact Cf.mono_left inf_le_left
end
/-- If a monotone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the supremum of the composition. -/
lemma map_csupr_of_continuous_at_of_monotone {f : α → β} {g : γ → α}
(Cf : continuous_at f (⨆ i, g i)) (Mf : monotone f) (H : bdd_above (range g)) :
f (⨆ i, g i) = ⨆ i, f (g i) :=
by rw [supr, map_cSup_of_continuous_at_of_monotone Cf Mf (range_nonempty _) H, ← range_comp, supr]
/-- If a monotone function is continuous at the infimum of a nonempty bounded below set `s`,
then it sends this infimum to the infimum of the image of `s`. -/
lemma map_cInf_of_continuous_at_of_monotone {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Mf : monotone f) (ne : s.nonempty) (H : bdd_below s) :
f (Inf s) = Inf (f '' s) :=
@map_cSup_of_continuous_at_of_monotone (order_dual α) (order_dual β) _ _ _ _ _ _ f s Cf
Mf.order_dual ne H
/-- A continuous monotone function sends indexed infimum to indexed infimum in conditionally
complete linear order, under a boundedness assumption. -/
lemma map_cinfi_of_continuous_at_of_monotone {f : α → β} {g : γ → α}
(Cf : continuous_at f (⨅ i, g i)) (Mf : monotone f) (H : bdd_below (range g)) :
f (⨅ i, g i) = ⨅ i, f (g i) :=
@map_csupr_of_continuous_at_of_monotone (order_dual α) (order_dual β) _ _ _ _ _ _ _ _ _ _
Cf Mf.order_dual H
/-- A bounded connected subset of a conditionally complete linear order includes the open interval
`(Inf s, Sup s)`. -/
lemma is_connected.Ioo_cInf_cSup_subset {s : set α} (hs : is_connected s) (hb : bdd_below s)
(ha : bdd_above s) :
Ioo (Inf s) (Sup s) ⊆ s :=
λ x hx, let ⟨y, ys, hy⟩ := (is_glb_lt_iff (is_glb_cInf hs.nonempty hb)).1 hx.1 in
let ⟨z, zs, hz⟩ := (lt_is_lub_iff (is_lub_cSup hs.nonempty ha)).1 hx.2 in
hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩
lemma eq_Icc_cInf_cSup_of_connected_bdd_closed {s : set α} (hc : is_connected s) (hb : bdd_below s)
(ha : bdd_above s) (hcl : is_closed s) :
s = Icc (Inf s) (Sup s) :=
subset.antisymm (subset_Icc_cInf_cSup hb ha) $
hc.Icc_subset (hcl.cInf_mem hc.nonempty hb) (hcl.cSup_mem hc.nonempty ha)
lemma is_preconnected.Ioi_cInf_subset {s : set α} (hs : is_preconnected s) (hb : bdd_below s)
(ha : ¬bdd_above s) :
Ioi (Inf s) ⊆ s :=
begin
have sne : s.nonempty := @nonempty_of_not_bdd_above α _ s ⟨Inf ∅⟩ ha,
intros x hx,
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := (is_glb_lt_iff (is_glb_cInf sne hb)).1 hx,
obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bdd_above_iff.1 ha x,
exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩
end
lemma is_preconnected.Iio_cSup_subset {s : set α} (hs : is_preconnected s) (hb : ¬bdd_below s)
(ha : bdd_above s) :
Iio (Sup s) ⊆ s :=
@is_preconnected.Ioi_cInf_subset (order_dual α) _ _ _ s hs ha hb
/-- A preconnected set in a conditionally complete linear order is either one of the intervals
`[Inf s, Sup s]`, `[Inf s, Sup s)`, `(Inf s, Sup s]`, `(Inf s, Sup s)`, `[Inf s, +∞)`,
`(Inf s, +∞)`, `(-∞, Sup s]`, `(-∞, Sup s)`, `(-∞, +∞)`, or `∅`. The converse statement requires
`α` to be densely ordererd. -/
lemma is_preconnected.mem_intervals {s : set α} (hs : is_preconnected s) :
s ∈ ({Icc (Inf s) (Sup s), Ico (Inf s) (Sup s), Ioc (Inf s) (Sup s), Ioo (Inf s) (Sup s),
Ici (Inf s), Ioi (Inf s), Iic (Sup s), Iio (Sup s), univ, ∅} : set (set α)) :=
begin
rcases s.eq_empty_or_nonempty with rfl|hne,
{ apply_rules [or.inr, mem_singleton] },
have hs' : is_connected s := ⟨hne, hs⟩,
by_cases hb : bdd_below s; by_cases ha : bdd_above s,
{ rcases mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset (hs'.Ioo_cInf_cSup_subset hb ha)
(subset_Icc_cInf_cSup hb ha) with hs|hs|hs|hs,
{ exact (or.inl hs) },
{ exact (or.inr $ or.inl hs) },
{ exact (or.inr $ or.inr $ or.inl hs) },
{ exact (or.inr $ or.inr $ or.inr $ or.inl hs) } },
{ refine (or.inr $ or.inr $ or.inr $ or.inr _),
cases mem_Ici_Ioi_of_subset_of_subset (hs.Ioi_cInf_subset hb ha) (λ x hx, cInf_le hb hx)
with hs hs,
{ exact or.inl hs },
{ exact or.inr (or.inl hs) } },
{ iterate 6 { apply or.inr },
cases mem_Iic_Iio_of_subset_of_subset (hs.Iio_cSup_subset hb ha) (λ x hx, le_cSup ha hx)
with hs hs,
{ exact or.inl hs },
{ exact or.inr (or.inl hs) } },
{ iterate 8 { apply or.inr },
exact or.inl (hs.eq_univ_of_unbounded hb ha) }
end
/-- A preconnected set is either one of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`,
`Iic`, `Iio`, or `univ`, or `∅`. The converse statement requires `α` to be densely ordererd. Though
one can represent `∅` as `(Inf s, Inf s)`, we include it into the list of possible cases to improve
readability. -/
lemma set_of_is_preconnected_subset_of_ordered :
{s : set α | is_preconnected s} ⊆
-- bounded intervals
(range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪
-- unbounded intervals and `univ`
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) :=
begin
intros s hs,
rcases hs.mem_intervals with hs|hs|hs|hs|hs|hs|hs|hs|hs|hs,
{ exact (or.inl $ or.inl $ or.inl $ or.inl ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inl $ or.inl $ or.inl $ or.inr ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inl $ or.inl $ or.inr ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inl $ or.inr ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inr $ or.inl $ or.inl $ or.inl $ or.inl ⟨Inf s, hs.symm⟩) },
{ exact (or.inr $ or.inl $ or.inl $ or.inl $ or.inr ⟨Inf s, hs.symm⟩) },
{ exact (or.inr $ or.inl $ or.inl $ or.inr ⟨Sup s, hs.symm⟩) },
{ exact (or.inr $ or.inl $ or.inr ⟨Sup s, hs.symm⟩) },
{ exact (or.inr $ or.inr $ or.inl hs) },
{ exact (or.inr $ or.inr $ or.inr hs) }
end
/-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and the set `s ∩ [a, b)` has no maximal point, then `b ∈ s`. -/
lemma is_closed.mem_of_ge_of_forall_exists_gt {a b : α} {s : set α} (hs : is_closed (s ∩ Icc a b))
(ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).nonempty) :
b ∈ s :=
begin
let S := s ∩ Icc a b,
replace ha : a ∈ S, from ⟨ha, left_mem_Icc.2 hab⟩,
have Sbd : bdd_above S, from ⟨b, λ z hz, hz.2.2⟩,
let c := Sup (s ∩ Icc a b),
have c_mem : c ∈ S, from hs.cSup_mem ⟨_, ha⟩ Sbd,
have c_le : c ≤ b, from cSup_le ⟨_, ha⟩ (λ x hx, hx.2.2),
cases eq_or_lt_of_le c_le with hc hc, from hc ▸ c_mem.1,
exfalso,
rcases hgt c ⟨c_mem.1, c_mem.2.1, hc⟩ with ⟨x, xs, cx, xb⟩,
exact not_lt_of_le (le_cSup Sbd ⟨xs, le_trans (le_cSup Sbd ha) (le_of_lt cx), xb⟩) cx
end
/-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `a ≤ x < y ≤ b`, `x ∈ s`, the set `s ∩ (x, y]`
is not empty, then `[a, b] ⊆ s`. -/
lemma is_closed.Icc_subset_of_forall_exists_gt {a b : α} {s : set α} (hs : is_closed (s ∩ Icc a b))
(ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).nonempty) :
Icc a b ⊆ s :=
begin
assume y hy,
have : is_closed (s ∩ Icc a y),
{ suffices : s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y,
{ rw this, exact is_closed_inter hs is_closed_Icc },
rw [inter_assoc],
congr,
exact (inter_eq_self_of_subset_right $ Icc_subset_Icc_right hy.2).symm },
exact is_closed.mem_of_ge_of_forall_exists_gt this ha hy.1
(λ x hx, hgt x ⟨hx.1, Ico_subset_Ico_right hy.2 hx.2⟩ y hx.2.2)
end
section densely_ordered
variables [densely_ordered α] {a b : α}
/-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `x ∈ s ∩ [a, b)` the set `s` includes some open
neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. -/
lemma is_closed.Icc_subset_of_forall_mem_nhds_within {a b : α} {s : set α}
(hs : is_closed (s ∩ Icc a b)) (ha : a ∈ s)
(hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[Ioi x] x) :
Icc a b ⊆ s :=
begin
apply hs.Icc_subset_of_forall_exists_gt ha,
rintros x ⟨hxs, hxab⟩ y hyxb,
have : s ∩ Ioc x y ∈ 𝓝[Ioi x] x,
from inter_mem_sets (hgt x ⟨hxs, hxab⟩) (Ioc_mem_nhds_within_Ioi ⟨le_refl _, hyxb⟩),
exact (nhds_within_Ioi_self_ne_bot' hxab.2).nonempty_of_mem this
end
/-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/
lemma is_preconnected_Icc : is_preconnected (Icc a b) :=
is_preconnected_closed_iff.2
begin
rintros s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩,
wlog hxy : x ≤ y := le_total x y using [x y s t, y x t s],
have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2,
by_contradiction hst,
suffices : Icc x y ⊆ s,
from hst ⟨y, xyab $ right_mem_Icc.2 hxy, this $ right_mem_Icc.2 hxy, hy.2⟩,
apply (is_closed_inter hs is_closed_Icc).Icc_subset_of_forall_mem_nhds_within hx.2,
rintros z ⟨zs, hz⟩,
have zt : z ∈ tᶜ, from λ zt, hst ⟨z, xyab $ Ico_subset_Icc_self hz, zs, zt⟩,
have : tᶜ ∩ Ioc z y ∈ 𝓝[Ioi z] z,
{ rw [← nhds_within_Ioc_eq_nhds_within_Ioi hz.2],
exact mem_nhds_within.2 ⟨tᶜ, ht, zt, subset.refl _⟩},
apply mem_sets_of_superset this,
have : Ioc z y ⊆ s ∪ t, from λ w hw, hab (xyab ⟨le_trans hz.1 (le_of_lt hw.1), hw.2⟩),
exact λ w ⟨wt, wzy⟩, (this wzy).elim id (λ h, (wt h).elim)
end
lemma is_preconnected_interval : is_preconnected (interval a b) := is_preconnected_Icc
lemma is_preconnected_iff_ord_connected {s : set α} :
is_preconnected s ↔ ord_connected s :=
⟨λ h x hx y hy, h.Icc_subset hx hy, λ h, is_preconnected_of_forall_pair $ λ x y hx hy,
⟨interval x y, h.interval_subset hx hy, left_mem_interval, right_mem_interval,
is_preconnected_interval⟩⟩
alias is_preconnected_iff_ord_connected ↔
is_preconnected.ord_connected set.ord_connected.is_preconnected
lemma is_preconnected_Ici : is_preconnected (Ici a) := ord_connected_Ici.is_preconnected
lemma is_preconnected_Iic : is_preconnected (Iic a) := ord_connected_Iic.is_preconnected
lemma is_preconnected_Iio : is_preconnected (Iio a) := ord_connected_Iio.is_preconnected
lemma is_preconnected_Ioi : is_preconnected (Ioi a) := ord_connected_Ioi.is_preconnected
lemma is_preconnected_Ioo : is_preconnected (Ioo a b) := ord_connected_Ioo.is_preconnected
lemma is_preconnected_Ioc : is_preconnected (Ioc a b) := ord_connected_Ioc.is_preconnected
lemma is_preconnected_Ico : is_preconnected (Ico a b) := ord_connected_Ico.is_preconnected
@[priority 100]
instance ordered_connected_space : preconnected_space α :=
⟨ord_connected_univ.is_preconnected⟩
/-- In a dense conditionally complete linear order, the set of preconnected sets is exactly
the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-∞, +∞)`,
or `∅`. Though one can represent `∅` as `(Inf s, Inf s)`, we include it into the list of
possible cases to improve readability. -/
lemma set_of_is_preconnected_eq_of_ordered :
{s : set α | is_preconnected s} =
-- bounded intervals
(range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪
-- unbounded intervals and `univ`
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) :=
begin
refine subset.antisymm set_of_is_preconnected_subset_of_ordered _,
simp only [subset_def, -mem_range, forall_range_iff, uncurry, or_imp_distrib, forall_and_distrib,
mem_union, mem_set_of_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true,
is_preconnected_Icc, is_preconnected_Ico, is_preconnected_Ioc,
is_preconnected_Ioo, is_preconnected_Ioi, is_preconnected_Iio, is_preconnected_Ici,
is_preconnected_Iic, is_preconnected_univ, is_preconnected_empty],
end
variables {δ : Type*} [linear_order δ] [topological_space δ] [order_closed_topology δ]
/--Intermediate Value Theorem for continuous functions on closed intervals, case `f a ≤ t ≤ f b`.-/
lemma intermediate_value_Icc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Icc (f a) (f b) ⊆ f '' (Icc a b) :=
is_preconnected_Icc.intermediate_value (left_mem_Icc.2 hab) (right_mem_Icc.2 hab) hf
/--Intermediate Value Theorem for continuous functions on closed intervals, case `f a ≥ t ≥ f b`.-/
lemma intermediate_value_Icc' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Icc (f b) (f a) ⊆ f '' (Icc a b) :=
is_preconnected_Icc.intermediate_value (right_mem_Icc.2 hab) (left_mem_Icc.2 hab) hf
/-- A continuous function which tendsto `at_top` `at_top` and to `at_bot` `at_bot` is surjective. -/
lemma surjective_of_continuous {f : α → δ} (hf : continuous f) (h_top : tendsto f at_top at_top)
(h_bot : tendsto f at_bot at_bot) :
function.surjective f :=
λ p, mem_range_of_exists_le_of_exists_ge hf
(h_bot.eventually (eventually_le_at_bot p)).exists
(h_top.eventually (eventually_ge_at_top p)).exists
/-- A continuous function which tendsto `at_bot` `at_top` and to `at_top` `at_bot` is surjective. -/
lemma surjective_of_continuous' {f : α → δ} (hf : continuous f) (h_top : tendsto f at_bot at_top)
(h_bot : tendsto f at_top at_bot) :
function.surjective f :=
@surjective_of_continuous (order_dual α) _ _ _ _ _ _ _ _ _ hf h_top h_bot
end densely_ordered
lemma is_compact.Inf_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
Inf s ∈ s :=
hs.is_closed.cInf_mem ne_s hs.bdd_below
lemma is_compact.Sup_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
Sup s ∈ s :=
@is_compact.Inf_mem (order_dual α) _ _ _ _ hs ne_s
lemma is_compact.is_glb_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_glb s (Inf s) :=
is_glb_cInf ne_s hs.bdd_below
lemma is_compact.is_lub_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_lub s (Sup s) :=
@is_compact.is_glb_Inf (order_dual α) _ _ _ _ hs ne_s
lemma is_compact.is_least_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_least s (Inf s) :=
⟨hs.Inf_mem ne_s, (hs.is_glb_Inf ne_s).1⟩
lemma is_compact.is_greatest_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_greatest s (Sup s) :=
@is_compact.is_least_Inf (order_dual α) _ _ _ _ hs ne_s
lemma is_compact.exists_is_least {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x, is_least s x :=
⟨_, hs.is_least_Inf ne_s⟩
lemma is_compact.exists_is_greatest {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x, is_greatest s x :=
⟨_, hs.is_greatest_Sup ne_s⟩
lemma is_compact.exists_is_glb {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x ∈ s, is_glb s x :=
⟨_, hs.Inf_mem ne_s, hs.is_glb_Inf ne_s⟩
lemma is_compact.exists_is_lub {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x ∈ s, is_lub s x :=
⟨_, hs.Sup_mem ne_s, hs.is_lub_Sup ne_s⟩
lemma is_compact.exists_Inf_image_eq {α : Type u} [topological_space α]
{s : set α} (hs : is_compact s) (ne_s : s.nonempty) {f : α → β} (hf : continuous_on f s) :
∃ x ∈ s, Inf (f '' s) = f x :=
let ⟨x, hxs, hx⟩ := (hs.image_of_continuous_on hf).Inf_mem (ne_s.image f)
in ⟨x, hxs, hx.symm⟩
lemma is_compact.exists_Sup_image_eq {α : Type u} [topological_space α]:
∀ {s : set α}, is_compact s → s.nonempty → ∀ {f : α → β}, continuous_on f s →
∃ x ∈ s, Sup (f '' s) = f x :=
@is_compact.exists_Inf_image_eq (order_dual β) _ _ _ _ _
lemma eq_Icc_of_connected_compact {s : set α} (h₁ : is_connected s) (h₂ : is_compact s) :
s = Icc (Inf s) (Sup s) :=
eq_Icc_cInf_cSup_of_connected_bdd_closed h₁ h₂.bdd_below h₂.bdd_above h₂.is_closed
/-- The extreme value theorem: a continuous function realizes its minimum on a compact set -/
lemma is_compact.exists_forall_le {α : Type u} [topological_space α]
{s : set α} (hs : is_compact s) (ne_s : s.nonempty) {f : α → β} (hf : continuous_on f s) :
∃x∈s, ∀y∈s, f x ≤ f y :=
begin
rcases hs.exists_Inf_image_eq ne_s hf with ⟨x, hxs, hx⟩,
refine ⟨x, hxs, λ y hy, _⟩,
rw ← hx,
exact ((hs.image_of_continuous_on hf).is_glb_Inf (ne_s.image f)).1 (mem_image_of_mem _ hy)
end
/-- The extreme value theorem: a continuous function realizes its maximum on a compact set -/
lemma is_compact.exists_forall_ge {α : Type u} [topological_space α]:
∀ {s : set α}, is_compact s → s.nonempty → ∀ {f : α → β}, continuous_on f s →
∃x∈s, ∀y∈s, f y ≤ f x :=
@is_compact.exists_forall_le (order_dual β) _ _ _ _ _
/-- The extreme value theorem: if a continuous function `f` tends to infinity away from compact
sets, then it has a global minimum. -/
lemma continuous.exists_forall_le {α : Type*} [topological_space α] [nonempty α] {f : α → β}
(hf : continuous f) (hlim : tendsto f (cocompact α) at_top) :
∃ x, ∀ y, f x ≤ f y :=
begin
inhabit α,
obtain ⟨s : set α, hsc : is_compact s, hsf : ∀ x ∉ s, f (default α) ≤ f x⟩ :=
(has_basis_cocompact.tendsto_iff at_top_basis).1 hlim (f $ default α) trivial,
obtain ⟨x, -, hx⟩ :=
(hsc.insert (default α)).exists_forall_le (nonempty_insert _ _) hf.continuous_on,
refine ⟨x, λ y, _⟩,
by_cases hy : y ∈ s,
exacts [hx y (or.inr hy), (hx _ (or.inl rfl)).trans (hsf y hy)]
end
/-- The extreme value theorem: if a continuous function `f` tends to negative infinity away from
compactx sets, then it has a global maximum. -/
lemma continuous.exists_forall_ge {α : Type*} [topological_space α] [nonempty α] {f : α → β}
(hf : continuous f) (hlim : tendsto f (cocompact α) at_bot) :
∃ x, ∀ y, f y ≤ f x :=
@continuous.exists_forall_le (order_dual β) _ _ _ _ _ _ _ hf hlim
end conditionally_complete_linear_order
section liminf_limsup
section order_closed_topology
variables [semilattice_sup α] [topological_space α] [order_topology α]
lemma is_bounded_le_nhds (a : α) : (𝓝 a).is_bounded (≤) :=
match forall_le_or_exists_lt_sup a with
| or.inl h := ⟨a, eventually_of_forall h⟩
| or.inr ⟨b, hb⟩ := ⟨b, ge_mem_nhds hb⟩
end
lemma filter.tendsto.is_bounded_under_le {f : filter β} {u : β → α} {a : α}
(h : tendsto u f (𝓝 a)) : f.is_bounded_under (≤) u :=
(is_bounded_le_nhds a).mono h
lemma is_cobounded_ge_nhds (a : α) : (𝓝 a).is_cobounded (≥) :=
(is_bounded_le_nhds a).is_cobounded_flip
lemma filter.tendsto.is_cobounded_under_ge {f : filter β} {u : β → α} {a : α}
[ne_bot f] (h : tendsto u f (𝓝 a)) : f.is_cobounded_under (≥) u :=
h.is_bounded_under_le.is_cobounded_flip
end order_closed_topology
section order_closed_topology
variables [semilattice_inf α] [topological_space α] [order_topology α]
lemma is_bounded_ge_nhds (a : α) : (𝓝 a).is_bounded (≥) :=
@is_bounded_le_nhds (order_dual α) _ _ _ a
lemma filter.tendsto.is_bounded_under_ge {f : filter β} {u : β → α} {a : α}
(h : tendsto u f (𝓝 a)) : f.is_bounded_under (≥) u :=
(is_bounded_ge_nhds a).mono h
lemma is_cobounded_le_nhds (a : α) : (𝓝 a).is_cobounded (≤) :=
(is_bounded_ge_nhds a).is_cobounded_flip
lemma filter.tendsto.is_cobounded_under_le {f : filter β} {u : β → α} {a : α}
[ne_bot f] (h : tendsto u f (𝓝 a)) : f.is_cobounded_under (≤) u :=
h.is_bounded_under_ge.is_cobounded_flip
end order_closed_topology
section conditionally_complete_linear_order
variables [conditionally_complete_linear_order α]
theorem lt_mem_sets_of_Limsup_lt {f : filter α} {b} (h : f.is_bounded (≤)) (l : f.Limsup < b) :
∀ᶠ a in f, a < b :=
let ⟨c, (h : ∀ᶠ a in f, a ≤ c), hcb⟩ := exists_lt_of_cInf_lt h l in
mem_sets_of_superset h $ assume a hac, lt_of_le_of_lt hac hcb
theorem gt_mem_sets_of_Liminf_gt : ∀ {f : filter α} {b}, f.is_bounded (≥) → b < f.Liminf →
∀ᶠ a in f, b < a :=
@lt_mem_sets_of_Limsup_lt (order_dual α) _
variables [topological_space α] [order_topology α]
/-- If the liminf and the limsup of a filter coincide, then this filter converges to
their common value, at least if the filter is eventually bounded above and below. -/
theorem le_nhds_of_Limsup_eq_Liminf {f : filter α} {a : α}
(hl : f.is_bounded (≤)) (hg : f.is_bounded (≥)) (hs : f.Limsup = a) (hi : f.Liminf = a) :
f ≤ 𝓝 a :=
tendsto_order.2 $ and.intro
(assume b hb, gt_mem_sets_of_Liminf_gt hg $ hi.symm ▸ hb)
(assume b hb, lt_mem_sets_of_Limsup_lt hl $ hs.symm ▸ hb)
theorem Limsup_nhds (a : α) : Limsup (𝓝 a) = a :=
cInf_intro (is_bounded_le_nhds a)
(assume a' (h : {n : α | n ≤ a'} ∈ 𝓝 a), show a ≤ a', from @mem_of_nhds α _ a _ h)
(assume b (hba : a < b), show ∃c (h : {n : α | n ≤ c} ∈ 𝓝 a), c < b, from
match dense_or_discrete a b with
| or.inl ⟨c, hac, hcb⟩ := ⟨c, ge_mem_nhds hac, hcb⟩
| or.inr ⟨_, h⟩ := ⟨a, (𝓝 a).sets_of_superset (gt_mem_nhds hba) h, hba⟩
end)
theorem Liminf_nhds : ∀ (a : α), Liminf (𝓝 a) = a :=
@Limsup_nhds (order_dual α) _ _ _
/-- If a filter is converging, its limsup coincides with its limit. -/
theorem Liminf_eq_of_le_nhds {f : filter α} {a : α} [ne_bot f] (h : f ≤ 𝓝 a) : f.Liminf = a :=
have hb_ge : is_bounded (≥) f, from (is_bounded_ge_nhds a).mono h,
have hb_le : is_bounded (≤) f, from (is_bounded_le_nhds a).mono h,
le_antisymm
(calc f.Liminf ≤ f.Limsup : Liminf_le_Limsup hb_le hb_ge
... ≤ (𝓝 a).Limsup :
Limsup_le_Limsup_of_le h hb_ge.is_cobounded_flip (is_bounded_le_nhds a)
... = a : Limsup_nhds a)
(calc a = (𝓝 a).Liminf : (Liminf_nhds a).symm
... ≤ f.Liminf :
Liminf_le_Liminf_of_le h (is_bounded_ge_nhds a) hb_le.is_cobounded_flip)
/-- If a filter is converging, its liminf coincides with its limit. -/
theorem Limsup_eq_of_le_nhds : ∀ {f : filter α} {a : α} [ne_bot f], f ≤ 𝓝 a → f.Limsup = a :=
@Liminf_eq_of_le_nhds (order_dual α) _ _ _
/-- If a function has a limit, then its limsup coincides with its limit. -/
theorem filter.tendsto.limsup_eq {f : filter β} {u : β → α} {a : α} [ne_bot f]
(h : tendsto u f (𝓝 a)) : limsup f u = a :=
Limsup_eq_of_le_nhds h
/-- If a function has a limit, then its liminf coincides with its limit. -/
theorem filter.tendsto.liminf_eq {f : filter β} {u : β → α} {a : α} [ne_bot f]
(h : tendsto u f (𝓝 a)) : liminf f u = a :=
Liminf_eq_of_le_nhds h
end conditionally_complete_linear_order
section complete_linear_order
variables [complete_linear_order α] [topological_space α] [order_topology α]
-- In complete_linear_order, the above theorems take a simpler form
/-- If the liminf and the limsup of a function coincide, then the limit of the function
exists and has the same value -/
theorem tendsto_of_liminf_eq_limsup {f : filter β} {u : β → α} {a : α}
(hinf : liminf f u = a) (hsup : limsup f u = a) : tendsto u f (𝓝 a) :=
le_nhds_of_Limsup_eq_Liminf is_bounded_le_of_top is_bounded_ge_of_bot hsup hinf
/-- If a number `a` is less than or equal to the `liminf` of a function `f` at some filter
and is greater than or equal to the `limsup` of `f`, then `f` tends to `a` along this filter. -/
theorem tendsto_of_le_liminf_of_limsup_le {f : filter β} {u : β → α} {a : α}
(hinf : a ≤ liminf f u) (hsup : limsup f u ≤ a) :
tendsto u f (𝓝 a) :=
if hf : f = ⊥ then hf.symm ▸ tendsto_bot
else by haveI : ne_bot f := hf; exact tendsto_of_liminf_eq_limsup
(le_antisymm (le_trans liminf_le_limsup hsup) hinf)
(le_antisymm hsup (le_trans hinf liminf_le_limsup))
end complete_linear_order
end liminf_limsup
end order_topology
lemma order_topology_of_nhds_abs
{α : Type*} [linear_ordered_add_comm_group α] [topological_space α]
(h_nhds : ∀a:α, 𝓝 a = (⨅r>0, 𝓟 {b | abs (a - b) < r})) : order_topology α :=
order_topology.mk $ eq_of_nhds_eq_nhds $ assume a:α, le_antisymm_iff.mpr
begin
simp [infi_and, topological_space.nhds_generate_from,
h_nhds, le_infi_iff, -le_principal_iff, and_comm],
refine ⟨λ s ha b hs, _, λ r hr, _⟩,
{ rcases hs with rfl | rfl,
{ refine infi_le_of_le (a - b)
(infi_le_of_le (lt_sub_left_of_add_lt $ by simpa using ha) $
principal_mono.mpr $ assume c (hc : abs (a - c) < a - b), _),
have : a + -c < a + -b :=
by simpa only [sub_eq_add_neg] using lt_of_le_of_lt (le_abs_self _) hc,
exact lt_of_neg_lt_neg (lt_of_add_lt_add_left this) },
{ refine infi_le_of_le (b - a)
(infi_le_of_le (lt_sub_left_of_add_lt $ by simpa using ha) $
principal_mono.mpr $ assume c (hc : abs (a - c) < b - a), _),
have : abs (c - a) < b - a, {rw abs_sub; simpa using hc},
have : c + -a < b + -a :=
by simpa only [sub_eq_add_neg] using lt_of_le_of_lt (le_abs_self _) this,
exact lt_of_add_lt_add_right this } },
{ have h : {b | abs (a - b) < r} = {b | a - r < b} ∩ {b | b < a + r},
from set.ext (assume b,
by simp [abs_lt, sub_lt, lt_sub_iff_add_lt, sub_lt_iff_lt_add']; cc),
rw [h, ← inf_principal],
apply le_inf _ _,
{ exact infi_le_of_le {b : α | a - r < b} (infi_le_of_le (sub_lt_self a hr) $
infi_le_of_le (a - r) $ infi_le _ (or.inl rfl)) },
{ exact infi_le_of_le {b : α | b < a + r} (infi_le_of_le (lt_add_of_pos_right _ hr) $
infi_le_of_le (a + r) $ infi_le _ (or.inr rfl)) } }
end
/-- $\lim_{x\to+\infty}|x|=+\infty$ -/
lemma tendsto_abs_at_top_at_top [linear_ordered_add_comm_group α] :
tendsto (abs : α → α) at_top at_top :=
tendsto_at_top_mono (λ n, le_abs_self _) tendsto_id
local notation `|` x `|` := abs x
lemma linear_ordered_add_comm_group.tendsto_nhds
[linear_ordered_add_comm_group α] [topological_space α] [order_topology α] {β : Type*}
(f : β → α) (x : filter β) (a : α) :
filter.tendsto f x (nhds a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε :=
begin
rw (show _, from @tendsto_order α), -- does not work without `show` for some reason
split,
{ rintros ⟨hyp_lt_a, hyp_gt_a⟩ ε ε_pos,
suffices : {b : β | f b - a < ε ∧ a - f b < ε} ∈ x, by simpa only [abs_sub_lt_iff],
have set1 : {b : β | a - f b < ε} ∈ x,
{ have : {b : β | a - ε < f b} ∈ x, from hyp_lt_a (a - ε) (sub_lt_self a ε_pos),
have : ∀ b, a - f b < ε ↔ a - ε < f b, by { intro _, exact sub_lt },
simpa only [this] },
have set2 : {b : β | f b - a < ε} ∈ x,
{ have : {b : β | a + ε > f b} ∈ x, from hyp_gt_a (a + ε) (lt_add_of_pos_right a ε_pos),
have : ∀ b, f b - a < ε ↔ a + ε > f b, by { intro _, exact sub_lt_iff_lt_add' },
simpa only [this] },
exact (x.inter_sets set2 set1) },
{ assume hyp_ε_pos,
split,
{ assume a' a'_lt_a,
let ε := a - a',
have : {b : β | |f b - a| < ε} ∈ x, from hyp_ε_pos ε (sub_pos.elim_right a'_lt_a),
have : {b : β | f b - a < ε ∧ a - f b < ε} ∈ x, by simpa only [abs_sub_lt_iff] using this,
have : {b : β | a - f b < ε} ∈ x, from x.sets_of_superset this (set.inter_subset_right _ _),
have : ∀ b, a' < f b ↔ a - f b < ε, by {intro b, rw [sub_lt, sub_sub_self] },
simpa only [this] },
{ assume a' a'_gt_a,
let ε := a' - a,
have : {b : β | |f b - a| < ε} ∈ x, from hyp_ε_pos ε (sub_pos.elim_right a'_gt_a),
have : {b : β | f b - a < ε ∧ a - f b < ε} ∈ x, by simpa only [abs_sub_lt_iff] using this,
have : {b : β | f b - a < ε} ∈ x, from x.sets_of_superset this (set.inter_subset_left _ _),
have : ∀ b, f b < a' ↔ f b - a < ε, by { intro b, simp [lt_sub_iff_add_lt] },
simpa only [this] }}
end
/-!
Here is a counter-example to a version of the following with `conditionally_complete_lattice α`.
Take `α = [0, 1) → ℝ` with the natural lattice structure, `ι = ℕ`. Put `f n x = -x^n`. Then
`⨆ n, f n = 0` while none of `f n` is strictly greater than the constant function `-0.5`.
-/
lemma tendsto_at_top_csupr {ι α : Type*} [preorder ι] [topological_space α]
[conditionally_complete_linear_order α] [order_topology α]
{f : ι → α} (h_mono : monotone f) (hbdd : bdd_above $ range f) :
tendsto f at_top (𝓝 (⨆i, f i)) :=
begin
by_cases hi : nonempty ι,
{ resetI,
rw tendsto_order,
split,
{ intros a h,
cases exists_lt_of_lt_csupr h with N hN,
apply eventually.mono (mem_at_top N),
exact λ i hi, lt_of_lt_of_le hN (h_mono hi) },
{ exact λ a h, eventually_of_forall (λ n, lt_of_le_of_lt (le_csupr hbdd n) h) } },
{ exact tendsto_of_not_nonempty hi }
end
lemma tendsto_at_top_cinfi {ι α : Type*} [preorder ι] [topological_space α]
[conditionally_complete_linear_order α] [order_topology α]
{f : ι → α} (h_mono : ∀ ⦃i j⦄, i ≤ j → f j ≤ f i) (hbdd : bdd_below $ range f) :
tendsto f at_top (𝓝 (⨅i, f i)) :=
@tendsto_at_top_csupr _ (order_dual α) _ _ _ _ _ @h_mono hbdd
lemma tendsto_at_top_supr {ι α : Type*} [preorder ι] [topological_space α]
[complete_linear_order α] [order_topology α] {f : ι → α} (h_mono : monotone f) :
tendsto f at_top (𝓝 (⨆i, f i)) :=
tendsto_at_top_csupr h_mono (order_top.bdd_above _)
lemma tendsto_at_top_infi {ι α : Type*} [preorder ι] [topological_space α]
[complete_linear_order α] [order_topology α] {f : ι → α} (h_mono : ∀ ⦃i j⦄, i ≤ j → f j ≤ f i) :
tendsto f at_top (𝓝 (⨅i, f i)) :=
tendsto_at_top_cinfi @h_mono (order_bot.bdd_below _)
lemma tendsto_of_monotone {ι α : Type*} [preorder ι] [topological_space α]
[conditionally_complete_linear_order α] [order_topology α] {f : ι → α} (h_mono : monotone f) :
tendsto f at_top at_top ∨ (∃ l, tendsto f at_top (𝓝 l)) :=
if H : bdd_above (range f) then or.inr ⟨_, tendsto_at_top_csupr h_mono H⟩
else or.inl $ tendsto_at_top_at_top_of_monotone' h_mono H
lemma supr_eq_of_tendsto {α β} [topological_space α] [complete_linear_order α] [order_topology α]
[nonempty β] [semilattice_sup β] {f : β → α} {a : α} (hf : monotone f) :
tendsto f at_top (𝓝 a) → supr f = a :=
tendsto_nhds_unique (tendsto_at_top_supr hf)
lemma infi_eq_of_tendsto {α} [topological_space α] [complete_linear_order α] [order_topology α]
[nonempty β] [semilattice_sup β] {f : β → α} {a : α} (hf : ∀n m, n ≤ m → f m ≤ f n) :
tendsto f at_top (𝓝 a) → infi f = a :=
tendsto_nhds_unique (tendsto_at_top_infi hf)
@[to_additive] lemma tendsto_inv_nhds_within_Ioi [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Ioi a] a) (𝓝[Iio (a⁻¹)] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Iio [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Iio a] a) (𝓝[Ioi (a⁻¹)] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Ioi_inv [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Ioi (a⁻¹)] (a⁻¹)) (𝓝[Iio a] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Ioi _ _ _ _ (a⁻¹)
@[to_additive] lemma tendsto_inv_nhds_within_Iio_inv [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Iio (a⁻¹)] (a⁻¹)) (𝓝[Ioi a] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Iio _ _ _ _ (a⁻¹)
@[to_additive] lemma tendsto_inv_nhds_within_Ici [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Ici a] a) (𝓝[Iic (a⁻¹)] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Iic [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Iic a] a) (𝓝[Ici (a⁻¹)] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Ici_inv [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Ici (a⁻¹)] (a⁻¹)) (𝓝[Iic a] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Ici _ _ _ _ (a⁻¹)
@[to_additive] lemma tendsto_inv_nhds_within_Iic_inv [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Iic (a⁻¹)] (a⁻¹)) (𝓝[Ici a] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Iic _ _ _ _ (a⁻¹)
lemma nhds_left_sup_nhds_right (a : α) [topological_space α] [linear_order α] :
𝓝[Iic a] a ⊔ 𝓝[Ici a] a = 𝓝 a :=
by rw [← nhds_within_union, Iic_union_Ici, nhds_within_univ]
lemma nhds_left'_sup_nhds_right (a : α) [topological_space α] [linear_order α] :
𝓝[Iio a] a ⊔ 𝓝[Ici a] a = 𝓝 a :=
by rw [← nhds_within_union, Iio_union_Ici, nhds_within_univ]
lemma nhds_left_sup_nhds_right' (a : α) [topological_space α] [linear_order α] :
𝓝[Iic a] a ⊔ 𝓝[Ioi a] a = 𝓝 a :=
by rw [← nhds_within_union, Iic_union_Ioi, nhds_within_univ]
lemma continuous_at_iff_continuous_left_right [topological_space α] [linear_order α]
[topological_space β] {a : α} {f : α → β} :
continuous_at f a ↔ continuous_within_at f (Iic a) a ∧ continuous_within_at f (Ici a) a :=
by simp only [continuous_within_at, continuous_at, ← tendsto_sup, nhds_left_sup_nhds_right]
lemma continuous_on_Icc_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α]
[order_topology α] [topological_space β] [regular_space β] {f : α → β} {a b : α}
{la lb : β} (hab : a < b) (hf : continuous_on f (Ioo a b))
(ha : tendsto f (𝓝[Ioi a] a) (𝓝 la)) (hb : tendsto f (𝓝[Iio b] b) (𝓝 lb)) :
continuous_on (extend_from (Ioo a b) f) (Icc a b) :=
begin
apply continuous_on_extend_from,
{ rw closure_Ioo hab, },
{ intros x x_in,
rcases mem_Ioo_or_eq_endpoints_of_mem_Icc x_in with rfl | rfl | h,
{ use la,
simpa [hab] },
{ use lb,
simpa [hab] },
{ use [f x, hf x h] } }
end
lemma eq_lim_at_left_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α]
[order_topology α] [topological_space β] [t2_space β] {f : α → β} {a b : α}
{la : β} (hab : a < b) (ha : tendsto f (𝓝[Ioi a] a) (𝓝 la)) :
extend_from (Ioo a b) f a = la :=
begin
apply extend_from_eq,
{ rw closure_Ioo hab,
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc] },
{ simpa [hab] }
end
lemma eq_lim_at_right_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α]
[order_topology α] [topological_space β] [t2_space β] {f : α → β} {a b : α}
{lb : β} (hab : a < b) (hb : tendsto f (𝓝[Iio b] b) (𝓝 lb)) :
extend_from (Ioo a b) f b = lb :=
begin
apply extend_from_eq,
{ rw closure_Ioo hab,
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc] },
{ simpa [hab] }
end
lemma continuous_on_Ico_extend_from_Ioo [topological_space α]
[linear_order α] [densely_ordered α] [order_topology α] [topological_space β]
[regular_space β] {f : α → β} {a b : α} {la : β} (hab : a < b) (hf : continuous_on f (Ioo a b))
(ha : tendsto f (𝓝[Ioi a] a) (𝓝 la)) :
continuous_on (extend_from (Ioo a b) f) (Ico a b) :=
begin
apply continuous_on_extend_from,
{ rw [closure_Ioo hab], exact Ico_subset_Icc_self, },
{ intros x x_in,
rcases mem_Ioo_or_eq_left_of_mem_Ico x_in with rfl | h,
{ use la,
simpa [hab] },
{ use [f x, hf x h] } }
end
lemma continuous_on_Ioc_extend_from_Ioo [topological_space α]
[linear_order α] [densely_ordered α] [order_topology α] [topological_space β]
[regular_space β] {f : α → β} {a b : α} {lb : β} (hab : a < b) (hf : continuous_on f (Ioo a b))
(hb : tendsto f (𝓝[Iio b] b) (𝓝 lb)) :
continuous_on (extend_from (Ioo a b) f) (Ioc a b) :=
begin
have := @continuous_on_Ico_extend_from_Ioo (order_dual α) _ _ _ _ _ _ _ f _ _ _ hab,
erw [dual_Ico, dual_Ioi, dual_Ioo] at this,
exact this hf hb
end
lemma continuous_within_at_Ioi_iff_Ici {α β : Type*} [topological_space α] [partial_order α]
[topological_space β] {a : α} {f : α → β} :
continuous_within_at f (Ioi a) a ↔ continuous_within_at f (Ici a) a :=
by simp only [← Ici_diff_left, continuous_within_at_diff_self]
lemma continuous_within_at_Iio_iff_Iic
{α β : Type*} [topological_space α] [linear_order α] [topological_space β] {a : α} {f : α → β} :
continuous_within_at f (Iio a) a ↔ continuous_within_at f (Iic a) a :=
begin
have := @continuous_within_at_Ioi_iff_Ici (order_dual α) _ _ _ _ _ f,
erw [dual_Ici, dual_Ioi] at this,
exact this,
end
lemma continuous_at_iff_continuous_left'_right' [topological_space α] [linear_order α]
[topological_space β] {a : α} {f : α → β} :
continuous_at f a ↔ continuous_within_at f (Iio a) a ∧ continuous_within_at f (Ioi a) a :=
by rw [continuous_within_at_Ioi_iff_Ici, continuous_within_at_Iio_iff_Iic,
continuous_at_iff_continuous_left_right]
/-!
### Continuity of monotone functions
In this section we prove the following fact: if `f` is a monotone function on a neighborhood of `a`
and the image of this neighborhood is a neighborhood of `f a`, then `f` is continuous at `a`, see
`continuous_at_of_mono_incr_on_of_image_mem_nhds`, as well as several similar facts.
-/
section linear_order
variables [linear_order α] [topological_space α] [order_topology α]
variables [linear_order β] [topological_space β] [order_topology β]
/-- If `f` is a function strictly monotonically increasing on a right neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `(f a, b]`, `b > f a`, then `f` is
continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` is required because otherwise the
function `f : ℝ → ℝ` given by `f x = if x ≤ 0 then x else x + 1` would be a counter-example at
`a = 0`. -/
lemma strict_mono_incr_on.continuous_at_right_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Ici a] a)
(hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
continuous_within_at f (Ici a) a :=
begin
have ha : a ∈ Ici a := left_mem_Ici,
have has : a ∈ s := mem_of_mem_nhds_within ha hs,
refine tendsto_order.2 ⟨λ b hb, _, λ b hb, _⟩,
{ filter_upwards [hs, self_mem_nhds_within],
intros x hxs hxa,
exact hb.trans_le ((h_mono.le_iff_le has hxs).2 hxa) },
{ rcases hfs b hb with ⟨c, hcs, hac, hcb⟩,
rw [h_mono.lt_iff_lt has hcs] at hac,
filter_upwards [hs, Ico_mem_nhds_within_Ici (left_mem_Ico.2 hac)],
rintros x hx ⟨hax, hxc⟩,
exact ((h_mono.lt_iff_lt hx hcs).2 hxc).trans_le hcb }
end
/-- If `f` is a function monotonically increasing function on a right neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `(f a, b)`, `b > f a`, then `f` is
continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b` cannot be replaced by the weaker
assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` we use for strictly monotone functions
because otherwise the function `ceil : ℝ → ℤ` would be a counter-example at `a = 0`. -/
lemma continuous_at_right_of_mono_incr_on_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝[Ici a] a)
(hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
continuous_within_at f (Ici a) a :=
begin
have ha : a ∈ Ici a := left_mem_Ici,
have has : a ∈ s := mem_of_mem_nhds_within ha hs,
refine tendsto_order.2 ⟨λ b hb, _, λ b hb, _⟩,
{ filter_upwards [hs, self_mem_nhds_within],
intros x hxs hxa,
exact hb.trans_le (h_mono _ has _ hxs hxa) },
{ rcases hfs b hb with ⟨c, hcs, hac, hcb⟩,
have : a < c, from not_le.1 (λ h, hac.not_le $ h_mono _ hcs _ has h),
filter_upwards [hs, Ico_mem_nhds_within_Ici (left_mem_Ico.2 this)],
rintros x hx ⟨hax, hxc⟩,
exact (h_mono _ hx _ hcs hxc.le).trans_lt hcb }
end
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a right
neighborhood of `a` and the closure of the image of this neighborhood under `f` is a right
neighborhood of `f a`, then `f` is continuous at `a` from the right. -/
lemma continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[Ici a] a) (hfs : closure (f '' s) ∈ 𝓝[Ici (f a)] (f a)) :
continuous_within_at f (Ici a) a :=
begin
refine continuous_at_right_of_mono_incr_on_of_exists_between h_mono hs (λ b hb, _),
rcases (mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset hb).1 hfs with ⟨b', ⟨hab', hbb'⟩, hb'⟩,
rcases exists_between hab' with ⟨c', hc'⟩,
rcases mem_closure_iff.1 (hb' ⟨hc'.1.le, hc'.2⟩) (Ioo (f a) b') is_open_Ioo hc'
with ⟨_, hc, ⟨c, hcs, rfl⟩⟩,
exact ⟨c, hcs, hc.1, hc.2.trans_le hbb'⟩
end
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a right
neighborhood of `a` and the image of this neighborhood under `f` is a right neighborhood of `f a`,
then `f` is continuous at `a` from the right. -/
lemma continuous_at_right_of_mono_incr_on_of_image_mem_nhds_within [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝[Ici a] a)
(hfs : f '' s ∈ 𝓝[Ici (f a)] (f a)) :
continuous_within_at f (Ici a) a :=
continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within h_mono hs $
mem_sets_of_superset hfs subset_closure
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
right neighborhood of `a` and the closure of the image of this neighborhood under `f` is a right
neighborhood of `f a`, then `f` is continuous at `a` from the right. -/
lemma strict_mono_incr_on.continuous_at_right_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Ici a] a)
(hfs : closure (f '' s) ∈ 𝓝[Ici (f a)] (f a)) :
continuous_within_at f (Ici a) a :=
continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within
(λ x hx y hy, (h_mono.le_iff_le hx hy).2) hs hfs
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
right neighborhood of `a` and the image of this neighborhood under `f` is a right neighborhood of
`f a`, then `f` is continuous at `a` from the right. -/
lemma strict_mono_incr_on.continuous_at_right_of_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Ici a] a)
(hfs : f '' s ∈ 𝓝[Ici (f a)] (f a)) :
continuous_within_at f (Ici a) a :=
h_mono.continuous_at_right_of_closure_image_mem_nhds_within hs
(mem_sets_of_superset hfs subset_closure)
/-- If a function `f` is strictly monotonically increasing on a right neighborhood of `a` and the
image of this neighborhood under `f` includes `Ioi (f a)`, then `f` is continuous at `a` from the
right. -/
lemma strict_mono_incr_on.continuous_at_right_of_surj_on {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Ici a] a) (hfs : surj_on f s (Ioi (f a))) :
continuous_within_at f (Ici a) a :=
h_mono.continuous_at_right_of_exists_between hs $ λ b hb, let ⟨c, hcs, hcb⟩ := hfs hb in
⟨c, hcs, hcb.symm ▸ hb, hcb.le⟩
/-- If `f` is a function strictly monotonically increasing on a left neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, then `f` is
continuous at `a` from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` is required because otherwise the
function `f : ℝ → ℝ` given by `f x = if x < 0 then x else x + 1` would be a counter-example at
`a = 0`. -/
lemma strict_mono_incr_on.continuous_at_left_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Iic a] a)
(hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) :
continuous_within_at f (Iic a) a :=
h_mono.dual.continuous_at_right_of_exists_between hs $
λ b hb, let ⟨c, hcs, hcb, hca⟩ := hfs b hb in ⟨c, hcs, hca, hcb⟩
/-- If `f` is a function monotonically increasing function on a left neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `(b, f a)`, `b < f a`, then `f` is
continuous at `a` from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)` cannot be replaced by the weaker
assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` we use for strictly monotone functions
because otherwise the function `floor : ℝ → ℤ` would be a counter-example at `a = 0`. -/
lemma continuous_at_left_of_mono_incr_on_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝[Iic a] a)
(hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) :
continuous_within_at f (Iic a) a :=
@continuous_at_right_of_mono_incr_on_of_exists_between (order_dual α) (order_dual β) _ _ _ _ _ _
f s a (λ x hx y hy, h_mono y hy x hx) hs $
λ b hb, let ⟨c, hcs, hcb, hca⟩ := hfs b hb in ⟨c, hcs, hca, hcb⟩
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a left
neighborhood of `a` and the closure of the image of this neighborhood under `f` is a left
neighborhood of `f a`, then `f` is continuous at `a` from the left -/
lemma continuous_at_left_of_mono_incr_on_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[Iic a] a) (hfs : closure (f '' s) ∈ 𝓝[Iic (f a)] (f a)) :
continuous_within_at f (Iic a) a :=
@continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within (order_dual α) (order_dual β)
_ _ _ _ _ _ _ f s a (λ x hx y hy, h_mono y hy x hx) hs hfs
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a left
neighborhood of `a` and the image of this neighborhood under `f` is a left neighborhood of `f a`,
then `f` is continuous at `a` from the left. -/
lemma continuous_at_left_of_mono_incr_on_of_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[Iic a] a) (hfs : f '' s ∈ 𝓝[Iic (f a)] (f a)) :
continuous_within_at f (Iic a) a :=
continuous_at_left_of_mono_incr_on_of_closure_image_mem_nhds_within h_mono hs
(mem_sets_of_superset hfs subset_closure)
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
left neighborhood of `a` and the closure of the image of this neighborhood under `f` is a left
neighborhood of `f a`, then `f` is continuous at `a` from the left. -/
lemma strict_mono_incr_on.continuous_at_left_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Iic a] a)
(hfs : closure (f '' s) ∈ 𝓝[Iic (f a)] (f a)) :
continuous_within_at f (Iic a) a :=
h_mono.dual.continuous_at_right_of_closure_image_mem_nhds_within hs hfs
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
left neighborhood of `a` and the image of this neighborhood under `f` is a left neighborhood of
`f a`, then `f` is continuous at `a` from the left. -/
lemma strict_mono_incr_on.continuous_at_left_of_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Iic a] a)
(hfs : f '' s ∈ 𝓝[Iic (f a)] (f a)) :
continuous_within_at f (Iic a) a :=
h_mono.dual.continuous_at_right_of_image_mem_nhds_within hs hfs
/-- If a function `f` is strictly monotonically increasing on a left neighborhood of `a` and the
image of this neighborhood under `f` includes `Iio (f a)`, then `f` is continuous at `a` from the
left. -/
lemma strict_mono_incr_on.continuous_at_left_of_surj_on {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Iic a] a) (hfs : surj_on f s (Iio (f a))) :
continuous_within_at f (Iic a) a :=
h_mono.dual.continuous_at_right_of_surj_on hs hfs
/-- If a function `f` is strictly monotonically increasing on a neighborhood of `a` and the image of
this neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, and every interval
`(f a, b]`, `b > f a`, then `f` is continuous at `a`. -/
lemma strict_mono_incr_on.continuous_at_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝 a)
(hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
continuous_at f a :=
continuous_at_iff_continuous_left_right.2
⟨h_mono.continuous_at_left_of_exists_between (mem_nhds_within_of_mem_nhds hs) hfs_l,
h_mono.continuous_at_right_of_exists_between (mem_nhds_within_of_mem_nhds hs) hfs_r⟩
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
neighborhood of `a` and the closure of the image of this neighborhood under `f` is a neighborhood of
`f a`, then `f` is continuous at `a`. -/
lemma strict_mono_incr_on.continuous_at_of_closure_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) :
continuous_at f a :=
continuous_at_iff_continuous_left_right.2
⟨h_mono.continuous_at_left_of_closure_image_mem_nhds_within (mem_nhds_within_of_mem_nhds hs)
(mem_nhds_within_of_mem_nhds hfs),
h_mono.continuous_at_right_of_closure_image_mem_nhds_within (mem_nhds_within_of_mem_nhds hs)
(mem_nhds_within_of_mem_nhds hfs)⟩
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
neighborhood of `a` and the image of this set under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. -/
lemma strict_mono_incr_on.continuous_at_of_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) :
continuous_at f a :=
h_mono.continuous_at_of_closure_image_mem_nhds hs (mem_sets_of_superset hfs subset_closure)
/-- If `f` is a function monotonically increasing function on a neighborhood of `a` and the image of
this neighborhood under `f` meets every interval `(b, f a)`, `b < f a`, and every interval `(f a,
b)`, `b > f a`, then `f` is continuous at `a`. -/
lemma continuous_at_of_mono_incr_on_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝 a)
(hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
continuous_at f a :=
continuous_at_iff_continuous_left_right.2
⟨continuous_at_left_of_mono_incr_on_of_exists_between h_mono
(mem_nhds_within_of_mem_nhds hs) hfs_l,
continuous_at_right_of_mono_incr_on_of_exists_between h_mono
(mem_nhds_within_of_mem_nhds hs) hfs_r⟩
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a neighborhood
of `a` and the closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then
`f` is continuous at `a`. -/
lemma continuous_at_of_mono_incr_on_of_closure_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) :
continuous_at f a :=
continuous_at_iff_continuous_left_right.2
⟨continuous_at_left_of_mono_incr_on_of_closure_image_mem_nhds_within h_mono
(mem_nhds_within_of_mem_nhds hs) (mem_nhds_within_of_mem_nhds hfs),
continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within h_mono
(mem_nhds_within_of_mem_nhds hs) (mem_nhds_within_of_mem_nhds hfs)⟩
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a neighborhood
of `a` and the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. -/
lemma continuous_at_of_mono_incr_on_of_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝 a)
(hfs : f '' s ∈ 𝓝 (f a)) :
continuous_at f a :=
continuous_at_of_mono_incr_on_of_closure_image_mem_nhds h_mono hs
(mem_sets_of_superset hfs subset_closure)
/-- A monotone function with densely ordered codomain and a dense range is continuous. -/
lemma monotone.continuous_of_dense_range [densely_ordered β] {f : α → β}
(h_mono : monotone f) (h_dense : dense_range f) :
continuous f :=
continuous_iff_continuous_at.mpr $ λ a,
continuous_at_of_mono_incr_on_of_closure_image_mem_nhds (λ x hx y hy hxy, h_mono hxy)
univ_mem_sets $ by simp only [image_univ, h_dense.closure_eq, univ_mem_sets]
/-- A monotone surjective function with a densely ordered codomain is surjective. -/
lemma monotone.continuous_of_surjective [densely_ordered β] {f : α → β} (h_mono : monotone f)
(h_surj : function.surjective f) :
continuous f :=
h_mono.continuous_of_dense_range h_surj.dense_range
end linear_order
/-!
### Continuity of order isomorphisms
In this section we prove that an `order_iso` is continuous, hence it is a `homeomorph`. We prove
this for an `order_iso` between to partial orders with order topology.
-/
namespace order_iso
variables [partial_order α] [partial_order β] [topological_space α] [topological_space β]
[order_topology α] [order_topology β]
protected lemma continuous (e : α ≃o β) : continuous e :=
begin
rw [‹order_topology β›.topology_eq_generate_intervals],
refine continuous_generated_from (λ s hs, _),
rcases hs with ⟨a, rfl|rfl⟩,
{ rw e.preimage_Ioi, apply is_open_lt' },
{ rw e.preimage_Iio, apply is_open_gt' }
end
/-- An order isomorphism between two linear order `order_topology` spaces is a homeomorphism. -/
def to_homeomorph (e : α ≃o β) : α ≃ₜ β :=
{ continuous_to_fun := e.continuous,
continuous_inv_fun := e.symm.continuous,
.. e }
@[simp] lemma coe_to_homeomorph (e : α ≃o β) : ⇑e.to_homeomorph = e := rfl
@[simp] lemma coe_to_homeomorph_symm (e : α ≃o β) : ⇑e.to_homeomorph.symm = e.symm := rfl
end order_iso
section conditionally_complete_linear_order
variables
[conditionally_complete_linear_order α] [densely_ordered α] [topological_space α]
[order_topology α] [conditionally_complete_linear_order β] [topological_space β]
[order_topology β]
/-- If `f : α → β` is strictly monotone and continuous, and tendsto `at_top` `at_top` and to
`at_bot` `at_bot`, then it is a homeomorphism. -/
noncomputable def homeomorph_of_strict_mono_continuous
(f : α → β) (h_mono : strict_mono f) (h_cont : continuous f) (h_top : tendsto f at_top at_top)
(h_bot : tendsto f at_bot at_bot) :
homeomorph α β :=
(h_mono.order_iso_of_surjective f (surjective_of_continuous h_cont h_top h_bot)).to_homeomorph
@[simp] lemma coe_homeomorph_of_strict_mono_continuous
(f : α → β) (h_mono : strict_mono f) (h_cont : continuous f) (h_top : tendsto f at_top at_top)
(h_bot : tendsto f at_bot at_bot) :
(homeomorph_of_strict_mono_continuous f h_mono h_cont h_top h_bot : α → β) = f := rfl
/- Now we prove a relative version of the above result. This (`Ioo` to `univ`) is provided as a
sample; there are at least 16 possible variations with open intervals (`univ` to `Ioo`, `Ioi` to
`univ`, ...), not to mention the possibilities with closed or half-closed intervals. -/
variables {a b : α}
/-- If `f : α → β` is strictly monotone and continuous on the interval `Ioo a b` of `α`, and tends
to `at_top` within `𝓝[Iio b] b` and to `at_bot` within `𝓝[Ioi a] a`, then it restricts to a
homeomorphism from `Ioo a b` to `β`. -/
noncomputable def homeomorph_of_strict_mono_continuous_Ioo
(f : α → β) (h : a < b)
(h_mono : ∀ ⦃x y : α⦄, a < x → y < b → x < y → f x < f y)
(h_cont : continuous_on f (Ioo a b))
(h_top : tendsto f (𝓝[Iio b] b) at_top)
(h_bot : tendsto f (𝓝[Ioi a] a) at_bot) :
homeomorph (Ioo a b) β :=
by haveI : inhabited (Ioo a b) := inhabited_of_nonempty (nonempty_Ioo_subtype h); exact
homeomorph_of_strict_mono_continuous
(restrict f (Ioo a b))
(λ x y, h_mono x.2.1 y.2.2)
(continuous_on_iff_continuous_restrict.mp h_cont)
(by rwa [restrict_eq f (Ioo a b), ← tendsto_map'_iff, map_coe_Ioo_at_top h])
(by rwa [restrict_eq f (Ioo a b), ← tendsto_map'_iff, map_coe_Ioo_at_bot h])
@[simp] lemma coe_homeomorph_of_strict_mono_continuous_Ioo
(f : α → β) (h : a < b)
(h_mono : ∀ ⦃x y : α⦄, a < x → y < b → x < y → f x < f y)
(h_cont : continuous_on f (Ioo a b))
(h_top : tendsto f (𝓝[Iio b] b) at_top)
(h_bot : tendsto f (𝓝[Ioi a] a) at_bot) :
(homeomorph_of_strict_mono_continuous_Ioo f h h_mono h_cont h_top h_bot : Ioo a b → β)
= restrict f (Ioo a b) :=
rfl
end conditionally_complete_linear_order
|
2d6ef90cfb9183fe6b262999a4c315b3b32d3ae1 | 64874bd1010548c7f5a6e3e8902efa63baaff785 | /library/init/relation.lean | 0477ab33009a242fa64166ba53bc8daac3b12201 | [
"Apache-2.0"
] | permissive | tjiaqi/lean | 4634d729795c164664d10d093f3545287c76628f | d0ce4cf62f4246b0600c07e074d86e51f2195e30 | refs/heads/master | 1,622,323,796,480 | 1,422,643,069,000 | 1,422,643,069,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,455 | lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module init.relation
Authors: Leonardo de Moura
-/
prelude
import init.logic
-- TODO(Leo): remove duplication between this file and algebra/relation.lean
-- We need some of the following definitions asap when "initializing" Lean.
variables {A B : Type} (R : B → B → Prop)
local infix `≺`:50 := R
definition reflexive := ∀x, x ≺ x
definition symmetric := ∀⦃x y⦄, x ≺ y → y ≺ x
definition transitive := ∀⦃x y z⦄, x ≺ y → y ≺ z → x ≺ z
definition irreflexive := ∀x, ¬ x ≺ x
definition anti_symmetric := ∀⦃x y⦄, x ≺ y → y ≺ x → x = y
definition empty_relation := λa₁ a₂ : A, false
definition subrelation (Q R : B → B → Prop) := ∀⦃x y⦄, Q x y → R x y
definition inv_image (f : A → B) : A → A → Prop :=
λa₁ a₂, f a₁ ≺ f a₂
theorem inv_image.trans (f : A → B) (H : transitive R) : transitive (inv_image R f) :=
λ (a₁ a₂ a₃ : A) (H₁ : inv_image R f a₁ a₂) (H₂ : inv_image R f a₂ a₃), H H₁ H₂
theorem inv_image.irreflexive (f : A → B) (H : irreflexive R) : irreflexive (inv_image R f) :=
λ (a : A) (H₁ : inv_image R f a a), H (f a) H₁
inductive tc {A : Type} (R : A → A → Prop) : A → A → Prop :=
base : ∀a b, R a b → tc R a b,
trans : ∀a b c, tc R a b → tc R b c → tc R a c
|
0e92d4da9bcae032adb0b4fa4a219969d0971fbf | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/data/string/defs.lean | 5dcca2d98d967f451bc5bc0480b75d7e9c58c576 | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 2,030 | lean | /-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Keeley Hoek, Floris van Doorn
-/
import data.list.defs
/-!
# Definitions for `string`
This file defines a bunch of functions for the `string` datatype.
-/
namespace string
/-- `s.split_on c` tokenizes `s : string` on `c : char`. -/
def split_on (s : string) (c : char) : list string :=
split (= c) s
/-- `string.map_tokens c f s` tokenizes `s : string` on `c : char`, maps `f` over each token, and
then reassembles the string by intercalating the separator token `c` over the mapped tokens. -/
def map_tokens (c : char) (f : string → string) : string → string :=
intercalate (singleton c) ∘ list.map f ∘ split (= c)
/-- Tests whether the first string is a prefix of the second string. -/
def is_prefix_of (x y : string) : bool :=
x.to_list.is_prefix_of y.to_list
/-- Tests whether the first string is a suffix of the second string. -/
def is_suffix_of (x y : string) : bool :=
x.to_list.is_suffix_of y.to_list
/-- `x.starts_with y` is true if `y` is a prefix of `x`, and is false otherwise. -/
abbreviation starts_with (x y : string) : bool :=
y.is_prefix_of x
/-- `x.ends_with y` is true if `y` is a suffix of `x`, and is false otherwise. -/
abbreviation ends_with (x y : string) : bool :=
y.is_suffix_of x
/-- `get_rest s t` returns `some r` if `s = t ++ r`.
If `t` is not a prefix of `s`, returns `none` -/
def get_rest (s t : string) : option string :=
list.as_string <$> s.to_list.get_rest t.to_list
/-- Removes the first `n` elements from the string `s` -/
def popn (s : string) (n : nat) : string :=
(s.mk_iterator.nextn n).next_to_string
/-- `is_nat s` is true iff `s` is a nonempty sequence of digits. -/
def is_nat (s : string) : bool :=
¬ s.is_empty ∧ s.to_list.all (λ c, to_bool c.is_digit)
/-- Produce the head character from the string `s`, if `s` is not empty, otherwise 'A'. -/
def head (s : string) : char :=
s.mk_iterator.curr
end string
|
3da18cdacb6352c484eddf1c4cd6ff1d5ee4f766 | 5ec8f5218a7c8e87dd0d70dc6b715b36d61a8d61 | /memory.lean | c938f8e170fc1f4ed6e308ae4af8f20a476f3275 | [] | no_license | mbrodersen/kremlin | f9f2f9dd77b9744fe0ffd5f70d9fa0f1f8bd8cec | d4665929ce9012e93a0b05fc7063b96256bab86f | refs/heads/master | 1,624,057,268,130 | 1,496,957,084,000 | 1,496,957,084,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 72,177 | lean |
/- This file develops the memory model that is used in the dynamic
semantics of all the languages used in the compiler.
It defines a type [mem] of memory states, the following 4 basic
operations over memory states, and their properties:
- [load]: read a memory chunk at a given address;
- [store]: store a memory chunk at a given address;
- [alloc]: allocate a fresh memory block;
- [free]: invalidate a memory block.
-/
/- Memory states are accessed by addresses [b, ofs]: pairs of a block
identifier [b] and a byte offset [ofs] within that block.
Each address is associated to permissions, also known as access rights.
The following permissions are expressible:
- Freeable (exclusive access): all operations permitted
- Writable: load, store and pointer comparison operations are permitted,
but freeing is not.
- Readable: only load and pointer comparison operations are permitted.
- Nonempty: valid, but only pointer comparisons are permitted.
- Empty: not yet allocated or previously freed; no operation permitted.
The first four cases are represented by the following type of permissions.
Being empty is represented by the absence of any permission.
-/
import .memdata data.nat.bquant data.nat.dvd
namespace memory
open maps memdata values ast memdata.memval word integers ast.memory_chunk
inductive permission : Type
| Freeable : permission
| Writable : permission
| Readable : permission
| Nonempty : permission
export permission
/- In the list, each permission implies the other permissions further down the
list. We reflect this fact by the following order over permissions. -/
inductive perm_order : permission → permission → Prop
| perm_refl (p) : perm_order p p
| perm_F_any (p) : perm_order Freeable p
| perm_W_R : perm_order Writable Readable
| perm_any_N (p) : perm_order p Nonempty
export perm_order
lemma perm_order_trans {p1 p2 p3}
(p12 : perm_order p1 p2) (p23 : perm_order p2 p3) : perm_order p1 p3 := sorry'
instance : decidable_rel perm_order :=
λ p1 p2, by cases p1; cases p2; {right, constructor} <|> {left, intro h, cases h}
/- Each address has not one, but two permissions associated
with it. The first is the current permission. It governs whether
operations (load, store, free, etc) over this address succeed or
not. The other is the maximal permission. It is always at least as
strong as the current permission. Once a block is allocated, the
maximal permission of an address within this block can only
decrease, as a result of [free] or [drop_perm] operations, or of
external calls. In contrast, the current permission of an address
can be temporarily lowered by an external call, then raised again by
another external call. -/
inductive perm_kind : Type
| Max : perm_kind
| Cur : perm_kind
open perm_kind
local notation a # b := PTree.get b a
def perm_order' : option permission → permission → Prop
| (some p') p := perm_order p' p
| none p := false
instance (op p) : decidable (perm_order' op p) :=
by cases op; dsimp [perm_order']; apply_instance
def perm_order'' : option permission → option permission → Prop
| _ none := true
| p1 (some p2) := perm_order' p1 p2
theorem perm_order''.refl (op) : perm_order'' op op :=
by { cases op, trivial, apply perm_refl }
structure mem : Type :=
(mem_contents : PTree (PTree memval)) /- [block → offset → memval] -/
(mem_access : PTree (ℕ → perm_kind → option permission))
/- [block → offset → kind → option permission] -/
(nextblock : block)
(access_max : ∀ (b : block) (ofs : ℕ), let m := (mem_access#b).get_or_else (λ_ _, none) in
perm_order'' (m ofs Max) (m ofs Cur))
(nextblock_noaccess : ∀ {b : block} (ofs : ℕ) (k : perm_kind), nextblock ≤ b →
(mem_access#b).get_or_else (λ_ _, none) ofs k = none)
/- * Validity of blocks and accesses -/
/- A block address is valid if it was previously allocated. It remains valid
even after being freed. -/
def valid_block (m : mem) (b : block) := b < m.nextblock
instance valid_block_dec (m b) : decidable (valid_block m b) := by delta valid_block; apply_instance
theorem valid_not_valid_diff {m b b'} (h1 : valid_block m b) : ¬ valid_block m b' → b ≠ b' :=
mt $ λ bb, by rw bb at h1; assumption
def mem.access (m : mem) (b : block) : ℕ → perm_kind → option permission :=
(m.mem_access#b).get_or_else (λ_ _, none)
def mem.get_block (m : mem) (b : block) : PTree memval :=
(m.mem_contents#b).get_or_else ∅
def mem.contents (m : mem) (b : block) (ofs : ℕ) : memval :=
(m.get_block b # num.succ' ofs).get_or_else Undef
/- Permissions -/
def perm (m : mem) (b : block) (ofs : ℕ) (k : perm_kind) : permission → Prop :=
perm_order' (m.access b ofs k)
theorem perm_implies {m b ofs k p1 p2} :
perm_order p1 p2 → perm m b ofs k p1 → perm m b ofs k p2 :=
by { delta perm, cases m.access b ofs k, apply λ_, id, apply λa b, perm_order_trans b a }
theorem perm_cur_max {m b ofs p} : perm m b ofs Cur p → perm m b ofs Max p := sorry'
theorem perm_cur {m b ofs k p} : perm m b ofs Cur p → perm m b ofs k p := sorry'
theorem perm_max {m b ofs k p} : perm m b ofs k p → perm m b ofs Max p := sorry'
theorem perm_valid_block {m b ofs k p} : perm m b ofs k p → valid_block m b := sorry'
instance perm_dec (m b ofs k p) : decidable (perm m b ofs k p) :=
by delta perm; apply_instance
def range_perm (m b lo hi k p) := ∀ ⦃ofs⦄, lo ≤ ofs → ofs < hi → perm m b ofs k p
theorem range_perm_implies {m b lo hi k p1 p2} :
perm_order p1 p2 → range_perm m b lo hi k p1 → range_perm m b lo hi k p2 :=
λ o r ofs h1 h2, perm_implies o (r h1 h2)
theorem range_perm_cur {m b lo hi k p} :
range_perm m b lo hi Cur p → range_perm m b lo hi k p :=
λ r ofs h1 h2, perm_cur (r h1 h2)
theorem range_perm_max {m b lo hi k p} :
range_perm m b lo hi k p → range_perm m b lo hi Max p :=
λ r ofs h1 h2, perm_max (r h1 h2)
instance range_perm_dec {m b lo hi k p} : decidable (range_perm m b lo hi k p) :=
decidable_lo_hi _ _ _
/- [valid_access m chunk b ofs p] holds if a memory access
of the given chunk is possible in [m] at address [b, ofs]
with current permissions [p].
This means:
- The range of bytes accessed all have current permission [p].
- The offset [ofs] is aligned.
-/
def valid_access (m) (chunk b ofs p) : Prop :=
range_perm m b ofs (ofs + memory_chunk.size chunk) Cur p ∧ chunk.align ∣ ofs
theorem valid_access_implies {m chunk b ofs p1 p2} :
perm_order p1 p2 → valid_access m chunk b ofs p1 → valid_access m chunk b ofs p2 :=
λ o, and.imp (range_perm_implies o) id
theorem valid_access_freeable_any {m chunk b ofs p} :
valid_access m chunk b ofs Freeable → valid_access m chunk b ofs p :=
valid_access_implies (by constructor)
lemma valid_access_perm {m chunk b ofs p} (k) :
valid_access m chunk b ofs p → perm m b ofs k p :=
λ ⟨h, _⟩, perm_cur $ h (le_refl _) (nat.lt_add_of_pos_right chunk.size_pos)
theorem valid_access_valid_block {m chunk b ofs} :
valid_access m chunk b ofs Nonempty → valid_block m b :=
λ h, decidable.by_contradiction $ λ hn,
have t : mem.access m b ofs Cur = none, from m.nextblock_noaccess ofs Cur (le_of_not_gt hn),
have p : _, from valid_access_perm Cur h,
by delta perm at p; rw t at p; exact p
lemma valid_access_compat {m} {chunk1 chunk2 : memory_chunk} {b ofs p} :
chunk1.size = chunk2.size → chunk2.align ≤ chunk1.align →
valid_access m chunk1 b ofs p → valid_access m chunk2 b ofs p := sorry'
instance valid_access_dec (m chunk b ofs p) : decidable (valid_access m chunk b ofs p) :=
by delta valid_access; apply_instance
/- [valid_pointer m b ofs] returns [tt] if the address [b, ofs]
is nonempty in [m] and [ff] if it is empty. -/
def valid_pointer (m : mem) (b : block) (ofs : ℕ) : bool := perm m b ofs Cur Nonempty
theorem valid_pointer_nonempty_perm {m b ofs} :
valid_pointer m b ofs ↔ perm m b ofs Cur Nonempty :=
to_bool_iff _
theorem valid_pointer_valid_access {m b ofs} :
valid_pointer m b ofs ↔ valid_access m Mint8unsigned b ofs Nonempty :=
valid_pointer_nonempty_perm.trans
⟨λh, ⟨λofs0 lo hi,
have ofs = ofs0, from le_antisymm lo (nat.le_of_lt_succ hi),
by rwa -show ofs = ofs0, from le_antisymm lo (nat.le_of_lt_succ hi),
one_dvd _⟩,
valid_access_perm _⟩
/- C allows pointers one past the last element of an array. These are not
valid according to the previously defined [valid_pointer]. The property
[weak_valid_pointer m b ofs] holds if address [b, ofs] is a valid pointer
in [m], or a pointer one past a valid block in [m]. -/
def weak_valid_pointer (m : mem) (b : block) (ofs : ℕ) :=
valid_pointer m b ofs || valid_pointer m b (ofs - 1)
lemma weak_valid_pointer_spec {m b ofs} :
weak_valid_pointer m b ofs ↔
valid_pointer m b ofs ∨ valid_pointer m b (ofs - 1) :=
bor_iff _ _
lemma valid_pointer_implies {m b ofs}
(v : valid_pointer m b ofs) : weak_valid_pointer m b ofs :=
weak_valid_pointer_spec.2 $ or.inl v
/- * Operations over memory stores -/
/- The initial store -/
protected def empty : mem :=
{ mem_contents := ∅,
mem_access := ∅,
nextblock := 1,
access_max := λ b ofs, by rw PTree.gempty; trivial,
nextblock_noaccess := λ b ofs k h, by rw PTree.gempty; trivial }
instance : has_emptyc mem := ⟨memory.empty⟩
/- Allocation of a fresh block with the given bounds. Return an updated
memory state and the address of the fresh block, which initially contains
undefined cells. Note that allocation never fails: we model an
infinite memory. -/
def mem.alloc (m : mem) (lo hi : ℕ) : mem :=
{ mem_contents := PTree.set m.nextblock ∅ m.mem_contents,
mem_access := PTree.set m.nextblock
(λ ofs k, if lo ≤ ofs ∧ ofs < hi then some Freeable else none) m.mem_access,
nextblock := m.nextblock.succ,
access_max := λ b ofs, by {
rw PTree.gsspec,
by_cases (b = m.nextblock); simp [h],
simp [option.get_or_else, perm_order''.refl],
apply m.access_max },
nextblock_noaccess := λ b ofs k (h : (_:ℕ)≤_), by {
rw PTree.gsspec,
rw pos_num.succ_to_nat at h,
by_cases (b = m.nextblock) with h'; simp [h'],
{ note := @not_le_of_gt nat _ _ _ h,
rw h' at this, exact absurd (le_refl _) this },
{ apply m.nextblock_noaccess,
exact le_of_lt h } } }
/- Freeing a block between the given bounds.
Return the updated memory state where the given range of the given block
has been invalidated: future reads and writes to this
range will fail. Requires freeable permission on the given range. -/
def unchecked_free (m : mem) (b : block) (lo hi : ℕ) : mem :=
{ mem_contents := m.mem_contents,
mem_access := option.cases_on (m.mem_access#b) m.mem_access $
λ old, PTree.set b (λ ofs k, if lo ≤ ofs ∧ ofs < hi then none else old ofs k) m.mem_access,
nextblock := m.nextblock,
access_max := λ b' ofs, by {
ginduction m.mem_access#b with h f; dsimp,
{ exact m.access_max b' ofs },
{ rw PTree.gsspec,
note := m.access_max b' ofs,
by_cases (b' = b) with bb; simp [bb, option.get_or_else]; simp [bb] at this,
by_cases (lo ≤ ofs ∧ ofs < hi) with lh; simp [lh]; simp [h, option.get_or_else] at this,
all_goals {exact this} } },
nextblock_noaccess := λ b' ofs k hl, by {
ginduction m.mem_access#b with h f; dsimp,
{ apply m.nextblock_noaccess ofs k hl },
{ rw PTree.gsspec,
note := m.nextblock_noaccess ofs k hl,
by_cases (b' = b) with bb; simp [bb, option.get_or_else],
{ simp [bb, h, option.get_or_else] at this,
by_cases (lo ≤ ofs ∧ ofs < hi) with lh; simp [lh, this] },
{ exact this } } } }
def free (m : mem) (b : block) (lo hi : ℕ) : option mem :=
if range_perm m b lo hi Cur Freeable then some (unchecked_free m b lo hi) else none
def free_list : mem → list (block × ℕ × ℕ) → option mem
| m [] := some m
| m ((b, lo, hi) :: l') := do m' ← free m b lo hi, free_list m' l'
/- Memory reads. -/
/- Reading N adjacent bytes in a block content. -/
def get1 : ℤ → PTree memval → memval
| (p : ℕ) c := (PTree.get (num.succ' p) c).get_or_else Undef
| _ c := Undef
def getN : ℕ → ℤ → PTree memval → list memval
| 0 p c := []
| (n + 1) p c := get1 p c :: getN n (p+1) c
/- [load chunk m b ofs] perform a read in memory state [m], at address
[b] and offset [ofs]. It returns the value of the memory chunk
at that address. [None] is returned if the accessed bytes
are not readable. -/
def load (chunk : memory_chunk) (m : mem) (b : block) (ofs : ℕ) : option val :=
if valid_access m chunk b ofs Readable
then some $ decode_val chunk $ getN chunk.size ofs (m.get_block b) else none
/- [loadv chunk m addr] is similar, but the address and offset are given
as a single value [addr], which must be a pointer value. -/
def loadv (chunk : memory_chunk) (m : mem) : val → option val
| (Vptr b ofs) := load chunk m b (unsigned ofs)
| _ := none
/- [load_bytes m b ofs n] reads [n] consecutive bytes starting at
location [(b, ofs)]. Returns [None] if the accessed locations are
not readable. -/
def load_bytes (m : mem) (b : block) (ofs n : ℕ) : option (list memval) :=
if range_perm m b ofs (ofs + n) Cur Readable then some $ getN n ofs (m.get_block b) else none
/- Memory stores. -/
/- Writing N adjacent bytes in a block content. -/
def setN' : list memval → pos_num → PTree memval → PTree memval
| [] p c := c
| (v :: vl') p c := setN' vl' p.succ (PTree.set p v c)
def setN (vl : list memval) (ofs : ℕ) : PTree memval → PTree memval :=
setN' vl (num.succ' ofs)
theorem setN_other {vl c p q} : (∀ r, p ≤ r → r < p + list.length vl → r ≠ q) →
get1 q (setN vl p c) = get1 q c := sorry'
theorem setN_outside {vl c p q} : q < p ∨ q ≥ p + list.length vl →
get1 q (setN vl p c) = get1 q c :=
by { intro h,
apply setN_other,
intros r h1 h2 hn,
rw hn at h1 h2,
cases h,
{ exact not_le_of_gt a h1 },
{ exact not_le_of_gt h2 a } }
theorem getN_setN_same {vl p c} : getN (list.length vl) ↑p (setN vl p c) = vl := sorry'
theorem getN.ext {c1 c2 n p} :
(∀ i, p ≤ i → i < p + ↑n → get1 i c1 = get1 i c2) →
getN n p c1 = getN n p c2 := sorry'
def interval_disjoint (x dx y dy : ℕ) := ∀ r, x ≤ r → r < x + dx → y ≤ r → r < y + dy → false
theorem getN_setN_disjoint {vl q c n p} :
interval_disjoint p n q (list.length vl) → getN n p (setN vl q c) = getN n p c := sorry'
theorem getN_setN_outside {vl q c n p} :
p + n ≤ q ∨ q + list.length vl ≤ p → getN n p (setN vl q c) = getN n p c :=
by { intro h,
apply getN_setN_disjoint,
intros r hx hdx hy hdy,
cases h,
{ exact not_le_of_gt hdx (le_trans a hy) },
{ exact not_le_of_gt hdy (le_trans a hx) } }
lemma setN_in {vl p q c} : p ≤ q → q < p + list.length vl → get1 q (setN vl p c) ∈ vl := sorry'
lemma getN_in {c q n p} : p ≤ q → q < p + ↑n → get1 q c ∈ getN n p c := sorry'
/- [store_bytes m b ofs bytes] stores the given list of bytes [bytes]
starting at location [(b, ofs)]. Returns updated memory state
or [None] if the accessed locations are not writable. -/
def store_bytes.val (m : mem) (b : block) (ofs : ℕ) (bytes : list memval) : mem :=
{ mem_contents := PTree.set b (setN bytes ofs (m.get_block b)) m.mem_contents,
mem_access := m.mem_access,
nextblock := m.nextblock,
access_max := m.access_max,
nextblock_noaccess := m.nextblock_noaccess }
def store_bytes (m : mem) (b : block) (ofs : ℕ) (bytes : list memval) : option mem :=
bnot (range_perm m b ofs (ofs + bytes.length) Cur Writable) |> store_bytes.val m b ofs bytes
/- [store chunk m b ofs v] perform a write in memory state [m].
Value [v] is stored at address [b] and offset [ofs].
Return the updated memory store, or [None] if the accessed bytes
are not writable. -/
def store (chunk : memory_chunk) (m : mem) (b : block) (ofs : ℕ) (v : val) : option mem :=
bnot (valid_access m chunk b ofs Writable) |> store_bytes.val m b ofs (encode_val chunk v)
/- [storev chunk m addr v] is similar, but the address and offset are given
as a single value [addr], which must be a pointer value. -/
def storev (chunk : memory_chunk) (m : mem) : val → val → option mem
| (Vptr b ofs) v := store chunk m b (unsigned ofs) v
| _ v := none
/- [drop_perm m b lo hi p] sets the max permissions of the byte range
[(b, lo) ... (b, hi - 1)] to [p]. These bytes must have current permissions
[Freeable] in the initial memory state [m].
Returns updated memory state, or [None] if insufficient permissions. -/
def drop_perm.val (m : mem) (b : block) (lo hi : ℕ) (p : permission)
(rp : range_perm m b lo hi Cur Freeable) : mem :=
{ mem_contents := m.mem_contents,
mem_access := option.cases_on (m.mem_access#b) m.mem_access $
λ old, PTree.set b (λ ofs k, if lo ≤ ofs ∧ ofs < hi then some p else old ofs k) m.mem_access,
nextblock := m.nextblock,
access_max := λ b' ofs, by {
ginduction m.mem_access#b with h f; dsimp,
{ exact m.access_max b' ofs },
{ rw PTree.gsspec,
note := m.access_max b' ofs,
by_cases (b' = b) with bb; simp [bb, option.get_or_else]; simp [bb] at this,
{ by_cases (lo ≤ ofs ∧ ofs < hi) with lh; simp [lh]; simp [h, option.get_or_else] at this,
{ apply perm_order''.refl },
{ assumption } },
{ exact this } } },
nextblock_noaccess := λ b' ofs k hb, by {
ginduction m.mem_access#b with h f; dsimp,
{ apply m.nextblock_noaccess ofs k hb },
{ rw PTree.gsspec,
by_cases (b' = b) with bb; simp [bb, option.get_or_else],
{ by_cases (lo ≤ ofs ∧ ofs < hi) with lh,
{ cases lh with hl hh,
apply absurd (rp hl hh),
note := m.nextblock_noaccess ofs Cur hb,
simp [bb, h, option.get_or_else] at this,
simp [perm, mem.access, h, option.get_or_else, this],
exact not_false },
{ note := m.nextblock_noaccess ofs k hb,
simp [bb, h, option.get_or_else] at this,
simp [lh, this] } },
{ exact m.nextblock_noaccess ofs k hb } } } }
def drop_perm (m : mem) (b : block) (lo hi : ℕ) (p : permission) : option mem :=
if rp : range_perm m b lo hi Cur Freeable then some (drop_perm.val m b lo hi p rp) else none
/- * Properties of the memory operations -/
/- Properties of the empty store. -/
theorem nextblock_empty : (∅ : mem).nextblock = 1 := rfl
theorem perm_empty {b ofs k p} : ¬ perm ∅ b ofs k p := id
theorem valid_access_empty {chunk b ofs p} : ¬ valid_access ∅ chunk b ofs p :=
mt (valid_access_perm Cur) perm_empty
/- ** Properties related to [load] -/
theorem valid_access_load {m chunk b ofs}
(h : valid_access m chunk b ofs Readable) : ∃ v, load chunk m b ofs = some v :=
⟨_, by unfold load; rw if_pos h; refl⟩
theorem load_valid_access {m chunk b ofs v}
(h : load chunk m b ofs = some v) : valid_access m chunk b ofs Readable :=
by unfold load at h; by_contradiction; simp [a] at h; contradiction
lemma load_result {chunk m b ofs v}
(h : load chunk m b ofs = some v) : v = decode_val chunk (getN chunk.size ofs (m.get_block b)) :=
by { simp [load, load_valid_access h] at h, injection h, exact h.symm }
theorem load_type {m chunk b ofs v} (h : load chunk m b ofs = some v) : v.has_type chunk.type :=
by rw load_result h; apply decode_val_type
theorem load_cast {m chunk b ofs v} (h : load chunk m b ofs = some v) : decode_val_cast_type v chunk :=
by rw load_result h; apply decode_val_cast
theorem load_int8_signed_unsigned {m b ofs} :
load Mint8signed m b ofs = sign_ext W8 <$> load Mint8unsigned m b ofs := sorry'
theorem load_int16_signed_unsigned {m b ofs} :
load Mint16signed m b ofs = sign_ext W16 <$> load Mint16unsigned m b ofs := sorry'
/- ** Properties related to [load_bytes] -/
theorem range_perm_load_bytes {m b ofs len}
(h : range_perm m b ofs (ofs + len) Cur Readable) : ∃ bytes, load_bytes m b ofs len = some bytes :=
⟨_, by unfold load_bytes; rw if_pos h; refl⟩
theorem load_bytes_range_perm {m b ofs len bytes}
(h : load_bytes m b ofs len = some bytes) : range_perm m b ofs (ofs + len) Cur Readable :=
by unfold load_bytes at h; by_contradiction; simp [a] at h; contradiction
theorem load_bytes_getN {m b ofs n bytes}
(h : load_bytes m b ofs n = some bytes) :
getN n ofs (mem.get_block m b) = bytes :=
by { note := load_bytes_range_perm h,
unfold load valid_access,
unfold load_bytes at h,
note := show some _ = _, by simp [this] at h; exact h,
injection this with this }
theorem load_bytes_load {chunk m b ofs bytes}
(h : load_bytes m b ofs (memory_chunk.size chunk) = some bytes) (al : chunk.align ∣ ofs) :
load chunk m b ofs = some (decode_val chunk bytes) :=
by { note := load_bytes_range_perm h,
unfold load valid_access,
rw if_pos,
{ rw load_bytes_getN h; refl },
{ constructor; assumption } }
theorem load_load_bytes {chunk m b ofs v} (h : load chunk m b ofs = some v) :
∃ bytes, load_bytes m b ofs chunk.size = some bytes ∧ v = decode_val chunk bytes :=
let ⟨rp, al⟩ := load_valid_access h, ⟨bytes, h'⟩ := range_perm_load_bytes rp in
⟨bytes, h', option.no_confusion (h.symm.trans (load_bytes_load h' al)) id⟩
lemma getN_length {c n p} : (getN n p c).length = n :=
by revert p; induction n; intro p; simph [getN]
theorem load_bytes_length {m b ofs n bytes}
(h : load_bytes m b ofs n = some bytes) : bytes.length = n :=
by rw [-load_bytes_getN h, getN_length]
theorem load_bytes_empty {m b ofs} : load_bytes m b ofs 0 = some [] :=
by {delta load_bytes, rw if_pos, refl, intros r h1 h2, exact absurd h1 (not_le_of_gt h2)}
lemma getN_concat {c n1 n2 p} : getN (n1 + n2) p c = getN n1 p c ++ getN n2 (p + n1) c := sorry'
theorem load_bytes_concat {m b ofs n1 n2 bytes1 bytes2} :
load_bytes m b ofs n1 = some bytes1 →
load_bytes m b (ofs + n1) n2 = some bytes2 →
load_bytes m b ofs (n1 + n2) = some (bytes1 ++ bytes2) := sorry'
theorem load_bytes_split {m b ofs n1 n2 bytes} :
load_bytes m b ofs (n1 + n2) = some bytes →
∃ bytes1, ∃ bytes2,
load_bytes m b ofs n1 = some bytes1
∧ load_bytes m b (ofs + n1) n2 = some bytes2
∧ bytes = bytes1 ++ bytes2 := sorry'
theorem load_rep {ch : memory_chunk} {m1 m2 : mem} {b ofs v1 v2} :
(∀ z < ch.size, get1 (ofs + z : ℕ) (m1.get_block b) = get1 (ofs + z) (m2.get_block b)) →
load ch m1 b ofs = some v1 → load ch m2 b ofs = some v2 → v1 = v2 := sorry'
theorem load_int64_split {m b ofs v} :
load Mint64 m b ofs = some v → ¬ archi.ptr64 →
∃ v1 v2,
load Mint32 m b ofs = some (if archi.big_endian then v1 else v2)
∧ load Mint32 m b (ofs + 4) = some (if archi.big_endian then v2 else v1)
∧ lessdef v (long_of_words v1 v2) := sorry'
lemma addressing_int64_split {i : ptrofs} : ¬ archi.ptr64 → 8 ∣ unsigned i →
unsigned (i + ptrofs.of_int (repr 4)) = unsigned i + 4 := sorry'
theorem loadv_int64_split {m a v} : loadv Mint64 m a = some v → ¬ archi.ptr64 →
∃ v1 v2,
loadv Mint32 m a = some (if archi.big_endian then v1 else v2)
∧ loadv Mint32 m (a + Vint (repr 4)) = some (if archi.big_endian then v2 else v1)
∧ lessdef v (long_of_words v1 v2) := sorry'
/- ** Properties related to [store_bytes]. -/
theorem range_perm_store_bytes {m1 b ofs bytes}
(h : range_perm m1 b ofs (ofs + list.length bytes) Cur Writable) :
{ m2 : mem // store_bytes m1 b ofs bytes = some m2 } :=
⟨_, by unfold store_bytes; rw to_bool_tt h; refl⟩
theorem store_bytes_store {m1 b ofs chunk v m2} :
store_bytes m1 b ofs (encode_val chunk v) = some m2 →
chunk.align ∣ ofs → store chunk m1 b ofs v = some m2 := sorry'
theorem store_store_bytes {m1 b ofs chunk v m2} :
store chunk m1 b ofs v = some m2 →
store_bytes m1 b ofs (encode_val chunk v) = some m2 := sorry'
section store_bytes
parameters {m1 m2 : mem} {b : block} {ofs : ℕ} {bytes : list memval}
parameter (STORE : store_bytes m1 b ofs bytes = some m2)
include STORE
lemma store_bytes_val_eq : m2 = store_bytes.val m1 b ofs bytes :=
by { unfold store_bytes at STORE,
cases to_bool (range_perm m1 b ofs (ofs + bytes.length) Cur Writable); try {contradiction},
simp [option.rhoare] at STORE, injection STORE, exact h.symm }
lemma store_bytes_access : m2.mem_access = m1.mem_access :=
by rw store_bytes_val_eq STORE; refl
lemma store_bytes_mem_contents :
m2.mem_contents = PTree.set b (setN bytes ofs (m1.get_block b)) m1.mem_contents :=
by rw store_bytes_val_eq STORE; refl
theorem perm_store_bytes {b' ofs' k p} : perm m2 b' ofs' k p ↔ perm m1 b' ofs' k p :=
by unfold perm mem.access; rw store_bytes_access STORE; exact iff.rfl
theorem nextblock_store_bytes : m2.nextblock = m1.nextblock :=
by rw store_bytes_val_eq STORE; refl
theorem store_bytes_valid_block {b'} : valid_block m2 b' ↔ valid_block m1 b' :=
by unfold valid_block; rw nextblock_store_bytes STORE; exact iff.rfl
theorem store_bytes_range_perm {b' lo hi k p} :
range_perm m2 b' lo hi k p ↔ range_perm m1 b' lo hi k p :=
forall_congr $ λ_, forall_congr $ λ_, forall_congr $ λ_, perm_store_bytes
theorem store_bytes_valid_access {chunk' b' ofs' p} :
valid_access m2 chunk' b' ofs' p ↔ valid_access m1 chunk' b' ofs' p :=
and_congr store_bytes_range_perm iff.rfl
theorem store_bytes_range_perm_1 : range_perm m1 b ofs (ofs + bytes.length) Cur Writable :=
by unfold store_bytes at STORE; by_contradiction; rw to_bool_ff a at STORE; contradiction
theorem store_bytes_range_perm_2 : range_perm m2 b ofs (ofs + bytes.length) Cur Writable :=
store_bytes_range_perm.2 store_bytes_range_perm_1
theorem load_store_bytes_same : load_bytes m2 b ofs bytes.length = some bytes :=
by { unfold load_bytes mem.get_block,
rw [if_pos, store_bytes_mem_contents STORE, PTree.gss],
{ simp [option.get_or_else], rw getN_setN_same },
{ apply range_perm_implies _ (store_bytes_range_perm_2 STORE),
exact dec_trivial } }
theorem load_bytes_store_bytes_disjoint {b' ofs' len} :
b' ≠ b ∨ interval_disjoint ofs' len ofs bytes.length →
load_bytes m2 b' ofs' len = load_bytes m1 b' ofs' len := sorry'
theorem load_bytes_store_bytes_other {b' ofs' len} :
b' ≠ b ∨ ofs' + len ≤ ofs ∨ ofs + bytes.length ≤ ofs' →
load_bytes m2 b' ofs' len = load_bytes m1 b' ofs' len := sorry'
theorem load_store_bytes_other {chunk : memory_chunk} {b' ofs'} :
b' ≠ b ∨ ofs' + chunk.size ≤ ofs ∨ ofs + bytes.length ≤ ofs' →
load chunk m2 b' ofs' = load chunk m1 b' ofs' := sorry'
end store_bytes
/- ** Properties related to [store] -/
def store_eq_store_bytes {m chunk b ofs v} (h : align chunk ∣ ofs) :
store chunk m b ofs v = store_bytes m b ofs (encode_val chunk v) :=
by { unfold store store_bytes,
rw to_bool_congr _,
rw encode_val_length,
apply and_iff_left h }
def valid_access_store {m1 chunk b ofs v}
(h : valid_access m1 chunk b ofs Writable) : { m2 : mem // store chunk m1 b ofs v = some m2 } :=
⟨_, by unfold store; rw to_bool_tt h; refl⟩
section store
parameters {chunk : memory_chunk} {m1 m2 : mem} {b : block} {ofs : ℕ} {v : val}
parameter (STORE : store chunk m1 b ofs v = some m2)
include STORE
theorem store_valid_access' : valid_access m1 chunk b ofs Writable :=
by unfold store at STORE; by_contradiction; rw to_bool_ff a at STORE; contradiction
lemma store_bytes_of_store : store_bytes m1 b ofs (encode_val chunk v) = some m2 :=
(store_eq_store_bytes $ store_valid_access'.right).symm.trans STORE
lemma store_val_eq : m2 = store_bytes.val m1 b ofs (encode_val chunk v) :=
store_bytes_val_eq store_bytes_of_store
--by {unfold store at STORE, cases to_bool (valid_access m1 chunk b ofs Writable); try {contradiction},
-- simp [option.rhoare] at STORE, injection STORE, exact h.symm}
lemma store_access : m2.mem_access = m1.mem_access :=
store_bytes_access store_bytes_of_store
lemma store_mem_contents :
m2.mem_contents = PTree.set b (setN (encode_val chunk v) ofs (m1.get_block b)) m1.mem_contents :=
store_bytes_mem_contents store_bytes_of_store
theorem perm_store {b' ofs' k p} : perm m2 b' ofs' k p ↔ perm m1 b' ofs' k p :=
perm_store_bytes store_bytes_of_store
theorem nextblock_store : m2.nextblock = m1.nextblock :=
nextblock_store_bytes store_bytes_of_store
theorem store_valid_block {b'} : valid_block m2 b' ↔ valid_block m1 b' :=
store_bytes_valid_block store_bytes_of_store
theorem store_valid_access {chunk' b' ofs' p} :
valid_access m2 chunk' b' ofs' p ↔ valid_access m1 chunk' b' ofs' p :=
store_bytes_valid_access store_bytes_of_store
theorem load_store_similar {chunk' : memory_chunk} :
chunk'.size = chunk.size → chunk'.align ≤ chunk.align →
∃ v', load chunk' m2 b ofs = some v' ∧ decode_encode_val v chunk chunk' v' := sorry'
theorem load_store_similar_2 {chunk' : memory_chunk} :
chunk'.size = chunk.size → chunk'.align ≤ chunk.align → chunk'.type = chunk.type →
load chunk' m2 b ofs = some (val.load_result chunk' v) :=
λ hs ha ht, let ⟨v', h1, ed⟩ := load_store_similar hs ha in
by rw [h1, decode_encode_val_similar ed ht.symm hs.symm]
theorem load_store_same : load chunk m2 b ofs = some (val.load_result chunk v) :=
load_store_similar_2 rfl (le_refl _) rfl
theorem load_store_other {chunk' : memory_chunk} {b' ofs'}
(h : b' ≠ b ∨ ofs' + chunk'.size ≤ ofs ∨ ofs + chunk.size ≤ ofs') :
load chunk' m2 b' ofs' = load chunk' m1 b' ofs' :=
by { apply load_store_bytes_other (store_bytes_of_store STORE), rwa encode_val_length }
theorem load_bytes_store_same : load_bytes m2 b ofs chunk.size = some (encode_val chunk v) :=
by { rw -encode_val_length, exact load_store_bytes_same (store_bytes_of_store STORE) }
theorem load_bytes_store_other {b' ofs' n}
(h : b' ≠ b ∨ ofs' + n ≤ ofs ∨ ofs + chunk.size ≤ ofs') :
load_bytes m2 b' ofs' n = load_bytes m1 b' ofs' n :=
by { apply load_bytes_store_bytes_other (store_bytes_of_store STORE), rwa encode_val_length }
end store
lemma load_store_overlap {chunk m1 b ofs v m2 chunk' ofs' v'} :
store chunk m1 b ofs v = some m2 → load chunk' m2 b ofs' = some v' →
ofs' + chunk'.size > ofs → ofs + chunk.size > ofs' →
∃ mv1 mvl mv1' mvl',
shape_encoding chunk v (mv1 :: mvl)
∧ shape_decoding chunk' (mv1' :: mvl') v'
∧ ((ofs' = ofs ∧ mv1' = mv1)
∨ (ofs' > ofs ∧ mv1' ∈ mvl)
∨ (ofs' < ofs ∧ mv1 ∈ mvl')) := sorry'
def compat_pointer_chunks : memory_chunk → memory_chunk → bool
| Mint32 Mint32 := tt
| Mint32 Many32 := tt
| Many32 Mint32 := tt
| Many32 Many32 := tt
| Mint64 Mint64 := tt
| Mint64 Many64 := tt
| Many64 Mint64 := tt
| Many64 Many64 := tt
| _ _ := ff
lemma compat_pointer_chunks_true {chunk1 chunk2} :
(chunk1 = Mint32 ∨ chunk1 = Many32 ∨ chunk1 = Mint64 ∨ chunk1 = Many64) →
(chunk2 = Mint32 ∨ chunk2 = Many32 ∨ chunk2 = Mint64 ∨ chunk2 = Many64) →
quantity_chunk chunk1 = quantity_chunk chunk2 → compat_pointer_chunks chunk1 chunk2 :=
by cases chunk1; cases chunk2; exact dec_trivial
theorem load_pointer_store {chunk m1 b ofs v m2 chunk' b' ofs' vb vo} :
store chunk m1 b ofs v = some m2 →
load chunk' m2 b' ofs' = some (Vptr vb vo) →
(v = Vptr vb vo ∧ compat_pointer_chunks chunk chunk' ∧ b' = b ∧ ofs' = ofs)
∨ (b' ≠ b ∨ ofs' + chunk'.size ≤ ofs ∨ ofs + chunk.size ≤ ofs') := sorry'
theorem load_store_pointer_overlap {chunk m1 b ofs vb vo m2 chunk' ofs' v} :
store chunk m1 b ofs (Vptr vb vo) = some m2 → load chunk' m2 b ofs' = some v →
ofs' ≠ ofs → ofs' + chunk'.size > ofs → ofs + chunk.size > ofs' → v = Vundef := sorry'
theorem load_store_pointer_mismatch {chunk m1 b ofs vb vo m2 chunk' v} :
store chunk m1 b ofs (Vptr vb vo) = some m2 → load chunk' m2 b ofs = some v →
¬ compat_pointer_chunks chunk chunk' → v = Vundef := sorry'
lemma store_similar_chunks {chunk1 chunk2 v1 v2 m b ofs} :
encode_val chunk1 v1 = encode_val chunk2 v2 → chunk1.align = chunk2.align →
store chunk1 m b ofs v1 = store chunk2 m b ofs v2 := sorry'
theorem store_signed_unsigned_8 {m b ofs v} :
store Mint8signed m b ofs v = store Mint8unsigned m b ofs v := sorry'
theorem store_signed_unsigned_16 {m b ofs v} :
store Mint16signed m b ofs v = store Mint16unsigned m b ofs v := sorry'
theorem store_int8_zero_ext {m b ofs n} :
store Mint8unsigned m b ofs (Vint (zero_ext 8 n)) =
store Mint8unsigned m b ofs (Vint n) := sorry'
theorem store_int8_sign_ext {m b ofs n} :
store Mint8signed m b ofs (Vint (sign_ext W8 n)) =
store Mint8signed m b ofs (Vint n) := sorry'
theorem store_int16_zero_ext {m b ofs n} :
store Mint16unsigned m b ofs (Vint (zero_ext 16 n)) =
store Mint16unsigned m b ofs (Vint n) := sorry'
theorem store_int16_sign_ext {m b ofs n} :
store Mint16signed m b ofs (Vint (sign_ext W16 n)) =
store Mint16signed m b ofs (Vint n) := sorry'
lemma setN_concat {bytes1 bytes2 ofs c} :
setN (bytes1 ++ bytes2) ofs c = setN bytes2 (ofs + bytes1.length) (setN bytes1 ofs c) := sorry'
theorem store_bytes_concat {m b ofs bytes1 m1 bytes2 m2} :
store_bytes m b ofs bytes1 = some m1 →
store_bytes m1 b (ofs + bytes1.length) bytes2 = some m2 →
store_bytes m b ofs (bytes1 ++ bytes2) = some m2 := sorry'
theorem store_bytes_split {m b ofs bytes1 bytes2 m2} :
store_bytes m b ofs (bytes1 ++ bytes2) = some m2 →
∃ m1, store_bytes m b ofs bytes1 = some m1
∧ store_bytes m1 b (ofs + bytes1.length) bytes2 = some m2 := sorry'
theorem store_int64_split {m b ofs v m'} :
store Mint64 m b ofs v = some m' → ¬ archi.ptr64 →
∃ m1, store Mint32 m b ofs (if archi.big_endian then hiword v else loword v) = some m1
∧ store Mint32 m1 b (ofs + 4) (if archi.big_endian then loword v else hiword v) = some m' := sorry'
theorem storev_int64_split {m a v m'} :
storev Mint64 m a v = some m' → archi.ptr64 = ff →
∃ m1, storev Mint32 m a (if archi.big_endian then hiword v else loword v) = some m1
∧ storev Mint32 m1 (a + Vint (repr 4)) (if archi.big_endian then loword v else hiword v) = some m' := sorry'
/- ** Properties related to [alloc]. -/
section alloc
parameters {m1 : mem} {lo hi : ℕ}
local notation `b` := m1.nextblock
private def m2 := m1.alloc lo hi
theorem nextblock_alloc : m2.nextblock = m1.nextblock.succ := rfl
theorem alloc_result : b = m1.nextblock := rfl
theorem valid_new_block : valid_block m2 b :=
show (_:ℕ)<_, by rw [nextblock_alloc, pos_num.succ_to_nat]; apply nat.lt_succ_self
theorem valid_block_alloc {b'} (h : valid_block m1 b') : valid_block m2 b' :=
lt_trans h valid_new_block
theorem fresh_block_alloc : ¬ valid_block m1 b := lt_irrefl _
theorem valid_block_alloc_inv {b'} (h : valid_block m2 b') : b' = b ∨ valid_block m1 b' :=
have (_:ℕ)<pos_num.succ _, from h,
or.symm $ lt_or_eq_of_le $ nat.le_of_lt_succ $ by rwa pos_num.succ_to_nat at this
theorem perm_alloc_1 {b' ofs k p} : perm m1 b' ofs k p → perm m2 b' ofs k p :=
begin
dsimp[perm, mem.access, m2, mem.alloc], rw PTree.gsspec,
by_cases (b' = m1.nextblock); simp [h],
{ simp [m1.nextblock_noaccess ofs k (le_refl _), perm_order'] },
{ exact id }
end
theorem perm_alloc_2 {ofs k} : lo ≤ ofs → ofs < hi → perm m2 b ofs k Freeable := sorry'
theorem perm_alloc_inv {b' ofs k p} : perm m2 b' ofs k p →
if b' = b then lo ≤ ofs ∧ ofs < hi else perm m1 b' ofs k p := sorry'
theorem perm_alloc_3 {ofs k p} (h : perm m2 b ofs k p) : lo ≤ ofs ∧ ofs < hi :=
by note := perm_alloc_inv h; simp at this; exact this
theorem perm_alloc_4 {b' ofs k p} (h1 : perm m2 b' ofs k p)
(h2 : b' ≠ b) : perm m1 b' ofs k p :=
by note := perm_alloc_inv h1; simp [h2] at this; exact this
theorem valid_access_alloc_other {chunk b' ofs p} :
valid_access m1 chunk b' ofs p →
valid_access m2 chunk b' ofs p :=
and_implies (λal ofs' h1 h2, perm_alloc_1 $ al h1 h2) id
theorem valid_access_alloc_same {chunk : memory_chunk} {ofs}
(hl : lo ≤ ofs) (hh : ofs + chunk.size ≤ hi) (ha : chunk.align ∣ ofs) :
valid_access m2 chunk b ofs Freeable :=
⟨λofs' h1 h2, perm_alloc_2 (le_trans hl h1) (lt_of_lt_of_le h2 hh), ha⟩
theorem valid_access_alloc_inv {chunk b' ofs p} (h : valid_access m2 chunk b' ofs p) :
if b' = b
then lo ≤ ofs ∧ ofs + chunk.size ≤ hi ∧ chunk.align ∣ ofs
else valid_access m1 chunk b' ofs p := sorry'
theorem load_alloc_unchanged (chunk b' ofs) (h : valid_block m1 b') :
load chunk m2 b' ofs = load chunk m1 b' ofs := sorry'
theorem load_alloc_other {chunk b' ofs v}
(h : load chunk m1 b' ofs = some v) : load chunk m2 b' ofs = some v :=
by rwa -load_alloc_unchanged at h; exact sorry'
theorem load_alloc_same {chunk ofs v} :
load chunk m2 b ofs = some v → v = Vundef := sorry'
theorem load_alloc_same' {chunk : memory_chunk} {ofs} :
lo ≤ ofs → ofs + chunk.size ≤ hi → chunk.align ∣ ofs →
load chunk m2 b ofs = some Vundef := sorry'
theorem load_bytes_alloc_unchanged {b' ofs n} :
valid_block m1 b' → load_bytes m2 b' ofs n = load_bytes m1 b' ofs n := sorry'
theorem load_bytes_alloc_same {n ofs bytes byte} :
load_bytes m2 b ofs n = some bytes →
byte ∈ bytes → byte = Undef := sorry'
end alloc
/- ** Properties related to [free]. -/
theorem range_perm_free {m1 b lo hi}
(h : range_perm m1 b lo hi Cur Freeable) : { m2 // free m1 b lo hi = some m2 } :=
by unfold free; simp [h]; exact ⟨_, rfl⟩
section free
parameters {m1 m2 : mem} {bf : block} {lo hi : ℕ}
parameter (FREE : free m1 bf lo hi = some m2)
include FREE
theorem free_range_perm : range_perm m1 bf lo hi Cur Freeable :=
by by_contradiction; simp [free, a] at FREE; contradiction
lemma free_result : m2 = unchecked_free m1 bf lo hi :=
by simp [free, free_range_perm FREE] at FREE; symmetry; injection FREE
theorem nextblock_free : m2.nextblock = m1.nextblock :=
by rw free_result FREE; refl
theorem valid_block_free {b} : valid_block m1 b ↔ valid_block m2 b :=
by rw free_result FREE; refl
theorem perm_free_1 {b ofs k p} :
b ≠ bf ∨ ofs < lo ∨ hi ≤ ofs →
perm m1 b ofs k p → perm m2 b ofs k p := sorry'
theorem perm_free_2 {ofs k p} : lo ≤ ofs → ofs < hi → ¬ perm m2 bf ofs k p := sorry'
theorem perm_free_3 {b ofs k p} : perm m2 b ofs k p → perm m1 b ofs k p := sorry'
theorem perm_free_inv {b ofs k p} :
perm m1 b ofs k p → (b = bf ∧ lo ≤ ofs ∧ ofs < hi) ∨ perm m2 b ofs k p := sorry'
theorem valid_access_free_1 {chunk b ofs p} :
valid_access m1 chunk b ofs p →
b ≠ bf ∨ lo ≥ hi ∨ ofs + chunk.size ≤ lo ∨ hi ≤ ofs →
valid_access m2 chunk b ofs p := sorry'
theorem valid_access_free_2 {chunk : memory_chunk} {ofs p} :
lo < hi → ofs + chunk.size > lo → ofs < hi →
¬ valid_access m2 chunk bf ofs p := sorry'
theorem valid_access_free_inv_1 {chunk b ofs p} :
valid_access m2 chunk b ofs p → valid_access m1 chunk b ofs p :=
and_implies (λal ofs' h1 h2, perm_free_3 $ al h1 h2) id
theorem valid_access_free_inv_2 {chunk ofs p} :
valid_access m2 chunk bf ofs p →
lo ≥ hi ∨ ofs + chunk.size ≤ lo ∨ hi ≤ ofs := sorry'
theorem load_free {chunk :memory_chunk} {b ofs} :
b ≠ bf ∨ lo ≥ hi ∨ ofs + chunk.size ≤ lo ∨ hi ≤ ofs →
load chunk m2 b ofs = load chunk m1 b ofs := sorry'
theorem load_free_2 {chunk b ofs v} :
load chunk m2 b ofs = some v → load chunk m1 b ofs = some v := sorry'
theorem load_bytes_free {b ofs n} :
b ≠ bf ∨ lo ≥ hi ∨ ofs + n ≤ lo ∨ hi ≤ ofs →
load_bytes m2 b ofs n = load_bytes m1 b ofs n := sorry'
theorem load_bytes_free_2 {b ofs n bytes} :
load_bytes m2 b ofs n = some bytes → load_bytes m1 b ofs n = some bytes := sorry'
end free
/- ** Properties related to [drop_perm] -/
theorem range_perm_drop_2 {m b lo hi p}
(h : range_perm m b lo hi Cur Freeable) : {m' // drop_perm m b lo hi p = some m'} :=
by unfold drop_perm; rw [dif_pos h]; exact ⟨_, rfl⟩
section drop
parameters {m m' : mem} {b : block} {lo hi : ℕ} {p : permission}
parameter (DROP : drop_perm m b lo hi p = some m')
include DROP
theorem range_perm_drop_1 : range_perm m b lo hi Cur Freeable :=
by by_contradiction; unfold drop_perm at DROP; rw [dif_neg a] at DROP; contradiction
theorem drop_eq_val : m' = drop_perm.val m b lo hi p range_perm_drop_1 :=
by unfold drop_perm at DROP;
rw [dif_pos (range_perm_drop_1 DROP)] at DROP; symmetry; injection DROP
theorem nextblock_drop : m'.nextblock = m.nextblock :=
by rw drop_eq_val DROP; refl
theorem drop_perm_valid_block {b'} : valid_block m' b' ↔ valid_block m b' :=
by rw drop_eq_val DROP; refl
theorem perm_mem_access {ofs k} (hl : lo ≤ ofs) (hh : ofs < hi) :
mem.access m' b ofs k = some p :=
begin
rw drop_eq_val DROP, dsimp [drop_perm.val],
dsimp [mem.access]; ginduction m.mem_access#b with ma f,
{ note := range_perm_drop_1 DROP hl hh,
dsimp [perm, mem.access] at this,
rw ma at this,
apply false.elim this },
{ rw PTree.gss, dsimp [option.get_or_else],
rw if_pos (and.intro hl hh) }
end
theorem perm_drop_2 {ofs k p'} (hl : lo ≤ ofs) (hh : ofs < hi) :
perm m' b ofs k p' ↔ perm_order p p' :=
by unfold perm; rw perm_mem_access DROP hl hh; refl
theorem perm_drop_1 {ofs k} (hl : lo ≤ ofs) (hh : ofs < hi) : perm m' b ofs k p :=
(perm_drop_2 hl hh).2 (perm_order.perm_refl _)
theorem perm_drop_3 {b' ofs k p'} : b' ≠ b ∨ ofs < lo ∨ hi ≤ ofs →
perm m b' ofs k p' → perm m' b' ofs k p' := sorry'
theorem perm_drop_4 {b' ofs k p'} : perm m' b' ofs k p' → perm m b' ofs k p' := sorry'
lemma range_perm_drop_3 {b' lo' hi' k p'}
(h : b' ≠ b ∨ hi' ≤ lo ∨ hi ≤ lo' ∨ perm_order p p') :
range_perm m b' lo' hi' k p' → range_perm m' b' lo' hi' k p' :=
λal ofs' hl hh, begin
by_cases b' ≠ b ∨ ofs' < lo ∨ hi ≤ ofs' with H,
{ exact perm_drop_3 DROP H (al hl hh) },
note bb : b' = b := decidable.by_contradiction (λh, H $ or.inl h),
note ol : lo ≤ ofs' := le_of_not_gt (λh, H $ or.inr $ or.inl h),
note oh : ofs' < hi := lt_of_not_ge (λh, H $ or.inr $ or.inr h),
rw bb, apply (perm_drop_2 DROP ol oh).2,
exact ((h.resolve_left (λn, n bb))
.resolve_left (λn, not_le_of_gt hh $ le_trans n ol))
.resolve_left (λn, not_le_of_gt oh $ le_trans n hl)
end
lemma range_perm_drop_4 {b' lo' hi' k p'} :
range_perm m' b' lo' hi' k p' → range_perm m b' lo' hi' k p' :=
λal ofs' hl hh, perm_drop_4 $ al hl hh
lemma valid_access_drop_1 {chunk : memory_chunk} {b' ofs p'}
(h : b' ≠ b ∨ ofs + chunk.size ≤ lo ∨ hi ≤ ofs ∨ perm_order p p') :
valid_access m chunk b' ofs p' → valid_access m' chunk b' ofs p' :=
and_implies (range_perm_drop_3 h) id
lemma valid_access_drop_2 {chunk b' ofs p'} :
valid_access m' chunk b' ofs p' → valid_access m chunk b' ofs p' :=
and_implies range_perm_drop_4 id
theorem load_drop {chunk : memory_chunk} {b' ofs}
(h : b' ≠ b ∨ ofs + chunk.size ≤ lo ∨ hi ≤ ofs ∨ perm_order p Readable) :
load chunk m' b' ofs = load chunk m b' ofs :=
begin
note := valid_access_drop_1 DROP h,
unfold load,
by_cases valid_access m chunk b' ofs Readable with va; simp [va],
{ simp [valid_access_drop_1 DROP h va], rw drop_eq_val DROP, refl },
{ simp [mt (valid_access_drop_2 DROP) va] }
end
theorem load_bytes_drop {b' ofs n}
(h : b' ≠ b ∨ ofs + n ≤ lo ∨ hi ≤ ofs ∨ perm_order p Readable) :
load_bytes m' b' ofs n = load_bytes m b' ofs n :=
begin
unfold load_bytes,
unfold load,
by_cases range_perm m' b' ofs (ofs + n) Cur Readable with va; simp [va],
{ simp [range_perm_drop_4 DROP va], rw drop_eq_val DROP, refl },
{ simp [mt (range_perm_drop_3 DROP h) va] }
end
end drop
/- * Generic injections -/
/- A memory state [m1] generically injects into another memory state [m2] via the
memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
is itself valid;
- the memory value associated in [m1] to an accessible address
must inject into [m2]'s memory value at the corresponding address.
-/
def perm_Z (m : mem) (b : block) (ofs : ℤ) (k : perm_kind) (p : permission) : Prop :=
ofs.ex_nat $ λ ofs', perm m b ofs' k p
def range_perm_Z (m b) (lo hi : ℤ) (k p) : Prop :=
lo.ex_nat $ λlo', hi.ex_nat $ λhi', range_perm m b lo' hi' k p
def valid_access_Z (m chunk b) (ofs : ℤ) (p) : Prop :=
ofs.ex_nat $ λ ofs', valid_access m chunk b ofs' p
structure mem_inj (f : meminj) (m1 m2 : mem) : Prop :=
(mi_perm : ∀ b1 b2 delta ofs k p,
f b1 = some (b2, delta) →
perm_Z m1 b1 ofs k p →
perm_Z m2 b2 (ofs + delta) k p)
(mi_align : ∀ b1 b2 delta (chunk : memory_chunk) ofs p,
f b1 = some (b2, delta) →
range_perm m1 b1 ofs (ofs + chunk.size) Max p →
↑chunk.align ∣ delta)
(mi_memval : ∀ b1 ofs b2 delta,
f b1 = some (b2, delta) →
perm m1 b1 ofs Cur Readable →
(↑ofs + delta).ex_nat (λ ofs',
memval_inject f (get1 ofs (m1.get_block b1)) (get1 ofs' (m2.get_block b2))))
/- Preservation of permissions -/
lemma range_perm_inj {f m1 m2 b1 lo hi k p b2 delta} :
mem_inj f m1 m2 →
range_perm_Z m1 b1 lo hi k p →
f b1 = some (b2, delta) →
range_perm_Z m2 b2 (lo + delta) (hi + delta) k p := sorry'
lemma valid_access_inj {f m1 m2 b1 b2 delta chunk ofs p} :
mem_inj f m1 m2 →
f b1 = some (b2, delta) →
valid_access_Z m1 chunk b1 ofs p →
valid_access_Z m2 chunk b2 (ofs + delta) p := sorry'
/- Preservation of loads. -/
lemma getN_inj {f m1 m2 b1 b2 delta} :
mem_inj f m1 m2 →
f b1 = some (b2, delta) →
∀ {n ofs},
range_perm m1 b1 ofs (ofs + n) Cur Readable →
(↑ofs + delta).ex_nat (λ ofs',
list.forall2 (memval_inject f)
(getN n ofs (m1.get_block b1))
(getN n ofs' (m2.get_block b2))) := sorry'
lemma load_inj {f m1 m2 chunk b1 ofs b2 delta v1} :
mem_inj f m1 m2 →
load chunk m1 b1 ofs = some v1 →
f b1 = some (b2, delta) →
(↑ofs + delta).ex_nat (λ ofs',
∃ v2, load chunk m2 b2 ofs' = some v2 ∧ inject f v1 v2) := sorry'
lemma load_bytes_inj {f m1 m2 len b1 ofs b2 delta bytes1} :
mem_inj f m1 m2 →
load_bytes m1 b1 ofs len = some bytes1 →
f b1 = some (b2, delta) →
(↑ofs + delta).ex_nat (λ ofs',
∃ bytes2, load_bytes m2 b2 ofs' len = some bytes2
∧ list.forall2 (memval_inject f) bytes1 bytes2) := sorry'
/- Preservation of stores. -/
lemma setN_inj (access : ℤ → Prop) {delta f vl1 vl2} :
list.forall2 (memval_inject f) vl1 vl2 →
∀ p c1 c2,
(∀ q, access q → memval_inject f (get1 q c1) (get1 (q + delta) c2)) →
(∀ q, access q → (↑p + delta).all_nat (λp',
memval_inject f (get1 q (setN vl1 p c1))
(get1 (q + delta) (setN vl2 p' c2)))) := sorry'
def meminj_no_overlap (f : meminj) (m : mem) : Prop :=
∀ b1 b1' delta1 b2 b2' delta2 ofs1 ofs2,
b1 ≠ b2 →
f b1 = some (b1', delta1) →
f b2 = some (b2', delta2) →
perm m b1 ofs1 Max Nonempty →
perm m b2 ofs2 Max Nonempty →
b1' ≠ b2' ∨ ↑ofs1 + delta1 ≠ ofs2 + delta2
lemma store_mapped_inj {f chunk m1 b1 ofs v1 n1 m2 b2 delta v2} :
mem_inj f m1 m2 →
store chunk m1 b1 ofs v1 = some n1 →
meminj_no_overlap f m1 →
f b1 = some (b2, delta) →
inject f v1 v2 →
(↑ofs + delta).ex_nat (λ ofs',
∃ n2,
store chunk m2 b2 ofs' v2 = some n2
∧ mem_inj f n1 n2) := sorry'
lemma store_unmapped_inj {f chunk m1 b1 ofs v1 n1 m2} :
mem_inj f m1 m2 →
store chunk m1 b1 ofs v1 = some n1 →
f b1 = none →
mem_inj f n1 m2 := sorry'
lemma store_outside_inj {f m1 m2 chunk b ofs v m2'} :
mem_inj f m1 m2 →
(∀ b' delta ofs',
f b' = some (b, delta) →
perm m1 b' ofs' Cur Readable →
↑ofs ≤ ↑ofs' + delta → ↑ofs' + delta < ↑ofs + memory_chunk.size chunk → false) →
store chunk m2 b ofs v = some m2' →
mem_inj f m1 m2' := sorry'
lemma store_bytes_mapped_inj {f m1 b1 ofs bytes1 n1 m2 b2 delta bytes2} :
mem_inj f m1 m2 →
store_bytes m1 b1 ofs bytes1 = some n1 →
meminj_no_overlap f m1 →
f b1 = some (b2, delta) →
list.forall2 (memval_inject f) bytes1 bytes2 →
(↑ofs + delta).ex_nat (λ ofs',
∃ n2,
store_bytes m2 b2 ofs' bytes2 = some n2
∧ mem_inj f n1 n2) := sorry'
lemma store_bytes_unmapped_inj {f m1 b1 ofs bytes1 n1 m2} :
mem_inj f m1 m2 →
store_bytes m1 b1 ofs bytes1 = some n1 →
f b1 = none →
mem_inj f n1 m2 := sorry'
lemma store_bytes_outside_inj {f m1 m2 b ofs bytes2 m2'} :
mem_inj f m1 m2 →
(∀ b' delta ofs',
f b' = some (b, delta) →
perm m1 b' ofs' Cur Readable →
↑ofs ≤ ↑ofs' + delta → ↑ofs' + delta < ↑ofs + list.length bytes2 → false) →
store_bytes m2 b ofs bytes2 = some m2' →
mem_inj f m1 m2' := sorry'
lemma store_bytes_empty_inj {f m1 b1 ofs1 m1' m2 b2 ofs2 m2'} :
mem_inj f m1 m2 →
store_bytes m1 b1 ofs1 [] = some m1' →
store_bytes m2 b2 ofs2 [] = some m2' →
mem_inj f m1' m2' := sorry'
/- Preservation of allocations -/
lemma alloc_right_inj {f m1 m2 lo hi} :
mem_inj f m1 m2 →
mem_inj f m1 (m2.alloc lo hi) := sorry'
lemma alloc_left_unmapped_inj {f m1 m2 lo hi} :
mem_inj f m1 m2 →
f m1.nextblock = none →
mem_inj f (m1.alloc lo hi) m2 := sorry'
def inj_offset_aligned (delta : ℤ) (size : ℕ) : Prop :=
∀ chunk : memory_chunk, chunk.size ≤ size → ↑chunk.align ∣ delta
lemma alloc_left_mapped_inj {f m1 m2 lo hi b2 delta} :
mem_inj f m1 m2 →
valid_block m2 b2 →
inj_offset_aligned delta (hi-lo) →
(∀ ofs k p, lo ≤ ofs → ofs < hi → perm_Z m2 b2 (↑ofs + delta) k p) →
f m1.nextblock = some (b2, delta) →
mem_inj f (m1.alloc lo hi) m2 := sorry'
lemma free_left_inj {f m1 m2 b lo hi m1'} :
mem_inj f m1 m2 →
free m1 b lo hi = some m1' →
mem_inj f m1' m2 := sorry'
lemma free_right_inj {f m1 m2 b lo hi m2'} :
mem_inj f m1 m2 →
free m2 b lo hi = some m2' →
(∀ b' delta ofs k p,
f b' = some (b, delta) →
perm m1 b' ofs k p → ↑lo ≤ ↑ofs + delta → ↑ofs + delta < hi → false) →
mem_inj f m1 m2' := sorry'
/- Preservation of [drop_perm] operations. -/
lemma drop_unmapped_inj {f m1 m2 b lo hi p m1'} :
mem_inj f m1 m2 →
drop_perm m1 b lo hi p = some m1' →
f b = none →
mem_inj f m1' m2 := sorry'
lemma drop_mapped_inj {f m1 m2 b1 b2 delta lo hi p m1'} :
mem_inj f m1 m2 →
drop_perm m1 b1 lo hi p = some m1' →
meminj_no_overlap f m1 →
f b1 = some (b2, delta) →
(↑lo + delta).all_nat (λlo', (↑hi + delta).all_nat $ λhi',
∃ m2', drop_perm m2 b2 lo' hi' p = some m2'
∧ mem_inj f m1' m2') := sorry'
lemma drop_outside_inj {f m1 m2 b lo hi p m2'} :
mem_inj f m1 m2 →
drop_perm m2 b lo hi p = some m2' →
(∀ b' delta ofs' k p,
f b' = some (b, delta) →
perm m1 b' ofs' k p →
↑lo ≤ ↑ofs' + delta → ↑ofs' + delta < hi → false) →
mem_inj f m1 m2' := sorry'
/- * Memory extensions -/
/- A store [m2] extends a store [m1] if [m2] can be obtained from [m1]
by increasing the sizes of the memory blocks of [m1] (decreasing
the low bounds, increasing the high bounds), and replacing some of
the [Vundef] values stored in [m1] by more defined values stored
in [m2] at the same locations. -/
structure extends' (m1 m2 : mem) : Prop :=
(next : m1.nextblock = m2.nextblock)
(inj : mem_inj inject_id m1 m2)
(perm_inv : ∀ b ofs k p, perm m1 b ofs Max Nonempty →
perm m2 b ofs k p → perm m1 b ofs k p)
theorem extends_refl (m) : extends' m m := sorry'
theorem load_extends {chunk m1 m2 b ofs v1} :
extends' m1 m2 →
load chunk m1 b ofs = some v1 →
∃ v2, load chunk m2 b ofs = some v2 ∧ v1.lessdef v2 := sorry'
theorem loadv_extends {chunk m1 m2 addr1 addr2 v1} :
extends' m1 m2 →
loadv chunk m1 addr1 = some v1 →
addr1.lessdef addr2 →
∃ v2, loadv chunk m2 addr2 = some v2 ∧ v1.lessdef v2 := sorry'
theorem load_bytes_extends {m1 m2 b ofs len bytes1} :
extends' m1 m2 →
load_bytes m1 b ofs len = some bytes1 →
∃ bytes2, load_bytes m2 b ofs len = some bytes2
∧ list.forall2 memval_lessdef bytes1 bytes2 := sorry'
theorem store_within_extends {chunk m1 m2 b ofs v1 m1' v2} :
extends' m1 m2 →
store chunk m1 b ofs v1 = some m1' →
v1.lessdef v2 →
∃ m2',
store chunk m2 b ofs v2 = some m2'
∧ extends' m1' m2' := sorry'
theorem store_outside_extends {chunk m1 m2 b ofs v m2'} :
extends' m1 m2 →
store chunk m2 b ofs v = some m2' →
(∀ ofs', perm m1 b ofs' Cur Readable → ofs ≤ ofs' → ofs' < ofs + chunk.size → false) →
extends' m1 m2' := sorry'
theorem storev_extends {chunk m1 m2 addr1 v1 m1' addr2 v2} :
extends' m1 m2 →
storev chunk m1 addr1 v1 = some m1' →
addr1.lessdef addr2 →
v1.lessdef v2 →
∃ m2',
storev chunk m2 addr2 v2 = some m2'
∧ extends' m1' m2' := sorry'
theorem store_bytes_within_extends {m1 m2 b ofs bytes1 m1' bytes2} :
extends' m1 m2 →
store_bytes m1 b ofs bytes1 = some m1' →
list.forall2 memval_lessdef bytes1 bytes2 →
∃ m2',
store_bytes m2 b ofs bytes2 = some m2'
∧ extends' m1' m2' := sorry'
theorem store_bytes_outside_extends {m1 m2 b ofs bytes2 m2'} :
extends' m1 m2 →
store_bytes m2 b ofs bytes2 = some m2' →
(∀ ofs', perm m1 b ofs' Cur Readable → ofs ≤ ofs' → ofs' < ofs + bytes2.length → false) →
extends' m1 m2' := sorry'
theorem alloc_extends {m1 m2 lo1 hi1 lo2 hi2} :
extends' m1 m2 →
lo2 ≤ lo1 → hi1 ≤ hi2 →
extends' (m1.alloc lo1 hi1) (m2.alloc lo2 hi2) := sorry'
theorem free_left_extends {m1 m2 b lo hi m1'} :
extends' m1 m2 →
free m1 b lo hi = some m1' →
extends' m1' m2 := sorry'
theorem free_right_extends {m1 m2 b lo hi m2'} :
extends' m1 m2 →
free m2 b lo hi = some m2' →
(∀ ofs k p, perm m1 b ofs k p → lo ≤ ofs → ofs < hi → false) →
extends' m1 m2' := sorry'
theorem free_parallel_extends {m1 m2 b lo hi m1'} :
extends' m1 m2 →
free m1 b lo hi = some m1' →
∃ m2',
free m2 b lo hi = some m2'
∧ extends' m1' m2' := sorry'
theorem valid_block_extends {m1 m2 b} :
extends' m1 m2 →
(valid_block m1 b ↔ valid_block m2 b) := sorry'
theorem perm_extends {m1 m2 b ofs k p} :
extends' m1 m2 → perm m1 b ofs k p → perm m2 b ofs k p := sorry'
theorem perm_extends_inv {m1 m2 b ofs k p} :
extends' m1 m2 → perm m1 b ofs Max Nonempty →
perm m2 b ofs k p → perm m1 b ofs k p := sorry'
theorem valid_access_extends {m1 m2 chunk b ofs p} :
extends' m1 m2 → valid_access m1 chunk b ofs p → valid_access m2 chunk b ofs p := sorry'
theorem valid_pointer_extends {m1 m2 b ofs} :
extends' m1 m2 → valid_pointer m1 b ofs → valid_pointer m2 b ofs := sorry'
theorem weak_valid_pointer_extends {m1 m2 b ofs} :
extends' m1 m2 →
weak_valid_pointer m1 b ofs → weak_valid_pointer m2 b ofs := sorry'
/- * Memory injections -/
/- A memory state [m1] injects into another memory state [m2] via the
memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
is itself valid;
- the memory value associated in [m1] to an accessible address
must inject into [m2]'s memory value at the corersponding address;
- unallocated blocks in [m1] must be mapped to [None] by [f];
- if [f b = Some(b', delta)], [b'] must be valid in [m2];
- distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2];
- the sizes of [m2]'s blocks are representable with unsigned machine integers;
- pointers that could be represented using unsigned machine integers remain
representable after the injection.
-/
structure inject (f : meminj) (m1 m2 : mem) : Prop :=
(mi_inj : mem_inj f m1 m2)
(mi_freeblocks : ∀ b, ¬ valid_block m1 b → f b = none)
(mi_mappedblocks : ∀ b b' delta, f b = some (b', delta) → valid_block m2 b')
(mi_no_overlap : meminj_no_overlap f m1)
(mi_representable : ∀ b b' delta (ofs : ptrofs),
f b = some (b', delta) →
perm m1 b (unsigned ofs) Max Nonempty ∨ perm m1 b (unsigned ofs - 1) Max Nonempty →
delta.ex_nat (λd, unsigned ofs + d ≤ max_unsigned ptrofs.wordsize))
(mi_perm_inv : ∀ b1 ofs b2 delta k p,
f b1 = some (b2, delta) →
perm m1 b1 ofs Max Nonempty →
perm_Z m2 b2 (ofs + delta) k p → perm m1 b1 ofs k p)
/- Preservation of access validity and pointer validity -/
theorem valid_block_inject_1 {f : meminj} {m1 m2 b1 b2 delta} :
f b1 = some (b2, delta) →
inject f m1 m2 →
valid_block m1 b1 := sorry'
theorem valid_block_inject_2 {f : meminj} {m1 m2 b1 b2 delta} :
f b1 = some (b2, delta) →
inject f m1 m2 →
valid_block m2 b2 := sorry'
theorem perm_inject {f : meminj} {m1 m2 b1 b2 delta ofs k p} :
f b1 = some (b2, delta) →
inject f m1 m2 →
perm m1 b1 ofs k p → perm_Z m2 b2 (ofs + delta) k p := sorry'
theorem perm_inject_inv {f m1 m2 b1 ofs b2 delta k p} :
inject f m1 m2 →
f b1 = some (b2, delta) →
perm m1 b1 ofs Max Nonempty →
perm_Z m2 b2 (ofs + delta) k p →
perm m1 b1 ofs k p := sorry'
theorem range_perm_inject {f : meminj} {m1 m2 b1 b2 delta lo hi k p} :
f b1 = some (b2, delta) →
inject f m1 m2 →
range_perm m1 b1 lo hi k p → range_perm_Z m2 b2 (lo + delta) (hi + delta) k p := sorry'
theorem valid_access_inject {f : meminj} {m1 m2 chunk b1 ofs b2 delta p} :
f b1 = some (b2, delta) →
inject f m1 m2 →
valid_access m1 chunk b1 ofs p →
valid_access_Z m2 chunk b2 (ofs + delta) p := sorry'
theorem valid_pointer_inject {f : meminj} {m1 m2 b1 ofs b2 delta} :
f b1 = some (b2, delta) →
inject f m1 m2 →
valid_pointer m1 b1 ofs →
(↑ofs + delta).ex_nat (λofs', valid_pointer m2 b2 ofs') := sorry'
theorem weak_valid_pointer_inject {f : meminj} {m1 m2 b1 ofs b2 delta} :
f b1 = some (b2, delta) →
inject f m1 m2 →
weak_valid_pointer m1 b1 ofs →
(↑ofs + delta).ex_nat (λofs', weak_valid_pointer m2 b2 ofs') := sorry'
/- The following lemmas establish the absence of machine integer overflow
during address computations. -/
lemma address_inject {f m1 m2 b1} {ofs1 : ptrofs} {b2 delta p} :
inject f m1 m2 →
perm m1 b1 (unsigned ofs1) Cur p →
f b1 = some (b2, delta) →
↑(unsigned (ofs1 + repr delta)) = ↑(unsigned ofs1) + delta := sorry'
lemma address_inject' {f m1 m2 chunk b1} {ofs1 : ptrofs} {b2 delta} :
inject f m1 m2 →
valid_access m1 chunk b1 (unsigned ofs1) Nonempty →
f b1 = some (b2, delta) →
↑(unsigned (ofs1 + repr delta)) = ↑(unsigned ofs1) + delta := sorry'
theorem weak_valid_pointer_inject_no_overflow {f m1 m2 b} {ofs : ptrofs} {b' delta} :
inject f m1 m2 →
weak_valid_pointer m1 b (unsigned ofs) →
f b = some (b', delta) →
unsigned ofs + unsigned (repr delta : ptrofs) ≤ max_unsigned ptrofs.wordsize := sorry'
theorem valid_pointer_inject_no_overflow {f m1 m2 b} {ofs : ptrofs} {b' delta} :
inject f m1 m2 →
valid_pointer m1 b (unsigned ofs) →
f b = some (b', delta) →
unsigned ofs + unsigned (repr delta : ptrofs) ≤ max_unsigned ptrofs.wordsize := sorry'
theorem valid_pointer_inject_val {f m1 m2 b ofs b' ofs'} :
inject f m1 m2 →
valid_pointer m1 b (unsigned ofs) →
val.inject f (Vptr b ofs) (Vptr b' ofs') →
valid_pointer m2 b' (unsigned ofs') := sorry'
theorem weak_valid_pointer_inject_val {f m1 m2 b ofs b' ofs'} :
inject f m1 m2 →
weak_valid_pointer m1 b (unsigned ofs) →
val.inject f (Vptr b ofs) (Vptr b' ofs') →
weak_valid_pointer m2 b' (unsigned ofs') := sorry'
theorem inject_no_overlap {f m1 m2 b1 b2 b1' b2' delta1 delta2 ofs1 ofs2} :
inject f m1 m2 →
b1 ≠ b2 →
f b1 = some (b1', delta1) →
f b2 = some (b2', delta2) →
perm m1 b1 ofs1 Max Nonempty →
perm m1 b2 ofs2 Max Nonempty →
b1' ≠ b2' ∨ ↑ofs1 + delta1 ≠ ofs2 + delta2 := sorry'
theorem different_pointers_inject {f m m' b1 b2} {ofs1 ofs2 : ptrofs} {b1' delta1 b2' delta2} :
inject f m m' →
b1 ≠ b2 →
valid_pointer m b1 (unsigned ofs1) →
valid_pointer m b2 (unsigned ofs2) →
f b1 = some (b1', delta1) →
f b2 = some (b2', delta2) →
b1' ≠ b2' ∨
unsigned (ofs1 + repr delta1) ≠
unsigned (ofs2 + repr delta2) := sorry'
theorem disjoint_or_equal_inject {f m m' b1 b1' delta1 b2 b2' delta2 ofs1 ofs2 sz} :
inject f m m' →
f b1 = some (b1', delta1) →
f b2 = some (b2', delta2) →
range_perm m b1 ofs1 (ofs1 + sz) Max Nonempty →
range_perm m b2 ofs2 (ofs2 + sz) Max Nonempty →
sz > 0 →
b1 ≠ b2 ∨ ofs1 = ofs2 ∨ ofs1 + sz ≤ ofs2 ∨ ofs2 + sz ≤ ofs1 →
b1' ≠ b2' ∨ ↑ofs1 + delta1 = ofs2 + delta2
∨ ↑ofs1 + delta1 + sz ≤ ofs2 + delta2
∨ ↑ofs2 + delta2 + sz ≤ ofs1 + delta1 := sorry'
theorem aligned_area_inject {f m m' b ofs al sz b' delta} :
inject f m m' →
al = 1 ∨ al = 2 ∨ al = 4 ∨ al = 8 → sz > 0 →
al ∣ sz →
range_perm m b ofs (ofs + sz) Cur Nonempty →
al ∣ ofs →
f b = some (b', delta) →
↑al ∣ ↑ofs + delta := sorry'
/- Preservation of loads -/
theorem load_inject {f m1 m2 chunk b1 ofs b2 delta v1} :
inject f m1 m2 →
load chunk m1 b1 ofs = some v1 →
f b1 = some (b2, delta) →
(↑ofs + delta).ex_nat (λ ofs',
∃ v2, load chunk m2 b2 ofs' = some v2 ∧ val.inject f v1 v2) := sorry'
theorem loadv_inject {f m1 m2 chunk a1 a2 v1} :
inject f m1 m2 →
loadv chunk m1 a1 = some v1 →
val.inject f a1 a2 →
∃ v2, loadv chunk m2 a2 = some v2 ∧ val.inject f v1 v2 := sorry'
theorem load_bytes_inject {f m1 m2 b1 ofs len b2 delta bytes1} :
inject f m1 m2 →
load_bytes m1 b1 ofs len = some bytes1 →
f b1 = some (b2, delta) →
(↑ofs + delta).ex_nat (λofs',
∃ bytes2, load_bytes m2 b2 ofs' len = some bytes2
∧ list.forall2 (memval_inject f) bytes1 bytes2) := sorry'
/- Preservation of stores -/
theorem store_mapped_inject {f chunk m1 b1 ofs v1 n1 m2 b2 delta v2} :
inject f m1 m2 →
store chunk m1 b1 ofs v1 = some n1 →
f b1 = some (b2, delta) →
val.inject f v1 v2 →
(↑ofs + delta).ex_nat (λofs', ∃ n2,
store chunk m2 b2 ofs' v2 = some n2
∧ inject f n1 n2) := sorry'
theorem store_unmapped_inject {f chunk m1 b1 ofs v1 n1 m2} :
inject f m1 m2 →
store chunk m1 b1 ofs v1 = some n1 →
f b1 = none →
inject f n1 m2 := sorry'
theorem store_outside_inject {f m1 m2} {chunk : memory_chunk} {b ofs v m2'} :
inject f m1 m2 →
(∀ b' delta ofs',
f b' = some (b, delta) →
perm m1 b' ofs' Cur Readable →
↑ofs ≤ ↑ofs' + delta → ↑ofs' + delta < ↑ofs + chunk.size → false) →
store chunk m2 b ofs v = some m2' →
inject f m1 m2' := sorry'
theorem storev_mapped_inject {f chunk m1 a1 v1 n1 m2 a2 v2} :
inject f m1 m2 →
storev chunk m1 a1 v1 = some n1 →
val.inject f a1 a2 →
val.inject f v1 v2 →
∃ n2, storev chunk m2 a2 v2 = some n2 ∧ inject f n1 n2 := sorry'
theorem store_bytes_mapped_inject {f m1 b1 ofs bytes1 n1 m2 b2 delta bytes2} :
inject f m1 m2 →
store_bytes m1 b1 ofs bytes1 = some n1 →
f b1 = some (b2, delta) →
list.forall2 (memval_inject f) bytes1 bytes2 →
(↑ofs + delta).ex_nat (λofs',
∃ n2,
store_bytes m2 b2 ofs' bytes2 = some n2
∧ inject f n1 n2) := sorry'
theorem store_bytes_unmapped_inject {f m1 b1 ofs bytes1 n1 m2} :
inject f m1 m2 →
store_bytes m1 b1 ofs bytes1 = some n1 →
f b1 = none →
inject f n1 m2 := sorry'
theorem store_bytes_outside_inject {f m1 m2 b ofs} {bytes2 : list memval} {m2'} :
inject f m1 m2 →
(∀ b' delta ofs',
f b' = some (b, delta) →
perm m1 b' ofs' Cur Readable →
↑ofs ≤ ↑ofs' + delta → ↑ofs' + delta < ↑ofs + bytes2.length → false) →
store_bytes m2 b ofs bytes2 = some m2' →
inject f m1 m2' := sorry'
theorem store_bytes_empty_inject {f m1 b1 ofs1 m1' m2 b2 ofs2 m2'} :
inject f m1 m2 →
store_bytes m1 b1 ofs1 [] = some m1' →
store_bytes m2 b2 ofs2 [] = some m2' →
inject f m1' m2' := sorry'
/- Preservation of allocations -/
theorem alloc_right_inject {f m1 m2 lo hi} :
inject f m1 m2 →
inject f m1 (m2.alloc lo hi) := sorry'
theorem alloc_left_unmapped_inject {f m1 m2 lo hi} :
inject f m1 m2 →
∃ f',
inject f' (m1.alloc lo hi) m2
∧ inject_incr f f'
∧ f' m1.nextblock = none
∧ (∀ b, b ≠ m1.nextblock → f' b = f b) := sorry'
theorem alloc_left_mapped_inject {f m1 m2 lo hi b2 delta} :
inject f m1 m2 →
valid_block m2 b2 →
delta ≤ max_unsigned ptrofs.wordsize →
(∀ ofs k p, perm m2 b2 ofs k p → delta = 0 ∨ ofs < max_unsigned ptrofs.wordsize) →
(∀ ofs k p, lo ≤ ofs → ofs < hi → perm m2 b2 (ofs + delta) k p) →
inj_offset_aligned delta (hi-lo) →
(∀ b delta' ofs k p,
f b = some (b2, delta') →
perm m1 b ofs k p →
(lo + delta : ℤ) ≤ ofs + delta' → ↑ofs + delta' < hi + delta → false) →
∃ f',
inject f' (m1.alloc lo hi) m2
∧ inject_incr f f'
∧ f' m1.nextblock = some (b2, delta)
∧ (∀ b, b ≠ m1.nextblock → f' b = f b) := sorry'
theorem alloc_parallel_inject {f m1 m2 lo1 hi1 lo2 hi2} :
inject f m1 m2 →
lo2 ≤ lo1 → hi1 ≤ hi2 →
∃ f',
inject f' (m1.alloc lo1 hi1) (m2.alloc lo2 hi2)
∧ inject_incr f f'
∧ f' m1.nextblock = some (m2.nextblock, 0)
∧ (∀ b, b ≠ m1.nextblock → f' b = f b) := sorry'
/- Preservation of [free] operations -/
lemma free_left_inject {f m1 m2 b lo hi m1'} :
inject f m1 m2 →
free m1 b lo hi = some m1' →
inject f m1' m2 := sorry'
lemma free_list_left_inject {f m2 l m1 m1'} :
inject f m1 m2 →
free_list m1 l = some m1' →
inject f m1' m2 := sorry'
lemma free_right_inject {f m1 m2 b lo hi m2'} :
inject f m1 m2 →
free m2 b lo hi = some m2' →
(∀ b1 delta ofs k p,
f b1 = some (b, delta) → perm m1 b1 ofs k p →
↑lo ≤ ↑ofs + delta → ↑ofs + delta < hi → false) →
inject f m1 m2' := sorry'
lemma perm_free_list {l m m' b ofs k p} :
free_list m l = some m' →
perm m' b ofs k p →
perm m b ofs k p ∧
(∀ lo hi, (b, lo, hi) ∈ l → lo ≤ ofs → ofs < hi → false) := sorry'
theorem free_inject {f m1 l m1' m2 b lo hi m2'} :
inject f m1 m2 →
free_list m1 l = some m1' →
free m2 b lo hi = some m2' →
(∀ b1 delta ofs k p,
f b1 = some (b, delta) →
perm m1 b1 ofs k p → ↑lo ≤ ↑ofs + delta → ↑ofs + delta < hi →
∃ lo1, ∃ hi1, (b1, lo1, hi1) ∈ l ∧ lo1 ≤ ofs ∧ ofs < hi1) →
inject f m1' m2' := sorry'
theorem free_parallel_inject {f m1 m2 b lo hi m1' b' delta} :
inject f m1 m2 →
free m1 b lo hi = some m1' →
f b = some (b', delta) →
(↑lo + delta).ex_nat (λ lo',
(↑hi + delta).ex_nat (λ hi',
∃ m2', free m2 b' lo' hi' = some m2' ∧ inject f m1' m2')) := sorry'
lemma drop_outside_inject : ∀ f m1 m2 b lo hi p m2',
inject f m1 m2 →
drop_perm m2 b lo hi p = some m2' →
(∀ b' delta ofs k p,
f b' = some (b, delta) →
perm m1 b' ofs k p → ↑lo ≤ ↑ofs + delta → ↑ofs + delta < hi → false) →
inject f m1 m2' := sorry'
/- Composing two memory injections. -/
lemma mem_inj_compose {f f' m1 m2 m3} :
mem_inj f m1 m2 → mem_inj f' m2 m3 → mem_inj (f.comp f') m1 m3 := sorry'
theorem inject_compose {f f' m1 m2 m3} :
inject f m1 m2 → inject f' m2 m3 →
inject (f.comp f') m1 m3 := sorry'
lemma val_lessdef_inject_compose {f v1 v2 v3} :
val.lessdef v1 v2 → val.inject f v2 v3 → val.inject f v1 v3 := sorry'
lemma val_inject_lessdef_compose {f v1 v2 v3} :
val.inject f v1 v2 → val.lessdef v2 v3 → val.inject f v1 v3 := sorry'
lemma extends_inject_compose {f m1 m2 m3} :
extends' m1 m2 → inject f m2 m3 → inject f m1 m3 := sorry'
lemma inject_extends_compose {f m1 m2 m3} :
inject f m1 m2 → extends' m2 m3 → inject f m1 m3 := sorry'
lemma extends_extends_compose {m1 m2 m3} :
extends' m1 m2 → extends' m2 m3 → extends' m1 m3 := sorry'
/- Injecting a memory into itself. -/
def flat_inj (thr : block) : meminj :=
λ (b : block), if b < thr then some (b, 0) else none
def inject_neutral (thr : block) (m : mem) :=
mem_inj (flat_inj thr) m m
theorem flat_inj_no_overlap {thr m} : meminj_no_overlap (flat_inj thr) m := sorry'
theorem neutral_inject {m : mem} :
inject_neutral m.nextblock m → inject (flat_inj m.nextblock) m m := sorry'
theorem empty_inject_neutral {thr} : inject_neutral thr ∅ := sorry'
theorem alloc_inject_neutral {thr m lo hi} :
inject_neutral thr m →
m.nextblock < thr →
inject_neutral thr (m.alloc lo hi) := sorry'
theorem store_inject_neutral {chunk m b ofs v m' thr} :
store chunk m b ofs v = some m' →
inject_neutral thr m →
b < thr →
val.inject (flat_inj thr) v v →
inject_neutral thr m' := sorry'
theorem drop_inject_neutral {m b lo hi p m' thr} :
drop_perm m b lo hi p = some m' →
inject_neutral thr m →
b < thr →
inject_neutral thr m' := sorry'
/- * Invariance properties between two memory states -/
section unchanged_on
parameter P : block → ℕ → Prop
structure unchanged_on (m_before m_after : mem) : Prop :=
(nextblock : m_before.nextblock ≤ m_after.nextblock)
(perm : ∀ b ofs k p,
P b ofs → valid_block m_before b →
(perm m_before b ofs k p ↔ perm m_after b ofs k p))
(unchanged_on_contents : ∀ b ofs,
P b ofs → memory.perm m_before b ofs Cur Readable →
get1 ofs (m_after.get_block b) =
get1 ofs (m_before.get_block b))
lemma unchanged_on_refl {m} : unchanged_on m m := sorry'
lemma valid_block_unchanged_on {m m' b} :
unchanged_on m m' → valid_block m b → valid_block m' b := sorry'
lemma perm_unchanged_on {m m' b ofs k p} :
unchanged_on m m' → P b ofs →
perm m b ofs k p → perm m' b ofs k p := sorry'
lemma perm_unchanged_on_2 {m m' b ofs k p} :
unchanged_on m m' → P b ofs → valid_block m b →
perm m' b ofs k p → perm m b ofs k p := sorry'
lemma unchanged_on_trans {m1 m2 m3} : unchanged_on m1 m2 → unchanged_on m2 m3 → unchanged_on m1 m3 := sorry'
lemma load_bytes_unchanged_on_1 {m m' b ofs n} :
unchanged_on m m' →
valid_block m b →
(∀ i, ofs ≤ i → i < ofs + n → P b i) →
load_bytes m' b ofs n = load_bytes m b ofs n := sorry'
lemma load_bytes_unchanged_on {m m' b ofs n bytes} :
unchanged_on m m' →
(∀ i, ofs ≤ i → i < ofs + n → P b i) →
load_bytes m b ofs n = some bytes →
load_bytes m' b ofs n = some bytes := sorry'
lemma load_unchanged_on_1 {m m' b ofs} {chunk : memory_chunk} :
unchanged_on m m' →
valid_block m b →
(∀ i, ofs ≤ i → i < ofs + chunk.size → P b i) →
load chunk m' b ofs = load chunk m b ofs := sorry'
lemma load_unchanged_on {m m' b ofs v} {chunk : memory_chunk} :
unchanged_on m m' →
(∀ i, ofs ≤ i → i < ofs + chunk.size → P b i) →
load chunk m b ofs = some v →
load chunk m' b ofs = some v := sorry'
lemma store_unchanged_on {chunk m b ofs v m'} :
store chunk m b ofs v = some m' →
(∀ i, ofs ≤ i → i < ofs + chunk.size → ¬ P b i) →
unchanged_on m m' := sorry'
lemma store_bytes_unchanged_on {m b ofs bytes m'} :
store_bytes m b ofs bytes = some m' →
(∀ i, ofs ≤ i → i < ofs + bytes.length → ¬ P b i) →
unchanged_on m m' := sorry'
lemma alloc_unchanged_on {m lo hi} :
unchanged_on m (m.alloc lo hi) := sorry'
lemma free_unchanged_on {m b lo hi m'} :
free m b lo hi = some m' →
(∀ i, lo ≤ i → i < hi → ¬ P b i) →
unchanged_on m m' := sorry'
lemma drop_perm_unchanged_on {m b lo hi p m'} :
drop_perm m b lo hi p = some m' →
(∀ i, lo ≤ i → i < hi → ¬ P b i) →
unchanged_on m m' := sorry'
end unchanged_on
lemma unchanged_on_implies (P Q : block → ℕ → Prop) {m m'} :
(∀ b ofs, Q b ofs → valid_block m b → P b ofs) →
unchanged_on P m m' → unchanged_on Q m m' := sorry'
end memory
|
a534e988c2d8f14392b2cf5b916df635e21a486d | f57749ca63d6416f807b770f67559503fdb21001 | /hott/init/bool.hlean | 0c19b95dea93d75478600fe0b69d4a9a1a897025 | [
"Apache-2.0"
] | permissive | aliassaf/lean | bd54e85bed07b1ff6f01396551867b2677cbc6ac | f9b069b6a50756588b309b3d716c447004203152 | refs/heads/master | 1,610,982,152,948 | 1,438,916,029,000 | 1,438,916,029,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 568 | hlean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import init.reserved_notation
namespace bool
definition cond {A : Type} (b : bool) (t e : A) :=
bool.rec_on b e t
definition bor (a b : bool) :=
bool.rec_on a (bool.rec_on b ff tt) tt
notation a || b := bor a b
definition band (a b : bool) :=
bool.rec_on a ff (bool.rec_on b ff tt)
notation a && b := band a b
definition bnot (a : bool) :=
bool.rec_on a tt ff
end bool
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3c2f36f8841bf217fae3c0762375ff300edab3f7 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/group_theory/divisible.lean | fac1e460efe546fcdc1d60a5a6a613e38a09ae2b | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 10,248 | lean | /-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import group_theory.subgroup.pointwise
import group_theory.group_action.pi
import group_theory.quotient_group
import algebra.group.pi
/-!
# Divisible Group and rootable group
In this file, we define a divisible add monoid and a rootable monoid with some basic properties.
## Main definition
* `divisible_by A α`: An additive monoid `A` is said to be divisible by `α` iff for all `n ≠ 0 ∈ α`
and `y ∈ A`, there is an `x ∈ A` such that `n • x = y`. In this file, we adopt a constructive
approach, i.e. we ask for an explicit `div : A → α → A` function such that `div a 0 = 0` and
`n • div a n = a` for all `n ≠ 0 ∈ α`.
* `rootable_by A α`: A monoid `A` is said to be rootable by `α` iff for all `n ≠ 0 ∈ α` and `y ∈ A`,
there is an `x ∈ A` such that `x^n = y`. In this file, we adopt a constructive approach, i.e. we
ask for an explicit `root : A → α → A` function such that `root a 0 = 1` and `(root a n)ⁿ = a` for
all `n ≠ 0 ∈ α`.
## Main results
For additive monoids and groups:
* `divisible_by_of_smul_right_surj` : the constructive definition of divisiblity is implied by
the condition that `n • x = a` has solutions for all `n ≠ 0` and `a ∈ A`.
* `smul_right_surj_of_divisible_by` : the constructive definition of divisiblity implies
the condition that `n • x = a` has solutions for all `n ≠ 0` and `a ∈ A`.
* `prod.divisible_by` : `A × B` is divisible for any two divisible additive monoids.
* `pi.divisible_by` : any product of divisble additive monoids is divisible.
* `add_group.divisible_by_int_of_divisible_by_nat` : for additive groups, int divisiblity is implied
by nat divisiblity.
* `add_group.divisible_by_nat_of_divisible_by_int` : for additive groups, nat divisiblity is implied
by int divisiblity.
* `add_comm_group.divisible_by_int_of_smul_top_eq_top`: the constructive definition of divisiblity
is implied by the condition that `n • A = A` for all `n ≠ 0`.
* `add_comm_group.smul_top_eq_top_of_divisible_by_int`: the constructive definition of divisiblity
implies the condition that `n • A = A` for all `n ≠ 0`.
* `divisible_by_int_of_char_zero` : any field of characteristic zero is divisible.
* `quotient_add_group.divisible_by` : quotient group of divisible group is divisible.
* `function.surjective.divisible_by` : if `A` is divisible and `A →+ B` is surjective, then `B`
is divisible.
and their multiplicative counterparts:
* `rootable_by_of_pow_left_surj` : the constructive definition of rootablity is implied by the
condition that `xⁿ = y` has solutions for all `n ≠ 0` and `a ∈ A`.
* `pow_left_surj_of_rootable_by` : the constructive definition of rootablity implies the
condition that `xⁿ = y` has solutions for all `n ≠ 0` and `a ∈ A`.
* `prod.rootable_by` : any product of two rootable monoids is rootable.
* `pi.rootable_by` : any product of rootable monoids is rootable.
* `group.rootable_by_int_of_rootable_by_nat` : in groups, int rootablity is implied by nat
rootablity.
* `group.rootable_by_nat_of_rootable_by_int` : in groups, nat rootablity is implied by int
rootablity.
* `quotient_group.rootable_by` : quotient group of rootable group is rootable.
* `function.surjective.rootable_by` : if `A` is rootable and `A →* B` is surjective, then `B` is
rootable.
TODO: Show that divisibility implies injectivity in the category of `AddCommGroup`.
-/
open_locale pointwise
section add_monoid
variables (A α : Type*) [add_monoid A] [has_smul α A] [has_zero α]
/--
An `add_monoid A` is `α`-divisible iff `n • x = a` has a solution for all `n ≠ 0 ∈ α` and `a ∈ A`.
Here we adopt a constructive approach where we ask an explicit `div : A → α → A` function such that
* `div a 0 = 0` for all `a ∈ A`
* `n • div a n = a` for all `n ≠ 0 ∈ α` and `a ∈ A`.
-/
class divisible_by :=
(div : A → α → A)
(div_zero : ∀ a, div a 0 = 0)
(div_cancel : ∀ {n : α} (a : A), n ≠ 0 → n • (div a n) = a)
end add_monoid
section monoid
variables (A α : Type*) [monoid A] [has_pow A α] [has_zero α]
/--
A `monoid A` is `α`-rootable iff `xⁿ = a` has a solution for all `n ≠ 0 ∈ α` and `a ∈ A`.
Here we adopt a constructive approach where we ask an explicit `root : A → α → A` function such that
* `root a 0 = 1` for all `a ∈ A`
* `(root a n)ⁿ = a` for all `n ≠ 0 ∈ α` and `a ∈ A`.
-/
@[to_additive]
class rootable_by :=
(root : A → α → A)
(root_zero : ∀ a, root a 0 = 1)
(root_cancel : ∀ {n : α} (a : A), n ≠ 0 → (root a n)^n = a)
@[to_additive smul_right_surj_of_divisible_by]
lemma pow_left_surj_of_rootable_by [rootable_by A α] {n : α} (hn : n ≠ 0) :
function.surjective (λ a, pow a n : A → A) :=
λ x, ⟨rootable_by.root x n, rootable_by.root_cancel _ hn⟩
/--
A `monoid A` is `α`-rootable iff the `pow _ n` function is surjective, i.e. the constructive version
implies the textbook approach.
-/
@[to_additive divisible_by_of_smul_right_surj
"An `add_monoid A` is `α`-divisible iff `n • _` is a surjective function, i.e. the constructive
version implies the textbook approach."]
noncomputable def rootable_by_of_pow_left_surj
(H : ∀ {n : α}, n ≠ 0 → function.surjective (λ a, a^n : A → A)) :
rootable_by A α :=
{ root := λ a n, @dite _ (n = 0) (classical.dec _) (λ _, (1 : A)) (λ hn, (H hn a).some),
root_zero := λ _, by classical; exact dif_pos rfl,
root_cancel := λ n a hn, by rw dif_neg hn; exact (H hn a).some_spec }
section pi
variables {ι β : Type*} (B : ι → Type*) [Π (i : ι), has_pow (B i) β]
variables [has_zero β] [Π (i : ι), monoid (B i)] [Π i, rootable_by (B i) β]
@[to_additive]
instance pi.rootable_by : rootable_by (Π i, B i) β :=
{ root := λ x n i, rootable_by.root (x i) n,
root_zero := λ x, funext $ λ i, rootable_by.root_zero _,
root_cancel := λ n x hn, funext $ λ i, rootable_by.root_cancel _ hn }
end pi
section prod
variables {β B B' : Type*} [has_pow B β] [has_pow B' β]
variables [has_zero β] [monoid B] [monoid B'] [rootable_by B β] [rootable_by B' β]
@[to_additive]
instance prod.rootable_by : rootable_by (B × B') β :=
{ root := λ p n, (rootable_by.root p.1 n, rootable_by.root p.2 n),
root_zero := λ p, prod.ext (rootable_by.root_zero _) (rootable_by.root_zero _),
root_cancel := λ n p hn, prod.ext (rootable_by.root_cancel _ hn) (rootable_by.root_cancel _ hn) }
end prod
end monoid
namespace add_comm_group
variables (A : Type*) [add_comm_group A]
lemma smul_top_eq_top_of_divisible_by_int [divisible_by A ℤ] {n : ℤ} (hn : n ≠ 0) :
n • (⊤ : add_subgroup A) = ⊤ :=
add_subgroup.map_top_of_surjective _ $ λ a, ⟨divisible_by.div a n, divisible_by.div_cancel _ hn⟩
/--
If for all `n ≠ 0 ∈ ℤ`, `n • A = A`, then `A` is divisible.
-/
noncomputable def divisible_by_int_of_smul_top_eq_top
(H : ∀ {n : ℤ} (hn : n ≠ 0), n • (⊤ : add_subgroup A) = ⊤) :
divisible_by A ℤ :=
{ div := λ a n, if hn : n = 0 then 0 else
(show a ∈ n • (⊤ : add_subgroup A), by rw [H hn]; trivial).some,
div_zero := λ a, dif_pos rfl,
div_cancel := λ n a hn, begin
rw [dif_neg hn],
generalize_proofs h1,
exact h1.some_spec.2,
end }
end add_comm_group
@[priority 100]
instance divisible_by_int_of_char_zero {𝕜} [division_ring 𝕜] [char_zero 𝕜] : divisible_by 𝕜 ℤ :=
{ div := λ q n, q / n,
div_zero := λ q, by norm_num,
div_cancel := λ n q hn,
by rw [zsmul_eq_mul, (int.cast_commute n _).eq, div_mul_cancel q (int.cast_ne_zero.mpr hn)] }
namespace group
variables (A : Type*) [group A]
/--
A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive add_group.divisible_by_int_of_divisible_by_nat
"An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootable_by_int_of_rootable_by_nat [rootable_by A ℕ] : rootable_by A ℤ :=
{ root := λ a z, match z with
| (n : ℕ) := rootable_by.root a n
| -[1+n] := (rootable_by.root a (n + 1))⁻¹
end,
root_zero := λ a, rootable_by.root_zero a,
root_cancel := λ n a hn, begin
induction n,
{ change (rootable_by.root a _) ^ _ = a,
norm_num,
rw [rootable_by.root_cancel],
rw [int.of_nat_eq_coe] at hn,
exact_mod_cast hn, },
{ change ((rootable_by.root a _) ⁻¹)^_ = a,
norm_num,
rw [rootable_by.root_cancel],
norm_num, }
end}
/--A group is `ℕ`-rootable if it is `ℤ`-rootable
-/
@[to_additive add_group.divisible_by_nat_of_divisible_by_int
"An additive group is `ℕ`-divisible if it `ℤ`-divisible."]
def rootable_by_nat_of_rootable_by_int [rootable_by A ℤ] : rootable_by A ℕ :=
{ root := λ a n, rootable_by.root a (n : ℤ),
root_zero := λ a, rootable_by.root_zero a,
root_cancel := λ n a hn, begin
have := rootable_by.root_cancel a (show (n : ℤ) ≠ 0, by exact_mod_cast hn),
norm_num at this,
exact this,
end }
end group
section hom
variables {α A B : Type*}
variables [has_zero α] [monoid A] [monoid B] [has_pow A α] [has_pow B α] [rootable_by A α]
variables (f : A → B)
/--
If `f : A → B` is a surjective homomorphism and `A` is `α`-rootable, then `B` is also `α`-rootable.
-/
@[to_additive "If `f : A → B` is a surjective homomorphism and
`A` is `α`-divisible, then `B` is also `α`-divisible."]
noncomputable def function.surjective.rootable_by (hf : function.surjective f)
(hpow : ∀ (a : A) (n : α), f (a ^ n) = f a ^ n) : rootable_by B α :=
rootable_by_of_pow_left_surj _ _ $ λ n hn x,
let ⟨y, hy⟩ := hf x in ⟨f $ rootable_by.root y n, (by rw [←hpow (rootable_by.root y n) n,
rootable_by.root_cancel _ hn, hy] : _ ^ _ = x)⟩
end hom
section quotient
variables (α : Type*) {A : Type*} [comm_group A] (B : subgroup A)
/-- Any quotient group of a rootable group is rootable. -/
@[to_additive quotient_add_group.divisible_by
"Any quotient group of a divisible group is divisible"]
noncomputable instance quotient_group.rootable_by [rootable_by A ℕ] : rootable_by (A ⧸ B) ℕ :=
quotient_group.mk_surjective.rootable_by _ $ λ _ _, rfl
end quotient
|
c6bec4f333ec72e9b17a24aacce27846cd2eda5f | 11e28114d9553ecd984ac4819661ffce3068bafe | /src/equate.lean | c2a9cc12edc5298ba867aa2fad12bb93bbbebd22 | [
"MIT"
] | permissive | EdAyers/lean-subtask | 9a26eb81f0c8576effed4ca94342ae1281445c59 | 04ac5a6c3bc3bfd190af4d6dcce444ddc8914e4b | refs/heads/master | 1,586,516,665,621 | 1,558,701,948,000 | 1,558,701,948,000 | 160,983,035 | 4 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 4,904 | lean | /- Author: E.W.Ayers © 2019 -/
import .engine .policy
import algebra.group_power
import tactic.ring
/- The `equate` tactic. -/
namespace robot
meta def equate_user_attr : user_attribute rule_table unit :=
{ name := `equate
, descr := "add the given lemma to equate's rule-table."
, cache_cfg :=
{ mk_cache := rule_table.of_names
, dependencies := []
}
}
meta def get_equate_rule_table : tactic rule_table :=
user_attribute.get_cache equate_user_attr
run_cmd attribute.register `robot.equate_user_attr
/-- Solves simple equalities using lemmas tagged with `equate`. -/
meta def equate (names : list name := []) := do
base ← get_equate_rule_table,
bonus ← rule_table.of_names names,
all ← rule_table.join bonus base,
tactic.timetac "time: " $ robot.run robot.simple_policy all
/- DEFAULT RULES: -/
attribute [equate] mul_assoc
attribute [equate] mul_comm
attribute [equate] mul_left_inv
attribute [equate] mul_right_inv
attribute [equate] one_mul
attribute [equate] mul_one
attribute [equate] inv_inv
@[equate] lemma mul_inv {G:Type*} [group G] {a b : G} : (a * b) ⁻¹ = b⁻¹ * a⁻¹ := by simp
attribute [equate] pow_succ
attribute [equate] pow_zero
@[equate] lemma nat_succ_is_nat_plus_one {n} : nat.succ n = n + 1 := rfl
attribute [equate] add_assoc
attribute [equate] add_left_neg
attribute [equate] add_right_neg
attribute [equate] zero_add
attribute [equate] add_zero
attribute [equate] add_comm
attribute [equate] neg_add
attribute [equate] neg_neg
@[equate] lemma one_plus_one_is_two : 1 + 1 = 2 := rfl
attribute [equate] sub_self
attribute [equate] sub_eq_add_neg
attribute [equate] zero_sub
attribute [equate] sub_zero
attribute [equate] sub_neg_eq_add
-- attribute [equate] add_sub_assoc -- [NOTE] commented out because it short-circuits an exampleproof
attribute [equate] left_distrib
attribute [equate] right_distrib
attribute [equate] mul_zero
attribute [equate] zero_mul
attribute [equate] neg_mul_eq_neg_mul
attribute [equate] neg_mul_eq_mul_neg
attribute [equate] neg_mul_neg
attribute [equate] neg_mul_comm
attribute [equate] mul_sub_left_distrib
attribute [equate] mul_sub_right_distrib
@[equate] lemma comp_def {α β γ : Type*} {f : α → β} {g : β → γ} {x : α} : (g ∘ f) x = g (f x) := rfl
/- Additional equalities for rings.
These are needed because `equate` does not unfold definitions.
-/
namespace ring_theory
universe u
variables {R : Type u} [comm_ring R] {x y z a b c : R}
@[equate] def P1 : x ^ 0 = 1 := rfl
@[equate] def P2 {n} : x ^ (nat.succ n) = x * x ^ n := rfl
@[equate] def X2 : 2 * x = x + x := by ring
end ring_theory
/- Some equalities for set theory. -/
namespace set_rules
universe u
variables {α : Type u} {A B C : set α}
def ext : (∀ a, a ∈ A ↔ a ∈ B) → A = B := begin intro, funext, rw <-iff_eq_eq, apply a x end
@[equate] def A1 : A \ B = A ∩ (- B) := rfl
@[equate] def A2 : - ( B ∩ C ) = -B ∪ -C := ext $ λ a, ⟨λ h, classical.by_cases (λ aB, classical.by_cases (λ aC, absurd (and.intro aB aC) h) or.inr ) or.inl,λ h, or.cases_on h (λ h ⟨ab,_⟩, absurd ab h) (λ h ⟨_,ac⟩, absurd ac h)⟩
@[equate] def A3 : - ( B ∪ C ) = - B ∩ - C := ext $ λ a, ⟨λ h, ⟨h ∘ or.inl,h ∘ or.inr⟩, λ ⟨x,y⟩ h₂, or.cases_on h₂ x y⟩
@[equate] def A4 : B ∩ C = C ∩ B := ext $ λ a, ⟨and.swap,and.swap⟩
@[equate] def A5 : B ∪ C = C ∪ B := ext $ λ a, ⟨or.swap, or.swap⟩
@[equate] def A6 : A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) := ext $ λ a, ⟨λ h, or.cases_on h (λ h, ⟨or.inl h, or.inl h⟩) (λ ⟨b,c⟩, ⟨or.inr b, or.inr c⟩),λ ⟨ab,ac⟩, or.cases_on ab or.inl $ λ b, or.cases_on ac or.inl $ λ c, or.inr ⟨b,c⟩⟩ -- [NOTE] use mathlib, don't be macochistic.
@[equate] def A7 : A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) := ext $ λ a, ⟨λ ⟨a,bc⟩,or.cases_on bc (λ b, or.inl ⟨a,b⟩) (λ c, or.inr ⟨a,c⟩), λ h, or.cases_on h (λ ⟨a,b⟩, ⟨a,or.inl b⟩) (λ ⟨a,c⟩, ⟨a,or.inr c⟩)⟩
@[equate] def A8 : (A ∩ B) ∩ C = A ∩ (B ∩ C) := ext $ λ a, and.assoc
@[equate] def A9 : (A ∪ B) ∪ C = A ∪ (B ∪ C) := ext $ λ a, or.assoc
@[equate] def A10 : A ∪ ∅ = A := ext $ λ _, or_false _
@[equate] def A11 : A ∩ ∅ = ∅ := ext $ λ _, and_false _
@[equate] def A12 : A ∪ set.univ = set.univ := ext $ λ _, or_true _
@[equate] def A13 : A ∩ set.univ = A := ext $ λ _, and_true _
@[equate] def A14 : A ∩ A = A := ext $ λ a, ⟨λ ⟨x,y⟩, x, λ x, ⟨x,x⟩⟩
@[equate] def A15 : A ∪ A = A := ext $ λ a, ⟨λ xy, or.rec_on xy id id, or.inl⟩
@[equate] def A16 : A \ A = ∅ := ext $ λ a, ⟨λ ⟨x,y⟩, y x, λ x, false.rec_on _ x⟩
end set_rules
namespace list_theory
end list_theory
-- run_cmd (get_equate_rule_table >>= tactic.trace)
end robot |
47c84d8699527f24a67d6d9aeba20cc388a7fb18 | 827a8a5c2041b1d7f55e128581f583dfbd65ecf6 | /truncation.hlean | c4a564ed5f3f7f93a854ee1116f1700f78538ba1 | [
"Apache-2.0"
] | permissive | fpvandoorn/leansnippets | 6af0499f6f3fd2c07e4b580734d77b67574e7c27 | 601bafbe07e9534af76f60994d6bdf741996ef93 | refs/heads/master | 1,590,063,910,882 | 1,545,093,878,000 | 1,545,093,878,000 | 36,044,957 | 2 | 2 | null | 1,442,619,708,000 | 1,432,256,875,000 | Lean | UTF-8 | Lean | false | false | 11,118 | hlean | import types.eq types.pi hit.colimit types.nat.hott homotopy.sphere
--open unit pi equiv sum
open eq quotient sphere sphere.ops nat seq_colim is_trunc sum pointed
-- /- HELPER LEMMAS -/
-- definition inv_con_con_eq_of_eq_con_con_inv {A : Type} {a₁ a₂ b₁ b₂ : A} {p : a₁ = b₁}
-- {q : a₁ = a₂} {r : a₂ = b₂} {s : b₁ = b₂} (H : q = p ⬝ s ⬝ r⁻¹) : p⁻¹ ⬝ q ⬝ r = s :=
-- begin
-- apply con_eq_of_eq_con_inv,
-- apply inv_con_eq_of_eq_con,
-- rewrite -con.assoc,
-- apply H
-- end
-- /-
-- Call a function f weakly constant if Πa a', f a = f a'
-- This theorem states that if f is weakly constant, then ap f is weakly constant.
-- -/
-- definition weakly_constant_ap {A B : Type} {f : A → B} {a a' : A} (p q : a = a')
-- (H : Π(a a' : A), f a = f a') : ap f p = ap f q :=
-- have H' : Π{b c : A} (r : b = c), (H a b)⁻¹ ⬝ H a c = ap f r, from
-- (λb c r, eq.rec_on r !con.left_inv),
-- !H'⁻¹ ⬝ !H'
/- definition of the one step n-truncation -/
namespace dtr
section
parameters {k : ℕ} {A : Type}
abbreviation B := A ⊎ (S k → A)
inductive R : B → B → Type :=
| Rmk : Π(s : S k → A) (x : S k), R (inl (s x)) (inr s)
definition dtr : Type := quotient R --Make explicit after #652
definition tr [constructor] (a : A) : dtr :=
class_of R (inl a)
definition aux (s : S k → A) : dtr :=
class_of R (inr s)
definition tr_eq (s : S k → A) (x : S k) : tr (s x) = aux s :=
!eq_of_rel !R.Rmk
protected definition rec [recursor] {P : dtr → Type} (Pt : Π(a : A), P (tr a))
(Pa : Π(s : S k → A), P (aux s))
(Pe : Π(s : S k → A) (x : S k), Pt (s x) =[tr_eq s x] Pa s) (x : dtr) : P x :=
begin
fapply (quotient.rec_on x),
{ intro a, cases a, apply Pt, apply Pa},
{ intro a a' H, cases H, apply Pe}
end
end
end dtr
attribute dtr.rec [unfold 7]
open dtr
section
parameters {X : Type} {k : ℕ}
/- basic constructors -/
private definition A [reducible] (n : ℕ) : Type := nat.rec_on n X (λn' X', @dtr k X')
private definition f [reducible] [constructor] ⦃n : ℕ⦄ (a : A n) : A (succ n) := tr a
private definition f_eq [reducible] ⦃n : ℕ⦄ (s : S k → A n) (x : S k) : f (s x) = aux s :=
tr_eq s x
private definition f_eq2 [reducible] ⦃n : ℕ⦄ (s : S k → A n) (x y : S k) : f (s x) = f (s y) :=
tr_eq s x ⬝ (tr_eq s y)⁻¹
private definition my_tr [reducible] : Type := @seq_colim A f
private definition i [reducible] {n : ℕ} (a : A n) : my_tr := inclusion f a
private definition g [reducible] {n : ℕ} (a : A n) : i (f a) = i a := glue f a
exit
-- defining the recursor
private definition rec {P : my_tr → Type} [Pt : Πx, is_trunc (k.-2.+1) (P x)]
(H : Π(a : X), P (@i 0 a)) (x : my_tr) : P x :=
begin
induction x,
{ induction n with n IH,
{ exact H a},
{ --▸@dtr k (A n) at a,
induction a,
{ exact (g a)⁻¹ ▸ IH a},
{ exact f_eq s base ▸ !g⁻¹ ▸ IH (s (base))},
{ apply pathover_of_tr_eq, apply map_sphere_constant_of_is_trunc}}},
{ induction n with n IH; all_goals esimp at *,
{ apply tr_pathover} ,
{ induction a; all_goals esimp at *,
{ clear IH, apply sorry},
{ apply sorry},
{ apply sorry}}
}
end
exit
-- private definition elim {P : Type} [Pt : is_trunc (k.-2.+1) P]
-- (H : X → P) (x : my_tr) : P :=
-- begin
-- induction x,
-- { induction n with n IH,
-- { exact H a},
-- { rewrite ▸@dtr k (A n) at a,
-- induction a,
-- { exact IH a},
-- { },
-- { apply pathover_of_eq, revert s x, }
-- }},
-- { }
-- end
/-
To prove: is_trunc (k-1) my_tr
-/
/- point operations -/
definition fr [reducible] [unfold-c 4] {n m : ℕ} (a : A n) (H : n ≤ m) : A m :=
begin
induction H with m H b,
{ exact a},
{ exact f b},
end
/- path operations -/
definition i_fr [unfold-c 4] {n m : ℕ} (a : A n) (H : n ≤ m) : i (fr a H) = i a :=
begin
induction H with m H IH,
{ reflexivity},
{ exact g (fr a H) ⬝ IH},
end
definition eq_same {n : ℕ} (a a' : A n) : i a = i a' :=
calc
i a = i (f a) : g
... = i (f a') : ap i (f_eq a a')
... = i a' : g
-- step (1), case n ≥ m
definition eq_ge {n m : ℕ} (a : A n) (b : A m) (H : n ≥ m) : i a = i b :=
calc
i a = i (fr b H) : eq_same
... = i b : i_fr
-- step (1), case n ≤ m
definition eq_le {n m : ℕ} (a : A n) (b : A m) (H : n ≤ m) : i a = i b :=
calc
i a = i (fr a H) : i_fr
... = i b : eq_same
-- step (1), combined
definition eq_constructors {n m : ℕ} (a : A n) (b : A m) : i a = i b :=
lt_ge_by_cases (λH, eq_le a b (le_of_lt H)) !eq_ge
-- some other path operations needed for 2-dimensional path operations
definition fr_step {n m : ℕ} (a : A n) (H : n ≤ m) : fr a (le.step H) = f (fr a H) := idp
definition fr_irrel {n m : ℕ} (a : A n) (H H' : n ≤ m) : fr a H = fr a H' :=
ap (fr a) !is_prop.elim
-- TODO: the proofs can probably be slightly simplified if H is expressed in terms of H2
definition fr_f {n m : ℕ} (a : A n) (H : n ≤ m) (H2 : succ n ≤ m) : fr a H = fr (f a) H2 :=
begin
induction H with m H IH,
{ exfalso, exact not_succ_le_self H2},
{ refine _ ⬝ ap (fr (f a)) (to_right_inv !le_equiv_succ_le_succ H2),
--add some unfold-c's in files
esimp [le_equiv_succ_le_succ,equiv_of_is_prop, is_equiv_of_is_prop],
revert H IH,
eapply le.rec_on (le_of_succ_le_succ H2),
{ intros, esimp [succ_le_succ], apply concat,
apply fr_irrel _ _ (le.step !le.refl),
reflexivity},
{ intros, rewrite [↑fr,↓fr a H,↓succ_le_succ a_1], exact ap (@f _) !IH}},
end
/- 2-dimensional path operations -/
theorem i_fr_step {n m : ℕ} (a : A n) (H : n ≤ m)
: i_fr a (le.step H) = g (fr a H) ⬝ i_fr a H := idp
-- make the next two theorems instances of general theorems about lt_ge_by_cases
theorem eq_constructors_le {n m : ℕ} (a : A n) (b : A m) (H : n ≤ m)
: eq_constructors a b = eq_le a b H :=
begin
unfold [eq_constructors,lt_ge_by_cases],
induction (lt_or_ge n m) with H2 H2;all_goals esimp,
{ rewrite [is_prop.elim H (le_of_lt H2)]},
{ let p := le.antisymm H H2, cases p,
rewrite [is_prop.elim H (le.refl m), is_prop.elim H2 (le.refl m)]; exact !idp_con⁻¹},
end
theorem eq_constructors_ge {n m : ℕ} (a : A n) (b : A m) (H : n ≥ m)
: eq_constructors a b = eq_ge a b H :=
begin
unfold [eq_constructors,lt_ge_by_cases],
induction (lt_or_ge n m) with H2 H2;all_goals esimp,
{ exfalso, apply lt.irrefl, exact lt_of_le_of_lt H H2},
{ rewrite [is_prop.elim H H2]},
end
theorem ap_i_ap_f {n : ℕ} {a a' : A n} (p : a = a')
: ap i (ap !f p) = !g ⬝ ap i p ⬝ !g⁻¹ :=
eq.rec_on p !con.right_inv⁻¹
theorem ap_i_eq_ap_i_same {n : ℕ} {a a' : A n} (p q : a = a') : ap i p = ap i q :=
!weakly_constant_ap eq_same
theorem ap_f_eq_f' {n : ℕ} (a a' : A n)
: ap i (f_eq (f a) (f a')) = g (f a) ⬝ ap i (f_eq a a') ⬝ (g (f a'))⁻¹ :=
!ap_i_eq_ap_i_same ⬝ !ap_i_ap_f
theorem ap_f_eq_f {n : ℕ} (a a' : A n)
: (g (f a))⁻¹ ⬝ ap i (f_eq (f a) (f a')) ⬝ g (f a') = ap i (f_eq a a') :=
inv_con_con_eq_of_eq_con_con_inv !ap_f_eq_f'
theorem eq_same_f {n : ℕ} (a a' : A n)
: (g a)⁻¹ ⬝ eq_same (f a) (f a') ⬝ g a' = eq_same a a' :=
begin
esimp [eq_same],
apply (ap (λx, _ ⬝ x ⬝ _)),
apply (ap_f_eq_f a a'),
end
theorem i_fr_g {n m : ℕ} (b : A n) (H1 : n ≤ m) (H2 : succ n ≤ m)
: ap i (fr_f b H1 H2) ⬝ i_fr (f b) H2 ⬝ g b = i_fr b H1 :=
begin
induction H1 with m H IH, exfalso, exact not_succ_le_self H2,
cases H2 with x H3, -- x is unused
{ rewrite [is_prop.elim H !le.refl,↑fr_f,
↑le_equiv_succ_le_succ,▸*],
-- some le.rec's are not reduced if previous line is replaced by "↑le_equiv_succ_le_succ,↑i_fr,↑fr,▸*], state,"
refine (_ ⬝ !idp_con), apply ap (λx, x ⬝ _), apply (ap (ap i)),
rewrite [is_prop_elim_self,↑fr_irrel,▸*,is_prop_elim_self]},
{ rewrite [↑i_fr,↓i_fr b H,↓i_fr (f b) H3,↓fr (f b) H3,↓fr b H, -IH H3,
-con.assoc,-con.assoc,-con.assoc],
apply ap (λx, x ⬝ _ ⬝ _), apply con_eq_of_eq_con_inv, rewrite [-ap_i_ap_f],
apply ap_i_eq_ap_i_same}
end -- this was the longest time I've every spent on 14 lines of text
definition eq_same_con {n : ℕ} (a : A n) {a' a'' : A n} (p : a' = a'')
: eq_same a a' = eq_same a a'' ⬝ (ap i p)⁻¹ :=
by induction p; reflexivity
-- step (2), n > m
theorem eq_gt_f {n m : ℕ} (a : A n) (b : A m) (H1 : n ≥ succ m) (H2 : n ≥ m)
: eq_ge a (f b) H1 ⬝ g b = eq_ge a b H2 :=
begin
esimp [eq_ge],
let lem := eq_inv_con_of_con_eq (!con.assoc⁻¹ ⬝ i_fr_g b H2 H1),
rewrite [con.assoc,lem,-con.assoc], apply ap (λx, x ⬝ _),
exact !eq_same_con⁻¹
end
-- step (2), n ≤ m
theorem eq_le_f {n m : ℕ} (a : A n) (b : A m) (H1 : n ≤ succ m) (H2 : n ≤ m)
: eq_le a (f b) H1 ⬝ g b = eq_le a b H2 :=
begin
rewrite [↑eq_le,is_prop.elim H1 (le.step H2),i_fr_step,con_inv,con.assoc,con.assoc],
clear H1,
apply ap (λx, _ ⬝ x),
rewrite [↑fr,↓fr a H2],
rewrite -con.assoc, exact !eq_same_f
end
-- step (2), combined
theorem eq_constructors_comp_right {n m : ℕ} (a : A n) (b : A m) :
eq_constructors a (f b) ⬝ g b = eq_constructors a b :=
begin
apply @lt_ge_by_cases m n,
{ intro H, let H2 := le.trans !le_succ H,
rewrite [eq_constructors_ge a (f b) H,eq_constructors_ge a b H2,eq_gt_f a b H H2]},
{ intro H2, let H := le.trans H2 !le_succ,
rewrite [eq_constructors_le a (f b) H,eq_constructors_le a b H2,eq_le_f a b H H2]}
end
-- induction on b
definition my_f_eq1 [reducible] (b : my_tr) {n : ℕ} (a : A n) : i a = b :=
begin
induction b with m b,
{ apply eq_constructors},
{ apply (equiv.to_inv !pathover_eq_equiv_r), apply eq_constructors_comp_right},
end
-- induction on a
theorem my_f_eq2 (a : my_tr) : Πb, a = b :=
begin
induction a,
{ intro b, apply my_f_eq1},
{ apply is_prop.elimo}
end
-- final result
theorem is_prop_my_tr : is_prop my_tr := is_prop.mk my_f_eq2
end
definition my_trunc.{u} (A : Type.{u}) : Type.{u} := @my_tr A
definition tr {A : Type} : A → my_trunc A := @i A 0
definition is_prop_my_trunc (A : Type) : is_prop (my_trunc A) := is_prop_my_tr
definition my_trunc.rec {A : Type} {P : my_trunc A → Type}
[Pt : Π(x : my_trunc A), is_prop (P x)]
(H : Π(a : A), P (tr a)) : Π(x : my_trunc A), P x := @rec A P Pt H
example {A : Type} {P : my_trunc A → Type} [Pt : Πaa, is_prop (P aa)]
(H : Πa, P (tr a)) (a : A) : (my_trunc.rec H) (tr a) = H a := by reflexivity
|
5c8911ce4c4489ebda84691726d3daecabb7ce0a | 7b89826c26634aa18c0110f1634f73027851edfe | /natural-number-game/src/world01/level04.lean | 4d5c5f8f8eaeb6b23603988b1032cfdfb2e420e1 | [
"MIT"
] | permissive | marcofavorito/leanings | b7642344d8c9012a1cec74a804c5884297880c4d | 581b83be66ff4f8dd946fb6a1bb045d2ddf91076 | refs/heads/master | 1,672,310,991,244 | 1,603,031,766,000 | 1,603,031,766,000 | 279,163,004 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 261 | lean | import mynat.add
namespace mynat
/-
add_zero (a : mynat) := a + 0 = a
add_succ (a b : mynat) := a + succ(b) = succ(a + b)
-/
lemma add_succ_zero (a : mynat) : a + succ(0) = succ(a) :=
begin [nat_num_game]
rw add_succ,
rw add_zero,
refl,
end
end mynat
|
ce552c1eb1c07940231897b044acae26e32d86a0 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/order/copy_auto.lean | 12d7f5a8d08b4bf59a173baef7a32764d1248c1e | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 4,273 | lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.order.conditionally_complete_lattice
import Mathlib.PostPort
universes u
namespace Mathlib
/-!
# Tooling to make copies of lattice structures
Sometimes it is useful to make a copy of a lattice structure
where one replaces the data parts with provably equal definitions
that have better definitional properties.
-/
/-- A function to create a provable equal copy of a bounded lattice
with possibly different definitional equalities. -/
def bounded_lattice.copy {α : Type u} (c : bounded_lattice α) (le : α → α → Prop)
(eq_le : le = bounded_lattice.le) (top : α) (eq_top : top = bounded_lattice.top) (bot : α)
(eq_bot : bot = bounded_lattice.bot) (sup : α → α → α) (eq_sup : sup = bounded_lattice.sup)
(inf : α → α → α) (eq_inf : inf = bounded_lattice.inf) : bounded_lattice α :=
bounded_lattice.mk sup le (lattice.lt._default le) sorry sorry sorry sorry sorry sorry inf sorry
sorry sorry top sorry bot sorry
/-- A function to create a provable equal copy of a distributive lattice
with possibly different definitional equalities. -/
def distrib_lattice.copy {α : Type u} (c : distrib_lattice α) (le : α → α → Prop)
(eq_le : le = distrib_lattice.le) (sup : α → α → α) (eq_sup : sup = distrib_lattice.sup)
(inf : α → α → α) (eq_inf : inf = distrib_lattice.inf) : distrib_lattice α :=
distrib_lattice.mk sup le (lattice.lt._default le) sorry sorry sorry sorry sorry sorry inf sorry
sorry sorry sorry
/-- A function to create a provable equal copy of a complete lattice
with possibly different definitional equalities. -/
def complete_lattice.copy {α : Type u} (c : complete_lattice α) (le : α → α → Prop)
(eq_le : le = complete_lattice.le) (top : α) (eq_top : top = complete_lattice.top) (bot : α)
(eq_bot : bot = complete_lattice.bot) (sup : α → α → α) (eq_sup : sup = complete_lattice.sup)
(inf : α → α → α) (eq_inf : inf = complete_lattice.inf) (Sup : set α → α)
(eq_Sup : Sup = complete_lattice.Sup) (Inf : set α → α) (eq_Inf : Inf = complete_lattice.Inf) :
complete_lattice α :=
complete_lattice.mk sup le bounded_lattice.lt sorry sorry sorry sorry sorry sorry inf sorry sorry
sorry top sorry bot sorry Sup Inf sorry sorry sorry sorry
/-- A function to create a provable equal copy of a complete distributive lattice
with possibly different definitional equalities. -/
def complete_distrib_lattice.copy {α : Type u} (c : complete_distrib_lattice α) (le : α → α → Prop)
(eq_le : le = complete_distrib_lattice.le) (top : α)
(eq_top : top = complete_distrib_lattice.top) (bot : α)
(eq_bot : bot = complete_distrib_lattice.bot) (sup : α → α → α)
(eq_sup : sup = complete_distrib_lattice.sup) (inf : α → α → α)
(eq_inf : inf = complete_distrib_lattice.inf) (Sup : set α → α)
(eq_Sup : Sup = complete_distrib_lattice.Sup) (Inf : set α → α)
(eq_Inf : Inf = complete_distrib_lattice.Inf) : complete_distrib_lattice α :=
complete_distrib_lattice.mk sup le complete_lattice.lt sorry sorry sorry sorry sorry sorry inf
sorry sorry sorry top sorry bot sorry Sup Inf sorry sorry sorry sorry sorry sorry
/-- A function to create a provable equal copy of a conditionally complete lattice
with possibly different definitional equalities. -/
def conditionally_complete_lattice.copy {α : Type u} (c : conditionally_complete_lattice α)
(le : α → α → Prop) (eq_le : le = conditionally_complete_lattice.le) (sup : α → α → α)
(eq_sup : sup = conditionally_complete_lattice.sup) (inf : α → α → α)
(eq_inf : inf = conditionally_complete_lattice.inf) (Sup : set α → α)
(eq_Sup : Sup = conditionally_complete_lattice.Sup) (Inf : set α → α)
(eq_Inf : Inf = conditionally_complete_lattice.Inf) : conditionally_complete_lattice α :=
conditionally_complete_lattice.mk sup le (lattice.lt._default le) sorry sorry sorry sorry sorry
sorry inf sorry sorry sorry Sup Inf sorry sorry sorry sorry
end Mathlib |
e23c4789421a25e5369e0ea6992407a27702472a | 618003631150032a5676f229d13a079ac875ff77 | /src/data/list/func.lean | a3b5b094b3c6a673979fa6d928f25e7ac934ec07 | [
"Apache-2.0"
] | permissive | awainverse/mathlib | 939b68c8486df66cfda64d327ad3d9165248c777 | ea76bd8f3ca0a8bf0a166a06a475b10663dec44a | refs/heads/master | 1,659,592,962,036 | 1,590,987,592,000 | 1,590,987,592,000 | 268,436,019 | 1 | 0 | Apache-2.0 | 1,590,990,500,000 | 1,590,990,500,000 | null | UTF-8 | Lean | false | false | 10,348 | lean | /-
Copyright (c) 2019 Seul Baek. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Seul Baek
-/
import data.nat.basic
open list
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w}
namespace list
namespace func
variables {a : α}
variables {as as1 as2 as3 : list α}
/- Definitions for using lists as finite
representations of functions with domain ℕ. -/
def neg [has_neg α] (as : list α) := as.map (λ a, -a)
variables [inhabited α] [inhabited β]
@[simp] def set (a : α) : list α → ℕ → list α
| (_::as) 0 := a::as
| [] 0 := [a]
| (h::as) (k+1) := h::(set as k)
| [] (k+1) := (default α)::(set ([] : list α) k)
localized "notation as ` {` m ` ↦ ` a `}` := list.func.set a as m" in list.func
@[simp] def get : ℕ → list α → α
| _ [] := default α
| 0 (a::as) := a
| (n+1) (a::as) := get n as
def equiv (as1 as2 : list α) : Prop :=
∀ (m : nat), get m as1 = get m as2
@[simp] def pointwise (f : α → β → γ) : list α → list β → list γ
| [] [] := []
| [] (b::bs) := map (f $ default α) (b::bs)
| (a::as) [] := map (λ x, f x $ default β) (a::as)
| (a::as) (b::bs) := (f a b)::(pointwise as bs)
def add {α : Type u} [has_zero α] [has_add α] : list α → list α → list α :=
@pointwise α α α ⟨0⟩ ⟨0⟩ (+)
def sub {α : Type u} [has_zero α] [has_sub α] : list α → list α → list α :=
@pointwise α α α ⟨0⟩ ⟨0⟩ (@has_sub.sub α _)
/- set -/
lemma length_set : ∀ {m : ℕ} {as : list α},
(as {m ↦ a}).length = _root_.max as.length (m+1)
| 0 [] := rfl
| 0 (a::as) := by {rw max_eq_left, refl, simp [nat.le_add_right]}
| (m+1) [] := by simp only [set, nat.zero_max, length, @length_set m]
| (m+1) (a::as) := by simp only [set, nat.max_succ_succ, length, @length_set m]
@[simp] lemma get_nil {k : ℕ} : get k [] = default α :=
by {cases k; refl}
lemma get_eq_default_of_le :
∀ (k : ℕ) {as : list α}, as.length ≤ k → get k as = default α
| 0 [] h1 := rfl
| 0 (a::as) h1 := by cases h1
| (k+1) [] h1 := rfl
| (k+1) (a::as) h1 :=
begin
apply get_eq_default_of_le k,
rw ← nat.succ_le_succ_iff, apply h1,
end
@[simp] lemma get_set {a : α} :
∀ {k : ℕ} {as : list α}, get k (as {k ↦ a}) = a
| 0 as := by {cases as; refl, }
| (k+1) as := by {cases as; simp [get_set]}
lemma eq_get_of_mem {a : α} :
∀ {as : list α}, a ∈ as → ∃ n : nat, ∀ d : α, a = (get n as)
| [] h := by cases h
| (b::as) h :=
begin
rw mem_cons_iff at h, cases h,
{ existsi 0, intro d, apply h },
{ cases eq_get_of_mem h with n h2,
existsi (n+1), apply h2 }
end
lemma mem_get_of_le :
∀ {n : ℕ} {as : list α}, n < as.length → get n as ∈ as
| _ [] h1 := by cases h1
| 0 (a::as) _ := or.inl rfl
| (n+1) (a::as) h1 :=
begin
apply or.inr, unfold get,
apply mem_get_of_le,
apply nat.lt_of_succ_lt_succ h1,
end
lemma mem_get_of_ne_zero :
∀ {n : ℕ} {as : list α},
get n as ≠ default α → get n as ∈ as
| _ [] h1 := begin exfalso, apply h1, rw get_nil end
| 0 (a::as) h1 := or.inl rfl
| (n+1) (a::as) h1 :=
begin
unfold get,
apply (or.inr (mem_get_of_ne_zero _)),
apply h1
end
lemma get_set_eq_of_ne {a : α} :
∀ {as : list α} (k : ℕ) (m : ℕ),
m ≠ k → get m (as {k ↦ a}) = get m as
| as 0 m h1 :=
by { cases m, contradiction, cases as;
simp only [set, get, get_nil] }
| as (k+1) m h1 :=
begin
cases as; cases m,
simp only [set, get],
{ have h3 : get m (nil {k ↦ a}) = default α,
{ rw [get_set_eq_of_ne k m, get_nil],
intro hc, apply h1, simp [hc] },
apply h3 },
simp only [set, get],
{ apply get_set_eq_of_ne k m,
intro hc, apply h1, simp [hc], }
end
lemma get_map {f : α → β} : ∀ {n : ℕ} {as : list α}, n < as.length →
get n (as.map f) = f (get n as)
| _ [] h := by cases h
| 0 (a::as) h := rfl
| (n+1) (a::as) h1 :=
begin
have h2 : n < length as,
{ rw [← nat.succ_le_iff, ← nat.lt_succ_iff],
apply h1 },
apply get_map h2,
end
lemma get_map' {f : α → β} {n : ℕ} {as : list α} :
f (default α) = (default β) →
get n (as.map f) = f (get n as) :=
begin
intro h1, by_cases h2 : n < as.length,
{ apply get_map h2, },
{ rw not_lt at h2,
rw [get_eq_default_of_le _ h2, get_eq_default_of_le, h1],
rw [length_map], apply h2 }
end
lemma forall_val_of_forall_mem {as : list α} {p : α → Prop} :
p (default α) → (∀ x ∈ as, p x) → (∀ n, p (get n as)) :=
begin
intros h1 h2 n,
by_cases h3 : n < as.length,
{ apply h2 _ (mem_get_of_le h3) },
{ rw not_lt at h3,
rw get_eq_default_of_le _ h3, apply h1 }
end
/- equiv -/
lemma equiv_refl : equiv as as := λ k, rfl
lemma equiv_symm : equiv as1 as2 → equiv as2 as1 :=
λ h1 k, (h1 k).symm
lemma equiv_trans :
equiv as1 as2 → equiv as2 as3 → equiv as1 as3 :=
λ h1 h2 k, eq.trans (h1 k) (h2 k)
lemma equiv_of_eq : as1 = as2 → equiv as1 as2 :=
begin intro h1, rw h1, apply equiv_refl end
lemma eq_of_equiv : ∀ {as1 as2 : list α}, as1.length = as2.length →
equiv as1 as2 → as1 = as2
| [] [] h1 h2 := rfl
| (_::_) [] h1 h2 := by cases h1
| [] (_::_) h1 h2 := by cases h1
| (a1::as1) (a2::as2) h1 h2 :=
begin
congr,
{ apply h2 0 },
have h3 : as1.length = as2.length,
{ simpa [add_left_inj, add_comm, length] using h1 },
apply eq_of_equiv h3,
intro m, apply h2 (m+1)
end
end func
-- We want to drop the `inhabited` instances for a moment,
-- so we close and open the namespace
namespace func
/- neg -/
@[simp] lemma get_neg [add_group α] {k : ℕ} {as : list α} :
@get α ⟨0⟩ k (neg as) = -(@get α ⟨0⟩ k as) :=
by {unfold neg, rw (@get_map' α α ⟨0⟩), apply neg_zero}
@[simp] lemma length_neg [has_neg α] (as : list α) : (neg as).length = as.length :=
by simp only [neg, length_map]
variables [inhabited α] [inhabited β]
/- pointwise -/
lemma nil_pointwise {f : α → β → γ} : ∀ bs : list β, pointwise f [] bs = bs.map (f $ default α)
| [] := rfl
| (b::bs) :=
by simp only [nil_pointwise bs, pointwise,
eq_self_iff_true, and_self, map]
lemma pointwise_nil {f : α → β → γ} : ∀ as : list α, pointwise f as [] = as.map (λ a, f a $ default β)
| [] := rfl
| (a::as) :=
by simp only [pointwise_nil as, pointwise,
eq_self_iff_true, and_self, list.map]
lemma get_pointwise [inhabited γ] {f : α → β → γ} (h1 : f (default α) (default β) = default γ) :
∀ (k : nat) (as : list α) (bs : list β),
get k (pointwise f as bs) = f (get k as) (get k bs)
| k [] [] := by simp only [h1, get_nil, pointwise, get]
| 0 [] (b::bs) :=
by simp only [get_pointwise, get_nil,
pointwise, get, nat.nat_zero_eq_zero, map]
| (k+1) [] (b::bs) :=
by { have : get k (map (f $ default α) bs) = f (default α) (get k bs),
{ simpa [nil_pointwise, get_nil] using (get_pointwise k [] bs) },
simpa [get, get_nil, pointwise, map] }
| 0 (a::as) [] :=
by simp only [get_pointwise, get_nil,
pointwise, get, nat.nat_zero_eq_zero, map]
| (k+1) (a::as) [] :=
by simpa [get, get_nil, pointwise, map, pointwise_nil, get_nil]
using get_pointwise k as []
| 0 (a::as) (b::bs) := by simp only [pointwise, get]
| (k+1) (a::as) (b::bs) :=
by simp only [pointwise, get, get_pointwise k]
lemma length_pointwise {f : α → β → γ} :
∀ {as : list α} {bs : list β},
(pointwise f as bs).length = _root_.max as.length bs.length
| [] [] := rfl
| [] (b::bs) :=
by simp only [pointwise, length, length_map,
max_eq_right (nat.zero_le (length bs + 1))]
| (a::as) [] :=
by simp only [pointwise, length, length_map,
max_eq_left (nat.zero_le (length as + 1))]
| (a::as) (b::bs) :=
by simp only [pointwise, length,
nat.max_succ_succ, @length_pointwise as bs]
end func
namespace func
/- add -/
@[simp] lemma get_add {α : Type u} [add_monoid α] {k : ℕ} {xs ys : list α} :
@get α ⟨0⟩ k (add xs ys) = ( @get α ⟨0⟩ k xs + @get α ⟨0⟩ k ys) :=
by {apply get_pointwise, apply zero_add}
@[simp] lemma length_add {α : Type u}
[has_zero α] [has_add α] {xs ys : list α} :
(add xs ys).length = _root_.max xs.length ys.length :=
@length_pointwise α α α ⟨0⟩ ⟨0⟩ _ _ _
@[simp] lemma nil_add {α : Type u} [add_monoid α]
(as : list α) : add [] as = as :=
begin
rw [add, @nil_pointwise α α α ⟨0⟩ ⟨0⟩],
apply eq.trans _ (map_id as),
congr, ext,
have : @default α ⟨0⟩ = 0 := rfl,
rw [this, zero_add], refl
end
@[simp] lemma add_nil {α : Type u} [add_monoid α]
(as : list α) : add as [] = as :=
begin
rw [add, @pointwise_nil α α α ⟨0⟩ ⟨0⟩],
apply eq.trans _ (map_id as),
congr, ext,
have : @default α ⟨0⟩ = 0 := rfl,
rw [this, add_zero], refl
end
lemma map_add_map {α : Type u} [add_monoid α] (f g : α → α) {as : list α} :
add (as.map f) (as.map g) = as.map (λ x, f x + g x) :=
begin
apply @eq_of_equiv _ (⟨0⟩ : inhabited α),
{ rw [length_map, length_add, max_eq_left, length_map],
apply le_of_eq,
rw [length_map, length_map] },
intros m,
rw [get_add],
by_cases h : m < length as,
{ repeat {rw [@get_map α α ⟨0⟩ ⟨0⟩ _ _ _ h]} },
rw not_lt at h,
repeat {rw [get_eq_default_of_le m]};
try {rw length_map, apply h},
apply zero_add
end
/- sub -/
@[simp] lemma get_sub {α : Type u}
[add_group α] {k : ℕ} {xs ys : list α} :
@get α ⟨0⟩ k (sub xs ys) = (@get α ⟨0⟩ k xs - @get α ⟨0⟩ k ys) :=
by {apply get_pointwise, apply sub_zero}
@[simp] lemma length_sub [has_zero α] [has_sub α] {xs ys : list α} :
(sub xs ys).length = _root_.max xs.length ys.length :=
@length_pointwise α α α ⟨0⟩ ⟨0⟩ _ _ _
@[simp] lemma nil_sub {α : Type} [add_group α]
(as : list α) : sub [] as = neg as :=
begin
rw [sub, nil_pointwise],
congr, ext,
have : @default α ⟨0⟩ = 0 := rfl,
rw [this, zero_sub]
end
@[simp] lemma sub_nil {α : Type} [add_group α]
(as : list α) : sub as [] = as :=
begin
rw [sub, pointwise_nil],
apply eq.trans _ (map_id as),
congr, ext,
have : @default α ⟨0⟩ = 0 := rfl,
rw [this, sub_zero], refl
end
end func
end list
|
da9e82a27ac6562e5f126c031bb84f55cabbb3e9 | 64874bd1010548c7f5a6e3e8902efa63baaff785 | /tests/lean/run/uuu.lean | ccff603e307cc9777da4174ffbad4d3780cf538c | [
"Apache-2.0"
] | permissive | tjiaqi/lean | 4634d729795c164664d10d093f3545287c76628f | d0ce4cf62f4246b0600c07e074d86e51f2195e30 | refs/heads/master | 1,622,323,796,480 | 1,422,643,069,000 | 1,422,643,069,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 5,407 | lean | prelude
-- Porting Vladimir's file to Lean
notation `assume` binders `,` r:(scoped f, f) := r
notation `take` binders `,` r:(scoped f, f) := r
inductive empty : Type
inductive unit : Type :=
tt : unit
definition tt := @unit.tt
inductive nat : Type :=
O : nat,
S : nat → nat
inductive paths {A : Type} (a : A) : A → Type :=
idpath : paths a a
definition idpath := @paths.idpath
inductive sum (A : Type) (B : Type) : Type :=
inl : A -> sum A B,
inr : B -> sum A B
definition coprod := sum
definition ii1fun {A : Type} (B : Type) (a : A) := sum.inl B a
definition ii2fun (A : Type) {B : Type} (b : B) := sum.inr A b
definition ii1 {A : Type} {B : Type} (a : A) := sum.inl B a
definition ii2 {A : Type} {B : Type} (b : B) := sum.inl A b
inductive total2 {T: Type} (P: T → Type) : Type :=
tpair : Π (t : T) (tp : P t), total2 P
definition tpair := @total2.tpair
definition pr1 {T : Type} {P : T → Type} (tp : total2 P) : T
:= total2.rec (λ a b, a) tp
definition pr2 {T : Type} {P : T → Type} (tp : total2 P) : P (pr1 tp)
:= total2.rec (λ a b, b) tp
inductive Phant (T : Type) : Type :=
phant : Phant T
definition fromempty {X : Type} : empty → X
:= λe, empty.rec (λe, X) e
definition tounit {X : Type} : X → unit
:= λx, tt
definition termfun {X : Type} (x : X) : unit → X
:= λt, x
definition idfun (T : Type) := λt : T, t
definition funcomp {X : Type} {Y : Type} {Z : Type} (f : X → Y) (g : Y → Z)
:= λx, g (f x)
infixl `∘`:60 := funcomp
definition iteration {T : Type} (f : T → T) (n : nat) : T → T
:= nat.rec (idfun T) (λ m fm, funcomp fm f) n
definition adjev {X : Type} {Y : Type} (x : X) (f : X → Y) := f x
definition adjev2 {X : Type} {Y : Type} (phi : ((X → Y) → Y ) → Y ) : X → Y
:= λx, phi (λf, f x)
definition dirprod (X : Type) (Y : Type) := total2 (λ x : X, Y)
definition dirprodpair {X : Type} {Y : Type} := @tpair _ (λ x : X, Y)
definition dirprodadj {X : Type} {Y : Type} {Z : Type} (f : dirprod X Y → Z ) : X → Y → Z
:= λx y, f (dirprodpair x y)
definition dirprodf {X : Type} {Y : Type} {X' : Type} {Y' : Type} (f : X → Y) (f' : X' → Y') (xx' : dirprod X X') : dirprod Y Y'
:= dirprodpair (f (pr1 xx')) (f' (pr2 xx'))
definition ddualand {X : Type} {Y : Type} {P : Type} (xp : (X → P) → P) (yp : (Y → P) → P) : (dirprod X Y → P) → P
:= λ X0,
let int1 := λ (ypp : (Y → P) → P) (x : X), yp (λ y : Y, X0 (dirprodpair x y)) in
xp (int1 yp)
definition neg (X : Type) : Type := X → empty
definition negf {X : Type} {Y : Type} (f : X → Y) : neg Y → neg X
:= λ (phi : Y → empty) (x : X), phi (f x)
definition dneg (X : Type) : Type := (X → empty) → empty
definition dnegf {X : Type} {Y : Type} (f : X → Y) : dneg X → dneg Y
:= negf (negf f)
definition todneg (X : Type) : X → dneg X
:= adjev
definition dnegnegtoneg {X : Type} : dneg (neg X) → neg X
:= adjev2
lemma dneganddnegl1 {X : Type} {Y : Type} (dnx : dneg X) (dny : dneg Y) : neg (X → neg Y)
:= take X2 : X → neg Y,
have X3 : dneg X → neg Y, from
take xx : dneg X, dnegnegtoneg (dnegf X2 xx),
dny (X3 dnx)
definition logeq (X : Type) (Y : Type) := dirprod (X → Y) (Y → X)
infix `<->`:25 := logeq
infix `↔`:25 := logeq
definition logeqnegs {X : Type} {Y : Type} (l : X ↔ Y) : (neg X) ↔ (neg Y)
:= dirprodpair (negf (pr2 l)) (negf (pr1 l))
infix `=`:50 := paths
definition pathscomp0 {X : Type} {a b c : X} (e1 : a = b) (e2 : b = c) : a = c
:= paths.rec e1 e2
definition pathscomp0rid {X : Type} {a b : X} (e1 : a = b) : pathscomp0 e1 (idpath b) = e1
:= idpath _
definition pathsinv0 {X : Type} {a b : X} (e : a = b) : b = a
:= paths.rec (idpath _) e
definition transport {A : Type} {a b : A} {P : A → Type} (H1 : a = b) (H2 : P a) : P b
:= paths.rec H2 H1
infixr `▸`:75 := transport
infixr `⬝`:75 := pathscomp0
postfix `⁻¹`:100 := pathsinv0
definition idinv {X : Type} (x : X) : (idpath x)⁻¹ = idpath x
:= idpath (idpath x)
definition idtrans {A : Type} (x : A) : (idpath x) ⬝ (idpath x) = (idpath x)
:= idpath (idpath x)
definition pathsinv0l {X : Type} {a b : X} (e : a = b) : e⁻¹ ⬝ e = idpath b
:= paths.rec (idinv a⁻¹ ▸ idtrans a) e
definition pathsinv0r {A : Type} {x y : A} (p : x = y) : p⁻¹ ⬝ p = idpath y
:= paths.rec (idinv x⁻¹ ▸ idtrans x) p
definition pathsinv0inv0 {A : Type} {x y : A} (p : x = y) : (p⁻¹)⁻¹ = p
:= paths.rec (idpath (idpath x)) p
definition pathsdirprod {X : Type} {Y : Type} {x1 x2 : X} {y1 y2 : Y} (ex : x1 = x2) (ey : y1 = y2 ) : dirprodpair x1 y1 = dirprodpair x2 y2
:= ex ▸ ey ▸ idpath (dirprodpair x1 y1)
definition maponpaths {T1 : Type} {T2 : Type} (f : T1 → T2) {t1 t2 : T1} (e : t1 = t2) : f t1 = f t2
:= e ▸ idpath (f t1)
definition ap {T1 : Type} {T2 : Type} := @maponpaths T1 T2
definition maponpathscomp0 {X : Type} {Y : Type} {x y z : X} (f : X → Y) (p : x = y) (q : y = z) : ap f (p ⬝ q) = (ap f p) ⬝ (ap f q)
:= paths.rec (idpath _) q
definition maponpathsinv0 {X : Type} {Y : Type} (f : X → Y) {x1 x2 : X} (e : x1 = x2 ) : ap f (e⁻¹) = (ap f e)⁻¹
:= paths.rec (idpath _) e
lemma maponpathsidfun {X : Type} {x x' : X} (e : x = x') : ap (idfun X) e = e
:= paths.rec (idpath _) e
lemma maponpathscomp {X : Type} {Y : Type} {Z : Type} {x x' : X} (f : X → Y) (g : Y → Z) (e : x = x') : ap g (ap f e) = ap (f ∘ g) e
:= paths.rec (idpath _) e
|
ab730e3b92659bec9c0f81e07bc06a88501c6cca | 9cba98daa30c0804090f963f9024147a50292fa0 | /old/src/physlib.lean | 11aee4c718f2ecd587bd26d8fb0dc52279f2cb46 | [] | no_license | kevinsullivan/phys | dcb192f7b3033797541b980f0b4a7e75d84cea1a | ebc2df3779d3605ff7a9b47eeda25c2a551e011f | refs/heads/master | 1,637,490,575,500 | 1,629,899,064,000 | 1,629,899,064,000 | 168,012,884 | 0 | 3 | null | 1,629,644,436,000 | 1,548,699,832,000 | Lean | UTF-8 | Lean | false | false | 76 | lean | import .classical_geometry
import .classical_time
import .classical_velocity |
a01a1d2b20d509723bfbf9c87569c80b7c96023b | fbf512ee44de430e3d3c5869751ad95325c938d7 | /simp_fol.lean | 00c48037ae36c65756f5c0f38c96757e5f0bedbc | [] | no_license | skbaek/clausify | 4858c9005fb86a5e410bfcaa77524f82d955f655 | d09b071bdcce7577c3fffacd0893b776285b1590 | refs/heads/master | 1,588,360,590,818 | 1,553,553,880,000 | 1,553,553,880,000 | 177,644,542 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 69 | lean | run_cmd mk_simp_attr `fol
meta def simp_fol := `[simp only with fol] |
8f9a8c76fc1ef2dcadf6f6a0f5f61a7e5f995065 | c3de33d4701e6113627153fe1103b255e752ed7d | /topology/topological_space.lean | 45eafa6dae8816a831d08ae67c03b0ee24b1090d | [] | no_license | jroesch/library_dev | 77d2b246ff47ab05d55cb9706a37d3de97038388 | 4faa0a45c6aa7eee6e661113c2072b8840bff79b | refs/heads/master | 1,611,281,606,352 | 1,495,661,644,000 | 1,495,661,644,000 | 92,340,430 | 0 | 0 | null | 1,495,663,344,000 | 1,495,663,344,000 | null | UTF-8 | Lean | false | false | 25,789 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
Theory of topological spaces.
-/
import algebra.lattice.filter
open set filter lattice
universes u v w
structure topological_space (α : Type u) :=
(open' : set α → Prop)
(open_univ : open' univ)
(open_inter : ∀s t, open' s → open' t → open' (s ∩ t))
(open_sUnion : ∀s, (∀t∈s, open' t) → open' (⋃₀ s))
attribute [class] topological_space
section topological_space
variables {α : Type u} {β : Type v} {ι : Sort w} {a a₁ a₂ : α} {s s₁ s₂ : set α}
lemma topological_space_eq {f g : topological_space α} (h' : f^.open' = g^.open') : f = g :=
begin
cases f with a, cases g with b,
assert h : a = b, assumption,
clear h',
subst h
end
section
variables [t : topological_space α]
include t
/- open -/
def open' (s : set α) : Prop := topological_space.open' t s
@[simp]
lemma open_univ : open' (univ : set α) := topological_space.open_univ t
lemma open_inter (h₁ : open' s₁) (h₂ : open' s₂) : open' (s₁ ∩ s₂) := topological_space.open_inter t s₁ s₂ h₁ h₂
lemma open_sUnion {s : set (set α)} (h : ∀t ∈ s, open' t) : open' (⋃₀ s) := topological_space.open_sUnion t s h
end
variables [topological_space α]
lemma open_Union {f : ι → set α} (h : ∀i, open' (f i)) : open' (⋃i, f i) :=
open_sUnion $ take t ⟨i, (heq : t = f i)⟩, heq^.symm ▸ h i
@[simp]
lemma open_empty : open' (∅ : set α) :=
have open' (⋃₀ ∅ : set α), from open_sUnion (take a, false.elim),
by simp at this; assumption
/- closed -/
def closed (s : set α) : Prop := open' (-s)
@[simp]
lemma closed_empty : closed (∅ : set α) := by simp [closed]
@[simp]
lemma closed_univ : closed (univ : set α) := by simp [closed]
lemma closed_union : closed s₁ → closed s₂ → closed (s₁ ∪ s₂) :=
by simp [closed]; exact open_inter
lemma closed_sInter {s : set (set α)} : (∀t ∈ s, closed t) → closed (⋂₀ s) :=
by simp [closed, compl_sInter]; exact take h, open_Union $ take t, open_Union $ take ht, h t ht
lemma closed_Inter {f : ι → set α} (h : ∀i, closed (f i)) : closed (⋂i, f i ) :=
closed_sInter $ take t ⟨i, (heq : t = f i)⟩, heq^.symm ▸ h i
@[simp]
lemma closed_compl_iff_open {s : set α} : open' (-s) ↔ closed s :=
by refl
@[simp]
lemma open_compl_iff_closed {s : set α} : closed (-s) ↔ open' s :=
by rw [-closed_compl_iff_open, compl_compl]
lemma open_diff {s t : set α} (h₁ : open' s) (h₂ : closed t) : open' (s - t) :=
open_inter h₁ $ closed_compl_iff_open^.mpr h₂
/- interior -/
def interior (s : set α) : set α := ⋃₀ {t | open' t ∧ t ⊆ s}
@[simp]
lemma open_interior {s : set α} : open' (interior s) :=
open_sUnion $ take t ⟨h₁, h₂⟩, h₁
lemma interior_subset {s : set α} : interior s ⊆ s :=
sUnion_subset $ take t ⟨h₁, h₂⟩, h₂
lemma interior_maximal {s t : set α} (h₁ : t ⊆ s) (h₂ : open' t) : t ⊆ interior s :=
subset_sUnion_of_mem ⟨h₂, h₁⟩
lemma interior_eq_of_open {s : set α} (h : open' s) : interior s = s :=
subset.antisymm interior_subset (interior_maximal (subset.refl s) h)
lemma interior_eq_iff_open {s : set α} : interior s = s ↔ open' s :=
⟨take h, h ▸ open_interior, interior_eq_of_open⟩
lemma subset_interior_iff_subset_of_open {s t : set α} (h₁ : open' s) :
s ⊆ interior t ↔ s ⊆ t :=
⟨take h, subset.trans h interior_subset, take h₂, interior_maximal h₂ h₁⟩
lemma interior_mono {s t : set α} (h : s ⊆ t) : interior s ⊆ interior t :=
interior_maximal (subset.trans interior_subset h) open_interior
@[simp]
lemma interior_empty : interior (∅ : set α) = ∅ :=
interior_eq_of_open open_empty
@[simp]
lemma interior_univ : interior (univ : set α) = univ :=
interior_eq_of_open open_univ
@[simp]
lemma interior_interior {s : set α} : interior (interior s) = interior s :=
interior_eq_of_open open_interior
@[simp]
lemma interior_inter {s t : set α} : interior (s ∩ t) = interior s ∩ interior t :=
subset.antisymm
(subset_inter (interior_mono $ inter_subset_left s t) (interior_mono $ inter_subset_right s t))
(interior_maximal (inter_subset_inter interior_subset interior_subset) $ by simp [open_inter])
lemma interior_union_closed_of_interior_empty {s t : set α} (h₁ : closed s) (h₂ : interior t = ∅) :
interior (s ∪ t) = interior s :=
have interior (s ∪ t) ⊆ s, from
take x ⟨u, ⟨(hu₁ : open' u), (hu₂ : u ⊆ s ∪ t)⟩, (hx₁ : x ∈ u)⟩,
classical.by_contradiction $ assume hx₂ : x ∉ s,
have u - s ⊆ t,
from take x ⟨h₁, h₂⟩, or.resolve_left (hu₂ h₁) h₂,
have u - s ⊆ interior t,
by simp [subset_interior_iff_subset_of_open, this, open_diff hu₁ h₁],
have u - s ⊆ ∅,
by rw [h₂] at this; assumption,
this ⟨hx₁, hx₂⟩,
subset.antisymm
(interior_maximal this open_interior)
(interior_mono $ subset_union_left _ _)
/- closure -/
def closure (s : set α) : set α := ⋂₀ {t | closed t ∧ s ⊆ t}
@[simp]
lemma closed_closure {s : set α} : closed (closure s) :=
closed_sInter $ take t ⟨h₁, h₂⟩, h₁
lemma subset_closure {s : set α} : s ⊆ closure s :=
subset_sInter $ take t ⟨h₁, h₂⟩, h₂
lemma closure_minimal {s t : set α} (h₁ : s ⊆ t) (h₂ : closed t) : closure s ⊆ t :=
sInter_subset_of_mem ⟨h₂, h₁⟩
lemma closure_eq_of_closed {s : set α} (h : closed s) : closure s = s :=
subset.antisymm (closure_minimal (subset.refl s) h) subset_closure
lemma closure_eq_iff_closed {s : set α} : closure s = s ↔ closed s :=
⟨take h, h ▸ closed_closure, closure_eq_of_closed⟩
lemma closure_subset_iff_subset_of_closed {s t : set α} (h₁ : closed t) :
closure s ⊆ t ↔ s ⊆ t :=
⟨subset.trans subset_closure, take h, closure_minimal h h₁⟩
lemma closure_mono {s t : set α} (h : s ⊆ t) : closure s ⊆ closure t :=
closure_minimal (subset.trans h subset_closure) closed_closure
@[simp]
lemma closure_empty : closure (∅ : set α) = ∅ :=
closure_eq_of_closed closed_empty
@[simp]
lemma closure_univ : closure (univ : set α) = univ :=
closure_eq_of_closed closed_univ
@[simp]
lemma closure_closure {s : set α} : closure (closure s) = closure s :=
closure_eq_of_closed closed_closure
@[simp]
lemma closure_union {s t : set α} : closure (s ∪ t) = closure s ∪ closure t :=
subset.antisymm
(closure_minimal (union_subset_union subset_closure subset_closure) $ by simp [closed_union])
(union_subset (closure_mono $ subset_union_left _ _) (closure_mono $ subset_union_right _ _))
lemma interior_subset_closure {s : set α} : interior s ⊆ closure s :=
subset.trans interior_subset subset_closure
lemma closure_eq_compl_interior_compl {s : set α} : closure s = - interior (- s) :=
begin
simp [interior, closure],
rw [compl_sUnion, compl_image_set_of],
simp [neg_subset_neg_iff_subset]
end
@[simp]
lemma interior_compl_eq {s : set α} : interior (- s) = - closure s :=
by simp [closure_eq_compl_interior_compl]
@[simp]
lemma closure_compl_eq {s : set α} : closure (- s) = - interior s :=
by simp [closure_eq_compl_interior_compl]
/- neighbourhood filter -/
def nhds (a : α) : filter α := (⨅ s ∈ {s : set α | a ∈ s ∧ open' s}, principal s)
lemma nhds_sets {a : α} : (nhds a)^.sets = {s | ∃t⊆s, open' t ∧ a ∈ t} :=
calc (nhds a)^.sets = (⋃s∈{s : set α| a ∈ s ∧ open' s}, (principal s)^.sets) : infi_sets_eq'
begin
simp,
exact take x ⟨hx₁, hx₂⟩ y ⟨hy₁, hy₂⟩, ⟨_, ⟨open_inter hx₁ hy₁, ⟨hx₂, hy₂⟩⟩,
⟨inter_subset_left _ _, inter_subset_right _ _⟩⟩
end
⟨univ, by simp⟩
... = {s | ∃t⊆s, open' t ∧ a ∈ t} :
le_antisymm
(supr_le $ take i, supr_le $ take ⟨hi₁, hi₂⟩ t ht, ⟨i, ht, hi₂, hi₁⟩)
(take t ⟨i, hi₁, hi₂, hi₃⟩, begin simp; exact ⟨i, hi₂, hi₁, hi₃⟩ end)
lemma map_nhds {a : α} {f : α → β} :
map f (nhds a) = (⨅ s ∈ {s : set α | a ∈ s ∧ open' s}, principal (image f s)) :=
calc map f (nhds a) = (⨅ s ∈ {s : set α | a ∈ s ∧ open' s}, map f (principal s)) :
map_binfi_eq
begin
simp,
exact take x ⟨hx₁, hx₂⟩ y ⟨hy₁, hy₂⟩, ⟨_, ⟨open_inter hx₁ hy₁, ⟨hx₂, hy₂⟩⟩,
⟨inter_subset_left _ _, inter_subset_right _ _⟩⟩
end
⟨univ, by simp⟩
... = _ : by simp
lemma mem_nhds_sets_iff {a : α} {s : set α} :
s ∈ (nhds a)^.sets ↔ ∃t⊆s, open' t ∧ a ∈ t :=
by simp [nhds_sets]
lemma mem_nhds_sets {a : α} {s : set α} (hs : open' s) (ha : a ∈ s) :
s ∈ (nhds a)^.sets :=
by simp [nhds_sets]; exact ⟨s, hs, subset.refl _, ha⟩
lemma return_le_nhds : return ≤ (nhds : α → filter α) :=
take a, le_infi $ take s, le_infi $ take ⟨h₁, _⟩, principal_mono^.mpr $ by simp [h₁]
@[simp]
lemma nhds_neq_bot {a : α} : nhds a ≠ ⊥ :=
suppose nhds a = ⊥,
have return a = (⊥ : filter α),
from lattice.bot_unique $ this ▸ return_le_nhds a,
return_neq_bot this
lemma interior_eq_nhds {s : set α} : interior s = {a | nhds a ≤ principal s} :=
set.ext $ by simp [interior, nhds_sets]
lemma open_iff_nhds {s : set α} : open' s ↔ (∀a∈s, nhds a ≤ principal s) :=
calc open' s ↔ interior s = s : by rw [interior_eq_iff_open]
... ↔ s ⊆ interior s : ⟨take h, by simph [subset.refl], subset.antisymm interior_subset⟩
... ↔ (∀a∈s, nhds a ≤ principal s) : by rw [interior_eq_nhds]; refl
lemma closure_eq_nhds {s : set α} : closure s = {a | nhds a ⊓ principal s ≠ ⊥} :=
calc closure s = - interior (- s) : closure_eq_compl_interior_compl
... = {a | ¬ nhds a ≤ principal (-s)} : by rw [interior_eq_nhds]; refl
... = {a | nhds a ⊓ principal s ≠ ⊥} : set.ext $ take a, not_congr
(inf_eq_bot_iff_le_compl
(show principal s ⊔ principal (-s) = ⊤, by simp [principal_univ])
(by simp))^.symm
lemma closed_iff_nhds {s : set α} : closed s ↔ (∀a, nhds a ⊓ principal s ≠ ⊥ → a ∈ s) :=
calc closed s ↔ closure s = s : by rw [closure_eq_iff_closed]
... ↔ closure s ⊆ s : ⟨take h, by simph [subset.refl], take h, subset.antisymm h subset_closure⟩
... ↔ (∀a, nhds a ⊓ principal s ≠ ⊥ → a ∈ s) : by rw [closure_eq_nhds]; refl
/- locally finite family [General Topology (Bourbaki, 1995)] -/
section locally_finite
def locally_finite (f : β → set α) :=
∀x:α, ∃t∈(nhds x)^.sets, finite {i | f i ∩ t ≠ ∅ }
theorem not_eq_empty_iff_exists {s : set α} : ¬ (s = ∅) ↔ ∃ x, x ∈ s :=
⟨exists_mem_of_ne_empty,
take ⟨x, (hx : x ∈ s)⟩ h_eq, by rw [h_eq] at hx; assumption⟩
lemma closed_Union_of_locally_finite {f : β → set α}
(h₁ : locally_finite f) (h₂ : ∀i, closed (f i)) : closed (⋃i, f i) :=
open_iff_nhds^.mpr $ take a, assume h : a ∉ (⋃i, f i),
have ∀i, a ∈ -f i,
from take i hi, by simp at h; exact h ⟨i, hi⟩,
have ∀i, - f i ∈ (nhds a).sets,
by rw [nhds_sets]; exact take i, ⟨- f i, subset.refl _, h₂ i, this i⟩,
let ⟨t, h_sets, (h_fin : finite {i | f i ∩ t ≠ ∅ })⟩ := h₁ a in
calc nhds a ≤ principal (t ∩ (⋂ i∈{i | f i ∩ t ≠ ∅ }, - f i)) :
begin
simp,
apply @filter.inter_mem_sets _ (nhds a) _ _ h_sets,
apply @filter.Inter_mem_sets _ _ (nhds a) _ _ h_fin,
exact take i h, this i
end
... ≤ principal (- ⋃i, f i) :
begin
simp,
intro x,
simp [not_eq_empty_iff_exists],
exact take ⟨xt, ht⟩ i xfi, ht i ⟨x, xt, xfi⟩ xfi
end
end locally_finite
section compact
def compact (s : set α) := ∀{f}, f ≠ ⊥ → f ≤ principal s → ∃a∈s, f ⊓ nhds a ≠ ⊥
lemma compact_adherence_nhdset {s t : set α} {f : filter α}
(hs : compact s) (hf₂ : f ≤ principal s) (ht₁ : open' t) (ht₂ : ∀a∈s, nhds a ⊓ f ≠ ⊥ → a ∈ t) :
t ∈ f.sets :=
classical.by_cases mem_sets_of_neq_bot $
suppose f ⊓ principal (- t) ≠ ⊥,
let ⟨a, ha, (hfa : f ⊓ principal (-t) ⊓ nhds a ≠ ⊥)⟩ := hs this $ inf_le_left_of_le hf₂ in
have a ∈ t,
from ht₂ a ha $ neq_bot_of_le_neq_bot hfa $ le_inf inf_le_right $ inf_le_left_of_le inf_le_left,
have nhds a ⊓ principal (-t) ≠ ⊥,
from neq_bot_of_le_neq_bot hfa $ le_inf inf_le_right $ inf_le_left_of_le inf_le_right,
have ∀s∈(nhds a ⊓ principal (-t)).sets, s ≠ ∅,
from forall_sets_neq_empty_iff_neq_bot.mpr this,
have false,
from this _ ⟨t, mem_nhds_sets ht₁ ‹a ∈ t ›, -t, subset.refl _, subset.refl _⟩ (by simp),
by contradiction
lemma compact_iff_ultrafilter_le_nhds {s : set α} :
compact s ↔ (∀f, ultrafilter f → f ≤ principal s → ∃a∈s, f ≤ nhds a) :=
⟨assume hs : compact s, take f hf hfs,
let ⟨a, ha, h⟩ := hs hf.left hfs in
⟨a, ha, le_of_ultrafilter hf h⟩,
assume hs : (∀f, ultrafilter f → f ≤ principal s → ∃a∈s, f ≤ nhds a),
take f hf hfs,
let ⟨a, ha, (h : ultrafilter_of f ≤ nhds a)⟩ :=
hs (ultrafilter_of f) (ultrafilter_ultrafilter_of hf) (le_trans ultrafilter_of_le hfs) in
have ultrafilter_of f ⊓ nhds a ≠ ⊥,
by simp [inf_of_le_left, h]; exact (ultrafilter_ultrafilter_of hf).left,
⟨a, ha, neq_bot_of_le_neq_bot this (inf_le_inf ultrafilter_of_le (le_refl _))⟩⟩
lemma finite_subcover_of_compact {s : set α} {c : set (set α)}
(hs : compact s) (hc₁ : ∀t∈c, open' t) (hc₂ : s ⊆ ⋃₀ c) : ∃c'⊆c, finite c' ∧ s ⊆ ⋃₀ c' :=
classical.by_contradiction $ take h,
have h : ∀{c'}, c' ⊆ c → finite c' → ¬ s ⊆ ⋃₀ c',
from take c' h₁ h₂ h₃, h ⟨c', h₁, h₂, h₃⟩,
let
f : filter α := (⨅c':{c' : set (set α) // c' ⊆ c ∧ finite c'}, principal (s - ⋃₀ c')),
⟨a, ha⟩ := @exists_mem_of_ne_empty α s
(take h', h (empty_subset _) finite.empty $ h'.symm ▸ empty_subset _)
in
have f ≠ ⊥, from infi_neq_bot_of_directed ⟨a⟩
(take ⟨c₁, hc₁, hc'₁⟩ ⟨c₂, hc₂, hc'₂⟩, ⟨⟨c₁ ∪ c₂, union_subset hc₁ hc₂, finite_union hc'₁ hc'₂⟩,
principal_mono.mpr $ diff_right_antimono $ sUnion_mono $ subset_union_left _ _,
principal_mono.mpr $ diff_right_antimono $ sUnion_mono $ subset_union_right _ _⟩)
(take ⟨c', hc'₁, hc'₂⟩, by simp [diff_neq_empty]; exact h hc'₁ hc'₂),
have f ≤ principal s, from infi_le_of_le ⟨∅, empty_subset _, finite.empty⟩ $
show principal (s - ⋃₀∅) ≤ principal s, by simp; exact subset.refl s,
let
⟨a, ha, (h : f ⊓ nhds a ≠ ⊥)⟩ := hs ‹f ≠ ⊥› this,
⟨t, ht₁, (ht₂ : a ∈ t)⟩ := hc₂ ha
in
have f ≤ principal (-t), from infi_le_of_le ⟨{t}, by simp [ht₁], finite_insert finite.empty⟩ $
principal_mono.mpr $ show s - ⋃₀{t} ⊆ - t, begin simp; exact take x ⟨_, hnt⟩, hnt end,
have closed (- t), from closed_compl_iff_open.mp $ by simp; exact hc₁ t ht₁,
have a ∈ - t, from closed_iff_nhds.mp this _ $ neq_bot_of_le_neq_bot h $
le_inf inf_le_right (inf_le_left_of_le $ ‹f ≤ principal (- t)›),
this ‹a ∈ t›
end compact
section separation
class t1_space (α : Type u) [topological_space α] :=
(t1 : ∀x, closed ({x} : set α))
class t2_space (α : Type u) [topological_space α] :=
(t2 : ∀x y, x ≠ y → ∃u v : set α, open' u ∧ open' v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅)
lemma eq_of_nhds_neq_bot [ht : t2_space α] {x y : α} (h : nhds x ⊓ nhds y ≠ ⊥) : x = y :=
classical.by_contradiction $ suppose x ≠ y,
let ⟨u, v, hu, hv, hx, hy, huv⟩ := t2_space.t2 _ x y this in
have h₁ : u ∈ (nhds x ⊓ nhds y).sets,
from @mem_inf_sets_of_left α (nhds x) (nhds y) _ $ mem_nhds_sets hu hx,
have h₂ : v ∈ (nhds x ⊓ nhds y).sets,
from @mem_inf_sets_of_right α (nhds x) (nhds y) _ $ mem_nhds_sets hv hy,
have u ∩ v ∈ (nhds x ⊓ nhds y).sets,
from @inter_mem_sets α (nhds x ⊓ nhds y) _ _ h₁ h₂,
h $ empty_in_sets_eq_bot.mp $ huv ▸ this
end separation
end topological_space
namespace topological_space
variables {α : Type u}
inductive generate_open (g : set (set α)) : set α → Prop
| basic : ∀s∈g, generate_open s
| univ : generate_open univ
| inter : ∀s t, generate_open s → generate_open t → generate_open (s ∩ t)
| sUnion : ∀k, (∀s∈k, generate_open s) → generate_open (⋃₀ k)
def generate_from (g : set (set α)) : topological_space α :=
{ topological_space .
open' := generate_open g,
open_univ := generate_open.univ g,
open_inter := generate_open.inter,
open_sUnion := generate_open.sUnion }
lemma nhds_generate_from {g : set (set α)} {a : α} :
@nhds α (generate_from g) a = (⨅s∈{s | a ∈ s ∧ s ∈ g}, principal s) :=
le_antisymm
(infi_le_infi $ take s, infi_le_infi_const $ take ⟨as, sg⟩, ⟨as, generate_open.basic _ sg⟩)
(le_infi $ take s, le_infi $ take ⟨as, hs⟩,
have ∀s, generate_open g s → a ∈ s → (⨅s∈{s | a ∈ s ∧ s ∈ g}, principal s) ≤ principal s,
begin
intros s hs,
induction hs,
case generate_open.basic s hs
{ exact take as, infi_le_of_le s $ infi_le _ ⟨as, hs⟩ },
case generate_open.univ
{ rw [principal_univ],
exact take _, le_top },
case generate_open.inter s t hs' ht' hs ht
{ exact take ⟨has, hat⟩, calc _ ≤ principal s ⊓ principal t : le_inf (hs has) (ht hat)
... = _ : by simp },
case generate_open.sUnion k hk' hk
{ intro h,
simp at h,
revert h,
exact take ⟨t, hat, htk⟩, calc _ ≤ principal t : hk t htk hat
... ≤ _ : begin simp; exact subset_sUnion_of_mem htk end },
end,
this s hs as)
end topological_space
section constructions
variables {α : Type u} {β : Type v}
instance : weak_order (topological_space α) :=
{ weak_order .
le := λt s, t^.open' ≤ s^.open',
le_antisymm := take t s h₁ h₂, topological_space_eq $ le_antisymm h₁ h₂,
le_refl := take t, le_refl t^.open',
le_trans := take a b c h₁ h₂, @le_trans _ _ a^.open' b^.open' c^.open' h₁ h₂ }
instance : has_Inf (topological_space α) :=
⟨take (tt : set (topological_space α)), { topological_space .
open' := λs, ∀t∈tt, topological_space.open' t s,
open_univ := take t h, t^.open_univ,
open_inter := take s₁ s₂ h₁ h₂ t ht, t^.open_inter s₁ s₂ (h₁ t ht) (h₂ t ht),
open_sUnion := take s h t ht, t^.open_sUnion _ $ take s' hss', h _ hss' _ ht }⟩
private lemma Inf_le {tt : set (topological_space α)} {t : topological_space α} (h : t ∈ tt) :
Inf tt ≤ t :=
take s hs, hs t h
private lemma le_Inf {tt : set (topological_space α)} {t : topological_space α} (h : ∀t'∈tt, t ≤ t') :
t ≤ Inf tt :=
take s hs t' ht', h t' ht' s hs
def topological_space.induced {α : Type u} {β : Type v} (f : α → β) (t : topological_space β) :
topological_space α :=
{ topological_space .
open' := λs, ∃s', t^.open' s' ∧ s = vimage f s',
open_univ := ⟨univ, by simp; exact t^.open_univ⟩,
open_inter := take s₁ s₂ ⟨s'₁, hs₁, eq₁⟩ ⟨s'₂, hs₂, eq₂⟩,
⟨s'₁ ∩ s'₂, by simp [eq₁, eq₂]; exact t^.open_inter _ _ hs₁ hs₂⟩,
open_sUnion := take s h,
begin
simp [classical.skolem] at h,
cases h with f hf,
apply exists.intro (⋃(x : set α) (h : x ∈ s), f x h),
simp [sUnion_eq_Union, (λx h, (hf x h)^.right^.symm)],
exact (@open_Union β _ t _ $ take i,
show open' (⋃h, f i h), from @open_Union β _ t _ $ take h, (hf i h)^.left)
end }
def topological_space.coinduced {α : Type u} {β : Type v} (f : α → β) (t : topological_space α) :
topological_space β :=
{ topological_space .
open' := λs, t^.open' (vimage f s),
open_univ := by simp; exact t^.open_univ,
open_inter := take s₁ s₂ h₁ h₂, by simp; exact t^.open_inter _ _ h₁ h₂,
open_sUnion := take s h, by rw [vimage_sUnion]; exact (@open_Union _ _ t _ $ take i,
show open' (⋃ (H : i ∈ s), vimage f i), from
@open_Union _ _ t _ $ take hi, h i hi) }
instance : has_inf (topological_space α) :=
⟨take t₁ t₂ : topological_space α, { topological_space .
open' := λs, t₁.open' s ∧ t₂.open' s,
open_univ := ⟨t₁^.open_univ, t₂^.open_univ⟩,
open_inter := take s₁ s₂ ⟨h₁₁, h₁₂⟩ ⟨h₂₁, h₂₂⟩, ⟨t₁.open_inter s₁ s₂ h₁₁ h₂₁, t₂.open_inter s₁ s₂ h₁₂ h₂₂⟩,
open_sUnion := take s h, ⟨t₁.open_sUnion _ $ take t ht, (h t ht).left, t₂.open_sUnion _ $ take t ht, (h t ht).right⟩ }⟩
instance : has_top (topological_space α) :=
⟨{topological_space .
open' := λs, true,
open_univ := trivial,
open_inter := take a b ha hb, trivial,
open_sUnion := take s h, trivial }⟩
instance {α : Type u} : complete_lattice (topological_space α) :=
{ topological_space.weak_order with
sup := λa b, Inf {x | a ≤ x ∧ b ≤ x},
le_sup_left := take a b, le_Inf $ take x, assume h : a ≤ x ∧ b ≤ x, h^.left,
le_sup_right := take a b, le_Inf $ take x, assume h : a ≤ x ∧ b ≤ x, h^.right,
sup_le := take a b c h₁ h₂, Inf_le $ show c ∈ {x | a ≤ x ∧ b ≤ x}, from ⟨h₁, h₂⟩,
inf := inf,
le_inf := take a b h h₁ h₂ s hs, ⟨h₁ s hs, h₂ s hs⟩,
inf_le_left := take a b s ⟨h₁, h₂⟩, h₁,
inf_le_right := take a b s ⟨h₁, h₂⟩, h₂,
top := top,
le_top := take a t ht, trivial,
bot := Inf univ,
bot_le := take a, Inf_le $ mem_univ a,
Sup := λtt, Inf {t | ∀t'∈tt, t' ≤ t},
le_Sup := take s f h, le_Inf $ take t ht, ht _ h,
Sup_le := take s f h, Inf_le $ take t ht, h _ ht,
Inf := Inf,
le_Inf := take s a, le_Inf,
Inf_le := take s a, Inf_le }
instance inhabited_topological_space {α : Type u} : inhabited (topological_space α) :=
⟨⊤⟩
lemma t2_space_top : @t2_space α ⊤ :=
⟨take x y hxy, ⟨{x}, {y}, trivial, trivial, mem_insert _ _, mem_insert _ _,
eq_empty_of_forall_not_mem $ by intros z hz; simp at hz; cc⟩⟩
lemma le_of_nhds_le_nhds {t₁ t₂ : topological_space α} (h : ∀x, @nhds α t₂ x ≤ @nhds α t₁ x) :
t₁ ≤ t₂ :=
take s, show @open' α t₁ s → @open' α t₂ s,
begin simp [open_iff_nhds]; exact take hs a ha, h _ $ hs _ ha end
lemma eq_of_nhds_eq_nhds {t₁ t₂ : topological_space α} (h : ∀x, @nhds α t₂ x = @nhds α t₁ x) :
t₁ = t₂ :=
le_antisymm
(le_of_nhds_le_nhds $ take x, le_of_eq $ h x)
(le_of_nhds_le_nhds $ take x, le_of_eq $ (h x).symm)
instance : topological_space empty := ⊤
instance : topological_space unit := ⊤
instance : topological_space bool := ⊤
instance : topological_space ℕ := ⊤
instance : topological_space ℤ := ⊤
instance sierpinski_space : topological_space Prop :=
topological_space.generate_from {{true}}
instance {p : α → Prop} [t : topological_space α] : topological_space (subtype p) :=
topological_space.induced subtype.val t
instance [t₁ : topological_space α] [t₂ : topological_space β] : topological_space (α × β) :=
topological_space.induced prod.fst t₁ ⊔ topological_space.induced prod.snd t₂
instance [t₁ : topological_space α] [t₂ : topological_space β] : topological_space (α ⊕ β) :=
topological_space.coinduced sum.inl t₁ ⊓ topological_space.coinduced sum.inr t₂
instance {β : α → Type v} [t₂ : Πa, topological_space (β a)] : topological_space (sigma β) :=
⨅a, topological_space.coinduced (sigma.mk a) (t₂ a)
instance topological_space_Pi {β : α → Type v} [t₂ : Πa, topological_space (β a)] : topological_space (Πa, β a) :=
⨆a, topological_space.induced (λf, f a) (t₂ a)
section
open topological_space
lemma generate_from_le {t : topological_space α} { g : set (set α) } (h : ∀s∈g, open' s) :
generate_from g ≤ t :=
take s (hs : generate_open g s), generate_open.rec_on hs h
open_univ
(take s t _ _ hs ht, open_inter hs ht)
(take k _ hk, open_sUnion hk)
lemma supr_eq_generate_from {ι : Sort w} { g : ι → topological_space α } :
supr g = generate_from (⋃i, {s | (g i).open' s}) :=
le_antisymm
(supr_le $ take i s open_s,
generate_open.basic _ $ by simp; exact ⟨i, open_s⟩)
(generate_from_le $ take s,
begin
simp,
exact take ⟨i, open_s⟩,
have g i ≤ supr g, from le_supr _ _,
this s open_s
end)
lemma sup_eq_generate_from { g₁ g₂ : topological_space α } :
g₁ ⊔ g₂ = generate_from {s | g₁.open' s ∨ g₂.open' s} :=
le_antisymm
(sup_le (take s, generate_open.basic _ ∘ or.inl) (take s, generate_open.basic _ ∘ or.inr))
(generate_from_le $ take s hs,
have h₁ : g₁ ≤ sup g₁ g₂, from le_sup_left,
have h₂ : g₂ ≤ sup g₁ g₂, from le_sup_right,
or.rec_on hs (h₁ s) (h₂ s))
lemma nhds_mono {t₁ t₂ : topological_space α} {a : α} (h : t₁ ≤ t₂) : @nhds α t₂ a ≤ @nhds α t₁ a :=
infi_le_infi $ take s, infi_le_infi2 $ take ⟨ha, hs⟩, ⟨⟨ha, h _ hs⟩, le_refl _⟩
lemma nhds_supr {ι : Sort w} {t : ι → topological_space α} {a : α} :
@nhds α (supr t) a = (⨅i, @nhds α (t i) a) :=
le_antisymm
(le_infi $ take i, nhds_mono $ le_supr _ _)
begin
rw [supr_eq_generate_from, nhds_generate_from],
simp,
exact (le_infi $ take s, le_infi $ take ⟨⟨i, hi⟩, hs⟩,
infi_le_of_le i $ le_principal_iff.mpr $ @mem_nhds_sets α (t i) _ _ hi hs)
end
end
end constructions
|
a20ca1f3460658b5231d0de8b5925485c7881edd | 4950bf76e5ae40ba9f8491647d0b6f228ddce173 | /src/algebra/ring/basic.lean | 883c93ad41e2a65adb4ef7266777355465cc4610 | [
"Apache-2.0"
] | permissive | ntzwq/mathlib | ca50b21079b0a7c6781c34b62199a396dd00cee2 | 36eec1a98f22df82eaccd354a758ef8576af2a7f | refs/heads/master | 1,675,193,391,478 | 1,607,822,996,000 | 1,607,822,996,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 38,211 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland
-/
import algebra.divisibility
import data.set.basic
/-!
# Properties and homomorphisms of semirings and rings
This file proves simple properties of semirings, rings and domains and their unit groups. It also
defines bundled homomorphisms of semirings and rings. As with monoid and groups, we use the same
structure `ring_hom a β`, a.k.a. `α →+* β`, for both homomorphism types.
The unbundled homomorphisms are defined in `deprecated/ring`. They are deprecated and the plan is to
slowly remove them from mathlib.
## Main definitions
ring_hom, nonzero, domain, integral_domain
## Notations
→+* for bundled ring homs (also use for semiring homs)
## Implementation notes
There's a coercion from bundled homs to fun, and the canonical
notation is to use the bundled hom as a function via this coercion.
There is no `semiring_hom` -- the idea is that `ring_hom` is used.
The constructor for a `ring_hom` between semirings needs a proof of `map_zero`, `map_one` and
`map_add` as well as `map_mul`; a separate constructor `ring_hom.mk'` will construct ring homs
between rings from monoid homs given only a proof that addition is preserved.
## Tags
`ring_hom`, `semiring_hom`, `semiring`, `comm_semiring`, `ring`, `comm_ring`, `domain`,
`integral_domain`, `nonzero`, `units`
-/
universes u v w x
variables {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
set_option old_structure_cmd true
open function
/-!
### `distrib` class
-/
/-- A typeclass stating that multiplication is left and right distributive
over addition. -/
@[protect_proj, ancestor has_mul has_add]
class distrib (R : Type*) extends has_mul R, has_add R :=
(left_distrib : ∀ a b c : R, a * (b + c) = (a * b) + (a * c))
(right_distrib : ∀ a b c : R, (a + b) * c = (a * c) + (b * c))
lemma left_distrib [distrib R] (a b c : R) : a * (b + c) = a * b + a * c :=
distrib.left_distrib a b c
alias left_distrib ← mul_add
lemma right_distrib [distrib R] (a b c : R) : (a + b) * c = a * c + b * c :=
distrib.right_distrib a b c
alias right_distrib ← add_mul
/-- Pullback a `distrib` instance along an injective function. -/
protected def function.injective.distrib {S} [has_mul R] [has_add R] [distrib S]
(f : R → S) (hf : injective f) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) :
distrib R :=
{ mul := (*),
add := (+),
left_distrib := λ x y z, hf $ by simp only [*, left_distrib],
right_distrib := λ x y z, hf $ by simp only [*, right_distrib] }
/-- Pushforward a `distrib` instance along a surjective function. -/
protected def function.surjective.distrib {S} [distrib R] [has_add S] [has_mul S]
(f : R → S) (hf : surjective f) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) :
distrib S :=
{ mul := (*),
add := (+),
left_distrib := hf.forall₃.2 $ λ x y z, by simp only [← add, ← mul, left_distrib],
right_distrib := hf.forall₃.2 $ λ x y z, by simp only [← add, ← mul, right_distrib] }
/-!
### Semirings
-/
/-- A semiring is a type with the following structures: additive commutative monoid
(`add_comm_monoid`), multiplicative monoid (`monoid`), distributive laws (`distrib`), and
multiplication by zero law (`mul_zero_class`). The actual definition extends `monoid_with_zero`
instead of `monoid` and `mul_zero_class`. -/
@[protect_proj, ancestor add_comm_monoid monoid_with_zero distrib]
class semiring (α : Type u) extends add_comm_monoid α, monoid_with_zero α, distrib α
section semiring
variables [semiring α]
/-- Pullback a `semiring` instance along an injective function. -/
protected def function.injective.semiring [has_zero β] [has_one β] [has_add β] [has_mul β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
semiring β :=
{ .. hf.monoid_with_zero f zero one mul, .. hf.add_comm_monoid f zero add,
.. hf.distrib f add mul }
/-- Pullback a `semiring` instance along an injective function. -/
protected def function.surjective.semiring [has_zero β] [has_one β] [has_add β] [has_mul β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
semiring β :=
{ .. hf.monoid_with_zero f zero one mul, .. hf.add_comm_monoid f zero add,
.. hf.distrib f add mul }
lemma one_add_one_eq_two : 1 + 1 = (2 : α) :=
by unfold bit0
theorem two_mul (n : α) : 2 * n = n + n :=
eq.trans (right_distrib 1 1 n) (by simp)
lemma distrib_three_right (a b c d : α) : (a + b + c) * d = a * d + b * d + c * d :=
by simp [right_distrib]
theorem mul_two (n : α) : n * 2 = n + n :=
(left_distrib n 1 1).trans (by simp)
theorem bit0_eq_two_mul (n : α) : bit0 n = 2 * n :=
(two_mul _).symm
@[to_additive] lemma mul_ite {α} [has_mul α] (P : Prop) [decidable P] (a b c : α) :
a * (if P then b else c) = if P then a * b else a * c :=
by split_ifs; refl
@[to_additive] lemma ite_mul {α} [has_mul α] (P : Prop) [decidable P] (a b c : α) :
(if P then a else b) * c = if P then a * c else b * c :=
by split_ifs; refl
-- We make `mul_ite` and `ite_mul` simp lemmas,
-- but not `add_ite` or `ite_add`.
-- The problem we're trying to avoid is dealing with
-- summations of the form `∑ x in s, (f x + ite P 1 0)`,
-- in which `add_ite` followed by `sum_ite` would needlessly slice up
-- the `f x` terms according to whether `P` holds at `x`.
-- There doesn't appear to be a corresponding difficulty so far with
-- `mul_ite` and `ite_mul`.
attribute [simp] mul_ite ite_mul
@[simp] lemma mul_boole {α} [semiring α] (P : Prop) [decidable P] (a : α) :
a * (if P then 1 else 0) = if P then a else 0 :=
by simp
@[simp] lemma boole_mul {α} [semiring α] (P : Prop) [decidable P] (a : α) :
(if P then 1 else 0) * a = if P then a else 0 :=
by simp
lemma ite_mul_zero_left {α : Type*} [mul_zero_class α] (P : Prop) [decidable P] (a b : α) :
ite P (a * b) 0 = ite P a 0 * b :=
by { by_cases h : P; simp [h], }
lemma ite_mul_zero_right {α : Type*} [mul_zero_class α] (P : Prop) [decidable P] (a b : α) :
ite P (a * b) 0 = a * ite P b 0 :=
by { by_cases h : P; simp [h], }
/-- An element `a` of a semiring is even if there exists `k` such `a = 2*k`. -/
def even (a : α) : Prop := ∃ k, a = 2*k
lemma even_iff_two_dvd {a : α} : even a ↔ 2 ∣ a := iff.rfl
/-- An element `a` of a semiring is odd if there exists `k` such `a = 2*k + 1`. -/
def odd (a : α) : Prop := ∃ k, a = 2*k + 1
end semiring
namespace add_monoid_hom
/-- Left multiplication by an element of a (semi)ring is an `add_monoid_hom` -/
def mul_left {R : Type*} [semiring R] (r : R) : R →+ R :=
{ to_fun := (*) r,
map_zero' := mul_zero r,
map_add' := mul_add r }
@[simp] lemma coe_mul_left {R : Type*} [semiring R] (r : R) : ⇑(mul_left r) = (*) r := rfl
/-- Right multiplication by an element of a (semi)ring is an `add_monoid_hom` -/
def mul_right {R : Type*} [semiring R] (r : R) : R →+ R :=
{ to_fun := λ a, a * r,
map_zero' := zero_mul r,
map_add' := λ _ _, add_mul _ _ r }
@[simp] lemma coe_mul_right {R : Type*} [semiring R] (r : R) :
⇑(mul_right r) = (* r) := rfl
lemma mul_right_apply {R : Type*} [semiring R] (a r : R) :
mul_right r a = a * r := rfl
end add_monoid_hom
/-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too.
This extends from both `monoid_hom` and `monoid_with_zero_hom` in order to put the fields in a
sensible order, even though `monoid_with_zero_hom` already extends `monoid_hom`. -/
structure ring_hom (α : Type*) (β : Type*) [semiring α] [semiring β]
extends monoid_hom α β, add_monoid_hom α β, monoid_with_zero_hom α β
infixr ` →+* `:25 := ring_hom
/-- Reinterpret a ring homomorphism `f : R →+* S` as a `monoid_with_zero_hom R S`.
The `simp`-normal form is `(f : monoid_with_zero_hom R S)`. -/
add_decl_doc ring_hom.to_monoid_with_zero_hom
/-- Reinterpret a ring homomorphism `f : R →+* S` as a monoid homomorphism `R →* S`.
The `simp`-normal form is `(f : R →* S)`. -/
add_decl_doc ring_hom.to_monoid_hom
/-- Reinterpret a ring homomorphism `f : R →+* S` as an additive monoid homomorphism `R →+ S`.
The `simp`-normal form is `(f : R →+ S)`. -/
add_decl_doc ring_hom.to_add_monoid_hom
namespace ring_hom
section coe
/-!
Throughout this section, some `semiring` arguments are specified with `{}` instead of `[]`.
See note [implicit instance arguments].
-/
variables {rα : semiring α} {rβ : semiring β}
include rα rβ
instance : has_coe_to_fun (α →+* β) := ⟨_, ring_hom.to_fun⟩
initialize_simps_projections ring_hom (to_fun → apply)
@[simp] lemma to_fun_eq_coe (f : α →+* β) : f.to_fun = f := rfl
@[simp] lemma coe_mk (f : α → β) (h₁ h₂ h₃ h₄) : ⇑(⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) = f := rfl
instance has_coe_monoid_hom : has_coe (α →+* β) (α →* β) := ⟨ring_hom.to_monoid_hom⟩
@[simp, norm_cast] lemma coe_monoid_hom (f : α →+* β) : ⇑(f : α →* β) = f := rfl
@[simp] lemma to_monoid_hom_eq_coe (f : α →+* β) : f.to_monoid_hom = f := rfl
@[simp] lemma coe_monoid_hom_mk (f : α → β) (h₁ h₂ h₃ h₄) :
((⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) : α →* β) = ⟨f, h₁, h₂⟩ :=
rfl
instance has_coe_add_monoid_hom : has_coe (α →+* β) (α →+ β) := ⟨ring_hom.to_add_monoid_hom⟩
@[simp, norm_cast] lemma coe_add_monoid_hom (f : α →+* β) : ⇑(f : α →+ β) = f := rfl
@[simp] lemma to_add_monoid_hom_eq_coe (f : α →+* β) : f.to_add_monoid_hom = f := rfl
@[simp] lemma coe_add_monoid_hom_mk (f : α → β) (h₁ h₂ h₃ h₄) :
((⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨f, h₃, h₄⟩ :=
rfl
end coe
variables [rα : semiring α] [rβ : semiring β]
section
include rα rβ
variables (f : α →+* β) {x y : α} {rα rβ}
theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x :=
congr_arg (λ h : α →+* β, h x) h
theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y :=
congr_arg (λ x : α, f x) h
theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g :=
by cases f; cases g; cases h; refl
@[ext] theorem ext ⦃f g : α →+* β⦄ (h : ∀ x, f x = g x) : f = g :=
coe_inj (funext h)
theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x :=
⟨λ h x, h ▸ rfl, λ h, ext h⟩
theorem coe_add_monoid_hom_injective : function.injective (coe : (α →+* β) → (α →+ β)) :=
λ f g h, ext (λ x, add_monoid_hom.congr_fun h x)
theorem coe_monoid_hom_injective : function.injective (coe : (α →+* β) → (α →* β)) :=
λ f g h, ext (λ x, monoid_hom.congr_fun h x)
/-- Ring homomorphisms map zero to zero. -/
@[simp] lemma map_zero (f : α →+* β) : f 0 = 0 := f.map_zero'
/-- Ring homomorphisms map one to one. -/
@[simp] lemma map_one (f : α →+* β) : f 1 = 1 := f.map_one'
/-- Ring homomorphisms preserve addition. -/
@[simp] lemma map_add (f : α →+* β) (a b : α) : f (a + b) = f a + f b := f.map_add' a b
/-- Ring homomorphisms preserve multiplication. -/
@[simp] lemma map_mul (f : α →+* β) (a b : α) : f (a * b) = f a * f b := f.map_mul' a b
/-- Ring homomorphisms preserve `bit0`. -/
@[simp] lemma map_bit0 (f : α →+* β) (a : α) : f (bit0 a) = bit0 (f a) := map_add _ _ _
/-- Ring homomorphisms preserve `bit1`. -/
@[simp] lemma map_bit1 (f : α →+* β) (a : α) : f (bit1 a) = bit1 (f a) :=
by simp [bit1]
/-- `f : R →+* S` has a trivial codomain iff `f 1 = 0`. -/
lemma codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 :=
by rw [map_one, eq_comm]
/-- `f : R →+* S` has a trivial codomain iff it has a trivial range. -/
lemma codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ (∀ x, f x = 0) :=
f.codomain_trivial_iff_map_one_eq_zero.trans
⟨λ h x, by rw [←mul_one x, map_mul, h, mul_zero], λ h, h 1⟩
/-- `f : R →+* S` has a trivial codomain iff its range is `{0}`. -/
lemma codomain_trivial_iff_range_eq_singleton_zero : (0 : β) = 1 ↔ set.range f = {0} :=
f.codomain_trivial_iff_range_trivial.trans
⟨ λ h, set.ext (λ y, ⟨λ ⟨x, hx⟩, by simp [←hx, h x], λ hy, ⟨0, by simpa using hy.symm⟩⟩),
λ h x, set.mem_singleton_iff.mp (h ▸ set.mem_range_self x)⟩
/-- `f : R →+* S` doesn't map `1` to `0` if `S` is nontrivial -/
lemma map_one_ne_zero [nontrivial β] : f 1 ≠ 0 :=
mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one
/-- If there is a homomorphism `f : R →+* S` and `S` is nontrivial, then `R` is nontrivial. -/
lemma domain_nontrivial [nontrivial β] : nontrivial α :=
⟨⟨1, 0, mt (λ h, show f 1 = 0, by rw [h, map_zero]) f.map_one_ne_zero⟩⟩
lemma is_unit_map (f : α →+* β) {a : α} (h : is_unit a) : is_unit (f a) :=
h.map (f.to_monoid_hom)
end
/-- The identity ring homomorphism from a semiring to itself. -/
def id (α : Type*) [semiring α] : α →+* α :=
by refine {to_fun := id, ..}; intros; refl
include rα
instance : inhabited (α →+* α) := ⟨id α⟩
@[simp] lemma id_apply (x : α) : ring_hom.id α x = x := rfl
variable {rγ : semiring γ}
include rβ rγ
/-- Composition of ring homomorphisms is a ring homomorphism. -/
def comp (hnp : β →+* γ) (hmn : α →+* β) : α →+* γ :=
{ to_fun := hnp ∘ hmn,
map_zero' := by simp,
map_one' := by simp,
map_add' := λ x y, by simp,
map_mul' := λ x y, by simp}
/-- Composition of semiring homomorphisms is associative. -/
lemma comp_assoc {δ} {rδ: semiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) :
(h.comp g).comp f = h.comp (g.comp f) := rfl
@[simp] lemma coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl
lemma comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x =
(hnp (hmn x)) := rfl
omit rγ
@[simp] lemma comp_id (f : α →+* β) : f.comp (id α) = f := ext $ λ x, rfl
@[simp] lemma id_comp (f : α →+* β) : (id β).comp f = f := ext $ λ x, rfl
omit rβ
instance : monoid (α →+* α) :=
{ one := id α,
mul := comp,
mul_one := comp_id,
one_mul := id_comp,
mul_assoc := λ f g h, comp_assoc _ _ _ }
lemma one_def : (1 : α →+* α) = id α := rfl
@[simp] lemma coe_one : ⇑(1 : α →+* α) = _root_.id := rfl
lemma mul_def (f g : α →+* α) : f * g = f.comp g := rfl
@[simp] lemma coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl
include rβ rγ
lemma cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨λ h, ring_hom.ext $ (forall_iff_forall_surj hf).1 (ext_iff.1 h), λ h, h ▸ rfl⟩
lemma cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨λ h, ring_hom.ext $ λ x, hg $ by rw [← comp_apply, h, comp_apply], λ h, h ▸ rfl⟩
omit rα rβ rγ
end ring_hom
/-- A commutative semiring is a `semiring` with commutative multiplication. In other words, it is a
type with the following structures: additive commutative monoid (`add_comm_monoid`), multiplicative
commutative monoid (`comm_monoid`), distributive laws (`distrib`), and multiplication by zero law
(`mul_zero_class`). -/
@[protect_proj, ancestor semiring comm_monoid]
class comm_semiring (α : Type u) extends semiring α, comm_monoid α
@[priority 100] -- see Note [lower instance priority]
instance comm_semiring.to_comm_monoid_with_zero [comm_semiring α] : comm_monoid_with_zero α :=
{ .. comm_semiring.to_comm_monoid α, .. comm_semiring.to_semiring α }
section comm_semiring
variables [comm_semiring α] [comm_semiring β] {a b c : α}
/-- Pullback a `semiring` instance along an injective function. -/
protected def function.injective.comm_semiring [has_zero γ] [has_one γ] [has_add γ] [has_mul γ]
(f : γ → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
comm_semiring γ :=
{ .. hf.semiring f zero one add mul, .. hf.comm_semigroup f mul }
/-- Pullback a `semiring` instance along an injective function. -/
protected def function.surjective.comm_semiring [has_zero γ] [has_one γ] [has_add γ] [has_mul γ]
(f : α → γ) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
comm_semiring γ :=
{ .. hf.semiring f zero one add mul, .. hf.comm_semigroup f mul }
lemma add_mul_self_eq (a b : α) : (a + b) * (a + b) = a*a + 2*a*b + b*b :=
calc (a + b)*(a + b) = a*a + (1+1)*a*b + b*b : by simp [add_mul, mul_add, mul_comm, add_assoc]
... = a*a + 2*a*b + b*b : by rw one_add_one_eq_two
theorem dvd_add (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b + c :=
dvd.elim h₁ (λ d hd, dvd.elim h₂ (λ e he, dvd.intro (d + e) (by simp [left_distrib, hd, he])))
@[simp] theorem two_dvd_bit0 : 2 ∣ bit0 a := ⟨a, bit0_eq_two_mul _⟩
lemma ring_hom.map_dvd (f : α →+* β) {a b : α} : a ∣ b → f a ∣ f b :=
λ ⟨z, hz⟩, ⟨f z, by rw [hz, f.map_mul]⟩
end comm_semiring
/-!
### Rings
-/
/-- A ring is a type with the following structures: additive commutative group (`add_comm_group`),
multiplicative monoid (`monoid`), and distributive laws (`distrib`). Equivalently, a ring is a
`semiring` with a negation operation making it an additive group. -/
@[protect_proj, ancestor add_comm_group monoid distrib]
class ring (α : Type u) extends add_comm_group α, monoid α, distrib α
section ring
variables [ring α] {a b c d e : α}
/- The instance from `ring` to `semiring` happens often in linear algebra, for which all the basic
definitions are given in terms of semirings, but many applications use rings or fields. We increase
a little bit its priority above 100 to try it quickly, but remaining below the default 1000 so that
more specific instances are tried first. -/
@[priority 200]
instance ring.to_semiring : semiring α :=
{ zero_mul := λ a, add_left_cancel $ show 0 * a + 0 * a = 0 * a + 0,
by rw [← add_mul, zero_add, add_zero],
mul_zero := λ a, add_left_cancel $ show a * 0 + a * 0 = a * 0 + 0,
by rw [← mul_add, add_zero, add_zero],
..‹ring α› }
/-- Pullback a `ring` instance along an injective function. -/
protected def function.injective.ring [has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) :
ring β :=
{ .. hf.add_comm_group f zero add neg, .. hf.monoid f one mul, .. hf.distrib f add mul }
/-- Pullback a `ring` instance along an injective function. -/
protected def function.surjective.ring [has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) :
ring β :=
{ .. hf.add_comm_group f zero add neg, .. hf.monoid f one mul, .. hf.distrib f add mul }
lemma neg_mul_eq_neg_mul (a b : α) : -(a * b) = -a * b :=
neg_eq_of_add_eq_zero
begin rw [← right_distrib, add_right_neg, zero_mul] end
lemma neg_mul_eq_mul_neg (a b : α) : -(a * b) = a * -b :=
neg_eq_of_add_eq_zero
begin rw [← left_distrib, add_right_neg, mul_zero] end
@[simp] lemma neg_mul_eq_neg_mul_symm (a b : α) : - a * b = - (a * b) :=
eq.symm (neg_mul_eq_neg_mul a b)
@[simp] lemma mul_neg_eq_neg_mul_symm (a b : α) : a * - b = - (a * b) :=
eq.symm (neg_mul_eq_mul_neg a b)
lemma neg_mul_neg (a b : α) : -a * -b = a * b :=
by simp
lemma neg_mul_comm (a b : α) : -a * b = a * -b :=
by simp
theorem neg_eq_neg_one_mul (a : α) : -a = -1 * a :=
by simp
lemma mul_sub_left_distrib (a b c : α) : a * (b - c) = a * b - a * c :=
by simpa only [sub_eq_add_neg, neg_mul_eq_mul_neg] using mul_add a b (-c)
alias mul_sub_left_distrib ← mul_sub
lemma mul_sub_right_distrib (a b c : α) : (a - b) * c = a * c - b * c :=
by simpa only [sub_eq_add_neg, neg_mul_eq_neg_mul] using add_mul a (-b) c
alias mul_sub_right_distrib ← sub_mul
/-- An element of a ring multiplied by the additive inverse of one is the element's additive
inverse. -/
lemma mul_neg_one (a : α) : a * -1 = -a := by simp
/-- The additive inverse of one multiplied by an element of a ring is the element's additive
inverse. -/
lemma neg_one_mul (a : α) : -1 * a = -a := by simp
/-- An iff statement following from right distributivity in rings and the definition
of subtraction. -/
theorem mul_add_eq_mul_add_iff_sub_mul_add_eq : a * e + c = b * e + d ↔ (a - b) * e + c = d :=
calc
a * e + c = b * e + d ↔ a * e + c = d + b * e : by simp [add_comm]
... ↔ a * e + c - b * e = d : iff.intro (λ h, begin rw h, simp end) (λ h,
begin rw ← h, simp end)
... ↔ (a - b) * e + c = d : begin simp [sub_mul, sub_add_eq_add_sub] end
/-- A simplification of one side of an equation exploiting right distributivity in rings
and the definition of subtraction. -/
theorem sub_mul_add_eq_of_mul_add_eq_mul_add : a * e + c = b * e + d → (a - b) * e + c = d :=
assume h,
calc
(a - b) * e + c = (a * e + c) - b * e : begin simp [sub_mul, sub_add_eq_add_sub] end
... = d : begin rw h, simp [@add_sub_cancel α] end
end ring
namespace units
variables [ring α] {a b : α}
/-- Each element of the group of units of a ring has an additive inverse. -/
instance : has_neg (units α) := ⟨λu, ⟨-↑u, -↑u⁻¹, by simp, by simp⟩ ⟩
/-- Representing an element of a ring's unit group as an element of the ring commutes with
mapping this element to its additive inverse. -/
@[simp, norm_cast] protected theorem coe_neg (u : units α) : (↑-u : α) = -u := rfl
@[simp, norm_cast] protected theorem coe_neg_one : ((-1 : units α) : α) = -1 := rfl
/-- Mapping an element of a ring's unit group to its inverse commutes with mapping this element
to its additive inverse. -/
@[simp] protected theorem neg_inv (u : units α) : (-u)⁻¹ = -u⁻¹ := rfl
/-- An element of a ring's unit group equals the additive inverse of its additive inverse. -/
@[simp] protected theorem neg_neg (u : units α) : - -u = u :=
units.ext $ neg_neg _
/-- Multiplication of elements of a ring's unit group commutes with mapping the first
argument to its additive inverse. -/
@[simp] protected theorem neg_mul (u₁ u₂ : units α) : -u₁ * u₂ = -(u₁ * u₂) :=
units.ext $ neg_mul_eq_neg_mul_symm _ _
/-- Multiplication of elements of a ring's unit group commutes with mapping the second argument
to its additive inverse. -/
@[simp] protected theorem mul_neg (u₁ u₂ : units α) : u₁ * -u₂ = -(u₁ * u₂) :=
units.ext $ (neg_mul_eq_mul_neg _ _).symm
/-- Multiplication of the additive inverses of two elements of a ring's unit group equals
multiplication of the two original elements. -/
@[simp] protected theorem neg_mul_neg (u₁ u₂ : units α) : -u₁ * -u₂ = u₁ * u₂ := by simp
/-- The additive inverse of an element of a ring's unit group equals the additive inverse of
one times the original element. -/
protected theorem neg_eq_neg_one_mul (u : units α) : -u = -1 * u := by simp
end units
namespace ring_hom
/-- Ring homomorphisms preserve additive inverse. -/
@[simp] theorem map_neg {α β} [ring α] [ring β] (f : α →+* β) (x : α) : f (-x) = -(f x) :=
(f : α →+ β).map_neg x
/-- Ring homomorphisms preserve subtraction. -/
@[simp] theorem map_sub {α β} [ring α] [ring β] (f : α →+* β) (x y : α) :
f (x - y) = (f x) - (f y) := (f : α →+ β).map_sub x y
/-- A ring homomorphism is injective iff its kernel is trivial. -/
theorem injective_iff {α β} [ring α] [semiring β] (f : α →+* β) :
function.injective f ↔ (∀ a, f a = 0 → a = 0) :=
(f : α →+ β).injective_iff
/-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/
def mk' {γ} [semiring α] [ring γ] (f : α →* γ) (map_add : ∀ a b : α, f (a + b) = f a + f b) :
α →+* γ :=
{ to_fun := f,
.. add_monoid_hom.mk' f map_add, .. f }
end ring_hom
/-- A commutative ring is a `ring` with commutative multiplication. -/
@[protect_proj, ancestor ring comm_semigroup]
class comm_ring (α : Type u) extends ring α, comm_semigroup α
@[priority 100] -- see Note [lower instance priority]
instance comm_ring.to_comm_semiring [s : comm_ring α] : comm_semiring α :=
{ mul_zero := mul_zero, zero_mul := zero_mul, ..s }
section comm_ring
variables [comm_ring α] {a b c : α}
/-- Pullback a `ring` instance along an injective function. -/
protected def function.injective.comm_ring [has_zero β] [has_one β] [has_add β] [has_mul β]
[has_neg β] (f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) :
comm_ring β :=
{ .. hf.ring f zero one add mul neg, .. hf.comm_semigroup f mul }
/-- Pullback a `ring` instance along an injective function. -/
protected def function.surjective.comm_ring [has_zero β] [has_one β] [has_add β] [has_mul β]
[has_neg β] (f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) :
comm_ring β :=
{ .. hf.ring f zero one add mul neg, .. hf.comm_semigroup f mul }
local attribute [simp] add_assoc add_comm add_left_comm mul_comm
theorem dvd_neg_of_dvd (h : a ∣ b) : (a ∣ -b) :=
dvd.elim h
(assume c, assume : b = a * c,
dvd.intro (-c) (by simp [this]))
theorem dvd_of_dvd_neg (h : a ∣ -b) : (a ∣ b) :=
let t := dvd_neg_of_dvd h in by rwa neg_neg at t
theorem dvd_neg_iff_dvd (a b : α) : (a ∣ -b) ↔ (a ∣ b) :=
⟨dvd_of_dvd_neg, dvd_neg_of_dvd⟩
theorem neg_dvd_of_dvd (h : a ∣ b) : -a ∣ b :=
dvd.elim h
(assume c, assume : b = a * c,
dvd.intro (-c) (by simp [this]))
theorem dvd_of_neg_dvd (h : -a ∣ b) : a ∣ b :=
let t := neg_dvd_of_dvd h in by rwa neg_neg at t
theorem neg_dvd_iff_dvd (a b : α) : (-a ∣ b) ↔ (a ∣ b) :=
⟨dvd_of_neg_dvd, neg_dvd_of_dvd⟩
theorem dvd_sub (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b - c :=
by { rw sub_eq_add_neg, exact dvd_add h₁ (dvd_neg_of_dvd h₂) }
theorem dvd_add_iff_left (h : a ∣ c) : a ∣ b ↔ a ∣ b + c :=
⟨λh₂, dvd_add h₂ h, λH, by have t := dvd_sub H h; rwa add_sub_cancel at t⟩
theorem dvd_add_iff_right (h : a ∣ b) : a ∣ c ↔ a ∣ b + c :=
by rw add_comm; exact dvd_add_iff_left h
theorem two_dvd_bit1 : 2 ∣ bit1 a ↔ (2 : α) ∣ 1 := (dvd_add_iff_right (@two_dvd_bit0 _ _ a)).symm
/-- Representation of a difference of two squares in a commutative ring as a product. -/
theorem mul_self_sub_mul_self (a b : α) : a * a - b * b = (a + b) * (a - b) :=
by rw [add_mul, mul_sub, mul_sub, mul_comm a b, sub_add_sub_cancel]
lemma mul_self_sub_one (a : α) : a * a - 1 = (a + 1) * (a - 1) :=
by rw [← mul_self_sub_mul_self, mul_one]
/-- An element a of a commutative ring divides the additive inverse of an element b iff a
divides b. -/
@[simp] lemma dvd_neg (a b : α) : (a ∣ -b) ↔ (a ∣ b) :=
⟨dvd_of_dvd_neg, dvd_neg_of_dvd⟩
/-- The additive inverse of an element a of a commutative ring divides another element b iff a
divides b. -/
@[simp] lemma neg_dvd (a b : α) : (-a ∣ b) ↔ (a ∣ b) :=
⟨dvd_of_neg_dvd, neg_dvd_of_dvd⟩
/-- If an element a divides another element c in a commutative ring, a divides the sum of another
element b with c iff a divides b. -/
theorem dvd_add_left (h : a ∣ c) : a ∣ b + c ↔ a ∣ b :=
(dvd_add_iff_left h).symm
/-- If an element a divides another element b in a commutative ring, a divides the sum of b and
another element c iff a divides c. -/
theorem dvd_add_right (h : a ∣ b) : a ∣ b + c ↔ a ∣ c :=
(dvd_add_iff_right h).symm
/-- An element a divides the sum a + b if and only if a divides b.-/
@[simp] lemma dvd_add_self_left {a b : α} : a ∣ a + b ↔ a ∣ b :=
dvd_add_right (dvd_refl a)
/-- An element a divides the sum b + a if and only if a divides b.-/
@[simp] lemma dvd_add_self_right {a b : α} : a ∣ b + a ↔ a ∣ b :=
dvd_add_left (dvd_refl a)
/-- Vieta's formula for a quadratic equation, relating the coefficients of the polynomial with
its roots. This particular version states that if we have a root `x` of a monic quadratic
polynomial, then there is another root `y` such that `x + y` is negative the `a_1` coefficient
and `x * y` is the `a_0` coefficient. -/
lemma Vieta_formula_quadratic {b c x : α} (h : x * x - b * x + c = 0) :
∃ y : α, y * y - b * y + c = 0 ∧ x + y = b ∧ x * y = c :=
begin
have : c = -(x * x - b * x) := (neg_eq_of_add_eq_zero h).symm,
have : c = x * (b - x), by subst this; simp [mul_sub, mul_comm],
refine ⟨b - x, _, by simp, by rw this⟩,
rw [this, sub_add, ← sub_mul, sub_self]
end
lemma dvd_mul_sub_mul {k a b x y : α} (hab : k ∣ a - b) (hxy : k ∣ x - y) :
k ∣ a * x - b * y :=
begin
convert dvd_add (dvd_mul_of_dvd_right hxy a) (dvd_mul_of_dvd_left hab y),
rw [mul_sub_left_distrib, mul_sub_right_distrib],
simp only [sub_eq_add_neg, add_assoc, neg_add_cancel_left],
end
lemma dvd_iff_dvd_of_dvd_sub {a b c : α} (h : a ∣ (b - c)) : (a ∣ b ↔ a ∣ c) :=
begin
split,
{ intro h',
convert dvd_sub h' h,
exact eq.symm (sub_sub_self b c) },
{ intro h',
convert dvd_add h h',
exact eq_add_of_sub_eq rfl }
end
end comm_ring
lemma succ_ne_self [ring α] [nontrivial α] (a : α) : a + 1 ≠ a :=
λ h, one_ne_zero ((add_right_inj a).mp (by simp [h]))
lemma pred_ne_self [ring α] [nontrivial α] (a : α) : a - 1 ≠ a :=
λ h, one_ne_zero (neg_injective ((add_right_inj a).mp (by simpa [sub_eq_add_neg] using h)))
/-- A domain is a ring with no zero divisors, i.e. satisfying
the condition `a * b = 0 ↔ a = 0 ∨ b = 0`. Alternatively, a domain
is an integral domain without assuming commutativity of multiplication. -/
@[protect_proj] class domain (α : Type u) extends ring α, nontrivial α :=
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀ a b : α, a * b = 0 → a = 0 ∨ b = 0)
section domain
variable [domain α]
@[priority 100] -- see Note [lower instance priority]
instance domain.to_no_zero_divisors : no_zero_divisors α :=
⟨domain.eq_zero_or_eq_zero_of_mul_eq_zero⟩
@[priority 100] -- see Note [lower instance priority]
instance domain.to_cancel_monoid_with_zero : cancel_monoid_with_zero α :=
{ mul_left_cancel_of_ne_zero := λ a b c ha,
by { rw [← sub_eq_zero, ← mul_sub], simp [ha, sub_eq_zero] },
mul_right_cancel_of_ne_zero := λ a b c hb,
by { rw [← sub_eq_zero, ← sub_mul], simp [hb, sub_eq_zero] },
.. (infer_instance : semiring α) }
/-- Pullback a `domain` instance along an injective function. -/
protected def function.injective.domain [has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) :
domain β :=
{ .. hf.ring f zero one add mul neg, .. pullback_nonzero f zero one,
.. hf.no_zero_divisors f zero mul }
end domain
/-!
### Integral domains
-/
/-- An integral domain is a commutative ring with no zero divisors, i.e. satisfying the condition
`a * b = 0 ↔ a = 0 ∨ b = 0`. Alternatively, an integral domain is a domain with commutative
multiplication. -/
@[protect_proj, ancestor comm_ring domain]
class integral_domain (α : Type u) extends comm_ring α, domain α
section integral_domain
variables [integral_domain α] {a b c d e : α}
@[priority 100] -- see Note [lower instance priority]
instance integral_domain.to_comm_cancel_monoid_with_zero : comm_cancel_monoid_with_zero α :=
{ ..comm_semiring.to_comm_monoid_with_zero, ..domain.to_cancel_monoid_with_zero }
/-- Pullback an `integral_domain` instance along an injective function. -/
protected def function.injective.integral_domain [has_zero β] [has_one β] [has_add β]
[has_mul β] [has_neg β] (f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) :
integral_domain β :=
{ .. hf.comm_ring f zero one add mul neg, .. hf.domain f zero one add mul neg }
lemma mul_self_eq_mul_self_iff {a b : α} : a * a = b * b ↔ a = b ∨ a = -b :=
by rw [← sub_eq_zero, mul_self_sub_mul_self, mul_eq_zero, or_comm, sub_eq_zero,
add_eq_zero_iff_eq_neg]
lemma mul_self_eq_one_iff {a : α} : a * a = 1 ↔ a = 1 ∨ a = -1 :=
by rw [← mul_self_eq_mul_self_iff, one_mul]
/-- In the unit group of an integral domain, a unit is its own inverse iff the unit is one or
one's additive inverse. -/
lemma units.inv_eq_self_iff (u : units α) : u⁻¹ = u ↔ u = 1 ∨ u = -1 :=
by { rw inv_eq_iff_mul_eq_one, simp only [units.ext_iff], push_cast, exact mul_self_eq_one_iff }
end integral_domain
namespace ring
variables [ring R]
open_locale classical
/-- Introduce a function `inverse` on a ring `R`, which sends `x` to `x⁻¹` if `x` is invertible and
to `0` otherwise. This definition is somewhat ad hoc, but one needs a fully (rather than partially)
defined inverse function for some purposes, including for calculus. -/
noncomputable def inverse : R → R :=
λ x, if h : is_unit x then (((classical.some h)⁻¹ : units R) : R) else 0
/-- By definition, if `x` is invertible then `inverse x = x⁻¹`. -/
@[simp] lemma inverse_unit (a : units R) : inverse (a : R) = (a⁻¹ : units R) :=
begin
simp [is_unit_unit, inverse],
exact units.inv_unique (classical.some_spec (is_unit_unit a)),
end
/-- By definition, if `x` is not invertible then `inverse x = 0`. -/
@[simp] lemma inverse_non_unit (x : R) (h : ¬(is_unit x)) : inverse x = 0 := dif_neg h
end ring
/-- A predicate to express that a ring is an integral domain.
This is mainly useful because such a predicate does not contain data,
and can therefore be easily transported along ring isomorphisms. -/
structure is_integral_domain (R : Type u) [ring R] extends nontrivial R : Prop :=
(mul_comm : ∀ (x y : R), x * y = y * x)
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀ x y : R, x * y = 0 → x = 0 ∨ y = 0)
-- The linter does not recognize that is_integral_domain.to_nontrivial is a structure
-- projection, disable it
attribute [nolint def_lemma doc_blame] is_integral_domain.to_nontrivial
/-- Every integral domain satisfies the predicate for integral domains. -/
lemma integral_domain.to_is_integral_domain (R : Type u) [integral_domain R] :
is_integral_domain R :=
{ .. (‹_› : integral_domain R) }
/-- If a ring satisfies the predicate for integral domains,
then it can be endowed with an `integral_domain` instance
whose data is definitionally equal to the existing data. -/
def is_integral_domain.to_integral_domain (R : Type u) [ring R] (h : is_integral_domain R) :
integral_domain R :=
{ .. (‹_› : ring R), .. (‹_› : is_integral_domain R) }
namespace semiconj_by
@[simp] lemma add_right [distrib R] {a x y x' y' : R}
(h : semiconj_by a x y) (h' : semiconj_by a x' y') :
semiconj_by a (x + x') (y + y') :=
by simp only [semiconj_by, left_distrib, right_distrib, h.eq, h'.eq]
@[simp] lemma add_left [distrib R] {a b x y : R}
(ha : semiconj_by a x y) (hb : semiconj_by b x y) :
semiconj_by (a + b) x y :=
by simp only [semiconj_by, left_distrib, right_distrib, ha.eq, hb.eq]
variables [ring R] {a b x y x' y' : R}
lemma neg_right (h : semiconj_by a x y) : semiconj_by a (-x) (-y) :=
by simp only [semiconj_by, h.eq, neg_mul_eq_neg_mul_symm, mul_neg_eq_neg_mul_symm]
@[simp] lemma neg_right_iff : semiconj_by a (-x) (-y) ↔ semiconj_by a x y :=
⟨λ h, neg_neg x ▸ neg_neg y ▸ h.neg_right, semiconj_by.neg_right⟩
lemma neg_left (h : semiconj_by a x y) : semiconj_by (-a) x y :=
by simp only [semiconj_by, h.eq, neg_mul_eq_neg_mul_symm, mul_neg_eq_neg_mul_symm]
@[simp] lemma neg_left_iff : semiconj_by (-a) x y ↔ semiconj_by a x y :=
⟨λ h, neg_neg a ▸ h.neg_left, semiconj_by.neg_left⟩
@[simp] lemma neg_one_right (a : R) : semiconj_by a (-1) (-1) :=
(one_right a).neg_right
@[simp] lemma neg_one_left (x : R) : semiconj_by (-1) x x :=
(semiconj_by.one_left x).neg_left
@[simp] lemma sub_right (h : semiconj_by a x y) (h' : semiconj_by a x' y') :
semiconj_by a (x - x') (y - y') :=
by simpa only [sub_eq_add_neg] using h.add_right h'.neg_right
@[simp] lemma sub_left (ha : semiconj_by a x y) (hb : semiconj_by b x y) :
semiconj_by (a - b) x y :=
by simpa only [sub_eq_add_neg] using ha.add_left hb.neg_left
end semiconj_by
namespace commute
@[simp] theorem add_right [distrib R] {a b c : R} :
commute a b → commute a c → commute a (b + c) :=
semiconj_by.add_right
@[simp] theorem add_left [distrib R] {a b c : R} :
commute a c → commute b c → commute (a + b) c :=
semiconj_by.add_left
variables [ring R] {a b c : R}
theorem neg_right : commute a b → commute a (- b) := semiconj_by.neg_right
@[simp] theorem neg_right_iff : commute a (-b) ↔ commute a b := semiconj_by.neg_right_iff
theorem neg_left : commute a b → commute (- a) b := semiconj_by.neg_left
@[simp] theorem neg_left_iff : commute (-a) b ↔ commute a b := semiconj_by.neg_left_iff
@[simp] theorem neg_one_right (a : R) : commute a (-1) := semiconj_by.neg_one_right a
@[simp] theorem neg_one_left (a : R): commute (-1) a := semiconj_by.neg_one_left a
@[simp] theorem sub_right : commute a b → commute a c → commute a (b - c) := semiconj_by.sub_right
@[simp] theorem sub_left : commute a c → commute b c → commute (a - b) c := semiconj_by.sub_left
end commute
|
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"Apache-2.0",
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"NCSA",
"LGPL-3.0-only",
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"BSD-3-Clause",
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"LicenseRef-scancode-pcre",
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"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | Kha/lean4 | 1005785d2c8797ae266a303968848e5f6ce2fe87 | b99e11346948023cd6c29d248cd8f3e3fb3474cf | refs/heads/master | 1,693,355,498,027 | 1,669,080,461,000 | 1,669,113,138,000 | 184,748,176 | 0 | 0 | Apache-2.0 | 1,665,995,520,000 | 1,556,884,930,000 | Lean | UTF-8 | Lean | false | false | 16,408 | lean | /-
Copyright (c) 2018 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Sebastian Ullrich, Leonardo de Moura
Message Type used by the Lean frontend
-/
import Lean.Data.Position
import Lean.Data.OpenDecl
import Lean.MetavarContext
import Lean.Environment
import Lean.Util.PPExt
namespace Lean
def mkErrorStringWithPos (fileName : String) (pos : Position) (msg : String) (endPos : Option Position := none) : String :=
let endPos := match endPos with
| some endPos => s!"-{endPos.line}:{endPos.column}"
| none => ""
s!"{fileName}:{pos.line}:{pos.column}{endPos}: {msg}"
inductive MessageSeverity where
| information | warning | error
deriving Inhabited, BEq
structure MessageDataContext where
env : Environment
mctx : MetavarContext
lctx : LocalContext
opts : Options
/-- A naming context is the information needed to shorten names in pretty printing.
It gives the current namespace and the list of open declarations.
-/
structure NamingContext where
currNamespace : Name
openDecls : List OpenDecl
/-- Structured message data. We use it for reporting errors, trace messages, etc. -/
inductive MessageData where
| ofFormat : Format → MessageData
| ofSyntax : Syntax → MessageData
| ofExpr : Expr → MessageData
| ofLevel : Level → MessageData
| ofName : Name → MessageData
| ofGoal : MVarId → MessageData
/-- `withContext ctx d` specifies the pretty printing context `(env, mctx, lctx, opts)` for the nested expressions in `d`. -/
| withContext : MessageDataContext → MessageData → MessageData
| withNamingContext : NamingContext → MessageData → MessageData
/-- Lifted `Format.nest` -/
| nest : Nat → MessageData → MessageData
/-- Lifted `Format.group` -/
| group : MessageData → MessageData
/-- Lifted `Format.compose` -/
| compose : MessageData → MessageData → MessageData
/-- Tagged sections. `Name` should be viewed as a "kind", and is used by `MessageData` inspector functions.
Example: an inspector that tries to find "definitional equality failures" may look for the tag "DefEqFailure". -/
| tagged : Name → MessageData → MessageData
| trace (cls : Name) (msg : MessageData) (children : Array MessageData)
(collapsed : Bool := false)
deriving Inhabited
namespace MessageData
/-- Determines whether the message contains any content. -/
def isEmpty : MessageData → Bool
| ofFormat f => f.isEmpty
| withContext _ m => m.isEmpty
| withNamingContext _ m => m.isEmpty
| nest _ m => m.isEmpty
| group m => m.isEmpty
| compose m₁ m₂ => m₁.isEmpty && m₂.isEmpty
| tagged _ m => m.isEmpty
| _ => false
/-- Instantiate metavariables occurring in nexted `ofExpr` constructors.
It uses the surrounding `MetavarContext` at `withContext` constructors.
-/
partial def instantiateMVars (msg : MessageData) : MessageData :=
visit msg {}
where
visit (msg : MessageData) (mctx : MetavarContext) : MessageData :=
match msg with
| ofExpr e => ofExpr <| instantiateMVarsCore mctx e |>.1
| withContext ctx msg => withContext ctx <| visit msg ctx.mctx
| withNamingContext ctx msg => withNamingContext ctx <| visit msg mctx
| nest n msg => nest n <| visit msg mctx
| group msg => group <| visit msg mctx
| compose msg₁ msg₂ => compose (visit msg₁ mctx) <| visit msg₂ mctx
| tagged n msg => tagged n <| visit msg mctx
| trace cls msg msgs c => trace cls (visit msg mctx) (msgs.map (visit · mctx)) c
| _ => msg
variable (p : Name → Bool) in
/-- Returns true when the message contains a `MessageData.tagged tag ..` constructor where `p tag` is true. -/
partial def hasTag : MessageData → Bool
| withContext _ msg => hasTag msg
| withNamingContext _ msg => hasTag msg
| nest _ msg => hasTag msg
| group msg => hasTag msg
| compose msg₁ msg₂ => hasTag msg₁ || hasTag msg₂
| tagged n msg => p n || hasTag msg
| trace cls msg msgs _ => p cls || hasTag msg || msgs.any hasTag
| _ => false
/-- An empty message. -/
def nil : MessageData :=
ofFormat Format.nil
/-- Whether the given message equals `MessageData.nil`. See also `MessageData.isEmpty`. -/
def isNil : MessageData → Bool
| ofFormat Format.nil => true
| _ => false
/-- Whether the message is a `MessageData.nest` constructor. -/
def isNest : MessageData → Bool
| nest _ _ => true
| _ => false
def mkPPContext (nCtx : NamingContext) (ctx : MessageDataContext) : PPContext := {
env := ctx.env, mctx := ctx.mctx, lctx := ctx.lctx, opts := ctx.opts,
currNamespace := nCtx.currNamespace, openDecls := nCtx.openDecls
}
partial def formatAux : NamingContext → Option MessageDataContext → MessageData → IO Format
| _, _, ofFormat fmt => return fmt
| _, _, ofLevel u => return format u
| _, _, ofName n => return format n
| nCtx, some ctx, ofSyntax s => ppTerm (mkPPContext nCtx ctx) ⟨s⟩ -- HACK: might not be a term
| _, none, ofSyntax s => return s.formatStx
| _, none, ofExpr e => return format (toString e)
| nCtx, some ctx, ofExpr e => ppExpr (mkPPContext nCtx ctx) e
| _, none, ofGoal mvarId => return "goal " ++ format (mkMVar mvarId)
| nCtx, some ctx, ofGoal mvarId => ppGoal (mkPPContext nCtx ctx) mvarId
| nCtx, _, withContext ctx d => formatAux nCtx ctx d
| _, ctx, withNamingContext nCtx d => formatAux nCtx ctx d
| nCtx, ctx, tagged _ d => formatAux nCtx ctx d
| nCtx, ctx, nest n d => Format.nest n <$> formatAux nCtx ctx d
| nCtx, ctx, compose d₁ d₂ => return (← formatAux nCtx ctx d₁) ++ (← formatAux nCtx ctx d₂)
| nCtx, ctx, group d => Format.group <$> formatAux nCtx ctx d
| nCtx, ctx, trace cls header children _ => do
let msg := f!"[{cls}] {(← formatAux nCtx ctx header).nest 2}"
let children ← children.mapM (formatAux nCtx ctx)
return .nest 2 (.joinSep (msg::children.toList) "\n")
protected def format (msgData : MessageData) : IO Format :=
formatAux { currNamespace := Name.anonymous, openDecls := [] } none msgData
protected def toString (msgData : MessageData) : IO String := do
return toString (← msgData.format)
instance : Append MessageData := ⟨compose⟩
instance : Coe String MessageData := ⟨ofFormat ∘ format⟩
instance : Coe Format MessageData := ⟨ofFormat⟩
instance : Coe Level MessageData := ⟨ofLevel⟩
instance : Coe Expr MessageData := ⟨ofExpr⟩
instance : Coe Name MessageData := ⟨ofName⟩
instance : Coe Syntax MessageData := ⟨ofSyntax⟩
instance : Coe MVarId MessageData := ⟨ofGoal⟩
instance : Coe (Option Expr) MessageData := ⟨fun o => match o with | none => "none" | some e => ofExpr e⟩
partial def arrayExpr.toMessageData (es : Array Expr) (i : Nat) (acc : MessageData) : MessageData :=
if h : i < es.size then
let e := es.get ⟨i, h⟩;
let acc := if i == 0 then acc ++ ofExpr e else acc ++ ", " ++ ofExpr e;
toMessageData es (i+1) acc
else
acc ++ "]"
instance : Coe (Array Expr) MessageData := ⟨fun es => arrayExpr.toMessageData es 0 "#["⟩
/-- Wrap the given message in `l` and `r`. See also `Format.bracket`. -/
def bracket (l : String) (f : MessageData) (r : String) : MessageData := group (nest l.length <| l ++ f ++ r)
/-- Wrap the given message in parentheses `()`. -/
def paren (f : MessageData) : MessageData := bracket "(" f ")"
/-- Wrap the given message in square brackets `[]`. -/
def sbracket (f : MessageData) : MessageData := bracket "[" f "]"
/-- Append the given list of messages with the given separarator. -/
def joinSep : List MessageData → MessageData → MessageData
| [], _ => Format.nil
| [a], _ => a
| a::as, sep => a ++ sep ++ joinSep as sep
/-- Write the given list of messages as a list, separating each item with `,\n` and surrounding with square brackets. -/
def ofList: List MessageData → MessageData
| [] => "[]"
| xs => sbracket <| joinSep xs (ofFormat "," ++ Format.line)
/-- See `MessageData.ofList`. -/
def ofArray (msgs : Array MessageData) : MessageData :=
ofList msgs.toList
instance : Coe (List MessageData) MessageData := ⟨ofList⟩
instance : Coe (List Expr) MessageData := ⟨fun es => ofList <| es.map ofExpr⟩
end MessageData
/-- A `Message` is a richly formatted piece of information emitted by Lean.
They are rendered by client editors in the infoview and in diagnostic windows. -/
structure Message where
fileName : String
pos : Position
endPos : Option Position := none
severity : MessageSeverity := MessageSeverity.error
caption : String := ""
/-- The content of the message. -/
data : MessageData
deriving Inhabited
namespace Message
protected def toString (msg : Message) (includeEndPos := false) : IO String := do
let mut str ← msg.data.toString
let endPos := if includeEndPos then msg.endPos else none
unless msg.caption == "" do
str := msg.caption ++ ":\n" ++ str
match msg.severity with
| MessageSeverity.information => pure ()
| MessageSeverity.warning => str := mkErrorStringWithPos msg.fileName msg.pos (endPos := endPos) "warning: " ++ str
| MessageSeverity.error => str := mkErrorStringWithPos msg.fileName msg.pos (endPos := endPos) "error: " ++ str
if str.isEmpty || str.back != '\n' then
str := str ++ "\n"
return str
end Message
/-- A persistent array of messages. -/
structure MessageLog where
msgs : PersistentArray Message := {}
deriving Inhabited
namespace MessageLog
def empty : MessageLog := ⟨{}⟩
def isEmpty (log : MessageLog) : Bool :=
log.msgs.isEmpty
def add (msg : Message) (log : MessageLog) : MessageLog :=
⟨log.msgs.push msg⟩
protected def append (l₁ l₂ : MessageLog) : MessageLog :=
⟨l₁.msgs ++ l₂.msgs⟩
instance : Append MessageLog :=
⟨MessageLog.append⟩
def hasErrors (log : MessageLog) : Bool :=
log.msgs.any fun m => match m.severity with
| MessageSeverity.error => true
| _ => false
def errorsToWarnings (log : MessageLog) : MessageLog :=
{ msgs := log.msgs.map (fun m => match m.severity with | MessageSeverity.error => { m with severity := MessageSeverity.warning } | _ => m) }
def getInfoMessages (log : MessageLog) : MessageLog :=
{ msgs := log.msgs.filter fun m => match m.severity with | MessageSeverity.information => true | _ => false }
def forM {m : Type → Type} [Monad m] (log : MessageLog) (f : Message → m Unit) : m Unit :=
log.msgs.forM f
def toList (log : MessageLog) : List Message :=
(log.msgs.foldl (fun acc msg => msg :: acc) []).reverse
end MessageLog
def MessageData.nestD (msg : MessageData) : MessageData :=
MessageData.nest 2 msg
def indentD (msg : MessageData) : MessageData :=
MessageData.nestD (Format.line ++ msg)
def indentExpr (e : Expr) : MessageData :=
indentD e
class AddMessageContext (m : Type → Type) where
addMessageContext : MessageData → m MessageData
export AddMessageContext (addMessageContext)
instance (m n) [MonadLift m n] [AddMessageContext m] : AddMessageContext n where
addMessageContext := fun msg => liftM (addMessageContext msg : m _)
def addMessageContextPartial {m} [Monad m] [MonadEnv m] [MonadOptions m] (msgData : MessageData) : m MessageData := do
let env ← getEnv
let opts ← getOptions
return MessageData.withContext { env := env, mctx := {}, lctx := {}, opts := opts } msgData
def addMessageContextFull {m} [Monad m] [MonadEnv m] [MonadMCtx m] [MonadLCtx m] [MonadOptions m] (msgData : MessageData) : m MessageData := do
let env ← getEnv
let mctx ← getMCtx
let lctx ← getLCtx
let opts ← getOptions
return MessageData.withContext { env := env, mctx := mctx, lctx := lctx, opts := opts } msgData
class ToMessageData (α : Type) where
toMessageData : α → MessageData
export ToMessageData (toMessageData)
def stringToMessageData (str : String) : MessageData :=
let lines := str.split (· == '\n')
let lines := lines.map (MessageData.ofFormat ∘ format)
MessageData.joinSep lines (MessageData.ofFormat Format.line)
instance [ToFormat α] : ToMessageData α := ⟨MessageData.ofFormat ∘ format⟩
instance : ToMessageData Expr := ⟨MessageData.ofExpr⟩
instance : ToMessageData Level := ⟨MessageData.ofLevel⟩
instance : ToMessageData Name := ⟨MessageData.ofName⟩
instance : ToMessageData String := ⟨stringToMessageData⟩
instance : ToMessageData Syntax := ⟨MessageData.ofSyntax⟩
instance : ToMessageData (TSyntax k) := ⟨(MessageData.ofSyntax ·)⟩
instance : ToMessageData Format := ⟨MessageData.ofFormat⟩
instance : ToMessageData MVarId := ⟨MessageData.ofGoal⟩
instance : ToMessageData MessageData := ⟨id⟩
instance [ToMessageData α] : ToMessageData (List α) := ⟨fun as => MessageData.ofList <| as.map toMessageData⟩
instance [ToMessageData α] : ToMessageData (Array α) := ⟨fun as => toMessageData as.toList⟩
instance [ToMessageData α] : ToMessageData (Subarray α) := ⟨fun as => toMessageData as.toArray.toList⟩
instance [ToMessageData α] : ToMessageData (Option α) := ⟨fun | none => "none" | some e => "some (" ++ toMessageData e ++ ")"⟩
instance : ToMessageData (Option Expr) := ⟨fun | none => "<not-available>" | some e => toMessageData e⟩
syntax:max "m!" interpolatedStr(term) : term
macro_rules
| `(m! $interpStr) => do interpStr.expandInterpolatedStr (← `(MessageData)) (← `(toMessageData))
def toMessageList (msgs : Array MessageData) : MessageData :=
indentD (MessageData.joinSep msgs.toList m!"\n\n")
namespace KernelException
private def mkCtx (env : Environment) (lctx : LocalContext) (opts : Options) (msg : MessageData) : MessageData :=
MessageData.withContext { env := env, mctx := {}, lctx := lctx, opts := opts } msg
def toMessageData (e : KernelException) (opts : Options) : MessageData :=
match e with
| unknownConstant env constName => mkCtx env {} opts m!"(kernel) unknown constant '{constName}'"
| alreadyDeclared env constName => mkCtx env {} opts m!"(kernel) constant has already been declared '{constName}'"
| declTypeMismatch env decl givenType =>
mkCtx env {} opts <|
let process (n : Name) (expectedType : Expr) : MessageData :=
m!"(kernel) declaration type mismatch, '{n}' has type{indentExpr givenType}\nbut it is expected to have type{indentExpr expectedType}";
match decl with
| Declaration.defnDecl { name := n, type := type, .. } => process n type
| Declaration.thmDecl { name := n, type := type, .. } => process n type
| _ => "(kernel) declaration type mismatch" -- TODO fix type checker, type mismatch for mutual decls does not have enough information
| declHasMVars env constName _ => mkCtx env {} opts m!"(kernel) declaration has metavariables '{constName}'"
| declHasFVars env constName _ => mkCtx env {} opts m!"(kernel) declaration has free variables '{constName}'"
| funExpected env lctx e => mkCtx env lctx opts m!"(kernel) function expected{indentExpr e}"
| typeExpected env lctx e => mkCtx env lctx opts m!"(kernel) type expected{indentExpr e}"
| letTypeMismatch env lctx n _ _ => mkCtx env lctx opts m!"(kernel) let-declaration type mismatch '{n}'"
| exprTypeMismatch env lctx e _ => mkCtx env lctx opts m!"(kernel) type mismatch at{indentExpr e}"
| appTypeMismatch env lctx e fnType argType =>
mkCtx env lctx opts m!"application type mismatch{indentExpr e}\nargument has type{indentExpr argType}\nbut function has type{indentExpr fnType}"
| invalidProj env lctx e => mkCtx env lctx opts m!"(kernel) invalid projection{indentExpr e}"
| other msg => m!"(kernel) {msg}"
end KernelException
end Lean
|
0f5fc4781209d74817f7461d32c4aecc9da71175 | e21db629d2e37a833531fdcb0b37ce4d71825408 | /src/default.lean | fb4d91a9654bf571de9285c448b8dd81d6792ab0 | [] | no_license | fischerman/GPU-transformation-verifier | 614a28cb4606a05a0eb27e8d4eab999f4f5ea60c | 75a5016f05382738ff93ce5859c4cfa47ccb63c1 | refs/heads/master | 1,586,985,789,300 | 1,579,290,514,000 | 1,579,290,514,000 | 165,031,073 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 45 | lean | import .parlang
import .mcl
import .syncablep |
649fce5a53015241da7a966e77b73d6edf7471b3 | 8eeb99d0fdf8125f5d39a0ce8631653f588ee817 | /src/measure_theory/measurable_space.lean | d5c30b52629c8ffd08c1d3da7b7301a59258c002 | [
"Apache-2.0"
] | permissive | jesse-michael-han/mathlib | a15c58378846011b003669354cbab7062b893cfe | fa6312e4dc971985e6b7708d99a5bc3062485c89 | refs/heads/master | 1,625,200,760,912 | 1,602,081,753,000 | 1,602,081,753,000 | 181,787,230 | 0 | 0 | null | 1,555,460,682,000 | 1,555,460,682,000 | null | UTF-8 | Lean | false | false | 54,273 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import data.set.disjointed
import data.set.countable
import data.indicator_function
import data.equiv.encodable.lattice
import order.filter.basic
/-!
# Measurable spaces and measurable functions
This file defines measurable spaces and the functions and isomorphisms
between them.
A measurable space is a set equipped with a σ-algebra, a collection of
subsets closed under complementation and countable union. A function
between measurable spaces is measurable if the preimage of each
measurable subset is measurable.
σ-algebras on a fixed set `α` form a complete lattice. Here we order
σ-algebras by writing `m₁ ≤ m₂` if every set which is `m₁`-measurable is
also `m₂`-measurable (that is, `m₁` is a subset of `m₂`). In particular, any
collection of subsets of `α` generates a smallest σ-algebra which
contains all of them. A function `f : α → β` induces a Galois connection
between the lattices of σ-algebras on `α` and `β`.
A measurable equivalence between measurable spaces is an equivalence
which respects the σ-algebras, that is, for which both directions of
the equivalence are measurable functions.
We say that a filter `f` is measurably generated if every set `s ∈ f` includes a measurable
set `t ∈ f`. This property is useful, e.g., to extract a measurable witness of `filter.eventually`.
## Main statements
The main theorem of this file is Dynkin's π-λ theorem, which appears
here as an induction principle `induction_on_inter`. Suppose `s` is a
collection of subsets of `α` such that the intersection of two members
of `s` belongs to `s` whenever it is nonempty. Let `m` be the σ-algebra
generated by `s`. In order to check that a predicate `C` holds on every
member of `m`, it suffices to check that `C` holds on the members of `s` and
that `C` is preserved by complementation and *disjoint* countable
unions.
## Implementation notes
Measurability of a function `f : α → β` between measurable spaces is
defined in terms of the Galois connection induced by f.
## References
* <https://en.wikipedia.org/wiki/Measurable_space>
* <https://en.wikipedia.org/wiki/Sigma-algebra>
* <https://en.wikipedia.org/wiki/Dynkin_system>
## Tags
measurable space, measurable function, dynkin system
-/
local attribute [instance] classical.prop_decidable
open set encodable function
open_locale classical filter
universes u v w x
variables {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} {ι : Sort x}
{s t u : set α}
/-- A measurable space is a space equipped with a σ-algebra. -/
structure measurable_space (α : Type u) :=
(is_measurable' : set α → Prop)
(is_measurable_empty : is_measurable' ∅)
(is_measurable_compl : ∀s, is_measurable' s → is_measurable' sᶜ)
(is_measurable_Union : ∀f:ℕ → set α, (∀i, is_measurable' (f i)) → is_measurable' (⋃i, f i))
attribute [class] measurable_space
section
variable [measurable_space α]
/-- `is_measurable s` means that `s` is measurable (in the ambient measure space on `α`) -/
def is_measurable : set α → Prop := ‹measurable_space α›.is_measurable'
@[simp] lemma is_measurable.empty : is_measurable (∅ : set α) :=
‹measurable_space α›.is_measurable_empty
lemma is_measurable.compl : is_measurable s → is_measurable sᶜ :=
‹measurable_space α›.is_measurable_compl s
lemma is_measurable.of_compl (h : is_measurable sᶜ) : is_measurable s :=
s.compl_compl ▸ h.compl
@[simp] lemma is_measurable.compl_iff : is_measurable sᶜ ↔ is_measurable s :=
⟨is_measurable.of_compl, is_measurable.compl⟩
@[simp] lemma is_measurable.univ : is_measurable (univ : set α) :=
by simpa using (@is_measurable.empty α _).compl
lemma subsingleton.is_measurable [subsingleton α] {s : set α} : is_measurable s :=
subsingleton.set_cases is_measurable.empty is_measurable.univ s
lemma is_measurable.congr {s t : set α} (hs : is_measurable s) (h : s = t) :
is_measurable t :=
by rwa ← h
lemma is_measurable.bUnion_decode2 [encodable β] ⦃f : β → set α⦄ (h : ∀ b, is_measurable (f b))
(n : ℕ) : is_measurable (⋃ b ∈ decode2 β n, f b) :=
encodable.Union_decode2_cases is_measurable.empty h
lemma is_measurable.Union [encodable β] ⦃f : β → set α⦄ (h : ∀b, is_measurable (f b)) :
is_measurable (⋃b, f b) :=
begin
rw ← encodable.Union_decode2,
exact ‹measurable_space α›.is_measurable_Union _ (is_measurable.bUnion_decode2 h)
end
lemma is_measurable.bUnion {f : β → set α} {s : set β} (hs : countable s)
(h : ∀b∈s, is_measurable (f b)) : is_measurable (⋃b∈s, f b) :=
begin
rw bUnion_eq_Union,
haveI := hs.to_encodable,
exact is_measurable.Union (by simpa using h)
end
lemma is_measurable.sUnion {s : set (set α)} (hs : countable s) (h : ∀t∈s, is_measurable t) :
is_measurable (⋃₀ s) :=
by { rw sUnion_eq_bUnion, exact is_measurable.bUnion hs h }
lemma is_measurable.Union_Prop {p : Prop} {f : p → set α} (hf : ∀b, is_measurable (f b)) :
is_measurable (⋃b, f b) :=
by { by_cases p; simp [h, hf, is_measurable.empty] }
lemma is_measurable.Inter [encodable β] {f : β → set α} (h : ∀b, is_measurable (f b)) :
is_measurable (⋂b, f b) :=
is_measurable.compl_iff.1 $
by { rw compl_Inter, exact is_measurable.Union (λ b, (h b).compl) }
lemma is_measurable.bInter {f : β → set α} {s : set β} (hs : countable s)
(h : ∀b∈s, is_measurable (f b)) : is_measurable (⋂b∈s, f b) :=
is_measurable.compl_iff.1 $
by { rw compl_bInter, exact is_measurable.bUnion hs (λ b hb, (h b hb).compl) }
lemma is_measurable.sInter {s : set (set α)} (hs : countable s) (h : ∀t∈s, is_measurable t) :
is_measurable (⋂₀ s) :=
by { rw sInter_eq_bInter, exact is_measurable.bInter hs h }
lemma is_measurable.Inter_Prop {p : Prop} {f : p → set α} (hf : ∀b, is_measurable (f b)) :
is_measurable (⋂b, f b) :=
by { by_cases p; simp [h, hf, is_measurable.univ] }
@[simp] lemma is_measurable.union {s₁ s₂ : set α} (h₁ : is_measurable s₁) (h₂ : is_measurable s₂) :
is_measurable (s₁ ∪ s₂) :=
by { rw union_eq_Union, exact is_measurable.Union (bool.forall_bool.2 ⟨h₂, h₁⟩) }
@[simp] lemma is_measurable.inter {s₁ s₂ : set α} (h₁ : is_measurable s₁) (h₂ : is_measurable s₂) :
is_measurable (s₁ ∩ s₂) :=
by { rw inter_eq_compl_compl_union_compl, exact (h₁.compl.union h₂.compl).compl }
@[simp] lemma is_measurable.diff {s₁ s₂ : set α} (h₁ : is_measurable s₁) (h₂ : is_measurable s₂) :
is_measurable (s₁ \ s₂) :=
h₁.inter h₂.compl
@[simp] lemma is_measurable.disjointed {f : ℕ → set α} (h : ∀i, is_measurable (f i)) (n) :
is_measurable (disjointed f n) :=
disjointed_induct (h n) (assume t i ht, is_measurable.diff ht $ h _)
@[simp] lemma is_measurable.const (p : Prop) : is_measurable {a : α | p} :=
by { by_cases p; simp [h, is_measurable.empty]; apply is_measurable.univ }
end
@[ext] lemma measurable_space.ext :
∀{m₁ m₂ : measurable_space α}, (∀s:set α, m₁.is_measurable' s ↔ m₂.is_measurable' s) → m₁ = m₂
| ⟨s₁, _, _, _⟩ ⟨s₂, _, _, _⟩ h :=
have s₁ = s₂, from funext $ assume x, propext $ h x,
by subst this
@[ext] lemma measurable_space.ext_iff {m₁ m₂ : measurable_space α} :
m₁ = m₂ ↔ (∀s:set α, m₁.is_measurable' s ↔ m₂.is_measurable' s) :=
⟨by { unfreezingI {rintro rfl}, intro s, refl }, measurable_space.ext⟩
/-- A typeclass mixin for `measurable_space`s such that each singleton is measurable. -/
class measurable_singleton_class (α : Type*) [measurable_space α] : Prop :=
(is_measurable_singleton : ∀ x, is_measurable ({x} : set α))
export measurable_singleton_class (is_measurable_singleton)
attribute [simp] is_measurable_singleton
section measurable_singleton_class
variables [measurable_space α] [measurable_singleton_class α]
lemma is_measurable_eq {a : α} : is_measurable {x | x = a} :=
is_measurable_singleton a
lemma is_measurable.insert {s : set α} (hs : is_measurable s) (a : α) :
is_measurable (insert a s) :=
(is_measurable_singleton a).union hs
@[simp] lemma is_measurable_insert {a : α} {s : set α} :
is_measurable (insert a s) ↔ is_measurable s :=
⟨λ h, if ha : a ∈ s then by rwa ← insert_eq_of_mem ha
else insert_diff_self_of_not_mem ha ▸ h.diff (is_measurable_singleton _),
λ h, h.insert a⟩
lemma set.finite.is_measurable {s : set α} (hs : finite s) : is_measurable s :=
finite.induction_on hs is_measurable.empty $ λ a s ha hsf hsm, hsm.insert _
protected lemma finset.is_measurable (s : finset α) : is_measurable (↑s : set α) :=
s.finite_to_set.is_measurable
end measurable_singleton_class
namespace measurable_space
section complete_lattice
instance : partial_order (measurable_space α) :=
{ le := λm₁ m₂, m₁.is_measurable' ≤ m₂.is_measurable',
le_refl := assume a b, le_refl _,
le_trans := assume a b c, le_trans,
le_antisymm := assume a b h₁ h₂, measurable_space.ext $ assume s, ⟨h₁ s, h₂ s⟩ }
/-- The smallest σ-algebra containing a collection `s` of basic sets -/
inductive generate_measurable (s : set (set α)) : set α → Prop
| basic : ∀u∈s, generate_measurable u
| empty : generate_measurable ∅
| compl : ∀s, generate_measurable s → generate_measurable sᶜ
| union : ∀f:ℕ → set α, (∀n, generate_measurable (f n)) → generate_measurable (⋃i, f i)
/-- Construct the smallest measure space containing a collection of basic sets -/
def generate_from (s : set (set α)) : measurable_space α :=
{ is_measurable' := generate_measurable s,
is_measurable_empty := generate_measurable.empty,
is_measurable_compl := generate_measurable.compl,
is_measurable_Union := generate_measurable.union }
lemma is_measurable_generate_from {s : set (set α)} {t : set α} (ht : t ∈ s) :
(generate_from s).is_measurable' t :=
generate_measurable.basic t ht
lemma generate_from_le {s : set (set α)} {m : measurable_space α} (h : ∀t∈s, m.is_measurable' t) :
generate_from s ≤ m :=
assume t (ht : generate_measurable s t), ht.rec_on h
(is_measurable_empty m)
(assume s _ hs, is_measurable_compl m s hs)
(assume f _ hf, is_measurable_Union m f hf)
lemma generate_from_le_iff {s : set (set α)} (m : measurable_space α) :
generate_from s ≤ m ↔ s ⊆ {t | m.is_measurable' t} :=
iff.intro
(assume h u hu, h _ $ is_measurable_generate_from hu)
(assume h, generate_from_le h)
/-- If `g` is a collection of subsets of `α` such that the `σ`-algebra generated from `g` contains the
same sets as `g`, then `g` was already a `σ`-algebra. -/
protected def mk_of_closure (g : set (set α)) (hg : {t | (generate_from g).is_measurable' t} = g) :
measurable_space α :=
{ is_measurable' := λs, s ∈ g,
is_measurable_empty := hg ▸ is_measurable_empty _,
is_measurable_compl := hg ▸ is_measurable_compl _,
is_measurable_Union := hg ▸ is_measurable_Union _ }
lemma mk_of_closure_sets {s : set (set α)}
{hs : {t | (generate_from s).is_measurable' t} = s} :
measurable_space.mk_of_closure s hs = generate_from s :=
measurable_space.ext $ assume t, show t ∈ s ↔ _, by { conv_lhs { rw [← hs] }, refl }
/-- We get a Galois insertion between `σ`-algebras on `α` and `set (set α)` by using `generate_from`
on one side and the collection of measurable sets on the other side. -/
def gi_generate_from : galois_insertion (@generate_from α) (λm, {t | @is_measurable α m t}) :=
{ gc := assume s, generate_from_le_iff,
le_l_u := assume m s, is_measurable_generate_from,
choice :=
λg hg, measurable_space.mk_of_closure g $ le_antisymm hg $ (generate_from_le_iff _).1 le_rfl,
choice_eq := assume g hg, mk_of_closure_sets }
instance : complete_lattice (measurable_space α) :=
gi_generate_from.lift_complete_lattice
instance : inhabited (measurable_space α) := ⟨⊤⟩
lemma is_measurable_bot_iff {s : set α} : @is_measurable α ⊥ s ↔ (s = ∅ ∨ s = univ) :=
let b : measurable_space α :=
{ is_measurable' := λs, s = ∅ ∨ s = univ,
is_measurable_empty := or.inl rfl,
is_measurable_compl := by simp [or_imp_distrib] {contextual := tt},
is_measurable_Union := assume f hf, classical.by_cases
(assume h : ∃i, f i = univ,
let ⟨i, hi⟩ := h in
or.inr $ eq_univ_of_univ_subset $ hi ▸ le_supr f i)
(assume h : ¬ ∃i, f i = univ,
or.inl $ eq_empty_of_subset_empty $ Union_subset $ assume i,
(hf i).elim (by simp {contextual := tt}) (assume hi, false.elim $ h ⟨i, hi⟩)) } in
have b = ⊥, from bot_unique $ assume s hs,
hs.elim (assume s, s.symm ▸ @is_measurable_empty _ ⊥) (assume s, s.symm ▸ @is_measurable.univ _ ⊥),
this ▸ iff.rfl
@[simp] theorem is_measurable_top {s : set α} : @is_measurable _ ⊤ s := trivial
@[simp] theorem is_measurable_inf {m₁ m₂ : measurable_space α} {s : set α} :
@is_measurable _ (m₁ ⊓ m₂) s ↔ @is_measurable _ m₁ s ∧ @is_measurable _ m₂ s :=
iff.rfl
@[simp] theorem is_measurable_Inf {ms : set (measurable_space α)} {s : set α} :
@is_measurable _ (Inf ms) s ↔ ∀ m ∈ ms, @is_measurable _ m s :=
show s ∈ (⋂m∈ms, {t | @is_measurable _ m t }) ↔ _, by simp
@[simp] theorem is_measurable_infi {ι} {m : ι → measurable_space α} {s : set α} :
@is_measurable _ (infi m) s ↔ ∀ i, @is_measurable _ (m i) s :=
show s ∈ (λm, {s | @is_measurable _ m s }) (infi m) ↔ _,
by { rw (@gi_generate_from α).gc.u_infi, simp }
theorem is_measurable_sup {m₁ m₂ : measurable_space α} {s : set α} :
@is_measurable _ (m₁ ⊔ m₂) s ↔ generate_measurable (m₁.is_measurable' ∪ m₂.is_measurable') s :=
iff.refl _
theorem is_measurable_Sup {ms : set (measurable_space α)} {s : set α} :
@is_measurable _ (Sup ms) s ↔ generate_measurable (⋃₀ (measurable_space.is_measurable' '' ms)) s :=
begin
change @is_measurable' _ (generate_from _) _ ↔ _,
dsimp [generate_from],
rw (show (⨆ (b : measurable_space α) (H : b ∈ ms), set_of (@is_measurable _ b)) = (⋃₀ (is_measurable' '' ms)),
{ ext,
simp only [exists_prop, mem_Union, sUnion_image, mem_set_of_eq],
refl, })
end
theorem is_measurable_supr {ι} {m : ι → measurable_space α} {s : set α} :
@is_measurable _ (supr m) s ↔ generate_measurable (⋃i, (m i).is_measurable') s :=
begin
convert @is_measurable_Sup _ (range m) s,
simp,
end
end complete_lattice
section functors
variables {m m₁ m₂ : measurable_space α} {m' : measurable_space β} {f : α → β} {g : β → α}
/-- The forward image of a measure space under a function. `map f m` contains the sets `s : set β`
whose preimage under `f` is measurable. -/
protected def map (f : α → β) (m : measurable_space α) : measurable_space β :=
{ is_measurable' := λs, m.is_measurable' $ f ⁻¹' s,
is_measurable_empty := m.is_measurable_empty,
is_measurable_compl := assume s hs, m.is_measurable_compl _ hs,
is_measurable_Union := assume f hf, by { rw [preimage_Union], exact m.is_measurable_Union _ hf }}
@[simp] lemma map_id : m.map id = m :=
measurable_space.ext $ assume s, iff.rfl
@[simp] lemma map_comp {f : α → β} {g : β → γ} : (m.map f).map g = m.map (g ∘ f) :=
measurable_space.ext $ assume s, iff.rfl
/-- The reverse image of a measure space under a function. `comap f m` contains the sets `s : set α`
such that `s` is the `f`-preimage of a measurable set in `β`. -/
protected def comap (f : α → β) (m : measurable_space β) : measurable_space α :=
{ is_measurable' := λs, ∃s', m.is_measurable' s' ∧ f ⁻¹' s' = s,
is_measurable_empty := ⟨∅, m.is_measurable_empty, rfl⟩,
is_measurable_compl := assume s ⟨s', h₁, h₂⟩, ⟨s'ᶜ, m.is_measurable_compl _ h₁, h₂ ▸ rfl⟩,
is_measurable_Union := assume s hs,
let ⟨s', hs'⟩ := classical.axiom_of_choice hs in
⟨⋃i, s' i, m.is_measurable_Union _ (λi, (hs' i).left), by simp [hs'] ⟩ }
@[simp] lemma comap_id : m.comap id = m :=
measurable_space.ext $ assume s, ⟨assume ⟨s', hs', h⟩, h ▸ hs', assume h, ⟨s, h, rfl⟩⟩
@[simp] lemma comap_comp {f : β → α} {g : γ → β} : (m.comap f).comap g = m.comap (f ∘ g) :=
measurable_space.ext $ assume s,
⟨assume ⟨t, ⟨u, h, hu⟩, ht⟩, ⟨u, h, ht ▸ hu ▸ rfl⟩, assume ⟨t, h, ht⟩, ⟨f ⁻¹' t, ⟨_, h, rfl⟩, ht⟩⟩
lemma comap_le_iff_le_map {f : α → β} : m'.comap f ≤ m ↔ m' ≤ m.map f :=
⟨assume h s hs, h _ ⟨_, hs, rfl⟩, assume h s ⟨t, ht, heq⟩, heq ▸ h _ ht⟩
lemma gc_comap_map (f : α → β) :
galois_connection (measurable_space.comap f) (measurable_space.map f) :=
assume f g, comap_le_iff_le_map
lemma map_mono (h : m₁ ≤ m₂) : m₁.map f ≤ m₂.map f := (gc_comap_map f).monotone_u h
lemma monotone_map : monotone (measurable_space.map f) := assume a b h, map_mono h
lemma comap_mono (h : m₁ ≤ m₂) : m₁.comap g ≤ m₂.comap g := (gc_comap_map g).monotone_l h
lemma monotone_comap : monotone (measurable_space.comap g) := assume a b h, comap_mono h
@[simp] lemma comap_bot : (⊥:measurable_space α).comap g = ⊥ := (gc_comap_map g).l_bot
@[simp] lemma comap_sup : (m₁ ⊔ m₂).comap g = m₁.comap g ⊔ m₂.comap g := (gc_comap_map g).l_sup
@[simp] lemma comap_supr {m : ι → measurable_space α} :(⨆i, m i).comap g = (⨆i, (m i).comap g) :=
(gc_comap_map g).l_supr
@[simp] lemma map_top : (⊤:measurable_space α).map f = ⊤ := (gc_comap_map f).u_top
@[simp] lemma map_inf : (m₁ ⊓ m₂).map f = m₁.map f ⊓ m₂.map f := (gc_comap_map f).u_inf
@[simp] lemma map_infi {m : ι → measurable_space α} : (⨅i, m i).map f = (⨅i, (m i).map f) :=
(gc_comap_map f).u_infi
lemma comap_map_le : (m.map f).comap f ≤ m := (gc_comap_map f).l_u_le _
lemma le_map_comap : m ≤ (m.comap g).map g := (gc_comap_map g).le_u_l _
end functors
lemma generate_from_le_generate_from {s t : set (set α)} (h : s ⊆ t) :
generate_from s ≤ generate_from t :=
gi_generate_from.gc.monotone_l h
lemma generate_from_sup_generate_from {s t : set (set α)} :
generate_from s ⊔ generate_from t = generate_from (s ∪ t) :=
(@gi_generate_from α).gc.l_sup.symm
lemma comap_generate_from {f : α → β} {s : set (set β)} :
(generate_from s).comap f = generate_from (preimage f '' s) :=
le_antisymm
(comap_le_iff_le_map.2 $ generate_from_le $ assume t hts,
generate_measurable.basic _ $ mem_image_of_mem _ $ hts)
(generate_from_le $ assume t ⟨u, hu, eq⟩, eq ▸ ⟨u, generate_measurable.basic _ hu, rfl⟩)
end measurable_space
section measurable_functions
open measurable_space
/-- A function `f` between measurable spaces is measurable if the preimage of every
measurable set is measurable. -/
def measurable [measurable_space α] [measurable_space β] (f : α → β) : Prop :=
∀ ⦃t : set β⦄, is_measurable t → is_measurable (f ⁻¹' t)
lemma measurable_iff_le_map {m₁ : measurable_space α} {m₂ : measurable_space β} {f : α → β} :
measurable f ↔ m₂ ≤ m₁.map f :=
iff.rfl
alias measurable_iff_le_map ↔ measurable.le_map measurable.of_le_map
lemma measurable_iff_comap_le {m₁ : measurable_space α} {m₂ : measurable_space β} {f : α → β} :
measurable f ↔ m₂.comap f ≤ m₁ :=
comap_le_iff_le_map.symm
alias measurable_iff_comap_le ↔ measurable.comap_le measurable.of_comap_le
lemma subsingleton.measurable [measurable_space α] [measurable_space β] [subsingleton α]
{f : α → β} : measurable f :=
λ s hs, @subsingleton.is_measurable α _ _ _
lemma measurable_id [measurable_space α] : measurable (@id α) := λ t, id
lemma measurable.comp [measurable_space α] [measurable_space β] [measurable_space γ]
{g : β → γ} {f : α → β} (hg : measurable g) (hf : measurable f) : measurable (g ∘ f) :=
λ t ht, hf (hg ht)
lemma measurable_from_top [measurable_space β] {f : α → β} :
@measurable _ _ ⊤ _ f :=
λ s hs, trivial
lemma measurable.mono {ma ma' : measurable_space α} {mb mb' : measurable_space β} {f : α → β}
(hf : @measurable α β ma mb f) (ha : ma ≤ ma') (hb : mb' ≤ mb) :
@measurable α β ma' mb' f :=
λ t ht, ha _ $ hf $ hb _ ht
lemma measurable_generate_from [measurable_space α] {s : set (set β)} {f : α → β}
(h : ∀t∈s, is_measurable (f ⁻¹' t)) : @measurable _ _ _ (generate_from s) f :=
measurable.of_le_map $ generate_from_le h
lemma measurable.piecewise [measurable_space α] [measurable_space β]
{s : set α} {_ : decidable_pred s} {f g : α → β}
(hs : is_measurable s) (hf : measurable f) (hg : measurable g) :
measurable (piecewise s f g) :=
begin
intros t ht,
simp only [piecewise_preimage],
exact (hs.inter $ hf ht).union (hs.compl.inter $ hg ht)
end
@[simp] lemma measurable_const {α β} [measurable_space α] [measurable_space β] {a : α} :
measurable (λb:β, a) :=
by { intros s hs, by_cases a ∈ s; simp [*, preimage] }
lemma measurable.indicator [measurable_space α] [measurable_space β] [has_zero β]
{s : set α} {f : α → β} (hf : measurable f) (hs : is_measurable s) :
measurable (s.indicator f) :=
hf.piecewise hs measurable_const
@[to_additive]
lemma measurable_one {α β} [measurable_space α] [has_one α] [measurable_space β] :
measurable (1 : β → α) := @measurable_const _ _ _ _ 1
end measurable_functions
section constructions
instance : measurable_space empty := ⊤
instance : measurable_space unit := ⊤
instance : measurable_space bool := ⊤
instance : measurable_space ℕ := ⊤
instance : measurable_space ℤ := ⊤
instance : measurable_space ℚ := ⊤
lemma measurable_to_encodable [encodable α] [measurable_space α] [measurable_space β] {f : β → α}
(h : ∀ y, is_measurable (f ⁻¹' {f y})) : measurable f :=
begin
assume s hs,
rw [← bUnion_preimage_singleton],
refine is_measurable.Union (λ y, is_measurable.Union_Prop $ λ hy, _),
by_cases hyf : y ∈ range f,
{ rcases hyf with ⟨y, rfl⟩,
apply h },
{ simp only [preimage_singleton_eq_empty.2 hyf, is_measurable.empty] }
end
lemma measurable_unit [measurable_space α] (f : unit → α) : measurable f :=
have f = (λu, f ()) := funext $ assume ⟨⟩, rfl,
by { rw this, exact measurable_const }
section nat
lemma measurable_from_nat [measurable_space α] {f : ℕ → α} : measurable f :=
measurable_from_top
lemma measurable_to_nat [measurable_space α] {f : α → ℕ} :
(∀ y, is_measurable (f ⁻¹' {f y})) → measurable f :=
measurable_to_encodable
lemma measurable_find_greatest' [measurable_space α] {p : α → ℕ → Prop}
{N} (hN : ∀ k ≤ N, is_measurable {x | nat.find_greatest (p x) N = k}) :
measurable (λ x, nat.find_greatest (p x) N) :=
measurable_to_nat $ λ x, hN _ nat.find_greatest_le
lemma measurable_find_greatest [measurable_space α] {p : α → ℕ → Prop}
{N} (hN : ∀ k ≤ N, is_measurable {x | p x k}) :
measurable (λ x, nat.find_greatest (p x) N) :=
begin
refine measurable_find_greatest' (λ k hk, _),
simp only [nat.find_greatest_eq_iff, set_of_and, set_of_forall, ← compl_set_of],
repeat { apply_rules [is_measurable.inter, is_measurable.const, is_measurable.Inter,
is_measurable.Inter_Prop, is_measurable.compl, hN]; try { intros } }
end
lemma measurable_find [measurable_space α] {p : α → ℕ → Prop} (hp : ∀ x, ∃ N, p x N)
(hm : ∀ k, is_measurable {x | p x k}) :
measurable (λ x, nat.find (hp x)) :=
begin
refine measurable_to_nat (λ x, _),
simp only [set.preimage, mem_singleton_iff, nat.find_eq_iff, set_of_and, set_of_forall,
← compl_set_of],
repeat { apply_rules [is_measurable.inter, hm, is_measurable.Inter, is_measurable.Inter_Prop,
is_measurable.compl]; try { intros } }
end
end nat
section subtype
instance {p : α → Prop} [m : measurable_space α] : measurable_space (subtype p) :=
m.comap (coe : _ → α)
lemma measurable_subtype_coe [measurable_space α] {p : α → Prop} :
measurable (coe : subtype p → α) :=
measurable_space.le_map_comap
lemma measurable.subtype_coe [measurable_space α] [measurable_space β] {p : β → Prop}
{f : α → subtype p} (hf : measurable f) : measurable (λa:α, (f a : β)) :=
measurable_subtype_coe.comp hf
lemma measurable.subtype_mk [measurable_space α] [measurable_space β] {p : β → Prop}
{f : α → β} (hf : measurable f) {h : ∀ x, p (f x)} :
measurable (λ x, (⟨f x, h x⟩ : subtype p)) :=
λ t ⟨s, hs⟩, hs.2 ▸ by simp only [← preimage_comp, (∘), subtype.coe_mk, hf hs.1]
lemma is_measurable.subtype_image [measurable_space α] {s : set α} {t : set s}
(hs : is_measurable s) : is_measurable t → is_measurable ((coe : s → α) '' t)
| ⟨u, (hu : is_measurable u), (eq : coe ⁻¹' u = t)⟩ :=
begin
rw [← eq, subtype.image_preimage_coe],
exact hu.inter hs
end
lemma measurable_of_measurable_union_cover
[measurable_space α] [measurable_space β]
{f : α → β} (s t : set α) (hs : is_measurable s) (ht : is_measurable t) (h : univ ⊆ s ∪ t)
(hc : measurable (λa:s, f a)) (hd : measurable (λa:t, f a)) :
measurable f :=
begin
intros u hu,
convert (hs.subtype_image (hc hu)).union (ht.subtype_image (hd hu)),
change f ⁻¹' u = coe '' (coe ⁻¹' (f ⁻¹' u) : set s) ∪ coe '' (coe ⁻¹' (f ⁻¹' u) : set t),
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, subtype.range_coe,
subtype.range_coe, ← inter_distrib_left, univ_subset_iff.1 h, inter_univ],
end
lemma measurable_of_measurable_on_compl_singleton [measurable_space α] [measurable_space β]
[measurable_singleton_class α]
{f : α → β} (a : α) (hf : measurable (set.restrict f {x | x ≠ a})) :
measurable f :=
measurable_of_measurable_union_cover _ _ is_measurable_eq is_measurable_eq.compl
(λ x hx, classical.em _)
(@subsingleton.measurable {x | x = a} _ _ _ ⟨λ x y, subtype.eq $ x.2.trans y.2.symm⟩ _) hf
end subtype
section prod
instance [m₁ : measurable_space α] [m₂ : measurable_space β] : measurable_space (α × β) :=
m₁.comap prod.fst ⊔ m₂.comap prod.snd
variables [measurable_space α] [measurable_space β] [measurable_space γ]
lemma measurable_fst : measurable (prod.fst : α × β → α) :=
measurable.of_comap_le le_sup_left
lemma measurable.fst {f : α → β × γ} (hf : measurable f) : measurable (λa:α, (f a).1) :=
measurable_fst.comp hf
lemma measurable_snd : measurable (prod.snd : α × β → β) :=
measurable.of_comap_le le_sup_right
lemma measurable.snd {f : α → β × γ} (hf : measurable f) : measurable (λa:α, (f a).2) :=
measurable_snd.comp hf
lemma measurable.prod {f : α → β × γ}
(hf₁ : measurable (λa, (f a).1)) (hf₂ : measurable (λa, (f a).2)) : measurable f :=
measurable.of_le_map $ sup_le
(by { rw [measurable_space.comap_le_iff_le_map, measurable_space.map_comp], exact hf₁ })
(by { rw [measurable_space.comap_le_iff_le_map, measurable_space.map_comp], exact hf₂ })
lemma measurable_prod {f : α → β × γ} : measurable f ↔
measurable (λa, (f a).1) ∧ measurable (λa, (f a).2) :=
⟨λ hf, ⟨measurable_fst.comp hf, measurable_snd.comp hf⟩, λ h, measurable.prod h.1 h.2⟩
lemma measurable.prod_mk {f : α → β} {g : α → γ} (hf : measurable f) (hg : measurable g) :
measurable (λa:α, (f a, g a)) :=
measurable.prod hf hg
lemma measurable_prod_mk_left {x : α} : measurable (@prod.mk _ β x) :=
measurable_const.prod_mk measurable_id
lemma measurable_prod_mk_right {y : β} : measurable (λ x : α, (x, y)) :=
measurable_id.prod_mk measurable_const
lemma measurable.of_uncurry_left {f : α → β → γ} (hf : measurable (uncurry f)) {x : α} :
measurable (f x) :=
hf.comp measurable_prod_mk_left
lemma measurable.of_uncurry_right {f : α → β → γ} (hf : measurable (uncurry f)) {y : β} :
measurable (λ x, f x y) :=
hf.comp measurable_prod_mk_right
lemma measurable_swap : measurable (prod.swap : α × β → β × α) :=
measurable.prod measurable_snd measurable_fst
lemma measurable_swap_iff {f : α × β → γ} : measurable (f ∘ prod.swap) ↔ measurable f :=
⟨λ hf, by { convert hf.comp measurable_swap, ext ⟨x, y⟩, refl }, λ hf, hf.comp measurable_swap⟩
lemma is_measurable.prod {s : set α} {t : set β} (hs : is_measurable s) (ht : is_measurable t) :
is_measurable (s.prod t) :=
is_measurable.inter (measurable_fst hs) (measurable_snd ht)
lemma is_measurable_prod_of_nonempty {s : set α} {t : set β} (h : (s.prod t).nonempty) :
is_measurable (s.prod t) ↔ is_measurable s ∧ is_measurable t :=
begin
rcases h with ⟨⟨x, y⟩, hx, hy⟩,
refine ⟨λ hst, _, λ h, h.1.prod h.2⟩,
have : is_measurable ((λ x, (x, y)) ⁻¹' s.prod t) := measurable_id.prod_mk measurable_const hst,
have : is_measurable (prod.mk x ⁻¹' s.prod t) := measurable_const.prod_mk measurable_id hst,
simp * at *
end
lemma is_measurable_prod {s : set α} {t : set β} :
is_measurable (s.prod t) ↔ (is_measurable s ∧ is_measurable t) ∨ s = ∅ ∨ t = ∅ :=
begin
cases (s.prod t).eq_empty_or_nonempty with h h,
{ simp [h, prod_eq_empty_iff.mp h] },
{ simp [←not_nonempty_iff_eq_empty, prod_nonempty_iff.mp h, is_measurable_prod_of_nonempty h] }
end
lemma is_measurable_swap_iff {s : set (α × β)} :
is_measurable (prod.swap ⁻¹' s) ↔ is_measurable s :=
⟨λ hs, by { convert measurable_swap hs, ext ⟨x, y⟩, refl }, λ hs, measurable_swap hs⟩
end prod
section pi
instance measurable_space.pi {α : Type u} {β : α → Type v} [m : Πa, measurable_space (β a)] :
measurable_space (Πa, β a) :=
⨆a, (m a).comap (λb, b a)
lemma measurable_pi_apply {α : Type u} {β : α → Type v} [Πa, measurable_space (β a)] (a : α) :
measurable (λf:Πa, β a, f a) :=
measurable.of_comap_le $ le_supr _ a
lemma measurable_pi_lambda {α : Type u} {β : α → Type v} {γ : Type w}
[Πa, measurable_space (β a)] [measurable_space γ]
(f : γ → Πa, β a) (hf : ∀a, measurable (λc, f c a)) :
measurable f :=
measurable.of_le_map $ supr_le $ assume a, measurable_space.comap_le_iff_le_map.2 (hf a)
end pi
instance [m₁ : measurable_space α] [m₂ : measurable_space β] : measurable_space (α ⊕ β) :=
m₁.map sum.inl ⊓ m₂.map sum.inr
section sum
variables [measurable_space α] [measurable_space β] [measurable_space γ]
lemma measurable_inl : measurable (@sum.inl α β) := measurable.of_le_map inf_le_left
lemma measurable_inr : measurable (@sum.inr α β) := measurable.of_le_map inf_le_right
lemma measurable_sum {f : α ⊕ β → γ}
(hl : measurable (f ∘ sum.inl)) (hr : measurable (f ∘ sum.inr)) : measurable f :=
measurable.of_comap_le $ le_inf
(measurable_space.comap_le_iff_le_map.2 $ hl)
(measurable_space.comap_le_iff_le_map.2 $ hr)
lemma measurable.sum_rec {f : α → γ} {g : β → γ}
(hf : measurable f) (hg : measurable g) : @measurable (α ⊕ β) γ _ _ (@sum.rec α β (λ_, γ) f g) :=
measurable_sum hf hg
lemma is_measurable.inl_image {s : set α} (hs : is_measurable s) :
is_measurable (sum.inl '' s : set (α ⊕ β)) :=
⟨show is_measurable (sum.inl ⁻¹' _), by { rwa [preimage_image_eq], exact (λ a b, sum.inl.inj) },
have sum.inr ⁻¹' (sum.inl '' s : set (α ⊕ β)) = ∅ :=
eq_empty_of_subset_empty $ assume x ⟨y, hy, eq⟩, by contradiction,
show is_measurable (sum.inr ⁻¹' _), by { rw [this], exact is_measurable.empty }⟩
lemma is_measurable_range_inl : is_measurable (range sum.inl : set (α ⊕ β)) :=
by { rw [← image_univ], exact is_measurable.univ.inl_image }
lemma is_measurable_inr_image {s : set β} (hs : is_measurable s) :
is_measurable (sum.inr '' s : set (α ⊕ β)) :=
⟨ have sum.inl ⁻¹' (sum.inr '' s : set (α ⊕ β)) = ∅ :=
eq_empty_of_subset_empty $ assume x ⟨y, hy, eq⟩, by contradiction,
show is_measurable (sum.inl ⁻¹' _), by { rw [this], exact is_measurable.empty },
show is_measurable (sum.inr ⁻¹' _), by { rwa [preimage_image_eq], exact λ a b, sum.inr.inj }⟩
lemma is_measurable_range_inr : is_measurable (range sum.inr : set (α ⊕ β)) :=
by { rw [← image_univ], exact is_measurable_inr_image is_measurable.univ }
end sum
instance {β : α → Type v} [m : Πa, measurable_space (β a)] : measurable_space (sigma β) :=
⨅a, (m a).map (sigma.mk a)
end constructions
/-- Equivalences between measurable spaces. Main application is the simplification of measurability
statements along measurable equivalences. -/
structure measurable_equiv (α β : Type*) [measurable_space α] [measurable_space β] extends α ≃ β :=
(measurable_to_fun : measurable to_fun)
(measurable_inv_fun : measurable inv_fun)
namespace measurable_equiv
instance (α β) [measurable_space α] [measurable_space β] : has_coe_to_fun (measurable_equiv α β) :=
⟨λ_, α → β, λe, e.to_equiv⟩
lemma coe_eq {α β} [measurable_space α] [measurable_space β] (e : measurable_equiv α β) :
(e : α → β) = e.to_equiv := rfl
/-- Any measurable space is equivalent to itself. -/
def refl (α : Type*) [measurable_space α] : measurable_equiv α α :=
{ to_equiv := equiv.refl α,
measurable_to_fun := measurable_id, measurable_inv_fun := measurable_id }
instance {α} [measurable_space α] : inhabited (measurable_equiv α α) := ⟨refl α⟩
/-- The composition of equivalences between measurable spaces. -/
def trans [measurable_space α] [measurable_space β] [measurable_space γ]
(ab : measurable_equiv α β) (bc : measurable_equiv β γ) :
measurable_equiv α γ :=
{ to_equiv := ab.to_equiv.trans bc.to_equiv,
measurable_to_fun := bc.measurable_to_fun.comp ab.measurable_to_fun,
measurable_inv_fun := ab.measurable_inv_fun.comp bc.measurable_inv_fun }
lemma trans_to_equiv {α β} [measurable_space α] [measurable_space β] [measurable_space γ]
(e : measurable_equiv α β) (f : measurable_equiv β γ) :
(e.trans f).to_equiv = e.to_equiv.trans f.to_equiv := rfl
/-- The inverse of an equivalence between measurable spaces. -/
def symm [measurable_space α] [measurable_space β] (ab : measurable_equiv α β) :
measurable_equiv β α :=
{ to_equiv := ab.to_equiv.symm,
measurable_to_fun := ab.measurable_inv_fun,
measurable_inv_fun := ab.measurable_to_fun }
lemma symm_to_equiv {α β} [measurable_space α] [measurable_space β] (e : measurable_equiv α β) :
e.symm.to_equiv = e.to_equiv.symm := rfl
/-- Equal measurable spaces are equivalent. -/
protected def cast {α β} [i₁ : measurable_space α] [i₂ : measurable_space β]
(h : α = β) (hi : i₁ == i₂) : measurable_equiv α β :=
{ to_equiv := equiv.cast h,
measurable_to_fun := by { substI h, substI hi, exact measurable_id },
measurable_inv_fun := by { substI h, substI hi, exact measurable_id }}
protected lemma measurable {α β} [measurable_space α] [measurable_space β]
(e : measurable_equiv α β) : measurable (e : α → β) :=
e.measurable_to_fun
protected lemma measurable_coe_iff {α β γ} [measurable_space α] [measurable_space β] [measurable_space γ]
{f : β → γ} (e : measurable_equiv α β) : measurable (f ∘ e) ↔ measurable f :=
iff.intro
(assume hfe,
have measurable (f ∘ (e.symm.trans e).to_equiv) := hfe.comp e.symm.measurable,
by rwa [trans_to_equiv, symm_to_equiv, equiv.symm_trans] at this)
(λh, h.comp e.measurable)
/-- Products of equivalent measurable spaces are equivalent. -/
def prod_congr [measurable_space α] [measurable_space β] [measurable_space γ] [measurable_space δ]
(ab : measurable_equiv α β) (cd : measurable_equiv γ δ) :
measurable_equiv (α × γ) (β × δ) :=
{ to_equiv := equiv.prod_congr ab.to_equiv cd.to_equiv,
measurable_to_fun := measurable.prod_mk
(ab.measurable_to_fun.comp (measurable.fst measurable_id))
(cd.measurable_to_fun.comp (measurable.snd measurable_id)),
measurable_inv_fun := measurable.prod_mk
(ab.measurable_inv_fun.comp (measurable.fst measurable_id))
(cd.measurable_inv_fun.comp (measurable.snd measurable_id)) }
/-- Products of measurable spaces are symmetric. -/
def prod_comm [measurable_space α] [measurable_space β] : measurable_equiv (α × β) (β × α) :=
{ to_equiv := equiv.prod_comm α β,
measurable_to_fun := measurable.prod_mk (measurable.snd measurable_id) (measurable.fst measurable_id),
measurable_inv_fun := measurable.prod_mk (measurable.snd measurable_id) (measurable.fst measurable_id) }
/-- Sums of measurable spaces are symmetric. -/
def sum_congr [measurable_space α] [measurable_space β] [measurable_space γ] [measurable_space δ]
(ab : measurable_equiv α β) (cd : measurable_equiv γ δ) :
measurable_equiv (α ⊕ γ) (β ⊕ δ) :=
{ to_equiv := equiv.sum_congr ab.to_equiv cd.to_equiv,
measurable_to_fun :=
begin
cases ab with ab' abm, cases ab', cases cd with cd' cdm, cases cd',
refine measurable_sum (measurable_inl.comp abm) (measurable_inr.comp cdm)
end,
measurable_inv_fun :=
begin
cases ab with ab' _ abm, cases ab', cases cd with cd' _ cdm, cases cd',
refine measurable_sum (measurable_inl.comp abm) (measurable_inr.comp cdm)
end }
/-- `set.prod s t ≃ (s × t)` as measurable spaces. -/
def set.prod [measurable_space α] [measurable_space β] (s : set α) (t : set β) :
measurable_equiv (s.prod t) (s × t) :=
{ to_equiv := equiv.set.prod s t,
measurable_to_fun := measurable.prod_mk
measurable_id.subtype_coe.fst.subtype_mk
measurable_id.subtype_coe.snd.subtype_mk,
measurable_inv_fun := measurable.subtype_mk $ measurable.prod_mk
measurable_id.fst.subtype_coe
measurable_id.snd.subtype_coe }
/-- `univ α ≃ α` as measurable spaces. -/
def set.univ (α : Type*) [measurable_space α] : measurable_equiv (univ : set α) α :=
{ to_equiv := equiv.set.univ α,
measurable_to_fun := measurable_id.subtype_coe,
measurable_inv_fun := measurable_id.subtype_mk }
/-- `{a} ≃ unit` as measurable spaces. -/
def set.singleton [measurable_space α] (a:α) : measurable_equiv ({a} : set α) unit :=
{ to_equiv := equiv.set.singleton a,
measurable_to_fun := measurable_const,
measurable_inv_fun := measurable_const }
/-- A set is equivalent to its image under a function `f` as measurable spaces,
if `f` is an injective measurable function that sends measurable sets to measurable sets. -/
noncomputable def set.image [measurable_space α] [measurable_space β]
(f : α → β) (s : set α)
(hf : function.injective f)
(hfm : measurable f) (hfi : ∀s, is_measurable s → is_measurable (f '' s)) :
measurable_equiv s (f '' s) :=
{ to_equiv := equiv.set.image f s hf,
measurable_to_fun := (hfm.comp measurable_id.subtype_coe).subtype_mk,
measurable_inv_fun :=
begin
rintro t ⟨u, hu, rfl⟩, simp [preimage_preimage, equiv.set.image_symm_preimage hf],
exact measurable_subtype_coe (hfi u hu)
end }
/-- The domain of `f` is equivalent to its range as measurable spaces,
if `f` is an injective measurable function that sends measurable sets to measurable sets. -/
noncomputable def set.range [measurable_space α] [measurable_space β]
(f : α → β) (hf : function.injective f) (hfm : measurable f)
(hfi : ∀s, is_measurable s → is_measurable (f '' s)) :
measurable_equiv α (range f) :=
(measurable_equiv.set.univ _).symm.trans $
(measurable_equiv.set.image f univ hf hfm hfi).trans $
measurable_equiv.cast (by rw image_univ) (by rw image_univ)
/-- `α` is equivalent to its image in `α ⊕ β` as measurable spaces. -/
def set.range_inl [measurable_space α] [measurable_space β] :
measurable_equiv (range sum.inl : set (α ⊕ β)) α :=
{ to_fun := λab, match ab with
| ⟨sum.inl a, _⟩ := a
| ⟨sum.inr b, p⟩ := have false, by { cases p, contradiction }, this.elim
end,
inv_fun := λa, ⟨sum.inl a, a, rfl⟩,
left_inv := by { rintro ⟨ab, a, rfl⟩, refl },
right_inv := assume a, rfl,
measurable_to_fun := assume s (hs : is_measurable s),
begin
refine ⟨_, hs.inl_image, set.ext _⟩,
rintros ⟨ab, a, rfl⟩,
simp [set.range_inl._match_1]
end,
measurable_inv_fun := measurable.subtype_mk measurable_inl }
/-- `β` is equivalent to its image in `α ⊕ β` as measurable spaces. -/
def set.range_inr [measurable_space α] [measurable_space β] :
measurable_equiv (range sum.inr : set (α ⊕ β)) β :=
{ to_fun := λab, match ab with
| ⟨sum.inr b, _⟩ := b
| ⟨sum.inl a, p⟩ := have false, by { cases p, contradiction }, this.elim
end,
inv_fun := λb, ⟨sum.inr b, b, rfl⟩,
left_inv := by { rintro ⟨ab, b, rfl⟩, refl },
right_inv := assume b, rfl,
measurable_to_fun := assume s (hs : is_measurable s),
begin
refine ⟨_, is_measurable_inr_image hs, set.ext _⟩,
rintros ⟨ab, b, rfl⟩,
simp [set.range_inr._match_1]
end,
measurable_inv_fun := measurable.subtype_mk measurable_inr }
/-- Products distribute over sums (on the right) as measurable spaces. -/
def sum_prod_distrib (α β γ) [measurable_space α] [measurable_space β] [measurable_space γ] :
measurable_equiv ((α ⊕ β) × γ) ((α × γ) ⊕ (β × γ)) :=
{ to_equiv := equiv.sum_prod_distrib α β γ,
measurable_to_fun :=
begin
refine measurable_of_measurable_union_cover
((range sum.inl).prod univ)
((range sum.inr).prod univ)
(is_measurable_range_inl.prod is_measurable.univ)
(is_measurable_range_inr.prod is_measurable.univ)
(by { rintro ⟨a|b, c⟩; simp [set.prod_eq] })
_
_,
{ refine (set.prod (range sum.inl) univ).symm.measurable_coe_iff.1 _,
refine (prod_congr set.range_inl (set.univ _)).symm.measurable_coe_iff.1 _,
dsimp [(∘)],
convert measurable_inl,
ext ⟨a, c⟩, refl },
{ refine (set.prod (range sum.inr) univ).symm.measurable_coe_iff.1 _,
refine (prod_congr set.range_inr (set.univ _)).symm.measurable_coe_iff.1 _,
dsimp [(∘)],
convert measurable_inr,
ext ⟨b, c⟩, refl }
end,
measurable_inv_fun :=
measurable_sum
((measurable_inl.comp (measurable.fst measurable_id)).prod_mk (measurable.snd measurable_id))
((measurable_inr.comp (measurable.fst measurable_id)).prod_mk (measurable.snd measurable_id)) }
/-- Products distribute over sums (on the left) as measurable spaces. -/
def prod_sum_distrib (α β γ) [measurable_space α] [measurable_space β] [measurable_space γ] :
measurable_equiv (α × (β ⊕ γ)) ((α × β) ⊕ (α × γ)) :=
prod_comm.trans $ (sum_prod_distrib _ _ _).trans $ sum_congr prod_comm prod_comm
/-- Products distribute over sums as measurable spaces. -/
def sum_prod_sum (α β γ δ)
[measurable_space α] [measurable_space β] [measurable_space γ] [measurable_space δ] :
measurable_equiv ((α ⊕ β) × (γ ⊕ δ)) (((α × γ) ⊕ (α × δ)) ⊕ ((β × γ) ⊕ (β × δ))) :=
(sum_prod_distrib _ _ _).trans $ sum_congr (prod_sum_distrib _ _ _) (prod_sum_distrib _ _ _)
end measurable_equiv
/-- A pi-system is a collection of subsets of `α` that is closed under intersections of sets that
are not disjoint. Usually it is also required that the collection is nonempty, but we don't do
that here. -/
def is_pi_system {α} (C : set (set α)) : Prop :=
∀ s t ∈ C, (s ∩ t : set α).nonempty → s ∩ t ∈ C
namespace measurable_space
/-- A Dynkin system is a collection of subsets of a type `α` that contains the empty set,
is closed under complementation and under countable union of pairwise disjoint sets.
The disjointness condition is the only difference with `σ`-algebras.
The main purpose of Dynkin systems is to provide a powerful induction rule for σ-algebras
generated by intersection stable set systems.
A Dynkin system is also known as a "λ-system" or a "d-system".
-/
structure dynkin_system (α : Type*) :=
(has : set α → Prop)
(has_empty : has ∅)
(has_compl : ∀{a}, has a → has aᶜ)
(has_Union_nat : ∀{f:ℕ → set α}, pairwise (disjoint on f) → (∀i, has (f i)) → has (⋃i, f i))
namespace dynkin_system
@[ext] lemma ext : ∀{d₁ d₂ : dynkin_system α}, (∀s:set α, d₁.has s ↔ d₂.has s) → d₁ = d₂
| ⟨s₁, _, _, _⟩ ⟨s₂, _, _, _⟩ h := have s₁ = s₂, from funext $ assume x, propext $ h x, by subst this
variable (d : dynkin_system α)
lemma has_compl_iff {a} : d.has aᶜ ↔ d.has a :=
⟨λ h, by simpa using d.has_compl h, λ h, d.has_compl h⟩
lemma has_univ : d.has univ :=
by simpa using d.has_compl d.has_empty
theorem has_Union {β} [encodable β] {f : β → set α}
(hd : pairwise (disjoint on f)) (h : ∀i, d.has (f i)) : d.has (⋃i, f i) :=
by { rw ← encodable.Union_decode2, exact
d.has_Union_nat (Union_decode2_disjoint_on hd)
(λ n, encodable.Union_decode2_cases d.has_empty h) }
theorem has_union {s₁ s₂ : set α}
(h₁ : d.has s₁) (h₂ : d.has s₂) (h : s₁ ∩ s₂ ⊆ ∅) : d.has (s₁ ∪ s₂) :=
by { rw union_eq_Union, exact
d.has_Union (pairwise_disjoint_on_bool.2 h) (bool.forall_bool.2 ⟨h₂, h₁⟩) }
lemma has_diff {s₁ s₂ : set α} (h₁ : d.has s₁) (h₂ : d.has s₂) (h : s₂ ⊆ s₁) : d.has (s₁ \ s₂) :=
begin
apply d.has_compl_iff.1,
simp [diff_eq, compl_inter],
exact d.has_union (d.has_compl h₁) h₂ (λ x ⟨h₁, h₂⟩, h₁ (h h₂)),
end
instance : partial_order (dynkin_system α) :=
{ le := λm₁ m₂, m₁.has ≤ m₂.has,
le_refl := assume a b, le_refl _,
le_trans := assume a b c, le_trans,
le_antisymm := assume a b h₁ h₂, ext $ assume s, ⟨h₁ s, h₂ s⟩ }
/-- Every measurable space (σ-algebra) forms a Dynkin system -/
def of_measurable_space (m : measurable_space α) : dynkin_system α :=
{ has := m.is_measurable',
has_empty := m.is_measurable_empty,
has_compl := m.is_measurable_compl,
has_Union_nat := assume f _ hf, m.is_measurable_Union f hf }
lemma of_measurable_space_le_of_measurable_space_iff {m₁ m₂ : measurable_space α} :
of_measurable_space m₁ ≤ of_measurable_space m₂ ↔ m₁ ≤ m₂ :=
iff.rfl
/-- The least Dynkin system containing a collection of basic sets.
This inductive type gives the underlying collection of sets. -/
inductive generate_has (s : set (set α)) : set α → Prop
| basic : ∀t∈s, generate_has t
| empty : generate_has ∅
| compl : ∀{a}, generate_has a → generate_has aᶜ
| Union : ∀{f:ℕ → set α}, pairwise (disjoint on f) →
(∀i, generate_has (f i)) → generate_has (⋃i, f i)
lemma generate_has_compl {C : set (set α)} {s : set α} : generate_has C sᶜ ↔ generate_has C s :=
by { refine ⟨_, generate_has.compl⟩, intro h, convert generate_has.compl h, simp }
/-- The least Dynkin system containing a collection of basic sets. -/
def generate (s : set (set α)) : dynkin_system α :=
{ has := generate_has s,
has_empty := generate_has.empty,
has_compl := assume a, generate_has.compl,
has_Union_nat := assume f, generate_has.Union }
lemma generate_has_def {C : set (set α)} : (generate C).has = generate_has C := rfl
instance : inhabited (dynkin_system α) := ⟨generate univ⟩
/-- If a Dynkin system is closed under binary intersection, then it forms a `σ`-algebra. -/
def to_measurable_space (h_inter : ∀s₁ s₂, d.has s₁ → d.has s₂ → d.has (s₁ ∩ s₂)) :=
{ measurable_space .
is_measurable' := d.has,
is_measurable_empty := d.has_empty,
is_measurable_compl := assume s h, d.has_compl h,
is_measurable_Union := assume f hf,
have ∀n, d.has (disjointed f n),
from assume n, disjointed_induct (hf n)
(assume t i h, h_inter _ _ h $ d.has_compl $ hf i),
have d.has (⋃n, disjointed f n), from d.has_Union disjoint_disjointed this,
by rwa [Union_disjointed] at this }
lemma of_measurable_space_to_measurable_space
(h_inter : ∀s₁ s₂, d.has s₁ → d.has s₂ → d.has (s₁ ∩ s₂)) :
of_measurable_space (d.to_measurable_space h_inter) = d :=
ext $ assume s, iff.rfl
/-- If `s` is in a Dynkin system `d`, we can form the new Dynkin system `{s ∩ t | t ∈ d}`. -/
def restrict_on {s : set α} (h : d.has s) : dynkin_system α :=
{ has := λt, d.has (t ∩ s),
has_empty := by simp [d.has_empty],
has_compl := assume t hts,
have tᶜ ∩ s = ((t ∩ s)ᶜ) \ sᶜ,
from set.ext $ assume x, by { by_cases x ∈ s; simp [h] },
by { rw [this], exact d.has_diff (d.has_compl hts) (d.has_compl h)
(compl_subset_compl.mpr $ inter_subset_right _ _) },
has_Union_nat := assume f hd hf,
begin
rw [inter_comm, inter_Union],
apply d.has_Union_nat,
{ exact λ i j h x ⟨⟨_, h₁⟩, _, h₂⟩, hd i j h ⟨h₁, h₂⟩ },
{ simpa [inter_comm] using hf },
end }
lemma generate_le {s : set (set α)} (h : ∀t∈s, d.has t) : generate s ≤ d :=
λ t ht, ht.rec_on h d.has_empty
(assume a _ h, d.has_compl h)
(assume f hd _ hf, d.has_Union hd hf)
lemma generate_has_subset_generate_measurable {C : set (set α)} {s : set α}
(hs : (generate C).has s) : (generate_from C).is_measurable' s :=
generate_le (of_measurable_space (generate_from C)) (λ t, is_measurable_generate_from) s hs
lemma generate_inter {s : set (set α)}
(hs : is_pi_system s) {t₁ t₂ : set α}
(ht₁ : (generate s).has t₁) (ht₂ : (generate s).has t₂) : (generate s).has (t₁ ∩ t₂) :=
have generate s ≤ (generate s).restrict_on ht₂,
from generate_le _ $ assume s₁ hs₁,
have (generate s).has s₁, from generate_has.basic s₁ hs₁,
have generate s ≤ (generate s).restrict_on this,
from generate_le _ $ assume s₂ hs₂,
show (generate s).has (s₂ ∩ s₁), from
(s₂ ∩ s₁).eq_empty_or_nonempty.elim
(λ h, h.symm ▸ generate_has.empty)
(λ h, generate_has.basic _ (hs _ _ hs₂ hs₁ h)),
have (generate s).has (t₂ ∩ s₁), from this _ ht₂,
show (generate s).has (s₁ ∩ t₂), by rwa [inter_comm],
this _ ht₁
/--
If we have a collection of sets closed under binary intersections, then the Dynkin system it
generates is equal to the σ-algebra it generates.
This result is known as the π-λ theorem.
A collection of sets closed under binary intersection is called a "π-system" if it is non-empty.
-/
lemma generate_from_eq {s : set (set α)} (hs : is_pi_system s) :
generate_from s = (generate s).to_measurable_space (assume t₁ t₂, generate_inter hs) :=
le_antisymm
(generate_from_le $ assume t ht, generate_has.basic t ht)
(of_measurable_space_le_of_measurable_space_iff.mp $
by { rw [of_measurable_space_to_measurable_space],
exact (generate_le _ $ assume t ht, is_measurable_generate_from ht) })
end dynkin_system
lemma induction_on_inter {C : set α → Prop} {s : set (set α)} [m : measurable_space α]
(h_eq : m = generate_from s)
(h_inter : is_pi_system s)
(h_empty : C ∅) (h_basic : ∀t∈s, C t) (h_compl : ∀t, is_measurable t → C t → C tᶜ)
(h_union : ∀f:ℕ → set α, pairwise (disjoint on f) →
(∀i, is_measurable (f i)) → (∀i, C (f i)) → C (⋃i, f i)) :
∀ ⦃t⦄, is_measurable t → C t :=
have eq : is_measurable = dynkin_system.generate_has s,
by { rw [h_eq, dynkin_system.generate_from_eq h_inter], refl },
assume t ht,
have dynkin_system.generate_has s t, by rwa [eq] at ht,
this.rec_on h_basic h_empty
(assume t ht, h_compl t $ by { rw [eq], exact ht })
(assume f hf ht, h_union f hf $ assume i, by { rw [eq], exact ht _ })
end measurable_space
namespace filter
variables [measurable_space α]
/-- A filter `f` is measurably generates if each `s ∈ f` includes a measurable `t ∈ f`. -/
class is_measurably_generated (f : filter α) : Prop :=
(exists_measurable_subset : ∀ ⦃s⦄, s ∈ f → ∃ t ∈ f, is_measurable t ∧ t ⊆ s)
instance is_measurably_generated_bot : is_measurably_generated (⊥ : filter α) :=
⟨λ _ _, ⟨∅, mem_bot_sets, is_measurable.empty, empty_subset _⟩⟩
instance is_measurably_generated_top : is_measurably_generated (⊤ : filter α) :=
⟨λ s hs, ⟨univ, univ_mem_sets, is_measurable.univ, λ x _, hs x⟩⟩
lemma eventually.exists_measurable_mem {f : filter α} [is_measurably_generated f]
{p : α → Prop} (h : ∀ᶠ x in f, p x) :
∃ s ∈ f, is_measurable s ∧ ∀ x ∈ s, p x :=
is_measurably_generated.exists_measurable_subset h
instance inf_is_measurably_generated (f g : filter α) [is_measurably_generated f]
[is_measurably_generated g] :
is_measurably_generated (f ⊓ g) :=
begin
refine ⟨_⟩,
rintros t ⟨sf, hsf, sg, hsg, ht⟩,
rcases is_measurably_generated.exists_measurable_subset hsf with ⟨s'f, hs'f, hmf, hs'sf⟩,
rcases is_measurably_generated.exists_measurable_subset hsg with ⟨s'g, hs'g, hmg, hs'sg⟩,
refine ⟨s'f ∩ s'g, inter_mem_inf_sets hs'f hs'g, hmf.inter hmg, _⟩,
exact subset.trans (inter_subset_inter hs'sf hs'sg) ht
end
lemma principal_is_measurably_generated_iff {s : set α} :
is_measurably_generated (𝓟 s) ↔ is_measurable s :=
begin
refine ⟨_, λ hs, ⟨λ t ht, ⟨s, mem_principal_self s, hs, ht⟩⟩⟩,
rintros ⟨hs⟩,
rcases hs (mem_principal_self s) with ⟨t, ht, htm, hts⟩,
have : t = s := subset.antisymm hts ht,
rwa ← this
end
alias principal_is_measurably_generated_iff ↔
_ is_measurable.principal_is_measurably_generated
instance infi_is_measurably_generated {f : ι → filter α} [∀ i, is_measurably_generated (f i)] :
is_measurably_generated (⨅ i, f i) :=
begin
refine ⟨λ s hs, _⟩,
rw [← equiv.plift.surjective.infi_comp, mem_infi_iff] at hs,
rcases hs with ⟨t, ht, ⟨V, hVf, hVs⟩⟩,
choose U hUf hU using λ i, is_measurably_generated.exists_measurable_subset (hVf i),
refine ⟨⋂ i : t, U i, _, _, _⟩,
{ rw [← equiv.plift.surjective.infi_comp, mem_infi_iff],
refine ⟨t, ht, U, hUf, subset.refl _⟩ },
{ haveI := ht.countable.to_encodable,
refine is_measurable.Inter (λ i, (hU i).1) },
{ exact subset.trans (Inter_subset_Inter $ λ i, (hU i).2) hVs }
end
end filter
|
2266741209b6e1c244fff70043d569befc83b852 | 969dbdfed67fda40a6f5a2b4f8c4a3c7dc01e0fb | /src/ring_theory/multiplicity.lean | d24cd1c1b16ea82fcae98cf356a1225bbbfc0c35 | [
"Apache-2.0"
] | permissive | SAAluthwela/mathlib | 62044349d72dd63983a8500214736aa7779634d3 | 83a4b8b990907291421de54a78988c024dc8a552 | refs/heads/master | 1,679,433,873,417 | 1,615,998,031,000 | 1,615,998,031,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 18,711 | lean | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import algebra.associated
import algebra.big_operators.basic
import ring_theory.valuation.basic
/-!
# Multiplicity of a divisor
For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves
several basic results on it.
## Main definitions
* `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest
number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers
`n`.
* `multiplicity.finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite.
-/
variables {α : Type*}
open nat roption
open_locale big_operators
/-- `multiplicity a b` returns the largest natural number `n` such that
`a ^ n ∣ b`, as an `enat` or natural with infinity. If `∀ n, a ^ n ∣ b`,
then it returns `⊤`-/
def multiplicity [comm_monoid α] [decidable_rel ((∣) : α → α → Prop)] (a b : α) : enat :=
enat.find $ λ n, ¬a ^ (n + 1) ∣ b
namespace multiplicity
section comm_monoid
variables [comm_monoid α]
/-- `multiplicity.finite a b` indicates that the multiplicity of `a` in `b` is finite. -/
@[reducible] def finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b
lemma finite_iff_dom [decidable_rel ((∣) : α → α → Prop)] {a b : α} :
finite a b ↔ (multiplicity a b).dom := iff.rfl
lemma finite_def {a b : α} : finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := iff.rfl
@[norm_cast]
theorem int.coe_nat_multiplicity (a b : ℕ) :
multiplicity (a : ℤ) (b : ℤ) = multiplicity a b :=
begin
apply roption.ext',
{ repeat {rw [← finite_iff_dom, finite_def]},
norm_cast },
{ intros h1 h2,
apply _root_.le_antisymm; { apply nat.find_le, norm_cast, simp }}
end
lemma not_finite_iff_forall {a b : α} : (¬ finite a b) ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨λ h n, nat.cases_on n (one_dvd _) (by simpa [finite, not_not] using h),
by simp [finite, multiplicity, not_not]; tauto⟩
lemma not_unit_of_finite {a b : α} (h : finite a b) : ¬is_unit a :=
let ⟨n, hn⟩ := h in mt (is_unit_iff_forall_dvd.1 ∘ is_unit.pow (n + 1)) $
λ h, hn (h b)
lemma finite_of_finite_mul_left {a b c : α} : finite a (b * c) → finite a c :=
λ ⟨n, hn⟩, ⟨n, λ h, hn (dvd.trans h (by simp [mul_pow]))⟩
lemma finite_of_finite_mul_right {a b c : α} : finite a (b * c) → finite a b :=
by rw mul_comm; exact finite_of_finite_mul_left
variable [decidable_rel ((∣) : α → α → Prop)]
lemma pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : enat) ≤ multiplicity a b → a ^ k ∣ b :=
nat.cases_on k (λ _, one_dvd _)
(λ k ⟨h₁, h₂⟩, by_contradiction (λ hk, (nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk)))
lemma pow_multiplicity_dvd {a b : α} (h : finite a b) : a ^ get (multiplicity a b) h ∣ b :=
pow_dvd_of_le_multiplicity (by rw enat.coe_get)
lemma is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b :=
λ h, by rw [enat.lt_coe_iff] at hm; exact nat.find_spec hm.fst (dvd.trans (pow_dvd_pow _ hm.snd) h)
lemma is_greatest' {a b : α} {m : ℕ} (h : finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b :=
is_greatest (by rwa [← enat.coe_lt_coe, enat.coe_get] at hm)
lemma unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : enat) = multiplicity a b :=
le_antisymm (le_of_not_gt (λ hk', is_greatest hk' hk)) $
have finite a b, from ⟨k, hsucc⟩,
by { rw [enat.le_coe_iff], exact ⟨this, nat.find_min' _ hsucc⟩ }
lemma unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬ a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ :=
by rw [← enat.coe_inj, enat.coe_get, unique hk hsucc]
lemma le_multiplicity_of_pow_dvd {a b : α}
{k : ℕ} (hk : a ^ k ∣ b) : (k : enat) ≤ multiplicity a b :=
le_of_not_gt $ λ hk', is_greatest hk' hk
lemma pow_dvd_iff_le_multiplicity {a b : α}
{k : ℕ} : a ^ k ∣ b ↔ (k : enat) ≤ multiplicity a b :=
⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩
lemma multiplicity_lt_iff_neg_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : enat) ↔ ¬ a ^ k ∣ b :=
by { rw [pow_dvd_iff_le_multiplicity, not_le] }
lemma eq_some_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : enat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b :=
⟨λ h, let ⟨h₁, h₂⟩ := eq_some_iff.1 h in
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest
(by { rw [enat.lt_coe_iff], exact ⟨h₁, lt_succ_self _⟩ })⟩,
λ h, eq_some_iff.2 ⟨⟨n, h.2⟩, eq.symm $ unique' h.1 h.2⟩⟩
lemma eq_top_iff {a b : α} :
multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(enat.find_eq_top_iff _).trans $
by { simp only [not_not], exact ⟨λ h n, nat.cases_on n (one_dvd _) (λ n, h _), λ h n, h _⟩ }
lemma one_right {a : α} (ha : ¬is_unit a) : multiplicity a 1 = 0 :=
eq_some_iff.2 ⟨dvd_refl _, mt is_unit_iff_dvd_one.2 $ by simpa⟩
@[simp] lemma get_one_right {a : α} (ha : finite a 1) : get (multiplicity a 1) ha = 0 :=
get_eq_iff_eq_some.2 (eq_some_iff.2 ⟨dvd_refl _,
by simpa [is_unit_iff_dvd_one.symm] using not_unit_of_finite ha⟩)
@[simp] lemma multiplicity_unit {a : α} (b : α) (ha : is_unit a) : multiplicity a b = ⊤ :=
eq_top_iff.2 (λ _, is_unit_iff_forall_dvd.1 (ha.pow _) _)
@[simp] lemma one_left (b : α) : multiplicity 1 b = ⊤ := by simp [eq_top_iff]
lemma multiplicity_eq_zero_of_not_dvd {a b : α} (ha : ¬a ∣ b) : multiplicity a b = 0 :=
eq_some_iff.2 (by simpa)
lemma eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬ finite a b :=
roption.eq_none_iff'
open_locale classical
lemma multiplicity_le_multiplicity_iff {a b c d : α} : multiplicity a b ≤ multiplicity c d ↔
(∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d) :=
⟨λ h n hab, (pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h)),
λ h, if hab : finite a b
then by rw [← enat.coe_get (finite_iff_dom.1 hab)];
exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _))
else
have ∀ n : ℕ, c ^ n ∣ d, from λ n, h n (not_finite_iff_forall.1 hab _),
by rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2
(not_finite_iff_forall.2 this)]⟩
lemma multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
multiplicity b c ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 $ λ n h, dvd_trans (pow_dvd_pow_of_dvd hdvd n) h
lemma eq_of_associated_left {a b c : α} (h : associated a b) :
multiplicity b c = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_left (dvd_of_associated h))
(multiplicity_le_multiplicity_of_dvd_left (dvd_of_associated h.symm))
lemma multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) :
multiplicity a b ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 $ λ n hb, dvd.trans hb h
lemma eq_of_associated_right {a b c : α} (h : associated b c) :
multiplicity a b = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_right (dvd_of_associated h))
(multiplicity_le_multiplicity_of_dvd_right (dvd_of_associated h.symm))
lemma dvd_of_multiplicity_pos {a b : α} (h : (0 : enat) < multiplicity a b) : a ∣ b :=
by rw [← pow_one a]; exact pow_dvd_of_le_multiplicity (enat.pos_iff_one_le.1 h)
lemma dvd_iff_multiplicity_pos {a b : α} : (0 : enat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos,
λ hdvd, lt_of_le_of_ne (zero_le _) (λ heq, is_greatest
(show multiplicity a b < 1, from heq ▸ enat.coe_lt_coe.mpr zero_lt_one)
(by rwa pow_one a))⟩
lemma finite_nat_iff {a b : ℕ} : finite a b ↔ (a ≠ 1 ∧ 0 < b) :=
begin
rw [← not_iff_not, not_finite_iff_forall, not_and_distrib, ne.def,
not_not, not_lt, nat.le_zero_iff],
exact ⟨λ h, or_iff_not_imp_right.2 (λ hb,
have ha : a ≠ 0, from λ ha, by simpa [ha] using h 1,
by_contradiction (λ ha1 : a ≠ 1,
have ha_gt_one : 1 < a, from
lt_of_not_ge (λ ha', by { clear h, revert ha ha1, dec_trivial! }),
not_lt_of_ge (le_of_dvd (nat.pos_of_ne_zero hb) (h b))
(lt_pow_self ha_gt_one b))),
λ h, by cases h; simp *⟩
end
end comm_monoid
section comm_monoid_with_zero
variable [comm_monoid_with_zero α]
lemma ne_zero_of_finite {a b : α} (h : finite a b) : b ≠ 0 :=
let ⟨n, hn⟩ := h in λ hb, by simpa [hb] using hn
variable [decidable_rel ((∣) : α → α → Prop)]
@[simp] protected lemma zero (a : α) : multiplicity a 0 = ⊤ :=
roption.eq_none_iff.2 (λ n ⟨⟨k, hk⟩, _⟩, hk (dvd_zero _))
@[simp] lemma multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 :=
begin
apply multiplicity.multiplicity_eq_zero_of_not_dvd,
rwa zero_dvd_iff,
end
end comm_monoid_with_zero
section comm_semiring
variables [comm_semiring α] [decidable_rel ((∣) : α → α → Prop)]
lemma min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(λ h, by rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact λ n hn, dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn))
(λ h, by rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact λ n hn, dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn)
end comm_semiring
section comm_ring
variables [comm_ring α] [decidable_rel ((∣) : α → α → Prop)]
open_locale classical
@[simp] protected lemma neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
roption.ext' (by simp only [multiplicity, enat.find, dvd_neg])
(λ h₁ h₂, enat.coe_inj.1 (by rw [enat.coe_get]; exact
eq.symm (unique ((dvd_neg _ _).2 (pow_multiplicity_dvd _))
(mt (dvd_neg _ _).1 (is_greatest' _ (lt_succ_self _))))))
lemma multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b :=
begin
apply le_antisymm,
{ apply enat.le_of_lt_add_one,
cases enat.ne_top_iff.mp (enat.ne_top_of_lt h) with k hk,
rw [hk], rw_mod_cast [multiplicity_lt_iff_neg_dvd], intro h_dvd,
rw [← dvd_add_iff_right] at h_dvd,
apply multiplicity.is_greatest _ h_dvd, rw [hk], apply_mod_cast nat.lt_succ_self,
rw [pow_dvd_iff_le_multiplicity, enat.coe_add, ← hk], exact enat.add_one_le_of_lt h },
{ convert min_le_multiplicity_add, rw [min_eq_right (le_of_lt h)] }
end
lemma multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b :=
by { rw [sub_eq_add_neg, multiplicity_add_of_gt]; rwa [multiplicity.neg] }
lemma multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) :=
begin
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with hab|hab|hab,
{ rw [add_comm, multiplicity_add_of_gt hab, min_eq_left], exact le_of_lt hab },
{ contradiction },
{ rw [multiplicity_add_of_gt hab, min_eq_right], exact le_of_lt hab},
end
end comm_ring
section comm_cancel_monoid_with_zero
variables [comm_cancel_monoid_with_zero α]
lemma finite_mul_aux {p : α} (hp : prime p) : ∀ {n m : ℕ} {a b : α},
¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n m := λ a b ha hb ⟨s, hs⟩,
have p ∣ a * b, from ⟨p ^ (n + m) * s,
by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩,
(hp.2.2 a b this).elim
(λ ⟨x, hx⟩, have hn0 : 0 < n,
from nat.pos_of_ne_zero (λ hn0, by clear _fun_match _fun_match; simpa [hx, hn0] using ha),
have wf : (n - 1) < n, from nat.sub_lt_self hn0 dec_trivial,
have hpx : ¬ p ^ (n - 1 + 1) ∣ x,
from λ ⟨y, hy⟩, ha (hx.symm ▸ ⟨y, mul_right_cancel' hp.1
$ by rw [nat.sub_add_cancel hn0] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩),
have 1 ≤ n + m, from le_trans hn0 (le_add_right n m),
finite_mul_aux hpx hb ⟨s, mul_right_cancel' hp.1 begin
rw [← nat.sub_add_comm hn0, nat.sub_add_cancel this],
clear _fun_match _fun_match finite_mul_aux,
simp [*, mul_comm, mul_assoc, mul_left_comm, pow_add] at *
end⟩)
(λ ⟨x, hx⟩, have hm0 : 0 < m,
from nat.pos_of_ne_zero (λ hm0, by clear _fun_match _fun_match; simpa [hx, hm0] using hb),
have wf : (m - 1) < m, from nat.sub_lt_self hm0 dec_trivial,
have hpx : ¬ p ^ (m - 1 + 1) ∣ x,
from λ ⟨y, hy⟩, hb (hx.symm ▸ ⟨y, mul_right_cancel' hp.1
$ by rw [nat.sub_add_cancel hm0] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩),
finite_mul_aux ha hpx ⟨s, mul_right_cancel' hp.1 begin
rw [add_assoc, nat.sub_add_cancel hm0],
clear _fun_match _fun_match finite_mul_aux,
simp [*, mul_comm, mul_assoc, mul_left_comm, pow_add] at *
end⟩)
lemma finite_mul {p a b : α} (hp : prime p) : finite p a → finite p b → finite p (a * b) :=
λ ⟨n, hn⟩ ⟨m, hm⟩, ⟨n + m, finite_mul_aux hp hn hm⟩
lemma finite_mul_iff {p a b : α} (hp : prime p) : finite p (a * b) ↔ finite p a ∧ finite p b :=
⟨λ h, ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩,
λ h, finite_mul hp h.1 h.2⟩
lemma finite_pow {p a : α} (hp : prime p) : Π {k : ℕ} (ha : finite p a), finite p (a ^ k)
| 0 ha := ⟨0, by simp [mt is_unit_iff_dvd_one.2 hp.2.1]⟩
| (k+1) ha := by rw [pow_succ]; exact finite_mul hp ha (finite_pow ha)
variable [decidable_rel ((∣) : α → α → Prop)]
@[simp] lemma multiplicity_self {a : α} (ha : ¬is_unit a) (ha0 : a ≠ 0) :
multiplicity a a = 1 :=
eq_some_iff.2 ⟨by simp, λ ⟨b, hb⟩, ha (is_unit_iff_dvd_one.2
⟨b, mul_left_cancel' ha0 $ by clear _fun_match;
simpa [pow_succ, mul_assoc] using hb⟩)⟩
@[simp] lemma get_multiplicity_self {a : α} (ha : finite a a) :
get (multiplicity a a) ha = 1 :=
roption.get_eq_iff_eq_some.2 (eq_some_iff.2
⟨by simp, λ ⟨b, hb⟩,
by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact mt is_unit_iff_dvd_one.2 (not_unit_of_finite ha)
⟨b, by clear _fun_match; simp * at *⟩⟩)
protected lemma mul' {p a b : α} (hp : prime p)
(h : (multiplicity p (a * b)).dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a,
from pow_multiplicity_dvd _,
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b,
from pow_multiplicity_dvd _,
have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2,
by simp [pow_add],
have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b,
by rw [hpoweq]; apply mul_dvd_mul; assumption,
have hsucc : ¬p ^ ((get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) + 1) ∣ a * b,
from λ h, not_or (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
(by exact succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h),
by rw [← enat.coe_inj, enat.coe_get, eq_some_iff];
exact ⟨hdiv, hsucc⟩
open_locale classical
protected lemma mul {p a b : α} (hp : prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : finite p a ∧ finite p b then
by rw [← enat.coe_get (finite_iff_dom.1 h.1), ← enat.coe_get (finite_iff_dom.1 h.2),
← enat.coe_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)),
← enat.coe_add, enat.coe_inj, multiplicity.mul' hp]; refl
else begin
rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)],
cases not_and_distrib.1 h with h h;
simp [eq_top_iff_not_finite.2 h]
end
lemma finset.prod {β : Type*} {p : α} (hp : prime p) (s : finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) :=
begin
classical,
induction s using finset.induction with a s has ih h,
{ simp only [finset.sum_empty, finset.prod_empty],
convert one_right hp.not_unit },
{ simp [has, ← ih],
convert multiplicity.mul hp }
end
protected lemma pow' {p a : α} (hp : prime p) (ha : finite p a) : ∀ {k : ℕ},
get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha
| 0 := by dsimp [pow_zero]; simp [one_right hp.not_unit]; refl
| (k+1) := by dsimp only [pow_succ];
erw [multiplicity.mul' hp, pow', add_mul, one_mul, add_comm]
lemma pow {p a : α} (hp : prime p) : ∀ {k : ℕ},
multiplicity p (a ^ k) = k •ℕ (multiplicity p a)
| 0 := by simp [one_right hp.not_unit]
| (succ k) := by simp [pow_succ, succ_nsmul, pow, multiplicity.mul hp]
lemma multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬ is_unit p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
by { rw [eq_some_iff], use dvd_refl _, rw [pow_dvd_pow_iff h0 hu], apply nat.not_succ_le_self }
lemma multiplicity_pow_self_of_prime {p : α} (hp : prime p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
multiplicity_pow_self hp.ne_zero hp.not_unit n
end comm_cancel_monoid_with_zero
section valuation
variables {R : Type*} [integral_domain R] {p : R} [decidable_rel (has_dvd.dvd : R → R → Prop)]
/-- `multiplicity` of a prime inan integral domain as an additive valuation to `enat`. -/
noncomputable def add_valuation (hp : prime p) : add_valuation R enat :=
add_valuation.of (multiplicity p) (multiplicity.zero _) (one_right hp.not_unit)
(λ _ _, min_le_multiplicity_add) (λ a b, multiplicity.mul hp)
@[simp]
lemma add_valuation_apply {hp : prime p} {r : R} : add_valuation hp r = multiplicity p r := rfl
end valuation
end multiplicity
section nat
open multiplicity
lemma multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b)
(hab : nat.coprime a b) : multiplicity p a = 0 :=
begin
rw [multiplicity_le_multiplicity_iff] at hle,
rw [← nonpos_iff_eq_zero, ← not_lt, enat.pos_iff_one_le, ← enat.coe_one,
← pow_dvd_iff_le_multiplicity],
assume h,
have := nat.dvd_gcd h (hle _ h),
rw [coprime.gcd_eq_one hab, nat.dvd_one, pow_one] at this,
exact hp this
end
end nat
|
8e1693f7b76bef9fafe9fbdbfb913603a4268c84 | 80cc5bf14c8ea85ff340d1d747a127dcadeb966f | /src/algebra/ordered_field.lean | 79920e542ae1fee0ebbc0a2f22b772e9fa62fd7d | [
"Apache-2.0"
] | permissive | lacker/mathlib | f2439c743c4f8eb413ec589430c82d0f73b2d539 | ddf7563ac69d42cfa4a1bfe41db1fed521bd795f | refs/heads/master | 1,671,948,326,773 | 1,601,479,268,000 | 1,601,479,268,000 | 298,686,743 | 0 | 0 | Apache-2.0 | 1,601,070,794,000 | 1,601,070,794,000 | null | UTF-8 | Lean | false | false | 21,412 | lean | /-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import algebra.ordered_ring
import algebra.field
/-!
### Linear ordered fields
A linear ordered field is a field equipped with a linear order such that
* addition respects the order: `a ≤ b → c + a ≤ c + b`;
* multiplication of positives is positive: `0 < a → 0 < b → 0 < a * b`;
* `0 < 1`.
### Main Definitions
* `linear_ordered_field`: the class of linear ordered fields.
* `discrete_linear_ordered_field`: the class of linear ordered fields where the inequality is
decidable.
-/
set_option old_structure_cmd true
variable {α : Type*}
/-- A linear ordered field is a field with a linear order respecting the operations. -/
@[protect_proj] class linear_ordered_field (α : Type*) extends linear_ordered_comm_ring α, field α
section linear_ordered_field
variables [linear_ordered_field α] {a b c d e : α}
/-!
### Lemmas about pos, nonneg, nonpos, neg
-/
@[simp] lemma inv_pos : 0 < a⁻¹ ↔ 0 < a :=
suffices ∀ a : α, 0 < a → 0 < a⁻¹,
from ⟨λ h, inv_inv' a ▸ this _ h, this a⟩,
assume a ha, flip lt_of_mul_lt_mul_left ha.le $ by simp [ne_of_gt ha, zero_lt_one]
@[simp] lemma inv_nonneg : 0 ≤ a⁻¹ ↔ 0 ≤ a :=
by simp only [le_iff_eq_or_lt, inv_pos, zero_eq_inv]
@[simp] lemma inv_lt_zero : a⁻¹ < 0 ↔ a < 0 :=
by simp only [← not_le, inv_nonneg]
@[simp] lemma inv_nonpos : a⁻¹ ≤ 0 ↔ a ≤ 0 :=
by simp only [← not_lt, inv_pos]
lemma one_div_pos : 0 < 1 / a ↔ 0 < a :=
inv_eq_one_div a ▸ inv_pos
lemma one_div_neg : 1 / a < 0 ↔ a < 0 :=
inv_eq_one_div a ▸ inv_lt_zero
lemma one_div_nonneg : 0 ≤ 1 / a ↔ 0 ≤ a :=
inv_eq_one_div a ▸ inv_nonneg
lemma one_div_nonpos : 1 / a ≤ 0 ↔ a ≤ 0 :=
inv_eq_one_div a ▸ inv_nonpos
lemma div_pos (ha : 0 < a) (hb : 0 < b) : 0 < a / b :=
mul_pos ha (inv_pos.2 hb)
lemma div_pos_of_neg_of_neg (ha : a < 0) (hb : b < 0) : 0 < a / b :=
mul_pos_of_neg_of_neg ha (inv_lt_zero.2 hb)
lemma div_neg_of_neg_of_pos (ha : a < 0) (hb : 0 < b) : a / b < 0 :=
mul_neg_of_neg_of_pos ha (inv_pos.2 hb)
lemma div_neg_of_pos_of_neg (ha : 0 < a) (hb : b < 0) : a / b < 0 :=
mul_neg_of_pos_of_neg ha (inv_lt_zero.2 hb)
lemma div_nonneg (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a / b :=
mul_nonneg ha (inv_nonneg.2 hb)
lemma div_nonneg_of_nonpos (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a / b :=
mul_nonneg_of_nonpos_of_nonpos ha (inv_nonpos.2 hb)
lemma div_nonpos_of_nonpos_of_nonneg (ha : a ≤ 0) (hb : 0 ≤ b) : a / b ≤ 0 :=
mul_nonpos_of_nonpos_of_nonneg ha (inv_nonneg.2 hb)
lemma div_nonpos_of_nonneg_of_nonpos (ha : 0 ≤ a) (hb : b ≤ 0) : a / b ≤ 0 :=
mul_nonpos_of_nonneg_of_nonpos ha (inv_nonpos.2 hb)
/-!
### Relating one division with another term.
-/
lemma le_div_iff (hc : 0 < c) : a ≤ b / c ↔ a * c ≤ b :=
⟨λ h, div_mul_cancel b (ne_of_lt hc).symm ▸ mul_le_mul_of_nonneg_right h hc.le,
λ h, calc
a = a * c * (1 / c) : mul_mul_div a (ne_of_lt hc).symm
... ≤ b * (1 / c) : mul_le_mul_of_nonneg_right h (one_div_pos.2 hc).le
... = b / c : (div_eq_mul_one_div b c).symm⟩
lemma le_div_iff' (hc : 0 < c) : a ≤ b / c ↔ c * a ≤ b :=
by rw [mul_comm, le_div_iff hc]
lemma div_le_iff (hb : 0 < b) : a / b ≤ c ↔ a ≤ c * b :=
⟨λ h, calc
a = a / b * b : by rw (div_mul_cancel _ (ne_of_lt hb).symm)
... ≤ c * b : mul_le_mul_of_nonneg_right h hb.le,
λ h, calc
a / b = a * (1 / b) : div_eq_mul_one_div a b
... ≤ (c * b) * (1 / b) : mul_le_mul_of_nonneg_right h (one_div_pos.2 hb).le
... = (c * b) / b : (div_eq_mul_one_div (c * b) b).symm
... = c : by refine (div_eq_iff (ne_of_gt hb)).mpr rfl⟩
lemma div_le_iff' (hb : 0 < b) : a / b ≤ c ↔ a ≤ b * c :=
by rw [mul_comm, div_le_iff hb]
lemma lt_div_iff (hc : 0 < c) : a < b / c ↔ a * c < b :=
lt_iff_lt_of_le_iff_le $ div_le_iff hc
lemma lt_div_iff' (hc : 0 < c) : a < b / c ↔ c * a < b :=
by rw [mul_comm, lt_div_iff hc]
lemma div_lt_iff (hc : 0 < c) : b / c < a ↔ b < a * c :=
lt_iff_lt_of_le_iff_le (le_div_iff hc)
lemma div_lt_iff' (hc : 0 < c) : b / c < a ↔ b < c * a :=
by rw [mul_comm, div_lt_iff hc]
lemma inv_mul_le_iff (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ b * c :=
begin
rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div],
exact div_le_iff' h,
end
lemma inv_mul_le_iff' (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ c * b :=
by rw [inv_mul_le_iff h, mul_comm]
lemma mul_inv_le_iff (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ b * c :=
by rw [mul_comm, inv_mul_le_iff h]
lemma mul_inv_le_iff' (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ c * b :=
by rw [mul_comm, inv_mul_le_iff' h]
lemma inv_mul_lt_iff (h : 0 < b) : b⁻¹ * a < c ↔ a < b * c :=
begin
rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div],
exact div_lt_iff' h,
end
lemma inv_mul_lt_iff' (h : 0 < b) : b⁻¹ * a < c ↔ a < c * b :=
by rw [inv_mul_lt_iff h, mul_comm]
lemma mul_inv_lt_iff (h : 0 < b) : a * b⁻¹ < c ↔ a < b * c :=
by rw [mul_comm, inv_mul_lt_iff h]
lemma mul_inv_lt_iff' (h : 0 < b) : a * b⁻¹ < c ↔ a < c * b :=
by rw [mul_comm, inv_mul_lt_iff' h]
lemma div_le_iff_of_neg (hc : c < 0) : b / c ≤ a ↔ a * c ≤ b :=
⟨λ h, div_mul_cancel b (ne_of_lt hc) ▸ mul_le_mul_of_nonpos_right h hc.le,
λ h, calc
a = a * c * (1 / c) : mul_mul_div a (ne_of_lt hc)
... ≥ b * (1 / c) : mul_le_mul_of_nonpos_right h (one_div_neg.2 hc).le
... = b / c : (div_eq_mul_one_div b c).symm⟩
lemma div_le_iff_of_neg' (hc : c < 0) : b / c ≤ a ↔ c * a ≤ b :=
by rw [mul_comm, div_le_iff_of_neg hc]
lemma le_div_iff_of_neg (hc : c < 0) : a ≤ b / c ↔ b ≤ a * c :=
by rw [← neg_neg c, mul_neg_eq_neg_mul_symm, div_neg, le_neg,
div_le_iff (neg_pos.2 hc), neg_mul_eq_neg_mul_symm]
lemma le_div_iff_of_neg' (hc : c < 0) : a ≤ b / c ↔ b ≤ c * a :=
by rw [mul_comm, le_div_iff_of_neg hc]
lemma div_lt_iff_of_neg (hc : c < 0) : b / c < a ↔ a * c < b :=
lt_iff_lt_of_le_iff_le $ le_div_iff_of_neg hc
lemma div_lt_iff_of_neg' (hc : c < 0) : b / c < a ↔ c * a < b :=
by rw [mul_comm, div_lt_iff_of_neg hc]
lemma lt_div_iff_of_neg (hc : c < 0) : a < b / c ↔ b < a * c :=
lt_iff_lt_of_le_iff_le $ div_le_iff_of_neg hc
lemma lt_div_iff_of_neg' (hc : c < 0) : a < b / c ↔ b < c * a :=
by rw [mul_comm, lt_div_iff_of_neg hc]
/-- One direction of `div_le_iff` where `b` is allowed to be `0` (but `c` must be nonnegative) -/
lemma div_le_iff_of_nonneg_of_le (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a / b ≤ c :=
by { rcases eq_or_lt_of_le hb with rfl|hb', simp [hc], rwa [div_le_iff hb'] }
/-!
### Bi-implications of inequalities using inversions
-/
lemma inv_le_inv_of_le (ha : 0 < a) (h : a ≤ b) : b⁻¹ ≤ a⁻¹ :=
by rwa [← one_div a, le_div_iff' ha, ← div_eq_mul_inv, div_le_iff (ha.trans_le h), one_mul]
/-- See `inv_le_inv_of_le` for the implication from right-to-left with one fewer assumption. -/
lemma inv_le_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a :=
by rw [← one_div, div_le_iff ha, ← div_eq_inv_mul, le_div_iff hb, one_mul]
lemma inv_le (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a :=
by rw [← inv_le_inv hb (inv_pos.2 ha), inv_inv']
lemma le_inv (ha : 0 < a) (hb : 0 < b) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ :=
by rw [← inv_le_inv (inv_pos.2 hb) ha, inv_inv']
lemma inv_lt_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b⁻¹ ↔ b < a :=
lt_iff_lt_of_le_iff_le (inv_le_inv hb ha)
lemma inv_lt (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b ↔ b⁻¹ < a :=
lt_iff_lt_of_le_iff_le (le_inv hb ha)
lemma lt_inv (ha : 0 < a) (hb : 0 < b) : a < b⁻¹ ↔ b < a⁻¹ :=
lt_iff_lt_of_le_iff_le (inv_le hb ha)
lemma inv_le_inv_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a :=
by rw [← one_div, div_le_iff_of_neg ha, ← div_eq_inv_mul, div_le_iff_of_neg hb, one_mul]
lemma inv_le_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a :=
by rw [← inv_le_inv_of_neg hb (inv_lt_zero.2 ha), inv_inv']
lemma le_inv_of_neg (ha : a < 0) (hb : b < 0) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ :=
by rw [← inv_le_inv_of_neg (inv_lt_zero.2 hb) ha, inv_inv']
lemma inv_lt_inv_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ < b⁻¹ ↔ b < a :=
lt_iff_lt_of_le_iff_le (inv_le_inv_of_neg hb ha)
lemma inv_lt_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ < b ↔ b⁻¹ < a :=
lt_iff_lt_of_le_iff_le (le_inv_of_neg hb ha)
lemma lt_inv_of_neg (ha : a < 0) (hb : b < 0) : a < b⁻¹ ↔ b < a⁻¹ :=
lt_iff_lt_of_le_iff_le (inv_le_of_neg hb ha)
lemma inv_lt_one (ha : 1 < a) : a⁻¹ < 1 :=
by rwa [inv_lt ((@zero_lt_one α _).trans ha) zero_lt_one, inv_one]
lemma one_lt_inv (h₁ : 0 < a) (h₂ : a < 1) : 1 < a⁻¹ :=
by rwa [lt_inv (@zero_lt_one α _) h₁, inv_one]
lemma inv_le_one (ha : 1 ≤ a) : a⁻¹ ≤ 1 :=
by rwa [inv_le ((@zero_lt_one α _).trans_le ha) zero_lt_one, inv_one]
lemma one_le_inv (h₁ : 0 < a) (h₂ : a ≤ 1) : 1 ≤ a⁻¹ :=
by rwa [le_inv (@zero_lt_one α _) h₁, inv_one]
/-!
### Relating two divisions.
-/
lemma div_le_div_of_le (hc : 0 ≤ c) (h : a ≤ b) : a / c ≤ b / c :=
begin
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],
exact mul_le_mul_of_nonneg_right h (one_div_nonneg.2 hc)
end
lemma div_le_div_of_le_left (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) : a / b ≤ a / c :=
begin
rw [div_eq_mul_inv, div_eq_mul_inv],
exact mul_le_mul_of_nonneg_left ((inv_le_inv (hc.trans_le h) hc).mpr h) ha
end
lemma div_le_div_of_le_of_nonneg (hab : a ≤ b) (hc : 0 ≤ c) : a / c ≤ b / c :=
mul_le_mul_of_nonneg_right hab (inv_nonneg.2 hc)
lemma div_le_div_of_nonpos_of_le (hc : c ≤ 0) (h : b ≤ a) : a / c ≤ b / c :=
begin
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],
exact mul_le_mul_of_nonpos_right h (one_div_nonpos.2 hc)
end
lemma div_lt_div_of_lt (hc : 0 < c) (h : a < b) : a / c < b / c :=
begin
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],
exact mul_lt_mul_of_pos_right h (one_div_pos.2 hc)
end
lemma div_lt_div_of_neg_of_lt (hc : c < 0) (h : b < a) : a / c < b / c :=
begin
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],
exact mul_lt_mul_of_neg_right h (one_div_neg.2 hc)
end
lemma div_le_div_right (hc : 0 < c) : a / c ≤ b / c ↔ a ≤ b :=
⟨le_imp_le_of_lt_imp_lt $ div_lt_div_of_lt hc, div_le_div_of_le $ hc.le⟩
lemma div_le_div_right_of_neg (hc : c < 0) : a / c ≤ b / c ↔ b ≤ a :=
⟨le_imp_le_of_lt_imp_lt $ div_lt_div_of_neg_of_lt hc, div_le_div_of_nonpos_of_le $ hc.le⟩
lemma div_lt_div_right (hc : 0 < c) : a / c < b / c ↔ a < b :=
lt_iff_lt_of_le_iff_le $ div_le_div_right hc
lemma div_lt_div_right_of_neg (hc : c < 0) : a / c < b / c ↔ b < a :=
lt_iff_lt_of_le_iff_le $ div_le_div_right_of_neg hc
lemma div_lt_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b < a / c ↔ c < b :=
(mul_lt_mul_left ha).trans (inv_lt_inv hb hc)
lemma div_le_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b ≤ a / c ↔ c ≤ b :=
le_iff_le_iff_lt_iff_lt.2 (div_lt_div_left ha hc hb)
lemma div_lt_div_iff (b0 : 0 < b) (d0 : 0 < d) :
a / b < c / d ↔ a * d < c * b :=
by rw [lt_div_iff d0, div_mul_eq_mul_div, div_lt_iff b0]
lemma div_le_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b :=
by rw [le_div_iff d0, div_mul_eq_mul_div, div_le_iff b0]
lemma div_le_div (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hbd : d ≤ b) : a / b ≤ c / d :=
by { rw div_le_div_iff (hd.trans_le hbd) hd, exact mul_le_mul hac hbd hd.le hc }
lemma div_lt_div (hac : a < c) (hbd : d ≤ b) (c0 : 0 ≤ c) (d0 : 0 < d) :
a / b < c / d :=
(div_lt_div_iff (d0.trans_le hbd) d0).2 (mul_lt_mul hac hbd d0 c0)
lemma div_lt_div' (hac : a ≤ c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d) :
a / b < c / d :=
(div_lt_div_iff (d0.trans hbd) d0).2 (mul_lt_mul' hac hbd d0.le c0)
lemma div_lt_div_of_lt_left (hb : 0 < b) (h : b < a) (hc : 0 < c) : c / a < c / b :=
(div_lt_div_left hc (hb.trans h) hb).mpr h
/-!
### Relating one division and involving `1`
-/
lemma one_le_div (hb : 0 < b) : 1 ≤ a / b ↔ b ≤ a :=
by rw [le_div_iff hb, one_mul]
lemma div_le_one (hb : 0 < b) : a / b ≤ 1 ↔ a ≤ b :=
by rw [div_le_iff hb, one_mul]
lemma one_lt_div (hb : 0 < b) : 1 < a / b ↔ b < a :=
by rw [lt_div_iff hb, one_mul]
lemma div_lt_one (hb : 0 < b) : a / b < 1 ↔ a < b :=
by rw [div_lt_iff hb, one_mul]
lemma one_le_div_of_neg (hb : b < 0) : 1 ≤ a / b ↔ a ≤ b :=
by rw [le_div_iff_of_neg hb, one_mul]
lemma div_le_one_of_neg (hb : b < 0) : a / b ≤ 1 ↔ b ≤ a :=
by rw [div_le_iff_of_neg hb, one_mul]
lemma one_lt_div_of_neg (hb : b < 0) : 1 < a / b ↔ a < b :=
by rw [lt_div_iff_of_neg hb, one_mul]
lemma div_lt_one_of_neg (hb : b < 0) : a / b < 1 ↔ b < a :=
by rw [div_lt_iff_of_neg hb, one_mul]
lemma one_div_le (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ b ↔ 1 / b ≤ a :=
by simpa using inv_le ha hb
lemma one_div_lt (ha : 0 < a) (hb : 0 < b) : 1 / a < b ↔ 1 / b < a :=
by simpa using inv_lt ha hb
lemma le_one_div (ha : 0 < a) (hb : 0 < b) : a ≤ 1 / b ↔ b ≤ 1 / a :=
by simpa using le_inv ha hb
lemma lt_one_div (ha : 0 < a) (hb : 0 < b) : a < 1 / b ↔ b < 1 / a :=
by simpa using lt_inv ha hb
lemma one_div_le_of_neg (ha : a < 0) (hb : b < 0) : 1 / a ≤ b ↔ 1 / b ≤ a :=
by simpa using inv_le_of_neg ha hb
lemma one_div_lt_of_neg (ha : a < 0) (hb : b < 0) : 1 / a < b ↔ 1 / b < a :=
by simpa using inv_lt_of_neg ha hb
lemma le_one_div_of_neg (ha : a < 0) (hb : b < 0) : a ≤ 1 / b ↔ b ≤ 1 / a :=
by simpa using le_inv_of_neg ha hb
lemma lt_one_div_of_neg (ha : a < 0) (hb : b < 0) : a < 1 / b ↔ b < 1 / a :=
by simpa using lt_inv_of_neg ha hb
/-!
### Relating two divisions, involving `1`
-/
lemma one_div_le_one_div_of_le (ha : 0 < a) (h : a ≤ b) : 1 / b ≤ 1 / a :=
by simpa using inv_le_inv_of_le ha h
lemma one_div_lt_one_div_of_lt (ha : 0 < a) (h : a < b) : 1 / b < 1 / a :=
by rwa [lt_div_iff' ha, ← div_eq_mul_one_div, div_lt_one (ha.trans h)]
lemma one_div_le_one_div_of_neg_of_le (hb : b < 0) (h : a ≤ b) : 1 / b ≤ 1 / a :=
by rwa [div_le_iff_of_neg' hb, ← div_eq_mul_one_div, div_le_one_of_neg (h.trans_lt hb)]
lemma one_div_lt_one_div_of_neg_of_lt (hb : b < 0) (h : a < b) : 1 / b < 1 / a :=
by rwa [div_lt_iff_of_neg' hb, ← div_eq_mul_one_div, div_lt_one_of_neg (h.trans hb)]
lemma le_of_one_div_le_one_div (ha : 0 < a) (h : 1 / a ≤ 1 / b) : b ≤ a :=
le_imp_le_of_lt_imp_lt (one_div_lt_one_div_of_lt ha) h
lemma lt_of_one_div_lt_one_div (ha : 0 < a) (h : 1 / a < 1 / b) : b < a :=
lt_imp_lt_of_le_imp_le (one_div_le_one_div_of_le ha) h
lemma le_of_neg_of_one_div_le_one_div (hb : b < 0) (h : 1 / a ≤ 1 / b) : b ≤ a :=
le_imp_le_of_lt_imp_lt (one_div_lt_one_div_of_neg_of_lt hb) h
lemma lt_of_neg_of_one_div_lt_one_div (hb : b < 0) (h : 1 / a < 1 / b) : b < a :=
lt_imp_lt_of_le_imp_le (one_div_le_one_div_of_neg_of_le hb) h
/-- For the single implications with fewer assumptions, see `one_div_le_one_div_of_le` and
`le_of_one_div_le_one_div` -/
lemma one_div_le_one_div (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ 1 / b ↔ b ≤ a :=
div_le_div_left zero_lt_one ha hb
/-- For the single implications with fewer assumptions, see `one_div_lt_one_div_of_lt` and
`lt_of_one_div_lt_one_div` -/
lemma one_div_lt_one_div (ha : 0 < a) (hb : 0 < b) : 1 / a < 1 / b ↔ b < a :=
div_lt_div_left zero_lt_one ha hb
/-- For the single implications with fewer assumptions, see `one_div_lt_one_div_of_neg_of_lt` and
`lt_of_one_div_lt_one_div` -/
lemma one_div_le_one_div_of_neg (ha : a < 0) (hb : b < 0) : 1 / a ≤ 1 / b ↔ b ≤ a :=
by simpa [one_div] using inv_le_inv_of_neg ha hb
/-- For the single implications with fewer assumptions, see `one_div_lt_one_div_of_lt` and
`lt_of_one_div_lt_one_div` -/
lemma one_div_lt_one_div_of_neg (ha : a < 0) (hb : b < 0) : 1 / a < 1 / b ↔ b < a :=
lt_iff_lt_of_le_iff_le (one_div_le_one_div_of_neg hb ha)
lemma one_lt_one_div (h1 : 0 < a) (h2 : a < 1) : 1 < 1 / a :=
by rwa [lt_one_div (@zero_lt_one α _) h1, one_div_one]
lemma one_le_one_div (h1 : 0 < a) (h2 : a ≤ 1) : 1 ≤ 1 / a :=
by rwa [le_one_div (@zero_lt_one α _) h1, one_div_one]
lemma one_div_lt_neg_one (h1 : a < 0) (h2 : -1 < a) : 1 / a < -1 :=
suffices 1 / a < 1 / -1, by rwa one_div_neg_one_eq_neg_one at this,
one_div_lt_one_div_of_neg_of_lt h1 h2
lemma one_div_le_neg_one (h1 : a < 0) (h2 : -1 ≤ a) : 1 / a ≤ -1 :=
suffices 1 / a ≤ 1 / -1, by rwa one_div_neg_one_eq_neg_one at this,
one_div_le_one_div_of_neg_of_le h1 h2
/-!
### Results about halving.
The equalities also hold in fields of characteristic `0`. -/
lemma add_halves (a : α) : a / 2 + a / 2 = a :=
by rw [div_add_div_same, ← two_mul, mul_div_cancel_left a two_ne_zero]
lemma sub_self_div_two (a : α) : a - a / 2 = a / 2 :=
suffices a / 2 + a / 2 - a / 2 = a / 2, by rwa add_halves at this,
by rw [add_sub_cancel]
lemma div_two_sub_self (a : α) : a / 2 - a = - (a / 2) :=
suffices a / 2 - (a / 2 + a / 2) = - (a / 2), by rwa add_halves at this,
by rw [sub_add_eq_sub_sub, sub_self, zero_sub]
lemma add_self_div_two (a : α) : (a + a) / 2 = a :=
by rw [← mul_two, mul_div_cancel a two_ne_zero]
lemma half_pos (h : 0 < a) : 0 < a / 2 := div_pos h two_pos
lemma one_half_pos : (0:α) < 1 / 2 := half_pos zero_lt_one
lemma div_two_lt_of_pos (h : 0 < a) : a / 2 < a :=
by { rw [div_lt_iff (@two_pos α _)], exact lt_mul_of_one_lt_right h one_lt_two }
lemma half_lt_self : 0 < a → a / 2 < a := div_two_lt_of_pos
lemma one_half_lt_one : (1 / 2 : α) < 1 := half_lt_self zero_lt_one
lemma add_sub_div_two_lt (h : a < b) : a + (b - a) / 2 < b :=
begin
rwa [← div_sub_div_same, sub_eq_add_neg, add_comm (b/2), ← add_assoc, ← sub_eq_add_neg,
← lt_sub_iff_add_lt, sub_self_div_two, sub_self_div_two, div_lt_div_right (@two_pos α _)]
end
/-!
### Miscellaneous lemmas
-/
lemma mul_sub_mul_div_mul_neg_iff (hc : c ≠ 0) (hd : d ≠ 0) :
(a * d - b * c) / (c * d) < 0 ↔ a / c < b / d :=
by rw [mul_comm b c, ← div_sub_div _ _ hc hd, sub_lt_zero]
alias mul_sub_mul_div_mul_neg_iff ↔ div_lt_div_of_mul_sub_mul_div_neg mul_sub_mul_div_mul_neg
lemma mul_sub_mul_div_mul_nonpos_iff (hc : c ≠ 0) (hd : d ≠ 0) :
(a * d - b * c) / (c * d) ≤ 0 ↔ a / c ≤ b / d :=
by rw [mul_comm b c, ← div_sub_div _ _ hc hd, sub_nonpos]
alias mul_sub_mul_div_mul_nonpos_iff ↔
div_le_div_of_mul_sub_mul_div_nonpos mul_sub_mul_div_mul_nonpos
lemma mul_le_mul_of_mul_div_le (h : a * (b / c) ≤ d) (hc : 0 < c) : b * a ≤ d * c :=
begin
rw [← mul_div_assoc] at h,
rwa [mul_comm b, ← div_le_iff hc],
end
lemma div_mul_le_div_mul_of_div_le_div (h : a / b ≤ c / d) (he : 0 ≤ e) :
a / (b * e) ≤ c / (d * e) :=
begin
rw [div_mul_eq_div_mul_one_div, div_mul_eq_div_mul_one_div],
exact mul_le_mul_of_nonneg_right h (one_div_nonneg.2 he)
end
lemma exists_add_lt_and_pos_of_lt (h : b < a) : ∃ c : α, b + c < a ∧ 0 < c :=
⟨(a - b) / 2, add_sub_div_two_lt h, div_pos (sub_pos_of_lt h) two_pos⟩
lemma le_of_forall_sub_le (h : ∀ ε > 0, b - ε ≤ a) : b ≤ a :=
begin
contrapose! h,
simpa only [and_comm ((0 : α) < _), lt_sub_iff_add_lt, gt_iff_lt]
using exists_add_lt_and_pos_of_lt h,
end
lemma monotone.div_const {β : Type*} [preorder β] {f : β → α} (hf : monotone f)
{c : α} (hc : 0 ≤ c) : monotone (λ x, (f x) / c) :=
hf.mul_const (inv_nonneg.2 hc)
lemma strict_mono.div_const {β : Type*} [preorder β] {f : β → α} (hf : strict_mono f)
{c : α} (hc : 0 < c) :
strict_mono (λ x, (f x) / c) :=
hf.mul_const (inv_pos.2 hc)
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_field.to_densely_ordered : densely_ordered α :=
{ dense := λ a₁ a₂ h, ⟨(a₁ + a₂) / 2,
calc a₁ = (a₁ + a₁) / 2 : (add_self_div_two a₁).symm
... < (a₁ + a₂) / 2 : div_lt_div_of_lt two_pos (add_lt_add_left h _),
calc (a₁ + a₂) / 2 < (a₂ + a₂) / 2 : div_lt_div_of_lt two_pos (add_lt_add_right h _)
... = a₂ : add_self_div_two a₂⟩ }
lemma mul_self_inj_of_nonneg (a0 : 0 ≤ a) (b0 : 0 ≤ b) : a * a = b * b ↔ a = b :=
mul_self_eq_mul_self_iff.trans $ or_iff_left_of_imp $
λ h, by { subst a, have : b = 0 := le_antisymm (neg_nonneg.1 a0) b0, rw [this, neg_zero] }
end linear_ordered_field
/-- A discrete linear ordered field is a field with a decidable linear order respecting the
operations. -/
@[protect_proj] class discrete_linear_ordered_field (α : Type*)
extends linear_ordered_field α, decidable_linear_ordered_comm_ring α
section discrete_linear_ordered_field
variables [discrete_linear_ordered_field α]
lemma abs_div (a b : α) : abs (a / b) = abs a / abs b :=
decidable.by_cases
(assume h : b = 0, by rw [h, abs_zero, div_zero, div_zero, abs_zero])
(assume h : b ≠ 0,
have h₁ : abs b ≠ 0, from mt eq_zero_of_abs_eq_zero h,
eq_div_of_mul_eq h₁ (show abs (a / b) * abs b = abs a, by rw [← abs_mul, div_mul_cancel _ h]))
lemma abs_one_div (a : α) : abs (1 / a) = 1 / abs a :=
by rw [abs_div, abs_of_nonneg (zero_le_one : 1 ≥ (0 : α))]
lemma abs_inv (a : α) : abs a⁻¹ = (abs a)⁻¹ :=
have h : abs (1 / a) = 1 / abs a,
by { rw [abs_div, abs_of_nonneg], exact zero_le_one },
by simp * at *
end discrete_linear_ordered_field
|
d4b45ac424157c1810011feafe452d5663fa3781 | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/combinatorics/simple_graph/matching.lean | 20810408767ee55d50d06e30d9b7551f5e2a7258 | [
"Apache-2.0"
] | permissive | robertylewis/mathlib | 3d16e3e6daf5ddde182473e03a1b601d2810952c | 1d13f5b932f5e40a8308e3840f96fc882fae01f0 | refs/heads/master | 1,651,379,945,369 | 1,644,276,960,000 | 1,644,276,960,000 | 98,875,504 | 0 | 0 | Apache-2.0 | 1,644,253,514,000 | 1,501,495,700,000 | Lean | UTF-8 | Lean | false | false | 4,577 | lean | /-
Copyright (c) 2020 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Arthur Paulino, Kyle Miller
-/
import combinatorics.simple_graph.degree_sum
import combinatorics.simple_graph.subgraph
/-!
# Matchings
A *matching* for a simple graph is a set of disjoint pairs of adjacent vertices, and the set of all
the vertices in a matching is called its *support* (and sometimes the vertices in the support are
said to be *saturated* by the matching). A *perfect matching* is a matching whose support contains
every vertex of the graph.
In this module, we represent a matching as a subgraph whose vertices are each incident to at most
one edge, and the edges of the subgraph represent the paired vertices.
## Main definitions
* `simple_graph.subgraph.is_matching`: `M.is_matching` means that `M` is a matching of its
underlying graph.
denoted `M.is_matching`.
* `simple_graph.subgraph.is_perfect_matching` defines when a subgraph `M` of a simple graph is a
perfect matching, denoted `M.is_perfect_matching`.
## TODO
* Define an `other` function and prove useful results about it (https://leanprover.zulipchat.com/#narrow/stream/252551-graph-theory/topic/matchings/near/266205863)
* Provide a bicoloring for matchings (https://leanprover.zulipchat.com/#narrow/stream/252551-graph-theory/topic/matchings/near/265495120)
* Tutte's Theorem
* Hall's Marriage Theorem (see combinatorics.hall)
-/
universe u
namespace simple_graph
variables {V : Type u} {G : simple_graph V} (M : subgraph G)
namespace subgraph
/--
The subgraph `M` of `G` is a matching if every vertex of `M` is incident to exactly one edge in `M`.
We say that the vertices in `M.support` are *matched* or *saturated*.
-/
def is_matching : Prop := ∀ ⦃v⦄, v ∈ M.verts → ∃! w, M.adj v w
/-- Given a vertex, returns the unique edge of the matching it is incident to. -/
noncomputable def is_matching.to_edge {M : subgraph G} (h : M.is_matching)
(v : M.verts) : M.edge_set :=
⟨⟦(v, (h v.property).some)⟧, (h v.property).some_spec.1⟩
lemma is_matching.to_edge_eq_of_adj {M : subgraph G} (h : M.is_matching) {v w : V}
(hv : v ∈ M.verts) (hvw : M.adj v w) : h.to_edge ⟨v, hv⟩ = ⟨⟦(v, w)⟧, hvw⟩ :=
begin
simp only [is_matching.to_edge, subtype.mk_eq_mk],
congr,
exact ((h (M.edge_vert hvw)).some_spec.2 w hvw).symm,
end
lemma is_matching.to_edge.surjective {M : subgraph G} (h : M.is_matching) :
function.surjective h.to_edge :=
begin
rintro ⟨e, he⟩,
refine sym2.ind (λ x y he, _) e he,
exact ⟨⟨x, M.edge_vert he⟩, h.to_edge_eq_of_adj _ he⟩,
end
lemma is_matching.to_edge_eq_to_edge_of_adj {M : subgraph G} {v w : V}
(h : M.is_matching) (hv : v ∈ M.verts) (hw : w ∈ M.verts) (ha : M.adj v w) :
h.to_edge ⟨v, hv⟩ = h.to_edge ⟨w, hw⟩ :=
by rw [h.to_edge_eq_of_adj hv ha, h.to_edge_eq_of_adj hw (M.symm ha),
subtype.mk_eq_mk, sym2.eq_swap]
/--
The subgraph `M` of `G` is a perfect matching on `G` if it's a matching and every vertex `G` is
matched.
-/
def is_perfect_matching : Prop := M.is_matching ∧ M.is_spanning
lemma is_matching.support_eq_verts {M : subgraph G} (h : M.is_matching) : M.support = M.verts :=
begin
refine M.support_subset_verts.antisymm (λ v hv, _),
obtain ⟨w, hvw, -⟩ := h hv,
exact ⟨_, hvw⟩,
end
lemma is_matching_iff_forall_degree {M : subgraph G} [Π (v : V), fintype (M.neighbor_set v)] :
M.is_matching ↔ ∀ (v : V), v ∈ M.verts → M.degree v = 1 :=
by simpa [degree_eq_one_iff_unique_adj]
lemma is_matching.even_card {M : subgraph G} [fintype M.verts] (h : M.is_matching) :
even M.verts.to_finset.card :=
begin
classical,
rw is_matching_iff_forall_degree at h,
use M.coe.edge_finset.card,
rw ← M.coe.sum_degrees_eq_twice_card_edges,
simp [h, finset.card_univ],
end
lemma is_perfect_matching_iff : M.is_perfect_matching ↔ ∀ v, ∃! w, M.adj v w :=
begin
refine ⟨_, λ hm, ⟨λ v hv, hm v, λ v, _⟩⟩,
{ rintro ⟨hm, hs⟩ v,
exact hm (hs v) },
{ obtain ⟨w, hw, -⟩ := hm v,
exact M.edge_vert hw }
end
lemma is_perfect_matching_iff_forall_degree {M : subgraph G} [Π v, fintype (M.neighbor_set v)] :
M.is_perfect_matching ↔ ∀ v, M.degree v = 1 :=
by simp [degree_eq_one_iff_unique_adj, is_perfect_matching_iff]
lemma is_perfect_matching.even_card {M : subgraph G} [fintype V] (h : M.is_perfect_matching) :
even (fintype.card V) :=
by { classical, simpa [h.2.card_verts] using is_matching.even_card h.1 }
end subgraph
end simple_graph
|
2d0671744cea3bf9938363b36fb72e6252edf5c5 | e19b506b59d98b1469dd0023680087d321d0293d | /src/command/where/command.lean | c766ef71d4cc5b253d2a9a376fe8b4b2f86d5ab6 | [] | no_license | khoek/lean-where | ea28c2a63b0c28159f9363ee5f3101348b2b4744 | d2550de8655ece1cafbb9bf1beafe4d1c4481913 | refs/heads/master | 1,586,277,232,465 | 1,542,797,636,000 | 1,542,797,636,000 | 158,533,733 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,548 | lean | reserve prefix `#where`:max
open lean.parser tactic
open interactive
namespace command.where
meta def sanitize_prefix (pfx : name) : name → name
| name.anonymous := name.anonymous
| (name.mk_string s p) :=
if p.get_prefix.get_prefix = name.anonymous then name.mk_string s pfx
else name.mk_string s name.anonymous
| (name.mk_numeral s p) :=
let s := "n" ++ to_string s in
if p.get_prefix.get_prefix = name.anonymous then name.mk_string s pfx
else name.mk_string s name.anonymous
meta def mk_user_fresh_name : tactic name := do
nm ← mk_fresh_name,
return $ (sanitize_prefix `user__ nm).append `user
meta def mk_flag (let_conts : option name := none) : lean.parser name := do
n ← mk_user_fresh_name,
(_, s) ← match let_conts with
| none := with_input command_like sformat!"def {n} : unit := ()"
| some v := with_input command_like sformat!"def {n} : unit := let {v} := {v} in ()"
end,
resolve_constant n
meta def get_namespace (flag : name) : lean.parser name := do
return $ flag.get_prefix.get_prefix.get_prefix
meta def trace_namespace (ns : name) : lean.parser unit := do
let str := match ns with
| name.anonymous := "[root namespace]"
| ns := to_string ns
end,
tactic.trace format!"namespace: {str}"
meta def strip_binders : expr → list (name × binder_info × expr)
| (expr.pi n bi t b) := (n, bi, t) :: strip_binders b
| _ := []
meta def get_includes (flag : name) : tactic (list (name × binder_info × expr)) :=
strip_binders <$> (mk_const flag >>= infer_type)
meta def binder_brackets : binder_info → string × string
| binder_info.implicit := ("{", "}")
| binder_info.strict_implicit := ("{", "}")
| binder_info.inst_implicit := ("[", "]")
| _ := ("(", ")")
meta def format_binder : name × binder_info × expr → tactic string
| (n, bi, e) := do
let (l, r) := binder_brackets bi,
e ← pp e,
return sformat!"{l}{n} : {e}{r}"
meta def format_variable_list (l : list (name × binder_info × expr)) : tactic string := do
string.intercalate "" <$> l.mmap format_binder
meta def trace_includes (flag : name) : lean.parser unit := do
l ← get_includes flag,
str ← format_variable_list l,
if l.length = 0 then skip
else tactic.trace format!"includes: {str}"
meta def get_all_in_namespace (ns : name) : tactic (list name) := do
e ← get_env,
return $ e.fold [] $ λ d l,
if ns.is_prefix_of d.to_name then d.to_name :: l else l
meta def erase_duplicates {α : Type} [decidable_eq α] : list α → list α
| [] := []
| (x :: t) := (x :: erase_duplicates (t.filter $ λ a, a ≠ x))
meta def fetch_potential_variable_names (ns : name) : tactic (list (name × binder_info × expr)) := do
l ← get_all_in_namespace ns,
l ← l.mmap (λ n, do
t ← mk_const n >>= infer_type, return $strip_binders t),
return $ erase_duplicates l.join
meta def is_variable_name : name × binder_info × expr → lean.parser bool
| (n, bi, e) :=
(mk_flag n >> return tt) <|> return ff
meta def trace_variables (ns : name) : lean.parser unit := do
l ← fetch_potential_variable_names ns,
l ← l.mfilter is_variable_name,
str ← format_variable_list l,
if l.length = 0 then skip
else tactic.trace format!"variables: {str}"
meta def trace_where : lean.parser unit := do
f ← mk_flag,
ns ← get_namespace f,
trace_namespace ns,
trace_includes f,
trace_variables ns
@[user_command]
meta def where_cmd (_ : decl_meta_info) (_ : parse $ tk "#where") : lean.parser unit := trace_where
end command.where
|
1669777540b72e1d754630dcc8d4b2e41c62519b | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/run/singleton_pair.lean | d9d0149963d30b3177a9918b3fa7c15ad6e761dd | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 99 | lean | example : {(0,1)} = (singleton (0, 1) : set (ℕ × ℕ)) := rfl
example : {(0,1)} = [(0,1)] := rfl |
41c85ed5fbf35875799e954ca02305dd4597d2e7 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/group_theory/group_action/opposite.lean | cd5674bf9bd1eb207d7d88ab0112db5c7bafe724 | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 5,821 | lean | /-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import algebra.group.opposite
import group_theory.group_action.defs
/-!
# Scalar actions on and by `Mᵐᵒᵖ`
This file defines the actions on the opposite type `has_smul R Mᵐᵒᵖ`, and actions by the opposite
type, `has_smul Rᵐᵒᵖ M`.
Note that `mul_opposite.has_smul` is provided in an earlier file as it is needed to provide the
`add_monoid.nsmul` and `add_comm_group.gsmul` fields.
-/
variables (α : Type*)
/-! ### Actions _on_ the opposite type
Actions on the opposite type just act on the underlying type.
-/
namespace mul_opposite
@[to_additive] instance (R : Type*) [monoid R] [mul_action R α] : mul_action R αᵐᵒᵖ :=
{ one_smul := λ x, unop_injective $ one_smul R (unop x),
mul_smul := λ r₁ r₂ x, unop_injective $ mul_smul r₁ r₂ (unop x),
.. mul_opposite.has_smul α R }
instance (R : Type*) [monoid R] [add_monoid α] [distrib_mul_action R α] :
distrib_mul_action R αᵐᵒᵖ :=
{ smul_add := λ r x₁ x₂, unop_injective $ smul_add r (unop x₁) (unop x₂),
smul_zero := λ r, unop_injective $ smul_zero r,
.. mul_opposite.mul_action α R }
instance (R : Type*) [monoid R] [monoid α] [mul_distrib_mul_action R α] :
mul_distrib_mul_action R αᵐᵒᵖ :=
{ smul_mul := λ r x₁ x₂, unop_injective $ smul_mul' r (unop x₂) (unop x₁),
smul_one := λ r, unop_injective $ smul_one r,
.. mul_opposite.mul_action α R }
@[to_additive]
instance {M N} [has_smul M N] [has_smul M α] [has_smul N α] [is_scalar_tower M N α] :
is_scalar_tower M N αᵐᵒᵖ :=
⟨λ x y z, unop_injective $ smul_assoc _ _ _⟩
@[to_additive] instance {M N} [has_smul M α] [has_smul N α] [smul_comm_class M N α] :
smul_comm_class M N αᵐᵒᵖ :=
⟨λ x y z, unop_injective $ smul_comm _ _ _⟩
instance (R : Type*) [has_smul R α] [has_smul Rᵐᵒᵖ α] [is_central_scalar R α] :
is_central_scalar R αᵐᵒᵖ :=
⟨λ r m, unop_injective $ op_smul_eq_smul _ _⟩
lemma op_smul_eq_op_smul_op {R : Type*} [has_smul R α] [has_smul Rᵐᵒᵖ α] [is_central_scalar R α]
(r : R) (a : α) : op (r • a) = op r • op a :=
(op_smul_eq_smul r (op a)).symm
lemma unop_smul_eq_unop_smul_unop {R : Type*} [has_smul R α] [has_smul Rᵐᵒᵖ α]
[is_central_scalar R α] (r : Rᵐᵒᵖ) (a : αᵐᵒᵖ) : unop (r • a) = unop r • unop a :=
(unop_smul_eq_smul r (unop a)).symm
end mul_opposite
/-! ### Actions _by_ the opposite type (right actions)
In `has_mul.to_has_smul` in another file, we define the left action `a₁ • a₂ = a₁ * a₂`. For the
multiplicative opposite, we define `mul_opposite.op a₁ • a₂ = a₂ * a₁`, with the multiplication
reversed.
-/
open mul_opposite
/-- Like `has_mul.to_has_smul`, but multiplies on the right.
See also `monoid.to_opposite_mul_action` and `monoid_with_zero.to_opposite_mul_action_with_zero`. -/
@[to_additive "Like `has_add.to_has_vadd`, but adds on the right.
See also `add_monoid.to_opposite_add_action`."]
instance has_mul.to_has_opposite_smul [has_mul α] : has_smul αᵐᵒᵖ α := ⟨λ c x, x * c.unop⟩
@[to_additive] lemma op_smul_eq_mul [has_mul α] {a a' : α} : op a • a' = a' * a := rfl
@[simp, to_additive] lemma mul_opposite.smul_eq_mul_unop [has_mul α] {a : αᵐᵒᵖ} {a' : α} :
a • a' = a' * a.unop := rfl
/-- The right regular action of a group on itself is transitive. -/
@[to_additive "The right regular action of an additive group on itself is transitive."]
instance mul_action.opposite_regular.is_pretransitive {G : Type*} [group G] :
mul_action.is_pretransitive Gᵐᵒᵖ G :=
⟨λ x y, ⟨op (x⁻¹ * y), mul_inv_cancel_left _ _⟩⟩
@[to_additive] instance semigroup.opposite_smul_comm_class [semigroup α] :
smul_comm_class αᵐᵒᵖ α α :=
{ smul_comm := λ x y z, (mul_assoc _ _ _) }
@[to_additive] instance semigroup.opposite_smul_comm_class' [semigroup α] :
smul_comm_class α αᵐᵒᵖ α :=
smul_comm_class.symm _ _ _
instance comm_semigroup.is_central_scalar [comm_semigroup α] : is_central_scalar α α :=
⟨λ r m, mul_comm _ _⟩
/-- Like `monoid.to_mul_action`, but multiplies on the right. -/
@[to_additive "Like `add_monoid.to_add_action`, but adds on the right."]
instance monoid.to_opposite_mul_action [monoid α] : mul_action αᵐᵒᵖ α :=
{ smul := (•),
one_smul := mul_one,
mul_smul := λ x y r, (mul_assoc _ _ _).symm }
@[to_additive]
instance is_scalar_tower.opposite_mid {M N} [has_mul N] [has_smul M N] [smul_comm_class M N N] :
is_scalar_tower M Nᵐᵒᵖ N :=
⟨λ x y z, mul_smul_comm _ _ _⟩
@[to_additive]
instance smul_comm_class.opposite_mid {M N} [has_mul N] [has_smul M N] [is_scalar_tower M N N] :
smul_comm_class M Nᵐᵒᵖ N :=
⟨λ x y z, by { induction y using mul_opposite.rec, simp [smul_mul_assoc] }⟩
-- The above instance does not create an unwanted diamond, the two paths to
-- `mul_action αᵐᵒᵖ αᵐᵒᵖ` are defeq.
example [monoid α] : monoid.to_mul_action αᵐᵒᵖ = mul_opposite.mul_action α αᵐᵒᵖ := rfl
/-- `monoid.to_opposite_mul_action` is faithful on cancellative monoids. -/
@[to_additive "`add_monoid.to_opposite_add_action` is faithful on cancellative monoids."]
instance left_cancel_monoid.to_has_faithful_opposite_scalar [left_cancel_monoid α] :
has_faithful_smul αᵐᵒᵖ α :=
⟨λ x y h, unop_injective $ mul_left_cancel (h 1)⟩
/-- `monoid.to_opposite_mul_action` is faithful on nontrivial cancellative monoids with zero. -/
instance cancel_monoid_with_zero.to_has_faithful_opposite_scalar
[cancel_monoid_with_zero α] [nontrivial α] : has_faithful_smul αᵐᵒᵖ α :=
⟨λ x y h, unop_injective $ mul_left_cancel₀ one_ne_zero (h 1)⟩
|
b09c9c1fc9a3bbdf1fbc463411ba0f9b205152e2 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/defInst.lean | c614d3528dd7f0c5be3ec9359a6dd456e71328c9 | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 624 | lean | def Foo := List Nat
def test (x : Nat) : Foo :=
[x, x+1, x+2]
#eval test 4
#check fun (x y : Foo) => x == y -- Error
deriving instance BEq, Repr for Foo
#eval test 4
#check fun (x y : Foo) => x == y
def Boo := List (String × String)
deriving BEq, Repr, DecidableEq
def mkBoo (s : String) : Boo :=
[(s, s)]
#eval mkBoo "hello"
#eval mkBoo "hell" == mkBoo "hello"
#eval mkBoo "hello" == mkBoo "hello"
#eval mkBoo "hello" = mkBoo "hello"
def M := ReaderT String (StateT Nat IO)
deriving Monad, MonadState, MonadReader
#print instMMonad
def action : M Unit := do
modify (· + 1)
dbg_trace "{← read}"
|
a555c59e10b48d54dc14a4043fadee63bafa5b1c | 31f556cdeb9239ffc2fad8f905e33987ff4feab9 | /stage0/src/Lean/Elab/Deriving/TypeName.lean | 17a0cc896a9c3fd509f6567602cbae48b265ba43 | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | tobiasgrosser/lean4 | ce0fd9cca0feba1100656679bf41f0bffdbabb71 | ebdbdc10436a4d9d6b66acf78aae7a23f5bd073f | refs/heads/master | 1,673,103,412,948 | 1,664,930,501,000 | 1,664,930,501,000 | 186,870,185 | 0 | 0 | Apache-2.0 | 1,665,129,237,000 | 1,557,939,901,000 | Lean | UTF-8 | Lean | false | false | 868 | lean | /-
Copyright (c) 2022 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner
-/
import Lean.Elab.Deriving.Basic
namespace Lean.Elab
open Command Std Parser Term
private def deriveTypeNameInstance (declNames : Array Name) : CommandElabM Bool := do
for declName in declNames do
let cinfo ← getConstInfo declName
unless cinfo.levelParams.isEmpty do
throwError m!"{mkConst declName} has universe level parameters"
elabCommand <| ← withFreshMacroScope `(
unsafe def instImpl : TypeName @$(mkCIdent declName) := .mk _ $(quote declName)
@[implementedBy instImpl] opaque inst : TypeName @$(mkCIdent declName)
instance : TypeName @$(mkCIdent declName) := inst
)
return true
initialize
registerDerivingHandler ``TypeName deriveTypeNameInstance
|
f4f475cbc8a3d0065825a6916fc961028128a731 | 4d2583807a5ac6caaffd3d7a5f646d61ca85d532 | /counterexamples/direct_sum_is_internal.lean | 89beb501705dcc84bb6faf2610a898e30c56c221 | [
"Apache-2.0"
] | permissive | AntoineChambert-Loir/mathlib | 64aabb896129885f12296a799818061bc90da1ff | 07be904260ab6e36a5769680b6012f03a4727134 | refs/heads/master | 1,693,187,631,771 | 1,636,719,886,000 | 1,636,719,886,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,874 | lean | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard
-/
import algebra.direct_sum.module
import tactic.fin_cases
/-!
# Not all complementary decompositions of a module over a semiring make up a direct sum
This shows that while `ℤ≤0` and `ℤ≥0` are complementary `ℕ`-submodules of `ℤ`, which in turn
implies as a collection they are `complete_lattice.independent` and that they span all of `ℤ`, they
do not form a decomposition into a direct sum.
This file demonstrates why `direct_sum.submodule_is_internal_of_independent_of_supr_eq_top` must
take `ring R` and not `semiring R`.
-/
lemma units_int.one_ne_neg_one : (1 : units ℤ) ≠ -1 := dec_trivial
/-- Submodules of positive and negative integers, keyed by sign. -/
def with_sign (i : units ℤ) : submodule ℕ ℤ :=
add_submonoid.to_nat_submodule $ show add_submonoid ℤ, from
{ carrier := {z | 0 ≤ i • z},
zero_mem' := show 0 ≤ i • (0 : ℤ), from (smul_zero _).ge,
add_mem' := λ x y (hx : 0 ≤ i • x) (hy : 0 ≤ i • y), show _ ≤ _, begin
rw smul_add,
exact add_nonneg hx hy
end }
local notation `ℤ≥0` := with_sign 1
local notation `ℤ≤0` := with_sign (-1)
lemma mem_with_sign_one {x : ℤ} : x ∈ ℤ≥0 ↔ 0 ≤ x :=
show _ ≤ _ ↔ _, by rw one_smul
lemma mem_with_sign_neg_one {x : ℤ} : x ∈ ℤ≤0 ↔ x ≤ 0 :=
show _ ≤ _ ↔ _, by rw [units.neg_smul, le_neg, one_smul, neg_zero]
/-- The two submodules are complements. -/
lemma with_sign.is_compl : is_compl (ℤ≥0) (ℤ≤0) :=
begin
split,
{ apply submodule.disjoint_def.2,
intros x hx hx',
exact le_antisymm (mem_with_sign_neg_one.mp hx') (mem_with_sign_one.mp hx), },
{ intros x hx,
obtain hp | hn := (le_refl (0 : ℤ)).le_or_le x,
exact submodule.mem_sup_left (mem_with_sign_one.mpr hp),
exact submodule.mem_sup_right (mem_with_sign_neg_one.mpr hn), }
end
def with_sign.independent : complete_lattice.independent with_sign :=
begin
intros i,
rw [←finset.sup_univ_eq_supr, units_int.univ, finset.sup_insert, finset.sup_singleton],
fin_cases i,
{ convert with_sign.is_compl.disjoint,
convert bot_sup_eq,
{ exact supr_neg (not_not_intro rfl), },
{ rw supr_pos units_int.one_ne_neg_one.symm } },
{ convert with_sign.is_compl.disjoint.symm,
convert sup_bot_eq,
{ exact supr_neg (not_not_intro rfl), },
{ rw supr_pos units_int.one_ne_neg_one } },
end
lemma with_sign.supr : supr with_sign = ⊤ :=
begin
rw [←finset.sup_univ_eq_supr, units_int.univ, finset.sup_insert, finset.sup_singleton],
exact with_sign.is_compl.sup_eq_top,
end
/-- But there is no embedding into `ℤ` from the direct sum. -/
lemma with_sign.not_injective :
¬function.injective (direct_sum.to_module ℕ (units ℤ) ℤ (λ i, (with_sign i).subtype)) :=
begin
intro hinj,
let p1 : ℤ≥0 := ⟨1, mem_with_sign_one.2 zero_le_one⟩,
let n1 : ℤ≤0 := ⟨-1, mem_with_sign_neg_one.2 $ neg_nonpos.2 zero_le_one⟩,
let z := direct_sum.lof ℕ _ (λ i, with_sign i) 1 p1 +
direct_sum.lof ℕ _ (λ i, with_sign i) (-1) n1,
have : z ≠ 0,
{ intro h,
dsimp [z, direct_sum.lof_eq_of, direct_sum.of] at h,
replace h := dfinsupp.ext_iff.mp h 1,
rw [dfinsupp.zero_apply, dfinsupp.add_apply, dfinsupp.single_eq_same,
dfinsupp.single_eq_of_ne (units_int.one_ne_neg_one.symm), add_zero, subtype.ext_iff,
submodule.coe_zero] at h,
apply zero_ne_one h.symm, },
apply hinj.ne this,
rw [linear_map.map_zero, linear_map.map_add, direct_sum.to_module_lof, direct_sum.to_module_lof],
simp,
end
/-- And so they do not represent an internal direct sum. -/
lemma with_sign.not_internal : ¬direct_sum.submodule_is_internal with_sign :=
with_sign.not_injective ∘ and.elim_left
|
b072e1f89b66d54e6fd093e7d74fede8d17b62a0 | 2a70b774d16dbdf5a533432ee0ebab6838df0948 | /_target/deps/mathlib/src/analysis/calculus/mean_value.lean | 4c184b423d7f3c86c77011a1b5502d1aae7f9812 | [
"Apache-2.0"
] | permissive | hjvromen/lewis | 40b035973df7c77ebf927afab7878c76d05ff758 | 105b675f73630f028ad5d890897a51b3c1146fb0 | refs/heads/master | 1,677,944,636,343 | 1,676,555,301,000 | 1,676,555,301,000 | 327,553,599 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 59,339 | lean | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import analysis.calculus.local_extr
import analysis.convex.topology
/-!
# The mean value inequality and equalities
In this file we prove the following facts:
* `convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s`
and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with
constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the
derivative from a fixed linear map.
* `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or
`∥f x∥ ≤ B x` from upper estimates on `f'` or `∥f'∥`, respectively. These lemmas differ by
their assumptions:
* `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`;
* `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative
or its norm is less than `B' x`;
* `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `∥f x∥ = B x`;
* `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`;
* name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]`
and has a right derivative at every point of `[a, b)`, and (2) the lemma has
a counterpart assuming that `B` is differentiable everywhere on `ℝ`
* `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above
by a constant `C`, then `∥f x - f a∥ ≤ C * ∥x - a∥`; several versions deal with
right derivative and derivative within `[a, b]` (`has_deriv_within_at` or `deriv_within`).
* `convex.is_const_of_fderiv_within_eq_zero` : if a function has derivative `0` on a convex set `s`,
then it is a constant on `s`.
* `exists_ratio_has_deriv_at_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` :
Cauchy's Mean Value Theorem.
* `exists_has_deriv_at_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem.
* `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain.
* `convex.image_sub_lt_mul_sub_of_deriv_lt`, `convex.mul_sub_lt_image_sub_of_lt_deriv`,
`convex.image_sub_le_mul_sub_of_deriv_le`, `convex.mul_sub_le_image_sub_of_le_deriv`,
if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`.
* `convex.mono_of_deriv_nonneg`, `convex.antimono_of_deriv_nonpos`,
`convex.strict_mono_of_deriv_pos`, `convex.strict_antimono_of_deriv_neg` :
if the derivative of a function is non-negative/non-positive/positive/negative, then
the function is monotone/monotonically decreasing/strictly monotone/strictly monotonically
decreasing.
* `convex_on_of_deriv_mono`, `convex_on_of_deriv2_nonneg` : if the derivative of a function
is increasing or its second derivative is nonnegative, then the original function is convex.
* `strict_fderiv_of_cont_diff` : a C^1 function over the reals is strictly differentiable. (This
is a corollary of the mean value inequality.)
-/
variables {E : Type*} [normed_group E] [normed_space ℝ E]
{F : Type*} [normed_group F] [normed_space ℝ F]
open metric set asymptotics continuous_linear_map filter
open_locale classical topological_space nnreal
/-! ### One-dimensional fencing inequalities -/
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded above by a function `f'`;
* we have `f' x < B' x` whenever `f x = B x`.
Then `f x ≤ B x` everywhere on `[a, b]`. -/
lemma image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
-- `hf'` actually says `liminf (z - x)⁻¹ * (f z - f x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r →
∃ᶠ z in 𝓝[Ioi x] x, (z - x)⁻¹ * (f z - f x) < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
begin
change Icc a b ⊆ {x | f x ≤ B x},
set s := {x | f x ≤ B x} ∩ Icc a b,
have A : continuous_on (λ x, (f x, B x)) (Icc a b), from hf.prod hB,
have : is_closed s,
{ simp only [s, inter_comm],
exact A.preimage_closed_of_closed is_closed_Icc order_closed_topology.is_closed_le' },
apply this.Icc_subset_of_forall_exists_gt ha,
rintros x ⟨hxB : f x ≤ B x, xab⟩ y hy,
cases hxB.lt_or_eq with hxB hxB,
{ -- If `f x < B x`, then all we need is continuity of both sides
refine nonempty_of_mem_sets (inter_mem_sets _ (Ioc_mem_nhds_within_Ioi ⟨le_rfl, hy⟩)),
have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x,
from A x (Ico_subset_Icc_self xab)
(mem_nhds_sets (is_open_lt continuous_fst continuous_snd) hxB),
have : ∀ᶠ x in 𝓝[Ioi x] x, f x < B x,
from nhds_within_le_of_mem (Icc_mem_nhds_within_Ioi xab) this,
exact this.mono (λ y, le_of_lt) },
{ rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩,
specialize hf' x xab r hfr,
have HB : ∀ᶠ z in 𝓝[Ioi x] x, r < (z - x)⁻¹ * (B z - B x),
from (has_deriv_within_at_iff_tendsto_slope' $ lt_irrefl x).1
(hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB),
obtain ⟨z, ⟨hfz, hzB⟩, hz⟩ :
∃ z, ((z - x)⁻¹ * (f z - f x) < r ∧ r < (z - x)⁻¹ * (B z - B x)) ∧ z ∈ Ioc x y,
from ((hf'.and_eventually HB).and_eventually (Ioc_mem_nhds_within_Ioi ⟨le_rfl, hy⟩)).exists,
refine ⟨z, _, hz⟩,
have := (hfz.trans hzB).le,
rwa [mul_le_mul_left (inv_pos.2 $ sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this }
end
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has derivative `B'` everywhere on `ℝ`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded above by a function `f'`;
* we have `f' x < B' x` whenever `f x = B x`.
Then `f x ≤ B x` everywhere on `[a, b]`. -/
lemma image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
-- `hf'` actually says `liminf (z - x)⁻¹ * (f z - f x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r →
∃ᶠ z in 𝓝[Ioi x] x, (z - x)⁻¹ * (f z - f x) < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, has_deriv_at B (B' x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha
(λ x hx, (hB x).continuous_at.continuous_within_at)
(λ x hx, (hB x).has_deriv_within_at) bound
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded above by `B'`.
Then `f x ≤ B x` everywhere on `[a, b]`. -/
lemma image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
-- `bound` actually says `liminf (z - x)⁻¹ * (f z - f x) ≤ B' x`
(bound : ∀ x ∈ Ico a b, ∀ r, B' x < r →
∃ᶠ z in 𝓝[Ioi x] x, (z - x)⁻¹ * (f z - f x) < r) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
begin
have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a),
{ intros x hx r hr,
apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound,
{ rwa [sub_self, mul_zero, add_zero] },
{ exact hB.add (continuous_on_const.mul
(continuous_id.continuous_on.sub continuous_on_const)) },
{ assume x hx,
exact (hB' x hx).add (((has_deriv_within_at_id x (Ici x)).sub_const a).const_mul r) },
{ assume x hx _,
rw [mul_one],
exact (lt_add_iff_pos_right _).2 hr },
exact hx },
assume x hx,
have : continuous_within_at (λ r, B x + r * (x - a)) (Ioi 0) 0,
from continuous_within_at_const.add (continuous_within_at_id.mul continuous_within_at_const),
convert continuous_within_at_const.closure_le _ this (Hr x hx); simp
end
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* we have `f' x < B' x` whenever `f x = B x`.
Then `f x ≤ B x` everywhere on `[a, b]`. -/
lemma image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_liminf_slope_right_lt_deriv_boundary' hf
(λ x hx r hr, (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has derivative `B'` everywhere on `ℝ`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* we have `f' x < B' x` whenever `f x = B x`.
Then `f x ≤ B x` everywhere on `[a, b]`. -/
lemma image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, has_deriv_at B (B' x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha
(λ x hx, (hB x).continuous_at.continuous_within_at)
(λ x hx, (hB x).has_deriv_within_at) bound
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has derivative `B'` everywhere on `ℝ`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* we have `f' x ≤ B' x` on `[a, b)`.
Then `f x ≤ B x` everywhere on `[a, b]`. -/
lemma image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, f' x ≤ B' x) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' $
assume x hx r hr, (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr)
/-! ### Vector-valued functions `f : ℝ → E` -/
section
variables {f : ℝ → E} {a b : ℝ}
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `∥f a∥ ≤ B a`;
* `B` has right derivative at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(∥f z∥ - ∥f x∥) / (z - x)`
is bounded above by a function `f'`;
* we have `f' x < B' x` whenever `∥f x∥ = B x`.
Then `∥f x∥ ≤ B x` everywhere on `[a, b]`. -/
lemma image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [normed_group E]
{f : ℝ → E} {f' : ℝ → ℝ} (hf : continuous_on f (Icc a b))
-- `hf'` actually says `liminf ∥z - x∥⁻¹ * (∥f z∥ - ∥f x∥) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r →
∃ᶠ z in 𝓝[Ioi x] x, (z - x)⁻¹ * (∥f z∥ - ∥f x∥) < r)
{B B' : ℝ → ℝ} (ha : ∥f a∥ ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, ∥f x∥ = B x → f' x < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → ∥f x∥ ≤ B x :=
image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuous_on hf) hf'
ha hB hB' bound
/-- General fencing theorem for continuous functions with an estimate on the norm of the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `∥f a∥ ≤ B a`;
* `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`;
* the norm of `f'` is strictly less than `B'` whenever `∥f x∥ = B x`.
Then `∥f x∥ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions
to make this theorem work for piecewise differentiable functions.
-/
lemma image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ∥f a∥ ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, ∥f x∥ = B x → ∥f' x∥ < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → ∥f x∥ ≤ B x :=
image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf
(λ x hx r hr, (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound
/-- General fencing theorem for continuous functions with an estimate on the norm of the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `∥f a∥ ≤ B a`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* `B` has derivative `B'` everywhere on `ℝ`;
* the norm of `f'` is strictly less than `B'` whenever `∥f x∥ = B x`.
Then `∥f x∥ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions
to make this theorem work for piecewise differentiable functions.
-/
lemma image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ∥f a∥ ≤ B a) (hB : ∀ x, has_deriv_at B (B' x) x)
(bound : ∀ x ∈ Ico a b, ∥f x∥ = B x → ∥f' x∥ < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → ∥f x∥ ≤ B x :=
image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha
(λ x hx, (hB x).continuous_at.continuous_within_at)
(λ x hx, (hB x).has_deriv_within_at) bound
/-- General fencing theorem for continuous functions with an estimate on the norm of the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `∥f a∥ ≤ B a`;
* `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`;
* we have `∥f' x∥ ≤ B x` everywhere on `[a, b)`.
Then `∥f x∥ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions
to make this theorem work for piecewise differentiable functions.
-/
lemma image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ∥f a∥ ≤ B a) (hB : continuous_on B (Icc a b))
(hB' : ∀ x ∈ Ico a b, has_deriv_within_at B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, ∥f' x∥ ≤ B' x) :
∀ ⦃x⦄, x ∈ Icc a b → ∥f x∥ ≤ B x :=
image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuous_on hf) ha hB hB' $
(λ x hx r hr, (hf' x hx).liminf_right_slope_norm_le (lt_of_le_of_lt (bound x hx) hr))
/-- General fencing theorem for continuous functions with an estimate on the norm of the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `∥f a∥ ≤ B a`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* `B` has derivative `B'` everywhere on `ℝ`;
* we have `∥f' x∥ ≤ B x` everywhere on `[a, b)`.
Then `∥f x∥ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions
to make this theorem work for piecewise differentiable functions.
-/
lemma image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ∥f a∥ ≤ B a) (hB : ∀ x, has_deriv_at B (B' x) x)
(bound : ∀ x ∈ Ico a b, ∥f' x∥ ≤ B' x) :
∀ ⦃x⦄, x ∈ Icc a b → ∥f x∥ ≤ B x :=
image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha
(λ x hx, (hB x).continuous_at.continuous_within_at)
(λ x hx, (hB x).has_deriv_within_at) bound
/-- A function on `[a, b]` with the norm of the right derivative bounded by `C`
satisfies `∥f x - f a∥ ≤ C * (x - a)`. -/
theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ}
(hf : continuous_on f (Icc a b))
(hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
(bound : ∀x ∈ Ico a b, ∥f' x∥ ≤ C) :
∀ x ∈ Icc a b, ∥f x - f a∥ ≤ C * (x - a) :=
begin
let g := λ x, f x - f a,
have hg : continuous_on g (Icc a b), from hf.sub continuous_on_const,
have hg' : ∀ x ∈ Ico a b, has_deriv_within_at g (f' x) (Ici x) x,
{ assume x hx,
simpa using (hf' x hx).sub (has_deriv_within_at_const _ _ _) },
let B := λ x, C * (x - a),
have hB : ∀ x, has_deriv_at B C x,
{ assume x,
simpa using (has_deriv_at_const x C).mul ((has_deriv_at_id x).sub (has_deriv_at_const x a)) },
convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound,
{ simp only [g, B] },
{ simp only [g, B], rw [sub_self, norm_zero, sub_self, mul_zero] }
end
/-- A function on `[a, b]` with the norm of the derivative within `[a, b]`
bounded by `C` satisfies `∥f x - f a∥ ≤ C * (x - a)`, `has_deriv_within_at`
version. -/
theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ}
(hf : ∀ x ∈ Icc a b, has_deriv_within_at f (f' x) (Icc a b) x)
(bound : ∀x ∈ Ico a b, ∥f' x∥ ≤ C) :
∀ x ∈ Icc a b, ∥f x - f a∥ ≤ C * (x - a) :=
begin
refine norm_image_sub_le_of_norm_deriv_right_le_segment
(λ x hx, (hf x hx).continuous_within_at) (λ x hx, _) bound,
exact (hf x $ Ico_subset_Icc_self hx).nhds_within (Icc_mem_nhds_within_Ici hx)
end
/-- A function on `[a, b]` with the norm of the derivative within `[a, b]`
bounded by `C` satisfies `∥f x - f a∥ ≤ C * (x - a)`, `deriv_within`
version. -/
theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : differentiable_on ℝ f (Icc a b))
(bound : ∀x ∈ Ico a b, ∥deriv_within f (Icc a b) x∥ ≤ C) :
∀ x ∈ Icc a b, ∥f x - f a∥ ≤ C * (x - a) :=
begin
refine norm_image_sub_le_of_norm_deriv_le_segment' _ bound,
exact λ x hx, (hf x hx).has_deriv_within_at
end
/-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]`
bounded by `C` satisfies `∥f 1 - f 0∥ ≤ C`, `has_deriv_within_at`
version. -/
theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ}
(hf : ∀ x ∈ Icc (0:ℝ) 1, has_deriv_within_at f (f' x) (Icc (0:ℝ) 1) x)
(bound : ∀x ∈ Ico (0:ℝ) 1, ∥f' x∥ ≤ C) :
∥f 1 - f 0∥ ≤ C :=
by simpa only [sub_zero, mul_one]
using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one)
/-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]`
bounded by `C` satisfies `∥f 1 - f 0∥ ≤ C`, `deriv_within` version. -/
theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ}
(hf : differentiable_on ℝ f (Icc (0:ℝ) 1))
(bound : ∀x ∈ Ico (0:ℝ) 1, ∥deriv_within f (Icc (0:ℝ) 1) x∥ ≤ C) :
∥f 1 - f 0∥ ≤ C :=
by simpa only [sub_zero, mul_one]
using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one)
theorem constant_of_has_deriv_right_zero (hcont : continuous_on f (Icc a b))
(hderiv : ∀ x ∈ Ico a b, has_deriv_within_at f 0 (Ici x) x) :
∀ x ∈ Icc a b, f x = f a :=
by simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using
λ x hx, norm_image_sub_le_of_norm_deriv_right_le_segment
hcont hderiv (λ y hy, by rw norm_le_zero_iff) x hx
theorem constant_of_deriv_within_zero (hdiff : differentiable_on ℝ f (Icc a b))
(hderiv : ∀ x ∈ Ico a b, deriv_within f (Icc a b) x = 0) :
∀ x ∈ Icc a b, f x = f a :=
begin
have H : ∀ x ∈ Ico a b, ∥deriv_within f (Icc a b) x∥ ≤ 0 :=
by simpa only [norm_le_zero_iff] using λ x hx, hderiv x hx,
simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using
λ x hx, norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx,
end
variables {f' g : ℝ → E}
/-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`,
then they are equal everywhere on `[a, b]`. -/
theorem eq_of_has_deriv_right_eq
(derivf : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
(derivg : ∀ x ∈ Ico a b, has_deriv_within_at g (f' x) (Ici x) x)
(fcont : continuous_on f (Icc a b)) (gcont : continuous_on g (Icc a b))
(hi : f a = g a) :
∀ y ∈ Icc a b, f y = g y :=
begin
simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢,
exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont)
(λ y hy, by simpa only [sub_self] using (derivf y hy).sub (derivg y hy)),
end
/-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere
on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/
theorem eq_of_deriv_within_eq (fdiff : differentiable_on ℝ f (Icc a b))
(gdiff : differentiable_on ℝ g (Icc a b))
(hderiv : eq_on (deriv_within f (Icc a b)) (deriv_within g (Icc a b)) (Ico a b))
(hi : f a = g a) :
∀ y ∈ Icc a b, f y = g y :=
begin
have A : ∀ y ∈ Ico a b, has_deriv_within_at f (deriv_within f (Icc a b) y) (Ici y) y :=
λ y hy, (fdiff y (mem_Icc_of_Ico hy)).has_deriv_within_at.nhds_within
(Icc_mem_nhds_within_Ici hy),
have B : ∀ y ∈ Ico a b, has_deriv_within_at g (deriv_within g (Icc a b) y) (Ici y) y :=
λ y hy, (gdiff y (mem_Icc_of_Ico hy)).has_deriv_within_at.nhds_within
(Icc_mem_nhds_within_Ici hy),
exact eq_of_has_deriv_right_eq A (λ y hy, (hderiv hy).symm ▸ B y hy) fdiff.continuous_on
gdiff.continuous_on hi
end
end
/-! ### Vector-valued functions `f : E → F` -/
/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then
the function is `C`-Lipschitz. Version with `has_fderiv_within`. -/
theorem convex.norm_image_sub_le_of_norm_has_fderiv_within_le
{f : E → F} {C : ℝ} {s : set E} {x y : E} {f' : E → (E →L[ℝ] F)}
(hf : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (bound : ∀x∈s, ∥f' x∥ ≤ C)
(hs : convex s) (xs : x ∈ s) (ys : y ∈ s) : ∥f y - f x∥ ≤ C * ∥y - x∥ :=
begin
/- By composition with `t ↦ x + t • (y-x)`, we reduce to a statement for functions defined
on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`.
We just have to check the differentiability of the composition and bounds on its derivative,
which is straightforward but tedious for lack of automation. -/
have C0 : 0 ≤ C := le_trans (norm_nonneg _) (bound x xs),
set g : ℝ → E := λ t, x + t • (y - x),
have Dg : ∀ t, has_deriv_at g (y-x) t,
{ assume t,
simpa only [one_smul] using ((has_deriv_at_id t).smul_const (y - x)).const_add x },
have segm : Icc 0 1 ⊆ g ⁻¹' s,
{ rw [← image_subset_iff, ← segment_eq_image'],
apply hs.segment_subset xs ys },
have : f x = f (g 0), by { simp only [g], rw [zero_smul, add_zero] },
rw this,
have : f y = f (g 1), by { simp only [g], rw [one_smul, add_sub_cancel'_right] },
rw this,
have D2: ∀ t ∈ Icc (0:ℝ) 1, has_deriv_within_at (f ∘ g)
((f' (g t) : E → F) (y-x)) (Icc (0:ℝ) 1) t,
{ intros t ht,
exact (hf (g t) $ segm ht).comp_has_deriv_within_at _
(Dg t).has_deriv_within_at segm },
apply norm_image_sub_le_of_norm_deriv_le_segment_01' D2,
assume t ht,
refine le_trans (le_op_norm _ _) (mul_le_mul_of_nonneg_right _ (norm_nonneg _)),
exact bound (g t) (segm $ Ico_subset_Icc_self ht)
end
/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`,
then the function is `C`-Lipschitz on `s`. Version with `has_fderiv_within` and `lipschitz_on_with`. -/
theorem convex.lipschitz_on_with_of_norm_has_fderiv_within_le
{f : E → F} {C : ℝ} {s : set E} {f' : E → (E →L[ℝ] F)}
(hf : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (bound : ∀x∈s, ∥f' x∥ ≤ C)
(hs : convex s) : lipschitz_on_with (nnreal.of_real C) f s :=
begin
rw lipschitz_on_with_iff_norm_sub_le,
intros x x_in y y_in,
convert hs.norm_image_sub_le_of_norm_has_fderiv_within_le hf bound y_in x_in,
exact nnreal.coe_of_real C ((norm_nonneg $ f' x).trans $ bound x x_in)
end
/-- The mean value theorem on a convex set: if the derivative of a function within this set is
bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv_within`. -/
theorem convex.norm_image_sub_le_of_norm_fderiv_within_le {f : E → F} {C : ℝ} {s : set E} {x y : E}
(hf : differentiable_on ℝ f s) (bound : ∀x∈s, ∥fderiv_within ℝ f s x∥ ≤ C)
(hs : convex s) (xs : x ∈ s) (ys : y ∈ s) : ∥f y - f x∥ ≤ C * ∥y - x∥ :=
hs.norm_image_sub_le_of_norm_has_fderiv_within_le (λ x hx, (hf x hx).has_fderiv_within_at)
bound xs ys
/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`,
then the function is `C`-Lipschitz on `s`. Version with `fderiv_within` and `lipschitz_on_with`. -/
theorem convex.lipschitz_on_with_of_norm_fderiv_within_le {f : E → F} {C : ℝ} {s : set E}
(hf : differentiable_on ℝ f s) (bound : ∀x∈s, ∥fderiv_within ℝ f s x∥ ≤ C)
(hs : convex s) : lipschitz_on_with (nnreal.of_real C) f s:=
hs.lipschitz_on_with_of_norm_has_fderiv_within_le (λ x hx, (hf x hx).has_fderiv_within_at) bound
/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`,
then the function is `C`-Lipschitz. Version with `fderiv`. -/
theorem convex.norm_image_sub_le_of_norm_fderiv_le {f : E → F} {C : ℝ} {s : set E} {x y : E}
(hf : ∀ x ∈ s, differentiable_at ℝ f x) (bound : ∀x∈s, ∥fderiv ℝ f x∥ ≤ C)
(hs : convex s) (xs : x ∈ s) (ys : y ∈ s) : ∥f y - f x∥ ≤ C * ∥y - x∥ :=
hs.norm_image_sub_le_of_norm_has_fderiv_within_le
(λ x hx, (hf x hx).has_fderiv_at.has_fderiv_within_at) bound xs ys
/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`,
then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `lipschitz_on_with`. -/
theorem convex.lipschitz_on_with_of_norm_fderiv_le {f : E → F} {C : ℝ} {s : set E}
(hf : ∀ x ∈ s, differentiable_at ℝ f x) (bound : ∀x∈s, ∥fderiv ℝ f x∥ ≤ C)
(hs : convex s) : lipschitz_on_with (nnreal.of_real C) f s :=
hs.lipschitz_on_with_of_norm_has_fderiv_within_le
(λ x hx, (hf x hx).has_fderiv_at.has_fderiv_within_at) bound
/-- Variant of the mean value inequality on a convex set, using a bound on the difference between
the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with
`has_fderiv_within`. -/
theorem convex.norm_image_sub_le_of_norm_has_fderiv_within_le'
{f : E → F} {C : ℝ} {s : set E} {x y : E} {f' : E → (E →L[ℝ] F)} {φ : E →L[ℝ] F}
(hf : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (bound : ∀x∈s, ∥f' x - φ∥ ≤ C)
(hs : convex s) (xs : x ∈ s) (ys : y ∈ s) : ∥f y - f x - φ (y - x)∥ ≤ C * ∥y - x∥ :=
begin
/- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem
applies, `convex.norm_image_sub_le_of_norm_has_fderiv_within_le`. Then, we just need to glue
together the pieces, expressing back `f` in terms of `g`. -/
let g := λy, f y - φ y,
have hg : ∀ x ∈ s, has_fderiv_within_at g (f' x - φ) s x :=
λ x xs, (hf x xs).sub φ.has_fderiv_within_at,
calc ∥f y - f x - φ (y - x)∥ = ∥f y - f x - (φ y - φ x)∥ : by simp
... = ∥(f y - φ y) - (f x - φ x)∥ : by abel
... = ∥g y - g x∥ : by simp
... ≤ C * ∥y - x∥ : convex.norm_image_sub_le_of_norm_has_fderiv_within_le hg bound hs xs ys,
end
/-- Variant of the mean value inequality on a convex set. Version with `fderiv_within`. -/
theorem convex.norm_image_sub_le_of_norm_fderiv_within_le' {f : E → F} {C : ℝ} {s : set E} {x y : E}
{φ : E →L[ℝ] F} (hf : differentiable_on ℝ f s) (bound : ∀x∈s, ∥fderiv_within ℝ f s x - φ∥ ≤ C)
(hs : convex s) (xs : x ∈ s) (ys : y ∈ s) : ∥f y - f x - φ (y - x)∥ ≤ C * ∥y - x∥ :=
hs.norm_image_sub_le_of_norm_has_fderiv_within_le' (λ x hx, (hf x hx).has_fderiv_within_at)
bound xs ys
/-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/
theorem convex.norm_image_sub_le_of_norm_fderiv_le' {f : E → F} {C : ℝ} {s : set E} {x y : E}
{φ : E →L[ℝ] F} (hf : ∀ x ∈ s, differentiable_at ℝ f x) (bound : ∀x∈s, ∥fderiv ℝ f x - φ∥ ≤ C)
(hs : convex s) (xs : x ∈ s) (ys : y ∈ s) : ∥f y - f x - φ (y - x)∥ ≤ C * ∥y - x∥ :=
hs.norm_image_sub_le_of_norm_has_fderiv_within_le'
(λ x hx, (hf x hx).has_fderiv_at.has_fderiv_within_at) bound xs ys
/-- If a function has zero Fréchet derivative at every point of a convex set,
then it is a constant on this set. -/
theorem convex.is_const_of_fderiv_within_eq_zero {s : set E} (hs : convex s)
{f : E → F} (hf : differentiable_on ℝ f s) (hf' : ∀ x ∈ s, fderiv_within ℝ f s x = 0)
{x y : E} (hx : x ∈ s) (hy : y ∈ s) :
f x = f y :=
have bound : ∀ x ∈ s, ∥fderiv_within ℝ f s x∥ ≤ 0,
from λ x hx, by simp only [hf' x hx, norm_zero],
by simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm]
using hs.norm_image_sub_le_of_norm_fderiv_within_le hf bound hx hy
theorem is_const_of_fderiv_eq_zero {f : E → F} (hf : differentiable ℝ f)
(hf' : ∀ x, fderiv ℝ f x = 0) (x y : E) :
f x = f y :=
convex_univ.is_const_of_fderiv_within_eq_zero hf.differentiable_on
(λ x _, by rw fderiv_within_univ; exact hf' x) trivial trivial
/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C`, then the function is `C`-Lipschitz. Version with `has_deriv_within`. -/
theorem convex.norm_image_sub_le_of_norm_has_deriv_within_le
{f f' : ℝ → F} {C : ℝ} {s : set ℝ} {x y : ℝ}
(hf : ∀ x ∈ s, has_deriv_within_at f (f' x) s x) (bound : ∀x∈s, ∥f' x∥ ≤ C)
(hs : convex s) (xs : x ∈ s) (ys : y ∈ s) : ∥f y - f x∥ ≤ C * ∥y - x∥ :=
convex.norm_image_sub_le_of_norm_has_fderiv_within_le (λ x hx, (hf x hx).has_fderiv_within_at)
(λ x hx, le_trans (by simp) (bound x hx)) hs xs ys
/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
Version with `has_deriv_within` and `lipschitz_on_with`. -/
theorem convex.lipschitz_on_with_of_norm_has_deriv_within_le
{f f' : ℝ → F} {C : ℝ} {s : set ℝ} (hs : convex s)
(hf : ∀ x ∈ s, has_deriv_within_at f (f' x) s x) (bound : ∀x∈s, ∥f' x∥ ≤ C) :
lipschitz_on_with (nnreal.of_real C) f s :=
convex.lipschitz_on_with_of_norm_has_fderiv_within_le (λ x hx, (hf x hx).has_fderiv_within_at)
(λ x hx, le_trans (by simp) (bound x hx)) hs
/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within
this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv_within` -/
theorem convex.norm_image_sub_le_of_norm_deriv_within_le
{f : ℝ → F} {C : ℝ} {s : set ℝ} {x y : ℝ}
(hf : differentiable_on ℝ f s) (bound : ∀x∈s, ∥deriv_within f s x∥ ≤ C)
(hs : convex s) (xs : x ∈ s) (ys : y ∈ s) : ∥f y - f x∥ ≤ C * ∥y - x∥ :=
hs.norm_image_sub_le_of_norm_has_deriv_within_le (λ x hx, (hf x hx).has_deriv_within_at)
bound xs ys
/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
Version with `deriv_within` and `lipschitz_on_with`. -/
theorem convex.lipschitz_on_with_of_norm_deriv_within_le
{f : ℝ → F} {C : ℝ} {s : set ℝ} (hs : convex s)
(hf : differentiable_on ℝ f s) (bound : ∀x∈s, ∥deriv_within f s x∥ ≤ C) :
lipschitz_on_with (nnreal.of_real C) f s :=
hs.lipschitz_on_with_of_norm_has_deriv_within_le (λ x hx, (hf x hx).has_deriv_within_at) bound
/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/
theorem convex.norm_image_sub_le_of_norm_deriv_le {f : ℝ → F} {C : ℝ} {s : set ℝ} {x y : ℝ}
(hf : ∀ x ∈ s, differentiable_at ℝ f x) (bound : ∀x∈s, ∥deriv f x∥ ≤ C)
(hs : convex s) (xs : x ∈ s) (ys : y ∈ s) : ∥f y - f x∥ ≤ C * ∥y - x∥ :=
hs.norm_image_sub_le_of_norm_has_deriv_within_le
(λ x hx, (hf x hx).has_deriv_at.has_deriv_within_at) bound xs ys
/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`.
Version with `deriv` and `lipschitz_on_with`. -/
theorem convex.lipschitz_on_with_of_norm_deriv_le {f : ℝ → F} {C : ℝ} {s : set ℝ}
(hf : ∀ x ∈ s, differentiable_at ℝ f x) (bound : ∀x∈s, ∥deriv f x∥ ≤ C)
(hs : convex s) : lipschitz_on_with (nnreal.of_real C) f s :=
hs.lipschitz_on_with_of_norm_has_deriv_within_le
(λ x hx, (hf x hx).has_deriv_at.has_deriv_within_at) bound
/-! ### Functions `[a, b] → ℝ`. -/
section interval
-- Declare all variables here to make sure they come in a correct order
variables (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : continuous_on f (Icc a b))
(hff' : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hfd : differentiable_on ℝ f (Ioo a b))
(g g' : ℝ → ℝ) (hgc : continuous_on g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, has_deriv_at g (g' x) x)
(hgd : differentiable_on ℝ g (Ioo a b))
include hab hfc hff' hgc hgg'
/-- Cauchy's Mean Value Theorem, `has_deriv_at` version. -/
lemma exists_ratio_has_deriv_at_eq_ratio_slope :
∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c :=
begin
let h := λ x, (g b - g a) * f x - (f b - f a) * g x,
have hI : h a = h b,
{ simp only [h], ring },
let h' := λ x, (g b - g a) * f' x - (f b - f a) * g' x,
have hhh' : ∀ x ∈ Ioo a b, has_deriv_at h (h' x) x,
from λ x hx, ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)),
have hhc : continuous_on h (Icc a b),
from (continuous_on_const.mul hfc).sub (continuous_on_const.mul hgc),
rcases exists_has_deriv_at_eq_zero h h' hab hhc hI hhh' with ⟨c, cmem, hc⟩,
exact ⟨c, cmem, sub_eq_zero.1 hc⟩
end
omit hfc hgc
/-- Cauchy's Mean Value Theorem, extended `has_deriv_at` version. -/
lemma exists_ratio_has_deriv_at_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
(hff' : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, has_deriv_at g (g' x) x)
(hfa : tendsto f (𝓝[Ioi a] a) (𝓝 lfa)) (hga : tendsto g (𝓝[Ioi a] a) (𝓝 lga))
(hfb : tendsto f (𝓝[Iio b] b) (𝓝 lfb)) (hgb : tendsto g (𝓝[Iio b] b) (𝓝 lgb)) :
∃ c ∈ Ioo a b, (lgb - lga) * (f' c) = (lfb - lfa) * (g' c) :=
begin
let h := λ x, (lgb - lga) * f x - (lfb - lfa) * g x,
have hha : tendsto h (𝓝[Ioi a] a) (𝓝 $ lgb * lfa - lfb * lga),
{ have : tendsto h (𝓝[Ioi a] a)(𝓝 $ (lgb - lga) * lfa - (lfb - lfa) * lga) :=
(tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga),
convert this using 2,
ring },
have hhb : tendsto h (𝓝[Iio b] b) (𝓝 $ lgb * lfa - lfb * lga),
{ have : tendsto h (𝓝[Iio b] b)(𝓝 $ (lgb - lga) * lfb - (lfb - lfa) * lgb) :=
(tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb),
convert this using 2,
ring },
let h' := λ x, (lgb - lga) * f' x - (lfb - lfa) * g' x,
have hhh' : ∀ x ∈ Ioo a b, has_deriv_at h (h' x) x,
{ intros x hx,
exact ((hff' x hx).const_mul _ ).sub (((hgg' x hx)).const_mul _) },
rcases exists_has_deriv_at_eq_zero' hab hha hhb hhh' with ⟨c, cmem, hc⟩,
exact ⟨c, cmem, sub_eq_zero.1 hc⟩
end
include hfc
omit hgg'
/-- Lagrange's Mean Value Theorem, `has_deriv_at` version -/
lemma exists_has_deriv_at_eq_slope : ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a) :=
begin
rcases exists_ratio_has_deriv_at_eq_ratio_slope f f' hab hfc hff'
id 1 continuous_id.continuous_on (λ x hx, has_deriv_at_id x) with ⟨c, cmem, hc⟩,
use [c, cmem],
simp only [_root_.id, pi.one_apply, mul_one] at hc,
rw [← hc, mul_div_cancel_left],
exact ne_of_gt (sub_pos.2 hab)
end
omit hff'
/-- Cauchy's Mean Value Theorem, `deriv` version. -/
lemma exists_ratio_deriv_eq_ratio_slope :
∃ c ∈ Ioo a b, (g b - g a) * (deriv f c) = (f b - f a) * (deriv g c) :=
exists_ratio_has_deriv_at_eq_ratio_slope f (deriv f) hab hfc
(λ x hx, ((hfd x hx).differentiable_at $ mem_nhds_sets is_open_Ioo hx).has_deriv_at)
g (deriv g) hgc (λ x hx, ((hgd x hx).differentiable_at $ mem_nhds_sets is_open_Ioo hx).has_deriv_at)
omit hfc
/-- Cauchy's Mean Value Theorem, extended `deriv` version. -/
lemma exists_ratio_deriv_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
(hdf : differentiable_on ℝ f $ Ioo a b) (hdg : differentiable_on ℝ g $ Ioo a b)
(hfa : tendsto f (𝓝[Ioi a] a) (𝓝 lfa)) (hga : tendsto g (𝓝[Ioi a] a) (𝓝 lga))
(hfb : tendsto f (𝓝[Iio b] b) (𝓝 lfb)) (hgb : tendsto g (𝓝[Iio b] b) (𝓝 lgb)) :
∃ c ∈ Ioo a b, (lgb - lga) * (deriv f c) = (lfb - lfa) * (deriv g c) :=
exists_ratio_has_deriv_at_eq_ratio_slope' _ _ hab _ _
(λ x hx, ((hdf x hx).differentiable_at $ Ioo_mem_nhds hx.1 hx.2).has_deriv_at)
(λ x hx, ((hdg x hx).differentiable_at $ Ioo_mem_nhds hx.1 hx.2).has_deriv_at)
hfa hga hfb hgb
/-- Lagrange's Mean Value Theorem, `deriv` version. -/
lemma exists_deriv_eq_slope : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) :=
exists_has_deriv_at_eq_slope f (deriv f) hab hfc
(λ x hx, ((hfd x hx).differentiable_at $ mem_nhds_sets is_open_Ioo hx).has_deriv_at)
end interval
/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `C < f'`, then
`f` grows faster than `C * x` on `D`, i.e., `C * (y - x) < f y - f x` whenever `x, y ∈ D`,
`x < y`. -/
theorem convex.mul_sub_lt_image_sub_of_lt_deriv {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
{C} (hf'_gt : ∀ x ∈ interior D, C < deriv f x) :
∀ x y ∈ D, x < y → C * (y - x) < f y - f x :=
begin
assume x y hx hy hxy,
have hxyD : Icc x y ⊆ D, from hD.ord_connected hx hy,
have hxyD' : Ioo x y ⊆ interior D,
from subset_sUnion_of_mem ⟨is_open_Ioo, subset.trans Ioo_subset_Icc_self hxyD⟩,
obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x),
from exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD'),
have : C < (f y - f x) / (y - x), by { rw [← ha], exact hf'_gt _ (hxyD' a_mem) },
exact (lt_div_iff (sub_pos.2 hxy)).1 this
end
/-- Let `f : ℝ → ℝ` be a differentiable function. If `C < f'`, then `f` grows faster than
`C * x`, i.e., `C * (y - x) < f y - f x` whenever `x < y`. -/
theorem mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : differentiable ℝ f)
{C} (hf'_gt : ∀ x, C < deriv f x) ⦃x y⦄ (hxy : x < y) :
C * (y - x) < f y - f x :=
convex_univ.mul_sub_lt_image_sub_of_lt_deriv hf.continuous.continuous_on hf.differentiable_on
(λ x _, hf'_gt x) x y trivial trivial hxy
/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `C ≤ f'`, then
`f` grows at least as fast as `C * x` on `D`, i.e., `C * (y - x) ≤ f y - f x` whenever `x, y ∈ D`,
`x ≤ y`. -/
theorem convex.mul_sub_le_image_sub_of_le_deriv {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
{C} (hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x) :
∀ x y ∈ D, x ≤ y → C * (y - x) ≤ f y - f x :=
begin
assume x y hx hy hxy,
cases eq_or_lt_of_le hxy with hxy' hxy', by rw [hxy', sub_self, sub_self, mul_zero],
have hxyD : Icc x y ⊆ D, from hD.ord_connected hx hy,
have hxyD' : Ioo x y ⊆ interior D,
from subset_sUnion_of_mem ⟨is_open_Ioo, subset.trans Ioo_subset_Icc_self hxyD⟩,
obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x),
from exists_deriv_eq_slope f hxy' (hf.mono hxyD) (hf'.mono hxyD'),
have : C ≤ (f y - f x) / (y - x), by { rw [← ha], exact hf'_ge _ (hxyD' a_mem) },
exact (le_div_iff (sub_pos.2 hxy')).1 this
end
/-- Let `f : ℝ → ℝ` be a differentiable function. If `C ≤ f'`, then `f` grows at least as fast
as `C * x`, i.e., `C * (y - x) ≤ f y - f x` whenever `x ≤ y`. -/
theorem mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : differentiable ℝ f)
{C} (hf'_ge : ∀ x, C ≤ deriv f x) ⦃x y⦄ (hxy : x ≤ y) :
C * (y - x) ≤ f y - f x :=
convex_univ.mul_sub_le_image_sub_of_le_deriv hf.continuous.continuous_on hf.differentiable_on
(λ x _, hf'_ge x) x y trivial trivial hxy
/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f' < C`, then
`f` grows slower than `C * x` on `D`, i.e., `f y - f x < C * (y - x)` whenever `x, y ∈ D`,
`x < y`. -/
theorem convex.image_sub_lt_mul_sub_of_deriv_lt {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
{C} (lt_hf' : ∀ x ∈ interior D, deriv f x < C) :
∀ x y ∈ D, x < y → f y - f x < C * (y - x) :=
begin
assume x y hx hy hxy,
have hf'_gt : ∀ x ∈ interior D, -C < deriv (λ y, -f y) x,
{ assume x hx,
rw [deriv.neg, neg_lt_neg_iff],
exact lt_hf' x hx },
simpa [-neg_lt_neg_iff]
using neg_lt_neg (hD.mul_sub_lt_image_sub_of_lt_deriv hf.neg hf'.neg hf'_gt x y hx hy hxy)
end
/-- Let `f : ℝ → ℝ` be a differentiable function. If `f' < C`, then `f` grows slower than
`C * x` on `D`, i.e., `f y - f x < C * (y - x)` whenever `x < y`. -/
theorem image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : differentiable ℝ f)
{C} (lt_hf' : ∀ x, deriv f x < C) ⦃x y⦄ (hxy : x < y) :
f y - f x < C * (y - x) :=
convex_univ.image_sub_lt_mul_sub_of_deriv_lt hf.continuous.continuous_on hf.differentiable_on
(λ x _, lt_hf' x) x y trivial trivial hxy
/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f' ≤ C`, then
`f` grows at most as fast as `C * x` on `D`, i.e., `f y - f x ≤ C * (y - x)` whenever `x, y ∈ D`,
`x ≤ y`. -/
theorem convex.image_sub_le_mul_sub_of_deriv_le {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
{C} (le_hf' : ∀ x ∈ interior D, deriv f x ≤ C) :
∀ x y ∈ D, x ≤ y → f y - f x ≤ C * (y - x) :=
begin
assume x y hx hy hxy,
have hf'_ge : ∀ x ∈ interior D, -C ≤ deriv (λ y, -f y) x,
{ assume x hx,
rw [deriv.neg, neg_le_neg_iff],
exact le_hf' x hx },
simpa [-neg_le_neg_iff]
using neg_le_neg (hD.mul_sub_le_image_sub_of_le_deriv hf.neg hf'.neg hf'_ge x y hx hy hxy)
end
/-- Let `f : ℝ → ℝ` be a differentiable function. If `f' ≤ C`, then `f` grows at most as fast
as `C * x`, i.e., `f y - f x ≤ C * (y - x)` whenever `x ≤ y`. -/
theorem image_sub_le_mul_sub_of_deriv_le {f : ℝ → ℝ} (hf : differentiable ℝ f)
{C} (le_hf' : ∀ x, deriv f x ≤ C) ⦃x y⦄ (hxy : x ≤ y) :
f y - f x ≤ C * (y - x) :=
convex_univ.image_sub_le_mul_sub_of_deriv_le hf.continuous.continuous_on hf.differentiable_on
(λ x _, le_hf' x) x y trivial trivial hxy
/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f'` is positive, then
`f` is a strictly monotonically increasing function on `D`. -/
theorem convex.strict_mono_of_deriv_pos {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'_pos : ∀ x ∈ interior D, 0 < deriv f x) :
∀ x y ∈ D, x < y → f x < f y :=
by simpa only [zero_mul, sub_pos] using hD.mul_sub_lt_image_sub_of_lt_deriv hf hf' hf'_pos
/-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is positive, then
`f` is a strictly monotonically increasing function. -/
theorem strict_mono_of_deriv_pos {f : ℝ → ℝ} (hf : differentiable ℝ f)
(hf'_pos : ∀ x, 0 < deriv f x) :
strict_mono f :=
λ x y hxy, convex_univ.strict_mono_of_deriv_pos hf.continuous.continuous_on hf.differentiable_on
(λ x _, hf'_pos x) x y trivial trivial hxy
/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonnegative, then
`f` is a monotonically increasing function on `D`. -/
theorem convex.mono_of_deriv_nonneg {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'_nonneg : ∀ x ∈ interior D, 0 ≤ deriv f x) :
∀ x y ∈ D, x ≤ y → f x ≤ f y :=
by simpa only [zero_mul, sub_nonneg] using hD.mul_sub_le_image_sub_of_le_deriv hf hf' hf'_nonneg
/-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonnegative, then
`f` is a monotonically increasing function. -/
theorem mono_of_deriv_nonneg {f : ℝ → ℝ} (hf : differentiable ℝ f) (hf' : ∀ x, 0 ≤ deriv f x) :
monotone f :=
λ x y hxy, convex_univ.mono_of_deriv_nonneg hf.continuous.continuous_on hf.differentiable_on
(λ x _, hf' x) x y trivial trivial hxy
/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f'` is negative, then
`f` is a strictly monotonically decreasing function on `D`. -/
theorem convex.strict_antimono_of_deriv_neg {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'_neg : ∀ x ∈ interior D, deriv f x < 0) :
∀ x y ∈ D, x < y → f y < f x :=
by simpa only [zero_mul, sub_lt_zero] using hD.image_sub_lt_mul_sub_of_deriv_lt hf hf' hf'_neg
/-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is negative, then
`f` is a strictly monotonically decreasing function. -/
theorem strict_antimono_of_deriv_neg {f : ℝ → ℝ} (hf : differentiable ℝ f)
(hf' : ∀ x, deriv f x < 0) :
∀ ⦃x y⦄, x < y → f y < f x :=
λ x y hxy, convex_univ.strict_antimono_of_deriv_neg hf.continuous.continuous_on hf.differentiable_on
(λ x _, hf' x) x y trivial trivial hxy
/-- Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonpositive, then
`f` is a monotonically decreasing function on `D`. -/
theorem convex.antimono_of_deriv_nonpos {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0) :
∀ x y ∈ D, x ≤ y → f y ≤ f x :=
by simpa only [zero_mul, sub_nonpos] using hD.image_sub_le_mul_sub_of_deriv_le hf hf' hf'_nonpos
/-- Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonpositive, then
`f` is a monotonically decreasing function. -/
theorem antimono_of_deriv_nonpos {f : ℝ → ℝ} (hf : differentiable ℝ f) (hf' : ∀ x, deriv f x ≤ 0) :
∀ ⦃x y⦄, x ≤ y → f y ≤ f x :=
λ x y hxy, convex_univ.antimono_of_deriv_nonpos hf.continuous.continuous_on hf.differentiable_on
(λ x _, hf' x) x y trivial trivial hxy
/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is differentiable on its interior,
and `f'` is monotone on the interior, then `f` is convex on `D`. -/
theorem convex_on_of_deriv_mono {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'_mono : ∀ x y ∈ interior D, x ≤ y → deriv f x ≤ deriv f y) :
convex_on D f :=
convex_on_real_of_slope_mono_adjacent hD
begin
intros x y z hx hz hxy hyz,
-- First we prove some trivial inclusions
have hxzD : Icc x z ⊆ D, from hD.ord_connected hx hz,
have hxyD : Icc x y ⊆ D, from subset.trans (Icc_subset_Icc_right $ le_of_lt hyz) hxzD,
have hxyD' : Ioo x y ⊆ interior D,
from subset_sUnion_of_mem ⟨is_open_Ioo, subset.trans Ioo_subset_Icc_self hxyD⟩,
have hyzD : Icc y z ⊆ D, from subset.trans (Icc_subset_Icc_left $ le_of_lt hxy) hxzD,
have hyzD' : Ioo y z ⊆ interior D,
from subset_sUnion_of_mem ⟨is_open_Ioo, subset.trans Ioo_subset_Icc_self hyzD⟩,
-- Then we apply MVT to both `[x, y]` and `[y, z]`
obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x),
from exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD'),
obtain ⟨b, ⟨hyb, hbz⟩, hb⟩ : ∃ b ∈ Ioo y z, deriv f b = (f z - f y) / (z - y),
from exists_deriv_eq_slope f hyz (hf.mono hyzD) (hf'.mono hyzD'),
rw [← ha, ← hb],
exact hf'_mono a b (hxyD' ⟨hxa, hay⟩) (hyzD' ⟨hyb, hbz⟩) (le_of_lt $ lt_trans hay hyb)
end
/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is differentiable on its interior,
and `f'` is antimonotone on the interior, then `f` is concave on `D`. -/
theorem concave_on_of_deriv_antimono {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'_mono : ∀ x y ∈ interior D, x ≤ y → deriv f y ≤ deriv f x) :
concave_on D f :=
begin
have : ∀ x y ∈ interior D, x ≤ y → deriv (-f) x ≤ deriv (-f) y,
{ intros x y hx hy hxy,
convert neg_le_neg (hf'_mono x y hx hy hxy);
convert deriv.neg },
exact (neg_convex_on_iff D f).mp (convex_on_of_deriv_mono hD
hf.neg hf'.neg this),
end
/-- If a function `f` is differentiable and `f'` is monotone on `ℝ` then `f` is convex. -/
theorem convex_on_univ_of_deriv_mono {f : ℝ → ℝ} (hf : differentiable ℝ f)
(hf'_mono : monotone (deriv f)) : convex_on univ f :=
convex_on_of_deriv_mono convex_univ hf.continuous.continuous_on hf.differentiable_on
(λ x y _ _ h, hf'_mono h)
/-- If a function `f` is differentiable and `f'` is antimonotone on `ℝ` then `f` is concave. -/
theorem concave_on_univ_of_deriv_antimono {f : ℝ → ℝ} (hf : differentiable ℝ f)
(hf'_antimono : ∀⦃a b⦄, a ≤ b → (deriv f) b ≤ (deriv f) a) : concave_on univ f :=
concave_on_of_deriv_antimono convex_univ hf.continuous.continuous_on hf.differentiable_on
(λ x y _ _ h, hf'_antimono h)
/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its interior,
and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
theorem convex_on_of_deriv2_nonneg {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'' : differentiable_on ℝ (deriv f) (interior D))
(hf''_nonneg : ∀ x ∈ interior D, 0 ≤ (deriv^[2] f x)) :
convex_on D f :=
convex_on_of_deriv_mono hD hf hf' $
assume x y hx hy hxy,
hD.interior.mono_of_deriv_nonneg hf''.continuous_on (by rwa [interior_interior])
(by rwa [interior_interior]) _ _ hx hy hxy
/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its interior,
and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
theorem concave_on_of_deriv2_nonpos {D : set ℝ} (hD : convex D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'' : differentiable_on ℝ (deriv f) (interior D))
(hf''_nonpos : ∀ x ∈ interior D, (deriv^[2] f x) ≤ 0) :
concave_on D f :=
concave_on_of_deriv_antimono hD hf hf' $
assume x y hx hy hxy,
hD.interior.antimono_of_deriv_nonpos hf''.continuous_on (by rwa [interior_interior])
(by rwa [interior_interior]) _ _ hx hy hxy
/-- If a function `f` is twice differentiable on a open convex set `D ⊆ ℝ` and
`f''` is nonnegative on `D`, then `f` is convex on `D`. -/
theorem convex_on_open_of_deriv2_nonneg {D : set ℝ} (hD : convex D) (hD₂ : is_open D) {f : ℝ → ℝ}
(hf' : differentiable_on ℝ f D) (hf'' : differentiable_on ℝ (deriv f) D)
(hf''_nonneg : ∀ x ∈ D, 0 ≤ (deriv^[2] f x)) : convex_on D f :=
convex_on_of_deriv2_nonneg hD hf'.continuous_on (by simpa [hD₂.interior_eq] using hf')
(by simpa [hD₂.interior_eq] using hf'') (by simpa [hD₂.interior_eq] using hf''_nonneg)
/-- If a function `f` is twice differentiable on a open convex set `D ⊆ ℝ` and
`f''` is nonpositive on `D`, then `f` is concave on `D`. -/
theorem concave_on_open_of_deriv2_nonpos {D : set ℝ} (hD : convex D) (hD₂ : is_open D) {f : ℝ → ℝ}
(hf' : differentiable_on ℝ f D) (hf'' : differentiable_on ℝ (deriv f) D)
(hf''_nonpos : ∀ x ∈ D, (deriv^[2] f x) ≤ 0) : concave_on D f :=
concave_on_of_deriv2_nonpos hD hf'.continuous_on (by simpa [hD₂.interior_eq] using hf')
(by simpa [hD₂.interior_eq] using hf'') (by simpa [hD₂.interior_eq] using hf''_nonpos)
/-- If a function `f` is twice differentiable on `ℝ`, and `f''` is nonnegative on `ℝ`,
then `f` is convex on `ℝ`. -/
theorem convex_on_univ_of_deriv2_nonneg {f : ℝ → ℝ} (hf' : differentiable ℝ f)
(hf'' : differentiable ℝ (deriv f)) (hf''_nonneg : ∀ x, 0 ≤ (deriv^[2] f x)) :
convex_on univ f :=
convex_on_open_of_deriv2_nonneg convex_univ is_open_univ hf'.differentiable_on
hf''.differentiable_on (λ x _, hf''_nonneg x)
/-- If a function `f` is twice differentiable on `ℝ`, and `f''` is nonpositive on `ℝ`,
then `f` is concave on `ℝ`. -/
theorem concave_on_univ_of_deriv2_nonpos {f : ℝ → ℝ} (hf' : differentiable ℝ f)
(hf'' : differentiable ℝ (deriv f)) (hf''_nonpos : ∀ x, (deriv^[2] f x) ≤ 0) :
concave_on univ f :=
concave_on_open_of_deriv2_nonpos convex_univ is_open_univ hf'.differentiable_on
hf''.differentiable_on (λ x _, hf''_nonpos x)
/-! ### Functions `f : E → ℝ` -/
/-- Lagrange's Mean Value Theorem, applied to convex domains. -/
theorem domain_mvt
{f : E → ℝ} {s : set E} {x y : E} {f' : E → (E →L[ℝ] ℝ)}
(hf : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (hs : convex s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment x y, f y - f x = f' z (y - x) :=
begin
have hIccIoo := @Ioo_subset_Icc_self ℝ _ 0 1,
-- parametrize segment
set g : ℝ → E := λ t, x + t • (y - x),
have hseg : ∀ t ∈ Icc (0:ℝ) 1, g t ∈ segment x y,
{ rw segment_eq_image',
simp only [mem_image, and_imp, add_right_inj],
intros t ht, exact ⟨t, ht, rfl⟩ },
have hseg' : Icc 0 1 ⊆ g ⁻¹' s,
{ rw ← image_subset_iff, unfold image, change ∀ _, _,
intros z Hz, rw mem_set_of_eq at Hz, rcases Hz with ⟨t, Ht, hgt⟩,
rw ← hgt, exact hs.segment_subset xs ys (hseg t Ht) },
-- derivative of pullback of f under parametrization
have hfg: ∀ t ∈ Icc (0:ℝ) 1, has_deriv_within_at (f ∘ g)
((f' (g t) : E → ℝ) (y-x)) (Icc (0:ℝ) 1) t,
{ intros t Ht,
have hg : has_deriv_at g (y-x) t,
{ have := ((has_deriv_at_id t).smul_const (y - x)).const_add x,
rwa one_smul at this },
exact (hf (g t) $ hseg' Ht).comp_has_deriv_within_at _ hg.has_deriv_within_at hseg' },
-- apply 1-variable mean value theorem to pullback
have hMVT : ∃ (t ∈ Ioo (0:ℝ) 1), ((f' (g t) : E → ℝ) (y-x)) = (f (g 1) - f (g 0)) / (1 - 0),
{ refine exists_has_deriv_at_eq_slope (f ∘ g) _ (by norm_num) _ _,
{ unfold continuous_on,
exact λ t Ht, (hfg t Ht).continuous_within_at },
{ refine λ t Ht, (hfg t $ hIccIoo Ht).has_deriv_at _,
refine mem_nhds_sets_iff.mpr _,
use (Ioo (0:ℝ) 1),
refine ⟨hIccIoo, _, Ht⟩,
simp [real.Ioo_eq_ball, is_open_ball] } },
-- reinterpret on domain
rcases hMVT with ⟨t, Ht, hMVT'⟩,
use g t, refine ⟨hseg t $ hIccIoo Ht, _⟩,
simp [g, hMVT'],
end
/-! ### Vector-valued functions `f : E → F`. Strict differentiability. -/
/-- Over the reals, a continuously differentiable function is strictly differentiable. -/
lemma strict_fderiv_of_cont_diff
{f : E → F} {s : set E} {x : E} {f' : E → (E →L[ℝ] F)}
(hf : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (hcont : continuous_on f' s) (hs : s ∈ 𝓝 x) :
has_strict_fderiv_at f (f' x) x :=
begin
-- turn little-o definition of strict_fderiv into an epsilon-delta statement
apply is_o_iff_forall_is_O_with.mpr,
intros c hc,
refine is_O_with.of_bound (eventually_iff.mpr (mem_nhds_iff.mpr _)),
-- the correct ε is the modulus of continuity of f', shrunk to be inside s
rcases (metric.continuous_on_iff.mp hcont x (mem_of_nhds hs) c hc) with ⟨ε₁, H₁, hcont'⟩,
rcases (mem_nhds_iff.mp hs) with ⟨ε₂, H₂, hε₂⟩,
refine ⟨min ε₁ ε₂, lt_min H₁ H₂, _⟩,
-- mess with ε construction
set t := ball x (min ε₁ ε₂),
have hts : t ⊆ s := λ _ hy, hε₂ (ball_subset_ball (min_le_right ε₁ ε₂) hy),
have Hf : ∀ y ∈ t, has_fderiv_within_at f (f' y) t y :=
λ y yt, has_fderiv_within_at.mono (hf y (hts yt)) hts,
have hconv := convex_ball x (min ε₁ ε₂),
-- simplify formulas involving the product E × E
rintros ⟨a, b⟩ h,
simp only [mem_set_of_eq, map_sub],
have hab : a ∈ t ∧ b ∈ t := by rwa [mem_ball, prod.dist_eq, max_lt_iff] at h,
-- exploit the choice of ε as the modulus of continuity of f'
have hf' : ∀ x' ∈ t, ∥f' x' - f' x∥ ≤ c,
{ intros x' H',
refine le_of_lt (hcont' x' (hts H') _),
exact ball_subset_ball (min_le_left ε₁ ε₂) H' },
-- apply mean value theorem
simpa using convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Hf hf' hconv hab.2 hab.1,
end
|
348c4617df4ee9d1b048154d1da6bafb737a1d03 | dc253be9829b840f15d96d986e0c13520b085033 | /homotopy/dsmash.hlean | bce9d1035dd09598719528bd6f79c671d4a488a2 | [
"Apache-2.0"
] | permissive | cmu-phil/Spectral | 4ce68e5c1ef2a812ffda5260e9f09f41b85ae0ea | 3b078f5f1de251637decf04bd3fc8aa01930a6b3 | refs/heads/master | 1,685,119,195,535 | 1,684,169,772,000 | 1,684,169,772,000 | 46,450,197 | 42 | 13 | null | 1,505,516,767,000 | 1,447,883,921,000 | Lean | UTF-8 | Lean | false | false | 84,840 | hlean | -- Authors: Floris van Doorn
/-
The "dependent" smash product.
Given A : Type* and B : A → Type* it is the cofiber of
A ∨ B pt → Σ(a : A), B a
However, we define it (equivalently) as the pushout of 2 ← A + B pt → Σ(a : A), B a.
-/
import .smash_adjoint ..pointed_binary
open bool pointed eq equiv is_equiv sum bool prod unit circle cofiber sigma.ops wedge sigma function prod.ops
namespace dsmash
variables {A : Type*} {B : A → Type*}
definition sigma_of_sum [unfold 3] (u : A + B pt) : Σa, B a :=
by induction u with a b; exact ⟨a, pt⟩; exact ⟨pt, b⟩
definition dsmash' (B : A → Type*) : Type := pushout.pushout (@sigma_of_sum A B) (@smash.bool_of_sum A (B pt))
protected definition mk' (a : A) (b : B a) : dsmash' B := pushout.inl ⟨a, b⟩
definition dsmash [constructor] (B : A → Type*) : Type* :=
pointed.MK (dsmash' B) (dsmash.mk' pt pt)
notation `⋀` := dsmash
protected definition mk (a : A) (b : B a) : ⋀ B := pushout.inl ⟨a, b⟩
definition auxl : ⋀ B := pushout.inr ff
definition auxr : ⋀ B := pushout.inr tt
definition gluel (a : A) : dsmash.mk a pt = auxl :> ⋀ B := pushout.glue (sum.inl a)
definition gluer (b : B pt) : dsmash.mk pt b = auxr :> ⋀ B := pushout.glue (sum.inr b)
definition gluel' (a a' : A) : dsmash.mk a pt = dsmash.mk a' pt :> ⋀ B :=
gluel a ⬝ (gluel a')⁻¹
definition gluer' (b b' : B pt) : dsmash.mk pt b = dsmash.mk pt b' :> ⋀ B :=
gluer b ⬝ (gluer b')⁻¹
definition glue (a : A) (b : B pt) : dsmash.mk a pt = dsmash.mk pt b :> ⋀ B :=
gluel' a pt ⬝ gluer' pt b
definition glue_pt_left (b : B pt) : glue (Point A) b = gluer' pt b :=
whisker_right _ !con.right_inv ⬝ !idp_con
definition glue_pt_right (a : A) : glue a (Point (B a)) = gluel' a pt :=
proof whisker_left _ !con.right_inv qed
definition ap_mk_left {a a' : A} (p : a = a') : ap (λa, dsmash.mk a (Point (B a))) p = gluel' a a' :=
!ap_is_constant
definition ap_mk_right {b b' : B pt} (p : b = b') : ap (dsmash.mk (Point A)) p = gluer' b b' :=
!ap_is_constant
protected definition rec {P : ⋀ B → Type} (Pmk : Πa b, P (dsmash.mk a b))
(Pl : P auxl) (Pr : P auxr) (Pgl : Πa, Pmk a pt =[gluel a] Pl)
(Pgr : Πb, Pmk pt b =[gluer b] Pr) (x : dsmash' B) : P x :=
begin
induction x with x b u,
{ induction x with a b, exact Pmk a b },
{ induction b, exact Pl, exact Pr },
{ induction u,
{ apply Pgl },
{ apply Pgr }}
end
theorem rec_gluel {P : ⋀ B → Type} {Pmk : Πa b, P (dsmash.mk a b)}
{Pl : P auxl} {Pr : P auxr} (Pgl : Πa, Pmk a pt =[gluel a] Pl)
(Pgr : Πb, Pmk pt b =[gluer b] Pr) (a : A) :
apd (dsmash.rec Pmk Pl Pr Pgl Pgr) (gluel a) = Pgl a :=
!pushout.rec_glue
theorem rec_gluer {P : ⋀ B → Type} {Pmk : Πa b, P (dsmash.mk a b)}
{Pl : P auxl} {Pr : P auxr} (Pgl : Πa, Pmk a pt =[gluel a] Pl)
(Pgr : Πb, Pmk pt b =[gluer b] Pr) (b : B pt) :
apd (dsmash.rec Pmk Pl Pr Pgl Pgr) (gluer b) = Pgr b :=
!pushout.rec_glue
theorem rec_glue {P : ⋀ B → Type} {Pmk : Πa b, P (dsmash.mk a b)}
{Pl : P auxl} {Pr : P auxr} (Pgl : Πa, Pmk a pt =[gluel a] Pl)
(Pgr : Π(b : B pt), Pmk pt b =[gluer b] Pr) (a : A) (b : B pt) :
apd (dsmash.rec Pmk Pl Pr Pgl Pgr) (dsmash.glue a b) =
(Pgl a ⬝o (Pgl pt)⁻¹ᵒ) ⬝o (Pgr pt ⬝o (Pgr b)⁻¹ᵒ) :=
by rewrite [↑glue, ↑gluel', ↑gluer', +apd_con, +apd_inv, +rec_gluel, +rec_gluer]
protected definition elim {P : Type} (Pmk : Πa b, P) (Pl Pr : P)
(Pgl : Πa : A, Pmk a pt = Pl) (Pgr : Πb : B pt, Pmk pt b = Pr) (x : dsmash' B) : P :=
dsmash.rec Pmk Pl Pr (λa, pathover_of_eq _ (Pgl a)) (λb, pathover_of_eq _ (Pgr b)) x
-- an elim where you are forced to make (Pgl pt) and (Pgl pt) to be reflexivity
protected definition elim' [reducible] {P : Type} (Pmk : Πa b, P)
(Pgl : Πa : A, Pmk a pt = Pmk pt pt) (Pgr : Πb : B pt, Pmk pt b = Pmk pt pt)
(ql : Pgl pt = idp) (qr : Pgr pt = idp) (x : dsmash' B) : P :=
dsmash.elim Pmk (Pmk pt pt) (Pmk pt pt) Pgl Pgr x
theorem elim_gluel {P : Type} {Pmk : Πa b, P} {Pl Pr : P}
(Pgl : Πa : A, Pmk a pt = Pl) (Pgr : Πb : B pt, Pmk pt b = Pr) (a : A) :
ap (dsmash.elim Pmk Pl Pr Pgl Pgr) (gluel a) = Pgl a :=
begin
apply inj_inv !(pathover_constant (@gluel A B a)),
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑dsmash.elim,rec_gluel],
end
theorem elim_gluer {P : Type} {Pmk : Πa b, P} {Pl Pr : P}
(Pgl : Πa : A, Pmk a pt = Pl) (Pgr : Πb : B pt, Pmk pt b = Pr) (b : B pt) :
ap (dsmash.elim Pmk Pl Pr Pgl Pgr) (gluer b) = Pgr b :=
begin
apply inj_inv !(pathover_constant (@gluer A B b)),
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑dsmash.elim,rec_gluer],
end
theorem elim_glue {P : Type} {Pmk : Πa b, P} {Pl Pr : P}
(Pgl : Πa : A, Pmk a pt = Pl) (Pgr : Πb : B pt, Pmk pt b = Pr) (a : A) (b : B pt) :
ap (dsmash.elim Pmk Pl Pr Pgl Pgr) (glue a b) = (Pgl a ⬝ (Pgl pt)⁻¹) ⬝ (Pgr pt ⬝ (Pgr b)⁻¹) :=
by rewrite [↑glue, ↑gluel', ↑gluer', +ap_con, +ap_inv, +elim_gluel, +elim_gluer]
end dsmash
open dsmash
attribute dsmash.mk dsmash.mk' dsmash.auxl dsmash.auxr [constructor]
attribute dsmash.elim' dsmash.rec dsmash.elim [unfold 9] [recursor 9]
namespace dsmash
variables {A A' C : Type*} {B : A → Type*} {D : C → Type*} {a a' : A} {b : B a} {b' : B a'}
definition mk_eq_mk (p : a = a') (q : b =[p] b') : dsmash.mk a b = dsmash.mk a' b' :=
ap pushout.inl (dpair_eq_dpair p q)
definition gluer2 (b : B a) (p : a = pt) : dsmash.mk a b = auxr :=
mk_eq_mk p !pathover_tr ⬝ gluer _
definition elim_gluel' {P : Type} {Pmk : Πa b, P} {Pl Pr : P}
(Pgl : Πa : A, Pmk a pt = Pl) (Pgr : Πb : B pt, Pmk pt b = Pr) (a a' : A) :
ap (dsmash.elim Pmk Pl Pr Pgl Pgr) (gluel' a a') = Pgl a ⬝ (Pgl a')⁻¹ :=
!ap_con ⬝ whisker_left _ !ap_inv ⬝ !elim_gluel ◾ !elim_gluel⁻²
definition elim_gluer' {P : Type} {Pmk : Πa b, P} {Pl Pr : P}
(Pgl : Πa : A, Pmk a pt = Pl) (Pgr : Πb : B pt, Pmk pt b = Pr) (b b' : B pt) :
ap (dsmash.elim Pmk Pl Pr Pgl Pgr) (gluer' b b') = Pgr b ⬝ (Pgr b')⁻¹ :=
!ap_con ⬝ whisker_left _ !ap_inv ⬝ !elim_gluer ◾ !elim_gluer⁻²
definition elim_gluel'_same {P : Type} {Pmk : Πa b, P} {Pl Pr : P}
(Pgl : Πa : A, Pmk a pt = Pl) (Pgr : Πb : B pt, Pmk pt b = Pr) (a : A) :
elim_gluel' Pgl Pgr a a =
ap02 (dsmash.elim Pmk Pl Pr Pgl Pgr) (con.right_inv (gluel a)) ⬝ (con.right_inv (Pgl a))⁻¹ :=
begin
refine _ ⬝ whisker_right _ (eq_top_of_square (!ap_con_right_inv_sq))⁻¹,
refine _ ⬝ whisker_right _ !con_idp⁻¹,
refine _ ⬝ !con.assoc⁻¹,
apply whisker_left,
apply eq_con_inv_of_con_eq, symmetry,
apply con_right_inv_natural
end
definition elim_gluer'_same {P : Type} {Pmk : Πa b, P} {Pl Pr : P}
(Pgl : Πa : A, Pmk a pt = Pl) (Pgr : Πb : B pt, Pmk pt b = Pr) (b : B pt) :
elim_gluer' Pgl Pgr b b =
ap02 (dsmash.elim Pmk Pl Pr Pgl Pgr) (con.right_inv (gluer b)) ⬝ (con.right_inv (Pgr b))⁻¹ :=
begin
refine _ ⬝ whisker_right _ (eq_top_of_square (!ap_con_right_inv_sq))⁻¹,
refine _ ⬝ whisker_right _ !con_idp⁻¹,
refine _ ⬝ !con.assoc⁻¹,
apply whisker_left,
apply eq_con_inv_of_con_eq, symmetry,
apply con_right_inv_natural
end
definition elim'_gluel'_pt {P : Type} {Pmk : Πa b, P}
(Pgl : Πa : A, Pmk a pt = Pmk pt pt) (Pgr : Πb : B pt, Pmk pt b = Pmk pt pt)
(a : A) (ql : Pgl pt = idp) (qr : Pgr pt = idp) :
ap (dsmash.elim' Pmk Pgl Pgr ql qr) (gluel' a pt) = Pgl a :=
!elim_gluel' ⬝ whisker_left _ ql⁻²
definition elim'_gluer'_pt {P : Type} {Pmk : Πa b, P}
(Pgl : Πa : A, Pmk a pt = Pmk pt pt) (Pgr : Πb : B pt, Pmk pt b = Pmk pt pt)
(b : B pt) (ql : Pgl pt = idp) (qr : Pgr pt = idp) :
ap (dsmash.elim' Pmk Pgl Pgr ql qr) (gluer' b pt) = Pgr b :=
!elim_gluer' ⬝ whisker_left _ qr⁻²
protected definition rec_eq {C : Type} {f g : ⋀ B → C}
(Pmk : Πa b, f (dsmash.mk a b) = g (dsmash.mk a b))
(Pl : f auxl = g auxl) (Pr : f auxr = g auxr)
(Pgl : Πa, square (Pmk a pt) Pl (ap f (gluel a)) (ap g (gluel a)))
(Pgr : Πb, square (Pmk pt b) Pr (ap f (gluer b)) (ap g (gluer b))) (x : dsmash' B) : f x = g x :=
begin
induction x with a b a b,
{ exact Pmk a b },
{ exact Pl },
{ exact Pr },
{ apply eq_pathover, apply Pgl },
{ apply eq_pathover, apply Pgr }
end
definition rec_eq_gluel {C : Type} {f g : ⋀ B → C}
{Pmk : Πa b, f (dsmash.mk a b) = g (dsmash.mk a b)}
{Pl : f auxl = g auxl} {Pr : f auxr = g auxr}
(Pgl : Πa, square (Pmk a pt) Pl (ap f (gluel a)) (ap g (gluel a)))
(Pgr : Πb, square (Pmk pt b) Pr (ap f (gluer b)) (ap g (gluer b))) (a : A) :
natural_square (dsmash.rec_eq Pmk Pl Pr Pgl Pgr) (gluel a) = Pgl a :=
begin
refine ap square_of_pathover !rec_gluel ⬝ _,
apply to_right_inv !eq_pathover_equiv_square
end
definition rec_eq_gluer {C : Type} {f g : ⋀ B → C}
{Pmk : Πa b, f (dsmash.mk a b) = g (dsmash.mk a b)}
{Pl : f auxl = g auxl} {Pr : f auxr = g auxr}
(Pgl : Πa, square (Pmk a pt) Pl (ap f (gluel a)) (ap g (gluel a)))
(Pgr : Πb, square (Pmk pt b) Pr (ap f (gluer b)) (ap g (gluer b))) (b : B pt) :
natural_square (dsmash.rec_eq Pmk Pl Pr Pgl Pgr) (gluer b) = Pgr b :=
begin
refine ap square_of_pathover !rec_gluer ⬝ _,
apply to_right_inv !eq_pathover_equiv_square
end
/- the functorial action of the dependent smash product -/
definition dsmash_functor' [unfold 7] (f : A →* C) (g : Πa, B a →* D (f a)) : ⋀ B → ⋀ D :=
begin
intro x, induction x,
{ exact dsmash.mk (f a) (g a b) },
{ exact auxl },
{ exact auxr },
{ exact ap (dsmash.mk (f a)) (respect_pt (g a)) ⬝ gluel (f a) },
{ exact gluer2 (g pt b) (respect_pt f) }
end
definition dsmash_functor [constructor] (f : A →* C) (g : Πa, B a →* D (f a)) : ⋀ B →* ⋀ D :=
begin
fapply pmap.mk,
{ exact dsmash_functor' f g },
{ exact mk_eq_mk (respect_pt f) (respect_pt (g pt) ⬝po apd (λa, Point (D a)) (respect_pt f)) },
end
infixr ` ⋀→ `:65 := dsmash_functor
definition pmap_of_map' [constructor] (A : Type*) {B : Type} (f : A → B) :
A →* pointed.MK B (f pt) :=
pmap.mk f idp
definition functor_gluel (f : A →* C) (g : Πa, B a →* D (f a)) (a : A) :
ap (f ⋀→ g) (gluel a) = ap (dsmash.mk (f a)) (respect_pt (g a)) ⬝ gluel (f a) :=
!elim_gluel
definition functor_gluer (f : A →* C) (g : Πa, B a →* D (f a)) (b : B pt) :
ap (f ⋀→ g) (gluer b) = gluer2 (g pt b) (respect_pt f) :=
!elim_gluer
-- definition functor_gluel2 {C : Type} {D : C → Type} (f : A → C) (g : Πa, B a → D (f a)) (a : A) :
-- ap (@dsmash_functor A (pointed.MK C (f pt)) B (λc, pointed.MK (D c) _) (pmap_of_map' A f) (λa, pmap_of_map' (B a) (g a))) _ = _ :=
-- begin
-- refine !elim_gluel ⬝ !idp_con
-- end
-- definition functor_gluer2 {C D : Type} (f : A → C) (g : B → D) (b : B) :
-- ap (pmap_of_map f pt ⋀→ pmap_of_map g pt) (gluer b) = gluer (g b) :=
-- begin
-- refine !elim_gluer ⬝ !idp_con
-- end
-- definition functor_gluel' (f : A →* C) (g : Πa, B a →* D (f a)) (a a' : A) :
-- ap (f ⋀→ g) (gluel' a a') = ap (dsmash.mk (f a)) (respect_pt g) ⬝
-- gluel' (f a) (f a') ⬝ (ap (dsmash.mk (f a')) (respect_pt g))⁻¹ :=
-- begin
-- refine !elim_gluel' ⬝ _,
-- refine whisker_left _ !con_inv ⬝ _,
-- refine !con.assoc⁻¹ ⬝ _, apply whisker_right,
-- apply con.assoc
-- end
-- definition functor_gluer' (f : A →* C) (g : Πa, B a →* D (f a)) (b b' : B) :
-- ap (f ⋀→ g) (gluer' b b') = ap (λc, dsmash.mk c (g b)) (respect_pt f) ⬝
-- gluer' (g b) (g b') ⬝ (ap (λc, dsmash.mk c (g b')) (respect_pt f))⁻¹ :=
-- begin
-- refine !elim_gluer' ⬝ _,
-- refine whisker_left _ !con_inv ⬝ _,
-- refine !con.assoc⁻¹ ⬝ _, apply whisker_right,
-- apply con.assoc
-- end
/- the statements of the above rules becomes easier if one of the functions respects the basepoint
by reflexivity -/
-- definition functor_gluel'2 {D : Type} (f : A →* C) (g : B → D) (a a' : A) :
-- ap (f ⋀→ (pmap_of_map g pt)) (gluel' a a') = gluel' (f a) (f a') :=
-- begin
-- refine !ap_con ⬝ whisker_left _ !ap_inv ⬝ _,
-- refine (!functor_gluel ⬝ !idp_con) ◾ (!functor_gluel ⬝ !idp_con)⁻²
-- end
-- definition functor_gluer'2 {C : Type} (f : A → C) (g : Πa, B a →* D (f a)) (b b' : B) :
-- ap (pmap_of_map f pt ⋀→ g) (gluer' b b') = gluer' (g b) (g b') :=
-- begin
-- refine !ap_con ⬝ whisker_left _ !ap_inv ⬝ _,
-- refine (!functor_gluer ⬝ !idp_con) ◾ (!functor_gluer ⬝ !idp_con)⁻²
-- end
-- definition functor_gluel'2 {C D : Type} (f : A → C) (g : B → D) (a a' : A) :
-- ap (pmap_of_map f pt ⋀→ pmap_of_map g pt) (gluel' a a') = gluel' (f a) (f a') :=
-- !ap_con ⬝ whisker_left _ !ap_inv ⬝ !functor_gluel2 ◾ !functor_gluel2⁻²
-- definition functor_gluer'2 {C D : Type} (f : A → C) (g : B → D) (b b' : B) :
-- ap (pmap_of_map f pt ⋀→ pmap_of_map g pt) (gluer' b b') = gluer' (g b) (g b') :=
-- !ap_con ⬝ whisker_left _ !ap_inv ⬝ !functor_gluer2 ◾ !functor_gluer2⁻²
-- lemma functor_gluel'2_same {C D : Type} (f : A → C) (g : B → D) (a : A) :
-- functor_gluel'2 f g a a =
-- ap02 (pmap_of_map f pt ⋀→ pmap_of_map g pt) (con.right_inv (gluel a)) ⬝
-- (con.right_inv (gluel (f a)))⁻¹ :=
-- begin
-- refine _ ⬝ whisker_right _ (eq_top_of_square (!ap_con_right_inv_sq))⁻¹,
-- refine _ ⬝ whisker_right _ !con_idp⁻¹,
-- refine _ ⬝ !con.assoc⁻¹,
-- apply whisker_left,
-- apply eq_con_inv_of_con_eq, symmetry,
-- apply con_right_inv_natural
-- end
-- lemma functor_gluer'2_same {C D : Type} (f : A → C) (g : B → D) (b : B) :
-- functor_gluer'2 (pmap_of_map f pt) g b b =
-- ap02 (pmap_of_map f pt ⋀→ pmap_of_map g pt) (con.right_inv (gluer b)) ⬝
-- (con.right_inv (gluer (g b)))⁻¹ :=
-- begin
-- refine _ ⬝ whisker_right _ (eq_top_of_square (!ap_con_right_inv_sq))⁻¹,
-- refine _ ⬝ whisker_right _ !con_idp⁻¹,
-- refine _ ⬝ !con.assoc⁻¹,
-- apply whisker_left,
-- apply eq_con_inv_of_con_eq, symmetry,
-- apply con_right_inv_natural
-- end
definition dsmash_functor_pid [constructor] (B : A → Type*) :
pid A ⋀→ (λa, pid (B a)) ~* pid (⋀ B) :=
begin
fapply phomotopy.mk,
{ intro x, induction x with a b a b,
{ reflexivity },
{ reflexivity },
{ reflexivity },
{ apply eq_pathover_id_right, apply hdeg_square, exact !functor_gluel ⬝ !idp_con },
{ apply eq_pathover_id_right, apply hdeg_square, exact !functor_gluer ⬝ !idp_con }},
{ reflexivity }
end
/- the functorial action of the dependent smash product respects pointed homotopies, and some computation rules for this pointed homotopy -/
definition dsmash_functor_phomotopy {f f' : A →* C} {g : Πa, B a →* D (f a)} {g' : Πa, B a →* D (f' a)}
(h₁ : f ~* f') (h₂ : Πa, ptransport D (h₁ a) ∘* g a ~* g' a) : f ⋀→ g ~* f' ⋀→ g' :=
begin
induction h₁ using phomotopy_rec_idp,
--induction h₂ using phomotopy_rec_idp,
exact sorry --reflexivity
end
infixr ` ⋀~ `:78 := dsmash_functor_phomotopy
/- a more explicit proof, if we ever need it -/
-- definition dsmash_functor_homotopy [unfold 11] {f f' : A →* C} {g g' : Πa, B a →* D (f a)}
-- (h₁ : f ~* f') (h₂ : g ~* g') : f ⋀→ g ~ f' ⋀→ g' :=
-- begin
-- intro x, induction x with a b a b,
-- { exact ap011 dsmash.mk (h₁ a) (h₂ b) },
-- { reflexivity },
-- { reflexivity },
-- { apply eq_pathover,
-- refine !functor_gluel ⬝ph _ ⬝hp !functor_gluel⁻¹,
-- refine _ ⬝v square_of_eq_top (ap_mk_left (h₁ a)),
-- exact ap011_ap_square_right dsmash.mk (h₁ a) (to_homotopy_pt h₂) },
-- { apply eq_pathover,
-- refine !functor_gluer ⬝ph _ ⬝hp !functor_gluer⁻¹,
-- refine _ ⬝v square_of_eq_top (ap_mk_right (h₂ b)),
-- exact ap011_ap_square_left dsmash.mk (h₂ b) (to_homotopy_pt h₁) },
-- end
-- definition dsmash_functor_phomotopy [constructor] {f f' : A →* C} {g g' : Πa, B a →* D (f a)}
-- (h₁ : f ~* f') (h₂ : g ~* g') : f ⋀→ g ~* f' ⋀→ g' :=
-- begin
-- apply phomotopy.mk (dsmash_functor_homotopy h₁ h₂),
-- induction h₁ with h₁ h₁₀, induction h₂ with h₂ h₂₀,
-- induction f with f f₀, induction g with g g₀,
-- induction f' with f' f'₀, induction g' with g' g'₀,
-- induction C with C c₀, induction D with D d₀, esimp at *,
-- induction h₁₀, induction h₂₀, induction f'₀, induction g'₀,
-- exact !ap_ap011⁻¹
-- end
-- definition dsmash_functor_phomotopy_refl (f : A →* C) (g : Πa, B a →* D (f a)) :
-- dsmash_functor_phomotopy (phomotopy.refl f) (phomotopy.refl g) = phomotopy.rfl :=
-- !phomotopy_rec_idp_refl ⬝ !phomotopy_rec_idp_refl
-- definition dsmash_functor_phomotopy_symm {f₁ f₂ : A →* C} {g₁ g₂ : Πa, B a →* D (f a)}
-- (h : f₁ ~* f₂) (k : g₁ ~* g₂) :
-- dsmash_functor_phomotopy h⁻¹* k⁻¹* = (dsmash_functor_phomotopy h k)⁻¹* :=
-- begin
-- induction h using phomotopy_rec_idp, induction k using phomotopy_rec_idp,
-- exact ap011 dsmash_functor_phomotopy !refl_symm !refl_symm ⬝ !dsmash_functor_phomotopy_refl ⬝
-- !refl_symm⁻¹ ⬝ !dsmash_functor_phomotopy_refl⁻¹⁻²**
-- end
-- definition dsmash_functor_phomotopy_trans {f₁ f₂ f₃ : A →* C} {g₁ g₂ g₃ : Πa, B a →* D (f a)}
-- (h₁ : f₁ ~* f₂) (h₂ : f₂ ~* f₃) (k₁ : g₁ ~* g₂) (k₂ : g₂ ~* g₃) :
-- dsmash_functor_phomotopy (h₁ ⬝* h₂) (k₁ ⬝* k₂) =
-- dsmash_functor_phomotopy h₁ k₁ ⬝* dsmash_functor_phomotopy h₂ k₂ :=
-- begin
-- induction h₁ using phomotopy_rec_idp, induction h₂ using phomotopy_rec_idp,
-- induction k₁ using phomotopy_rec_idp, induction k₂ using phomotopy_rec_idp,
-- refine ap011 dsmash_functor_phomotopy !trans_refl !trans_refl ⬝ !trans_refl⁻¹ ⬝ idp ◾** _,
-- exact !dsmash_functor_phomotopy_refl⁻¹
-- end
-- definition dsmash_functor_phomotopy_trans_right {f₁ f₂ : A →* C} {g₁ g₂ g₃ : Πa, B a →* D (f a)}
-- (h₁ : f₁ ~* f₂) (k₁ : g₁ ~* g₂) (k₂ : g₂ ~* g₃) :
-- dsmash_functor_phomotopy h₁ (k₁ ⬝* k₂) =
-- dsmash_functor_phomotopy h₁ k₁ ⬝* dsmash_functor_phomotopy phomotopy.rfl k₂ :=
-- begin
-- refine ap (λx, dsmash_functor_phomotopy x _) !trans_refl⁻¹ ⬝ !dsmash_functor_phomotopy_trans,
-- end
-- definition dsmash_functor_phomotopy_phsquare {f₁ f₂ f₃ f₄ : A →* C} {g₁ g₂ g₃ g₄ : Πa, B a →* D (f a)}
-- {h₁ : f₁ ~* f₂} {h₂ : f₃ ~* f₄} {h₃ : f₁ ~* f₃} {h₄ : f₂ ~* f₄}
-- {k₁ : g₁ ~* g₂} {k₂ : g₃ ~* g₄} {k₃ : g₁ ~* g₃} {k₄ : g₂ ~* g₄}
-- (p : phsquare h₁ h₂ h₃ h₄) (q : phsquare k₁ k₂ k₃ k₄) :
-- phsquare (dsmash_functor_phomotopy h₁ k₁)
-- (dsmash_functor_phomotopy h₂ k₂)
-- (dsmash_functor_phomotopy h₃ k₃)
-- (dsmash_functor_phomotopy h₄ k₄) :=
-- !dsmash_functor_phomotopy_trans⁻¹ ⬝ ap011 dsmash_functor_phomotopy p q ⬝
-- !dsmash_functor_phomotopy_trans
-- definition dsmash_functor_eq_of_phomotopy (f : A →* C) {g g' : Πa, B a →* D (f a)}
-- (p : g ~* g') : ap (dsmash_functor f) (eq_of_phomotopy p) =
-- eq_of_phomotopy (dsmash_functor_phomotopy phomotopy.rfl p) :=
-- begin
-- induction p using phomotopy_rec_idp,
-- refine ap02 _ !eq_of_phomotopy_refl ⬝ _,
-- refine !eq_of_phomotopy_refl⁻¹ ⬝ _,
-- apply ap eq_of_phomotopy,
-- exact !dsmash_functor_phomotopy_refl⁻¹
-- end
-- definition dsmash_functor_eq_of_phomotopy_left (g : Πa, B a →* D (f a)) {f f' : A →* C}
-- (p : f ~* f') : ap (λf, dsmash_functor f g) (eq_of_phomotopy p) =
-- eq_of_phomotopy (dsmash_functor_phomotopy p phomotopy.rfl) :=
-- begin
-- induction p using phomotopy_rec_idp,
-- refine ap02 _ !eq_of_phomotopy_refl ⬝ _,
-- refine !eq_of_phomotopy_refl⁻¹ ⬝ _,
-- apply ap eq_of_phomotopy,
-- exact !dsmash_functor_phomotopy_refl⁻¹
-- end
/- the functorial action preserves compositions, the interchange law -/
-- definition dsmash_functor_pcompose_homotopy [unfold 11] {C D E F : Type}
-- (f' : C → E) (f : A → C) (g' : D → F) (g : B → D) :
-- (pmap_of_map f' (f pt) ∘* pmap_of_map f pt) ⋀→ (pmap_of_map g' (g pt) ∘* pmap_of_map g pt) ~
-- (pmap_of_map f' (f pt) ⋀→ pmap_of_map g' (g pt)) ∘* (pmap_of_map f pt ⋀→ pmap_of_map g pt) :=
-- begin
-- intro x, induction x with a b a b,
-- { reflexivity },
-- { reflexivity },
-- { reflexivity },
-- { apply eq_pathover, refine !functor_gluel2 ⬝ph _, esimp,
-- refine _ ⬝hp (ap_compose (_ ⋀→ _) _ _)⁻¹,
-- refine _ ⬝hp ap02 _ !functor_gluel2⁻¹, refine _ ⬝hp !functor_gluel2⁻¹, exact hrfl },
-- { apply eq_pathover, refine !functor_gluer2 ⬝ph _, esimp,
-- refine _ ⬝hp (ap_compose (_ ⋀→ _) _ _)⁻¹,
-- refine _ ⬝hp ap02 _ !functor_gluer2⁻¹, refine _ ⬝hp !functor_gluer2⁻¹, exact hrfl }
-- end
-- definition dsmash_functor_pcompose (f' : C →* E) (f : A →* C) (g' : D →* F) (g : Πa, B a →* D (f a)) :
-- (f' ∘* f) ⋀→ (g' ∘* g) ~* f' ⋀→ g' ∘* f ⋀→ g :=
-- begin
-- induction C with C, induction D with D, induction E with E, induction F with F,
-- induction f with f f₀, induction f' with f' f'₀, induction g with g g₀,
-- induction g' with g' g'₀, esimp at *,
-- induction f₀, induction f'₀, induction g₀, induction g'₀,
-- fapply phomotopy.mk,
-- { rexact dsmash_functor_pcompose_homotopy f' f g' g },
-- { reflexivity }
-- end
-- definition dsmash_functor_split (f : A →* C) (g : Πa, B a →* D (f a)) :
-- f ⋀→ g ~* f ⋀→ pid D ∘* pid A ⋀→ g :=
-- dsmash_functor_phomotopy !pcompose_pid⁻¹* !pid_pcompose⁻¹* ⬝* !dsmash_functor_pcompose
-- definition dsmash_functor_split_rev (f : A →* C) (g : Πa, B a →* D (f a)) :
-- f ⋀→ g ~* pid C ⋀→ g ∘* f ⋀→ pid B :=
-- dsmash_functor_phomotopy !pid_pcompose⁻¹* !pcompose_pid⁻¹* ⬝* !dsmash_functor_pcompose
/- An alternative proof which doesn't start by applying inductions, so which is more explicit -/
-- definition dsmash_functor_pcompose_homotopy [unfold 11] (f' : C →* E) (f : A →* C) (g' : D →* F)
-- (g : Πa, B a →* D (f a)) : (f' ∘* f) ⋀→ (g' ∘* g) ~ (f' ⋀→ g') ∘* (f ⋀→ g) :=
-- begin
-- intro x, induction x with a b a b,
-- { reflexivity },
-- { reflexivity },
-- { reflexivity },
-- { apply eq_pathover, exact abstract begin apply hdeg_square,
-- refine !functor_gluel ⬝ _ ⬝ (ap_compose (f' ⋀→ g') _ _)⁻¹,
-- refine whisker_right _ !ap_con ⬝ !con.assoc ⬝ _ ⬝ ap02 _ !functor_gluel⁻¹,
-- refine (!ap_compose'⁻¹ ⬝ !ap_compose') ◾ proof !functor_gluel⁻¹ qed ⬝ !ap_con⁻¹ end end },
-- { apply eq_pathover, exact abstract begin apply hdeg_square,
-- refine !functor_gluer ⬝ _ ⬝ (ap_compose (f' ⋀→ g') _ _)⁻¹,
-- refine whisker_right _ !ap_con ⬝ !con.assoc ⬝ _ ⬝ ap02 _ !functor_gluer⁻¹,
-- refine (!ap_compose'⁻¹ ⬝ !ap_compose') ◾ proof !functor_gluer⁻¹ qed ⬝ !ap_con⁻¹ end end }
-- end
-- definition dsmash_functor_pcompose [constructor] (f' : C →* E) (f : A →* C) (g' : D →* F) (g : Πa, B a →* D (f a)) :
-- (f' ∘* f) ⋀→ (g' ∘* g) ~* f' ⋀→ g' ∘* f ⋀→ g :=
-- begin
-- fapply phomotopy.mk,
-- { exact dsmash_functor_pcompose_homotopy f' f g' g },
-- { exact abstract begin induction C, induction D, induction E, induction F,
-- induction f with f f₀, induction f' with f' f'₀, induction g with g g₀,
-- induction g' with g' g'₀, esimp at *,
-- induction f₀, induction f'₀, induction g₀, induction g'₀, reflexivity end end }
-- end
-- definition dsmash_functor_pid_pcompose [constructor] (A : Type*) (g' : C →* D) (g : B →* C)
-- : pid A ⋀→ (g' ∘* g) ~* pid A ⋀→ g' ∘* pid A ⋀→ g :=
-- dsmash_functor_phomotopy !pid_pcompose⁻¹* phomotopy.rfl ⬝* !dsmash_functor_pcompose
-- definition dsmash_functor_pcompose_pid [constructor] (B : Type*) (f' : C →* D) (f : A →* C)
-- : (f' ∘* f) ⋀→ pid B ~* f' ⋀→ (pid B) ∘* f ⋀→ (pid B) :=
-- dsmash_functor_phomotopy phomotopy.rfl !pid_pcompose⁻¹* ⬝* !dsmash_functor_pcompose
/- composing commutes with applying homotopies -/
-- definition dsmash_functor_pcompose_phomotopy {f₂ f₂' : C →* E} {f f' : A →* C} {g₂ g₂' : D →* F}
-- {g g' : Πa, B a →* D (f a)} (h₂ : f₂ ~* f₂') (h₁ : f ~* f') (k₂ : g₂ ~* g₂') (k₁ : g ~* g') :
-- phsquare (dsmash_functor_pcompose f₂ f g₂ g)
-- (dsmash_functor_pcompose f₂' f' g₂' g')
-- (dsmash_functor_phomotopy (h₂ ◾* h₁) (k₂ ◾* k₁))
-- (dsmash_functor_phomotopy h₂ k₂ ◾* dsmash_functor_phomotopy h₁ k₁) :=
-- begin
-- induction h₁ using phomotopy_rec_idp, induction h₂ using phomotopy_rec_idp,
-- induction k₁ using phomotopy_rec_idp, induction k₂ using phomotopy_rec_idp,
-- refine (ap011 dsmash_functor_phomotopy !pcompose2_refl !pcompose2_refl ⬝
-- !dsmash_functor_phomotopy_refl) ⬝ph** phvrfl ⬝hp**
-- (ap011 pcompose2 !dsmash_functor_phomotopy_refl !dsmash_functor_phomotopy_refl ⬝
-- !pcompose2_refl)⁻¹,
-- end
-- definition dsmash_functor_pid_pcompose_phomotopy_right (g₂ : D →* E) {g g' : Πa, B a →* D (f a)}
-- (k : g ~* g') :
-- phsquare (dsmash_functor_pid_pcompose A g₂ g)
-- (dsmash_functor_pid_pcompose A g₂ g')
-- (dsmash_functor_phomotopy phomotopy.rfl (pwhisker_left g₂ k))
-- (pwhisker_left (pid A ⋀→ g₂) (dsmash_functor_phomotopy phomotopy.rfl k)) :=
-- begin
-- refine dsmash_functor_phomotopy_phsquare _ _ ⬝h** !dsmash_functor_pcompose_phomotopy ⬝hp**
-- ((ap (pwhisker_right _) !dsmash_functor_phomotopy_refl) ◾** idp ⬝ !pcompose2_refl_left),
-- exact (!pcompose2_refl ⬝ph** phvrfl)⁻¹ʰ**,
-- exact (phhrfl ⬝hp** !pcompose2_refl_left⁻¹)
-- end
section
variables {A₀₀ A₂₀ A₀₂ A₂₂ : Type*} {B₀₀ B₂₀ B₀₂ B₂₂ : Type*}
{f₁₀ : A₀₀ →* A₂₀} {f₀₁ : A₀₀ →* A₀₂} {f₂₁ : A₂₀ →* A₂₂} {f₁₂ : A₀₂ →* A₂₂}
{g₁₀ : B₀₀ →* B₂₀} {g₀₁ : B₀₀ →* B₀₂} {g₂₁ : B₂₀ →* B₂₂} {g₁₂ : B₀₂ →* B₂₂}
/- applying the functorial action of dsmash to squares of pointed maps -/
-- definition dsmash_functor_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : psquare g₁₀ g₁₂ g₀₁ g₂₁) :
-- psquare (f₁₀ ⋀→ g₁₀) (f₁₂ ⋀→ g₁₂) (f₀₁ ⋀→ g₀₁) (f₂₁ ⋀→ g₂₁) :=
-- !dsmash_functor_pcompose⁻¹* ⬝* dsmash_functor_phomotopy p q ⬝* !dsmash_functor_pcompose
end
/- f ∧ g is a pointed equivalence if f and g are -/
definition dsmash_functor_using_pushout [unfold 7] (f : A →* C) (g : Πa, B a →* D (f a)) :
⋀ B → ⋀ D :=
begin
refine pushout.functor (f +→ (transport D (respect_pt f) ∘ g pt)) (sigma_functor f g) id _ _,
{ intro v, induction v with a b,
exact ap (dpair _) (respect_pt (g a)),
exact sigma_eq (respect_pt f) !pathover_tr },
{ intro v, induction v with a b: reflexivity }
end
definition dsmash_functor_homotopy_pushout_functor (f : A →* C) (g : Πa, B a →* D (f a)) :
f ⋀→ g ~ dsmash_functor_using_pushout f g :=
begin
intro x, induction x,
{ reflexivity },
{ reflexivity },
{ reflexivity },
{ apply eq_pathover, refine !elim_gluel ⬝ph _ ⬝hp !pushout.elim_glue⁻¹,
apply hdeg_square, esimp, apply whisker_right, apply ap_compose },
{ apply eq_pathover, refine !elim_gluer ⬝ph _ ⬝hp !pushout.elim_glue⁻¹,
apply hdeg_square, reflexivity }
end
definition dsmash_pequiv [constructor] (f : A ≃* C) (g : Πa, B a ≃* D (f a)) : ⋀ B ≃* ⋀ D :=
begin
fapply pequiv_of_pmap (f ⋀→ g),
refine homotopy_closed _ (dsmash_functor_homotopy_pushout_functor f g)⁻¹ʰᵗʸ _,
apply pushout.is_equiv_functor,
{ apply is_equiv_sum_functor, apply is_equiv_compose, apply pequiv.to_is_equiv,
exact to_is_equiv (equiv_ap _ _) },
apply is_equiv_sigma_functor, intro a, apply pequiv.to_is_equiv
end
infixr ` ⋀≃ `:80 := dsmash_pequiv
definition dsmash_pequiv_left [constructor] (B : C → Type*) (f : A ≃* C) :
⋀ (B ∘ f) ≃* ⋀ B :=
f ⋀≃ λa, pequiv.rfl
definition dsmash_pequiv_right [constructor] {D : A → Type*} (g : Πa, B a ≃* D a) : ⋀ B ≃* ⋀ D :=
pequiv.rfl ⋀≃ g
/- f ∧ g is constant if f is constant -/
-- definition dsmash_functor_pconst_left_homotopy [unfold 6] {B' : Type} (f : B → B') (x : ⋀ B) :
-- (pconst A C ⋀→ pmap_of_map f pt) x = pt :=
-- begin
-- induction x with a b a b,
-- { exact gluer' (f b) pt },
-- { exact (gluel pt)⁻¹ },
-- { exact (gluer pt)⁻¹ᵖ },
-- { apply eq_pathover, note x := functor_gluel2 (λx : A, Point A') f a,
-- refine x ⬝ph _, refine _ ⬝hp !ap_constant⁻¹, apply square_of_eq,
-- rexact con.right_inv (gluer (f pt)) ⬝ (con.right_inv (gluel pt))⁻¹ },
-- { apply eq_pathover, note x := functor_gluer2 (λx : A, Point A') f b,
-- refine x ⬝ph _, refine _ ⬝hp !ap_constant⁻¹, apply square_of_eq, reflexivity }
-- end
-- definition dsmash_functor_pconst_left (f : B →* B') : pconst A A' ⋀→ f ~* pconst (⋀ B) (A' ∧ B') :=
-- begin
-- induction B' with B', induction f with f f₀, esimp at *, induction f₀,
-- fapply phomotopy.mk,
-- { exact dsmash_functor_pconst_left_homotopy f },
-- { rexact con.right_inv (gluer (f pt)) }
-- end
-- definition dsmash_functor_pconst_left_phomotopy {f f' : B →* B'} (p : f ~* f') :
-- phomotopy.refl (pconst A A') ⋀~ p ⬝* dsmash_functor_pconst_left f' = dsmash_functor_pconst_left f :=
-- begin
-- induction p using phomotopy_rec_idp,
-- exact !dsmash_functor_phomotopy_refl ◾** idp ⬝ !refl_trans
-- end
-- /- This makes dsmash_functor into a pointed map (B →* B') →* (⋀ B →* ⋀ B') -/
-- definition dsmash_functor_left [constructor] (A A' B : Type*) :
-- ppmap A A' →* ppmap (⋀ B) (A' ∧ B) :=
-- pmap.mk (λf, f ⋀→ pid B) (eq_of_phomotopy (dsmash_functor_pconst_left (pid B)))
/- We want to show that dsmash_functor_left is natural in A, B and C.
For this we need two coherence rules. Given the function h := (f' ∘ f) ⋀→ (g' ∘ g) and suppose
that either f' or f is constant. There are two ways to show that h is constant: either by using
exchange, or directly. We need to show that these two proofs result in the same pointed
homotopy. First we do the case where f is constant -/
-- private definition my_squarel {A : Type} {a₁ a₂ a₃ : A} (p₁ : a₁ = a₃) (p₂ : a₂ = a₃) :
-- square (p₁ ⬝ p₂⁻¹) p₂⁻¹ p₁ idp :=
-- proof square_of_eq idp qed
-- private definition my_squarer {A : Type} {a₁ a₂ a₃ : A} (p₁ : a₁ = a₃) (p₂ : a₁ = a₂) :
-- square (p₁ ⬝ p₁⁻¹) p₂⁻¹ p₂ idp :=
-- proof square_of_eq (con.right_inv p₁ ⬝ (con.right_inv p₂)⁻¹) qed
-- private definition my_cube_fillerl {B C : Type} {g : B → C} {fa₁ fa₂ b₀ : B}
-- {pa₁ : fa₁ = b₀} {pa₂ : fa₂ = b₀} {qa₁ : g (fa₁) = g b₀} {qa₂ : g (fa₂) = g b₀}
-- (ra₁ : ap g (pa₁) = qa₁) (ra₂ : ap g (pa₂) = qa₂) :
-- cube (hrfl ⬝hp (ra₁)⁻¹) hrfl
-- (my_squarel (qa₁) (qa₂)) (aps g (my_squarel (pa₁) (pa₂)))
-- (hrfl ⬝hp (!ap_con ⬝ whisker_left _ !ap_inv ⬝ (ra₁) ◾ (ra₂)⁻²)⁻¹)
-- (hrfl ⬝hp (ra₂)⁻²⁻¹ ⬝hp !ap_inv⁻¹) :=
-- begin
-- induction ra₂, induction ra₁, induction pa₂, induction pa₁, exact idc
-- end
-- private definition my_cube_fillerr {B C : Type} {g : B → C} {b₀ bl br : B}
-- {pl : b₀ = bl} {pr : b₀ = br} {ql : g b₀ = g bl} {qr : g b₀ = g br}
-- (sl : ap g pl = ql) (sr : ap g pr = qr) :
-- cube (hrfl ⬝hp sr⁻¹) hrfl
-- (my_squarer ql qr) (aps g (my_squarer pl pr))
-- (hrfl ⬝hp (!ap_con ⬝ whisker_left _ !ap_inv ⬝ sl ◾ sl⁻²)⁻¹)
-- (hrfl ⬝hp sr⁻²⁻¹ ⬝hp !ap_inv⁻¹) :=
-- begin
-- induction sr, induction sl, induction pr, induction pl, exact idc
-- end
-- definition dsmash_functor_pcompose_pconst2_homotopy {A A' A'' B B' B'' : Type}
-- (a₀ : A) (b₀ : B) (a₀' : A') (f' : A' → A'') (g' : B' → B'') (g : B → B')
-- (x : pointed.MK A a₀ ∧ pointed.MK B b₀) :
-- square (dsmash_functor_pcompose_homotopy f' (λ a, a₀') g' g x)
-- idp
-- (dsmash_functor_pconst_left_homotopy (λ a, g' (g a)) x)
-- (ap (dsmash_functor' (pmap.mk f' (refl (f' a₀'))) (pmap.mk g' (refl (g' (g b₀)))))
-- (dsmash_functor_pconst_left_homotopy g x)) :=
-- begin
-- induction x with a b a b,
-- { refine _ ⬝hp (functor_gluer'2 f' g' (g b) (g b₀))⁻¹, exact hrfl },
-- { refine _ ⬝hp !ap_inv⁻¹, refine _ ⬝hp !functor_gluel2⁻²⁻¹, exact hrfl },
-- { refine _ ⬝hp !ap_inv⁻¹, refine _ ⬝hp !functor_gluer2⁻²⁻¹, exact hrfl },
-- { exact abstract begin apply square_pathover,
-- refine !rec_eq_gluel ⬝p1 _ ⬝1p !natural_square_refl⁻¹,
-- refine !rec_eq_gluel ⬝p2 _ ⬝2p !natural_square_ap_fn⁻¹,
-- apply whisker001, apply whisker021,
-- apply move201, refine _ ⬝1p !eq_hconcat_hdeg_square⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine ap (hconcat_eq _) !ap_inv ⬝p1 _ ⬝2p (ap (aps _) !rec_eq_gluel ⬝ !aps_eq_hconcat)⁻¹,
-- apply whisker021, refine _ ⬝2p !aps_hconcat_eq⁻¹, apply move221,
-- refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝1p ap hdeg_square (eq_bot_of_square (transpose !ap02_ap_constant)),
-- apply my_cube_fillerr end end },
-- { exact abstract begin apply square_pathover,
-- refine !rec_eq_gluer ⬝p1 _ ⬝1p !natural_square_refl⁻¹,
-- refine !rec_eq_gluer ⬝p2 _ ⬝2p !natural_square_ap_fn⁻¹,
-- apply whisker001, apply whisker021,
-- apply move201, refine _ ⬝1p !eq_hconcat_hdeg_square⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine ap (hconcat_eq _) !ap_inv ⬝p1 _ ⬝2p (ap (aps _) !rec_eq_gluer ⬝ !aps_eq_hconcat)⁻¹,
-- apply whisker021, refine _ ⬝2p !aps_hconcat_eq⁻¹, apply move221,
-- refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝1p ap hdeg_square (eq_bot_of_square (transpose !ap02_ap_constant)),
-- apply my_cube_fillerl end end }
-- end
-- definition dsmash_functor_pcompose_pconst2 (f' : A' →* A'') (g' : B' →* B'') (g : B →* B') :
-- phsquare (dsmash_functor_pcompose f' (pconst A A') g' g)
-- (dsmash_functor_pconst_left (g' ∘* g))
-- (dsmash_functor_phomotopy (pcompose_pconst f') phomotopy.rfl)
-- (pwhisker_left (f' ⋀→ g') (dsmash_functor_pconst_left g) ⬝*
-- pcompose_pconst (f' ⋀→ g')) :=
-- begin
-- induction A with A a₀, induction B with B b₀,
-- induction A' with A' a₀', induction B' with B' b₀',
-- induction A'' with A'' a₀'', induction B'' with B'' b₀'',
-- induction f' with f' f'₀, induction g' with g' g₀', induction g with g g₀,
-- esimp at *, induction f'₀, induction g₀', induction g₀,
-- refine !dsmash_functor_phomotopy_refl ⬝ph** _, refine _ ⬝ !refl_trans⁻¹,
-- fapply phomotopy_eq,
-- { intro x, refine eq_of_square _ ⬝ !con_idp,
-- exact dsmash_functor_pcompose_pconst2_homotopy a₀ b₀ a₀' f' g' g x },
-- { refine _ ⬝ !idp_con⁻¹,
-- refine whisker_right _ (!whisker_right_idp ⬝ !eq_of_square_hrfl_hconcat_eq) ⬝ _,
-- refine !con.assoc ⬝ _, apply con_eq_of_eq_inv_con,
-- refine whisker_right _ _ ⬝ _, rotate 1, rexact functor_gluer'2_same f' g' (g b₀),
-- refine !inv_con_cancel_right ⬝ _,
-- refine !whisker_left_idp⁻¹ ⬝ _,
-- refine !con_idp ⬝ _,
-- apply whisker_left,
-- apply ap (whisker_left idp),
-- exact (!idp_con ⬝ !idp_con ⬝ !whisker_right_idp ⬝ !idp_con)⁻¹ }
-- end
-- /- a version where the left maps are identities -/
-- definition dsmash_functor_pcompose_pconst2_pid (f' : A' →* A'') :
-- phsquare (dsmash_functor_pcompose_pid B f' (pconst A A'))
-- (dsmash_functor_pconst_left (pid B))
-- (pcompose_pconst f' ⋀~ phomotopy.rfl)
-- (pwhisker_left (f' ⋀→ pid B) (dsmash_functor_pconst_left (pid B)) ⬝*
-- pcompose_pconst (f' ⋀→ pid B)) :=
-- (!dsmash_functor_phomotopy_refl ◾** idp ⬝ !refl_trans) ⬝pv**
-- dsmash_functor_pcompose_pconst2 f' (pid B) (pid B)
-- /- a small rewrite of the previous -/
-- -- definition dsmash_functor_pcompose_pid_pconst' (f' : A' →* A'') :
-- -- pwhisker_left (f' ⋀→ pid B) (dsmash_functor_pconst_left (pid B)) ⬝*
-- -- pcompose_pconst (f' ⋀→ pid B) =
-- -- (dsmash_functor_pcompose_pid B f' (pconst A A'))⁻¹* ⬝*
-- -- (pcompose_pconst f' ⋀~ phomotopy.rfl ⬝*
-- -- dsmash_functor_pconst_left (pid B)) :=
-- -- begin
-- -- apply eq_symm_trans_of_trans_eq,
-- -- exact dsmash_functor_pcompose_pid_pconst f'
-- -- end
-- /- if f' is constant -/
-- definition dsmash_functor_pcompose_pconst1_homotopy [unfold 13] {A A' A'' B B' B'' : Type}
-- (a₀ : A) (b₀ : B) (a₀'' : A'') (f : A → A') (g' : B' → B'') (g : B → B')
-- (x : pointed.MK A a₀ ∧ pointed.MK B b₀) :
-- square (dsmash_functor_pcompose_homotopy (λa', a₀'') f g' g x)
-- idp
-- (dsmash_functor_pconst_left_homotopy (λ a, g' (g a)) x)
-- (dsmash_functor_pconst_left_homotopy g'
-- ((pmap_of_map f a₀ ⋀→ pmap_of_map g b₀) x)) :=
-- begin
-- induction x with a b a b,
-- { exact hrfl },
-- { exact hrfl },
-- { exact hrfl },
-- { exact abstract begin apply square_pathover,
-- refine !rec_eq_gluel ⬝p1 _ ⬝1p !natural_square_refl⁻¹,
-- refine !rec_eq_gluel ⬝p2 _ ⬝2p
-- (natural_square_compose (dsmash_functor_pconst_left_homotopy g') _ _)⁻¹ᵖ,
-- apply whisker001, apply whisker021,
-- apply move201, refine _ ⬝1p !eq_hconcat_hdeg_square⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine ap (hconcat_eq _) !ap_inv ⬝p1 _ ⬝2p (natural_square_eq2 _ !functor_gluel2)⁻¹ᵖ,
-- apply whisker021,
-- refine _ ⬝1p ap hdeg_square (eq_of_square (!ap_constant_compose⁻¹ʰ) ⬝ !idp_con)⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝2p !rec_eq_gluel⁻¹, apply whisker021,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝1p ap hdeg_square (eq_bot_of_square (transpose !ap02_constant)),
-- exact rfl2 end end },
-- { exact abstract begin apply square_pathover,
-- refine !rec_eq_gluer ⬝p1 _ ⬝1p !natural_square_refl⁻¹,
-- refine !rec_eq_gluer ⬝p2 _ ⬝2p
-- (natural_square_compose (dsmash_functor_pconst_left_homotopy g') _ _)⁻¹ᵖ,
-- apply whisker001, apply whisker021,
-- apply move201, refine _ ⬝1p !eq_hconcat_hdeg_square⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine ap (hconcat_eq _) !ap_inv ⬝p1 _ ⬝2p (natural_square_eq2 _ !functor_gluer2)⁻¹ᵖ,
-- apply whisker021,
-- refine _ ⬝1p ap hdeg_square (eq_of_square (!ap_constant_compose⁻¹ʰ) ⬝ !idp_con)⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝2p !rec_eq_gluer⁻¹, apply whisker021,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝1p ap hdeg_square (eq_bot_of_square (transpose !ap02_constant)),
-- exact rfl2 end end },
-- end
-- definition dsmash_functor_pcompose_pconst1 (f : A →* A') (g' : B' →* B'') (g : B →* B') :
-- phsquare (dsmash_functor_pcompose (pconst A' A'') f g' g)
-- (dsmash_functor_pconst_left (g' ∘* g))
-- (pconst_pcompose f ⋀~ phomotopy.rfl)
-- (pwhisker_right (f ⋀→ g) (dsmash_functor_pconst_left g') ⬝*
-- pconst_pcompose (f ⋀→ g)) :=
-- begin
-- induction A with A a₀, induction B with B b₀,
-- induction A' with A' a₀', induction B' with B' b₀',
-- induction A'' with A'' a₀'', induction B'' with B'' b₀'',
-- induction f with f f₀, induction g' with g' g₀', induction g with g g₀,
-- esimp at *, induction f₀, induction g₀', induction g₀,
-- refine !dsmash_functor_phomotopy_refl ⬝ph** _, refine _ ⬝ !refl_trans⁻¹,
-- fapply phomotopy_eq,
-- { intro x, refine eq_of_square (dsmash_functor_pcompose_pconst1_homotopy a₀ b₀ a₀'' f g' g x) },
-- { refine whisker_right _ (!whisker_right_idp ⬝ !eq_of_square_hrfl) ⬝ _,
-- have H : Π{A : Type} {a a' : A} (p : a = a'),
-- idp_con (p ⬝ p⁻¹) ⬝ con.right_inv p = idp ⬝
-- whisker_left idp (idp ⬝ (idp ⬝ proof whisker_right idp (idp_con (p ⬝ p⁻¹ᵖ))⁻¹ᵖ qed ⬝
-- whisker_left idp (con.right_inv p))), by intros; induction p; reflexivity,
-- rexact H (gluer (g' (g b₀))) }
-- end
-- /- a version where the left maps are identities -/
-- definition dsmash_functor_pcompose_pconst1_pid (f : A →* A') :
-- phsquare (dsmash_functor_pcompose_pid B (pconst A' A'') f)
-- (dsmash_functor_pconst_left (pid B))
-- (pconst_pcompose f ⋀~ phomotopy.rfl)
-- (pwhisker_right (f ⋀→ pid B) (dsmash_functor_pconst_left (pid B)) ⬝*
-- pconst_pcompose (f ⋀→ pid B)) :=
-- (!dsmash_functor_phomotopy_refl ◾** idp ⬝ !refl_trans) ⬝pv**
-- dsmash_functor_pcompose_pconst1 f (pid B) (pid B)
-- /- Using these lemmas we show that dsmash_functor_left is natural in all arguments -/
-- definition dsmash_functor_left_natural_left (B C : Type*) (f : A' →* A) :
-- psquare (dsmash_functor_left A B C) (dsmash_functor_left A' B C)
-- (ppcompose_right f) (ppcompose_right (f ⋀→ pid C)) :=
-- begin
-- refine _⁻¹*,
-- fapply phomotopy_mk_ppmap,
-- { intro g, exact dsmash_functor_pcompose_pid C g f },
-- { refine idp ◾** (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !pcompose_right_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy) ⬝ _ ,
-- refine _ ⬝ (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !dsmash_functor_eq_of_phomotopy_left ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy)⁻¹,
-- apply dsmash_functor_pcompose_pconst1_pid }
-- end
-- definition dsmash_functor_left_natural_middle (A C : Type*) (f : B →* B') :
-- psquare (dsmash_functor_left A B C) (dsmash_functor_left A B' C)
-- (ppcompose_left f) (ppcompose_left (f ⋀→ pid C)) :=
-- begin
-- refine _⁻¹*,
-- fapply phomotopy_mk_ppmap,
-- { exact dsmash_functor_pcompose_pid C f },
-- { refine idp ◾** (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !pcompose_left_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy) ⬝ _ ,
-- refine _ ⬝ (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !dsmash_functor_eq_of_phomotopy_left ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy)⁻¹,
-- apply dsmash_functor_pcompose_pconst2_pid }
-- end
-- definition dsmash_functor_left_natural_right (A B : Type*) (f : C →* C') :
-- psquare (dsmash_functor_left A B C) (ppcompose_right (pid A ⋀→ f))
-- (dsmash_functor_left A B C') (ppcompose_left (pid B ⋀→ f)) :=
-- begin
-- refine _⁻¹*,
-- fapply phomotopy_mk_ppmap,
-- { intro g, exact dsmash_functor_psquare proof phomotopy.rfl qed proof phomotopy.rfl qed },
-- { esimp,
-- refine idp ◾** (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !pcompose_left_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy) ⬝ _ ,
-- refine _ ⬝ (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !pcompose_right_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy)⁻¹,
-- apply eq_of_phsquare,
-- refine (phmove_bot_of_left _ !dsmash_functor_pcompose_pconst1⁻¹ʰ**) ⬝h**
-- (!dsmash_functor_phomotopy_refl ⬝pv** !phhrfl) ⬝h** !dsmash_functor_pcompose_pconst2 ⬝vp** _,
-- refine !trans_assoc ⬝ !trans_assoc ⬝ idp ◾** _ ⬝ !trans_refl,
-- refine idp ◾** !refl_trans ⬝ !trans_left_inv }
-- end
-- definition dsmash_functor_left_natural_middle_phomotopy (A C : Type*) {f f' : B →* B'}
-- (p : f ~* f') : dsmash_functor_left_natural_middle A C f =
-- ppcompose_left_phomotopy p ⬝ph* dsmash_functor_left_natural_middle A C f' ⬝hp*
-- ppcompose_left_phomotopy (p ⋀~ phomotopy.rfl) :=
-- begin
-- induction p using phomotopy_rec_idp,
-- symmetry,
-- refine !ppcompose_left_phomotopy_refl ◾ph* idp ◾hp*
-- (ap ppcompose_left_phomotopy !dsmash_functor_phomotopy_refl ⬝
-- !ppcompose_left_phomotopy_refl) ⬝ _,
-- exact !rfl_phomotopy_hconcat ⬝ !hconcat_phomotopy_rfl
-- end
-- /- the following is not really used, but a symmetric version of the natural equivalence
-- dsmash_functor_left -/
-- /- f ∧ g is constant if g is constant -/
-- definition dsmash_functor_pconst_right_homotopy [unfold 6] {C : Type} (f : A → C) (x : ⋀ B) :
-- (pmap_of_map f pt ⋀→ pconst B D) x = pt :=
-- begin
-- induction x with a b a b,
-- { exact gluel' (f a) pt },
-- { exact (gluel pt)⁻¹ },
-- { exact (gluer pt)⁻¹ },
-- { apply eq_pathover, note x := functor_gluel2 f (λx : B, Point D) a, esimp [pconst] at *,
-- refine x ⬝ph _, refine _ ⬝hp !ap_constant⁻¹, apply square_of_eq, reflexivity },
-- { apply eq_pathover, note x := functor_gluer2 f (λx : B, Point D) b, esimp [pconst] at *,
-- refine x ⬝ph _, refine _ ⬝hp !ap_constant⁻¹, apply square_of_eq,
-- rexact con.right_inv (gluel (f pt)) ⬝ (con.right_inv (gluer pt))⁻¹ }
-- end
-- definition dsmash_functor_pconst_right (f : A →* C) :
-- f ⋀→ (pconst B D) ~* pconst (⋀ B) (⋀ D) :=
-- begin
-- induction C with C, induction f with f f₀, esimp at *, induction f₀,
-- fapply phomotopy.mk,
-- { exact dsmash_functor_pconst_right_homotopy f },
-- { rexact con.right_inv (gluel (f pt)) }
-- end
-- definition dsmash_functor_pconst_right_phomotopy {f f' : A →* C} (p : f ~* f') :
-- dsmash_functor_phomotopy p (phomotopy.refl (pconst B D)) ⬝* dsmash_functor_pconst_right f' =
-- dsmash_functor_pconst_right f :=
-- begin
-- induction p using phomotopy_rec_idp,
-- exact !dsmash_functor_phomotopy_refl ◾** idp ⬝ !refl_trans
-- end
-- /- This makes dsmash_functor into a pointed map (B →* B') →* (⋀ B →* ⋀ B') -/
-- definition dsmash_functor_right [constructor] (B : A → Type*) (C : Type*) :
-- ppmap B C →* ppmap (⋀ B) (A ∧ C) :=
-- pmap.mk (dsmash_functor (pid A)) (eq_of_phomotopy (dsmash_functor_pconst_right (pid A)))
-- /- We want to show that dsmash_functor_right is natural in A, B and C.
-- For this we need two coherence rules. Given the function h := (f' ∘ f) ⋀→ (g' ∘ g) and suppose
-- that either g' or g is constant. There are two ways to show that h is constant: either by using
-- exchange, or directly. We need to show that these two proofs result in the same pointed
-- homotopy. First we do the case where g is constant -/
-- definition dsmash_functor_pcompose_pconst4_homotopy {A B C D E F : Type}
-- (a₀ : A) (b₀ : B) (d₀ : D) (f' : C → E) (f : A → C) (g : D → F)
-- (x : pointed.MK A a₀ ∧ pointed.MK B b₀) :
-- square (dsmash_functor_pcompose_homotopy f' f g (λ a, d₀) x)
-- idp
-- (dsmash_functor_pconst_right_homotopy (λ a, f' (f a)) x)
-- (ap (dsmash_functor' (pmap.mk f' (refl (f' (f a₀)))) (pmap.mk g (refl (g d₀))))
-- (dsmash_functor_pconst_right_homotopy f x)) :=
-- begin
-- induction x with a b a b,
-- { refine _ ⬝hp (functor_gluel'2 f' g (f a) (f a₀))⁻¹, exact hrfl },
-- { refine _ ⬝hp !ap_inv⁻¹, refine _ ⬝hp !functor_gluel2⁻²⁻¹, exact hrfl },
-- { refine _ ⬝hp !ap_inv⁻¹, refine _ ⬝hp !functor_gluer2⁻²⁻¹, exact hrfl },
-- { exact abstract begin apply square_pathover,
-- refine !rec_eq_gluel ⬝p1 _ ⬝1p !natural_square_refl⁻¹,
-- refine !rec_eq_gluel ⬝p2 _ ⬝2p !natural_square_ap_fn⁻¹,
-- apply whisker001, apply whisker021,
-- apply move201, refine _ ⬝1p !eq_hconcat_hdeg_square⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine ap (hconcat_eq _) !ap_inv ⬝p1 _ ⬝2p (ap (aps _) !rec_eq_gluel ⬝ !aps_eq_hconcat)⁻¹,
-- apply whisker021, refine _ ⬝2p !aps_hconcat_eq⁻¹, apply move221,
-- refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝1p ap hdeg_square (eq_bot_of_square (transpose !ap02_ap_constant)),
-- apply my_cube_fillerl end end },
-- { exact abstract begin apply square_pathover,
-- refine !rec_eq_gluer ⬝p1 _ ⬝1p !natural_square_refl⁻¹,
-- refine !rec_eq_gluer ⬝p2 _ ⬝2p !natural_square_ap_fn⁻¹,
-- apply whisker001, apply whisker021,
-- apply move201, refine _ ⬝1p !eq_hconcat_hdeg_square⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine ap (hconcat_eq _) !ap_inv ⬝p1 _ ⬝2p (ap (aps _) !rec_eq_gluer ⬝ !aps_eq_hconcat)⁻¹,
-- apply whisker021, refine _ ⬝2p !aps_hconcat_eq⁻¹, apply move221,
-- refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝1p ap hdeg_square (eq_bot_of_square (transpose !ap02_ap_constant)),
-- apply my_cube_fillerr end end }
-- end
-- definition dsmash_functor_pcompose_pconst4 (f' : C →* E) (f : A →* C) (g : D →* F) :
-- phsquare (dsmash_functor_pcompose f' f g (pconst B D))
-- (dsmash_functor_pconst_right (f' ∘* f))
-- (dsmash_functor_phomotopy phomotopy.rfl (pcompose_pconst g))
-- (pwhisker_left (f' ⋀→ g) (dsmash_functor_pconst_right f) ⬝*
-- pcompose_pconst (f' ⋀→ g)) :=
-- begin
-- induction A with A a₀, induction B with B b₀,
-- induction E with E e₀, induction C with C c₀, induction F with F x₀, induction D with D d₀,
-- induction f' with f' f'₀, induction f with f f₀, induction g with g g₀,
-- esimp at *, induction f'₀, induction f₀, induction g₀,
-- refine !dsmash_functor_phomotopy_refl ⬝ph** _, refine _ ⬝ !refl_trans⁻¹,
-- fapply phomotopy_eq,
-- { intro x, refine eq_of_square _ ⬝ !con_idp,
-- exact dsmash_functor_pcompose_pconst4_homotopy a₀ b₀ d₀ f' f g x, },
-- { refine _ ⬝ !idp_con⁻¹,
-- refine whisker_right _ (!whisker_right_idp ⬝ !eq_of_square_hrfl_hconcat_eq) ⬝ _,
-- refine !con.assoc ⬝ _, apply con_eq_of_eq_inv_con,
-- refine whisker_right _ _ ⬝ _, rotate 1, rexact functor_gluel'2_same f' g (f a₀),
-- refine !inv_con_cancel_right ⬝ _,
-- refine !whisker_left_idp⁻¹ ⬝ _,
-- refine !con_idp ⬝ _,
-- apply whisker_left,
-- apply ap (whisker_left idp),
-- exact (!idp_con ⬝ !idp_con ⬝ !whisker_right_idp ⬝ !idp_con)⁻¹ }
-- end
-- /- a version where the left maps are identities -/
-- definition dsmash_functor_pcompose_pconst4_pid (g : D →* F) :
-- phsquare (dsmash_functor_pid_pcompose A g (pconst B D))
-- (dsmash_functor_pconst_right (pid A))
-- (dsmash_functor_phomotopy phomotopy.rfl (pcompose_pconst g))
-- (pwhisker_left (pid A ⋀→ g) (dsmash_functor_pconst_right (pid A)) ⬝*
-- pcompose_pconst (pid A ⋀→ g)) :=
-- (!dsmash_functor_phomotopy_refl ◾** idp ⬝ !refl_trans) ⬝pv**
-- dsmash_functor_pcompose_pconst4 (pid A) (pid A) g
-- /- a small rewrite of the previous -/
-- -- definition dsmash_functor_pid_pcompose_pconst' (g : D →* F) :
-- -- pwhisker_left (pid A ⋀→ g) (dsmash_functor_pconst_right (pid A)) ⬝*
-- -- pcompose_pconst (pid A ⋀→ g) =
-- -- (dsmash_functor_pid_pcompose A g (pconst B D))⁻¹* ⬝*
-- -- (dsmash_functor_phomotopy phomotopy.rfl (pcompose_pconst g) ⬝*
-- -- dsmash_functor_pconst_right (pid A)) :=
-- -- begin
-- -- apply eq_symm_trans_of_trans_eq,
-- -- exact dsmash_functor_pid_pcompose_pconst g
-- -- end
-- /- if g' is constant -/
-- definition dsmash_functor_pcompose_pconst3_homotopy [unfold 13] {A B C D E F : Type}
-- (a₀ : A) (b₀ : B) (x₀ : F) (f' : C → E) (f : A → C) (g : B → D)
-- (x : pointed.MK A a₀ ∧ pointed.MK B b₀) :
-- square (dsmash_functor_pcompose_homotopy f' f (λ a, x₀) g x)
-- idp
-- (dsmash_functor_pconst_right_homotopy (λ a, f' (f a)) x)
-- (dsmash_functor_pconst_right_homotopy f'
-- (dsmash_functor (pmap_of_map f a₀) (pmap_of_map g b₀) x)) :=
-- begin
-- induction x with a b a b,
-- { exact hrfl },
-- { exact hrfl },
-- { exact hrfl },
-- { exact abstract begin apply square_pathover,
-- refine !rec_eq_gluel ⬝p1 _ ⬝1p !natural_square_refl⁻¹,
-- refine !rec_eq_gluel ⬝p2 _ ⬝2p
-- (natural_square_compose (dsmash_functor_pconst_right_homotopy f') _ _)⁻¹ᵖ,
-- apply whisker001, apply whisker021,
-- apply move201, refine _ ⬝1p !eq_hconcat_hdeg_square⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine ap (hconcat_eq _) !ap_inv ⬝p1 _ ⬝2p (natural_square_eq2 _ !functor_gluel2)⁻¹ᵖ,
-- apply whisker021,
-- refine _ ⬝1p ap hdeg_square (eq_of_square (!ap_constant_compose⁻¹ʰ) ⬝ !idp_con)⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝2p !rec_eq_gluel⁻¹, apply whisker021,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝1p ap hdeg_square (eq_bot_of_square (transpose !ap02_constant)),
-- exact rfl2 end end },
-- { exact abstract begin apply square_pathover,
-- refine !rec_eq_gluer ⬝p1 _ ⬝1p !natural_square_refl⁻¹,
-- refine !rec_eq_gluer ⬝p2 _ ⬝2p
-- (natural_square_compose (dsmash_functor_pconst_right_homotopy f') _ _)⁻¹ᵖ,
-- apply whisker001, apply whisker021,
-- apply move201, refine _ ⬝1p !eq_hconcat_hdeg_square⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine ap (hconcat_eq _) !ap_inv ⬝p1 _ ⬝2p (natural_square_eq2 _ !functor_gluer2)⁻¹ᵖ,
-- apply whisker021,
-- refine _ ⬝1p ap hdeg_square (eq_of_square (!ap_constant_compose⁻¹ʰ) ⬝ !idp_con)⁻¹,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝2p !rec_eq_gluer⁻¹, apply whisker021,
-- apply move221, refine _ ⬝1p !hdeg_square_hconcat_eq⁻¹,
-- refine _ ⬝1p ap hdeg_square (eq_bot_of_square (transpose !ap02_constant)),
-- exact rfl2 end end },
-- end
-- definition dsmash_functor_pcompose_pconst3 (f' : C →* E) (f : A →* C) (g : Πa, B a →* D (f a)) :
-- phsquare (dsmash_functor_pcompose f' f (pconst D F) g)
-- (dsmash_functor_pconst_right (f' ∘* f))
-- (dsmash_functor_phomotopy phomotopy.rfl (pconst_pcompose g))
-- (pwhisker_right (f ⋀→ g) (dsmash_functor_pconst_right f') ⬝*
-- pconst_pcompose (f ⋀→ g)) :=
-- begin
-- induction A with A a₀, induction B with B b₀,
-- induction E with E e₀, induction C with C c₀, induction F with F x₀, induction D with D d₀,
-- induction f' with f' f'₀, induction f with f f₀, induction g with g g₀,
-- esimp at *, induction f'₀, induction f₀, induction g₀,
-- refine !dsmash_functor_phomotopy_refl ⬝ph** _, refine _ ⬝ !refl_trans⁻¹,
-- fapply phomotopy_eq,
-- { intro x, refine eq_of_square (dsmash_functor_pcompose_pconst3_homotopy a₀ b₀ x₀ f' f g x) },
-- { refine whisker_right _ (!whisker_right_idp ⬝ !eq_of_square_hrfl) ⬝ _,
-- have H : Π{A : Type} {a a' : A} (p : a = a'),
-- idp_con (p ⬝ p⁻¹) ⬝ con.right_inv p = idp ⬝
-- whisker_left idp (idp ⬝ (idp ⬝ proof whisker_right idp (idp_con (p ⬝ p⁻¹ᵖ))⁻¹ᵖ qed ⬝
-- whisker_left idp (con.right_inv p))), by intros; induction p; reflexivity,
-- rexact H (gluel (f' (f a₀))) }
-- end
-- /- a version where the left maps are identities -/
-- definition dsmash_functor_pcompose_pconst3_pid (g : Πa, B a →* D (f a)) :
-- phsquare (dsmash_functor_pid_pcompose A (pconst D F) g)
-- (dsmash_functor_pconst_right (pid A))
-- (dsmash_functor_phomotopy phomotopy.rfl (pconst_pcompose g))
-- (pwhisker_right (pid A ⋀→ g) (dsmash_functor_pconst_right (pid A)) ⬝*
-- pconst_pcompose (pid A ⋀→ g)) :=
-- (!dsmash_functor_phomotopy_refl ◾** idp ⬝ !refl_trans) ⬝pv**
-- dsmash_functor_pcompose_pconst3 (pid A) (pid A) g
-- /- Using these lemmas we show that dsmash_functor_right is natural in all arguments -/
-- definition dsmash_functor_right_natural_right (A B : Type*) (f : C →* C') :
-- psquare (dsmash_functor_right A B C) (dsmash_functor_right A B C')
-- (ppcompose_left f) (ppcompose_left (pid A ⋀→ f)) :=
-- begin
-- refine _⁻¹*,
-- fapply phomotopy_mk_ppmap,
-- { exact dsmash_functor_pid_pcompose A f },
-- { refine idp ◾** (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !pcompose_left_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy) ⬝ _ ,
-- refine _ ⬝ (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !dsmash_functor_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy)⁻¹,
-- apply dsmash_functor_pcompose_pconst4_pid }
-- end
-- definition dsmash_functor_right_natural_middle (A C : Type*) (f : B' →* B) :
-- psquare (dsmash_functor_right A B C) (dsmash_functor_right A B' C)
-- (ppcompose_right f) (ppcompose_right (pid A ⋀→ f)) :=
-- begin
-- refine _⁻¹*,
-- fapply phomotopy_mk_ppmap,
-- { intro g, exact dsmash_functor_pid_pcompose A g f },
-- { refine idp ◾** (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !pcompose_right_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy) ⬝ _ ,
-- refine _ ⬝ (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !dsmash_functor_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy)⁻¹,
-- apply dsmash_functor_pcompose_pconst3_pid }
-- end
-- definition dsmash_functor_right_natural_left (B C : Type*) (f : A →* A') :
-- psquare (dsmash_functor_right A B C) (ppcompose_right (f ⋀→ (pid B)))
-- (dsmash_functor_right A' B C) (ppcompose_left (f ⋀→ (pid C))) :=
-- begin
-- refine _⁻¹*,
-- fapply phomotopy_mk_ppmap,
-- { intro g, exact dsmash_functor_psquare proof phomotopy.rfl qed proof phomotopy.rfl qed },
-- { esimp,
-- refine idp ◾** (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !pcompose_left_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy) ⬝ _ ,
-- refine _ ⬝ (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !pcompose_right_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy)⁻¹,
-- apply eq_of_phsquare,
-- refine (phmove_bot_of_left _ !dsmash_functor_pcompose_pconst3⁻¹ʰ**) ⬝h**
-- (!dsmash_functor_phomotopy_refl ⬝pv** !phhrfl) ⬝h** !dsmash_functor_pcompose_pconst4 ⬝vp** _,
-- refine !trans_assoc ⬝ !trans_assoc ⬝ idp ◾** _ ⬝ !trans_refl,
-- refine idp ◾** !refl_trans ⬝ !trans_left_inv }
-- end
-- /- ⋀ B ≃* pcofiber (pprod_of_wedge A B) -/
-- variables (A B)
-- open pushout
-- definition dsmash_equiv_cofiber : ⋀ B ≃ cofiber (@prod_of_wedge A B) :=
-- begin
-- unfold [dsmash, cofiber, dsmash'], symmetry,
-- fapply pushout_vcompose_equiv wedge_of_sum,
-- { symmetry, refine equiv_unit_of_is_contr _ _, apply is_contr_pushout_wedge_of_sum },
-- { intro x, reflexivity },
-- { apply prod_of_wedge_of_sum }
-- end
-- definition dsmash_punit_pequiv [constructor] : dsmash A punit ≃* punit :=
-- begin
-- apply pequiv_punit_of_is_contr,
-- apply is_contr.mk (dsmash.mk pt ⋆), intro x,
-- induction x,
-- { induction b, exact gluel' pt a },
-- { exact gluel pt },
-- { exact gluer pt },
-- { apply eq_pathover_constant_left_id_right, apply square_of_eq_top,
-- exact whisker_right _ !idp_con⁻¹ },
-- { apply eq_pathover_constant_left_id_right, induction b,
-- refine !con.right_inv ⬝pv _, exact square_of_eq idp },
-- end
-- definition pprod_of_wedge [constructor] : wedge A B →* A ×* B :=
-- begin
-- fconstructor,
-- { exact prod_of_wedge },
-- { reflexivity }
-- end
-- definition dsmash_pequiv_pcofiber [constructor] : ⋀ B ≃* pcofiber (pprod_of_wedge A B) :=
-- begin
-- apply pequiv_of_equiv (dsmash_equiv_cofiber A B),
-- exact cofiber.glue pt
-- end
-- variables {A B}
-- /- commutativity -/
-- definition dsmash_flip' [unfold 3] (x : ⋀ B) : dsmash B A :=
-- begin
-- induction x,
-- { exact dsmash.mk b a },
-- { exact auxr },
-- { exact auxl },
-- { exact gluer a },
-- { exact gluel b }
-- end
-- definition dsmash_flip_dsmash_flip' [unfold 3] (x : ⋀ B) : dsmash_flip' (dsmash_flip' x) = x :=
-- begin
-- induction x,
-- { reflexivity },
-- { reflexivity },
-- { reflexivity },
-- { apply eq_pathover_id_right,
-- refine ap_compose dsmash_flip' _ _ ⬝ ap02 _ !elim_gluel ⬝ !elim_gluer ⬝ph _,
-- apply hrfl },
-- { apply eq_pathover_id_right,
-- refine ap_compose dsmash_flip' _ _ ⬝ ap02 _ !elim_gluer ⬝ !elim_gluel ⬝ph _,
-- apply hrfl }
-- end
-- variables (A B)
-- definition dsmash_flip [constructor] : ⋀ B →* dsmash B A :=
-- pmap.mk dsmash_flip' idp
-- definition dsmash_flip_dsmash_flip [constructor] :
-- dsmash_flip B A ∘* dsmash_flip A B ~* pid (⋀ B) :=
-- phomotopy.mk dsmash_flip_dsmash_flip' idp
-- definition dsmash_comm [constructor] : ⋀ B ≃* dsmash B A :=
-- begin
-- apply pequiv.MK, do 2 apply dsmash_flip_dsmash_flip
-- end
-- variables {A B}
-- definition dsmash_flip_dsmash_functor' [unfold 7] (f : A →* C) (g : Πa, B a →* D (f a)) : hsquare
-- dsmash_flip' dsmash_flip' (dsmash_functor' f g) (dsmash_functor' g f) :=
-- begin
-- intro x, induction x,
-- { reflexivity },
-- { reflexivity },
-- { reflexivity },
-- { apply eq_pathover,
-- refine ap_compose (dsmash_functor' _ _) _ _ ⬝ ap02 _ !elim_gluel ⬝ !functor_gluer ⬝ph _
-- ⬝hp (ap_compose dsmash_flip' _ _ ⬝ ap02 _ !functor_gluel)⁻¹ᵖ,
-- refine _ ⬝hp (!ap_con ⬝ !ap_compose' ◾ !elim_gluel)⁻¹, exact hrfl },
-- { apply eq_pathover,
-- refine ap_compose (dsmash_functor' _ _) _ _ ⬝ ap02 _ !elim_gluer ⬝ !functor_gluel ⬝ph _
-- ⬝hp (ap_compose dsmash_flip' _ _ ⬝ ap02 _ !functor_gluer)⁻¹ᵖ,
-- refine _ ⬝hp (!ap_con ⬝ !ap_compose' ◾ !elim_gluer)⁻¹, exact hrfl },
-- end
-- definition dsmash_flip_dsmash_functor (f : A →* C) (g : Πa, B a →* D (f a)) :
-- psquare (dsmash_flip A B) (dsmash_flip C D) (f ⋀→ g) (g ⋀→ f) :=
-- begin
-- apply phomotopy.mk (dsmash_flip_dsmash_functor' f g), refine !idp_con ⬝ _ ⬝ !idp_con⁻¹,
-- refine !ap_ap011 ⬝ _, apply ap011_flip,
-- end
definition pinr [constructor] (B : A → Type*) (a : A) : B a →* ⋀ B :=
begin
fapply pmap.mk,
{ intro b, exact dsmash.mk a b },
{ exact gluel' a pt }
end
definition pinl [constructor] (b : Πa, B a) : A →* ⋀ B :=
begin
fapply pmap.mk,
{ intro a, exact dsmash.mk a (b a) },
{ exact gluer' (b pt) pt }
end
-- definition pinr_phomotopy {a a' : A} (p : a = a') : pinr B a ~* pinr B a' :=
-- begin
-- fapply phomotopy.mk,
-- { exact ap010 (pmap.to_fun ∘ pinr B) p },
-- { induction p, apply idp_con }
-- end
definition dsmash_pmap_unit_pt [constructor] (B : A → Type*) : pinr B pt ~* pconst (B pt) (⋀ B) :=
begin
fapply phomotopy.mk,
{ intro b, exact gluer' b pt },
{ rexact con.right_inv (gluer pt) ⬝ (con.right_inv (gluel pt))⁻¹ }
end
definition dsmash_pmap_unit [constructor] (B : A → Type*) : Π*(a : A), B a →** ⋀ B :=
begin
fapply ppi.mk,
{ exact pinr B },
{ apply eq_of_phomotopy, exact dsmash_pmap_unit_pt B }
end
/- The unit is natural in the first argument -/
definition dsmash_functor_pid_pinr' [constructor] (B : A' → Type*) (f : A →* A') (a : A) :
pinr B (f a) ~* (f ⋀→ λa, !pid) ∘* pinr (B ∘ f) a :=
begin
fapply phomotopy.mk,
{ intro b, reflexivity },
{ refine !idp_con ⬝ _,
induction A' with A' a₀', induction f with f f₀, esimp at *,
induction f₀, exact sorry }
end
-- definition dsmash_pmap_unit_pt_natural [constructor] (B : A → Type*) (f : A →* A') :
-- dsmash_functor_pid_pinr' B f pt ⬝*
-- pwhisker_left (f ⋀→ λa, !pid) (dsmash_pmap_unit_pt A B) ⬝*
-- pcompose_pconst (f ⋀→ λa, !pid) =
-- _ /-pinr_phomotopy (respect_pt f)-/ ⬝* dsmash_pmap_unit_pt A' B :=
-- begin
-- induction f with f f₀, induction A' with A' a₀', esimp at *,
-- induction f₀, refine _ ⬝ !refl_trans⁻¹,
-- refine !trans_refl ⬝ _,
-- fapply phomotopy_eq',
-- { intro b, refine !idp_con ⬝ _,
-- rexact functor_gluer'2 f (pid B) b pt },
-- { refine whisker_right_idp _ ⬝ph _,
-- refine ap (λx, _ ⬝ x) _ ⬝ph _,
-- rotate 1, rexact (functor_gluer'2_same f (pid B) pt),
-- refine whisker_right _ !idp_con ⬝pv _,
-- refine !con.assoc⁻¹ ⬝ph _, apply whisker_bl,
-- refine whisker_left _ !to_homotopy_pt_mk ⬝pv _,
-- refine !con.assoc⁻¹ ⬝ whisker_right _ _ ⬝pv _,
-- rotate 1, esimp, apply whisker_left_idp_con,
-- refine !con.assoc ⬝pv _, apply whisker_tl,
-- refine whisker_right _ !idp_con ⬝pv _,
-- refine whisker_right _ !whisker_right_idp ⬝pv _,
-- refine whisker_right _ (!idp_con ⬝ !ap02_con) ⬝ !con.assoc ⬝pv _,
-- apply whisker_tl,
-- apply vdeg_square,
-- refine whisker_right _ !ap_inv ⬝ _, apply inv_con_eq_of_eq_con,
-- rexact functor_gluel'2_same (pmap_of_map f pt) (pmap_of_map id (Point B)) pt }
-- end
-- definition dsmash_pmap_unit_natural (B : A' → Type*) (f : A →* A') :
-- psquare (dsmash_pmap_unit (B ∘ f)) (dsmash_pmap_unit B) f _ := --(ppcompose_left (f ⋀→ pid B))
-- begin
-- apply ptranspose,
-- induction A with A a₀, induction B with B b₀, induction A' with A' a₀',
-- induction f with f f₀, esimp at *, induction f₀, fapply phomotopy_mk_ppmap,
-- { intro a, exact dsmash_functor_pid_pinr' _ (pmap_of_map f a₀) a },
-- { refine ap (λx, _ ⬝* phomotopy_of_eq x) !respect_pt_pcompose ⬝ _
-- ⬝ ap phomotopy_of_eq !respect_pt_pcompose⁻¹,
-- esimp, refine _ ⬝ ap phomotopy_of_eq !idp_con⁻¹,
-- refine _ ⬝ !phomotopy_of_eq_of_phomotopy⁻¹,
-- refine ap (λx, _ ⬝* phomotopy_of_eq (x ⬝ _)) !pcompose_left_eq_of_phomotopy ⬝ _,
-- refine ap (λx, _ ⬝* x) (!phomotopy_of_eq_con ⬝
-- !phomotopy_of_eq_of_phomotopy ◾** !phomotopy_of_eq_of_phomotopy ⬝ !trans_refl) ⬝ _,
-- refine _ ⬝ dsmash_pmap_unit_pt_natural _ (pmap_of_map f a₀) ⬝ _,
-- { exact !trans_refl⁻¹ },
-- { exact !refl_trans }}
-- end
/- The unit is also dinatural in the first argument, but that's easier to prove after the adjunction.
We don't need it for the adjunction -/
/- The unit-counit laws -/
-- definition dsmash_pmap_unit_counit (B : A → Type*) :
-- dsmash_pmap_counit B (⋀ B) ∘* dsmash_pmap_unit A B ⋀→ pid B ~* pid (⋀ B) :=
-- begin
-- fapply phomotopy.mk,
-- { intro x,
-- induction x with a b a b,
-- { reflexivity },
-- { exact gluel pt },
-- { exact gluer pt },
-- { apply eq_pathover_id_right,
-- refine ap_compose dsmash_pmap_counit_map _ _ ⬝ ap02 _ !functor_gluel ⬝ph _,
-- refine !ap_con ⬝ !ap_compose' ◾ !elim_gluel ⬝ph _,
-- refine !idp_con ⬝ph _,
-- apply square_of_eq, refine !idp_con ⬝ !inv_con_cancel_right⁻¹ },
-- { apply eq_pathover_id_right,
-- refine ap_compose dsmash_pmap_counit_map _ _ ⬝ ap02 _ !functor_gluer ⬝ph _,
-- refine !ap_con ⬝ !ap_compose' ◾ !elim_gluer ⬝ph _,
-- refine !ap_eq_of_phomotopy ⬝ph _,
-- apply square_of_eq, refine !idp_con ⬝ !inv_con_cancel_right⁻¹ }},
-- { refine _ ⬝ !ap_compose, refine _ ⬝ (ap_is_constant respect_pt _)⁻¹,
-- rexact (con.right_inv (gluel pt))⁻¹ }
-- end
-- definition dsmash_pmap_counit_unit_pt [constructor] (f : A →* B) :
-- dsmash_pmap_counit A B ∘* pinr A f ~* f :=
-- begin
-- fapply phomotopy.mk,
-- { intro a, reflexivity },
-- { refine !idp_con ⬝ !elim_gluel'⁻¹ }
-- end
-- definition dsmash_pmap_counit_unit (B : A → Type*) :
-- ppcompose_left (dsmash_pmap_counit A B) ∘* dsmash_pmap_unit (ppmap A B) A ~* pid (ppmap A B) :=
-- begin
-- fapply phomotopy_mk_ppmap,
-- { intro f, exact dsmash_pmap_counit_unit_pt f },
-- { refine !trans_refl ⬝ _,
-- refine _ ⬝ ap (λx, phomotopy_of_eq (x ⬝ _)) !pcompose_left_eq_of_phomotopy⁻¹,
-- refine _ ⬝ !phomotopy_of_eq_con⁻¹,
-- refine _ ⬝ !phomotopy_of_eq_of_phomotopy⁻¹ ◾** !phomotopy_of_eq_of_phomotopy⁻¹,
-- refine _ ⬝ !trans_refl⁻¹,
-- fapply phomotopy_eq,
-- { intro a, esimp, refine !elim_gluer'⁻¹ },
-- { esimp, refine whisker_right _ !whisker_right_idp ⬝ _ ⬝ !idp_con⁻¹,
-- refine whisker_right _ !elim_gluer'_same⁻² ⬝ _ ⬝ !elim_gluel'_same⁻¹⁻²,
-- apply inv_con_eq_of_eq_con, refine !idp_con ⬝ _, esimp,
-- refine _ ⬝ !ap02_con ⬝ whisker_left _ !ap_inv,
-- refine !whisker_right_idp ⬝ _,
-- exact !idp_con }}
-- end
/- The underlying (unpointed) functions of the equivalence A →* (B →* C) ≃* ⋀ B →* C) -/
-- definition dsmash_elim [constructor] (f : Π*(a : A), ppmap (B a) C) : ⋀ B →* C :=
-- smash.smash_pmap_counit _ C ∘* _ /-f ⋀→ pid B_-/
-- definition dsmash_elim_inv [constructor] (g : ⋀ B →* C) : A →* ppmap B C :=
-- ppcompose_left g ∘* dsmash_pmap_unit A B
-- /- They are inverses, constant on the constant function and natural -/
-- definition dsmash_elim_left_inv (f : A →* ppmap B C) : dsmash_elim_inv (dsmash_elim f) ~* f :=
-- begin
-- refine !pwhisker_right !ppcompose_left_pcompose ⬝* _,
-- refine !passoc ⬝* _,
-- refine !pwhisker_left !dsmash_pmap_unit_natural ⬝* _,
-- refine !passoc⁻¹* ⬝* _,
-- refine !pwhisker_right !dsmash_pmap_counit_unit ⬝* _,
-- apply pid_pcompose
-- end
-- definition dsmash_elim_right_inv (g : ⋀ B →* C) : dsmash_elim (dsmash_elim_inv g) ~* g :=
-- begin
-- refine !pwhisker_left !dsmash_functor_pcompose_pid ⬝* _,
-- refine !passoc⁻¹* ⬝* _,
-- refine !pwhisker_right !dsmash_pmap_counit_natural_right⁻¹* ⬝* _,
-- refine !passoc ⬝* _,
-- refine !pwhisker_left !dsmash_pmap_unit_counit ⬝* _,
-- apply pcompose_pid
-- end
-- definition dsmash_elim_pconst (B : A → Type*) (C : Type*) :
-- dsmash_elim (pconst A (ppmap B C)) ~* pconst (⋀ B) C :=
-- begin
-- refine pwhisker_left _ (dsmash_functor_pconst_left (pid B)) ⬝* !pcompose_pconst
-- end
-- definition dsmash_elim_inv_pconst (B : A → Type*) (C : Type*) :
-- dsmash_elim_inv (pconst (⋀ B) C) ~* pconst A (ppmap B C) :=
-- begin
-- fapply phomotopy_mk_ppmap,
-- { intro f, apply pconst_pcompose },
-- { esimp, refine !trans_refl ⬝ _,
-- refine _ ⬝ (!phomotopy_of_eq_con ⬝ (ap phomotopy_of_eq !pcompose_left_eq_of_phomotopy ⬝
-- !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy)⁻¹,
-- apply pconst_pcompose_phomotopy_pconst }
-- end
-- definition dsmash_elim_natural_right (f : C →* C') (g : A →* ppmap B C) :
-- f ∘* dsmash_elim g ~* dsmash_elim (ppcompose_left f ∘* g) :=
-- begin
-- refine _ ⬝* pwhisker_left _ !dsmash_functor_pcompose_pid⁻¹*,
-- refine !passoc⁻¹* ⬝* pwhisker_right _ _ ⬝* !passoc,
-- apply dsmash_pmap_counit_natural_right
-- end
-- definition dsmash_elim_inv_natural_right {A B C C' : Type*} (f : C →* C')
-- (g : ⋀ B →* C) : ppcompose_left f ∘* dsmash_elim_inv g ~* dsmash_elim_inv (f ∘* g) :=
-- begin
-- refine !passoc⁻¹* ⬝* pwhisker_right _ _,
-- exact !ppcompose_left_pcompose⁻¹*
-- end
-- definition dsmash_elim_natural_left (f : A →* A') (g : B →* B') (h : A' →* ppmap B' C) :
-- dsmash_elim h ∘* (f ⋀→ g) ~* dsmash_elim (ppcompose_right g ∘* h ∘* f) :=
-- begin
-- refine !dsmash_functor_pcompose_pid ⬝ph* _,
-- refine _ ⬝v* !dsmash_pmap_counit_natural_left,
-- refine dsmash_functor_psquare !pid_pcompose⁻¹* (phrefl g)
-- end
-- definition dsmash_elim_phomotopy {f f' : A →* ppmap B C} (p : f ~* f') :
-- dsmash_elim f ~* dsmash_elim f' :=
-- begin
-- apply pwhisker_left,
-- exact dsmash_functor_phomotopy p phomotopy.rfl
-- end
-- definition dsmash_elim_inv_phomotopy {f f' : ⋀ B →* C} (p : f ~* f') :
-- dsmash_elim_inv f ~* dsmash_elim_inv f' :=
-- pwhisker_right _ (ppcompose_left_phomotopy p)
-- definition dsmash_elim_eq_of_phomotopy {f f' : A →* ppmap B C} (p : f ~* f') :
-- ap dsmash_elim (eq_of_phomotopy p) = eq_of_phomotopy (dsmash_elim_phomotopy p) :=
-- begin
-- induction p using phomotopy_rec_idp,
-- refine ap02 _ !eq_of_phomotopy_refl ⬝ _,
-- refine !eq_of_phomotopy_refl⁻¹ ⬝ _,
-- apply ap eq_of_phomotopy,
-- refine _ ⬝ ap (pwhisker_left _) !dsmash_functor_phomotopy_refl⁻¹,
-- refine !pwhisker_left_refl⁻¹
-- end
-- definition dsmash_elim_inv_eq_of_phomotopy {f f' : ⋀ B →* C} (p : f ~* f') :
-- ap dsmash_elim_inv (eq_of_phomotopy p) = eq_of_phomotopy (dsmash_elim_inv_phomotopy p) :=
-- begin
-- induction p using phomotopy_rec_idp,
-- refine ap02 _ !eq_of_phomotopy_refl ⬝ _,
-- refine !eq_of_phomotopy_refl⁻¹ ⬝ _,
-- apply ap eq_of_phomotopy,
-- refine _ ⬝ ap (pwhisker_right _) !ppcompose_left_phomotopy_refl⁻¹,
-- refine !pwhisker_right_refl⁻¹
-- end
/- The pointed maps of the equivalence A →* (B →* C) ≃* ⋀ B →* C -/
-- definition dsmash_pelim : (Π*(a : A), ppmap (B a) C) →* ppmap (⋀ B) C :=
-- ppcompose_left (smash.smash_pmap_counit (B pt) C) ∘* sorry
-- -- ppcompose_left (smash_pmap_counit B C) ∘* dsmash_functor_left A (ppmap B C) B
-- open smash
-- definition smash_pelim_inv (B : A → Type*) (C : Type*) :
-- ppmap (A ∧ B) C →* ppmap A (ppmap B C) :=
-- ppcompose_right (smash_pmap_unit A B) ∘* pppcompose B (A ∧ B) C
-- -- definition smash_pelim_inv : ppmap (⋀ B) C →* Π*a, ppmap (B a) C :=
-- -- _ ∘* pppcompose _ (⋀ B) C --ppcompose_right (smash_pmap_unit A B) ∘* pppcompose B (A ∧ B) C
-- /- The forward function is natural in all three arguments -/
-- definition dsmash_pelim_natural_left (B C : Type*) (f : A' →* A) :
-- psquare (dsmash_pelim A B C) (dsmash_pelim A' B C)
-- (ppcompose_right f) (ppcompose_right (f ⋀→ pid B)) :=
-- dsmash_functor_left_natural_left (ppmap B C) B f ⬝h* !ppcompose_left_ppcompose_right
-- definition dsmash_pelim_natural_middle (A C : Type*) (f : B' →* B) :
-- psquare (dsmash_pelim A B C) (dsmash_pelim A B' C)
-- (ppcompose_left (ppcompose_right f)) (ppcompose_right (pid A ⋀→ f)) :=
-- pwhisker_tl _ !ppcompose_left_ppcompose_right ⬝*
-- (!dsmash_functor_left_natural_right⁻¹* ⬝pv*
-- dsmash_functor_left_natural_middle _ _ (ppcompose_right f) ⬝h*
-- ppcompose_left_psquare !dsmash_pmap_counit_natural_left)
-- definition dsmash_pelim_natural_right (B : A → Type*) (f : C →* C') :
-- psquare (dsmash_pelim A B C) (dsmash_pelim A B C')
-- (ppcompose_left (ppcompose_left f)) (ppcompose_left f) :=
-- dsmash_functor_left_natural_middle _ _ (ppcompose_left f) ⬝h*
-- ppcompose_left_psquare (dsmash_pmap_counit_natural_right _ f)
-- definition dsmash_pelim_natural_lm (C : Type*) (f : A' →* A) (g : B' →* B) :
-- psquare (dsmash_pelim A B C) (dsmash_pelim A' B' C)
-- (ppcompose_left (ppcompose_right g) ∘* ppcompose_right f) (ppcompose_right (f ⋀→ g)) :=
-- dsmash_pelim_natural_left B C f ⬝v* dsmash_pelim_natural_middle A' C g ⬝hp*
-- ppcompose_right_phomotopy (dsmash_functor_split f g) ⬝* !ppcompose_right_pcompose
-- definition dsmash_pelim_pid (B C : Type*) :
-- dsmash_pelim (ppmap B C) B C !pid ~* dsmash_pmap_counit B C :=
-- pwhisker_left _ !dsmash_functor_pid ⬝* !pcompose_pid
-- definition dsmash_pelim_inv_pid (B : A → Type*) :
-- dsmash_pelim_inv A B (⋀ B) !pid ~* dsmash_pmap_unit A B :=
-- pwhisker_right _ !ppcompose_left_pid ⬝* !pid_pcompose
-- /- The equivalence (note: the forward function of dsmash_adjoint_pmap is dsmash_pelim_inv) -/
-- definition is_equiv_dsmash_elim [constructor] (B : A → Type*) (C : Type*) : is_equiv (@dsmash_elim A B C) :=
-- begin
-- fapply adjointify,
-- { exact dsmash_elim_inv },
-- { intro g, apply eq_of_phomotopy, exact dsmash_elim_right_inv g },
-- { intro f, apply eq_of_phomotopy, exact dsmash_elim_left_inv f }
-- end
-- definition dsmash_adjoint_pmap_inv [constructor] (B : A → Type*) (C : Type*) :
-- ppmap A (ppmap B C) ≃* ppmap (⋀ B) C :=
-- pequiv_of_pmap (dsmash_pelim A B C) (is_equiv_dsmash_elim A B C)
definition dsmash_pelim_fn_fn [constructor] (f : ⋀ B →* C) (a : A) : B a →* C :=
pmap.mk (λb, f (dsmash.mk a b)) (ap f (gluel' a pt) ⬝ respect_pt f)
definition dsmash_pelim_fn [constructor] (f : ⋀ B →* C) : dbpmap B (λa, C) :=
begin
fapply dbpmap.mk (dsmash_pelim_fn_fn f),
{ intro b, exact ap f (gluer' b pt) ⬝ respect_pt f },
{ apply whisker_right, apply ap02, exact !con.right_inv ⬝ !con.right_inv⁻¹ }
end
definition dsmash_pelim_pmap [constructor] (B : A → Type*) (C : Type*) :
ppmap (⋀ B) C →* dbppmap B (λa, C) :=
begin
apply pmap.mk dsmash_pelim_fn,
fapply dbpmap_eq,
{ intro a, exact phomotopy.mk homotopy.rfl !ap_constant⁻¹ },
{ intro b, exact !ap_constant ⬝pv ids },
{ esimp, esimp [whisker_right], exact sorry }
end
definition dsmash_pelim_pequiv [constructor] (B : A → Type*) (C : Type*) :
ppmap (⋀ B) C ≃* dbppmap B (λa, C) :=
sorry
set_option pp.binder_types true
open is_trunc
/- we could also use pushout_arrow_equiv -/
definition dsmash_arrow_equiv [constructor] (B : A → Type*) (C : Type) :
(⋀ B → C) ≃ Σ(f : Πa, B a → C) (c₁ : C) (c₀ : C), (Πa, f a pt = c₀) × (Πb, f pt b = c₁) :=
begin
fapply equiv.MK,
{ intro f, exact ⟨λa b, f (dsmash.mk a b), f auxr, f auxl, (λa, ap f (gluel a), λb, ap f (gluer b))⟩ },
{ intro x, exact dsmash.elim x.1 x.2.2.1 x.2.1 x.2.2.2.1 x.2.2.2.2 },
{ intro x, induction x with f x, induction x with c₁ x, induction x with c₀ x, induction x with p₁ p₂,
apply ap (dpair _), apply ap (dpair _), apply ap (dpair _), esimp,
exact pair_eq (eq_of_homotopy (elim_gluel _ _)) (eq_of_homotopy (elim_gluer _ _)) },
{ intro f, apply eq_of_homotopy, intro x, induction x, reflexivity, reflexivity, reflexivity,
apply eq_pathover, apply hdeg_square, apply elim_gluel,
apply eq_pathover, apply hdeg_square, apply elim_gluer }
end
definition dsmash_arrow_equiv_inv_pt [constructor]
(x : Σ(f : Πa, B a → C) (c₁ : C) (c₀ : C), (Πa, f a pt = c₀) × (Πb, f pt b = c₁)) :
to_inv (dsmash_arrow_equiv B C) x pt = x.1 pt pt :=
by reflexivity
open pi
definition dsmash_pelim_equiv (B : A → Type*) (C : Type*) :
ppmap (⋀ B) C ≃ dbppmap B (λa, C) :=
begin
refine !pmap.sigma_char ⬝e _,
refine sigma_equiv_sigma_left !dsmash_arrow_equiv ⬝e _,
refine sigma_equiv_sigma_right (λx, equiv_eq_closed_left _ (dsmash_arrow_equiv_inv_pt x)) ⬝e _,
refine !sigma_assoc_equiv ⬝e _,
refine sigma_equiv_sigma_right (λf, !sigma_assoc_equiv ⬝e
sigma_equiv_sigma_right (λc₁, !sigma_assoc_equiv ⬝e
sigma_equiv_sigma_right (λc₂, sigma_equiv_sigma_left !sigma.equiv_prod⁻¹ᵉ ⬝e
!sigma_assoc_equiv) ⬝e
sigma_assoc_equiv_left _ (sigma_homotopy_constant_equiv pt (λa, f a pt))⁻¹ᵉ ⬝e
sigma_equiv_sigma_right (λh₁, !sigma_comm_equiv) ⬝e !sigma_comm_equiv) ⬝e
sigma_assoc_equiv_left _ (sigma_homotopy_constant_equiv pt (f pt))⁻¹ᵉ ⬝e
sigma_equiv_sigma_right (λh₂,
sigma_equiv_sigma_right (λr₂,
sigma_equiv_sigma_right (λh₁, !comm_equiv_constant) ⬝e !sigma_comm_equiv) ⬝e
!sigma_comm_equiv) ⬝e
!sigma_comm_equiv ⬝e
sigma_equiv_sigma_right (λr,
sigma_equiv_sigma_left (pi_equiv_pi_right (λb, equiv_eq_closed_right _ r)) ⬝e
sigma_equiv_sigma_right (λh₂,
sigma_equiv_sigma !eq_equiv_con_inv_eq_idp⁻¹ᵉ (λr₂,
sigma_equiv_sigma_left (pi_equiv_pi_right (λa, equiv_eq_closed_right _ r)) ⬝e
sigma_equiv_sigma_right (λh₁, !eq_equiv_con_inv_eq_idp⁻¹ᵉ)) ⬝e
!sigma_comm_equiv ⬝e
sigma_equiv_sigma_right (λh₁, !comm_equiv_constant)) ⬝e
!sigma_comm_equiv) ⬝e
!sigma_comm_equiv ⬝e sigma_equiv_sigma_right (λh₁,
!sigma_comm_equiv ⬝e sigma_equiv_sigma_right (λh₂,
!sigma_sigma_eq_right))) ⬝e _,
refine !sigma_assoc_equiv' ⬝e _,
refine sigma_equiv_sigma_left (@sigma_pi_equiv_pi_sigma _ _ (λa f, f pt = pt) ⬝e
pi_equiv_pi_right (λa, !pmap.sigma_char⁻¹ᵉ)) ⬝e _,
exact !dbpmap.sigma_char⁻¹ᵉ
end
definition dsmash_pelim_equiv' (B : A → Type*) (C : Type*) :
ppmap (⋀ B) C ≃ Π*a, B a →** C :=
dsmash_pelim_equiv B C ⬝e (ppi_equiv_dbpmap B (λa, C))⁻¹ᵉ
end dsmash
|
e0acbe899a9531d2828d40d83ab249ac49cefe9f | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/measure_theory/function/lp_space.lean | 48991e413c3b58c27a5fd8082438b732512fadb9 | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 127,210 | lean | /-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import analysis.normed_space.indicator_function
import analysis.normed.group.hom
import measure_theory.function.ess_sup
import measure_theory.function.ae_eq_fun
import measure_theory.integral.mean_inequalities
import topology.continuous_function.compact
/-!
# ℒp space and Lp space
This file describes properties of almost everywhere strongly measurable functions with finite
seminorm, denoted by `snorm f p μ` and defined for `p:ℝ≥0∞` as `0` if `p=0`,
`(∫ ∥f a∥^p ∂μ) ^ (1/p)` for `0 < p < ∞` and `ess_sup ∥f∥ μ` for `p=∞`.
The Prop-valued `mem_ℒp f p μ` states that a function `f : α → E` has finite seminorm.
The space `Lp E p μ` is the subtype of elements of `α →ₘ[μ] E` (see ae_eq_fun) such that
`snorm f p μ` is finite. For `1 ≤ p`, `snorm` defines a norm and `Lp` is a complete metric space.
## Main definitions
* `snorm' f p μ` : `(∫ ∥f a∥^p ∂μ) ^ (1/p)` for `f : α → F` and `p : ℝ`, where `α` is a measurable
space and `F` is a normed group.
* `snorm_ess_sup f μ` : seminorm in `ℒ∞`, equal to the essential supremum `ess_sup ∥f∥ μ`.
* `snorm f p μ` : for `p : ℝ≥0∞`, seminorm in `ℒp`, equal to `0` for `p=0`, to `snorm' f p μ`
for `0 < p < ∞` and to `snorm_ess_sup f μ` for `p = ∞`.
* `mem_ℒp f p μ` : property that the function `f` is almost everywhere strongly measurable and has
finite `p`-seminorm for the measure `μ` (`snorm f p μ < ∞`)
* `Lp E p μ` : elements of `α →ₘ[μ] E` (see ae_eq_fun) such that `snorm f p μ` is finite. Defined
as an `add_subgroup` of `α →ₘ[μ] E`.
Lipschitz functions vanishing at zero act by composition on `Lp`. We define this action, and prove
that it is continuous. In particular,
* `continuous_linear_map.comp_Lp` defines the action on `Lp` of a continuous linear map.
* `Lp.pos_part` is the positive part of an `Lp` function.
* `Lp.neg_part` is the negative part of an `Lp` function.
When `α` is a topological space equipped with a finite Borel measure, there is a bounded linear map
from the normed space of bounded continuous functions (`α →ᵇ E`) to `Lp E p μ`. We construct this
as `bounded_continuous_function.to_Lp`.
## Notations
* `α →₁[μ] E` : the type `Lp E 1 μ`.
* `α →₂[μ] E` : the type `Lp E 2 μ`.
## Implementation
Since `Lp` is defined as an `add_subgroup`, dot notation does not work. Use `Lp.measurable f` to
say that the coercion of `f` to a genuine function is measurable, instead of the non-working
`f.measurable`.
To prove that two `Lp` elements are equal, it suffices to show that their coercions to functions
coincide almost everywhere (this is registered as an `ext` rule). This can often be done using
`filter_upwards`. For instance, a proof from first principles that `f + (g + h) = (f + g) + h`
could read (in the `Lp` namespace)
```
example (f g h : Lp E p μ) : (f + g) + h = f + (g + h) :=
begin
ext1,
filter_upwards [coe_fn_add (f + g) h, coe_fn_add f g, coe_fn_add f (g + h), coe_fn_add g h]
with _ ha1 ha2 ha3 ha4,
simp only [ha1, ha2, ha3, ha4, add_assoc],
end
```
The lemma `coe_fn_add` states that the coercion of `f + g` coincides almost everywhere with the sum
of the coercions of `f` and `g`. All such lemmas use `coe_fn` in their name, to distinguish the
function coercion from the coercion to almost everywhere defined functions.
-/
noncomputable theory
open topological_space measure_theory filter
open_locale nnreal ennreal big_operators topological_space measure_theory
variables {α E F G : Type*} {m m0 : measurable_space α} {p : ℝ≥0∞} {q : ℝ} {μ ν : measure α}
[normed_add_comm_group E] [normed_add_comm_group F] [normed_add_comm_group G]
namespace measure_theory
section ℒp
/-!
### ℒp seminorm
We define the ℒp seminorm, denoted by `snorm f p μ`. For real `p`, it is given by an integral
formula (for which we use the notation `snorm' f p μ`), and for `p = ∞` it is the essential
supremum (for which we use the notation `snorm_ess_sup f μ`).
We also define a predicate `mem_ℒp f p μ`, requesting that a function is almost everywhere
measurable and has finite `snorm f p μ`.
This paragraph is devoted to the basic properties of these definitions. It is constructed as
follows: for a given property, we prove it for `snorm'` and `snorm_ess_sup` when it makes sense,
deduce it for `snorm`, and translate it in terms of `mem_ℒp`.
-/
section ℒp_space_definition
/-- `(∫ ∥f a∥^q ∂μ) ^ (1/q)`, which is a seminorm on the space of measurable functions for which
this quantity is finite -/
def snorm' {m : measurable_space α} (f : α → F) (q : ℝ) (μ : measure α) : ℝ≥0∞ :=
(∫⁻ a, ∥f a∥₊^q ∂μ) ^ (1/q)
/-- seminorm for `ℒ∞`, equal to the essential supremum of `∥f∥`. -/
def snorm_ess_sup {m : measurable_space α} (f : α → F) (μ : measure α) :=
ess_sup (λ x, (∥f x∥₊ : ℝ≥0∞)) μ
/-- `ℒp` seminorm, equal to `0` for `p=0`, to `(∫ ∥f a∥^p ∂μ) ^ (1/p)` for `0 < p < ∞` and to
`ess_sup ∥f∥ μ` for `p = ∞`. -/
def snorm {m : measurable_space α} (f : α → F) (p : ℝ≥0∞) (μ : measure α) : ℝ≥0∞ :=
if p = 0 then 0 else (if p = ∞ then snorm_ess_sup f μ else snorm' f (ennreal.to_real p) μ)
lemma snorm_eq_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = snorm' f (ennreal.to_real p) μ :=
by simp [snorm, hp_ne_zero, hp_ne_top]
lemma snorm_eq_lintegral_rpow_nnnorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = (∫⁻ x, ∥f x∥₊ ^ p.to_real ∂μ) ^ (1 / p.to_real) :=
by rw [snorm_eq_snorm' hp_ne_zero hp_ne_top, snorm']
lemma snorm_one_eq_lintegral_nnnorm {f : α → F} : snorm f 1 μ = ∫⁻ x, ∥f x∥₊ ∂μ :=
by simp_rw [snorm_eq_lintegral_rpow_nnnorm one_ne_zero ennreal.coe_ne_top, ennreal.one_to_real,
one_div_one, ennreal.rpow_one]
@[simp] lemma snorm_exponent_top {f : α → F} : snorm f ∞ μ = snorm_ess_sup f μ := by simp [snorm]
/-- The property that `f:α→E` is ae strongly measurable and `(∫ ∥f a∥^p ∂μ)^(1/p)` is finite
if `p < ∞`, or `ess_sup f < ∞` if `p = ∞`. -/
def mem_ℒp {α} {m : measurable_space α}
(f : α → E) (p : ℝ≥0∞) (μ : measure α . volume_tac) : Prop :=
ae_strongly_measurable f μ ∧ snorm f p μ < ∞
lemma mem_ℒp.ae_strongly_measurable {f : α → E} {p : ℝ≥0∞} (h : mem_ℒp f p μ) :
ae_strongly_measurable f μ := h.1
lemma lintegral_rpow_nnnorm_eq_rpow_snorm' {f : α → F} (hq0_lt : 0 < q) :
∫⁻ a, ∥f a∥₊ ^ q ∂μ = (snorm' f q μ) ^ q :=
begin
rw [snorm', ←ennreal.rpow_mul, one_div, inv_mul_cancel, ennreal.rpow_one],
exact (ne_of_lt hq0_lt).symm,
end
end ℒp_space_definition
section top
lemma mem_ℒp.snorm_lt_top {f : α → E} (hfp : mem_ℒp f p μ) : snorm f p μ < ∞ := hfp.2
lemma mem_ℒp.snorm_ne_top {f : α → E} (hfp : mem_ℒp f p μ) : snorm f p μ ≠ ∞ := ne_of_lt (hfp.2)
lemma lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top {f : α → F} (hq0_lt : 0 < q)
(hfq : snorm' f q μ < ∞) :
∫⁻ a, ∥f a∥₊ ^ q ∂μ < ∞ :=
begin
rw lintegral_rpow_nnnorm_eq_rpow_snorm' hq0_lt,
exact ennreal.rpow_lt_top_of_nonneg (le_of_lt hq0_lt) (ne_of_lt hfq),
end
lemma lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top {f : α → F} (hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ∞) (hfp : snorm f p μ < ∞) :
∫⁻ a, ∥f a∥₊ ^ p.to_real ∂μ < ∞ :=
begin
apply lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top,
{ exact ennreal.to_real_pos hp_ne_zero hp_ne_top },
{ simpa [snorm_eq_snorm' hp_ne_zero hp_ne_top] using hfp }
end
lemma snorm_lt_top_iff_lintegral_rpow_nnnorm_lt_top {f : α → F} (hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ∞) :
snorm f p μ < ∞ ↔ ∫⁻ a, ∥f a∥₊ ^ p.to_real ∂μ < ∞ :=
⟨lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top hp_ne_zero hp_ne_top,
begin
intros h,
have hp' := ennreal.to_real_pos hp_ne_zero hp_ne_top,
have : 0 < 1 / p.to_real := div_pos zero_lt_one hp',
simpa [snorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top] using
ennreal.rpow_lt_top_of_nonneg (le_of_lt this) (ne_of_lt h)
end⟩
end top
section zero
@[simp] lemma snorm'_exponent_zero {f : α → F} : snorm' f 0 μ = 1 :=
by rw [snorm', div_zero, ennreal.rpow_zero]
@[simp] lemma snorm_exponent_zero {f : α → F} : snorm f 0 μ = 0 :=
by simp [snorm]
lemma mem_ℒp_zero_iff_ae_strongly_measurable {f : α → E} :
mem_ℒp f 0 μ ↔ ae_strongly_measurable f μ :=
by simp [mem_ℒp, snorm_exponent_zero]
@[simp] lemma snorm'_zero (hp0_lt : 0 < q) : snorm' (0 : α → F) q μ = 0 :=
by simp [snorm', hp0_lt]
@[simp] lemma snorm'_zero' (hq0_ne : q ≠ 0) (hμ : μ ≠ 0) : snorm' (0 : α → F) q μ = 0 :=
begin
cases le_or_lt 0 q with hq0 hq_neg,
{ exact snorm'_zero (lt_of_le_of_ne hq0 hq0_ne.symm), },
{ simp [snorm', ennreal.rpow_eq_zero_iff, hμ, hq_neg], },
end
@[simp] lemma snorm_ess_sup_zero : snorm_ess_sup (0 : α → F) μ = 0 :=
begin
simp_rw [snorm_ess_sup, pi.zero_apply, nnnorm_zero, ennreal.coe_zero, ←ennreal.bot_eq_zero],
exact ess_sup_const_bot,
end
@[simp] lemma snorm_zero : snorm (0 : α → F) p μ = 0 :=
begin
by_cases h0 : p = 0,
{ simp [h0], },
by_cases h_top : p = ∞,
{ simp only [h_top, snorm_exponent_top, snorm_ess_sup_zero], },
rw ←ne.def at h0,
simp [snorm_eq_snorm' h0 h_top, ennreal.to_real_pos h0 h_top],
end
@[simp] lemma snorm_zero' : snorm (λ x : α, (0 : F)) p μ = 0 :=
by convert snorm_zero
lemma zero_mem_ℒp : mem_ℒp (0 : α → E) p μ :=
⟨ae_strongly_measurable_zero, by { rw snorm_zero, exact ennreal.coe_lt_top, } ⟩
lemma zero_mem_ℒp' : mem_ℒp (λ x : α, (0 : E)) p μ :=
by convert zero_mem_ℒp
variables [measurable_space α]
lemma snorm'_measure_zero_of_pos {f : α → F} (hq_pos : 0 < q) :
snorm' f q (0 : measure α) = 0 :=
by simp [snorm', hq_pos]
lemma snorm'_measure_zero_of_exponent_zero {f : α → F} : snorm' f 0 (0 : measure α) = 1 :=
by simp [snorm']
lemma snorm'_measure_zero_of_neg {f : α → F} (hq_neg : q < 0) : snorm' f q (0 : measure α) = ∞ :=
by simp [snorm', hq_neg]
@[simp] lemma snorm_ess_sup_measure_zero {f : α → F} : snorm_ess_sup f (0 : measure α) = 0 :=
by simp [snorm_ess_sup]
@[simp] lemma snorm_measure_zero {f : α → F} : snorm f p (0 : measure α) = 0 :=
begin
by_cases h0 : p = 0,
{ simp [h0], },
by_cases h_top : p = ∞,
{ simp [h_top], },
rw ←ne.def at h0,
simp [snorm_eq_snorm' h0 h_top, snorm', ennreal.to_real_pos h0 h_top],
end
end zero
section const
lemma snorm'_const (c : F) (hq_pos : 0 < q) :
snorm' (λ x : α , c) q μ = (∥c∥₊ : ℝ≥0∞) * (μ set.univ) ^ (1/q) :=
begin
rw [snorm', lintegral_const, ennreal.mul_rpow_of_nonneg _ _ (by simp [hq_pos.le] : 0 ≤ 1 / q)],
congr,
rw ←ennreal.rpow_mul,
suffices hq_cancel : q * (1/q) = 1, by rw [hq_cancel, ennreal.rpow_one],
rw [one_div, mul_inv_cancel (ne_of_lt hq_pos).symm],
end
lemma snorm'_const' [is_finite_measure μ] (c : F) (hc_ne_zero : c ≠ 0) (hq_ne_zero : q ≠ 0) :
snorm' (λ x : α , c) q μ = (∥c∥₊ : ℝ≥0∞) * (μ set.univ) ^ (1/q) :=
begin
rw [snorm', lintegral_const, ennreal.mul_rpow_of_ne_top _ (measure_ne_top μ set.univ)],
{ congr,
rw ←ennreal.rpow_mul,
suffices hp_cancel : q * (1/q) = 1, by rw [hp_cancel, ennreal.rpow_one],
rw [one_div, mul_inv_cancel hq_ne_zero], },
{ rw [ne.def, ennreal.rpow_eq_top_iff, not_or_distrib, not_and_distrib, not_and_distrib],
split,
{ left,
rwa [ennreal.coe_eq_zero, nnnorm_eq_zero], },
{ exact or.inl ennreal.coe_ne_top, }, },
end
lemma snorm_ess_sup_const (c : F) (hμ : μ ≠ 0) :
snorm_ess_sup (λ x : α, c) μ = (∥c∥₊ : ℝ≥0∞) :=
by rw [snorm_ess_sup, ess_sup_const _ hμ]
lemma snorm'_const_of_is_probability_measure (c : F) (hq_pos : 0 < q) [is_probability_measure μ] :
snorm' (λ x : α , c) q μ = (∥c∥₊ : ℝ≥0∞) :=
by simp [snorm'_const c hq_pos, measure_univ]
lemma snorm_const (c : F) (h0 : p ≠ 0) (hμ : μ ≠ 0) :
snorm (λ x : α , c) p μ = (∥c∥₊ : ℝ≥0∞) * (μ set.univ) ^ (1/(ennreal.to_real p)) :=
begin
by_cases h_top : p = ∞,
{ simp [h_top, snorm_ess_sup_const c hμ], },
simp [snorm_eq_snorm' h0 h_top, snorm'_const, ennreal.to_real_pos h0 h_top],
end
lemma snorm_const' (c : F) (h0 : p ≠ 0) (h_top: p ≠ ∞) :
snorm (λ x : α , c) p μ = (∥c∥₊ : ℝ≥0∞) * (μ set.univ) ^ (1/(ennreal.to_real p)) :=
begin
simp [snorm_eq_snorm' h0 h_top, snorm'_const, ennreal.to_real_pos h0 h_top],
end
lemma snorm_const_lt_top_iff {p : ℝ≥0∞} {c : F} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
snorm (λ x : α, c) p μ < ∞ ↔ c = 0 ∨ μ set.univ < ∞ :=
begin
have hp : 0 < p.to_real, from ennreal.to_real_pos hp_ne_zero hp_ne_top,
by_cases hμ : μ = 0,
{ simp only [hμ, measure.coe_zero, pi.zero_apply, or_true, with_top.zero_lt_top,
snorm_measure_zero], },
by_cases hc : c = 0,
{ simp only [hc, true_or, eq_self_iff_true, with_top.zero_lt_top, snorm_zero'], },
rw snorm_const' c hp_ne_zero hp_ne_top,
by_cases hμ_top : μ set.univ = ∞,
{ simp [hc, hμ_top, hp], },
rw ennreal.mul_lt_top_iff,
simp only [true_and, one_div, ennreal.rpow_eq_zero_iff, hμ, false_or, or_false,
ennreal.coe_lt_top, nnnorm_eq_zero, ennreal.coe_eq_zero,
measure_theory.measure.measure_univ_eq_zero, hp, inv_lt_zero, hc, and_false, false_and,
_root_.inv_pos, or_self, hμ_top, ne.lt_top hμ_top, iff_true],
exact ennreal.rpow_lt_top_of_nonneg (inv_nonneg.mpr hp.le) hμ_top,
end
lemma mem_ℒp_const (c : E) [is_finite_measure μ] : mem_ℒp (λ a:α, c) p μ :=
begin
refine ⟨ae_strongly_measurable_const, _⟩,
by_cases h0 : p = 0,
{ simp [h0], },
by_cases hμ : μ = 0,
{ simp [hμ], },
rw snorm_const c h0 hμ,
refine ennreal.mul_lt_top ennreal.coe_ne_top _,
refine (ennreal.rpow_lt_top_of_nonneg _ (measure_ne_top μ set.univ)).ne,
simp,
end
lemma mem_ℒp_top_const (c : E) : mem_ℒp (λ a:α, c) ∞ μ :=
begin
refine ⟨ae_strongly_measurable_const, _⟩,
by_cases h : μ = 0,
{ simp only [h, snorm_measure_zero, with_top.zero_lt_top] },
{ rw snorm_const _ ennreal.top_ne_zero h,
simp only [ennreal.top_to_real, div_zero, ennreal.rpow_zero, mul_one, ennreal.coe_lt_top] }
end
lemma mem_ℒp_const_iff {p : ℝ≥0∞} {c : E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
mem_ℒp (λ x : α, c) p μ ↔ c = 0 ∨ μ set.univ < ∞ :=
begin
rw ← snorm_const_lt_top_iff hp_ne_zero hp_ne_top,
exact ⟨λ h, h.2, λ h, ⟨ae_strongly_measurable_const, h⟩⟩,
end
end const
lemma snorm'_mono_ae {f : α → F} {g : α → G} (hq : 0 ≤ q) (h : ∀ᵐ x ∂μ, ∥f x∥ ≤ ∥g x∥) :
snorm' f q μ ≤ snorm' g q μ :=
begin
rw [snorm'],
refine ennreal.rpow_le_rpow _ (one_div_nonneg.2 hq),
refine lintegral_mono_ae (h.mono $ λ x hx, _),
exact ennreal.rpow_le_rpow (ennreal.coe_le_coe.2 hx) hq
end
lemma snorm'_congr_norm_ae {f g : α → F} (hfg : ∀ᵐ x ∂μ, ∥f x∥ = ∥g x∥) :
snorm' f q μ = snorm' g q μ :=
begin
have : (λ x, (∥f x∥₊ ^ q : ℝ≥0∞)) =ᵐ[μ] (λ x, ∥g x∥₊ ^ q),
from hfg.mono (λ x hx, by { simp only [← coe_nnnorm, nnreal.coe_eq] at hx, simp [hx] }),
simp only [snorm', lintegral_congr_ae this]
end
lemma snorm'_congr_ae {f g : α → F} (hfg : f =ᵐ[μ] g) : snorm' f q μ = snorm' g q μ :=
snorm'_congr_norm_ae (hfg.fun_comp _)
lemma snorm_ess_sup_congr_ae {f g : α → F} (hfg : f =ᵐ[μ] g) :
snorm_ess_sup f μ = snorm_ess_sup g μ :=
ess_sup_congr_ae (hfg.fun_comp (coe ∘ nnnorm))
lemma snorm_mono_ae {f : α → F} {g : α → G} (h : ∀ᵐ x ∂μ, ∥f x∥ ≤ ∥g x∥) :
snorm f p μ ≤ snorm g p μ :=
begin
simp only [snorm],
split_ifs,
{ exact le_rfl },
{ refine ess_sup_mono_ae (h.mono $ λ x hx, _),
exact_mod_cast hx },
{ exact snorm'_mono_ae ennreal.to_real_nonneg h }
end
lemma snorm_mono_ae_real {f : α → F} {g : α → ℝ} (h : ∀ᵐ x ∂μ, ∥f x∥ ≤ g x) :
snorm f p μ ≤ snorm g p μ :=
snorm_mono_ae $ h.mono (λ x hx, hx.trans ((le_abs_self _).trans (real.norm_eq_abs _).symm.le))
lemma snorm_mono {f : α → F} {g : α → G} (h : ∀ x, ∥f x∥ ≤ ∥g x∥) :
snorm f p μ ≤ snorm g p μ :=
snorm_mono_ae (eventually_of_forall (λ x, h x))
lemma snorm_mono_real {f : α → F} {g : α → ℝ} (h : ∀ x, ∥f x∥ ≤ g x) :
snorm f p μ ≤ snorm g p μ :=
snorm_mono_ae_real (eventually_of_forall (λ x, h x))
lemma snorm_ess_sup_le_of_ae_bound {f : α → F} {C : ℝ} (hfC : ∀ᵐ x ∂μ, ∥f x∥ ≤ C) :
snorm_ess_sup f μ ≤ ennreal.of_real C:=
begin
simp_rw [snorm_ess_sup, ← of_real_norm_eq_coe_nnnorm],
refine ess_sup_le_of_ae_le (ennreal.of_real C) (hfC.mono (λ x hx, _)),
exact ennreal.of_real_le_of_real hx,
end
lemma snorm_ess_sup_lt_top_of_ae_bound {f : α → F} {C : ℝ} (hfC : ∀ᵐ x ∂μ, ∥f x∥ ≤ C) :
snorm_ess_sup f μ < ∞ :=
(snorm_ess_sup_le_of_ae_bound hfC).trans_lt ennreal.of_real_lt_top
lemma snorm_le_of_ae_bound {f : α → F} {C : ℝ} (hfC : ∀ᵐ x ∂μ, ∥f x∥ ≤ C) :
snorm f p μ ≤ ((μ set.univ) ^ p.to_real⁻¹) * (ennreal.of_real C) :=
begin
by_cases hμ : μ = 0,
{ simp [hμ] },
haveI : μ.ae.ne_bot := ae_ne_bot.mpr hμ,
by_cases hp : p = 0,
{ simp [hp] },
have hC : 0 ≤ C, from le_trans (norm_nonneg _) hfC.exists.some_spec,
have hC' : ∥C∥ = C := by rw [real.norm_eq_abs, abs_eq_self.mpr hC],
have : ∀ᵐ x ∂μ, ∥f x∥ ≤ ∥(λ _, C) x∥, from hfC.mono (λ x hx, hx.trans (le_of_eq hC'.symm)),
convert snorm_mono_ae this,
rw [snorm_const _ hp hμ, mul_comm, ← of_real_norm_eq_coe_nnnorm, hC', one_div]
end
lemma snorm_congr_norm_ae {f : α → F} {g : α → G} (hfg : ∀ᵐ x ∂μ, ∥f x∥ = ∥g x∥) :
snorm f p μ = snorm g p μ :=
le_antisymm (snorm_mono_ae $ eventually_eq.le hfg)
(snorm_mono_ae $ (eventually_eq.symm hfg).le)
@[simp] lemma snorm'_norm {f : α → F} : snorm' (λ a, ∥f a∥) q μ = snorm' f q μ :=
by simp [snorm']
@[simp] lemma snorm_norm (f : α → F) : snorm (λ x, ∥f x∥) p μ = snorm f p μ :=
snorm_congr_norm_ae $ eventually_of_forall $ λ x, norm_norm _
lemma snorm'_norm_rpow (f : α → F) (p q : ℝ) (hq_pos : 0 < q) :
snorm' (λ x, ∥f x∥ ^ q) p μ = (snorm' f (p * q) μ) ^ q :=
begin
simp_rw snorm',
rw [← ennreal.rpow_mul, ←one_div_mul_one_div],
simp_rw one_div,
rw [mul_assoc, inv_mul_cancel hq_pos.ne.symm, mul_one],
congr,
ext1 x,
simp_rw ← of_real_norm_eq_coe_nnnorm,
rw [real.norm_eq_abs, abs_eq_self.mpr (real.rpow_nonneg_of_nonneg (norm_nonneg _) _),
mul_comm, ← ennreal.of_real_rpow_of_nonneg (norm_nonneg _) hq_pos.le, ennreal.rpow_mul],
end
lemma snorm_norm_rpow (f : α → F) (hq_pos : 0 < q) :
snorm (λ x, ∥f x∥ ^ q) p μ = (snorm f (p * ennreal.of_real q) μ) ^ q :=
begin
by_cases h0 : p = 0,
{ simp [h0, ennreal.zero_rpow_of_pos hq_pos], },
by_cases hp_top : p = ∞,
{ simp only [hp_top, snorm_exponent_top, ennreal.top_mul, hq_pos.not_le, ennreal.of_real_eq_zero,
if_false, snorm_exponent_top, snorm_ess_sup],
have h_rpow : ess_sup (λ (x : α), (∥(∥f x∥ ^ q)∥₊ : ℝ≥0∞)) μ
= ess_sup (λ (x : α), (↑∥f x∥₊) ^ q) μ,
{ congr,
ext1 x,
nth_rewrite 1 ← nnnorm_norm,
rw [ennreal.coe_rpow_of_nonneg _ hq_pos.le, ennreal.coe_eq_coe],
ext,
push_cast,
rw real.norm_rpow_of_nonneg (norm_nonneg _), },
rw h_rpow,
have h_rpow_mono := ennreal.strict_mono_rpow_of_pos hq_pos,
have h_rpow_surj := (ennreal.rpow_left_bijective hq_pos.ne.symm).2,
let iso := h_rpow_mono.order_iso_of_surjective _ h_rpow_surj,
exact (iso.ess_sup_apply (λ x, (∥f x∥₊ : ℝ≥0∞)) μ).symm, },
rw [snorm_eq_snorm' h0 hp_top, snorm_eq_snorm' _ _],
swap, { refine mul_ne_zero h0 _, rwa [ne.def, ennreal.of_real_eq_zero, not_le], },
swap, { exact ennreal.mul_ne_top hp_top ennreal.of_real_ne_top, },
rw [ennreal.to_real_mul, ennreal.to_real_of_real hq_pos.le],
exact snorm'_norm_rpow f p.to_real q hq_pos,
end
lemma snorm_congr_ae {f g : α → F} (hfg : f =ᵐ[μ] g) : snorm f p μ = snorm g p μ :=
snorm_congr_norm_ae $ hfg.mono (λ x hx, hx ▸ rfl)
lemma mem_ℒp_congr_ae {f g : α → E} (hfg : f =ᵐ[μ] g) : mem_ℒp f p μ ↔ mem_ℒp g p μ :=
by simp only [mem_ℒp, snorm_congr_ae hfg, ae_strongly_measurable_congr hfg]
lemma mem_ℒp.ae_eq {f g : α → E} (hfg : f =ᵐ[μ] g) (hf_Lp : mem_ℒp f p μ) : mem_ℒp g p μ :=
(mem_ℒp_congr_ae hfg).1 hf_Lp
lemma mem_ℒp.of_le {f : α → E} {g : α → F}
(hg : mem_ℒp g p μ) (hf : ae_strongly_measurable f μ) (hfg : ∀ᵐ x ∂μ, ∥f x∥ ≤ ∥g x∥) :
mem_ℒp f p μ :=
⟨hf, (snorm_mono_ae hfg).trans_lt hg.snorm_lt_top⟩
alias mem_ℒp.of_le ← mem_ℒp.mono
lemma mem_ℒp.mono' {f : α → E} {g : α → ℝ} (hg : mem_ℒp g p μ)
(hf : ae_strongly_measurable f μ) (h : ∀ᵐ a ∂μ, ∥f a∥ ≤ g a) : mem_ℒp f p μ :=
hg.mono hf $ h.mono $ λ x hx, le_trans hx (le_abs_self _)
lemma mem_ℒp.congr_norm {f : α → E} {g : α → F} (hf : mem_ℒp f p μ)
(hg : ae_strongly_measurable g μ) (h : ∀ᵐ a ∂μ, ∥f a∥ = ∥g a∥) :
mem_ℒp g p μ :=
hf.mono hg $ eventually_eq.le $ eventually_eq.symm h
lemma mem_ℒp_congr_norm {f : α → E} {g : α → F}
(hf : ae_strongly_measurable f μ) (hg : ae_strongly_measurable g μ) (h : ∀ᵐ a ∂μ, ∥f a∥ = ∥g a∥) :
mem_ℒp f p μ ↔ mem_ℒp g p μ :=
⟨λ h2f, h2f.congr_norm hg h, λ h2g, h2g.congr_norm hf $ eventually_eq.symm h⟩
lemma mem_ℒp_top_of_bound {f : α → E} (hf : ae_strongly_measurable f μ) (C : ℝ)
(hfC : ∀ᵐ x ∂μ, ∥f x∥ ≤ C) :
mem_ℒp f ∞ μ :=
⟨hf, by { rw snorm_exponent_top, exact snorm_ess_sup_lt_top_of_ae_bound hfC, }⟩
lemma mem_ℒp.of_bound [is_finite_measure μ] {f : α → E} (hf : ae_strongly_measurable f μ)
(C : ℝ) (hfC : ∀ᵐ x ∂μ, ∥f x∥ ≤ C) :
mem_ℒp f p μ :=
(mem_ℒp_const C).of_le hf (hfC.mono (λ x hx, le_trans hx (le_abs_self _)))
@[mono] lemma snorm'_mono_measure (f : α → F) (hμν : ν ≤ μ) (hq : 0 ≤ q) :
snorm' f q ν ≤ snorm' f q μ :=
begin
simp_rw snorm',
suffices h_integral_mono : (∫⁻ a, (∥f a∥₊ : ℝ≥0∞) ^ q ∂ν) ≤ ∫⁻ a, ∥f a∥₊ ^ q ∂μ,
from ennreal.rpow_le_rpow h_integral_mono (by simp [hq]),
exact lintegral_mono' hμν le_rfl,
end
@[mono] lemma snorm_ess_sup_mono_measure (f : α → F) (hμν : ν ≪ μ) :
snorm_ess_sup f ν ≤ snorm_ess_sup f μ :=
by { simp_rw snorm_ess_sup, exact ess_sup_mono_measure hμν, }
@[mono] lemma snorm_mono_measure (f : α → F) (hμν : ν ≤ μ) :
snorm f p ν ≤ snorm f p μ :=
begin
by_cases hp0 : p = 0,
{ simp [hp0], },
by_cases hp_top : p = ∞,
{ simp [hp_top, snorm_ess_sup_mono_measure f (measure.absolutely_continuous_of_le hμν)], },
simp_rw snorm_eq_snorm' hp0 hp_top,
exact snorm'_mono_measure f hμν ennreal.to_real_nonneg,
end
lemma mem_ℒp.mono_measure {f : α → E} (hμν : ν ≤ μ) (hf : mem_ℒp f p μ) :
mem_ℒp f p ν :=
⟨hf.1.mono_measure hμν, (snorm_mono_measure f hμν).trans_lt hf.2⟩
lemma mem_ℒp.restrict (s : set α) {f : α → E} (hf : mem_ℒp f p μ) :
mem_ℒp f p (μ.restrict s) :=
hf.mono_measure measure.restrict_le_self
lemma snorm'_smul_measure {p : ℝ} (hp : 0 ≤ p) {f : α → F} (c : ℝ≥0∞) :
snorm' f p (c • μ) = c ^ (1 / p) * snorm' f p μ :=
by { rw [snorm', lintegral_smul_measure, ennreal.mul_rpow_of_nonneg, snorm'], simp [hp], }
lemma snorm_ess_sup_smul_measure {f : α → F} {c : ℝ≥0∞} (hc : c ≠ 0) :
snorm_ess_sup f (c • μ) = snorm_ess_sup f μ :=
by { simp_rw [snorm_ess_sup], exact ess_sup_smul_measure hc, }
/-- Use `snorm_smul_measure_of_ne_top` instead. -/
private lemma snorm_smul_measure_of_ne_zero_of_ne_top {p : ℝ≥0∞} (hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ∞) {f : α → F} (c : ℝ≥0∞) :
snorm f p (c • μ) = c ^ (1 / p).to_real • snorm f p μ :=
begin
simp_rw snorm_eq_snorm' hp_ne_zero hp_ne_top,
rw snorm'_smul_measure ennreal.to_real_nonneg,
congr,
simp_rw one_div,
rw ennreal.to_real_inv,
end
lemma snorm_smul_measure_of_ne_zero {p : ℝ≥0∞} {f : α → F} {c : ℝ≥0∞} (hc : c ≠ 0) :
snorm f p (c • μ) = c ^ (1 / p).to_real • snorm f p μ :=
begin
by_cases hp0 : p = 0,
{ simp [hp0], },
by_cases hp_top : p = ∞,
{ simp [hp_top, snorm_ess_sup_smul_measure hc], },
exact snorm_smul_measure_of_ne_zero_of_ne_top hp0 hp_top c,
end
lemma snorm_smul_measure_of_ne_top {p : ℝ≥0∞} (hp_ne_top : p ≠ ∞) {f : α → F} (c : ℝ≥0∞) :
snorm f p (c • μ) = c ^ (1 / p).to_real • snorm f p μ :=
begin
by_cases hp0 : p = 0,
{ simp [hp0], },
{ exact snorm_smul_measure_of_ne_zero_of_ne_top hp0 hp_ne_top c, },
end
lemma snorm_one_smul_measure {f : α → F} (c : ℝ≥0∞) :
snorm f 1 (c • μ) = c * snorm f 1 μ :=
by { rw @snorm_smul_measure_of_ne_top _ _ _ μ _ 1 (@ennreal.coe_ne_top 1) f c, simp, }
lemma mem_ℒp.of_measure_le_smul {μ' : measure α} (c : ℝ≥0∞) (hc : c ≠ ∞)
(hμ'_le : μ' ≤ c • μ) {f : α → E} (hf : mem_ℒp f p μ) :
mem_ℒp f p μ' :=
begin
refine ⟨hf.1.mono' (measure.absolutely_continuous_of_le_smul hμ'_le), _⟩,
refine (snorm_mono_measure f hμ'_le).trans_lt _,
by_cases hc0 : c = 0,
{ simp [hc0], },
rw [snorm_smul_measure_of_ne_zero hc0, smul_eq_mul],
refine ennreal.mul_lt_top _ hf.2.ne,
simp [hc, hc0],
end
lemma mem_ℒp.smul_measure {f : α → E} {c : ℝ≥0∞} (hf : mem_ℒp f p μ) (hc : c ≠ ∞) :
mem_ℒp f p (c • μ) :=
hf.of_measure_le_smul c hc le_rfl
include m
lemma snorm_one_add_measure (f : α → F) (μ ν : measure α) :
snorm f 1 (μ + ν) = snorm f 1 μ + snorm f 1 ν :=
by { simp_rw snorm_one_eq_lintegral_nnnorm, rw lintegral_add_measure _ μ ν, }
lemma snorm_le_add_measure_right (f : α → F) (μ ν : measure α) {p : ℝ≥0∞} :
snorm f p μ ≤ snorm f p (μ + ν) :=
snorm_mono_measure f $ measure.le_add_right $ le_refl _
lemma snorm_le_add_measure_left (f : α → F) (μ ν : measure α) {p : ℝ≥0∞} :
snorm f p ν ≤ snorm f p (μ + ν) :=
snorm_mono_measure f $ measure.le_add_left $ le_refl _
omit m
lemma mem_ℒp.left_of_add_measure {f : α → E} (h : mem_ℒp f p (μ + ν)) : mem_ℒp f p μ :=
h.mono_measure $ measure.le_add_right $ le_refl _
lemma mem_ℒp.right_of_add_measure {f : α → E} (h : mem_ℒp f p (μ + ν)) : mem_ℒp f p ν :=
h.mono_measure $ measure.le_add_left $ le_refl _
lemma mem_ℒp.norm {f : α → E} (h : mem_ℒp f p μ) : mem_ℒp (λ x, ∥f x∥) p μ :=
h.of_le h.ae_strongly_measurable.norm (eventually_of_forall (λ x, by simp))
lemma mem_ℒp_norm_iff {f : α → E} (hf : ae_strongly_measurable f μ) :
mem_ℒp (λ x, ∥f x∥) p μ ↔ mem_ℒp f p μ :=
⟨λ h, ⟨hf, by { rw ← snorm_norm, exact h.2, }⟩, λ h, h.norm⟩
lemma snorm'_eq_zero_of_ae_zero {f : α → F} (hq0_lt : 0 < q) (hf_zero : f =ᵐ[μ] 0) :
snorm' f q μ = 0 :=
by rw [snorm'_congr_ae hf_zero, snorm'_zero hq0_lt]
lemma snorm'_eq_zero_of_ae_zero' (hq0_ne : q ≠ 0) (hμ : μ ≠ 0) {f : α → F} (hf_zero : f =ᵐ[μ] 0) :
snorm' f q μ = 0 :=
by rw [snorm'_congr_ae hf_zero, snorm'_zero' hq0_ne hμ]
lemma ae_eq_zero_of_snorm'_eq_zero {f : α → E} (hq0 : 0 ≤ q) (hf : ae_strongly_measurable f μ)
(h : snorm' f q μ = 0) : f =ᵐ[μ] 0 :=
begin
rw [snorm', ennreal.rpow_eq_zero_iff] at h,
cases h,
{ rw lintegral_eq_zero_iff' (hf.ennnorm.pow_const q) at h,
refine h.left.mono (λ x hx, _),
rw [pi.zero_apply, ennreal.rpow_eq_zero_iff] at hx,
cases hx,
{ cases hx with hx _,
rwa [←ennreal.coe_zero, ennreal.coe_eq_coe, nnnorm_eq_zero] at hx, },
{ exact absurd hx.left ennreal.coe_ne_top, }, },
{ exfalso,
rw [one_div, inv_lt_zero] at h,
exact hq0.not_lt h.right },
end
lemma snorm'_eq_zero_iff (hq0_lt : 0 < q) {f : α → E} (hf : ae_strongly_measurable f μ) :
snorm' f q μ = 0 ↔ f =ᵐ[μ] 0 :=
⟨ae_eq_zero_of_snorm'_eq_zero (le_of_lt hq0_lt) hf, snorm'_eq_zero_of_ae_zero hq0_lt⟩
lemma coe_nnnorm_ae_le_snorm_ess_sup {m : measurable_space α} (f : α → F) (μ : measure α) :
∀ᵐ x ∂μ, (∥f x∥₊ : ℝ≥0∞) ≤ snorm_ess_sup f μ :=
ennreal.ae_le_ess_sup (λ x, (∥f x∥₊ : ℝ≥0∞))
@[simp] lemma snorm_ess_sup_eq_zero_iff {f : α → F} : snorm_ess_sup f μ = 0 ↔ f =ᵐ[μ] 0 :=
by simp [eventually_eq, snorm_ess_sup]
lemma snorm_eq_zero_iff {f : α → E} (hf : ae_strongly_measurable f μ) (h0 : p ≠ 0) :
snorm f p μ = 0 ↔ f =ᵐ[μ] 0 :=
begin
by_cases h_top : p = ∞,
{ rw [h_top, snorm_exponent_top, snorm_ess_sup_eq_zero_iff], },
rw snorm_eq_snorm' h0 h_top,
exact snorm'_eq_zero_iff (ennreal.to_real_pos h0 h_top) hf,
end
lemma snorm'_add_le {f g : α → E}
(hf : ae_strongly_measurable f μ) (hg : ae_strongly_measurable g μ) (hq1 : 1 ≤ q) :
snorm' (f + g) q μ ≤ snorm' f q μ + snorm' g q μ :=
calc (∫⁻ a, ↑∥(f + g) a∥₊ ^ q ∂μ) ^ (1 / q)
≤ (∫⁻ a, (((λ a, (∥f a∥₊ : ℝ≥0∞))
+ (λ a, (∥g a∥₊ : ℝ≥0∞))) a) ^ q ∂μ) ^ (1 / q) :
begin
refine ennreal.rpow_le_rpow _ (by simp [le_trans zero_le_one hq1] : 0 ≤ 1 / q),
refine lintegral_mono (λ a, ennreal.rpow_le_rpow _ (le_trans zero_le_one hq1)),
simp [←ennreal.coe_add, nnnorm_add_le],
end
... ≤ snorm' f q μ + snorm' g q μ :
ennreal.lintegral_Lp_add_le hf.ennnorm hg.ennnorm hq1
lemma snorm_ess_sup_add_le {f g : α → F} :
snorm_ess_sup (f + g) μ ≤ snorm_ess_sup f μ + snorm_ess_sup g μ :=
begin
refine le_trans (ess_sup_mono_ae (eventually_of_forall (λ x, _)))
(ennreal.ess_sup_add_le _ _),
simp_rw [pi.add_apply, ←ennreal.coe_add, ennreal.coe_le_coe],
exact nnnorm_add_le _ _,
end
lemma snorm_add_le
{f g : α → E} (hf : ae_strongly_measurable f μ) (hg : ae_strongly_measurable g μ) (hp1 : 1 ≤ p) :
snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ :=
begin
by_cases hp0 : p = 0,
{ simp [hp0], },
by_cases hp_top : p = ∞,
{ simp [hp_top, snorm_ess_sup_add_le], },
have hp1_real : 1 ≤ p.to_real,
by rwa [← ennreal.one_to_real, ennreal.to_real_le_to_real ennreal.one_ne_top hp_top],
repeat { rw snorm_eq_snorm' hp0 hp_top, },
exact snorm'_add_le hf hg hp1_real,
end
lemma snorm_sub_le
{f g : α → E} (hf : ae_strongly_measurable f μ) (hg : ae_strongly_measurable g μ) (hp1 : 1 ≤ p) :
snorm (f - g) p μ ≤ snorm f p μ + snorm g p μ :=
calc snorm (f - g) p μ = snorm (f + - g) p μ : by rw sub_eq_add_neg
-- We cannot use snorm_add_le on f and (-g) because we don't have `ae_measurable (-g) μ`, since
-- we don't suppose `[borel_space E]`.
... = snorm (λ x, ∥f x + - g x∥) p μ : (snorm_norm (f + - g)).symm
... ≤ snorm (λ x, ∥f x∥ + ∥- g x∥) p μ : by
{ refine snorm_mono_real (λ x, _), rw norm_norm, exact norm_add_le _ _, }
... = snorm (λ x, ∥f x∥ + ∥g x∥) p μ : by simp_rw norm_neg
... ≤ snorm (λ x, ∥f x∥) p μ + snorm (λ x, ∥g x∥) p μ : snorm_add_le hf.norm hg.norm hp1
... = snorm f p μ + snorm g p μ : by rw [← snorm_norm f, ← snorm_norm g]
lemma snorm_add_lt_top_of_one_le {f g : α → E} (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ)
(hq1 : 1 ≤ p) : snorm (f + g) p μ < ∞ :=
lt_of_le_of_lt (snorm_add_le hf.1 hg.1 hq1) (ennreal.add_lt_top.mpr ⟨hf.2, hg.2⟩)
lemma snorm'_add_lt_top_of_le_one
{f g : α → E} (hf : ae_strongly_measurable f μ)
(hf_snorm : snorm' f q μ < ∞) (hg_snorm : snorm' g q μ < ∞) (hq_pos : 0 < q) (hq1 : q ≤ 1) :
snorm' (f + g) q μ < ∞ :=
calc (∫⁻ a, ↑∥(f + g) a∥₊ ^ q ∂μ) ^ (1 / q)
≤ (∫⁻ a, (((λ a, (∥f a∥₊ : ℝ≥0∞))
+ (λ a, (∥g a∥₊ : ℝ≥0∞))) a) ^ q ∂μ) ^ (1 / q) :
begin
refine ennreal.rpow_le_rpow _ (by simp [hq_pos.le] : 0 ≤ 1 / q),
refine lintegral_mono (λ a, ennreal.rpow_le_rpow _ hq_pos.le),
simp [←ennreal.coe_add, nnnorm_add_le],
end
... ≤ (∫⁻ a, (∥f a∥₊ : ℝ≥0∞) ^ q + (∥g a∥₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) :
begin
refine ennreal.rpow_le_rpow (lintegral_mono (λ a, _)) (by simp [hq_pos.le] : 0 ≤ 1 / q),
exact ennreal.rpow_add_le_add_rpow _ _ hq_pos.le hq1,
end
... < ∞ :
begin
refine ennreal.rpow_lt_top_of_nonneg (by simp [hq_pos.le] : 0 ≤ 1 / q) _,
rw [lintegral_add_left' (hf.ennnorm.pow_const q), ennreal.add_ne_top],
exact ⟨(lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top hq_pos hf_snorm).ne,
(lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top hq_pos hg_snorm).ne⟩,
end
lemma snorm_add_lt_top {f g : α → E} (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ) :
snorm (f + g) p μ < ∞ :=
begin
by_cases h0 : p = 0,
{ simp [h0], },
rw ←ne.def at h0,
cases le_total 1 p with hp1 hp1,
{ exact snorm_add_lt_top_of_one_le hf hg hp1, },
have hp_top : p ≠ ∞, from (lt_of_le_of_lt hp1 ennreal.coe_lt_top).ne,
have hp_pos : 0 < p.to_real,
{ rw [← ennreal.zero_to_real, @ennreal.to_real_lt_to_real 0 p ennreal.coe_ne_top hp_top],
exact ((zero_le p).lt_of_ne h0.symm), },
have hp1_real : p.to_real ≤ 1,
{ rwa [← ennreal.one_to_real, @ennreal.to_real_le_to_real p 1 hp_top ennreal.coe_ne_top], },
rw snorm_eq_snorm' h0 hp_top,
rw [mem_ℒp, snorm_eq_snorm' h0 hp_top] at hf hg,
exact snorm'_add_lt_top_of_le_one hf.1 hf.2 hg.2 hp_pos hp1_real,
end
section map_measure
variables {β : Type*} {mβ : measurable_space β} {f : α → β} {g : β → E}
include mβ
lemma snorm_ess_sup_map_measure
(hg : ae_strongly_measurable g (measure.map f μ)) (hf : ae_measurable f μ) :
snorm_ess_sup g (measure.map f μ) = snorm_ess_sup (g ∘ f) μ :=
ess_sup_map_measure hg.ennnorm hf
lemma snorm_map_measure (hg : ae_strongly_measurable g (measure.map f μ)) (hf : ae_measurable f μ) :
snorm g p (measure.map f μ) = snorm (g ∘ f) p μ :=
begin
by_cases hp_zero : p = 0,
{ simp only [hp_zero, snorm_exponent_zero], },
by_cases hp_top : p = ∞,
{ simp_rw [hp_top, snorm_exponent_top],
exact snorm_ess_sup_map_measure hg hf, },
simp_rw snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top,
rw lintegral_map' (hg.ennnorm.pow_const p.to_real) hf,
end
lemma mem_ℒp_map_measure_iff
(hg : ae_strongly_measurable g (measure.map f μ)) (hf : ae_measurable f μ) :
mem_ℒp g p (measure.map f μ) ↔ mem_ℒp (g ∘ f) p μ :=
by simp [mem_ℒp, snorm_map_measure hg hf, hg.comp_ae_measurable hf, hg]
lemma _root_.measurable_embedding.snorm_ess_sup_map_measure {g : β → F}
(hf : measurable_embedding f) :
snorm_ess_sup g (measure.map f μ) = snorm_ess_sup (g ∘ f) μ :=
hf.ess_sup_map_measure
lemma _root_.measurable_embedding.snorm_map_measure {g : β → F} (hf : measurable_embedding f) :
snorm g p (measure.map f μ) = snorm (g ∘ f) p μ :=
begin
by_cases hp_zero : p = 0,
{ simp only [hp_zero, snorm_exponent_zero], },
by_cases hp : p = ∞,
{ simp_rw [hp, snorm_exponent_top],
exact hf.ess_sup_map_measure, },
{ simp_rw snorm_eq_lintegral_rpow_nnnorm hp_zero hp,
rw hf.lintegral_map, },
end
lemma _root_.measurable_embedding.mem_ℒp_map_measure_iff {g : β → F}
(hf : measurable_embedding f) :
mem_ℒp g p (measure.map f μ) ↔ mem_ℒp (g ∘ f) p μ :=
by simp_rw [mem_ℒp, hf.ae_strongly_measurable_map_iff, hf.snorm_map_measure]
lemma _root_.measurable_equiv.mem_ℒp_map_measure_iff (f : α ≃ᵐ β) {g : β → F} :
mem_ℒp g p (measure.map f μ) ↔ mem_ℒp (g ∘ f) p μ :=
f.measurable_embedding.mem_ℒp_map_measure_iff
omit mβ
end map_measure
section trim
lemma snorm'_trim (hm : m ≤ m0) {f : α → E} (hf : strongly_measurable[m] f) :
snorm' f q (ν.trim hm) = snorm' f q ν :=
begin
simp_rw snorm',
congr' 1,
refine lintegral_trim hm _,
refine @measurable.pow_const _ _ _ _ _ _ _ m _ (@measurable.coe_nnreal_ennreal _ m _ _) _,
apply @strongly_measurable.measurable,
exact (@strongly_measurable.nnnorm α m _ _ _ hf),
end
lemma limsup_trim (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : measurable[m] f) :
(ν.trim hm).ae.limsup f = ν.ae.limsup f :=
begin
simp_rw limsup_eq,
suffices h_set_eq : {a : ℝ≥0∞ | ∀ᵐ n ∂(ν.trim hm), f n ≤ a} = {a : ℝ≥0∞ | ∀ᵐ n ∂ν, f n ≤ a},
by rw h_set_eq,
ext1 a,
suffices h_meas_eq : ν {x | ¬ f x ≤ a} = ν.trim hm {x | ¬ f x ≤ a},
by simp_rw [set.mem_set_of_eq, ae_iff, h_meas_eq],
refine (trim_measurable_set_eq hm _).symm,
refine @measurable_set.compl _ _ m (@measurable_set_le ℝ≥0∞ _ _ _ _ m _ _ _ _ _ hf _),
exact @measurable_const _ _ _ m _,
end
lemma ess_sup_trim (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : measurable[m] f) :
ess_sup f (ν.trim hm) = ess_sup f ν :=
by { simp_rw ess_sup, exact limsup_trim hm hf, }
lemma snorm_ess_sup_trim (hm : m ≤ m0) {f : α → E} (hf : strongly_measurable[m] f) :
snorm_ess_sup f (ν.trim hm) = snorm_ess_sup f ν :=
ess_sup_trim _ (@strongly_measurable.ennnorm _ m _ _ _ hf)
lemma snorm_trim (hm : m ≤ m0) {f : α → E} (hf : strongly_measurable[m] f) :
snorm f p (ν.trim hm) = snorm f p ν :=
begin
by_cases h0 : p = 0,
{ simp [h0], },
by_cases h_top : p = ∞,
{ simpa only [h_top, snorm_exponent_top] using snorm_ess_sup_trim hm hf, },
simpa only [snorm_eq_snorm' h0 h_top] using snorm'_trim hm hf,
end
lemma snorm_trim_ae (hm : m ≤ m0) {f : α → E} (hf : ae_strongly_measurable f (ν.trim hm)) :
snorm f p (ν.trim hm) = snorm f p ν :=
begin
rw [snorm_congr_ae hf.ae_eq_mk, snorm_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk)],
exact snorm_trim hm hf.strongly_measurable_mk,
end
lemma mem_ℒp_of_mem_ℒp_trim (hm : m ≤ m0) {f : α → E} (hf : mem_ℒp f p (ν.trim hm)) :
mem_ℒp f p ν :=
⟨ae_strongly_measurable_of_ae_strongly_measurable_trim hm hf.1,
(le_of_eq (snorm_trim_ae hm hf.1).symm).trans_lt hf.2⟩
end trim
@[simp] lemma snorm'_neg {f : α → F} : snorm' (-f) q μ = snorm' f q μ := by simp [snorm']
@[simp] lemma snorm_neg {f : α → F} : snorm (-f) p μ = snorm f p μ :=
begin
by_cases h0 : p = 0,
{ simp [h0], },
by_cases h_top : p = ∞,
{ simp [h_top, snorm_ess_sup], },
simp [snorm_eq_snorm' h0 h_top],
end
section borel_space
-- variable [borel_space E]
lemma mem_ℒp.neg {f : α → E} (hf : mem_ℒp f p μ) : mem_ℒp (-f) p μ :=
⟨ae_strongly_measurable.neg hf.1, by simp [hf.right]⟩
lemma mem_ℒp_neg_iff {f : α → E} : mem_ℒp (-f) p μ ↔ mem_ℒp f p μ :=
⟨λ h, neg_neg f ▸ h.neg, mem_ℒp.neg⟩
lemma snorm'_le_snorm'_mul_rpow_measure_univ {p q : ℝ} (hp0_lt : 0 < p) (hpq : p ≤ q)
{f : α → E} (hf : ae_strongly_measurable f μ) :
snorm' f p μ ≤ snorm' f q μ * (μ set.univ) ^ (1/p - 1/q) :=
begin
have hq0_lt : 0 < q, from lt_of_lt_of_le hp0_lt hpq,
by_cases hpq_eq : p = q,
{ rw [hpq_eq, sub_self, ennreal.rpow_zero, mul_one],
exact le_rfl, },
have hpq : p < q, from lt_of_le_of_ne hpq hpq_eq,
let g := λ a : α, (1 : ℝ≥0∞),
have h_rw : ∫⁻ a, ↑∥f a∥₊^p ∂ μ = ∫⁻ a, (∥f a∥₊ * (g a))^p ∂ μ,
from lintegral_congr (λ a, by simp),
repeat {rw snorm'},
rw h_rw,
let r := p * q / (q - p),
have hpqr : 1/p = 1/q + 1/r,
{ field_simp [(ne_of_lt hp0_lt).symm,
(ne_of_lt hq0_lt).symm],
ring, },
calc (∫⁻ (a : α), (↑∥f a∥₊ * g a) ^ p ∂μ) ^ (1/p)
≤ (∫⁻ (a : α), ↑∥f a∥₊ ^ q ∂μ) ^ (1/q) * (∫⁻ (a : α), (g a) ^ r ∂μ) ^ (1/r) :
ennreal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr μ hf.ennnorm ae_measurable_const
... = (∫⁻ (a : α), ↑∥f a∥₊ ^ q ∂μ) ^ (1/q) * μ set.univ ^ (1/p - 1/q) :
by simp [hpqr],
end
lemma snorm'_le_snorm_ess_sup_mul_rpow_measure_univ (hq_pos : 0 < q) {f : α → F} :
snorm' f q μ ≤ snorm_ess_sup f μ * (μ set.univ) ^ (1/q) :=
begin
have h_le : ∫⁻ (a : α), ↑∥f a∥₊ ^ q ∂μ ≤ ∫⁻ (a : α), (snorm_ess_sup f μ) ^ q ∂μ,
{ refine lintegral_mono_ae _,
have h_nnnorm_le_snorm_ess_sup := coe_nnnorm_ae_le_snorm_ess_sup f μ,
refine h_nnnorm_le_snorm_ess_sup.mono (λ x hx, ennreal.rpow_le_rpow hx (le_of_lt hq_pos)), },
rw [snorm', ←ennreal.rpow_one (snorm_ess_sup f μ)],
nth_rewrite 1 ←mul_inv_cancel (ne_of_lt hq_pos).symm,
rw [ennreal.rpow_mul, one_div,
←ennreal.mul_rpow_of_nonneg _ _ (by simp [hq_pos.le] : 0 ≤ q⁻¹)],
refine ennreal.rpow_le_rpow _ (by simp [hq_pos.le]),
rwa lintegral_const at h_le,
end
lemma snorm_le_snorm_mul_rpow_measure_univ {p q : ℝ≥0∞} (hpq : p ≤ q) {f : α → E}
(hf : ae_strongly_measurable f μ) :
snorm f p μ ≤ snorm f q μ * (μ set.univ) ^ (1/p.to_real - 1/q.to_real) :=
begin
by_cases hp0 : p = 0,
{ simp [hp0, zero_le], },
rw ← ne.def at hp0,
have hp0_lt : 0 < p, from lt_of_le_of_ne (zero_le _) hp0.symm,
have hq0_lt : 0 < q, from lt_of_lt_of_le hp0_lt hpq,
by_cases hq_top : q = ∞,
{ simp only [hq_top, div_zero, one_div, ennreal.top_to_real, sub_zero, snorm_exponent_top,
inv_zero],
by_cases hp_top : p = ∞,
{ simp only [hp_top, ennreal.rpow_zero, mul_one, ennreal.top_to_real, sub_zero, inv_zero,
snorm_exponent_top],
exact le_rfl, },
rw snorm_eq_snorm' hp0 hp_top,
have hp_pos : 0 < p.to_real, from ennreal.to_real_pos hp0_lt.ne' hp_top,
refine (snorm'_le_snorm_ess_sup_mul_rpow_measure_univ hp_pos).trans (le_of_eq _),
congr,
exact one_div _, },
have hp_lt_top : p < ∞, from hpq.trans_lt (lt_top_iff_ne_top.mpr hq_top),
have hp_pos : 0 < p.to_real, from ennreal.to_real_pos hp0_lt.ne' hp_lt_top.ne,
rw [snorm_eq_snorm' hp0_lt.ne.symm hp_lt_top.ne, snorm_eq_snorm' hq0_lt.ne.symm hq_top],
have hpq_real : p.to_real ≤ q.to_real, by rwa ennreal.to_real_le_to_real hp_lt_top.ne hq_top,
exact snorm'_le_snorm'_mul_rpow_measure_univ hp_pos hpq_real hf,
end
lemma snorm'_le_snorm'_of_exponent_le {m : measurable_space α} {p q : ℝ} (hp0_lt : 0 < p)
(hpq : p ≤ q) (μ : measure α) [is_probability_measure μ] {f : α → E}
(hf : ae_strongly_measurable f μ) :
snorm' f p μ ≤ snorm' f q μ :=
begin
have h_le_μ := snorm'_le_snorm'_mul_rpow_measure_univ hp0_lt hpq hf,
rwa [measure_univ, ennreal.one_rpow, mul_one] at h_le_μ,
end
lemma snorm'_le_snorm_ess_sup (hq_pos : 0 < q) {f : α → F} [is_probability_measure μ] :
snorm' f q μ ≤ snorm_ess_sup f μ :=
le_trans (snorm'_le_snorm_ess_sup_mul_rpow_measure_univ hq_pos) (le_of_eq (by simp [measure_univ]))
lemma snorm_le_snorm_of_exponent_le {p q : ℝ≥0∞} (hpq : p ≤ q) [is_probability_measure μ]
{f : α → E} (hf : ae_strongly_measurable f μ) :
snorm f p μ ≤ snorm f q μ :=
(snorm_le_snorm_mul_rpow_measure_univ hpq hf).trans (le_of_eq (by simp [measure_univ]))
lemma snorm'_lt_top_of_snorm'_lt_top_of_exponent_le {p q : ℝ} [is_finite_measure μ] {f : α → E}
(hf : ae_strongly_measurable f μ) (hfq_lt_top : snorm' f q μ < ∞)
(hp_nonneg : 0 ≤ p) (hpq : p ≤ q) :
snorm' f p μ < ∞ :=
begin
cases le_or_lt p 0 with hp_nonpos hp_pos,
{ rw le_antisymm hp_nonpos hp_nonneg,
simp, },
have hq_pos : 0 < q, from lt_of_lt_of_le hp_pos hpq,
calc snorm' f p μ
≤ snorm' f q μ * (μ set.univ) ^ (1/p - 1/q) :
snorm'_le_snorm'_mul_rpow_measure_univ hp_pos hpq hf
... < ∞ :
begin
rw ennreal.mul_lt_top_iff,
refine or.inl ⟨hfq_lt_top, ennreal.rpow_lt_top_of_nonneg _ (measure_ne_top μ set.univ)⟩,
rwa [le_sub, sub_zero, one_div, one_div, inv_le_inv hq_pos hp_pos],
end
end
variables (μ)
lemma pow_mul_meas_ge_le_snorm {f : α → E}
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (hf : ae_strongly_measurable f μ) (ε : ℝ≥0∞) :
(ε * μ {x | ε ≤ ∥f x∥₊ ^ p.to_real}) ^ (1 / p.to_real) ≤ snorm f p μ :=
begin
rw snorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top,
exact ennreal.rpow_le_rpow (mul_meas_ge_le_lintegral₀ (hf.ennnorm.pow_const _) ε)
(one_div_nonneg.2 ennreal.to_real_nonneg),
end
lemma mul_meas_ge_le_pow_snorm {f : α → E}
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (hf : ae_strongly_measurable f μ) (ε : ℝ≥0∞) :
ε * μ {x | ε ≤ ∥f x∥₊ ^ p.to_real} ≤ snorm f p μ ^ p.to_real :=
begin
have : 1 / p.to_real * p.to_real = 1,
{ refine one_div_mul_cancel _,
rw [ne, ennreal.to_real_eq_zero_iff],
exact not_or hp_ne_zero hp_ne_top },
rw [← ennreal.rpow_one (ε * μ {x | ε ≤ ∥f x∥₊ ^ p.to_real}), ← this, ennreal.rpow_mul],
exact ennreal.rpow_le_rpow (pow_mul_meas_ge_le_snorm μ hp_ne_zero hp_ne_top hf ε)
ennreal.to_real_nonneg,
end
/-- A version of Markov's inequality using Lp-norms. -/
lemma mul_meas_ge_le_pow_snorm' {f : α → E}
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (hf : ae_strongly_measurable f μ) (ε : ℝ≥0∞) :
ε ^ p.to_real * μ {x | ε ≤ ∥f x∥₊} ≤ snorm f p μ ^ p.to_real :=
begin
convert mul_meas_ge_le_pow_snorm μ hp_ne_zero hp_ne_top hf (ε ^ p.to_real),
ext x,
rw ennreal.rpow_le_rpow_iff (ennreal.to_real_pos hp_ne_zero hp_ne_top),
end
lemma meas_ge_le_mul_pow_snorm {f : α → E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf : ae_strongly_measurable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
μ {x | ε ≤ ∥f x∥₊} ≤ ε⁻¹ ^ p.to_real * snorm f p μ ^ p.to_real :=
begin
by_cases ε = ∞,
{ simp [h] },
have hεpow : ε ^ p.to_real ≠ 0 := (ennreal.rpow_pos (pos_iff_ne_zero.2 hε) h).ne.symm,
have hεpow' : ε ^ p.to_real ≠ ∞ := (ennreal.rpow_ne_top_of_nonneg ennreal.to_real_nonneg h),
rw [ennreal.inv_rpow, ← ennreal.mul_le_mul_left hεpow hεpow', ← mul_assoc,
ennreal.mul_inv_cancel hεpow hεpow', one_mul],
exact mul_meas_ge_le_pow_snorm' μ hp_ne_zero hp_ne_top hf ε,
end
variables {μ}
lemma mem_ℒp.mem_ℒp_of_exponent_le {p q : ℝ≥0∞} [is_finite_measure μ] {f : α → E}
(hfq : mem_ℒp f q μ) (hpq : p ≤ q) :
mem_ℒp f p μ :=
begin
cases hfq with hfq_m hfq_lt_top,
by_cases hp0 : p = 0,
{ rwa [hp0, mem_ℒp_zero_iff_ae_strongly_measurable], },
rw ←ne.def at hp0,
refine ⟨hfq_m, _⟩,
by_cases hp_top : p = ∞,
{ have hq_top : q = ∞,
by rwa [hp_top, top_le_iff] at hpq,
rw [hp_top],
rwa hq_top at hfq_lt_top, },
have hp_pos : 0 < p.to_real, from ennreal.to_real_pos hp0 hp_top,
by_cases hq_top : q = ∞,
{ rw snorm_eq_snorm' hp0 hp_top,
rw [hq_top, snorm_exponent_top] at hfq_lt_top,
refine lt_of_le_of_lt (snorm'_le_snorm_ess_sup_mul_rpow_measure_univ hp_pos) _,
refine ennreal.mul_lt_top hfq_lt_top.ne _,
exact (ennreal.rpow_lt_top_of_nonneg (by simp [hp_pos.le]) (measure_ne_top μ set.univ)).ne },
have hq0 : q ≠ 0,
{ by_contra hq_eq_zero,
have hp_eq_zero : p = 0, from le_antisymm (by rwa hq_eq_zero at hpq) (zero_le _),
rw [hp_eq_zero, ennreal.zero_to_real] at hp_pos,
exact (lt_irrefl _) hp_pos, },
have hpq_real : p.to_real ≤ q.to_real, by rwa ennreal.to_real_le_to_real hp_top hq_top,
rw snorm_eq_snorm' hp0 hp_top,
rw snorm_eq_snorm' hq0 hq_top at hfq_lt_top,
exact snorm'_lt_top_of_snorm'_lt_top_of_exponent_le hfq_m hfq_lt_top (le_of_lt hp_pos) hpq_real,
end
section has_measurable_add
-- variable [has_measurable_add₂ E]
lemma snorm'_sum_le {ι} {f : ι → α → E} {s : finset ι}
(hfs : ∀ i, i ∈ s → ae_strongly_measurable (f i) μ) (hq1 : 1 ≤ q) :
snorm' (∑ i in s, f i) q μ ≤ ∑ i in s, snorm' (f i) q μ :=
finset.le_sum_of_subadditive_on_pred (λ (f : α → E), snorm' f q μ)
(λ f, ae_strongly_measurable f μ) (snorm'_zero (zero_lt_one.trans_le hq1))
(λ f g hf hg, snorm'_add_le hf hg hq1) (λ f g hf hg, hf.add hg) _ hfs
lemma snorm_sum_le {ι} {f : ι → α → E} {s : finset ι}
(hfs : ∀ i, i ∈ s → ae_strongly_measurable (f i) μ) (hp1 : 1 ≤ p) :
snorm (∑ i in s, f i) p μ ≤ ∑ i in s, snorm (f i) p μ :=
finset.le_sum_of_subadditive_on_pred (λ (f : α → E), snorm f p μ)
(λ f, ae_strongly_measurable f μ) snorm_zero (λ f g hf hg, snorm_add_le hf hg hp1)
(λ f g hf hg, hf.add hg) _ hfs
lemma mem_ℒp.add {f g : α → E} (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ) : mem_ℒp (f + g) p μ :=
⟨ae_strongly_measurable.add hf.1 hg.1, snorm_add_lt_top hf hg⟩
lemma mem_ℒp.sub {f g : α → E} (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ) : mem_ℒp (f - g) p μ :=
by { rw sub_eq_add_neg, exact hf.add hg.neg }
lemma mem_ℒp_finset_sum {ι} (s : finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, mem_ℒp (f i) p μ) :
mem_ℒp (λ a, ∑ i in s, f i a) p μ :=
begin
haveI : decidable_eq ι := classical.dec_eq _,
revert hf,
refine finset.induction_on s _ _,
{ simp only [zero_mem_ℒp', finset.sum_empty, implies_true_iff], },
{ intros i s his ih hf,
simp only [his, finset.sum_insert, not_false_iff],
exact (hf i (s.mem_insert_self i)).add (ih (λ j hj, hf j (finset.mem_insert_of_mem hj))), },
end
lemma mem_ℒp_finset_sum' {ι} (s : finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, mem_ℒp (f i) p μ) :
mem_ℒp (∑ i in s, f i) p μ :=
begin
convert mem_ℒp_finset_sum s hf,
ext x,
simp,
end
end has_measurable_add
end borel_space
section normed_space
variables {𝕜 : Type*} [normed_field 𝕜] [normed_space 𝕜 E] [normed_space 𝕜 F]
lemma snorm'_const_smul {f : α → F} (c : 𝕜) (hq_pos : 0 < q) :
snorm' (c • f) q μ = (∥c∥₊ : ℝ≥0∞) * snorm' f q μ :=
begin
rw snorm',
simp_rw [pi.smul_apply, nnnorm_smul, ennreal.coe_mul,
ennreal.mul_rpow_of_nonneg _ _ hq_pos.le],
suffices h_integral : ∫⁻ a, ↑(∥c∥₊) ^ q * ↑∥f a∥₊ ^ q ∂μ
= (∥c∥₊ : ℝ≥0∞)^q * ∫⁻ a, ∥f a∥₊ ^ q ∂μ,
{ apply_fun (λ x, x ^ (1/q)) at h_integral,
rw [h_integral, ennreal.mul_rpow_of_nonneg _ _ (by simp [hq_pos.le] : 0 ≤ 1 / q)],
congr,
simp_rw [←ennreal.rpow_mul, one_div, mul_inv_cancel hq_pos.ne.symm, ennreal.rpow_one], },
rw lintegral_const_mul',
rw ennreal.coe_rpow_of_nonneg _ hq_pos.le,
exact ennreal.coe_ne_top,
end
lemma snorm_ess_sup_const_smul {f : α → F} (c : 𝕜) :
snorm_ess_sup (c • f) μ = (∥c∥₊ : ℝ≥0∞) * snorm_ess_sup f μ :=
by simp_rw [snorm_ess_sup, pi.smul_apply, nnnorm_smul, ennreal.coe_mul, ennreal.ess_sup_const_mul]
lemma snorm_const_smul {f : α → F} (c : 𝕜) :
snorm (c • f) p μ = (∥c∥₊ : ℝ≥0∞) * snorm f p μ :=
begin
by_cases h0 : p = 0,
{ simp [h0], },
by_cases h_top : p = ∞,
{ simp [h_top, snorm_ess_sup_const_smul], },
repeat { rw snorm_eq_snorm' h0 h_top, },
rw ←ne.def at h0,
exact snorm'_const_smul c (ennreal.to_real_pos h0 h_top),
end
lemma mem_ℒp.const_smul {f : α → E} (hf : mem_ℒp f p μ) (c : 𝕜) :
mem_ℒp (c • f) p μ :=
⟨ae_strongly_measurable.const_smul hf.1 c,
(snorm_const_smul c).le.trans_lt (ennreal.mul_lt_top ennreal.coe_ne_top hf.2.ne)⟩
lemma mem_ℒp.const_mul {f : α → 𝕜} (hf : mem_ℒp f p μ) (c : 𝕜) :
mem_ℒp (λ x, c * f x) p μ :=
hf.const_smul c
lemma snorm'_smul_le_mul_snorm' {p q r : ℝ}
{f : α → E} (hf : ae_strongly_measurable f μ) {φ : α → 𝕜} (hφ : ae_strongly_measurable φ μ)
(hp0_lt : 0 < p) (hpq : p < q) (hpqr : 1/p = 1/q + 1/r) :
snorm' (φ • f) p μ ≤ snorm' φ q μ * snorm' f r μ :=
begin
simp_rw [snorm', pi.smul_apply', nnnorm_smul, ennreal.coe_mul],
exact ennreal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr μ hφ.ennnorm
hf.ennnorm,
end
lemma snorm_smul_le_snorm_top_mul_snorm (p : ℝ≥0∞)
{f : α → E} (hf : ae_strongly_measurable f μ) (φ : α → 𝕜) :
snorm (φ • f) p μ ≤ snorm φ ∞ μ * snorm f p μ :=
begin
by_cases hp_top : p = ∞,
{ simp_rw [hp_top, snorm_exponent_top, snorm_ess_sup, pi.smul_apply', nnnorm_smul,
ennreal.coe_mul],
exact ennreal.ess_sup_mul_le _ _, },
by_cases hp_zero : p = 0,
{ simp only [hp_zero, snorm_exponent_zero, mul_zero, le_zero_iff], },
simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top, snorm_exponent_top, snorm_ess_sup],
calc (∫⁻ x, ↑∥(φ • f) x∥₊ ^ p.to_real ∂μ) ^ (1 / p.to_real)
= (∫⁻ x, ↑∥φ x∥₊ ^ p.to_real * ↑∥f x∥₊ ^ p.to_real ∂μ) ^ (1 / p.to_real) :
begin
congr,
ext1 x,
rw [pi.smul_apply', nnnorm_smul, ennreal.coe_mul,
ennreal.mul_rpow_of_nonneg _ _ (ennreal.to_real_nonneg)],
end
... ≤ (∫⁻ x, (ess_sup (λ x, ↑∥φ x∥₊) μ) ^ p.to_real * ↑∥f x∥₊ ^ p.to_real ∂μ) ^ (1 / p.to_real) :
begin
refine ennreal.rpow_le_rpow _ _,
swap, { rw one_div_nonneg, exact ennreal.to_real_nonneg, },
refine lintegral_mono_ae _,
filter_upwards [@ennreal.ae_le_ess_sup _ _ μ (λ x, ↑∥φ x∥₊)] with x hx,
refine ennreal.mul_le_mul _ le_rfl,
exact ennreal.rpow_le_rpow hx ennreal.to_real_nonneg,
end
... = ess_sup (λ x, ↑∥φ x∥₊) μ * (∫⁻ x, ↑∥f x∥₊ ^ p.to_real ∂μ) ^ (1 / p.to_real) :
begin
rw lintegral_const_mul'',
swap, { exact hf.nnnorm.ae_measurable.coe_nnreal_ennreal.pow ae_measurable_const, },
rw ennreal.mul_rpow_of_nonneg,
swap, { rw one_div_nonneg, exact ennreal.to_real_nonneg, },
rw [← ennreal.rpow_mul, one_div, mul_inv_cancel, ennreal.rpow_one],
rw [ne.def, ennreal.to_real_eq_zero_iff, auto.not_or_eq],
exact ⟨hp_zero, hp_top⟩,
end
end
lemma snorm_smul_le_snorm_mul_snorm_top (p : ℝ≥0∞)
(f : α → E) {φ : α → 𝕜} (hφ : ae_strongly_measurable φ μ) :
snorm (φ • f) p μ ≤ snorm φ p μ * snorm f ∞ μ :=
begin
rw ← snorm_norm,
simp_rw [pi.smul_apply', norm_smul],
have : (λ x, ∥φ x∥ * ∥f x∥) = (λ x, ∥f x∥) • (λ x, ∥φ x∥),
{ rw [smul_eq_mul, mul_comm], refl, },
rw this,
have h := snorm_smul_le_snorm_top_mul_snorm p hφ.norm (λ x, ∥f x∥),
refine h.trans_eq _,
simp_rw snorm_norm,
rw mul_comm,
end
/-- Hölder's inequality, as an inequality on the `ℒp` seminorm of a scalar product `φ • f`. -/
lemma snorm_smul_le_mul_snorm {p q r : ℝ≥0∞}
{f : α → E} (hf : ae_strongly_measurable f μ) {φ : α → 𝕜} (hφ : ae_strongly_measurable φ μ)
(hpqr : 1/p = 1/q + 1/r) :
snorm (φ • f) p μ ≤ snorm φ q μ * snorm f r μ :=
begin
by_cases hp_zero : p = 0,
{ simp [hp_zero], },
have hq_ne_zero : q ≠ 0,
{ intro hq_zero,
simp only [hq_zero, hp_zero, one_div, ennreal.inv_zero, ennreal.top_add,
ennreal.inv_eq_top] at hpqr,
exact hpqr, },
have hr_ne_zero : r ≠ 0,
{ intro hr_zero,
simp only [hr_zero, hp_zero, one_div, ennreal.inv_zero, ennreal.add_top,
ennreal.inv_eq_top] at hpqr,
exact hpqr, },
by_cases hq_top : q = ∞,
{ have hpr : p = r,
{ simpa only [hq_top, one_div, ennreal.div_top, zero_add, inv_inj] using hpqr, },
rw [← hpr, hq_top],
exact snorm_smul_le_snorm_top_mul_snorm p hf φ, },
by_cases hr_top : r = ∞,
{ have hpq : p = q,
{ simpa only [hr_top, one_div, ennreal.div_top, add_zero, inv_inj] using hpqr, },
rw [← hpq, hr_top],
exact snorm_smul_le_snorm_mul_snorm_top p f hφ, },
have hpq : p < q,
{ suffices : 1 / q < 1 / p,
{ rwa [one_div, one_div, ennreal.inv_lt_inv] at this, },
rw hpqr,
refine ennreal.lt_add_right _ _,
{ simp only [hq_ne_zero, one_div, ne.def, ennreal.inv_eq_top, not_false_iff], },
{ simp only [hr_top, one_div, ne.def, ennreal.inv_eq_zero, not_false_iff], }, },
rw [snorm_eq_snorm' hp_zero (hpq.trans_le le_top).ne, snorm_eq_snorm' hq_ne_zero hq_top,
snorm_eq_snorm' hr_ne_zero hr_top],
refine snorm'_smul_le_mul_snorm' hf hφ _ _ _,
{ exact ennreal.to_real_pos hp_zero (hpq.trans_le le_top).ne, },
{ exact ennreal.to_real_strict_mono hq_top hpq, },
rw [← ennreal.one_to_real, ← ennreal.to_real_div, ← ennreal.to_real_div, ← ennreal.to_real_div,
hpqr, ennreal.to_real_add],
{ simp only [hq_ne_zero, one_div, ne.def, ennreal.inv_eq_top, not_false_iff], },
{ simp only [hr_ne_zero, one_div, ne.def, ennreal.inv_eq_top, not_false_iff], },
end
lemma mem_ℒp.smul {p q r : ℝ≥0∞} {f : α → E} {φ : α → 𝕜}
(hf : mem_ℒp f r μ) (hφ : mem_ℒp φ q μ) (hpqr : 1/p = 1/q + 1/r) :
mem_ℒp (φ • f) p μ :=
⟨hφ.1.smul hf.1, (snorm_smul_le_mul_snorm hf.1 hφ.1 hpqr).trans_lt
(ennreal.mul_lt_top hφ.snorm_ne_top hf.snorm_ne_top)⟩
end normed_space
section monotonicity
lemma snorm_le_mul_snorm_aux_of_nonneg {f : α → F} {g : α → G} {c : ℝ}
(h : ∀ᵐ x ∂μ, ∥f x∥ ≤ c * ∥g x∥) (hc : 0 ≤ c) (p : ℝ≥0∞) :
snorm f p μ ≤ (ennreal.of_real c) * snorm g p μ :=
begin
lift c to ℝ≥0 using hc,
rw [ennreal.of_real_coe_nnreal, ← c.nnnorm_eq, ← snorm_norm g, ← snorm_const_smul (c : ℝ)],
swap, apply_instance,
refine snorm_mono_ae _,
simpa
end
lemma snorm_le_mul_snorm_aux_of_neg {f : α → F} {g : α → G} {c : ℝ}
(h : ∀ᵐ x ∂μ, ∥f x∥ ≤ c * ∥g x∥) (hc : c < 0) (p : ℝ≥0∞) :
snorm f p μ = 0 ∧ snorm g p μ = 0 :=
begin
suffices : f =ᵐ[μ] 0 ∧ g =ᵐ[μ] 0,
by simp [snorm_congr_ae this.1, snorm_congr_ae this.2],
refine ⟨h.mono $ λ x hx, _, h.mono $ λ x hx, _⟩,
{ refine norm_le_zero_iff.1 (hx.trans _),
exact mul_nonpos_of_nonpos_of_nonneg hc.le (norm_nonneg _) },
{ refine norm_le_zero_iff.1 (nonpos_of_mul_nonneg_right _ hc),
exact (norm_nonneg _).trans hx }
end
lemma snorm_le_mul_snorm_of_ae_le_mul {f : α → F} {g : α → G} {c : ℝ}
(h : ∀ᵐ x ∂μ, ∥f x∥ ≤ c * ∥g x∥) (p : ℝ≥0∞) :
snorm f p μ ≤ (ennreal.of_real c) * snorm g p μ :=
begin
cases le_or_lt 0 c with hc hc,
{ exact snorm_le_mul_snorm_aux_of_nonneg h hc p },
{ simp [snorm_le_mul_snorm_aux_of_neg h hc p] }
end
lemma mem_ℒp.of_le_mul {f : α → E} {g : α → F} {c : ℝ}
(hg : mem_ℒp g p μ) (hf : ae_strongly_measurable f μ) (hfg : ∀ᵐ x ∂μ, ∥f x∥ ≤ c * ∥g x∥) :
mem_ℒp f p μ :=
begin
simp only [mem_ℒp, hf, true_and],
apply lt_of_le_of_lt (snorm_le_mul_snorm_of_ae_le_mul hfg p),
simp [lt_top_iff_ne_top, hg.snorm_ne_top],
end
end monotonicity
lemma snorm_indicator_ge_of_bdd_below (hp : p ≠ 0) (hp' : p ≠ ∞)
{f : α → F} (C : ℝ≥0) {s : set α} (hs : measurable_set s)
(hf : ∀ᵐ x ∂μ, x ∈ s → C ≤ ∥s.indicator f x∥₊) :
C • μ s ^ (1 / p.to_real) ≤ snorm (s.indicator f) p μ :=
begin
rw [ennreal.smul_def, smul_eq_mul, snorm_eq_lintegral_rpow_nnnorm hp hp',
ennreal.le_rpow_one_div_iff (ennreal.to_real_pos hp hp'),
ennreal.mul_rpow_of_nonneg _ _ ennreal.to_real_nonneg,
← ennreal.rpow_mul, one_div_mul_cancel (ennreal.to_real_pos hp hp').ne.symm, ennreal.rpow_one,
← set_lintegral_const, ← lintegral_indicator _ hs],
refine lintegral_mono_ae _,
filter_upwards [hf] with x hx,
rw nnnorm_indicator_eq_indicator_nnnorm,
by_cases hxs : x ∈ s,
{ simp only [set.indicator_of_mem hxs] at ⊢ hx,
exact ennreal.rpow_le_rpow (ennreal.coe_le_coe.2 (hx hxs)) ennreal.to_real_nonneg },
{ simp [set.indicator_of_not_mem hxs] },
end
section is_R_or_C
variables {𝕜 : Type*} [is_R_or_C 𝕜] {f : α → 𝕜}
lemma mem_ℒp.re (hf : mem_ℒp f p μ) : mem_ℒp (λ x, is_R_or_C.re (f x)) p μ :=
begin
have : ∀ x, ∥is_R_or_C.re (f x)∥ ≤ 1 * ∥f x∥,
by { intro x, rw one_mul, exact is_R_or_C.norm_re_le_norm (f x), },
exact hf.of_le_mul hf.1.re (eventually_of_forall this),
end
lemma mem_ℒp.im (hf : mem_ℒp f p μ) : mem_ℒp (λ x, is_R_or_C.im (f x)) p μ :=
begin
have : ∀ x, ∥is_R_or_C.im (f x)∥ ≤ 1 * ∥f x∥,
by { intro x, rw one_mul, exact is_R_or_C.norm_im_le_norm (f x), },
exact hf.of_le_mul hf.1.im (eventually_of_forall this),
end
end is_R_or_C
section inner_product
variables {E' 𝕜 : Type*} [is_R_or_C 𝕜] [inner_product_space 𝕜 E']
local notation `⟪`x`, `y`⟫` := @inner 𝕜 E' _ x y
lemma mem_ℒp.const_inner (c : E') {f : α → E'} (hf : mem_ℒp f p μ) :
mem_ℒp (λ a, ⟪c, f a⟫) p μ :=
hf.of_le_mul (ae_strongly_measurable.inner ae_strongly_measurable_const hf.1)
(eventually_of_forall (λ x, norm_inner_le_norm _ _))
lemma mem_ℒp.inner_const {f : α → E'} (hf : mem_ℒp f p μ) (c : E') :
mem_ℒp (λ a, ⟪f a, c⟫) p μ :=
hf.of_le_mul (ae_strongly_measurable.inner hf.1 ae_strongly_measurable_const)
(eventually_of_forall (λ x, by { rw mul_comm, exact norm_inner_le_norm _ _, }))
end inner_product
section liminf
variables [measurable_space E] [opens_measurable_space E] {R : ℝ≥0}
lemma ae_bdd_liminf_at_top_rpow_of_snorm_bdd {p : ℝ≥0∞}
{f : ℕ → α → E} (hfmeas : ∀ n, measurable (f n)) (hbdd : ∀ n, snorm (f n) p μ ≤ R) :
∀ᵐ x ∂μ, liminf at_top (λ n, (∥f n x∥₊ ^ p.to_real : ℝ≥0∞)) < ∞ :=
begin
by_cases hp0 : p.to_real = 0,
{ simp only [hp0, ennreal.rpow_zero],
refine eventually_of_forall (λ x, _),
rw liminf_const (1 : ℝ≥0∞),
exacts [ennreal.one_lt_top, at_top_ne_bot] },
have hp : p ≠ 0 := λ h, by simpa [h] using hp0,
have hp' : p ≠ ∞ := λ h, by simpa [h] using hp0,
refine ae_lt_top
(measurable_liminf (λ n, (hfmeas n).nnnorm.coe_nnreal_ennreal.pow_const p.to_real))
(lt_of_le_of_lt (lintegral_liminf_le
(λ n, (hfmeas n).nnnorm.coe_nnreal_ennreal.pow_const p.to_real))
(lt_of_le_of_lt _ (ennreal.rpow_lt_top_of_nonneg
ennreal.to_real_nonneg ennreal.coe_ne_top : ↑R ^ p.to_real < ∞))).ne,
simp_rw snorm_eq_lintegral_rpow_nnnorm hp hp' at hbdd,
simp_rw [liminf_eq, eventually_at_top],
exact Sup_le (λ b ⟨a, ha⟩, (ha a le_rfl).trans
((ennreal.rpow_one_div_le_iff (ennreal.to_real_pos hp hp')).1 (hbdd _))),
end
lemma ae_bdd_liminf_at_top_of_snorm_bdd {p : ℝ≥0∞} (hp : p ≠ 0)
{f : ℕ → α → E} (hfmeas : ∀ n, measurable (f n)) (hbdd : ∀ n, snorm (f n) p μ ≤ R) :
∀ᵐ x ∂μ, liminf at_top (λ n, (∥f n x∥₊ : ℝ≥0∞)) < ∞ :=
begin
by_cases hp' : p = ∞,
{ subst hp',
simp_rw snorm_exponent_top at hbdd,
have : ∀ n, ∀ᵐ x ∂μ, (∥f n x∥₊ : ℝ≥0∞) < R + 1 :=
λ n, ae_lt_of_ess_sup_lt (lt_of_le_of_lt (hbdd n) $
ennreal.lt_add_right ennreal.coe_ne_top one_ne_zero),
rw ← ae_all_iff at this,
filter_upwards [this] with x hx using lt_of_le_of_lt
(liminf_le_of_frequently_le' $ frequently_of_forall $ λ n, (hx n).le)
(ennreal.add_lt_top.2 ⟨ennreal.coe_lt_top, ennreal.one_lt_top⟩) },
filter_upwards [ae_bdd_liminf_at_top_rpow_of_snorm_bdd hfmeas hbdd] with x hx,
have hppos : 0 < p.to_real := ennreal.to_real_pos hp hp',
have : liminf at_top (λ n, (∥f n x∥₊ ^ p.to_real : ℝ≥0∞)) =
liminf at_top (λ n, (∥f n x∥₊ : ℝ≥0∞)) ^ p.to_real,
{ change liminf at_top (λ n, ennreal.order_iso_rpow p.to_real hppos (∥f n x∥₊ : ℝ≥0∞)) =
ennreal.order_iso_rpow p.to_real hppos (liminf at_top (λ n, (∥f n x∥₊ : ℝ≥0∞))),
refine (order_iso.liminf_apply (ennreal.order_iso_rpow p.to_real _) _ _ _ _).symm;
is_bounded_default },
rw this at hx,
rw [← ennreal.rpow_one (liminf at_top (λ n, ∥f n x∥₊)), ← mul_inv_cancel hppos.ne.symm,
ennreal.rpow_mul],
exact ennreal.rpow_lt_top_of_nonneg (inv_nonneg.2 hppos.le) hx.ne,
end
end liminf
end ℒp
/-!
### Lp space
The space of equivalence classes of measurable functions for which `snorm f p μ < ∞`.
-/
@[simp] lemma snorm_ae_eq_fun {α E : Type*} [measurable_space α] {μ : measure α}
[normed_add_comm_group E] {p : ℝ≥0∞} {f : α → E} (hf : ae_strongly_measurable f μ) :
snorm (ae_eq_fun.mk f hf) p μ = snorm f p μ :=
snorm_congr_ae (ae_eq_fun.coe_fn_mk _ _)
lemma mem_ℒp.snorm_mk_lt_top {α E : Type*} [measurable_space α] {μ : measure α}
[normed_add_comm_group E] {p : ℝ≥0∞} {f : α → E} (hfp : mem_ℒp f p μ) :
snorm (ae_eq_fun.mk f hfp.1) p μ < ∞ :=
by simp [hfp.2]
/-- Lp space -/
def Lp {α} (E : Type*) {m : measurable_space α} [normed_add_comm_group E]
(p : ℝ≥0∞) (μ : measure α . volume_tac) : add_subgroup (α →ₘ[μ] E) :=
{ carrier := {f | snorm f p μ < ∞},
zero_mem' := by simp [snorm_congr_ae ae_eq_fun.coe_fn_zero, snorm_zero],
add_mem' := λ f g hf hg, by simp [snorm_congr_ae (ae_eq_fun.coe_fn_add _ _),
snorm_add_lt_top ⟨f.ae_strongly_measurable, hf⟩ ⟨g.ae_strongly_measurable, hg⟩],
neg_mem' := λ f hf,
by rwa [set.mem_set_of_eq, snorm_congr_ae (ae_eq_fun.coe_fn_neg _), snorm_neg] }
localized "notation (name := measure_theory.L1)
α ` →₁[`:25 μ `] ` E := measure_theory.Lp E 1 μ" in measure_theory
localized "notation (name := measure_theory.L2)
α ` →₂[`:25 μ `] ` E := measure_theory.Lp E 2 μ" in measure_theory
namespace mem_ℒp
/-- make an element of Lp from a function verifying `mem_ℒp` -/
def to_Lp (f : α → E) (h_mem_ℒp : mem_ℒp f p μ) : Lp E p μ :=
⟨ae_eq_fun.mk f h_mem_ℒp.1, h_mem_ℒp.snorm_mk_lt_top⟩
lemma coe_fn_to_Lp {f : α → E} (hf : mem_ℒp f p μ) : hf.to_Lp f =ᵐ[μ] f :=
ae_eq_fun.coe_fn_mk _ _
lemma to_Lp_congr {f g : α → E} (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ) (hfg : f =ᵐ[μ] g) :
hf.to_Lp f = hg.to_Lp g :=
by simp [to_Lp, hfg]
@[simp] lemma to_Lp_eq_to_Lp_iff {f g : α → E} (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ) :
hf.to_Lp f = hg.to_Lp g ↔ f =ᵐ[μ] g :=
by simp [to_Lp]
@[simp] lemma to_Lp_zero (h : mem_ℒp (0 : α → E) p μ) : h.to_Lp 0 = 0 := rfl
lemma to_Lp_add {f g : α → E} (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ) :
(hf.add hg).to_Lp (f + g) = hf.to_Lp f + hg.to_Lp g := rfl
lemma to_Lp_neg {f : α → E} (hf : mem_ℒp f p μ) : hf.neg.to_Lp (-f) = - hf.to_Lp f := rfl
lemma to_Lp_sub {f g : α → E} (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ) :
(hf.sub hg).to_Lp (f - g) = hf.to_Lp f - hg.to_Lp g := rfl
end mem_ℒp
namespace Lp
instance : has_coe_to_fun (Lp E p μ) (λ _, α → E) := ⟨λ f, ((f : α →ₘ[μ] E) : α → E)⟩
@[ext] lemma ext {f g : Lp E p μ} (h : f =ᵐ[μ] g) : f = g :=
begin
cases f,
cases g,
simp only [subtype.mk_eq_mk],
exact ae_eq_fun.ext h
end
lemma ext_iff {f g : Lp E p μ} : f = g ↔ f =ᵐ[μ] g :=
⟨λ h, by rw h, λ h, ext h⟩
lemma mem_Lp_iff_snorm_lt_top {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ snorm f p μ < ∞ := iff.refl _
lemma mem_Lp_iff_mem_ℒp {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ mem_ℒp f p μ :=
by simp [mem_Lp_iff_snorm_lt_top, mem_ℒp, f.strongly_measurable.ae_strongly_measurable]
protected lemma antitone [is_finite_measure μ] {p q : ℝ≥0∞} (hpq : p ≤ q) : Lp E q μ ≤ Lp E p μ :=
λ f hf, (mem_ℒp.mem_ℒp_of_exponent_le ⟨f.ae_strongly_measurable, hf⟩ hpq).2
@[simp] lemma coe_fn_mk {f : α →ₘ[μ] E} (hf : snorm f p μ < ∞) :
((⟨f, hf⟩ : Lp E p μ) : α → E) = f := rfl
@[simp] lemma coe_mk {f : α →ₘ[μ] E} (hf : snorm f p μ < ∞) :
((⟨f, hf⟩ : Lp E p μ) : α →ₘ[μ] E) = f := rfl
@[simp] lemma to_Lp_coe_fn (f : Lp E p μ) (hf : mem_ℒp f p μ) : hf.to_Lp f = f :=
by { cases f, simp [mem_ℒp.to_Lp] }
lemma snorm_lt_top (f : Lp E p μ) : snorm f p μ < ∞ := f.prop
lemma snorm_ne_top (f : Lp E p μ) : snorm f p μ ≠ ∞ := (snorm_lt_top f).ne
@[measurability]
protected lemma strongly_measurable (f : Lp E p μ) : strongly_measurable f :=
f.val.strongly_measurable
@[measurability]
protected lemma ae_strongly_measurable (f : Lp E p μ) : ae_strongly_measurable f μ :=
f.val.ae_strongly_measurable
protected lemma mem_ℒp (f : Lp E p μ) : mem_ℒp f p μ := ⟨Lp.ae_strongly_measurable f, f.prop⟩
variables (E p μ)
lemma coe_fn_zero : ⇑(0 : Lp E p μ) =ᵐ[μ] 0 := ae_eq_fun.coe_fn_zero
variables {E p μ}
lemma coe_fn_neg (f : Lp E p μ) : ⇑(-f) =ᵐ[μ] -f := ae_eq_fun.coe_fn_neg _
lemma coe_fn_add (f g : Lp E p μ) : ⇑(f + g) =ᵐ[μ] f + g := ae_eq_fun.coe_fn_add _ _
lemma coe_fn_sub (f g : Lp E p μ) : ⇑(f - g) =ᵐ[μ] f - g := ae_eq_fun.coe_fn_sub _ _
lemma mem_Lp_const (α) {m : measurable_space α} (μ : measure α) (c : E) [is_finite_measure μ] :
@ae_eq_fun.const α _ _ μ _ c ∈ Lp E p μ :=
(mem_ℒp_const c).snorm_mk_lt_top
instance : has_norm (Lp E p μ) := { norm := λ f, ennreal.to_real (snorm f p μ) }
instance : has_dist (Lp E p μ) := { dist := λ f g, ∥f - g∥}
instance : has_edist (Lp E p μ) := { edist := λ f g, snorm (f - g) p μ }
lemma norm_def (f : Lp E p μ) : ∥f∥ = ennreal.to_real (snorm f p μ) := rfl
@[simp] lemma norm_to_Lp (f : α → E) (hf : mem_ℒp f p μ) :
∥hf.to_Lp f∥ = ennreal.to_real (snorm f p μ) :=
by rw [norm_def, snorm_congr_ae (mem_ℒp.coe_fn_to_Lp hf)]
lemma dist_def (f g : Lp E p μ) : dist f g = (snorm (f - g) p μ).to_real :=
begin
simp_rw [dist, norm_def],
congr' 1,
apply snorm_congr_ae (coe_fn_sub _ _),
end
lemma edist_def (f g : Lp E p μ) : edist f g = snorm (f - g) p μ :=
rfl
@[simp] lemma edist_to_Lp_to_Lp (f g : α → E) (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ) :
edist (hf.to_Lp f) (hg.to_Lp g) = snorm (f - g) p μ :=
by { rw edist_def, exact snorm_congr_ae (hf.coe_fn_to_Lp.sub hg.coe_fn_to_Lp) }
@[simp] lemma edist_to_Lp_zero (f : α → E) (hf : mem_ℒp f p μ) :
edist (hf.to_Lp f) 0 = snorm f p μ :=
by { convert edist_to_Lp_to_Lp f 0 hf zero_mem_ℒp, simp }
@[simp] lemma norm_zero : ∥(0 : Lp E p μ)∥ = 0 :=
begin
change (snorm ⇑(0 : α →ₘ[μ] E) p μ).to_real = 0,
simp [snorm_congr_ae ae_eq_fun.coe_fn_zero, snorm_zero]
end
lemma norm_eq_zero_iff {f : Lp E p μ} (hp : 0 < p) : ∥f∥ = 0 ↔ f = 0 :=
begin
refine ⟨λ hf, _, λ hf, by simp [hf]⟩,
rw [norm_def, ennreal.to_real_eq_zero_iff] at hf,
cases hf,
{ rw snorm_eq_zero_iff (Lp.ae_strongly_measurable f) hp.ne.symm at hf,
exact subtype.eq (ae_eq_fun.ext (hf.trans ae_eq_fun.coe_fn_zero.symm)), },
{ exact absurd hf (snorm_ne_top f), },
end
lemma eq_zero_iff_ae_eq_zero {f : Lp E p μ} : f = 0 ↔ f =ᵐ[μ] 0 :=
begin
split,
{ assume h,
rw h,
exact ae_eq_fun.coe_fn_const _ _ },
{ assume h,
ext1,
filter_upwards [h, ae_eq_fun.coe_fn_const α (0 : E)] with _ ha h'a,
rw ha,
exact h'a.symm, },
end
@[simp] lemma norm_neg {f : Lp E p μ} : ∥-f∥ = ∥f∥ :=
by rw [norm_def, norm_def, snorm_congr_ae (coe_fn_neg _), snorm_neg]
lemma norm_le_mul_norm_of_ae_le_mul {c : ℝ} {f : Lp E p μ} {g : Lp F p μ}
(h : ∀ᵐ x ∂μ, ∥f x∥ ≤ c * ∥g x∥) : ∥f∥ ≤ c * ∥g∥ :=
begin
by_cases pzero : p = 0,
{ simp [pzero, norm_def] },
cases le_or_lt 0 c with hc hc,
{ have := snorm_le_mul_snorm_aux_of_nonneg h hc p,
rw [← ennreal.to_real_le_to_real, ennreal.to_real_mul, ennreal.to_real_of_real hc] at this,
{ exact this },
{ exact (Lp.mem_ℒp _).snorm_ne_top },
{ simp [(Lp.mem_ℒp _).snorm_ne_top] } },
{ have := snorm_le_mul_snorm_aux_of_neg h hc p,
simp only [snorm_eq_zero_iff (Lp.ae_strongly_measurable _) pzero, ← eq_zero_iff_ae_eq_zero]
at this,
simp [this] }
end
lemma norm_le_norm_of_ae_le {f : Lp E p μ} {g : Lp F p μ} (h : ∀ᵐ x ∂μ, ∥f x∥ ≤ ∥g x∥) :
∥f∥ ≤ ∥g∥ :=
begin
rw [norm_def, norm_def, ennreal.to_real_le_to_real (snorm_ne_top _) (snorm_ne_top _)],
exact snorm_mono_ae h
end
lemma mem_Lp_of_ae_le_mul {c : ℝ} {f : α →ₘ[μ] E} {g : Lp F p μ} (h : ∀ᵐ x ∂μ, ∥f x∥ ≤ c * ∥g x∥) :
f ∈ Lp E p μ :=
mem_Lp_iff_mem_ℒp.2 $ mem_ℒp.of_le_mul (Lp.mem_ℒp g) f.ae_strongly_measurable h
lemma mem_Lp_of_ae_le {f : α →ₘ[μ] E} {g : Lp F p μ} (h : ∀ᵐ x ∂μ, ∥f x∥ ≤ ∥g x∥) :
f ∈ Lp E p μ :=
mem_Lp_iff_mem_ℒp.2 $ mem_ℒp.of_le (Lp.mem_ℒp g) f.ae_strongly_measurable h
lemma mem_Lp_of_ae_bound [is_finite_measure μ] {f : α →ₘ[μ] E} (C : ℝ) (hfC : ∀ᵐ x ∂μ, ∥f x∥ ≤ C) :
f ∈ Lp E p μ :=
mem_Lp_iff_mem_ℒp.2 $ mem_ℒp.of_bound f.ae_strongly_measurable _ hfC
lemma norm_le_of_ae_bound [is_finite_measure μ] {f : Lp E p μ} {C : ℝ} (hC : 0 ≤ C)
(hfC : ∀ᵐ x ∂μ, ∥f x∥ ≤ C) :
∥f∥ ≤ (measure_univ_nnreal μ) ^ (p.to_real)⁻¹ * C :=
begin
by_cases hμ : μ = 0,
{ by_cases hp : p.to_real⁻¹ = 0,
{ simpa [hp, hμ, norm_def] using hC },
{ simp [hμ, norm_def, real.zero_rpow hp] } },
let A : ℝ≥0 := (measure_univ_nnreal μ) ^ (p.to_real)⁻¹ * ⟨C, hC⟩,
suffices : snorm f p μ ≤ A,
{ exact ennreal.to_real_le_coe_of_le_coe this },
convert snorm_le_of_ae_bound hfC,
rw [← coe_measure_univ_nnreal μ, ennreal.coe_rpow_of_ne_zero (measure_univ_nnreal_pos hμ).ne',
ennreal.coe_mul],
congr,
rw max_eq_left hC
end
instance [hp : fact (1 ≤ p)] : normed_add_comm_group (Lp E p μ) :=
{ edist := edist,
edist_dist := λ f g, by
rw [edist_def, dist_def, ←snorm_congr_ae (coe_fn_sub _ _),
ennreal.of_real_to_real (snorm_ne_top (f - g))],
.. normed_add_comm_group.of_core (Lp E p μ)
{ norm_eq_zero_iff := λ f, norm_eq_zero_iff (ennreal.zero_lt_one.trans_le hp.1),
triangle := begin
assume f g,
simp only [norm_def],
rw ← ennreal.to_real_add (snorm_ne_top f) (snorm_ne_top g),
suffices h_snorm : snorm ⇑(f + g) p μ ≤ snorm ⇑f p μ + snorm ⇑g p μ,
{ rwa ennreal.to_real_le_to_real (snorm_ne_top (f + g)),
exact ennreal.add_ne_top.mpr ⟨snorm_ne_top f, snorm_ne_top g⟩, },
rw [snorm_congr_ae (coe_fn_add _ _)],
exact snorm_add_le (Lp.ae_strongly_measurable f) (Lp.ae_strongly_measurable g) hp.1,
end,
norm_neg := by simp } }
-- check no diamond is created
example [fact (1 ≤ p)] :
pseudo_emetric_space.to_has_edist = (Lp.has_edist : has_edist (Lp E p μ)) :=
rfl
section normed_space
variables {𝕜 : Type*} [normed_field 𝕜] [normed_space 𝕜 E]
lemma mem_Lp_const_smul (c : 𝕜) (f : Lp E p μ) : c • ↑f ∈ Lp E p μ :=
begin
rw [mem_Lp_iff_snorm_lt_top, snorm_congr_ae (ae_eq_fun.coe_fn_smul _ _), snorm_const_smul,
ennreal.mul_lt_top_iff],
exact or.inl ⟨ennreal.coe_lt_top, f.prop⟩,
end
variables (E p μ 𝕜)
/-- The `𝕜`-submodule of elements of `α →ₘ[μ] E` whose `Lp` norm is finite. This is `Lp E p μ`,
with extra structure. -/
def Lp_submodule : submodule 𝕜 (α →ₘ[μ] E) :=
{ smul_mem' := λ c f hf, by simpa using mem_Lp_const_smul c ⟨f, hf⟩,
.. Lp E p μ }
variables {E p μ 𝕜}
lemma coe_Lp_submodule : (Lp_submodule E p μ 𝕜).to_add_subgroup = Lp E p μ := rfl
instance : module 𝕜 (Lp E p μ) :=
{ .. (Lp_submodule E p μ 𝕜).module }
lemma coe_fn_smul (c : 𝕜) (f : Lp E p μ) : ⇑(c • f) =ᵐ[μ] c • f := ae_eq_fun.coe_fn_smul _ _
lemma norm_const_smul (c : 𝕜) (f : Lp E p μ) : ∥c • f∥ = ∥c∥ * ∥f∥ :=
by rw [norm_def, snorm_congr_ae (coe_fn_smul _ _), snorm_const_smul c,
ennreal.to_real_mul, ennreal.coe_to_real, coe_nnnorm, norm_def]
instance [fact (1 ≤ p)] : normed_space 𝕜 (Lp E p μ) :=
{ norm_smul_le := λ _ _, by simp [norm_const_smul] }
end normed_space
end Lp
namespace mem_ℒp
variables {𝕜 : Type*} [normed_field 𝕜] [normed_space 𝕜 E]
lemma to_Lp_const_smul {f : α → E} (c : 𝕜) (hf : mem_ℒp f p μ) :
(hf.const_smul c).to_Lp (c • f) = c • hf.to_Lp f := rfl
end mem_ℒp
/-! ### Indicator of a set as an element of Lᵖ
For a set `s` with `(hs : measurable_set s)` and `(hμs : μ s < ∞)`, we build
`indicator_const_Lp p hs hμs c`, the element of `Lp` corresponding to `s.indicator (λ x, c)`.
-/
section indicator
variables {s : set α} {hs : measurable_set s} {c : E} {f : α → E} {hf : ae_strongly_measurable f μ}
lemma snorm_ess_sup_indicator_le (s : set α) (f : α → G) :
snorm_ess_sup (s.indicator f) μ ≤ snorm_ess_sup f μ :=
begin
refine ess_sup_mono_ae (eventually_of_forall (λ x, _)),
rw [ennreal.coe_le_coe, nnnorm_indicator_eq_indicator_nnnorm],
exact set.indicator_le_self s _ x,
end
lemma snorm_ess_sup_indicator_const_le (s : set α) (c : G) :
snorm_ess_sup (s.indicator (λ x : α , c)) μ ≤ ∥c∥₊ :=
begin
by_cases hμ0 : μ = 0,
{ rw [hμ0, snorm_ess_sup_measure_zero, ennreal.coe_nonneg],
exact zero_le', },
{ exact (snorm_ess_sup_indicator_le s (λ x, c)).trans (snorm_ess_sup_const c hμ0).le, },
end
lemma snorm_ess_sup_indicator_const_eq (s : set α) (c : G) (hμs : μ s ≠ 0) :
snorm_ess_sup (s.indicator (λ x : α , c)) μ = ∥c∥₊ :=
begin
refine le_antisymm (snorm_ess_sup_indicator_const_le s c) _,
by_contra' h,
have h' := ae_iff.mp (ae_lt_of_ess_sup_lt h),
push_neg at h',
refine hμs (measure_mono_null (λ x hx_mem, _) h'),
rw [set.mem_set_of_eq, set.indicator_of_mem hx_mem],
exact le_rfl,
end
variables (hs)
lemma snorm_indicator_le {E : Type*} [normed_add_comm_group E] (f : α → E) :
snorm (s.indicator f) p μ ≤ snorm f p μ :=
begin
refine snorm_mono_ae (eventually_of_forall (λ x, _)),
suffices : ∥s.indicator f x∥₊ ≤ ∥f x∥₊,
{ exact nnreal.coe_mono this },
rw nnnorm_indicator_eq_indicator_nnnorm,
exact s.indicator_le_self _ x,
end
variables {hs}
lemma snorm_indicator_const {c : G} (hs : measurable_set s) (hp : p ≠ 0) (hp_top : p ≠ ∞) :
snorm (s.indicator (λ x, c)) p μ = ∥c∥₊ * (μ s) ^ (1 / p.to_real) :=
begin
have hp_pos : 0 < p.to_real, from ennreal.to_real_pos hp hp_top,
rw snorm_eq_lintegral_rpow_nnnorm hp hp_top,
simp_rw [nnnorm_indicator_eq_indicator_nnnorm, ennreal.coe_indicator],
have h_indicator_pow : (λ a : α, s.indicator (λ (x : α), (∥c∥₊ : ℝ≥0∞)) a ^ p.to_real)
= s.indicator (λ (x : α), ↑∥c∥₊ ^ p.to_real),
{ rw set.comp_indicator_const (∥c∥₊ : ℝ≥0∞) (λ x, x ^ p.to_real) _,
simp [hp_pos], },
rw [h_indicator_pow, lintegral_indicator _ hs, set_lintegral_const, ennreal.mul_rpow_of_nonneg],
{ rw [← ennreal.rpow_mul, mul_one_div_cancel hp_pos.ne.symm, ennreal.rpow_one], },
{ simp [hp_pos.le], },
end
lemma snorm_indicator_const' {c : G} (hs : measurable_set s) (hμs : μ s ≠ 0) (hp : p ≠ 0) :
snorm (s.indicator (λ _, c)) p μ = ∥c∥₊ * (μ s) ^ (1 / p.to_real) :=
begin
by_cases hp_top : p = ∞,
{ simp [hp_top, snorm_ess_sup_indicator_const_eq s c hμs], },
{ exact snorm_indicator_const hs hp hp_top, },
end
lemma mem_ℒp.indicator (hs : measurable_set s) (hf : mem_ℒp f p μ) :
mem_ℒp (s.indicator f) p μ :=
⟨hf.ae_strongly_measurable.indicator hs, lt_of_le_of_lt (snorm_indicator_le f) hf.snorm_lt_top⟩
lemma snorm_ess_sup_indicator_eq_snorm_ess_sup_restrict {f : α → F} (hs : measurable_set s) :
snorm_ess_sup (s.indicator f) μ = snorm_ess_sup f (μ.restrict s) :=
begin
simp_rw [snorm_ess_sup, nnnorm_indicator_eq_indicator_nnnorm, ennreal.coe_indicator],
by_cases hs_null : μ s = 0,
{ rw measure.restrict_zero_set hs_null,
simp only [ess_sup_measure_zero, ennreal.ess_sup_eq_zero_iff, ennreal.bot_eq_zero],
have hs_empty : s =ᵐ[μ] (∅ : set α), by { rw ae_eq_set, simpa using hs_null, },
refine (indicator_ae_eq_of_ae_eq_set hs_empty).trans _,
rw set.indicator_empty,
refl, },
rw ess_sup_indicator_eq_ess_sup_restrict (eventually_of_forall (λ x, _)) hs hs_null,
rw pi.zero_apply,
exact zero_le _,
end
lemma snorm_indicator_eq_snorm_restrict {f : α → F} (hs : measurable_set s) :
snorm (s.indicator f) p μ = snorm f p (μ.restrict s) :=
begin
by_cases hp_zero : p = 0,
{ simp only [hp_zero, snorm_exponent_zero], },
by_cases hp_top : p = ∞,
{ simp_rw [hp_top, snorm_exponent_top],
exact snorm_ess_sup_indicator_eq_snorm_ess_sup_restrict hs, },
simp_rw snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top,
suffices : ∫⁻ x, ∥s.indicator f x∥₊ ^ p.to_real ∂μ = ∫⁻ x in s, ∥f x∥₊ ^ p.to_real ∂μ,
by rw this,
rw ← lintegral_indicator _ hs,
congr,
simp_rw [nnnorm_indicator_eq_indicator_nnnorm, ennreal.coe_indicator],
have h_zero : (λ x, x ^ p.to_real) (0 : ℝ≥0∞) = 0,
by simp [ennreal.to_real_pos hp_zero hp_top],
exact (set.indicator_comp_of_zero h_zero).symm,
end
lemma mem_ℒp_indicator_iff_restrict (hs : measurable_set s) :
mem_ℒp (s.indicator f) p μ ↔ mem_ℒp f p (μ.restrict s) :=
by simp [mem_ℒp, ae_strongly_measurable_indicator_iff hs, snorm_indicator_eq_snorm_restrict hs]
lemma mem_ℒp_indicator_const (p : ℝ≥0∞) (hs : measurable_set s) (c : E) (hμsc : c = 0 ∨ μ s ≠ ∞) :
mem_ℒp (s.indicator (λ _, c)) p μ :=
begin
rw mem_ℒp_indicator_iff_restrict hs,
by_cases hp_zero : p = 0,
{ rw hp_zero, exact mem_ℒp_zero_iff_ae_strongly_measurable.mpr ae_strongly_measurable_const, },
by_cases hp_top : p = ∞,
{ rw hp_top,
exact mem_ℒp_top_of_bound ae_strongly_measurable_const (∥c∥)
(eventually_of_forall (λ x, le_rfl)), },
rw [mem_ℒp_const_iff hp_zero hp_top, measure.restrict_apply_univ],
cases hμsc,
{ exact or.inl hμsc, },
{ exact or.inr hμsc.lt_top, },
end
end indicator
section indicator_const_Lp
open set function
variables {s : set α} {hs : measurable_set s} {hμs : μ s ≠ ∞} {c : E}
/-- Indicator of a set as an element of `Lp`. -/
def indicator_const_Lp (p : ℝ≥0∞) (hs : measurable_set s) (hμs : μ s ≠ ∞) (c : E) : Lp E p μ :=
mem_ℒp.to_Lp (s.indicator (λ _, c)) (mem_ℒp_indicator_const p hs c (or.inr hμs))
lemma indicator_const_Lp_coe_fn : ⇑(indicator_const_Lp p hs hμs c) =ᵐ[μ] s.indicator (λ _, c) :=
mem_ℒp.coe_fn_to_Lp (mem_ℒp_indicator_const p hs c (or.inr hμs))
lemma indicator_const_Lp_coe_fn_mem :
∀ᵐ (x : α) ∂μ, x ∈ s → indicator_const_Lp p hs hμs c x = c :=
indicator_const_Lp_coe_fn.mono (λ x hx hxs, hx.trans (set.indicator_of_mem hxs _))
lemma indicator_const_Lp_coe_fn_nmem :
∀ᵐ (x : α) ∂μ, x ∉ s → indicator_const_Lp p hs hμs c x = 0 :=
indicator_const_Lp_coe_fn.mono (λ x hx hxs, hx.trans (set.indicator_of_not_mem hxs _))
lemma norm_indicator_const_Lp (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
∥indicator_const_Lp p hs hμs c∥ = ∥c∥ * (μ s).to_real ^ (1 / p.to_real) :=
by rw [Lp.norm_def, snorm_congr_ae indicator_const_Lp_coe_fn,
snorm_indicator_const hs hp_ne_zero hp_ne_top, ennreal.to_real_mul, ennreal.to_real_rpow,
ennreal.coe_to_real, coe_nnnorm]
lemma norm_indicator_const_Lp_top (hμs_ne_zero : μ s ≠ 0) : ∥indicator_const_Lp ∞ hs hμs c∥ = ∥c∥ :=
by rw [Lp.norm_def, snorm_congr_ae indicator_const_Lp_coe_fn,
snorm_indicator_const' hs hμs_ne_zero ennreal.top_ne_zero, ennreal.top_to_real, div_zero,
ennreal.rpow_zero, mul_one, ennreal.coe_to_real, coe_nnnorm]
lemma norm_indicator_const_Lp' (hp_pos : p ≠ 0) (hμs_pos : μ s ≠ 0) :
∥indicator_const_Lp p hs hμs c∥ = ∥c∥ * (μ s).to_real ^ (1 / p.to_real) :=
begin
by_cases hp_top : p = ∞,
{ rw [hp_top, ennreal.top_to_real, div_zero, real.rpow_zero, mul_one],
exact norm_indicator_const_Lp_top hμs_pos, },
{ exact norm_indicator_const_Lp hp_pos hp_top, },
end
@[simp] lemma indicator_const_empty :
indicator_const_Lp p measurable_set.empty (by simp : μ ∅ ≠ ∞) c = 0 :=
begin
rw Lp.eq_zero_iff_ae_eq_zero,
convert indicator_const_Lp_coe_fn,
simp [set.indicator_empty'],
end
lemma mem_ℒp_add_of_disjoint {f g : α → E}
(h : disjoint (support f) (support g)) (hf : strongly_measurable f) (hg : strongly_measurable g) :
mem_ℒp (f + g) p μ ↔ mem_ℒp f p μ ∧ mem_ℒp g p μ :=
begin
borelize E,
refine ⟨λ hfg, ⟨_, _⟩, λ h, h.1.add h.2⟩,
{ rw ← indicator_add_eq_left h, exact hfg.indicator (measurable_set_support hf.measurable) },
{ rw ← indicator_add_eq_right h, exact hfg.indicator (measurable_set_support hg.measurable) }
end
/-- The indicator of a disjoint union of two sets is the sum of the indicators of the sets. -/
lemma indicator_const_Lp_disjoint_union {s t : set α} (hs : measurable_set s)
(ht : measurable_set t) (hμs : μ s ≠ ∞) (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (c : E) :
(indicator_const_Lp p (hs.union ht) ((measure_union_le s t).trans_lt
(lt_top_iff_ne_top.mpr (ennreal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne c)
= indicator_const_Lp p hs hμs c + indicator_const_Lp p ht hμt c :=
begin
ext1,
refine indicator_const_Lp_coe_fn.trans (eventually_eq.trans _ (Lp.coe_fn_add _ _).symm),
refine eventually_eq.trans _
(eventually_eq.add indicator_const_Lp_coe_fn.symm indicator_const_Lp_coe_fn.symm),
rw set.indicator_union_of_disjoint (set.disjoint_iff_inter_eq_empty.mpr hst) _,
end
end indicator_const_Lp
lemma mem_ℒp.norm_rpow_div {f : α → E}
(hf : mem_ℒp f p μ) (q : ℝ≥0∞) :
mem_ℒp (λ (x : α), ∥f x∥ ^ q.to_real) (p/q) μ :=
begin
refine ⟨(hf.1.norm.ae_measurable.pow_const q.to_real).ae_strongly_measurable, _⟩,
by_cases q_top : q = ∞, { simp [q_top] },
by_cases q_zero : q = 0,
{ simp [q_zero],
by_cases p_zero : p = 0, { simp [p_zero] },
rw ennreal.div_zero p_zero,
exact (mem_ℒp_top_const (1 : ℝ)).2 },
rw snorm_norm_rpow _ (ennreal.to_real_pos q_zero q_top),
apply ennreal.rpow_lt_top_of_nonneg ennreal.to_real_nonneg,
rw [ennreal.of_real_to_real q_top, div_eq_mul_inv, mul_assoc,
ennreal.inv_mul_cancel q_zero q_top, mul_one],
exact hf.2.ne
end
lemma mem_ℒp_norm_rpow_iff {q : ℝ≥0∞} {f : α → E} (hf : ae_strongly_measurable f μ)
(q_zero : q ≠ 0) (q_top : q ≠ ∞) :
mem_ℒp (λ (x : α), ∥f x∥ ^ q.to_real) (p/q) μ ↔ mem_ℒp f p μ :=
begin
refine ⟨λ h, _, λ h, h.norm_rpow_div q⟩,
apply (mem_ℒp_norm_iff hf).1,
convert h.norm_rpow_div (q⁻¹),
{ ext x,
rw [real.norm_eq_abs, real.abs_rpow_of_nonneg (norm_nonneg _), ← real.rpow_mul (abs_nonneg _),
ennreal.to_real_inv, mul_inv_cancel, abs_of_nonneg (norm_nonneg _), real.rpow_one],
simp [ennreal.to_real_eq_zero_iff, not_or_distrib, q_zero, q_top] },
{ rw [div_eq_mul_inv, inv_inv, div_eq_mul_inv, mul_assoc, ennreal.inv_mul_cancel q_zero q_top,
mul_one] }
end
lemma mem_ℒp.norm_rpow {f : α → E}
(hf : mem_ℒp f p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
mem_ℒp (λ (x : α), ∥f x∥ ^ p.to_real) 1 μ :=
begin
convert hf.norm_rpow_div p,
rw [div_eq_mul_inv, ennreal.mul_inv_cancel hp_ne_zero hp_ne_top],
end
end measure_theory
open measure_theory
/-!
### Composition on `L^p`
We show that Lipschitz functions vanishing at zero act by composition on `L^p`, and specialize
this to the composition with continuous linear maps, and to the definition of the positive
part of an `L^p` function.
-/
section composition
variables {g : E → F} {c : ℝ≥0}
lemma lipschitz_with.comp_mem_ℒp {α E F} {K} [measurable_space α] {μ : measure α}
[normed_add_comm_group E] [normed_add_comm_group F] {f : α → E} {g : E → F}
(hg : lipschitz_with K g) (g0 : g 0 = 0) (hL : mem_ℒp f p μ) : mem_ℒp (g ∘ f) p μ :=
begin
have : ∀ᵐ x ∂μ, ∥g (f x)∥ ≤ K * ∥f x∥,
{ apply filter.eventually_of_forall (λ x, _),
rw [← dist_zero_right, ← dist_zero_right, ← g0],
apply hg.dist_le_mul },
exact hL.of_le_mul (hg.continuous.comp_ae_strongly_measurable hL.1) this,
end
lemma measure_theory.mem_ℒp.of_comp_antilipschitz_with {α E F} {K'}
[measurable_space α] {μ : measure α} [normed_add_comm_group E] [normed_add_comm_group F]
{f : α → E} {g : E → F} (hL : mem_ℒp (g ∘ f) p μ)
(hg : uniform_continuous g) (hg' : antilipschitz_with K' g) (g0 : g 0 = 0) : mem_ℒp f p μ :=
begin
have A : ∀ᵐ x ∂μ, ∥f x∥ ≤ K' * ∥g (f x)∥,
{ apply filter.eventually_of_forall (λ x, _),
rw [← dist_zero_right, ← dist_zero_right, ← g0],
apply hg'.le_mul_dist },
have B : ae_strongly_measurable f μ :=
((hg'.uniform_embedding hg).embedding.ae_strongly_measurable_comp_iff.1 hL.1),
exact hL.of_le_mul B A,
end
namespace lipschitz_with
lemma mem_ℒp_comp_iff_of_antilipschitz {α E F} {K K'} [measurable_space α] {μ : measure α}
[normed_add_comm_group E] [normed_add_comm_group F]
{f : α → E} {g : E → F} (hg : lipschitz_with K g) (hg' : antilipschitz_with K' g) (g0 : g 0 = 0) :
mem_ℒp (g ∘ f) p μ ↔ mem_ℒp f p μ :=
⟨λ h, h.of_comp_antilipschitz_with hg.uniform_continuous hg' g0, λ h, hg.comp_mem_ℒp g0 h⟩
/-- When `g` is a Lipschitz function sending `0` to `0` and `f` is in `Lp`, then `g ∘ f` is well
defined as an element of `Lp`. -/
def comp_Lp (hg : lipschitz_with c g) (g0 : g 0 = 0) (f : Lp E p μ) : Lp F p μ :=
⟨ae_eq_fun.comp g hg.continuous (f : α →ₘ[μ] E),
begin
suffices : ∀ᵐ x ∂μ, ∥ae_eq_fun.comp g hg.continuous (f : α →ₘ[μ] E) x∥ ≤ c * ∥f x∥,
{ exact Lp.mem_Lp_of_ae_le_mul this },
filter_upwards [ae_eq_fun.coe_fn_comp g hg.continuous (f : α →ₘ[μ] E)] with a ha,
simp only [ha],
rw [← dist_zero_right, ← dist_zero_right, ← g0],
exact hg.dist_le_mul (f a) 0,
end⟩
lemma coe_fn_comp_Lp (hg : lipschitz_with c g) (g0 : g 0 = 0) (f : Lp E p μ) :
hg.comp_Lp g0 f =ᵐ[μ] g ∘ f :=
ae_eq_fun.coe_fn_comp _ _ _
@[simp] lemma comp_Lp_zero (hg : lipschitz_with c g) (g0 : g 0 = 0) :
hg.comp_Lp g0 (0 : Lp E p μ) = 0 :=
begin
rw Lp.eq_zero_iff_ae_eq_zero,
apply (coe_fn_comp_Lp _ _ _).trans,
filter_upwards [Lp.coe_fn_zero E p μ] with _ ha,
simp [ha, g0],
end
lemma norm_comp_Lp_sub_le (hg : lipschitz_with c g) (g0 : g 0 = 0) (f f' : Lp E p μ) :
∥hg.comp_Lp g0 f - hg.comp_Lp g0 f'∥ ≤ c * ∥f - f'∥ :=
begin
apply Lp.norm_le_mul_norm_of_ae_le_mul,
filter_upwards [hg.coe_fn_comp_Lp g0 f, hg.coe_fn_comp_Lp g0 f',
Lp.coe_fn_sub (hg.comp_Lp g0 f) (hg.comp_Lp g0 f'), Lp.coe_fn_sub f f'] with a ha1 ha2 ha3 ha4,
simp [ha1, ha2, ha3, ha4, ← dist_eq_norm],
exact hg.dist_le_mul (f a) (f' a)
end
lemma norm_comp_Lp_le (hg : lipschitz_with c g) (g0 : g 0 = 0) (f : Lp E p μ) :
∥hg.comp_Lp g0 f∥ ≤ c * ∥f∥ :=
by simpa using hg.norm_comp_Lp_sub_le g0 f 0
lemma lipschitz_with_comp_Lp [fact (1 ≤ p)] (hg : lipschitz_with c g) (g0 : g 0 = 0) :
lipschitz_with c (hg.comp_Lp g0 : Lp E p μ → Lp F p μ) :=
lipschitz_with.of_dist_le_mul $ λ f g, by simp [dist_eq_norm, norm_comp_Lp_sub_le]
lemma continuous_comp_Lp [fact (1 ≤ p)] (hg : lipschitz_with c g) (g0 : g 0 = 0) :
continuous (hg.comp_Lp g0 : Lp E p μ → Lp F p μ) :=
(lipschitz_with_comp_Lp hg g0).continuous
end lipschitz_with
namespace continuous_linear_map
variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] [normed_space 𝕜 E] [normed_space 𝕜 F]
/-- Composing `f : Lp ` with `L : E →L[𝕜] F`. -/
def comp_Lp (L : E →L[𝕜] F) (f : Lp E p μ) : Lp F p μ :=
L.lipschitz.comp_Lp (map_zero L) f
lemma coe_fn_comp_Lp (L : E →L[𝕜] F) (f : Lp E p μ) :
∀ᵐ a ∂μ, (L.comp_Lp f) a = L (f a) :=
lipschitz_with.coe_fn_comp_Lp _ _ _
lemma coe_fn_comp_Lp' (L : E →L[𝕜] F) (f : Lp E p μ) :
L.comp_Lp f =ᵐ[μ] λ a, L (f a) :=
L.coe_fn_comp_Lp f
lemma comp_mem_ℒp (L : E →L[𝕜] F) (f : Lp E p μ) : mem_ℒp (L ∘ f) p μ :=
(Lp.mem_ℒp (L.comp_Lp f)).ae_eq (L.coe_fn_comp_Lp' f)
lemma comp_mem_ℒp' (L : E →L[𝕜] F) {f : α → E} (hf : mem_ℒp f p μ) : mem_ℒp (L ∘ f) p μ :=
(L.comp_mem_ℒp (hf.to_Lp f)).ae_eq (eventually_eq.fun_comp (hf.coe_fn_to_Lp) _)
section is_R_or_C
variables {K : Type*} [is_R_or_C K]
lemma _root_.measure_theory.mem_ℒp.of_real
{f : α → ℝ} (hf : mem_ℒp f p μ) : mem_ℒp (λ x, (f x : K)) p μ :=
(@is_R_or_C.of_real_clm K _).comp_mem_ℒp' hf
lemma _root_.measure_theory.mem_ℒp_re_im_iff {f : α → K} :
mem_ℒp (λ x, is_R_or_C.re (f x)) p μ ∧ mem_ℒp (λ x, is_R_or_C.im (f x)) p μ ↔
mem_ℒp f p μ :=
begin
refine ⟨_, λ hf, ⟨hf.re, hf.im⟩⟩,
rintro ⟨hre, him⟩,
convert hre.of_real.add (him.of_real.const_mul is_R_or_C.I),
{ ext1 x,
rw [pi.add_apply, mul_comm, is_R_or_C.re_add_im] },
all_goals { apply_instance }
end
end is_R_or_C
lemma add_comp_Lp (L L' : E →L[𝕜] F) (f : Lp E p μ) :
(L + L').comp_Lp f = L.comp_Lp f + L'.comp_Lp f :=
begin
ext1,
refine (coe_fn_comp_Lp' (L + L') f).trans _,
refine eventually_eq.trans _ (Lp.coe_fn_add _ _).symm,
refine eventually_eq.trans _
(eventually_eq.add (L.coe_fn_comp_Lp' f).symm (L'.coe_fn_comp_Lp' f).symm),
refine eventually_of_forall (λ x, _),
refl,
end
lemma smul_comp_Lp {𝕜'} [normed_field 𝕜'] [normed_space 𝕜' F] [smul_comm_class 𝕜 𝕜' F]
(c : 𝕜') (L : E →L[𝕜] F) (f : Lp E p μ) :
(c • L).comp_Lp f = c • L.comp_Lp f :=
begin
ext1,
refine (coe_fn_comp_Lp' (c • L) f).trans _,
refine eventually_eq.trans _ (Lp.coe_fn_smul _ _).symm,
refine (L.coe_fn_comp_Lp' f).mono (λ x hx, _),
rw [pi.smul_apply, hx],
refl,
end
lemma norm_comp_Lp_le (L : E →L[𝕜] F) (f : Lp E p μ) : ∥L.comp_Lp f∥ ≤ ∥L∥ * ∥f∥ :=
lipschitz_with.norm_comp_Lp_le _ _ _
variables (μ p)
/-- Composing `f : Lp E p μ` with `L : E →L[𝕜] F`, seen as a `𝕜`-linear map on `Lp E p μ`. -/
def comp_Lpₗ (L : E →L[𝕜] F) : (Lp E p μ) →ₗ[𝕜] (Lp F p μ) :=
{ to_fun := λ f, L.comp_Lp f,
map_add' := begin
intros f g,
ext1,
filter_upwards [Lp.coe_fn_add f g, coe_fn_comp_Lp L (f + g), coe_fn_comp_Lp L f,
coe_fn_comp_Lp L g, Lp.coe_fn_add (L.comp_Lp f) (L.comp_Lp g)],
assume a ha1 ha2 ha3 ha4 ha5,
simp only [ha1, ha2, ha3, ha4, ha5, map_add, pi.add_apply],
end,
map_smul' := begin
intros c f,
dsimp,
ext1,
filter_upwards [Lp.coe_fn_smul c f, coe_fn_comp_Lp L (c • f), Lp.coe_fn_smul c (L.comp_Lp f),
coe_fn_comp_Lp L f] with _ ha1 ha2 ha3 ha4,
simp only [ha1, ha2, ha3, ha4, smul_hom_class.map_smul, pi.smul_apply],
end }
/-- Composing `f : Lp E p μ` with `L : E →L[𝕜] F`, seen as a continuous `𝕜`-linear map on
`Lp E p μ`. See also the similar
* `linear_map.comp_left` for functions,
* `continuous_linear_map.comp_left_continuous` for continuous functions,
* `continuous_linear_map.comp_left_continuous_bounded` for bounded continuous functions,
* `continuous_linear_map.comp_left_continuous_compact` for continuous functions on compact spaces.
-/
def comp_LpL [fact (1 ≤ p)] (L : E →L[𝕜] F) : (Lp E p μ) →L[𝕜] (Lp F p μ) :=
linear_map.mk_continuous (L.comp_Lpₗ p μ) ∥L∥ L.norm_comp_Lp_le
variables {μ p}
lemma coe_fn_comp_LpL [fact (1 ≤ p)] (L : E →L[𝕜] F) (f : Lp E p μ) :
L.comp_LpL p μ f =ᵐ[μ] λ a, L (f a) :=
L.coe_fn_comp_Lp f
lemma add_comp_LpL [fact (1 ≤ p)] (L L' : E →L[𝕜] F) :
(L + L').comp_LpL p μ = L.comp_LpL p μ + L'.comp_LpL p μ :=
by { ext1 f, exact add_comp_Lp L L' f }
lemma smul_comp_LpL [fact (1 ≤ p)] (c : 𝕜) (L : E →L[𝕜] F) :
(c • L).comp_LpL p μ = c • (L.comp_LpL p μ) :=
by { ext1 f, exact smul_comp_Lp c L f }
/-- TODO: written in an "apply" way because of a missing `has_smul` instance. -/
lemma smul_comp_LpL_apply [fact (1 ≤ p)] {𝕜'} [normed_field 𝕜'] [normed_space 𝕜' F]
[smul_comm_class 𝕜 𝕜' F] (c : 𝕜') (L : E →L[𝕜] F) (f : Lp E p μ) :
(c • L).comp_LpL p μ f = c • (L.comp_LpL p μ f) :=
smul_comp_Lp c L f
lemma norm_compLpL_le [fact (1 ≤ p)] (L : E →L[𝕜] F) :
∥L.comp_LpL p μ∥ ≤ ∥L∥ :=
linear_map.mk_continuous_norm_le _ (norm_nonneg _) _
end continuous_linear_map
namespace measure_theory
lemma indicator_const_Lp_eq_to_span_singleton_comp_Lp {s : set α} [normed_space ℝ F]
(hs : measurable_set s) (hμs : μ s ≠ ∞) (x : F) :
indicator_const_Lp 2 hs hμs x =
(continuous_linear_map.to_span_singleton ℝ x).comp_Lp (indicator_const_Lp 2 hs hμs (1 : ℝ)) :=
begin
ext1,
refine indicator_const_Lp_coe_fn.trans _,
have h_comp_Lp := (continuous_linear_map.to_span_singleton ℝ x).coe_fn_comp_Lp
(indicator_const_Lp 2 hs hμs (1 : ℝ)),
rw ← eventually_eq at h_comp_Lp,
refine eventually_eq.trans _ h_comp_Lp.symm,
refine (@indicator_const_Lp_coe_fn _ _ _ 2 μ _ s hs hμs (1 : ℝ)).mono (λ y hy, _),
dsimp only,
rw hy,
simp_rw [continuous_linear_map.to_span_singleton_apply],
by_cases hy_mem : y ∈ s; simp [hy_mem, continuous_linear_map.lsmul_apply],
end
namespace Lp
section pos_part
lemma lipschitz_with_pos_part : lipschitz_with 1 (λ (x : ℝ), max x 0) :=
lipschitz_with.of_dist_le_mul $ λ x y, by simp [real.dist_eq, abs_max_sub_max_le_abs]
lemma _root_.measure_theory.mem_ℒp.pos_part {f : α → ℝ} (hf : mem_ℒp f p μ) :
mem_ℒp (λ x, max (f x) 0) p μ :=
lipschitz_with_pos_part.comp_mem_ℒp (max_eq_right le_rfl) hf
lemma _root_.measure_theory.mem_ℒp.neg_part {f : α → ℝ} (hf : mem_ℒp f p μ) :
mem_ℒp (λ x, max (-f x) 0) p μ :=
lipschitz_with_pos_part.comp_mem_ℒp (max_eq_right le_rfl) hf.neg
/-- Positive part of a function in `L^p`. -/
def pos_part (f : Lp ℝ p μ) : Lp ℝ p μ :=
lipschitz_with_pos_part.comp_Lp (max_eq_right le_rfl) f
/-- Negative part of a function in `L^p`. -/
def neg_part (f : Lp ℝ p μ) : Lp ℝ p μ := pos_part (-f)
@[norm_cast]
lemma coe_pos_part (f : Lp ℝ p μ) : (pos_part f : α →ₘ[μ] ℝ) = (f : α →ₘ[μ] ℝ).pos_part := rfl
lemma coe_fn_pos_part (f : Lp ℝ p μ) : ⇑(pos_part f) =ᵐ[μ] λ a, max (f a) 0 :=
ae_eq_fun.coe_fn_pos_part _
lemma coe_fn_neg_part_eq_max (f : Lp ℝ p μ) : ∀ᵐ a ∂μ, neg_part f a = max (- f a) 0 :=
begin
rw neg_part,
filter_upwards [coe_fn_pos_part (-f), coe_fn_neg f] with _ h₁ h₂,
rw [h₁, h₂, pi.neg_apply],
end
lemma coe_fn_neg_part (f : Lp ℝ p μ) : ∀ᵐ a ∂μ, neg_part f a = - min (f a) 0 :=
(coe_fn_neg_part_eq_max f).mono $ assume a h,
by rw [h, ← max_neg_neg, neg_zero]
lemma continuous_pos_part [fact (1 ≤ p)] : continuous (λf : Lp ℝ p μ, pos_part f) :=
lipschitz_with.continuous_comp_Lp _ _
lemma continuous_neg_part [fact (1 ≤ p)] : continuous (λf : Lp ℝ p μ, neg_part f) :=
have eq : (λf : Lp ℝ p μ, neg_part f) = (λf : Lp ℝ p μ, pos_part (-f)) := rfl,
by { rw eq, exact continuous_pos_part.comp continuous_neg }
end pos_part
end Lp
end measure_theory
end composition
/-!
## `L^p` is a complete space
We show that `L^p` is a complete space for `1 ≤ p`.
-/
section complete_space
namespace measure_theory
namespace Lp
lemma snorm'_lim_eq_lintegral_liminf {ι} [nonempty ι] [linear_order ι] {f : ι → α → G} {p : ℝ}
(hp_nonneg : 0 ≤ p) {f_lim : α → G}
(h_lim : ∀ᵐ (x : α) ∂μ, tendsto (λ n, f n x) at_top (𝓝 (f_lim x))) :
snorm' f_lim p μ = (∫⁻ a, at_top.liminf (λ m, (∥f m a∥₊ : ℝ≥0∞)^p) ∂μ) ^ (1/p) :=
begin
suffices h_no_pow : (∫⁻ a, ∥f_lim a∥₊ ^ p ∂μ)
= (∫⁻ a, at_top.liminf (λ m, (∥f m a∥₊ : ℝ≥0∞)^p) ∂μ),
{ rw [snorm', h_no_pow], },
refine lintegral_congr_ae (h_lim.mono (λ a ha, _)),
rw tendsto.liminf_eq,
simp_rw [ennreal.coe_rpow_of_nonneg _ hp_nonneg, ennreal.tendsto_coe],
refine ((nnreal.continuous_rpow_const hp_nonneg).tendsto (∥f_lim a∥₊)).comp _,
exact (continuous_nnnorm.tendsto (f_lim a)).comp ha,
end
lemma snorm'_lim_le_liminf_snorm' {E} [normed_add_comm_group E] {f : ℕ → α → E} {p : ℝ}
(hp_pos : 0 < p) (hf : ∀ n, ae_strongly_measurable (f n) μ) {f_lim : α → E}
(h_lim : ∀ᵐ (x : α) ∂μ, tendsto (λ n, f n x) at_top (𝓝 (f_lim x))) :
snorm' f_lim p μ ≤ at_top.liminf (λ n, snorm' (f n) p μ) :=
begin
rw snorm'_lim_eq_lintegral_liminf hp_pos.le h_lim,
rw [←ennreal.le_rpow_one_div_iff (by simp [hp_pos] : 0 < 1 / p), one_div_one_div],
refine (lintegral_liminf_le' (λ m, ((hf m).ennnorm.pow_const _))).trans_eq _,
have h_pow_liminf : at_top.liminf (λ n, snorm' (f n) p μ) ^ p
= at_top.liminf (λ n, (snorm' (f n) p μ) ^ p),
{ have h_rpow_mono := ennreal.strict_mono_rpow_of_pos hp_pos,
have h_rpow_surj := (ennreal.rpow_left_bijective hp_pos.ne.symm).2,
refine (h_rpow_mono.order_iso_of_surjective _ h_rpow_surj).liminf_apply _ _ _ _,
all_goals { is_bounded_default }, },
rw h_pow_liminf,
simp_rw [snorm', ← ennreal.rpow_mul, one_div, inv_mul_cancel hp_pos.ne.symm, ennreal.rpow_one],
end
lemma snorm_exponent_top_lim_eq_ess_sup_liminf {ι} [nonempty ι] [linear_order ι] {f : ι → α → G}
{f_lim : α → G}
(h_lim : ∀ᵐ (x : α) ∂μ, tendsto (λ n, f n x) at_top (𝓝 (f_lim x))) :
snorm f_lim ∞ μ = ess_sup (λ x, at_top.liminf (λ m, (∥f m x∥₊ : ℝ≥0∞))) μ :=
begin
rw [snorm_exponent_top, snorm_ess_sup],
refine ess_sup_congr_ae (h_lim.mono (λ x hx, _)),
rw tendsto.liminf_eq,
rw ennreal.tendsto_coe,
exact (continuous_nnnorm.tendsto (f_lim x)).comp hx,
end
lemma snorm_exponent_top_lim_le_liminf_snorm_exponent_top {ι} [nonempty ι] [countable ι]
[linear_order ι] {f : ι → α → F} {f_lim : α → F}
(h_lim : ∀ᵐ (x : α) ∂μ, tendsto (λ n, f n x) at_top (𝓝 (f_lim x))) :
snorm f_lim ∞ μ ≤ at_top.liminf (λ n, snorm (f n) ∞ μ) :=
begin
rw snorm_exponent_top_lim_eq_ess_sup_liminf h_lim,
simp_rw [snorm_exponent_top, snorm_ess_sup],
exact ennreal.ess_sup_liminf_le (λ n, (λ x, (∥f n x∥₊ : ℝ≥0∞))),
end
lemma snorm_lim_le_liminf_snorm {E} [normed_add_comm_group E]
{f : ℕ → α → E} (hf : ∀ n, ae_strongly_measurable (f n) μ) (f_lim : α → E)
(h_lim : ∀ᵐ (x : α) ∂μ, tendsto (λ n, f n x) at_top (𝓝 (f_lim x))) :
snorm f_lim p μ ≤ at_top.liminf (λ n, snorm (f n) p μ) :=
begin
by_cases hp0 : p = 0,
{ simp [hp0], },
rw ← ne.def at hp0,
by_cases hp_top : p = ∞,
{ simp_rw [hp_top],
exact snorm_exponent_top_lim_le_liminf_snorm_exponent_top h_lim, },
simp_rw snorm_eq_snorm' hp0 hp_top,
have hp_pos : 0 < p.to_real, from ennreal.to_real_pos hp0 hp_top,
exact snorm'_lim_le_liminf_snorm' hp_pos hf h_lim,
end
/-! ### `Lp` is complete iff Cauchy sequences of `ℒp` have limits in `ℒp` -/
lemma tendsto_Lp_iff_tendsto_ℒp' {ι} {fi : filter ι} [fact (1 ≤ p)]
(f : ι → Lp E p μ) (f_lim : Lp E p μ) :
fi.tendsto f (𝓝 f_lim) ↔ fi.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0) :=
begin
rw tendsto_iff_dist_tendsto_zero,
simp_rw dist_def,
rw [← ennreal.zero_to_real, ennreal.tendsto_to_real_iff (λ n, _) ennreal.zero_ne_top],
rw snorm_congr_ae (Lp.coe_fn_sub _ _).symm,
exact Lp.snorm_ne_top _,
end
lemma tendsto_Lp_iff_tendsto_ℒp {ι} {fi : filter ι} [fact (1 ≤ p)]
(f : ι → Lp E p μ) (f_lim : α → E) (f_lim_ℒp : mem_ℒp f_lim p μ) :
fi.tendsto f (𝓝 (f_lim_ℒp.to_Lp f_lim)) ↔ fi.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0) :=
begin
rw tendsto_Lp_iff_tendsto_ℒp',
suffices h_eq : (λ n, snorm (f n - mem_ℒp.to_Lp f_lim f_lim_ℒp) p μ)
= (λ n, snorm (f n - f_lim) p μ),
by rw h_eq,
exact funext (λ n, snorm_congr_ae (eventually_eq.rfl.sub (mem_ℒp.coe_fn_to_Lp f_lim_ℒp))),
end
lemma tendsto_Lp_iff_tendsto_ℒp'' {ι} {fi : filter ι} [fact (1 ≤ p)]
(f : ι → α → E) (f_ℒp : ∀ n, mem_ℒp (f n) p μ) (f_lim : α → E) (f_lim_ℒp : mem_ℒp f_lim p μ) :
fi.tendsto (λ n, (f_ℒp n).to_Lp (f n)) (𝓝 (f_lim_ℒp.to_Lp f_lim))
↔ fi.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0) :=
begin
convert Lp.tendsto_Lp_iff_tendsto_ℒp' _ _,
ext1 n,
apply snorm_congr_ae,
filter_upwards [((f_ℒp n).sub f_lim_ℒp).coe_fn_to_Lp,
Lp.coe_fn_sub ((f_ℒp n).to_Lp (f n)) (f_lim_ℒp.to_Lp f_lim)] with _ hx₁ hx₂,
rw ← hx₂,
exact hx₁.symm,
end
lemma tendsto_Lp_of_tendsto_ℒp {ι} {fi : filter ι} [hp : fact (1 ≤ p)]
{f : ι → Lp E p μ} (f_lim : α → E) (f_lim_ℒp : mem_ℒp f_lim p μ)
(h_tendsto : fi.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0)) :
fi.tendsto f (𝓝 (f_lim_ℒp.to_Lp f_lim)) :=
(tendsto_Lp_iff_tendsto_ℒp f f_lim f_lim_ℒp).mpr h_tendsto
lemma cauchy_seq_Lp_iff_cauchy_seq_ℒp {ι} [nonempty ι] [semilattice_sup ι] [hp : fact (1 ≤ p)]
(f : ι → Lp E p μ) :
cauchy_seq f ↔ tendsto (λ (n : ι × ι), snorm (f n.fst - f n.snd) p μ) at_top (𝓝 0) :=
begin
simp_rw [cauchy_seq_iff_tendsto_dist_at_top_0, dist_def],
rw [← ennreal.zero_to_real, ennreal.tendsto_to_real_iff (λ n, _) ennreal.zero_ne_top],
rw snorm_congr_ae (Lp.coe_fn_sub _ _).symm,
exact snorm_ne_top _,
end
lemma complete_space_Lp_of_cauchy_complete_ℒp [hp : fact (1 ≤ p)]
(H : ∀ (f : ℕ → α → E) (hf : ∀ n, mem_ℒp (f n) p μ) (B : ℕ → ℝ≥0∞) (hB : ∑' i, B i < ∞)
(h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N),
∃ (f_lim : α → E) (hf_lim_meas : mem_ℒp f_lim p μ),
at_top.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0)) :
complete_space (Lp E p μ) :=
begin
let B := λ n : ℕ, ((1:ℝ) / 2) ^ n,
have hB_pos : ∀ n, 0 < B n, from λ n, pow_pos (div_pos zero_lt_one zero_lt_two) n,
refine metric.complete_of_convergent_controlled_sequences B hB_pos (λ f hf, _),
rsuffices ⟨f_lim, hf_lim_meas, h_tendsto⟩ : ∃ (f_lim : α → E) (hf_lim_meas : mem_ℒp f_lim p μ),
at_top.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0),
{ exact ⟨hf_lim_meas.to_Lp f_lim, tendsto_Lp_of_tendsto_ℒp f_lim hf_lim_meas h_tendsto⟩, },
have hB : summable B, from summable_geometric_two,
cases hB with M hB,
let B1 := λ n, ennreal.of_real (B n),
have hB1_has : has_sum B1 (ennreal.of_real M),
{ have h_tsum_B1 : ∑' i, B1 i = (ennreal.of_real M),
{ change (∑' (n : ℕ), ennreal.of_real (B n)) = ennreal.of_real M,
rw ←hB.tsum_eq,
exact (ennreal.of_real_tsum_of_nonneg (λ n, le_of_lt (hB_pos n)) hB.summable).symm, },
have h_sum := (@ennreal.summable _ B1).has_sum,
rwa h_tsum_B1 at h_sum, },
have hB1 : ∑' i, B1 i < ∞, by {rw hB1_has.tsum_eq, exact ennreal.of_real_lt_top, },
let f1 : ℕ → α → E := λ n, f n,
refine H f1 (λ n, Lp.mem_ℒp (f n)) B1 hB1 (λ N n m hn hm, _),
specialize hf N n m hn hm,
rw dist_def at hf,
simp_rw [f1, B1],
rwa ennreal.lt_of_real_iff_to_real_lt,
rw snorm_congr_ae (Lp.coe_fn_sub _ _).symm,
exact Lp.snorm_ne_top _,
end
/-! ### Prove that controlled Cauchy sequences of `ℒp` have limits in `ℒp` -/
private lemma snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' {f : ℕ → α → E}
(hf : ∀ n, ae_strongly_measurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p)
{B : ℕ → ℝ≥0∞} (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm' (f n - f m) p μ < B N) (n : ℕ) :
snorm' (λ x, ∑ i in finset.range (n + 1), ∥f (i + 1) x - f i x∥) p μ ≤ ∑' i, B i :=
begin
let f_norm_diff := λ i x, ∥f (i + 1) x - f i x∥,
have hgf_norm_diff : ∀ n, (λ x, ∑ i in finset.range (n + 1), ∥f (i + 1) x - f i x∥)
= ∑ i in finset.range (n + 1), f_norm_diff i,
from λ n, funext (λ x, by simp [f_norm_diff]),
rw hgf_norm_diff,
refine (snorm'_sum_le (λ i _, ((hf (i+1)).sub (hf i)).norm) hp1).trans _,
simp_rw [←pi.sub_apply, snorm'_norm],
refine (finset.sum_le_sum _).trans (sum_le_tsum _ (λ m _, zero_le _) ennreal.summable),
exact λ m _, (h_cau m (m + 1) m (nat.le_succ m) (le_refl m)).le,
end
private lemma lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum {f : ℕ → α → E}
(hf : ∀ n, ae_strongly_measurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (n : ℕ)
(hn : snorm' (λ x, ∑ i in finset.range (n + 1), ∥f (i + 1) x - f i x∥) p μ ≤ ∑' i, B i) :
∫⁻ a, (∑ i in finset.range (n + 1), ∥f (i + 1) a - f i a∥₊ : ℝ≥0∞)^p ∂μ
≤ (∑' i, B i) ^ p :=
begin
have hp_pos : 0 < p := zero_lt_one.trans_le hp1,
rw [←one_div_one_div p, @ennreal.le_rpow_one_div_iff _ _ (1/p) (by simp [hp_pos]),
one_div_one_div p],
simp_rw snorm' at hn,
have h_nnnorm_nonneg :
(λ a, (∥∑ i in finset.range (n + 1), ∥f (i + 1) a - f i a∥∥₊ : ℝ≥0∞) ^ p)
= λ a, (∑ i in finset.range (n + 1), (∥f (i + 1) a - f i a∥₊ : ℝ≥0∞)) ^ p,
{ ext1 a,
congr,
simp_rw ←of_real_norm_eq_coe_nnnorm,
rw ←ennreal.of_real_sum_of_nonneg,
{ rw real.norm_of_nonneg _,
exact finset.sum_nonneg (λ x hx, norm_nonneg _), },
{ exact λ x hx, norm_nonneg _, }, },
change (∫⁻ a, (λ x, ↑∥∑ i in finset.range (n + 1), ∥f (i+1) x - f i x∥∥₊^p) a ∂μ)^(1/p)
≤ ∑' i, B i at hn,
rwa h_nnnorm_nonneg at hn,
end
private lemma lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum {f : ℕ → α → E}
(hf : ∀ n, ae_strongly_measurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞}
(h : ∀ n, ∫⁻ a, (∑ i in finset.range (n + 1), ∥f (i + 1) a - f i a∥₊ : ℝ≥0∞)^p ∂μ
≤ (∑' i, B i) ^ p) :
(∫⁻ a, (∑' i, ∥f (i + 1) a - f i a∥₊ : ℝ≥0∞)^p ∂μ) ^ (1/p) ≤ ∑' i, B i :=
begin
have hp_pos : 0 < p := zero_lt_one.trans_le hp1,
suffices h_pow : ∫⁻ a, (∑' i, ∥f (i + 1) a - f i a∥₊ : ℝ≥0∞)^p ∂μ ≤ (∑' i, B i) ^ p,
by rwa [←ennreal.le_rpow_one_div_iff (by simp [hp_pos] : 0 < 1 / p), one_div_one_div],
have h_tsum_1 : ∀ g : ℕ → ℝ≥0∞,
∑' i, g i = at_top.liminf (λ n, ∑ i in finset.range (n + 1), g i),
by { intro g, rw [ennreal.tsum_eq_liminf_sum_nat, ← liminf_nat_add _ 1], },
simp_rw h_tsum_1 _,
rw ← h_tsum_1,
have h_liminf_pow : ∫⁻ a, at_top.liminf (λ n, ∑ i in finset.range (n + 1),
(∥f (i + 1) a - f i a∥₊))^p ∂μ
= ∫⁻ a, at_top.liminf (λ n, (∑ i in finset.range (n + 1), (∥f (i + 1) a - f i a∥₊))^p) ∂μ,
{ refine lintegral_congr (λ x, _),
have h_rpow_mono := ennreal.strict_mono_rpow_of_pos (zero_lt_one.trans_le hp1),
have h_rpow_surj := (ennreal.rpow_left_bijective hp_pos.ne.symm).2,
refine (h_rpow_mono.order_iso_of_surjective _ h_rpow_surj).liminf_apply _ _ _ _,
all_goals { is_bounded_default }, },
rw h_liminf_pow,
refine (lintegral_liminf_le' _).trans _,
{ exact λ n, (finset.ae_measurable_sum (finset.range (n+1))
(λ i _, ((hf (i+1)).sub (hf i)).ennnorm)).pow_const _, },
{ exact liminf_le_of_frequently_le' (frequently_of_forall h), },
end
private lemma tsum_nnnorm_sub_ae_lt_top
{f : ℕ → α → E} (hf : ∀ n, ae_strongly_measurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞}
(hB : ∑' i, B i ≠ ∞)
(h : (∫⁻ a, (∑' i, ∥f (i + 1) a - f i a∥₊ : ℝ≥0∞)^p ∂μ) ^ (1/p) ≤ ∑' i, B i) :
∀ᵐ x ∂μ, (∑' i, ∥f (i + 1) x - f i x∥₊ : ℝ≥0∞) < ∞ :=
begin
have hp_pos : 0 < p := zero_lt_one.trans_le hp1,
have h_integral : ∫⁻ a, (∑' i, ∥f (i + 1) a - f i a∥₊ : ℝ≥0∞)^p ∂μ < ∞,
{ have h_tsum_lt_top : (∑' i, B i) ^ p < ∞,
from ennreal.rpow_lt_top_of_nonneg hp_pos.le hB,
refine lt_of_le_of_lt _ h_tsum_lt_top,
rwa [←ennreal.le_rpow_one_div_iff (by simp [hp_pos] : 0 < 1 / p), one_div_one_div] at h, },
have rpow_ae_lt_top : ∀ᵐ x ∂μ, (∑' i, ∥f (i + 1) x - f i x∥₊ : ℝ≥0∞)^p < ∞,
{ refine ae_lt_top' (ae_measurable.pow_const _ _) h_integral.ne,
exact ae_measurable.ennreal_tsum (λ n, ((hf (n+1)).sub (hf n)).ennnorm), },
refine rpow_ae_lt_top.mono (λ x hx, _),
rwa [←ennreal.lt_rpow_one_div_iff hp_pos,
ennreal.top_rpow_of_pos (by simp [hp_pos] : 0 < 1 / p)] at hx,
end
lemma ae_tendsto_of_cauchy_snorm' [complete_space E] {f : ℕ → α → E} {p : ℝ}
(hf : ∀ n, ae_strongly_measurable (f n) μ) (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
(h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm' (f n - f m) p μ < B N) :
∀ᵐ x ∂μ, ∃ l : E, at_top.tendsto (λ n, f n x) (𝓝 l) :=
begin
have h_summable : ∀ᵐ x ∂μ, summable (λ (i : ℕ), f (i + 1) x - f i x),
{ have h1 : ∀ n, snorm' (λ x, ∑ i in finset.range (n + 1), ∥f (i + 1) x - f i x∥) p μ
≤ ∑' i, B i,
from snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau,
have h2 : ∀ n, ∫⁻ a, (∑ i in finset.range (n + 1), ∥f (i + 1) a - f i a∥₊ : ℝ≥0∞)^p ∂μ
≤ (∑' i, B i) ^ p,
from λ n, lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hf hp1 n (h1 n),
have h3 : (∫⁻ a, (∑' i, ∥f (i + 1) a - f i a∥₊ : ℝ≥0∞)^p ∂μ) ^ (1/p) ≤ ∑' i, B i,
from lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2,
have h4 : ∀ᵐ x ∂μ, (∑' i, ∥f (i + 1) x - f i x∥₊ : ℝ≥0∞) < ∞,
from tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3,
exact h4.mono (λ x hx, summable_of_summable_nnnorm
(ennreal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx))), },
have h : ∀ᵐ x ∂μ, ∃ l : E,
at_top.tendsto (λ n, ∑ i in finset.range n, (f (i + 1) x - f i x)) (𝓝 l),
{ refine h_summable.mono (λ x hx, _),
let hx_sum := hx.has_sum.tendsto_sum_nat,
exact ⟨∑' i, (f (i + 1) x - f i x), hx_sum⟩, },
refine h.mono (λ x hx, _),
cases hx with l hx,
have h_rw_sum : (λ n, ∑ i in finset.range n, (f (i + 1) x - f i x)) = λ n, f n x - f 0 x,
{ ext1 n,
change ∑ (i : ℕ) in finset.range n, ((λ m, f m x) (i + 1) - (λ m, f m x) i) = f n x - f 0 x,
rw finset.sum_range_sub, },
rw h_rw_sum at hx,
have hf_rw : (λ n, f n x) = λ n, f n x - f 0 x + f 0 x, by { ext1 n, abel, },
rw hf_rw,
exact ⟨l + f 0 x, tendsto.add_const _ hx⟩,
end
lemma ae_tendsto_of_cauchy_snorm [complete_space E] {f : ℕ → α → E}
(hf : ∀ n, ae_strongly_measurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
(h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
∀ᵐ x ∂μ, ∃ l : E, at_top.tendsto (λ n, f n x) (𝓝 l) :=
begin
by_cases hp_top : p = ∞,
{ simp_rw [hp_top] at *,
have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (∥(f n - f m) x∥₊ : ℝ≥0∞) < B N,
{ simp_rw ae_all_iff,
exact λ N n m hnN hmN, ae_lt_of_ess_sup_lt (h_cau N n m hnN hmN), },
simp_rw [snorm_exponent_top, snorm_ess_sup] at h_cau,
refine h_cau_ae.mono (λ x hx, cauchy_seq_tendsto_of_complete _),
refine cauchy_seq_of_le_tendsto_0 (λ n, (B n).to_real) _ _,
{ intros n m N hnN hmN,
specialize hx N n m hnN hmN,
rw [dist_eq_norm, ←ennreal.to_real_of_real (norm_nonneg _),
ennreal.to_real_le_to_real ennreal.of_real_ne_top
(ennreal.ne_top_of_tsum_ne_top hB N)],
rw ←of_real_norm_eq_coe_nnnorm at hx,
exact hx.le, },
{ rw ← ennreal.zero_to_real,
exact tendsto.comp (ennreal.tendsto_to_real ennreal.zero_ne_top)
(ennreal.tendsto_at_top_zero_of_tsum_ne_top hB), }, },
have hp1 : 1 ≤ p.to_real,
{ rw [← ennreal.of_real_le_iff_le_to_real hp_top, ennreal.of_real_one],
exact hp, },
have h_cau' : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm' (f n - f m) (p.to_real) μ < B N,
{ intros N n m hn hm,
specialize h_cau N n m hn hm,
rwa snorm_eq_snorm' (ennreal.zero_lt_one.trans_le hp).ne.symm hp_top at h_cau, },
exact ae_tendsto_of_cauchy_snorm' hf hp1 hB h_cau',
end
lemma cauchy_tendsto_of_tendsto {f : ℕ → α → E} (hf : ∀ n, ae_strongly_measurable (f n) μ)
(f_lim : α → E) {B : ℕ → ℝ≥0∞}
(hB : ∑' i, B i ≠ ∞) (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N)
(h_lim : ∀ᵐ (x : α) ∂μ, tendsto (λ n, f n x) at_top (𝓝 (f_lim x))) :
at_top.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0) :=
begin
rw ennreal.tendsto_at_top_zero,
intros ε hε,
have h_B : ∃ (N : ℕ), B N ≤ ε,
{ suffices h_tendsto_zero : ∃ (N : ℕ), ∀ n : ℕ, N ≤ n → B n ≤ ε,
from ⟨h_tendsto_zero.some, h_tendsto_zero.some_spec _ le_rfl⟩,
exact (ennreal.tendsto_at_top_zero.mp (ennreal.tendsto_at_top_zero_of_tsum_ne_top hB))
ε hε, },
cases h_B with N h_B,
refine ⟨N, λ n hn, _⟩,
have h_sub : snorm (f n - f_lim) p μ ≤ at_top.liminf (λ m, snorm (f n - f m) p μ),
{ refine snorm_lim_le_liminf_snorm (λ m, (hf n).sub (hf m)) (f n - f_lim) _,
refine h_lim.mono (λ x hx, _),
simp_rw sub_eq_add_neg,
exact tendsto.add tendsto_const_nhds (tendsto.neg hx), },
refine h_sub.trans _,
refine liminf_le_of_frequently_le' (frequently_at_top.mpr _),
refine λ N1, ⟨max N N1, le_max_right _ _, _⟩,
exact (h_cau N n (max N N1) hn (le_max_left _ _)).le.trans h_B,
end
lemma mem_ℒp_of_cauchy_tendsto (hp : 1 ≤ p) {f : ℕ → α → E} (hf : ∀ n, mem_ℒp (f n) p μ)
(f_lim : α → E) (h_lim_meas : ae_strongly_measurable f_lim μ)
(h_tendsto : at_top.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0)) :
mem_ℒp f_lim p μ :=
begin
refine ⟨h_lim_meas, _⟩,
rw ennreal.tendsto_at_top_zero at h_tendsto,
cases (h_tendsto 1 ennreal.zero_lt_one) with N h_tendsto_1,
specialize h_tendsto_1 N (le_refl N),
have h_add : f_lim = f_lim - f N + f N, by abel,
rw h_add,
refine lt_of_le_of_lt (snorm_add_le (h_lim_meas.sub (hf N).1) (hf N).1 hp) _,
rw ennreal.add_lt_top,
split,
{ refine lt_of_le_of_lt _ ennreal.one_lt_top,
have h_neg : f_lim - f N = -(f N - f_lim), by simp,
rwa [h_neg, snorm_neg], },
{ exact (hf N).2, },
end
lemma cauchy_complete_ℒp [complete_space E] (hp : 1 ≤ p)
{f : ℕ → α → E} (hf : ∀ n, mem_ℒp (f n) p μ) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
(h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
∃ (f_lim : α → E) (hf_lim_meas : mem_ℒp f_lim p μ),
at_top.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0) :=
begin
obtain ⟨f_lim, h_f_lim_meas, h_lim⟩ : ∃ (f_lim : α → E) (hf_lim_meas : strongly_measurable f_lim),
∀ᵐ x ∂μ, tendsto (λ n, f n x) at_top (nhds (f_lim x)),
from exists_strongly_measurable_limit_of_tendsto_ae (λ n, (hf n).1)
(ae_tendsto_of_cauchy_snorm (λ n, (hf n).1) hp hB h_cau),
have h_tendsto' : at_top.tendsto (λ n, snorm (f n - f_lim) p μ) (𝓝 0),
from cauchy_tendsto_of_tendsto (λ m, (hf m).1) f_lim hB h_cau h_lim,
have h_ℒp_lim : mem_ℒp f_lim p μ,
from mem_ℒp_of_cauchy_tendsto hp hf f_lim h_f_lim_meas.ae_strongly_measurable h_tendsto',
exact ⟨f_lim, h_ℒp_lim, h_tendsto'⟩,
end
/-! ### `Lp` is complete for `1 ≤ p` -/
instance [complete_space E] [hp : fact (1 ≤ p)] : complete_space (Lp E p μ) :=
complete_space_Lp_of_cauchy_complete_ℒp $
λ f hf B hB h_cau, cauchy_complete_ℒp hp.elim hf hB.ne h_cau
end Lp
end measure_theory
end complete_space
/-! ### Continuous functions in `Lp` -/
open_locale bounded_continuous_function
open bounded_continuous_function
section
variables [topological_space α] [borel_space α] [second_countable_topology_either α E]
variables (E p μ)
/-- An additive subgroup of `Lp E p μ`, consisting of the equivalence classes which contain a
bounded continuous representative. -/
def measure_theory.Lp.bounded_continuous_function : add_subgroup (Lp E p μ) :=
add_subgroup.add_subgroup_of
((continuous_map.to_ae_eq_fun_add_hom μ).comp (to_continuous_map_add_hom α E)).range
(Lp E p μ)
variables {E p μ}
/-- By definition, the elements of `Lp.bounded_continuous_function E p μ` are the elements of
`Lp E p μ` which contain a bounded continuous representative. -/
lemma measure_theory.Lp.mem_bounded_continuous_function_iff {f : (Lp E p μ)} :
f ∈ measure_theory.Lp.bounded_continuous_function E p μ
↔ ∃ f₀ : (α →ᵇ E), f₀.to_continuous_map.to_ae_eq_fun μ = (f : α →ₘ[μ] E) :=
add_subgroup.mem_add_subgroup_of
namespace bounded_continuous_function
variables [is_finite_measure μ]
/-- A bounded continuous function on a finite-measure space is in `Lp`. -/
lemma mem_Lp (f : α →ᵇ E) :
f.to_continuous_map.to_ae_eq_fun μ ∈ Lp E p μ :=
begin
refine Lp.mem_Lp_of_ae_bound (∥f∥) _,
filter_upwards [f.to_continuous_map.coe_fn_to_ae_eq_fun μ] with x _,
convert f.norm_coe_le_norm x
end
/-- The `Lp`-norm of a bounded continuous function is at most a constant (depending on the measure
of the whole space) times its sup-norm. -/
lemma Lp_norm_le (f : α →ᵇ E) :
∥(⟨f.to_continuous_map.to_ae_eq_fun μ, mem_Lp f⟩ : Lp E p μ)∥
≤ (measure_univ_nnreal μ) ^ (p.to_real)⁻¹ * ∥f∥ :=
begin
apply Lp.norm_le_of_ae_bound (norm_nonneg f),
{ refine (f.to_continuous_map.coe_fn_to_ae_eq_fun μ).mono _,
intros x hx,
convert f.norm_coe_le_norm x },
{ apply_instance }
end
variables (p μ)
/-- The normed group homomorphism of considering a bounded continuous function on a finite-measure
space as an element of `Lp`. -/
def to_Lp_hom [fact (1 ≤ p)] : normed_add_group_hom (α →ᵇ E) (Lp E p μ) :=
{ bound' := ⟨_, Lp_norm_le⟩,
.. add_monoid_hom.cod_restrict
((continuous_map.to_ae_eq_fun_add_hom μ).comp (to_continuous_map_add_hom α E))
(Lp E p μ)
mem_Lp }
lemma range_to_Lp_hom [fact (1 ≤ p)] :
((to_Lp_hom p μ).range : add_subgroup (Lp E p μ))
= measure_theory.Lp.bounded_continuous_function E p μ :=
begin
symmetry,
convert add_monoid_hom.add_subgroup_of_range_eq_of_le
((continuous_map.to_ae_eq_fun_add_hom μ).comp (to_continuous_map_add_hom α E))
(by { rintros - ⟨f, rfl⟩, exact mem_Lp f } : _ ≤ Lp E p μ),
end
variables (𝕜 : Type*)
/-- The bounded linear map of considering a bounded continuous function on a finite-measure space
as an element of `Lp`. -/
def to_Lp [normed_field 𝕜] [normed_space 𝕜 E] [fact (1 ≤ p)] :
(α →ᵇ E) →L[𝕜] (Lp E p μ) :=
linear_map.mk_continuous
(linear_map.cod_restrict
(Lp.Lp_submodule E p μ 𝕜)
((continuous_map.to_ae_eq_fun_linear_map μ).comp (to_continuous_map_linear_map α E 𝕜))
mem_Lp)
_
Lp_norm_le
variables {𝕜}
lemma range_to_Lp [normed_field 𝕜] [normed_space 𝕜 E] [fact (1 ≤ p)] :
(((to_Lp p μ 𝕜).range : submodule 𝕜 (Lp E p μ)).to_add_subgroup)
= measure_theory.Lp.bounded_continuous_function E p μ :=
range_to_Lp_hom p μ
variables {p}
lemma coe_fn_to_Lp [normed_field 𝕜] [normed_space 𝕜 E] [fact (1 ≤ p)] (f : α →ᵇ E) :
to_Lp p μ 𝕜 f =ᵐ[μ] f :=
ae_eq_fun.coe_fn_mk f _
lemma to_Lp_norm_le [nontrivially_normed_field 𝕜] [normed_space 𝕜 E] [fact (1 ≤ p)] :
∥(to_Lp p μ 𝕜 : (α →ᵇ E) →L[𝕜] (Lp E p μ))∥ ≤ (measure_univ_nnreal μ) ^ (p.to_real)⁻¹ :=
linear_map.mk_continuous_norm_le _ ((measure_univ_nnreal μ) ^ (p.to_real)⁻¹).coe_nonneg _
end bounded_continuous_function
namespace continuous_map
variables [compact_space α] [is_finite_measure μ]
variables (𝕜 : Type*) (p μ) [fact (1 ≤ p)]
/-- The bounded linear map of considering a continuous function on a compact finite-measure
space `α` as an element of `Lp`. By definition, the norm on `C(α, E)` is the sup-norm, transferred
from the space `α →ᵇ E` of bounded continuous functions, so this construction is just a matter of
transferring the structure from `bounded_continuous_function.to_Lp` along the isometry. -/
def to_Lp [normed_field 𝕜] [normed_space 𝕜 E] :
C(α, E) →L[𝕜] (Lp E p μ) :=
(bounded_continuous_function.to_Lp p μ 𝕜).comp
(linear_isometry_bounded_of_compact α E 𝕜).to_linear_isometry.to_continuous_linear_map
variables {𝕜}
lemma range_to_Lp [normed_field 𝕜] [normed_space 𝕜 E] :
((to_Lp p μ 𝕜).range : submodule 𝕜 (Lp E p μ)).to_add_subgroup
= measure_theory.Lp.bounded_continuous_function E p μ :=
begin
refine set_like.ext' _,
have := (linear_isometry_bounded_of_compact α E 𝕜).surjective,
convert function.surjective.range_comp this (bounded_continuous_function.to_Lp p μ 𝕜),
rw ← bounded_continuous_function.range_to_Lp p μ,
refl,
end
variables {p}
lemma coe_fn_to_Lp [normed_field 𝕜] [normed_space 𝕜 E] (f : C(α, E)) :
to_Lp p μ 𝕜 f =ᵐ[μ] f :=
ae_eq_fun.coe_fn_mk f _
lemma to_Lp_def [normed_field 𝕜] [normed_space 𝕜 E] (f : C(α, E)) :
to_Lp p μ 𝕜 f
= bounded_continuous_function.to_Lp p μ 𝕜 (linear_isometry_bounded_of_compact α E 𝕜 f) :=
rfl
@[simp] lemma to_Lp_comp_to_continuous_map [normed_field 𝕜] [normed_space 𝕜 E] (f : α →ᵇ E) :
to_Lp p μ 𝕜 f.to_continuous_map
= bounded_continuous_function.to_Lp p μ 𝕜 f :=
rfl
@[simp] lemma coe_to_Lp [normed_field 𝕜] [normed_space 𝕜 E] (f : C(α, E)) :
(to_Lp p μ 𝕜 f : α →ₘ[μ] E) = f.to_ae_eq_fun μ :=
rfl
variables [nontrivially_normed_field 𝕜] [normed_space 𝕜 E]
lemma to_Lp_norm_eq_to_Lp_norm_coe :
∥(to_Lp p μ 𝕜 : C(α, E) →L[𝕜] (Lp E p μ))∥
= ∥(bounded_continuous_function.to_Lp p μ 𝕜 : (α →ᵇ E) →L[𝕜] (Lp E p μ))∥ :=
continuous_linear_map.op_norm_comp_linear_isometry_equiv _ _
/-- Bound for the operator norm of `continuous_map.to_Lp`. -/
lemma to_Lp_norm_le :
∥(to_Lp p μ 𝕜 : C(α, E) →L[𝕜] (Lp E p μ))∥ ≤ (measure_univ_nnreal μ) ^ (p.to_real)⁻¹ :=
by { rw to_Lp_norm_eq_to_Lp_norm_coe, exact bounded_continuous_function.to_Lp_norm_le μ }
end continuous_map
end
namespace measure_theory
namespace Lp
lemma pow_mul_meas_ge_le_norm (f : Lp E p μ)
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (ε : ℝ≥0∞) :
(ε * μ {x | ε ≤ ∥f x∥₊ ^ p.to_real}) ^ (1 / p.to_real) ≤ (ennreal.of_real ∥f∥) :=
(ennreal.of_real_to_real (snorm_ne_top f)).symm ▸
pow_mul_meas_ge_le_snorm μ hp_ne_zero hp_ne_top (Lp.ae_strongly_measurable f) ε
lemma mul_meas_ge_le_pow_norm (f : Lp E p μ)
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (ε : ℝ≥0∞) :
ε * μ {x | ε ≤ ∥f x∥₊ ^ p.to_real} ≤ (ennreal.of_real ∥f∥) ^ p.to_real :=
(ennreal.of_real_to_real (snorm_ne_top f)).symm ▸
mul_meas_ge_le_pow_snorm μ hp_ne_zero hp_ne_top (Lp.ae_strongly_measurable f) ε
/-- A version of Markov's inequality with elements of Lp. -/
lemma mul_meas_ge_le_pow_norm' (f : Lp E p μ)
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (ε : ℝ≥0∞) :
ε ^ p.to_real * μ {x | ε ≤ ∥f x∥₊} ≤ (ennreal.of_real ∥f∥) ^ p.to_real :=
(ennreal.of_real_to_real (snorm_ne_top f)).symm ▸
mul_meas_ge_le_pow_snorm' μ hp_ne_zero hp_ne_top (Lp.ae_strongly_measurable f) ε
lemma meas_ge_le_mul_pow_norm (f : Lp E p μ)
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
μ {x | ε ≤ ∥f x∥₊} ≤ ε⁻¹ ^ p.to_real * (ennreal.of_real ∥f∥) ^ p.to_real :=
(ennreal.of_real_to_real (snorm_ne_top f)).symm ▸
meas_ge_le_mul_pow_snorm μ hp_ne_zero hp_ne_top (Lp.ae_strongly_measurable f) hε
end Lp
end measure_theory
|
7940f184d410fcfc775c047edd73bfe9fac7a61c | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/linear_algebra/affine_space/midpoint.lean | 24a4ee2123c5714a7c6e8d62fa9329904e4ec093 | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 6,443 | lean | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Yury Kudryashov
-/
import algebra.invertible
import linear_algebra.affine_space.affine_equiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midpoint of the segment `[x, y]`. We define it for `x` and `y`
in a module over a ring `R` with invertible `2`.
* `add_monoid_hom.of_map_midpoint`: construct an `add_monoid_hom` given a map `f` such that
`f` sends zero to zero and midpoints to midpoints.
## Main theorems
* `midpoint_eq_iff`: `z` is the midpoint of `[x, y]` if and only if `x + y = z + z`,
* `midpoint_unique`: `midpoint R x y` does not depend on `R`;
* `midpoint x y` is linear both in `x` and `y`;
* `point_reflection_midpoint_left`, `point_reflection_midpoint_right`:
`equiv.point_reflection (midpoint R x y)` swaps `x` and `y`.
We do not mark most lemmas as `@[simp]` because it is hard to tell which side is simpler.
## Tags
midpoint, add_monoid_hom
-/
open affine_map affine_equiv
section
variables (R : Type*) {V V' P P' : Type*} [ring R] [invertible (2:R)]
[add_comm_group V] [semimodule R V] [add_torsor V P]
[add_comm_group V'] [semimodule R V'] [add_torsor V' P']
include V
/-- `midpoint x y` is the midpoint of the segment `[x, y]`. -/
def midpoint (x y : P) : P := line_map x y (⅟2:R)
variables {R} {x y z : P}
include V'
@[simp] lemma affine_map.map_midpoint (f : P →ᵃ[R] P') (a b : P) :
f (midpoint R a b) = midpoint R (f a) (f b) :=
f.apply_line_map a b _
@[simp] lemma affine_equiv.map_midpoint (f : P ≃ᵃ[R] P') (a b : P) :
f (midpoint R a b) = midpoint R (f a) (f b) :=
f.apply_line_map a b _
omit V'
@[simp] lemma affine_equiv.point_reflection_midpoint_left (x y : P) :
point_reflection R (midpoint R x y) x = y :=
by rw [midpoint, point_reflection_apply, line_map_apply, vadd_vsub,
vadd_assoc, ← add_smul, ← two_mul, mul_inv_of_self, one_smul, vsub_vadd]
lemma midpoint_comm (x y : P) : midpoint R x y = midpoint R y x :=
by rw [midpoint, ← line_map_apply_one_sub, one_sub_inv_of_two, midpoint]
@[simp] lemma affine_equiv.point_reflection_midpoint_right (x y : P) :
point_reflection R (midpoint R x y) y = x :=
by rw [midpoint_comm, affine_equiv.point_reflection_midpoint_left]
lemma midpoint_vsub_midpoint (p₁ p₂ p₃ p₄ : P) :
midpoint R p₁ p₂ -ᵥ midpoint R p₃ p₄ = midpoint R (p₁ -ᵥ p₃) (p₂ -ᵥ p₄) :=
line_map_vsub_line_map _ _ _ _ _
lemma midpoint_vadd_midpoint (v v' : V) (p p' : P) :
midpoint R v v' +ᵥ midpoint R p p' = midpoint R (v +ᵥ p) (v' +ᵥ p') :=
line_map_vadd_line_map _ _ _ _ _
lemma midpoint_eq_iff {x y z : P} : midpoint R x y = z ↔ point_reflection R z x = y :=
eq_comm.trans ((injective_point_reflection_left_of_module R x).eq_iff'
(affine_equiv.point_reflection_midpoint_left x y)).symm
variable (R)
lemma midpoint_eq_midpoint_iff_vsub_eq_vsub {x x' y y' : P} :
midpoint R x y = midpoint R x' y' ↔ x -ᵥ x' = y' -ᵥ y :=
by rw [← @vsub_eq_zero_iff_eq V, midpoint_vsub_midpoint, midpoint_eq_iff, point_reflection_apply,
vsub_eq_sub, zero_sub, vadd_eq_add, add_zero, neg_eq_iff_neg_eq, neg_vsub_eq_vsub_rev, eq_comm]
lemma midpoint_eq_iff' {x y z : P} : midpoint R x y = z ↔ equiv.point_reflection z x = y :=
midpoint_eq_iff
/-- `midpoint` does not depend on the ring `R`. -/
lemma midpoint_unique (R' : Type*) [ring R'] [invertible (2:R')] [semimodule R' V] (x y : P) :
midpoint R x y = midpoint R' x y :=
(midpoint_eq_iff' R).2 $ (midpoint_eq_iff' R').1 rfl
@[simp] lemma midpoint_self (x : P) : midpoint R x x = x :=
line_map_same_apply _ _
@[simp] lemma midpoint_add_self (x y : V) : midpoint R x y + midpoint R x y = x + y :=
calc midpoint R x y +ᵥ midpoint R x y = midpoint R x y +ᵥ midpoint R y x : by rw midpoint_comm
... = x + y : by rw [midpoint_vadd_midpoint, vadd_eq_add, vadd_eq_add, add_comm, midpoint_self]
lemma midpoint_zero_add (x y : V) : midpoint R 0 (x + y) = midpoint R x y :=
(midpoint_eq_midpoint_iff_vsub_eq_vsub R).2 $ by simp [sub_add_eq_sub_sub_swap]
end
lemma line_map_inv_two {R : Type*} {V P : Type*} [division_ring R] [char_zero R]
[add_comm_group V] [semimodule R V] [add_torsor V P] (a b : P) :
line_map a b (2⁻¹:R) = midpoint R a b :=
rfl
lemma line_map_one_half {R : Type*} {V P : Type*} [division_ring R] [char_zero R]
[add_comm_group V] [semimodule R V] [add_torsor V P] (a b : P) :
line_map a b (1/2:R) = midpoint R a b :=
by rw [one_div, line_map_inv_two]
lemma homothety_inv_of_two {R : Type*} {V P : Type*} [comm_ring R] [invertible (2:R)]
[add_comm_group V] [semimodule R V] [add_torsor V P] (a b : P) :
homothety a (⅟2:R) b = midpoint R a b :=
rfl
lemma homothety_inv_two {k : Type*} {V P : Type*} [field k] [char_zero k]
[add_comm_group V] [semimodule k V] [add_torsor V P] (a b : P) :
homothety a (2⁻¹:k) b = midpoint k a b :=
rfl
lemma homothety_one_half {k : Type*} {V P : Type*} [field k] [char_zero k]
[add_comm_group V] [semimodule k V] [add_torsor V P] (a b : P) :
homothety a (1/2:k) b = midpoint k a b :=
by rw [one_div, homothety_inv_two]
@[simp] lemma pi_midpoint_apply {k ι : Type*} {V : Π i : ι, Type*} {P : Π i : ι, Type*} [field k]
[invertible (2:k)] [Π i, add_comm_group (V i)] [Π i, semimodule k (V i)]
[Π i, add_torsor (V i) (P i)] (f g : Π i, P i) (i : ι) :
midpoint k f g i = midpoint k (f i) (g i) := rfl
namespace add_monoid_hom
variables (R R' : Type*) {E F : Type*}
[ring R] [invertible (2:R)] [add_comm_group E] [semimodule R E]
[ring R'] [invertible (2:R')] [add_comm_group F] [semimodule R' F]
/-- A map `f : E → F` sending zero to zero and midpoints to midpoints is an `add_monoid_hom`. -/
def of_map_midpoint (f : E → F) (h0 : f 0 = 0)
(hm : ∀ x y, f (midpoint R x y) = midpoint R' (f x) (f y)) :
E →+ F :=
{ to_fun := f,
map_zero' := h0,
map_add' := λ x y,
calc f (x + y) = f 0 + f (x + y) : by rw [h0, zero_add]
... = midpoint R' (f 0) (f (x + y)) + midpoint R' (f 0) (f (x + y)) :
(midpoint_add_self _ _ _).symm
... = f (midpoint R x y) + f (midpoint R x y) : by rw [← hm, midpoint_zero_add]
... = f x + f y : by rw [hm, midpoint_add_self] }
@[simp] lemma coe_of_map_midpoint (f : E → F) (h0 : f 0 = 0)
(hm : ∀ x y, f (midpoint R x y) = midpoint R' (f x) (f y)) :
⇑(of_map_midpoint R R' f h0 hm) = f := rfl
end add_monoid_hom
|
b1e99ab3c9f7d6e83c05f5c3421f19c66cb63f70 | 57aec6ee746bc7e3a3dd5e767e53bd95beb82f6d | /src/Std/Data/RBTree.lean | 24fbab418e2f78a1bb10308d3260c88ba705a6a8 | [
"Apache-2.0"
] | permissive | collares/lean4 | 861a9269c4592bce49b71059e232ff0bfe4594cc | 52a4f535d853a2c7c7eea5fee8a4fa04c682c1ee | refs/heads/master | 1,691,419,031,324 | 1,618,678,138,000 | 1,618,678,138,000 | 358,989,750 | 0 | 0 | Apache-2.0 | 1,618,696,333,000 | 1,618,696,333,000 | null | UTF-8 | Lean | false | false | 3,293 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Std.Data.RBMap
namespace Std
universes u v w
def RBTree (α : Type u) (lt : α → α → Bool) : Type u :=
RBMap α Unit lt
instance : Inhabited (RBTree α p) where
default := RBMap.empty
@[inline] def mkRBTree (α : Type u) (lt : α → α → Bool) : RBTree α lt :=
mkRBMap α Unit lt
instance (α : Type u) (lt : α → α → Bool) : EmptyCollection (RBTree α lt) :=
⟨mkRBTree α lt⟩
namespace RBTree
variable {α : Type u} {β : Type v} {lt : α → α → Bool}
@[inline] def empty : RBTree α lt :=
RBMap.empty
@[inline] def depth (f : Nat → Nat → Nat) (t : RBTree α lt) : Nat :=
RBMap.depth f t
@[inline] def fold (f : β → α → β) (init : β) (t : RBTree α lt) : β :=
RBMap.fold (fun r a _ => f r a) init t
@[inline] def revFold (f : β → α → β) (init : β) (t : RBTree α lt) : β :=
RBMap.revFold (fun r a _ => f r a) init t
@[inline] def foldM {m : Type v → Type w} [Monad m] (f : β → α → m β) (init : β) (t : RBTree α lt) : m β :=
RBMap.foldM (fun r a _ => f r a) init t
@[inline] def forM {m : Type v → Type w} [Monad m] (f : α → m PUnit) (t : RBTree α lt) : m PUnit :=
t.foldM (fun _ a => f a) ⟨⟩
@[inline] protected def forIn [Monad m] (t : RBTree α lt) (init : σ) (f : α → σ → m (ForInStep σ)) : m σ :=
t.val.forIn init (fun a _ acc => f a acc)
instance : ForIn m (RBTree α lt) α where
forIn := RBTree.forIn
@[inline] def isEmpty (t : RBTree α lt) : Bool :=
RBMap.isEmpty t
@[specialize] def toList (t : RBTree α lt) : List α :=
t.revFold (fun as a => a::as) []
@[inline] protected def min (t : RBTree α lt) : Option α :=
match RBMap.min t with
| some ⟨a, _⟩ => some a
| none => none
@[inline] protected def max (t : RBTree α lt) : Option α :=
match RBMap.max t with
| some ⟨a, _⟩ => some a
| none => none
instance [Repr α] : Repr (RBTree α lt) where
reprPrec t prec := Repr.addAppParen ("Std.rbtreeOf " ++ repr t.toList) prec
@[inline] def insert (t : RBTree α lt) (a : α) : RBTree α lt :=
RBMap.insert t a ()
@[inline] def erase (t : RBTree α lt) (a : α) : RBTree α lt :=
RBMap.erase t a
@[specialize] def ofList : List α → RBTree α lt
| [] => mkRBTree ..
| x::xs => (ofList xs).insert x
@[inline] def find? (t : RBTree α lt) (a : α) : Option α :=
match RBMap.findCore? t a with
| some ⟨a, _⟩ => some a
| none => none
@[inline] def contains (t : RBTree α lt) (a : α) : Bool :=
(t.find? a).isSome
def fromList (l : List α) (lt : α → α → Bool) : RBTree α lt :=
l.foldl insert (mkRBTree α lt)
@[inline] def all (t : RBTree α lt) (p : α → Bool) : Bool :=
RBMap.all t (fun a _ => p a)
@[inline] def any (t : RBTree α lt) (p : α → Bool) : Bool :=
RBMap.any t (fun a _ => p a)
def subset (t₁ t₂ : RBTree α lt) : Bool :=
t₁.all fun a => (t₂.find? a).toBool
def seteq (t₁ t₂ : RBTree α lt) : Bool :=
subset t₁ t₂ && subset t₂ t₁
end RBTree
def rbtreeOf {α : Type u} (l : List α) (lt : α → α → Bool) : RBTree α lt :=
RBTree.fromList l lt
end Std
|
83a3d41ca7d9ed59978735189dbcc09f4c9f61f3 | 1b8f093752ba748c5ca0083afef2959aaa7dace5 | /src/category_theory/limits/products.lean | ef5ac9f518237110de4d896413f6302a1f7d5b9d | [] | no_license | khoek/lean-category-theory | 7ec4cda9cc64a5a4ffeb84712ac7d020dbbba386 | 63dcb598e9270a3e8b56d1769eb4f825a177cd95 | refs/heads/master | 1,585,251,725,759 | 1,539,344,445,000 | 1,539,344,445,000 | 145,281,070 | 0 | 0 | null | 1,534,662,376,000 | 1,534,662,376,000 | null | UTF-8 | Lean | false | false | 9,496 | lean | -- Copyright (c) 2018 Scott Morrison. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Scott Morrison, Reid Barton, Mario Carneiro
import category_theory.limits.shape
open category_theory
namespace category_theory.limits
universes u v w
variables {C : Type u} [𝒞 : category.{u v} C]
include 𝒞
section product
variables {β : Type v} {f : β → C}
structure is_product (t : fan f) :=
(lift : ∀ (s : fan f), s.X ⟶ t.X)
(fac' : ∀ (s : fan f), ∀ b, (lift s) ≫ t.π b = s.π b . obviously)
(uniq' : ∀ (s : fan f) (m : s.X ⟶ t.X) (w : ∀ b, m ≫ t.π b = s.π b), m = lift s . obviously)
restate_axiom is_product.fac'
attribute [simp,search] is_product.fac
restate_axiom is_product.uniq'
attribute [search,back'] is_product.uniq
@[extensionality] lemma is_product.ext {t : fan f} (P Q : is_product t) : P = Q :=
begin tactic.unfreeze_local_instances, cases P, cases Q, congr, obviously end
instance is_product_subsingleton {t : fan f} : subsingleton (is_product t) := by obviously
lemma is_product.uniq'' {t : fan f} (h : is_product t) {X' : C} (m : X' ⟶ t.X) : m = h.lift { X := X', π := λ b, m ≫ t.π b } :=
h.uniq { X := X', π := λ b, m ≫ t.π b } m (by obviously)
-- TODO provide alternative constructor using uniq'' instead of uniq'.
lemma is_product.univ {t : fan f} (h : is_product t) (s : fan f) (φ : s.X ⟶ t.X) : (∀ b, φ ≫ t.π b = s.π b) ↔ (φ = h.lift s) :=
by obviously
def is_product.of_lift_univ {t : fan f}
(lift : Π (s : fan f), s.X ⟶ t.X)
(univ : Π (s : fan f) (φ : s.X ⟶ t.X), (∀ b, φ ≫ t.π b = s.π b) ↔ (φ = lift s)) : is_product t :=
{ lift := lift,
fac' := λ s b, ((univ s (lift s)).mpr (eq.refl (lift s))) b,
uniq' := begin obviously, apply univ_s_m.mp, obviously, end }
end product
section coproduct
variables {β : Type v} {f : β → C}
structure is_coproduct (t : cofan f) :=
(desc : ∀ (s : cofan f), t.X ⟶ s.X)
(fac' : ∀ (s : cofan f), ∀ b, t.ι b ≫ (desc s) = s.ι b . obviously)
(uniq' : ∀ (s : cofan f) (m : t.X ⟶ s.X) (w : ∀ b, t.ι b ≫ m = s.ι b), m = desc s . obviously)
restate_axiom is_coproduct.fac'
attribute [simp,search] is_coproduct.fac
restate_axiom is_coproduct.uniq'
attribute [search, back'] is_coproduct.uniq
@[extensionality] lemma is_coproduct.ext {t : cofan f} (P Q : is_coproduct t) : P = Q :=
begin tactic.unfreeze_local_instances, cases P, cases Q, congr, obviously end
instance is_coproduct_subsingleton {t : cofan f} : subsingleton (is_coproduct t) := by obviously
lemma is_coproduct.uniq'' {t : cofan f} (h : is_coproduct t) {X' : C} (m : t.X ⟶ X') : m = h.desc { X := X', ι := λ b, t.ι b ≫ m } :=
h.uniq { X := X', ι := λ b, t.ι b ≫ m } m (by obviously)
-- TODO provide alternative constructor using uniq'' instead of uniq'.
lemma is_coproduct.univ {t : cofan f} (h : is_coproduct t) (s : cofan f) (φ : t.X ⟶ s.X) : (∀ b, t.ι b ≫ φ = s.ι b) ↔ (φ = h.desc s) :=
by obviously
def is_coproduct.of_desc_univ {t :cofan f}
(desc : Π (s : cofan f), t.X ⟶ s.X)
(univ : Π (s : cofan f) (φ : t.X ⟶ s.X), (∀ b, t.ι b ≫ φ = s.ι b) ↔ (φ = desc s)) : is_coproduct t :=
{ desc := desc,
fac' := λ s b, ((univ s (desc s)).mpr (eq.refl (desc s))) b,
uniq' := begin obviously, apply univ_s_m.mp, obviously, end }
end coproduct
variable (C)
class has_products :=
(prod : Π {β : Type v} (f : β → C), fan.{u v} f)
(is_product : Π {β : Type v} (f : β → C), is_product (prod f) . obviously)
class has_coproducts :=
(coprod : Π {β : Type v} (f : β → C), cofan.{u v} f)
(is_coproduct : Π {β : Type v} (f : β → C), is_coproduct (coprod f) . obviously)
variable {C}
section
variables [has_products.{u v} C] {β : Type v}
def pi.fan (f : β → C) := has_products.prod.{u v} f
def pi (f : β → C) : C := (pi.fan f).X
def pi.π (f : β → C) (b : β) : pi f ⟶ f b := (pi.fan f).π b
def pi.universal_property (f : β → C) : is_product (pi.fan f) := has_products.is_product.{u v} C f
@[simp] def pi.fan_π (f : β → C) (b : β) : (pi.fan f).π b = @pi.π C _ _ _ f b := rfl
def pi.lift {f : β → C} {P : C} (p : Π b, P ⟶ f b) : P ⟶ pi f :=
(pi.universal_property f).lift ⟨ ⟨ P ⟩, p ⟩
-- @[simp] lemma pi.universal_property_lift (f : β → C) {P : C} (p : Π b, P ⟶ f b) :
-- (pi.universal_property f).lift ⟨ ⟨ P ⟩, p ⟩ = pi.lift p := rfl
@[simp,search] def pi.lift_π {f : β → C} {P : C} (p : Π b, P ⟶ f b) (b : β) : pi.lift p ≫ pi.π f b = p b :=
by erw is_product.fac
def pi.map {f : β → C} {g : β → C} (k : Π b, f b ⟶ g b) : (pi f) ⟶ (pi g) :=
pi.lift (λ b, pi.π f b ≫ k b)
@[simp] def pi.map_π {f : β → C} {g : β → C} (k : Π b, f b ⟶ g b) (b : β) : pi.map k ≫ pi.π g b = pi.π f b ≫ k b :=
by erw is_product.fac
def pi.pre {α} (f : α → C) (h : β → α) : pi f ⟶ pi (f ∘ h) :=
pi.lift (λ g, pi.π f (h g))
@[simp] def pi.pre_π {α} (f : α → C) (h : β → α) (b : β) : pi.pre f h ≫ pi.π (f ∘ h) b = pi.π f (h b) :=
by erw is_product.fac
section
variables {D : Type u} [𝒟 : category.{u v} D] [has_products.{u v} D]
include 𝒟
def pi.post (f : β → C) (G : C ⥤ D) : G (pi f) ⟶ (pi (G.obj ∘ f)) :=
@is_product.lift _ _ _ _ (pi.fan (G.obj ∘ f)) (pi.universal_property _) { X := _, π := λ b, G.map (pi.π f b) }
@[simp] def pi.post_π (f : β → C) (G : C ⥤ D) (b : β) : pi.post f G ≫ pi.π _ b = G.map (pi.π f b) :=
by erw is_product.fac
end
@[extensionality] lemma pi.hom_ext (f : β → C) {X : C} (g h : X ⟶ pi f) (w : ∀ b, g ≫ pi.π f b = h ≫ pi.π f b) : g = h :=
begin
rw is_product.uniq (pi.universal_property f) { X := X, π := λ b, g ≫ pi.π f b } g,
rw is_product.uniq (pi.universal_property f) { X := X, π := λ b, g ≫ pi.π f b } h,
obviously,
end
@[simp] def pi.lift_map
{f : β → C} {g : β → C} {P : C} (p : Π b, P ⟶ f b) (k : Π b, f b ⟶ g b) :
pi.lift p ≫ pi.map k = pi.lift (λ b, p b ≫ k b) :=
by obviously
@[simp] def pi.map_map {f1 : β → C} {f2 : β → C} {f3 : β → C}
(k1 : Π b, f1 b ⟶ f2 b) (k2 : Π b, f2 b ⟶ f3 b) :
pi.map k1 ≫ pi.map k2 = pi.map (λ b, k1 b ≫ k2 b) :=
by obviously.
@[simp] def pi.lift_pre {α : Type v} {f : β → C} {P : C} (p : Π b, P ⟶ f b) (h : α → β) :
pi.lift p ≫ pi.pre _ h = pi.lift (λ a, p (h a)) :=
by obviously
-- TODO lemmas describing interactions:
-- map_pre, pre_pre, lift_post, map_post, pre_post, post_post
end
section
variables [has_coproducts.{u v} C] {β : Type v}
def Sigma.cofan (f : β → C) := has_coproducts.coprod.{u v} f
def Sigma (f : β → C) : C := (Sigma.cofan f).X
def Sigma.ι (f : β → C) (b : β) : f b ⟶ Sigma f := (Sigma.cofan f).ι b
def Sigma.universal_property (f : β → C) : is_coproduct (Sigma.cofan f) := has_coproducts.is_coproduct.{u v} C f
@[simp] def Sigma.cofan_ι (f : β → C) (b : β) : (Sigma.cofan f).ι b = @Sigma.ι C _ _ _ f b := rfl
def Sigma.desc {f : β → C} {P : C} (p : Π b, f b ⟶ P) : Sigma f ⟶ P :=
(Sigma.universal_property f).desc ⟨ ⟨ P ⟩, p ⟩
@[simp,search] def Sigma.lift_ι {f : β → C} {P : C} (p : Π b, f b ⟶ P) (b : β) : Sigma.ι f b ≫ Sigma.desc p = p b :=
by erw is_coproduct.fac
def Sigma.map {f : β → C} {g : β → C} (k : Π b, f b ⟶ g b) : (Sigma f) ⟶ (Sigma g) :=
Sigma.desc (λ b, k b ≫ Sigma.ι g b)
@[simp] def Sigma.map_ι {f : β → C} {g : β → C} (k : Π b, f b ⟶ g b) (b : β) : Sigma.ι f b ≫ Sigma.map k = k b ≫ Sigma.ι g b :=
by erw is_coproduct.fac
def Sigma.pre {α} (f : α → C) (h : β → α) : Sigma (f ∘ h) ⟶ Sigma f :=
Sigma.desc (λ g, Sigma.ι f (h g))
@[simp] def Sigma.pre_ι {α} (f : α → C) (h : β → α) (b : β) : Sigma.ι (f ∘ h) b ≫ Sigma.pre f h = Sigma.ι f (h b) :=
by erw is_coproduct.fac
section
variables {D : Type u} [𝒟 : category.{u v} D] [has_coproducts.{u v} D]
include 𝒟
def Sigma.post (f : β → C) (G : C ⥤ D) : (Sigma (G.obj ∘ f)) ⟶ G (Sigma f) :=
@is_coproduct.desc _ _ _ _ (Sigma.cofan (G.obj ∘ f)) (Sigma.universal_property _) { X := _, ι := λ b, G.map (Sigma.ι f b) }
@[simp] def Sigma.post_π (f : β → C) (G : C ⥤ D) (b : β) : Sigma.ι _ b ≫ Sigma.post f G = G.map (Sigma.ι f b) :=
by erw is_coproduct.fac
end
@[extensionality] lemma Sigma.hom_ext (f : β → C) {X : C} (g h : Sigma f ⟶ X) (w : ∀ b, Sigma.ι f b ≫ g = Sigma.ι f b ≫ h) : g = h :=
begin
rw is_coproduct.uniq (Sigma.universal_property f) { X := X, ι := λ b, Sigma.ι f b ≫ g } g,
rw is_coproduct.uniq (Sigma.universal_property f) { X := X, ι := λ b, Sigma.ι f b ≫ g } h,
obviously
end
@[simp] def Sigma.desc_map
{f : β → C} {g : β → C} {P : C} (k : Π b, f b ⟶ g b) (p : Π b, g b ⟶ P) :
Sigma.map k ≫ Sigma.desc p = Sigma.desc (λ b, k b ≫ p b) :=
by obviously
@[simp] def Sigma.map_map {f1 : β → C} {f2 : β → C} {f3 : β → C}
(k1 : Π b, f1 b ⟶ f2 b) (k2 : Π b, f2 b ⟶ f3 b) :
Sigma.map k1 ≫ Sigma.map k2 = Sigma.map (λ b, k1 b ≫ k2 b) :=
by obviously.
@[simp] def Sigma.desc_pre {α : Type v} {f : β → C} {P : C} (p : Π b, f b ⟶ P) (h : α → β) :
Sigma.pre _ h ≫ Sigma.desc p = Sigma.desc (λ a, p (h a)) :=
by obviously
-- TODO lemmas describing interactions:
-- desc_pre, map_pre, pre_pre, desc_post, map_post, pre_post, post_post
end
end category_theory.limits
|
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Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
Infinite sum over a topological monoid
This sum is known as unconditionally convergent, as it sums to the same value under all possible
permutations. For Euclidean spaces (finite dimensional Banach spaces) this is equivalent to absolute
convergence.
Note: There are summable sequences which are not unconditionally convergent! The other way holds
generally, see `tendsto_sum_nat_of_is_sum`.
Reference:
* Bourbaki: General Topology (1995), Chapter 3 §5 (Infinite sums in commutative groups)
-/
import logic.function algebra.big_operators data.set data.finset
topology.metric_space.basic topology.algebra.uniform_group topology.algebra.ring
topology.algebra.ordered topology.instances.real
noncomputable theory
open lattice finset filter function classical
local attribute [instance] prop_decidable
def option.cases_on' {α β} : option α → β → (α → β) → β
| none n s := n
| (some a) n s := s a
variables {α : Type*} {β : Type*} {γ : Type*}
section is_sum
variables [add_comm_monoid α] [topological_space α] [topological_add_monoid α]
/-- Infinite sum on a topological monoid
The `at_top` filter on `finset α` is the limit of all finite sets towards the entire type. So we sum
up bigger and bigger sets. This sum operation is still invariant under reordering, and a absolute
sum operator.
This is based on Mario Carneiro's infinite sum in Metamath.
-/
def is_sum (f : β → α) (a : α) : Prop := tendsto (λs:finset β, s.sum f) at_top (nhds a)
/-- `has_sum f` means that `f` has some (infinite) sum. Use `tsum` to get the value. -/
def has_sum (f : β → α) : Prop := ∃a, is_sum f a
/-- `tsum f` is the sum of `f` it exists, or 0 otherwise -/
def tsum (f : β → α) := if h : has_sum f then classical.some h else 0
notation `∑` binders `, ` r:(scoped f, tsum f) := r
variables {f g : β → α} {a b : α} {s : finset β}
lemma is_sum_tsum (ha : has_sum f) : is_sum f (∑b, f b) :=
by simp [ha, tsum]; exact some_spec ha
lemma has_sum_spec (ha : is_sum f a) : has_sum f := ⟨a, ha⟩
lemma is_sum_zero : is_sum (λb, 0 : β → α) 0 :=
by simp [is_sum, tendsto_const_nhds]
lemma has_sum_zero : has_sum (λb, 0 : β → α) := has_sum_spec is_sum_zero
lemma is_sum_add (hf : is_sum f a) (hg : is_sum g b) : is_sum (λb, f b + g b) (a + b) :=
by simp [is_sum, sum_add_distrib]; exact tendsto_add hf hg
lemma has_sum_add (hf : has_sum f) (hg : has_sum g) : has_sum (λb, f b + g b) :=
has_sum_spec $ is_sum_add (is_sum_tsum hf)(is_sum_tsum hg)
lemma is_sum_sum {f : γ → β → α} {a : γ → α} {s : finset γ} :
(∀i∈s, is_sum (f i) (a i)) → is_sum (λb, s.sum $ λi, f i b) (s.sum a) :=
finset.induction_on s (by simp [is_sum_zero]) (by simp [is_sum_add] {contextual := tt})
lemma has_sum_sum {f : γ → β → α} {s : finset γ} (hf : ∀i∈s, has_sum (f i)) :
has_sum (λb, s.sum $ λi, f i b) :=
has_sum_spec $ is_sum_sum $ assume i hi, is_sum_tsum $ hf i hi
lemma is_sum_sum_of_ne_finset_zero (hf : ∀b∉s, f b = 0) : is_sum f (s.sum f) :=
tendsto_infi' s $ tendsto.congr'
(assume t (ht : s ⊆ t), show s.sum f = t.sum f, from sum_subset ht $ assume x _, hf _)
tendsto_const_nhds
lemma has_sum_sum_of_ne_finset_zero (hf : ∀b∉s, f b = 0) : has_sum f :=
has_sum_spec $ is_sum_sum_of_ne_finset_zero hf
lemma is_sum_ite_eq (b : β) (a : α) : is_sum (λb', if b' = b then a else 0) a :=
suffices
is_sum (λb', if b' = b then a else 0) (({b} : finset β).sum (λb', if b' = b then a else 0)), from
by simpa,
is_sum_sum_of_ne_finset_zero $ assume b' hb,
have b' ≠ b, by simpa using hb,
by rw [if_neg this]
lemma is_sum_of_iso {j : γ → β} {i : β → γ}
(hf : is_sum f a) (h₁ : ∀x, i (j x) = x) (h₂ : ∀x, j (i x) = x) : is_sum (f ∘ j) a :=
have ∀x y, j x = j y → x = y,
from assume x y h,
have i (j x) = i (j y), by rw [h],
by rwa [h₁, h₁] at this,
have (λs:finset γ, s.sum (f ∘ j)) = (λs:finset β, s.sum f) ∘ (λs:finset γ, s.image j),
from funext $ assume s, (sum_image $ assume x _ y _, this x y).symm,
show tendsto (λs:finset γ, s.sum (f ∘ j)) at_top (nhds a),
by rw [this]; apply (tendsto_finset_image_at_top_at_top h₂).comp hf
lemma is_sum_iff_is_sum_of_iso {j : γ → β} (i : β → γ)
(h₁ : ∀x, i (j x) = x) (h₂ : ∀x, j (i x) = x) :
is_sum (f ∘ j) a ↔ is_sum f a :=
iff.intro
(assume hfj,
have is_sum ((f ∘ j) ∘ i) a, from is_sum_of_iso hfj h₂ h₁,
by simp [(∘), h₂] at this; assumption)
(assume hf, is_sum_of_iso hf h₁ h₂)
lemma is_sum_hom (g : α → γ) [add_comm_monoid γ] [topological_space γ] [topological_add_monoid γ]
[is_add_monoid_hom g] (h₃ : continuous g) (hf : is_sum f a) :
is_sum (g ∘ f) (g a) :=
have (λs:finset β, s.sum (g ∘ f)) = g ∘ (λs:finset β, s.sum f),
from funext $ assume s, sum_hom g,
show tendsto (λs:finset β, s.sum (g ∘ f)) at_top (nhds (g a)),
by rw [this]; exact hf.comp (continuous_iff_continuous_at.mp h₃ a)
lemma tendsto_sum_nat_of_is_sum {f : ℕ → α} (h : is_sum f a) :
tendsto (λn:ℕ, (range n).sum f) at_top (nhds a) :=
suffices map (λ (n : ℕ), sum (range n) f) at_top ≤ map (λ (s : finset ℕ), sum s f) at_top,
from le_trans this h,
assume s (hs : {t : finset ℕ | t.sum f ∈ s} ∈ at_top.sets),
let ⟨t, ht⟩ := mem_at_top_sets.mp hs, ⟨n, hn⟩ := @exists_nat_subset_range t in
mem_at_top_sets.mpr ⟨n, assume n' hn', ht _ $ finset.subset.trans hn $ range_subset.mpr hn'⟩
lemma is_sum_sigma [regular_space α] {γ : β → Type*} {f : (Σ b:β, γ b) → α} {g : β → α} {a : α}
(hf : ∀b, is_sum (λc, f ⟨b, c⟩) (g b)) (ha : is_sum f a) : is_sum g a :=
assume s' hs',
let
⟨s, hs, hss', hsc⟩ := nhds_is_closed hs',
⟨u, hu⟩ := mem_at_top_sets.mp $ ha $ hs,
fsts := u.image sigma.fst,
snds := λb, u.bind (λp, (if h : p.1 = b then {cast (congr_arg γ h) p.2} else ∅ : finset (γ b)))
in
have u_subset : u ⊆ fsts.sigma snds,
from subset_iff.mpr $ assume ⟨b, c⟩ hu,
have hb : b ∈ fsts, from finset.mem_image.mpr ⟨_, hu, rfl⟩,
have hc : c ∈ snds b, from mem_bind.mpr ⟨_, hu, by simp; refl⟩,
by simp [mem_sigma, hb, hc] ,
mem_at_top_sets.mpr $ exists.intro fsts $ assume bs (hbs : fsts ⊆ bs),
have h : ∀cs : Π b ∈ bs, finset (γ b),
(⋂b (hb : b ∈ bs), (λp:Πb, finset (γ b), p b) ⁻¹' {cs' | cs b hb ⊆ cs' }) ∩
(λp, bs.sum (λb, (p b).sum (λc, f ⟨b, c⟩))) ⁻¹' s ≠ ∅,
from assume cs,
let cs' := λb, (if h : b ∈ bs then cs b h else ∅) ∪ snds b in
have sum_eq : bs.sum (λb, (cs' b).sum (λc, f ⟨b, c⟩)) = (bs.sigma cs').sum f,
from sum_sigma.symm,
have (bs.sigma cs').sum f ∈ s,
from hu _ $ finset.subset.trans u_subset $ sigma_mono hbs $
assume b, @finset.subset_union_right (γ b) _ _ _,
set.ne_empty_iff_exists_mem.mpr $ exists.intro cs' $
by simp [sum_eq, this]; { intros b hb, simp [cs', hb, finset.subset_union_right] },
have tendsto (λp:(Πb:β, finset (γ b)), bs.sum (λb, (p b).sum (λc, f ⟨b, c⟩)))
(⨅b (h : b ∈ bs), at_top.comap (λp, p b)) (nhds (bs.sum g)),
from tendsto_finset_sum bs $
assume c hc, tendsto_infi' c $ tendsto_infi' hc $ tendsto_comap.comp (hf c),
have bs.sum g ∈ s,
from mem_of_closed_of_tendsto' this hsc $ forall_sets_neq_empty_iff_neq_bot.mp $
by simp [mem_inf_sets, exists_imp_distrib, and_imp, forall_and_distrib,
filter.mem_infi_sets_finset, mem_comap_sets, skolem, mem_at_top_sets,
and_comm];
from
assume s₁ s₂ s₃ hs₁ hs₃ p hs₂ p' hp cs hp',
have (⋂b (h : b ∈ bs), (λp:(Πb, finset (γ b)), p b) ⁻¹' {cs' | cs b h ⊆ cs' }) ≤ (⨅b∈bs, p b),
from infi_le_infi $ assume b, infi_le_infi $ assume hb,
le_trans (set.preimage_mono $ hp' b hb) (hp b hb),
neq_bot_of_le_neq_bot (h _) (le_trans (set.inter_subset_inter (le_trans this hs₂) hs₃) hs₁),
hss' this
lemma has_sum_sigma [regular_space α] {γ : β → Type*} {f : (Σb:β, γ b) → α}
(hf : ∀b, has_sum (λc, f ⟨b, c⟩)) (ha : has_sum f) : has_sum (λb, ∑c, f ⟨b, c⟩):=
has_sum_spec $ is_sum_sigma (assume b, is_sum_tsum $ hf b) (is_sum_tsum ha)
end is_sum
section is_sum_iff_is_sum_of_iso_ne_zero
variables [add_comm_monoid α] [topological_space α] [topological_add_monoid α]
variables {f : β → α} {g : γ → α} {a : α}
lemma is_sum_of_is_sum
(h_eq : ∀u:finset γ, ∃v:finset β, ∀v', v ⊆ v' → ∃u', u ⊆ u' ∧ u'.sum g = v'.sum f)
(hf : is_sum g a) : is_sum f a :=
suffices at_top.map (λs:finset β, s.sum f) ≤ at_top.map (λs:finset γ, s.sum g),
from le_trans this hf,
by rw [map_at_top_eq, map_at_top_eq];
from (le_infi $ assume b, let ⟨v, hv⟩ := h_eq b in infi_le_of_le v $
by simp [set.image_subset_iff]; exact hv)
lemma is_sum_iff_is_sum
(h₁ : ∀u:finset γ, ∃v:finset β, ∀v', v ⊆ v' → ∃u', u ⊆ u' ∧ u'.sum g = v'.sum f)
(h₂ : ∀v:finset β, ∃u:finset γ, ∀u', u ⊆ u' → ∃v', v ⊆ v' ∧ v'.sum f = u'.sum g) :
is_sum f a ↔ is_sum g a :=
⟨is_sum_of_is_sum h₂, is_sum_of_is_sum h₁⟩
variables
(i : Π⦃c⦄, g c ≠ 0 → β) (hi : ∀⦃c⦄ (h : g c ≠ 0), f (i h) ≠ 0)
(j : Π⦃b⦄, f b ≠ 0 → γ) (hj : ∀⦃b⦄ (h : f b ≠ 0), g (j h) ≠ 0)
(hji : ∀⦃c⦄ (h : g c ≠ 0), j (hi h) = c)
(hij : ∀⦃b⦄ (h : f b ≠ 0), i (hj h) = b)
(hgj : ∀⦃b⦄ (h : f b ≠ 0), g (j h) = f b)
include hi hj hji hij hgj
lemma is_sum_of_is_sum_ne_zero : is_sum g a → is_sum f a :=
have j_inj : ∀x y (hx : f x ≠ 0) (hy : f y ≠ 0), (j hx = j hy ↔ x = y),
from assume x y hx hy,
⟨assume h,
have i (hj hx) = i (hj hy), by simp [h],
by rwa [hij, hij] at this; assumption,
by simp {contextual := tt}⟩,
let ii : finset γ → finset β := λu, u.bind $ λc, if h : g c = 0 then ∅ else {i h} in
let jj : finset β → finset γ := λv, v.bind $ λb, if h : f b = 0 then ∅ else {j h} in
is_sum_of_is_sum $ assume u, exists.intro (ii u) $
assume v hv, exists.intro (u ∪ jj v) $ and.intro (subset_union_left _ _) $
have ∀c:γ, c ∈ u ∪ jj v → c ∉ jj v → g c = 0,
from assume c hc hnc, classical.by_contradiction $ assume h : g c ≠ 0,
have c ∈ u,
from (finset.mem_union.1 hc).resolve_right hnc,
have i h ∈ v,
from hv $ by simp [mem_bind]; existsi c; simp [h, this],
have j (hi h) ∈ jj v,
by simp [mem_bind]; existsi i h; simp [h, hi, this],
by rw [hji h] at this; exact hnc this,
calc (u ∪ jj v).sum g = (jj v).sum g : (sum_subset (subset_union_right _ _) this).symm
... = v.sum _ : sum_bind $ by intros x hx y hy hxy; by_cases f x = 0; by_cases f y = 0; simp [*]
... = v.sum f : sum_congr rfl $ by intros x hx; by_cases f x = 0; simp [*]
lemma is_sum_iff_is_sum_of_ne_zero : is_sum f a ↔ is_sum g a :=
iff.intro
(is_sum_of_is_sum_ne_zero j hj i hi hij hji $ assume b hb, by rw [←hgj (hi _), hji])
(is_sum_of_is_sum_ne_zero i hi j hj hji hij hgj)
lemma has_sum_iff_has_sum_ne_zero : has_sum g ↔ has_sum f :=
exists_congr $
assume a, is_sum_iff_is_sum_of_ne_zero j hj i hi hij hji $
assume b hb, by rw [←hgj (hi _), hji]
end is_sum_iff_is_sum_of_iso_ne_zero
section is_sum_iff_is_sum_of_bij_ne_zero
variables [add_comm_monoid α] [topological_space α] [topological_add_monoid α]
variables {f : β → α} {g : γ → α} {a : α}
(i : Π⦃c⦄, g c ≠ 0 → β)
(h₁ : ∀⦃c₁ c₂⦄ (h₁ : g c₁ ≠ 0) (h₂ : g c₂ ≠ 0), i h₁ = i h₂ → c₁ = c₂)
(h₂ : ∀⦃b⦄, f b ≠ 0 → ∃c (h : g c ≠ 0), i h = b)
(h₃ : ∀⦃c⦄ (h : g c ≠ 0), f (i h) = g c)
include i h₁ h₂ h₃
lemma is_sum_iff_is_sum_of_ne_zero_bij : is_sum f a ↔ is_sum g a :=
have hi : ∀⦃c⦄ (h : g c ≠ 0), f (i h) ≠ 0,
from assume c h, by simp [h₃, h],
let j : Π⦃b⦄, f b ≠ 0 → γ := λb h, some $ h₂ h in
have hj : ∀⦃b⦄ (h : f b ≠ 0), ∃(h : g (j h) ≠ 0), i h = b,
from assume b h, some_spec $ h₂ h,
have hj₁ : ∀⦃b⦄ (h : f b ≠ 0), g (j h) ≠ 0,
from assume b h, let ⟨h₁, _⟩ := hj h in h₁,
have hj₂ : ∀⦃b⦄ (h : f b ≠ 0), i (hj₁ h) = b,
from assume b h, let ⟨h₁, h₂⟩ := hj h in h₂,
is_sum_iff_is_sum_of_ne_zero i hi j hj₁
(assume c h, h₁ (hj₁ _) h $ hj₂ _) hj₂ (assume b h, by rw [←h₃ (hj₁ _), hj₂])
lemma has_sum_iff_has_sum_ne_zero_bij : has_sum f ↔ has_sum g :=
exists_congr $
assume a, is_sum_iff_is_sum_of_ne_zero_bij @i h₁ h₂ h₃
end is_sum_iff_is_sum_of_bij_ne_zero
section tsum
variables [add_comm_monoid α] [topological_space α] [topological_add_monoid α] [t2_space α]
variables {f g : β → α} {a a₁ a₂ : α}
lemma is_sum_unique : is_sum f a₁ → is_sum f a₂ → a₁ = a₂ := tendsto_nhds_unique at_top_ne_bot
lemma tsum_eq_is_sum (ha : is_sum f a) : (∑b, f b) = a := is_sum_unique (is_sum_tsum ⟨a, ha⟩) ha
lemma is_sum_iff_of_has_sum (h : has_sum f) : is_sum f a ↔ (∑b, f b) = a :=
iff.intro tsum_eq_is_sum (assume eq, eq ▸ is_sum_tsum h)
@[simp] lemma tsum_zero : (∑b:β, 0:α) = 0 := tsum_eq_is_sum is_sum_zero
lemma tsum_add (hf : has_sum f) (hg : has_sum g) : (∑b, f b + g b) = (∑b, f b) + (∑b, g b) :=
tsum_eq_is_sum $ is_sum_add (is_sum_tsum hf) (is_sum_tsum hg)
lemma tsum_sum {f : γ → β → α} {s : finset γ} (hf : ∀i∈s, has_sum (f i)) :
(∑b, s.sum (λi, f i b)) = s.sum (λi, ∑b, f i b) :=
tsum_eq_is_sum $ is_sum_sum $ assume i hi, is_sum_tsum $ hf i hi
lemma tsum_eq_sum {f : β → α} {s : finset β} (hf : ∀b∉s, f b = 0) :
(∑b, f b) = s.sum f :=
tsum_eq_is_sum $ is_sum_sum_of_ne_finset_zero hf
lemma tsum_fintype [fintype β] (f : β → α) : (∑b, f b) = finset.univ.sum f :=
tsum_eq_sum $ λ a h, h.elim (mem_univ _)
lemma tsum_eq_single {f : β → α} (b : β) (hf : ∀b' ≠ b, f b' = 0) :
(∑b, f b) = f b :=
calc (∑b, f b) = (finset.singleton b).sum f : tsum_eq_sum $ by simp [hf] {contextual := tt}
... = f b : by simp
lemma tsum_sigma [regular_space α] {γ : β → Type*} {f : (Σb:β, γ b) → α}
(h₁ : ∀b, has_sum (λc, f ⟨b, c⟩)) (h₂ : has_sum f) : (∑p, f p) = (∑b c, f ⟨b, c⟩):=
(tsum_eq_is_sum $ is_sum_sigma (assume b, is_sum_tsum $ h₁ b) $ is_sum_tsum h₂).symm
@[simp] lemma tsum_ite_eq (b : β) (a : α) : (∑b', if b' = b then a else 0) = a :=
tsum_eq_is_sum (is_sum_ite_eq b a)
lemma tsum_eq_tsum_of_is_sum_iff_is_sum {f : β → α} {g : γ → α}
(h : ∀{a}, is_sum f a ↔ is_sum g a) : (∑b, f b) = (∑c, g c) :=
by_cases
(assume : ∃a, is_sum f a,
let ⟨a, hfa⟩ := this in
have hga : is_sum g a, from h.mp hfa,
by rw [tsum_eq_is_sum hfa, tsum_eq_is_sum hga])
(assume hf : ¬ has_sum f,
have hg : ¬ has_sum g, from assume ⟨a, hga⟩, hf ⟨a, h.mpr hga⟩,
by simp [tsum, hf, hg])
lemma tsum_eq_tsum_of_ne_zero {f : β → α} {g : γ → α}
(i : Π⦃c⦄, g c ≠ 0 → β) (hi : ∀⦃c⦄ (h : g c ≠ 0), f (i h) ≠ 0)
(j : Π⦃b⦄, f b ≠ 0 → γ) (hj : ∀⦃b⦄ (h : f b ≠ 0), g (j h) ≠ 0)
(hji : ∀⦃c⦄ (h : g c ≠ 0), j (hi h) = c)
(hij : ∀⦃b⦄ (h : f b ≠ 0), i (hj h) = b)
(hgj : ∀⦃b⦄ (h : f b ≠ 0), g (j h) = f b) :
(∑i, f i) = (∑j, g j) :=
tsum_eq_tsum_of_is_sum_iff_is_sum $ assume a, is_sum_iff_is_sum_of_ne_zero i hi j hj hji hij hgj
lemma tsum_eq_tsum_of_ne_zero_bij {f : β → α} {g : γ → α}
(i : Π⦃c⦄, g c ≠ 0 → β)
(h₁ : ∀⦃c₁ c₂⦄ (h₁ : g c₁ ≠ 0) (h₂ : g c₂ ≠ 0), i h₁ = i h₂ → c₁ = c₂)
(h₂ : ∀⦃b⦄, f b ≠ 0 → ∃c (h : g c ≠ 0), i h = b)
(h₃ : ∀⦃c⦄ (h : g c ≠ 0), f (i h) = g c) :
(∑i, f i) = (∑j, g j) :=
tsum_eq_tsum_of_is_sum_iff_is_sum $ assume a, is_sum_iff_is_sum_of_ne_zero_bij i h₁ h₂ h₃
lemma tsum_eq_tsum_of_iso (j : γ → β) (i : β → γ)
(h₁ : ∀x, i (j x) = x) (h₂ : ∀x, j (i x) = x) :
(∑c, f (j c)) = (∑b, f b) :=
tsum_eq_tsum_of_is_sum_iff_is_sum $ assume a, is_sum_iff_is_sum_of_iso i h₁ h₂
lemma tsum_equiv (j : γ ≃ β) : (∑c, f (j c)) = (∑b, f b) :=
tsum_eq_tsum_of_iso j j.symm (by simp) (by simp)
end tsum
section topological_group
variables [add_comm_group α] [topological_space α] [topological_add_group α]
variables {f g : β → α} {a a₁ a₂ : α}
lemma is_sum_neg : is_sum f a → is_sum (λb, - f b) (- a) :=
is_sum_hom has_neg.neg continuous_neg'
lemma has_sum_neg (hf : has_sum f) : has_sum (λb, - f b) :=
has_sum_spec $ is_sum_neg $ is_sum_tsum $ hf
lemma is_sum_sub (hf : is_sum f a₁) (hg : is_sum g a₂) : is_sum (λb, f b - g b) (a₁ - a₂) :=
by simp; exact is_sum_add hf (is_sum_neg hg)
lemma has_sum_sub (hf : has_sum f) (hg : has_sum g) : has_sum (λb, f b - g b) :=
has_sum_spec $ is_sum_sub (is_sum_tsum hf) (is_sum_tsum hg)
section tsum
variables [t2_space α]
lemma tsum_neg (hf : has_sum f) : (∑b, - f b) = - (∑b, f b) :=
tsum_eq_is_sum $ is_sum_neg $ is_sum_tsum $ hf
lemma tsum_sub (hf : has_sum f) (hg : has_sum g) : (∑b, f b - g b) = (∑b, f b) - (∑b, g b) :=
tsum_eq_is_sum $ is_sum_sub (is_sum_tsum hf) (is_sum_tsum hg)
end tsum
end topological_group
section topological_semiring
variables [semiring α] [topological_space α] [topological_semiring α]
variables {f g : β → α} {a a₁ a₂ : α}
lemma is_sum_mul_left (a₂) : is_sum f a₁ → is_sum (λb, a₂ * f b) (a₂ * a₁) :=
is_sum_hom _ (continuous_mul continuous_const continuous_id)
lemma is_sum_mul_right (a₂) (hf : is_sum f a₁) : is_sum (λb, f b * a₂) (a₁ * a₂) :=
@is_sum_hom _ _ _ _ _ _ f a₁ (λa, a * a₂) _ _ _ _
(continuous_mul continuous_id continuous_const) hf
lemma has_sum_mul_left (a) (hf : has_sum f) : has_sum (λb, a * f b) :=
has_sum_spec $ is_sum_mul_left _ $ is_sum_tsum hf
lemma has_sum_mul_right (a) (hf : has_sum f) : has_sum (λb, f b * a) :=
has_sum_spec $ is_sum_mul_right _ $ is_sum_tsum hf
section tsum
variables [t2_space α]
lemma tsum_mul_left (a) (hf : has_sum f) : (∑b, a * f b) = a * (∑b, f b) :=
tsum_eq_is_sum $ is_sum_mul_left _ $ is_sum_tsum hf
lemma tsum_mul_right (a) (hf : has_sum f) : (∑b, f b * a) = (∑b, f b) * a :=
tsum_eq_is_sum $ is_sum_mul_right _ $ is_sum_tsum hf
end tsum
end topological_semiring
section order_topology
variables [ordered_comm_monoid α] [topological_space α] [ordered_topology α]
[topological_add_monoid α]
variables {f g : β → α} {a a₁ a₂ : α}
lemma is_sum_le (h : ∀b, f b ≤ g b) (hf : is_sum f a₁) (hg : is_sum g a₂) : a₁ ≤ a₂ :=
le_of_tendsto_of_tendsto at_top_ne_bot hf hg $ univ_mem_sets' $
assume s, sum_le_sum' $ assume b _, h b
lemma is_sum_le_inj {g : γ → α} (i : β → γ) (hi : injective i) (hs : ∀c∉set.range i, 0 ≤ g c)
(h : ∀b, f b ≤ g (i b)) (hf : is_sum f a₁) (hg : is_sum g a₂) : a₁ ≤ a₂ :=
have is_sum (λc, (partial_inv i c).cases_on' 0 f) a₁,
begin
refine (is_sum_iff_is_sum_of_ne_zero_bij (λb _, i b) _ _ _).2 hf,
{ assume c₁ c₂ h₁ h₂ eq, exact hi eq },
{ assume c hc,
cases eq : partial_inv i c with b; rw eq at hc,
{ contradiction },
{ rw [partial_inv_of_injective hi] at eq,
exact ⟨b, hc, eq⟩ } },
{ assume c hc, rw [partial_inv_left hi, option.cases_on'] }
end,
begin
refine is_sum_le (assume c, _) this hg,
by_cases c ∈ set.range i,
{ rcases h with ⟨b, rfl⟩,
rw [partial_inv_left hi, option.cases_on'],
exact h _ },
{ have : partial_inv i c = none := dif_neg h,
rw [this, option.cases_on'],
exact hs _ h }
end
lemma tsum_le_tsum (h : ∀b, f b ≤ g b) (hf : has_sum f) (hg : has_sum g) : (∑b, f b) ≤ (∑b, g b) :=
is_sum_le h (is_sum_tsum hf) (is_sum_tsum hg)
end order_topology
section uniform_group
variables [add_comm_group α] [uniform_space α] [complete_space α] [uniform_add_group α]
variables (f g : β → α) {a a₁ a₂ : α}
lemma has_sum_iff_cauchy : has_sum f ↔ cauchy (map (λ (s : finset β), sum s f) at_top) :=
(cauchy_map_iff_exists_tendsto at_top_ne_bot).symm
lemma has_sum_iff_vanishing :
has_sum f ↔ ∀e∈(nhds (0:α)).sets, (∃s:finset β, ∀t, disjoint t s → t.sum f ∈ e) :=
begin
simp only [has_sum_iff_cauchy, cauchy_map_iff, and_iff_right at_top_ne_bot,
prod_at_top_at_top_eq, uniformity_eq_comap_nhds_zero α, tendsto_comap_iff, (∘)],
rw [tendsto_at_top' (_ : finset β × finset β → α)],
split,
{ assume h e he,
rcases h e he with ⟨⟨s₁, s₂⟩, h⟩,
use [s₁ ∪ s₂],
assume t ht,
have : (s₁ ∪ s₂) ∩ t = ∅ := finset.disjoint_iff_inter_eq_empty.1 ht.symm,
specialize h (s₁ ∪ s₂, (s₁ ∪ s₂) ∪ t) ⟨le_sup_left, le_sup_left_of_le le_sup_right⟩,
simpa only [finset.sum_union this, add_sub_cancel'] using h },
{ assume h e he,
rcases exists_nhds_half_neg he with ⟨d, hd, hde⟩,
rcases h d hd with ⟨s, h⟩,
use [(s, s)],
rintros ⟨t₁, t₂⟩ ⟨ht₁, ht₂⟩,
have : t₂.sum f - t₁.sum f = (t₂ \ s).sum f - (t₁ \ s).sum f,
{ simp only [(finset.sum_sdiff ht₁).symm, (finset.sum_sdiff ht₂).symm,
add_sub_add_right_eq_sub] },
simp only [this],
exact hde _ _ (h _ finset.sdiff_disjoint) (h _ finset.sdiff_disjoint) }
end
/- TODO: generalize to monoid with a uniform continuous subtraction operator: `(a + b) - b = a` -/
lemma has_sum_of_has_sum_of_sub (hf : has_sum f) (h : ∀b, g b = 0 ∨ g b = f b) : has_sum g :=
(has_sum_iff_vanishing g).2 $
assume e he,
let ⟨s, hs⟩ := (has_sum_iff_vanishing f).1 hf e he in
⟨s, assume t ht,
have eq : (t.filter (λb, g b = f b)).sum f = t.sum g :=
calc (t.filter (λb, g b = f b)).sum f = (t.filter (λb, g b = f b)).sum g :
finset.sum_congr rfl (assume b hb, (finset.mem_filter.1 hb).2.symm)
... = t.sum g :
begin
refine finset.sum_subset (finset.filter_subset _) _,
assume b hbt hb,
simp only [(∉), finset.mem_filter, and_iff_right hbt] at hb,
exact (h b).resolve_right hb
end,
eq ▸ hs _ $ finset.disjoint_of_subset_left (finset.filter_subset _) ht⟩
lemma has_sum_comp_of_has_sum_of_injective {i : γ → β} (hf : has_sum f) (hi : injective i) :
has_sum (f ∘ i) :=
suffices has_sum (λb, if b ∈ set.range i then f b else 0),
begin
refine (has_sum_iff_has_sum_ne_zero_bij (λc _, i c) _ _ _).1 this,
{ assume c₁ c₂ hc₁ hc₂ eq, exact hi eq },
{ assume b hb,
split_ifs at hb,
{ rcases h with ⟨c, rfl⟩,
exact ⟨c, hb, rfl⟩ },
{ contradiction } },
{ assume c hc, exact if_pos (set.mem_range_self _) }
end,
has_sum_of_has_sum_of_sub _ _ hf $ assume b, by by_cases b ∈ set.range i; simp [h]
end uniform_group
section cauchy_seq
open finset.Ico filter
lemma cauchy_seq_of_has_sum_dist [metric_space α] {f : ℕ → α}
(h : has_sum (λn, dist (f n) (f n.succ))) : cauchy_seq f :=
begin
let d := λn, dist (f n) (f (n+1)),
refine metric.cauchy_seq_iff'.2 (λε εpos, _),
rcases (has_sum_iff_vanishing _).1 h {x : ℝ | x < ε} (gt_mem_nhds εpos) with ⟨s, hs⟩,
have : ∃N:ℕ, ∀x ∈ s, x < N,
{ by_cases h : s = ∅,
{ use 0, simp [h]},
{ use s.max' h + 1,
exact λx hx, lt_of_le_of_lt (s.le_max' h x hx) (nat.lt_succ_self _) }},
rcases this with ⟨N, hN⟩,
refine ⟨N, λn hn, _⟩,
have : ∀n, n ≥ N → dist (f N) (f n) ≤ (Ico N n).sum d,
{ apply nat.le_induction,
{ simp },
{ assume n hn hrec,
calc dist (f N) (f (n+1)) ≤ dist (f N) (f n) + d n : dist_triangle _ _ _
... ≤ (Ico N n).sum d + d n : add_le_add hrec (le_refl _)
... = (Ico N (n+1)).sum d : by rw [succ_top hn, sum_insert, add_comm]; simp }},
calc dist (f n) (f N) ≤ (Ico N n).sum d : by rw dist_comm; apply this n hn
... < ε : hs _ (finset.disjoint_iff_ne.2
(λa ha b hb, ne_of_gt (lt_of_lt_of_le (hN _ hb) (mem.1 ha).1)))
end
end cauchy_seq
|
7e7879cd88545ee8bf5c1b109a6b4f92e3f71390 | b00eb947a9c4141624aa8919e94ce6dcd249ed70 | /src/Lean/Data/JsonRpc.lean | 811e10add4cc2959110e0b4f52be8f588c5daa35 | [
"Apache-2.0"
] | permissive | gebner/lean4-old | a4129a041af2d4d12afb3a8d4deedabde727719b | ee51cdfaf63ee313c914d83264f91f414a0e3b6e | refs/heads/master | 1,683,628,606,745 | 1,622,651,300,000 | 1,622,654,405,000 | 142,608,821 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 9,708 | lean | /-
Copyright (c) 2020 Marc Huisinga. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Marc Huisinga, Wojciech Nawrocki
-/
import Init.Control
import Init.System.IO
import Std.Data.RBTree
import Lean.Data.Json
/-! Implementation of JSON-RPC 2.0 (https://www.jsonrpc.org/specification)
for use in the LSP server. -/
namespace Lean.JsonRpc
open Json
open Std (RBNode)
inductive RequestID where
| str (s : String)
| num (n : JsonNumber)
| null
deriving Inhabited, BEq, Ord
instance : ToString RequestID where
toString
| RequestID.str s => s!"\"s\""
| RequestID.num n => toString n
| RequestID.null => "null"
/-- Error codes defined by JSON-RPC and LSP. -/
inductive ErrorCode where
| parseError
| invalidRequest
| methodNotFound
| invalidParams
| internalError
| serverErrorStart
| serverErrorEnd
| serverNotInitialized
| unknownErrorCode
-- LSP-specific codes below.
| requestCancelled
| contentModified
deriving Inhabited, BEq
instance : FromJson ErrorCode := ⟨fun
| num (-32700 : Int) => ErrorCode.parseError
| num (-32600 : Int) => ErrorCode.invalidRequest
| num (-32601 : Int) => ErrorCode.methodNotFound
| num (-32602 : Int) => ErrorCode.invalidParams
| num (-32603 : Int) => ErrorCode.internalError
| num (-32099 : Int) => ErrorCode.serverErrorStart
| num (-32000 : Int) => ErrorCode.serverErrorEnd
| num (-32002 : Int) => ErrorCode.serverNotInitialized
| num (-32001 : Int) => ErrorCode.unknownErrorCode
| num (-32800 : Int) => ErrorCode.requestCancelled
| num (-32801 : Int) => ErrorCode.contentModified
| _ => none⟩
instance : ToJson ErrorCode := ⟨fun
| ErrorCode.parseError => (-32700 : Int)
| ErrorCode.invalidRequest => (-32600 : Int)
| ErrorCode.methodNotFound => (-32601 : Int)
| ErrorCode.invalidParams => (-32602 : Int)
| ErrorCode.internalError => (-32603 : Int)
| ErrorCode.serverErrorStart => (-32099 : Int)
| ErrorCode.serverErrorEnd => (-32000 : Int)
| ErrorCode.serverNotInitialized => (-32002 : Int)
| ErrorCode.unknownErrorCode => (-32001 : Int)
| ErrorCode.requestCancelled => (-32800 : Int)
| ErrorCode.contentModified => (-32801 : Int)⟩
/- Uses separate constructors for notifications and errors because client and server
behavior is expected to be wildly different for both. -/
inductive Message where
| request (id : RequestID) (method : String) (params? : Option Structured)
| notification (method : String) (params? : Option Structured)
| response (id : RequestID) (result : Json)
| responseError (id : RequestID) (code : ErrorCode) (message : String) (data? : Option Json)
def Batch := Array Message
-- Compound type with simplified APIs for passing around
-- jsonrpc data
structure Request (α : Type u) where
id : RequestID
method : String
param : α
deriving Inhabited, BEq
instance [ToJson α] : Coe (Request α) Message :=
⟨fun r => Message.request r.id r.method (toStructured? r.param)⟩
structure Notification (α : Type u) where
method : String
param : α
deriving Inhabited, BEq
instance [ToJson α] : Coe (Notification α) Message :=
⟨fun r => Message.notification r.method (toStructured? r.param)⟩
structure Response (α : Type u) where
id : RequestID
result : α
deriving Inhabited, BEq
instance [ToJson α] : Coe (Response α) Message :=
⟨fun r => Message.response r.id (toJson r.result)⟩
structure ResponseError (α : Type u) where
id : RequestID
code : ErrorCode
message : String
data? : Option α := none
deriving Inhabited, BEq
instance [ToJson α] : Coe (ResponseError α) Message :=
⟨fun r => Message.responseError r.id r.code r.message (r.data?.map toJson)⟩
instance : Coe String RequestID := ⟨RequestID.str⟩
instance : Coe JsonNumber RequestID := ⟨RequestID.num⟩
private def RequestID.lt : RequestID → RequestID → Bool
| RequestID.str a, RequestID.str b => a < b
| RequestID.num a, RequestID.num b => a < b
| RequestID.null, RequestID.num _ => true
| RequestID.null, RequestID.str _ => true
| RequestID.num _, RequestID.str _ => true
| _, _ /- str < *, num < null, null < null -/ => false
private def RequestID.ltProp : LT RequestID :=
⟨fun a b => RequestID.lt a b = true⟩
instance : LT RequestID :=
RequestID.ltProp
instance (a b : RequestID) : Decidable (a < b) :=
inferInstanceAs (Decidable (RequestID.lt a b = true))
instance : FromJson RequestID := ⟨fun j =>
match j with
| str s => RequestID.str s
| num n => RequestID.num n
| _ => none⟩
instance : ToJson RequestID := ⟨fun rid =>
match rid with
| RequestID.str s => s
| RequestID.num n => num n
| RequestID.null => null⟩
instance : ToJson Message := ⟨fun m =>
mkObj $ ⟨"jsonrpc", "2.0"⟩ :: match m with
| Message.request id method params? =>
[ ⟨"id", toJson id⟩,
⟨"method", method⟩
] ++ opt "params" params?
| Message.notification method params? =>
⟨"method", method⟩ ::
opt "params" params?
| Message.response id result =>
[ ⟨"id", toJson id⟩,
⟨"result", result⟩]
| Message.responseError id code message data? =>
[ ⟨"id", toJson id⟩,
⟨"error", mkObj $ [
⟨"code", toJson code⟩,
⟨"message", message⟩
] ++ opt "data" data?⟩
]⟩
instance : FromJson Message where
fromJson? j := OptionM.run do
let "2.0" ← j.getObjVal? "jsonrpc" | none
(do let id ← j.getObjValAs? RequestID "id"
let method ← j.getObjValAs? String "method"
let params? := j.getObjValAs? Structured "params"
pure (Message.request id method params?)) <|>
(do let method ← j.getObjValAs? String "method"
let params? := j.getObjValAs? Structured "params"
pure (Message.notification method params?)) <|>
(do let id ← j.getObjValAs? RequestID "id"
let result ← j.getObjVal? "result"
pure (Message.response id result)) <|>
(do let id ← j.getObjValAs? RequestID "id"
let err ← j.getObjVal? "error"
let code ← err.getObjValAs? ErrorCode "code"
let message ← err.getObjValAs? String "message"
let data? := err.getObjVal? "data"
pure (Message.responseError id code message data?))
-- TODO(WN): temporary until we have deriving FromJson
instance [FromJson α] : FromJson (Notification α) where
fromJson? j := OptionM.run do
let msg : Message ← fromJson? j
if let Message.notification method params? := msg then
let params ← params?
let param : α ← fromJson? (toJson params)
pure $ ⟨method, param⟩
else none
end Lean.JsonRpc
namespace IO.FS.Stream
open Lean
open Lean.JsonRpc
section
def readMessage (h : FS.Stream) (nBytes : Nat) : IO Message := do
let j ← h.readJson nBytes
match fromJson? j with
| some m => pure m
| none => throw $ userError s!"JSON '{j.compress}' did not have the format of a JSON-RPC message"
def readRequestAs (h : FS.Stream) (nBytes : Nat) (expectedMethod : String) (α) [FromJson α] : IO (Request α) := do
let m ← h.readMessage nBytes
match m with
| Message.request id method params? =>
if method = expectedMethod then
let j := toJson params?
match fromJson? j with
| some v => pure ⟨id, expectedMethod, v⟩
| none => throw $ userError s!"Unexpected param '{j.compress}' for method '{expectedMethod}'"
else
throw $ userError s!"Expected method '{expectedMethod}', got method '{method}'"
| _ => throw $ userError s!"Expected JSON-RPC request, got: '{(toJson m).compress}'"
def readNotificationAs (h : FS.Stream) (nBytes : Nat) (expectedMethod : String) (α) [FromJson α] : IO (Notification α) := do
let m ← h.readMessage nBytes
match m with
| Message.notification method params? =>
if method = expectedMethod then
let j := toJson params?
match fromJson? j with
| some v => pure ⟨expectedMethod, v⟩
| none => throw $ userError s!"Unexpected param '{j.compress}' for method '{expectedMethod}'"
else
throw $ userError s!"Expected method '{expectedMethod}', got method '{method}'"
| _ => throw $ userError s!"Expected JSON-RPC notification, got: '{(toJson m).compress}'"
def readResponseAs (h : FS.Stream) (nBytes : Nat) (expectedID : RequestID) (α) [FromJson α] : IO (Response α) := do
let m ← h.readMessage nBytes
match m with
| Message.response id result =>
if id == expectedID then
match fromJson? result with
| some v => pure ⟨expectedID, v⟩
| none => throw $ userError s!"Unexpected result '{result.compress}'"
else
throw $ userError s!"Expected id {expectedID}, got id {id}"
| _ => throw $ userError s!"Expected JSON-RPC response, got: '{(toJson m).compress}'"
end
section
variable [ToJson α]
def writeMessage (h : FS.Stream) (m : Message) : IO Unit :=
h.writeJson (toJson m)
def writeRequest (h : FS.Stream) (r : Request α) : IO Unit :=
h.writeMessage r
def writeNotification (h : FS.Stream) (n : Notification α) : IO Unit :=
h.writeMessage n
def writeResponse (h : FS.Stream) (r : Response α) : IO Unit :=
h.writeMessage r
def writeResponseError (h : FS.Stream) (e : ResponseError Unit) : IO Unit :=
h.writeMessage (Message.responseError e.id e.code e.message none)
def writeResponseErrorWithData (h : FS.Stream) (e : ResponseError α) : IO Unit :=
h.writeMessage e
end
end IO.FS.Stream
|
006c3ddee7a6ba68a1fc96dabd5a85b80b592770 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/analysis/subadditive.lean | 56d0adc2ebdf3f1971ea6d3a680d05bd50a18d84 | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 4,717 | lean | /-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import topology.instances.real
import order.filter.archimedean
/-!
# Convergence of subadditive sequences
A subadditive sequence `u : ℕ → ℝ` is a sequence satisfying `u (m + n) ≤ u m + u n` for all `m, n`.
We define this notion as `subadditive u`, and prove in `subadditive.tendsto_lim` that, if `u n / n`
is bounded below, then it converges to a limit (that we denote by `subadditive.lim` for
convenience). This result is known as Fekete's lemma in the literature.
-/
noncomputable theory
open set filter
open_locale topological_space
/-- A real-valued sequence is subadditive if it satisfies the inequality `u (m + n) ≤ u m + u n`
for all `m, n`. -/
def subadditive (u : ℕ → ℝ) : Prop :=
∀ m n, u (m + n) ≤ u m + u n
namespace subadditive
variables {u : ℕ → ℝ} (h : subadditive u)
include h
/-- The limit of a bounded-below subadditive sequence. The fact that the sequence indeed tends to
this limit is given in `subadditive.tendsto_lim` -/
@[irreducible, nolint unused_arguments]
protected def lim := Inf ((λ (n : ℕ), u n / n) '' (Ici 1))
lemma lim_le_div (hbdd : bdd_below (range (λ n, u n / n))) {n : ℕ} (hn : n ≠ 0) :
h.lim ≤ u n / n :=
begin
rw subadditive.lim,
apply cInf_le _ _,
{ rcases hbdd with ⟨c, hc⟩,
exact ⟨c, λ x hx, hc (image_subset_range _ _ hx)⟩ },
{ apply mem_image_of_mem,
exact zero_lt_iff.2 hn }
end
lemma apply_mul_add_le (k n r) : u (k * n + r) ≤ k * u n + u r :=
begin
induction k with k IH, { simp only [nat.cast_zero, zero_mul, zero_add] },
calc
u ((k+1) * n + r)
= u (n + (k * n + r)) : by { congr' 1, ring }
... ≤ u n + u (k * n + r) : h _ _
... ≤ u n + (k * u n + u r) : add_le_add_left IH _
... = (k+1) * u n + u r : by ring
end
lemma eventually_div_lt_of_div_lt {L : ℝ} {n : ℕ} (hn : n ≠ 0) (hL : u n / n < L) :
∀ᶠ p in at_top, u p / p < L :=
begin
have I : ∀ (i : ℕ), 0 < i → (i : ℝ) ≠ 0,
{ assume i hi, simp only [hi.ne', ne.def, nat.cast_eq_zero, not_false_iff] },
obtain ⟨w, nw, wL⟩ : ∃ w, u n / n < w ∧ w < L := exists_between hL,
obtain ⟨x, hx⟩ : ∃ x, ∀ i < n, u i - i * w ≤ x,
{ obtain ⟨x, hx⟩ : bdd_above (↑(finset.image (λ i, u i - i * w) (finset.range n))) :=
finset.bdd_above _,
refine ⟨x, λ i hi, _⟩,
simp only [upper_bounds, mem_image, and_imp, forall_exists_index, mem_set_of_eq,
forall_apply_eq_imp_iff₂, finset.mem_range, finset.mem_coe, finset.coe_image] at hx,
exact hx _ hi },
have A : ∀ (p : ℕ), u p ≤ p * w + x,
{ assume p,
let s := p / n,
let r := p % n,
have hp : p = s * n + r, by rw [mul_comm, nat.div_add_mod],
calc u p = u (s * n + r) : by rw hp
... ≤ s * u n + u r : h.apply_mul_add_le _ _ _
... = s * n * (u n / n) + u r : by { field_simp [I _ hn.bot_lt], ring }
... ≤ s * n * w + u r : add_le_add_right
(mul_le_mul_of_nonneg_left nw.le (mul_nonneg (nat.cast_nonneg _) (nat.cast_nonneg _))) _
... = (s * n + r) * w + (u r - r * w) : by ring
... = p * w + (u r - r * w) : by { rw hp, simp only [nat.cast_add, nat.cast_mul] }
... ≤ p * w + x : add_le_add_left (hx _ (nat.mod_lt _ hn.bot_lt)) _ },
have B : ∀ᶠ p in at_top, u p / p ≤ w + x / p,
{ refine eventually_at_top.2 ⟨1, λ p hp, _⟩,
simp only [I p hp, ne.def, not_false_iff] with field_simps,
refine div_le_div_of_le_of_nonneg _ (nat.cast_nonneg _),
rw mul_comm,
exact A _ },
have C : ∀ᶠ (p : ℕ) in at_top, w + x / p < L,
{ have : tendsto (λ (p : ℕ), w + x / p) at_top (𝓝 (w + 0)) :=
tendsto_const_nhds.add (tendsto_const_nhds.div_at_top tendsto_coe_nat_at_top_at_top),
rw add_zero at this,
exact (tendsto_order.1 this).2 _ wL },
filter_upwards [B, C] with _ hp h'p using hp.trans_lt h'p,
end
/-- Fekete's lemma: a subadditive sequence which is bounded below converges. -/
theorem tendsto_lim (hbdd : bdd_below (range (λ n, u n / n))) :
tendsto (λ n, u n / n) at_top (𝓝 h.lim) :=
begin
refine tendsto_order.2 ⟨λ l hl, _, λ L hL, _⟩,
{ refine eventually_at_top.2
⟨1, λ n hn, hl.trans_le (h.lim_le_div hbdd ((zero_lt_one.trans_le hn).ne'))⟩ },
{ obtain ⟨n, npos, hn⟩ : ∃ (n : ℕ), 0 < n ∧ u n / n < L,
{ rw subadditive.lim at hL,
rcases exists_lt_of_cInf_lt (by simp) hL with ⟨x, hx, xL⟩,
rcases (mem_image _ _ _).1 hx with ⟨n, hn, rfl⟩,
exact ⟨n, zero_lt_one.trans_le hn, xL⟩ },
exact h.eventually_div_lt_of_div_lt npos.ne' hn }
end
end subadditive
|
52d73d2054a6832356d6ef58dc727a39ce26dfb0 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/category_theory/limits/concrete_category.lean | 78bd6cbd28a39861a344a8695e4e4d07589ea6de | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,149 | lean | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.category_theory.limits.limits
import Mathlib.category_theory.concrete_category.basic
import Mathlib.PostPort
universes v u
namespace Mathlib
/-!
# Facts about (co)limits of functors into concrete categories
-/
namespace category_theory.limits
-- We now prove a lemma about naturality of cones over functors into bundled categories.
namespace cone
/-- Naturality of a cone over functors to a concrete category. -/
@[simp] theorem w_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] {F : J ⥤ C} (s : cone F) {j : J} {j' : J} (f : j ⟶ j') (x : ↥(X s)) : coe_fn (functor.map F f) (coe_fn (nat_trans.app (π s) j) x) = coe_fn (nat_trans.app (π s) j') x := sorry
@[simp] theorem w_forget_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J ⥤ C) (s : cone (F ⋙ forget C)) {j : J} {j' : J} (f : j ⟶ j') (x : X s) : coe_fn (functor.map F f) (nat_trans.app (π s) j x) = nat_trans.app (π s) j' x :=
congr_fun (w s f) x
end cone
namespace cocone
/-- Naturality of a cocone over functors into a concrete category. -/
@[simp] theorem w_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] {F : J ⥤ C} (s : cocone F) {j : J} {j' : J} (f : j ⟶ j') (x : ↥(functor.obj F j)) : coe_fn (nat_trans.app (ι s) j') (coe_fn (functor.map F f) x) = coe_fn (nat_trans.app (ι s) j) x := sorry
@[simp] theorem w_forget_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J ⥤ C) (s : cocone (F ⋙ forget C)) {j : J} {j' : J} (f : j ⟶ j') (x : ↥(functor.obj F j)) : nat_trans.app (ι s) j' (coe_fn (functor.map F f) x) = nat_trans.app (ι s) j x :=
congr_fun (w s f) x
end cocone
@[simp] theorem limit.lift_π_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J ⥤ C) [has_limit F] (s : cone F) (j : J) (x : ↥(cone.X s)) : coe_fn (limit.π F j) (coe_fn (limit.lift F s) x) = coe_fn (nat_trans.app (cone.π s) j) x := sorry
@[simp] theorem limit.w_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J ⥤ C) [has_limit F] {j : J} {j' : J} (f : j ⟶ j') (x : ↥(limit F)) : coe_fn (functor.map F f) (coe_fn (limit.π F j) x) = coe_fn (limit.π F j') x := sorry
@[simp] theorem colimit.ι_desc_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J ⥤ C) [has_colimit F] (s : cocone F) (j : J) (x : ↥(functor.obj F j)) : coe_fn (colimit.desc F s) (coe_fn (colimit.ι F j) x) = coe_fn (nat_trans.app (cocone.ι s) j) x := sorry
@[simp] theorem colimit.w_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J ⥤ C) [has_colimit F] {j : J} {j' : J} (f : j ⟶ j') (x : ↥(functor.obj F j)) : coe_fn (colimit.ι F j') (coe_fn (functor.map F f) x) = coe_fn (colimit.ι F j) x := sorry
|
ee798d0b62e77f994f02f20598d1531350aeb60f | 4aca55eba10c989f0d58647d3c2f371e7da44355 | /src/finite_field.lean | ec9416fb95f857fa6c2272d80f20d8ac7e47c936 | [] | no_license | eric-wieser/l534zhan-my_project | f9fc75fb5454405e1a2fa9b56cf96c355f6f2336 | febc91e76b7b00fe2517f258ca04d27b7f35fcf3 | refs/heads/master | 1,689,218,910,420 | 1,630,439,440,000 | 1,630,439,440,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 29,312 | lean | /-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Lu-Ming Zhang.
-/
import field_theory.finite.basic
import group_theory.quotient_group
import monoid_hom
import set_finset_fintype
/-!
# Finite fields
This file supplements basic results about finite field that are missing in mathlib.
As well, this files defines quadratic residues and the quadratic character,
and contains basic results about them.
Throughout most of this file, `F` denotes a finite field,
and `q` is notation for the cardinality of `F`, and `p` is the character of `F`.
## Main results
1. `finite_field.is_quad_residue`: a proposition predicating whether a given `a : F` is a quadratic residue in `F`.
`finite_field.is_quad_residue a` is defined to be `a ≠ 0 ∧ ∃ b, a = b^2`.
2. `finite_field.is_non_residue`: a proposition predicating whether a given `a : F` is a quadratic non-residue in `F`.
`finite_field.is_non_residue a` is defined to be `a ≠ 0 ∧ ¬ ∃ b, a = b^2`.
3. `finite_field.quad_residues`: the subtype of `F` containing quadratic residues.
4. `finite_field.quad_residues_set`: the set containing quadratic residues of `F`.
5. `finite_field.non_residues`: the subtype of `F` containing quadratic non-residues.
6. `finite_field.non_residues_set`: the set containing quadratic non-residues of `F`.
7. `finite_field.sq`: the square function from `units F` to `units F`, defined as a group homomorphism.
8. `finite_filed.non_residue_mul`: is the map `_ * b` given a non-residue `b` defined on `{a : F // is_quad_residue a}`.
9. `finite_field.quad_char`: defines the quadratic character of `a` in a finite field `F`.
- `χ a = 0 ` if `a = 0`
- `χ a = 1` if `a ≠ 0` and `a` is a square
- `χ a = -1` otherwise
## Notation
Throughout most of this file, `F` denotes a finite field,
and `q` is notation for the cardinality of `F`, and `p` is the character of `F`.
`χ a` denotes the quadratic character of `a : F`.
-/
--local attribute [instance] set_fintype
--local attribute [-instance] set.fintype_univ
set_option pp.beta true
open_locale big_operators
open finset fintype monoid_hom nat.modeq function
namespace finite_field
variables {F : Type*} [field F] [fintype F] {p : ℕ} [char_p F p]
local notation `q` := fintype.card F -- declares `q` as a notation
/- ## basic -/
section basic
/-- `|F| = |units F| + 1`. The `+` version of `finite_field.card_units`. -/
lemma card_units' :
fintype.card F = fintype.card (units F) + 1 :=
begin
simp [card_units],
have h : 0 < fintype.card F := fintype.card_pos_iff.2 ⟨0⟩,
rw [nat.sub_add_cancel h]
end
variables (p)
/-- If `n` is a prime number and `↑n = 0` in `F`, then `p = n`. -/
lemma char_eq_of_prime_eq_zero
{n : ℕ} (h₁ : nat.prime n) (h₂ : ↑n = (0 : F)) : p = n :=
begin
have g₁ : nat.prime p, {obtain ⟨n, h1, h2⟩:= finite_field.card F p, assumption},
have g₂ := (char_p.cast_eq_zero_iff F p n).1 h₂,
exact (nat.prime_dvd_prime_iff_eq g₁ h₁).1 g₂,
end
lemma char_ne_two_of :
q ≡ 1 [MOD 4] ∨ q ≡ 3 [MOD 4] → p ≠ 2 :=
begin
obtain ⟨n, p_prime, g⟩:= finite_field.card F p,
rw g,
rintro (h | h),
any_goals
{ rw nat.modeq_iff_dvd at h,
rintro hp, rw hp at h,
rcases h with ⟨k, h⟩,
simp at h,
have : 2 ∣ 4 * k + 2 ^ ↑n,
{ use (2 * k + (2 : ℤ) ^ n.1.pred),
simp [mul_add], congr' 1,
ring,
have : (2 : ℤ) * 2 ^ (n : ℕ).pred = 2 ^ ((n : ℕ).pred + 1), {refl},
convert this.symm,
clear this,
convert (nat.succ_pred_eq_of_pos _).symm,
exact n.2 },
},
{ have g := eq_add_of_sub_eq h, rw ← g at this, revert this, dec_trivial },
{ have g := eq_add_of_sub_eq h, rw ← g at this, revert this, dec_trivial },
end
lemma char_ne_two_of' :
q ≡ 3 [MOD 4] → p ≠ 2 ∧ ¬ q ≡ 1 [MOD 4] :=
begin
intros h,
refine ⟨char_ne_two_of p (or.inr h), _⟩,
intro h',
have g := h'.symm.trans h,
simp [nat.modeq_iff_dvd] at g,
rcases g with ⟨k, g⟩,
norm_num at g,
have : 2 / 4 = (4 * k) / 4, {rw ← g},
norm_num at this,
simp [← this] at g,
assumption
end
/-- For non-zero `a : F`, `to_unit` converts `a` to an instance of `units F`. -/
@[simp] def to_unit {a : F} (h : a ≠ 0): units F :=
by refine {val := a, inv := a⁻¹, ..}; simp* at *
/-- The order of `a : units F` equals the order of `a` coerced in `F`. -/
lemma order_of_unit_eq (a : units F) : order_of a = order_of (a : F) :=
begin
convert (@order_of_injective (units F) _ F _ (units.coe_hom F) _ a).symm,
intros i j h,
simp[*, units.ext_iff] at *,
end
variables {p}
/-- If the character of `F` is not `2`, `-1` is not equal to `1` in `F`. -/
lemma neg_one_ne_one_of (hp: p ≠ 2) : (-1 : F) ≠ 1 :=
begin
intros h,
have h' := calc ↑(2 : ℕ) = (2 : F) : by simp
... = (1 : F) + 1 : by ring
... = (-1 : F) + 1 : by nth_rewrite 0 ←h
... = 0 : by ring,
have hp' := char_eq_of_prime_eq_zero p nat.prime_two h',
contradiction
end
/-- If the character of `F` is not `2`, `-1` is not equal to `1` in `units F`. -/
lemma neg_one_ne_one_of' (hp: p ≠ 2) : (-1 : units F) ≠ 1 :=
by simp [units.ext_iff]; exact neg_one_ne_one_of hp
/-- For `x : F`, `x^2 = 1 ↔ x = 1 ∨ x = -1`. -/
lemma sq_eq_one_iff_eq_one_or_eq_neg_one (x : F) :
x^2 = 1 ↔ x = 1 ∨ x = -1 :=
begin
rw [sub_eq_zero.symm],
have h: x ^ 2 - 1 = (x - 1) * (x + 1), {ring},
rw [h, mul_eq_zero, sub_eq_zero, add_eq_zero_iff_eq_neg]
end
/-- For `x : units F`, `x^2 = 1 ↔ x = 1 ∨ x = -1`. -/
lemma sq_eq_one_iff_eq_one_or_eq_neg_one' (x : units F) :
x^2 = 1 ↔ x = 1 ∨ x = -1 :=
begin
simp only [units.ext_iff],
convert sq_eq_one_iff_eq_one_or_eq_neg_one (x : F),
simp,
end
/-- If the character of `F` is not `2`, `-1` is the only order 2 element in `F`. -/
lemma order_of_eq_two_iff (hp: p ≠ 2) (x : F) : order_of x = 2 ↔ x = -1 :=
begin
have g := pow_order_of_eq_one x, -- g: x ^ order_of x = 1
split, -- splits into `order_of x = 2 → x = -1` and `x = -1 → order_of x = 2`
any_goals {intros h},
{ rw [h, sq_eq_one_iff_eq_one_or_eq_neg_one] at g, -- g : x = 1 ∨ x = -1
have : x ≠ 1, { intro hx, rw ←order_of_eq_one_iff at hx, linarith },
exact or_iff_not_imp_left.1 g this },
{ have hx : x^2 = 1, {rw h, ring},
have inst : fact (nat.prime 2),
{exact ⟨nat.prime_two⟩},
resetI, -- resets the instance cache.
convert order_of_eq_prime hx _,
rw h,
exact neg_one_ne_one_of hp }
end
/-- The "units" version of `order_of_eq_two_iff`. -/
lemma order_of_eq_two_iff' (hp: p ≠ 2) (x : units F) : order_of x = 2 ↔ x = -1 :=
by simp [units.ext_iff]; rw [← order_of_eq_two_iff hp (x : F), order_of_unit_eq x]
lemma card_eq_one_mod_n_iff_n_divide_card_units (n : ℕ):
q ≡ 1 [MOD n] ↔ n ∣ fintype.card (units F) :=
begin
rw [finite_field.card_units, nat.modeq.comm],
convert nat.modeq_iff_dvd' (card_pos_iff.mpr ⟨(0 : F)⟩),
end
/-- `-1` is a square in `units F` iff the cardinal `q ≡ 1 [MOD 4]`. -/
theorem neg_one_eq_sq_iff' (hp: p ≠ 2) :
(∃ a : units F, -1 = a^2) ↔ q ≡ 1 [MOD 4] :=
begin
-- rewrites the RHS to `4 ∣ fintype.card (units F)`
rw card_eq_one_mod_n_iff_n_divide_card_units,
split, -- splits the goal into two directions
-- the `→` direction
{ rintros ⟨a, h'⟩,
-- h: a ^ 4 = 1
have h := calc a^4 = a^2 * a^2 : by group
... = 1: by simp [← h'],
-- h₀: order_of a ∣ 4
have h₀ := order_of_dvd_of_pow_eq_one h,
-- au: order_of a ≤ 4
have au := nat.le_of_dvd dec_trivial h₀,
-- g₁ : a ≠ 1
have g₁ : a ≠ 1,
{ rintro rfl, simp at h',
exact absurd h' (neg_one_ne_one_of' hp) },
-- h₁ : order_of a ≠ 1
have h₁ := mt order_of_eq_one_iff.1 g₁,
-- g₂ : a ≠ -1
have g₂ : a ≠ -1,
{ rintro rfl, simp [pow_two] at h',
exact absurd h' (neg_one_ne_one_of' hp) },
-- h₂ : order_of a ≠ 2
have h₂ := mt (order_of_eq_two_iff' hp a).1 g₂,
-- ha : order_of a = 4
have ha : order_of a = 4,
{ revert h₀ h₁ h₂,
interval_cases (order_of a),
any_goals {rw h_1, norm_num} },
simp [← ha, order_of_dvd_card_univ] },
-- the `←` direction
{ rintro ⟨k, hF⟩, -- hF : |units F| = 4 * k
-- `hg'` is a proof that `g` generates `units F`
obtain ⟨g, hg'⟩ := is_cyclic.exists_generator (units F),
-- hg : g ^ |units F| = 1
have hg := @pow_card_eq_one (units F) g _ _,
have eq : 4 * k = k * 2 * 2, {ring},
-- rewrite `hg` to `hg : g ^ (k * 2) = 1 ∨ g ^ (k * 2) = -1`
rw [hF, eq, pow_mul, sq_eq_one_iff_eq_one_or_eq_neg_one'] at hg,
rcases hg with (hg | hg), -- splits into two cases
-- case `g ^ (k * 2) = 1`
{ have hk₁ := card_pos_iff.mpr ⟨(1 : units F)⟩,
have o₁ := order_of_eq_card_of_forall_mem_gpowers hg',
have o₂ := order_of_dvd_of_pow_eq_one hg,
have le := nat.le_of_dvd (by linarith) o₂,
rw [o₁, hF] at *,
exfalso, linarith },
-- case `g ^ (k * 2) = -1`
{ use g ^ k, simp [← hg, pow_mul] } },
end
/-- `-1` is a square in `F` iff the cardinal `q ≡ 1 [MOD 4]`. -/
lemma neg_one_eq_sq_iff (hp: p ≠ 2) :
(∃ a : F, -1 = a^2) ↔ fintype.card F ≡ 1 [MOD 4] :=
begin
rw [←neg_one_eq_sq_iff' hp],
-- the current goal is
-- `(∃ (a : F), -1 = a ^ 2) ↔ ∃ (a : units F), -1 = a ^ 2`
split, -- splits into two directions
any_goals {rintros ⟨a, ha⟩},
-- the `→` direction
{ have ha' : a ≠ 0,
{rintros rfl, simp* at *},
use (to_unit ha'),
simpa [units.ext_iff] },
-- the `←` direction
{ use a, simp [units.ext_iff] at ha, assumption },
assumption
end
variables (F)
lemma disjoint_units_zero : disjoint {a : F | a ≠ 0} {0} :=
by simp
lemma units_union_zero : {a : F | a ≠ 0} ∪ {0} = @set.univ F :=
by {tidy, tauto}
end basic
/- ## end basic -/
/- ## quad_residues -/
section quad_residues
variables {F p}
/-- A proposition predicating whether a given `a : F` is a quadratic residue in `F`.
`finite_field.is_quad_residue a` is defined to be `a ≠ 0 ∧ ∃ b, a = b^2`. -/
--@[derive decidable]
def is_quad_residue (a : F) : Prop := a ≠ 0 ∧ ∃ b, a = b^2
/-- A proposition predicating whether a given `a : F` is a quadratic non-residue in `F`.
`finite_field.is_non_residue a` is defined to be `a ≠ 0 ∧ ¬ ∃ b, a = b^2`. -/
--@[derive decidable]
def is_non_residue (a : F) : Prop := a ≠ 0 ∧ ¬ ∃ b, a = b^2
instance [decidable_eq F] (a : F) : decidable (is_quad_residue a) :=
by {unfold is_quad_residue, apply_instance}
instance [decidable_eq F] (a : F) : decidable (is_non_residue a) :=
by {unfold is_non_residue, apply_instance}
variables {F p}
lemma not_residue_iff_is_non_residue
{a : F} (h : a ≠ 0) :
¬ is_quad_residue a ↔ is_non_residue a :=
by simp [is_quad_residue, is_non_residue, *] at *
lemma is_non_residue_of_not_residue
{a : F} (h : a ≠ 0) (g : ¬ is_quad_residue a) :
is_non_residue a :=
(not_residue_iff_is_non_residue h).1 g
lemma not_non_residue_iff_is_residue
{a : F} (h : a ≠ 0) :
¬ is_non_residue a ↔ is_quad_residue a :=
by simp [is_quad_residue, is_non_residue, *] at *
lemma is_residue_of_not_non_residue
{a : F} (h : a ≠ 0) (g : ¬ is_non_residue a) :
is_quad_residue a :=
(not_non_residue_iff_is_residue h).1 g
lemma residue_or_non_residue
{a : F} (h : a ≠ 0) :
is_quad_residue a ∨ is_non_residue a :=
begin
by_cases g: is_quad_residue a,
exact or.inl g,
exact or.inr (is_non_residue_of_not_residue h g),
end
/-- `-1` is a residue if `q ≡ 1 [MOD 4]`. -/
lemma neg_one_is_residue_of (hF : q ≡ 1 [MOD 4]) :
is_quad_residue (-1 : F) :=
begin
obtain ⟨p, inst⟩ := char_p.exists F, -- derive the char p of F
resetI, -- resets the instance cache
have hp := char_ne_two_of p (or.inl hF), -- hp: p ≠ 2
have h := (neg_one_eq_sq_iff hp).2 hF, -- h : -1 is a square
refine ⟨by tidy, h⟩
end
/-- `-1` is a non-residue if `q ≡ 3 [MOD 4]`. -/
lemma neg_one_is_non_residue_of (hF : q ≡ 3 [MOD 4]) :
is_non_residue (-1 : F) :=
begin
obtain ⟨p, inst⟩ := char_p.exists F, -- derive the char p of F
resetI, -- resets the instance cache
-- hp: p ≠ 2, hF': ¬fintype.card F ≡ 1 [MOD 4]
obtain ⟨hp, hF'⟩ := char_ne_two_of' p hF,
-- h: ¬∃ (a : F), -1 = a ^ 2
have h := mt (neg_one_eq_sq_iff hp).1 hF',
refine ⟨by tidy, h⟩
end
variable (F)
@[simp] lemma eq_residues :
{j // ¬j = 0 ∧ ∃ (b : F), j = b ^ 2} =
{a : F // is_quad_residue a} := rfl
@[simp] lemma eq_non_residues :
{j // ¬j = 0 ∧ ¬∃ (x : F), j = x ^ 2} =
{a : F // is_non_residue a} := rfl
@[simp] lemma eq_non_residues' :
{j // ¬j = 0 ∧ ∀ (x : F), ¬j = x ^ 2} =
{a : F // is_non_residue a} := by simp [is_non_residue]
/-- `|F| = 1 + |{a : F // is_quad_residue a}| + |{a : F // is_non_residue a}|` -/
lemma eq_one_add_card_residues_add_card_non_residues
[decidable_eq F]:
q = 1 + fintype.card {a : F // is_quad_residue a} +
fintype.card {a : F // is_non_residue a} :=
begin
rw [fintype.card_split (λ a : F, a = 0),
fintype.card_split' (λ a : F, a ≠ 0) (λ a, ∃ b, a = b^2)],
simp [add_assoc],
end
/- ### sq_function -/
section sq_function
open quotient_group
variables (F) -- re-declares `F` as an explicit variable
/-- `sq` is the square function from `units F` to `units F`,
defined as a group homomorphism. -/
def sq : (units F) →* (units F) :=
⟨λ a, a * a, by simp, (λ x y, by simp [units.ext_iff]; ring)⟩
/-- `|units F| = |(sq F).range| * |(sq F).ker|` -/
theorem sq.iso [decidable_eq F] :
fintype.card (units F) = fintype.card (sq F).range * fintype.card (sq F).ker :=
begin
have iso := quotient_ker_equiv_range (sq F),
have eq := fintype.card_congr (mul_equiv.to_equiv iso),
rw [subgroup.card_eq_card_quotient_mul_card_subgroup (sq F).ker, eq]
end
/-- `sq.range_equiv` constructs the natural equivalence between
the `(sq F).range` and `{a : F // is_quad_residue a}`. -/
def sq.range_equiv : (sq F).range ≃ {a : F // is_quad_residue a} :=
⟨
λ a, ⟨a, by {have h := a.2, simp [sq] at h, rcases h with ⟨b, h⟩,
simp [is_quad_residue, ← h], use b, ring }⟩,
λ a, ⟨to_unit (a.2).1, by {obtain ⟨h, ⟨b, hab⟩⟩:= a.2,
have hb : b ≠ 0, {rintros rfl, simp* at *},
use to_unit hb, simp [units.ext_iff, sq, ← pow_two, ← hab] }⟩,
λ a, by simp [units.ext_iff],
λ a, by simp
⟩
/-- `|(sq F).range| = |{a : F // is_quad_residue a}|` -/
theorem sq.card_range_eq [decidable_eq F] :
fintype.card (sq F).range = fintype.card {a : F // is_quad_residue a} :=
by apply fintype.card_congr (sq.range_equiv F)
lemma sq.ker_carrier_eq :
(sq F).ker.carrier = {1, -1} :=
begin
simp [ker, subgroup.comap, set.preimage, sq],
ext, simp,
convert sq_eq_one_iff_eq_one_or_eq_neg_one' x,
simp [npow_rec],
end
lemma sq.card_ker_carrier_eq [decidable_eq F] (hp: p ≠ 2) :
fintype.card (sq F).ker.carrier = 2 :=
begin
simp [sq.ker_carrier_eq F],
convert @set.card_insert _ (1 : units F) {-1} _ _ _; simp,
exact ne.symm (neg_one_ne_one_of' hp),
end
/-- `|(sq F).ker| = 2` if `p ≠ 2` -/
theorem sq.card_ker_eq [decidable_eq F] (hp: p ≠ 2) :
fintype.card (sq F).ker = 2 :=
by rw [←sq.card_ker_carrier_eq F hp]; refl
end sq_function
/- ### end sq_function -/
variable (F)
/-- `|units F| = |{a : F // is_quad_residue a}| * 2` -/
theorem card_units_eq_card_residues_mul_two [decidable_eq F] (hp: p ≠ 2) :
fintype.card (units F) = fintype.card {a : F // is_quad_residue a} * 2 :=
by rwa [sq.iso, sq.card_range_eq, sq.card_ker_eq F hp]
/-- `|{a : F // is_quad_residue a}| = |{a : F // is_non_residue a}|`-/
theorem card_residues_eq_card_non_residues
[decidable_eq F] (hp : p ≠ 2):
fintype.card {a : F // is_quad_residue a} =
fintype.card {a : F // is_non_residue a} :=
begin
have eq := eq_one_add_card_residues_add_card_non_residues F,
simp [card_units', card_units_eq_card_residues_mul_two F hp] at eq,
linarith
end
/- `card_units_eq_card_residues_mul_two F hp` is a built API that proves
`|F| = 1 + |{a : F // is_quad_residue a}| + |{a : F // is_non_residue a}|` -/
/-- unfolded version of `card_residues_eq_card_non_residues` -/
theorem card_residues_eq_card_non_residues'
[decidable_eq F] (hp : p ≠ 2):
(@univ {a : F // is_quad_residue a} _).card =
(@univ {a : F // is_non_residue a} _).card :=
by convert card_residues_eq_card_non_residues F hp
variable {F} -- re-declares `F` as an implicit variable
example : (0 : F)⁻¹ = 0 := by simp
/-- `a⁻¹` is a residue if and only if `a` is. -/
theorem inv_is_residue_iff {a : F} :
is_quad_residue a⁻¹ ↔ is_quad_residue a :=
begin
split, -- splits into two directions
any_goals {rintro ⟨h, b, g⟩, refine ⟨by tidy, b⁻¹, by simp [←g]⟩},
end
/-- `a⁻¹` is a non residue if and only if `a` is. -/
theorem inv_is_non_residue_iff {a : F} :
is_non_residue a⁻¹ ↔ is_non_residue a :=
begin
by_cases h : a = 0,
{simp* at *}, -- when `a = 0`
have h' : a⁻¹ ≠ 0 := by simp [h],
simp [←not_residue_iff_is_non_residue, *, inv_is_residue_iff]
end
theorem residue_mul_residue_is_residue
{a b : F} (ha : is_quad_residue a) (hb : is_quad_residue b) :
is_quad_residue (a * b) :=
begin
obtain ⟨ha, c, rfl⟩ := ha,
obtain ⟨hb, d, rfl⟩ := hb,
refine ⟨mul_ne_zero ha hb, c*d, _⟩,
ring
end
theorem non_residue_mul_residue_is_non_residue
{a b : F} (ha : is_non_residue a) (hb : is_quad_residue b) :
is_non_residue (a * b) :=
begin
obtain ⟨hb, c, rfl⟩ := hb,
refine ⟨mul_ne_zero ha.1 hb, _⟩,
rintro ⟨d, h⟩,
convert ha.2 ⟨(d * c⁻¹), _⟩,
field_simp [← h],
end
theorem residue_mul_non_residue_is_non_residue
{a b : F} (ha : is_quad_residue a) (hb : is_non_residue b):
is_non_residue (a * b) :=
by simp [mul_comm a, non_residue_mul_residue_is_non_residue hb ha]
/-- `finite_filed.non_residue_mul` is the map `a * _` given a non-residue `a`
defined on `{b : F // is_quad_residue b}`. -/
def non_residue_mul {a: F} (ha : is_non_residue a) :
{b : F // is_quad_residue b} → {b : F // is_non_residue b} :=
λ b, ⟨a * b, non_residue_mul_residue_is_non_residue ha b.2⟩
open function
/-- proves that `a * _` is injective from residues for a non-residue `a` -/
lemma is_non_residue.mul_is_injective
{a: F} (ha : is_non_residue a) :
injective (non_residue_mul ha):=
begin
intros b₁ b₂ h,
simp [non_residue_mul] at h,
ext,
convert or.resolve_right h ha.1,
end
/-- proves that `a * _` is surjective onto non-residues for a non-residue `a` -/
lemma is_non_residue.mul_is_surjective
[decidable_eq F] (hp : p ≠ 2) {a: F} (ha : is_non_residue a) :
surjective (non_residue_mul ha):=
begin
by_contra, -- prove by contradtiction
have lt := card_lt_of_injective_not_surjective
(non_residue_mul ha) (ha.mul_is_injective) h,
have eq := card_residues_eq_card_non_residues F hp,
linarith
end
theorem non_residue_mul_non_residue_is_residue
[decidable_eq F] (hp : p ≠ 2)
{a b : F} (ha : is_non_residue a) (hb : is_non_residue b):
is_quad_residue (a * b) :=
begin
by_contra h, -- prove by contradtiction
-- rw `h` to `is_non_residue (a * b)`
rw [not_residue_iff_is_non_residue (mul_ne_zero ha.1 hb.1)] at h,
-- surj : `a * _` is surjective onto non-residues from residues
have surj := ha.mul_is_surjective hp,
simp [function.surjective] at surj,
-- in particular, non-residue `a * b` is in the range of `a * _`
specialize surj (a * b) h,
-- say `a * b' = a * b` and `hb' : is_quad_residue b'`
rcases surj with ⟨b', hb', eq⟩,
simp [non_residue_mul, ha.1] at eq, -- `eq: b' = b`
rw [eq] at hb',
exact absurd hb'.2 hb.2,
end
end quad_residues
/- ## end quad_residues -/
/- ## quad_char -/
section quad_char
variables {F} [decidable_eq F]
/-- `finite_field.quad_char` defines the quadratic character of `a` in a finite field `F`. -/
def quad_char (a : F) : ℚ :=
if a = 0 then 0 else
if ∃ b : F, a = b^2 then 1 else
-1.
notation `χ` := quad_char -- declare the notation for `quad_char`
@[simp] lemma quad_char_zero_eq_zero : χ (0 : F) = 0 := by simp [quad_char]
@[simp] lemma quad_char_one_eq_one : χ (1 : F) = 1 :=
by {simp [quad_char], intros h, specialize h 1, simp* at *}
@[simp] lemma quad_char_ne_zero_of_ne_zero {a : F} (h : a ≠ 0) :
χ a ≠ 0 :=
by by_cases (∃ (b : F), a = b ^ 2); simp [quad_char, *] at *
@[simp] lemma quad_char_eq_zero_iff_eq_zero (a : F) :
χ a = 0 ↔ a = 0 :=
begin
split,
{contrapose, exact quad_char_ne_zero_of_ne_zero},
{rintro rfl, exact quad_char_zero_eq_zero}
end
@[simp] lemma quad_char_i_sub_j_eq_zero_iff_i_eq_j (i j : F) :
χ (i - j) = 0 ↔ i = j :=
by simp [sub_eq_zero]
@[simp] lemma quad_char_eq_one_or_neg_one_of_char_ne_zero {a : F} (h : χ a ≠ 0) :
(χ a = 1) ∨ (χ a = -1) :=
by by_cases (∃ (b : F), a = b ^ 2); simp [quad_char, *] at *
@[simp] lemma quad_char_eq_neg_one_or_one_of_ne_zero' {a : F} (h : χ a ≠ 0) :
(χ a = -1) ∨ (χ a = 1) :=
or.swap (quad_char_eq_one_or_neg_one_of_char_ne_zero h)
@[simp] lemma quad_char_eq_one_or_neg_one_of_ne_zero {a : F} (h : a ≠ 0) :
(χ a = 1) ∨ (χ a = -1) :=
quad_char_eq_one_or_neg_one_of_char_ne_zero (quad_char_ne_zero_of_ne_zero h)
@[simp] lemma quad_char_suqare_eq_one_of_ne_zero {a : F} (h : a ≠ 0) :
χ a * χ a = 1 :=
by by_cases (∃ (b : F), a = b ^ 2); simp [quad_char, *] at *
@[simp] lemma quad_char_eq_one_of {a : F} (h : a ≠ 0) (h' : ∃ (b : F), a = b ^ 2) :
χ a = 1 :=
by simp [quad_char, *] at *
@[simp] lemma quad_char_eq_neg_one_of {a : F} (h : a ≠ 0) (h' : ¬ ∃ (b : F), a = b ^ 2) :
χ a = -1 :=
by simp [quad_char, *] at *
@[simp] lemma quad_char_eq_one_of_quad_residue {a : F} (h : is_quad_residue a) :
χ a = 1 := quad_char_eq_one_of h.1 h.2
@[simp] lemma quad_char_eq_neg_one_of_non_residue {a : F} (h : is_non_residue a) :
χ a = -1 := quad_char_eq_neg_one_of h.1 h.2
@[simp] lemma nonzero_quad_char_eq_one_or_neg_one' {a b : F} (h : a ≠ b) :
(χ (a - b) = 1) ∨ (χ (a - b) = -1) :=
by {have h':= sub_ne_zero.mpr h, simp* at *}
variables {F p}
theorem quad_char_inv (a : F) : χ a⁻¹ = χ a :=
begin
by_cases ha: a = 0,
{simp [ha]}, -- case `a=0`
-- splits into cases if `a` is a residue
obtain (ga | ga) := residue_or_non_residue ha,
-- case 1: `a` is a residue
{have g':= inv_is_residue_iff.2 ga, simp*},
-- case 2: `a` is a non-residue
{have g':= inv_is_non_residue_iff.2 ga, simp*},
end
theorem quad_char_mul (hp : p ≠ 2) (a b : F) : χ (a * b) = χ a * χ b :=
begin
by_cases ha: a = 0,
any_goals {by_cases hb : b = 0},
any_goals {simp*}, -- closes cases when `a = 0` or `b =0`
-- splits into cases if `a` is a residue
obtain (ga | ga) := residue_or_non_residue ha,
-- splits into cases if `b` is a residue
any_goals {obtain (gb | gb) := residue_or_non_residue hb},
-- case 1 : `a` is, `b` is
{ have g:= residue_mul_residue_is_residue ga gb, simp* },
-- case 2 : `a` is, `b` is not
{ have g:= residue_mul_non_residue_is_non_residue ga gb, simp* },
-- case 1 : `a` is not, `b` is
{ have g:= non_residue_mul_residue_is_non_residue ga gb, simp* },
-- case 4 : `a` is not, `b` is not
{ have g:= non_residue_mul_non_residue_is_residue hp ga gb, simp* },
end
/-- `χ (-1) = 1` if `q ≡ 1 [MOD 4]`. -/
@[simp] theorem char_neg_one_eq_one_of (hF : q ≡ 1 [MOD 4]) :
χ (-1 : F) = 1 :=
by simp [neg_one_is_residue_of hF]
/-- `χ (-1) = -1` if `q ≡ 3 [MOD 4]`. -/
@[simp] theorem char_neg_one_eq_neg_one_of (hF : q ≡ 3 [MOD 4]) :
χ (-1 : F) = -1 :=
by simp [neg_one_is_non_residue_of hF]
/-- `χ (-i) = χ i` if `q ≡ 1 [MOD 4]`. -/
theorem quad_char_is_sym_of (hF : q ≡ 1 [MOD 4]) (i : F) :
χ (-i) = χ i :=
begin
obtain ⟨p, inst⟩ := char_p.exists F, -- derive the char p of F
resetI, -- resets the instance cache
have hp := char_ne_two_of p (or.inl hF), -- hp: p ≠ 2
have h := char_neg_one_eq_one_of hF, -- h: χ (-1) = 1
-- χ (-i) = 1 * χ (-i) = χ (-1) * χ (-i) = χ ((-1) * (-i))
rw [← one_mul (χ (-i)), ← h, ← quad_char_mul hp],
simp, assumption
end
/-- another form of `quad_char_is_sym_of` -/
theorem quad_char_is_sym_of' (hF : q ≡ 1 [MOD 4]) (i j : F) :
χ (j - i) = χ (i - j) :=
by convert quad_char_is_sym_of hF (i - j); ring
/-- `χ (-i) = - χ i` if `q ≡ 3 [MOD 4]`. -/
theorem quad_char_is_skewsym_of (hF : q ≡ 3 [MOD 4]) (i : F) :
χ (-i) = - χ i :=
begin
obtain ⟨p, inst⟩ := char_p.exists F, -- derive the char p of F
resetI, -- resets the instance cache
have hp := char_ne_two_of p (or.inr hF), -- hp: p ≠ 2
have h := char_neg_one_eq_neg_one_of hF, -- h: χ (-1) = 1
rw [← neg_one_mul (χ i), ← h, ← quad_char_mul hp],
simp, assumption
end
/-- another form of `quad_char_is_skewsym_of` -/
theorem quad_char_is_skewsym_of' (hF : q ≡ 3 [MOD 4]) (i j : F) :
χ (j - i) = - χ (i - j) :=
by convert quad_char_is_skewsym_of hF (i - j); ring
variable (F) -- use `F` as an explicit parameter
/-- `∑ a : {a : F // a ≠ 0}, χ (a : F) = 0` if `p ≠ 2`. -/
lemma quad_char.sum_in_non_zeros_eq_zero (hp : p ≠ 2):
∑ a : {a : F // a ≠ 0}, χ (a : F) = 0 :=
begin
simp [fintype.sum_split' (λ a : F, a ≠ 0) (λ a : F, ∃ b, a = b^2)],
suffices h :
∑ (j : {j // j ≠ 0 ∧ ∃ (b : F), j = b ^ 2}), χ (j : F) =
∑ a : {a : F // is_quad_residue a} , 1,
suffices g :
∑ (j : {j // j ≠ 0 ∧ ¬∃ (b : F), j = b ^ 2}), χ (j : F) =
∑ a : {a : F // is_non_residue a} , -1,
simp [h, g, sum_neg_distrib,
card_residues_eq_card_non_residues' F hp],
any_goals
{apply fintype.sum_congr, intros a, have := a.2, simp* at *},
end
/-- `∑ (a : F), χ a = 0` if `p ≠ 2`. -/
theorem quad_char.sum_eq_zero (hp : p ≠ 2):
∑ (a : F), χ a = 0 :=
by simp [fintype.sum_split (λ b, b = (0 : F)),
quad_char.sum_in_non_zeros_eq_zero F hp, default]
variable {F} -- use `F` as an implicit parameter
/-- another form of `quad_char.sum_eq_zero` -/
@[simp] lemma quad_char.sum_eq_zero_reindex_1 (hp : p ≠ 2) {a : F}:
∑ (b : F), χ (a - b) = 0 :=
begin
rw ← quad_char.sum_eq_zero F hp,
refine fintype.sum_equiv ((equiv.sub_right a).trans (equiv.neg _)) _ _ (by simp),
end
/-- another form of `quad_char.sum_eq_zero` -/
@[simp] lemma quad_char.sum_eq_zero_reindex_2 (hp : p ≠ 2) {b : F}:
∑ (a : F), χ (a - b) = 0 :=
begin
rw ← quad_char.sum_eq_zero F hp,
refine fintype.sum_equiv (equiv.sub_right b) _ _ (by simp),
end
variable {F} -- use `F` as an implicit parameter
/-- helper of `quad_char.sum_mul'`: reindex the terms in the summation -/
lemma quad_char.sum_mul'_aux {b : F} (hb : b ≠ 0) :
∑ (a : F) in filter (λ (a : F), ¬a = 0) univ, χ (1 + a⁻¹ * b) =
∑ (c : F) in filter (λ (c : F), ¬c = 1) univ, χ (c) :=
begin
refine finset.sum_bij (λ a ha, 1 + a⁻¹ * b) (λ a ha, _)
(λ a ha, rfl) (λ a₁ a₂ h1 h2 h, _) (λ c hc, _),
{ simp at ha, simp* },
{ simp at h1 h2, field_simp at h, rw (mul_right_inj' hb).1 h.symm },
{ simp at hc, use b * (c - 1)⁻¹,
simp [*, mul_inv_rev', sub_ne_zero.2 hc] }
end
/-- If `b ≠ 0` and `p ≠ 2`, `∑ a : F, χ (a) * χ (a + b) = -1`. -/
theorem quad_char.sum_mul' {b : F} (hb : b ≠ 0) (hp : p ≠ 2):
∑ a : F, χ (a) * χ (a + b) = -1 :=
begin
rw [finset.sum_split _ (λ a, a = (0 : F))],
simp,
have h : ∑ (a : F) in filter (λ (a : F), ¬a = 0) univ, χ a * χ (a + b) =
∑ (a : F) in filter (λ (a : F), ¬a = 0) univ, χ (1 + a⁻¹ * b),
{ apply finset.sum_congr rfl,
intros a ha, simp at ha,
simp [←quad_char_inv a, ←quad_char_mul hp a⁻¹],
field_simp },
rw [h, quad_char.sum_mul'_aux hb],
have g:= @finset.sum_split _ _ _ (@finset.univ F _) (χ) (λ a : F, a = 1) _,
simp [quad_char.sum_eq_zero F hp] at g,
linarith
end
/-- another form of `quad_char.sum_mul'` -/
theorem quad_char.sum_mul {b c : F} (hbc : b ≠ c) (hp : p ≠ 2):
∑ a : F, χ (b - a) * χ (c - a) = -1 :=
begin
rw ← quad_char.sum_mul' (sub_ne_zero.2 (ne.symm hbc)) hp,
refine fintype.sum_equiv ((equiv.sub_right b).trans (equiv.neg _)) _ _ (by simp),
end
end quad_char
/- ## end quad_char -/
end finite_field |
ea671f5bcf861aa2f37bd91807fb435f47458c3f | a7eef317ddec01b9fc6cfbb876fe7ac00f205ac7 | /src/order/basic.lean | 152d6ee84263f3a15cb2db3672c61654a3cb4041 | [
"Apache-2.0"
] | permissive | kmill/mathlib | ea5a007b67ae4e9e18dd50d31d8aa60f650425ee | 1a419a9fea7b959317eddd556e1bb9639f4dcc05 | refs/heads/master | 1,668,578,197,719 | 1,593,629,163,000 | 1,593,629,163,000 | 276,482,939 | 0 | 0 | null | 1,593,637,960,000 | 1,593,637,959,000 | null | UTF-8 | Lean | false | false | 18,678 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import data.set.basic
open function
/-!
# Basic definitions about `≤` and `<`
## Definitions
### Predicates on functions
- `monotone f`: a function between two types equipped with `≤` is monotone
if `a ≤ b` implies `f a ≤ f b`.
- `strict_mono f` : a function between two types equipped with `<` is strictly monotone
if `a < b` implies `f a < f b`.
- `order_dual α` : a type tag reversing the meaning of all inequalities.
### Transfering orders
- `order.preimage`, `preorder.lift`: transfer a (pre)order on `β` to an order on `α`
using a function `f : α → β`.
- `partial_order.lift`, `linear_order.lift`, `decidable_linear_order.lift`:
transfer a partial (resp., linear, decidable linear) order on `β` to a partial
(resp., linear, decidable linear) order on `α` using an injective function `f`.
### Extra classes
- `no_top_order`, `no_bot_order`: an order without a maximal/minimal element.
- `densely_ordered`: an order with no gaps, i.e. for any two elements `a<b` there exists
`c`, `a<c<b`.
## Main theorems
- `monotone_of_monotone_nat`: if `f : ℕ → α` and `f n ≤ f (n + 1)` for all `n`, then
`f` is monotone;
- `strict_mono.nat`: if `f : ℕ → α` and `f n < f (n + 1)` for all `n`, then f is strictly monotone.
## TODO
- expand module docs
- automatic construction of dual definitions / theorems
## Tags
preorder, order, partial order, linear order, monotone, strictly monotone
-/
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w} {r : α → α → Prop}
@[nolint ge_or_gt] -- see Note [nolint_ge]
theorem ge_of_eq [preorder α] {a b : α} : a = b → a ≥ b :=
λ h, h ▸ le_refl a
theorem preorder.ext {α} {A B : preorder α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
begin
casesI A, casesI B, congr,
{ funext x y, exact propext (H x y) },
{ funext x y,
dsimp [(≤)] at A_lt_iff_le_not_le B_lt_iff_le_not_le H,
simp [A_lt_iff_le_not_le, B_lt_iff_le_not_le, H] },
end
theorem partial_order.ext {α} {A B : partial_order α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
by { haveI this := preorder.ext H,
casesI A, casesI B, injection this, congr' }
theorem linear_order.ext {α} {A B : linear_order α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
by { haveI this := partial_order.ext H,
casesI A, casesI B, injection this, congr' }
/-- Given an order `R` on `β` and a function `f : α → β`,
the preimage order on `α` is defined by `x ≤ y ↔ f x ≤ f y`.
It is the unique order on `α` making `f` an order embedding
(assuming `f` is injective). -/
@[simp] def order.preimage {α β} (f : α → β) (s : β → β → Prop) (x y : α) := s (f x) (f y)
infix ` ⁻¹'o `:80 := order.preimage
/-- The preimage of a decidable order is decidable. -/
instance order.preimage.decidable {α β} (f : α → β) (s : β → β → Prop) [H : decidable_rel s] :
decidable_rel (f ⁻¹'o s) :=
λ x y, H _ _
section monotone
variables [preorder α] [preorder β] [preorder γ]
/-- A function between preorders is monotone if
`a ≤ b` implies `f a ≤ f b`. -/
def monotone (f : α → β) := ∀⦃a b⦄, a ≤ b → f a ≤ f b
theorem monotone_id : @monotone α α _ _ id := assume x y h, h
theorem monotone_const {b : β} : monotone (λ(a:α), b) := assume x y h, le_refl b
protected theorem monotone.comp {g : β → γ} {f : α → β} (m_g : monotone g) (m_f : monotone f) :
monotone (g ∘ f) :=
assume a b h, m_g (m_f h)
protected theorem monotone.iterate {f : α → α} (hf : monotone f) (n : ℕ) : monotone (f^[n]) :=
nat.rec_on n monotone_id (λ n ihn, ihn.comp hf)
lemma monotone_of_monotone_nat {f : ℕ → α} (hf : ∀n, f n ≤ f (n + 1)) :
monotone f | n m h :=
begin
induction h,
{ refl },
{ transitivity, assumption, exact hf _ }
end
lemma reflect_lt {α β} [linear_order α] [preorder β] {f : α → β} (hf : monotone f)
{x x' : α} (h : f x < f x') : x < x' :=
by { rw [← not_le], intro h', apply not_le_of_lt h, exact hf h' }
end monotone
/-- A function `f` is strictly monotone if `a < b` implies `f a < f b`. -/
def strict_mono [has_lt α] [has_lt β] (f : α → β) : Prop :=
∀ ⦃a b⦄, a < b → f a < f b
lemma strict_mono_id [has_lt α] : strict_mono (id : α → α) := λ a b, id
/-- A function `f` is strictly monotone increasing on `t` if `x < y` for `x,y ∈ t` implies
`f x < f y`. -/
def strict_mono_incr_on [has_lt α] [has_lt β] (f : α → β) (t : set α) : Prop :=
∀ (x ∈ t) (y ∈ t), x < y → f x < f y
/-- A function `f` is strictly monotone decreasing on `t` if `x < y` for `x,y ∈ t` implies
`f y < f x`. -/
def strict_mono_decr_on [has_lt α] [has_lt β] (f : α → β) (t : set α) : Prop :=
∀ (x ∈ t) (y ∈ t), x < y → f y < f x
namespace strict_mono
open ordering function
lemma comp [has_lt α] [has_lt β] [has_lt γ] {g : β → γ} {f : α → β}
(hg : strict_mono g) (hf : strict_mono f) :
strict_mono (g ∘ f) :=
λ a b h, hg (hf h)
protected theorem iterate [has_lt α] {f : α → α} (hf : strict_mono f) (n : ℕ) :
strict_mono (f^[n]) :=
nat.rec_on n strict_mono_id (λ n ihn, ihn.comp hf)
lemma id_le {φ : ℕ → ℕ} (h : strict_mono φ) : ∀ n, n ≤ φ n :=
λ n, nat.rec_on n (nat.zero_le _) (λ n hn, nat.succ_le_of_lt (lt_of_le_of_lt hn $ h $ nat.lt_succ_self n))
section
variables [linear_order α] [preorder β] {f : α → β}
lemma lt_iff_lt (H : strict_mono f) {a b} :
f a < f b ↔ a < b :=
⟨λ h, ((lt_trichotomy b a)
.resolve_left $ λ h', lt_asymm h $ H h')
.resolve_left $ λ e, ne_of_gt h $ congr_arg _ e, @H _ _⟩
lemma injective (H : strict_mono f) : injective f
| a b e := ((lt_trichotomy a b)
.resolve_left $ λ h, ne_of_lt (H h) e)
.resolve_right $ λ h, ne_of_gt (H h) e
theorem compares (H : strict_mono f) {a b} :
∀ {o}, compares o (f a) (f b) ↔ compares o a b
| lt := H.lt_iff_lt
| eq := ⟨λ h, H.injective h, congr_arg _⟩
| gt := H.lt_iff_lt
lemma le_iff_le (H : strict_mono f) {a b} :
f a ≤ f b ↔ a ≤ b :=
⟨λ h, le_of_not_gt $ λ h', not_le_of_lt (H h') h,
λ h, (lt_or_eq_of_le h).elim (λ h', le_of_lt (H h')) (λ h', h' ▸ le_refl _)⟩
end
protected lemma nat {β} [preorder β] {f : ℕ → β} (h : ∀n, f n < f (n+1)) : strict_mono f :=
by { intros n m hnm, induction hnm with m' hnm' ih, apply h, exact lt.trans ih (h _) }
-- `preorder α` isn't strong enough: if the preorder on α is an equivalence relation,
-- then `strict_mono f` is vacuously true.
lemma monotone [partial_order α] [preorder β] {f : α → β} (H : strict_mono f) : monotone f :=
λ a b h, (lt_or_eq_of_le h).rec (le_of_lt ∘ (@H _ _)) (by rintro rfl; refl)
end strict_mono
section
open function
lemma injective_of_lt_imp_ne [linear_order α] {f : α → β} (h : ∀ x y, x < y → f x ≠ f y) : injective f :=
begin
intros x y k,
contrapose k,
rw [←ne.def, ne_iff_lt_or_gt] at k,
cases k,
{ apply h _ _ k },
{ rw eq_comm,
apply h _ _ k }
end
variables [partial_order α] [partial_order β] {f : α → β}
lemma strict_mono_of_monotone_of_injective (h₁ : monotone f) (h₂ : injective f) :
strict_mono f :=
λ a b h,
begin
rw lt_iff_le_and_ne at ⊢ h,
exact ⟨h₁ h.1, λ e, h.2 (h₂ e)⟩
end
end
/-- Type tag for a set with dual order: `≤` means `≥` and `<` means `>`. -/
def order_dual (α : Type*) := α
namespace order_dual
instance (α : Type*) [h : nonempty α] : nonempty (order_dual α) := h
instance (α : Type*) [has_le α] : has_le (order_dual α) := ⟨λx y:α, y ≤ x⟩
instance (α : Type*) [has_lt α] : has_lt (order_dual α) := ⟨λx y:α, y < x⟩
-- `dual_le` and `dual_lt` should not be simp lemmas:
-- they cause a loop since `α` and `order_dual α` are definitionally equal
lemma dual_le [has_le α] {a b : α} :
@has_le.le (order_dual α) _ a b ↔ @has_le.le α _ b a := iff.rfl
lemma dual_lt [has_lt α] {a b : α} :
@has_lt.lt (order_dual α) _ a b ↔ @has_lt.lt α _ b a := iff.rfl
instance (α : Type*) [preorder α] : preorder (order_dual α) :=
{ le_refl := le_refl,
le_trans := assume a b c hab hbc, le_trans hbc hab,
lt_iff_le_not_le := λ _ _, lt_iff_le_not_le,
.. order_dual.has_le α,
.. order_dual.has_lt α }
instance (α : Type*) [partial_order α] : partial_order (order_dual α) :=
{ le_antisymm := assume a b hab hba, @le_antisymm α _ a b hba hab, .. order_dual.preorder α }
instance (α : Type*) [linear_order α] : linear_order (order_dual α) :=
{ le_total := assume a b:α, le_total b a, .. order_dual.partial_order α }
instance (α : Type*) [decidable_linear_order α] : decidable_linear_order (order_dual α) :=
{ decidable_le := show decidable_rel (λa b:α, b ≤ a), by apply_instance,
decidable_lt := show decidable_rel (λa b:α, b < a), by apply_instance,
.. order_dual.linear_order α }
instance : Π [inhabited α], inhabited (order_dual α) := id
end order_dual
/- order instances on the function space -/
instance pi.preorder {ι : Type u} {α : ι → Type v} [∀i, preorder (α i)] : preorder (Πi, α i) :=
{ le := λx y, ∀i, x i ≤ y i,
le_refl := assume a i, le_refl (a i),
le_trans := assume a b c h₁ h₂ i, le_trans (h₁ i) (h₂ i) }
instance pi.partial_order {ι : Type u} {α : ι → Type v} [∀i, partial_order (α i)] :
partial_order (Πi, α i) :=
{ le_antisymm := λf g h1 h2, funext (λb, le_antisymm (h1 b) (h2 b)),
..pi.preorder }
theorem comp_le_comp_left_of_monotone [preorder α] [preorder β]
{f : β → α} {g h : γ → β} (m_f : monotone f) (le_gh : g ≤ h) :
has_le.le.{max w u} (f ∘ g) (f ∘ h) :=
assume x, m_f (le_gh x)
section monotone
variables [preorder α] [preorder γ]
protected theorem monotone.order_dual {f : α → γ} (hf : monotone f) :
@monotone (order_dual α) (order_dual γ) _ _ f :=
λ x y hxy, hf hxy
theorem monotone_lam {f : α → β → γ} (m : ∀b, monotone (λa, f a b)) : monotone f :=
assume a a' h b, m b h
theorem monotone_app (f : β → α → γ) (b : β) (m : monotone (λa b, f b a)) : monotone (f b) :=
assume a a' h, m h b
end monotone
theorem strict_mono.order_dual [has_lt α] [has_lt β] {f : α → β} (hf : strict_mono f) :
@strict_mono (order_dual α) (order_dual β) _ _ f :=
λ x y hxy, hf hxy
/-- Transfer a `preorder` on `β` to a `preorder` on `α` using a function `f : α → β`. -/
def preorder.lift {α β} [preorder β] (f : α → β) : preorder α :=
{ le := λx y, f x ≤ f y,
le_refl := λ a, le_refl _,
le_trans := λ a b c, le_trans,
lt := λx y, f x < f y,
lt_iff_le_not_le := λ a b, lt_iff_le_not_le }
/-- Transfer a `partial_order` on `β` to a `partial_order` on `α` using an injective
function `f : α → β`. -/
def partial_order.lift {α β} [partial_order β] (f : α → β) (inj : injective f) :
partial_order α :=
{ le_antisymm := λ a b h₁ h₂, inj (le_antisymm h₁ h₂), .. preorder.lift f }
/-- Transfer a `linear_order` on `β` to a `linear_order` on `α` using an injective
function `f : α → β`. -/
def linear_order.lift {α β} [linear_order β] (f : α → β) (inj : injective f) :
linear_order α :=
{ le_total := λx y, le_total (f x) (f y), .. partial_order.lift f inj }
/-- Transfer a `decidable_linear_order` on `β` to a `decidable_linear_order` on `α` using
an injective function `f : α → β`. -/
def decidable_linear_order.lift {α β} [decidable_linear_order β] (f : α → β) (inj : injective f) :
decidable_linear_order α :=
{ decidable_le := λ x y, show decidable (f x ≤ f y), by apply_instance,
decidable_lt := λ x y, show decidable (f x < f y), by apply_instance,
decidable_eq := λ x y, decidable_of_iff _ ⟨@inj x y, congr_arg f⟩,
.. linear_order.lift f inj }
instance subtype.preorder {α} [preorder α] (p : α → Prop) : preorder (subtype p) :=
preorder.lift subtype.val
instance subtype.partial_order {α} [partial_order α] (p : α → Prop) :
partial_order (subtype p) :=
partial_order.lift subtype.val subtype.val_injective
instance subtype.linear_order {α} [linear_order α] (p : α → Prop) : linear_order (subtype p) :=
linear_order.lift subtype.val subtype.val_injective
instance subtype.decidable_linear_order {α} [decidable_linear_order α] (p : α → Prop) :
decidable_linear_order (subtype p) :=
decidable_linear_order.lift subtype.val subtype.val_injective
instance prod.has_le (α : Type u) (β : Type v) [has_le α] [has_le β] : has_le (α × β) :=
⟨λp q, p.1 ≤ q.1 ∧ p.2 ≤ q.2⟩
instance prod.preorder (α : Type u) (β : Type v) [preorder α] [preorder β] : preorder (α × β) :=
{ le_refl := assume ⟨a, b⟩, ⟨le_refl a, le_refl b⟩,
le_trans := assume ⟨a, b⟩ ⟨c, d⟩ ⟨e, f⟩ ⟨hac, hbd⟩ ⟨hce, hdf⟩,
⟨le_trans hac hce, le_trans hbd hdf⟩,
.. prod.has_le α β }
/-- The pointwise partial order on a product.
(The lexicographic ordering is defined in order/lexicographic.lean, and the instances are
available via the type synonym `lex α β = α × β`.) -/
instance prod.partial_order (α : Type u) (β : Type v) [partial_order α] [partial_order β] :
partial_order (α × β) :=
{ le_antisymm := assume ⟨a, b⟩ ⟨c, d⟩ ⟨hac, hbd⟩ ⟨hca, hdb⟩,
prod.ext (le_antisymm hac hca) (le_antisymm hbd hdb),
.. prod.preorder α β }
/-!
### Additional order classes
-/
/-- order without a top element; somtimes called cofinal -/
class no_top_order (α : Type u) [preorder α] : Prop :=
(no_top : ∀a:α, ∃a', a < a')
lemma no_top [preorder α] [no_top_order α] : ∀a:α, ∃a', a < a' :=
no_top_order.no_top
/-- order without a bottom element; somtimes called coinitial or dense -/
class no_bot_order (α : Type u) [preorder α] : Prop :=
(no_bot : ∀a:α, ∃a', a' < a)
lemma no_bot [preorder α] [no_bot_order α] : ∀a:α, ∃a', a' < a :=
no_bot_order.no_bot
instance order_dual.no_top_order (α : Type u) [preorder α] [no_bot_order α] :
no_top_order (order_dual α) :=
⟨λ a, @no_bot α _ _ a⟩
instance order_dual.no_bot_order (α : Type u) [preorder α] [no_top_order α] :
no_bot_order (order_dual α) :=
⟨λ a, @no_top α _ _ a⟩
/-- An order is dense if there is an element between any pair of distinct elements. -/
class densely_ordered (α : Type u) [preorder α] : Prop :=
(dense : ∀a₁ a₂:α, a₁ < a₂ → ∃a, a₁ < a ∧ a < a₂)
lemma dense [preorder α] [densely_ordered α] : ∀{a₁ a₂:α}, a₁ < a₂ → ∃a, a₁ < a ∧ a < a₂ :=
densely_ordered.dense
instance order_dual.densely_ordered (α : Type u) [preorder α] [densely_ordered α] :
densely_ordered (order_dual α) :=
⟨λ a₁ a₂ ha, (@dense α _ _ _ _ ha).imp $ λ a, and.symm⟩
lemma le_of_forall_le_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
(h : ∀a₃>a₂, a₁ ≤ a₃) :
a₁ ≤ a₂ :=
le_of_not_gt $ assume ha,
let ⟨a, ha₁, ha₂⟩ := dense ha in
lt_irrefl a $ lt_of_lt_of_le ‹a < a₁› (h _ ‹a₂ < a›)
lemma eq_of_le_of_forall_le_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
(h₁ : a₂ ≤ a₁) (h₂ : ∀a₃>a₂, a₁ ≤ a₃) : a₁ = a₂ :=
le_antisymm (le_of_forall_le_of_dense h₂) h₁
lemma le_of_forall_ge_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
(h : ∀a₃<a₁, a₂ ≥ a₃) :
a₁ ≤ a₂ :=
le_of_not_gt $ assume ha,
let ⟨a, ha₁, ha₂⟩ := dense ha in
lt_irrefl a $ lt_of_le_of_lt (h _ ‹a < a₁›) ‹a₂ < a›
lemma eq_of_le_of_forall_ge_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
(h₁ : a₂ ≤ a₁) (h₂ : ∀a₃<a₁, a₂ ≥ a₃) : a₁ = a₂ :=
le_antisymm (le_of_forall_ge_of_dense h₂) h₁
@[nolint ge_or_gt] -- see Note [nolint_ge]
lemma dense_or_discrete [linear_order α] (a₁ a₂ : α) :
(∃a, a₁ < a ∧ a < a₂) ∨ ((∀a>a₁, a ≥ a₂) ∧ (∀a<a₂, a ≤ a₁)) :=
classical.or_iff_not_imp_left.2 $ assume h,
⟨assume a ha₁, le_of_not_gt $ assume ha₂, h ⟨a, ha₁, ha₂⟩,
assume a ha₂, le_of_not_gt $ assume ha₁, h ⟨a, ha₁, ha₂⟩⟩
variables {s : β → β → Prop} {t : γ → γ → Prop}
/-- Any `linear_order` is a noncomputable `decidable_linear_order`. This is not marked
as an instance to avoid a loop. -/
noncomputable def classical.DLO (α) [LO : linear_order α] : decidable_linear_order α :=
{ decidable_le := classical.dec_rel _, ..LO }
variable (r)
local infix ` ≼ ` : 50 := r
/-- A family of elements of α is directed (with respect to a relation `≼` on α)
if there is a member of the family `≼`-above any pair in the family. -/
def directed {ι : Sort v} (f : ι → α) := ∀x y, ∃z, f x ≼ f z ∧ f y ≼ f z
/-- A subset of α is directed if there is an element of the set `≼`-above any
pair of elements in the set. -/
def directed_on (s : set α) := ∀ (x ∈ s) (y ∈ s), ∃z ∈ s, x ≼ z ∧ y ≼ z
theorem directed_on_iff_directed {s} : @directed_on α r s ↔ directed r (coe : s → α) :=
by simp [directed, directed_on]; refine ball_congr (λ x hx, by simp; refl)
theorem directed_on_image {s} {f : β → α} :
directed_on r (f '' s) ↔ directed_on (f ⁻¹'o r) s :=
by simp only [directed_on, set.ball_image_iff, set.bex_image_iff, order.preimage]
theorem directed_on.mono {s : set α} (h : directed_on r s)
{r' : α → α → Prop} (H : ∀ {a b}, r a b → r' a b) :
directed_on r' s :=
λ x hx y hy, let ⟨z, zs, xz, yz⟩ := h x hx y hy in ⟨z, zs, H xz, H yz⟩
theorem directed_comp {ι} (f : ι → β) (g : β → α) :
directed r (g ∘ f) ↔ directed (g ⁻¹'o r) f := iff.rfl
variable {r}
theorem directed.mono {s : α → α → Prop} {ι} {f : ι → α}
(H : ∀ a b, r a b → s a b) (h : directed r f) : directed s f :=
λ a b, let ⟨c, h₁, h₂⟩ := h a b in ⟨c, H _ _ h₁, H _ _ h₂⟩
theorem directed.mono_comp {ι} {rb : β → β → Prop} {g : α → β} {f : ι → α}
(hg : ∀ ⦃x y⦄, x ≼ y → rb (g x) (g y)) (hf : directed r f) :
directed rb (g ∘ f) :=
(directed_comp rb f g).2 $ hf.mono hg
section prio
set_option default_priority 100 -- see Note [default priority]
/-- A `preorder` is a `directed_order` if for any two elements `i`, `j`
there is an element `k` such that `i ≤ k` and `j ≤ k`. -/
class directed_order (α : Type u) extends preorder α :=
(directed : ∀ i j : α, ∃ k, i ≤ k ∧ j ≤ k)
end prio
|
b75c9c0a21c7b12dc222b97b236731364cd267b6 | 1546f9083f4babf70df0329497d1ee05adc8c665 | /src/monoidal_categories_reboot/examples/types.lean | 5f3374e66ce8f0563be081b2db3d085631e90a52 | [
"Apache-2.0"
] | permissive | khoek/monoidal-categories-reboot | 0899b0d4552ff039388042059c91f7207c6c34e5 | ed3df8ecce5d4e3d95cb858911bad12bb632cf8a | refs/heads/master | 1,588,877,903,131 | 1,554,987,273,000 | 1,554,987,273,000 | 180,791,863 | 0 | 0 | null | 1,554,987,295,000 | 1,554,987,295,000 | null | UTF-8 | Lean | false | false | 1,704 | lean | -- Copyright (c) 2018 Michael Jendrusch. All rights reserved.
import category_theory.category
import category_theory.functor
import category_theory.products
import category_theory.natural_isomorphism
import ..monoidal_category
import ..braided_monoidal_category
open category_theory
open tactic
universes u v
namespace category_theory.monoidal
section
open monoidal_category
open braided_monoidal_category
def types_left_unitor (α : Type u) : punit × α → α := λ X, X.2
def types_left_unitor_inv (α : Type u) : α → punit × α := λ X, ⟨punit.star, X⟩
def types_right_unitor (α : Type u) : α × punit → α := λ X, X.1
def types_right_unitor_inv (α : Type u) : α → α × punit := λ X, ⟨X, punit.star⟩
def types_associator (α β γ : Type u) : (α × β) × γ → α × (β × γ) :=
λ X, ⟨X.1.1, ⟨X.1.2, X.2⟩⟩
def types_associator_inv (α β γ : Type u) : α × (β × γ) → (α × β) × γ :=
λ X, ⟨⟨X.1, X.2.1⟩, X.2.2⟩
def types_braiding (α β : Type u) : α × β → β × α :=
λ X, ⟨X.2, X.1⟩
def types_braiding_inv := types_braiding
instance types : symmetric_monoidal_category.{(u+1) u} (Type u) :=
{ tensor_obj := λ X Y, X × Y,
tensor_hom := λ _ _ _ _ f g, prod.map f g,
tensor_unit := punit,
left_unitor := λ X,
{ hom := types_left_unitor X,
inv := types_left_unitor_inv X },
right_unitor := λ X,
{ hom := types_right_unitor X,
inv := types_right_unitor_inv X },
associator := λ X Y Z,
{ hom := types_associator X Y Z,
inv := types_associator_inv X Y Z},
braiding := λ X Y,
{ hom := types_braiding X Y,
inv := types_braiding_inv Y X } }
end
end category_theory.monoidal
|
a53107c3e96a59b0deaf82de98c1aef1f05d4c79 | 6b2a480f27775cba4f3ae191b1c1387a29de586e | /group_rep1/morphism/projector.lean | d18fb171c76e04276f5fe7558a974c5ff71bde4f | [] | no_license | Or7ando/group_representation | a681de2e19d1930a1e1be573d6735a2f0b8356cb | 9b576984f17764ebf26c8caa2a542d248f1b50d2 | refs/heads/master | 1,662,413,107,324 | 1,590,302,389,000 | 1,590,302,389,000 | 258,130,829 | 0 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 2,918 | lean | import morphism.Reynold_operator
import sub_module
import homothetic
import Projection.test_linear_algebra
open Reynold
universes u v w w'
variables {G : Type u} [group G][fintype G] {R : Type v}[comm_ring R]
variables {M : Type w}[add_comm_group M] [module R M] ( ρ : group_representation G R M)
(W : submodule R M)(hyp : stability.stable_submodule ρ W)
lemma fact_minus_one (p : M→ₗ[R]M )(hyp : is_projector p)(x : M) : p (p x) = p x :=
begin
change (has_mul.mul p p) x = p x,
unfold is_projector at hyp,
rw hyp,
end
lemma fact0 (g : G) (x : M) : ρ g ((ρ g⁻¹) x) = x := begin
change ((ρ g) * (ρ g⁻¹ )) x = x,
erw ← ρ.map_mul,
erw mul_inv_self,
erw ρ.map_one, exact rfl,
end
lemma fact1 (p : M→ₗ[R]M )(hyp : is_projector p) (g : G) : is_projector (mixte_conj ρ ρ p g) :=
begin
unfold is_projector,
unfold mixte_conj, erw mul_assoc,
erw ← mul_assoc ↑(ρ g), erw ← mul_assoc ↑(ρ g),
ext,simp,
erw fact0,
erw fact_minus_one,
assumption,
end
lemma range_mixte_conj (p : M→ₗ[R]M )(g : G) : linear_map.range (mixte_conj ρ ρ p g) =
submodule.map (ρ g⁻¹ : M →ₗ[R]M ) (linear_map.range p) := begin
apply le_antisymm,
unfold mixte_conj, rw submodule.le_def',
intros x, intros hyp, rw submodule.mem_map,rw linear_map.mem_range at hyp,
rcases hyp with ⟨y, hyp_y ⟩,
use p (ρ g y),split,
exact image_in_range p _,
rw ← hyp_y,
exact rfl,
rw submodule.le_def' at *, intros x, intros hyp_x, rw linear_map.mem_range,
rcases hyp_x with ⟨z, hyp_z⟩ , rcases hyp_z,
erw linear_map.mem_range at hyp_z_left,
rcases hyp_z_left with ⟨y,hyp_u⟩ , use (ρ g⁻¹ ) y,
dunfold mixte_conj,
rw linear_map.comp_apply, erw fact0, rw ← hyp_z_right, rw ← hyp_u, exact rfl,
end
lemma range_grall(hyp : stability.stable_submodule ρ W) ( g : G) : submodule.map (ρ g : M →ₗ[R] M) W = W := begin
apply le_antisymm,
{rw submodule.le_def' ,intros x, intro hyp', rcases hyp', rw ← hyp'_h.2,
unfold stability.stable_submodule at hyp,
exact ((hyp g) ⟨ hyp'_w,hyp'_h.1⟩ ),},
{rw submodule.le_def',intros x, intro certifa, rw submodule.mem_map,
use (ρ g⁻¹ ) x, split,{exact (hyp g⁻¹ )⟨x,certifa⟩},{erw fact0},}
end
theorem pre_mask (hyp : stability.stable_submodule ρ W) (Hyp : has_projector W) (a : R ) (inv : a * (fintype.card G) = 1 ) :
∃ F : ρ ⟶ ρ,is_projector F.ℓ ∧ linear_map.range (F.ℓ) = W :=
begin
rcases Hyp with ⟨p,hyp_p⟩,
use a • (ℛ ρ ρ p),
rw homothetic.smul_ext, --- amélioration de la class homothetic
erw reynold_ext,
apply @sum_proj R _ M _ _ G _ W a _,
intros g,
apply fact1, exact hyp_p.1,
intros g,
rw range_mixte_conj,rw hyp_p.2,rw range_grall, assumption,
assumption,
end |
51effa689bf3fe2e5c49f78ca61b538c9e08d0cd | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/Lean3Lib/init/data/unsigned/ops_auto.lean | 49179fcd4a7fcbe0b8a95c5a848eb0fac87caa65 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,017 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.data.unsigned.basic
import Mathlib.Lean3Lib.init.data.fin.ops
namespace Mathlib
namespace unsigned
def of_nat (n : ℕ) : unsigned := fin.of_nat n
protected instance has_zero : HasZero unsigned := { zero := fin.of_nat 0 }
protected instance has_one : HasOne unsigned := { one := fin.of_nat 1 }
protected instance has_add : Add unsigned := { add := fin.add }
protected instance has_sub : Sub unsigned := { sub := fin.sub }
protected instance has_mul : Mul unsigned := { mul := fin.mul }
protected instance has_mod : Mod unsigned := { mod := fin.mod }
protected instance has_div : Div unsigned := { div := fin.div }
protected instance has_lt : HasLess unsigned := { Less := fin.lt }
end unsigned
protected instance unsigned.has_le : HasLessEq unsigned := { LessEq := fin.le }
end Mathlib |
6985d0c2336687cd2a59a3e83115cbef396c522a | f09e92753b1d3d2eb3ce2cfb5288a7f5d1d4bd89 | /src/for_mathlib/topological_rings.lean | 27c9f10931d90a6933e3ae44acc981a2f40a2ff8 | [
"Apache-2.0"
] | permissive | PatrickMassot/lean-perfectoid-spaces | 7f63c581db26461b5a92d968e7563247e96a5597 | 5f70b2020b3c6d508431192b18457fa988afa50d | refs/heads/master | 1,625,797,721,782 | 1,547,308,357,000 | 1,547,309,364,000 | 136,658,414 | 0 | 1 | Apache-2.0 | 1,528,486,100,000 | 1,528,486,100,000 | null | UTF-8 | Lean | false | false | 884 | lean | import analysis.topology.topological_structures
import ring_theory.subring
import ring_theory.ideal_operations
universe u
variables {A : Type u} [comm_ring A] [topological_space A] [topological_ring A]
instance subring_has_zero (R : Type u) [comm_ring R] (S : set R) [HS : is_subring S] : has_zero S :=
⟨⟨0, is_add_submonoid.zero_mem S⟩⟩
instance topological_subring (A₀ : set A) [is_subring A₀] : topological_ring A₀ :=
{ continuous_neg := sorry,
continuous_add := sorry,
continuous_mul := sorry
}
def is_ideal_adic (J : ideal A) : Prop :=
(∀ n : ℕ, is_open (J^n : ideal A).carrier) ∧ (∀ S : set A, (0 : A) ∈ S → is_open S → ∃ n : ℕ, (J^n : ideal A).carrier ⊆ S)
notation `is-`J`-adic` := is_ideal_adic J
def is_adic (A₀ : set A) [is_subring A₀] : Prop := ∃ (J : ideal A₀),
(by haveI := topological_subring A₀; exact is-J-adic)
|
3d88188093a2b51bffb15f46642608e51f57bf5c | 690889011852559ee5ac4dfea77092de8c832e7e | /src/data/seq/computation.lean | d0660601032740b19ae5d1459e9c3c7a69dc6957 | [
"Apache-2.0"
] | permissive | williamdemeo/mathlib | f6df180148f8acc91de9ba5e558976ab40a872c7 | 1fa03c29f9f273203bbffb79d10d31f696b3d317 | refs/heads/master | 1,584,785,260,929 | 1,572,195,914,000 | 1,572,195,913,000 | 138,435,193 | 0 | 0 | Apache-2.0 | 1,529,789,739,000 | 1,529,789,739,000 | null | UTF-8 | Lean | false | false | 37,490 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import data.stream logic.relator tactic.basic
universes u v w
/-
coinductive computation (α : Type u) : Type u
| return : α → computation α
| think : computation α → computation α
-/
/-- `computation α` is the type of unbounded computations returning `α`.
An element of `computation α` is an infinite sequence of `option α` such
that if `f n = some a` for some `n` then it is constantly `some a` after that. -/
def computation (α : Type u) : Type u :=
{ f : stream (option α) // ∀ {n a}, f n = some a → f (n+1) = some a }
namespace computation
variables {α : Type u} {β : Type v} {γ : Type w}
-- constructors
/-- `return a` is the computation that immediately terminates with result `a`. -/
def return (a : α) : computation α := ⟨stream.const (some a), λn a', id⟩
instance : has_coe α (computation α) := ⟨return⟩
/-- `think c` is the computation that delays for one "tick" and then performs
computation `c`. -/
def think (c : computation α) : computation α :=
⟨none :: c.1, λn a h, by {cases n with n, contradiction, exact c.2 h}⟩
/-- `thinkN c n` is the computation that delays for `n` ticks and then performs
computation `c`. -/
def thinkN (c : computation α) : ℕ → computation α
| 0 := c
| (n+1) := think (thinkN n)
-- check for immediate result
/-- `head c` is the first step of computation, either `some a` if `c = return a`
or `none` if `c = think c'`. -/
def head (c : computation α) : option α := c.1.head
-- one step of computation
/-- `tail c` is the remainder of computation, either `c` if `c = return a`
or `c'` if `c = think c'`. -/
def tail (c : computation α) : computation α :=
⟨c.1.tail, λ n a, let t := c.2 in t⟩
/-- `empty α` is the computation that never returns, an infinite sequence of
`think`s. -/
def empty (α) : computation α := ⟨stream.const none, λn a', id⟩
/-- `run_for c n` evaluates `c` for `n` steps and returns the result, or `none`
if it did not terminate after `n` steps. -/
def run_for : computation α → ℕ → option α := subtype.val
/-- `destruct c` is the destructor for `computation α` as a coinductive type.
It returns `inl a` if `c = return a` and `inr c'` if `c = think c'`. -/
def destruct (c : computation α) : α ⊕ computation α :=
match c.1 0 with
| none := sum.inr (tail c)
| some a := sum.inl a
end
/-- `run c` is an unsound meta function that runs `c` to completion, possibly
resulting in an infinite loop in the VM. -/
meta def run : computation α → α | c :=
match destruct c with
| sum.inl a := a
| sum.inr ca := run ca
end
theorem destruct_eq_ret {s : computation α} {a : α} :
destruct s = sum.inl a → s = return a :=
begin
dsimp [destruct],
induction f0 : s.1 0; intro h,
{ contradiction },
{ apply subtype.eq, funext n,
induction n with n IH,
{ injection h with h', rwa h' at f0 },
{ exact s.2 IH } }
end
theorem destruct_eq_think {s : computation α} {s'} :
destruct s = sum.inr s' → s = think s' :=
begin
dsimp [destruct],
induction f0 : s.1 0 with a'; intro h,
{ injection h with h', rw ←h',
cases s with f al,
apply subtype.eq, dsimp [think, tail],
rw ←f0, exact (stream.eta f).symm },
{ contradiction }
end
@[simp] theorem destruct_ret (a : α) : destruct (return a) = sum.inl a := rfl
@[simp] theorem destruct_think : ∀ s : computation α, destruct (think s) = sum.inr s
| ⟨f, al⟩ := rfl
@[simp] theorem destruct_empty : destruct (empty α) = sum.inr (empty α) := rfl
@[simp] theorem head_ret (a : α) : head (return a) = some a := rfl
@[simp] theorem head_think (s : computation α) : head (think s) = none := rfl
@[simp] theorem head_empty : head (empty α) = none := rfl
@[simp] theorem tail_ret (a : α) : tail (return a) = return a := rfl
@[simp] theorem tail_think (s : computation α) : tail (think s) = s :=
by cases s with f al; apply subtype.eq; dsimp [tail, think]; rw [stream.tail_cons]
@[simp] theorem tail_empty : tail (empty α) = empty α := rfl
theorem think_empty : empty α = think (empty α) :=
destruct_eq_think destruct_empty
def cases_on {C : computation α → Sort v} (s : computation α)
(h1 : ∀ a, C (return a)) (h2 : ∀ s, C (think s)) : C s := begin
induction H : destruct s with v v,
{ rw destruct_eq_ret H, apply h1 },
{ cases v with a s', rw destruct_eq_think H, apply h2 }
end
def corec.F (f : β → α ⊕ β) : α ⊕ β → option α × (α ⊕ β)
| (sum.inl a) := (some a, sum.inl a)
| (sum.inr b) := (match f b with
| sum.inl a := some a
| sum.inr b' := none
end, f b)
/-- `corec f b` is the corecursor for `computation α` as a coinductive type.
If `f b = inl a` then `corec f b = return a`, and if `f b = inl b'` then
`corec f b = think (corec f b')`. -/
def corec (f : β → α ⊕ β) (b : β) : computation α :=
begin
refine ⟨stream.corec' (corec.F f) (sum.inr b), λn a' h, _⟩,
rw stream.corec'_eq,
change stream.corec' (corec.F f) (corec.F f (sum.inr b)).2 n = some a',
revert h, generalize : sum.inr b = o, revert o,
induction n with n IH; intro o,
{ change (corec.F f o).1 = some a' → (corec.F f (corec.F f o).2).1 = some a',
cases o with a b; intro h, { exact h },
dsimp [corec.F] at h, dsimp [corec.F],
cases f b with a b', { exact h },
{ contradiction } },
{ rw [stream.corec'_eq (corec.F f) (corec.F f o).2,
stream.corec'_eq (corec.F f) o],
exact IH (corec.F f o).2 }
end
/-- left map of `⊕` -/
def lmap (f : α → β) : α ⊕ γ → β ⊕ γ
| (sum.inl a) := sum.inl (f a)
| (sum.inr b) := sum.inr b
/-- right map of `⊕` -/
def rmap (f : β → γ) : α ⊕ β → α ⊕ γ
| (sum.inl a) := sum.inl a
| (sum.inr b) := sum.inr (f b)
attribute [simp] lmap rmap
@[simp] lemma corec_eq (f : β → α ⊕ β) (b : β) :
destruct (corec f b) = rmap (corec f) (f b) :=
begin
dsimp [corec, destruct],
change stream.corec' (corec.F f) (sum.inr b) 0 with corec.F._match_1 (f b),
induction h : f b with a b', { refl },
dsimp [corec.F, destruct],
apply congr_arg, apply subtype.eq,
dsimp [corec, tail],
rw [stream.corec'_eq, stream.tail_cons],
dsimp [corec.F], rw h
end
section bisim
variable (R : computation α → computation α → Prop)
local infix ~ := R
def bisim_o : α ⊕ computation α → α ⊕ computation α → Prop
| (sum.inl a) (sum.inl a') := a = a'
| (sum.inr s) (sum.inr s') := R s s'
| _ _ := false
attribute [simp] bisim_o
def is_bisimulation := ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ → bisim_o R (destruct s₁) (destruct s₂)
-- If two computations are bisimilar, then they are equal
theorem eq_of_bisim (bisim : is_bisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂ :=
begin
apply subtype.eq,
apply stream.eq_of_bisim (λx y, ∃ s s' : computation α, s.1 = x ∧ s'.1 = y ∧ R s s'),
dsimp [stream.is_bisimulation],
intros t₁ t₂ e,
exact match t₁, t₂, e with ._, ._, ⟨s, s', rfl, rfl, r⟩ :=
suffices head s = head s' ∧ R (tail s) (tail s'), from
and.imp id (λr, ⟨tail s, tail s',
by cases s; refl, by cases s'; refl, r⟩) this,
begin
have := bisim r, revert r this,
apply cases_on s _ _; intros; apply cases_on s' _ _; intros; intros r this,
{ constructor, dsimp at this, rw this, assumption },
{ rw [destruct_ret, destruct_think] at this,
exact false.elim this },
{ rw [destruct_ret, destruct_think] at this,
exact false.elim this },
{ simp at this, simp [*] }
end
end,
exact ⟨s₁, s₂, rfl, rfl, r⟩
end
end bisim
-- It's more of a stretch to use ∈ for this relation, but it
-- asserts that the computation limits to the given value.
protected def mem (a : α) (s : computation α) := some a ∈ s.1
instance : has_mem α (computation α) := ⟨computation.mem⟩
theorem le_stable (s : computation α) {a m n} (h : m ≤ n) :
s.1 m = some a → s.1 n = some a :=
by {cases s with f al, induction h with n h IH, exacts [id, λ h2, al (IH h2)]}
theorem mem_unique :
relator.left_unique ((∈) : α → computation α → Prop) :=
λa s b ⟨m, ha⟩ ⟨n, hb⟩, by injection
(le_stable s (le_max_left m n) ha.symm).symm.trans
(le_stable s (le_max_right m n) hb.symm)
/-- `terminates s` asserts that the computation `s` eventually terminates with some value. -/
@[class] def terminates (s : computation α) : Prop := ∃ a, a ∈ s
theorem terminates_of_mem {s : computation α} {a : α} : a ∈ s → terminates s :=
exists.intro a
theorem terminates_def (s : computation α) : terminates s ↔ ∃ n, (s.1 n).is_some :=
⟨λ⟨a, n, h⟩, ⟨n, by {dsimp [stream.nth] at h, rw ←h, exact rfl}⟩,
λ⟨n, h⟩, ⟨option.get h, n, (option.eq_some_of_is_some h).symm⟩⟩
theorem ret_mem (a : α) : a ∈ return a :=
exists.intro 0 rfl
theorem eq_of_ret_mem {a a' : α} (h : a' ∈ return a) : a' = a :=
mem_unique h (ret_mem _)
instance ret_terminates (a : α) : terminates (return a) :=
terminates_of_mem (ret_mem _)
theorem think_mem {s : computation α} {a} : a ∈ s → a ∈ think s
| ⟨n, h⟩ := ⟨n+1, h⟩
instance think_terminates (s : computation α) :
∀ [terminates s], terminates (think s)
| ⟨a, n, h⟩ := ⟨a, n+1, h⟩
theorem of_think_mem {s : computation α} {a} : a ∈ think s → a ∈ s
| ⟨n, h⟩ := by {cases n with n', contradiction, exact ⟨n', h⟩}
theorem of_think_terminates {s : computation α} :
terminates (think s) → terminates s
| ⟨a, h⟩ := ⟨a, of_think_mem h⟩
theorem not_mem_empty (a : α) : a ∉ empty α :=
λ ⟨n, h⟩, by clear _fun_match; contradiction
theorem not_terminates_empty : ¬ terminates (empty α) :=
λ ⟨a, h⟩, not_mem_empty a h
theorem eq_empty_of_not_terminates {s} (H : ¬ terminates s) : s = empty α :=
begin
apply subtype.eq, funext n,
induction h : s.val n, {refl},
refine absurd _ H, exact ⟨_, _, h.symm⟩
end
theorem thinkN_mem {s : computation α} {a} : ∀ n, a ∈ thinkN s n ↔ a ∈ s
| 0 := iff.rfl
| (n+1) := iff.trans ⟨of_think_mem, think_mem⟩ (thinkN_mem n)
instance thinkN_terminates (s : computation α) :
∀ [terminates s] n, terminates (thinkN s n)
| ⟨a, h⟩ n := ⟨a, (thinkN_mem n).2 h⟩
theorem of_thinkN_terminates (s : computation α) (n) :
terminates (thinkN s n) → terminates s
| ⟨a, h⟩ := ⟨a, (thinkN_mem _).1 h⟩
/-- `promises s a`, or `s ~> a`, asserts that although the computation `s`
may not terminate, if it does, then the result is `a`. -/
def promises (s : computation α) (a : α) : Prop := ∀ ⦃a'⦄, a' ∈ s → a = a'
infix ` ~> `:50 := promises
theorem mem_promises {s : computation α} {a : α} : a ∈ s → s ~> a :=
λ h a', mem_unique h
theorem empty_promises (a : α) : empty α ~> a :=
λ a' h, absurd h (not_mem_empty _)
section get
variables (s : computation α) [h : terminates s]
include s h
/-- `length s` gets the number of steps of a terminating computation -/
def length : ℕ := nat.find ((terminates_def _).1 h)
/-- `get s` returns the result of a terminating computation -/
def get : α := option.get (nat.find_spec $ (terminates_def _).1 h)
theorem get_mem : get s ∈ s :=
exists.intro (length s) (option.eq_some_of_is_some _).symm
theorem get_eq_of_mem {a} : a ∈ s → get s = a :=
mem_unique (get_mem _)
theorem mem_of_get_eq {a} : get s = a → a ∈ s :=
by intro h; rw ←h; apply get_mem
@[simp] theorem get_think : get (think s) = get s :=
get_eq_of_mem _ $ let ⟨n, h⟩ := get_mem s in ⟨n+1, h⟩
@[simp] theorem get_thinkN (n) : get (thinkN s n) = get s :=
get_eq_of_mem _ $ (thinkN_mem _).2 (get_mem _)
theorem get_promises : s ~> get s := λ a, get_eq_of_mem _
theorem mem_of_promises {a} (p : s ~> a) : a ∈ s :=
by unfreezeI; cases h with a' h; rw p h; exact h
theorem get_eq_of_promises {a} : s ~> a → get s = a :=
get_eq_of_mem _ ∘ mem_of_promises _
end get
/-- `results s a n` completely characterizes a terminating computation:
it asserts that `s` terminates after exactly `n` steps, with result `a`. -/
def results (s : computation α) (a : α) (n : ℕ) :=
∃ (h : a ∈ s), @length _ s (terminates_of_mem h) = n
theorem results_of_terminates (s : computation α) [T : terminates s] :
results s (get s) (length s) :=
⟨get_mem _, rfl⟩
theorem results_of_terminates' (s : computation α) [T : terminates s] {a} (h : a ∈ s) :
results s a (length s) :=
by rw ←get_eq_of_mem _ h; apply results_of_terminates
theorem results.mem {s : computation α} {a n} : results s a n → a ∈ s
| ⟨m, _⟩ := m
theorem results.terminates {s : computation α} {a n} (h : results s a n) : terminates s :=
terminates_of_mem h.mem
theorem results.length {s : computation α} {a n} [T : terminates s] :
results s a n → length s = n
| ⟨_, h⟩ := h
theorem results.val_unique {s : computation α} {a b m n}
(h1 : results s a m) (h2 : results s b n) : a = b :=
mem_unique h1.mem h2.mem
theorem results.len_unique {s : computation α} {a b m n}
(h1 : results s a m) (h2 : results s b n) : m = n :=
by haveI := h1.terminates; haveI := h2.terminates; rw [←h1.length, h2.length]
theorem exists_results_of_mem {s : computation α} {a} (h : a ∈ s) : ∃ n, results s a n :=
by haveI := terminates_of_mem h; exact ⟨_, results_of_terminates' s h⟩
@[simp] theorem get_ret (a : α) : get (return a) = a :=
get_eq_of_mem _ ⟨0, rfl⟩
@[simp] theorem length_ret (a : α) : length (return a) = 0 :=
let h := computation.ret_terminates a in
nat.eq_zero_of_le_zero $ nat.find_min' ((terminates_def (return a)).1 h) rfl
theorem results_ret (a : α) : results (return a) a 0 :=
⟨_, length_ret _⟩
@[simp] theorem length_think (s : computation α) [h : terminates s] :
length (think s) = length s + 1 :=
begin
apply le_antisymm,
{ exact nat.find_min' _ (nat.find_spec ((terminates_def _).1 h)) },
{ have : (option.is_some ((think s).val (length (think s))) : Prop) :=
nat.find_spec ((terminates_def _).1 s.think_terminates),
cases length (think s) with n,
{ contradiction },
{ apply nat.succ_le_succ, apply nat.find_min', apply this } }
end
theorem results_think {s : computation α} {a n}
(h : results s a n) : results (think s) a (n + 1) :=
by haveI := h.terminates; exact ⟨think_mem h.mem, by rw [length_think, h.length]⟩
theorem of_results_think {s : computation α} {a n}
(h : results (think s) a n) : ∃ m, results s a m ∧ n = m + 1 :=
begin
haveI := of_think_terminates h.terminates,
have := results_of_terminates' _ (of_think_mem h.mem),
exact ⟨_, this, results.len_unique h (results_think this)⟩,
end
@[simp] theorem results_think_iff {s : computation α} {a n} :
results (think s) a (n + 1) ↔ results s a n :=
⟨λ h, let ⟨n', r, e⟩ := of_results_think h in by injection e with h'; rwa h',
results_think⟩
theorem results_thinkN {s : computation α} {a m} :
∀ n, results s a m → results (thinkN s n) a (m + n)
| 0 h := h
| (n+1) h := results_think (results_thinkN n h)
theorem results_thinkN_ret (a : α) (n) : results (thinkN (return a) n) a n :=
by have := results_thinkN n (results_ret a); rwa zero_add at this
@[simp] theorem length_thinkN (s : computation α) [h : terminates s] (n) :
length (thinkN s n) = length s + n :=
(results_thinkN n (results_of_terminates _)).length
theorem eq_thinkN {s : computation α} {a n} (h : results s a n) :
s = thinkN (return a) n :=
begin
revert s,
induction n with n IH; intro s;
apply cases_on s (λ a', _) (λ s, _); intro h,
{ rw ←eq_of_ret_mem h.mem, refl },
{ cases of_results_think h with n h, cases h, contradiction },
{ have := h.len_unique (results_ret _), contradiction },
{ rw IH (results_think_iff.1 h), refl }
end
theorem eq_thinkN' (s : computation α) [h : terminates s] :
s = thinkN (return (get s)) (length s) :=
eq_thinkN (results_of_terminates _)
def mem_rec_on {C : computation α → Sort v} {a s} (M : a ∈ s)
(h1 : C (return a)) (h2 : ∀ s, C s → C (think s)) : C s :=
begin
haveI T := terminates_of_mem M,
rw [eq_thinkN' s, get_eq_of_mem s M],
generalize : length s = n,
induction n with n IH, exacts [h1, h2 _ IH]
end
def terminates_rec_on {C : computation α → Sort v} (s) [terminates s]
(h1 : ∀ a, C (return a)) (h2 : ∀ s, C s → C (think s)) : C s :=
mem_rec_on (get_mem s) (h1 _) h2
/-- Map a function on the result of a computation. -/
def map (f : α → β) : computation α → computation β
| ⟨s, al⟩ := ⟨s.map (λo, option.cases_on o none (some ∘ f)),
λn b, begin
dsimp [stream.map, stream.nth],
induction e : s n with a; intro h,
{ contradiction }, { rw [al e, ←h] }
end⟩
def bind.G : β ⊕ computation β → β ⊕ computation α ⊕ computation β
| (sum.inl b) := sum.inl b
| (sum.inr cb') := sum.inr $ sum.inr cb'
def bind.F (f : α → computation β) :
computation α ⊕ computation β → β ⊕ computation α ⊕ computation β
| (sum.inl ca) :=
match destruct ca with
| sum.inl a := bind.G $ destruct (f a)
| sum.inr ca' := sum.inr $ sum.inl ca'
end
| (sum.inr cb) := bind.G $ destruct cb
/-- Compose two computations into a monadic `bind` operation. -/
def bind (c : computation α) (f : α → computation β) : computation β :=
corec (bind.F f) (sum.inl c)
instance : has_bind computation := ⟨@bind⟩
theorem has_bind_eq_bind {β} (c : computation α) (f : α → computation β) :
c >>= f = bind c f := rfl
/-- Flatten a computation of computations into a single computation. -/
def join (c : computation (computation α)) : computation α := c >>= id
@[simp] theorem map_ret (f : α → β) (a) : map f (return a) = return (f a) := rfl
@[simp] theorem map_think (f : α → β) : ∀ s, map f (think s) = think (map f s)
| ⟨s, al⟩ := by apply subtype.eq; dsimp [think, map]; rw stream.map_cons
@[simp] theorem destruct_map (f : α → β) (s) : destruct (map f s) = lmap f (rmap (map f) (destruct s)) :=
by apply s.cases_on; intro; simp
@[simp] theorem map_id : ∀ (s : computation α), map id s = s
| ⟨f, al⟩ := begin
apply subtype.eq; simp [map, function.comp],
have e : (@option.rec α (λ_, option α) none some) = id,
{ funext x, cases x; refl },
simp [e, stream.map_id]
end
theorem map_comp (f : α → β) (g : β → γ) :
∀ (s : computation α), map (g ∘ f) s = map g (map f s)
| ⟨s, al⟩ := begin
apply subtype.eq; dsimp [map],
rw stream.map_map,
apply congr_arg (λ f : _ → option γ, stream.map f s),
funext x, cases x with x; refl
end
@[simp] theorem ret_bind (a) (f : α → computation β) :
bind (return a) f = f a :=
begin
apply eq_of_bisim (λc₁ c₂,
c₁ = bind (return a) f ∧ c₂ = f a ∨
c₁ = corec (bind.F f) (sum.inr c₂)),
{ intros c₁ c₂ h,
exact match c₁, c₂, h with
| ._, ._, or.inl ⟨rfl, rfl⟩ := begin
simp [bind, bind.F],
cases destruct (f a) with b cb; simp [bind.G]
end
| ._, c, or.inr rfl := begin
simp [bind.F],
cases destruct c with b cb; simp [bind.G]
end end },
{ simp }
end
@[simp] theorem think_bind (c) (f : α → computation β) :
bind (think c) f = think (bind c f) :=
destruct_eq_think $ by simp [bind, bind.F]
@[simp] theorem bind_ret (f : α → β) (s) : bind s (return ∘ f) = map f s :=
begin
apply eq_of_bisim (λc₁ c₂, c₁ = c₂ ∨
∃ s, c₁ = bind s (return ∘ f) ∧ c₂ = map f s),
{ intros c₁ c₂ h,
exact match c₁, c₂, h with
| _, _, or.inl (eq.refl c) := begin cases destruct c with b cb; simp end
| _, _, or.inr ⟨s, rfl, rfl⟩ := begin
apply cases_on s; intros s; simp,
exact or.inr ⟨s, rfl, rfl⟩
end end },
{ exact or.inr ⟨s, rfl, rfl⟩ }
end
@[simp] theorem bind_ret' (s : computation α) : bind s return = s :=
by rw bind_ret; change (λ x : α, x) with @id α; rw map_id
@[simp] theorem bind_assoc (s : computation α) (f : α → computation β) (g : β → computation γ) :
bind (bind s f) g = bind s (λ (x : α), bind (f x) g) :=
begin
apply eq_of_bisim (λc₁ c₂, c₁ = c₂ ∨
∃ s, c₁ = bind (bind s f) g ∧ c₂ = bind s (λ (x : α), bind (f x) g)),
{ intros c₁ c₂ h,
exact match c₁, c₂, h with
| _, _, or.inl (eq.refl c) := by cases destruct c with b cb; simp
| ._, ._, or.inr ⟨s, rfl, rfl⟩ := begin
apply cases_on s; intros s; simp,
{ generalize : f s = fs,
apply cases_on fs; intros t; simp,
{ cases destruct (g t) with b cb; simp } },
{ exact or.inr ⟨s, rfl, rfl⟩ }
end end },
{ exact or.inr ⟨s, rfl, rfl⟩ }
end
theorem results_bind {s : computation α} {f : α → computation β} {a b m n}
(h1 : results s a m) (h2 : results (f a) b n) : results (bind s f) b (n + m) :=
begin
have := h1.mem, revert m,
apply mem_rec_on this _ (λ s IH, _); intros m h1,
{ rw [ret_bind], rw h1.len_unique (results_ret _), exact h2 },
{ rw [think_bind], cases of_results_think h1 with m' h, cases h with h1 e,
rw e, exact results_think (IH h1) }
end
theorem mem_bind {s : computation α} {f : α → computation β} {a b}
(h1 : a ∈ s) (h2 : b ∈ f a) : b ∈ bind s f :=
let ⟨m, h1⟩ := exists_results_of_mem h1,
⟨n, h2⟩ := exists_results_of_mem h2 in (results_bind h1 h2).mem
instance terminates_bind (s : computation α) (f : α → computation β)
[terminates s] [terminates (f (get s))] :
terminates (bind s f) :=
terminates_of_mem (mem_bind (get_mem s) (get_mem (f (get s))))
@[simp] theorem get_bind (s : computation α) (f : α → computation β)
[terminates s] [terminates (f (get s))] :
get (bind s f) = get (f (get s)) :=
get_eq_of_mem _ (mem_bind (get_mem s) (get_mem (f (get s))))
@[simp] theorem length_bind (s : computation α) (f : α → computation β)
[T1 : terminates s] [T2 : terminates (f (get s))] :
length (bind s f) = length (f (get s)) + length s :=
(results_of_terminates _).len_unique $
results_bind (results_of_terminates _) (results_of_terminates _)
theorem of_results_bind {s : computation α} {f : α → computation β} {b k} :
results (bind s f) b k →
∃ a m n, results s a m ∧ results (f a) b n ∧ k = n + m :=
begin
induction k with n IH generalizing s;
apply cases_on s (λ a, _) (λ s', _); intro e,
{ simp [thinkN] at e, refine ⟨a, _, _, results_ret _, e, rfl⟩ },
{ have := congr_arg head (eq_thinkN e), contradiction },
{ simp at e, refine ⟨a, _, n+1, results_ret _, e, rfl⟩ },
{ simp at e, exact let ⟨a, m, n', h1, h2, e'⟩ := IH e in
by rw e'; exact ⟨a, m.succ, n', results_think h1, h2, rfl⟩ }
end
theorem exists_of_mem_bind {s : computation α} {f : α → computation β} {b}
(h : b ∈ bind s f) : ∃ a ∈ s, b ∈ f a :=
let ⟨k, h⟩ := exists_results_of_mem h,
⟨a, m, n, h1, h2, e⟩ := of_results_bind h in ⟨a, h1.mem, h2.mem⟩
theorem bind_promises {s : computation α} {f : α → computation β} {a b}
(h1 : s ~> a) (h2 : f a ~> b) : bind s f ~> b :=
λ b' bB, begin
rcases exists_of_mem_bind bB with ⟨a', a's, ba'⟩,
rw ←h1 a's at ba', exact h2 ba'
end
instance : monad computation :=
{ map := @map,
pure := @return,
bind := @bind }
instance : is_lawful_monad computation :=
{ id_map := @map_id,
bind_pure_comp_eq_map := @bind_ret,
pure_bind := @ret_bind,
bind_assoc := @bind_assoc }
theorem has_map_eq_map {β} (f : α → β) (c : computation α) : f <$> c = map f c := rfl
@[simp] theorem return_def (a) : (_root_.return a : computation α) = return a := rfl
@[simp] theorem map_ret' {α β} : ∀ (f : α → β) (a), f <$> return a = return (f a) := map_ret
@[simp] theorem map_think' {α β} : ∀ (f : α → β) s, f <$> think s = think (f <$> s) := map_think
theorem mem_map (f : α → β) {a} {s : computation α} (m : a ∈ s) : f a ∈ map f s :=
by rw ←bind_ret; apply mem_bind m; apply ret_mem
theorem exists_of_mem_map {f : α → β} {b : β} {s : computation α} (h : b ∈ map f s) :
∃ a, a ∈ s ∧ f a = b :=
by rw ←bind_ret at h; exact
let ⟨a, as, fb⟩ := exists_of_mem_bind h in ⟨a, as, mem_unique (ret_mem _) fb⟩
instance terminates_map (f : α → β) (s : computation α) [terminates s] : terminates (map f s) :=
by rw ←bind_ret; apply_instance
theorem terminates_map_iff (f : α → β) (s : computation α) :
terminates (map f s) ↔ terminates s :=
⟨λ⟨a, h⟩, let ⟨b, h1, _⟩ := exists_of_mem_map h in ⟨_, h1⟩, @computation.terminates_map _ _ _ _⟩
-- Parallel computation
/-- `c₁ <|> c₂` calculates `c₁` and `c₂` simultaneously, returning
the first one that gives a result. -/
def orelse (c₁ c₂ : computation α) : computation α :=
@computation.corec α (computation α × computation α)
(λ⟨c₁, c₂⟩, match destruct c₁ with
| sum.inl a := sum.inl a
| sum.inr c₁' := match destruct c₂ with
| sum.inl a := sum.inl a
| sum.inr c₂' := sum.inr (c₁', c₂')
end
end) (c₁, c₂)
instance : alternative computation :=
{ orelse := @orelse, failure := @empty, ..computation.monad }
@[simp] theorem ret_orelse (a : α) (c₂ : computation α) :
(return a <|> c₂) = return a :=
destruct_eq_ret $ by unfold has_orelse.orelse; simp [orelse]
@[simp] theorem orelse_ret (c₁ : computation α) (a : α) :
(think c₁ <|> return a) = return a :=
destruct_eq_ret $ by unfold has_orelse.orelse; simp [orelse]
@[simp] theorem orelse_think (c₁ c₂ : computation α) :
(think c₁ <|> think c₂) = think (c₁ <|> c₂) :=
destruct_eq_think $ by unfold has_orelse.orelse; simp [orelse]
@[simp] theorem empty_orelse (c) : (empty α <|> c) = c :=
begin
apply eq_of_bisim (λc₁ c₂, (empty α <|> c₂) = c₁) _ rfl,
intros s' s h, rw ←h,
apply cases_on s; intros s; rw think_empty; simp,
rw ←think_empty,
end
@[simp] theorem orelse_empty (c : computation α) : (c <|> empty α) = c :=
begin
apply eq_of_bisim (λc₁ c₂, (c₂ <|> empty α) = c₁) _ rfl,
intros s' s h, rw ←h,
apply cases_on s; intros s; rw think_empty; simp,
rw←think_empty,
end
/-- `c₁ ~ c₂` asserts that `c₁` and `c₂` either both terminate with the same result,
or both loop forever. -/
def equiv (c₁ c₂ : computation α) : Prop := ∀ a, a ∈ c₁ ↔ a ∈ c₂
infix ~ := equiv
@[refl] theorem equiv.refl (s : computation α) : s ~ s := λ_, iff.rfl
@[symm] theorem equiv.symm {s t : computation α} : s ~ t → t ~ s :=
λh a, (h a).symm
@[trans] theorem equiv.trans {s t u : computation α} : s ~ t → t ~ u → s ~ u :=
λh1 h2 a, (h1 a).trans (h2 a)
theorem equiv.equivalence : equivalence (@equiv α) :=
⟨@equiv.refl _, @equiv.symm _, @equiv.trans _⟩
theorem equiv_of_mem {s t : computation α} {a} (h1 : a ∈ s) (h2 : a ∈ t) : s ~ t :=
λa', ⟨λma, by rw mem_unique ma h1; exact h2,
λma, by rw mem_unique ma h2; exact h1⟩
theorem terminates_congr {c₁ c₂ : computation α}
(h : c₁ ~ c₂) : terminates c₁ ↔ terminates c₂ :=
exists_congr h
theorem promises_congr {c₁ c₂ : computation α}
(h : c₁ ~ c₂) (a) : c₁ ~> a ↔ c₂ ~> a :=
forall_congr (λa', imp_congr (h a') iff.rfl)
theorem get_equiv {c₁ c₂ : computation α} (h : c₁ ~ c₂)
[terminates c₁] [terminates c₂] : get c₁ = get c₂ :=
get_eq_of_mem _ $ (h _).2 $ get_mem _
theorem think_equiv (s : computation α) : think s ~ s :=
λ a, ⟨of_think_mem, think_mem⟩
theorem thinkN_equiv (s : computation α) (n) : thinkN s n ~ s :=
λ a, thinkN_mem n
theorem bind_congr {s1 s2 : computation α} {f1 f2 : α → computation β}
(h1 : s1 ~ s2) (h2 : ∀ a, f1 a ~ f2 a) : bind s1 f1 ~ bind s2 f2 :=
λ b, ⟨λh, let ⟨a, ha, hb⟩ := exists_of_mem_bind h in
mem_bind ((h1 a).1 ha) ((h2 a b).1 hb),
λh, let ⟨a, ha, hb⟩ := exists_of_mem_bind h in
mem_bind ((h1 a).2 ha) ((h2 a b).2 hb)⟩
theorem equiv_ret_of_mem {s : computation α} {a} (h : a ∈ s) : s ~ return a :=
equiv_of_mem h (ret_mem _)
/-- `lift_rel R ca cb` is a generalization of `equiv` to relations other than
equality. It asserts that if `ca` terminates with `a`, then `cb` terminates with
some `b` such that `R a b`, and if `cb` terminates with `b` then `ca` terminates
with some `a` such that `R a b`. -/
def lift_rel (R : α → β → Prop) (ca : computation α) (cb : computation β) : Prop :=
(∀ {a}, a ∈ ca → ∃ {b}, b ∈ cb ∧ R a b) ∧
∀ {b}, b ∈ cb → ∃ {a}, a ∈ ca ∧ R a b
theorem lift_rel.swap (R : α → β → Prop) (ca : computation α) (cb : computation β) :
lift_rel (function.swap R) cb ca ↔ lift_rel R ca cb :=
and_comm _ _
theorem lift_eq_iff_equiv (c₁ c₂ : computation α) : lift_rel (=) c₁ c₂ ↔ c₁ ~ c₂ :=
⟨λ⟨h1, h2⟩ a,
⟨λ a1, let ⟨b, b2, ab⟩ := h1 a1 in by rwa ab,
λ a2, let ⟨b, b1, ab⟩ := h2 a2 in by rwa ←ab⟩,
λe, ⟨λ a a1, ⟨a, (e _).1 a1, rfl⟩, λ a a2, ⟨a, (e _).2 a2, rfl⟩⟩⟩
theorem lift_rel.refl (R : α → α → Prop) (H : reflexive R) : reflexive (lift_rel R) :=
λ s, ⟨λ a as, ⟨a, as, H a⟩, λ b bs, ⟨b, bs, H b⟩⟩
theorem lift_rel.symm (R : α → α → Prop) (H : symmetric R) : symmetric (lift_rel R) :=
λ s1 s2 ⟨l, r⟩,
⟨λ a a2, let ⟨b, b1, ab⟩ := r a2 in ⟨b, b1, H ab⟩,
λ a a1, let ⟨b, b2, ab⟩ := l a1 in ⟨b, b2, H ab⟩⟩
theorem lift_rel.trans (R : α → α → Prop) (H : transitive R) : transitive (lift_rel R) :=
λ s1 s2 s3 ⟨l1, r1⟩ ⟨l2, r2⟩,
⟨λ a a1, let ⟨b, b2, ab⟩ := l1 a1, ⟨c, c3, bc⟩ := l2 b2 in ⟨c, c3, H ab bc⟩,
λ c c3, let ⟨b, b2, bc⟩ := r2 c3, ⟨a, a1, ab⟩ := r1 b2 in ⟨a, a1, H ab bc⟩⟩
theorem lift_rel.equiv (R : α → α → Prop) : equivalence R → equivalence (lift_rel R)
| ⟨refl, symm, trans⟩ :=
⟨lift_rel.refl R refl, lift_rel.symm R symm, lift_rel.trans R trans⟩
theorem lift_rel.imp {R S : α → β → Prop} (H : ∀ {a b}, R a b → S a b) (s t) :
lift_rel R s t → lift_rel S s t | ⟨l, r⟩ :=
⟨λ a as, let ⟨b, bt, ab⟩ := l as in ⟨b, bt, H ab⟩,
λ b bt, let ⟨a, as, ab⟩ := r bt in ⟨a, as, H ab⟩⟩
theorem terminates_of_lift_rel {R : α → β → Prop} {s t} :
lift_rel R s t → (terminates s ↔ terminates t) | ⟨l, r⟩ :=
⟨λ ⟨a, as⟩, let ⟨b, bt, ab⟩ := l as in ⟨b, bt⟩,
λ ⟨b, bt⟩, let ⟨a, as, ab⟩ := r bt in ⟨a, as⟩⟩
theorem rel_of_lift_rel {R : α → β → Prop} {ca cb} :
lift_rel R ca cb → ∀ {a b}, a ∈ ca → b ∈ cb → R a b
| ⟨l, r⟩ a b ma mb :=
let ⟨b', mb', ab'⟩ := l ma in by rw mem_unique mb mb'; exact ab'
theorem lift_rel_of_mem {R : α → β → Prop} {a b ca cb}
(ma : a ∈ ca) (mb : b ∈ cb) (ab : R a b) : lift_rel R ca cb :=
⟨λ a' ma', by rw mem_unique ma' ma; exact ⟨b, mb, ab⟩,
λ b' mb', by rw mem_unique mb' mb; exact ⟨a, ma, ab⟩⟩
theorem exists_of_lift_rel_left {R : α → β → Prop} {ca cb}
(H : lift_rel R ca cb) {a} (h : a ∈ ca) : ∃ {b}, b ∈ cb ∧ R a b :=
H.left h
theorem exists_of_lift_rel_right {R : α → β → Prop} {ca cb}
(H : lift_rel R ca cb) {b} (h : b ∈ cb) : ∃ {a}, a ∈ ca ∧ R a b :=
H.right h
theorem lift_rel_def {R : α → β → Prop} {ca cb} : lift_rel R ca cb ↔
(terminates ca ↔ terminates cb) ∧ ∀ {a b}, a ∈ ca → b ∈ cb → R a b :=
⟨λh, ⟨terminates_of_lift_rel h, λ a b ma mb,
let ⟨b', mb', ab⟩ := h.left ma in by rwa mem_unique mb mb'⟩,
λ⟨l, r⟩,
⟨λ a ma, let ⟨b, mb⟩ := l.1 ⟨_, ma⟩ in ⟨b, mb, r ma mb⟩,
λ b mb, let ⟨a, ma⟩ := l.2 ⟨_, mb⟩ in ⟨a, ma, r ma mb⟩⟩⟩
theorem lift_rel_bind {δ} (R : α → β → Prop) (S : γ → δ → Prop)
{s1 : computation α} {s2 : computation β}
{f1 : α → computation γ} {f2 : β → computation δ}
(h1 : lift_rel R s1 s2) (h2 : ∀ {a b}, R a b → lift_rel S (f1 a) (f2 b))
: lift_rel S (bind s1 f1) (bind s2 f2) :=
let ⟨l1, r1⟩ := h1 in
⟨λ c cB,
let ⟨a, a1, c₁⟩ := exists_of_mem_bind cB,
⟨b, b2, ab⟩ := l1 a1,
⟨l2, r2⟩ := h2 ab,
⟨d, d2, cd⟩ := l2 c₁ in
⟨_, mem_bind b2 d2, cd⟩,
λ d dB,
let ⟨b, b1, d1⟩ := exists_of_mem_bind dB,
⟨a, a2, ab⟩ := r1 b1,
⟨l2, r2⟩ := h2 ab,
⟨c, c₂, cd⟩ := r2 d1 in
⟨_, mem_bind a2 c₂, cd⟩⟩
@[simp] theorem lift_rel_return_left (R : α → β → Prop) (a : α) (cb : computation β) :
lift_rel R (return a) cb ↔ ∃ {b}, b ∈ cb ∧ R a b :=
⟨λ⟨l, r⟩, l (ret_mem _),
λ⟨b, mb, ab⟩,
⟨λ a' ma', by rw eq_of_ret_mem ma'; exact ⟨b, mb, ab⟩,
λ b' mb', ⟨_, ret_mem _, by rw mem_unique mb' mb; exact ab⟩⟩⟩
@[simp] theorem lift_rel_return_right (R : α → β → Prop) (ca : computation α) (b : β) :
lift_rel R ca (return b) ↔ ∃ {a}, a ∈ ca ∧ R a b :=
by rw [lift_rel.swap, lift_rel_return_left]
@[simp] theorem lift_rel_return (R : α → β → Prop) (a : α) (b : β) :
lift_rel R (return a) (return b) ↔ R a b :=
by rw [lift_rel_return_left]; exact
⟨λ⟨b', mb', ab'⟩, by rwa eq_of_ret_mem mb' at ab',
λab, ⟨_, ret_mem _, ab⟩⟩
@[simp] theorem lift_rel_think_left (R : α → β → Prop) (ca : computation α) (cb : computation β) :
lift_rel R (think ca) cb ↔ lift_rel R ca cb :=
and_congr (forall_congr $ λb, imp_congr ⟨of_think_mem, think_mem⟩ iff.rfl)
(forall_congr $ λb, imp_congr iff.rfl $
exists_congr $ λ b, and_congr ⟨of_think_mem, think_mem⟩ iff.rfl)
@[simp] theorem lift_rel_think_right (R : α → β → Prop) (ca : computation α) (cb : computation β) :
lift_rel R ca (think cb) ↔ lift_rel R ca cb :=
by rw [←lift_rel.swap R, ←lift_rel.swap R]; apply lift_rel_think_left
theorem lift_rel_mem_cases {R : α → β → Prop} {ca cb}
(Ha : ∀ a ∈ ca, lift_rel R ca cb)
(Hb : ∀ b ∈ cb, lift_rel R ca cb) : lift_rel R ca cb :=
⟨λ a ma, (Ha _ ma).left ma, λ b mb, (Hb _ mb).right mb⟩
theorem lift_rel_congr {R : α → β → Prop} {ca ca' : computation α} {cb cb' : computation β}
(ha : ca ~ ca') (hb : cb ~ cb') : lift_rel R ca cb ↔ lift_rel R ca' cb' :=
and_congr
(forall_congr $ λ a, imp_congr (ha _) $ exists_congr $ λ b, and_congr (hb _) iff.rfl)
(forall_congr $ λ b, imp_congr (hb _) $ exists_congr $ λ a, and_congr (ha _) iff.rfl)
theorem lift_rel_map {δ} (R : α → β → Prop) (S : γ → δ → Prop)
{s1 : computation α} {s2 : computation β}
{f1 : α → γ} {f2 : β → δ}
(h1 : lift_rel R s1 s2) (h2 : ∀ {a b}, R a b → S (f1 a) (f2 b))
: lift_rel S (map f1 s1) (map f2 s2) :=
by rw [←bind_ret, ←bind_ret]; apply lift_rel_bind _ _ h1; simp; exact @h2
theorem map_congr (R : α → α → Prop) (S : β → β → Prop)
{s1 s2 : computation α} {f : α → β}
(h1 : s1 ~ s2) : map f s1 ~ map f s2 :=
by rw [←lift_eq_iff_equiv];
exact lift_rel_map eq _ ((lift_eq_iff_equiv _ _).2 h1) (λ a b, congr_arg _)
def lift_rel_aux (R : α → β → Prop)
(C : computation α → computation β → Prop) :
α ⊕ computation α → β ⊕ computation β → Prop
| (sum.inl a) (sum.inl b) := R a b
| (sum.inl a) (sum.inr cb) := ∃ {b}, b ∈ cb ∧ R a b
| (sum.inr ca) (sum.inl b) := ∃ {a}, a ∈ ca ∧ R a b
| (sum.inr ca) (sum.inr cb) := C ca cb
attribute [simp] lift_rel_aux
@[simp] lemma lift_rel_aux.ret_left (R : α → β → Prop)
(C : computation α → computation β → Prop) (a cb) :
lift_rel_aux R C (sum.inl a) (destruct cb) ↔ ∃ {b}, b ∈ cb ∧ R a b :=
begin
apply cb.cases_on (λ b, _) (λ cb, _),
{ exact ⟨λ h, ⟨_, ret_mem _, h⟩, λ ⟨b', mb, h⟩,
by rw [mem_unique (ret_mem _) mb]; exact h⟩ },
{ rw [destruct_think],
exact ⟨λ ⟨b, h, r⟩, ⟨b, think_mem h, r⟩,
λ ⟨b, h, r⟩, ⟨b, of_think_mem h, r⟩⟩ }
end
theorem lift_rel_aux.swap (R : α → β → Prop) (C) (a b) :
lift_rel_aux (function.swap R) (function.swap C) b a = lift_rel_aux R C a b :=
by cases a with a ca; cases b with b cb; simp only [lift_rel_aux]
@[simp] lemma lift_rel_aux.ret_right (R : α → β → Prop)
(C : computation α → computation β → Prop) (b ca) :
lift_rel_aux R C (destruct ca) (sum.inl b) ↔ ∃ {a}, a ∈ ca ∧ R a b :=
by rw [←lift_rel_aux.swap, lift_rel_aux.ret_left]
theorem lift_rel_rec.lem {R : α → β → Prop} (C : computation α → computation β → Prop)
(H : ∀ {ca cb}, C ca cb → lift_rel_aux R C (destruct ca) (destruct cb))
(ca cb) (Hc : C ca cb) (a) (ha : a ∈ ca) : lift_rel R ca cb :=
begin
revert cb, refine mem_rec_on ha _ (λ ca' IH, _);
intros cb Hc; have h := H Hc,
{ simp at h, simp [h] },
{ have h := H Hc, simp, revert h, apply cb.cases_on (λ b, _) (λ cb', _);
intro h; simp at h; simp [h], exact IH _ h }
end
theorem lift_rel_rec {R : α → β → Prop} (C : computation α → computation β → Prop)
(H : ∀ {ca cb}, C ca cb → lift_rel_aux R C (destruct ca) (destruct cb))
(ca cb) (Hc : C ca cb) : lift_rel R ca cb :=
lift_rel_mem_cases (lift_rel_rec.lem C @H ca cb Hc) (λ b hb,
(lift_rel.swap _ _ _).2 $
lift_rel_rec.lem (function.swap C)
(λ cb ca h, cast (lift_rel_aux.swap _ _ _ _).symm $ H h)
cb ca Hc b hb)
end computation
|
ce19572cfcd0d317d7535db9b60d87a19e6c07ca | 82b86ba2ae0d5aed0f01f49c46db5afec0eb2bd7 | /tests/lean/match4.lean | 9026e68e2ea0be4602445c1f2740de85b323ea75 | [
"Apache-2.0"
] | permissive | banksonian/lean4 | 3a2e6b0f1eb63aa56ff95b8d07b2f851072d54dc | 78da6b3aa2840693eea354a41e89fc5b212a5011 | refs/heads/master | 1,673,703,624,165 | 1,605,123,551,000 | 1,605,123,551,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,114 | lean |
def f1 (x : Nat × Nat) : Nat :=
match x with
| { fst := x, snd := y } => x - y
#eval f1 (20, 15)
def g1 (h : Nat × Nat × Nat → Nat) : Nat :=
h (1, 2, 3)
def f2 (w : Nat) : Nat :=
g1 fun (x, y, z) => x + y + z + w
#eval f2 10
def g2 (h : Nat × Nat → Nat × Nat → Nat) : Nat :=
h (1, 2) (20, 40)
def f3 (a : Nat) : Nat :=
g2 fun (x, y) (z, w) => a*(y - x) + (w - z)
#eval f3 100
def f4 (x : Nat × Nat) : Nat :=
let (a, b) := x;
a + b
#eval f4 (10, 20)
def f5 (x y : Nat) : Nat :=
let h : Nat → Nat → Nat
| 0, b => b
| a, b => a*b;
h x y
#eval f5 0 10
#eval f5 20 10
def f6 (x : Nat × Nat) : Nat :=
match x with
| { fst := x, .. } => x * 10
#eval f6 (5, 20)
def Vector (α : Type) (n : Nat) := { a : Array α // a.size = n }
def mkVec {α : Type} (n : Nat) (a : α) : Vector α n :=
⟨mkArray n a, rfl⟩
structure S :=
(n : Nat)
(y : Vector Nat n)
(z : Vector Nat n)
(h : y = z)
(m : { v : Nat // v = y.val.size })
def f7 (s : S) : Nat :=
match s with
| { n := n, m := m, .. } => n + m.val
#eval f7 { n := 10, y := mkVec 10 0, z := mkVec 10 0, h := rfl, m := ⟨10, rfl⟩ }
|
717460a0e25384ffdfa8dad0fc31a4d823b715ad | 94e33a31faa76775069b071adea97e86e218a8ee | /src/analysis/seminorm.lean | fd31357a301a1f9188ef32b77bf9db4acc46d2ce | [
"Apache-2.0"
] | permissive | urkud/mathlib | eab80095e1b9f1513bfb7f25b4fa82fa4fd02989 | 6379d39e6b5b279df9715f8011369a301b634e41 | refs/heads/master | 1,658,425,342,662 | 1,658,078,703,000 | 1,658,078,703,000 | 186,910,338 | 0 | 0 | Apache-2.0 | 1,568,512,083,000 | 1,557,958,709,000 | Lean | UTF-8 | Lean | false | false | 36,082 | lean | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import analysis.locally_convex.basic
import data.real.pointwise
import data.real.sqrt
import topology.algebra.filter_basis
import topology.algebra.module.locally_convex
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For an addditive group:
* `add_group_seminorm`: A function `f` from an add_group `G` to the reals that preserves zero,
takes nonnegative values, is subadditive and such that `f (-x) = f x` for all `x ∈ G`.
For a module over a normed ring:
* `seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `norm_seminorm 𝕜 E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option old_structure_cmd true
open normed_field set
open_locale big_operators nnreal pointwise topological_space
variables {R R' 𝕜 E F G ι : Type*}
/-- A seminorm on an add_group `G` is a function A function `f : G → ℝ` that preserves zero, takes
nonnegative values, is subadditive and such that `f (-x) = f x` for all `x ∈ G`. -/
structure add_group_seminorm (G : Type*) [add_group G]
extends zero_hom G ℝ :=
(nonneg' : ∀ r, 0 ≤ to_fun r)
(add_le' : ∀ r s, to_fun (r + s) ≤ to_fun r + to_fun s)
(neg' : ∀ r, to_fun (- r) = to_fun r)
attribute [nolint doc_blame] add_group_seminorm.to_zero_hom
namespace add_group_seminorm
variables [add_group E]
instance zero_hom_class : zero_hom_class (add_group_seminorm E) E ℝ :=
{ coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_zero := λ f, f.map_zero' }
/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`. -/
instance : has_coe_to_fun (add_group_seminorm E) (λ _, E → ℝ) := ⟨λ p, p.to_fun⟩
@[ext] lemma ext {p q : add_group_seminorm E} (h : ∀ x, (p : E → ℝ) x = q x) : p = q :=
fun_like.ext p q h
instance : has_zero (add_group_seminorm E) :=
⟨{ to_fun := 0,
nonneg' := λ r, le_refl _,
map_zero' := pi.zero_apply _,
add_le' := λ _ _, eq.ge (zero_add _),
neg' := λ x, rfl}⟩
@[simp] lemma coe_zero : ⇑(0 : add_group_seminorm E) = 0 := rfl
@[simp] lemma zero_apply (x : E) : (0 : add_group_seminorm E) x = 0 := rfl
instance : inhabited (add_group_seminorm E) := ⟨0⟩
variables (p : add_group_seminorm E) (x y : E) (r : ℝ)
protected lemma nonneg : 0 ≤ p x := p.nonneg' _
@[simp] protected lemma map_zero : p 0 = 0 := p.map_zero'
protected lemma add_le : p (x + y) ≤ p x + p y := p.add_le' _ _
@[simp] protected lemma neg : p (- x) = p x := p.neg' _
/-- Any action on `ℝ` which factors through `ℝ≥0` applies to an `add_group_seminorm`. -/
instance [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ] :
has_smul R (add_group_seminorm E) :=
{ smul := λ r p,
{ to_fun := λ x, r • p x,
nonneg' := λ x, begin
simp only [←smul_one_smul ℝ≥0 r (_ : ℝ), nnreal.smul_def, smul_eq_mul],
exact mul_nonneg (nnreal.coe_nonneg _) (p.nonneg _)
end,
map_zero' := by simp only [←smul_one_smul ℝ≥0 r (_ : ℝ), nnreal.smul_def, smul_eq_mul,
p.map_zero, mul_zero],
add_le' := λ _ _, begin
simp only [←smul_one_smul ℝ≥0 r (_ : ℝ), nnreal.smul_def, smul_eq_mul],
exact (mul_le_mul_of_nonneg_left (p.add_le _ _) (nnreal.coe_nonneg _)).trans_eq
(mul_add _ _ _),
end,
neg' := λ x, by rw p.neg }}
instance [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
[has_smul R' ℝ] [has_smul R' ℝ≥0] [is_scalar_tower R' ℝ≥0 ℝ]
[has_smul R R'] [is_scalar_tower R R' ℝ] :
is_scalar_tower R R' (add_group_seminorm E) :=
{ smul_assoc := λ r a p, ext $ λ x, smul_assoc r a (p x) }
@[simp] lemma coe_smul [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p : add_group_seminorm E) : ⇑(r • p) = r • p := rfl
@[simp] lemma smul_apply [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p : add_group_seminorm E) (x : E) : (r • p) x = r • p x := rfl
instance : has_add (add_group_seminorm E) :=
{ add := λ p q,
{ to_fun := λ x, p x + q x,
nonneg' := λ x, add_nonneg (p.nonneg _) (q.nonneg _),
map_zero' := by rw [p.map_zero, q.map_zero, zero_add],
add_le' := λ _ _, has_le.le.trans_eq (add_le_add (p.add_le _ _) (q.add_le _ _))
(add_add_add_comm _ _ _ _),
neg' := λ x, by rw [p.neg, q.neg] }}
@[simp] lemma coe_add (p q : add_group_seminorm E) : ⇑(p + q) = p + q := rfl
@[simp] lemma add_apply (p q : add_group_seminorm E) (x : E) : (p + q) x = p x + q x := rfl
-- TODO: define `has_Sup` too, from the skeleton at
-- https://github.com/leanprover-community/mathlib/pull/11329#issuecomment-1008915345
noncomputable instance : has_sup (add_group_seminorm E) :=
{ sup := λ p q,
{ to_fun := p ⊔ q,
nonneg' := λ x, begin
simp only [pi.sup_apply, le_sup_iff],
exact or.intro_left _ (p.nonneg _),
end,
map_zero' := begin
simp only [pi.sup_apply],
rw [← p.map_zero, sup_eq_left, p.map_zero, q.map_zero],
end,
add_le' := λ x y, sup_le
((p.add_le x y).trans $ add_le_add le_sup_left le_sup_left)
((q.add_le x y).trans $ add_le_add le_sup_right le_sup_right),
neg' := λ x, by rw [pi.sup_apply, pi.sup_apply, p.neg, q.neg] }}
@[simp] lemma coe_sup (p q : add_group_seminorm E) : ⇑(p ⊔ q) = p ⊔ q := rfl
lemma sup_apply (p q : add_group_seminorm E) (x : E) : (p ⊔ q) x = p x ⊔ q x := rfl
lemma smul_sup [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p q : add_group_seminorm E) :
r • (p ⊔ q) = r • p ⊔ r • q :=
have real.smul_max : ∀ x y : ℝ, r • max x y = max (r • x) (r • y),
from λ x y, by simpa only [←smul_eq_mul, ←nnreal.smul_def, smul_one_smul ℝ≥0 r (_ : ℝ)]
using mul_max_of_nonneg x y (r • 1 : ℝ≥0).prop,
ext $ λ x, real.smul_max _ _
instance : partial_order (add_group_seminorm E) := partial_order.lift _ fun_like.coe_injective
lemma le_def (p q : add_group_seminorm E) : p ≤ q ↔ (p : E → ℝ) ≤ q := iff.rfl
lemma lt_def (p q : add_group_seminorm E) : p < q ↔ (p : E → ℝ) < q := iff.rfl
noncomputable instance : semilattice_sup (add_group_seminorm E) :=
function.injective.semilattice_sup _ fun_like.coe_injective coe_sup
section add_comm_group
variable [add_comm_group G]
variables (q : add_group_seminorm G)
protected lemma sub_le (x y : G) : q (x - y) ≤ q x + q y :=
calc
q (x - y)
= q (x + -y) : by rw sub_eq_add_neg
... ≤ q x + q (-y) : q.add_le x (-y)
... = q x + q y : by rw q.neg
lemma sub_rev (x y : G) : q (x - y) = q (y - x) :=
by rw [←neg_sub, q.neg]
/-- The direct path from 0 to y is shorter than the path with x "inserted" in between. -/
lemma le_insert (x y : G) : q y ≤ q x + q (x - y) :=
calc q y = q (x - (x - y)) : by rw sub_sub_cancel
... ≤ q x + q (x - y) : q.sub_le _ _
/-- The direct path from 0 to x is shorter than the path with y "inserted" in between. -/
lemma le_insert' (x y : G) : q x ≤ q y + q (x - y) :=
by { rw sub_rev, exact le_insert _ _ _ }
private lemma bdd_below_range_add (x : G) (p q : add_group_seminorm G) :
bdd_below (range (λ (u : G), p u + q (x - u))) :=
by { use 0, rintro _ ⟨x, rfl⟩, exact add_nonneg (p.nonneg _) (q.nonneg _) }
noncomputable instance : has_inf (add_group_seminorm G) :=
{ inf := λ p q,
{ to_fun := λ x, ⨅ u : G, p u + q (x-u),
map_zero' := cinfi_eq_of_forall_ge_of_forall_gt_exists_lt
(λ x, add_nonneg (p.nonneg _) (q.nonneg _))
(λ r hr, ⟨0, by simpa [sub_zero, p.map_zero, q.map_zero, add_zero] using hr⟩),
nonneg' := λ x, le_cinfi (λ x, add_nonneg (p.nonneg _) (q.nonneg _)),
add_le' := λ x y, begin
refine le_cinfi_add_cinfi (λ u v, _),
apply cinfi_le_of_le (bdd_below_range_add _ _ _) (v+u), dsimp only,
convert add_le_add (p.add_le v u) (q.add_le (y-v) (x-u)) using 1,
{ rw show x + y - (v + u) = y - v + (x - u), by abel },
{ abel },
end,
neg' := λ x, begin
have : (⨅ (u : G), p u + q (x - u) : ℝ) = ⨅ (u : G), p (- u) + q (x + u),
{ apply function.surjective.infi_congr (λ (x : G), -x) neg_surjective,
{ intro u,
simp only [neg_neg, add_right_inj, sub_eq_add_neg] }},
rw this,
apply congr_arg,
ext u,
rw [p.neg, sub_eq_add_neg, ← neg_add_rev, add_comm u, q.neg],
end }}
@[simp] lemma inf_apply (p q : add_group_seminorm G) (x : G) :
(p ⊓ q) x = ⨅ u : G, p u + q (x-u) := rfl
noncomputable instance : lattice (add_group_seminorm G) :=
{ inf := (⊓),
inf_le_left := λ p q x, begin
apply cinfi_le_of_le (bdd_below_range_add _ _ _) x,
simp only [sub_self, map_zero, add_zero],
end,
inf_le_right := λ p q x, begin
apply cinfi_le_of_le (bdd_below_range_add _ _ _) (0:G),
simp only [sub_self, map_zero, zero_add, sub_zero],
end,
le_inf := λ a b c hab hac x,
le_cinfi $ λ u, le_trans (a.le_insert' _ _) (add_le_add (hab _) (hac _)),
..add_group_seminorm.semilattice_sup }
end add_comm_group
section comp
variables [add_group F] [add_group G]
/-- Composition of an add_group_seminorm with an add_monoid_hom is an add_group_seminorm. -/
def comp (p : add_group_seminorm F) (f : E →+ F) : add_group_seminorm E :=
{ to_fun := λ x, p (f x),
nonneg' := λ x, p.nonneg _,
map_zero' := by rw [f.map_zero, p.map_zero],
add_le' := λ _ _, by apply eq.trans_le (congr_arg p (f.map_add _ _)) (p.add_le _ _),
neg' := λ x, by rw [map_neg, p.neg] }
@[simp] lemma coe_comp (p : add_group_seminorm F) (f : E →+ F) : ⇑(p.comp f) = p ∘ f := rfl
@[simp] lemma comp_apply (p : add_group_seminorm F) (f : E →+ F) (x : E) :
(p.comp f) x = p (f x) := rfl
@[simp] lemma comp_id (p : add_group_seminorm E) : p.comp (add_monoid_hom.id _) = p :=
ext $ λ _, rfl
@[simp] lemma comp_zero (p : add_group_seminorm F) : p.comp (0 : E →+ F) = 0 :=
ext $ λ _, map_zero p
@[simp] lemma zero_comp (f : E →+ F) : (0 : add_group_seminorm F).comp f = 0 :=
ext $ λ _, rfl
lemma comp_comp (p : add_group_seminorm G) (g : F →+ G) (f : E →+ F) :
p.comp (g.comp f) = (p.comp g).comp f :=
ext $ λ _, rfl
lemma add_comp (p q : add_group_seminorm F) (f : E →+ F) : (p + q).comp f = p.comp f + q.comp f :=
ext $ λ _, rfl
lemma comp_add_le {A B : Type*} [add_comm_group A] [add_comm_group B]
(p : add_group_seminorm B) (f g : A →+ B) : p.comp (f + g) ≤ p.comp f + p.comp g :=
λ _, p.add_le _ _
lemma comp_mono {p : add_group_seminorm F} {q : add_group_seminorm F} (f : E →+ F) (hp : p ≤ q) :
p.comp f ≤ q.comp f := λ _, hp _
end comp
end add_group_seminorm
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure seminorm (𝕜 : Type*) (E : Type*) [semi_normed_ring 𝕜] [add_group E] [has_smul 𝕜 E]
extends add_group_seminorm E :=
(smul' : ∀ (a : 𝕜) (x : E), to_fun (a • x) = ∥a∥ * to_fun x)
attribute [nolint doc_blame] seminorm.to_add_group_seminorm
private lemma map_zero.of_smul {𝕜 : Type*} {E : Type*} [semi_normed_ring 𝕜] [add_group E]
[smul_with_zero 𝕜 E] {f : E → ℝ} (smul : ∀ (a : 𝕜) (x : E), f (a • x) = ∥a∥ * f x) : f 0 = 0 :=
calc f 0 = f ((0 : 𝕜) • 0) : by rw zero_smul
... = 0 : by rw [smul, norm_zero, zero_mul]
private lemma neg.of_smul {𝕜 : Type*} {E : Type*} [semi_normed_ring 𝕜] [add_comm_group E]
[module 𝕜 E] {f : E → ℝ} (smul : ∀ (a : 𝕜) (x : E), f (a • x) = ∥a∥ * f x) (x : E) :
f (-x) = f x :=
by rw [←neg_one_smul 𝕜, smul, norm_neg, ← smul, one_smul]
private lemma nonneg.of {𝕜 : Type*} {E : Type*} [semi_normed_ring 𝕜] [add_comm_group E] [module 𝕜 E]
{f : E → ℝ} (add_le : ∀ (x y : E), f (x + y) ≤ f x + f y)
(smul : ∀ (a : 𝕜) (x : E), f (a • x) = ∥a∥ * f x) (x : E) : 0 ≤ f x :=
have h: 0 ≤ 2 * f x, from
calc 0 = f (x + (- x)) : by rw [add_neg_self, map_zero.of_smul smul]
... ≤ f x + f (-x) : add_le _ _
... = 2 * f x : by rw [neg.of_smul smul, two_mul],
nonneg_of_mul_nonneg_right h zero_lt_two
/-- Alternative constructor for a `seminorm` on an `add_comm_group E` that is a module over a
`semi_norm_ring 𝕜`. -/
def seminorm.of {𝕜 : Type*} {E : Type*} [semi_normed_ring 𝕜] [add_comm_group E] [module 𝕜 E]
(f : E → ℝ) (add_le : ∀ (x y : E), f (x + y) ≤ f x + f y)
(smul : ∀ (a : 𝕜) (x : E), f (a • x) = ∥a∥ * f x) : seminorm 𝕜 E :=
{ to_fun := f,
map_zero' := map_zero.of_smul smul,
nonneg' := nonneg.of add_le smul,
add_le' := add_le,
smul' := smul,
neg' := neg.of_smul smul }
namespace seminorm
section semi_normed_ring
variables [semi_normed_ring 𝕜]
section add_group
variables [add_group E]
section has_smul
variables [has_smul 𝕜 E]
instance zero_hom_class : zero_hom_class (seminorm 𝕜 E) E ℝ :=
{ coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_zero := λ f, f.map_zero' }
/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`. -/
instance : has_coe_to_fun (seminorm 𝕜 E) (λ _, E → ℝ) := ⟨λ p, p.to_fun⟩
@[ext] lemma ext {p q : seminorm 𝕜 E} (h : ∀ x, (p : E → ℝ) x = q x) : p = q := fun_like.ext p q h
instance : has_zero (seminorm 𝕜 E) :=
⟨{ smul' := λ _ _, (mul_zero _).symm,
..add_group_seminorm.has_zero.zero }⟩
@[simp] lemma coe_zero : ⇑(0 : seminorm 𝕜 E) = 0 := rfl
@[simp] lemma zero_apply (x : E) : (0 : seminorm 𝕜 E) x = 0 := rfl
instance : inhabited (seminorm 𝕜 E) := ⟨0⟩
variables (p : seminorm 𝕜 E) (c : 𝕜) (x y : E) (r : ℝ)
protected lemma nonneg : 0 ≤ p x := p.nonneg' _
protected lemma map_zero : p 0 = 0 := p.map_zero'
protected lemma smul : p (c • x) = ∥c∥ * p x := p.smul' _ _
protected lemma add_le : p (x + y) ≤ p x + p y := p.add_le' _ _
/-- Any action on `ℝ` which factors through `ℝ≥0` applies to a seminorm. -/
instance [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ] :
has_smul R (seminorm 𝕜 E) :=
{ smul := λ r p,
{ to_fun := λ x, r • p x,
smul' := λ _ _, begin
simp only [←smul_one_smul ℝ≥0 r (_ : ℝ), nnreal.smul_def, smul_eq_mul],
rw [p.smul, mul_left_comm],
end,
..(r • p.to_add_group_seminorm) }}
instance [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
[has_smul R' ℝ] [has_smul R' ℝ≥0] [is_scalar_tower R' ℝ≥0 ℝ]
[has_smul R R'] [is_scalar_tower R R' ℝ] :
is_scalar_tower R R' (seminorm 𝕜 E) :=
{ smul_assoc := λ r a p, ext $ λ x, smul_assoc r a (p x) }
lemma coe_smul [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p : seminorm 𝕜 E) : ⇑(r • p) = r • p := rfl
@[simp] lemma smul_apply [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p : seminorm 𝕜 E) (x : E) : (r • p) x = r • p x := rfl
instance : has_add (seminorm 𝕜 E) :=
{ add := λ p q,
{ to_fun := λ x, p x + q x,
smul' := λ a x, by simp only [p.smul, q.smul, mul_add],
..(p.to_add_group_seminorm + q.to_add_group_seminorm) }}
lemma coe_add (p q : seminorm 𝕜 E) : ⇑(p + q) = p + q := rfl
@[simp] lemma add_apply (p q : seminorm 𝕜 E) (x : E) : (p + q) x = p x + q x := rfl
instance : add_monoid (seminorm 𝕜 E) :=
fun_like.coe_injective.add_monoid _ rfl coe_add (λ p n, coe_smul n p)
instance : ordered_cancel_add_comm_monoid (seminorm 𝕜 E) :=
fun_like.coe_injective.ordered_cancel_add_comm_monoid _ rfl coe_add (λ p n, coe_smul n p)
instance [monoid R] [mul_action R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ] :
mul_action R (seminorm 𝕜 E) :=
fun_like.coe_injective.mul_action _ coe_smul
variables (𝕜 E)
/-- `coe_fn` as an `add_monoid_hom`. Helper definition for showing that `seminorm 𝕜 E` is
a module. -/
@[simps]
def coe_fn_add_monoid_hom : add_monoid_hom (seminorm 𝕜 E) (E → ℝ) := ⟨coe_fn, coe_zero, coe_add⟩
lemma coe_fn_add_monoid_hom_injective : function.injective (coe_fn_add_monoid_hom 𝕜 E) :=
show @function.injective (seminorm 𝕜 E) (E → ℝ) coe_fn, from fun_like.coe_injective
variables {𝕜 E}
instance [monoid R] [distrib_mul_action R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ] :
distrib_mul_action R (seminorm 𝕜 E) :=
(coe_fn_add_monoid_hom_injective 𝕜 E).distrib_mul_action _ coe_smul
instance [semiring R] [module R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ] :
module R (seminorm 𝕜 E) :=
(coe_fn_add_monoid_hom_injective 𝕜 E).module R _ coe_smul
-- TODO: define `has_Sup` too, from the skeleton at
-- https://github.com/leanprover-community/mathlib/pull/11329#issuecomment-1008915345
noncomputable instance : has_sup (seminorm 𝕜 E) :=
{ sup := λ p q,
{ to_fun := p ⊔ q,
smul' := λ x v, (congr_arg2 max (p.smul x v) (q.smul x v)).trans $
(mul_max_of_nonneg _ _ $ norm_nonneg x).symm,
..(p.to_add_group_seminorm ⊔ q.to_add_group_seminorm) } }
@[simp] lemma coe_sup (p q : seminorm 𝕜 E) : ⇑(p ⊔ q) = p ⊔ q := rfl
lemma sup_apply (p q : seminorm 𝕜 E) (x : E) : (p ⊔ q) x = p x ⊔ q x := rfl
lemma smul_sup [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p q : seminorm 𝕜 E) :
r • (p ⊔ q) = r • p ⊔ r • q :=
have real.smul_max : ∀ x y : ℝ, r • max x y = max (r • x) (r • y),
from λ x y, by simpa only [←smul_eq_mul, ←nnreal.smul_def, smul_one_smul ℝ≥0 r (_ : ℝ)]
using mul_max_of_nonneg x y (r • 1 : ℝ≥0).prop,
ext $ λ x, real.smul_max _ _
instance : partial_order (seminorm 𝕜 E) :=
partial_order.lift _ fun_like.coe_injective
lemma le_def (p q : seminorm 𝕜 E) : p ≤ q ↔ (p : E → ℝ) ≤ q := iff.rfl
lemma lt_def (p q : seminorm 𝕜 E) : p < q ↔ (p : E → ℝ) < q := iff.rfl
noncomputable instance : semilattice_sup (seminorm 𝕜 E) :=
function.injective.semilattice_sup _ fun_like.coe_injective coe_sup
end has_smul
end add_group
section module
variables [add_comm_group E] [add_comm_group F] [add_comm_group G]
variables [module 𝕜 E] [module 𝕜 F] [module 𝕜 G]
variables [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : seminorm 𝕜 F) (f : E →ₗ[𝕜] F) : seminorm 𝕜 E :=
{ to_fun := λ x, p (f x),
smul' := λ _ _, (congr_arg p (f.map_smul _ _)).trans (p.smul _ _),
..(p.to_add_group_seminorm.comp f.to_add_monoid_hom) }
lemma coe_comp (p : seminorm 𝕜 F) (f : E →ₗ[𝕜] F) : ⇑(p.comp f) = p ∘ f := rfl
@[simp] lemma comp_apply (p : seminorm 𝕜 F) (f : E →ₗ[𝕜] F) (x : E) : (p.comp f) x = p (f x) := rfl
@[simp] lemma comp_id (p : seminorm 𝕜 E) : p.comp linear_map.id = p :=
ext $ λ _, rfl
@[simp] lemma comp_zero (p : seminorm 𝕜 F) : p.comp (0 : E →ₗ[𝕜] F) = 0 :=
ext $ λ _, map_zero p
@[simp] lemma zero_comp (f : E →ₗ[𝕜] F) : (0 : seminorm 𝕜 F).comp f = 0 :=
ext $ λ _, rfl
lemma comp_comp (p : seminorm 𝕜 G) (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) :
p.comp (g.comp f) = (p.comp g).comp f :=
ext $ λ _, rfl
lemma add_comp (p q : seminorm 𝕜 F) (f : E →ₗ[𝕜] F) : (p + q).comp f = p.comp f + q.comp f :=
ext $ λ _, rfl
lemma comp_add_le (p : seminorm 𝕜 F) (f g : E →ₗ[𝕜] F) : p.comp (f + g) ≤ p.comp f + p.comp g :=
λ _, p.add_le _ _
lemma smul_comp (p : seminorm 𝕜 F) (f : E →ₗ[𝕜] F) (c : R) : (c • p).comp f = c • (p.comp f) :=
ext $ λ _, rfl
lemma comp_mono {p : seminorm 𝕜 F} {q : seminorm 𝕜 F} (f : E →ₗ[𝕜] F) (hp : p ≤ q) :
p.comp f ≤ q.comp f := λ _, hp _
/-- The composition as an `add_monoid_hom`. -/
@[simps] def pullback (f : E →ₗ[𝕜] F) : add_monoid_hom (seminorm 𝕜 F) (seminorm 𝕜 E) :=
⟨λ p, p.comp f, zero_comp f, λ p q, add_comp p q f⟩
section
variables (p : seminorm 𝕜 E)
@[simp]
protected lemma neg (x : E) : p (-x) = p x :=
by rw [←neg_one_smul 𝕜, seminorm.smul, norm_neg, ←seminorm.smul, one_smul]
protected lemma sub_le (x y : E) : p (x - y) ≤ p x + p y :=
calc
p (x - y)
= p (x + -y) : by rw sub_eq_add_neg
... ≤ p x + p (-y) : p.add_le x (-y)
... = p x + p y : by rw p.neg
lemma sub_rev (x y : E) : p (x - y) = p (y - x) := by rw [←neg_sub, p.neg]
/-- The direct path from 0 to y is shorter than the path with x "inserted" in between. -/
lemma le_insert (x y : E) : p y ≤ p x + p (x - y) :=
calc p y = p (x - (x - y)) : by rw sub_sub_cancel
... ≤ p x + p (x - y) : p.sub_le _ _
/-- The direct path from 0 to x is shorter than the path with y "inserted" in between. -/
lemma le_insert' (x y : E) : p x ≤ p y + p (x - y) := by { rw sub_rev, exact le_insert _ _ _ }
end
instance : order_bot (seminorm 𝕜 E) := ⟨0, seminorm.nonneg⟩
@[simp] lemma coe_bot : ⇑(⊥ : seminorm 𝕜 E) = 0 := rfl
lemma bot_eq_zero : (⊥ : seminorm 𝕜 E) = 0 := rfl
lemma smul_le_smul {p q : seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q :=
begin
simp_rw [le_def, pi.le_def, coe_smul],
intros x,
simp_rw [pi.smul_apply, nnreal.smul_def, smul_eq_mul],
exact mul_le_mul hab (hpq x) (p.nonneg x) (nnreal.coe_nonneg b),
end
lemma finset_sup_apply (p : ι → seminorm 𝕜 E) (s : finset ι) (x : E) :
s.sup p x = ↑(s.sup (λ i, ⟨p i x, (p i).nonneg x⟩) : ℝ≥0) :=
begin
induction s using finset.cons_induction_on with a s ha ih,
{ rw [finset.sup_empty, finset.sup_empty, coe_bot, _root_.bot_eq_zero, pi.zero_apply,
nonneg.coe_zero] },
{ rw [finset.sup_cons, finset.sup_cons, coe_sup, sup_eq_max, pi.sup_apply, sup_eq_max,
nnreal.coe_max, subtype.coe_mk, ih] }
end
lemma finset_sup_le_sum (p : ι → seminorm 𝕜 E) (s : finset ι) : s.sup p ≤ ∑ i in s, p i :=
begin
classical,
refine finset.sup_le_iff.mpr _,
intros i hi,
rw [finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left],
exact bot_le,
end
lemma finset_sup_apply_le {p : ι → seminorm 𝕜 E} {s : finset ι} {x : E} {a : ℝ} (ha : 0 ≤ a)
(h : ∀ i, i ∈ s → p i x ≤ a) : s.sup p x ≤ a :=
begin
lift a to ℝ≥0 using ha,
rw [finset_sup_apply, nnreal.coe_le_coe],
exact finset.sup_le h,
end
lemma finset_sup_apply_lt {p : ι → seminorm 𝕜 E} {s : finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a :=
begin
lift a to ℝ≥0 using ha.le,
rw [finset_sup_apply, nnreal.coe_lt_coe, finset.sup_lt_iff],
{ exact h },
{ exact nnreal.coe_pos.mpr ha },
end
end module
end semi_normed_ring
section semi_normed_comm_ring
variables [semi_normed_comm_ring 𝕜] [add_comm_group E] [add_comm_group F] [module 𝕜 E] [module 𝕜 F]
lemma comp_smul (p : seminorm 𝕜 F) (f : E →ₗ[𝕜] F) (c : 𝕜) :
p.comp (c • f) = ∥c∥₊ • p.comp f :=
ext $ λ _, by rw [comp_apply, smul_apply, linear_map.smul_apply, p.smul, nnreal.smul_def,
coe_nnnorm, smul_eq_mul, comp_apply]
lemma comp_smul_apply (p : seminorm 𝕜 F) (f : E →ₗ[𝕜] F) (c : 𝕜) (x : E) :
p.comp (c • f) x = ∥c∥ * p (f x) := p.smul _ _
end semi_normed_comm_ring
section normed_field
variables [normed_field 𝕜] [add_comm_group E] [module 𝕜 E]
private lemma bdd_below_range_add (x : E) (p q : seminorm 𝕜 E) :
bdd_below (range (λ (u : E), p u + q (x - u))) :=
by { use 0, rintro _ ⟨x, rfl⟩, exact add_nonneg (p.nonneg _) (q.nonneg _) }
noncomputable instance : has_inf (seminorm 𝕜 E) :=
{ inf := λ p q,
{ to_fun := λ x, ⨅ u : E, p u + q (x-u),
smul' :=
begin
intros a x,
obtain rfl | ha := eq_or_ne a 0,
{ rw [norm_zero, zero_mul, zero_smul],
refine cinfi_eq_of_forall_ge_of_forall_gt_exists_lt
(λ i, add_nonneg (p.nonneg _) (q.nonneg _))
(λ x hx, ⟨0, by rwa [map_zero, sub_zero, map_zero, add_zero]⟩) },
simp_rw [real.mul_infi_of_nonneg (norm_nonneg a), mul_add, ←p.smul, ←q.smul, smul_sub],
refine function.surjective.infi_congr ((•) a⁻¹ : E → E) (λ u, ⟨a • u, inv_smul_smul₀ ha u⟩)
(λ u, _),
rw smul_inv_smul₀ ha
end,
..(p.to_add_group_seminorm ⊓ q.to_add_group_seminorm) }}
@[simp] lemma inf_apply (p q : seminorm 𝕜 E) (x : E) : (p ⊓ q) x = ⨅ u : E, p u + q (x-u) := rfl
noncomputable instance : lattice (seminorm 𝕜 E) :=
{ inf := (⊓),
inf_le_left := λ p q x, begin
apply cinfi_le_of_le (bdd_below_range_add _ _ _) x,
simp only [sub_self, map_zero, add_zero],
end,
inf_le_right := λ p q x, begin
apply cinfi_le_of_le (bdd_below_range_add _ _ _) (0:E),
simp only [sub_self, map_zero, zero_add, sub_zero],
end,
le_inf := λ a b c hab hac x,
le_cinfi $ λ u, le_trans (a.le_insert' _ _) (add_le_add (hab _) (hac _)),
..seminorm.semilattice_sup }
lemma smul_inf [has_smul R ℝ] [has_smul R ℝ≥0] [is_scalar_tower R ℝ≥0 ℝ]
(r : R) (p q : seminorm 𝕜 E) :
r • (p ⊓ q) = r • p ⊓ r • q :=
begin
ext,
simp_rw [smul_apply, inf_apply, smul_apply, ←smul_one_smul ℝ≥0 r (_ : ℝ), nnreal.smul_def,
smul_eq_mul, real.mul_infi_of_nonneg (subtype.prop _), mul_add],
end
end normed_field
/-! ### Seminorm ball -/
section semi_normed_ring
variables [semi_normed_ring 𝕜]
section add_comm_group
variables [add_comm_group E]
section has_smul
variables [has_smul 𝕜 E] (p : seminorm 𝕜 E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < `r`. -/
def ball (x : E) (r : ℝ) := { y : E | p (y - x) < r }
variables {x y : E} {r : ℝ}
@[simp] lemma mem_ball : y ∈ ball p x r ↔ p (y - x) < r := iff.rfl
lemma mem_ball_zero : y ∈ ball p 0 r ↔ p y < r := by rw [mem_ball, sub_zero]
lemma ball_zero_eq : ball p 0 r = { y : E | p y < r } := set.ext $ λ x, p.mem_ball_zero
@[simp] lemma ball_zero' (x : E) (hr : 0 < r) : ball (0 : seminorm 𝕜 E) x r = set.univ :=
begin
rw [set.eq_univ_iff_forall, ball],
simp [hr],
end
lemma ball_smul (p : seminorm 𝕜 E) {c : nnreal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).ball x r = p.ball x (r / c) :=
by { ext, rw [mem_ball, mem_ball, smul_apply, nnreal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (nnreal.coe_pos.mpr hc)] }
lemma ball_sup (p : seminorm 𝕜 E) (q : seminorm 𝕜 E) (e : E) (r : ℝ) :
ball (p ⊔ q) e r = ball p e r ∩ ball q e r :=
by simp_rw [ball, ←set.set_of_and, coe_sup, pi.sup_apply, sup_lt_iff]
lemma ball_finset_sup' (p : ι → seminorm 𝕜 E) (s : finset ι) (H : s.nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H (λ i, ball (p i) e r) :=
begin
induction H using finset.nonempty.cons_induction with a a s ha hs ih,
{ classical, simp },
{ rw [finset.sup'_cons hs, finset.inf'_cons hs, ball_sup, inf_eq_inter, ih] },
end
lemma ball_mono {p : seminorm 𝕜 E} {r₁ r₂ : ℝ} (h : r₁ ≤ r₂) : p.ball x r₁ ⊆ p.ball x r₂ :=
λ _ (hx : _ < _), hx.trans_le h
lemma ball_antitone {p q : seminorm 𝕜 E} (h : q ≤ p) : p.ball x r ⊆ q.ball x r :=
λ _, (h _).trans_lt
lemma ball_add_ball_subset (p : seminorm 𝕜 E) (r₁ r₂ : ℝ) (x₁ x₂ : E):
p.ball (x₁ : E) r₁ + p.ball (x₂ : E) r₂ ⊆ p.ball (x₁ + x₂) (r₁ + r₂) :=
begin
rintros x ⟨y₁, y₂, hy₁, hy₂, rfl⟩,
rw [mem_ball, add_sub_add_comm],
exact (p.add_le _ _).trans_lt (add_lt_add hy₁ hy₂),
end
end has_smul
section module
variables [module 𝕜 E]
variables [add_comm_group F] [module 𝕜 F]
lemma ball_comp (p : seminorm 𝕜 F) (f : E →ₗ[𝕜] F) (x : E) (r : ℝ) :
(p.comp f).ball x r = f ⁻¹' (p.ball (f x) r) :=
begin
ext,
simp_rw [ball, mem_preimage, comp_apply, set.mem_set_of_eq, map_sub],
end
variables (p : seminorm 𝕜 E)
lemma ball_zero_eq_preimage_ball {r : ℝ} :
p.ball 0 r = p ⁻¹' (metric.ball 0 r) :=
begin
ext x,
simp only [mem_ball, sub_zero, mem_preimage, mem_ball_zero_iff],
rw real.norm_of_nonneg,
exact p.nonneg _,
end
@[simp] lemma ball_bot {r : ℝ} (x : E) (hr : 0 < r) :
ball (⊥ : seminorm 𝕜 E) x r = set.univ :=
ball_zero' x hr
/-- Seminorm-balls at the origin are balanced. -/
lemma balanced_ball_zero (r : ℝ) : balanced 𝕜 (ball p 0 r) :=
begin
rintro a ha x ⟨y, hy, hx⟩,
rw [mem_ball_zero, ←hx, p.smul],
calc _ ≤ p y : mul_le_of_le_one_left (p.nonneg _) ha
... < r : by rwa mem_ball_zero at hy,
end
lemma ball_finset_sup_eq_Inter (p : ι → seminorm 𝕜 E) (s : finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
ball (s.sup p) x r = ⋂ (i ∈ s), ball (p i) x r :=
begin
lift r to nnreal using hr.le,
simp_rw [ball, Inter_set_of, finset_sup_apply, nnreal.coe_lt_coe,
finset.sup_lt_iff (show ⊥ < r, from hr), ←nnreal.coe_lt_coe, subtype.coe_mk],
end
lemma ball_finset_sup (p : ι → seminorm 𝕜 E) (s : finset ι) (x : E) {r : ℝ} (hr : 0 < r) :
ball (s.sup p) x r = s.inf (λ i, ball (p i) x r) :=
begin
rw finset.inf_eq_infi,
exact ball_finset_sup_eq_Inter _ _ _ hr,
end
lemma ball_smul_ball (p : seminorm 𝕜 E) (r₁ r₂ : ℝ) :
metric.ball (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂) :=
begin
rw set.subset_def,
intros x hx,
rw set.mem_smul at hx,
rcases hx with ⟨a, y, ha, hy, hx⟩,
rw [←hx, mem_ball_zero, seminorm.smul],
exact mul_lt_mul'' (mem_ball_zero_iff.mp ha) (p.mem_ball_zero.mp hy) (norm_nonneg a) (p.nonneg y),
end
@[simp] lemma ball_eq_emptyset (p : seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ :=
begin
ext,
rw [seminorm.mem_ball, set.mem_empty_eq, iff_false, not_lt],
exact hr.trans (p.nonneg _),
end
end module
end add_comm_group
end semi_normed_ring
section normed_field
variables [normed_field 𝕜] [add_comm_group E] [module 𝕜 E] (p : seminorm 𝕜 E) {A B : set E}
{a : 𝕜} {r : ℝ} {x : E}
lemma smul_ball_zero {p : seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ∥k∥) :
k • p.ball 0 r = p.ball 0 (∥k∥ * r) :=
begin
ext,
rw [set.mem_smul_set, seminorm.mem_ball_zero],
split; intro h,
{ rcases h with ⟨y, hy, h⟩,
rw [←h, seminorm.smul],
rw seminorm.mem_ball_zero at hy,
exact (mul_lt_mul_left hk).mpr hy },
refine ⟨k⁻¹ • x, _, _⟩,
{ rw [seminorm.mem_ball_zero, seminorm.smul, norm_inv, ←(mul_lt_mul_left hk),
←mul_assoc, ←(div_eq_mul_inv ∥k∥ ∥k∥), div_self (ne_of_gt hk), one_mul],
exact h},
rw [←smul_assoc, smul_eq_mul, ←div_eq_mul_inv, div_self (norm_pos_iff.mp hk), one_smul],
end
lemma ball_zero_absorbs_ball_zero (p : seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) :=
begin
by_cases hr₂ : r₂ ≤ 0,
{ rw ball_eq_emptyset p hr₂, exact absorbs_empty },
rw [not_le] at hr₂,
rcases exists_between hr₁ with ⟨r, hr, hr'⟩,
refine ⟨r₂/r, div_pos hr₂ hr, _⟩,
simp_rw set.subset_def,
intros a ha x hx,
have ha' : 0 < ∥a∥ := lt_of_lt_of_le (div_pos hr₂ hr) ha,
rw [smul_ball_zero ha', p.mem_ball_zero],
rw p.mem_ball_zero at hx,
rw div_le_iff hr at ha,
exact hx.trans (lt_of_le_of_lt ha ((mul_lt_mul_left ha').mpr hr')),
end
/-- Seminorm-balls at the origin are absorbent. -/
protected lemma absorbent_ball_zero (hr : 0 < r) : absorbent 𝕜 (ball p (0 : E) r) :=
begin
rw absorbent_iff_nonneg_lt,
rintro x,
have hxr : 0 ≤ p x/r := div_nonneg (p.nonneg _) hr.le,
refine ⟨p x/r, hxr, λ a ha, _⟩,
have ha₀ : 0 < ∥a∥ := hxr.trans_lt ha,
refine ⟨a⁻¹ • x, _, smul_inv_smul₀ (norm_pos_iff.1 ha₀) x⟩,
rwa [mem_ball_zero, p.smul, norm_inv, inv_mul_lt_iff ha₀, ←div_lt_iff hr],
end
/-- Seminorm-balls containing the origin are absorbent. -/
protected lemma absorbent_ball (hpr : p x < r) : absorbent 𝕜 (ball p x r) :=
begin
refine (p.absorbent_ball_zero $ sub_pos.2 hpr).subset (λ y hy, _),
rw p.mem_ball_zero at hy,
exact p.mem_ball.2 ((p.sub_le _ _).trans_lt $ add_lt_of_lt_sub_right hy),
end
lemma symmetric_ball_zero (r : ℝ) (hx : x ∈ ball p 0 r) : -x ∈ ball p 0 r :=
balanced_ball_zero p r (-1) (by rw [norm_neg, norm_one]) ⟨x, hx, by rw [neg_smul, one_smul]⟩
@[simp]
lemma neg_ball (p : seminorm 𝕜 E) (r : ℝ) (x : E) :
-ball p x r = ball p (-x) r :=
by { ext, rw [mem_neg, mem_ball, mem_ball, ←neg_add', sub_neg_eq_add, p.neg], }
@[simp]
lemma smul_ball_preimage (p : seminorm 𝕜 E) (y : E) (r : ℝ) (a : 𝕜) (ha : a ≠ 0) :
((•) a) ⁻¹' p.ball y r = p.ball (a⁻¹ • y) (r / ∥a∥) :=
set.ext $ λ _, by rw [mem_preimage, mem_ball, mem_ball,
lt_div_iff (norm_pos_iff.mpr ha), mul_comm, ←p.smul, smul_sub, smul_inv_smul₀ ha]
end normed_field
section convex
variables [normed_field 𝕜] [add_comm_group E] [normed_space ℝ 𝕜] [module 𝕜 E]
section has_smul
variables [has_smul ℝ E] [is_scalar_tower ℝ 𝕜 E] (p : seminorm 𝕜 E)
/-- A seminorm is convex. Also see `convex_on_norm`. -/
protected lemma convex_on : convex_on ℝ univ p :=
begin
refine ⟨convex_univ, λ x y _ _ a b ha hb hab, _⟩,
calc p (a • x + b • y) ≤ p (a • x) + p (b • y) : p.add_le _ _
... = ∥a • (1 : 𝕜)∥ * p x + ∥b • (1 : 𝕜)∥ * p y
: by rw [←p.smul, ←p.smul, smul_one_smul, smul_one_smul]
... = a * p x + b * p y
: by rw [norm_smul, norm_smul, norm_one, mul_one, mul_one, real.norm_of_nonneg ha,
real.norm_of_nonneg hb],
end
end has_smul
section module
variables [module ℝ E] [is_scalar_tower ℝ 𝕜 E] (p : seminorm 𝕜 E) (x : E) (r : ℝ)
/-- Seminorm-balls are convex. -/
lemma convex_ball : convex ℝ (ball p x r) :=
begin
convert (p.convex_on.translate_left (-x)).convex_lt r,
ext y,
rw [preimage_univ, sep_univ, p.mem_ball, sub_eq_add_neg],
refl,
end
end module
end convex
end seminorm
/-! ### The norm as a seminorm -/
section norm_seminorm
variables (𝕜) (E) [normed_field 𝕜] [semi_normed_group E] [normed_space 𝕜 E] {r : ℝ}
/-- The norm of a seminormed group as an add_monoid seminorm. -/
def norm_add_group_seminorm : add_group_seminorm E :=
⟨norm, norm_zero, norm_nonneg, norm_add_le, norm_neg⟩
@[simp] lemma coe_norm_add_group_seminorm : ⇑(norm_add_group_seminorm E) = norm := rfl
/-- The norm of a seminormed group as a seminorm. -/
def norm_seminorm : seminorm 𝕜 E :=
{ smul' := norm_smul,
..(norm_add_group_seminorm E)}
@[simp] lemma coe_norm_seminorm : ⇑(norm_seminorm 𝕜 E) = norm := rfl
@[simp] lemma ball_norm_seminorm : (norm_seminorm 𝕜 E).ball = metric.ball :=
by { ext x r y, simp only [seminorm.mem_ball, metric.mem_ball, coe_norm_seminorm, dist_eq_norm] }
variables {𝕜 E} {x : E}
/-- Balls at the origin are absorbent. -/
lemma absorbent_ball_zero (hr : 0 < r) : absorbent 𝕜 (metric.ball (0 : E) r) :=
by { rw ←ball_norm_seminorm 𝕜, exact (norm_seminorm _ _).absorbent_ball_zero hr }
/-- Balls containing the origin are absorbent. -/
lemma absorbent_ball (hx : ∥x∥ < r) : absorbent 𝕜 (metric.ball x r) :=
by { rw ←ball_norm_seminorm 𝕜, exact (norm_seminorm _ _).absorbent_ball hx }
/-- Balls at the origin are balanced. -/
lemma balanced_ball_zero : balanced 𝕜 (metric.ball (0 : E) r) :=
by { rw ←ball_norm_seminorm 𝕜, exact (norm_seminorm _ _).balanced_ball_zero r }
end norm_seminorm
|
7852bd61fb40621c026ee6f47ad57c7a34b359e6 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/analysis/normed/group/completion.lean | 44087d460df0ce5d8da9f64541aab16fa347052c | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 1,449 | lean | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import analysis.normed.group.basic
import topology.algebra.group_completion
import topology.metric_space.completion
/-!
# Completion of a normed group
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
In this file we prove that the completion of a (semi)normed group is a normed group.
## Tags
normed group, completion
-/
noncomputable theory
namespace uniform_space
namespace completion
variables (E : Type*)
instance [uniform_space E] [has_norm E] :
has_norm (completion E) :=
{ norm := completion.extension has_norm.norm }
@[simp] lemma norm_coe {E} [seminormed_add_comm_group E] (x : E) :
‖(x : completion E)‖ = ‖x‖ :=
completion.extension_coe uniform_continuous_norm x
instance [seminormed_add_comm_group E] : normed_add_comm_group (completion E) :=
{ dist_eq :=
begin
intros x y,
apply completion.induction_on₂ x y; clear x y,
{ refine is_closed_eq (completion.uniform_continuous_extension₂ _).continuous _,
exact continuous.comp completion.continuous_extension continuous_sub },
{ intros x y,
rw [← completion.coe_sub, norm_coe, completion.dist_eq, dist_eq_norm] }
end,
.. completion.add_comm_group,
.. completion.metric_space }
end completion
end uniform_space
|
86b3ad1548fcfb3d9861c2975499f77aeb14f8d3 | 367134ba5a65885e863bdc4507601606690974c1 | /src/category_theory/limits/shapes/images.lean | 34a0ac8272bb2a84f8e45e052e4272fd5bd239d5 | [
"Apache-2.0"
] | permissive | kodyvajjha/mathlib | 9bead00e90f68269a313f45f5561766cfd8d5cad | b98af5dd79e13a38d84438b850a2e8858ec21284 | refs/heads/master | 1,624,350,366,310 | 1,615,563,062,000 | 1,615,563,062,000 | 162,666,963 | 0 | 0 | Apache-2.0 | 1,545,367,651,000 | 1,545,367,651,000 | null | UTF-8 | Lean | false | false | 25,343 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import category_theory.limits.shapes.equalizers
import category_theory.limits.shapes.pullbacks
import category_theory.limits.shapes.strong_epi
/-!
# Categorical images
We define the categorical image of `f` as a factorisation `f = e ≫ m` through a monomorphism `m`,
so that `m` factors through the `m'` in any other such factorisation.
## Main definitions
* A `mono_factorisation` is a factorisation `f = e ≫ m`, where `m` is a monomorphism
* `is_image F` means that a given mono factorisation `F` has the universal property of the image.
* `has_image f` means that we have chosen an image for the morphism `f : X ⟶ Y`.
* In this case, `image f` is the image object, `image.ι f : image f ⟶ Y` is the monomorphism `m`
of the factorisation and `factor_thru_image f : X ⟶ image f` is the morphism `e`.
* `has_images C` means that every morphism in `C` has an image.
* Let `f : X ⟶ Y` and `g : P ⟶ Q` be morphisms in `C`, which we will represent as objects of the
arrow category `arrow C`. Then `sq : f ⟶ g` is a commutative square in `C`. If `f` and `g` have
images, then `has_image_map sq` represents the fact that there is a morphism
`i : image f ⟶ image g` making the diagram
X ----→ image f ----→ Y
| | |
| | |
↓ ↓ ↓
P ----→ image g ----→ Q
commute, where the top row is the image factorisation of `f`, the bottom row is the image
factorisation of `g`, and the outer rectangle is the commutative square `sq`.
* If a category `has_images`, then `has_image_maps` means that every commutative square admits an
image map.
* If a category `has_images`, then `has_strong_epi_images` means that the morphism to the image is
always a strong epimorphism.
## Main statements
* When `C` has equalizers, the morphism `e` appearing in an image factorisation is an epimorphism.
* When `C` has strong epi images, then these images admit image maps.
## Future work
* TODO: coimages, and abelian categories.
* TODO: connect this with existing working in the group theory and ring theory libraries.
-/
noncomputable theory
universes v u
open category_theory
open category_theory.limits.walking_parallel_pair
namespace category_theory.limits
variables {C : Type u} [category.{v} C]
variables {X Y : C} (f : X ⟶ Y)
/-- A factorisation of a morphism `f = e ≫ m`, with `m` monic. -/
structure mono_factorisation (f : X ⟶ Y) :=
(I : C)
(m : I ⟶ Y)
[m_mono : mono m]
(e : X ⟶ I)
(fac' : e ≫ m = f . obviously)
restate_axiom mono_factorisation.fac'
attribute [simp, reassoc] mono_factorisation.fac
attribute [instance] mono_factorisation.m_mono
attribute [instance] mono_factorisation.m_mono
namespace mono_factorisation
/-- The obvious factorisation of a monomorphism through itself. -/
def self [mono f] : mono_factorisation f :=
{ I := X,
m := f,
e := 𝟙 X }
-- I'm not sure we really need this, but the linter says that an inhabited instance
-- ought to exist...
instance [mono f] : inhabited (mono_factorisation f) := ⟨self f⟩
/-- The morphism `m` in a factorisation `f = e ≫ m` through a monomorphism is uniquely
determined. -/
@[ext]
lemma ext
{F F' : mono_factorisation f} (hI : F.I = F'.I) (hm : F.m = (eq_to_hom hI) ≫ F'.m) : F = F' :=
begin
cases F, cases F',
cases hI,
simp at hm,
dsimp at F_fac' F'_fac',
congr,
{ assumption },
{ resetI, apply (cancel_mono F_m).1,
rw [F_fac', hm, F'_fac'], }
end
end mono_factorisation
variable {f}
/-- Data exhibiting that a given factorisation through a mono is initial. -/
structure is_image (F : mono_factorisation f) :=
(lift : Π (F' : mono_factorisation f), F.I ⟶ F'.I)
(lift_fac' : Π (F' : mono_factorisation f), lift F' ≫ F'.m = F.m . obviously)
restate_axiom is_image.lift_fac'
attribute [simp, reassoc] is_image.lift_fac
@[simp, reassoc] lemma is_image.fac_lift {F : mono_factorisation f} (hF : is_image F)
(F' : mono_factorisation f) : F.e ≫ hF.lift F' = F'.e :=
(cancel_mono F'.m).1 $ by simp
variable (f)
namespace is_image
/-- The trivial factorisation of a monomorphism satisfies the universal property. -/
@[simps]
def self [mono f] : is_image (mono_factorisation.self f) :=
{ lift := λ F', F'.e }
instance [mono f] : inhabited (is_image (mono_factorisation.self f)) :=
⟨self f⟩
variable {f}
/-- Two factorisations through monomorphisms satisfying the universal property
must factor through isomorphic objects. -/
-- TODO this is another good candidate for a future `unique_up_to_canonical_iso`.
@[simps]
def iso_ext {F F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F') : F.I ≅ F'.I :=
{ hom := hF.lift F',
inv := hF'.lift F,
hom_inv_id' := (cancel_mono F.m).1 (by simp),
inv_hom_id' := (cancel_mono F'.m).1 (by simp) }
variables {F F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F')
lemma iso_ext_hom_m : (iso_ext hF hF').hom ≫ F'.m = F.m := by simp
lemma iso_ext_inv_m : (iso_ext hF hF').inv ≫ F.m = F'.m := by simp
lemma e_iso_ext_hom : F.e ≫ (iso_ext hF hF').hom = F'.e := by simp
lemma e_iso_ext_inv : F'.e ≫ (iso_ext hF hF').inv = F.e := by simp
end is_image
/-- Data exhibiting that a morphism `f` has an image. -/
structure image_factorisation (f : X ⟶ Y) :=
(F : mono_factorisation f)
(is_image : is_image F)
instance inhabited_image_factorisation (f : X ⟶ Y) [mono f] : inhabited (image_factorisation f) :=
⟨⟨_, is_image.self f⟩⟩
/-- `has_image f` means that there exists an image factorisation of `f`. -/
class has_image (f : X ⟶ Y) : Prop :=
mk' :: (exists_image : nonempty (image_factorisation f))
lemma has_image.mk {f : X ⟶ Y} (F : image_factorisation f) : has_image f :=
⟨nonempty.intro F⟩
section
variable [has_image f]
/-- The chosen factorisation of `f` through a monomorphism. -/
def image.mono_factorisation : mono_factorisation f :=
(classical.choice (has_image.exists_image)).F
/-- The witness of the universal property for the chosen factorisation of `f` through
a monomorphism. -/
def image.is_image : is_image (image.mono_factorisation f) :=
(classical.choice (has_image.exists_image)).is_image
/-- The categorical image of a morphism. -/
def image : C := (image.mono_factorisation f).I
/-- The inclusion of the image of a morphism into the target. -/
def image.ι : image f ⟶ Y := (image.mono_factorisation f).m
@[simp] lemma image.as_ι : (image.mono_factorisation f).m = image.ι f := rfl
instance : mono (image.ι f) := (image.mono_factorisation f).m_mono
/-- The map from the source to the image of a morphism. -/
def factor_thru_image : X ⟶ image f := (image.mono_factorisation f).e
/-- Rewrite in terms of the `factor_thru_image` interface. -/
@[simp]
lemma as_factor_thru_image : (image.mono_factorisation f).e = factor_thru_image f := rfl
@[simp, reassoc]
lemma image.fac : factor_thru_image f ≫ image.ι f = f := (image.mono_factorisation f).fac'
variable {f}
/-- Any other factorisation of the morphism `f` through a monomorphism receives a map from the
image. -/
def image.lift (F' : mono_factorisation f) : image f ⟶ F'.I := (image.is_image f).lift F'
@[simp, reassoc]
lemma image.lift_fac (F' : mono_factorisation f) : image.lift F' ≫ F'.m = image.ι f :=
(image.is_image f).lift_fac' F'
@[simp, reassoc]
lemma image.fac_lift (F' : mono_factorisation f) : factor_thru_image f ≫ image.lift F' = F'.e :=
(image.is_image f).fac_lift F'
@[simp, reassoc]
lemma is_image.lift_ι {F : mono_factorisation f} (hF : is_image F) :
hF.lift (image.mono_factorisation f) ≫ image.ι f = F.m :=
hF.lift_fac _
-- TODO we could put a category structure on `mono_factorisation f`,
-- with the morphisms being `g : I ⟶ I'` commuting with the `m`s
-- (they then automatically commute with the `e`s)
-- and show that an `image_of f` gives an initial object there
-- (uniqueness of the lift comes for free).
instance lift_mono (F' : mono_factorisation f) : mono (image.lift F') :=
by { apply mono_of_mono _ F'.m, simpa using mono_factorisation.m_mono _ }
lemma has_image.uniq
(F' : mono_factorisation f) (l : image f ⟶ F'.I) (w : l ≫ F'.m = image.ι f) :
l = image.lift F' :=
(cancel_mono F'.m).1 (by simp [w])
end
section
variables (C)
/-- `has_images` represents a choice of image for every morphism -/
class has_images :=
(has_image : Π {X Y : C} (f : X ⟶ Y), has_image f)
attribute [instance, priority 100] has_images.has_image
end
section
variables (f) [has_image f]
/-- The image of a monomorphism is isomorphic to the source. -/
def image_mono_iso_source [mono f] : image f ≅ X :=
is_image.iso_ext (image.is_image f) (is_image.self f)
@[simp, reassoc]
lemma image_mono_iso_source_inv_ι [mono f] : (image_mono_iso_source f).inv ≫ image.ι f = f :=
by simp [image_mono_iso_source]
@[simp, reassoc]
lemma image_mono_iso_source_hom_self [mono f] : (image_mono_iso_source f).hom ≫ f = image.ι f :=
begin
conv { to_lhs, congr, skip, rw ←image_mono_iso_source_inv_ι f, },
rw [←category.assoc, iso.hom_inv_id, category.id_comp],
end
-- This is the proof that `factor_thru_image f` is an epimorphism
-- from https://en.wikipedia.org/wiki/Image_%28category_theory%29, which is in turn taken from:
-- Mitchell, Barry (1965), Theory of categories, MR 0202787, p.12, Proposition 10.1
@[ext]
lemma image.ext {W : C} {g h : image f ⟶ W} [has_limit (parallel_pair g h)]
(w : factor_thru_image f ≫ g = factor_thru_image f ≫ h) :
g = h :=
begin
let q := equalizer.ι g h,
let e' := equalizer.lift _ w,
let F' : mono_factorisation f :=
{ I := equalizer g h,
m := q ≫ image.ι f,
m_mono := by apply mono_comp,
e := e' },
let v := image.lift F',
have t₀ : v ≫ q ≫ image.ι f = image.ι f := image.lift_fac F',
have t : v ≫ q = 𝟙 (image f) :=
(cancel_mono_id (image.ι f)).1 (by { convert t₀ using 1, rw category.assoc }),
-- The proof from wikipedia next proves `q ≫ v = 𝟙 _`,
-- and concludes that `equalizer g h ≅ image f`,
-- but this isn't necessary.
calc g = 𝟙 (image f) ≫ g : by rw [category.id_comp]
... = v ≫ q ≫ g : by rw [←t, category.assoc]
... = v ≫ q ≫ h : by rw [equalizer.condition g h]
... = 𝟙 (image f) ≫ h : by rw [←category.assoc, t]
... = h : by rw [category.id_comp]
end
instance [Π {Z : C} (g h : image f ⟶ Z), has_limit (parallel_pair g h)] :
epi (factor_thru_image f) :=
⟨λ Z g h w, image.ext f w⟩
lemma epi_image_of_epi {X Y : C} (f : X ⟶ Y) [has_image f] [E : epi f] : epi (image.ι f) :=
begin
rw ←image.fac f at E,
resetI,
exact epi_of_epi (factor_thru_image f) (image.ι f),
end
lemma epi_of_epi_image {X Y : C} (f : X ⟶ Y) [has_image f]
[epi (image.ι f)] [epi (factor_thru_image f)] : epi f :=
by { rw [←image.fac f], apply epi_comp, }
end
section
variables {f} {f' : X ⟶ Y} [has_image f] [has_image f']
/--
An equation between morphisms gives a comparison map between the images
(which momentarily we prove is an iso).
-/
def image.eq_to_hom (h : f = f') : image f ⟶ image f' :=
image.lift
{ I := image f',
m := image.ι f',
e := factor_thru_image f', }.
instance (h : f = f') : is_iso (image.eq_to_hom h) :=
{ inv := image.eq_to_hom h.symm,
hom_inv_id' := (cancel_mono (image.ι f)).1 (by simp [image.eq_to_hom]),
inv_hom_id' := (cancel_mono (image.ι f')).1 (by simp [image.eq_to_hom]), }
/-- An equation between morphisms gives an isomorphism between the images. -/
def image.eq_to_iso (h : f = f') : image f ≅ image f' := as_iso (image.eq_to_hom h)
/--
As long as the category has equalizers,
the image inclusion maps commute with `image.eq_to_iso`.
-/
lemma image.eq_fac [has_equalizers C] (h : f = f') :
image.ι f = (image.eq_to_iso h).hom ≫ image.ι f' :=
by { ext, simp [image.eq_to_iso, image.eq_to_hom], }
end
section
variables {Z : C} (g : Y ⟶ Z)
/-- The comparison map `image (f ≫ g) ⟶ image g`. -/
def image.pre_comp [has_image g] [has_image (f ≫ g)] : image (f ≫ g) ⟶ image g :=
image.lift
{ I := image g,
m := image.ι g,
e := f ≫ factor_thru_image g }
@[simp, reassoc]
lemma image.factor_thru_image_pre_comp [has_image g] [has_image (f ≫ g)] :
factor_thru_image (f ≫ g) ≫ image.pre_comp f g = f ≫ factor_thru_image g :=
by simp [image.pre_comp]
/--
The two step comparison map
`image (f ≫ (g ≫ h)) ⟶ image (g ≫ h) ⟶ image h`
agrees with the one step comparison map
`image (f ≫ (g ≫ h)) ≅ image ((f ≫ g) ≫ h) ⟶ image h`.
-/
lemma image.pre_comp_comp {W : C} (h : Z ⟶ W)
[has_image (g ≫ h)] [has_image (f ≫ g ≫ h)]
[has_image h] [has_image ((f ≫ g) ≫ h)] :
image.pre_comp f (g ≫ h) ≫ image.pre_comp g h =
image.eq_to_hom (category.assoc f g h).symm ≫ (image.pre_comp (f ≫ g) h) :=
begin
apply (cancel_mono (image.ι h)).1,
simp [image.pre_comp, image.eq_to_hom],
end
variables [has_equalizers C]
/--
`image.pre_comp f g` is an isomorphism when `f` is an isomorphism
(we need `C` to have equalizers to prove this).
-/
instance image.is_iso_precomp_iso (f : X ≅ Y) [has_image g] [has_image (f.hom ≫ g)] :
is_iso (image.pre_comp f.hom g) :=
{ inv := image.lift
{ I := image (f.hom ≫ g),
m := image.ι (f.hom ≫ g),
e := f.inv ≫ factor_thru_image (f.hom ≫ g) },
hom_inv_id' := by { ext, simp [image.pre_comp], },
inv_hom_id' := by { ext, simp [image.pre_comp], }, }
-- Note that in general we don't have the other comparison map you might expect
-- `image f ⟶ image (f ≫ g)`.
/-- Postcomposing by an isomorphism induces an isomorphism on the image. -/
def image.post_comp_is_iso [is_iso g] [has_image f] [has_image (f ≫ g)] :
image f ≅ image (f ≫ g) :=
{ hom := image.lift
{ I := _,
m := image.ι (f ≫ g) ≫ inv g,
m_mono := mono_comp _ _,
e := factor_thru_image (f ≫ g) },
inv := image.lift
{ I := _,
m := image.ι f ≫ g,
m_mono := mono_comp _ _,
e := factor_thru_image f } }
@[simp, reassoc] lemma image.post_comp_is_iso_hom_comp_image_ι [is_iso g] [has_image f]
[has_image (f ≫ g)] : (image.post_comp_is_iso f g).hom ≫ image.ι (f ≫ g) = image.ι f ≫ g :=
by { ext, simp [image.post_comp_is_iso] }
@[simp, reassoc] lemma image.post_comp_is_iso_inv_comp_image_ι [is_iso g] [has_image f]
[has_image (f ≫ g)] : (image.post_comp_is_iso f g).inv ≫ image.ι f = image.ι (f ≫ g) ≫ inv g :=
by { ext, simp [image.post_comp_is_iso] }
end
end category_theory.limits
namespace category_theory.limits
variables {C : Type u} [category.{v} C]
section
instance {X Y : C} (f : X ⟶ Y) [has_image f] : has_image (arrow.mk f).hom :=
show has_image f, by apply_instance
end
section has_image_map
/-- An image map is a morphism `image f → image g` fitting into a commutative square and satisfying
the obvious commutativity conditions. -/
structure image_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g) :=
(map : image f.hom ⟶ image g.hom)
(map_ι' : map ≫ image.ι g.hom = image.ι f.hom ≫ sq.right . obviously)
instance inhabited_image_map {f : arrow C} [has_image f.hom] : inhabited (image_map (𝟙 f)) :=
⟨⟨𝟙 _, by tidy⟩⟩
restate_axiom image_map.map_ι'
attribute [simp, reassoc] image_map.map_ι
@[simp, reassoc]
lemma image_map.factor_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)
(m : image_map sq) :
factor_thru_image f.hom ≫ m.map = sq.left ≫ factor_thru_image g.hom :=
(cancel_mono (image.ι g.hom)).1 $ by simp
/-- To give an image map for a commutative square with `f` at the top and `g` at the bottom, it
suffices to give a map between any mono factorisation of `f` and any image factorisation of
`g`. -/
def image_map.transport {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)
(F : mono_factorisation f.hom) {F' : mono_factorisation g.hom} (hF' : is_image F')
{map : F.I ⟶ F'.I} (map_ι : map ≫ F'.m = F.m ≫ sq.right) : image_map sq :=
{ map := image.lift F ≫ map ≫ hF'.lift (image.mono_factorisation g.hom),
map_ι' := by simp [map_ι] }
/-- `has_image_map sq` means that there is an `image_map` for the square `sq`. -/
class has_image_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g) : Prop :=
mk' :: (has_image_map : nonempty (image_map sq))
lemma has_image_map.mk {f g : arrow C} [has_image f.hom] [has_image g.hom] {sq : f ⟶ g}
(m : image_map sq) : has_image_map sq :=
⟨nonempty.intro m⟩
lemma has_image_map.transport {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)
(F : mono_factorisation f.hom) {F' : mono_factorisation g.hom} (hF' : is_image F')
(map : F.I ⟶ F'.I) (map_ι : map ≫ F'.m = F.m ≫ sq.right) : has_image_map sq :=
has_image_map.mk $ image_map.transport sq F hF' map_ι
/-- Obtain an `image_map` from a `has_image_map` instance. -/
def has_image_map.image_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)
[has_image_map sq] : image_map sq :=
classical.choice $ @has_image_map.has_image_map _ _ _ _ _ _ sq _
variables {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)
section
local attribute [ext] image_map
instance : subsingleton (image_map sq) :=
subsingleton.intro $ λ a b, image_map.ext a b $ (cancel_mono (image.ι g.hom)).1 $
by simp only [image_map.map_ι]
end
variable [has_image_map sq]
/-- The map on images induced by a commutative square. -/
abbreviation image.map : image f.hom ⟶ image g.hom :=
(has_image_map.image_map sq).map
lemma image.factor_map :
factor_thru_image f.hom ≫ image.map sq = sq.left ≫ factor_thru_image g.hom :=
by simp
lemma image.map_ι : image.map sq ≫ image.ι g.hom = image.ι f.hom ≫ sq.right :=
by simp
lemma image.map_hom_mk'_ι {X Y P Q : C} {k : X ⟶ Y} [has_image k] {l : P ⟶ Q} [has_image l]
{m : X ⟶ P} {n : Y ⟶ Q} (w : m ≫ l = k ≫ n) [has_image_map (arrow.hom_mk' w)] :
image.map (arrow.hom_mk' w) ≫ image.ι l = image.ι k ≫ n :=
image.map_ι _
section
variables {h : arrow C} [has_image h.hom] (sq' : g ⟶ h)
variables [has_image_map sq']
/-- Image maps for composable commutative squares induce an image map in the composite square. -/
def image_map_comp : image_map (sq ≫ sq') :=
{ map := image.map sq ≫ image.map sq' }
@[simp]
lemma image.map_comp [has_image_map (sq ≫ sq')] :
image.map (sq ≫ sq') = image.map sq ≫ image.map sq' :=
show (has_image_map.image_map (sq ≫ sq')).map = (image_map_comp sq sq').map, by congr
end
section
variables (f)
/-- The identity `image f ⟶ image f` fits into the commutative square represented by the identity
morphism `𝟙 f` in the arrow category. -/
def image_map_id : image_map (𝟙 f) :=
{ map := 𝟙 (image f.hom) }
@[simp]
lemma image.map_id [has_image_map (𝟙 f)] : image.map (𝟙 f) = 𝟙 (image f.hom) :=
show (has_image_map.image_map (𝟙 f)).map = (image_map_id f).map, by congr
end
end has_image_map
section
variables (C) [has_images C]
/-- If a category `has_image_maps`, then all commutative squares induce morphisms on images. -/
class has_image_maps :=
(has_image_map : Π {f g : arrow C} (st : f ⟶ g), has_image_map st)
attribute [instance, priority 100] has_image_maps.has_image_map
end
section has_image_maps
variables [has_images C] [has_image_maps C]
/-- The functor from the arrow category of `C` to `C` itself that maps a morphism to its image
and a commutative square to the induced morphism on images. -/
@[simps]
def im : arrow C ⥤ C :=
{ obj := λ f, image f.hom,
map := λ _ _ st, image.map st }
end has_image_maps
section strong_epi_mono_factorisation
/-- A strong epi-mono factorisation is a decomposition `f = e ≫ m` with `e` a strong epimorphism
and `m` a monomorphism. -/
structure strong_epi_mono_factorisation {X Y : C} (f : X ⟶ Y) extends mono_factorisation f :=
[e_strong_epi : strong_epi e]
attribute [instance] strong_epi_mono_factorisation.e_strong_epi
/-- Satisfying the inhabited linter -/
instance strong_epi_mono_factorisation_inhabited {X Y : C} (f : X ⟶ Y) [strong_epi f] :
inhabited (strong_epi_mono_factorisation f) :=
⟨⟨⟨Y, 𝟙 Y, f, by simp⟩⟩⟩
/-- A mono factorisation coming from a strong epi-mono factorisation always has the universal
property of the image. -/
def strong_epi_mono_factorisation.to_mono_is_image {X Y : C} {f : X ⟶ Y}
(F : strong_epi_mono_factorisation f) : is_image F.to_mono_factorisation :=
{ lift := λ G, arrow.lift $ arrow.hom_mk' $
show G.e ≫ G.m = F.e ≫ F.m, by rw [F.to_mono_factorisation.fac, G.fac] }
variable (C)
/-- A category has strong epi-mono factorisations if every morphism admits a strong epi-mono
factorisation. -/
class has_strong_epi_mono_factorisations : Prop :=
mk' :: (has_fac : Π {X Y : C} (f : X ⟶ Y), nonempty (strong_epi_mono_factorisation f))
variable {C}
lemma has_strong_epi_mono_factorisations.mk
(d : Π {X Y : C} (f : X ⟶ Y), strong_epi_mono_factorisation f) :
has_strong_epi_mono_factorisations C :=
⟨λ X Y f, nonempty.intro $ d f⟩
@[priority 100]
instance has_images_of_has_strong_epi_mono_factorisations
[has_strong_epi_mono_factorisations C] : has_images C :=
{ has_image := λ X Y f,
let F' := classical.choice (has_strong_epi_mono_factorisations.has_fac f) in
has_image.mk { F := F'.to_mono_factorisation,
is_image := F'.to_mono_is_image } }
end strong_epi_mono_factorisation
section has_strong_epi_images
variables (C) [has_images C]
/-- A category has strong epi images if it has all images and `factor_thru_image f` is a strong
epimorphism for all `f`. -/
class has_strong_epi_images : Prop :=
(strong_factor_thru_image : Π {X Y : C} (f : X ⟶ Y), strong_epi (factor_thru_image f))
attribute [instance] has_strong_epi_images.strong_factor_thru_image
end has_strong_epi_images
section has_strong_epi_images
/-- If there is a single strong epi-mono factorisation of `f`, then every image factorisation is a
strong epi-mono factorisation. -/
lemma strong_epi_of_strong_epi_mono_factorisation {X Y : C} {f : X ⟶ Y}
(F : strong_epi_mono_factorisation f) {F' : mono_factorisation f} (hF' : is_image F') :
strong_epi F'.e :=
by { rw ←is_image.e_iso_ext_hom F.to_mono_is_image hF', apply strong_epi_comp }
lemma strong_epi_factor_thru_image_of_strong_epi_mono_factorisation {X Y : C} {f : X ⟶ Y}
[has_image f] (F : strong_epi_mono_factorisation f) : strong_epi (factor_thru_image f) :=
strong_epi_of_strong_epi_mono_factorisation F $ image.is_image f
/-- If we constructed our images from strong epi-mono factorisations, then these images are
strong epi images. -/
@[priority 100]
instance has_strong_epi_images_of_has_strong_epi_mono_factorisations
[has_strong_epi_mono_factorisations C] : has_strong_epi_images C :=
{ strong_factor_thru_image := λ X Y f,
strong_epi_factor_thru_image_of_strong_epi_mono_factorisation $
classical.choice $ has_strong_epi_mono_factorisations.has_fac f }
end has_strong_epi_images
section has_strong_epi_images
variables [has_images C]
/-- A category with strong epi images has image maps. -/
@[priority 100]
instance has_image_maps_of_has_strong_epi_images [has_strong_epi_images C] :
has_image_maps C :=
{ has_image_map := λ f g st, has_image_map.mk
{ map := arrow.lift $ arrow.hom_mk' $ show (st.left ≫ factor_thru_image g.hom) ≫ image.ι g.hom =
factor_thru_image f.hom ≫ (image.ι f.hom ≫ st.right), by simp } }
/-- If a category has images, equalizers and pullbacks, then images are automatically strong epi
images. -/
@[priority 100]
instance has_strong_epi_images_of_has_pullbacks_of_has_equalizers [has_pullbacks C]
[has_equalizers C] : has_strong_epi_images C :=
{ strong_factor_thru_image := λ X Y f,
{ epi := by apply_instance,
has_lift := λ A B x y h h_mono w, arrow.has_lift.mk
{ lift := image.lift
{ I := pullback h y,
m := pullback.snd ≫ image.ι f,
m_mono := by exactI mono_comp _ _,
e := pullback.lift _ _ w } ≫ pullback.fst,
fac_left := by simp,
fac_right := by tidy } } }
end has_strong_epi_images
variables [has_strong_epi_mono_factorisations.{v} C]
variables {X Y : C} {f : X ⟶ Y}
/--
If `C` has strong epi mono factorisations, then the image is unique up to isomorphism, in that if
`f` factors as a strong epi followed by a mono, this factorisation is essentially the image
factorisation.
-/
def image.iso_strong_epi_mono {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f) [strong_epi e]
[mono m] :
I' ≅ image f :=
is_image.iso_ext {strong_epi_mono_factorisation . I := I', m := m, e := e}.to_mono_is_image $
image.is_image f
@[simp]
lemma image.iso_strong_epi_mono_hom_comp_ι {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f)
[strong_epi e] [mono m] :
(image.iso_strong_epi_mono e m comm).hom ≫ image.ι f = m :=
is_image.lift_fac _ _
@[simp]
lemma image.iso_strong_epi_mono_inv_comp_mono {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f)
[strong_epi e] [mono m] :
(image.iso_strong_epi_mono e m comm).inv ≫ m = image.ι f :=
image.lift_fac _
end category_theory.limits
|
fae66139cddd276f049897d6231af47af983b9a7 | dbd2d342e674b01c9097c761d49e297e3ce18292 | /src/check.lean | 62b1133c4c5d39ac6bb8f7f234a1ccd509e82d41 | [] | no_license | cipher1024/lean-depot | b4c3b096fb2d9b669d171fe9b2395f0822821dbe | 4b2b879f429eac70dc4e9fa210a3d92f1db2b7f5 | refs/heads/master | 1,599,422,553,618 | 1,574,715,661,000 | 1,574,715,661,000 | 220,534,909 | 1 | 0 | null | 1,574,715,992,000 | 1,573,241,547,000 | Lean | UTF-8 | Lean | false | false | 21,095 | lean | import system.io leanpkg.git leanpkg.main
def io.cmd' (args : io.process.spawn_args) : io unit :=
leanpkg.exec_cmd args
-- do child ← io.proc.spawn args,
-- exitv ← io.proc.wait child,
-- when (exitv ≠ 0) $ io.fail $ "process exited with status " ++ repr exitv,
-- return ()
open io io.proc io.fs
def git_checkout_hash (hash : string) : io unit :=
io.cmd' {cmd := "git", args := ["checkout","-f","--detach",hash]}
def git_pull_master : io unit :=
io.cmd' {cmd := "git", args := ["checkout","pull","-f","origin","master"]}
def git_checkout_tag (tag : string) : io unit :=
io.cmd' {cmd := "git", args := ["checkout",sformat!"tags/{tag}","-f","--detach"]}
def git_fetch : io unit :=
io.cmd' { cmd := "git", args := ["fetch"] }
def git_remote : io unit :=
io.cmd' { cmd := "git", args := ["remote","-v"] }
def string.is_prefix_of_aux : string.iterator → string.iterator → ℕ → bool
| i j 0 := tt
| i j (nat.succ n) :=
i.curr = j.curr ∧ string.is_prefix_of_aux i.next j.next n
def string.is_prefix_of (x y : string) : bool :=
x.length ≤ y.length ∧ string.is_prefix_of_aux x.mk_iterator y.mk_iterator x.length
def string.drop (x : string) (n : ℕ) : string :=
(x.mk_iterator.nextn n).next_to_string
def string.replace_prefix (x y z : string) : string :=
if y.is_prefix_of x
then z ++ x.drop y.length
else x
def add_token (repo : string) : io string :=
do some s ← io.env.get "GITHUB_TOKEN" | pure repo,
return $ (repo.replace_prefix "https://github.com" (sformat!"https://{s}@github.com"))
.replace_prefix "https://www.github.com" (sformat!"https://{s}@www.github.com")
def git_clone (repo : string) : option string → io unit
| none :=
do put_str_ln sformat!"clone {repo}",
io.cmd' { cmd := "git", args := ["clone",repo] }
| (some dir) :=
do put_str_ln sformat!"clone {repo}",
io.cmd' { cmd := "git", args := ["clone",repo,dir] }
def list_dir (dir : string) : io (list string) :=
do ls ← (list.filter (≠ "") ∘ string.split (= '\n')) <$> io.cmd { cmd := "ls", args := [dir] },
return $ ls.map $ λ fn, sformat!"{dir}/{fn}"
def read_lines (fn : string) : io (list string) :=
do h ← mk_file_handle fn io.mode.read ff,
r ← iterate [] $ λ r,
do { done ← is_eof h,
if done
then return none
else do
c ← get_line h,
return $ some (c.to_string :: r) },
return r.reverse
def list.take_while {α} (p : α → Prop) [decidable_pred p] : list α → list α
| [] := []
| (x :: xs) :=
if p x then x :: list.take_while xs
else []
def trim : string → string :=
list.as_string ∘ list.take_while (not ∘ char.is_whitespace) ∘ list.drop_while char.is_whitespace ∘ string.to_list
def split_path (p : string) : list string :=
(p.split (= '/')).filter (λ s, ¬ s.is_empty)
-- #eval -- split_path "a/b"
-- do xs <- read_lines "pkgs/list",
-- let xs := (xs.filter (≠ "")).map trim,
-- print $ xs.map $ list.ilast ∘ split_path
-- open native
-- def dep_versions' (l : leanpkg.source) (m : rbmap string (list string)) : rbmap string (list string) :=
open leanpkg.source
@[reducible]
def vmap := rbmap string $ rbmap (string × string) unit
-- #exit
variable conv_url : rbmap string string
def canonicalize_url (s : string) : string :=
let xs := list.reverse (s.to_list.reverse.drop_while (= '/')),
ys := if ".git".to_list.is_suffix_of xs
then list.as_string $ xs.take (xs.length - 4)
else list.as_string xs in
(conv_url.find ys).get_or_else ys
def dep_versions (l : leanpkg.dependency) (m : vmap) : vmap :=
match l.src with
| (path p) := m
| (git url sha br) :=
let url := canonicalize_url conv_url url,
xs := (m.find url).get_or_else (mk_rbmap _ _) in
m.insert url (xs.insert (sha, br.get_or_else "") ())
end
instance {α β} [has_lt α] [has_to_string α] [has_to_string β] : has_to_string (rbmap α β) :=
⟨ to_string ∘ rbtree.to_list ⟩
def mathlib_sha := ("https://github.com/leanprover-community/mathlib",
["019b2364cadb1dd8b30c9f13c602d61c19d3b6ea",
"14e10bbda42f928bdc6f9664e8d9826e598a68fe",
"257fd84fe98927ff4a5ffe044f68c4e9d235cc75",
"2c84af163dbc83c6eb6a94a03d5c25f25b6b8623",
"410ae5d9ec48843f0c2bf6787faafaa83c766623",
"5329bf3af9ecea916d32362dc28fd391075a63a1",
"58909bd424209739a2214961eefaa012fb8a18d2",
"9b0fd36245ae958aeda9e10d79650801f63cd3ae",
"b1920f599efe036cefbd02916aba9f687793c8c8",
"d60d1616958787ccb9842dc943534f90ea0bab64"])
-- #eval mathlib_sha.2.length
structure package :=
(config_file dir : string)
(url : string)
(alternate_urls : list string)
(commit : option string)
(update : bool)
(lean_support : list string)
(description : string)
(author : option string )
(left_over : list (string × toml.value))
-- (pkg_left_over : list (string × toml.value))
instance : has_to_string package :=
{ to_string := λ p,
sformat!"{{ config_file := {repr p.config_file},
url := {repr p.url},
alternate_urls := {repr p.alternate_urls},
dir := {repr p.dir},
commit := {repr p.commit},
lean_support := {repr p.lean_support},
update := {p.update},
description := {p.description},
author := {p.author} }" }
structure app_args :=
(tag : string)
(lean_version : string)
(dir : string)
-- (mathlib : bool)
def usage {α} : io α :=
io.fail "usage: check (--makefile LEAN_VERSION TAG | --update)"
-- def snapshots' (args : app_args) (ps : list package) (ms : list leanpkg.manifest) :
-- (string × list (string × string × leanpkg.manifest)) :=
-- (args.tag, ps.zip_with (λ p m, (p.url, p.dir, m)) ms)
-- list.map (prod.map id prod.fst) <$> (m.to_list.mmap $ λ ⟨p,v⟩, (prod.mk p ∘ prod.fst) <$> v.to_list)
-- def snapshots (m : vmap) : list (ℕ × list (string × string)) :=
-- list.enum $
-- list.map (prod.map id prod.fst) <$> (m.to_list.mmap $ λ ⟨p,v⟩, (prod.mk p ∘ prod.fst) <$> v.to_list)
def dir_part : list string → string
| [] := ""
| [x] := x
| [x,y] := sformat!"{x}/{y}"
| (x :: xs) := dir_part xs
-- def list_packages : io (list package) :=
-- do xs <- read_lines "pkgs/list",
-- let xs := ((xs.filter (≠ "")).map trim).filter (λ s, ¬ string.is_prefix_of "--" s),
-- return $ xs.map $ λ d, ⟨d,dir_part (split_path d),none⟩
def split_ext : string → list string := string.split (= '.')
def package.name (p : package) : string :=
".".intercalate (split_ext (split_path p.config_file).ilast).init
def parse_file (fn : string) : io toml.value :=
do cnts ← io.fs.read_file fn,
match parser.run toml.File cnts with
| sum.inl err :=
io.fail $ "toml parse error in " ++ fn ++ "\n\n" ++ err
| sum.inr res := return res
end
def split_off : toml.value → string → option (toml.value × list (string × toml.value))
| (toml.value.table cs) k :=
match cs.partition (λ c : string × toml.value, c.fst = k) with
| ([],_) := none
| ((k,v) ::_, xs) := some (v,xs)
end
| _ _ := none
class is_toml_value (α : Type*) :=
(read_toml : toml.value → option α)
export is_toml_value (read_toml)
instance : is_toml_value string :=
{ read_toml := λ v, match v with
| (toml.value.str s) := some s
| _ := none
end }
instance : is_toml_value bool :=
{ read_toml := λ v, match v with
| (toml.value.bool s) := some s
| _ := none
end }
instance {α} [is_toml_value α] : is_toml_value (list α) :=
{ read_toml := λ v, match v with
| (toml.value.array s) := s.mmap (read_toml _)
| _ := none
end }
def lookup_toml (α) [is_toml_value α] (v : toml.value) (key : string) : option α :=
v.lookup key >>= read_toml _
def lookup_optional_toml (α) [is_toml_value α] (v : toml.value) (key : string) : option (option α) :=
match v.lookup key with
| (some v) := some <$> read_toml _ v
| none := some none
end
def split_off' (α) [is_toml_value α] (v : toml.value) (key : string) : option (α × toml.value) :=
split_off v key >>= λ ⟨v,xs⟩,
prod.mk <$> read_toml _ v <*> pure (toml.value.table xs)
def split_off_optional (α) [is_toml_value α] (v : toml.value) (key : string) : option (option α × toml.value) :=
match split_off v key with
| (some (v,xs)) := prod.mk <$> (some <$> read_toml _ v) <*> pure (toml.value.table xs)
| none := some (none,v)
end
-- #exit
def package.to_toml (p : package) : toml.value :=
toml.value.table $
let sha := match p.commit with
| some sha := [("commit",toml.value.str sha)]
| none := []
end,
author := match p.author with
| some sha := [("author",toml.value.str sha)]
| none := []
end,
alternate_urls := match p.alternate_urls with
| [] := []
| xs := [("alternate_urls",toml.value.array $ toml.value.str <$> xs)]
end in
("package", toml.value.table $
[ ("url", toml.value.str p.url),
("description", toml.value.str p.description),
("auto_update", toml.value.bool p.update),
("lean_support", toml.value.array $ toml.value.str <$> p.lean_support) ]
++ alternate_urls ++ sha ++ author ) :: p.left_over
def package.from_toml (fn : string) (t : toml.value) : option package :=
do (pkg,xs) ← split_off t "package",
url ← lookup_toml string pkg "url",
alt_url ← lookup_optional_toml (list string) pkg "alternate_urls" ,
desc ← lookup_toml string pkg "description",
update ← lookup_toml bool pkg "auto_update",
lean_support ← lookup_toml (list string) pkg "lean_support",
commit ← lookup_optional_toml string pkg "commit",
author ← lookup_optional_toml string pkg "author",
pure { config_file := fn, url := url,
alternate_urls := alt_url.get_or_else [],
update := update,
commit := commit, left_over := xs,
lean_support := lean_support,
author := author,
description := desc,
dir := dir_part (split_path url) }
def parse_package_file (fn : string) : io package :=
do val ← parse_file fn,
some pkg ← pure $ package.from_toml fn val | io.fail sformat!"invalid file format for {fn}",
pure pkg
def rewrite_package_file (fn : package) : io unit :=
leanpkg.write_file fn.config_file (repr fn.to_toml)
-- #eval parse_package_file "pkgs/mathlib.toml" >>= print
def list_packages' (v : option string := none) : io (list package) :=
do xs <- list_dir "pkgs",
let xs := xs.filter $ λ fn, (split_ext fn).ilast = "toml",
xs ← xs.mmap parse_package_file,
match v with
| some v := return $ xs.filter $ λ p, v ∈ p.lean_support
| none := pure xs
end
def with_cwd {α} (d : string) (m : io α) : io α :=
do d' ← env.get_cwd,
mkdir d tt,
finally (env.set_cwd d *> m) (env.set_cwd d')
def mk_local (d : leanpkg.dependency) : leanpkg.dependency :=
{ src := match d.src with
| (path p) := path p
| (git u c b) := path $ "../../" ++ dir_part (split_path $ canonicalize_url conv_url u)
end
.. d }
-- #exit
-- (xs : list package) (ys : list leanpkg.manifest) :
-- def checkout_snapshot :
-- string × list (string × string × leanpkg.manifest) →
-- io (string × list string)
-- | (tag,m) :=
-- do { let sd := sformat!"build/{tag}",
-- mkdir sd tt,
-- put_str_ln sd,
-- -- let m := rbmap.from_list m,
-- xs ← m.mmap $ λ ⟨url,dir,(m : leanpkg.manifest)⟩,
-- with_cwd sd $
-- do { ex ← dir_exists dir,
-- put_str_ln dir,
-- when (¬ ex) $ git_clone url dir,
-- git_fetch,
-- env.set_cwd dir,
-- catch (git_checkout_tag tag) (λ _ : io.error, (return () : io unit)),
-- -- put_str_ln r.url,
-- -- env.get_cwd >>= put_str_ln,
-- leanpkg.write_manifest
-- { dependencies := m.dependencies.map mk_local,
-- lean_version := "3.4.2", .. m },
-- return sformat!"{sd}/{dir}"},
-- return (sd, xs) }
-- #exit
def update_package_desc :
list (package × leanpkg.manifest) → io unit | ps :=
ps.mmap' $ λ p, rewrite_package_file p.1
def checkout_snapshot' (args : app_args) :
list (package × leanpkg.manifest) →
io (string)
| ps :=
do { let sd := sformat!"{args.dir}/build/{args.tag}",
mkdir sd tt,
put_str_ln sd,
-- let m := rbmap.from_list m,
ps.mmap' $ λ ⟨p,(m : leanpkg.manifest)⟩,
with_cwd sd $
do { let dir := p.dir,
ex ← dir_exists dir,
put_str_ln dir,
-- d ← env.get_cwd,
-- put_str_ln sformat!"> cwd: {d}",
when (¬ ex) $ git_clone p.url dir,
env.set_cwd dir,
-- io.cmd' { cmd := "ls", args := ["-la"] },
-- put_str_ln $ repr
-- put_str_ln sformat!"> cwd (2): {d}, {dir}",
d ← env.get_cwd,
-- put_str_ln sformat!"> cwd (3): {d}",
-- git_remote,
git_fetch,
-- git_remote,
match p.commit with
| some sha := git_checkout_hash sha
| none := catch (git_checkout_tag args.tag) (λ _ : io.error, (return () : io unit))
end,
-- put_str_ln r.url,
-- env.get_cwd >>= put_str_ln,
let version_string : string :=
if "nightly".is_prefix_of args.lean_version ∨
"3.5.".is_prefix_of args.lean_version
then "leanprover-community/lean:" ++ args.lean_version
else args.lean_version,
leanpkg.write_manifest
{ dependencies := m.dependencies.map (mk_local conv_url),
lean_version := version_string, .. m },
return sformat!"{sd}/{dir}"},
return (sd) }
def dep_to_target_name (m : rbmap string (package × leanpkg.manifest)) : leanpkg.dependency → option string
| { src := (leanpkg.source.git url _ _), .. } := m.find (canonicalize_url conv_url url) >>= λ ⟨p,_⟩, some sformat!"{p.dir}.pkg"
| { src := (leanpkg.source.path _), .. } := none
def write_Makefile (dir : string) (ps : list $ package × leanpkg.manifest) : io unit :=
with_cwd dir $
do let m : rbmap string (package × leanpkg.manifest) :=
rbmap.from_list (ps.map $ λ ⟨p,q⟩, (p.url, p, q)) ,
h ← mk_file_handle "Makefile" mode.write,
let deps := ps.map $ λ p, sformat!"{p.1.dir}.pkg",
let all_tgt := sformat!"all: {\" \".intercalate deps}\n\n",
let init_tgt := sformat!"
.PHONY: init all
init:
\t@echo [failure] > failure.toml
\t@echo \"\" > snapshot.toml
\n",
write h (all_tgt ++ init_tgt).to_char_buffer,
ps.mmap' $ λ p : package × _,
do { some sha ← pure p.1.commit | pure (),
let deps := p.2.dependencies.filter_map $ dep_to_target_name conv_url m,
let git := list.map (toml.value.escape ∘ repr) $ p.1.url :: p.1.alternate_urls,
let echo := sformat!"echo \"{p.1.name} = {{ git = {git}, rev = \\\"{sha}\\\" } \"",
-- let header := sformat!"\n",
let cmds := sformat!"
.PHONY: {p.1.dir}.pkg
{p.1.dir}.pkg: init {\" \".intercalate deps}
\tcd {p.1.dir} && time leanpkg test || ({echo} >> ../../failure.toml && false)
\t@echo \"[snapshot.{p.1.name}]\" >> snapshot.toml
\t@echo \"git = {git}\" >> snapshot.toml
\t@echo \"rev = \\\"{sha}\\\"\" >> snapshot.toml
\t@echo \"desc = \\\"{p.1.description}\\\"\" >> snapshot.toml
\t@echo \"\" >> snapshot.toml\n\n",
write h cmds.to_char_buffer },
close h
def setup_snapshots (n : app_args) : io unit :=
do xs ← list_packages' n.lean_version,
-- print xs,
mkdir "build/root" tt,
ys ← with_cwd "build/root" $
do { xs.mmap $ λ r, do {
let dir := r.dir,
ex ← dir_exists dir,
when (¬ ex) $ git_clone r.url r.dir,
r ← if r.update ∧ r.commit.is_none
then do c ← leanpkg.git_head_revision r.dir,
pure { commit := c, .. r }
else pure r,
prod.mk r <$> leanpkg.manifest.from_file (dir ++ "/" ++ leanpkg.leanpkg_toml_fn) } },
-- let zs := (ys.bind leanpkg.manifest.dependencies).foldl (flip dep_versions) (mk_rbmap _ _),
-- let s := snapshots' n xs ys,
-- let s : list (ℕ × list (string × string)) := snapshots zs,
-- xs.mmap' $ io.put_str_ln ∘ to_string,
update_package_desc ys,
let conv_url : rbmap string string := rbmap.from_list (ys.bind $ λ ⟨p,q⟩, p.alternate_urls.map $ λ q, (q, p.url)),
dir ← checkout_snapshot' conv_url n ys,
write_Makefile conv_url dir ys,
pure ()
-- return ds.join,
def try (cmd : io unit) : io bool :=
io.catch (tt <$ cmd) (λ _, pure ff)
def make (s : string) : io bool :=
do put_str_ln sformat!">> build: {s}",
env.get_cwd >>= put_str_ln,
m ← leanpkg.manifest.from_file sformat!"{s}/{leanpkg.leanpkg_toml_fn}",
try $ with_cwd s $ do
leanpkg.configure >> leanpkg.make []
def build_result_file (xs : list (string × list (string × bool))) : toml.value :=
toml.value.table $ xs.map $ prod.map id $ toml.value.table ∘ list.map (prod.map id toml.value.bool)
def spawn_piped (p1 p2 : process.spawn_args) : io unit :=
do h1 ← proc.spawn { stdout := process.stdio.piped .. p1 },
h2 ← proc.spawn { stdin := process.stdio.piped .. p2 },
io.iterate () (λ x : unit,
do done ← is_eof h1.stdout,
if done
then close h2.stdin >> return none
else do
c ← read h1.stdout 1024,
write h2.stdin c,
return $ some ()),
wait h1, wait h2,
return ()
def send_mail (addr subject : string) (attachment : string) : io unit :=
do -- spawn_piped { cmd := "uuencode", args := [attachment,(split_path attachment).ilast] }
-- { cmd := "mail", args := ["-s",subject,addr] }
cmd' { cmd := "sh", args := ["send_mail.sh",attachment,(split_path attachment).ilast,subject,addr] }
-- uuencode /usr/bin/xxx.c MyFile.c | mail -s "mailing my c file" youremail@yourdomain.com
def list.head' {α} : list α → option α
| [] := none
| (x :: xs) := x
def select_snapshot : io (option string) :=
list.head' <$> cmdline_args
def parse_args : io (option app_args) :=
do xs ← cmdline_args | usage,
match xs with
| ["--makefile",vers,tag] :=
pure (some { tag := tag, lean_version := vers, dir := "." })
| ["--makefile",vers,tag,p] :=
pure (some { tag := tag, lean_version := vers, dir := p })
| ["--update"] := pure none
| _ := usage
end
def update_pkgs : io unit :=
do xs ← list_packages',
-- print xs,
mkdir "build/root" tt,
let xs := xs.filter $ λ p, p.update,
ys ← with_cwd "build/root" $
do { xs.mmap $ λ r, do {
let dir := r.dir,
ex ← dir_exists dir,
if ¬ ex
then git_clone r.url r.dir
else git_pull_master,
c ← leanpkg.git_head_revision r.dir,
let r := { commit := c, .. r },
prod.mk r <$> leanpkg.manifest.from_file (dir ++ "/" ++ leanpkg.leanpkg_toml_fn) } },
update_package_desc ys,
return ()
def main : io unit :=
do some opt ← parse_args | update_pkgs,
setup_snapshots opt,
-- mkdir "snapshot" tt,
-- let snap_name := "snapshot-lean-3.4.2--2019-10",
-- io.cmd { cmd := "git",args := ["clone", "--single-branch", "--branch", "snapshot", "http://gitlab.com/simon.hudon/lean-depot", "snapshot"] },
-- io.cmd { cmd := "cp", args := [sformat!"build/{snap_name}/snapshot.toml",
-- sformat!"snapshot/{snap_name}.toml"] },
-- io.cmd { cmd := "cp", args := [sformat!"build/{snap_name}/failure.toml",
-- sformat!"snapshot/{snap_name}-failure.toml"] },
-- io.cmd { cmd := "git",
-- args := ["add",sformat!"snapshot/{snap_name}-failure.toml",
-- sformat!"snapshot/{snap_name}.toml"],
-- cwd := "snapshot" },
-- print s,
pure ()
-- list_dir "pkgs" >>= print
-- def main' : io unit :=
-- do -- cmd' { cmd := "elan", args := ["default","3.4.2"] },
-- args ← parse_args,
-- (s,xs) ← setup_snapshots args,
-- -- env.set_cwd "/Users/simon/lean/lean-depot",
-- -- _,
-- s ← prod.mk s <$> xs.mmap (λ y, prod.mk y <$> make y),
-- leanpkg.write_file "results.toml" (repr $ build_result_file [s]),
-- cmd' { cmd := "cat", args := ["results.toml"] },
-- pure ()
-- #eval main
-- #eval env.set_cwd "/Users/simon/lean/lean-depot"
-- #eval do
-- env.set_cwd "/Users/simon/lean/lean-depot",
-- main
-- #eval do
-- env.set_cwd "/Users/simon/lean/lean-depot",
-- send_mail "simon.hudon@gmail.com" "mail sent from Lean" "leanpkg.toml"
|
78c18232c172539afaf9df7127afb32c9601a39d | d9ed0fce1c218297bcba93e046cb4e79c83c3af8 | /library/init/data/to_string.lean | 7d1a46b9edc3fa9c7e8dffed713e438a61a2168e | [
"Apache-2.0"
] | permissive | leodemoura/lean_clone | 005c63aa892a6492f2d4741ee3c2cb07a6be9d7f | cc077554b584d39bab55c360bc12a6fe7957afe6 | refs/heads/master | 1,610,506,475,484 | 1,482,348,354,000 | 1,482,348,543,000 | 77,091,586 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,129 | lean | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
prelude
import init.data.string.basic init.data.bool.basic init.data.subtype.basic
import init.data.unsigned init.data.prod init.data.sum.basic
open sum subtype nat
universe variables u v
class has_to_string (α : Type u) :=
(to_string : α → string)
def to_string {α : Type u} [has_to_string α] : α → string :=
has_to_string.to_string
instance : has_to_string bool :=
⟨λ b, cond b "tt" "ff"⟩
instance {p : Prop} : has_to_string (decidable p) :=
-- Remark: type class inference will not consider local instance `b` in the new elaborator
⟨λ b : decidable p, @ite p b _ "tt" "ff"⟩
protected def list.to_string_aux {α : Type u} [has_to_string α] : bool → list α → string
| b [] := ""
| tt (x::xs) := to_string x ++ list.to_string_aux ff xs
| ff (x::xs) := ", " ++ to_string x ++ list.to_string_aux ff xs
protected def list.to_string {α : Type u} [has_to_string α] : list α → string
| [] := "[]"
| (x::xs) := "[" ++ list.to_string_aux tt (x::xs) ++ "]"
instance {α : Type u} [has_to_string α] : has_to_string (list α) :=
⟨list.to_string⟩
instance : has_to_string unit :=
⟨λ u, "star"⟩
instance {α : Type u} [has_to_string α] : has_to_string (option α) :=
⟨λ o, match o with | none := "none" | (some a) := "(some " ++ to_string a ++ ")" end⟩
instance {α : Type u} {β : Type v} [has_to_string α] [has_to_string β] : has_to_string (α ⊕ β) :=
⟨λ s, match s with | (inl a) := "(inl " ++ to_string a ++ ")" | (inr b) := "(inr " ++ to_string b ++ ")" end⟩
instance {α : Type u} {β : Type v} [has_to_string α] [has_to_string β] : has_to_string (α × β) :=
⟨λ p, "(" ++ to_string p.1 ++ ", " ++ to_string p.2 ++ ")"⟩
instance {α : Type u} {β : α → Type v} [has_to_string α] [s : ∀ x, has_to_string (β x)] : has_to_string (sigma β) :=
⟨λ p, "⟨" ++ to_string p.1 ++ ", " ++ to_string p.2 ++ "⟩"⟩
instance {α : Type u} {p : α → Prop} [has_to_string α] : has_to_string (subtype p) :=
⟨λ s, to_string (elt_of s)⟩
def char.quote_core (c : char) : string :=
if c = #"\n" then "\\n"
else if c = #"\\" then "\\\\"
else if c = #"\"" then "\\\""
else [c]
instance : has_to_string char :=
⟨λ c, "#\"" ++ char.quote_core c ++ "\""⟩
def string.quote_aux : string → string
| [] := ""
| (x::xs) := string.quote_aux xs ++ char.quote_core x
def string.quote : string → string
| [] := "\"\""
| (x::xs) := "\"" ++ string.quote_aux (x::xs) ++ "\""
instance : has_to_string string :=
⟨string.quote⟩
/- Remark: the code generator replaces this definition with one that display natural numbers in decimal notation -/
protected def nat.to_string : nat → string
| 0 := "zero"
| (succ a) := "(succ " ++ nat.to_string a ++ ")"
instance : has_to_string nat :=
⟨nat.to_string⟩
instance (n : nat) : has_to_string (fin n) :=
⟨λ f, to_string (fin.val f)⟩
instance : has_to_string unsigned :=
⟨λ n, to_string (fin.val n)⟩
|
ff1ed75e9296750376248cf14ea735ab76e66be8 | 82e44445c70db0f03e30d7be725775f122d72f3e | /src/algebra/category/Group/colimits.lean | 505fd5c8340377b3033173ee01cbf5b38cac2784 | [
"Apache-2.0"
] | permissive | stjordanis/mathlib | 51e286d19140e3788ef2c470bc7b953e4991f0c9 | 2568d41bca08f5d6bf39d915434c8447e21f42ee | refs/heads/master | 1,631,748,053,501 | 1,627,938,886,000 | 1,627,938,886,000 | 228,728,358 | 0 | 0 | Apache-2.0 | 1,576,630,588,000 | 1,576,630,587,000 | null | UTF-8 | Lean | false | false | 9,699 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import algebra.category.Group.preadditive
import group_theory.quotient_group
import category_theory.limits.concrete_category
import category_theory.limits.shapes.kernels
import category_theory.limits.shapes.concrete_category
/-!
# The category of additive commutative groups has all colimits.
This file uses a "pre-automated" approach, just as for `Mon/colimits.lean`.
It is a very uniform approach, that conceivably could be synthesised directly
by a tactic that analyses the shape of `add_comm_group` and `monoid_hom`.
TODO:
In fact, in `AddCommGroup` there is a much nicer model of colimits as quotients
of finitely supported functions, and we really should implement this as well (or instead).
-/
universes u v
open category_theory
open category_theory.limits
-- [ROBOT VOICE]:
-- You should pretend for now that this file was automatically generated.
-- It follows the same template as colimits in Mon.
namespace AddCommGroup.colimits
/-!
We build the colimit of a diagram in `AddCommGroup` by constructing the
free group on the disjoint union of all the abelian groups in the diagram,
then taking the quotient by the abelian group laws within each abelian group,
and the identifications given by the morphisms in the diagram.
-/
variables {J : Type v} [small_category J] (F : J ⥤ AddCommGroup.{v})
/--
An inductive type representing all group expressions (without relations)
on a collection of types indexed by the objects of `J`.
-/
inductive prequotient
-- There's always `of`
| of : Π (j : J) (x : F.obj j), prequotient
-- Then one generator for each operation
| zero : prequotient
| neg : prequotient → prequotient
| add : prequotient → prequotient → prequotient
instance : inhabited (prequotient F) := ⟨prequotient.zero⟩
open prequotient
/--
The relation on `prequotient` saying when two expressions are equal
because of the abelian group laws, or
because one element is mapped to another by a morphism in the diagram.
-/
inductive relation : prequotient F → prequotient F → Prop
-- Make it an equivalence relation:
| refl : Π (x), relation x x
| symm : Π (x y) (h : relation x y), relation y x
| trans : Π (x y z) (h : relation x y) (k : relation y z), relation x z
-- There's always a `map` relation
| map : Π (j j' : J) (f : j ⟶ j') (x : F.obj j), relation (of j' (F.map f x)) (of j x)
-- Then one relation per operation, describing the interaction with `of`
| zero : Π (j), relation (of j 0) zero
| neg : Π (j) (x : F.obj j), relation (of j (-x)) (neg (of j x))
| add : Π (j) (x y : F.obj j), relation (of j (x + y)) (add (of j x) (of j y))
-- Then one relation per argument of each operation
| neg_1 : Π (x x') (r : relation x x'), relation (neg x) (neg x')
| add_1 : Π (x x' y) (r : relation x x'), relation (add x y) (add x' y)
| add_2 : Π (x y y') (r : relation y y'), relation (add x y) (add x y')
-- And one relation per axiom
| zero_add : Π (x), relation (add zero x) x
| add_zero : Π (x), relation (add x zero) x
| add_left_neg : Π (x), relation (add (neg x) x) zero
| add_comm : Π (x y), relation (add x y) (add y x)
| add_assoc : Π (x y z), relation (add (add x y) z) (add x (add y z))
/--
The setoid corresponding to group expressions modulo abelian group relations and identifications.
-/
def colimit_setoid : setoid (prequotient F) :=
{ r := relation F, iseqv := ⟨relation.refl, relation.symm, relation.trans⟩ }
attribute [instance] colimit_setoid
/--
The underlying type of the colimit of a diagram in `AddCommGroup`.
-/
@[derive inhabited]
def colimit_type : Type v := quotient (colimit_setoid F)
instance : add_comm_group (colimit_type F) :=
{ zero :=
begin
exact quot.mk _ zero
end,
neg :=
begin
fapply @quot.lift,
{ intro x,
exact quot.mk _ (neg x) },
{ intros x x' r,
apply quot.sound,
exact relation.neg_1 _ _ r },
end,
add :=
begin
fapply @quot.lift _ _ ((colimit_type F) → (colimit_type F)),
{ intro x,
fapply @quot.lift,
{ intro y,
exact quot.mk _ (add x y) },
{ intros y y' r,
apply quot.sound,
exact relation.add_2 _ _ _ r } },
{ intros x x' r,
funext y,
induction y,
dsimp,
apply quot.sound,
{ exact relation.add_1 _ _ _ r },
{ refl } },
end,
zero_add := λ x,
begin
induction x,
dsimp,
apply quot.sound,
apply relation.zero_add,
refl,
end,
add_zero := λ x,
begin
induction x,
dsimp,
apply quot.sound,
apply relation.add_zero,
refl,
end,
add_left_neg := λ x,
begin
induction x,
dsimp,
apply quot.sound,
apply relation.add_left_neg,
refl,
end,
add_comm := λ x y,
begin
induction x,
induction y,
dsimp,
apply quot.sound,
apply relation.add_comm,
refl,
refl,
end,
add_assoc := λ x y z,
begin
induction x,
induction y,
induction z,
dsimp,
apply quot.sound,
apply relation.add_assoc,
refl,
refl,
refl,
end, }
@[simp] lemma quot_zero : quot.mk setoid.r zero = (0 : colimit_type F) := rfl
@[simp] lemma quot_neg (x) :
quot.mk setoid.r (neg x) = (-(quot.mk setoid.r x) : colimit_type F) := rfl
@[simp] lemma quot_add (x y) :
quot.mk setoid.r (add x y) = ((quot.mk setoid.r x) + (quot.mk setoid.r y) : colimit_type F) := rfl
/-- The bundled abelian group giving the colimit of a diagram. -/
def colimit : AddCommGroup := AddCommGroup.of (colimit_type F)
/-- The function from a given abelian group in the diagram to the colimit abelian group. -/
def cocone_fun (j : J) (x : F.obj j) : colimit_type F :=
quot.mk _ (of j x)
/-- The group homomorphism from a given abelian group in the diagram to the colimit abelian
group. -/
def cocone_morphism (j : J) : F.obj j ⟶ colimit F :=
{ to_fun := cocone_fun F j,
map_zero' := by apply quot.sound; apply relation.zero,
map_add' := by intros; apply quot.sound; apply relation.add }
@[simp] lemma cocone_naturality {j j' : J} (f : j ⟶ j') :
F.map f ≫ (cocone_morphism F j') = cocone_morphism F j :=
begin
ext,
apply quot.sound,
apply relation.map,
end
@[simp] lemma cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j):
(cocone_morphism F j') (F.map f x) = (cocone_morphism F j) x :=
by { rw ←cocone_naturality F f, refl }
/-- The cocone over the proposed colimit abelian group. -/
def colimit_cocone : cocone F :=
{ X := colimit F,
ι :=
{ app := cocone_morphism F } }.
/-- The function from the free abelian group on the diagram to the cone point of any other
cocone. -/
@[simp] def desc_fun_lift (s : cocone F) : prequotient F → s.X
| (of j x) := (s.ι.app j) x
| zero := 0
| (neg x) := -(desc_fun_lift x)
| (add x y) := desc_fun_lift x + desc_fun_lift y
/-- The function from the colimit abelian group to the cone point of any other cocone. -/
def desc_fun (s : cocone F) : colimit_type F → s.X :=
begin
fapply quot.lift,
{ exact desc_fun_lift F s },
{ intros x y r,
induction r; try { dsimp },
-- refl
{ refl },
-- symm
{ exact r_ih.symm },
-- trans
{ exact eq.trans r_ih_h r_ih_k },
-- map
{ simp, },
-- zero
{ simp, },
-- neg
{ simp, },
-- add
{ simp, },
-- neg_1
{ rw r_ih, },
-- add_1
{ rw r_ih, },
-- add_2
{ rw r_ih, },
-- zero_add
{ rw zero_add, },
-- add_zero
{ rw add_zero, },
-- add_left_neg
{ rw add_left_neg, },
-- add_comm
{ rw add_comm, },
-- add_assoc
{ rw add_assoc, },
}
end
/-- The group homomorphism from the colimit abelian group to the cone point of any other cocone. -/
def desc_morphism (s : cocone F) : colimit F ⟶ s.X :=
{ to_fun := desc_fun F s,
map_zero' := rfl,
map_add' := λ x y, by { induction x; induction y; refl }, }
/-- Evidence that the proposed colimit is the colimit. -/
def colimit_cocone_is_colimit : is_colimit (colimit_cocone F) :=
{ desc := λ s, desc_morphism F s,
uniq' := λ s m w,
begin
ext,
induction x,
induction x,
{ have w' := congr_fun (congr_arg (λ f : F.obj x_j ⟶ s.X, (f : F.obj x_j → s.X)) (w x_j)) x_x,
erw w',
refl, },
{ simp *, },
{ simp *, },
{ simp *, },
refl
end }.
instance has_colimits_AddCommGroup : has_colimits AddCommGroup :=
{ has_colimits_of_shape := λ J 𝒥, by exactI
{ has_colimit := λ F, has_colimit.mk
{ cocone := colimit_cocone F,
is_colimit := colimit_cocone_is_colimit F } } }
end AddCommGroup.colimits
namespace AddCommGroup
open quotient_add_group
/--
The categorical cokernel of a morphism in `AddCommGroup`
agrees with the usual group-theoretical quotient.
-/
noncomputable def cokernel_iso_quotient {G H : AddCommGroup} (f : G ⟶ H) :
cokernel f ≅ AddCommGroup.of (quotient (add_monoid_hom.range f)) :=
{ hom := cokernel.desc f (mk' _)
(by { ext, apply quotient.sound, fsplit, exact -x,
simp only [add_zero, add_monoid_hom.map_neg], }),
inv := quotient_add_group.lift _ (cokernel.π f)
(by { intros x H_1, cases H_1, induction H_1_h,
simp only [cokernel.condition_apply, zero_apply]}),
-- obviously can take care of the next goals, but it is really slow
hom_inv_id' := begin
ext1, simp only [coequalizer_as_cokernel, category.comp_id, cokernel.π_desc_assoc], ext1, refl,
end,
inv_hom_id' := begin
ext1, induction x,
{ simp only [colimit.ι_desc_apply, id_apply, lift_quot_mk,
cofork.of_π_ι_app, comp_apply], refl },
{ refl }
end, }
end AddCommGroup
|
7a16aa0d791d1bed74bdaaed1f48daccd5d78fa3 | ac89c256db07448984849346288e0eeffe8b20d0 | /stage0/src/Lean/Meta/Tactic/Contradiction.lean | 0d9729f986f207f8362fb2e8e4148642f439a74f | [
"Apache-2.0"
] | permissive | chepinzhang/lean4 | 002cc667f35417a418f0ebc9cb4a44559bb0ccac | 24fe2875c68549b5481f07c57eab4ad4a0ae5305 | refs/heads/master | 1,688,942,838,326 | 1,628,801,942,000 | 1,628,801,995,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,604 | lean | /-
Copyright (c) 2021 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Meta.MatchUtil
import Lean.Meta.Tactic.Assumption
import Lean.Meta.Tactic.Cases
namespace Lean.Meta
def elimEmptyInductive (mvarId : MVarId) (fvarId : FVarId) (searchDepth : Nat) : MetaM Bool :=
match searchDepth with
| 0 => return false
| searchDepth + 1 =>
withMVarContext mvarId do
let localDecl ← getLocalDecl fvarId
let type ← whnfD localDecl.type
matchConstInduct type.getAppFn (fun _ => pure false) fun info _ => do
if info.ctors.length == 0 || info.numIndices > 0 then
-- We only consider inductives with no constructors and indexed families
commitWhen do
let subgoals ← try cases mvarId fvarId catch _ => return false
for subgoal in subgoals do
-- If one of the fields is uninhabited, then we are done
let mut found := false
for field in subgoal.fields do
let field := subgoal.subst.apply field
if field.isFVar then
if (← elimEmptyInductive subgoal.mvarId field.fvarId! searchDepth) then
found := true
break
unless found == true do -- TODO: check why we need true here
return false
return true
else
return false
def contradictionCore (mvarId : MVarId) (useDecide : Bool) (searchDepth : Nat) : MetaM Bool := do
withMVarContext mvarId do
checkNotAssigned mvarId `contradiction
for localDecl in (← getLCtx) do
unless localDecl.isAuxDecl do
-- (h : ¬ p) (h' : p)
if let some p ← matchNot? localDecl.type then
if let some pFVarId ← findLocalDeclWithType? p then
assignExprMVar mvarId (← mkAbsurd (← getMVarType mvarId) (mkFVar pFVarId) localDecl.toExpr)
return true
-- (h : <empty-inductive-type>)
if (← elimEmptyInductive mvarId localDecl.fvarId searchDepth) then
return true
-- (h : x ≠ x)
if let some (_, lhs, rhs) ← matchNe? localDecl.type then
if (← isDefEq lhs rhs) then
assignExprMVar mvarId (← mkAbsurd (← getMVarType mvarId) (← mkEqRefl lhs) localDecl.toExpr)
return true
-- (h : ctor₁ ... = ctor₂ ...)
if let some (_, lhs, rhs) ← matchEq? localDecl.type then
if let some lhsCtor ← matchConstructorApp? lhs then
if let some rhsCtor ← matchConstructorApp? rhs then
if lhsCtor.name != rhsCtor.name then
assignExprMVar mvarId (← mkNoConfusion (← getMVarType mvarId) localDecl.toExpr)
return true
-- (h : p) s.t. `decide p` evaluates to `false`
if useDecide && !localDecl.type.hasFVar && !localDecl.type.hasMVar then
try
let d ← mkDecide localDecl.type
let r ← withDefault <| whnf d
if r.isConstOf ``false then
let hn := mkAppN (mkConst ``of_decide_eq_false) <| d.getAppArgs.push (← mkEqRefl d)
assignExprMVar mvarId (← mkAbsurd (← getMVarType mvarId) localDecl.toExpr hn)
return true
catch _ =>
pure ()
return false
def contradiction (mvarId : MVarId) (useDecide : Bool := true) (searchDepth : Nat := 2) : MetaM Unit :=
unless (← contradictionCore mvarId useDecide searchDepth) do
throwTacticEx `contradiction mvarId ""
end Lean.Meta
|
d5470dae54b4a666a68344fd473e139bb03c8fd7 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/model_theory/order.lean | f483e7782e1962b61f7ad7e97226645c7a6bb62e | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 9,163 | lean | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import model_theory.semantics
/-!
# Ordered First-Ordered Structures
This file defines ordered first-order languages and structures, as well as their theories.
## Main Definitions
* `first_order.language.order` is the language consisting of a single relation representing `≤`.
* `first_order.language.order_Structure` is the structure on an ordered type, assigning the symbol
representing `≤` to the actual relation `≤`.
* `first_order.language.is_ordered` points out a specific symbol in a language as representing `≤`.
* `first_order.language.is_ordered_structure` indicates that a structure over a
* `first_order.language.Theory.linear_order` and similar define the theories of preorders,
partial orders, and linear orders.
* `first_order.language.Theory.DLO` defines the theory of dense linear orders without endpoints, a
particularly useful example in model theory.
## Main Results
* `partial_order`s model the theory of partial orders, `linear_order`s model the theory of
linear orders, and dense linear orders without endpoints model `Theory.DLO`.
-/
universes u v w w'
namespace first_order
namespace language
open_locale first_order
open Structure
variables {L : language.{u v}} {α : Type w} {M : Type w'} {n : ℕ}
/-- The language consisting of a single relation representing `≤`. -/
protected def order : language :=
language.mk₂ empty empty empty empty unit
namespace order
instance Structure [has_le M] : language.order.Structure M :=
Structure.mk₂ empty.elim empty.elim empty.elim empty.elim (λ _, (≤))
instance : is_relational (language.order) := language.is_relational_mk₂
instance : subsingleton (language.order.relations n) :=
language.subsingleton_mk₂_relations
end order
/-- A language is ordered if it has a symbol representing `≤`. -/
class is_ordered (L : language.{u v}) := (le_symb : L.relations 2)
export is_ordered (le_symb)
section is_ordered
variables [is_ordered L]
/-- Joins two terms `t₁, t₂` in a formula representing `t₁ ≤ t₂`. -/
def term.le (t₁ t₂ : L.term (α ⊕ fin n)) : L.bounded_formula α n :=
le_symb.bounded_formula₂ t₁ t₂
/-- Joins two terms `t₁, t₂` in a formula representing `t₁ < t₂`. -/
def term.lt (t₁ t₂ : L.term (α ⊕ fin n)) : L.bounded_formula α n :=
(t₁.le t₂) ⊓ ∼ (t₂.le t₁)
variable (L)
/-- The language homomorphism sending the unique symbol `≤` of `language.order` to `≤` in an ordered
language. -/
def order_Lhom : language.order →ᴸ L :=
Lhom.mk₂ empty.elim empty.elim empty.elim empty.elim (λ _, le_symb)
end is_ordered
instance : is_ordered language.order := ⟨unit.star⟩
@[simp] lemma order_Lhom_le_symb [L.is_ordered] :
(order_Lhom L).on_relation le_symb = (le_symb : L.relations 2) := rfl
@[simp]
lemma order_Lhom_order : order_Lhom language.order = Lhom.id language.order :=
Lhom.funext (subsingleton.elim _ _) (subsingleton.elim _ _)
instance : is_ordered (L.sum language.order) := ⟨sum.inr is_ordered.le_symb⟩
/-- The theory of preorders. -/
protected def Theory.preorder : language.order.Theory :=
{le_symb.reflexive, le_symb.transitive}
/-- The theory of partial orders. -/
protected def Theory.partial_order : language.order.Theory :=
{le_symb.reflexive, le_symb.antisymmetric, le_symb.transitive}
/-- The theory of linear orders. -/
protected def Theory.linear_order : language.order.Theory :=
{le_symb.reflexive, le_symb.antisymmetric, le_symb.transitive, le_symb.total}
/-- A sentence indicating that an order has no top element:
$\forall x, \exists y, \neg y \le x$. -/
protected def sentence.no_top_order : language.order.sentence := ∀' ∃' ∼ ((&1).le &0)
/-- A sentence indicating that an order has no bottom element:
$\forall x, \exists y, \neg x \le y$. -/
protected def sentence.no_bot_order : language.order.sentence := ∀' ∃' ∼ ((&0).le &1)
/-- A sentence indicating that an order is dense:
$\forall x, \forall y, x < y \to \exists z, x < z \wedge z < y$. -/
protected def sentence.densely_ordered : language.order.sentence :=
∀' ∀' (((&0).lt &1) ⟹ (∃' (((&0).lt &2) ⊓ ((&2).lt &1))))
/-- The theory of dense linear orders without endpoints. -/
protected def Theory.DLO : language.order.Theory :=
Theory.linear_order ∪ {sentence.no_top_order, sentence.no_bot_order, sentence.densely_ordered}
variables (L M)
/-- A structure is ordered if its language has a `≤` symbol whose interpretation is -/
abbreviation is_ordered_structure [is_ordered L] [has_le M] [L.Structure M] : Prop :=
Lhom.is_expansion_on (order_Lhom L) M
variables {L M}
@[simp] lemma is_ordered_structure_iff [is_ordered L] [has_le M] [L.Structure M] :
L.is_ordered_structure M ↔ Lhom.is_expansion_on (order_Lhom L) M := iff.rfl
instance is_ordered_structure_has_le [has_le M] :
is_ordered_structure language.order M :=
begin
rw [is_ordered_structure_iff, order_Lhom_order],
exact Lhom.id_is_expansion_on M,
end
instance model_preorder [preorder M] :
M ⊨ Theory.preorder :=
begin
simp only [Theory.preorder, Theory.model_iff, set.mem_insert_iff, set.mem_singleton_iff,
forall_eq_or_imp, relations.realize_reflexive, rel_map_apply₂, forall_eq,
relations.realize_transitive],
exact ⟨le_refl, λ _ _ _, le_trans⟩
end
instance model_partial_order [partial_order M] :
M ⊨ Theory.partial_order :=
begin
simp only [Theory.partial_order, Theory.model_iff, set.mem_insert_iff, set.mem_singleton_iff,
forall_eq_or_imp, relations.realize_reflexive, rel_map_apply₂, relations.realize_antisymmetric,
forall_eq, relations.realize_transitive],
exact ⟨le_refl, λ _ _, le_antisymm, λ _ _ _, le_trans⟩,
end
instance model_linear_order [linear_order M] :
M ⊨ Theory.linear_order :=
begin
simp only [Theory.linear_order, Theory.model_iff, set.mem_insert_iff, set.mem_singleton_iff,
forall_eq_or_imp, relations.realize_reflexive, rel_map_apply₂, relations.realize_antisymmetric,
relations.realize_transitive, forall_eq, relations.realize_total],
exact ⟨le_refl, λ _ _, le_antisymm, λ _ _ _, le_trans, le_total⟩,
end
section is_ordered_structure
variables [is_ordered L] [L.Structure M]
@[simp] lemma rel_map_le_symb [has_le M] [L.is_ordered_structure M] {a b : M} :
rel_map (le_symb : L.relations 2) ![a, b] ↔ a ≤ b :=
begin
rw [← order_Lhom_le_symb, Lhom.is_expansion_on.map_on_relation],
refl,
end
@[simp] lemma term.realize_le [has_le M] [L.is_ordered_structure M]
{t₁ t₂ : L.term (α ⊕ fin n)} {v : α → M} {xs : fin n → M} :
(t₁.le t₂).realize v xs ↔ t₁.realize (sum.elim v xs) ≤ t₂.realize (sum.elim v xs) :=
by simp [term.le]
@[simp] lemma term.realize_lt [preorder M] [L.is_ordered_structure M]
{t₁ t₂ : L.term (α ⊕ fin n)} {v : α → M} {xs : fin n → M} :
(t₁.lt t₂).realize v xs ↔ t₁.realize (sum.elim v xs) < t₂.realize (sum.elim v xs) :=
by simp [term.lt, lt_iff_le_not_le]
end is_ordered_structure
section has_le
variables [has_le M]
theorem realize_no_top_order_iff : M ⊨ sentence.no_top_order ↔ no_top_order M :=
begin
simp only [sentence.no_top_order, sentence.realize, formula.realize, bounded_formula.realize_all,
bounded_formula.realize_ex, bounded_formula.realize_not, realize, term.realize_le,
sum.elim_inr],
refine ⟨λ h, ⟨λ a, h a⟩, _⟩,
introsI h a,
exact exists_not_le a,
end
@[simp] lemma realize_no_top_order [h : no_top_order M] : M ⊨ sentence.no_top_order :=
realize_no_top_order_iff.2 h
theorem realize_no_bot_order_iff : M ⊨ sentence.no_bot_order ↔ no_bot_order M :=
begin
simp only [sentence.no_bot_order, sentence.realize, formula.realize, bounded_formula.realize_all,
bounded_formula.realize_ex, bounded_formula.realize_not, realize, term.realize_le,
sum.elim_inr],
refine ⟨λ h, ⟨λ a, h a⟩, _⟩,
introsI h a,
exact exists_not_ge a,
end
@[simp] lemma realize_no_bot_order [h : no_bot_order M] : M ⊨ sentence.no_bot_order :=
realize_no_bot_order_iff.2 h
end has_le
theorem realize_densely_ordered_iff [preorder M] :
M ⊨ sentence.densely_ordered ↔ densely_ordered M :=
begin
simp only [sentence.densely_ordered, sentence.realize, formula.realize,
bounded_formula.realize_imp, bounded_formula.realize_all, realize, term.realize_lt,
sum.elim_inr, bounded_formula.realize_ex, bounded_formula.realize_inf],
refine ⟨λ h, ⟨λ a b ab, h a b ab⟩, _⟩,
introsI h a b ab,
exact exists_between ab,
end
@[simp] lemma realize_densely_ordered [preorder M] [h : densely_ordered M] :
M ⊨ sentence.densely_ordered :=
realize_densely_ordered_iff.2 h
instance model_DLO [linear_order M] [densely_ordered M] [no_top_order M] [no_bot_order M] :
M ⊨ Theory.DLO :=
begin
simp only [Theory.DLO, set.union_insert, set.union_singleton, Theory.model_iff,
set.mem_insert_iff, forall_eq_or_imp, realize_no_top_order, realize_no_bot_order,
realize_densely_ordered, true_and],
rw ← Theory.model_iff,
apply_instance,
end
end language
end first_order
|
427aead678c9a188b3ecfa0a01c7f981a604ab15 | dd0f5513e11c52db157d2fcc8456d9401a6cd9da | /11_Tactic-Style_Proofs.org.9.lean | 13e0b0f8c4b32611ac633f548e2e7a78868c6e7e | [] | no_license | cjmazey/lean-tutorial | ba559a49f82aa6c5848b9bf17b7389bf7f4ba645 | 381f61c9fcac56d01d959ae0fa6e376f2c4e3b34 | refs/heads/master | 1,610,286,098,832 | 1,447,124,923,000 | 1,447,124,923,000 | 43,082,433 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 152 | lean | import standard
example (A : Type) : A → A :=
begin
intro a,
exact a
end
example (A : Type) : ∀ x : A, x = x :=
begin
intro x,
exact eq.refl x
end
|
3ac90868bd1c590e187e2fef6c52d532fc28f868 | d8870c0d7a56f998464cfef82fdac792d2fd9dac | /homework/teamwork4.lean | 5747b2ec678227c5c51ee896ca7f51655113a8a6 | [] | no_license | williamdemeo/math2001-spring2019 | 565b70405e7b92c5f93a295d2114729786350b78 | 1b3bf3e0bf53684b064b3f96210ce1022850ebf0 | refs/heads/master | 1,587,030,510,505 | 1,558,394,100,000 | 1,558,394,100,000 | 165,477,693 | 2 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 3,451 | lean | -- Math 2001 Teamwork 4 (Optional)
-- You may submit individual solutions to this assignment for extra credit.
-- You may work on a team, but you must submit your own tw4-lastname.lean file if you want it to be graded and counted.
-- Due Date: Fri 3 May 2019
-- Below you will be asked to prove some facts about natural numbers from Sections 17.4 and 17.5.
-- First, here are some examples. (You can use these theorems in your own proofs).
open nat
variables m n : ℕ
example : m + 0 = m := add_zero m
example : m + succ n = succ (m + n) := add_succ m n
-- Similarly, we have the defining equations for the predecessor function and multiplication:
#check @pred_zero
#check @pred_succ
#check @mul_zero
#check @mul_succ
-- Here are five propositions proved in 17.4:
namespace hidden
theorem succ_pred (n : ℕ) : n ≠ 0 → succ (pred n) = n :=
nat.rec_on n
(assume H : 0 ≠ 0,
show succ (pred 0) = 0, from absurd rfl H)
(assume n,
assume ih,
assume H : succ n ≠ 0,
show succ (pred (succ n)) = succ n,
by rewrite pred_succ)
theorem zero_add (n : nat) : 0 + n = n :=
nat.rec_on n
(show 0 + 0 = 0, from rfl)
(assume n,
assume ih : 0 + n = n,
show 0 + succ n = succ n, from calc
0 + succ n = succ (0 + n) : rfl
... = succ n : by rw ih)
theorem succ_add (m n : nat) : succ m + n = succ (m + n) :=
nat.rec_on n
(show succ m + 0 = succ (m + 0), from rfl)
(assume n,
assume ih : succ m + n = succ (m + n),
show succ m + succ n = succ (m + succ n), from calc
succ m + succ n = succ (succ m + n) : rfl
... = succ (succ (m + n)) : by rw ih
... = succ (m + succ n) : rfl)
theorem add_assoc (m n k : nat) : m + n + k = m + (n + k) :=
nat.rec_on k
(show m + n + 0 = m + (n + 0), by rw [add_zero, add_zero])
(assume k,
assume ih : m + n + k = m + (n + k),
show m + n + succ k = m + (n + (succ k)), from calc
m + n + succ k = succ (m + n + k) : by rw add_succ
... = succ (m + (n + k)) : by rw ih
... = m + (n + succ k) : by rw add_succ)
theorem add_comm (m n : nat) : m + n = n + m :=
nat.rec_on n
(show m + 0 = 0 + m, by rewrite [add_zero, zero_add])
(assume n,
assume ih : m + n = n + m,
show m + succ n = succ n + m, from calc
m + succ n = succ (m + n) : by rw add_succ
... = succ (n + m) : by rw ih
... = succ n + m : by rw succ_add)
--EXERCISES--
--1. Prove the following identities from 17.4 by replacing each `sorry` with a proof.
--1.a.
theorem mul_distr : ∀ m n k : nat, m * (n + k) = m * n + m * k := sorry
--1.b.
example : ∀ n : nat, 0 * n = 0 := sorry
--1.c.
example : ∀ n : nat, 1 * n = n := sorry
--1.d.
example : ∀ m n k : nat, (m * n) * k = m * (n * k) := sorry
--1.e.
example : ∀ m n : nat, m * n= n * m := sorry
--2. Prove the remaining identities from 17.5 by replacing each `sorry` with a proof.
open nat
--2.a.
example : ∀ m n k : nat, n ≤ m → n + k ≤ m + k := sorry
--2.b.
example : ∀ m n k : nat, n + k ≤ m + k → n ≤ m := sorry
--2.c.
example : ∀ m n k : nat, n ≤ m → n * k ≤ m * k := sorry
--2.d.
example : ∀ m n k : nat, m ≥ n → m = n ∨ m ≥ n+1 := sorry
--2.e.
example : ∀ n : nat, 0 ≤ n := sorry
end hidden
|
a17ab867c41571a90bdb090aaf28aec580a0aeab | 367134ba5a65885e863bdc4507601606690974c1 | /src/combinatorics/composition.lean | adccba57837cb845829007fa4b012a9e5d936bc9 | [
"Apache-2.0"
] | permissive | kodyvajjha/mathlib | 9bead00e90f68269a313f45f5561766cfd8d5cad | b98af5dd79e13a38d84438b850a2e8858ec21284 | refs/heads/master | 1,624,350,366,310 | 1,615,563,062,000 | 1,615,563,062,000 | 162,666,963 | 0 | 0 | Apache-2.0 | 1,545,367,651,000 | 1,545,367,651,000 | null | UTF-8 | Lean | false | false | 36,600 | lean | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import data.fintype.card
import data.finset.sort
import algebra.big_operators.order
/-!
# Compositions
A composition of a natural number `n` is a decomposition `n = i₀ + ... + i_{k-1}` of `n` into a sum
of positive integers. Combinatorially, it corresponds to a decomposition of `{0, ..., n-1}` into
non-empty blocks of consecutive integers, where the `iⱼ` are the lengths of the blocks.
This notion is closely related to that of a partition of `n`, but in a composition of `n` the
order of the `iⱼ`s matters.
We implement two different structures covering these two viewpoints on compositions. The first
one, made of a list of positive integers summing to `n`, is the main one and is called
`composition n`. The second one is useful for combinatorial arguments (for instance to show that
the number of compositions of `n` is `2^(n-1)`). It is given by a subset of `{0, ..., n}`
containing `0` and `n`, where the elements of the subset (other than `n`) correspond to the leftmost
points of each block. The main API is built on `composition n`, and we provide an equivalence
between the two types.
## Main functions
* `c : composition n` is a structure, made of a list of integers which are all positive and
add up to `n`.
* `composition_card` states that the cardinality of `composition n` is exactly
`2^(n-1)`, which is proved by constructing an equiv with `composition_as_set n` (see below), which
is itself in bijection with the subsets of `fin (n-1)` (this holds even for `n = 0`, where `-` is
nat subtraction).
Let `c : composition n` be a composition of `n`. Then
* `c.blocks` is the list of blocks in `c`.
* `c.length` is the number of blocks in the composition.
* `c.blocks_fun : fin c.length → ℕ` is the realization of `c.blocks` as a function on
`fin c.length`. This is the main object when using compositions to understand the composition of
analytic functions.
* `c.size_up_to : ℕ → ℕ` is the sum of the size of the blocks up to `i`.;
* `c.embedding i : fin (c.blocks_fun i) → fin n` is the increasing embedding of the `i`-th block in
`fin n`;
* `c.index j`, for `j : fin n`, is the index of the block containing `j`.
* `composition.ones n` is the composition of `n` made of ones, i.e., `[1, ..., 1]`.
* `composition.single n (hn : 0 < n)` is the composition of `n` made of a single block of size `n`.
Compositions can also be used to split lists. Let `l` be a list of length `n` and `c` a composition
of `n`.
* `l.split_wrt_composition c` is a list of lists, made of the slices of `l` corresponding to the
blocks of `c`.
* `join_split_wrt_composition` states that splitting a list and then joining it gives back the
original list.
* `split_wrt_composition_join` states that joining a list of lists, and then splitting it back
according to the right composition, gives back the original list of lists.
We turn to the second viewpoint on compositions, that we realize as a finset of `fin (n+1)`.
`c : composition_as_set n` is a structure made of a finset of `fin (n+1)` called `c.boundaries`
and proofs that it contains `0` and `n`. (Taking a finset of `fin n` containing `0` would not
make sense in the edge case `n = 0`, while the previous description works in all cases).
The elements of this set (other than `n`) correspond to leftmost points of blocks.
Thus, there is an equiv between `composition n` and `composition_as_set n`. We
only construct basic API on `composition_as_set` (notably `c.length` and `c.blocks`) to be able
to construct this equiv, called `composition_equiv n`. Since there is a straightforward equiv
between `composition_as_set n` and finsets of `{1, ..., n-1}` (obtained by removing `0` and `n`
from a `composition_as_set` and called `composition_as_set_equiv n`), we deduce that
`composition_as_set n` and `composition n` are both fintypes of cardinality `2^(n - 1)`
(see `composition_as_set_card` and `composition_card`).
## Implementation details
The main motivation for this structure and its API is in the construction of the composition of
formal multilinear series, and the proof that the composition of analytic functions is analytic.
The representation of a composition as a list is very handy as lists are very flexible and already
have a well-developed API.
## Tags
Composition, partition
## References
<https://en.wikipedia.org/wiki/Composition_(combinatorics)>
-/
open list
open_locale big_operators
variable {n : ℕ}
/-- A composition of `n` is a list of positive integers summing to `n`. -/
@[ext] structure composition (n : ℕ) :=
(blocks : list ℕ)
(blocks_pos : ∀ {i}, i ∈ blocks → 0 < i)
(blocks_sum : blocks.sum = n)
/-- Combinatorial viewpoint on a composition of `n`, by seeing it as non-empty blocks of
consecutive integers in `{0, ..., n-1}`. We register every block by its left end-point, yielding
a finset containing `0`. As this does not make sense for `n = 0`, we add `n` to this finset, and
get a finset of `{0, ..., n}` containing `0` and `n`. This is the data in the structure
`composition_as_set n`. -/
@[ext] structure composition_as_set (n : ℕ) :=
(boundaries : finset (fin n.succ))
(zero_mem : (0 : fin n.succ) ∈ boundaries)
(last_mem : fin.last n ∈ boundaries)
instance {n : ℕ} : inhabited (composition_as_set n) :=
⟨⟨finset.univ, finset.mem_univ _, finset.mem_univ _⟩⟩
/-!
### Compositions
A composition of an integer `n` is a decomposition `n = i₀ + ... + i_{k-1}` of `n` into a sum of
positive integers.
-/
namespace composition
variables (c : composition n)
instance (n : ℕ) : has_to_string (composition n) :=
⟨λ c, to_string c.blocks⟩
/-- The length of a composition, i.e., the number of blocks in the composition. -/
@[reducible] def length : ℕ := c.blocks.length
lemma blocks_length : c.blocks.length = c.length := rfl
/-- The blocks of a composition, seen as a function on `fin c.length`. When composing analytic
functions using compositions, this is the main player. -/
def blocks_fun : fin c.length → ℕ := λ i, nth_le c.blocks i i.2
lemma of_fn_blocks_fun : of_fn c.blocks_fun = c.blocks :=
of_fn_nth_le _
lemma sum_blocks_fun : ∑ i, c.blocks_fun i = n :=
by conv_rhs { rw [← c.blocks_sum, ← of_fn_blocks_fun, sum_of_fn] }
lemma blocks_fun_mem_blocks (i : fin c.length) : c.blocks_fun i ∈ c.blocks :=
nth_le_mem _ _ _
@[simp] lemma one_le_blocks {i : ℕ} (h : i ∈ c.blocks) : 1 ≤ i :=
c.blocks_pos h
@[simp] lemma one_le_blocks' {i : ℕ} (h : i < c.length) : 1 ≤ nth_le c.blocks i h:=
c.one_le_blocks (nth_le_mem (blocks c) i h)
@[simp] lemma blocks_pos' (i : ℕ) (h : i < c.length) : 0 < nth_le c.blocks i h:=
c.one_le_blocks' h
lemma one_le_blocks_fun (i : fin c.length) : 1 ≤ c.blocks_fun i :=
c.one_le_blocks (c.blocks_fun_mem_blocks i)
lemma length_le : c.length ≤ n :=
begin
conv_rhs { rw ← c.blocks_sum },
exact length_le_sum_of_one_le _ (λ i hi, c.one_le_blocks hi)
end
lemma length_pos_of_pos (h : 0 < n) : 0 < c.length :=
begin
apply length_pos_of_sum_pos,
convert h,
exact c.blocks_sum
end
/-- The sum of the sizes of the blocks in a composition up to `i`. -/
def size_up_to (i : ℕ) : ℕ := (c.blocks.take i).sum
@[simp] lemma size_up_to_zero : c.size_up_to 0 = 0 := by simp [size_up_to]
lemma size_up_to_of_length_le (i : ℕ) (h : c.length ≤ i) : c.size_up_to i = n :=
begin
dsimp [size_up_to],
convert c.blocks_sum,
exact take_all_of_le h
end
@[simp] lemma size_up_to_length : c.size_up_to c.length = n :=
c.size_up_to_of_length_le c.length (le_refl _)
lemma size_up_to_le (i : ℕ) : c.size_up_to i ≤ n :=
begin
conv_rhs { rw [← c.blocks_sum, ← sum_take_add_sum_drop _ i] },
exact nat.le_add_right _ _
end
lemma size_up_to_succ {i : ℕ} (h : i < c.length) :
c.size_up_to (i+1) = c.size_up_to i + c.blocks.nth_le i h :=
by { simp only [size_up_to], rw sum_take_succ _ _ h }
lemma size_up_to_succ' (i : fin c.length) :
c.size_up_to ((i : ℕ) + 1) = c.size_up_to i + c.blocks_fun i :=
c.size_up_to_succ i.2
lemma size_up_to_strict_mono {i : ℕ} (h : i < c.length) : c.size_up_to i < c.size_up_to (i+1) :=
by { rw c.size_up_to_succ h, simp }
lemma monotone_size_up_to : monotone c.size_up_to :=
monotone_sum_take _
/-- The `i`-th boundary of a composition, i.e., the leftmost point of the `i`-th block. We include
a virtual point at the right of the last block, to make for a nice equiv with
`composition_as_set n`. -/
def boundary : fin (c.length + 1) ↪o fin (n+1) :=
order_embedding.of_strict_mono (λ i, ⟨c.size_up_to i, nat.lt_succ_of_le (c.size_up_to_le i)⟩) $
fin.strict_mono_iff_lt_succ.2 $ λ i hi, c.size_up_to_strict_mono $
lt_of_add_lt_add_right hi
@[simp] lemma boundary_zero : c.boundary 0 = 0 :=
by simp [boundary, fin.ext_iff]
@[simp] lemma boundary_last : c.boundary (fin.last c.length) = fin.last n :=
by simp [boundary, fin.ext_iff]
/-- The boundaries of a composition, i.e., the leftmost point of all the blocks. We include
a virtual point at the right of the last block, to make for a nice equiv with
`composition_as_set n`. -/
def boundaries : finset (fin (n+1)) :=
finset.univ.map c.boundary.to_embedding
lemma card_boundaries_eq_succ_length : c.boundaries.card = c.length + 1 :=
by simp [boundaries]
/-- To `c : composition n`, one can associate a `composition_as_set n` by registering the leftmost
point of each block, and adding a virtual point at the right of the last block. -/
def to_composition_as_set : composition_as_set n :=
{ boundaries := c.boundaries,
zero_mem := begin
simp only [boundaries, finset.mem_univ, exists_prop_of_true, finset.mem_map],
exact ⟨0, rfl⟩,
end,
last_mem := begin
simp only [boundaries, finset.mem_univ, exists_prop_of_true, finset.mem_map],
exact ⟨fin.last c.length, c.boundary_last⟩,
end }
/-- The canonical increasing bijection between `fin (c.length + 1)` and `c.boundaries` is
exactly `c.boundary`. -/
lemma order_emb_of_fin_boundaries :
c.boundaries.order_emb_of_fin c.card_boundaries_eq_succ_length = c.boundary :=
begin
refine (finset.order_emb_of_fin_unique' _ _).symm,
exact λ i, (finset.mem_map' _).2 (finset.mem_univ _)
end
/-- Embedding the `i`-th block of a composition (identified with `fin (c.blocks_fun i)`) into
`fin n` at the relevant position. -/
def embedding (i : fin c.length) : fin (c.blocks_fun i) ↪o fin n :=
(fin.nat_add $ c.size_up_to i).trans $ fin.cast_le $
calc c.size_up_to i + c.blocks_fun i = c.size_up_to (i + 1) : (c.size_up_to_succ _).symm
... ≤ c.size_up_to c.length : monotone_sum_take _ i.2
... = n : c.size_up_to_length
@[simp] lemma coe_embedding (i : fin c.length) (j : fin (c.blocks_fun i)) :
(c.embedding i j : ℕ) = c.size_up_to i + j := rfl
/--
`index_exists` asserts there is some `i` with `j < c.size_up_to (i+1)`.
In the next definition `index` we use `nat.find` to produce the minimal such index.
-/
lemma index_exists {j : ℕ} (h : j < n) :
∃ i : ℕ, j < c.size_up_to i.succ ∧ i < c.length :=
begin
have n_pos : 0 < n := lt_of_le_of_lt (zero_le j) h,
have : 0 < c.blocks.sum, by rwa [← c.blocks_sum] at n_pos,
have length_pos : 0 < c.blocks.length := length_pos_of_sum_pos (blocks c) this,
refine ⟨c.length.pred, _, nat.pred_lt (ne_of_gt length_pos)⟩,
have : c.length.pred.succ = c.length := nat.succ_pred_eq_of_pos length_pos,
simp [this, h]
end
/-- `c.index j` is the index of the block in the composition `c` containing `j`. -/
def index (j : fin n) : fin c.length :=
⟨nat.find (c.index_exists j.2), (nat.find_spec (c.index_exists j.2)).2⟩
lemma lt_size_up_to_index_succ (j : fin n) : (j : ℕ) < c.size_up_to (c.index j).succ :=
(nat.find_spec (c.index_exists j.2)).1
lemma size_up_to_index_le (j : fin n) : c.size_up_to (c.index j) ≤ j :=
begin
by_contradiction H,
set i := c.index j with hi,
push_neg at H,
have i_pos : (0 : ℕ) < i,
{ by_contradiction i_pos,
push_neg at i_pos,
revert H, simp [nonpos_iff_eq_zero.1 i_pos, c.size_up_to_zero] },
let i₁ := (i : ℕ).pred,
have i₁_lt_i : i₁ < i := nat.pred_lt (ne_of_gt i_pos),
have i₁_succ : i₁.succ = i := nat.succ_pred_eq_of_pos i_pos,
have := nat.find_min (c.index_exists j.2) i₁_lt_i,
simp [lt_trans i₁_lt_i (c.index j).2, i₁_succ] at this,
exact nat.lt_le_antisymm H this
end
/-- Mapping an element `j` of `fin n` to the element in the block containing it, identified with
`fin (c.blocks_fun (c.index j))` through the canonical increasing bijection. -/
def inv_embedding (j : fin n) : fin (c.blocks_fun (c.index j)) :=
⟨j - c.size_up_to (c.index j),
begin
rw [nat.sub_lt_right_iff_lt_add, add_comm, ← size_up_to_succ'],
{ exact lt_size_up_to_index_succ _ _ },
{ exact size_up_to_index_le _ _ }
end⟩
@[simp] lemma coe_inv_embedding (j : fin n) :
(c.inv_embedding j : ℕ) = j - c.size_up_to (c.index j) := rfl
lemma embedding_comp_inv (j : fin n) :
c.embedding (c.index j) (c.inv_embedding j) = j :=
begin
rw fin.ext_iff,
apply nat.add_sub_cancel' (c.size_up_to_index_le j),
end
lemma mem_range_embedding_iff {j : fin n} {i : fin c.length} :
j ∈ set.range (c.embedding i) ↔
c.size_up_to i ≤ j ∧ (j : ℕ) < c.size_up_to (i : ℕ).succ :=
begin
split,
{ assume h,
rcases set.mem_range.2 h with ⟨k, hk⟩,
rw fin.ext_iff at hk,
change c.size_up_to i + k = (j : ℕ) at hk,
rw ← hk,
simp [size_up_to_succ', k.is_lt] },
{ assume h,
apply set.mem_range.2,
refine ⟨⟨j - c.size_up_to i, _⟩, _⟩,
{ rw [nat.sub_lt_left_iff_lt_add, ← size_up_to_succ'],
{ exact h.2 },
{ exact h.1 } },
{ rw fin.ext_iff,
exact nat.add_sub_cancel' h.1 } }
end
/-- The embeddings of different blocks of a composition are disjoint. -/
lemma disjoint_range {i₁ i₂ : fin c.length} (h : i₁ ≠ i₂) :
disjoint (set.range (c.embedding i₁)) (set.range (c.embedding i₂)) :=
begin
classical,
wlog h' : i₁ ≤ i₂ using i₁ i₂,
by_contradiction d,
obtain ⟨x, hx₁, hx₂⟩ :
∃ x : fin n, (x ∈ set.range (c.embedding i₁) ∧ x ∈ set.range (c.embedding i₂)) :=
set.not_disjoint_iff.1 d,
have : i₁ < i₂ := lt_of_le_of_ne h' h,
have A : (i₁ : ℕ).succ ≤ i₂ := nat.succ_le_of_lt this,
apply lt_irrefl (x : ℕ),
calc (x : ℕ) < c.size_up_to (i₁ : ℕ).succ : (c.mem_range_embedding_iff.1 hx₁).2
... ≤ c.size_up_to (i₂ : ℕ) : monotone_sum_take _ A
... ≤ x : (c.mem_range_embedding_iff.1 hx₂).1
end
lemma mem_range_embedding (j : fin n) :
j ∈ set.range (c.embedding (c.index j)) :=
begin
have : c.embedding (c.index j) (c.inv_embedding j)
∈ set.range (c.embedding (c.index j)) := set.mem_range_self _,
rwa c.embedding_comp_inv j at this
end
lemma mem_range_embedding_iff' {j : fin n} {i : fin c.length} :
j ∈ set.range (c.embedding i) ↔ i = c.index j :=
begin
split,
{ rw ← not_imp_not,
assume h,
exact set.disjoint_right.1 (c.disjoint_range h) (c.mem_range_embedding j) },
{ assume h,
rw h,
exact c.mem_range_embedding j }
end
lemma index_embedding (i : fin c.length) (j : fin (c.blocks_fun i)) :
c.index (c.embedding i j) = i :=
begin
symmetry,
rw ← mem_range_embedding_iff',
apply set.mem_range_self
end
lemma inv_embedding_comp (i : fin c.length) (j : fin (c.blocks_fun i)) :
(c.inv_embedding (c.embedding i j) : ℕ) = j :=
by simp_rw [coe_inv_embedding, index_embedding, coe_embedding, nat.add_sub_cancel_left]
/-- Equivalence between the disjoint union of the blocks (each of them seen as
`fin (c.blocks_fun i)`) with `fin n`. -/
def blocks_fin_equiv : (Σ i : fin c.length, fin (c.blocks_fun i)) ≃ fin n :=
{ to_fun := λ x, c.embedding x.1 x.2,
inv_fun := λ j, ⟨c.index j, c.inv_embedding j⟩,
left_inv := λ x, begin
rcases x with ⟨i, y⟩,
dsimp,
congr, { exact c.index_embedding _ _ },
rw fin.heq_ext_iff,
{ exact c.inv_embedding_comp _ _ },
{ rw c.index_embedding }
end,
right_inv := λ j, c.embedding_comp_inv j }
lemma blocks_fun_congr {n₁ n₂ : ℕ} (c₁ : composition n₁) (c₂ : composition n₂)
(i₁ : fin c₁.length) (i₂ : fin c₂.length) (hn : n₁ = n₂)
(hc : c₁.blocks = c₂.blocks) (hi : (i₁ : ℕ) = i₂) :
c₁.blocks_fun i₁ = c₂.blocks_fun i₂ :=
by { cases hn, rw ← composition.ext_iff at hc, cases hc, congr, rwa fin.ext_iff }
/-- Two compositions (possibly of different integers) coincide if and only if they have the
same sequence of blocks. -/
lemma sigma_eq_iff_blocks_eq {c : Σ n, composition n} {c' : Σ n, composition n} :
c = c' ↔ c.2.blocks = c'.2.blocks :=
begin
refine ⟨λ H, by rw H, λ H, _⟩,
rcases c with ⟨n, c⟩,
rcases c' with ⟨n', c'⟩,
have : n = n', by { rw [← c.blocks_sum, ← c'.blocks_sum, H] },
induction this,
simp only [true_and, eq_self_iff_true, heq_iff_eq],
ext1,
exact H
end
/-! ### The composition `composition.ones` -/
/-- The composition made of blocks all of size `1`. -/
def ones (n : ℕ) : composition n :=
⟨repeat (1 : ℕ) n, λ i hi, by simp [list.eq_of_mem_repeat hi], by simp⟩
instance {n : ℕ} : inhabited (composition n) :=
⟨composition.ones n⟩
@[simp] lemma ones_length (n : ℕ) : (ones n).length = n :=
list.length_repeat 1 n
@[simp] lemma ones_blocks (n : ℕ) : (ones n).blocks = repeat (1 : ℕ) n := rfl
@[simp] lemma ones_blocks_fun (n : ℕ) (i : fin (ones n).length) :
(ones n).blocks_fun i = 1 :=
by simp [blocks_fun, ones, blocks, i.2]
@[simp] lemma ones_size_up_to (n : ℕ) (i : ℕ) : (ones n).size_up_to i = min i n :=
by simp [size_up_to, ones_blocks, take_repeat]
@[simp] lemma ones_embedding (i : fin (ones n).length) (h : 0 < (ones n).blocks_fun i) :
(ones n).embedding i ⟨0, h⟩ = ⟨i, lt_of_lt_of_le i.2 (ones n).length_le⟩ :=
by { ext, simpa using i.2.le }
lemma eq_ones_iff {c : composition n} :
c = ones n ↔ ∀ i ∈ c.blocks, i = 1 :=
begin
split,
{ rintro rfl,
exact λ i, eq_of_mem_repeat },
{ assume H,
ext1,
have A : c.blocks = repeat 1 c.blocks.length := eq_repeat_of_mem H,
have : c.blocks.length = n, by { conv_rhs { rw [← c.blocks_sum, A] }, simp },
rw [A, this, ones_blocks] },
end
lemma ne_ones_iff {c : composition n} :
c ≠ ones n ↔ ∃ i ∈ c.blocks, 1 < i :=
begin
refine (not_congr eq_ones_iff).trans _,
have : ∀ j ∈ c.blocks, j = 1 ↔ j ≤ 1 := λ j hj, by simp [le_antisymm_iff, c.one_le_blocks hj],
simp [this] {contextual := tt}
end
lemma eq_ones_iff_length {c : composition n} :
c = ones n ↔ c.length = n :=
begin
split,
{ rintro rfl,
exact ones_length n },
{ contrapose,
assume H length_n,
apply lt_irrefl n,
calc
n = ∑ (i : fin c.length), 1 : by simp [length_n]
... < ∑ (i : fin c.length), c.blocks_fun i :
begin
obtain ⟨i, hi, i_blocks⟩ : ∃ i ∈ c.blocks, 1 < i := ne_ones_iff.1 H,
rw [← of_fn_blocks_fun, mem_of_fn c.blocks_fun, set.mem_range] at hi,
obtain ⟨j : fin c.length, hj : c.blocks_fun j = i⟩ := hi,
rw ← hj at i_blocks,
exact finset.sum_lt_sum (λ i hi, by simp [blocks_fun]) ⟨j, finset.mem_univ _, i_blocks⟩,
end
... = n : c.sum_blocks_fun }
end
lemma eq_ones_iff_le_length {c : composition n} :
c = ones n ↔ n ≤ c.length :=
by simp [eq_ones_iff_length, le_antisymm_iff, c.length_le]
/-! ### The composition `composition.single` -/
/-- The composition made of a single block of size `n`. -/
def single (n : ℕ) (h : 0 < n) : composition n :=
⟨[n], by simp [h], by simp⟩
@[simp] lemma single_length {n : ℕ} (h : 0 < n) : (single n h).length = 1 := rfl
@[simp] lemma single_blocks {n : ℕ} (h : 0 < n) : (single n h).blocks = [n] := rfl
@[simp] lemma single_blocks_fun {n : ℕ} (h : 0 < n) (i : fin (single n h).length) :
(single n h).blocks_fun i = n :=
by simp [blocks_fun, single, blocks, i.2]
@[simp] lemma single_embedding {n : ℕ} (h : 0 < n) (i : fin n) :
(single n h).embedding ⟨0, single_length h ▸ zero_lt_one⟩ i = i :=
by { ext, simp }
lemma eq_single_iff_length {n : ℕ} (h : 0 < n) {c : composition n} :
c = single n h ↔ c.length = 1 :=
begin
split,
{ assume H,
rw H,
exact single_length h },
{ assume H,
ext1,
have A : c.blocks.length = 1 := H ▸ c.blocks_length,
have B : c.blocks.sum = n := c.blocks_sum,
rw eq_cons_of_length_one A at B ⊢,
simpa [single_blocks] using B }
end
lemma ne_single_iff {n : ℕ} (hn : 0 < n) {c : composition n} :
c ≠ single n hn ↔ ∀ i, c.blocks_fun i < n :=
begin
rw ← not_iff_not,
push_neg,
split,
{ rintros rfl,
exact ⟨⟨0, by simp⟩, by simp⟩ },
{ rintros ⟨i, hi⟩,
rw eq_single_iff_length,
have : ∀ j : fin c.length, j = i,
{ intros j,
by_contradiction ji,
apply lt_irrefl ∑ k, c.blocks_fun k,
calc ∑ k, c.blocks_fun k ≤ ∑ k in {i}, c.blocks_fun k : by simp [c.sum_blocks_fun, hi]
... < ∑ k, c.blocks_fun k : begin
have : j ∈ finset.univ \ {i}, by { rw [finset.mem_sdiff, finset.mem_singleton], simp [ji] },
refine finset.sum_lt_sum_of_subset (finset.subset_univ _) this (c.one_le_blocks_fun j) _,
exact λ k hk, zero_le_one.trans (c.one_le_blocks_fun k)
end },
simpa using fintype.card_eq_one_of_forall_eq this }
end
end composition
/-!
### Splitting a list
Given a list of length `n` and a composition `c` of `n`, one can split `l` into `c.length` sublists
of respective lengths `c.blocks_fun 0`, ..., `c.blocks_fun (c.length-1)`. This is inverse to the
join operation.
-/
namespace list
variable {α : Type*}
/-- Auxiliary for `list.split_wrt_composition`. -/
def split_wrt_composition_aux : list α → list ℕ → list (list α)
| l [] := []
| l (n :: ns) :=
let (l₁, l₂) := l.split_at n in
l₁ :: split_wrt_composition_aux l₂ ns
/-- Given a list of length `n` and a composition `[i₁, ..., iₖ]` of `n`, split `l` into a list of
`k` lists corresponding to the blocks of the composition, of respective lengths `i₁`, ..., `iₖ`.
This makes sense mostly when `n = l.length`, but this is not necessary for the definition. -/
def split_wrt_composition (l : list α) (c : composition n) : list (list α) :=
split_wrt_composition_aux l c.blocks
local attribute [simp] split_wrt_composition_aux.equations._eqn_1
local attribute [simp]
lemma split_wrt_composition_aux_cons (l : list α) (n ns) :
l.split_wrt_composition_aux (n :: ns) = take n l :: (drop n l).split_wrt_composition_aux ns :=
by simp [split_wrt_composition_aux]
lemma length_split_wrt_composition_aux (l : list α) (ns) :
length (l.split_wrt_composition_aux ns) = ns.length :=
by induction ns generalizing l; simp *
/-- When one splits a list along a composition `c`, the number of sublists thus created is
`c.length`. -/
@[simp] lemma length_split_wrt_composition (l : list α) (c : composition n) :
length (l.split_wrt_composition c) = c.length :=
length_split_wrt_composition_aux _ _
lemma map_length_split_wrt_composition_aux {ns : list ℕ} :
∀ {l : list α}, ns.sum ≤ l.length → map length (l.split_wrt_composition_aux ns) = ns :=
begin
induction ns with n ns IH; intros l h; simp at h ⊢,
have := le_trans (nat.le_add_right _ _) h,
rw IH, {simp [this]},
rwa [length_drop, nat.le_sub_left_iff_add_le this]
end
/-- When one splits a list along a composition `c`, the lengths of the sublists thus created are
given by the block sizes in `c`. -/
lemma map_length_split_wrt_composition (l : list α) (c : composition l.length) :
map length (l.split_wrt_composition c) = c.blocks :=
map_length_split_wrt_composition_aux (le_of_eq c.blocks_sum)
lemma length_pos_of_mem_split_wrt_composition {l l' : list α} {c : composition l.length}
(h : l' ∈ l.split_wrt_composition c) : 0 < length l' :=
begin
have : l'.length ∈ (l.split_wrt_composition c).map list.length :=
list.mem_map_of_mem list.length h,
rw map_length_split_wrt_composition at this,
exact c.blocks_pos this
end
lemma sum_take_map_length_split_wrt_composition
(l : list α) (c : composition l.length) (i : ℕ) :
(((l.split_wrt_composition c).map length).take i).sum = c.size_up_to i :=
by { congr, exact map_length_split_wrt_composition l c }
lemma nth_le_split_wrt_composition_aux (l : list α) (ns : list ℕ) {i : ℕ} (hi) :
nth_le (l.split_wrt_composition_aux ns) i hi =
(l.take (ns.take (i+1)).sum).drop (ns.take i).sum :=
begin
induction ns with n ns IH generalizing l i, {cases hi},
cases i; simp [IH],
rw [add_comm n, drop_add, drop_take],
end
/-- The `i`-th sublist in the splitting of a list `l` along a composition `c`, is the slice of `l`
between the indices `c.size_up_to i` and `c.size_up_to (i+1)`, i.e., the indices in the `i`-th
block of the composition. -/
lemma nth_le_split_wrt_composition (l : list α) (c : composition n)
{i : ℕ} (hi : i < (l.split_wrt_composition c).length) :
nth_le (l.split_wrt_composition c) i hi = (l.take (c.size_up_to (i+1))).drop (c.size_up_to i) :=
nth_le_split_wrt_composition_aux _ _ _
theorem join_split_wrt_composition_aux {ns : list ℕ} :
∀ {l : list α}, ns.sum = l.length → (l.split_wrt_composition_aux ns).join = l :=
begin
induction ns with n ns IH; intros l h; simp at h ⊢,
{ exact (length_eq_zero.1 h.symm).symm },
rw IH, {simp},
rwa [length_drop, ← h, nat.add_sub_cancel_left]
end
/-- If one splits a list along a composition, and then joins the sublists, one gets back the
original list. -/
@[simp] theorem join_split_wrt_composition (l : list α) (c : composition l.length) :
(l.split_wrt_composition c).join = l :=
join_split_wrt_composition_aux c.blocks_sum
/-- If one joins a list of lists and then splits the join along the right composition, one gets
back the original list of lists. -/
@[simp] theorem split_wrt_composition_join (L : list (list α)) (c : composition L.join.length)
(h : map length L = c.blocks) : split_wrt_composition (join L) c = L :=
by simp only [eq_self_iff_true, and_self, eq_iff_join_eq, join_split_wrt_composition,
map_length_split_wrt_composition, h]
end list
/-!
### Compositions as sets
Combinatorial viewpoints on compositions, seen as finite subsets of `fin (n+1)` containing `0` and
`n`, where the points of the set (other than `n`) correspond to the leftmost points of each block.
-/
/-- Bijection between compositions of `n` and subsets of `{0, ..., n-2}`, defined by
considering the restriction of the subset to `{1, ..., n-1}` and shifting to the left by one. -/
def composition_as_set_equiv (n : ℕ) : composition_as_set n ≃ finset (fin (n - 1)) :=
{ to_fun := λ c, {i : fin (n-1) |
(⟨1 + (i : ℕ), begin
apply (add_lt_add_left i.is_lt 1).trans_le,
rw [nat.succ_eq_add_one, add_comm],
exact add_le_add (nat.sub_le n 1) (le_refl 1)
end ⟩ : fin n.succ) ∈ c.boundaries}.to_finset,
inv_fun := λ s,
{ boundaries := {i : fin n.succ | (i = 0) ∨ (i = fin.last n)
∨ (∃ (j : fin (n-1)) (hj : j ∈ s), (i : ℕ) = j + 1)}.to_finset,
zero_mem := by simp,
last_mem := by simp },
left_inv := begin
assume c,
ext i,
simp only [exists_prop, add_comm, set.mem_to_finset, true_or, or_true, set.mem_set_of_eq],
split,
{ rintro (rfl | rfl | ⟨j, hj1, hj2⟩),
{ exact c.zero_mem },
{ exact c.last_mem },
{ convert hj1, rwa fin.ext_iff } },
{ simp only [or_iff_not_imp_left],
assume i_mem i_ne_zero i_ne_last,
simp [fin.ext_iff] at i_ne_zero i_ne_last,
have A : (1 + (i-1) : ℕ) = (i : ℕ),
by { rw add_comm, exact nat.succ_pred_eq_of_pos (pos_iff_ne_zero.mpr i_ne_zero) },
refine ⟨⟨i - 1, _⟩, _, _⟩,
{ have : (i : ℕ) < n + 1 := i.2,
simp [nat.lt_succ_iff_lt_or_eq, i_ne_last] at this,
exact nat.pred_lt_pred i_ne_zero this },
{ convert i_mem,
rw fin.ext_iff,
simp only [fin.coe_mk, A] },
{ simp [A] } },
end,
right_inv := begin
assume s,
ext i,
have : 1 + (i : ℕ) ≠ n,
{ apply ne_of_lt,
convert add_lt_add_left i.is_lt 1,
rw add_comm,
apply (nat.succ_pred_eq_of_pos _).symm,
exact (zero_le i.val).trans_lt (i.2.trans_le (nat.sub_le n 1)) },
simp only [fin.ext_iff, exists_prop, fin.coe_zero, add_comm,
set.mem_to_finset, set.mem_set_of_eq, fin.coe_last],
erw [set.mem_set_of_eq],
simp only [this, false_or, add_right_inj, add_eq_zero_iff, one_ne_zero, false_and, fin.coe_mk],
split,
{ rintros ⟨j, js, hj⟩, convert js, exact (fin.ext_iff _ _).2 hj },
{ assume h, exact ⟨i, h, rfl⟩ }
end }
instance composition_as_set_fintype (n : ℕ) : fintype (composition_as_set n) :=
fintype.of_equiv _ (composition_as_set_equiv n).symm
lemma composition_as_set_card (n : ℕ) : fintype.card (composition_as_set n) = 2 ^ (n - 1) :=
begin
have : fintype.card (finset (fin (n-1))) = 2 ^ (n - 1), by simp,
rw ← this,
exact fintype.card_congr (composition_as_set_equiv n)
end
namespace composition_as_set
variables (c : composition_as_set n)
lemma boundaries_nonempty : c.boundaries.nonempty :=
⟨0, c.zero_mem⟩
lemma card_boundaries_pos : 0 < finset.card c.boundaries :=
finset.card_pos.mpr c.boundaries_nonempty
/-- Number of blocks in a `composition_as_set`. -/
def length : ℕ := finset.card c.boundaries - 1
lemma card_boundaries_eq_succ_length : c.boundaries.card = c.length + 1 :=
(nat.sub_eq_iff_eq_add c.card_boundaries_pos).mp rfl
lemma length_lt_card_boundaries : c.length < c.boundaries.card :=
by { rw c.card_boundaries_eq_succ_length, exact lt_add_one _ }
lemma lt_length (i : fin c.length) : (i : ℕ) + 1 < c.boundaries.card :=
nat.add_lt_of_lt_sub_right i.2
lemma lt_length' (i : fin c.length) : (i : ℕ) < c.boundaries.card :=
lt_of_le_of_lt (nat.le_succ i) (c.lt_length i)
/-- Canonical increasing bijection from `fin c.boundaries.card` to `c.boundaries`. -/
def boundary : fin c.boundaries.card ↪o fin (n + 1) := c.boundaries.order_emb_of_fin rfl
@[simp] lemma boundary_zero : (c.boundary ⟨0, c.card_boundaries_pos⟩ : fin (n + 1)) = 0 :=
begin
rw [boundary, finset.order_emb_of_fin_zero rfl c.card_boundaries_pos],
exact le_antisymm (finset.min'_le _ _ c.zero_mem) (fin.zero_le _),
end
@[simp] lemma boundary_length : c.boundary ⟨c.length, c.length_lt_card_boundaries⟩ = fin.last n :=
begin
convert finset.order_emb_of_fin_last rfl c.card_boundaries_pos,
exact le_antisymm (finset.le_max' _ _ c.last_mem) (fin.le_last _)
end
/-- Size of the `i`-th block in a `composition_as_set`, seen as a function on `fin c.length`. -/
def blocks_fun (i : fin c.length) : ℕ :=
(c.boundary ⟨(i : ℕ) + 1, c.lt_length i⟩) - (c.boundary ⟨i, c.lt_length' i⟩)
lemma blocks_fun_pos (i : fin c.length) : 0 < c.blocks_fun i :=
begin
have : (⟨i, c.lt_length' i⟩ : fin c.boundaries.card) < ⟨i + 1, c.lt_length i⟩ :=
nat.lt_succ_self _,
exact nat.lt_sub_left_of_add_lt ((c.boundaries.order_emb_of_fin rfl).strict_mono this)
end
/-- List of the sizes of the blocks in a `composition_as_set`. -/
def blocks (c : composition_as_set n) : list ℕ :=
of_fn c.blocks_fun
@[simp] lemma blocks_length : c.blocks.length = c.length :=
length_of_fn _
lemma blocks_partial_sum {i : ℕ} (h : i < c.boundaries.card) :
(c.blocks.take i).sum = c.boundary ⟨i, h⟩ :=
begin
induction i with i IH, { simp },
have A : i < c.blocks.length,
{ rw c.card_boundaries_eq_succ_length at h,
simp [blocks, nat.lt_of_succ_lt_succ h] },
have B : i < c.boundaries.card := lt_of_lt_of_le A (by simp [blocks, length, nat.sub_le]),
rw [sum_take_succ _ _ A, IH B],
simp only [blocks, blocks_fun, nth_le_of_fn'],
apply nat.add_sub_cancel',
simp
end
lemma mem_boundaries_iff_exists_blocks_sum_take_eq {j : fin (n+1)} :
j ∈ c.boundaries ↔ ∃ i < c.boundaries.card, (c.blocks.take i).sum = j :=
begin
split,
{ assume hj,
rcases (c.boundaries.order_iso_of_fin rfl).surjective ⟨j, hj⟩ with ⟨i, hi⟩,
rw [subtype.ext_iff, subtype.coe_mk] at hi,
refine ⟨i.1, i.2, _⟩,
rw [← hi, c.blocks_partial_sum i.2],
refl },
{ rintros ⟨i, hi, H⟩,
convert (c.boundaries.order_iso_of_fin rfl ⟨i, hi⟩).2,
have : c.boundary ⟨i, hi⟩ = j, by rwa [fin.ext_iff, ← c.blocks_partial_sum hi],
exact this.symm }
end
lemma blocks_sum : c.blocks.sum = n :=
begin
have : c.blocks.take c.length = c.blocks := take_all_of_le (by simp [blocks]),
rw [← this, c.blocks_partial_sum c.length_lt_card_boundaries, c.boundary_length],
refl
end
/-- Associating a `composition n` to a `composition_as_set n`, by registering the sizes of the
blocks as a list of positive integers. -/
def to_composition : composition n :=
{ blocks := c.blocks,
blocks_pos := by simp only [blocks, forall_mem_of_fn_iff, blocks_fun_pos c, forall_true_iff],
blocks_sum := c.blocks_sum }
end composition_as_set
/-!
### Equivalence between compositions and compositions as sets
In this section, we explain how to go back and forth between a `composition` and a
`composition_as_set`, by showing that their `blocks` and `length` and `boundaries` correspond to
each other, and construct an equivalence between them called `composition_equiv`.
-/
@[simp] lemma composition.to_composition_as_set_length (c : composition n) :
c.to_composition_as_set.length = c.length :=
by simp [composition.to_composition_as_set, composition_as_set.length,
c.card_boundaries_eq_succ_length]
@[simp] lemma composition_as_set.to_composition_length (c : composition_as_set n) :
c.to_composition.length = c.length :=
by simp [composition_as_set.to_composition, composition.length, composition.blocks]
@[simp] lemma composition.to_composition_as_set_blocks (c : composition n) :
c.to_composition_as_set.blocks = c.blocks :=
begin
let d := c.to_composition_as_set,
change d.blocks = c.blocks,
have length_eq : d.blocks.length = c.blocks.length,
{ convert c.to_composition_as_set_length,
simp [composition_as_set.blocks] },
suffices H : ∀ (i ≤ d.blocks.length), (d.blocks.take i).sum = (c.blocks.take i).sum,
from eq_of_sum_take_eq length_eq H,
assume i hi,
have i_lt : i < d.boundaries.card,
{ convert nat.lt_succ_iff.2 hi,
convert d.card_boundaries_eq_succ_length,
exact length_of_fn _ },
have i_lt' : i < c.boundaries.card := i_lt,
have i_lt'' : i < c.length + 1, by rwa c.card_boundaries_eq_succ_length at i_lt',
have A : d.boundaries.order_emb_of_fin rfl ⟨i, i_lt⟩
= c.boundaries.order_emb_of_fin c.card_boundaries_eq_succ_length ⟨i, i_lt''⟩ := rfl,
have B : c.size_up_to i = c.boundary ⟨i, i_lt''⟩ := rfl,
rw [d.blocks_partial_sum i_lt, composition_as_set.boundary, ← composition.size_up_to, B,
A, c.order_emb_of_fin_boundaries]
end
@[simp] lemma composition_as_set.to_composition_blocks (c : composition_as_set n) :
c.to_composition.blocks = c.blocks := rfl
@[simp] lemma composition_as_set.to_composition_boundaries (c : composition_as_set n) :
c.to_composition.boundaries = c.boundaries :=
begin
ext j,
simp [c.mem_boundaries_iff_exists_blocks_sum_take_eq, c.card_boundaries_eq_succ_length,
composition.boundary, fin.ext_iff, composition.size_up_to, exists_prop, finset.mem_univ,
take, exists_prop_of_true, finset.mem_image, composition_as_set.to_composition_blocks,
composition.boundaries],
split,
{ rintros ⟨i, hi⟩,
refine ⟨i.1, _, hi⟩,
convert i.2,
simp },
{ rintros ⟨i, i_lt, hi⟩,
have : i < c.to_composition.length + 1, by simpa using i_lt,
exact ⟨⟨i, this⟩, hi⟩ }
end
@[simp] lemma composition.to_composition_as_set_boundaries (c : composition n) :
c.to_composition_as_set.boundaries = c.boundaries := rfl
/-- Equivalence between `composition n` and `composition_as_set n`. -/
def composition_equiv (n : ℕ) : composition n ≃ composition_as_set n :=
{ to_fun := λ c, c.to_composition_as_set,
inv_fun := λ c, c.to_composition,
left_inv := λ c, by { ext1, exact c.to_composition_as_set_blocks },
right_inv := λ c, by { ext1, exact c.to_composition_boundaries } }
instance composition_fintype (n : ℕ) : fintype (composition n) :=
fintype.of_equiv _ (composition_equiv n).symm
lemma composition_card (n : ℕ) : fintype.card (composition n) = 2 ^ (n - 1) :=
begin
rw ← composition_as_set_card n,
exact fintype.card_congr (composition_equiv n)
end
|
f74828f79efd8061b4c5b8952620359d028213d1 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/topology/category/Top/limits.lean | b92f7cf3f362ed1e8c2e61280629ac3d9f1f5711 | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 3,920 | lean | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro
-/
import topology.category.Top.basic
import category_theory.limits.types
import category_theory.limits.preserves.basic
open topological_space
open category_theory
open category_theory.limits
universe u
noncomputable theory
namespace Top
variables {J : Type u} [small_category J]
local notation `forget` := forget Top
/--
A choice of limit cone for a functor `F : J ⥤ Top`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `types.limit_cone`).
-/
def limit_cone (F : J ⥤ Top.{u}) : cone F :=
{ X := ⟨(types.limit_cone (F ⋙ forget)).X, ⨅j, (F.obj j).str.induced ((types.limit_cone (F ⋙ forget)).π.app j)⟩,
π :=
{ app := λ j, ⟨(types.limit_cone (F ⋙ forget)).π.app j, continuous_iff_le_induced.mpr (infi_le _ _)⟩,
naturality' := λ j j' f, continuous_map.coe_inj ((types.limit_cone (F ⋙ forget)).π.naturality f) } }
/--
The chosen cone `Top.limit_cone F` for a functor `F : J ⥤ Top` is a limit cone.
Generally you should just use `limit.is_limit F`, unless you need the actual definition
(which is in terms of `types.limit_cone_is_limit`).
-/
def limit_cone_is_limit (F : J ⥤ Top.{u}) : is_limit (limit_cone F) :=
by { refine is_limit.of_faithful forget (types.limit_cone_is_limit _) (λ s, ⟨_, _⟩) (λ s, rfl),
exact continuous_iff_coinduced_le.mpr (le_infi $ λ j,
coinduced_le_iff_le_induced.mp $ continuous_iff_coinduced_le.mp (s.π.app j).continuous) }
instance Top_has_limits : has_limits.{u} Top.{u} :=
{ has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit.mk { cone := limit_cone F, is_limit := limit_cone_is_limit F } } }
instance forget_preserves_limits : preserves_limits (forget : Top.{u} ⥤ Type u) :=
{ preserves_limits_of_shape := λ J 𝒥,
{ preserves_limit := λ F,
by exactI preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F) (types.limit_cone_is_limit (F ⋙ forget)) } }
/--
A choice of colimit cocone for a functor `F : J ⥤ Top`.
Generally you should just use `colimit.coone F`, unless you need the actual definition
(which is in terms of `types.colimit_cocone`).
-/
def colimit_cocone (F : J ⥤ Top.{u}) : cocone F :=
{ X := ⟨(types.colimit_cocone (F ⋙ forget)).X, ⨆ j, (F.obj j).str.coinduced ((types.colimit_cocone (F ⋙ forget)).ι.app j)⟩,
ι :=
{ app := λ j, ⟨(types.colimit_cocone (F ⋙ forget)).ι.app j, continuous_iff_coinduced_le.mpr (le_supr _ j)⟩,
naturality' := λ j j' f, continuous_map.coe_inj ((types.colimit_cocone (F ⋙ forget)).ι.naturality f) } }
/--
The chosen cocone `Top.colimit_cocone F` for a functor `F : J ⥤ Top` is a colimit cocone.
Generally you should just use `colimit.is_colimit F`, unless you need the actual definition
(which is in terms of `types.colimit_cocone_is_colimit`).
-/
def colimit_cocone_is_colimit (F : J ⥤ Top.{u}) : is_colimit (colimit_cocone F) :=
by { refine is_colimit.of_faithful forget (types.colimit_cocone_is_colimit _) (λ s, ⟨_, _⟩) (λ s, rfl),
exact continuous_iff_le_induced.mpr (supr_le $ λ j,
coinduced_le_iff_le_induced.mp $ continuous_iff_coinduced_le.mp (s.ι.app j).continuous) }
instance Top_has_colimits : has_colimits.{u} Top.{u} :=
{ has_colimits_of_shape := λ J 𝒥, by exactI
{ has_colimit := λ F, has_colimit.mk { cocone := colimit_cocone F, is_colimit := colimit_cocone_is_colimit F } } }
instance forget_preserves_colimits : preserves_colimits (forget : Top.{u} ⥤ Type u) :=
{ preserves_colimits_of_shape := λ J 𝒥,
{ preserves_colimit := λ F,
by exactI preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit F) (types.colimit_cocone_is_colimit (F ⋙ forget)) } }
end Top
|
9c01cb98c198a8101f975d7d479e3c835f9c11a8 | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /hott/algebra/homotopy_group.hlean | 8844ba18c0227b608674bbf260f2bea7cbc5e9ef | [
"Apache-2.0"
] | permissive | soonhokong/lean-osx | 4a954262c780e404c1369d6c06516161d07fcb40 | 3670278342d2f4faa49d95b46d86642d7875b47c | refs/heads/master | 1,611,410,334,552 | 1,474,425,686,000 | 1,474,425,686,000 | 12,043,103 | 5 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 5,122 | hlean | /-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
homotopy groups of a pointed space
-/
import .trunc_group .hott types.trunc
open nat eq pointed trunc is_trunc algebra
namespace eq
definition phomotopy_group [constructor] (n : ℕ) (A : Type*) : Set* :=
ptrunc 0 (Ω[n] A)
definition homotopy_group [reducible] (n : ℕ) (A : Type*) : Type :=
phomotopy_group n A
notation `π*[`:95 n:0 `] `:0 A:95 := phomotopy_group n A
notation `π[`:95 n:0 `] `:0 A:95 := homotopy_group n A
definition group_homotopy_group [instance] [constructor] (n : ℕ) (A : Type*)
: group (π[succ n] A) :=
trunc_group concat inverse idp con.assoc idp_con con_idp con.left_inv
definition comm_group_homotopy_group [constructor] (n : ℕ) (A : Type*)
: comm_group (π[succ (succ n)] A) :=
trunc_comm_group concat inverse idp con.assoc idp_con con_idp con.left_inv eckmann_hilton
local attribute comm_group_homotopy_group [instance]
definition ghomotopy_group [constructor] (n : ℕ) (A : Type*) : Group :=
Group.mk (π[succ n] A) _
definition cghomotopy_group [constructor] (n : ℕ) (A : Type*) : CommGroup :=
CommGroup.mk (π[succ (succ n)] A) _
definition fundamental_group [constructor] (A : Type*) : Group :=
ghomotopy_group zero A
notation `πg[`:95 n:0 ` +1] `:0 A:95 := ghomotopy_group n A
notation `πag[`:95 n:0 ` +2] `:0 A:95 := cghomotopy_group n A
prefix `π₁`:95 := fundamental_group
definition phomotopy_group_pequiv [constructor] (n : ℕ) {A B : Type*} (H : A ≃* B)
: π*[n] A ≃* π*[n] B :=
ptrunc_pequiv 0 (iterated_loop_space_pequiv n H)
set_option pp.coercions true
set_option pp.numerals false
definition phomotopy_group_pequiv_loop_ptrunc [constructor] (k : ℕ) (A : Type*) :
π*[k] A ≃* Ω[k] (ptrunc k A) :=
begin
refine !iterated_loop_ptrunc_pequiv⁻¹ᵉ* ⬝e* _,
rewrite [trunc_index.zero_add]
end
open equiv unit
theorem trivial_homotopy_of_is_set (A : Type*) [H : is_set A] (n : ℕ) : πg[n+1] A = G0 :=
begin
apply trivial_group_of_is_contr,
apply is_trunc_trunc_of_is_trunc,
apply is_contr_loop_of_is_trunc,
apply is_trunc_succ_succ_of_is_set
end
definition phomotopy_group_succ_out (A : Type*) (n : ℕ) : π*[n + 1] A = π₁ Ω[n] A := idp
definition phomotopy_group_succ_in (A : Type*) (n : ℕ) : π*[n + 1] A = π*[n] Ω A :=
ap (ptrunc 0) (loop_space_succ_eq_in A n)
definition ghomotopy_group_succ_out (A : Type*) (n : ℕ) : πg[n +1] A = π₁ Ω[n] A := idp
definition ghomotopy_group_succ_in (A : Type*) (n : ℕ) : πg[succ n +1] A = πg[n +1] Ω A :=
begin
fapply Group_eq,
{ apply equiv_of_eq, exact ap (ptrunc 0) (loop_space_succ_eq_in A (succ n))},
{ exact abstract [irreducible] begin refine trunc.rec _, intro p, refine trunc.rec _, intro q,
rewrite [▸*,-+tr_eq_cast_ap, +trunc_transport], refine !trunc_transport ⬝ _, apply ap tr,
apply loop_space_succ_eq_in_concat end end},
end
definition homotopy_group_add (A : Type*) (n m : ℕ) : πg[n+m +1] A = πg[n +1] Ω[m] A :=
begin
revert A, induction m with m IH: intro A,
{ reflexivity},
{ esimp [iterated_ploop_space, nat.add], refine !ghomotopy_group_succ_in ⬝ _, refine !IH ⬝ _,
exact ap (ghomotopy_group n) !loop_space_succ_eq_in⁻¹}
end
theorem trivial_homotopy_add_of_is_set_loop_space {A : Type*} {n : ℕ} (m : ℕ)
(H : is_set (Ω[n] A)) : πg[m+n+1] A = G0 :=
!homotopy_group_add ⬝ !trivial_homotopy_of_is_set
theorem trivial_homotopy_le_of_is_set_loop_space {A : Type*} {n : ℕ} (m : ℕ) (H1 : n ≤ m)
(H2 : is_set (Ω[n] A)) : πg[m+1] A = G0 :=
obtain (k : ℕ) (p : n + k = m), from le.elim H1,
ap (λx, πg[x+1] A) (p⁻¹ ⬝ add.comm n k) ⬝ trivial_homotopy_add_of_is_set_loop_space k H2
definition phomotopy_group_functor [constructor] (n : ℕ) {A B : Type*} (f : A →* B)
: π*[n] A →* π*[n] B :=
ptrunc_functor 0 (apn n f)
definition homotopy_group_functor (n : ℕ) {A B : Type*} (f : A →* B) : π[n] A → π[n] B :=
phomotopy_group_functor n f
notation `π→*[`:95 n:0 `] `:0 f:95 := phomotopy_group_functor n f
notation `π→[`:95 n:0 `] `:0 f:95 := homotopy_group_functor n f
definition tinverse [constructor] {X : Type*} : π*[1] X →* π*[1] X :=
ptrunc_functor 0 pinverse
definition ptrunc_functor_pinverse [constructor] {X : Type*}
: ptrunc_functor 0 (@pinverse X) ~* @tinverse X :=
begin
fapply phomotopy.mk,
{ reflexivity},
{ reflexivity}
end
definition phomotopy_group_functor_mul [constructor] (n : ℕ) {A B : Type*} (g : A →* B)
(p q : πg[n+1] A) :
(π→[n + 1] g) (p *[πg[n+1] A] q) = (π→[n + 1] g) p *[πg[n+1] B] (π→[n + 1] g) q :=
begin
unfold [ghomotopy_group, homotopy_group] at *,
refine @trunc.rec _ _ _ (λq, !is_trunc_eq) _ p, clear p, intro p,
refine @trunc.rec _ _ _ (λq, !is_trunc_eq) _ q, clear q, intro q,
apply ap tr, apply apn_con
end
end eq
|
88e61c71f0e1b28946ae10e90967644508f0eeed | 097294e9b80f0d9893ac160b9c7219aa135b51b9 | /assignments/hw2.lean | edfe5d819b72e4e6a36f27a7a9295a2df22aa828 | [] | no_license | AbigailCastro17/CS2102-Discrete-Math | cf296251be9418ce90206f5e66bde9163e21abf9 | d741e4d2d6a9b2e0c8380e51706218b8f608cee4 | refs/heads/main | 1,682,891,087,358 | 1,621,401,341,000 | 1,621,401,341,000 | 368,749,959 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,890 | lean | /-
UVa CS2102/Sullivan, Spring 2020, Homework #2
This homework assignment is due by noon on Tuesday,
Feb 4. Submit your result through the HW#2 tab under
the Assignments category on Collab. Do so by uploading
a completed version of this file.
The goal of this assignment is to develop and evaluate
your ability to write simple abstract data types in Lean,
comprising both inductive data definitions and definitions
of functions, in several syntactic styles, that operate
on values of such types.
-/
/- [49 points]
#1. In the space after this comment, first define a data
type, dm_bool, as we did in class, with values dm_tt and
dm_ff. We will take the values of this type to represent
the Boolean algebra truth values, true and false. Then
define functions operating on values of type dm_bool that
implement the Boolean functions, not, and, or, nand, xor,
implies, an equiv (iff).
Note: The heads-up announcement of a few days ago mis-stated
the types of these operations as involving values of type
bool. You must use dm_bool throughout. The point is that you
are now seeing how to specify/implement Boolean algebra, not
just to use Boolean algebra functions built in to a language.
Precede each of your function definitions with a comment
presenting the "truth table" for the function to be
defined. Then *after* each function definition, use Lean's
#eval or #reduce command to test it for all possible
combinations of argument values. For example, you should
have four test cases for each binary function, for each of
the four combinations of two Boolean values. You may use
resources such as Wikipedia to learn the truth tables for
each of these functions if you don't already know them.
-/
-- Answers here
/- [51 points]
#2. In a separate file called months.lean, define a new
abstract data type. It will define a data type, months,
the values of which are the names of the months. Use all
lower case names for months, e.g., january.
-/
-- Answer here
/-
Complete your "month" ADT definition with definitions of
two functions, next_month and is_winter_month, as follows.
However, to practice writing functions in different ways,
write each of the two functions in each of the following
styles, adding "prime" marks to the function names so as
to avoid naming conflicts:
- lambda expression (with a match/with expression)
- C-style (with a match/with expression)
- by cases (no explicit match/with expression needed)
-/
/-
A. Given a month as an argument, return the next month in
the sequence of months of the year. E.g., the function
application, (next_month december), will return january.
-/
-- Answer here
/-
B. Given a month as an argument, return the dm_bool
value, dm_tt (representing "true"), if the given month
is a winter month (december, january, or february),
and dm_ff otherwise. Do not use more than four pattern
matching rules.
-/
|
c15df2c4321920ec4bffbde2f84b79c1bfd3c8b4 | 78630e908e9624a892e24ebdd21260720d29cf55 | /src/logic_first_order/fol_08.lean | beaba7c4db1ae4278e8ea926edacf192e98259ee | [
"CC0-1.0"
] | permissive | tomasz-lisowski/lean-logic-examples | 84e612466776be0a16c23a0439ff8ef6114ddbe1 | 2b2ccd467b49c3989bf6c92ec0358a8d6ee68c5d | refs/heads/master | 1,683,334,199,431 | 1,621,938,305,000 | 1,621,938,305,000 | 365,041,573 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 351 | lean | namespace fol_08
variable A : Type
variable f : A → A
variable P : A → Prop
variable h : ∀ x, P x → P (f x)
include h
theorem fol_08 : ∀ y, P y → P (f (f y)) :=
assume y (h1: P y),
have h2: P y → P (f y), from h y,
have h3: P (f y), from h2 h1,
have h4: P (f y) → P (f (f y)), from h (f y),
show P (f (f y)), from h4 h3
end fol_08 |
56bcbaa1a6b14787c4e1c9f0361507b5714fda28 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/measure_theory/measure/vector_measure.lean | c6600b51af6429f06ac6ad7b370b1853857f733e | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 49,929 | lean | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import measure_theory.measure.measure_space
import analysis.complex.basic
/-!
# Vector valued measures
This file defines vector valued measures, which are σ-additive functions from a set to a add monoid
`M` such that it maps the empty set and non-measurable sets to zero. In the case
that `M = ℝ`, we called the vector measure a signed measure and write `signed_measure α`.
Similarly, when `M = ℂ`, we call the measure a complex measure and write `complex_measure α`.
## Main definitions
* `measure_theory.vector_measure` is a vector valued, σ-additive function that maps the empty
and non-measurable set to zero.
* `measure_theory.vector_measure.map` is the pushforward of a vector measure along a function.
* `measure_theory.vector_measure.restrict` is the restriction of a vector measure on some set.
## Notation
* `v ≤[i] w` means that the vector measure `v` restricted on the set `i` is less than or equal
to the vector measure `w` restricted on `i`, i.e. `v.restrict i ≤ w.restrict i`.
## Implementation notes
We require all non-measurable sets to be mapped to zero in order for the extensionality lemma
to only compare the underlying functions for measurable sets.
We use `has_sum` instead of `tsum` in the definition of vector measures in comparison to `measure`
since this provides summablity.
## Tags
vector measure, signed measure, complex measure
-/
noncomputable theory
open_locale classical big_operators nnreal ennreal measure_theory
namespace measure_theory
variables {α β : Type*} {m : measurable_space α}
/-- A vector measure on a measurable space `α` is a σ-additive `M`-valued function (for some `M`
an add monoid) such that the empty set and non-measurable sets are mapped to zero. -/
structure vector_measure (α : Type*) [measurable_space α]
(M : Type*) [add_comm_monoid M] [topological_space M] :=
(measure_of' : set α → M)
(empty' : measure_of' ∅ = 0)
(not_measurable' ⦃i : set α⦄ : ¬ measurable_set i → measure_of' i = 0)
(m_Union' ⦃f : ℕ → set α⦄ :
(∀ i, measurable_set (f i)) → pairwise (disjoint on f) →
has_sum (λ i, measure_of' (f i)) (measure_of' (⋃ i, f i)))
/-- A `signed_measure` is a `ℝ`-vector measure. -/
abbreviation signed_measure (α : Type*) [measurable_space α] := vector_measure α ℝ
/-- A `complex_measure` is a `ℂ`-vector_measure. -/
abbreviation complex_measure (α : Type*) [measurable_space α] := vector_measure α ℂ
open set measure_theory
namespace vector_measure
section
variables {M : Type*} [add_comm_monoid M] [topological_space M]
include m
instance : has_coe_to_fun (vector_measure α M) (λ _, set α → M) :=
⟨vector_measure.measure_of'⟩
initialize_simps_projections vector_measure (measure_of' → apply)
@[simp]
lemma measure_of_eq_coe (v : vector_measure α M) : v.measure_of' = v := rfl
@[simp]
lemma empty (v : vector_measure α M) : v ∅ = 0 := v.empty'
lemma not_measurable (v : vector_measure α M)
{i : set α} (hi : ¬ measurable_set i) : v i = 0 := v.not_measurable' hi
lemma m_Union (v : vector_measure α M) {f : ℕ → set α}
(hf₁ : ∀ i, measurable_set (f i)) (hf₂ : pairwise (disjoint on f)) :
has_sum (λ i, v (f i)) (v (⋃ i, f i)) :=
v.m_Union' hf₁ hf₂
lemma of_disjoint_Union_nat [t2_space M] (v : vector_measure α M) {f : ℕ → set α}
(hf₁ : ∀ i, measurable_set (f i)) (hf₂ : pairwise (disjoint on f)) :
v (⋃ i, f i) = ∑' i, v (f i) :=
(v.m_Union hf₁ hf₂).tsum_eq.symm
lemma coe_injective : @function.injective (vector_measure α M) (set α → M) coe_fn :=
λ v w h, by { cases v, cases w, congr' }
lemma ext_iff' (v w : vector_measure α M) :
v = w ↔ ∀ i : set α, v i = w i :=
by rw [← coe_injective.eq_iff, function.funext_iff]
lemma ext_iff (v w : vector_measure α M) :
v = w ↔ ∀ i : set α, measurable_set i → v i = w i :=
begin
split,
{ rintro rfl _ _, refl },
{ rw ext_iff',
intros h i,
by_cases hi : measurable_set i,
{ exact h i hi },
{ simp_rw [not_measurable _ hi] } }
end
@[ext] lemma ext {s t : vector_measure α M}
(h : ∀ i : set α, measurable_set i → s i = t i) : s = t :=
(ext_iff s t).2 h
variables [t2_space M] {v : vector_measure α M} {f : ℕ → set α}
lemma has_sum_of_disjoint_Union [encodable β] {f : β → set α}
(hf₁ : ∀ i, measurable_set (f i)) (hf₂ : pairwise (disjoint on f)) :
has_sum (λ i, v (f i)) (v (⋃ i, f i)) :=
begin
set g := λ i : ℕ, ⋃ (b : β) (H : b ∈ encodable.decode₂ β i), f b with hg,
have hg₁ : ∀ i, measurable_set (g i),
{ exact λ _, measurable_set.Union (λ b, measurable_set.Union_Prop $ λ _, hf₁ b) },
have hg₂ : pairwise (disjoint on g),
{ exact encodable.Union_decode₂_disjoint_on hf₂ },
have := v.of_disjoint_Union_nat hg₁ hg₂,
rw [hg, encodable.Union_decode₂] at this,
have hg₃ : (λ (i : β), v (f i)) = (λ i, v (g (encodable.encode i))),
{ ext, rw hg, simp only,
congr, ext y, simp only [exists_prop, mem_Union, option.mem_def],
split,
{ intro hy,
refine ⟨x, (encodable.decode₂_is_partial_inv _ _).2 rfl, hy⟩ },
{ rintro ⟨b, hb₁, hb₂⟩,
rw (encodable.decode₂_is_partial_inv _ _) at hb₁,
rwa ← encodable.encode_injective hb₁ } },
rw [summable.has_sum_iff, this, ← tsum_Union_decode₂],
{ exact v.empty },
{ rw hg₃, change summable ((λ i, v (g i)) ∘ encodable.encode),
rw function.injective.summable_iff encodable.encode_injective,
{ exact (v.m_Union hg₁ hg₂).summable },
{ intros x hx,
convert v.empty,
simp only [Union_eq_empty, option.mem_def, not_exists, mem_range] at ⊢ hx,
intros i hi,
exact false.elim ((hx i) ((encodable.decode₂_is_partial_inv _ _).1 hi)) } }
end
lemma of_disjoint_Union [encodable β] {f : β → set α}
(hf₁ : ∀ i, measurable_set (f i)) (hf₂ : pairwise (disjoint on f)) :
v (⋃ i, f i) = ∑' i, v (f i) :=
(has_sum_of_disjoint_Union hf₁ hf₂).tsum_eq.symm
lemma of_union {A B : set α}
(h : disjoint A B) (hA : measurable_set A) (hB : measurable_set B) :
v (A ∪ B) = v A + v B :=
begin
rw [union_eq_Union, of_disjoint_Union, tsum_fintype, fintype.sum_bool, cond, cond],
exacts [λ b, bool.cases_on b hB hA, pairwise_disjoint_on_bool.2 h]
end
lemma of_add_of_diff {A B : set α} (hA : measurable_set A) (hB : measurable_set B)
(h : A ⊆ B) : v A + v (B \ A) = v B :=
begin
rw [← of_union disjoint_diff hA (hB.diff hA), union_diff_cancel h],
apply_instance,
end
lemma of_diff {M : Type*} [add_comm_group M]
[topological_space M] [t2_space M] {v : vector_measure α M}
{A B : set α} (hA : measurable_set A) (hB : measurable_set B)
(h : A ⊆ B) : v (B \ A) = v B - (v A) :=
begin
rw [← of_add_of_diff hA hB h, add_sub_cancel'],
apply_instance,
end
lemma of_diff_of_diff_eq_zero {A B : set α}
(hA : measurable_set A) (hB : measurable_set B) (h' : v (B \ A) = 0) :
v (A \ B) + v B = v A :=
begin
symmetry,
calc v A = v (A \ B ∪ A ∩ B) : by simp only [set.diff_union_inter]
... = v (A \ B) + v (A ∩ B) :
by { rw of_union,
{ rw disjoint.comm,
exact set.disjoint_of_subset_left (A.inter_subset_right B) set.disjoint_diff },
{ exact hA.diff hB },
{ exact hA.inter hB } }
... = v (A \ B) + v (A ∩ B ∪ B \ A) :
by { rw [of_union, h', add_zero],
{ exact set.disjoint_of_subset_left (A.inter_subset_left B) set.disjoint_diff },
{ exact hA.inter hB },
{ exact hB.diff hA } }
... = v (A \ B) + v B :
by { rw [set.union_comm, set.inter_comm, set.diff_union_inter] }
end
lemma of_Union_nonneg {M : Type*} [topological_space M]
[ordered_add_comm_monoid M] [order_closed_topology M]
{v : vector_measure α M} (hf₁ : ∀ i, measurable_set (f i))
(hf₂ : pairwise (disjoint on f)) (hf₃ : ∀ i, 0 ≤ v (f i)) :
0 ≤ v (⋃ i, f i) :=
(v.of_disjoint_Union_nat hf₁ hf₂).symm ▸ tsum_nonneg hf₃
lemma of_Union_nonpos {M : Type*} [topological_space M]
[ordered_add_comm_monoid M] [order_closed_topology M]
{v : vector_measure α M} (hf₁ : ∀ i, measurable_set (f i))
(hf₂ : pairwise (disjoint on f)) (hf₃ : ∀ i, v (f i) ≤ 0) :
v (⋃ i, f i) ≤ 0 :=
(v.of_disjoint_Union_nat hf₁ hf₂).symm ▸ tsum_nonpos hf₃
lemma of_nonneg_disjoint_union_eq_zero {s : signed_measure α} {A B : set α}
(h : disjoint A B) (hA₁ : measurable_set A) (hB₁ : measurable_set B)
(hA₂ : 0 ≤ s A) (hB₂ : 0 ≤ s B)
(hAB : s (A ∪ B) = 0) : s A = 0 :=
begin
rw of_union h hA₁ hB₁ at hAB,
linarith,
apply_instance,
end
lemma of_nonpos_disjoint_union_eq_zero {s : signed_measure α} {A B : set α}
(h : disjoint A B) (hA₁ : measurable_set A) (hB₁ : measurable_set B)
(hA₂ : s A ≤ 0) (hB₂ : s B ≤ 0)
(hAB : s (A ∪ B) = 0) : s A = 0 :=
begin
rw of_union h hA₁ hB₁ at hAB,
linarith,
apply_instance,
end
end
section has_scalar
variables {M : Type*} [add_comm_monoid M] [topological_space M]
variables {R : Type*} [semiring R] [distrib_mul_action R M] [has_continuous_const_smul R M]
include m
/-- Given a real number `r` and a signed measure `s`, `smul r s` is the signed
measure corresponding to the function `r • s`. -/
def smul (r : R) (v : vector_measure α M) : vector_measure α M :=
{ measure_of' := r • v,
empty' := by rw [pi.smul_apply, empty, smul_zero],
not_measurable' := λ _ hi, by rw [pi.smul_apply, v.not_measurable hi, smul_zero],
m_Union' := λ _ hf₁ hf₂, has_sum.const_smul (v.m_Union hf₁ hf₂) }
instance : has_scalar R (vector_measure α M) := ⟨smul⟩
@[simp] lemma coe_smul (r : R) (v : vector_measure α M) : ⇑(r • v) = r • v := rfl
lemma smul_apply (r : R) (v : vector_measure α M) (i : set α) :
(r • v) i = r • v i := rfl
end has_scalar
section add_comm_monoid
variables {M : Type*} [add_comm_monoid M] [topological_space M]
include m
instance : has_zero (vector_measure α M) :=
⟨⟨0, rfl, λ _ _, rfl, λ _ _ _, has_sum_zero⟩⟩
instance : inhabited (vector_measure α M) := ⟨0⟩
@[simp] lemma coe_zero : ⇑(0 : vector_measure α M) = 0 := rfl
lemma zero_apply (i : set α) : (0 : vector_measure α M) i = 0 := rfl
variables [has_continuous_add M]
/-- The sum of two vector measure is a vector measure. -/
def add (v w : vector_measure α M) : vector_measure α M :=
{ measure_of' := v + w,
empty' := by simp,
not_measurable' := λ _ hi,
by simp [v.not_measurable hi, w.not_measurable hi],
m_Union' := λ f hf₁ hf₂,
has_sum.add (v.m_Union hf₁ hf₂) (w.m_Union hf₁ hf₂) }
instance : has_add (vector_measure α M) := ⟨add⟩
@[simp] lemma coe_add (v w : vector_measure α M) : ⇑(v + w) = v + w := rfl
lemma add_apply (v w : vector_measure α M) (i : set α) : (v + w) i = v i + w i := rfl
instance : add_comm_monoid (vector_measure α M) :=
function.injective.add_comm_monoid _ coe_injective coe_zero coe_add (λ _ _, coe_smul _ _)
/-- `coe_fn` is an `add_monoid_hom`. -/
@[simps]
def coe_fn_add_monoid_hom : vector_measure α M →+ (set α → M) :=
{ to_fun := coe_fn, map_zero' := coe_zero, map_add' := coe_add }
end add_comm_monoid
section add_comm_group
variables {M : Type*} [add_comm_group M] [topological_space M] [topological_add_group M]
include m
/-- The negative of a vector measure is a vector measure. -/
def neg (v : vector_measure α M) : vector_measure α M :=
{ measure_of' := -v,
empty' := by simp,
not_measurable' := λ _ hi, by simp [v.not_measurable hi],
m_Union' := λ f hf₁ hf₂, has_sum.neg $ v.m_Union hf₁ hf₂ }
instance : has_neg (vector_measure α M) := ⟨neg⟩
@[simp] lemma coe_neg (v : vector_measure α M) : ⇑(-v) = - v := rfl
lemma neg_apply (v : vector_measure α M) (i : set α) :(-v) i = - v i := rfl
/-- The difference of two vector measure is a vector measure. -/
def sub (v w : vector_measure α M) : vector_measure α M :=
{ measure_of' := v - w,
empty' := by simp,
not_measurable' := λ _ hi,
by simp [v.not_measurable hi, w.not_measurable hi],
m_Union' := λ f hf₁ hf₂,
has_sum.sub (v.m_Union hf₁ hf₂)
(w.m_Union hf₁ hf₂) }
instance : has_sub (vector_measure α M) := ⟨sub⟩
@[simp] lemma coe_sub (v w : vector_measure α M) : ⇑(v - w) = v - w := rfl
lemma sub_apply (v w : vector_measure α M) (i : set α) : (v - w) i = v i - w i := rfl
instance : add_comm_group (vector_measure α M) :=
function.injective.add_comm_group _ coe_injective coe_zero coe_add coe_neg coe_sub
(λ _ _, coe_smul _ _) (λ _ _, coe_smul _ _)
end add_comm_group
section distrib_mul_action
variables {M : Type*} [add_comm_monoid M] [topological_space M]
variables {R : Type*} [semiring R] [distrib_mul_action R M] [has_continuous_const_smul R M]
include m
instance [has_continuous_add M] : distrib_mul_action R (vector_measure α M) :=
function.injective.distrib_mul_action coe_fn_add_monoid_hom coe_injective coe_smul
end distrib_mul_action
section module
variables {M : Type*} [add_comm_monoid M] [topological_space M]
variables {R : Type*} [semiring R] [module R M] [has_continuous_const_smul R M]
include m
instance [has_continuous_add M] : module R (vector_measure α M) :=
function.injective.module R coe_fn_add_monoid_hom coe_injective coe_smul
end module
end vector_measure
namespace measure
include m
/-- A finite measure coerced into a real function is a signed measure. -/
@[simps]
def to_signed_measure (μ : measure α) [hμ : is_finite_measure μ] : signed_measure α :=
{ measure_of' := λ i : set α, if measurable_set i then (μ.measure_of i).to_real else 0,
empty' := by simp [μ.empty],
not_measurable' := λ _ hi, if_neg hi,
m_Union' :=
begin
intros _ hf₁ hf₂,
rw [μ.m_Union hf₁ hf₂, ennreal.tsum_to_real_eq, if_pos (measurable_set.Union hf₁),
summable.has_sum_iff],
{ congr, ext n, rw if_pos (hf₁ n) },
{ refine @summable_of_nonneg_of_le _ (ennreal.to_real ∘ μ ∘ f) _ _ _ _,
{ intro, split_ifs,
exacts [ennreal.to_real_nonneg, le_rfl] },
{ intro, split_ifs,
exacts [le_rfl, ennreal.to_real_nonneg] },
exact summable_measure_to_real hf₁ hf₂ },
{ intros a ha,
apply ne_of_lt hμ.measure_univ_lt_top,
rw [eq_top_iff, ← ha, outer_measure.measure_of_eq_coe, coe_to_outer_measure],
exact measure_mono (set.subset_univ _) }
end }
lemma to_signed_measure_apply_measurable {μ : measure α} [is_finite_measure μ]
{i : set α} (hi : measurable_set i) :
μ.to_signed_measure i = (μ i).to_real :=
if_pos hi
-- Without this lemma, `singular_part_neg` in `measure_theory.decomposition.lebesgue` is
-- extremely slow
lemma to_signed_measure_congr {μ ν : measure α} [is_finite_measure μ] [is_finite_measure ν]
(h : μ = ν) : μ.to_signed_measure = ν.to_signed_measure :=
by { congr, exact h }
lemma to_signed_measure_eq_to_signed_measure_iff
{μ ν : measure α} [is_finite_measure μ] [is_finite_measure ν] :
μ.to_signed_measure = ν.to_signed_measure ↔ μ = ν :=
begin
refine ⟨λ h, _, λ h, _⟩,
{ ext1 i hi,
have : μ.to_signed_measure i = ν.to_signed_measure i,
{ rw h },
rwa [to_signed_measure_apply_measurable hi, to_signed_measure_apply_measurable hi,
ennreal.to_real_eq_to_real] at this;
{ exact measure_ne_top _ _ } },
{ congr, assumption }
end
@[simp] lemma to_signed_measure_zero :
(0 : measure α).to_signed_measure = 0 :=
by { ext i hi, simp }
@[simp] lemma to_signed_measure_add (μ ν : measure α) [is_finite_measure μ] [is_finite_measure ν] :
(μ + ν).to_signed_measure = μ.to_signed_measure + ν.to_signed_measure :=
begin
ext i hi,
rw [to_signed_measure_apply_measurable hi, add_apply,
ennreal.to_real_add (ne_of_lt (measure_lt_top _ _ )) (ne_of_lt (measure_lt_top _ _)),
vector_measure.add_apply, to_signed_measure_apply_measurable hi,
to_signed_measure_apply_measurable hi],
all_goals { apply_instance }
end
@[simp] lemma to_signed_measure_smul (μ : measure α) [is_finite_measure μ] (r : ℝ≥0) :
(r • μ).to_signed_measure = r • μ.to_signed_measure :=
begin
ext i hi,
rw [to_signed_measure_apply_measurable hi, vector_measure.smul_apply,
to_signed_measure_apply_measurable hi, coe_smul, pi.smul_apply,
ennreal.to_real_smul],
end
/-- A measure is a vector measure over `ℝ≥0∞`. -/
@[simps]
def to_ennreal_vector_measure (μ : measure α) : vector_measure α ℝ≥0∞ :=
{ measure_of' := λ i : set α, if measurable_set i then μ i else 0,
empty' := by simp [μ.empty],
not_measurable' := λ _ hi, if_neg hi,
m_Union' := λ _ hf₁ hf₂,
begin
rw summable.has_sum_iff ennreal.summable,
{ rw [if_pos (measurable_set.Union hf₁), measure_theory.measure_Union hf₂ hf₁],
exact tsum_congr (λ n, if_pos (hf₁ n)) },
end }
lemma to_ennreal_vector_measure_apply_measurable
{μ : measure α} {i : set α} (hi : measurable_set i) :
μ.to_ennreal_vector_measure i = μ i :=
if_pos hi
@[simp] lemma to_ennreal_vector_measure_zero :
(0 : measure α).to_ennreal_vector_measure = 0 :=
by { ext i hi, simp }
@[simp] lemma to_ennreal_vector_measure_add (μ ν : measure α) :
(μ + ν).to_ennreal_vector_measure = μ.to_ennreal_vector_measure + ν.to_ennreal_vector_measure :=
begin
refine measure_theory.vector_measure.ext (λ i hi, _),
rw [to_ennreal_vector_measure_apply_measurable hi, add_apply, vector_measure.add_apply,
to_ennreal_vector_measure_apply_measurable hi, to_ennreal_vector_measure_apply_measurable hi]
end
lemma to_signed_measure_sub_apply {μ ν : measure α} [is_finite_measure μ] [is_finite_measure ν]
{i : set α} (hi : measurable_set i) :
(μ.to_signed_measure - ν.to_signed_measure) i = (μ i).to_real - (ν i).to_real :=
begin
rw [vector_measure.sub_apply, to_signed_measure_apply_measurable hi,
measure.to_signed_measure_apply_measurable hi, sub_eq_add_neg]
end
end measure
namespace vector_measure
open measure
section
/-- A vector measure over `ℝ≥0∞` is a measure. -/
def ennreal_to_measure {m : measurable_space α} (v : vector_measure α ℝ≥0∞) : measure α :=
of_measurable (λ s _, v s) v.empty (λ f hf₁ hf₂, v.of_disjoint_Union_nat hf₁ hf₂)
lemma ennreal_to_measure_apply {m : measurable_space α} {v : vector_measure α ℝ≥0∞}
{s : set α} (hs : measurable_set s) : ennreal_to_measure v s = v s :=
by rw [ennreal_to_measure, of_measurable_apply _ hs]
/-- The equiv between `vector_measure α ℝ≥0∞` and `measure α` formed by
`measure_theory.vector_measure.ennreal_to_measure` and
`measure_theory.measure.to_ennreal_vector_measure`. -/
@[simps] def equiv_measure [measurable_space α] : vector_measure α ℝ≥0∞ ≃ measure α :=
{ to_fun := ennreal_to_measure,
inv_fun := to_ennreal_vector_measure,
left_inv := λ _, ext (λ s hs,
by rw [to_ennreal_vector_measure_apply_measurable hs, ennreal_to_measure_apply hs]),
right_inv := λ _, measure.ext (λ s hs,
by rw [ennreal_to_measure_apply hs, to_ennreal_vector_measure_apply_measurable hs]) }
end
section
variables [measurable_space α] [measurable_space β]
variables {M : Type*} [add_comm_monoid M] [topological_space M]
variables (v : vector_measure α M)
/-- The pushforward of a vector measure along a function. -/
def map (v : vector_measure α M) (f : α → β) :
vector_measure β M :=
if hf : measurable f then
{ measure_of' := λ s, if measurable_set s then v (f ⁻¹' s) else 0,
empty' := by simp,
not_measurable' := λ i hi, if_neg hi,
m_Union' :=
begin
intros g hg₁ hg₂,
convert v.m_Union (λ i, hf (hg₁ i)) (λ i j hij x hx, hg₂ i j hij hx),
{ ext i, rw if_pos (hg₁ i) },
{ rw [preimage_Union, if_pos (measurable_set.Union hg₁)] }
end } else 0
lemma map_not_measurable {f : α → β} (hf : ¬ measurable f) : v.map f = 0 :=
dif_neg hf
lemma map_apply {f : α → β} (hf : measurable f) {s : set β} (hs : measurable_set s) :
v.map f s = v (f ⁻¹' s) :=
by { rw [map, dif_pos hf], exact if_pos hs }
@[simp] lemma map_id : v.map id = v :=
ext (λ i hi, by rw [map_apply v measurable_id hi, preimage_id])
@[simp] lemma map_zero (f : α → β) : (0 : vector_measure α M).map f = 0 :=
begin
by_cases hf : measurable f,
{ ext i hi,
rw [map_apply _ hf hi, zero_apply, zero_apply] },
{ exact dif_neg hf }
end
section
variables {N : Type*} [add_comm_monoid N] [topological_space N]
/-- Given a vector measure `v` on `M` and a continuous add_monoid_hom `f : M → N`, `f ∘ v` is a
vector measure on `N`. -/
def map_range (v : vector_measure α M) (f : M →+ N) (hf : continuous f) : vector_measure α N :=
{ measure_of' := λ s, f (v s),
empty' := by rw [empty, add_monoid_hom.map_zero],
not_measurable' := λ i hi, by rw [not_measurable v hi, add_monoid_hom.map_zero],
m_Union' := λ g hg₁ hg₂, has_sum.map (v.m_Union hg₁ hg₂) f hf }
@[simp] lemma map_range_apply {f : M →+ N} (hf : continuous f) {s : set α} :
v.map_range f hf s = f (v s) :=
rfl
@[simp] lemma map_range_id :
v.map_range (add_monoid_hom.id M) continuous_id = v :=
by { ext, refl }
@[simp] lemma map_range_zero {f : M →+ N} (hf : continuous f) :
map_range (0 : vector_measure α M) f hf = 0 :=
by { ext, simp }
section has_continuous_add
variables [has_continuous_add M] [has_continuous_add N]
@[simp] lemma map_range_add {v w : vector_measure α M} {f : M →+ N} (hf : continuous f) :
(v + w).map_range f hf = v.map_range f hf + w.map_range f hf :=
by { ext, simp }
/-- Given a continuous add_monoid_hom `f : M → N`, `map_range_hom` is the add_monoid_hom mapping the
vector measure `v` on `M` to the vector measure `f ∘ v` on `N`. -/
def map_range_hom (f : M →+ N) (hf : continuous f) : vector_measure α M →+ vector_measure α N :=
{ to_fun := λ v, v.map_range f hf,
map_zero' := map_range_zero hf,
map_add' := λ _ _, map_range_add hf }
end has_continuous_add
section module
variables {R : Type*} [semiring R] [module R M] [module R N]
variables [has_continuous_add M] [has_continuous_add N]
[has_continuous_const_smul R M] [has_continuous_const_smul R N]
/-- Given a continuous linear map `f : M → N`, `map_rangeₗ` is the linear map mapping the
vector measure `v` on `M` to the vector measure `f ∘ v` on `N`. -/
def map_rangeₗ (f : M →ₗ[R] N) (hf : continuous f) : vector_measure α M →ₗ[R] vector_measure α N :=
{ to_fun := λ v, v.map_range f.to_add_monoid_hom hf,
map_add' := λ _ _, map_range_add hf,
map_smul' := by { intros, ext, simp } }
end module
end
/-- The restriction of a vector measure on some set. -/
def restrict (v : vector_measure α M) (i : set α) :
vector_measure α M :=
if hi : measurable_set i then
{ measure_of' := λ s, if measurable_set s then v (s ∩ i) else 0,
empty' := by simp,
not_measurable' := λ i hi, if_neg hi,
m_Union' :=
begin
intros f hf₁ hf₂,
convert v.m_Union (λ n, (hf₁ n).inter hi)
(hf₂.mono $ λ i j, disjoint.mono inf_le_left inf_le_left),
{ ext n, rw if_pos (hf₁ n) },
{ rw [Union_inter, if_pos (measurable_set.Union hf₁)] }
end } else 0
lemma restrict_not_measurable {i : set α} (hi : ¬ measurable_set i) :
v.restrict i = 0 :=
dif_neg hi
lemma restrict_apply {i : set α} (hi : measurable_set i)
{j : set α} (hj : measurable_set j) : v.restrict i j = v (j ∩ i) :=
by { rw [restrict, dif_pos hi], exact if_pos hj }
lemma restrict_eq_self {i : set α} (hi : measurable_set i)
{j : set α} (hj : measurable_set j) (hij : j ⊆ i) : v.restrict i j = v j :=
by rw [restrict_apply v hi hj, inter_eq_left_iff_subset.2 hij]
@[simp] lemma restrict_empty : v.restrict ∅ = 0 :=
ext (λ i hi, by rw [restrict_apply v measurable_set.empty hi, inter_empty, v.empty, zero_apply])
@[simp] lemma restrict_univ : v.restrict univ = v :=
ext (λ i hi, by rw [restrict_apply v measurable_set.univ hi, inter_univ])
@[simp] lemma restrict_zero {i : set α} :
(0 : vector_measure α M).restrict i = 0 :=
begin
by_cases hi : measurable_set i,
{ ext j hj, rw [restrict_apply 0 hi hj], refl },
{ exact dif_neg hi }
end
section has_continuous_add
variables [has_continuous_add M]
lemma map_add (v w : vector_measure α M) (f : α → β) :
(v + w).map f = v.map f + w.map f :=
begin
by_cases hf : measurable f,
{ ext i hi,
simp [map_apply _ hf hi] },
{ simp [map, dif_neg hf] }
end
/-- `vector_measure.map` as an additive monoid homomorphism. -/
@[simps] def map_gm (f : α → β) : vector_measure α M →+ vector_measure β M :=
{ to_fun := λ v, v.map f,
map_zero' := map_zero f,
map_add' := λ _ _, map_add _ _ f }
lemma restrict_add (v w : vector_measure α M) (i : set α) :
(v + w).restrict i = v.restrict i + w.restrict i :=
begin
by_cases hi : measurable_set i,
{ ext j hj,
simp [restrict_apply _ hi hj] },
{ simp [restrict_not_measurable _ hi] }
end
/--`vector_measure.restrict` as an additive monoid homomorphism. -/
@[simps] def restrict_gm (i : set α) : vector_measure α M →+ vector_measure α M :=
{ to_fun := λ v, v.restrict i,
map_zero' := restrict_zero,
map_add' := λ _ _, restrict_add _ _ i }
end has_continuous_add
end
section
variables [measurable_space β]
variables {M : Type*} [add_comm_monoid M] [topological_space M]
variables {R : Type*} [semiring R] [distrib_mul_action R M] [has_continuous_const_smul R M]
include m
@[simp] lemma map_smul {v : vector_measure α M} {f : α → β} (c : R) :
(c • v).map f = c • v.map f :=
begin
by_cases hf : measurable f,
{ ext i hi,
simp [map_apply _ hf hi] },
{ simp only [map, dif_neg hf],
-- `smul_zero` does not work since we do not require `has_continuous_add`
ext i hi, simp }
end
@[simp] lemma restrict_smul {v :vector_measure α M} {i : set α} (c : R) :
(c • v).restrict i = c • v.restrict i :=
begin
by_cases hi : measurable_set i,
{ ext j hj,
simp [restrict_apply _ hi hj] },
{ simp only [restrict_not_measurable _ hi],
-- `smul_zero` does not work since we do not require `has_continuous_add`
ext j hj, simp }
end
end
section
variables [measurable_space β]
variables {M : Type*} [add_comm_monoid M] [topological_space M]
variables {R : Type*} [semiring R] [module R M]
[has_continuous_const_smul R M] [has_continuous_add M]
include m
/-- `vector_measure.map` as a linear map. -/
@[simps] def mapₗ (f : α → β) : vector_measure α M →ₗ[R] vector_measure β M :=
{ to_fun := λ v, v.map f,
map_add' := λ _ _, map_add _ _ f,
map_smul' := λ _ _, map_smul _ }
/-- `vector_measure.restrict` as an additive monoid homomorphism. -/
@[simps] def restrictₗ (i : set α) : vector_measure α M →ₗ[R] vector_measure α M :=
{ to_fun := λ v, v.restrict i,
map_add' := λ _ _, restrict_add _ _ i,
map_smul' := λ _ _, restrict_smul _ }
end
section
variables {M : Type*} [topological_space M] [add_comm_monoid M] [partial_order M]
include m
/-- Vector measures over a partially ordered monoid is partially ordered.
This definition is consistent with `measure.partial_order`. -/
instance : partial_order (vector_measure α M) :=
{ le := λ v w, ∀ i, measurable_set i → v i ≤ w i,
le_refl := λ v i hi, le_rfl,
le_trans := λ u v w h₁ h₂ i hi, le_trans (h₁ i hi) (h₂ i hi),
le_antisymm := λ v w h₁ h₂, ext (λ i hi, le_antisymm (h₁ i hi) (h₂ i hi)) }
variables {u v w : vector_measure α M}
lemma le_iff : v ≤ w ↔ ∀ i, measurable_set i → v i ≤ w i :=
iff.rfl
lemma le_iff' : v ≤ w ↔ ∀ i, v i ≤ w i :=
begin
refine ⟨λ h i, _, λ h i hi, h i⟩,
by_cases hi : measurable_set i,
{ exact h i hi },
{ rw [v.not_measurable hi, w.not_measurable hi] }
end
end
localized "notation v ` ≤[`:50 i:50 `] `:0 w:50 :=
measure_theory.vector_measure.restrict v i ≤ measure_theory.vector_measure.restrict w i"
in measure_theory
section
variables {M : Type*} [topological_space M] [add_comm_monoid M] [partial_order M]
variables (v w : vector_measure α M)
lemma restrict_le_restrict_iff {i : set α} (hi : measurable_set i) :
v ≤[i] w ↔ ∀ ⦃j⦄, measurable_set j → j ⊆ i → v j ≤ w j :=
⟨λ h j hj₁ hj₂, (restrict_eq_self v hi hj₁ hj₂) ▸ (restrict_eq_self w hi hj₁ hj₂) ▸ h j hj₁,
λ h, le_iff.1 (λ j hj, (restrict_apply v hi hj).symm ▸ (restrict_apply w hi hj).symm ▸
h (hj.inter hi) (set.inter_subset_right j i))⟩
lemma subset_le_of_restrict_le_restrict {i : set α}
(hi : measurable_set i) (hi₂ : v ≤[i] w) {j : set α} (hj : j ⊆ i) :
v j ≤ w j :=
begin
by_cases hj₁ : measurable_set j,
{ exact (restrict_le_restrict_iff _ _ hi).1 hi₂ hj₁ hj },
{ rw [v.not_measurable hj₁, w.not_measurable hj₁] },
end
lemma restrict_le_restrict_of_subset_le {i : set α}
(h : ∀ ⦃j⦄, measurable_set j → j ⊆ i → v j ≤ w j) : v ≤[i] w :=
begin
by_cases hi : measurable_set i,
{ exact (restrict_le_restrict_iff _ _ hi).2 h },
{ rw [restrict_not_measurable v hi, restrict_not_measurable w hi],
exact le_rfl },
end
lemma restrict_le_restrict_subset {i j : set α}
(hi₁ : measurable_set i) (hi₂ : v ≤[i] w) (hij : j ⊆ i) : v ≤[j] w :=
restrict_le_restrict_of_subset_le v w (λ k hk₁ hk₂,
subset_le_of_restrict_le_restrict v w hi₁ hi₂ (set.subset.trans hk₂ hij))
lemma le_restrict_empty : v ≤[∅] w :=
begin
intros j hj,
rw [restrict_empty, restrict_empty]
end
lemma le_restrict_univ_iff_le : v ≤[univ] w ↔ v ≤ w :=
begin
split,
{ intros h s hs,
have := h s hs,
rwa [restrict_apply _ measurable_set.univ hs, inter_univ,
restrict_apply _ measurable_set.univ hs, inter_univ] at this },
{ intros h s hs,
rw [restrict_apply _ measurable_set.univ hs, inter_univ,
restrict_apply _ measurable_set.univ hs, inter_univ],
exact h s hs }
end
end
section
variables {M : Type*} [topological_space M] [ordered_add_comm_group M] [topological_add_group M]
variables (v w : vector_measure α M)
lemma neg_le_neg {i : set α} (hi : measurable_set i) (h : v ≤[i] w) : -w ≤[i] -v :=
begin
intros j hj₁,
rw [restrict_apply _ hi hj₁, restrict_apply _ hi hj₁, neg_apply, neg_apply],
refine neg_le_neg _,
rw [← restrict_apply _ hi hj₁, ← restrict_apply _ hi hj₁],
exact h j hj₁,
end
@[simp]
lemma neg_le_neg_iff {i : set α} (hi : measurable_set i) : -w ≤[i] -v ↔ v ≤[i] w :=
⟨λ h, neg_neg v ▸ neg_neg w ▸ neg_le_neg _ _ hi h, λ h, neg_le_neg _ _ hi h⟩
end
section
variables {M : Type*} [topological_space M] [ordered_add_comm_monoid M] [order_closed_topology M]
variables (v w : vector_measure α M) {i j : set α}
lemma restrict_le_restrict_Union {f : ℕ → set α}
(hf₁ : ∀ n, measurable_set (f n)) (hf₂ : ∀ n, v ≤[f n] w) :
v ≤[⋃ n, f n] w :=
begin
refine restrict_le_restrict_of_subset_le v w (λ a ha₁ ha₂, _),
have ha₃ : (⋃ n, a ∩ disjointed f n) = a,
{ rwa [← inter_Union, Union_disjointed, inter_eq_left_iff_subset] },
have ha₄ : pairwise (disjoint on (λ n, a ∩ disjointed f n)),
{ exact (disjoint_disjointed _).mono (λ i j, disjoint.mono inf_le_right inf_le_right) },
rw [← ha₃, v.of_disjoint_Union_nat _ ha₄, w.of_disjoint_Union_nat _ ha₄],
refine tsum_le_tsum (λ n, (restrict_le_restrict_iff v w (hf₁ n)).1 (hf₂ n) _ _) _ _,
{ exact (ha₁.inter (measurable_set.disjointed hf₁ n)) },
{ exact set.subset.trans (set.inter_subset_right _ _) (disjointed_subset _ _) },
{ refine (v.m_Union (λ n, _) _).summable,
{ exact ha₁.inter (measurable_set.disjointed hf₁ n) },
{ exact (disjoint_disjointed _).mono (λ i j, disjoint.mono inf_le_right inf_le_right) } },
{ refine (w.m_Union (λ n, _) _).summable,
{ exact ha₁.inter (measurable_set.disjointed hf₁ n) },
{ exact (disjoint_disjointed _).mono (λ i j, disjoint.mono inf_le_right inf_le_right) } },
{ intro n, exact (ha₁.inter (measurable_set.disjointed hf₁ n)) },
{ exact λ n, ha₁.inter (measurable_set.disjointed hf₁ n) }
end
lemma restrict_le_restrict_encodable_Union [encodable β] {f : β → set α}
(hf₁ : ∀ b, measurable_set (f b)) (hf₂ : ∀ b, v ≤[f b] w) :
v ≤[⋃ b, f b] w :=
begin
rw ← encodable.Union_decode₂,
refine restrict_le_restrict_Union v w _ _,
{ intro n, measurability },
{ intro n,
cases encodable.decode₂ β n with b,
{ simp },
{ simp [hf₂ b] } }
end
lemma restrict_le_restrict_union
(hi₁ : measurable_set i) (hi₂ : v ≤[i] w)
(hj₁ : measurable_set j) (hj₂ : v ≤[j] w) :
v ≤[i ∪ j] w :=
begin
rw union_eq_Union,
refine restrict_le_restrict_encodable_Union v w _ _,
{ measurability },
{ rintro (_ | _); simpa }
end
end
section
variables {M : Type*} [topological_space M] [ordered_add_comm_monoid M]
variables (v w : vector_measure α M) {i j : set α}
lemma nonneg_of_zero_le_restrict (hi₂ : 0 ≤[i] v) :
0 ≤ v i :=
begin
by_cases hi₁ : measurable_set i,
{ exact (restrict_le_restrict_iff _ _ hi₁).1 hi₂ hi₁ set.subset.rfl },
{ rw v.not_measurable hi₁ },
end
lemma nonpos_of_restrict_le_zero (hi₂ : v ≤[i] 0) :
v i ≤ 0 :=
begin
by_cases hi₁ : measurable_set i,
{ exact (restrict_le_restrict_iff _ _ hi₁).1 hi₂ hi₁ set.subset.rfl },
{ rw v.not_measurable hi₁ }
end
lemma zero_le_restrict_not_measurable (hi : ¬ measurable_set i) :
0 ≤[i] v :=
begin
rw [restrict_zero, restrict_not_measurable _ hi],
exact le_rfl,
end
lemma restrict_le_zero_of_not_measurable (hi : ¬ measurable_set i) :
v ≤[i] 0 :=
begin
rw [restrict_zero, restrict_not_measurable _ hi],
exact le_rfl,
end
lemma measurable_of_not_zero_le_restrict (hi : ¬ 0 ≤[i] v) : measurable_set i :=
not.imp_symm (zero_le_restrict_not_measurable _) hi
lemma measurable_of_not_restrict_le_zero (hi : ¬ v ≤[i] 0) : measurable_set i :=
not.imp_symm (restrict_le_zero_of_not_measurable _) hi
lemma zero_le_restrict_subset (hi₁ : measurable_set i) (hij : j ⊆ i) (hi₂ : 0 ≤[i] v):
0 ≤[j] v :=
restrict_le_restrict_of_subset_le _ _
(λ k hk₁ hk₂, (restrict_le_restrict_iff _ _ hi₁).1 hi₂ hk₁ (set.subset.trans hk₂ hij))
lemma restrict_le_zero_subset (hi₁ : measurable_set i) (hij : j ⊆ i) (hi₂ : v ≤[i] 0):
v ≤[j] 0 :=
restrict_le_restrict_of_subset_le _ _
(λ k hk₁ hk₂, (restrict_le_restrict_iff _ _ hi₁).1 hi₂ hk₁ (set.subset.trans hk₂ hij))
end
section
variables {M : Type*} [topological_space M] [linear_ordered_add_comm_monoid M]
variables (v w : vector_measure α M) {i j : set α}
include m
lemma exists_pos_measure_of_not_restrict_le_zero (hi : ¬ v ≤[i] 0) :
∃ j : set α, measurable_set j ∧ j ⊆ i ∧ 0 < v j :=
begin
have hi₁ : measurable_set i := measurable_of_not_restrict_le_zero _ hi,
rw [restrict_le_restrict_iff _ _ hi₁] at hi,
push_neg at hi,
obtain ⟨j, hj₁, hj₂, hj⟩ := hi,
exact ⟨j, hj₁, hj₂, hj⟩,
end
end
section
variables {M : Type*} [topological_space M] [add_comm_monoid M] [partial_order M]
[covariant_class M M (+) (≤)] [has_continuous_add M]
include m
instance covariant_add_le :
covariant_class (vector_measure α M) (vector_measure α M) (+) (≤) :=
⟨λ u v w h i hi, add_le_add_left (h i hi) _⟩
end
section
variables {L M N : Type*}
variables [add_comm_monoid L] [topological_space L] [add_comm_monoid M] [topological_space M]
[add_comm_monoid N] [topological_space N]
include m
/-- A vector measure `v` is absolutely continuous with respect to a measure `μ` if for all sets
`s`, `μ s = 0`, we have `v s = 0`. -/
def absolutely_continuous (v : vector_measure α M) (w : vector_measure α N) :=
∀ ⦃s : set α⦄, w s = 0 → v s = 0
localized "infix ` ≪ᵥ `:50 := measure_theory.vector_measure.absolutely_continuous"
in measure_theory
open_locale measure_theory
namespace absolutely_continuous
variables {v : vector_measure α M} {w : vector_measure α N}
lemma mk (h : ∀ ⦃s : set α⦄, measurable_set s → w s = 0 → v s = 0) : v ≪ᵥ w :=
begin
intros s hs,
by_cases hmeas : measurable_set s,
{ exact h hmeas hs },
{ exact not_measurable v hmeas }
end
lemma eq {w : vector_measure α M} (h : v = w) : v ≪ᵥ w :=
λ s hs, h.symm ▸ hs
@[refl] lemma refl (v : vector_measure α M) : v ≪ᵥ v :=
eq rfl
@[trans] lemma trans {u : vector_measure α L} {v : vector_measure α M} {w : vector_measure α N}
(huv : u ≪ᵥ v) (hvw : v ≪ᵥ w) : u ≪ᵥ w :=
λ _ hs, huv $ hvw hs
lemma zero (v : vector_measure α N) : (0 : vector_measure α M) ≪ᵥ v :=
λ s _, vector_measure.zero_apply s
lemma neg_left {M : Type*} [add_comm_group M] [topological_space M] [topological_add_group M]
{v : vector_measure α M} {w : vector_measure α N} (h : v ≪ᵥ w) : -v ≪ᵥ w :=
λ s hs, by { rw [neg_apply, h hs, neg_zero] }
lemma neg_right {N : Type*} [add_comm_group N] [topological_space N] [topological_add_group N]
{v : vector_measure α M} {w : vector_measure α N} (h : v ≪ᵥ w) : v ≪ᵥ -w :=
begin
intros s hs,
rw [neg_apply, neg_eq_zero] at hs,
exact h hs
end
lemma add [has_continuous_add M] {v₁ v₂ : vector_measure α M} {w : vector_measure α N}
(hv₁ : v₁ ≪ᵥ w) (hv₂ : v₂ ≪ᵥ w) : v₁ + v₂ ≪ᵥ w :=
λ s hs, by { rw [add_apply, hv₁ hs, hv₂ hs, zero_add] }
lemma sub {M : Type*} [add_comm_group M] [topological_space M] [topological_add_group M]
{v₁ v₂ : vector_measure α M} {w : vector_measure α N} (hv₁ : v₁ ≪ᵥ w) (hv₂ : v₂ ≪ᵥ w) :
v₁ - v₂ ≪ᵥ w :=
λ s hs, by { rw [sub_apply, hv₁ hs, hv₂ hs, zero_sub, neg_zero] }
lemma smul {R : Type*} [semiring R] [distrib_mul_action R M] [has_continuous_const_smul R M]
{r : R} {v : vector_measure α M} {w : vector_measure α N} (h : v ≪ᵥ w) :
r • v ≪ᵥ w :=
λ s hs, by { rw [smul_apply, h hs, smul_zero] }
lemma map [measure_space β] (h : v ≪ᵥ w) (f : α → β) :
v.map f ≪ᵥ w.map f :=
begin
by_cases hf : measurable f,
{ refine mk (λ s hs hws, _),
rw map_apply _ hf hs at hws ⊢,
exact h hws },
{ intros s hs,
rw [map_not_measurable v hf, zero_apply] }
end
lemma ennreal_to_measure {μ : vector_measure α ℝ≥0∞} :
(∀ ⦃s : set α⦄, μ.ennreal_to_measure s = 0 → v s = 0) ↔ v ≪ᵥ μ :=
begin
split; intro h,
{ refine mk (λ s hmeas hs, h _),
rw [← hs, ennreal_to_measure_apply hmeas] },
{ intros s hs,
by_cases hmeas : measurable_set s,
{ rw ennreal_to_measure_apply hmeas at hs,
exact h hs },
{ exact not_measurable v hmeas } },
end
end absolutely_continuous
/-- Two vector measures `v` and `w` are said to be mutually singular if there exists a measurable
set `s`, such that for all `t ⊆ s`, `v t = 0` and for all `t ⊆ sᶜ`, `w t = 0`.
We note that we do not require the measurability of `t` in the definition since this makes it easier
to use. This is equivalent to the definition which requires measurability. To prove
`mutually_singular` with the measurability condition, use
`measure_theory.vector_measure.mutually_singular.mk`. -/
def mutually_singular (v : vector_measure α M) (w : vector_measure α N) : Prop :=
∃ (s : set α), measurable_set s ∧ (∀ t ⊆ s, v t = 0) ∧ (∀ t ⊆ sᶜ, w t = 0)
localized "infix ` ⊥ᵥ `:60 := measure_theory.vector_measure.mutually_singular" in measure_theory
namespace mutually_singular
variables {v v₁ v₂ : vector_measure α M} {w w₁ w₂ : vector_measure α N}
lemma mk (s : set α) (hs : measurable_set s)
(h₁ : ∀ t ⊆ s, measurable_set t → v t = 0)
(h₂ : ∀ t ⊆ sᶜ, measurable_set t → w t = 0) : v ⊥ᵥ w :=
begin
refine ⟨s, hs, λ t hst, _, λ t hst, _⟩;
by_cases ht : measurable_set t,
{ exact h₁ t hst ht },
{ exact not_measurable v ht },
{ exact h₂ t hst ht },
{ exact not_measurable w ht }
end
lemma symm (h : v ⊥ᵥ w) : w ⊥ᵥ v :=
let ⟨s, hmeas, hs₁, hs₂⟩ := h in
⟨sᶜ, hmeas.compl, hs₂, λ t ht, hs₁ _ (compl_compl s ▸ ht : t ⊆ s)⟩
lemma zero_right : v ⊥ᵥ (0 : vector_measure α N) :=
⟨∅, measurable_set.empty, λ t ht, (subset_empty_iff.1 ht).symm ▸ v.empty, λ _ _, zero_apply _⟩
lemma zero_left : (0 : vector_measure α M) ⊥ᵥ w :=
zero_right.symm
lemma add_left [t2_space N] [has_continuous_add M] (h₁ : v₁ ⊥ᵥ w) (h₂ : v₂ ⊥ᵥ w) : v₁ + v₂ ⊥ᵥ w :=
begin
obtain ⟨u, hmu, hu₁, hu₂⟩ := h₁,
obtain ⟨v, hmv, hv₁, hv₂⟩ := h₂,
refine mk (u ∩ v) (hmu.inter hmv) (λ t ht hmt, _) (λ t ht hmt, _),
{ rw [add_apply, hu₁ _ (subset_inter_iff.1 ht).1, hv₁ _ (subset_inter_iff.1 ht).2, zero_add] },
{ rw compl_inter at ht,
rw [(_ : t = (uᶜ ∩ t) ∪ (vᶜ \ uᶜ ∩ t)),
of_union _ (hmu.compl.inter hmt) ((hmv.compl.diff hmu.compl).inter hmt),
hu₂, hv₂, add_zero],
{ exact subset.trans (inter_subset_left _ _) (diff_subset _ _) },
{ exact inter_subset_left _ _ },
{ apply_instance },
{ exact disjoint.mono (inter_subset_left _ _) (inter_subset_left _ _) disjoint_diff },
{ apply subset.antisymm;
intros x hx,
{ by_cases hxu' : x ∈ uᶜ,
{ exact or.inl ⟨hxu', hx⟩ },
rcases ht hx with (hxu | hxv),
exacts [false.elim (hxu' hxu), or.inr ⟨⟨hxv, hxu'⟩, hx⟩] },
{ rcases hx; exact hx.2 } } },
end
lemma add_right [t2_space M] [has_continuous_add N] (h₁ : v ⊥ᵥ w₁) (h₂ : v ⊥ᵥ w₂) : v ⊥ᵥ w₁ + w₂ :=
(add_left h₁.symm h₂.symm).symm
lemma smul_right {R : Type*} [semiring R] [distrib_mul_action R N] [has_continuous_const_smul R N]
(r : R) (h : v ⊥ᵥ w) : v ⊥ᵥ r • w :=
let ⟨s, hmeas, hs₁, hs₂⟩ := h in
⟨s, hmeas, hs₁, λ t ht, by simp only [coe_smul, pi.smul_apply, hs₂ t ht, smul_zero]⟩
lemma smul_left {R : Type*} [semiring R] [distrib_mul_action R M] [has_continuous_const_smul R M]
(r : R) (h : v ⊥ᵥ w) : r • v ⊥ᵥ w :=
(smul_right r h.symm).symm
lemma neg_left {M : Type*} [add_comm_group M] [topological_space M] [topological_add_group M]
{v : vector_measure α M} {w : vector_measure α N} (h : v ⊥ᵥ w) : -v ⊥ᵥ w :=
begin
obtain ⟨u, hmu, hu₁, hu₂⟩ := h,
refine ⟨u, hmu, λ s hs, _, hu₂⟩,
rw [neg_apply v s, neg_eq_zero],
exact hu₁ s hs
end
lemma neg_right {N : Type*} [add_comm_group N] [topological_space N] [topological_add_group N]
{v : vector_measure α M} {w : vector_measure α N} (h : v ⊥ᵥ w) : v ⊥ᵥ -w :=
h.symm.neg_left.symm
@[simp]
lemma neg_left_iff {M : Type*} [add_comm_group M] [topological_space M] [topological_add_group M]
{v : vector_measure α M} {w : vector_measure α N} :
-v ⊥ᵥ w ↔ v ⊥ᵥ w :=
⟨λ h, neg_neg v ▸ h.neg_left, neg_left⟩
@[simp]
lemma neg_right_iff {N : Type*} [add_comm_group N] [topological_space N] [topological_add_group N]
{v : vector_measure α M} {w : vector_measure α N} :
v ⊥ᵥ -w ↔ v ⊥ᵥ w :=
⟨λ h, neg_neg w ▸ h.neg_right, neg_right⟩
end mutually_singular
section trim
omit m
/-- Restriction of a vector measure onto a sub-σ-algebra. -/
@[simps] def trim {m n : measurable_space α} (v : vector_measure α M) (hle : m ≤ n) :
@vector_measure α m M _ _ :=
{ measure_of' := λ i, if measurable_set[m] i then v i else 0,
empty' := by rw [if_pos measurable_set.empty, v.empty],
not_measurable' := λ i hi, by rw if_neg hi,
m_Union' := λ f hf₁ hf₂,
begin
have hf₁' : ∀ k, measurable_set[n] (f k) := λ k, hle _ (hf₁ k),
convert v.m_Union hf₁' hf₂,
{ ext n, rw if_pos (hf₁ n) },
{ rw if_pos (@measurable_set.Union _ _ m _ _ hf₁) }
end }
variables {n : measurable_space α} {v : vector_measure α M}
lemma trim_eq_self : v.trim le_rfl = v :=
begin
ext1 i hi,
exact if_pos hi,
end
@[simp] lemma zero_trim (hle : m ≤ n) :
(0 : vector_measure α M).trim hle = 0 :=
begin
ext1 i hi,
exact if_pos hi,
end
lemma trim_measurable_set_eq (hle : m ≤ n) {i : set α} (hi : measurable_set[m] i) :
v.trim hle i = v i :=
if_pos hi
lemma restrict_trim (hle : m ≤ n) {i : set α} (hi : measurable_set[m] i) :
@vector_measure.restrict α m M _ _ (v.trim hle) i = (v.restrict i).trim hle :=
begin
ext j hj,
rw [restrict_apply, trim_measurable_set_eq hle hj, restrict_apply, trim_measurable_set_eq],
all_goals { measurability }
end
end trim
end
end vector_measure
namespace signed_measure
open vector_measure
open_locale measure_theory
include m
/-- The underlying function for `signed_measure.to_measure_of_zero_le`. -/
def to_measure_of_zero_le' (s : signed_measure α) (i : set α) (hi : 0 ≤[i] s)
(j : set α) (hj : measurable_set j) : ℝ≥0∞ :=
@coe ℝ≥0 ℝ≥0∞ _ ⟨s.restrict i j, le_trans (by simp) (hi j hj)⟩
/-- Given a signed measure `s` and a positive measurable set `i`, `to_measure_of_zero_le`
provides the measure, mapping measurable sets `j` to `s (i ∩ j)`. -/
def to_measure_of_zero_le (s : signed_measure α) (i : set α)
(hi₁ : measurable_set i) (hi₂ : 0 ≤[i] s) : measure α :=
measure.of_measurable (s.to_measure_of_zero_le' i hi₂)
(by { simp_rw [to_measure_of_zero_le', s.restrict_apply hi₁ measurable_set.empty,
set.empty_inter i, s.empty], refl })
begin
intros f hf₁ hf₂,
have h₁ : ∀ n, measurable_set (i ∩ f n) := λ n, hi₁.inter (hf₁ n),
have h₂ : pairwise (disjoint on λ (n : ℕ), i ∩ f n),
{ rintro n m hnm x ⟨⟨_, hx₁⟩, _, hx₂⟩,
exact hf₂ n m hnm ⟨hx₁, hx₂⟩ },
simp only [to_measure_of_zero_le', s.restrict_apply hi₁ (measurable_set.Union hf₁),
set.inter_comm, set.inter_Union, s.of_disjoint_Union_nat h₁ h₂,
ennreal.some_eq_coe, id.def],
have h : ∀ n, 0 ≤ s (i ∩ f n) :=
λ n, s.nonneg_of_zero_le_restrict
(s.zero_le_restrict_subset hi₁ (inter_subset_left _ _) hi₂),
rw [nnreal.coe_tsum_of_nonneg h, ennreal.coe_tsum],
{ refine tsum_congr (λ n, _),
simp_rw [s.restrict_apply hi₁ (hf₁ n), set.inter_comm] },
{ exact (nnreal.summable_coe_of_nonneg h).2 (s.m_Union h₁ h₂).summable }
end
variables (s : signed_measure α) {i j : set α}
lemma to_measure_of_zero_le_apply (hi : 0 ≤[i] s)
(hi₁ : measurable_set i) (hj₁ : measurable_set j) :
s.to_measure_of_zero_le i hi₁ hi j =
@coe ℝ≥0 ℝ≥0∞ _ ⟨s (i ∩ j), nonneg_of_zero_le_restrict s
(zero_le_restrict_subset s hi₁ (set.inter_subset_left _ _) hi)⟩ :=
by simp_rw [to_measure_of_zero_le, measure.of_measurable_apply _ hj₁, to_measure_of_zero_le',
s.restrict_apply hi₁ hj₁, set.inter_comm]
/-- Given a signed measure `s` and a negative measurable set `i`, `to_measure_of_le_zero`
provides the measure, mapping measurable sets `j` to `-s (i ∩ j)`. -/
def to_measure_of_le_zero (s : signed_measure α) (i : set α) (hi₁ : measurable_set i)
(hi₂ : s ≤[i] 0) : measure α :=
to_measure_of_zero_le (-s) i hi₁ $ (@neg_zero (vector_measure α ℝ) _) ▸ neg_le_neg _ _ hi₁ hi₂
lemma to_measure_of_le_zero_apply (hi : s ≤[i] 0)
(hi₁ : measurable_set i) (hj₁ : measurable_set j) :
s.to_measure_of_le_zero i hi₁ hi j =
@coe ℝ≥0 ℝ≥0∞ _ ⟨-s (i ∩ j), neg_apply s (i ∩ j) ▸ nonneg_of_zero_le_restrict _
(zero_le_restrict_subset _ hi₁ (set.inter_subset_left _ _)
((@neg_zero (vector_measure α ℝ) _) ▸ neg_le_neg _ _ hi₁ hi))⟩ :=
begin
erw [to_measure_of_zero_le_apply],
{ simp },
{ assumption },
end
/-- `signed_measure.to_measure_of_zero_le` is a finite measure. -/
instance to_measure_of_zero_le_finite (hi : 0 ≤[i] s) (hi₁ : measurable_set i) :
is_finite_measure (s.to_measure_of_zero_le i hi₁ hi) :=
{ measure_univ_lt_top :=
begin
rw [to_measure_of_zero_le_apply s hi hi₁ measurable_set.univ],
exact ennreal.coe_lt_top,
end }
/-- `signed_measure.to_measure_of_le_zero` is a finite measure. -/
instance to_measure_of_le_zero_finite (hi : s ≤[i] 0) (hi₁ : measurable_set i) :
is_finite_measure (s.to_measure_of_le_zero i hi₁ hi) :=
{ measure_univ_lt_top :=
begin
rw [to_measure_of_le_zero_apply s hi hi₁ measurable_set.univ],
exact ennreal.coe_lt_top,
end }
lemma to_measure_of_zero_le_to_signed_measure (hs : 0 ≤[univ] s) :
(s.to_measure_of_zero_le univ measurable_set.univ hs).to_signed_measure = s :=
begin
ext i hi,
simp [measure.to_signed_measure_apply_measurable hi, to_measure_of_zero_le_apply _ _ _ hi],
end
lemma to_measure_of_le_zero_to_signed_measure (hs : s ≤[univ] 0) :
(s.to_measure_of_le_zero univ measurable_set.univ hs).to_signed_measure = -s :=
begin
ext i hi,
simp [measure.to_signed_measure_apply_measurable hi, to_measure_of_le_zero_apply _ _ _ hi],
end
end signed_measure
namespace measure
open vector_measure
variables (μ : measure α) [is_finite_measure μ]
lemma zero_le_to_signed_measure : 0 ≤ μ.to_signed_measure :=
begin
rw ← le_restrict_univ_iff_le,
refine restrict_le_restrict_of_subset_le _ _ (λ j hj₁ _, _),
simp only [measure.to_signed_measure_apply_measurable hj₁, coe_zero, pi.zero_apply,
ennreal.to_real_nonneg, vector_measure.coe_zero]
end
lemma to_signed_measure_to_measure_of_zero_le :
μ.to_signed_measure.to_measure_of_zero_le univ measurable_set.univ
((le_restrict_univ_iff_le _ _).2 (zero_le_to_signed_measure μ)) = μ :=
begin
refine measure.ext (λ i hi, _),
lift μ i to ℝ≥0 using (measure_lt_top _ _).ne with m hm,
simp [signed_measure.to_measure_of_zero_le_apply _ _ _ hi,
measure.to_signed_measure_apply_measurable hi, ← hm],
end
end measure
end measure_theory
|
09cd0ded46fb6b5ab4268f1ff860271ad5c0c509 | 4727251e0cd73359b15b664c3170e5d754078599 | /test/free_algebra.lean | 4e3b18edf7702ece325a1d8a71773d3e8430a7f1 | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 1,181 | lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import linear_algebra.exterior_algebra.basic
import linear_algebra.clifford_algebra
/-!
Tests that the ring instances for `free_algebra` and derived quotient types actually work.
There is some discussion about this in
https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/algebra.2Esemiring_to_ring.20breaks.20semimodule.20typeclass.20lookup/near/212580241
In essence, the use of `attribute [irreducible] the_type` was breaking instance resolution on that
type.
-/
variables {S : Type*} {M : Type*}
section free
variables [comm_ring S]
example : (1 : free_algebra S M) - (1 : free_algebra S M) = 0 := by rw sub_self
end free
section exterior
variables [comm_ring S] [add_comm_monoid M] [module S M]
example : (1 : exterior_algebra S M) - (1 : exterior_algebra S M) = 0 := by rw sub_self
end exterior
section clifford
variables [comm_ring S] [add_comm_group M] [module S M] (Q : quadratic_form S M)
example : (1 : clifford_algebra Q) - (1 : clifford_algebra Q) = 0 := by rw sub_self
end clifford
|
cde764c5cd71bcad811dc3bd6d8c5f8932f84159 | 7cef822f3b952965621309e88eadf618da0c8ae9 | /src/linear_algebra/dual.lean | acc111910e9176ca717f2107b068c095d2b4f7c4 | [
"Apache-2.0"
] | permissive | rmitta/mathlib | 8d90aee30b4db2b013e01f62c33f297d7e64a43d | 883d974b608845bad30ae19e27e33c285200bf84 | refs/heads/master | 1,585,776,832,544 | 1,576,874,096,000 | 1,576,874,096,000 | 153,663,165 | 0 | 2 | Apache-2.0 | 1,544,806,490,000 | 1,539,884,365,000 | Lean | UTF-8 | Lean | false | false | 10,943 | lean | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle
-/
import linear_algebra.tensor_product
import linear_algebra.finite_dimensional
import tactic.apply_fun
noncomputable theory
/-!
# Dual vector spaces
The dual space of an R-module M is the R-module of linear maps `M → R`.
## Main definitions
* `dual R M` defines the dual space of M over R.
* Given a basis for a K-vector space `V`, `is_basis.to_dual` produces a map from `V` to `dual K V`.
* Given families of vectors `e` and `ε`, `dual_pair e ε` states that these families have the
characteristic properties of a basis and a dual.
## Main results
* `to_dual_equiv` : the dual space is linearly equivalent to the primal space.
* `dual_pair.is_basis` and `dual_pair.eq_dual`: if `e` and `ε` form a dual pair, `e` is a basis and
`ε` is its dual basis.
## Notation
We sometimes use `V'` as local notation for `dual K V`.
## Implementation details
Because of unresolved type class issues, the following local instance can be of use:
```
private def help_tcs : has_scalar K V := mul_action.to_has_scalar _ _
local attribute [instance] help_tcs
```
-/
namespace module
variables (R : Type*) (M : Type*)
variables [comm_ring R] [add_comm_group M] [module R M]
/-- The dual space of an R-module M is the R-module of linear maps `M → R`. -/
@[derive [add_comm_group, module R]] def dual := M →ₗ[R] R
namespace dual
instance : has_coe_to_fun (dual R M) := ⟨_, linear_map.to_fun⟩
/-- Maps a module M to the dual of the dual of M. See `vector_space.eval_range` and
`vector_space.eval_equiv`. -/
def eval : M →ₗ[R] (dual R (dual R M)) := linear_map.id.flip
lemma eval_apply (v : M) (a : dual R M) : (eval R M v) a = a v :=
begin
dunfold eval,
rw [linear_map.flip_apply, linear_map.id_apply]
end
end dual
end module
namespace is_basis
universes u v w
variables {K : Type u} {V : Type v} {ι : Type w}
variables [discrete_field K] [add_comm_group V] [vector_space K V]
open vector_space module module.dual submodule linear_map cardinal function
instance dual.vector_space : vector_space K (dual K V) := { ..module.dual.inst K V }
variables [de : decidable_eq ι]
variables {B : ι → V} (h : is_basis K B)
include de h
/-- The linear map from a vector space equipped with basis to its dual vector space,
taking basis elements to corresponding dual basis elements. -/
def to_dual : V →ₗ[K] module.dual K V :=
h.constr $ λ v, h.constr $ λ w, if w = v then 1 else 0
lemma to_dual_apply (i j : ι) :
(h.to_dual (B i)) (B j) = if i = j then 1 else 0 :=
by erw [constr_basis h, constr_basis h]; ac_refl
def to_dual_flip (v : V) : (V →ₗ[K] K) := (linear_map.flip h.to_dual).to_fun v
omit de h
/-- Evaluation of finitely supported functions at a fixed point `i`, as a `K`-linear map. -/
def eval_finsupp_at (i : ι) : (ι →₀ K) →ₗ[K] K :=
{ to_fun := λ f, f i,
add := by intros; rw finsupp.add_apply,
smul := by intros; rw finsupp.smul_apply }
include h
set_option class.instance_max_depth 50
def coord_fun (i : ι) : (V →ₗ[K] K) := (eval_finsupp_at i).comp h.repr
lemma coord_fun_eq_repr (v : V) (i : ι) : h.coord_fun i v = h.repr v i := rfl
include de
lemma to_dual_swap_eq_to_dual (v w : V) : h.to_dual_flip v w = h.to_dual w v := rfl
lemma to_dual_eq_repr (v : V) (i : ι) : (h.to_dual v) (B i) = h.repr v i :=
begin
rw [←coord_fun_eq_repr, ←to_dual_swap_eq_to_dual],
apply congr_fun,
dsimp,
congr',
apply h.ext,
{ intros,
rw [to_dual_swap_eq_to_dual, to_dual_apply],
{ split_ifs with hx,
{ rwa [hx, coord_fun_eq_repr, repr_eq_single, finsupp.single_apply, if_pos rfl] },
{ rw [coord_fun_eq_repr, repr_eq_single, finsupp.single_apply], symmetry, convert if_neg hx } } }
end
lemma to_dual_inj (v : V) (a : h.to_dual v = 0) : v = 0 :=
begin
rw [← mem_bot K, ← h.repr_ker, mem_ker],
apply finsupp.ext,
intro b,
rw [←to_dual_eq_repr _ _ _, a],
refl
end
theorem to_dual_ker : h.to_dual.ker = ⊥ :=
ker_eq_bot'.mpr h.to_dual_inj
theorem to_dual_range [fin : fintype ι] : h.to_dual.range = ⊤ :=
begin
rw eq_top_iff',
intro f,
rw linear_map.mem_range,
let lin_comb : ι →₀ K := finsupp.on_finset fin.elems (λ i, f.to_fun (B i)) _,
{ use finsupp.total ι V K B lin_comb,
apply h.ext,
{ intros i,
rw [h.to_dual_eq_repr _ i, repr_total h],
{ simpa },
{ rw [finsupp.mem_supported],
exact λ _ _, set.mem_univ _ } } },
{ intros a _,
apply fin.complete }
end
/-- Maps a basis for `V` to a basis for the dual space. -/
def dual_basis : ι → dual K V := λ i, h.to_dual (B i)
theorem dual_lin_independent : linear_independent K h.dual_basis :=
begin
apply linear_independent.image h.1,
rw to_dual_ker,
exact disjoint_bot_right
end
/-- A vector space is linearly equivalent to its dual space. -/
def to_dual_equiv [fintype ι] : V ≃ₗ[K] (dual K V) :=
linear_equiv.of_bijective h.to_dual h.to_dual_ker h.to_dual_range
theorem dual_basis_is_basis [fintype ι] : is_basis K h.dual_basis :=
h.to_dual_equiv.is_basis h
@[simp] lemma to_dual_to_dual [fintype ι] :
(h.dual_basis_is_basis.to_dual).comp h.to_dual = eval K V :=
begin
apply @is_basis.ext _ _ _ _ _ _ _ _ _ _ _ _ h,
intros i,
apply @is_basis.ext _ _ _ _ _ _ _ _ _ _ _ _ h.dual_basis_is_basis,
intros j,
dunfold eval,
rw [linear_map.flip_apply, linear_map.id_apply, linear_map.comp_apply],
apply eq.trans (to_dual_apply h.dual_basis_is_basis i j),
{ dunfold dual_basis,
rw to_dual_apply,
split_ifs with h₁ h₂; try {refl},
{ exfalso, apply h₂ h₁.symm },
{ exfalso, apply ne.symm h₁ (by assumption) } }
end
theorem dual_dim_eq [fintype ι] :
cardinal.lift.{v u} (dim K V) = dim K (dual K V) :=
begin
have := linear_equiv.dim_eq_lift h.to_dual_equiv,
simp only [cardinal.lift_umax] at this,
rw [this, ← cardinal.lift_umax],
apply cardinal.lift_id,
end
end is_basis
namespace vector_space
universes u v
variables {K : Type u} {V : Type v}
variables [discrete_field K] [add_comm_group V] [vector_space K V]
open module module.dual submodule linear_map cardinal is_basis
theorem eval_ker : (eval K V).ker = ⊥ :=
begin
haveI := classical.dec_eq K,
haveI := classical.dec_eq V,
haveI := classical.dec_eq (dual K V),
rw ker_eq_bot',
intros v h,
rw linear_map.ext_iff at h,
by_contradiction H,
rcases exists_subset_is_basis (linear_independent_singleton H) with ⟨b, hv, hb⟩,
swap 4, assumption,
have hv' : v = (λ (i : b), i.val) ⟨v, hv (set.mem_singleton v)⟩ := rfl,
let hx := h (hb.to_dual v),
erw [eval_apply, hv', to_dual_apply, if_pos rfl, zero_apply _] at hx,
exact one_ne_zero hx
end
theorem dual_dim_eq (h : dim K V < omega) :
cardinal.lift.{v u} (dim K V) = dim K (dual K V) :=
begin
letI := classical.dec_eq (dual K V),
letI := classical.dec_eq V,
rcases exists_is_basis_fintype h with ⟨b, hb, ⟨hf⟩⟩,
resetI,
exact hb.dual_dim_eq
end
set_option class.instance_max_depth 70
lemma eval_range (h : dim K V < omega) : (eval K V).range = ⊤ :=
begin
haveI := classical.dec_eq (dual K V),
haveI := classical.dec_eq (dual K (dual K V)),
letI := classical.dec_eq V,
rcases exists_is_basis_fintype h with ⟨b, hb, ⟨hf⟩⟩,
resetI,
rw [← hb.to_dual_to_dual, range_comp, hb.to_dual_range, map_top, to_dual_range _],
apply_instance
end
/-- A vector space is linearly equivalent to the dual of its dual space. -/
def eval_equiv (h : dim K V < omega) : V ≃ₗ[K] dual K (dual K V) :=
linear_equiv.of_bijective (eval K V) eval_ker (eval_range h)
end vector_space
section dual_pair
open vector_space module module.dual linear_map function
universes u v w
variables {K : Type u} {V : Type v} {ι : Type w} [decidable_eq ι]
variables [discrete_field K] [add_comm_group V] [vector_space K V]
local notation `V'` := dual K V
/-- `e` and `ε` have characteristic properties of a basis and its dual -/
structure dual_pair (e : ι → V) (ε : ι → V') :=
(eval : ∀ i j : ι, ε i (e j) = if i = j then 1 else 0)
(total : ∀ {v : V}, (∀ i, ε i v = 0) → v = 0)
[finite : ∀ v : V, fintype {i | ε i v ≠ 0}]
end dual_pair
namespace dual_pair
open vector_space module module.dual linear_map function
universes u v w
variables {K : Type u} {V : Type v} {ι : Type w} [dι : decidable_eq ι]
variables [discrete_field K] [add_comm_group V] [vector_space K V]
variables {e : ι → V} {ε : ι → dual K V} (h : dual_pair e ε)
include h
/-- The coefficients of `v` on the basis `e` -/
def coeffs (v : V) : ι →₀ K :=
{ to_fun := λ i, ε i v,
support := by { haveI := h.finite v, exact {i : ι | ε i v ≠ 0}.to_finset },
mem_support_to_fun := by {intro i, rw set.mem_to_finset, exact iff.rfl } }
@[simp] lemma coeffs_apply (v : V) (i : ι) : h.coeffs v i = ε i v := rfl
omit h
private def help_tcs : has_scalar K V := mul_action.to_has_scalar _ _
local attribute [instance] help_tcs
/-- linear combinations of elements of `e`.
This is a convenient abbreviation for `finsupp.total _ V K e l` -/
def lc (e : ι → V) (l : ι →₀ K) : V := l.sum (λ (i : ι) (a : K), a • (e i))
include h
lemma dual_lc (l : ι →₀ K) (i : ι) : ε i (dual_pair.lc e l) = l i :=
begin
erw linear_map.map_sum,
simp only [h.eval, map_smul, smul_eq_mul],
rw finset.sum_eq_single i,
{ simp },
{ intros q q_in q_ne,
simp [q_ne.symm] },
{ intro p_not_in,
simp [finsupp.not_mem_support_iff.1 p_not_in] },
end
@[simp]
lemma coeffs_lc (l : ι →₀ K) : h.coeffs (dual_pair.lc e l) = l :=
by { ext i, rw [h.coeffs_apply, h.dual_lc] }
/-- For any v : V n, \sum_{p ∈ Q n} (ε p v) • e p = v -/
lemma decomposition (v : V) : dual_pair.lc e (h.coeffs v) = v :=
begin
refine eq_of_sub_eq_zero (h.total _),
intros i,
simp [-sub_eq_add_neg, linear_map.map_sub, h.dual_lc, sub_eq_zero_iff_eq]
end
lemma mem_of_mem_span {H : set ι} {x : V} (hmem : x ∈ submodule.span K (e '' H)) :
∀ i : ι, ε i x ≠ 0 → i ∈ H :=
begin
intros i hi,
rcases (finsupp.mem_span_iff_total _).mp hmem with ⟨l, supp_l, sum_l⟩,
change dual_pair.lc e l = x at sum_l,
rw finsupp.mem_supported' at supp_l,
apply classical.by_contradiction,
intro i_not,
apply hi,
rw ← sum_l,
simpa [h.dual_lc] using supp_l i i_not
end
lemma is_basis : is_basis K e :=
begin
split,
{ rw linear_independent_iff,
intros l H,
change dual_pair.lc e l = 0 at H,
ext i,
apply_fun ε i at H,
simpa [h.dual_lc] using H },
{ rw submodule.eq_top_iff',
intro v,
rw [← set.image_univ, finsupp.mem_span_iff_total],
exact ⟨h.coeffs v, by simp, h.decomposition v⟩ },
end
lemma eq_dual : ε = is_basis.dual_basis h.is_basis :=
begin
funext i,
refine h.is_basis.ext (λ _, _),
erw [is_basis.to_dual_apply, h.eval]
end
end dual_pair
|
3f316cd3f99c8b43567317983c21873d1b59316f | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/algebra/algebra/bilinear.lean | 1d3bb7a750620a921c7cea0b6ec76da504c66050 | [
"Apache-2.0"
] | permissive | robertylewis/mathlib | 3d16e3e6daf5ddde182473e03a1b601d2810952c | 1d13f5b932f5e40a8308e3840f96fc882fae01f0 | refs/heads/master | 1,651,379,945,369 | 1,644,276,960,000 | 1,644,276,960,000 | 98,875,504 | 0 | 0 | Apache-2.0 | 1,644,253,514,000 | 1,501,495,700,000 | Lean | UTF-8 | Lean | false | false | 5,367 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import algebra.algebra.basic
import linear_algebra.tensor_product
import algebra.iterate_hom
/-!
# Facts about algebras involving bilinear maps and tensor products
We move a few basic statements about algebras out of `algebra.algebra.basic`,
in order to avoid importing `linear_algebra.bilinear_map` and
`linear_algebra.tensor_product` unnecessarily.
-/
universes u v w
namespace algebra
open_locale tensor_product
open module
section
variables (R A : Type*) [comm_semiring R] [semiring A] [algebra R A]
/-- The multiplication in an algebra is a bilinear map.
A weaker version of this for semirings exists as `add_monoid_hom.mul`. -/
def lmul : A →ₐ[R] (End R A) :=
{ map_one' := by { ext a, exact one_mul a },
map_mul' := by { intros a b, ext c, exact mul_assoc a b c },
map_zero' := by { ext a, exact zero_mul a },
commutes' := by { intro r, ext a, dsimp, rw [smul_def] },
.. (show A →ₗ[R] A →ₗ[R] A, from linear_map.mk₂ R (*)
(λ x y z, add_mul x y z)
(λ c x y, by rw [smul_def, smul_def, mul_assoc _ x y])
(λ x y z, mul_add x y z)
(λ c x y, by rw [smul_def, smul_def, left_comm])) }
variables {R A}
@[simp] lemma lmul_apply (p q : A) : lmul R A p q = p * q := rfl
variables (R)
/-- The multiplication on the left in an algebra is a linear map. -/
def lmul_left (r : A) : A →ₗ[R] A :=
lmul R A r
@[simp] lemma lmul_left_to_add_monoid_hom (r : A) :
(lmul_left R r : A →+ A) = add_monoid_hom.mul_left r :=
fun_like.coe_injective rfl
/-- The multiplication on the right in an algebra is a linear map. -/
def lmul_right (r : A) : A →ₗ[R] A :=
(lmul R A).to_linear_map.flip r
@[simp] lemma lmul_right_to_add_monoid_hom (r : A) :
(lmul_right R r : A →+ A) = add_monoid_hom.mul_right r :=
fun_like.coe_injective rfl
/-- Simultaneous multiplication on the left and right is a linear map. -/
def lmul_left_right (vw: A × A) : A →ₗ[R] A :=
(lmul_right R vw.2).comp (lmul_left R vw.1)
lemma commute_lmul_left_right (a b : A) :
commute (lmul_left R a) (lmul_right R b) :=
by { ext c, exact (mul_assoc a c b).symm, }
/-- The multiplication map on an algebra, as an `R`-linear map from `A ⊗[R] A` to `A`. -/
def lmul' : A ⊗[R] A →ₗ[R] A :=
tensor_product.lift (lmul R A).to_linear_map
variables {R A}
@[simp] lemma lmul'_apply {x y : A} : lmul' R (x ⊗ₜ y) = x * y :=
by simp only [algebra.lmul', tensor_product.lift.tmul, alg_hom.to_linear_map_apply, lmul_apply]
@[simp] lemma lmul_left_apply (p q : A) : lmul_left R p q = p * q := rfl
@[simp] lemma lmul_right_apply (p q : A) : lmul_right R p q = q * p := rfl
@[simp] lemma lmul_left_right_apply (vw : A × A) (p : A) :
lmul_left_right R vw p = vw.1 * p * vw.2 := rfl
@[simp] lemma lmul_left_one : lmul_left R (1:A) = linear_map.id :=
by { ext, simp only [linear_map.id_coe, one_mul, id.def, lmul_left_apply] }
@[simp] lemma lmul_left_mul (a b : A) :
lmul_left R (a * b) = (lmul_left R a).comp (lmul_left R b) :=
by { ext, simp only [lmul_left_apply, linear_map.comp_apply, mul_assoc] }
@[simp] lemma lmul_right_one : lmul_right R (1:A) = linear_map.id :=
by { ext, simp only [linear_map.id_coe, mul_one, id.def, lmul_right_apply] }
@[simp] lemma lmul_right_mul (a b : A) :
lmul_right R (a * b) = (lmul_right R b).comp (lmul_right R a) :=
by { ext, simp only [lmul_right_apply, linear_map.comp_apply, mul_assoc] }
@[simp] lemma lmul_left_zero_eq_zero :
lmul_left R (0 : A) = 0 :=
(lmul R A).map_zero
@[simp] lemma lmul_right_zero_eq_zero :
lmul_right R (0 : A) = 0 :=
(lmul R A).to_linear_map.flip.map_zero
@[simp] lemma lmul_left_eq_zero_iff (a : A) :
lmul_left R a = 0 ↔ a = 0 :=
begin
split; intros h,
{ rw [← mul_one a, ← lmul_left_apply a 1, h, linear_map.zero_apply], },
{ rw h, exact lmul_left_zero_eq_zero, },
end
@[simp] lemma lmul_right_eq_zero_iff (a : A) :
lmul_right R a = 0 ↔ a = 0 :=
begin
split; intros h,
{ rw [← one_mul a, ← lmul_right_apply a 1, h, linear_map.zero_apply], },
{ rw h, exact lmul_right_zero_eq_zero, },
end
@[simp] lemma pow_lmul_left (a : A) (n : ℕ) :
(lmul_left R a) ^ n = lmul_left R (a ^ n) :=
((lmul R A).map_pow a n).symm
@[simp] lemma pow_lmul_right (a : A) (n : ℕ) :
(lmul_right R a) ^ n = lmul_right R (a ^ n) :=
linear_map.coe_injective $ ((lmul_right R a).coe_pow n).symm ▸ (mul_right_iterate a n)
end
section
variables {R A : Type*} [comm_semiring R] [ring A] [algebra R A]
lemma lmul_left_injective [no_zero_divisors A] {x : A} (hx : x ≠ 0) :
function.injective (lmul_left R x) :=
by { letI : is_domain A := { exists_pair_ne := ⟨x, 0, hx⟩, ..‹ring A›, ..‹no_zero_divisors A› },
exact mul_right_injective₀ hx }
lemma lmul_right_injective [no_zero_divisors A] {x : A} (hx : x ≠ 0) :
function.injective (lmul_right R x) :=
by { letI : is_domain A := { exists_pair_ne := ⟨x, 0, hx⟩, ..‹ring A›, ..‹no_zero_divisors A› },
exact mul_left_injective₀ hx }
lemma lmul_injective [no_zero_divisors A] {x : A} (hx : x ≠ 0) :
function.injective (lmul R A x) :=
by { letI : is_domain A := { exists_pair_ne := ⟨x, 0, hx⟩, ..‹ring A›, ..‹no_zero_divisors A› },
exact mul_right_injective₀ hx }
end
end algebra
|
9db5bb0b95adcb39080204dcd5a2fb736b2eb0ff | 1fd908b06e3f9c1252cb2285ada1102623a67f72 | /init/pathover.lean | 4299f88501edd4382b18a18470098bca693a35ba | [
"Apache-2.0"
] | permissive | avigad/hott3 | 609a002849182721e7c7ae536d9f1e2956d6d4d3 | f64750cd2de7a81e87d4828246d1369d59f16f43 | refs/heads/master | 1,629,027,243,322 | 1,510,946,717,000 | 1,510,946,717,000 | 103,570,461 | 0 | 0 | null | 1,505,415,620,000 | 1,505,415,620,000 | null | UTF-8 | Lean | false | false | 20,235 | lean | /-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
Basic theorems about pathovers
-/
import .path .equiv
universes u v l
hott_theory
namespace hott
open hott.equiv hott.is_equiv function
variables {A : Type _} {A' : Type _} {B : A → Type _} {B' : A → Type _} {B'' : A' → Type _} {C : Π⦃a⦄, B a → Type _}
{a a₂ a₃ a₄ : A} {p p' : a = a₂} {p₂ : a₂ = a₃} {p₃ : a₃ = a₄} {p₁₃ : a = a₃}
{b b' : B a} {b₂ b₂' : B a₂} {b₃ : B a₃} {b₄ : B a₄}
{c : C b} {c₂ : C b₂}
namespace eq
inductive pathover (B : A → Type.{l}) (b : B a) : Π{a₂ : A}, a = a₂ → B a₂ → Type.{l}
| idpatho : pathover (refl a) b
notation b ` =[`:50 p:0 `] `:0 b₂:50 := pathover _ b p b₂
notation b ` =[`:50 p:0 `; `:0 B `] `:0 b₂:50 := pathover B b p b₂
@[hott, refl, reducible]
def idpo : b =[refl a] b :=
pathover.idpatho b
@[hott, reducible] def idpatho (b : B a) : b =[refl a] b :=
pathover.idpatho b
/- equivalences with equality using transport -/
@[hott] def pathover_of_tr_eq (r : p ▸ b = b₂) : b =[p] b₂ :=
by induction p; induction r; constructor
@[hott] def pathover_of_eq_tr (r : b = p⁻¹ ▸ b₂) : b =[p] b₂ :=
begin induction p, change b = b₂ at r, induction r, constructor end
@[hott] def tr_eq_of_pathover (r : b =[p] b₂) : p ▸ b = b₂ :=
by induction r; reflexivity
@[hott] def eq_tr_of_pathover (r : b =[p] b₂) : b = p⁻¹ ▸ b₂ :=
by induction r; reflexivity
@[hott] def pathover_equiv_tr_eq (p : a = a₂) (b : B a) (b₂ : B a₂)
: (b =[p] b₂) ≃ (p ▸ b = b₂) :=
begin
fapply equiv.MK,
{ exact tr_eq_of_pathover},
{ exact pathover_of_tr_eq},
{ intro r, induction p, induction r, apply idp},
{ intro r, induction r, apply idp},
end
@[hott] def pathover_equiv_eq_tr (p : a = a₂) (b : B a) (b₂ : B a₂)
: (b =[p] b₂) ≃ (b = p⁻¹ ▸ b₂) :=
begin
fapply equiv.MK,
{ exact eq_tr_of_pathover},
{ exact pathover_of_eq_tr},
{ intro r, induction p, change b = b₂ at r, induction r, apply idp},
{ intro r, induction r, apply idp},
end
@[hott] def pathover_tr (p : a = a₂) (b : B a) : b =[p] p ▸ b :=
by induction p; constructor
@[hott] def tr_pathover (p : a = a₂) (b : B a₂) : p⁻¹ ▸ b =[p] b :=
by induction p; constructor
@[hott] def concato (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) : b =[p ⬝ p₂] b₃ :=
by induction r₂; exact r
@[hott] def inverseo (r : b =[p] b₂) : b₂ =[p⁻¹] b :=
by induction r; constructor
@[hott] def concato_eq (r : b =[p] b₂) (q : b₂ = b₂') : b =[p] b₂' :=
by induction q; exact r
@[hott] def eq_concato (q : b = b') (r : b' =[p] b₂) : b =[p] b₂ :=
by induction q; exact r
@[hott] def change_path (q : p = p') (r : b =[p] b₂) : b =[p'] b₂ :=
q ▸ r
@[hott, hsimp] def change_path_idp (r : b =[p] b₂) : change_path idp r = r :=
by reflexivity
-- infix ` ⬝ ` := concato
infix ` ⬝o `:72 := concato
infix ` ⬝op `:73 := concato_eq
infix ` ⬝po `:73 := eq_concato
-- postfix `⁻¹` := inverseo
postfix `⁻¹ᵒ`:(max+10) := inverseo
@[hott] def pathover_cancel_right (q : b =[p ⬝ p₂] b₃) (r : b₃ =[p₂⁻¹] b₂) : b =[p] b₂ :=
change_path (con_inv_cancel_right _ _) (q ⬝o r)
@[hott] def pathover_cancel_right' (q : b =[p₁₃ ⬝ p₂⁻¹] b₂) (r : b₂ =[p₂] b₃) : b =[p₁₃] b₃ :=
change_path (inv_con_cancel_right _ _) (q ⬝o r)
@[hott] def pathover_cancel_left (q : b₂ =[p⁻¹] b) (r : b =[p ⬝ p₂] b₃) : b₂ =[p₂] b₃ :=
change_path (inv_con_cancel_left _ _) (q ⬝o r)
@[hott] def pathover_cancel_left' (q : b =[p] b₂) (r : b₂ =[p⁻¹ ⬝ p₁₃] b₃) : b =[p₁₃] b₃ :=
change_path (con_inv_cancel_left _ _) (q ⬝o r)
/- Some of the theorems analogous to theorems for = in init.path -/
@[hott, hsimp] def idpo_invo : idpo⁻¹ᵒ = idpo :> b =[idp] b :=
by refl
@[hott, hsimp] def cono_idpo (r : b =[p] b₂) : r ⬝o idpo = r :=
by refl
@[hott] def idpo_cono (r : b =[p] b₂) : idpo ⬝o r =[idp_con p; λ x, b =[x] b₂] r :=
by induction r; refl
@[hott] def cono.assoc' (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) (r₃ : b₃ =[p₃] b₄) :
r ⬝o (r₂ ⬝o r₃) =[con.assoc' _ _ _; λ x, b =[x] b₄] ((r ⬝o r₂) ⬝o r₃) :=
by induction r₃; induction r₂; induction r; refl
@[hott] def cono.assoc (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) (r₃ : b₃ =[p₃] b₄) :
(r ⬝o r₂) ⬝o r₃ =[con.assoc _ _ _; λ x, b =[x] b₄] r ⬝o (r₂ ⬝o r₃) :=
by induction r₃; induction r₂; induction r; refl
@[hott] def cono.right_inv (r : b =[p] b₂) : r ⬝o r⁻¹ᵒ =[con.right_inv _; λ x, b =[x] b] idpo :=
by induction r; refl
@[hott] def cono.left_inv (r : b =[p] b₂) : r⁻¹ᵒ ⬝o r =[con.left_inv _; λ x, b₂ =[x] b₂] idpo :=
by induction r; refl
@[hott] def eq_of_pathover {a' a₂' : A'} (q : a' =[p; λ _, A'] a₂') : a' = a₂' :=
by induction q; refl
@[hott] def pathover_of_eq (p : a = a₂) {a' a₂' : A'} (q : a' = a₂') : a' =[p; λ _, A'] a₂' :=
by induction p; induction q; constructor
@[hott] def pathover_constant (p : a = a₂) (a' a₂' : A') : a' =[p; λ _, A'] a₂' ≃ a' = a₂' :=
begin
fapply equiv.MK,
{ exact eq_of_pathover},
{ exact pathover_of_eq p},
abstract { intro r, induction p, induction r, refl},
abstract { intro r, induction r, refl},
end
@[hott]
def pathover_of_eq_tr_constant_inv (p : a = a₂) (a' : A')
: pathover_of_eq p (tr_constant p a')⁻¹ = pathover_tr p a' :=
by induction p; constructor
@[hott, elab_simple]
def eq_of_pathover_idp {b' : B a} (q : b =[idpath a] b') : b = b' :=
tr_eq_of_pathover q
variable (B)
@[hott]
def pathover_idp_of_eq {b' : B a} (q : b = b') : b =[idpath a] b' :=
pathover_of_tr_eq q
@[hott]
def pathover_idp (b : B a) (b' : B a) : b =[idpath a] b' ≃ b = b' :=
begin
fapply equiv.MK,
{exact eq_of_pathover_idp},
{exact pathover_idp_of_eq B},
{intros, refine to_right_inv (pathover_equiv_tr_eq _ _ _) _ },
{intro r, refine to_left_inv (pathover_equiv_tr_eq _ _ _) r, }
end
variable {B}
@[hott, hsimp]
def eq_of_pathover_idp_pathover_of_eq {A X : Type _} (x : X) {a a' : A} (p : a = a') :
eq_of_pathover_idp (pathover_of_eq (idpath x) p) = p :=
by induction p; refl
variable (B)
@[hott, hsimp] def idpo_concato_eq (r : b = b') : eq_of_pathover_idp (@idpo A B a b ⬝op r) = r :=
by induction r; reflexivity
variable {B}
-- def pathover_idp (b : B a) (b' : B a) : b =[idpath a] b' ≃ b = b' :=
-- pathover_equiv_tr_eq idp b b'
-- def eq_of_pathover_idp [reducible] {b' : B a} (q : b =[idpath a] b') : b = b' :=
-- to_fun !pathover_idp q
-- def pathover_idp_of_eq [reducible] {b' : B a} (q : b = b') : b =[idpath a] b' :=
-- to_inv !pathover_idp q
@[hott, elab_as_eliminator, reducible, recursor] def idp_rec_on {P : Π⦃b₂ : B a⦄, b =[idpath a] b₂ → Sort u}
{b₂ : B a} (r : b =[idpath a] b₂) (H : P idpo) : P r :=
by exact
have H2 : P (pathover_idp_of_eq B (eq_of_pathover_idp r)), from
eq.rec_on (eq_of_pathover_idp r) H,
have H3 : pathover_idp_of_eq B (eq_of_pathover_idp r) = r,
from to_left_inv (pathover_idp B b b₂) r,
H3 ▸ H2
@[hott] def rec_on_right {P : Π⦃b₂ : B a₂⦄, b =[p] b₂ → Type _}
{b₂ : B a₂} (r : b =[p] b₂) (H : P (pathover_tr _ _)) : P r :=
by induction r; exact H
@[hott] def rec_on_left {P : Π⦃b : B a⦄, b =[p] b₂ → Type _}
{b : B a} (r : b =[p] b₂) (H : P (tr_pathover _ _)) : P r :=
by induction r; exact H
--pathover with fibration B' ∘ f
@[hott] def pathover_ap (B' : A' → Type _) (f : A → A') {p : a = a₂}
{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[p; B' ∘ f] b₂) : b =[ap f p] b₂ :=
by induction q; constructor
@[hott] def pathover_of_pathover_ap (B' : A' → Type u) (f : A → A') {p : a = a₂}
{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[ap f p] b₂) : b =[p; B' ∘ f] b₂ :=
begin
induction p, induction q using hott.eq.idp_rec_on, exact idpo
end
@[hott] def pathover_compose (B' : A' → Type _) (f : A → A') (p : a = a₂)
(b : B' (f a)) (b₂ : B' (f a₂)) : b =[p; B' ∘ f] b₂ ≃ b =[ap f p] b₂ :=
begin
fapply equiv.MK,
{ exact pathover_ap B' f},
{ exact pathover_of_pathover_ap B' f},
{ intro q, induction p, induction q using hott.eq.idp_rec_on, refl},
{ intro q, induction q, refl},
end
@[hott] def pathover_of_pathover_tr (q : b =[p ⬝ p₂] p₂ ▸ b₂) : b =[p] b₂ :=
pathover_cancel_right q (pathover_tr _ _)⁻¹ᵒ
@[hott] def pathover_tr_of_pathover (q : b =[p₁₃ ⬝ p₂⁻¹] b₂) : b =[p₁₃] p₂ ▸ b₂ :=
pathover_cancel_right' q (pathover_tr _ _)
@[hott] def pathover_of_tr_pathover (q : p ▸ b =[p⁻¹ ⬝ p₁₃] b₃) : b =[p₁₃] b₃ :=
pathover_cancel_left' (pathover_tr _ _) q
@[hott] def tr_pathover_of_pathover (q : b =[p ⬝ p₂] b₃) : p ▸ b =[p₂] b₃ :=
pathover_cancel_left (pathover_tr _ _)⁻¹ᵒ q
@[hott] def pathover_tr_of_eq (q : b = b') : b =[p] p ▸ b' :=
by induction q;apply pathover_tr
@[hott] def tr_pathover_of_eq (q : b₂ = b₂') : p⁻¹ ▸ b₂ =[p] b₂' :=
by induction q;apply tr_pathover
@[hott] def eq_of_parallel_po_right (q : b =[p] b₂) (q' : b =[p] b₂') : b₂ = b₂' :=
begin
apply @eq_of_pathover_idp A B, apply change_path (con.left_inv p),
exact q⁻¹ᵒ ⬝o q'
end
@[hott] def eq_of_parallel_po_left (q : b =[p] b₂) (q' : b' =[p] b₂) : b = b' :=
begin
apply @eq_of_pathover_idp A B, apply change_path (con.right_inv p),
exact q ⬝o q'⁻¹ᵒ
end
variable (C)
@[hott, elab_simple] def transporto (r : b =[p] b₂) (c : C b) : C b₂ :=
by induction r;exact c
infix ` ▸o `:75 := transporto _
@[hott] def fn_tro_eq_tro_fn {C' : Π ⦃a : A⦄, B a → Type _} (q : b =[p] b₂)
(f : Π⦃a : A⦄ (b : B a), C b → C' b) (c : C b) : f b₂ (q ▸o c) = q ▸o (f b c) :=
by induction q; reflexivity
variable {C}
/- various variants of ap for pathovers -/
@[hott] def apd (f : Πa, B a) (p : a = a₂) : f a =[p] f a₂ :=
by induction p; constructor
@[hott] def apd_idp (f : Πa, B a) : apd f idp = @idpo A B a (f a) :=
by reflexivity
@[hott] def apo {f : A → A'} (g : Πa, B a → B'' (f a)) (q : b =[p] b₂) :
g a b =[p; B'' ∘ f] g a₂ b₂ :=
by induction q; constructor
@[hott] def apd011 (f : Πa, B a → A') (Ha : a = a₂) (Hb : b =[Ha] b₂)
: f a b = f a₂ b₂ :=
by induction Hb; reflexivity
@[hott] def apd0111 (f : Πa b, C b → A') (Ha : a = a₂) (Hb : b =[Ha] b₂)
(Hc : c =[apd011 C Ha Hb; id] c₂) : f a b c = f a₂ b₂ c₂ :=
by induction Hb; induction Hc using hott.eq.idp_rec_on; refl
@[hott] def apod11 {f : Πb, C b} {g : Πb₂, C b₂} (r : f =[p; λ a, Π b : B a, C b] g)
{b : B a} {b₂ : B a₂} (q : b =[p] b₂) : f b =[apd011 C p q; id] g b₂ :=
by induction r; induction q using hott.eq.idp_rec_on; constructor
@[hott] def apdo10 {f : Πb, C b} {g : Πb₂, C b₂} (r : f =[p; λ a, Π b : B a, C b] g)
(b : B a) : f b =[apd011 C p (pathover_tr _ _); id] g (p ▸ b) :=
by induction r; constructor
@[hott] def apo10 {f : B a → B' a} {g : B a₂ → B' a₂} (r : f =[p; λ a, B a → B' a] g)
(b : B a) : f b =[p] g (p ▸ b) :=
by induction r; constructor
@[hott] def apo10_constant_right {f : B a → A'} {g : B a₂ → A'} (r : f =[p; λ a, B a → A'] g)
(b : B a) : f b = g (p ▸ b) :=
by induction r; constructor
@[hott] def apo10_constant_left {f : A' → B a} {g : A' → B a₂} (r : f =[p; λ a, A' → B a] g)
(a' : A') : f a' =[p] g a' :=
by induction r; constructor
@[hott] def apo11 {f : B a → B' a} {g : B a₂ → B' a₂} (r : f =[p; λ a, B a → B' a] g)
(q : b =[p] b₂) : f b =[p] g b₂ :=
by induction q; exact apo10 r b
@[hott] def apo011 {A : Type _} {B C D : A → Type _} {a a' : A} {p : a = a'} {b : B a} {b' : B a'}
{c : C a} {c' : C a'} (f : Π⦃a⦄, B a → C a → D a) (q : b =[p] b') (r : c =[p] c') :
f b c =[p] f b' c' :=
begin induction q, induction r using hott.eq.idp_rec_on, exact idpo end
@[hott] def apdo011 {A : Type _} {B : A → Type _} {C : Π⦃a⦄, B a → Type _}
(f : Π⦃a⦄ (b : B a), C b) {a a' : A} (p : a = a') {b : B a} {b' : B a'} (q : b =[p] b')
: f b =[apd011 C p q; id] f b' :=
by induction q; constructor
@[hott] def apdo0111 {A : Type _} {B : A → Type _} {C C' : Π⦃a⦄, B a → Type _}
(f : Π⦃a⦄ {b : B a}, C b → C' b) {a a' : A} (p : a = a') {b : B a} {b' : B a'} (q : b =[p] b')
{c : C b} {c' : C b'} (r : c =[apd011 C p q; id] c')
: f c =[apd011 C' p q; id] f c' :=
begin
induction q, dsimp [apd011] at r,
induction r using hott.eq.idp_rec_on, refl
end
@[hott] def apo11_constant_right {f : B a → A'} {g : B a₂ → A'}
(q : f =[p; λ a, B a → A'] g) (r : b =[p] b₂) : f b = g b₂ :=
eq_of_pathover (apo11 q r)
/- properties about these "ap"s, transporto and pathover_ap -/
@[hott] def apd_con (f : Πa, B a) (p : a = a₂) (q : a₂ = a₃)
: apd f (p ⬝ q) = apd f p ⬝o apd f q :=
by induction p; induction q; reflexivity
@[hott] def apd_inv (f : Πa, B a) (p : a = a₂) : apd f p⁻¹ = (apd f p)⁻¹ᵒ :=
by induction p; reflexivity
@[hott] def apd_eq_pathover_of_eq_ap (f : A → A') (p : a = a₂) :
apd f p = pathover_of_eq p (ap f p) :=
eq.rec_on p idp
@[hott] def apo_invo (f : Πa, B a → B' a) {Ha : a = a₂} (Hb : b =[Ha] b₂)
: (apo f Hb)⁻¹ᵒ = apo f Hb⁻¹ᵒ :=
by induction Hb; reflexivity
@[hott] def apd011_inv (f : Πa, B a → A') (Ha : a = a₂) (Hb : b =[Ha] b₂)
: (apd011 f Ha Hb)⁻¹ = (apd011 f Ha⁻¹ Hb⁻¹ᵒ) :=
by induction Hb; reflexivity
@[hott] def cast_apd011 (q : b =[p] b₂) (c : C b) : cast (apd011 C p q) c = q ▸o c :=
by induction q; reflexivity
@[hott] def apd_compose1 (g : Πa, B a → B' a) (f : Πa, B a) (p : a = a₂)
: apd (g ∘' f) p = apo g (apd f p) :=
by induction p; reflexivity
@[hott] def apd_compose2 (g : Πa', B'' a') (f : A → A') (p : a = a₂)
: apd (λa, g (f a)) p = pathover_of_pathover_ap B'' f (apd g (ap f p)) :=
by induction p; reflexivity
@[hott] def apo_tro (C : Π⦃a⦄, B' a → Type _) (f : Π⦃a⦄, B a → B' a) (q : b =[p] b₂)
(c : C (f b)) : apo f q ▸o c = q ▸o c :=
by induction q; reflexivity
@[hott] def pathover_ap_tro {B' : A' → Type _} (C : Π⦃a'⦄, B' a' → Type _) (f : A → A')
{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[p; B' ∘ f] b₂) (c : C b) :
pathover_ap B' f q ▸o c = q ▸o c :=
by induction q; reflexivity
@[hott] def pathover_ap_invo_tro {B' : A' → Type _} (C : Π⦃a'⦄, B' a' → Type _) (f : A → A')
{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[p; B' ∘ f] b₂) (c : C b₂)
: (pathover_ap B' f q)⁻¹ᵒ ▸o c = q⁻¹ᵒ ▸o c :=
by induction q; reflexivity
@[hott, elab_simple] def pathover_tro (q : b =[p] b₂) (c : C b) : c =[apd011 C p q; id] q ▸o c :=
by induction q; constructor
@[hott] def pathover_ap_invo {B' : A' → Type _} (f : A → A') {p : a = a₂}
{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[p; B' ∘ f] b₂)
: pathover_ap B' f q⁻¹ᵒ =[ap_inv f p; λ x, b₂ =[x] b] (pathover_ap B' f q)⁻¹ᵒ :=
by induction q; exact idpo
@[hott] def tro_invo_tro {A : Type _} {B : A → Type _} (C : Π⦃a⦄, B a → Type _)
{a a' : A} {p : a = a'} {b : B a} {b' : B a'} (q : b =[p] b') (c : C b') :
q ▸o (q⁻¹ᵒ ▸o c) = c :=
by induction q; reflexivity
@[hott] def invo_tro_tro {A : Type _} {B : A → Type _} (C : Π⦃a⦄, B a → Type _)
{a a' : A} {p : a = a'} {b : B a} {b' : B a'} (q : b =[p] b') (c : C b) :
q⁻¹ᵒ ▸o (q ▸o c) = c :=
by induction q; reflexivity
@[hott] def cono_tro {A : Type _} {B : A → Type _} (C : Π⦃a⦄, B a → Type _)
{a₁ a₂ a₃ : A} {p₁ : a₁ = a₂} {p₂ : a₂ = a₃} {b₁ : B a₁} {b₂ : B a₂} {b₃ : B a₃}
(q₁ : b₁ =[p₁] b₂) (q₂ : b₂ =[p₂] b₃) (c : C b₁) :
transporto C (q₁ ⬝o q₂) c = transporto C q₂ (transporto C q₁ c) :=
by induction q₂; reflexivity
@[hott] def is_equiv_transporto {A : Type _} {B : A → Type _} (C : Π⦃a⦄, B a → Type _)
{a a' : A} {p : a = a'} {b : B a} {b' : B a'} (q : b =[p] b') : is_equiv (transporto C q) :=
begin
fapply adjointify,
{ exact transporto C q⁻¹ᵒ},
{ exact tro_invo_tro C q},
{ exact invo_tro_tro C q}
end
@[hott] def equiv_apd011 {A : Type _} {B : A → Type _} (C : Π⦃a⦄, B a → Type _)
{a a' : A} {p : a = a'} {b : B a} {b' : B a'} (q : b =[p] b') : C b ≃ C b' :=
equiv.mk (transporto C q) (is_equiv_transporto _ _)
/- some cancellation laws for concato_eq an variants -/
@[hott] def cono.right_inv_eq (q : b = b') :
pathover_idp_of_eq B q ⬝op q⁻¹ = (idpo : b =[refl a] b) :=
by induction q;constructor
@[hott] def cono.right_inv_eq' (q : b = b') :
q ⬝po (pathover_idp_of_eq B q⁻¹) = (idpo : b =[refl a] b) :=
by induction q;constructor
@[hott] def cono.left_inv_eq (q : b = b') :
pathover_idp_of_eq B q⁻¹ ⬝op q = (idpo : b' =[refl a] b') :=
by induction q;constructor
@[hott] def cono.left_inv_eq' (q : b = b') :
q⁻¹ ⬝po pathover_idp_of_eq B q = (idpo : b' =[refl a] b') :=
by induction q;constructor
@[hott] def pathover_of_fn_pathover_fn (f : Π{a}, B a ≃ B' a) (r : f.to_fun b =[p] f.to_fun b₂) : b =[p] b₂ :=
(left_inv f.to_fun b)⁻¹ ⬝po apo (λa, to_fun f⁻¹ᵉ) r ⬝op left_inv f.to_fun b₂
/- a pathover in a pathover type where the only thing which varies is the path is the same as
an equality with a change_path -/
@[hott] def change_path_of_pathover (s : p = p') (r : b =[p] b₂) (r' : b =[p'] b₂)
(q : r =[s; λ p, b =[p] b₂] r') : change_path s r = r' :=
by induction s; induction q using hott.eq.idp_rec_on; reflexivity
@[hott] def pathover_of_change_path (s : p = p') (r : b =[p] b₂) (r' : b =[p'] b₂)
(q : change_path s r = r') : r =[s; λ p, b =[p] b₂] r' :=
by induction s; induction q; constructor
@[hott] def pathover_pathover_path (s : p = p') (r : b =[p] b₂) (r' : b =[p'] b₂) :
(r =[s; λ p, b =[p] b₂] r') ≃ change_path s r = r' :=
begin
fapply equiv.MK,
{ apply change_path_of_pathover},
{ apply pathover_of_change_path},
{ intro q, induction s, induction q, reflexivity},
{ intro q, induction s, induction q using hott.eq.idp_rec_on, reflexivity},
end
/- variants of inverse2 and concat2 -/
@[hott] def inverseo2 {r r' : b =[p] b₂} (s : r = r') : r⁻¹ᵒ = r'⁻¹ᵒ :=
by induction s; reflexivity
@[hott] def concato2 {r r' : b =[p] b₂} {r₂ r₂' : b₂ =[p₂] b₃}
(s : r = r') (s₂ : r₂ = r₂') : r ⬝o r₂ = r' ⬝o r₂' :=
by induction s; induction s₂; reflexivity
infixl ` ◾o `:75 := concato2
postfix [parsing_only] `⁻²ᵒ`:(max+10) := inverseo2 --this notation is abusive, should we use it?
-- find a better name for this
@[hott] def fn_tro_eq_tro_fn2 (q : b =[p] b₂)
{k : A → A} {l : Π⦃a⦄, B a → B (k a)} (m : Π⦃a⦄ {b : B a}, C b → C (l b))
(c : C b) :
m (q ▸o c) = (pathover_ap B k (apo l q)) ▸o (m c) :=
by induction q; reflexivity
@[hott] def apd0111_precompose (f : Π⦃a⦄ {b : B a}, C b → A')
{k : A → A} {l : Π⦃a⦄, B a → B (k a)} (m : Π⦃a⦄ {b : B a}, C b → C (l b))
{q : b =[p] b₂} (c : C b)
: apd0111 (λa b (c : C b), f (m c)) p q (pathover_tro q c) ⬝ ap (@f _ _) (fn_tro_eq_tro_fn2 q m c) =
apd0111 f (ap k p) (pathover_ap B k (apo l q)) (pathover_tro _ (m c)) :=
by induction q; reflexivity
end eq
end hott |
1fe84d15de04b424de34265d10a8602cc118e5c8 | f57749ca63d6416f807b770f67559503fdb21001 | /library/algebra/ordered_ring.lean | 037566b91bc859bc5a9daf35ec52d58cee839852 | [
"Apache-2.0"
] | permissive | aliassaf/lean | bd54e85bed07b1ff6f01396551867b2677cbc6ac | f9b069b6a50756588b309b3d716c447004203152 | refs/heads/master | 1,610,982,152,948 | 1,438,916,029,000 | 1,438,916,029,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 23,898 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
Here an "ordered_ring" is partially ordered ring, which is ordered with respect to both a weak
order and an associated strict order. Our numeric structures (int, rat, and real) will be instances
of "linear_ordered_comm_ring". This development is modeled after Isabelle's library.
-/
import algebra.ordered_group algebra.ring
open eq eq.ops
namespace algebra
variable {A : Type}
private definition absurd_a_lt_a {B : Type} {a : A} [s : strict_order A] (H : a < a) : B :=
absurd H (lt.irrefl a)
/- semiring structures -/
structure ordered_semiring [class] (A : Type)
extends semiring A, ordered_cancel_comm_monoid A :=
(mul_le_mul_of_nonneg_left: ∀a b c, le a b → le zero c → le (mul c a) (mul c b))
(mul_le_mul_of_nonneg_right: ∀a b c, le a b → le zero c → le (mul a c) (mul b c))
(mul_lt_mul_of_pos_left: ∀a b c, lt a b → lt zero c → lt (mul c a) (mul c b))
(mul_lt_mul_of_pos_right: ∀a b c, lt a b → lt zero c → lt (mul a c) (mul b c))
section
variable [s : ordered_semiring A]
variables (a b c d e : A)
include s
theorem mul_le_mul_of_nonneg_left {a b c : A} (Hab : a ≤ b) (Hc : 0 ≤ c) :
c * a ≤ c * b := !ordered_semiring.mul_le_mul_of_nonneg_left Hab Hc
theorem mul_le_mul_of_nonneg_right {a b c : A} (Hab : a ≤ b) (Hc : 0 ≤ c) :
a * c ≤ b * c := !ordered_semiring.mul_le_mul_of_nonneg_right Hab Hc
-- TODO: there are four variations, depending on which variables we assume to be nonneg
theorem mul_le_mul {a b c d : A} (Hac : a ≤ c) (Hbd : b ≤ d) (nn_b : 0 ≤ b) (nn_c : 0 ≤ c) :
a * b ≤ c * d :=
calc
a * b ≤ c * b : mul_le_mul_of_nonneg_right Hac nn_b
... ≤ c * d : mul_le_mul_of_nonneg_left Hbd nn_c
theorem mul_nonneg {a b : A} (Ha : a ≥ 0) (Hb : b ≥ 0) : a * b ≥ 0 :=
begin
have H : 0 * b ≤ a * b, from mul_le_mul_of_nonneg_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_nonpos_of_nonneg_of_nonpos {a b : A} (Ha : a ≥ 0) (Hb : b ≤ 0) : a * b ≤ 0 :=
begin
have H : a * b ≤ a * 0, from mul_le_mul_of_nonneg_left Hb Ha,
rewrite mul_zero at H,
exact H
end
theorem mul_nonpos_of_nonpos_of_nonneg {a b : A} (Ha : a ≤ 0) (Hb : b ≥ 0) : a * b ≤ 0 :=
begin
have H : a * b ≤ 0 * b, from mul_le_mul_of_nonneg_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_lt_mul_of_pos_left {a b c : A} (Hab : a < b) (Hc : 0 < c) :
c * a < c * b := !ordered_semiring.mul_lt_mul_of_pos_left Hab Hc
theorem mul_lt_mul_of_pos_right {a b c : A} (Hab : a < b) (Hc : 0 < c) :
a * c < b * c := !ordered_semiring.mul_lt_mul_of_pos_right Hab Hc
-- TODO: once again, there are variations
theorem mul_lt_mul {a b c d : A} (Hac : a < c) (Hbd : b ≤ d) (pos_b : 0 < b) (nn_c : 0 ≤ c) :
a * b < c * d :=
calc
a * b < c * b : mul_lt_mul_of_pos_right Hac pos_b
... ≤ c * d : mul_le_mul_of_nonneg_left Hbd nn_c
theorem mul_pos {a b : A} (Ha : a > 0) (Hb : b > 0) : a * b > 0 :=
begin
have H : 0 * b < a * b, from mul_lt_mul_of_pos_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_neg_of_pos_of_neg {a b : A} (Ha : a > 0) (Hb : b < 0) : a * b < 0 :=
begin
have H : a * b < a * 0, from mul_lt_mul_of_pos_left Hb Ha,
rewrite mul_zero at H,
exact H
end
theorem mul_neg_of_neg_of_pos {a b : A} (Ha : a < 0) (Hb : b > 0) : a * b < 0 :=
begin
have H : a * b < 0 * b, from mul_lt_mul_of_pos_right Ha Hb,
rewrite zero_mul at H,
exact H
end
end
structure linear_ordered_semiring [class] (A : Type)
extends ordered_semiring A, linear_strong_order_pair A :=
(zero_lt_one : lt zero one)
section
variable [s : linear_ordered_semiring A]
variables {a b c : A}
include s
theorem zero_lt_one : 0 < (1:A) := linear_ordered_semiring.zero_lt_one A
theorem lt_of_mul_lt_mul_left (H : c * a < c * b) (Hc : c ≥ 0) : a < b :=
lt_of_not_ge
(assume H1 : b ≤ a,
have H2 : c * b ≤ c * a, from mul_le_mul_of_nonneg_left H1 Hc,
not_lt_of_ge H2 H)
theorem lt_of_mul_lt_mul_right (H : a * c < b * c) (Hc : c ≥ 0) : a < b :=
lt_of_not_ge
(assume H1 : b ≤ a,
have H2 : b * c ≤ a * c, from mul_le_mul_of_nonneg_right H1 Hc,
not_lt_of_ge H2 H)
theorem le_of_mul_le_mul_left (H : c * a ≤ c * b) (Hc : c > 0) : a ≤ b :=
le_of_not_gt
(assume H1 : b < a,
have H2 : c * b < c * a, from mul_lt_mul_of_pos_left H1 Hc,
not_le_of_gt H2 H)
theorem le_of_mul_le_mul_right (H : a * c ≤ b * c) (Hc : c > 0) : a ≤ b :=
le_of_not_gt
(assume H1 : b < a,
have H2 : b * c < a * c, from mul_lt_mul_of_pos_right H1 Hc,
not_le_of_gt H2 H)
theorem le_iff_mul_le_mul_left (a b : A) {c : A} (H : c > 0) : a ≤ b ↔ c * a ≤ c * b :=
iff.intro
(assume H', mul_le_mul_of_nonneg_left H' (le_of_lt H))
(assume H', le_of_mul_le_mul_left H' H)
theorem le_iff_mul_le_mul_right (a b : A) {c : A} (H : c > 0) : a ≤ b ↔ a * c ≤ b * c :=
iff.intro
(assume H', mul_le_mul_of_nonneg_right H' (le_of_lt H))
(assume H', le_of_mul_le_mul_right H' H)
theorem pos_of_mul_pos_left (H : 0 < a * b) (H1 : 0 ≤ a) : 0 < b :=
lt_of_not_ge
(assume H2 : b ≤ 0,
have H3 : a * b ≤ 0, from mul_nonpos_of_nonneg_of_nonpos H1 H2,
not_lt_of_ge H3 H)
theorem pos_of_mul_pos_right (H : 0 < a * b) (H1 : 0 ≤ b) : 0 < a :=
lt_of_not_ge
(assume H2 : a ≤ 0,
have H3 : a * b ≤ 0, from mul_nonpos_of_nonpos_of_nonneg H2 H1,
not_lt_of_ge H3 H)
theorem nonneg_of_mul_nonneg_left (H : 0 ≤ a * b) (H1 : 0 < a) : 0 ≤ b :=
le_of_not_gt
(assume H2 : b < 0,
not_le_of_gt (mul_neg_of_pos_of_neg H1 H2) H)
theorem nonneg_of_mul_nonneg_right (H : 0 ≤ a * b) (H1 : 0 < b) : 0 ≤ a :=
le_of_not_gt
(assume H2 : a < 0,
not_le_of_gt (mul_neg_of_neg_of_pos H2 H1) H)
theorem neg_of_mul_neg_left (H : a * b < 0) (H1 : 0 ≤ a) : b < 0 :=
lt_of_not_ge
(assume H2 : b ≥ 0,
not_lt_of_ge (mul_nonneg H1 H2) H)
theorem neg_of_mul_neg_right (H : a * b < 0) (H1 : 0 ≤ b) : a < 0 :=
lt_of_not_ge
(assume H2 : a ≥ 0,
not_lt_of_ge (mul_nonneg H2 H1) H)
theorem nonpos_of_mul_nonpos_left (H : a * b ≤ 0) (H1 : 0 < a) : b ≤ 0 :=
le_of_not_gt
(assume H2 : b > 0,
not_le_of_gt (mul_pos H1 H2) H)
theorem nonpos_of_mul_nonpos_right (H : a * b ≤ 0) (H1 : 0 < b) : a ≤ 0 :=
le_of_not_gt
(assume H2 : a > 0,
not_le_of_gt (mul_pos H2 H1) H)
end
structure decidable_linear_ordered_semiring [class] (A : Type)
extends linear_ordered_semiring A, decidable_linear_order A
/- ring structures -/
structure ordered_ring [class] (A : Type)
extends ring A, ordered_comm_group A, zero_ne_one_class A :=
(mul_nonneg : ∀a b, le zero a → le zero b → le zero (mul a b))
(mul_pos : ∀a b, lt zero a → lt zero b → lt zero (mul a b))
theorem ordered_ring.mul_le_mul_of_nonneg_left [s : ordered_ring A] {a b c : A}
(Hab : a ≤ b) (Hc : 0 ≤ c) : c * a ≤ c * b :=
have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
assert H2 : 0 ≤ c * (b - a), from ordered_ring.mul_nonneg _ _ Hc H1,
begin
rewrite mul_sub_left_distrib at H2,
exact (iff.mp !sub_nonneg_iff_le H2)
end
theorem ordered_ring.mul_le_mul_of_nonneg_right [s : ordered_ring A] {a b c : A}
(Hab : a ≤ b) (Hc : 0 ≤ c) : a * c ≤ b * c :=
have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
assert H2 : 0 ≤ (b - a) * c, from ordered_ring.mul_nonneg _ _ H1 Hc,
begin
rewrite mul_sub_right_distrib at H2,
exact (iff.mp !sub_nonneg_iff_le H2)
end
theorem ordered_ring.mul_lt_mul_of_pos_left [s : ordered_ring A] {a b c : A}
(Hab : a < b) (Hc : 0 < c) : c * a < c * b :=
have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
assert H2 : 0 < c * (b - a), from ordered_ring.mul_pos _ _ Hc H1,
begin
rewrite mul_sub_left_distrib at H2,
exact (iff.mp !sub_pos_iff_lt H2)
end
theorem ordered_ring.mul_lt_mul_of_pos_right [s : ordered_ring A] {a b c : A}
(Hab : a < b) (Hc : 0 < c) : a * c < b * c :=
have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
assert H2 : 0 < (b - a) * c, from ordered_ring.mul_pos _ _ H1 Hc,
begin
rewrite mul_sub_right_distrib at H2,
exact (iff.mp !sub_pos_iff_lt H2)
end
definition ordered_ring.to_ordered_semiring [trans-instance] [coercion] [reducible]
[s : ordered_ring A] :
ordered_semiring A :=
⦃ ordered_semiring, s,
mul_zero := mul_zero,
zero_mul := zero_mul,
add_left_cancel := @add.left_cancel A s,
add_right_cancel := @add.right_cancel A s,
le_of_add_le_add_left := @le_of_add_le_add_left A s,
mul_le_mul_of_nonneg_left := @ordered_ring.mul_le_mul_of_nonneg_left A s,
mul_le_mul_of_nonneg_right := @ordered_ring.mul_le_mul_of_nonneg_right A s,
mul_lt_mul_of_pos_left := @ordered_ring.mul_lt_mul_of_pos_left A s,
mul_lt_mul_of_pos_right := @ordered_ring.mul_lt_mul_of_pos_right A s,
lt_of_add_lt_add_left := @lt_of_add_lt_add_left A s⦄
section
variable [s : ordered_ring A]
variables {a b c : A}
include s
theorem mul_le_mul_of_nonpos_left (H : b ≤ a) (Hc : c ≤ 0) : c * a ≤ c * b :=
have Hc' : -c ≥ 0, from iff.mpr !neg_nonneg_iff_nonpos Hc,
assert H1 : -c * b ≤ -c * a, from mul_le_mul_of_nonneg_left H Hc',
have H2 : -(c * b) ≤ -(c * a),
begin
rewrite [-*neg_mul_eq_neg_mul at H1],
exact H1
end,
iff.mp !neg_le_neg_iff_le H2
theorem mul_le_mul_of_nonpos_right (H : b ≤ a) (Hc : c ≤ 0) : a * c ≤ b * c :=
have Hc' : -c ≥ 0, from iff.mpr !neg_nonneg_iff_nonpos Hc,
assert H1 : b * -c ≤ a * -c, from mul_le_mul_of_nonneg_right H Hc',
have H2 : -(b * c) ≤ -(a * c),
begin
rewrite [-*neg_mul_eq_mul_neg at H1],
exact H1
end,
iff.mp !neg_le_neg_iff_le H2
theorem mul_nonneg_of_nonpos_of_nonpos (Ha : a ≤ 0) (Hb : b ≤ 0) : 0 ≤ a * b :=
begin
have H : 0 * b ≤ a * b, from mul_le_mul_of_nonpos_right Ha Hb,
rewrite zero_mul at H,
exact H
end
theorem mul_lt_mul_of_neg_left (H : b < a) (Hc : c < 0) : c * a < c * b :=
have Hc' : -c > 0, from iff.mpr !neg_pos_iff_neg Hc,
assert H1 : -c * b < -c * a, from mul_lt_mul_of_pos_left H Hc',
have H2 : -(c * b) < -(c * a),
begin
rewrite [-*neg_mul_eq_neg_mul at H1],
exact H1
end,
iff.mp !neg_lt_neg_iff_lt H2
theorem mul_lt_mul_of_neg_right (H : b < a) (Hc : c < 0) : a * c < b * c :=
have Hc' : -c > 0, from iff.mpr !neg_pos_iff_neg Hc,
assert H1 : b * -c < a * -c, from mul_lt_mul_of_pos_right H Hc',
have H2 : -(b * c) < -(a * c),
begin
rewrite [-*neg_mul_eq_mul_neg at H1],
exact H1
end,
iff.mp !neg_lt_neg_iff_lt H2
theorem mul_pos_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a * b :=
begin
have H : 0 * b < a * b, from mul_lt_mul_of_neg_right Ha Hb,
rewrite zero_mul at H,
exact H
end
end
-- TODO: we can eliminate mul_pos_of_pos, but now it is not worth the effort to redeclare the
-- class instance
structure linear_ordered_ring [class] (A : Type)
extends ordered_ring A, linear_strong_order_pair A :=
(zero_lt_one : lt zero one)
definition linear_ordered_ring.to_linear_ordered_semiring [trans-instance] [coercion] [reducible]
[s : linear_ordered_ring A] :
linear_ordered_semiring A :=
⦃ linear_ordered_semiring, s,
mul_zero := mul_zero,
zero_mul := zero_mul,
add_left_cancel := @add.left_cancel A s,
add_right_cancel := @add.right_cancel A s,
le_of_add_le_add_left := @le_of_add_le_add_left A s,
mul_le_mul_of_nonneg_left := @mul_le_mul_of_nonneg_left A s,
mul_le_mul_of_nonneg_right := @mul_le_mul_of_nonneg_right A s,
mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left A s,
mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right A s,
le_total := linear_ordered_ring.le_total,
lt_of_add_lt_add_left := @lt_of_add_lt_add_left A s ⦄
structure linear_ordered_comm_ring [class] (A : Type) extends linear_ordered_ring A, comm_monoid A
theorem linear_ordered_comm_ring.eq_zero_or_eq_zero_of_mul_eq_zero [s : linear_ordered_comm_ring A]
{a b : A} (H : a * b = 0) : a = 0 ∨ b = 0 :=
lt.by_cases
(assume Ha : 0 < a,
lt.by_cases
(assume Hb : 0 < b,
begin
have H1 : 0 < a * b, from mul_pos Ha Hb,
rewrite H at H1,
apply absurd_a_lt_a H1
end)
(assume Hb : 0 = b, or.inr (Hb⁻¹))
(assume Hb : 0 > b,
begin
have H1 : 0 > a * b, from mul_neg_of_pos_of_neg Ha Hb,
rewrite H at H1,
apply absurd_a_lt_a H1
end))
(assume Ha : 0 = a, or.inl (Ha⁻¹))
(assume Ha : 0 > a,
lt.by_cases
(assume Hb : 0 < b,
begin
have H1 : 0 > a * b, from mul_neg_of_neg_of_pos Ha Hb,
rewrite H at H1,
apply absurd_a_lt_a H1
end)
(assume Hb : 0 = b, or.inr (Hb⁻¹))
(assume Hb : 0 > b,
begin
have H1 : 0 < a * b, from mul_pos_of_neg_of_neg Ha Hb,
rewrite H at H1,
apply absurd_a_lt_a H1
end))
-- Linearity implies no zero divisors. Doesn't need commutativity.
definition linear_ordered_comm_ring.to_integral_domain [trans-instance] [coercion] [reducible]
[s: linear_ordered_comm_ring A] : integral_domain A :=
⦃ integral_domain, s,
eq_zero_or_eq_zero_of_mul_eq_zero :=
@linear_ordered_comm_ring.eq_zero_or_eq_zero_of_mul_eq_zero A s ⦄
section
variable [s : linear_ordered_ring A]
variables (a b c : A)
include s
theorem mul_self_nonneg : a * a ≥ 0 :=
or.elim (le.total 0 a)
(assume H : a ≥ 0, mul_nonneg H H)
(assume H : a ≤ 0, mul_nonneg_of_nonpos_of_nonpos H H)
theorem zero_le_one : 0 ≤ (1:A) := one_mul 1 ▸ mul_self_nonneg 1
theorem pos_and_pos_or_neg_and_neg_of_mul_pos {a b : A} (Hab : a * b > 0) :
(a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0) :=
lt.by_cases
(assume Ha : 0 < a,
lt.by_cases
(assume Hb : 0 < b, or.inl (and.intro Ha Hb))
(assume Hb : 0 = b,
begin
rewrite [-Hb at Hab, mul_zero at Hab],
apply absurd_a_lt_a Hab
end)
(assume Hb : b < 0,
absurd Hab (lt.asymm (mul_neg_of_pos_of_neg Ha Hb))))
(assume Ha : 0 = a,
begin
rewrite [-Ha at Hab, zero_mul at Hab],
apply absurd_a_lt_a Hab
end)
(assume Ha : a < 0,
lt.by_cases
(assume Hb : 0 < b,
absurd Hab (lt.asymm (mul_neg_of_neg_of_pos Ha Hb)))
(assume Hb : 0 = b,
begin
rewrite [-Hb at Hab, mul_zero at Hab],
apply absurd_a_lt_a Hab
end)
(assume Hb : b < 0, or.inr (and.intro Ha Hb)))
theorem gt_of_mul_lt_mul_neg_left {a b c : A} (H : c * a < c * b) (Hc : c ≤ 0) : a > b :=
have nhc : -c ≥ 0, from neg_nonneg_of_nonpos Hc,
have H2 : -(c * b) < -(c * a), from iff.mpr (neg_lt_neg_iff_lt _ _) H,
have H3 : (-c) * b < (-c) * a, from calc
(-c) * b = - (c * b) : neg_mul_eq_neg_mul
... < -(c * a) : H2
... = (-c) * a : neg_mul_eq_neg_mul,
lt_of_mul_lt_mul_left H3 nhc
theorem zero_gt_neg_one : -1 < (0:A) :=
neg_zero ▸ (neg_lt_neg zero_lt_one)
theorem le_of_mul_le_of_ge_one {a b c : A} (H : a * c ≤ b) (Hb : b ≥ 0) (Hc : c ≥ 1) : a ≤ b :=
have H' : a * c ≤ b * c, from calc
a * c ≤ b : H
... = b * 1 : mul_one
... ≤ b * c : mul_le_mul_of_nonneg_left Hc Hb,
le_of_mul_le_mul_right H' (lt_of_lt_of_le zero_lt_one Hc)
end
/- TODO: Isabelle's library has all kinds of cancelation rules for the simplifier.
Search on mult_le_cancel_right1 in Rings.thy. -/
structure decidable_linear_ordered_comm_ring [class] (A : Type) extends linear_ordered_comm_ring A,
decidable_linear_ordered_comm_group A
section
variable [s : decidable_linear_ordered_comm_ring A]
variables {a b c : A}
include s
definition sign (a : A) : A := lt.cases a 0 (-1) 0 1
theorem sign_of_neg (H : a < 0) : sign a = -1 := lt.cases_of_lt H
theorem sign_zero : sign 0 = (0:A) := lt.cases_of_eq rfl
theorem sign_of_pos (H : a > 0) : sign a = 1 := lt.cases_of_gt H
theorem sign_one : sign 1 = (1:A) := sign_of_pos zero_lt_one
theorem sign_neg_one : sign (-1) = -(1:A) := sign_of_neg (neg_neg_of_pos zero_lt_one)
theorem sign_sign (a : A) : sign (sign a) = sign a :=
lt.by_cases
(assume H : a > 0,
calc
sign (sign a) = sign 1 : by rewrite (sign_of_pos H)
... = 1 : by rewrite sign_one
... = sign a : by rewrite (sign_of_pos H))
(assume H : 0 = a,
calc
sign (sign a) = sign (sign 0) : by rewrite H
... = sign 0 : by rewrite sign_zero at {1}
... = sign a : by rewrite -H)
(assume H : a < 0,
calc
sign (sign a) = sign (-1) : by rewrite (sign_of_neg H)
... = -1 : by rewrite sign_neg_one
... = sign a : by rewrite (sign_of_neg H))
theorem pos_of_sign_eq_one (H : sign a = 1) : a > 0 :=
lt.by_cases
(assume H1 : 0 < a, H1)
(assume H1 : 0 = a,
begin
rewrite [-H1 at H, sign_zero at H],
apply absurd H zero_ne_one
end)
(assume H1 : 0 > a,
have H2 : -1 = 1, from (sign_of_neg H1)⁻¹ ⬝ H,
absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
theorem eq_zero_of_sign_eq_zero (H : sign a = 0) : a = 0 :=
lt.by_cases
(assume H1 : 0 < a,
absurd (H⁻¹ ⬝ sign_of_pos H1) zero_ne_one)
(assume H1 : 0 = a, H1⁻¹)
(assume H1 : 0 > a,
have H2 : 0 = -1, from H⁻¹ ⬝ sign_of_neg H1,
have H3 : 1 = 0, from eq_neg_of_eq_neg H2 ⬝ neg_zero,
absurd (H3⁻¹) zero_ne_one)
theorem neg_of_sign_eq_neg_one (H : sign a = -1) : a < 0 :=
lt.by_cases
(assume H1 : 0 < a,
have H2 : -1 = 1, from H⁻¹ ⬝ (sign_of_pos H1),
absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
(assume H1 : 0 = a,
have H2 : (0:A) = -1,
begin
rewrite [-H1 at H, sign_zero at H],
exact H
end,
have H3 : 1 = 0, from eq_neg_of_eq_neg H2 ⬝ neg_zero,
absurd (H3⁻¹) zero_ne_one)
(assume H1 : 0 > a, H1)
theorem sign_neg (a : A) : sign (-a) = -(sign a) :=
lt.by_cases
(assume H1 : 0 < a,
calc
sign (-a) = -1 : sign_of_neg (neg_neg_of_pos H1)
... = -(sign a) : by rewrite (sign_of_pos H1))
(assume H1 : 0 = a,
calc
sign (-a) = sign (-0) : by rewrite H1
... = sign 0 : by rewrite neg_zero
... = 0 : by rewrite sign_zero
... = -0 : by rewrite neg_zero
... = -(sign 0) : by rewrite sign_zero
... = -(sign a) : by rewrite -H1)
(assume H1 : 0 > a,
calc
sign (-a) = 1 : sign_of_pos (neg_pos_of_neg H1)
... = -(-1) : by rewrite neg_neg
... = -(sign a) : sign_of_neg H1)
theorem sign_mul (a b : A) : sign (a * b) = sign a * sign b :=
lt.by_cases
(assume z_lt_a : 0 < a,
lt.by_cases
(assume z_lt_b : 0 < b,
by rewrite [sign_of_pos z_lt_a, sign_of_pos z_lt_b,
sign_of_pos (mul_pos z_lt_a z_lt_b), one_mul])
(assume z_eq_b : 0 = b, by rewrite [-z_eq_b, mul_zero, *sign_zero, mul_zero])
(assume z_gt_b : 0 > b,
by rewrite [sign_of_pos z_lt_a, sign_of_neg z_gt_b,
sign_of_neg (mul_neg_of_pos_of_neg z_lt_a z_gt_b), one_mul]))
(assume z_eq_a : 0 = a, by rewrite [-z_eq_a, zero_mul, *sign_zero, zero_mul])
(assume z_gt_a : 0 > a,
lt.by_cases
(assume z_lt_b : 0 < b,
by rewrite [sign_of_neg z_gt_a, sign_of_pos z_lt_b,
sign_of_neg (mul_neg_of_neg_of_pos z_gt_a z_lt_b), mul_one])
(assume z_eq_b : 0 = b, by rewrite [-z_eq_b, mul_zero, *sign_zero, mul_zero])
(assume z_gt_b : 0 > b,
by rewrite [sign_of_neg z_gt_a, sign_of_neg z_gt_b,
sign_of_pos (mul_pos_of_neg_of_neg z_gt_a z_gt_b),
neg_mul_neg, one_mul]))
theorem abs_eq_sign_mul (a : A) : abs a = sign a * a :=
lt.by_cases
(assume H1 : 0 < a,
calc
abs a = a : abs_of_pos H1
... = 1 * a : by rewrite one_mul
... = sign a * a : by rewrite (sign_of_pos H1))
(assume H1 : 0 = a,
calc
abs a = abs 0 : by rewrite H1
... = 0 : by rewrite abs_zero
... = 0 * a : by rewrite zero_mul
... = sign 0 * a : by rewrite sign_zero
... = sign a * a : by rewrite H1)
(assume H1 : a < 0,
calc
abs a = -a : abs_of_neg H1
... = -1 * a : by rewrite neg_eq_neg_one_mul
... = sign a * a : by rewrite (sign_of_neg H1))
theorem eq_sign_mul_abs (a : A) : a = sign a * abs a :=
lt.by_cases
(assume H1 : 0 < a,
calc
a = abs a : abs_of_pos H1
... = 1 * abs a : by rewrite one_mul
... = sign a * abs a : by rewrite (sign_of_pos H1))
(assume H1 : 0 = a,
calc
a = 0 : H1⁻¹
... = 0 * abs a : by rewrite zero_mul
... = sign 0 * abs a : by rewrite sign_zero
... = sign a * abs a : by rewrite H1)
(assume H1 : a < 0,
calc
a = -(-a) : by rewrite neg_neg
... = -abs a : by rewrite (abs_of_neg H1)
... = -1 * abs a : by rewrite neg_eq_neg_one_mul
... = sign a * abs a : by rewrite (sign_of_neg H1))
theorem abs_dvd_iff (a b : A) : abs a ∣ b ↔ a ∣ b :=
abs.by_cases !iff.refl !neg_dvd_iff_dvd
theorem dvd_abs_iff (a b : A) : a ∣ abs b ↔ a ∣ b :=
abs.by_cases !iff.refl !dvd_neg_iff_dvd
theorem abs_mul (a b : A) : abs (a * b) = abs a * abs b :=
or.elim (le.total 0 a)
(assume H1 : 0 ≤ a,
or.elim (le.total 0 b)
(assume H2 : 0 ≤ b,
calc
abs (a * b) = a * b : abs_of_nonneg (mul_nonneg H1 H2)
... = abs a * b : by rewrite (abs_of_nonneg H1)
... = abs a * abs b : by rewrite (abs_of_nonneg H2))
(assume H2 : b ≤ 0,
calc
abs (a * b) = -(a * b) : abs_of_nonpos (mul_nonpos_of_nonneg_of_nonpos H1 H2)
... = a * -b : by rewrite neg_mul_eq_mul_neg
... = abs a * -b : by rewrite (abs_of_nonneg H1)
... = abs a * abs b : by rewrite (abs_of_nonpos H2)))
(assume H1 : a ≤ 0,
or.elim (le.total 0 b)
(assume H2 : 0 ≤ b,
calc
abs (a * b) = -(a * b) : abs_of_nonpos (mul_nonpos_of_nonpos_of_nonneg H1 H2)
... = -a * b : by rewrite neg_mul_eq_neg_mul
... = abs a * b : by rewrite (abs_of_nonpos H1)
... = abs a * abs b : by rewrite (abs_of_nonneg H2))
(assume H2 : b ≤ 0,
calc
abs (a * b) = a * b : abs_of_nonneg (mul_nonneg_of_nonpos_of_nonpos H1 H2)
... = -a * -b : by rewrite neg_mul_neg
... = abs a * -b : by rewrite (abs_of_nonpos H1)
... = abs a * abs b : by rewrite (abs_of_nonpos H2)))
theorem abs_mul_self (a : A) : abs a * abs a = a * a :=
abs.by_cases rfl !neg_mul_neg
end
/- TODO: Multiplication and one, starting with mult_right_le_one_le. -/
end algebra
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c360cbff907681573a45549b6f2f8a972454f9ae | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/measure_theory/content_auto.lean | 5104b7c7dbc32eaac8d90894c4907571e1675c94 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 14,320 | lean | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.measure_theory.measure_space
import Mathlib.measure_theory.borel_space
import Mathlib.topology.opens
import Mathlib.topology.compacts
import Mathlib.PostPort
universes w
namespace Mathlib
/-!
# Contents
In this file we work with *contents*. A content `λ` is a function from a certain class of subsets
(such as the the compact subsets) to `ennreal` (or `ℝ≥0`) that is
* additive: If `K₁` and `K₂` are disjoint sets in the domain of `λ`,
then `λ(K₁ ∪ K₂) = λ(K₁) + λ(K₂)`;
* subadditive: If `K₁` and `K₂` are in the domain of `λ`, then `λ(K₁ ∪ K₂) ≤ λ(K₁) + λ(K₂)`;
* monotone: If `K₁ ⊆ K₂` are in the domain of `λ`, then `λ(K₁) ≤ λ(K₂)`.
We show that:
* Given a content `λ` on compact sets, we get a countably subadditive map that vanishes at `∅`.
In Halmos (1950) this is called the *inner content* `λ*` of `λ`.
* Given an inner content, we define an outer measure.
We don't explicitly define the type of contents.
In this file we only work on contents on compact sets, and inner contents on open sets, and both
contents and inner contents map into the extended nonnegative reals. However, in other applications
other choices can be made, and it is not a priori clear what the best interface should be.
## Main definitions
* `measure_theory.inner_content`: define an inner content from an content
* `measure_theory.outer_measure.of_content`: construct an outer measure from a content
## References
* Paul Halmos (1950), Measure Theory, §53
* https://en.wikipedia.org/wiki/Content_(measure_theory)
-/
namespace measure_theory
/-- Constructing the inner content of a content. From a content defined on the compact sets, we
obtain a function defined on all open sets, by taking the supremum of the content of all compact
subsets. -/
def inner_content {G : Type w} [topological_space G] (μ : topological_space.compacts G → ennreal)
(U : topological_space.opens G) : ennreal :=
supr fun (K : topological_space.compacts G) => supr fun (h : subtype.val K ⊆ ↑U) => μ K
theorem le_inner_content {G : Type w} [topological_space G]
{μ : topological_space.compacts G → ennreal} (K : topological_space.compacts G)
(U : topological_space.opens G) (h2 : subtype.val K ⊆ ↑U) : μ K ≤ inner_content μ U :=
le_supr_of_le K (le_supr (fun (h2 : subtype.val K ⊆ ↑U) => μ K) h2)
theorem inner_content_le {G : Type w} [topological_space G]
{μ : topological_space.compacts G → ennreal}
(h : ∀ (K₁ K₂ : topological_space.compacts G), subtype.val K₁ ⊆ subtype.val K₂ → μ K₁ ≤ μ K₂)
(U : topological_space.opens G) (K : topological_space.compacts G) (h2 : ↑U ⊆ subtype.val K) :
inner_content μ U ≤ μ K :=
bsupr_le
fun (K' : topological_space.compacts G) (hK' : subtype.val K' ⊆ ↑U) =>
h K' K (set.subset.trans hK' h2)
theorem inner_content_of_is_compact {G : Type w} [topological_space G]
{μ : topological_space.compacts G → ennreal}
(h : ∀ (K₁ K₂ : topological_space.compacts G), subtype.val K₁ ⊆ subtype.val K₂ → μ K₁ ≤ μ K₂)
{K : set G} (h1K : is_compact K) (h2K : is_open K) :
inner_content μ { val := K, property := h2K } = μ { val := K, property := h1K } :=
sorry
theorem inner_content_empty {G : Type w} [topological_space G]
{μ : topological_space.compacts G → ennreal} (h : μ ⊥ = 0) : inner_content μ ∅ = 0 :=
sorry
/-- This is "unbundled", because that it required for the API of `induced_outer_measure`. -/
theorem inner_content_mono {G : Type w} [topological_space G]
{μ : topological_space.compacts G → ennreal} {U : set G} {V : set G} (hU : is_open U)
(hV : is_open V) (h2 : U ⊆ V) :
inner_content μ { val := U, property := hU } ≤ inner_content μ { val := V, property := hV } :=
supr_le_supr
fun (K : topological_space.compacts G) =>
supr_le_supr_const
fun (hK : subtype.val K ⊆ ↑{ val := U, property := hU }) => set.subset.trans hK h2
theorem inner_content_exists_compact {G : Type w} [topological_space G]
{μ : topological_space.compacts G → ennreal} {U : topological_space.opens G}
(hU : inner_content μ U < ⊤) {ε : nnreal} (hε : 0 < ε) :
∃ (K : topological_space.compacts G), subtype.val K ⊆ ↑U ∧ inner_content μ U ≤ μ K + ↑ε :=
sorry
/-- The inner content of a supremum of opens is at most the sum of the individual inner
contents. -/
theorem inner_content_Sup_nat {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} (h1 : μ ⊥ = 0)
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂)
(U : ℕ → topological_space.opens G) :
inner_content μ (supr fun (i : ℕ) => U i) ≤ tsum fun (i : ℕ) => inner_content μ (U i) :=
sorry
/-- The inner content of a union of sets is at most the sum of the individual inner contents.
This is the "unbundled" version of `inner_content_Sup_nat`.
It required for the API of `induced_outer_measure`. -/
theorem inner_content_Union_nat {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} (h1 : μ ⊥ = 0)
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂) {U : ℕ → set G}
(hU : ∀ (i : ℕ), is_open (U i)) :
inner_content μ { val := set.Union fun (i : ℕ) => U i, property := is_open_Union hU } ≤
tsum fun (i : ℕ) => inner_content μ { val := U i, property := hU i } :=
sorry
theorem inner_content_comap {G : Type w} [topological_space G]
{μ : topological_space.compacts G → ennreal} (f : G ≃ₜ G)
(h :
∀ {K : topological_space.compacts G},
μ (topological_space.compacts.map (⇑f) (homeomorph.continuous f) K) = μ K)
(U : topological_space.opens G) :
inner_content μ (topological_space.opens.comap (homeomorph.continuous f) U) =
inner_content μ U :=
sorry
theorem is_mul_left_invariant_inner_content {G : Type w} [topological_space G] [group G]
[topological_group G] {μ : topological_space.compacts G → ennreal}
(h :
∀ (g : G) {K : topological_space.compacts G},
μ (topological_space.compacts.map (fun (b : G) => g * b) (continuous_mul_left g) K) = μ K)
(g : G) (U : topological_space.opens G) :
inner_content μ (topological_space.opens.comap (continuous_mul_left g) U) = inner_content μ U :=
sorry
-- @[to_additive] (fails for now)
theorem inner_content_pos_of_is_mul_left_invariant {G : Type w} [topological_space G] [t2_space G]
[group G] [topological_group G] {μ : topological_space.compacts G → ennreal} (h1 : μ ⊥ = 0)
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂)
(h3 :
∀ (g : G) {K : topological_space.compacts G},
μ (topological_space.compacts.map (fun (b : G) => g * b) (continuous_mul_left g) K) = μ K)
(K : topological_space.compacts G) (hK : 0 < μ K) (U : topological_space.opens G)
(hU : set.nonempty ↑U) : 0 < inner_content μ U :=
sorry
theorem inner_content_mono' {G : Type w} [topological_space G]
{μ : topological_space.compacts G → ennreal} {U : set G} {V : set G} (hU : is_open U)
(hV : is_open V) (h2 : U ⊆ V) :
inner_content μ { val := U, property := hU } ≤ inner_content μ { val := V, property := hV } :=
supr_le_supr
fun (K : topological_space.compacts G) =>
supr_le_supr_const
fun (hK : subtype.val K ⊆ ↑{ val := U, property := hU }) => set.subset.trans hK h2
namespace outer_measure
/-- Extending a content on compact sets to an outer measure on all sets. -/
def of_content {G : Type w} [topological_space G] [t2_space G]
(μ : topological_space.compacts G → ennreal) (h1 : μ ⊥ = 0) : outer_measure G :=
induced_outer_measure
(fun (U : set G) (hU : is_open U) => inner_content μ { val := U, property := hU }) is_open_empty
(inner_content_empty h1)
theorem of_content_opens {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂)
(U : topological_space.opens G) : coe_fn (of_content μ h1) ↑U = inner_content μ U :=
induced_outer_measure_eq' (fun (_x : ℕ → set G) => is_open_Union) (inner_content_Union_nat h1 h2)
inner_content_mono (subtype.property U)
theorem of_content_le {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂)
(h : ∀ (K₁ K₂ : topological_space.compacts G), subtype.val K₁ ⊆ subtype.val K₂ → μ K₁ ≤ μ K₂)
(U : topological_space.opens G) (K : topological_space.compacts G) (hUK : ↑U ⊆ subtype.val K) :
coe_fn (of_content μ h1) ↑U ≤ μ K :=
has_le.le.trans (eq.le (of_content_opens h2 U)) (inner_content_le h U K hUK)
theorem le_of_content_compacts {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂)
(K : topological_space.compacts G) : μ K ≤ coe_fn (of_content μ h1) (subtype.val K) :=
sorry
theorem of_content_eq_infi {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂) (A : set G) :
coe_fn (of_content μ h1) A =
infi
fun (U : set G) =>
infi
fun (hU : is_open U) =>
infi fun (h : A ⊆ U) => inner_content μ { val := U, property := hU } :=
induced_outer_measure_eq_infi is_open_Union (inner_content_Union_nat h1 h2) inner_content_mono A
theorem of_content_interior_compacts {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂)
(h3 : ∀ (K₁ K₂ : topological_space.compacts G), subtype.val K₁ ⊆ subtype.val K₂ → μ K₁ ≤ μ K₂)
(K : topological_space.compacts G) :
coe_fn (of_content μ h1) (interior (subtype.val K)) ≤ μ K :=
le_trans (le_of_eq (of_content_opens h2 (topological_space.opens.interior (subtype.val K))))
(inner_content_le h3 (topological_space.opens.interior (subtype.val K)) K interior_subset)
theorem of_content_exists_compact {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂)
{U : topological_space.opens G} (hU : coe_fn (of_content μ h1) ↑U < ⊤) {ε : nnreal}
(hε : 0 < ε) :
∃ (K : topological_space.compacts G),
subtype.val K ⊆ ↑U ∧
coe_fn (of_content μ h1) ↑U ≤ coe_fn (of_content μ h1) (subtype.val K) + ↑ε :=
sorry
theorem of_content_exists_open {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂) {A : set G}
(hA : coe_fn (of_content μ h1) A < ⊤) {ε : nnreal} (hε : 0 < ε) :
∃ (U : topological_space.opens G),
A ⊆ ↑U ∧ coe_fn (of_content μ h1) ↑U ≤ coe_fn (of_content μ h1) A + ↑ε :=
sorry
theorem of_content_preimage {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂) (f : G ≃ₜ G)
(h :
∀ {K : topological_space.compacts G},
μ (topological_space.compacts.map (⇑f) (homeomorph.continuous f) K) = μ K)
(A : set G) : coe_fn (of_content μ h1) (⇑f ⁻¹' A) = coe_fn (of_content μ h1) A :=
sorry
theorem is_mul_left_invariant_of_content {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂) [group G]
[topological_group G]
(h :
∀ (g : G) {K : topological_space.compacts G},
μ (topological_space.compacts.map (fun (b : G) => g * b) (continuous_mul_left g) K) = μ K)
(g : G) (A : set G) :
coe_fn (of_content μ h1) ((fun (h : G) => g * h) ⁻¹' A) = coe_fn (of_content μ h1) A :=
sorry
theorem of_content_caratheodory {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂) (A : set G) :
measurable_space.is_measurable' (outer_measure.caratheodory (of_content μ h1)) A ↔
∀ (U : topological_space.opens G),
coe_fn (of_content μ h1) (↑U ∩ A) + coe_fn (of_content μ h1) (↑U \ A) ≤
coe_fn (of_content μ h1) ↑U :=
sorry
-- @[to_additive] (fails for now)
theorem of_content_pos_of_is_mul_left_invariant {G : Type w} [topological_space G] [t2_space G]
{μ : topological_space.compacts G → ennreal} {h1 : μ ⊥ = 0}
(h2 : ∀ (K₁ K₂ : topological_space.compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂) [group G]
[topological_group G]
(h3 :
∀ (g : G) {K : topological_space.compacts G},
μ (topological_space.compacts.map (fun (b : G) => g * b) (continuous_mul_left g) K) = μ K)
(K : topological_space.compacts G) (hK : 0 < μ K) {U : set G} (h1U : is_open U)
(h2U : set.nonempty U) : 0 < coe_fn (of_content μ h1) U :=
sorry
end Mathlib |
ed95a631c16dae45309fff48532564c5a5e72c63 | 3d2a7f1582fe5bae4d0bdc2fe86e997521239a65 | /misc/bear.lean | 64cf625ed57287cf443b5f1f42c6f0292b1a18a2 | [] | no_license | own-pt/common-sense-lean | e4fa643ae010459de3d1bf673be7cbc7062563c9 | f672210aecb4172f5bae265e43e6867397e13b1c | refs/heads/master | 1,622,065,660,261 | 1,589,487,533,000 | 1,589,487,533,000 | 254,167,782 | 3 | 2 | null | 1,589,487,535,000 | 1,586,370,214,000 | Lean | UTF-8 | Lean | false | false | 742 | lean | /-
;; A. Bear and Big Brother Problem
;; https://codeforces.com/problemset/problem/791/A
(loop for x = 4 then (* x 3)
for y = 9 then (* y 2)
until (> x y)
collect (cons x y))
Theorem?
A) exists p in CL | p: range(1,10) x range(1,10) -> nat and for any a, b
in range(1,10) x range(1,10) it holds that
p(a, b) = inf set(n in nat | 2^n a > 3^n b)
B) there exists such p that has size < 1kb, runs in stack < 1kb and
terminates in less than 1k evaluation steps
C) Mario Carneiro: well, you could define that as a lean function and
prove that it has a given value.
https://leanprover.zulipchat.com/#narrow/stream/187764-Lean-for.20teaching/topic/from.20algorithms.20to.20proofs
-/
def compute : ℕ → ℕ → ℕ :=
|
6dad41137d612c6fa55e30b1d695c39524b9116b | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/ring_theory/prime.lean | d6b42a534d26db7af0082e6239aeefb9bb9929b3 | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 3,298 | lean | /-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import algebra.associated
import algebra.big_operators.basic
/-!
# Prime elements in rings
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file contains lemmas about prime elements of commutative rings.
-/
section cancel_comm_monoid_with_zero
variables {R : Type*} [cancel_comm_monoid_with_zero R]
open finset
open_locale big_operators
/-- If `x * y = a * ∏ i in s, p i` where `p i` is always prime, then
`x` and `y` can both be written as a divisor of `a` multiplied by
a product over a subset of `s` -/
lemma mul_eq_mul_prime_prod {α : Type*} [decidable_eq α] {x y a : R} {s : finset α}
{p : α → R} (hp : ∀ i ∈ s, prime (p i)) (hx : x * y = a * ∏ i in s, p i) :
∃ (t u : finset α) (b c : R),
t ∪ u = s
∧ disjoint t u
∧ a = b * c
∧ x = b * ∏ i in t, p i
∧ y = c * ∏ i in u, p i :=
begin
induction s using finset.induction with i s his ih generalizing x y a,
{ exact ⟨∅, ∅, x, y, by simp [hx]⟩ },
{ rw [prod_insert his, ← mul_assoc] at hx,
have hpi : prime (p i), { exact hp i (mem_insert_self _ _) },
rcases ih (λ i hi, hp i (mem_insert_of_mem hi)) hx with
⟨t, u, b, c, htus, htu, hbc, rfl, rfl⟩,
have hit : i ∉ t, from λ hit, his (htus ▸ mem_union_left _ hit),
have hiu : i ∉ u, from λ hiu, his (htus ▸ mem_union_right _ hiu),
obtain ⟨d, rfl⟩ | ⟨d, rfl⟩ : p i ∣ b ∨ p i ∣ c,
from hpi.dvd_or_dvd ⟨a, by rw [← hbc, mul_comm]⟩,
{ rw [mul_assoc, mul_comm a, mul_right_inj' hpi.ne_zero] at hbc,
exact ⟨insert i t, u, d, c, by rw [insert_union, htus],
disjoint_insert_left.2 ⟨hiu, htu⟩,
by simp [hbc, prod_insert hit, mul_assoc, mul_comm, mul_left_comm]⟩ },
{ rw [← mul_assoc, mul_right_comm b, mul_left_inj' hpi.ne_zero] at hbc,
exact ⟨t, insert i u, b, d, by rw [union_insert, htus],
disjoint_insert_right.2 ⟨hit, htu⟩,
by simp [← hbc, prod_insert hiu, mul_assoc, mul_comm, mul_left_comm]⟩ } }
end
/-- If ` x * y = a * p ^ n` where `p` is prime, then `x` and `y` can both be written
as the product of a power of `p` and a divisor of `a`. -/
lemma mul_eq_mul_prime_pow {x y a p : R} {n : ℕ} (hp : prime p) (hx : x * y = a * p ^ n) :
∃ (i j : ℕ) (b c : R),
i + j = n
∧ a = b * c
∧ x = b * p ^ i
∧ y = c * p ^ j :=
begin
rcases mul_eq_mul_prime_prod (λ _ _, hp)
(show x * y = a * (range n).prod (λ _, p), by simpa) with
⟨t, u, b, c, htus, htu, rfl, rfl, rfl⟩,
exact ⟨t.card, u.card, b, c, by rw [← card_disjoint_union htu, htus, card_range], by simp⟩,
end
end cancel_comm_monoid_with_zero
section comm_ring
variables {α : Type*} [comm_ring α]
lemma prime.neg {p : α} (hp : prime p) : prime (-p) :=
begin
obtain ⟨h1, h2, h3⟩ := hp,
exact ⟨neg_ne_zero.mpr h1, by rwa is_unit.neg_iff, by simpa [neg_dvd] using h3⟩
end
lemma prime.abs [linear_order α] {p : α} (hp : prime p) : prime (abs p) :=
begin
obtain h|h := abs_choice p; rw h,
{ exact hp },
{ exact hp.neg }
end
end comm_ring
|
4170d3843147f3d0d8e08c97cbae439cc4256e63 | 5ae26df177f810c5006841e9c73dc56e01b978d7 | /src/topology/metric_space/basic.lean | f05816048dad14d0f3e22611df3ba3d5551196e3 | [
"Apache-2.0"
] | permissive | ChrisHughes24/mathlib | 98322577c460bc6b1fe5c21f42ce33ad1c3e5558 | a2a867e827c2a6702beb9efc2b9282bd801d5f9a | refs/heads/master | 1,583,848,251,477 | 1,565,164,247,000 | 1,565,164,247,000 | 129,409,993 | 0 | 1 | Apache-2.0 | 1,565,164,817,000 | 1,523,628,059,000 | Lean | UTF-8 | Lean | false | false | 57,353 | lean | /-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Metric spaces.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
Many definitions and theorems expected on metric spaces are already introduced on uniform spaces and
topological spaces. For example:
open and closed sets, compactness, completeness, continuity and uniform continuity
-/
import data.real.nnreal topology.metric_space.emetric_space topology.algebra.ordered
open lattice set filter classical topological_space
noncomputable theory
local notation `𝓤` := uniformity
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w}
/-- Construct a uniform structure from a distance function and metric space axioms -/
def uniform_space_of_dist
(dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : uniform_space α :=
uniform_space.of_core {
uniformity := (⨅ ε>0, principal {p:α×α | dist p.1 p.2 < ε}),
refl := le_infi $ assume ε, le_infi $
by simp [set.subset_def, id_rel, dist_self, (>)] {contextual := tt},
comp := le_infi $ assume ε, le_infi $ assume h, lift'_le
(mem_infi_sets (ε / 2) $ mem_infi_sets (div_pos_of_pos_of_pos h two_pos) (subset.refl _)) $
have ∀ (a b c : α), dist a c < ε / 2 → dist c b < ε / 2 → dist a b < ε,
from assume a b c hac hcb,
calc dist a b ≤ dist a c + dist c b : dist_triangle _ _ _
... < ε / 2 + ε / 2 : add_lt_add hac hcb
... = ε : by rw [div_add_div_same, add_self_div_two],
by simpa [comp_rel],
symm := tendsto_infi.2 $ assume ε, tendsto_infi.2 $ assume h,
tendsto_infi' ε $ tendsto_infi' h $ tendsto_principal_principal.2 $ by simp [dist_comm] }
/-- The distance function (given an ambient metric space on `α`), which returns
a nonnegative real number `dist x y` given `x y : α`. -/
class has_dist (α : Type*) := (dist : α → α → ℝ)
export has_dist (dist)
/-- Metric space
Each metric space induces a canonical `uniform_space` and hence a canonical `topological_space`.
This is enforced in the type class definition, by extending the `uniform_space` structure. When
instantiating a `metric_space` structure, the uniformity fields are not necessary, they will be
filled in by default. In the same way, each metric space induces an emetric space structure.
It is included in the structure, but filled in by default.
When one instantiates a metric space structure, for instance a product structure,
this makes it possible to use a uniform structure and an edistance that are exactly
the ones for the uniform spaces product and the emetric spaces products, thereby
ensuring that everything in defeq in diamonds.-/
class metric_space (α : Type u) extends has_dist α : Type u :=
(dist_self : ∀ x : α, dist x x = 0)
(eq_of_dist_eq_zero : ∀ {x y : α}, dist x y = 0 → x = y)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
(edist : α → α → ennreal := λx y, ennreal.of_real (dist x y))
(edist_dist : ∀ x y : α, edist x y = ennreal.of_real (dist x y) . control_laws_tac)
(to_uniform_space : uniform_space α := uniform_space_of_dist dist dist_self dist_comm dist_triangle)
(uniformity_dist : 𝓤 α = ⨅ ε>0, principal {p:α×α | dist p.1 p.2 < ε} . control_laws_tac)
variables [metric_space α]
instance metric_space.to_uniform_space' : uniform_space α :=
metric_space.to_uniform_space α
instance metric_space.to_has_edist : has_edist α := ⟨metric_space.edist⟩
@[simp] theorem dist_self (x : α) : dist x x = 0 := metric_space.dist_self x
theorem eq_of_dist_eq_zero {x y : α} : dist x y = 0 → x = y :=
metric_space.eq_of_dist_eq_zero
theorem dist_comm (x y : α) : dist x y = dist y x := metric_space.dist_comm x y
theorem edist_dist (x y : α) : edist x y = ennreal.of_real (dist x y) :=
metric_space.edist_dist _ x y
@[simp] theorem dist_eq_zero {x y : α} : dist x y = 0 ↔ x = y :=
iff.intro eq_of_dist_eq_zero (assume : x = y, this ▸ dist_self _)
@[simp] theorem zero_eq_dist {x y : α} : 0 = dist x y ↔ x = y :=
by rw [eq_comm, dist_eq_zero]
theorem dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z :=
metric_space.dist_triangle x y z
theorem dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y :=
by rw dist_comm z; apply dist_triangle
theorem dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z :=
by rw dist_comm y; apply dist_triangle
lemma dist_triangle4 (x y z w : α) :
dist x w ≤ dist x y + dist y z + dist z w :=
calc
dist x w ≤ dist x z + dist z w : dist_triangle x z w
... ≤ (dist x y + dist y z) + dist z w : add_le_add_right (metric_space.dist_triangle x y z) _
lemma dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) :=
by rw [add_left_comm, dist_comm x₁, ← add_assoc]; apply dist_triangle4
lemma dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ :=
by rw [add_right_comm, dist_comm y₁]; apply dist_triangle4
theorem swap_dist : function.swap (@dist α _) = dist :=
by funext x y; exact dist_comm _ _
theorem abs_dist_sub_le (x y z : α) : abs (dist x z - dist y z) ≤ dist x y :=
abs_sub_le_iff.2
⟨sub_le_iff_le_add.2 (dist_triangle _ _ _),
sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩
theorem dist_nonneg {x y : α} : 0 ≤ dist x y :=
have 2 * dist x y ≥ 0,
from calc 2 * dist x y = dist x y + dist y x : by rw [dist_comm x y, two_mul]
... ≥ 0 : by rw ← dist_self x; apply dist_triangle,
nonneg_of_mul_nonneg_left this two_pos
@[simp] theorem dist_le_zero {x y : α} : dist x y ≤ 0 ↔ x = y :=
by simpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y
@[simp] theorem dist_pos {x y : α} : 0 < dist x y ↔ x ≠ y :=
by simpa [-dist_le_zero] using not_congr (@dist_le_zero _ _ x y)
@[simp] theorem abs_dist {a b : α} : abs (dist a b) = dist a b :=
abs_of_nonneg dist_nonneg
theorem eq_of_forall_dist_le {x y : α} (h : ∀ε, ε > 0 → dist x y ≤ ε) : x = y :=
eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h)
def nndist (a b : α) : nnreal := ⟨dist a b, dist_nonneg⟩
/--Express `nndist` in terms of `edist`-/
lemma nndist_edist (x y : α) : nndist x y = (edist x y).to_nnreal :=
by simp [nndist, edist_dist, nnreal.of_real, max_eq_left dist_nonneg, ennreal.of_real]
/--Express `edist` in terms of `nndist`-/
lemma edist_nndist (x y : α) : edist x y = ↑(nndist x y) :=
by { rw [edist_dist, nndist, ennreal.of_real_eq_coe_nnreal] }
/--In a metric space, the extended distance is always finite-/
lemma edist_ne_top (x y : α) : edist x y ≠ ⊤ :=
by rw [edist_dist x y]; apply ennreal.coe_ne_top
/--`nndist x x` vanishes-/
@[simp] lemma nndist_self (a : α) : nndist a a = 0 := (nnreal.coe_eq_zero _).1 (dist_self a)
/--Express `dist` in terms of `nndist`-/
lemma dist_nndist (x y : α) : dist x y = ↑(nndist x y) := rfl
/--Express `nndist` in terms of `dist`-/
lemma nndist_dist (x y : α) : nndist x y = nnreal.of_real (dist x y) :=
by rw [dist_nndist, nnreal.of_real_coe]
/--Deduce the equality of points with the vanishing of the nonnegative distance-/
theorem eq_of_nndist_eq_zero {x y : α} : nndist x y = 0 → x = y :=
by simp only [nnreal.eq_iff.symm, (dist_nndist _ _).symm, imp_self, nnreal.coe_zero, dist_eq_zero]
theorem nndist_comm (x y : α) : nndist x y = nndist y x :=
by simpa [nnreal.eq_iff.symm] using dist_comm x y
/--Characterize the equality of points with the vanishing of the nonnegative distance-/
@[simp] theorem nndist_eq_zero {x y : α} : nndist x y = 0 ↔ x = y :=
by simp only [nnreal.eq_iff.symm, (dist_nndist _ _).symm, imp_self, nnreal.coe_zero, dist_eq_zero]
@[simp] theorem zero_eq_nndist {x y : α} : 0 = nndist x y ↔ x = y :=
by simp only [nnreal.eq_iff.symm, (dist_nndist _ _).symm, imp_self, nnreal.coe_zero, zero_eq_dist]
/--Triangle inequality for the nonnegative distance-/
theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z :=
by simpa [nnreal.coe_le] using dist_triangle x y z
theorem nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y :=
by simpa [nnreal.coe_le] using dist_triangle_left x y z
theorem nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z :=
by simpa [nnreal.coe_le] using dist_triangle_right x y z
/--Express `dist` in terms of `edist`-/
lemma dist_edist (x y : α) : dist x y = (edist x y).to_real :=
by rw [edist_dist, ennreal.to_real_of_real (dist_nonneg)]
namespace metric
/- instantiate metric space as a topology -/
variables {x y z : α} {ε ε₁ ε₂ : ℝ} {s : set α}
/-- `ball x ε` is the set of all points `y` with `dist y x < ε` -/
def ball (x : α) (ε : ℝ) : set α := {y | dist y x < ε}
@[simp] theorem mem_ball : y ∈ ball x ε ↔ dist y x < ε := iff.rfl
theorem mem_ball' : y ∈ ball x ε ↔ dist x y < ε := by rw dist_comm; refl
/-- `closed_ball x ε` is the set of all points `y` with `dist y x ≤ ε` -/
def closed_ball (x : α) (ε : ℝ) := {y | dist y x ≤ ε}
@[simp] theorem mem_closed_ball : y ∈ closed_ball x ε ↔ dist y x ≤ ε := iff.rfl
theorem ball_subset_closed_ball : ball x ε ⊆ closed_ball x ε :=
assume y, by simp; intros h; apply le_of_lt h
theorem pos_of_mem_ball (hy : y ∈ ball x ε) : ε > 0 :=
lt_of_le_of_lt dist_nonneg hy
theorem mem_ball_self (h : ε > 0) : x ∈ ball x ε :=
show dist x x < ε, by rw dist_self; assumption
theorem mem_closed_ball_self (h : ε ≥ 0) : x ∈ closed_ball x ε :=
show dist x x ≤ ε, by rw dist_self; assumption
theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε :=
by simp [dist_comm]
theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ :=
λ y (yx : _ < ε₁), lt_of_lt_of_le yx h
theorem closed_ball_subset_closed_ball {α : Type u} [metric_space α] {ε₁ ε₂ : ℝ} {x : α} (h : ε₁ ≤ ε₂) :
closed_ball x ε₁ ⊆ closed_ball x ε₂ :=
λ y (yx : _ ≤ ε₁), le_trans yx h
theorem ball_disjoint (h : ε₁ + ε₂ ≤ dist x y) : ball x ε₁ ∩ ball y ε₂ = ∅ :=
eq_empty_iff_forall_not_mem.2 $ λ z ⟨h₁, h₂⟩,
not_lt_of_le (dist_triangle_left x y z)
(lt_of_lt_of_le (add_lt_add h₁ h₂) h)
theorem ball_disjoint_same (h : ε ≤ dist x y / 2) : ball x ε ∩ ball y ε = ∅ :=
ball_disjoint $ by rwa [← two_mul, ← le_div_iff' two_pos]
theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ :=
λ z zx, by rw ← add_sub_cancel'_right ε₁ ε₂; exact
lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)
theorem ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε :=
ball_subset $ by rw sub_self_div_two; exact le_of_lt h
theorem exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε :=
⟨_, sub_pos.2 h, ball_subset $ by rw sub_sub_self⟩
theorem ball_eq_empty_iff_nonpos : ε ≤ 0 ↔ ball x ε = ∅ :=
(eq_empty_iff_forall_not_mem.trans
⟨λ h, le_of_not_gt $ λ ε0, h _ $ mem_ball_self ε0,
λ ε0 y h, not_lt_of_le ε0 $ pos_of_mem_ball h⟩).symm
theorem uniformity_dist : 𝓤 α = (⨅ ε>0, principal {p:α×α | dist p.1 p.2 < ε}) :=
metric_space.uniformity_dist _
theorem uniformity_dist' : 𝓤 α = (⨅ε:{ε:ℝ // ε>0}, principal {p:α×α | dist p.1 p.2 < ε.val}) :=
by simp [infi_subtype]; exact uniformity_dist
theorem mem_uniformity_dist {s : set (α×α)} :
s ∈ 𝓤 α ↔ (∃ε>0, ∀{a b:α}, dist a b < ε → (a, b) ∈ s) :=
begin
rw [uniformity_dist', mem_infi],
simp [subset_def],
exact assume ⟨r, hr⟩ ⟨p, hp⟩, ⟨⟨min r p, lt_min hr hp⟩, by simp [lt_min_iff, (≥)] {contextual := tt}⟩,
exact ⟨⟨1, zero_lt_one⟩⟩
end
theorem dist_mem_uniformity {ε:ℝ} (ε0 : 0 < ε) :
{p:α×α | dist p.1 p.2 < ε} ∈ 𝓤 α :=
mem_uniformity_dist.2 ⟨ε, ε0, λ a b, id⟩
theorem uniform_continuous_iff [metric_space β] {f : α → β} :
uniform_continuous f ↔ ∀ ε > 0, ∃ δ > 0,
∀{a b:α}, dist a b < δ → dist (f a) (f b) < ε :=
uniform_continuous_def.trans
⟨λ H ε ε0, mem_uniformity_dist.1 $ H _ $ dist_mem_uniformity ε0,
λ H r ru,
let ⟨ε, ε0, hε⟩ := mem_uniformity_dist.1 ru, ⟨δ, δ0, hδ⟩ := H _ ε0 in
mem_uniformity_dist.2 ⟨δ, δ0, λ a b h, hε (hδ h)⟩⟩
theorem uniform_embedding_iff [metric_space β] {f : α → β} :
uniform_embedding f ↔ function.injective f ∧ uniform_continuous f ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
uniform_embedding_def'.trans $ and_congr iff.rfl $ and_congr iff.rfl
⟨λ H δ δ0, let ⟨t, tu, ht⟩ := H _ (dist_mem_uniformity δ0),
⟨ε, ε0, hε⟩ := mem_uniformity_dist.1 tu in
⟨ε, ε0, λ a b h, ht _ _ (hε h)⟩,
λ H s su, let ⟨δ, δ0, hδ⟩ := mem_uniformity_dist.1 su, ⟨ε, ε0, hε⟩ := H _ δ0 in
⟨_, dist_mem_uniformity ε0, λ a b h, hδ (hε h)⟩⟩
theorem totally_bounded_iff {s : set α} :
totally_bounded s ↔ ∀ ε > 0, ∃t : set α, finite t ∧ s ⊆ ⋃y∈t, ball y ε :=
⟨λ H ε ε0, H _ (dist_mem_uniformity ε0),
λ H r ru, let ⟨ε, ε0, hε⟩ := mem_uniformity_dist.1 ru,
⟨t, ft, h⟩ := H ε ε0 in
⟨t, ft, subset.trans h $ Union_subset_Union $ λ y, Union_subset_Union $ λ yt z, hε⟩⟩
/-- A metric space space is totally bounded if one can reconstruct up to any ε>0 any element of the
space from finitely many data. -/
lemma totally_bounded_of_finite_discretization {α : Type u} [metric_space α] {s : set α}
(H : ∀ε > (0 : ℝ), ∃ (β : Type u) [fintype β] (F : s → β),
∀x y, F x = F y → dist (x:α) y < ε) :
totally_bounded s :=
begin
classical, by_cases hs : s = ∅,
{ rw hs, exact totally_bounded_empty },
rcases exists_mem_of_ne_empty hs with ⟨x0, hx0⟩,
haveI : inhabited s := ⟨⟨x0, hx0⟩⟩,
refine totally_bounded_iff.2 (λ ε ε0, _),
rcases H ε ε0 with ⟨β, fβ, F, hF⟩,
let Finv := function.inv_fun F,
refine ⟨range (subtype.val ∘ Finv), finite_range _, λ x xs, _⟩,
let x' := Finv (F ⟨x, xs⟩),
have : F x' = F ⟨x, xs⟩ := function.inv_fun_eq ⟨⟨x, xs⟩, rfl⟩,
simp only [set.mem_Union, set.mem_range],
exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
end
protected lemma cauchy_iff {f : filter α} :
cauchy f ↔ f ≠ ⊥ ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x y ∈ t, dist x y < ε :=
cauchy_iff.trans $ and_congr iff.rfl
⟨λ H ε ε0, let ⟨t, tf, ts⟩ := H _ (dist_mem_uniformity ε0) in
⟨t, tf, λ x y xt yt, @ts (x, y) ⟨xt, yt⟩⟩,
λ H r ru, let ⟨ε, ε0, hε⟩ := mem_uniformity_dist.1 ru,
⟨t, tf, h⟩ := H ε ε0 in
⟨t, tf, λ ⟨x, y⟩ ⟨hx, hy⟩, hε (h x y hx hy)⟩⟩
theorem nhds_eq : nhds x = (⨅ε:{ε:ℝ // ε>0}, principal (ball x ε.val)) :=
begin
rw [nhds_eq_uniformity, uniformity_dist', lift'_infi],
{ apply congr_arg, funext ε,
rw [lift'_principal],
{ simp [ball, dist_comm] },
{ exact monotone_preimage } },
{ exact ⟨⟨1, zero_lt_one⟩⟩ },
{ intros, refl }
end
theorem mem_nhds_iff : s ∈ nhds x ↔ ∃ε>0, ball x ε ⊆ s :=
begin
rw [nhds_eq, mem_infi],
{ simp },
{ intros y z, cases y with y hy, cases z with z hz,
refine ⟨⟨min y z, lt_min hy hz⟩, _⟩,
simp [ball_subset_ball, min_le_left, min_le_right, (≥)] },
{ exact ⟨⟨1, zero_lt_one⟩⟩ }
end
theorem is_open_iff : is_open s ↔ ∀x∈s, ∃ε>0, ball x ε ⊆ s :=
by simp [is_open_iff_nhds, mem_nhds_iff]
theorem is_open_ball : is_open (ball x ε) :=
is_open_iff.2 $ λ y, exists_ball_subset_ball
theorem ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ nhds x :=
mem_nhds_sets is_open_ball (mem_ball_self ε0)
theorem tendsto_nhds_nhds [metric_space β] {f : α → β} {a b} :
tendsto f (nhds a) (nhds b) ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, dist x a < δ → dist (f x) b < ε :=
⟨λ H ε ε0, mem_nhds_iff.1 (H (ball_mem_nhds _ ε0)),
λ H s hs,
let ⟨ε, ε0, hε⟩ := mem_nhds_iff.1 hs, ⟨δ, δ0, hδ⟩ := H _ ε0 in
mem_nhds_iff.2 ⟨δ, δ0, λ x h, hε (hδ h)⟩⟩
theorem continuous_iff [metric_space β] {f : α → β} :
continuous f ↔
∀b (ε > 0), ∃ δ > 0, ∀a, dist a b < δ → dist (f a) (f b) < ε :=
continuous_iff_continuous_at.trans $ forall_congr $ λ b, tendsto_nhds_nhds
theorem exists_delta_of_continuous [metric_space β] {f : α → β} {ε : ℝ}
(hf : continuous f) (hε : ε > 0) (b : α) :
∃ δ > 0, ∀a, dist a b ≤ δ → dist (f a) (f b) < ε :=
let ⟨δ, δ_pos, hδ⟩ := continuous_iff.1 hf b ε hε in
⟨δ / 2, half_pos δ_pos, assume a ha, hδ a $ lt_of_le_of_lt ha $ div_two_lt_of_pos δ_pos⟩
theorem tendsto_nhds {f : filter β} {u : β → α} {a : α} :
tendsto u f (nhds a) ↔ ∀ ε > 0, ∃ n ∈ f, ∀x ∈ n, dist (u x) a < ε :=
by simp only [metric.nhds_eq, tendsto_infi, subtype.forall, tendsto_principal, mem_ball];
exact forall_congr (assume ε, forall_congr (assume hε, exists_sets_subset_iff.symm))
theorem continuous_iff' [topological_space β] {f : β → α} :
continuous f ↔ ∀a (ε > 0), ∃ n ∈ nhds a, ∀b ∈ n, dist (f b) (f a) < ε :=
continuous_iff_continuous_at.trans $ forall_congr $ λ b, tendsto_nhds
theorem tendsto_at_top [nonempty β] [semilattice_sup β] {u : β → α} {a : α} :
tendsto u at_top (nhds a) ↔ ∀ε>0, ∃N, ∀n≥N, dist (u n) a < ε :=
by simp only [metric.nhds_eq, tendsto_infi, subtype.forall, tendsto_at_top_principal]; refl
end metric
open metric
instance metric_space.to_separated : separated α :=
separated_def.2 $ λ x y h, eq_of_forall_dist_le $
λ ε ε0, le_of_lt (h _ (dist_mem_uniformity ε0))
/-Instantiate a metric space as an emetric space. Before we can state the instance,
we need to show that the uniform structure coming from the edistance and the
distance coincide. -/
/-- Expressing the uniformity in terms of `edist` -/
protected lemma metric.mem_uniformity_edist {s : set (α×α)} :
s ∈ 𝓤 α ↔ (∃ε>0, ∀{a b:α}, edist a b < ε → (a, b) ∈ s) :=
begin
refine mem_uniformity_dist.trans ⟨_, _⟩; rintro ⟨ε, ε0, Hε⟩,
{ refine ⟨ennreal.of_real ε, _, λ a b, _⟩,
{ rwa [gt, ennreal.of_real_pos] },
{ rw [edist_dist, ennreal.of_real_lt_of_real_iff ε0],
exact Hε } },
{ rcases ennreal.lt_iff_exists_real_btwn.1 ε0 with ⟨ε', _, ε0', hε⟩,
rw [ennreal.of_real_pos] at ε0',
refine ⟨ε', ε0', λ a b h, Hε (lt_trans _ hε)⟩,
rwa [edist_dist, ennreal.of_real_lt_of_real_iff ε0'] }
end
protected theorem metric.uniformity_edist' : 𝓤 α = (⨅ε:{ε:ennreal // ε>0}, principal {p:α×α | edist p.1 p.2 < ε.val}) :=
begin
ext s, rw mem_infi,
{ simp [metric.mem_uniformity_edist, subset_def] },
{ rintro ⟨r, hr⟩ ⟨p, hp⟩, use ⟨min r p, lt_min hr hp⟩,
simp [lt_min_iff, (≥)] {contextual := tt} },
{ exact ⟨⟨1, ennreal.zero_lt_one⟩⟩ }
end
theorem uniformity_edist : 𝓤 α = (⨅ ε>0, principal {p:α×α | edist p.1 p.2 < ε}) :=
by simpa [infi_subtype] using @metric.uniformity_edist' α _
/-- A metric space induces an emetric space -/
instance metric_space.to_emetric_space : emetric_space α :=
{ edist := edist,
edist_self := by simp [edist_dist],
eq_of_edist_eq_zero := assume x y h, by simpa [edist_dist] using h,
edist_comm := by simp only [edist_dist, dist_comm]; simp,
edist_triangle := assume x y z, begin
simp only [edist_dist, (ennreal.of_real_add _ _).symm, dist_nonneg],
rw ennreal.of_real_le_of_real_iff _,
{ exact dist_triangle _ _ _ },
{ simpa using add_le_add (dist_nonneg : 0 ≤ dist x y) dist_nonneg }
end,
uniformity_edist := uniformity_edist,
..‹metric_space α› }
/-- Balls defined using the distance or the edistance coincide -/
lemma metric.emetric_ball {x : α} {ε : ℝ} : emetric.ball x (ennreal.of_real ε) = ball x ε :=
begin
classical, by_cases h : 0 < ε,
{ ext y, by simp [edist_dist, ennreal.of_real_lt_of_real_iff h] },
{ have h' : ε ≤ 0, by simpa using h,
have A : ball x ε = ∅, by simpa [ball_eq_empty_iff_nonpos.symm],
have B : emetric.ball x (ennreal.of_real ε) = ∅,
by simp [ennreal.of_real_eq_zero.2 h', emetric.ball_eq_empty_iff],
rwa [A, B] }
end
/-- Closed balls defined using the distance or the edistance coincide -/
lemma metric.emetric_closed_ball {x : α} {ε : ℝ} (h : 0 ≤ ε) :
emetric.closed_ball x (ennreal.of_real ε) = closed_ball x ε :=
by ext y; simp [edist_dist]; rw ennreal.of_real_le_of_real_iff h
def metric_space.replace_uniformity {α} [U : uniform_space α] (m : metric_space α)
(H : @uniformity _ U = @uniformity _ (metric_space.to_uniform_space α)) :
metric_space α :=
{ dist := @dist _ m.to_has_dist,
dist_self := dist_self,
eq_of_dist_eq_zero := @eq_of_dist_eq_zero _ _,
dist_comm := dist_comm,
dist_triangle := dist_triangle,
edist := edist,
edist_dist := edist_dist,
to_uniform_space := U,
uniformity_dist := H.trans (metric_space.uniformity_dist α) }
/-- One gets a metric space from an emetric space if the edistance
is everywhere finite. We set it up so that the edist and the uniformity are
defeq in the metric space and the emetric space -/
def emetric_space.to_metric_space {α : Type u} [e : emetric_space α] (h : ∀x y: α, edist x y ≠ ⊤) :
metric_space α :=
let m : metric_space α :=
{ dist := λx y, ennreal.to_real (edist x y),
eq_of_dist_eq_zero := λx y hxy, by simpa [dist, ennreal.to_real_eq_zero_iff, h x y] using hxy,
dist_self := λx, by simp,
dist_comm := λx y, by simp [emetric_space.edist_comm],
dist_triangle := λx y z, begin
rw [← ennreal.to_real_add (h _ _) (h _ _), ennreal.to_real_le_to_real (h _ _)],
{ exact edist_triangle _ _ _ },
{ simp [ennreal.add_eq_top, h] }
end,
edist := λx y, edist x y,
edist_dist := λx y, by simp [ennreal.of_real_to_real, h] } in
metric_space.replace_uniformity m (by rw [uniformity_edist, uniformity_edist']; refl)
section real
/-- Instantiate the reals as a metric space. -/
instance real.metric_space : metric_space ℝ :=
{ dist := λx y, abs (x - y),
dist_self := by simp [abs_zero],
eq_of_dist_eq_zero := by simp [add_neg_eq_zero],
dist_comm := assume x y, abs_sub _ _,
dist_triangle := assume x y z, abs_sub_le _ _ _ }
theorem real.dist_eq (x y : ℝ) : dist x y = abs (x - y) := rfl
theorem real.dist_0_eq_abs (x : ℝ) : dist x 0 = abs x :=
by simp [real.dist_eq]
instance : orderable_topology ℝ :=
orderable_topology_of_nhds_abs $ λ x, begin
simp only [show ∀ r, {b : ℝ | abs (x - b) < r} = ball x r,
by simp [-sub_eq_add_neg, abs_sub, ball, real.dist_eq]],
apply le_antisymm,
{ simp [le_infi_iff],
exact λ ε ε0, mem_nhds_sets (is_open_ball) (mem_ball_self ε0) },
{ intros s h,
rcases mem_nhds_iff.1 h with ⟨ε, ε0, ss⟩,
exact mem_infi_sets _ (mem_infi_sets ε0 (mem_principal_sets.2 ss)) },
end
lemma closed_ball_Icc {x r : ℝ} : closed_ball x r = Icc (x-r) (x+r) :=
by ext y; rw [mem_closed_ball, dist_comm, real.dist_eq,
abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add', sub_le]
lemma squeeze_zero {α} {f g : α → ℝ} {t₀ : filter α} (hf : ∀t, 0 ≤ f t) (hft : ∀t, f t ≤ g t)
(g0 : tendsto g t₀ (nhds 0)) : tendsto f t₀ (nhds 0) :=
begin
apply tendsto_of_tendsto_of_tendsto_of_le_of_le (tendsto_const_nhds) g0;
simp [*]; exact filter.univ_mem_sets
end
theorem metric.uniformity_eq_comap_nhds_zero :
𝓤 α = comap (λp:α×α, dist p.1 p.2) (nhds (0 : ℝ)) :=
begin
simp only [uniformity_dist', nhds_eq, comap_infi, comap_principal],
congr, funext ε,
rw [principal_eq_iff_eq],
ext ⟨a, b⟩,
simp [real.dist_0_eq_abs]
end
lemma cauchy_seq_iff_tendsto_dist_at_top_0 [inhabited β] [semilattice_sup β] {u : β → α} :
cauchy_seq u ↔ tendsto (λ (n : β × β), dist (u n.1) (u n.2)) at_top (nhds 0) :=
by rw [cauchy_seq_iff_prod_map, metric.uniformity_eq_comap_nhds_zero, ← map_le_iff_le_comap,
filter.map_map, tendsto, prod.map_def]
end real
section cauchy_seq
variables [inhabited β] [semilattice_sup β]
/-- In a metric space, Cauchy sequences are characterized by the fact that, eventually,
the distance between its elements is arbitrarily small -/
theorem metric.cauchy_seq_iff {u : β → α} :
cauchy_seq u ↔ ∀ε>0, ∃N, ∀m n≥N, dist (u m) (u n) < ε :=
begin
unfold cauchy_seq,
rw metric.cauchy_iff,
simp only [true_and, exists_prop, filter.mem_at_top_sets, filter.at_top_ne_bot,
filter.mem_map, ne.def, filter.map_eq_bot_iff, not_false_iff, set.mem_set_of_eq],
split,
{ intros H ε εpos,
rcases H ε εpos with ⟨t, ⟨N, hN⟩, ht⟩,
exact ⟨N, λm n hm hn, ht _ _ (hN _ hm) (hN _ hn)⟩ },
{ intros H ε εpos,
rcases H (ε/2) (half_pos εpos) with ⟨N, hN⟩,
existsi ball (u N) (ε/2),
split,
{ exact ⟨N, λx hx, hN _ _ hx (le_refl N)⟩ },
{ exact λx y hx hy, calc
dist x y ≤ dist x (u N) + dist y (u N) : dist_triangle_right _ _ _
... < ε/2 + ε/2 : add_lt_add hx hy
... = ε : add_halves _ } }
end
/-- A variation around the metric characterization of Cauchy sequences -/
theorem metric.cauchy_seq_iff' {u : β → α} :
cauchy_seq u ↔ ∀ε>0, ∃N, ∀n≥N, dist (u n) (u N) < ε :=
begin
rw metric.cauchy_seq_iff,
split,
{ intros H ε εpos,
rcases H ε εpos with ⟨N, hN⟩,
exact ⟨N, λn hn, hN _ _ hn (le_refl N)⟩ },
{ intros H ε εpos,
rcases H (ε/2) (half_pos εpos) with ⟨N, hN⟩,
exact ⟨N, λ m n hm hn, calc
dist (u m) (u n) ≤ dist (u m) (u N) + dist (u n) (u N) : dist_triangle_right _ _ _
... < ε/2 + ε/2 : add_lt_add (hN _ hm) (hN _ hn)
... = ε : add_halves _⟩ }
end
/-- A Cauchy sequence on the natural numbers is bounded. -/
theorem cauchy_seq_bdd {u : ℕ → α} (hu : cauchy_seq u) :
∃ R > 0, ∀ m n, dist (u m) (u n) < R :=
begin
rcases metric.cauchy_seq_iff'.1 hu 1 zero_lt_one with ⟨N, hN⟩,
suffices : ∃ R > 0, ∀ n, dist (u n) (u N) < R,
{ rcases this with ⟨R, R0, H⟩,
exact ⟨_, add_pos R0 R0, λ m n,
lt_of_le_of_lt (dist_triangle_right _ _ _) (add_lt_add (H m) (H n))⟩ },
let R := finset.sup (finset.range N) (λ n, nndist (u n) (u N)),
refine ⟨↑R + 1, add_pos_of_nonneg_of_pos R.2 zero_lt_one, λ n, _⟩,
cases le_or_lt N n,
{ exact lt_of_lt_of_le (hN _ h) (le_add_of_nonneg_left R.2) },
{ have : _ ≤ R := finset.le_sup (finset.mem_range.2 h),
exact lt_of_le_of_lt this (lt_add_of_pos_right _ zero_lt_one) }
end
/-- Yet another metric characterization of Cauchy sequences on integers. This one is often the
most efficient. -/
lemma cauchy_seq_iff_le_tendsto_0 {s : ℕ → α} : cauchy_seq s ↔ ∃ b : ℕ → ℝ,
(∀ n, 0 ≤ b n) ∧
(∀ n m N : ℕ, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧
tendsto b at_top (nhds 0) :=
⟨λ hs, begin
/- `s` is a Cauchy sequence. The sequence `b` will be constructed by taking
the supremum of the distances between `s n` and `s m` for `n m ≥ N`.
First, we prove that all these distances are bounded, as otherwise the Sup
would not make sense. -/
let S := λ N, (λ(p : ℕ × ℕ), dist (s p.1) (s p.2)) '' {p | p.1 ≥ N ∧ p.2 ≥ N},
have hS : ∀ N, ∃ x, ∀ y ∈ S N, y ≤ x,
{ rcases cauchy_seq_bdd hs with ⟨R, R0, hR⟩,
refine λ N, ⟨R, _⟩, rintro _ ⟨⟨m, n⟩, _, rfl⟩,
exact le_of_lt (hR m n) },
have bdd : bdd_above (range (λ(p : ℕ × ℕ), dist (s p.1) (s p.2))),
{ rcases cauchy_seq_bdd hs with ⟨R, R0, hR⟩,
use R, rintro _ ⟨⟨m, n⟩, rfl⟩, exact le_of_lt (hR m n) },
-- Prove that it bounds the distances of points in the Cauchy sequence
have ub : ∀ m n N, N ≤ m → N ≤ n → dist (s m) (s n) ≤ real.Sup (S N) :=
λ m n N hm hn, real.le_Sup _ (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩,
have S0m : ∀ n, (0:ℝ) ∈ S n := λ n, ⟨⟨n, n⟩, ⟨le_refl _, le_refl _⟩, dist_self _⟩,
have S0 := λ n, real.le_Sup _ (hS n) (S0m n),
-- Prove that it tends to `0`, by using the Cauchy property of `s`
refine ⟨λ N, real.Sup (S N), S0, ub, metric.tendsto_at_top.2 (λ ε ε0, _)⟩,
refine (metric.cauchy_seq_iff.1 hs (ε/2) (half_pos ε0)).imp (λ N hN n hn, _),
rw [real.dist_0_eq_abs, abs_of_nonneg (S0 n)],
refine lt_of_le_of_lt (real.Sup_le_ub _ ⟨_, S0m _⟩ _) (half_lt_self ε0),
rintro _ ⟨⟨m', n'⟩, ⟨hm', hn'⟩, rfl⟩,
exact le_of_lt (hN _ _ (le_trans hn hm') (le_trans hn hn'))
end,
λ ⟨b, _, b_bound, b_lim⟩, metric.cauchy_seq_iff.2 $ λ ε ε0,
(metric.tendsto_at_top.1 b_lim ε ε0).imp $ λ N hN m n hm hn,
calc dist (s m) (s n) ≤ b N : b_bound m n N hm hn
... ≤ abs (b N) : le_abs_self _
... = dist (b N) 0 : by rw real.dist_0_eq_abs; refl
... < ε : (hN _ (le_refl N)) ⟩
end cauchy_seq
def metric_space.induced {α β} (f : α → β) (hf : function.injective f)
(m : metric_space β) : metric_space α :=
{ dist := λ x y, dist (f x) (f y),
dist_self := λ x, dist_self _,
eq_of_dist_eq_zero := λ x y h, hf (dist_eq_zero.1 h),
dist_comm := λ x y, dist_comm _ _,
dist_triangle := λ x y z, dist_triangle _ _ _,
edist := λ x y, edist (f x) (f y),
edist_dist := λ x y, edist_dist _ _,
to_uniform_space := uniform_space.comap f m.to_uniform_space,
uniformity_dist := begin
apply @uniformity_dist_of_mem_uniformity _ _ _ _ _ (λ x y, dist (f x) (f y)),
refine λ s, mem_comap_sets.trans _,
split; intro H,
{ rcases H with ⟨r, ru, rs⟩,
rcases mem_uniformity_dist.1 ru with ⟨ε, ε0, hε⟩,
refine ⟨ε, ε0, λ a b h, rs (hε _)⟩, exact h },
{ rcases H with ⟨ε, ε0, hε⟩,
exact ⟨_, dist_mem_uniformity ε0, λ ⟨a, b⟩, hε⟩ }
end }
instance subtype.metric_space {p : α → Prop} [t : metric_space α] : metric_space (subtype p) :=
metric_space.induced subtype.val (λ x y, subtype.eq) t
theorem subtype.dist_eq {p : α → Prop} [t : metric_space α] (x y : subtype p) :
dist x y = dist x.1 y.1 := rfl
section nnreal
instance : metric_space nnreal := by unfold nnreal; apply_instance
end nnreal
section prod
instance prod.metric_space_max [metric_space β] : metric_space (α × β) :=
{ dist := λ x y, max (dist x.1 y.1) (dist x.2 y.2),
dist_self := λ x, by simp,
eq_of_dist_eq_zero := λ x y h, begin
cases max_le_iff.1 (le_of_eq h) with h₁ h₂,
exact prod.ext_iff.2 ⟨dist_le_zero.1 h₁, dist_le_zero.1 h₂⟩
end,
dist_comm := λ x y, by simp [dist_comm],
dist_triangle := λ x y z, max_le
(le_trans (dist_triangle _ _ _) (add_le_add (le_max_left _ _) (le_max_left _ _)))
(le_trans (dist_triangle _ _ _) (add_le_add (le_max_right _ _) (le_max_right _ _))),
edist := λ x y, max (edist x.1 y.1) (edist x.2 y.2),
edist_dist := assume x y, begin
have : monotone ennreal.of_real := assume x y h, ennreal.of_real_le_of_real h,
rw [edist_dist, edist_dist, (max_distrib_of_monotone this).symm]
end,
uniformity_dist := begin
refine uniformity_prod.trans _,
simp [uniformity_dist, comap_infi],
rw ← infi_inf_eq, congr, funext,
rw ← infi_inf_eq, congr, funext,
simp [inf_principal, ext_iff, max_lt_iff]
end,
to_uniform_space := prod.uniform_space }
lemma prod.dist_eq [metric_space β] {x y : α × β} :
dist x y = max (dist x.1 y.1) (dist x.2 y.2) := rfl
end prod
theorem uniform_continuous_dist' : uniform_continuous (λp:α×α, dist p.1 p.2) :=
metric.uniform_continuous_iff.2 (λ ε ε0, ⟨ε/2, half_pos ε0,
begin
suffices,
{ intros p q h, cases p with p₁ p₂, cases q with q₁ q₂,
cases max_lt_iff.1 h with h₁ h₂, clear h,
dsimp at h₁ h₂ ⊢,
rw real.dist_eq,
refine abs_sub_lt_iff.2 ⟨_, _⟩,
{ revert p₁ p₂ q₁ q₂ h₁ h₂, exact this },
{ apply this; rwa dist_comm } },
intros p₁ p₂ q₁ q₂ h₁ h₂,
have := add_lt_add
(abs_sub_lt_iff.1 (lt_of_le_of_lt (abs_dist_sub_le p₁ q₁ p₂) h₁)).1
(abs_sub_lt_iff.1 (lt_of_le_of_lt (abs_dist_sub_le p₂ q₂ q₁) h₂)).1,
rwa [add_halves, dist_comm p₂, sub_add_sub_cancel, dist_comm q₂] at this
end⟩)
theorem uniform_continuous_dist [uniform_space β] {f g : β → α}
(hf : uniform_continuous f) (hg : uniform_continuous g) :
uniform_continuous (λb, dist (f b) (g b)) :=
uniform_continuous_dist'.comp (hf.prod_mk hg)
theorem continuous_dist' : continuous (λp:α×α, dist p.1 p.2) :=
uniform_continuous_dist'.continuous
theorem continuous_dist [topological_space β] {f g : β → α}
(hf : continuous f) (hg : continuous g) : continuous (λb, dist (f b) (g b)) :=
continuous_dist'.comp (hf.prod_mk hg)
theorem tendsto_dist {f g : β → α} {x : filter β} {a b : α}
(hf : tendsto f x (nhds a)) (hg : tendsto g x (nhds b)) :
tendsto (λx, dist (f x) (g x)) x (nhds (dist a b)) :=
have tendsto (λp:α×α, dist p.1 p.2) (nhds (a, b)) (nhds (dist a b)),
from continuous_iff_continuous_at.mp continuous_dist' (a, b),
tendsto.comp (by rw [nhds_prod_eq] at this; exact this) (hf.prod_mk hg)
lemma nhds_comap_dist (a : α) : (nhds (0 : ℝ)).comap (λa', dist a' a) = nhds a :=
have h₁ : ∀ε, (λa', dist a' a) ⁻¹' ball 0 ε ⊆ ball a ε,
by simp [subset_def, real.dist_0_eq_abs],
have h₂ : tendsto (λa', dist a' a) (nhds a) (nhds (dist a a)),
from tendsto_dist tendsto_id tendsto_const_nhds,
le_antisymm
(by simp [h₁, nhds_eq, infi_le_infi, principal_mono,
-le_principal_iff, -le_infi_iff])
(by simpa [map_le_iff_le_comap.symm, tendsto] using h₂)
lemma tendsto_iff_dist_tendsto_zero {f : β → α} {x : filter β} {a : α} :
(tendsto f x (nhds a)) ↔ (tendsto (λb, dist (f b) a) x (nhds 0)) :=
by rw [← nhds_comap_dist a, tendsto_comap_iff]
lemma uniform_continuous_nndist' : uniform_continuous (λp:α×α, nndist p.1 p.2) :=
uniform_continuous_subtype_mk uniform_continuous_dist' _
lemma continuous_nndist' : continuous (λp:α×α, nndist p.1 p.2) :=
uniform_continuous_nndist'.continuous
lemma continuous_nndist [topological_space β] {f g : β → α}
(hf : continuous f) (hg : continuous g) : continuous (λb, nndist (f b) (g b)) :=
continuous_nndist'.comp (hf.prod_mk hg)
lemma tendsto_nndist' (a b :α) :
tendsto (λp:α×α, nndist p.1 p.2) (filter.prod (nhds a) (nhds b)) (nhds (nndist a b)) :=
by rw [← nhds_prod_eq]; exact continuous_iff_continuous_at.1 continuous_nndist' _
namespace metric
variables {x y z : α} {ε ε₁ ε₂ : ℝ} {s : set α}
theorem is_closed_ball : is_closed (closed_ball x ε) :=
is_closed_le (continuous_dist continuous_id continuous_const) continuous_const
/-- ε-characterization of the closure in metric spaces-/
theorem mem_closure_iff' {α : Type u} [metric_space α] {s : set α} {a : α} :
a ∈ closure s ↔ ∀ε>0, ∃b ∈ s, dist a b < ε :=
⟨begin
intros ha ε hε,
have A : ball a ε ∩ s ≠ ∅ := mem_closure_iff.1 ha _ is_open_ball (mem_ball_self hε),
cases ne_empty_iff_exists_mem.1 A with b hb,
simp,
exact ⟨b, ⟨hb.2, by have B := hb.1; simpa [mem_ball'] using B⟩⟩
end,
begin
intros H,
apply mem_closure_iff.2,
intros o ho ao,
rcases is_open_iff.1 ho a ao with ⟨ε, ⟨εpos, hε⟩⟩,
rcases H ε εpos with ⟨b, ⟨bs, bdist⟩⟩,
have B : b ∈ o ∩ s := ⟨hε (by simpa [dist_comm]), bs⟩,
apply ne_empty_of_mem B
end⟩
lemma mem_closure_range_iff {α : Type u} [metric_space α] {e : β → α} {a : α} :
a ∈ closure (range e) ↔ ∀ε>0, ∃ k : β, dist a (e k) < ε :=
iff.intro
( assume ha ε hε,
let ⟨b, ⟨hb, hab⟩⟩ := metric.mem_closure_iff'.1 ha ε hε in
let ⟨k, hk⟩ := mem_range.1 hb in
⟨k, by { rw hk, exact hab }⟩ )
( assume h, metric.mem_closure_iff'.2 (assume ε hε,
let ⟨k, hk⟩ := h ε hε in
⟨e k, ⟨mem_range.2 ⟨k, rfl⟩, hk⟩⟩) )
lemma mem_closure_range_iff_nat {α : Type u} [metric_space α] {e : β → α} {a : α} :
a ∈ closure (range e) ↔ ∀n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) :=
⟨assume ha n, mem_closure_range_iff.1 ha (1 / ((n : ℝ) + 1)) nat.one_div_pos_of_nat,
assume h, mem_closure_range_iff.2 $ assume ε hε,
let ⟨n, hn⟩ := exists_nat_one_div_lt hε in
let ⟨k, hk⟩ := h n in
⟨k, calc dist a (e k) < 1 / ((n : ℝ) + 1) : hk ... < ε : hn⟩⟩
theorem mem_of_closed' {α : Type u} [metric_space α] {s : set α} (hs : is_closed s)
{a : α} : a ∈ s ↔ ∀ε>0, ∃b ∈ s, dist a b < ε :=
by simpa only [closure_eq_of_is_closed hs] using @mem_closure_iff' _ _ s a
end metric
section pi
open finset lattice
variables {π : β → Type*} [fintype β] [∀b, metric_space (π b)]
instance has_dist_pi : has_dist (Πb, π b) :=
⟨λf g, ((finset.sup univ (λb, nndist (f b) (g b)) : nnreal) : ℝ)⟩
lemma dist_pi_def (f g : Πb, π b) :
dist f g = (finset.sup univ (λb, nndist (f b) (g b)) : nnreal) := rfl
instance metric_space_pi : metric_space (Πb, π b) :=
{ dist := dist,
dist_self := assume f, (nnreal.coe_eq_zero _).2 $ bot_unique $ finset.sup_le $ by simp,
dist_comm := assume f g, nnreal.eq_iff.2 $ by congr; ext a; exact nndist_comm _ _,
dist_triangle := assume f g h, show dist f h ≤ (dist f g) + (dist g h), from
begin
simp only [dist_pi_def, (nnreal.coe_add _ _).symm, nnreal.coe_le.symm,
finset.sup_le_iff],
assume b hb,
exact le_trans (nndist_triangle _ (g b) _) (add_le_add (le_sup hb) (le_sup hb))
end,
eq_of_dist_eq_zero := assume f g eq0,
begin
simp only [dist_pi_def, nnreal.coe_eq_zero, nnreal.bot_eq_zero.symm, eq_bot_iff,
finset.sup_le_iff] at eq0,
exact (funext $ assume b, eq_of_nndist_eq_zero $ bot_unique $ eq0 b $ mem_univ b),
end,
edist := λ f g, finset.sup univ (λb, edist (f b) (g b)),
edist_dist := assume x y, begin
have A : sup univ (λ (b : β), ((nndist (x b) (y b)) : ennreal)) = ↑(sup univ (λ (b : β), nndist (x b) (y b))),
{ refine eq.symm (comp_sup_eq_sup_comp _ _ _),
exact (assume x y h, ennreal.coe_le_coe.2 h), refl },
simp [dist, edist_nndist, ennreal.of_real, A]
end }
end pi
section compact
/-- Any compact set in a metric space can be covered by finitely many balls of a given positive
radius -/
lemma finite_cover_balls_of_compact {α : Type u} [metric_space α] {s : set α}
(hs : compact s) {e : ℝ} (he : e > 0) :
∃t ⊆ s, finite t ∧ s ⊆ ⋃x∈t, ball x e :=
begin
apply compact_elim_finite_subcover_image hs,
{ simp [is_open_ball] },
{ intros x xs,
simp,
exact ⟨x, ⟨xs, by simpa⟩⟩ }
end
end compact
section proper_space
open metric
/-- A metric space is proper if all closed balls are compact. -/
class proper_space (α : Type u) [metric_space α] : Prop :=
(compact_ball : ∀x:α, ∀r, compact (closed_ball x r))
/- A compact metric space is proper -/
instance proper_of_compact [metric_space α] [compact_space α] : proper_space α :=
⟨assume x r, compact_of_is_closed_subset compact_univ is_closed_ball (subset_univ _)⟩
/-- A proper space is locally compact -/
instance locally_compact_of_proper [metric_space α] [proper_space α] :
locally_compact_space α :=
begin
apply locally_compact_of_compact_nhds,
intros x,
existsi closed_ball x 1,
split,
{ apply mem_nhds_iff.2,
existsi (1 : ℝ),
simp,
exact ⟨zero_lt_one, ball_subset_closed_ball⟩ },
{ apply proper_space.compact_ball }
end
/-- A proper space is complete -/
instance complete_of_proper {α : Type u} [metric_space α] [proper_space α] : complete_space α :=
⟨begin
intros f hf,
/- We want to show that the Cauchy filter `f` is converging. It suffices to find a closed
ball (therefore compact by properness) where it is nontrivial. -/
have A : ∃ t ∈ f, ∀ x y ∈ t, dist x y < 1 := (metric.cauchy_iff.1 hf).2 1 zero_lt_one,
rcases A with ⟨t, ⟨t_fset, ht⟩⟩,
rcases inhabited_of_mem_sets hf.1 t_fset with ⟨x, xt⟩,
have : t ⊆ closed_ball x 1 := by intros y yt; simp [dist_comm]; apply le_of_lt (ht x y xt yt),
have : closed_ball x 1 ∈ f := f.sets_of_superset t_fset this,
rcases (compact_iff_totally_bounded_complete.1 (proper_space.compact_ball x 1)).2 f hf (le_principal_iff.2 this)
with ⟨y, _, hy⟩,
exact ⟨y, hy⟩
end⟩
/-- A proper metric space is separable, and therefore second countable. Indeed, any ball is
compact, and therefore admits a countable dense subset. Taking a countable union over the balls
centered at a fixed point and with integer radius, one obtains a countable set which is
dense in the whole space. -/
instance second_countable_of_proper [metric_space α] [proper_space α] :
second_countable_topology α :=
begin
/- We show that the space admits a countable dense subset. The case where the space is empty
is special, and trivial. -/
have A : (univ : set α) = ∅ → ∃(s : set α), countable s ∧ closure s = (univ : set α) :=
assume H, ⟨∅, ⟨by simp, by simp; exact H.symm⟩⟩,
have B : (univ : set α) ≠ ∅ → ∃(s : set α), countable s ∧ closure s = (univ : set α) :=
begin
/- When the space is not empty, we take a point `x` in the space, and then a countable set
`T r` which is dense in the closed ball `closed_ball x r` for each `r`. Then the set
`t = ⋃ T n` (where the union is over all integers `n`) is countable, as a countable union
of countable sets, and dense in the space by construction. -/
assume non_empty,
rcases ne_empty_iff_exists_mem.1 non_empty with ⟨x, x_univ⟩,
choose T a using show ∀ (r:ℝ), ∃ t ⊆ closed_ball x r, (countable (t : set α) ∧ closed_ball x r = closure t),
from assume r, emetric.countable_closure_of_compact (proper_space.compact_ball _ _),
let t := (⋃n:ℕ, T (n : ℝ)),
have T₁ : countable t := by finish [countable_Union],
have T₂ : closure t ⊆ univ := by simp,
have T₃ : univ ⊆ closure t :=
begin
intros y y_univ,
rcases exists_nat_gt (dist y x) with ⟨n, n_large⟩,
have h : y ∈ closed_ball x (n : ℝ) := by simp; apply le_of_lt n_large,
have h' : closed_ball x (n : ℝ) = closure (T (n : ℝ)) := by finish,
have : y ∈ closure (T (n : ℝ)) := by rwa h' at h,
show y ∈ closure t, from mem_of_mem_of_subset this (by apply closure_mono; apply subset_Union (λ(n:ℕ), T (n:ℝ))),
end,
exact ⟨t, ⟨T₁, subset.antisymm T₂ T₃⟩⟩
end,
haveI : separable_space α := ⟨by_cases A B⟩,
apply emetric.second_countable_of_separable,
end
end proper_space
namespace metric
section second_countable
open topological_space
/-- A metric space is second countable if, for every ε > 0, there is a countable set which is ε-dense. -/
lemma second_countable_of_almost_dense_set
(H : ∀ε > (0 : ℝ), ∃ s : set α, countable s ∧ (∀x, ∃y ∈ s, dist x y ≤ ε)) :
second_countable_topology α :=
begin
choose T T_dense using H,
have I1 : ∀n:ℕ, (n:ℝ) + 1 > 0 :=
λn, lt_of_lt_of_le zero_lt_one (le_add_of_nonneg_left (nat.cast_nonneg _)),
have I : ∀n:ℕ, (n+1 : ℝ)⁻¹ > 0 := λn, inv_pos'.2 (I1 n),
let t := ⋃n:ℕ, T (n+1)⁻¹ (I n),
have count_t : countable t := by finish [countable_Union],
have clos_t : closure t = univ,
{ refine subset.antisymm (subset_univ _) (λx xuniv, mem_closure_iff'.2 (λε εpos, _)),
rcases exists_nat_gt ε⁻¹ with ⟨n, hn⟩,
have : ε⁻¹ < n + 1 := lt_of_lt_of_le hn (le_add_of_nonneg_right zero_le_one),
have nε : ((n:ℝ)+1)⁻¹ < ε := (inv_lt (I1 n) εpos).2 this,
rcases (T_dense (n+1)⁻¹ (I n)).2 x with ⟨y, yT, Dxy⟩,
have : y ∈ t := mem_of_mem_of_subset yT (by apply subset_Union (λ (n:ℕ), T (n+1)⁻¹ (I n))),
exact ⟨y, this, lt_of_le_of_lt Dxy nε⟩ },
haveI : separable_space α := ⟨⟨t, ⟨count_t, clos_t⟩⟩⟩,
exact emetric.second_countable_of_separable α
end
/-- A metric space space is second countable if one can reconstruct up to any ε>0 any element of the
space from countably many data. -/
lemma second_countable_of_countable_discretization {α : Type u} [metric_space α]
(H : ∀ε > (0 : ℝ), ∃ (β : Type u) [encodable β] (F : α → β), ∀x y, F x = F y → dist x y ≤ ε) :
second_countable_topology α :=
begin
classical, by_cases hs : (univ : set α) = ∅,
{ haveI : compact_space α := ⟨by rw hs; exact compact_of_finite (set.finite_empty)⟩, by apply_instance },
rcases exists_mem_of_ne_empty hs with ⟨x0, hx0⟩,
letI : inhabited α := ⟨x0⟩,
refine second_countable_of_almost_dense_set (λε ε0, _),
rcases H ε ε0 with ⟨β, fβ, F, hF⟩,
let Finv := function.inv_fun F,
refine ⟨range Finv, ⟨countable_range _, λx, _⟩⟩,
let x' := Finv (F x),
have : F x' = F x := function.inv_fun_eq ⟨x, rfl⟩,
exact ⟨x', mem_range_self _, hF _ _ this.symm⟩
end
end second_countable
end metric
lemma lebesgue_number_lemma_of_metric
{s : set α} {ι} {c : ι → set α} (hs : compact s)
(hc₁ : ∀ i, is_open (c i)) (hc₂ : s ⊆ ⋃ i, c i) :
∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i :=
let ⟨n, en, hn⟩ := lebesgue_number_lemma hs hc₁ hc₂,
⟨δ, δ0, hδ⟩ := mem_uniformity_dist.1 en in
⟨δ, δ0, assume x hx, let ⟨i, hi⟩ := hn x hx in
⟨i, assume y hy, hi (hδ (mem_ball'.mp hy))⟩⟩
lemma lebesgue_number_lemma_of_metric_sUnion
{s : set α} {c : set (set α)} (hs : compact s)
(hc₁ : ∀ t ∈ c, is_open t) (hc₂ : s ⊆ ⋃₀ c) :
∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t :=
by rw sUnion_eq_Union at hc₂;
simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
namespace metric
/-- Boundedness of a subset of a metric space. We formulate the definition to work
even in the empty space. -/
def bounded (s : set α) : Prop :=
∃C, ∀x y ∈ s, dist x y ≤ C
section bounded
variables {x : α} {s t : set α} {r : ℝ}
@[simp] lemma bounded_empty : bounded (∅ : set α) :=
⟨0, by simp⟩
lemma bounded_iff_mem_bounded : bounded s ↔ ∀ x ∈ s, bounded s :=
⟨λ h _ _, h, λ H, begin
classical, by_cases s = ∅,
{ subst s, exact ⟨0, by simp⟩ },
{ rcases exists_mem_of_ne_empty h with ⟨x, hx⟩,
exact H x hx }
end⟩
/-- Subsets of a bounded set are also bounded -/
lemma bounded.subset (incl : s ⊆ t) : bounded t → bounded s :=
Exists.imp $ λ C hC x y hx hy, hC x y (incl hx) (incl hy)
/-- Closed balls are bounded -/
lemma bounded_closed_ball : bounded (closed_ball x r) :=
⟨r + r, λ y z hy hz, begin
simp only [mem_closed_ball] at *,
calc dist y z ≤ dist y x + dist z x : dist_triangle_right _ _ _
... ≤ r + r : add_le_add hy hz
end⟩
/-- Open balls are bounded -/
lemma bounded_ball : bounded (ball x r) :=
bounded_closed_ball.subset ball_subset_closed_ball
/-- Given a point, a bounded subset is included in some ball around this point -/
lemma bounded_iff_subset_ball (c : α) : bounded s ↔ ∃r, s ⊆ closed_ball c r :=
begin
split; rintro ⟨C, hC⟩,
{ classical, by_cases s = ∅,
{ subst s, exact ⟨0, by simp⟩ },
{ rcases exists_mem_of_ne_empty h with ⟨x, hx⟩,
exact ⟨C + dist x c, λ y hy, calc
dist y c ≤ dist y x + dist x c : dist_triangle _ _ _
... ≤ C + dist x c : add_le_add_right (hC y x hy hx) _⟩ } },
{ exact bounded_closed_ball.subset hC }
end
/-- The union of two bounded sets is bounded iff each of the sets is bounded -/
@[simp] lemma bounded_union :
bounded (s ∪ t) ↔ bounded s ∧ bounded t :=
⟨λh, ⟨h.subset (by simp), h.subset (by simp)⟩,
begin
rintro ⟨hs, ht⟩,
refine bounded_iff_mem_bounded.2 (λ x _, _),
rw bounded_iff_subset_ball x at hs ht ⊢,
rcases hs with ⟨Cs, hCs⟩, rcases ht with ⟨Ct, hCt⟩,
exact ⟨max Cs Ct, union_subset
(subset.trans hCs $ closed_ball_subset_closed_ball $ le_max_left _ _)
(subset.trans hCt $ closed_ball_subset_closed_ball $ le_max_right _ _)⟩,
end⟩
/-- A finite union of bounded sets is bounded -/
lemma bounded_bUnion {I : set β} {s : β → set α} (H : finite I) :
bounded (⋃i∈I, s i) ↔ ∀i ∈ I, bounded (s i) :=
finite.induction_on H (by simp) $ λ x I _ _ IH,
by simp [or_imp_distrib, forall_and_distrib, IH]
/-- A compact set is bounded -/
lemma bounded_of_compact {s : set α} (h : compact s) : bounded s :=
-- We cover the compact set by finitely many balls of radius 1,
-- and then argue that a finite union of bounded sets is bounded
let ⟨t, ht, fint, subs⟩ := finite_cover_balls_of_compact h zero_lt_one in
bounded.subset subs $ (bounded_bUnion fint).2 $ λ i hi, bounded_ball
/-- A finite set is bounded -/
lemma bounded_of_finite {s : set α} (h : finite s) : bounded s :=
bounded_of_compact $ compact_of_finite h
/-- A singleton is bounded -/
lemma bounded_singleton {x : α} : bounded ({x} : set α) :=
bounded_of_finite $ finite_singleton _
/-- Characterization of the boundedness of the range of a function -/
lemma bounded_range_iff {f : β → α} : bounded (range f) ↔ ∃C, ∀x y, dist (f x) (f y) ≤ C :=
exists_congr $ λ C, ⟨
λ H x y, H _ _ ⟨x, rfl⟩ ⟨y, rfl⟩,
by rintro H _ _ ⟨x, rfl⟩ ⟨y, rfl⟩; exact H x y⟩
/-- In a compact space, all sets are bounded -/
lemma bounded_of_compact_space [compact_space α] : bounded s :=
(bounded_of_compact compact_univ).subset (subset_univ _)
/-- In a proper space, a set is compact if and only if it is closed and bounded -/
lemma compact_iff_closed_bounded [proper_space α] :
compact s ↔ is_closed s ∧ bounded s :=
⟨λ h, ⟨closed_of_compact _ h, bounded_of_compact h⟩, begin
rintro ⟨hc, hb⟩,
classical, by_cases s = ∅, {simp [h, compact_empty]},
rcases exists_mem_of_ne_empty h with ⟨x, hx⟩,
rcases (bounded_iff_subset_ball x).1 hb with ⟨r, hr⟩,
exact compact_of_is_closed_subset (proper_space.compact_ball x r) hc hr
end⟩
end bounded
section diam
variables {s : set α} {x y : α}
/-- The diameter of a set in a metric space. To get controllable behavior even when the diameter
should be infinite, we express it in terms of the emetric.diameter -/
def diam (s : set α) : ℝ := ennreal.to_real (emetric.diam s)
/-- The diameter of a set is always nonnegative -/
lemma diam_nonneg : 0 ≤ diam s :=
by simp [diam]
/-- The empty set has zero diameter -/
@[simp] lemma diam_empty : diam (∅ : set α) = 0 :=
by simp [diam]
/-- A singleton has zero diameter -/
@[simp] lemma diam_singleton : diam ({x} : set α) = 0 :=
by simp [diam]
/-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
lemma bounded_iff_diam_ne_top : bounded s ↔ emetric.diam s ≠ ⊤ :=
begin
classical, by_cases hs : s = ∅,
{ simp [hs] },
{ rcases ne_empty_iff_exists_mem.1 hs with ⟨x, hx⟩,
split,
{ assume bs,
rcases (bounded_iff_subset_ball x).1 bs with ⟨r, hr⟩,
have r0 : 0 ≤ r := by simpa [closed_ball] using hr hx,
have : emetric.diam s < ⊤ := calc
emetric.diam s ≤ emetric.diam (emetric.closed_ball x (ennreal.of_real r)) :
by rw emetric_closed_ball r0; exact emetric.diam_mono hr
... ≤ 2 * (ennreal.of_real r) : emetric.diam_closed_ball
... < ⊤ : begin apply ennreal.lt_top_iff_ne_top.2, simp [ennreal.mul_eq_top], end,
exact ennreal.lt_top_iff_ne_top.1 this },
{ assume ds,
have : s ⊆ closed_ball x (ennreal.to_real (emetric.diam s)),
{ rw [← emetric_closed_ball ennreal.to_real_nonneg, ennreal.of_real_to_real ds],
exact λy hy, emetric.edist_le_diam_of_mem hy hx },
exact bounded.subset this (bounded_closed_ball) }}
end
/-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `emetric.diam`.
This lemma makes it possible to avoid side conditions in some situations -/
lemma diam_eq_zero_of_unbounded (h : ¬(bounded s)) : diam s = 0 :=
begin
simp only [bounded_iff_diam_ne_top, not_not, ne.def] at h,
simp [diam, h]
end
/-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
lemma diam_mono {s t : set α} (h : s ⊆ t) (ht : bounded t) : diam s ≤ diam t :=
begin
unfold diam,
rw ennreal.to_real_le_to_real (bounded_iff_diam_ne_top.1 (bounded.subset h ht)) (bounded_iff_diam_ne_top.1 ht),
exact emetric.diam_mono h
end
/-- The distance between two points in a set is controlled by the diameter of the set. -/
lemma dist_le_diam_of_mem (h : bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
begin
rw [diam, dist_edist],
rw ennreal.to_real_le_to_real (edist_ne_top _ _) (bounded_iff_diam_ne_top.1 h),
exact emetric.edist_le_diam_of_mem hx hy
end
/-- If the distance between any two points in a set is bounded by some constant, this constant
bounds the diameter. -/
lemma diam_le_of_forall_dist_le {d : real} (hd : d ≥ 0) (h : ∀x y ∈ s, dist x y ≤ d) : diam s ≤ d :=
begin
have I : emetric.diam s ≤ ennreal.of_real d,
{ refine emetric.diam_le_of_forall_edist_le (λx y hx hy, _),
rw [edist_dist],
exact ennreal.of_real_le_of_real (h x y hx hy) },
have A : emetric.diam s ≠ ⊤ :=
ennreal.lt_top_iff_ne_top.1 (lt_of_le_of_lt I (ennreal.lt_top_iff_ne_top.2 (by simp))),
rw [← ennreal.to_real_of_real hd, diam, ennreal.to_real_le_to_real A],
{ exact I },
{ simp }
end
/-- The diameter of a union is controlled by the sum of the diameters, and the distance between
any two points in each of the sets. This lemma is true without any side condition, since it is
obviously true if `s ∪ t` is unbounded. -/
lemma diam_union {t : set α} (xs : x ∈ s) (yt : y ∈ t) : diam (s ∪ t) ≤ diam s + dist x y + diam t :=
have I1 : ¬(bounded (s ∪ t)) → diam (s ∪ t) ≤ diam s + dist x y + diam t := λh, calc
diam (s ∪ t) = 0 + 0 + 0 : by simp [diam_eq_zero_of_unbounded h]
... ≤ diam s + dist x y + diam t : add_le_add (add_le_add diam_nonneg dist_nonneg) diam_nonneg,
have I2 : (bounded (s ∪ t)) → diam (s ∪ t) ≤ diam s + dist x y + diam t := λh,
begin
have : bounded s := bounded.subset (subset_union_left _ _) h,
have : bounded t := bounded.subset (subset_union_right _ _) h,
have A : ∀a ∈ s, ∀b ∈ t, dist a b ≤ diam s + dist x y + diam t := λa ha b hb, calc
dist a b ≤ dist a x + dist x y + dist y b : dist_triangle4 _ _ _ _
... ≤ diam s + dist x y + diam t :
add_le_add (add_le_add (dist_le_diam_of_mem ‹bounded s› ha xs) (le_refl _)) (dist_le_diam_of_mem ‹bounded t› yt hb),
have B : ∀a b ∈ s ∪ t, dist a b ≤ diam s + dist x y + diam t := λa b ha hb,
begin
cases (mem_union _ _ _).1 ha with h'a h'a; cases (mem_union _ _ _).1 hb with h'b h'b,
{ calc dist a b ≤ diam s : dist_le_diam_of_mem ‹bounded s› h'a h'b
... = diam s + (0 + 0) : by simp
... ≤ diam s + (dist x y + diam t) : add_le_add (le_refl _) (add_le_add dist_nonneg diam_nonneg)
... = diam s + dist x y + diam t : by simp only [add_comm, eq_self_iff_true, add_left_comm] },
{ exact A a h'a b h'b },
{ have Z := A b h'b a h'a, rwa [dist_comm] at Z },
{ calc dist a b ≤ diam t : dist_le_diam_of_mem ‹bounded t› h'a h'b
... = (0 + 0) + diam t : by simp
... ≤ (diam s + dist x y) + diam t : add_le_add (add_le_add diam_nonneg dist_nonneg) (le_refl _) }
end,
have C : 0 ≤ diam s + dist x y + diam t := calc
0 = 0 + 0 + 0 : by simp
... ≤ diam s + dist x y + diam t : add_le_add (add_le_add diam_nonneg dist_nonneg) diam_nonneg,
exact diam_le_of_forall_dist_le C B
end,
classical.by_cases I2 I1
/-- If two sets intersect, the diameter of the union is bounded by the sum of the diameters. -/
lemma diam_union' {t : set α} (h : s ∩ t ≠ ∅) : diam (s ∪ t) ≤ diam s + diam t :=
begin
rcases ne_empty_iff_exists_mem.1 h with ⟨x, ⟨xs, xt⟩⟩,
simpa using diam_union xs xt
end
/-- The diameter of a closed ball of radius `r` is at most `2 r`. -/
lemma diam_closed_ball {r : ℝ} (h : r ≥ 0) : diam (closed_ball x r) ≤ 2 * r :=
diam_le_of_forall_dist_le (mul_nonneg (by norm_num) h) $ λa b ha hb, calc
dist a b ≤ dist a x + dist b x : dist_triangle_right _ _ _
... ≤ r + r : add_le_add ha hb
... = 2 * r : by simp [mul_two, mul_comm]
/-- The diameter of a ball of radius `r` is at most `2 r`. -/
lemma diam_ball {r : ℝ} (h : r ≥ 0) : diam (ball x r) ≤ 2 * r :=
le_trans (diam_mono ball_subset_closed_ball bounded_closed_ball) (diam_closed_ball h)
end diam
end metric
|
00a728793bbe34d557d4462bfaf7fed4e63a1a79 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/data/matrix/char_p.lean | 2a041013febb93f624b89da2ea2c7415d4e7869f | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 567 | lean | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import data.matrix.basic
import algebra.char_p
/-!
# Matrices in prime characteristic
-/
open matrix
variables {n : Type*} [fintype n] {R : Type*} [ring R]
instance matrix.char_p [decidable_eq n] [nonempty n] (p : ℕ) [char_p R p] :
char_p (matrix n n R) p :=
⟨begin
intro k,
rw [← char_p.cast_eq_zero_iff R p k, ← nat.cast_zero, ← (scalar n).map_nat_cast],
convert scalar_inj,
simpa
end⟩
|
d35580424fa612c143a8d97cd17506475915c777 | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/run/simp_refl_lemma_perf_issue.lean | 93daf4a3352b52eb24bd3febda44d6e98e28c44e | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 397 | lean | def foo (a : nat) (b : nat) := b
def g (a : nat) := foo (1000000000000 + 1) a
def f (a : nat) := foo (1 + 1000000000000) a
lemma foo_lemma (a b : nat) : foo b a = foo (1 + 1000000000000) a :=
rfl
lemma f_fold_lemma (a : nat) : foo (1 + 1000000000000) a = f a :=
rfl
lemma ex (a : nat) (p : nat → Prop) (h : p (f a)) : p (g a) :=
begin
simp only [g, foo_lemma, f_fold_lemma],
exact h,
end
|
60256ba1f68d9f19c332d1025ec6cfb86627a06a | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/data/finset/lattice.lean | bfba21f64515908a7dcf01f2eb7262e5dfd0c104 | [
"Apache-2.0"
] | permissive | dexmagic/mathlib | ff48eefc56e2412429b31d4fddd41a976eb287ce | 7a5d15a955a92a90e1d398b2281916b9c41270b2 | refs/heads/master | 1,693,481,322,046 | 1,633,360,193,000 | 1,633,360,193,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 44,898 | lean | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import data.finset.fold
import data.multiset.lattice
import order.order_dual
import order.complete_lattice
/-!
# Lattice operations on finsets
-/
variables {α β γ : Type*}
namespace finset
open multiset order_dual
/-! ### sup -/
section sup
variables [semilattice_sup_bot α]
/-- Supremum of a finite set: `sup {a, b, c} f = f a ⊔ f b ⊔ f c` -/
def sup (s : finset β) (f : β → α) : α := s.fold (⊔) ⊥ f
variables {s s₁ s₂ : finset β} {f : β → α}
lemma sup_def : s.sup f = (s.1.map f).sup := rfl
@[simp] lemma sup_empty : (∅ : finset β).sup f = ⊥ :=
fold_empty
@[simp] lemma sup_cons {b : β} (h : b ∉ s) : (cons b s h).sup f = f b ⊔ s.sup f :=
fold_cons h
@[simp] lemma sup_insert [decidable_eq β] {b : β} : (insert b s : finset β).sup f = f b ⊔ s.sup f :=
fold_insert_idem
lemma sup_image [decidable_eq β] (s : finset γ) (f : γ → β) (g : β → α):
(s.image f).sup g = s.sup (g ∘ f) :=
fold_image_idem
@[simp] lemma sup_map (s : finset γ) (f : γ ↪ β) (g : β → α) :
(s.map f).sup g = s.sup (g ∘ f) :=
fold_map
@[simp] lemma sup_singleton {b : β} : ({b} : finset β).sup f = f b :=
sup_singleton
lemma sup_union [decidable_eq β] : (s₁ ∪ s₂).sup f = s₁.sup f ⊔ s₂.sup f :=
finset.induction_on s₁ (by rw [empty_union, sup_empty, bot_sup_eq]) $ λ a s has ih,
by rw [insert_union, sup_insert, sup_insert, ih, sup_assoc]
theorem sup_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀a∈s₂, f a = g a) : s₁.sup f = s₂.sup g :=
by subst hs; exact finset.fold_congr hfg
@[simp] lemma sup_le_iff {a : α} : s.sup f ≤ a ↔ (∀b ∈ s, f b ≤ a) :=
begin
apply iff.trans multiset.sup_le,
simp only [multiset.mem_map, and_imp, exists_imp_distrib],
exact ⟨λ k b hb, k _ _ hb rfl, λ k a' b hb h, h ▸ k _ hb⟩,
end
@[simp] lemma sup_bUnion [decidable_eq β] (s : finset γ) (t : γ → finset β) :
(s.bUnion t).sup f = s.sup (λ x, (t x).sup f) :=
eq_of_forall_ge_iff $ λ c, by simp [@forall_swap _ β]
lemma sup_const {s : finset β} (h : s.nonempty) (c : α) : s.sup (λ _, c) = c :=
eq_of_forall_ge_iff $ λ b, sup_le_iff.trans h.forall_const
lemma sup_le {a : α} : (∀b ∈ s, f b ≤ a) → s.sup f ≤ a :=
sup_le_iff.2
lemma le_sup {b : β} (hb : b ∈ s) : f b ≤ s.sup f :=
sup_le_iff.1 (le_refl _) _ hb
lemma sup_mono_fun {g : β → α} (h : ∀b∈s, f b ≤ g b) : s.sup f ≤ s.sup g :=
sup_le (λ b hb, le_trans (h b hb) (le_sup hb))
lemma sup_mono (h : s₁ ⊆ s₂) : s₁.sup f ≤ s₂.sup f :=
sup_le $ assume b hb, le_sup (h hb)
@[simp] lemma sup_lt_iff [is_total α (≤)] {a : α} (ha : ⊥ < a) : s.sup f < a ↔ (∀ b ∈ s, f b < a) :=
⟨(λ hs b hb, lt_of_le_of_lt (le_sup hb) hs), finset.cons_induction_on s (λ _, ha)
(λ c t hc, by simpa only [sup_cons, sup_lt_iff, mem_cons, forall_eq_or_imp] using and.imp_right)⟩
@[simp] lemma le_sup_iff [is_total α (≤)] {a : α} (ha : ⊥ < a) : a ≤ s.sup f ↔ ∃ b ∈ s, a ≤ f b :=
⟨finset.cons_induction_on s (λ h, absurd h (not_le_of_lt ha))
(λ c t hc ih, by simpa using @or.rec _ _ (∃ b, (b = c ∨ b ∈ t) ∧ a ≤ f b)
(λ h, ⟨c, or.inl rfl, h⟩) (λ h, let ⟨b, hb, hle⟩ := ih h in ⟨b, or.inr hb, hle⟩)),
(λ ⟨b, hb, hle⟩, trans hle (le_sup hb))⟩
@[simp] lemma lt_sup_iff [is_total α (≤)] {a : α} : a < s.sup f ↔ ∃ b ∈ s, a < f b :=
⟨finset.cons_induction_on s (λ h, absurd h not_lt_bot)
(λ c t hc ih, by simpa using @or.rec _ _ (∃ b, (b = c ∨ b ∈ t) ∧ a < f b)
(λ h, ⟨c, or.inl rfl, h⟩) (λ h, let ⟨b, hb, hlt⟩ := ih h in ⟨b, or.inr hb, hlt⟩)),
(λ ⟨b, hb, hlt⟩, lt_of_lt_of_le hlt (le_sup hb))⟩
lemma comp_sup_eq_sup_comp [semilattice_sup_bot γ] {s : finset β}
{f : β → α} (g : α → γ) (g_sup : ∀ x y, g (x ⊔ y) = g x ⊔ g y) (bot : g ⊥ = ⊥) :
g (s.sup f) = s.sup (g ∘ f) :=
finset.cons_induction_on s bot (λ c t hc ih, by rw [sup_cons, sup_cons, g_sup, ih])
lemma comp_sup_eq_sup_comp_of_is_total [is_total α (≤)] {γ : Type} [semilattice_sup_bot γ]
(g : α → γ) (mono_g : monotone g) (bot : g ⊥ = ⊥) : g (s.sup f) = s.sup (g ∘ f) :=
comp_sup_eq_sup_comp g mono_g.map_sup bot
/-- Computating `sup` in a subtype (closed under `sup`) is the same as computing it in `α`. -/
lemma sup_coe {P : α → Prop}
{Pbot : P ⊥} {Psup : ∀{{x y}}, P x → P y → P (x ⊔ y)}
(t : finset β) (f : β → {x : α // P x}) :
(@sup _ _ (subtype.semilattice_sup_bot Pbot Psup) t f : α) = t.sup (λ x, f x) :=
by { rw [comp_sup_eq_sup_comp coe]; intros; refl }
@[simp] lemma sup_to_finset {α β} [decidable_eq β]
(s : finset α) (f : α → multiset β) :
(s.sup f).to_finset = s.sup (λ x, (f x).to_finset) :=
comp_sup_eq_sup_comp multiset.to_finset to_finset_union rfl
theorem subset_range_sup_succ (s : finset ℕ) : s ⊆ range (s.sup id).succ :=
λ n hn, mem_range.2 $ nat.lt_succ_of_le $ le_sup hn
theorem exists_nat_subset_range (s : finset ℕ) : ∃n : ℕ, s ⊆ range n :=
⟨_, s.subset_range_sup_succ⟩
lemma sup_induction {p : α → Prop} (hb : p ⊥) (hp : ∀ (a₁ a₂ : α), p a₁ → p a₂ → p (a₁ ⊔ a₂))
(hs : ∀ b ∈ s, p (f b)) : p (s.sup f) :=
begin
induction s using finset.cons_induction with c s hc ih,
{ exact hb, },
{ rw sup_cons,
apply hp,
{ exact hs c (mem_cons.2 (or.inl rfl)), },
{ exact ih (λ b h, hs b (mem_cons.2 (or.inr h))), }, },
end
lemma sup_le_of_le_directed {α : Type*} [semilattice_sup_bot α] (s : set α)
(hs : s.nonempty) (hdir : directed_on (≤) s) (t : finset α):
(∀ x ∈ t, ∃ y ∈ s, x ≤ y) → ∃ x, x ∈ s ∧ t.sup id ≤ x :=
begin
classical,
apply finset.induction_on t,
{ simpa only [forall_prop_of_true, and_true, forall_prop_of_false, bot_le, not_false_iff,
sup_empty, forall_true_iff, not_mem_empty], },
{ intros a r har ih h,
have incs : ↑r ⊆ ↑(insert a r), by { rw finset.coe_subset, apply finset.subset_insert, },
-- x ∈ s is above the sup of r
obtain ⟨x, ⟨hxs, hsx_sup⟩⟩ := ih (λ x hx, h x $ incs hx),
-- y ∈ s is above a
obtain ⟨y, hys, hay⟩ := h a (finset.mem_insert_self a r),
-- z ∈ s is above x and y
obtain ⟨z, hzs, ⟨hxz, hyz⟩⟩ := hdir x hxs y hys,
use [z, hzs],
rw [sup_insert, id.def, _root_.sup_le_iff],
exact ⟨le_trans hay hyz, le_trans hsx_sup hxz⟩, },
end
-- If we acquire sublattices
-- the hypotheses should be reformulated as `s : subsemilattice_sup_bot`
lemma sup_mem
(s : set α) (w₁ : ⊥ ∈ s) (w₂ : ∀ x y ∈ s, x ⊔ y ∈ s)
{ι : Type*} (t : finset ι) (p : ι → α) (h : ∀ i ∈ t, p i ∈ s) :
t.sup p ∈ s :=
@sup_induction _ _ _ _ _ (∈ s) w₁ w₂ h
end sup
lemma sup_eq_supr [complete_lattice β] (s : finset α) (f : α → β) : s.sup f = (⨆a∈s, f a) :=
le_antisymm
(finset.sup_le $ assume a ha, le_supr_of_le a $ le_supr _ ha)
(supr_le $ assume a, supr_le $ assume ha, le_sup ha)
lemma sup_id_eq_Sup [complete_lattice α] (s : finset α) : s.sup id = Sup s :=
by simp [Sup_eq_supr, sup_eq_supr]
lemma sup_eq_Sup_image [complete_lattice β] (s : finset α) (f : α → β) : s.sup f = Sup (f '' s) :=
begin
classical,
rw [←finset.coe_image, ←sup_id_eq_Sup, sup_image, function.comp.left_id],
end
/-! ### inf -/
section inf
variables [semilattice_inf_top α]
/-- Infimum of a finite set: `inf {a, b, c} f = f a ⊓ f b ⊓ f c` -/
def inf (s : finset β) (f : β → α) : α := s.fold (⊓) ⊤ f
variables {s s₁ s₂ : finset β} {f : β → α}
lemma inf_def : s.inf f = (s.1.map f).inf := rfl
@[simp] lemma inf_empty : (∅ : finset β).inf f = ⊤ :=
fold_empty
@[simp] lemma inf_cons {b : β} (h : b ∉ s) : (cons b s h).inf f = f b ⊓ s.inf f :=
@sup_cons (order_dual α) _ _ _ _ _ h
@[simp] lemma inf_insert [decidable_eq β] {b : β} : (insert b s : finset β).inf f = f b ⊓ s.inf f :=
fold_insert_idem
lemma inf_image [decidable_eq β] (s : finset γ) (f : γ → β) (g : β → α):
(s.image f).inf g = s.inf (g ∘ f) :=
fold_image_idem
@[simp] lemma inf_map (s : finset γ) (f : γ ↪ β) (g : β → α) :
(s.map f).inf g = s.inf (g ∘ f) :=
fold_map
@[simp] lemma inf_singleton {b : β} : ({b} : finset β).inf f = f b :=
inf_singleton
lemma inf_union [decidable_eq β] : (s₁ ∪ s₂).inf f = s₁.inf f ⊓ s₂.inf f :=
@sup_union (order_dual α) _ _ _ _ _ _
theorem inf_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀a∈s₂, f a = g a) : s₁.inf f = s₂.inf g :=
by subst hs; exact finset.fold_congr hfg
@[simp] lemma inf_bUnion [decidable_eq β] (s : finset γ) (t : γ → finset β) :
(s.bUnion t).inf f = s.inf (λ x, (t x).inf f) :=
@sup_bUnion (order_dual α) _ _ _ _ _ _ _
lemma inf_const {s : finset β} (h : s.nonempty) (c : α) : s.inf (λ _, c) = c :=
@sup_const (order_dual α) _ _ _ h _
lemma le_inf_iff {a : α} : a ≤ s.inf f ↔ ∀ b ∈ s, a ≤ f b :=
@sup_le_iff (order_dual α) _ _ _ _ _
lemma inf_le {b : β} (hb : b ∈ s) : s.inf f ≤ f b :=
le_inf_iff.1 (le_refl _) _ hb
lemma le_inf {a : α} : (∀b ∈ s, a ≤ f b) → a ≤ s.inf f :=
le_inf_iff.2
lemma inf_mono_fun {g : β → α} (h : ∀b∈s, f b ≤ g b) : s.inf f ≤ s.inf g :=
le_inf (λ b hb, le_trans (inf_le hb) (h b hb))
lemma inf_mono (h : s₁ ⊆ s₂) : s₂.inf f ≤ s₁.inf f :=
le_inf $ assume b hb, inf_le (h hb)
@[simp] lemma lt_inf_iff [is_total α (≤)] {a : α} (ha : a < ⊤) : a < s.inf f ↔ (∀ b ∈ s, a < f b) :=
@sup_lt_iff (order_dual α) _ _ _ _ _ _ ha
@[simp] lemma inf_le_iff [is_total α (≤)] {a : α} (ha : a < ⊤) : s.inf f ≤ a ↔ (∃ b ∈ s, f b ≤ a) :=
@le_sup_iff (order_dual α) _ _ _ _ _ _ ha
@[simp] lemma inf_lt_iff [is_total α (≤)] {a : α} : s.inf f < a ↔ (∃ b ∈ s, f b < a) :=
@lt_sup_iff (order_dual α) _ _ _ _ _ _
lemma comp_inf_eq_inf_comp [semilattice_inf_top γ] {s : finset β}
{f : β → α} (g : α → γ) (g_inf : ∀ x y, g (x ⊓ y) = g x ⊓ g y) (top : g ⊤ = ⊤) :
g (s.inf f) = s.inf (g ∘ f) :=
@comp_sup_eq_sup_comp (order_dual α) _ (order_dual γ) _ _ _ _ _ g_inf top
lemma comp_inf_eq_inf_comp_of_is_total [h : is_total α (≤)] {γ : Type} [semilattice_inf_top γ]
(g : α → γ) (mono_g : monotone g) (top : g ⊤ = ⊤) : g (s.inf f) = s.inf (g ∘ f) :=
comp_inf_eq_inf_comp g mono_g.map_inf top
/-- Computating `inf` in a subtype (closed under `inf`) is the same as computing it in `α`. -/
lemma inf_coe {P : α → Prop}
{Ptop : P ⊤} {Pinf : ∀{{x y}}, P x → P y → P (x ⊓ y)}
(t : finset β) (f : β → {x : α // P x}) :
(@inf _ _ (subtype.semilattice_inf_top Ptop Pinf) t f : α) = t.inf (λ x, f x) :=
@sup_coe (order_dual α) _ _ _ Ptop Pinf t f
lemma inf_induction {p : α → Prop} (ht : p ⊤) (hp : ∀ (a₁ a₂ : α), p a₁ → p a₂ → p (a₁ ⊓ a₂))
(hs : ∀ b ∈ s, p (f b)) : p (s.inf f) :=
@sup_induction (order_dual α) _ _ _ _ _ ht hp hs
lemma inf_mem
(s : set α) (w₁ : ⊤ ∈ s) (w₂ : ∀ x y ∈ s, x ⊓ y ∈ s)
{ι : Type*} (t : finset ι) (p : ι → α) (h : ∀ i ∈ t, p i ∈ s) :
t.inf p ∈ s :=
@inf_induction _ _ _ _ _ (∈ s) w₁ w₂ h
end inf
lemma inf_eq_infi [complete_lattice β] (s : finset α) (f : α → β) : s.inf f = (⨅a∈s, f a) :=
@sup_eq_supr _ (order_dual β) _ _ _
lemma inf_id_eq_Inf [complete_lattice α] (s : finset α) : s.inf id = Inf s :=
@sup_id_eq_Sup (order_dual α) _ _
lemma inf_eq_Inf_image [complete_lattice β] (s : finset α) (f : α → β) : s.inf f = Inf (f '' s) :=
@sup_eq_Sup_image _ (order_dual β) _ _ _
section sup'
variables [semilattice_sup α]
lemma sup_of_mem {s : finset β} (f : β → α) {b : β} (h : b ∈ s) :
∃ (a : α), s.sup (coe ∘ f : β → with_bot α) = ↑a :=
Exists.imp (λ a, Exists.fst) (@le_sup (with_bot α) _ _ _ _ _ h (f b) rfl)
/-- Given nonempty finset `s` then `s.sup' H f` is the supremum of its image under `f` in (possibly
unbounded) join-semilattice `α`, where `H` is a proof of nonemptiness. If `α` has a bottom element
you may instead use `finset.sup` which does not require `s` nonempty. -/
def sup' (s : finset β) (H : s.nonempty) (f : β → α) : α :=
option.get $ let ⟨b, hb⟩ := H in option.is_some_iff_exists.2 (sup_of_mem f hb)
variables {s : finset β} (H : s.nonempty) (f : β → α)
@[simp] lemma coe_sup' : ((s.sup' H f : α) : with_bot α) = s.sup (coe ∘ f) :=
by rw [sup', ←with_bot.some_eq_coe, option.some_get]
@[simp] lemma sup'_cons {b : β} {hb : b ∉ s} {h : (cons b s hb).nonempty} :
(cons b s hb).sup' h f = f b ⊔ s.sup' H f :=
by { rw ←with_bot.coe_eq_coe, simp only [coe_sup', sup_cons, with_bot.coe_sup], }
@[simp] lemma sup'_insert [decidable_eq β] {b : β} {h : (insert b s).nonempty} :
(insert b s).sup' h f = f b ⊔ s.sup' H f :=
by { rw ←with_bot.coe_eq_coe, simp only [coe_sup', sup_insert, with_bot.coe_sup], }
@[simp] lemma sup'_singleton {b : β} {h : ({b} : finset β).nonempty} :
({b} : finset β).sup' h f = f b := rfl
lemma sup'_le {a : α} (hs : ∀ b ∈ s, f b ≤ a) : s.sup' H f ≤ a :=
by { rw [←with_bot.coe_le_coe, coe_sup'], exact sup_le (λ b h, with_bot.coe_le_coe.2 $ hs b h), }
lemma le_sup' {b : β} (h : b ∈ s) : f b ≤ s.sup' ⟨b, h⟩ f :=
by { rw [←with_bot.coe_le_coe, coe_sup'], exact le_sup h, }
@[simp] lemma sup'_const (a : α) : s.sup' H (λ b, a) = a :=
begin
apply le_antisymm,
{ apply sup'_le, intros, apply le_refl, },
{ apply le_sup' (λ b, a) H.some_spec, }
end
@[simp] lemma sup'_le_iff {a : α} : s.sup' H f ≤ a ↔ ∀ b ∈ s, f b ≤ a :=
iff.intro (λ h b hb, trans (le_sup' f hb) h) (sup'_le H f)
@[simp] lemma sup'_lt_iff [is_total α (≤)] {a : α} : s.sup' H f < a ↔ (∀ b ∈ s, f b < a) :=
begin
rw [←with_bot.coe_lt_coe, coe_sup', sup_lt_iff (with_bot.bot_lt_coe a)],
exact ball_congr (λ b hb, with_bot.coe_lt_coe),
end
@[simp] lemma le_sup'_iff [is_total α (≤)] {a : α} : a ≤ s.sup' H f ↔ (∃ b ∈ s, a ≤ f b) :=
begin
rw [←with_bot.coe_le_coe, coe_sup', le_sup_iff (with_bot.bot_lt_coe a)],
exact bex_congr (λ b hb, with_bot.coe_le_coe),
end
@[simp] lemma lt_sup'_iff [is_total α (≤)] {a : α} : a < s.sup' H f ↔ (∃ b ∈ s, a < f b) :=
begin
rw [←with_bot.coe_lt_coe, coe_sup', lt_sup_iff],
exact bex_congr (λ b hb, with_bot.coe_lt_coe),
end
lemma sup'_bUnion [decidable_eq β] {s : finset γ} (Hs : s.nonempty) {t : γ → finset β}
(Ht : ∀ b, (t b).nonempty) :
(s.bUnion t).sup' (Hs.bUnion (λ b _, Ht b)) f = s.sup' Hs (λ b, (t b).sup' (Ht b) f) :=
eq_of_forall_ge_iff $ λ c, by simp [@forall_swap _ β]
lemma comp_sup'_eq_sup'_comp [semilattice_sup γ] {s : finset β} (H : s.nonempty)
{f : β → α} (g : α → γ) (g_sup : ∀ x y, g (x ⊔ y) = g x ⊔ g y) :
g (s.sup' H f) = s.sup' H (g ∘ f) :=
begin
rw [←with_bot.coe_eq_coe, coe_sup'],
let g' : with_bot α → with_bot γ := with_bot.rec_bot_coe ⊥ (λ x, ↑(g x)),
show g' ↑(s.sup' H f) = s.sup (λ a, g' ↑(f a)),
rw coe_sup',
refine comp_sup_eq_sup_comp g' _ rfl,
intros f₁ f₂,
cases f₁,
{ rw [with_bot.none_eq_bot, bot_sup_eq], exact bot_sup_eq.symm, },
{ cases f₂, refl,
exact congr_arg coe (g_sup f₁ f₂), },
end
lemma sup'_induction {p : α → Prop} (hp : ∀ (a₁ a₂ : α), p a₁ → p a₂ → p (a₁ ⊔ a₂))
(hs : ∀ b ∈ s, p (f b)) : p (s.sup' H f) :=
begin
show @with_bot.rec_bot_coe α (λ _, Prop) true p ↑(s.sup' H f),
rw coe_sup',
refine sup_induction trivial _ hs,
intros a₁ a₂ h₁ h₂,
cases a₁,
{ rw [with_bot.none_eq_bot, bot_sup_eq], exact h₂, },
{ cases a₂, exact h₁, exact hp a₁ a₂ h₁ h₂, },
end
lemma exists_mem_eq_sup' [is_total α (≤)] : ∃ b, b ∈ s ∧ s.sup' H f = f b :=
begin
induction s using finset.cons_induction with c s hc ih,
{ exact false.elim (not_nonempty_empty H), },
{ rcases s.eq_empty_or_nonempty with rfl | hs,
{ exact ⟨c, mem_singleton_self c, rfl⟩, },
{ rcases ih hs with ⟨b, hb, h'⟩,
rw [sup'_cons hs, h'],
cases total_of (≤) (f b) (f c) with h h,
{ exact ⟨c, mem_cons.2 (or.inl rfl), sup_eq_left.2 h⟩, },
{ exact ⟨b, mem_cons.2 (or.inr hb), sup_eq_right.2 h⟩, }, }, },
end
lemma sup'_mem
(s : set α) (w : ∀ x y ∈ s, x ⊔ y ∈ s)
{ι : Type*} (t : finset ι) (H : t.nonempty) (p : ι → α) (h : ∀ i ∈ t, p i ∈ s) :
t.sup' H p ∈ s :=
sup'_induction H p w h
@[congr] lemma sup'_congr {t : finset β} {f g : β → α} (h₁ : s = t) (h₂ : ∀ x ∈ s, f x = g x) :
s.sup' H f = t.sup' (h₁ ▸ H) g :=
begin
subst s,
refine eq_of_forall_ge_iff (λ c, _),
simp only [sup'_le_iff, h₂] { contextual := tt }
end
end sup'
section inf'
variables [semilattice_inf α]
lemma inf_of_mem {s : finset β} (f : β → α) {b : β} (h : b ∈ s) :
∃ (a : α), s.inf (coe ∘ f : β → with_top α) = ↑a :=
@sup_of_mem (order_dual α) _ _ _ f _ h
/-- Given nonempty finset `s` then `s.inf' H f` is the infimum of its image under `f` in (possibly
unbounded) meet-semilattice `α`, where `H` is a proof of nonemptiness. If `α` has a top element you
may instead use `finset.inf` which does not require `s` nonempty. -/
def inf' (s : finset β) (H : s.nonempty) (f : β → α) : α :=
@sup' (order_dual α) _ _ s H f
variables {s : finset β} (H : s.nonempty) (f : β → α)
@[simp] lemma coe_inf' : ((s.inf' H f : α) : with_top α) = s.inf (coe ∘ f) :=
@coe_sup' (order_dual α) _ _ _ H f
@[simp] lemma inf'_cons {b : β} {hb : b ∉ s} {h : (cons b s hb).nonempty} :
(cons b s hb).inf' h f = f b ⊓ s.inf' H f :=
@sup'_cons (order_dual α) _ _ _ H f _ _ _
@[simp] lemma inf'_insert [decidable_eq β] {b : β} {h : (insert b s).nonempty} :
(insert b s).inf' h f = f b ⊓ s.inf' H f :=
@sup'_insert (order_dual α) _ _ _ H f _ _ _
@[simp] lemma inf'_singleton {b : β} {h : ({b} : finset β).nonempty} :
({b} : finset β).inf' h f = f b := rfl
lemma le_inf' {a : α} (hs : ∀ b ∈ s, a ≤ f b) : a ≤ s.inf' H f :=
@sup'_le (order_dual α) _ _ _ H f _ hs
lemma inf'_le {b : β} (h : b ∈ s) : s.inf' ⟨b, h⟩ f ≤ f b :=
@le_sup' (order_dual α) _ _ _ f _ h
@[simp] lemma inf'_const (a : α) : s.inf' H (λ b, a) = a :=
@sup'_const (order_dual α) _ _ _ _ _
@[simp] lemma le_inf'_iff {a : α} : a ≤ s.inf' H f ↔ ∀ b ∈ s, a ≤ f b :=
@sup'_le_iff (order_dual α) _ _ _ H f _
@[simp] lemma lt_inf'_iff [is_total α (≤)] {a : α} : a < s.inf' H f ↔ (∀ b ∈ s, a < f b) :=
@sup'_lt_iff (order_dual α) _ _ _ H f _ _
@[simp] lemma inf'_le_iff [is_total α (≤)] {a : α} : s.inf' H f ≤ a ↔ (∃ b ∈ s, f b ≤ a) :=
@le_sup'_iff (order_dual α) _ _ _ H f _ _
@[simp] lemma inf'_lt_iff [is_total α (≤)] {a : α} : s.inf' H f < a ↔ (∃ b ∈ s, f b < a) :=
@lt_sup'_iff (order_dual α) _ _ _ H f _ _
lemma inf'_bUnion [decidable_eq β] {s : finset γ} (Hs : s.nonempty) {t : γ → finset β}
(Ht : ∀ b, (t b).nonempty) :
(s.bUnion t).inf' (Hs.bUnion (λ b _, Ht b)) f = s.inf' Hs (λ b, (t b).inf' (Ht b) f) :=
@sup'_bUnion (order_dual α) _ _ _ _ _ _ Hs _ Ht
lemma comp_inf'_eq_inf'_comp [semilattice_inf γ] {s : finset β} (H : s.nonempty)
{f : β → α} (g : α → γ) (g_inf : ∀ x y, g (x ⊓ y) = g x ⊓ g y) :
g (s.inf' H f) = s.inf' H (g ∘ f) :=
@comp_sup'_eq_sup'_comp (order_dual α) _ (order_dual γ) _ _ _ H f g g_inf
lemma inf'_induction {p : α → Prop} (hp : ∀ (a₁ a₂ : α), p a₁ → p a₂ → p (a₁ ⊓ a₂))
(hs : ∀ b ∈ s, p (f b)) : p (s.inf' H f) :=
@sup'_induction (order_dual α) _ _ _ H f _ hp hs
lemma exists_mem_eq_inf' [is_total α (≤)] : ∃ b, b ∈ s ∧ s.inf' H f = f b :=
@exists_mem_eq_sup' (order_dual α) _ _ _ H f _
lemma inf'_mem (s : set α) (w : ∀ x y ∈ s, x ⊓ y ∈ s)
{ι : Type*} (t : finset ι) (H : t.nonempty) (p : ι → α) (h : ∀ i ∈ t, p i ∈ s) :
t.inf' H p ∈ s :=
inf'_induction H p w h
@[congr] lemma inf'_congr {t : finset β} {f g : β → α} (h₁ : s = t) (h₂ : ∀ x ∈ s, f x = g x) :
s.inf' H f = t.inf' (h₁ ▸ H) g :=
@sup'_congr (order_dual α) _ _ _ H _ _ _ h₁ h₂
end inf'
section sup
variable [semilattice_sup_bot α]
lemma sup'_eq_sup {s : finset β} (H : s.nonempty) (f : β → α) : s.sup' H f = s.sup f :=
le_antisymm (sup'_le H f (λ b, le_sup)) (sup_le (λ b, le_sup' f))
lemma sup_closed_of_sup_closed {s : set α} (t : finset α) (htne : t.nonempty) (h_subset : ↑t ⊆ s)
(h : ∀⦃a b⦄, a ∈ s → b ∈ s → a ⊔ b ∈ s) : t.sup id ∈ s :=
sup'_eq_sup htne id ▸ sup'_induction _ _ h h_subset
lemma exists_mem_eq_sup [is_total α (≤)] (s : finset β) (h : s.nonempty) (f : β → α) :
∃ b, b ∈ s ∧ s.sup f = f b :=
sup'_eq_sup h f ▸ exists_mem_eq_sup' h f
end sup
section inf
variable [semilattice_inf_top α]
lemma inf'_eq_inf {s : finset β} (H : s.nonempty) (f : β → α) : s.inf' H f = s.inf f :=
@sup'_eq_sup (order_dual α) _ _ _ H f
lemma inf_closed_of_inf_closed {s : set α} (t : finset α) (htne : t.nonempty) (h_subset : ↑t ⊆ s)
(h : ∀⦃a b⦄, a ∈ s → b ∈ s → a ⊓ b ∈ s) : t.inf id ∈ s :=
@sup_closed_of_sup_closed (order_dual α) _ _ t htne h_subset h
lemma exists_mem_eq_inf [is_total α (≤)] (s : finset β) (h : s.nonempty) (f : β → α) :
∃ a, a ∈ s ∧ s.inf f = f a :=
@exists_mem_eq_sup (order_dual α) _ _ _ _ h f
end inf
section sup
variables {C : β → Type*} [Π (b : β), semilattice_sup_bot (C b)]
@[simp]
protected lemma sup_apply (s : finset α) (f : α → Π (b : β), C b) (b : β) :
s.sup f b = s.sup (λ a, f a b) :=
comp_sup_eq_sup_comp (λ x : Π b : β, C b, x b) (λ i j, rfl) rfl
end sup
section inf
variables {C : β → Type*} [Π (b : β), semilattice_inf_top (C b)]
@[simp]
protected lemma inf_apply (s : finset α) (f : α → Π (b : β), C b) (b : β) :
s.inf f b = s.inf (λ a, f a b) :=
@finset.sup_apply _ _ (λ b, order_dual (C b)) _ s f b
end inf
section sup'
variables {C : β → Type*} [Π (b : β), semilattice_sup (C b)]
@[simp]
protected lemma sup'_apply {s : finset α} (H : s.nonempty) (f : α → Π (b : β), C b) (b : β) :
s.sup' H f b = s.sup' H (λ a, f a b) :=
comp_sup'_eq_sup'_comp H (λ x : Π b : β, C b, x b) (λ i j, rfl)
end sup'
section inf'
variables {C : β → Type*} [Π (b : β), semilattice_inf (C b)]
@[simp]
protected lemma inf'_apply {s : finset α} (H : s.nonempty) (f : α → Π (b : β), C b) (b : β) :
s.inf' H f b = s.inf' H (λ a, f a b) :=
@finset.sup'_apply _ _ (λ b, order_dual (C b)) _ _ H f b
end inf'
/-! ### max and min of finite sets -/
section max_min
variables [linear_order α]
/-- Let `s` be a finset in a linear order. Then `s.max` is the maximum of `s` if `s` is not empty,
and `none` otherwise. It belongs to `option α`. If you want to get an element of `α`, see
`s.max'`. -/
protected def max : finset α → option α :=
fold (option.lift_or_get max) none some
theorem max_eq_sup_with_bot (s : finset α) :
s.max = @sup (with_bot α) α _ s some := rfl
@[simp] theorem max_empty : (∅ : finset α).max = none := rfl
@[simp] theorem max_insert {a : α} {s : finset α} :
(insert a s).max = option.lift_or_get max (some a) s.max := fold_insert_idem
@[simp] theorem max_singleton {a : α} : finset.max {a} = some a :=
by { rw [← insert_emptyc_eq], exact max_insert }
theorem max_of_mem {s : finset α} {a : α} (h : a ∈ s) : ∃ b, b ∈ s.max :=
(@le_sup (with_bot α) _ _ _ _ _ h _ rfl).imp $ λ b, Exists.fst
theorem max_of_nonempty {s : finset α} (h : s.nonempty) : ∃ a, a ∈ s.max :=
let ⟨a, ha⟩ := h in max_of_mem ha
theorem max_eq_none {s : finset α} : s.max = none ↔ s = ∅ :=
⟨λ h, s.eq_empty_or_nonempty.elim id
(λ H, let ⟨a, ha⟩ := max_of_nonempty H in by rw h at ha; cases ha),
λ h, h.symm ▸ max_empty⟩
theorem mem_of_max {s : finset α} : ∀ {a : α}, a ∈ s.max → a ∈ s :=
finset.induction_on s (λ _ H, by cases H)
(λ b s _ (ih : ∀ {a}, a ∈ s.max → a ∈ s) a (h : a ∈ (insert b s).max),
begin
by_cases p : b = a,
{ induction p, exact mem_insert_self b s },
{ cases option.lift_or_get_choice max_choice (some b) s.max with q q;
rw [max_insert, q] at h,
{ cases h, cases p rfl },
{ exact mem_insert_of_mem (ih h) } }
end)
theorem le_max_of_mem {s : finset α} {a b : α} (h₁ : a ∈ s) (h₂ : b ∈ s.max) : a ≤ b :=
by rcases @le_sup (with_bot α) _ _ _ _ _ h₁ _ rfl with ⟨b', hb, ab⟩;
cases h₂.symm.trans hb; assumption
/-- Let `s` be a finset in a linear order. Then `s.min` is the minimum of `s` if `s` is not empty,
and `none` otherwise. It belongs to `option α`. If you want to get an element of `α`, see
`s.min'`. -/
protected def min : finset α → option α :=
fold (option.lift_or_get min) none some
theorem min_eq_inf_with_top (s : finset α) :
s.min = @inf (with_top α) α _ s some := rfl
@[simp] theorem min_empty : (∅ : finset α).min = none := rfl
@[simp] theorem min_insert {a : α} {s : finset α} :
(insert a s).min = option.lift_or_get min (some a) s.min :=
fold_insert_idem
@[simp] theorem min_singleton {a : α} : finset.min {a} = some a :=
by { rw ← insert_emptyc_eq, exact min_insert }
theorem min_of_mem {s : finset α} {a : α} (h : a ∈ s) : ∃ b, b ∈ s.min :=
(@inf_le (with_top α) _ _ _ _ _ h _ rfl).imp $ λ b, Exists.fst
theorem min_of_nonempty {s : finset α} (h : s.nonempty) : ∃ a, a ∈ s.min :=
let ⟨a, ha⟩ := h in min_of_mem ha
theorem min_eq_none {s : finset α} : s.min = none ↔ s = ∅ :=
⟨λ h, s.eq_empty_or_nonempty.elim id
(λ H, let ⟨a, ha⟩ := min_of_nonempty H in by rw h at ha; cases ha),
λ h, h.symm ▸ min_empty⟩
theorem mem_of_min {s : finset α} : ∀ {a : α}, a ∈ s.min → a ∈ s :=
@mem_of_max (order_dual α) _ s
theorem min_le_of_mem {s : finset α} {a b : α} (h₁ : b ∈ s) (h₂ : a ∈ s.min) : a ≤ b :=
by rcases @inf_le (with_top α) _ _ _ _ _ h₁ _ rfl with ⟨b', hb, ab⟩;
cases h₂.symm.trans hb; assumption
/-- Given a nonempty finset `s` in a linear order `α `, then `s.min' h` is its minimum, as an
element of `α`, where `h` is a proof of nonemptiness. Without this assumption, use instead `s.min`,
taking values in `option α`. -/
def min' (s : finset α) (H : s.nonempty) : α :=
@option.get _ s.min $
let ⟨k, hk⟩ := H in
let ⟨b, hb⟩ := min_of_mem hk in by simp at hb; simp [hb]
/-- Given a nonempty finset `s` in a linear order `α `, then `s.max' h` is its maximum, as an
element of `α`, where `h` is a proof of nonemptiness. Without this assumption, use instead `s.max`,
taking values in `option α`. -/
def max' (s : finset α) (H : s.nonempty) : α :=
@option.get _ s.max $
let ⟨k, hk⟩ := H in
let ⟨b, hb⟩ := max_of_mem hk in by simp at hb; simp [hb]
variables (s : finset α) (H : s.nonempty)
theorem min'_mem : s.min' H ∈ s := mem_of_min $ by simp [min']
theorem min'_le (x) (H2 : x ∈ s) : s.min' ⟨x, H2⟩ ≤ x := min_le_of_mem H2 $ option.get_mem _
theorem le_min' (x) (H2 : ∀ y ∈ s, x ≤ y) : x ≤ s.min' H := H2 _ $ min'_mem _ _
theorem is_least_min' : is_least ↑s (s.min' H) := ⟨min'_mem _ _, min'_le _⟩
@[simp] lemma le_min'_iff {x} : x ≤ s.min' H ↔ ∀ y ∈ s, x ≤ y :=
le_is_glb_iff (is_least_min' s H).is_glb
/-- `{a}.min' _` is `a`. -/
@[simp] lemma min'_singleton (a : α) :
({a} : finset α).min' (singleton_nonempty _) = a :=
by simp [min']
theorem max'_mem : s.max' H ∈ s := mem_of_max $ by simp [max']
theorem le_max' (x) (H2 : x ∈ s) : x ≤ s.max' ⟨x, H2⟩ := le_max_of_mem H2 $ option.get_mem _
theorem max'_le (x) (H2 : ∀ y ∈ s, y ≤ x) : s.max' H ≤ x := H2 _ $ max'_mem _ _
theorem is_greatest_max' : is_greatest ↑s (s.max' H) := ⟨max'_mem _ _, le_max' _⟩
@[simp] lemma max'_le_iff {x} : s.max' H ≤ x ↔ ∀ y ∈ s, y ≤ x :=
is_lub_le_iff (is_greatest_max' s H).is_lub
@[simp] lemma max'_lt_iff {x} : s.max' H < x ↔ ∀ y ∈ s, y < x :=
⟨λ Hlt y hy, (s.le_max' y hy).trans_lt Hlt, λ H, H _ $ s.max'_mem _⟩
@[simp] lemma lt_min'_iff {x} : x < s.min' H ↔ ∀ y ∈ s, x < y :=
@max'_lt_iff (order_dual α) _ _ H _
lemma max'_eq_sup' : s.max' H = s.sup' H id :=
eq_of_forall_ge_iff $ λ a, (max'_le_iff _ _).trans (sup'_le_iff _ _).symm
lemma min'_eq_inf' : s.min' H = s.inf' H id :=
@max'_eq_sup' (order_dual α) _ s H
/-- `{a}.max' _` is `a`. -/
@[simp] lemma max'_singleton (a : α) :
({a} : finset α).max' (singleton_nonempty _) = a :=
by simp [max']
theorem min'_lt_max' {i j} (H1 : i ∈ s) (H2 : j ∈ s) (H3 : i ≠ j) :
s.min' ⟨i, H1⟩ < s.max' ⟨i, H1⟩ :=
is_glb_lt_is_lub_of_ne (s.is_least_min' _).is_glb (s.is_greatest_max' _).is_lub H1 H2 H3
/--
If there's more than 1 element, the min' is less than the max'. An alternate version of
`min'_lt_max'` which is sometimes more convenient.
-/
lemma min'_lt_max'_of_card (h₂ : 1 < card s) :
s.min' (finset.card_pos.mp $ lt_trans zero_lt_one h₂) <
s.max' (finset.card_pos.mp $ lt_trans zero_lt_one h₂) :=
begin
rcases one_lt_card.1 h₂ with ⟨a, ha, b, hb, hab⟩,
exact s.min'_lt_max' ha hb hab
end
lemma max'_eq_of_dual_min' {s : finset α} (hs : s.nonempty) :
max' s hs = of_dual (min' (image to_dual s) (nonempty.image hs to_dual)) :=
begin
rw [of_dual, to_dual, equiv.coe_fn_mk, equiv.coe_fn_symm_mk, id.def],
simp_rw (@image_id (order_dual α) (s : finset (order_dual α))),
refl,
end
lemma min'_eq_of_dual_max' {s : finset α} (hs : s.nonempty) :
min' s hs = of_dual (max' (image to_dual s) (nonempty.image hs to_dual)) :=
begin
rw [of_dual, to_dual, equiv.coe_fn_mk, equiv.coe_fn_symm_mk, id.def],
simp_rw (@image_id (order_dual α) (s : finset (order_dual α))),
refl,
end
@[simp] lemma of_dual_max_eq_min_of_dual {a b : α} :
of_dual (max a b) = min (of_dual a) (of_dual b) := rfl
@[simp] lemma of_dual_min_eq_max_of_dual {a b : α} :
of_dual (min a b) = max (of_dual a) (of_dual b) := rfl
lemma max'_subset {s t : finset α} (H : s.nonempty) (hst : s ⊆ t) :
s.max' H ≤ t.max' (H.mono hst) :=
le_max' _ _ (hst (s.max'_mem H))
lemma min'_subset {s t : finset α} (H : s.nonempty) (hst : s ⊆ t) :
t.min' (H.mono hst) ≤ s.min' H :=
min'_le _ _ (hst (s.min'_mem H))
lemma max'_insert (a : α) (s : finset α) (H : s.nonempty) :
(insert a s).max' (s.insert_nonempty a) = max (s.max' H) a :=
(is_greatest_max' _ _).unique $
by { rw [coe_insert, max_comm], exact (is_greatest_max' _ _).insert _ }
lemma min'_insert (a : α) (s : finset α) (H : s.nonempty) :
(insert a s).min' (s.insert_nonempty a) = min (s.min' H) a :=
(is_least_min' _ _).unique $
by { rw [coe_insert, min_comm], exact (is_least_min' _ _).insert _ }
lemma lt_max'_of_mem_erase_max' [decidable_eq α] {a : α} (ha : a ∈ s.erase (s.max' H)) :
a < s.max' H :=
lt_of_le_of_ne (le_max' _ _ (mem_of_mem_erase ha)) $ ne_of_mem_of_not_mem ha $ not_mem_erase _ _
lemma min'_lt_of_mem_erase_min' [decidable_eq α] {a : α} (ha : a ∈ s.erase (s.min' H)) :
s.min' H < a :=
@lt_max'_of_mem_erase_max' (order_dual α) _ s H _ a ha
@[simp] lemma max'_image [linear_order β]
{f : α → β} (hf : monotone f) (s : finset α) (h : (s.image f).nonempty) :
(s.image f).max' h = f (s.max' ((nonempty.image_iff f).mp h)) :=
begin
refine le_antisymm (max'_le _ _ _ (λ y hy, _))
(le_max' _ _ (mem_image.mpr ⟨_, max'_mem _ _, rfl⟩)),
obtain ⟨x, hx, rfl⟩ := mem_image.mp hy,
exact hf (le_max' _ _ hx)
end
@[simp] lemma min'_image [linear_order β]
{f : α → β} (hf : monotone f) (s : finset α) (h : (s.image f).nonempty) :
(s.image f).min' h = f (s.min' ((nonempty.image_iff f).mp h)) :=
begin
refine le_antisymm (min'_le _ _ (mem_image.mpr ⟨_, min'_mem _ _, rfl⟩))
(le_min' _ _ _ (λ y hy, _)),
obtain ⟨x, hx, rfl⟩ := mem_image.mp hy,
exact hf (min'_le _ _ hx)
end
/-- Induction principle for `finset`s in a linearly ordered type: a predicate is true on all
`s : finset α` provided that:
* it is true on the empty `finset`,
* for every `s : finset α` and an element `a` strictly greater than all elements of `s`, `p s`
implies `p (insert a s)`. -/
@[elab_as_eliminator]
lemma induction_on_max [decidable_eq α] {p : finset α → Prop} (s : finset α) (h0 : p ∅)
(step : ∀ a s, (∀ x ∈ s, x < a) → p s → p (insert a s)) : p s :=
begin
induction s using finset.strong_induction_on with s ihs,
rcases s.eq_empty_or_nonempty with rfl|hne,
{ exact h0 },
{ have H : s.max' hne ∈ s, from max'_mem s hne,
rw ← insert_erase H,
exact step _ _ (λ x, s.lt_max'_of_mem_erase_max' hne) (ihs _ $ erase_ssubset H) }
end
/-- Induction principle for `finset`s in a linearly ordered type: a predicate is true on all
`s : finset α` provided that:
* it is true on the empty `finset`,
* for every `s : finset α` and an element `a` strictly less than all elements of `s`, `p s`
implies `p (insert a s)`. -/
@[elab_as_eliminator]
lemma induction_on_min [decidable_eq α] {p : finset α → Prop} (s : finset α) (h0 : p ∅)
(step : ∀ a s, (∀ x ∈ s, a < x) → p s → p (insert a s)) : p s :=
@induction_on_max (order_dual α) _ _ _ s h0 step
end max_min
section exists_max_min
variables [linear_order α]
lemma exists_max_image (s : finset β) (f : β → α) (h : s.nonempty) :
∃ x ∈ s, ∀ x' ∈ s, f x' ≤ f x :=
begin
cases max_of_nonempty (h.image f) with y hy,
rcases mem_image.mp (mem_of_max hy) with ⟨x, hx, rfl⟩,
exact ⟨x, hx, λ x' hx', le_max_of_mem (mem_image_of_mem f hx') hy⟩,
end
lemma exists_min_image (s : finset β) (f : β → α) (h : s.nonempty) :
∃ x ∈ s, ∀ x' ∈ s, f x ≤ f x' :=
@exists_max_image (order_dual α) β _ s f h
end exists_max_min
end finset
namespace multiset
lemma count_sup [decidable_eq β] (s : finset α) (f : α → multiset β) (b : β) :
count b (s.sup f) = s.sup (λa, count b (f a)) :=
begin
letI := classical.dec_eq α,
refine s.induction _ _,
{ exact count_zero _ },
{ assume i s his ih,
rw [finset.sup_insert, sup_eq_union, count_union, finset.sup_insert, ih],
refl }
end
lemma mem_sup {α β} [decidable_eq β] {s : finset α} {f : α → multiset β}
{x : β} : x ∈ s.sup f ↔ ∃ v ∈ s, x ∈ f v :=
begin
classical,
apply s.induction_on,
{ simp },
{ intros a s has hxs,
rw [finset.sup_insert, multiset.sup_eq_union, multiset.mem_union],
split,
{ intro hxi,
cases hxi with hf hf,
{ refine ⟨a, _, hf⟩,
simp only [true_or, eq_self_iff_true, finset.mem_insert] },
{ rcases hxs.mp hf with ⟨v, hv, hfv⟩,
refine ⟨v, _, hfv⟩,
simp only [hv, or_true, finset.mem_insert] } },
{ rintros ⟨v, hv, hfv⟩,
rw [finset.mem_insert] at hv,
rcases hv with rfl | hv,
{ exact or.inl hfv },
{ refine or.inr (hxs.mpr ⟨v, hv, hfv⟩) } } },
end
end multiset
namespace finset
lemma mem_sup {α β} [decidable_eq β] {s : finset α} {f : α → finset β}
{x : β} : x ∈ s.sup f ↔ ∃ v ∈ s, x ∈ f v :=
begin
change _ ↔ ∃ v ∈ s, x ∈ (f v).val,
rw [←multiset.mem_sup, ←multiset.mem_to_finset, sup_to_finset],
simp_rw [val_to_finset],
end
lemma sup_eq_bUnion {α β} [decidable_eq β] (s : finset α) (t : α → finset β) :
s.sup t = s.bUnion t :=
by { ext, rw [mem_sup, mem_bUnion], }
end finset
section lattice
variables {ι : Type*} {ι' : Sort*} [complete_lattice α]
/-- Supremum of `s i`, `i : ι`, is equal to the supremum over `t : finset ι` of suprema
`⨆ i ∈ t, s i`. This version assumes `ι` is a `Type*`. See `supr_eq_supr_finset'` for a version
that works for `ι : Sort*`. -/
lemma supr_eq_supr_finset (s : ι → α) :
(⨆i, s i) = (⨆t:finset ι, ⨆i∈t, s i) :=
begin
classical,
exact le_antisymm
(supr_le $ assume b, le_supr_of_le {b} $ le_supr_of_le b $ le_supr_of_le
(by simp) $ le_refl _)
(supr_le $ assume t, supr_le $ assume b, supr_le $ assume hb, le_supr _ _)
end
/-- Supremum of `s i`, `i : ι`, is equal to the supremum over `t : finset ι` of suprema
`⨆ i ∈ t, s i`. This version works for `ι : Sort*`. See `supr_eq_supr_finset` for a version
that assumes `ι : Type*` but has no `plift`s. -/
lemma supr_eq_supr_finset' (s : ι' → α) :
(⨆i, s i) = (⨆t:finset (plift ι'), ⨆i∈t, s (plift.down i)) :=
by rw [← supr_eq_supr_finset, ← equiv.plift.surjective.supr_comp]; refl
/-- Infimum of `s i`, `i : ι`, is equal to the infimum over `t : finset ι` of infima
`⨆ i ∈ t, s i`. This version assumes `ι` is a `Type*`. See `infi_eq_infi_finset'` for a version
that works for `ι : Sort*`. -/
lemma infi_eq_infi_finset (s : ι → α) :
(⨅i, s i) = (⨅t:finset ι, ⨅i∈t, s i) :=
@supr_eq_supr_finset (order_dual α) _ _ _
/-- Infimum of `s i`, `i : ι`, is equal to the infimum over `t : finset ι` of infima
`⨆ i ∈ t, s i`. This version works for `ι : Sort*`. See `infi_eq_infi_finset` for a version
that assumes `ι : Type*` but has no `plift`s. -/
lemma infi_eq_infi_finset' (s : ι' → α) :
(⨅i, s i) = (⨅t:finset (plift ι'), ⨅i∈t, s (plift.down i)) :=
@supr_eq_supr_finset' (order_dual α) _ _ _
end lattice
namespace set
variables {ι : Type*} {ι' : Sort*}
/-- Union of an indexed family of sets `s : ι → set α` is equal to the union of the unions
of finite subfamilies. This version assumes `ι : Type*`. See also `Union_eq_Union_finset'` for
a version that works for `ι : Sort*`. -/
lemma Union_eq_Union_finset (s : ι → set α) :
(⋃i, s i) = (⋃t:finset ι, ⋃i∈t, s i) :=
supr_eq_supr_finset s
/-- Union of an indexed family of sets `s : ι → set α` is equal to the union of the unions
of finite subfamilies. This version works for `ι : Sort*`. See also `Union_eq_Union_finset` for
a version that assumes `ι : Type*` but avoids `plift`s in the right hand side. -/
lemma Union_eq_Union_finset' (s : ι' → set α) :
(⋃i, s i) = (⋃t:finset (plift ι'), ⋃i∈t, s (plift.down i)) :=
supr_eq_supr_finset' s
/-- Intersection of an indexed family of sets `s : ι → set α` is equal to the intersection of the
intersections of finite subfamilies. This version assumes `ι : Type*`. See also
`Inter_eq_Inter_finset'` for a version that works for `ι : Sort*`. -/
lemma Inter_eq_Inter_finset (s : ι → set α) :
(⋂i, s i) = (⋂t:finset ι, ⋂i∈t, s i) :=
infi_eq_infi_finset s
/-- Intersection of an indexed family of sets `s : ι → set α` is equal to the intersection of the
intersections of finite subfamilies. This version works for `ι : Sort*`. See also
`Inter_eq_Inter_finset` for a version that assumes `ι : Type*` but avoids `plift`s in the right
hand side. -/
lemma Inter_eq_Inter_finset' (s : ι' → set α) :
(⋂i, s i) = (⋂t:finset (plift ι'), ⋂i∈t, s (plift.down i)) :=
infi_eq_infi_finset' s
end set
namespace finset
open function
/-! ### Interaction with big lattice/set operations -/
section lattice
lemma supr_coe [has_Sup β] (f : α → β) (s : finset α) :
(⨆ x ∈ (↑s : set α), f x) = ⨆ x ∈ s, f x :=
rfl
lemma infi_coe [has_Inf β] (f : α → β) (s : finset α) :
(⨅ x ∈ (↑s : set α), f x) = ⨅ x ∈ s, f x :=
rfl
variables [complete_lattice β]
theorem supr_singleton (a : α) (s : α → β) : (⨆ x ∈ ({a} : finset α), s x) = s a :=
by simp
theorem infi_singleton (a : α) (s : α → β) : (⨅ x ∈ ({a} : finset α), s x) = s a :=
by simp
lemma supr_option_to_finset (o : option α) (f : α → β) :
(⨆ x ∈ o.to_finset, f x) = ⨆ x ∈ o, f x :=
by simp
lemma infi_option_to_finset (o : option α) (f : α → β) :
(⨅ x ∈ o.to_finset, f x) = ⨅ x ∈ o, f x :=
@supr_option_to_finset _ (order_dual β) _ _ _
variables [decidable_eq α]
theorem supr_union {f : α → β} {s t : finset α} :
(⨆ x ∈ s ∪ t, f x) = (⨆x∈s, f x) ⊔ (⨆x∈t, f x) :=
by simp [supr_or, supr_sup_eq]
theorem infi_union {f : α → β} {s t : finset α} :
(⨅ x ∈ s ∪ t, f x) = (⨅ x ∈ s, f x) ⊓ (⨅ x ∈ t, f x) :=
@supr_union α (order_dual β) _ _ _ _ _
lemma supr_insert (a : α) (s : finset α) (t : α → β) :
(⨆ x ∈ insert a s, t x) = t a ⊔ (⨆ x ∈ s, t x) :=
by { rw insert_eq, simp only [supr_union, finset.supr_singleton] }
lemma infi_insert (a : α) (s : finset α) (t : α → β) :
(⨅ x ∈ insert a s, t x) = t a ⊓ (⨅ x ∈ s, t x) :=
@supr_insert α (order_dual β) _ _ _ _ _
lemma supr_finset_image {f : γ → α} {g : α → β} {s : finset γ} :
(⨆ x ∈ s.image f, g x) = (⨆ y ∈ s, g (f y)) :=
by rw [← supr_coe, coe_image, supr_image, supr_coe]
lemma sup_finset_image {β γ : Type*} [semilattice_sup_bot β]
(f : γ → α) (g : α → β) (s : finset γ) :
(s.image f).sup g = s.sup (g ∘ f) :=
begin
classical,
induction s using finset.induction_on with a s' ha ih; simp *
end
lemma infi_finset_image {f : γ → α} {g : α → β} {s : finset γ} :
(⨅ x ∈ s.image f, g x) = (⨅ y ∈ s, g (f y)) :=
by rw [← infi_coe, coe_image, infi_image, infi_coe]
lemma supr_insert_update {x : α} {t : finset α} (f : α → β) {s : β} (hx : x ∉ t) :
(⨆ (i ∈ insert x t), function.update f x s i) = (s ⊔ ⨆ (i ∈ t), f i) :=
begin
simp only [finset.supr_insert, update_same],
rcongr i hi, apply update_noteq, rintro rfl, exact hx hi
end
lemma infi_insert_update {x : α} {t : finset α} (f : α → β) {s : β} (hx : x ∉ t) :
(⨅ (i ∈ insert x t), update f x s i) = (s ⊓ ⨅ (i ∈ t), f i) :=
@supr_insert_update α (order_dual β) _ _ _ _ f _ hx
lemma supr_bUnion (s : finset γ) (t : γ → finset α) (f : α → β) :
(⨆ y ∈ s.bUnion t, f y) = ⨆ (x ∈ s) (y ∈ t x), f y :=
by simp [@supr_comm _ α, supr_and]
lemma infi_bUnion (s : finset γ) (t : γ → finset α) (f : α → β) :
(⨅ y ∈ s.bUnion t, f y) = ⨅ (x ∈ s) (y ∈ t x), f y :=
@supr_bUnion _ (order_dual β) _ _ _ _ _ _
end lattice
theorem set_bUnion_coe (s : finset α) (t : α → set β) :
(⋃ x ∈ (↑s : set α), t x) = ⋃ x ∈ s, t x :=
rfl
theorem set_bInter_coe (s : finset α) (t : α → set β) :
(⋂ x ∈ (↑s : set α), t x) = ⋂ x ∈ s, t x :=
rfl
theorem set_bUnion_singleton (a : α) (s : α → set β) :
(⋃ x ∈ ({a} : finset α), s x) = s a :=
supr_singleton a s
theorem set_bInter_singleton (a : α) (s : α → set β) :
(⋂ x ∈ ({a} : finset α), s x) = s a :=
infi_singleton a s
@[simp] lemma set_bUnion_preimage_singleton (f : α → β) (s : finset β) :
(⋃ y ∈ s, f ⁻¹' {y}) = f ⁻¹' s :=
set.bUnion_preimage_singleton f s
lemma set_bUnion_option_to_finset (o : option α) (f : α → set β) :
(⋃ x ∈ o.to_finset, f x) = ⋃ x ∈ o, f x :=
supr_option_to_finset o f
lemma set_bInter_option_to_finset (o : option α) (f : α → set β) :
(⋂ x ∈ o.to_finset, f x) = ⋂ x ∈ o, f x :=
infi_option_to_finset o f
variables [decidable_eq α]
lemma set_bUnion_union (s t : finset α) (u : α → set β) :
(⋃ x ∈ s ∪ t, u x) = (⋃ x ∈ s, u x) ∪ (⋃ x ∈ t, u x) :=
supr_union
lemma set_bInter_inter (s t : finset α) (u : α → set β) :
(⋂ x ∈ s ∪ t, u x) = (⋂ x ∈ s, u x) ∩ (⋂ x ∈ t, u x) :=
infi_union
lemma set_bUnion_insert (a : α) (s : finset α) (t : α → set β) :
(⋃ x ∈ insert a s, t x) = t a ∪ (⋃ x ∈ s, t x) :=
supr_insert a s t
lemma set_bInter_insert (a : α) (s : finset α) (t : α → set β) :
(⋂ x ∈ insert a s, t x) = t a ∩ (⋂ x ∈ s, t x) :=
infi_insert a s t
lemma set_bUnion_finset_image {f : γ → α} {g : α → set β} {s : finset γ} :
(⋃x ∈ s.image f, g x) = (⋃y ∈ s, g (f y)) :=
supr_finset_image
lemma set_bInter_finset_image {f : γ → α} {g : α → set β} {s : finset γ} :
(⋂ x ∈ s.image f, g x) = (⋂ y ∈ s, g (f y)) :=
infi_finset_image
lemma set_bUnion_insert_update {x : α} {t : finset α} (f : α → set β) {s : set β} (hx : x ∉ t) :
(⋃ (i ∈ insert x t), @update _ _ _ f x s i) = (s ∪ ⋃ (i ∈ t), f i) :=
supr_insert_update f hx
lemma set_bInter_insert_update {x : α} {t : finset α} (f : α → set β) {s : set β} (hx : x ∉ t) :
(⋂ (i ∈ insert x t), @update _ _ _ f x s i) = (s ∩ ⋂ (i ∈ t), f i) :=
infi_insert_update f hx
lemma set_bUnion_bUnion (s : finset γ) (t : γ → finset α) (f : α → set β) :
(⋃ y ∈ s.bUnion t, f y) = ⋃ (x ∈ s) (y ∈ t x), f y :=
supr_bUnion s t f
lemma set_bInter_bUnion (s : finset γ) (t : γ → finset α) (f : α → set β) :
(⋂ y ∈ s.bUnion t, f y) = ⋂ (x ∈ s) (y ∈ t x), f y :=
infi_bUnion s t f
end finset
|
b08202b8fa188f998cea5b9a827be98dbad3b8f3 | 271e26e338b0c14544a889c31c30b39c989f2e0f | /src/Init/Lean/Compiler/IR/Checker.lean | 6d5985b82b9b858aa469c6792da6af53c9e48d0f | [
"Apache-2.0"
] | permissive | dgorokho/lean4 | 805f99b0b60c545b64ac34ab8237a8504f89d7d4 | e949a052bad59b1c7b54a82d24d516a656487d8a | refs/heads/master | 1,607,061,363,851 | 1,578,006,086,000 | 1,578,006,086,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 6,389 | lean | /-
Copyright (c) 2019 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.Lean.Compiler.IR.CompilerM
namespace Lean
namespace IR
namespace Checker
structure CheckerContext :=
(env : Environment) (localCtx : LocalContext := {}) (decls : Array Decl)
structure CheckerState :=
(foundVars : IndexSet := {})
abbrev M := ReaderT CheckerContext (ExceptT String (StateT CheckerState Id))
def markIndex (i : Index) : M Unit := do
s ← get;
when (s.foundVars.contains i) $ throw ("variable / joinpoint index " ++ toString i ++ " has already been used");
modify $ fun s => { foundVars := s.foundVars.insert i, .. s }
def markVar (x : VarId) : M Unit :=
markIndex x.idx
def markJP (j : JoinPointId) : M Unit :=
markIndex j.idx
def getDecl (c : Name) : M Decl := do
ctx ← read;
match findEnvDecl' ctx.env c ctx.decls with
| none => throw ("unknown declaration '" ++ toString c ++ "'")
| some d => pure d
def checkVar (x : VarId) : M Unit := do
ctx ← read;
unless (ctx.localCtx.isLocalVar x.idx || ctx.localCtx.isParam x.idx) $ throw ("unknown variable '" ++ toString x ++ "'")
def checkJP (j : JoinPointId) : M Unit := do
ctx ← read;
unless (ctx.localCtx.isJP j.idx) $ throw ("unknown join point '" ++ toString j ++ "'")
def checkArg (a : Arg) : M Unit :=
match a with
| Arg.var x => checkVar x
| other => pure ()
def checkArgs (as : Array Arg) : M Unit :=
as.forM checkArg
@[inline] def checkEqTypes (ty₁ ty₂ : IRType) : M Unit :=
unless (ty₁ == ty₂) $ throw ("unexpected type")
@[inline] def checkType (ty : IRType) (p : IRType → Bool) : M Unit :=
unless (p ty) $ throw ("unexpected type")
def checkObjType (ty : IRType) : M Unit := checkType ty IRType.isObj
def checkScalarType (ty : IRType) : M Unit := checkType ty IRType.isScalar
def getType (x : VarId) : M IRType := do
ctx ← read;
match ctx.localCtx.getType x with
| some ty => pure ty
| none => throw ("unknown variable '" ++ toString x ++ "'")
@[inline] def checkVarType (x : VarId) (p : IRType → Bool) : M Unit := do
ty ← getType x; checkType ty p
def checkObjVar (x : VarId) : M Unit :=
checkVarType x IRType.isObj
def checkScalarVar (x : VarId) : M Unit :=
checkVarType x IRType.isScalar
def checkFullApp (c : FunId) (ys : Array Arg) : M Unit := do
decl ← getDecl c;
unless (ys.size == decl.params.size) $
throw ("incorrect number of arguments to '" ++ toString c ++ "', " ++ toString ys.size ++ " provided, " ++ toString decl.params.size ++ " expected");
checkArgs ys
def checkPartialApp (c : FunId) (ys : Array Arg) : M Unit := do
decl ← getDecl c;
unless (ys.size < decl.params.size) $
throw ("too many arguments too partial application '" ++ toString c ++ "', num. args: " ++ toString ys.size ++ ", arity: " ++ toString decl.params.size);
checkArgs ys
def checkExpr (ty : IRType) : Expr → M Unit
| Expr.pap f ys => checkPartialApp f ys *> checkObjType ty -- partial applications should always produce a closure object
| Expr.ap x ys => checkObjVar x *> checkArgs ys
| Expr.fap f ys => checkFullApp f ys
| Expr.ctor c ys => when (!ty.isStruct && !ty.isUnion && c.isRef) (checkObjType ty) *> checkArgs ys
| Expr.reset _ x => checkObjVar x *> checkObjType ty
| Expr.reuse x i u ys => checkObjVar x *> checkArgs ys *> checkObjType ty
| Expr.box xty x => checkObjType ty *> checkScalarVar x *> checkVarType x (fun t => t == xty)
| Expr.unbox x => checkScalarType ty *> checkObjVar x
| Expr.proj i x => do xType ← getType x;
match xType with
| IRType.object => checkObjType ty
| IRType.tobject => checkObjType ty
| IRType.struct _ tys => if h : i < tys.size then checkEqTypes (tys.get ⟨i,h⟩) ty else throw "invalid proj index"
| IRType.union _ tys => if h : i < tys.size then checkEqTypes (tys.get ⟨i,h⟩) ty else throw "invalid proj index"
| other => throw "unexpected type"
| Expr.uproj _ x => checkObjVar x *> checkType ty (fun t => t == IRType.usize)
| Expr.sproj _ _ x => checkObjVar x *> checkScalarType ty
| Expr.isShared x => checkObjVar x *> checkType ty (fun t => t == IRType.uint8)
| Expr.isTaggedPtr x => checkObjVar x *> checkType ty (fun t => t == IRType.uint8)
| Expr.lit (LitVal.str _) => checkObjType ty
| Expr.lit _ => pure ()
@[inline] def withParams (ps : Array Param) (k : M Unit) : M Unit := do
ctx ← read;
localCtx ← ps.foldlM (fun (ctx : LocalContext) p => do
markVar p.x;
pure $ ctx.addParam p) ctx.localCtx;
adaptReader (fun _ => { localCtx := localCtx, .. ctx }) k
partial def checkFnBody : FnBody → M Unit
| FnBody.vdecl x t v b => do
checkExpr t v;
markVar x;
ctx ← read;
adaptReader (fun (ctx : CheckerContext) => { localCtx := ctx.localCtx.addLocal x t v, .. ctx }) (checkFnBody b)
| FnBody.jdecl j ys v b => do
markJP j;
withParams ys (checkFnBody v);
ctx ← read;
adaptReader (fun (ctx : CheckerContext) => { localCtx := ctx.localCtx.addJP j ys v, .. ctx }) (checkFnBody b)
| FnBody.set x _ y b => checkVar x *> checkArg y *> checkFnBody b
| FnBody.uset x _ y b => checkVar x *> checkVar y *> checkFnBody b
| FnBody.sset x _ _ y _ b => checkVar x *> checkVar y *> checkFnBody b
| FnBody.setTag x _ b => checkVar x *> checkFnBody b
| FnBody.inc x _ _ _ b => checkVar x *> checkFnBody b
| FnBody.dec x _ _ _ b => checkVar x *> checkFnBody b
| FnBody.del x b => checkVar x *> checkFnBody b
| FnBody.mdata _ b => checkFnBody b
| FnBody.jmp j ys => checkJP j *> checkArgs ys
| FnBody.ret x => checkArg x
| FnBody.case _ x _ alts => checkVar x *> alts.forM (fun alt => checkFnBody alt.body)
| FnBody.unreachable => pure ()
def checkDecl : Decl → M Unit
| Decl.fdecl f xs t b => withParams xs (checkFnBody b)
| Decl.extern f xs t _ => withParams xs (pure ())
end Checker
def checkDecl (decls : Array Decl) (decl : Decl) : CompilerM Unit := do
env ← getEnv;
match (Checker.checkDecl decl { env := env, decls := decls }).run' {} with
| Except.error msg => throw ("IR check failed at '" ++ toString decl.name ++ "', error: " ++ msg)
| other => pure ()
def checkDecls (decls : Array Decl) : CompilerM Unit :=
decls.forM (checkDecl decls)
end IR
end Lean
|
b7e99a636d2026a757cecc3c6c57baa5f581c7e7 | 1b8f093752ba748c5ca0083afef2959aaa7dace5 | /src/category_theory/currying.lean | 47a1d766194d0e03e327b256fd4280c7e23017cd | [] | no_license | khoek/lean-category-theory | 7ec4cda9cc64a5a4ffeb84712ac7d020dbbba386 | 63dcb598e9270a3e8b56d1769eb4f825a177cd95 | refs/heads/master | 1,585,251,725,759 | 1,539,344,445,000 | 1,539,344,445,000 | 145,281,070 | 0 | 0 | null | 1,534,662,376,000 | 1,534,662,376,000 | null | UTF-8 | Lean | false | false | 1,901 | lean | -- Copyright (c) 2017 Scott Morrison. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Scott Morrison
import category_theory.products.bifunctors
import category_theory.equivalence
-- FIXME why do we need this here?
@[obviously] meta def obviously_2 := tactic.tidy { tactics := extended_tidy_tactics }
namespace category_theory
universes u₁ v₁ u₂ v₂ u₃ v₃
variables {C : Type u₁} [𝒞 : category.{u₁ v₁} C] {D : Type u₂} [𝒟 : category.{u₂ v₂} D] {E : Type u₃} [ℰ : category.{u₃ v₃} E]
include 𝒞 𝒟 ℰ
def uncurry : (C ⥤ (D ⥤ E)) ⥤ ((C × D) ⥤ E) :=
{ obj := λ F, { obj := λ X, (F X.1) X.2,
map' := λ X Y f, ((F.map f.1) X.2) ≫ ((F Y.1).map f.2) },
map' := λ F G T, { app := λ X, (T X.1) X.2 } }.
def curry : ((C × D) ⥤ E) ⥤ (C ⥤ (D ⥤ E)) :=
{ obj := λ F, { obj := λ X, { obj := λ Y, F (X, Y),
map' := λ Y Y' g, F.map (𝟙 X, g) },
map' := λ X X' f, { app := λ Y, F.map (f, 𝟙 Y) } },
map' := λ F G T, { app := λ X, { app := λ Y, T (X, Y) } } }.
-- We need the @s here in order for the coercions to work. :-(
@[simp] lemma uncurry.obj_map {F : C ⥤ (D ⥤ E)} {X Y : C × D} {f : X ⟶ Y} : (@uncurry C _ D _ E _ F).map f = ((F.map f.1) X.2) ≫ ((F Y.1).map f.2) := rfl
@[simp] lemma curry.obj_obj_map {F : (C × D) ⥤ E} {X : C} {Y Y' : D} {g : Y ⟶ Y'} : ((@curry C _ D _ E _ F) X).map g = F.map (𝟙 X, g) := rfl
@[simp] lemma curry.obj_map_app {F : (C × D) ⥤ E} {X X' : C} {f : X ⟶ X'} {Y} : ((@curry C _ D _ E _ F).map f) Y = F.map (f, 𝟙 Y) := rfl
local attribute [back] category.id -- this is usually a bad idea, but just what we needed here
def currying : equivalence (C ⥤ (D ⥤ E)) ((C × D) ⥤ E) :=
{ functor := uncurry,
inverse := curry }
end category_theory |
350f9124fc4df5e02da6800e18550a2a9b5d40af | a0e23cfdd129a671bf3154ee1a8a3a72bf4c7940 | /src/Lean/Elab/PatternVar.lean | 555c1b8f135ff70a591ea8586679b99272909025 | [
"Apache-2.0"
] | permissive | WojciechKarpiel/lean4 | 7f89706b8e3c1f942b83a2c91a3a00b05da0e65b | f6e1314fa08293dea66a329e05b6c196a0189163 | refs/heads/master | 1,686,633,402,214 | 1,625,821,189,000 | 1,625,821,258,000 | 384,640,886 | 0 | 0 | Apache-2.0 | 1,625,903,617,000 | 1,625,903,026,000 | null | UTF-8 | Lean | false | false | 13,848 | lean | /-
Copyright (c) 2021 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Meta.Match.MatchPatternAttr
import Lean.Elab.Arg
import Lean.Elab.MatchAltView
namespace Lean.Elab.Term
open Meta
inductive PatternVar where
| localVar (userName : Name)
-- anonymous variables (`_`) are encoded using metavariables
| anonymousVar (mvarId : MVarId)
instance : ToString PatternVar := ⟨fun
| PatternVar.localVar x => toString x
| PatternVar.anonymousVar mvarId => s!"?m{mvarId}"⟩
/--
Create an auxiliary Syntax node wrapping a fresh metavariable id.
We use this kind of Syntax for representing `_` occurring in patterns.
The metavariables are created before we elaborate the patterns into `Expr`s. -/
private def mkMVarSyntax : TermElabM Syntax := do
let mvarId ← mkFreshId
return Syntax.node `MVarWithIdKind #[Syntax.node mvarId #[]]
/-- Given a syntax node constructed using `mkMVarSyntax`, return its MVarId -/
def getMVarSyntaxMVarId (stx : Syntax) : MVarId :=
stx[0].getKind
/-
Patterns define new local variables.
This module collect them and preprocess `_` occurring in patterns.
Recall that an `_` may represent anonymous variables or inaccessible terms
that are implied by typing constraints. Thus, we represent them with fresh named holes `?x`.
After we elaborate the pattern, if the metavariable remains unassigned, we transform it into
a regular pattern variable. Otherwise, it becomes an inaccessible term.
Macros occurring in patterns are expanded before the `collectPatternVars` method is executed.
The following kinds of Syntax are handled by this module
- Constructor applications
- Applications of functions tagged with the `[matchPattern]` attribute
- Identifiers
- Anonymous constructors
- Structure instances
- Inaccessible terms
- Named patterns
- Tuple literals
- Type ascriptions
- Literals: num, string and char
-/
namespace CollectPatternVars
structure State where
found : NameSet := {}
vars : Array PatternVar := #[]
abbrev M := StateRefT State TermElabM
private def throwCtorExpected {α} : M α :=
throwError "invalid pattern, constructor or constant marked with '[matchPattern]' expected"
private def getNumExplicitCtorParams (ctorVal : ConstructorVal) : TermElabM Nat :=
forallBoundedTelescope ctorVal.type ctorVal.numParams fun ps _ => do
let mut result := 0
for p in ps do
let localDecl ← getLocalDecl p.fvarId!
if localDecl.binderInfo.isExplicit then
result := result+1
pure result
private def throwInvalidPattern {α} : M α :=
throwError "invalid pattern"
/-
An application in a pattern can be
1- A constructor application
The elaborator assumes fields are accessible and inductive parameters are not accessible.
2- A regular application `(f ...)` where `f` is tagged with `[matchPattern]`.
The elaborator assumes implicit arguments are not accessible and explicit ones are accessible.
-/
structure Context where
funId : Syntax
ctorVal? : Option ConstructorVal -- It is `some`, if constructor application
explicit : Bool
ellipsis : Bool
paramDecls : Array (Name × BinderInfo) -- parameters names and binder information
paramDeclIdx : Nat := 0
namedArgs : Array NamedArg
args : List Arg
newArgs : Array Syntax := #[]
deriving Inhabited
private def isDone (ctx : Context) : Bool :=
ctx.paramDeclIdx ≥ ctx.paramDecls.size
private def finalize (ctx : Context) : M Syntax := do
if ctx.namedArgs.isEmpty && ctx.args.isEmpty then
let fStx ← `(@$(ctx.funId):ident)
return Syntax.mkApp fStx ctx.newArgs
else
throwError "too many arguments"
private def isNextArgAccessible (ctx : Context) : Bool :=
let i := ctx.paramDeclIdx
match ctx.ctorVal? with
| some ctorVal => i ≥ ctorVal.numParams -- For constructor applications only fields are accessible
| none =>
if h : i < ctx.paramDecls.size then
-- For `[matchPattern]` applications, only explicit parameters are accessible.
let d := ctx.paramDecls.get ⟨i, h⟩
d.2.isExplicit
else
false
private def getNextParam (ctx : Context) : (Name × BinderInfo) × Context :=
let i := ctx.paramDeclIdx
let d := ctx.paramDecls[i]
(d, { ctx with paramDeclIdx := ctx.paramDeclIdx + 1 })
private def processVar (idStx : Syntax) : M Syntax := do
unless idStx.isIdent do
throwErrorAt idStx "identifier expected"
let id := idStx.getId
unless id.eraseMacroScopes.isAtomic do
throwError "invalid pattern variable, must be atomic"
if (← get).found.contains id then
throwError "invalid pattern, variable '{id}' occurred more than once"
modify fun s => { s with vars := s.vars.push (PatternVar.localVar id), found := s.found.insert id }
return idStx
private def nameToPattern : Name → TermElabM Syntax
| Name.anonymous => `(Name.anonymous)
| Name.str p s _ => do let p ← nameToPattern p; `(Name.str $p $(quote s) _)
| Name.num p n _ => do let p ← nameToPattern p; `(Name.num $p $(quote n) _)
private def quotedNameToPattern (stx : Syntax) : TermElabM Syntax :=
match stx[0].isNameLit? with
| some val => nameToPattern val
| none => throwIllFormedSyntax
private def doubleQuotedNameToPattern (stx : Syntax) : TermElabM Syntax := do
match stx[1].isNameLit? with
| some val => nameToPattern (← resolveGlobalConstNoOverloadWithInfo stx[1] val)
| none => throwIllFormedSyntax
partial def collect (stx : Syntax) : M Syntax := withRef stx <| withFreshMacroScope do
let k := stx.getKind
if k == identKind then
processId stx
else if k == ``Lean.Parser.Term.app then
processCtorApp stx
else if k == ``Lean.Parser.Term.anonymousCtor then
let elems ← stx[1].getArgs.mapSepElemsM collect
return stx.setArg 1 <| mkNullNode elems
else if k == ``Lean.Parser.Term.structInst then
/-
```
leading_parser "{" >> optional (atomic (termParser >> " with "))
>> manyIndent (group (structInstField >> optional ", "))
>> optional ".."
>> optional (" : " >> termParser)
>> " }"
```
-/
let withMod := stx[1]
unless withMod.isNone do
throwErrorAt withMod "invalid struct instance pattern, 'with' is not allowed in patterns"
let fields ← stx[2].getArgs.mapM fun p => do
-- p is of the form (group (structInstField >> optional ", "))
let field := p[0]
-- leading_parser structInstLVal >> " := " >> termParser
let newVal ← collect field[2]
let field := field.setArg 2 newVal
pure <| field.setArg 0 field
return stx.setArg 2 <| mkNullNode fields
else if k == ``Lean.Parser.Term.hole then
let r ← mkMVarSyntax
modify fun s => { s with vars := s.vars.push <| PatternVar.anonymousVar <| getMVarSyntaxMVarId r }
return r
else if k == ``Lean.Parser.Term.paren then
let arg := stx[1]
if arg.isNone then
return stx -- `()`
else
let t := arg[0]
let s := arg[1]
if s.isNone || s[0].getKind == ``Lean.Parser.Term.typeAscription then
-- Ignore `s`, since it empty or it is a type ascription
let t ← collect t
let arg := arg.setArg 0 t
return stx.setArg 1 arg
else
return stx
else if k == ``Lean.Parser.Term.explicitUniv then
processCtor stx[0]
else if k == ``Lean.Parser.Term.namedPattern then
/- Recall that
def namedPattern := check... >> trailing_parser "@" >> termParser -/
let id := stx[0]
discard <| processVar id
let pat := stx[2]
let pat ← collect pat
`(_root_.namedPattern $id $pat)
else if k == ``Lean.Parser.Term.binop then
let lhs ← collect stx[2]
let rhs ← collect stx[3]
return stx.setArg 2 lhs |>.setArg 3 rhs
else if k == ``Lean.Parser.Term.inaccessible then
return stx
else if k == strLitKind then
return stx
else if k == numLitKind then
return stx
else if k == scientificLitKind then
return stx
else if k == charLitKind then
return stx
else if k == ``Lean.Parser.Term.quotedName then
/- Quoted names have an elaboration function associated with them, and they will not be macro expanded.
Note that macro expansion is not a good option since it produces a term using the smart constructors `Name.mkStr`, `Name.mkNum`
instead of the constructors `Name.str` and `Name.num` -/
quotedNameToPattern stx
else if k == ``Lean.Parser.Term.doubleQuotedName then
/- Similar to previous case -/
doubleQuotedNameToPattern stx
else if k == choiceKind then
throwError "invalid pattern, notation is ambiguous"
else
throwInvalidPattern
where
processCtorApp (stx : Syntax) : M Syntax := do
let (f, namedArgs, args, ellipsis) ← expandApp stx true
processCtorAppCore f namedArgs args ellipsis
processCtor (stx : Syntax) : M Syntax := do
processCtorAppCore stx #[] #[] false
/- Check whether `stx` is a pattern variable or constructor-like (i.e., constructor or constant tagged with `[matchPattern]` attribute) -/
processId (stx : Syntax) : M Syntax := do
match (← resolveId? stx "pattern" (withInfo := true)) with
| none => processVar stx
| some f => match f with
| Expr.const fName _ _ =>
match (← getEnv).find? fName with
| some (ConstantInfo.ctorInfo _) => processCtor stx
| some _ =>
if hasMatchPatternAttribute (← getEnv) fName then
processCtor stx
else
processVar stx
| none => throwCtorExpected
| _ => processVar stx
pushNewArg (accessible : Bool) (ctx : Context) (arg : Arg) : M Context := do
match arg with
| Arg.stx stx =>
let stx ← if accessible then collect stx else pure stx
return { ctx with newArgs := ctx.newArgs.push stx }
| _ => unreachable!
processExplicitArg (accessible : Bool) (ctx : Context) : M Context := do
match ctx.args with
| [] =>
if ctx.ellipsis then
pushNewArg accessible ctx (Arg.stx (← `(_)))
else
throwError "explicit parameter is missing, unused named arguments {ctx.namedArgs.map fun narg => narg.name}"
| arg::args =>
pushNewArg accessible { ctx with args := args } arg
processImplicitArg (accessible : Bool) (ctx : Context) : M Context := do
if ctx.explicit then
processExplicitArg accessible ctx
else
pushNewArg accessible ctx (Arg.stx (← `(_)))
processCtorAppContext (ctx : Context) : M Syntax := do
if isDone ctx then
finalize ctx
else
let accessible := isNextArgAccessible ctx
let (d, ctx) := getNextParam ctx
match ctx.namedArgs.findIdx? fun namedArg => namedArg.name == d.1 with
| some idx =>
let arg := ctx.namedArgs[idx]
let ctx := { ctx with namedArgs := ctx.namedArgs.eraseIdx idx }
let ctx ← pushNewArg accessible ctx arg.val
processCtorAppContext ctx
| none =>
let ctx ← match d.2 with
| BinderInfo.implicit => processImplicitArg accessible ctx
| BinderInfo.instImplicit => processImplicitArg accessible ctx
| _ => processExplicitArg accessible ctx
processCtorAppContext ctx
processCtorAppCore (f : Syntax) (namedArgs : Array NamedArg) (args : Array Arg) (ellipsis : Bool) : M Syntax := do
let args := args.toList
let (fId, explicit) ← match f with
| `($fId:ident) => pure (fId, false)
| `(@$fId:ident) => pure (fId, true)
| _ => throwError "identifier expected"
let some (Expr.const fName _ _) ← resolveId? fId "pattern" (withInfo := true) | throwCtorExpected
let fInfo ← getConstInfo fName
let paramDecls ← forallTelescopeReducing fInfo.type fun xs _ => xs.mapM fun x => do
let d ← getFVarLocalDecl x
return (d.userName, d.binderInfo)
match fInfo with
| ConstantInfo.ctorInfo val =>
processCtorAppContext
{ funId := fId, explicit := explicit, ctorVal? := val, paramDecls := paramDecls, namedArgs := namedArgs, args := args, ellipsis := ellipsis }
| _ =>
if hasMatchPatternAttribute (← getEnv) fName then
processCtorAppContext
{ funId := fId, explicit := explicit, ctorVal? := none, paramDecls := paramDecls, namedArgs := namedArgs, args := args, ellipsis := ellipsis }
else
throwCtorExpected
def main (alt : MatchAltView) : M MatchAltView := do
let patterns ← alt.patterns.mapM fun p => do
trace[Elab.match] "collecting variables at pattern: {p}"
collect p
return { alt with patterns := patterns }
end CollectPatternVars
def collectPatternVars (alt : MatchAltView) : TermElabM (Array PatternVar × MatchAltView) := do
let (alt, s) ← (CollectPatternVars.main alt).run {}
return (s.vars, alt)
/- Return the pattern variables in the given pattern.
Remark: this method is not used by the main `match` elaborator, but in the precheck hook and other macros (e.g., at `Do.lean`). -/
def getPatternVars (patternStx : Syntax) : TermElabM (Array PatternVar) := do
let patternStx ← liftMacroM <| expandMacros patternStx
let (_, s) ← (CollectPatternVars.collect patternStx).run {}
return s.vars
def getPatternsVars (patterns : Array Syntax) : TermElabM (Array PatternVar) := do
let collect : CollectPatternVars.M Unit := do
for pattern in patterns do
discard <| CollectPatternVars.collect (← liftMacroM <| expandMacros pattern)
let (_, s) ← collect.run {}
return s.vars
def getPatternVarNames (pvars : Array PatternVar) : Array Name :=
pvars.filterMap fun
| PatternVar.localVar x => some x
| _ => none
end Lean.Elab.Term
|
de13e663977fed2dd02f0ab16519e40254f0d9a5 | 947fa6c38e48771ae886239b4edce6db6e18d0fb | /src/linear_algebra/affine_space/affine_equiv.lean | 646f3b099e4444dc43fa9a9aa509e2cf1f060a38 | [
"Apache-2.0"
] | permissive | ramonfmir/mathlib | c5dc8b33155473fab97c38bd3aa6723dc289beaa | 14c52e990c17f5a00c0cc9e09847af16fabbed25 | refs/heads/master | 1,661,979,343,526 | 1,660,830,384,000 | 1,660,830,384,000 | 182,072,989 | 0 | 0 | null | 1,555,585,876,000 | 1,555,585,876,000 | null | UTF-8 | Lean | false | false | 17,545 | lean | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import linear_algebra.affine_space.affine_map
import algebra.invertible
/-!
# Affine equivalences
In this file we define `affine_equiv k P₁ P₂` (notation: `P₁ ≃ᵃ[k] P₂`) to be the type of affine
equivalences between `P₁` and `P₂, i.e., equivalences such that both forward and inverse maps are
affine maps.
We define the following equivalences:
* `affine_equiv.refl k P`: the identity map as an `affine_equiv`;
* `e.symm`: the inverse map of an `affine_equiv` as an `affine_equiv`;
* `e.trans e'`: composition of two `affine_equiv`s; note that the order follows `mathlib`'s
`category_theory` convention (apply `e`, then `e'`), not the convention used in function
composition and compositions of bundled morphisms.
We equip `affine_equiv k P P` with a `group` structure with multiplication corresponding to
composition in `affine_equiv.group`.
## Tags
affine space, affine equivalence
-/
open function set
open_locale affine
/-- An affine equivalence is an equivalence between affine spaces such that both forward
and inverse maps are affine.
We define it using an `equiv` for the map and a `linear_equiv` for the linear part in order
to allow affine equivalences with good definitional equalities. -/
@[nolint has_nonempty_instance]
structure affine_equiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [ring k]
[add_comm_group V₁] [module k V₁] [add_torsor V₁ P₁]
[add_comm_group V₂] [module k V₂] [add_torsor V₂ P₂] extends P₁ ≃ P₂ :=
(linear : V₁ ≃ₗ[k] V₂)
(map_vadd' : ∀ (p : P₁) (v : V₁), to_equiv (v +ᵥ p) = linear v +ᵥ to_equiv p)
notation P₁ ` ≃ᵃ[`:25 k:25 `] `:0 P₂:0 := affine_equiv k P₁ P₂
variables {k P₁ P₂ P₃ P₄ V₁ V₂ V₃ V₄ : Type*} [ring k]
[add_comm_group V₁] [module k V₁] [add_torsor V₁ P₁]
[add_comm_group V₂] [module k V₂] [add_torsor V₂ P₂]
[add_comm_group V₃] [module k V₃] [add_torsor V₃ P₃]
[add_comm_group V₄] [module k V₄] [add_torsor V₄ P₄]
namespace affine_equiv
include V₁ V₂
instance : has_coe_to_fun (P₁ ≃ᵃ[k] P₂) (λ _, P₁ → P₂) := ⟨λ e, e.to_fun⟩
instance : has_coe (P₁ ≃ᵃ[k] P₂) (P₁ ≃ P₂) := ⟨affine_equiv.to_equiv⟩
variables {k P₁}
@[simp] lemma map_vadd (e : P₁ ≃ᵃ[k] P₂) (p : P₁) (v : V₁) : e (v +ᵥ p) = e.linear v +ᵥ e p :=
e.map_vadd' p v
@[simp] lemma coe_to_equiv (e : P₁ ≃ᵃ[k] P₂) : ⇑e.to_equiv = e := rfl
/-- Reinterpret an `affine_equiv` as an `affine_map`. -/
def to_affine_map (e : P₁ ≃ᵃ[k] P₂) : P₁ →ᵃ[k] P₂ := { to_fun := e, .. e }
instance : has_coe (P₁ ≃ᵃ[k] P₂) (P₁ →ᵃ[k] P₂) := ⟨to_affine_map⟩
@[simp] lemma coe_to_affine_map (e : P₁ ≃ᵃ[k] P₂) :
(e.to_affine_map : P₁ → P₂) = (e : P₁ → P₂) :=
rfl
@[simp] lemma to_affine_map_mk (f : P₁ ≃ P₂) (f' : V₁ ≃ₗ[k] V₂) (h) :
to_affine_map (mk f f' h) = ⟨f, f', h⟩ :=
rfl
@[norm_cast, simp] lemma coe_coe (e : P₁ ≃ᵃ[k] P₂) : ((e : P₁ →ᵃ[k] P₂) : P₁ → P₂) = e := rfl
@[simp] lemma linear_to_affine_map (e : P₁ ≃ᵃ[k] P₂) : e.to_affine_map.linear = e.linear := rfl
lemma to_affine_map_injective : injective (to_affine_map : (P₁ ≃ᵃ[k] P₂) → (P₁ →ᵃ[k] P₂)) :=
begin
rintros ⟨e, el, h⟩ ⟨e', el', h'⟩ H,
simp only [to_affine_map_mk, equiv.coe_inj, linear_equiv.to_linear_map_inj] at H,
congr,
exacts [H.1, H.2]
end
@[simp] lemma to_affine_map_inj {e e' : P₁ ≃ᵃ[k] P₂} :
e.to_affine_map = e'.to_affine_map ↔ e = e' :=
to_affine_map_injective.eq_iff
@[ext] lemma ext {e e' : P₁ ≃ᵃ[k] P₂} (h : ∀ x, e x = e' x) : e = e' :=
to_affine_map_injective $ affine_map.ext h
lemma coe_fn_injective : @injective (P₁ ≃ᵃ[k] P₂) (P₁ → P₂) coe_fn :=
λ e e' H, ext $ congr_fun H
@[simp, norm_cast] lemma coe_fn_inj {e e' : P₁ ≃ᵃ[k] P₂} : (e : P₁ → P₂) = e' ↔ e = e' :=
coe_fn_injective.eq_iff
lemma to_equiv_injective : injective (to_equiv : (P₁ ≃ᵃ[k] P₂) → (P₁ ≃ P₂)) :=
λ e e' H, ext $ equiv.ext_iff.1 H
@[simp] lemma to_equiv_inj {e e' : P₁ ≃ᵃ[k] P₂} : e.to_equiv = e'.to_equiv ↔ e = e' :=
to_equiv_injective.eq_iff
@[simp] lemma coe_mk (e : P₁ ≃ P₂) (e' : V₁ ≃ₗ[k] V₂) (h) :
((⟨e, e', h⟩ : P₁ ≃ᵃ[k] P₂) : P₁ → P₂) = e :=
rfl
/-- Construct an affine equivalence by verifying the relation between the map and its linear part at
one base point. Namely, this function takes a map `e : P₁ → P₂`, a linear equivalence
`e' : V₁ ≃ₗ[k] V₂`, and a point `p` such that for any other point `p'` we have
`e p' = e' (p' -ᵥ p) +ᵥ e p`. -/
def mk' (e : P₁ → P₂) (e' : V₁ ≃ₗ[k] V₂) (p : P₁) (h : ∀ p' : P₁, e p' = e' (p' -ᵥ p) +ᵥ e p) :
P₁ ≃ᵃ[k] P₂ :=
{ to_fun := e,
inv_fun := λ q' : P₂, e'.symm (q' -ᵥ e p) +ᵥ p,
left_inv := λ p', by simp [h p'],
right_inv := λ q', by simp [h (e'.symm (q' -ᵥ e p) +ᵥ p)],
linear := e',
map_vadd' := λ p' v, by { simp [h p', h (v +ᵥ p'), vadd_vsub_assoc, vadd_vadd] } }
@[simp] lemma coe_mk' (e : P₁ ≃ P₂) (e' : V₁ ≃ₗ[k] V₂) (p h) : ⇑(mk' e e' p h) = e := rfl
@[simp] lemma linear_mk' (e : P₁ ≃ P₂) (e' : V₁ ≃ₗ[k] V₂) (p h) :
(mk' e e' p h).linear = e' := rfl
/-- Inverse of an affine equivalence as an affine equivalence. -/
@[symm] def symm (e : P₁ ≃ᵃ[k] P₂) : P₂ ≃ᵃ[k] P₁ :=
{ to_equiv := e.to_equiv.symm,
linear := e.linear.symm,
map_vadd' := λ v p, e.to_equiv.symm.apply_eq_iff_eq_symm_apply.2 $
by simpa using (e.to_equiv.apply_symm_apply v).symm }
@[simp] lemma symm_to_equiv (e : P₁ ≃ᵃ[k] P₂) : e.to_equiv.symm = e.symm.to_equiv := rfl
@[simp] lemma symm_linear (e : P₁ ≃ᵃ[k] P₂) : e.linear.symm = e.symm.linear := rfl
/-- See Note [custom simps projection] -/
def simps.apply (e : P₁ ≃ᵃ[k] P₂) : P₁ → P₂ := e
/-- See Note [custom simps projection] -/
def simps.symm_apply (e : P₁ ≃ᵃ[k] P₂) : P₂ → P₁ := e.symm
initialize_simps_projections affine_equiv
(to_equiv_to_fun → apply, to_equiv_inv_fun → symm_apply, linear → linear as_prefix, -to_equiv)
protected lemma bijective (e : P₁ ≃ᵃ[k] P₂) : bijective e := e.to_equiv.bijective
protected lemma surjective (e : P₁ ≃ᵃ[k] P₂) : surjective e := e.to_equiv.surjective
protected lemma injective (e : P₁ ≃ᵃ[k] P₂) : injective e := e.to_equiv.injective
@[simp] lemma range_eq (e : P₁ ≃ᵃ[k] P₂) : range e = univ := e.surjective.range_eq
@[simp] lemma apply_symm_apply (e : P₁ ≃ᵃ[k] P₂) (p : P₂) : e (e.symm p) = p :=
e.to_equiv.apply_symm_apply p
@[simp] lemma symm_apply_apply (e : P₁ ≃ᵃ[k] P₂) (p : P₁) : e.symm (e p) = p :=
e.to_equiv.symm_apply_apply p
lemma apply_eq_iff_eq_symm_apply (e : P₁ ≃ᵃ[k] P₂) {p₁ p₂} : e p₁ = p₂ ↔ p₁ = e.symm p₂ :=
e.to_equiv.apply_eq_iff_eq_symm_apply
@[simp] lemma apply_eq_iff_eq (e : P₁ ≃ᵃ[k] P₂) {p₁ p₂ : P₁} : e p₁ = e p₂ ↔ p₁ = p₂ :=
e.to_equiv.apply_eq_iff_eq
variables (k P₁)
omit V₂
/-- Identity map as an `affine_equiv`. -/
@[refl] def refl : P₁ ≃ᵃ[k] P₁ :=
{ to_equiv := equiv.refl P₁,
linear := linear_equiv.refl k V₁,
map_vadd' := λ _ _, rfl }
@[simp] lemma coe_refl : ⇑(refl k P₁) = id := rfl
@[simp] lemma coe_refl_to_affine_map : ↑(refl k P₁) = affine_map.id k P₁ := rfl
@[simp] lemma refl_apply (x : P₁) : refl k P₁ x = x := rfl
@[simp] lemma to_equiv_refl : (refl k P₁).to_equiv = equiv.refl P₁ := rfl
@[simp] lemma linear_refl : (refl k P₁).linear = linear_equiv.refl k V₁ := rfl
@[simp] lemma symm_refl : (refl k P₁).symm = refl k P₁ := rfl
variables {k P₁}
include V₂ V₃
/-- Composition of two `affine_equiv`alences, applied left to right. -/
@[trans] def trans (e : P₁ ≃ᵃ[k] P₂) (e' : P₂ ≃ᵃ[k] P₃) : P₁ ≃ᵃ[k] P₃ :=
{ to_equiv := e.to_equiv.trans e'.to_equiv,
linear := e.linear.trans e'.linear,
map_vadd' := λ p v, by simp only [linear_equiv.trans_apply, coe_to_equiv, (∘),
equiv.coe_trans, map_vadd] }
@[simp] lemma coe_trans (e : P₁ ≃ᵃ[k] P₂) (e' : P₂ ≃ᵃ[k] P₃) : ⇑(e.trans e') = e' ∘ e := rfl
@[simp]
lemma trans_apply (e : P₁ ≃ᵃ[k] P₂) (e' : P₂ ≃ᵃ[k] P₃) (p : P₁) : e.trans e' p = e' (e p) := rfl
include V₄
lemma trans_assoc (e₁ : P₁ ≃ᵃ[k] P₂) (e₂ : P₂ ≃ᵃ[k] P₃) (e₃ : P₃ ≃ᵃ[k] P₄) :
(e₁.trans e₂).trans e₃ = e₁.trans (e₂.trans e₃) :=
ext $ λ _, rfl
omit V₃ V₄
@[simp] lemma trans_refl (e : P₁ ≃ᵃ[k] P₂) : e.trans (refl k P₂) = e :=
ext $ λ _, rfl
@[simp] lemma refl_trans (e : P₁ ≃ᵃ[k] P₂) : (refl k P₁).trans e = e :=
ext $ λ _, rfl
@[simp] lemma self_trans_symm (e : P₁ ≃ᵃ[k] P₂) : e.trans e.symm = refl k P₁ :=
ext e.symm_apply_apply
@[simp] lemma symm_trans_self (e : P₁ ≃ᵃ[k] P₂) : e.symm.trans e = refl k P₂ :=
ext e.apply_symm_apply
@[simp] lemma apply_line_map (e : P₁ ≃ᵃ[k] P₂) (a b : P₁) (c : k) :
e (affine_map.line_map a b c) = affine_map.line_map (e a) (e b) c :=
e.to_affine_map.apply_line_map a b c
omit V₂
instance : group (P₁ ≃ᵃ[k] P₁) :=
{ one := refl k P₁,
mul := λ e e', e'.trans e,
inv := symm,
mul_assoc := λ e₁ e₂ e₃, trans_assoc _ _ _,
one_mul := trans_refl,
mul_one := refl_trans,
mul_left_inv := self_trans_symm }
lemma one_def : (1 : P₁ ≃ᵃ[k] P₁) = refl k P₁ := rfl
@[simp] lemma coe_one : ⇑(1 : P₁ ≃ᵃ[k] P₁) = id := rfl
lemma mul_def (e e' : P₁ ≃ᵃ[k] P₁) : e * e' = e'.trans e := rfl
@[simp] lemma coe_mul (e e' : P₁ ≃ᵃ[k] P₁) : ⇑(e * e') = e ∘ e' := rfl
lemma inv_def (e : P₁ ≃ᵃ[k] P₁) : e⁻¹ = e.symm := rfl
/-- `affine_equiv.linear` on automorphisms is a `monoid_hom`. -/
@[simps] def linear_hom : (P₁ ≃ᵃ[k] P₁) →* (V₁ ≃ₗ[k] V₁) :=
{ to_fun := linear,
map_one' := rfl,
map_mul' := λ _ _, rfl }
/-- The group of `affine_equiv`s are equivalent to the group of units of `affine_map`.
This is the affine version of `linear_map.general_linear_group.general_linear_equiv`. -/
@[simps]
def equiv_units_affine_map : (P₁ ≃ᵃ[k] P₁) ≃* (P₁ →ᵃ[k] P₁)ˣ :=
{ to_fun := λ e, ⟨e, e.symm, congr_arg coe e.symm_trans_self, congr_arg coe e.self_trans_symm⟩,
inv_fun := λ u,
{ to_fun := (u : P₁ →ᵃ[k] P₁), inv_fun := (↑(u⁻¹) : P₁ →ᵃ[k] P₁),
left_inv := affine_map.congr_fun u.inv_mul,
right_inv := affine_map.congr_fun u.mul_inv,
linear := linear_map.general_linear_group.general_linear_equiv _ _ $
units.map (by exact affine_map.linear_hom) u,
map_vadd' := λ _ _, (u : P₁ →ᵃ[k] P₁).map_vadd _ _ },
left_inv := λ e, affine_equiv.ext $ λ x, rfl,
right_inv := λ u, units.ext $ affine_map.ext $ λ x, rfl,
map_mul' := λ e₁ e₂, rfl }
variable (k)
/-- The map `v ↦ v +ᵥ b` as an affine equivalence between a module `V` and an affine space `P` with
tangent space `V`. -/
@[simps]
def vadd_const (b : P₁) : V₁ ≃ᵃ[k] P₁ :=
{ to_equiv := equiv.vadd_const b,
linear := linear_equiv.refl _ _,
map_vadd' := λ p v, add_vadd _ _ _ }
/-- `p' ↦ p -ᵥ p'` as an equivalence. -/
def const_vsub (p : P₁) : P₁ ≃ᵃ[k] V₁ :=
{ to_equiv := equiv.const_vsub p,
linear := linear_equiv.neg k,
map_vadd' := λ p' v, by simp [vsub_vadd_eq_vsub_sub, neg_add_eq_sub] }
@[simp] lemma coe_const_vsub (p : P₁) : ⇑(const_vsub k p) = (-ᵥ) p := rfl
@[simp] lemma coe_const_vsub_symm (p : P₁) : ⇑(const_vsub k p).symm = λ v, -v +ᵥ p := rfl
variable (P₁)
/-- The map `p ↦ v +ᵥ p` as an affine automorphism of an affine space.
Note that there is no need for an `affine_map.const_vadd` as it is always an equivalence.
This is roughly to `distrib_mul_action.to_linear_equiv` as `+ᵥ` is to `•`. -/
@[simps apply linear]
def const_vadd (v : V₁) : P₁ ≃ᵃ[k] P₁ :=
{ to_equiv := equiv.const_vadd P₁ v,
linear := linear_equiv.refl _ _,
map_vadd' := λ p w, vadd_comm _ _ _ }
@[simp] lemma const_vadd_zero : const_vadd k P₁ 0 = affine_equiv.refl _ _ := ext $ zero_vadd _
@[simp] lemma const_vadd_add (v w : V₁) :
const_vadd k P₁ (v + w) = (const_vadd k P₁ w).trans (const_vadd k P₁ v) := ext $ add_vadd _ _
@[simp] lemma const_vadd_symm (v : V₁) : (const_vadd k P₁ v).symm = const_vadd k P₁ (-v) :=
ext $ λ _, rfl
/-- A more bundled version of `affine_equiv.const_vadd`. -/
@[simps]
def const_vadd_hom : multiplicative V₁ →* P₁ ≃ᵃ[k] P₁ :=
{ to_fun := λ v, const_vadd k P₁ v.to_add,
map_one' := const_vadd_zero _ _,
map_mul' := const_vadd_add _ _ }
lemma const_vadd_nsmul (n : ℕ) (v : V₁) : const_vadd k P₁ (n • v) = (const_vadd k P₁ v)^n :=
(const_vadd_hom k P₁).map_pow _ _
lemma const_vadd_zsmul (z : ℤ) (v : V₁) : const_vadd k P₁ (z • v) = (const_vadd k P₁ v)^z :=
(const_vadd_hom k P₁).map_zpow _ _
section homothety
omit V₁
variables {R V P : Type*} [comm_ring R] [add_comm_group V] [module R V] [affine_space V P]
include V
/-- Fixing a point in affine space, homothety about this point gives a group homomorphism from (the
centre of) the units of the scalars into the group of affine equivalences. -/
def homothety_units_mul_hom (p : P) : Rˣ →* P ≃ᵃ[R] P :=
equiv_units_affine_map.symm.to_monoid_hom.comp $ units.map (affine_map.homothety_hom p)
@[simp] lemma coe_homothety_units_mul_hom_apply (p : P) (t : Rˣ) :
(homothety_units_mul_hom p t : P → P) = affine_map.homothety p (t : R) :=
rfl
@[simp] lemma coe_homothety_units_mul_hom_apply_symm (p : P) (t : Rˣ) :
((homothety_units_mul_hom p t).symm : P → P) = affine_map.homothety p (↑t⁻¹ : R) :=
rfl
@[simp] lemma coe_homothety_units_mul_hom_eq_homothety_hom_coe (p : P) :
(coe : (P ≃ᵃ[R] P) → P →ᵃ[R] P) ∘ homothety_units_mul_hom p =
(affine_map.homothety_hom p) ∘ (coe : Rˣ → R) :=
funext $ λ _, rfl
end homothety
variable {P₁}
open function
/-- Point reflection in `x` as a permutation. -/
def point_reflection (x : P₁) : P₁ ≃ᵃ[k] P₁ := (const_vsub k x).trans (vadd_const k x)
lemma point_reflection_apply (x y : P₁) : point_reflection k x y = x -ᵥ y +ᵥ x := rfl
@[simp] lemma point_reflection_symm (x : P₁) : (point_reflection k x).symm = point_reflection k x :=
to_equiv_injective $ equiv.point_reflection_symm x
@[simp] lemma to_equiv_point_reflection (x : P₁) :
(point_reflection k x).to_equiv = equiv.point_reflection x :=
rfl
@[simp] lemma point_reflection_self (x : P₁) : point_reflection k x x = x := vsub_vadd _ _
lemma point_reflection_involutive (x : P₁) : involutive (point_reflection k x : P₁ → P₁) :=
equiv.point_reflection_involutive x
/-- `x` is the only fixed point of `point_reflection x`. This lemma requires
`x + x = y + y ↔ x = y`. There is no typeclass to use here, so we add it as an explicit argument. -/
lemma point_reflection_fixed_iff_of_injective_bit0 {x y : P₁} (h : injective (bit0 : V₁ → V₁)) :
point_reflection k x y = y ↔ y = x :=
equiv.point_reflection_fixed_iff_of_injective_bit0 h
lemma injective_point_reflection_left_of_injective_bit0 (h : injective (bit0 : V₁ → V₁)) (y : P₁) :
injective (λ x : P₁, point_reflection k x y) :=
equiv.injective_point_reflection_left_of_injective_bit0 h y
lemma injective_point_reflection_left_of_module [invertible (2:k)]:
∀ y, injective (λ x : P₁, point_reflection k x y) :=
injective_point_reflection_left_of_injective_bit0 k $ λ x y h,
by rwa [bit0, bit0, ← two_smul k x, ← two_smul k y,
(is_unit_of_invertible (2:k)).smul_left_cancel] at h
lemma point_reflection_fixed_iff_of_module [invertible (2:k)] {x y : P₁} :
point_reflection k x y = y ↔ y = x :=
((injective_point_reflection_left_of_module k y).eq_iff' (point_reflection_self k y)).trans eq_comm
end affine_equiv
namespace linear_equiv
/-- Interpret a linear equivalence between modules as an affine equivalence. -/
def to_affine_equiv (e : V₁ ≃ₗ[k] V₂) : V₁ ≃ᵃ[k] V₂ :=
{ to_equiv := e.to_equiv,
linear := e,
map_vadd' := λ p v, e.map_add v p }
@[simp] lemma coe_to_affine_equiv (e : V₁ ≃ₗ[k] V₂) : ⇑e.to_affine_equiv = e := rfl
end linear_equiv
namespace affine_map
open affine_equiv
include V₁
lemma line_map_vadd (v v' : V₁) (p : P₁) (c : k) :
line_map v v' c +ᵥ p = line_map (v +ᵥ p) (v' +ᵥ p) c :=
(vadd_const k p).apply_line_map v v' c
lemma line_map_vsub (p₁ p₂ p₃ : P₁) (c : k) :
line_map p₁ p₂ c -ᵥ p₃ = line_map (p₁ -ᵥ p₃) (p₂ -ᵥ p₃) c :=
(vadd_const k p₃).symm.apply_line_map p₁ p₂ c
lemma vsub_line_map (p₁ p₂ p₃ : P₁) (c : k) :
p₁ -ᵥ line_map p₂ p₃ c = line_map (p₁ -ᵥ p₂) (p₁ -ᵥ p₃) c :=
(const_vsub k p₁).apply_line_map p₂ p₃ c
lemma vadd_line_map (v : V₁) (p₁ p₂ : P₁) (c : k) :
v +ᵥ line_map p₁ p₂ c = line_map (v +ᵥ p₁) (v +ᵥ p₂) c :=
(const_vadd k P₁ v).apply_line_map p₁ p₂ c
variables {R' : Type*} [comm_ring R'] [module R' V₁]
lemma homothety_neg_one_apply (c p : P₁) :
homothety c (-1:R') p = point_reflection R' c p :=
by simp [homothety_apply, point_reflection_apply]
end affine_map
|
5c7a21e3139f9267b0ea1d0d16c6cc68af912481 | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /tests/lean/eta_tac.lean | 449e7f844622c3812cc0362921155dce2920cd16 | [
"Apache-2.0"
] | permissive | soonhokong/lean-osx | 4a954262c780e404c1369d6c06516161d07fcb40 | 3670278342d2f4faa49d95b46d86642d7875b47c | refs/heads/master | 1,611,410,334,552 | 1,474,425,686,000 | 1,474,425,686,000 | 12,043,103 | 5 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 342 | lean | open tactic
set_option pp.binder_types true
set_option pp.implicit true
set_option pp.notation false
example (a : nat) : true :=
by do
mk_const `add >>= eta_expand >>= trace,
mk_const `nat.succ >>= eta_expand >>= trace,
to_expr `(add a) >>= eta_expand >>= trace,
to_expr `(λ x : nat, add x) >>= eta_expand >>= trace,
constructor
|
261b2da1f0843530d468f9707b3f71a6f642caec | b7f22e51856f4989b970961f794f1c435f9b8f78 | /tests/lean/nonexhaustive.lean | 0fea2f38536b6b09cd3528b6f44a2674e5d9f14e | [
"Apache-2.0"
] | permissive | soonhokong/lean | cb8aa01055ffe2af0fb99a16b4cda8463b882cd1 | 38607e3eb57f57f77c0ac114ad169e9e4262e24f | refs/heads/master | 1,611,187,284,081 | 1,450,766,737,000 | 1,476,122,547,000 | 11,513,992 | 2 | 0 | null | 1,401,763,102,000 | 1,374,182,235,000 | C++ | UTF-8 | Lean | false | false | 297 | lean | import data.examples.vector
open nat vector
variable {A : Type}
definition foo : Π {n : nat}, vector A n → nat
| foo nil := 0
| foo (a :: b :: v) := 0
set_option pp.implicit false
definition foo : Π {n : nat}, vector A n → nat
| foo nil := 0
| foo (a :: b :: v) := 0
|
924647f0c9bc1a3eeae27e7d90a90ca756a4a7c1 | 69d4931b605e11ca61881fc4f66db50a0a875e39 | /src/tactic/lint/type_classes.lean | 7b0ff5cba5b4baaed2315d5f213bb67d95be3ae5 | [
"Apache-2.0"
] | permissive | abentkamp/mathlib | d9a75d291ec09f4637b0f30cc3880ffb07549ee5 | 5360e476391508e092b5a1e5210bd0ed22dc0755 | refs/heads/master | 1,682,382,954,948 | 1,622,106,077,000 | 1,622,106,077,000 | 149,285,665 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 17,985 | lean | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Robert Y. Lewis, Gabriel Ebner
-/
import tactic.lint.basic
/-!
# Linters about type classes
This file defines several linters checking the correct usage of type classes
and the appropriate definition of instances:
* `instance_priority` ensures that blanket instances have low priority
* `has_inhabited_instances` checks that every type has an `inhabited` instance
* `impossible_instance` checks that there are no instances which can never apply
* `incorrect_type_class_argument` checks that only type classes are used in
instance-implicit arguments
* `dangerous_instance` checks for instances that generate subproblems with metavariables
* `fails_quickly` checks that type class resolution finishes quickly
* `class_structure` checks that every `class` is a structure, i.e. `@[class] def` is forbidden
* `has_coe_variable` checks that there is no instance of type `has_coe α t`
* `inhabited_nonempty` checks whether `[inhabited α]` arguments could be generalized
to `[nonempty α]`
* `decidable_classical` checks propositions for `[decidable_... p]` hypotheses that are not used
in the statement, and could thus be removed by using `classical` in the proof.
* `linter.has_coe_to_fun` checks whether necessary `has_coe_to_fun` instances are declared
-/
open tactic
/-- Pretty prints a list of arguments of a declaration. Assumes `l` is a list of argument positions
and binders (or any other element that can be pretty printed).
`l` can be obtained e.g. by applying `list.indexes_values` to a list obtained by
`get_pi_binders`. -/
meta def print_arguments {α} [has_to_tactic_format α] (l : list (ℕ × α)) : tactic string := do
fs ← l.mmap (λ ⟨n, b⟩, (λ s, to_fmt "argument " ++ to_fmt (n+1) ++ ": " ++ s) <$> pp b),
return $ fs.to_string_aux tt
/-- checks whether an instance that always applies has priority ≥ 1000. -/
private meta def instance_priority (d : declaration) : tactic (option string) := do
let nm := d.to_name,
b ← is_instance nm,
/- return `none` if `d` is not an instance -/
if ¬ b then return none else do
(is_persistent, prio) ← has_attribute `instance nm,
/- return `none` if `d` is has low priority -/
if prio < 1000 then return none else do
(_, tp) ← open_pis d.type,
tp ← whnf tp transparency.none,
let (fn, args) := tp.get_app_fn_args,
cls ← get_decl fn.const_name,
let (pi_args, _) := cls.type.pi_binders,
guard (args.length = pi_args.length),
/- List all the arguments of the class that block type-class inference from firing
(if they are metavariables). These are all the arguments except instance-arguments and
out-params. -/
let relevant_args := (args.zip pi_args).filter_map $ λ⟨e, ⟨_, info, tp⟩⟩,
if info = binder_info.inst_implicit ∨ tp.get_app_fn.is_constant_of `out_param
then none else some e,
let always_applies := relevant_args.all expr.is_local_constant ∧ relevant_args.nodup,
if always_applies then return $ some "set priority below 1000" else return none
/--
There are places where typeclass arguments are specified with implicit `{}` brackets instead of
the usual `[]` brackets. This is done when the instances can be inferred because they are implicit
arguments to the type of one of the other arguments. When they can be inferred from these other
arguments, it is faster to use this method than to use type class inference.
For example, when writing lemmas about `(f : α →+* β)`, it is faster to specify the fact that `α`
and `β` are `semiring`s as `{rα : semiring α} {rβ : semiring β}` rather than the usual
`[semiring α] [semiring β]`.
-/
library_note "implicit instance arguments"
/--
Certain instances always apply during type-class resolution. For example, the instance
`add_comm_group.to_add_group {α} [add_comm_group α] : add_group α` applies to all type-class
resolution problems of the form `add_group _`, and type-class inference will then do an
exhaustive search to find a commutative group. These instances take a long time to fail.
Other instances will only apply if the goal has a certain shape. For example
`int.add_group : add_group ℤ` or
`add_group.prod {α β} [add_group α] [add_group β] : add_group (α × β)`. Usually these instances
will fail quickly, and when they apply, they are almost the desired instance.
For this reason, we want the instances of the second type (that only apply in specific cases) to
always have higher priority than the instances of the first type (that always apply).
See also #1561.
Therefore, if we create an instance that always applies, we set the priority of these instances to
100 (or something similar, which is below the default value of 1000).
-/
library_note "lower instance priority"
/-- A linter object for checking instance priorities of instances that always apply.
This is in the default linter set. -/
@[linter] meta def linter.instance_priority : linter :=
{ test := instance_priority,
no_errors_found := "All instance priorities are good",
errors_found := "DANGEROUS INSTANCE PRIORITIES.
The following instances always apply, and therefore should have a priority < 1000.
If you don't know what priority to choose, use priority 100.
See note [lower instance priority] for instructions to change the priority.",
auto_decls := tt }
/-- Reports declarations of types that do not have an associated `inhabited` instance. -/
private meta def has_inhabited_instance (d : declaration) : tactic (option string) := do
tt ← pure d.is_trusted | pure none,
ff ← has_attribute' `reducible d.to_name | pure none,
ff ← has_attribute' `class d.to_name | pure none,
(_, ty) ← open_pis d.type,
ty ← whnf ty,
if ty = `(Prop) then pure none else do
`(Sort _) ← whnf ty | pure none,
insts ← attribute.get_instances `instance,
insts_tys ← insts.mmap $ λ i, expr.pi_codomain <$> declaration.type <$> get_decl i,
let inhabited_insts := insts_tys.filter (λ i,
i.app_fn.const_name = ``inhabited ∨ i.app_fn.const_name = `unique),
let inhabited_tys := inhabited_insts.map (λ i, i.app_arg.get_app_fn.const_name),
if d.to_name ∈ inhabited_tys then
pure none
else
pure "inhabited instance missing"
/-- A linter for missing `inhabited` instances. -/
@[linter]
meta def linter.has_inhabited_instance : linter :=
{ test := has_inhabited_instance,
auto_decls := ff,
no_errors_found := "No types have missing inhabited instances",
errors_found := "TYPES ARE MISSING INHABITED INSTANCES",
is_fast := ff }
attribute [nolint has_inhabited_instance] pempty
/-- Checks whether an instance can never be applied. -/
private meta def impossible_instance (d : declaration) : tactic (option string) := do
tt ← is_instance d.to_name | return none,
(binders, _) ← get_pi_binders_nondep d.type,
let bad_arguments := binders.filter $ λ nb, nb.2.info ≠ binder_info.inst_implicit,
_ :: _ ← return bad_arguments | return none,
(λ s, some $ "Impossible to infer " ++ s) <$> print_arguments bad_arguments
/-- A linter object for `impossible_instance`. -/
@[linter] meta def linter.impossible_instance : linter :=
{ test := impossible_instance,
auto_decls := tt,
no_errors_found := "All instances are applicable",
errors_found := "IMPOSSIBLE INSTANCES FOUND.
These instances have an argument that cannot be found during type-class resolution, and therefore can never succeed. Either mark the arguments with square brackets (if it is a class), or don't make it an instance" }
/-- Checks whether an instance can never be applied. -/
private meta def incorrect_type_class_argument (d : declaration) : tactic (option string) := do
(binders, _) ← get_pi_binders d.type,
let instance_arguments := binders.indexes_values $
λ b : binder, b.info = binder_info.inst_implicit,
/- the head of the type should either unfold to a class, or be a local constant.
A local constant is allowed, because that could be a class when applied to the
proper arguments. -/
bad_arguments ← instance_arguments.mfilter (λ ⟨_, b⟩, do
(_, head) ← open_pis b.type,
if head.get_app_fn.is_local_constant then return ff else do
bnot <$> is_class head),
_ :: _ ← return bad_arguments | return none,
(λ s, some $ "These are not classes. " ++ s) <$> print_arguments bad_arguments
/-- A linter object for `incorrect_type_class_argument`. -/
@[linter] meta def linter.incorrect_type_class_argument : linter :=
{ test := incorrect_type_class_argument,
auto_decls := tt,
no_errors_found := "All declarations have correct type-class arguments",
errors_found := "INCORRECT TYPE-CLASS ARGUMENTS.
Some declarations have non-classes between [square brackets]" }
/-- Checks whether an instance is dangerous: it creates a new type-class problem with metavariable
arguments. -/
private meta def dangerous_instance (d : declaration) : tactic (option string) := do
tt ← is_instance d.to_name | return none,
(local_constants, target) ← open_pis d.type,
let instance_arguments := local_constants.indexes_values $
λ e : expr, e.local_binding_info = binder_info.inst_implicit,
let bad_arguments := local_constants.indexes_values $ λ x,
!target.has_local_constant x &&
(x.local_binding_info ≠ binder_info.inst_implicit) &&
instance_arguments.any (λ nb, nb.2.local_type.has_local_constant x),
let bad_arguments : list (ℕ × binder) := bad_arguments.map $ λ ⟨n, e⟩, ⟨n, e.to_binder⟩,
_ :: _ ← return bad_arguments | return none,
(λ s, some $ "The following arguments become metavariables. " ++ s) <$> print_arguments bad_arguments
/-- A linter object for `dangerous_instance`. -/
@[linter] meta def linter.dangerous_instance : linter :=
{ test := dangerous_instance,
no_errors_found := "No dangerous instances",
errors_found := "DANGEROUS INSTANCES FOUND.\nThese instances are recursive, and create a new type-class problem which will have metavariables.
Possible solution: remove the instance attribute or make it a local instance instead.
Currently this linter does not check whether the metavariables only occur in arguments marked with `out_param`, in which case this linter gives a false positive.",
auto_decls := tt }
/-- Applies expression `e` to local constants, but lifts all the arguments that are `Sort`-valued to
`Type`-valued sorts. -/
meta def apply_to_fresh_variables (e : expr) : tactic expr := do
t ← infer_type e,
(xs, b) ← open_pis t,
xs.mmap' $ λ x, try $ do {
u ← mk_meta_univ,
tx ← infer_type x,
ttx ← infer_type tx,
unify ttx (expr.sort u.succ) },
return $ e.app_of_list xs
/-- Tests whether type-class inference search for a class will end quickly when applied to
variables. This tactic succeeds if `mk_instance` succeeds quickly or fails quickly with the error
message that it cannot find an instance. It fails if the tactic takes too long, or if any other
error message is raised.
We make sure that we apply the tactic to variables living in `Type u` instead of `Sort u`,
because many instances only apply in that special case, and we do want to catch those loops. -/
meta def fails_quickly (max_steps : ℕ) (d : declaration) : tactic (option string) := do
e ← mk_const d.to_name,
tt ← is_class e | return none,
e' ← apply_to_fresh_variables e,
sum.inr msg ← retrieve_or_report_error $ tactic.try_for max_steps $
succeeds_or_fails_with_msg (mk_instance e')
$ λ s, "tactic.mk_instance failed to generate instance for".is_prefix_of s | return none,
return $ some $
if msg = "try_for tactic failed, timeout" then "type-class inference timed out" else msg
/-- A linter object for `fails_quickly`. If we want to increase the maximum number of steps
type-class inference is allowed to take, we can increase the number `3000` in the definition.
As of 5 Mar 2020 the longest trace (for `is_add_hom`) takes 2900-3000 "heartbeats". -/
@[linter] meta def linter.fails_quickly : linter :=
{ test := fails_quickly 3000,
auto_decls := tt,
no_errors_found := "No type-class searches timed out",
errors_found := "TYPE CLASS SEARCHES TIMED OUT.
For the following classes, there is an instance that causes a loop, or an excessively long search.",
is_fast := ff }
/-- Checks that all uses of the `@[class]` attribute apply to structures or inductive types.
This is future-proofing for lean 4, which no longer supports `@[class] def`. -/
private meta def class_structure (n : name) : tactic (option string) := do
is_class ← has_attribute' `class n,
if is_class then do
env ← get_env,
pure $ if env.is_inductive n then none else
"is a non-structure or inductive type marked @[class]"
else pure none
/-- A linter object for `class_structure`. -/
@[linter] meta def linter.class_structure : linter :=
{ test := λ d, class_structure d.to_name,
auto_decls := tt,
no_errors_found := "All classes are structures",
errors_found := "USE OF @[class] def IS DISALLOWED" }
/--
Tests whether there is no instance of type `has_coe α t` where `α` is a variable,
or `has_coe t α` where `α` does not occur in `t`.
See note [use has_coe_t].
-/
private meta def has_coe_variable (d : declaration) : tactic (option string) := do
tt ← is_instance d.to_name | return none,
`(has_coe %%a %%b) ← return d.type.pi_codomain | return none,
if a.is_var then
return $ some $ "illegal instance, first argument is variable"
else if b.is_var ∧ ¬ b.occurs a then
return $ some $ "illegal instance, second argument is variable not occurring in first argument"
else
return none
/-- A linter object for `has_coe_variable`. -/
@[linter] meta def linter.has_coe_variable : linter :=
{ test := has_coe_variable,
auto_decls := tt,
no_errors_found := "No invalid `has_coe` instances",
errors_found := "INVALID `has_coe` INSTANCES.
Make the following declarations instances of the class `has_coe_t` instead of `has_coe`." }
/-- Checks whether a declaration is prop-valued and takes an `inhabited _` argument that is unused
elsewhere in the type. In this case, that argument can be replaced with `nonempty _`. -/
private meta def inhabited_nonempty (d : declaration) : tactic (option string) :=
do tt ← is_prop d.type | return none,
(binders, _) ← get_pi_binders_nondep d.type,
let inhd_binders := binders.filter $ λ pr, pr.2.type.is_app_of `inhabited,
if inhd_binders.length = 0 then return none
else (λ s, some $ "The following `inhabited` instances should be `nonempty`. " ++ s) <$>
print_arguments inhd_binders
/-- A linter object for `inhabited_nonempty`. -/
@[linter] meta def linter.inhabited_nonempty : linter :=
{ test := inhabited_nonempty,
auto_decls := ff,
no_errors_found := "No uses of `inhabited` arguments should be replaced with `nonempty`",
errors_found := "USES OF `inhabited` SHOULD BE REPLACED WITH `nonempty`." }
/-- Checks whether a declaration is `Prop`-valued and takes a `decidable* _` hypothesis that is unused
elsewhere in the type. In this case, that hypothesis can be replaced with `classical` in the proof.
Theorems in the `decidable` namespace are exempt from the check. -/
private meta def decidable_classical (d : declaration) : tactic (option string) :=
do tt ← is_prop d.type | return none,
ff ← pure $ (`decidable).is_prefix_of d.to_name | return none,
(binders, _) ← get_pi_binders_nondep d.type,
let deceq_binders := binders.filter $ λ pr, pr.2.type.is_app_of `decidable_eq
∨ pr.2.type.is_app_of `decidable_pred ∨ pr.2.type.is_app_of `decidable_rel
∨ pr.2.type.is_app_of `decidable,
if deceq_binders.length = 0 then return none
else (λ s, some $ "The following `decidable` hypotheses should be replaced with
`classical` in the proof. " ++ s) <$>
print_arguments deceq_binders
/-- A linter object for `decidable_classical`. -/
@[linter] meta def linter.decidable_classical : linter :=
{ test := decidable_classical,
auto_decls := ff,
no_errors_found := "No uses of `decidable` arguments should be replaced with `classical`",
errors_found := "USES OF `decidable` SHOULD BE REPLACED WITH `classical` IN THE PROOF." }
/- The file `logic/basic.lean` emphasizes the differences between what holds under classical
and non-classical logic. It makes little sense to make all these lemmas classical, so we add them
to the list of lemmas which are not checked by the linter `decidable_classical`. -/
attribute [nolint decidable_classical] dec_em dec_em' not.decidable_imp_symm
private meta def has_coe_to_fun_linter (d : declaration) : tactic (option string) :=
retrieve $ do
tt ← return d.is_trusted | pure none,
mk_meta_var d.type >>= set_goals ∘ pure,
args ← unfreezing intros,
expr.sort _ ← target | pure none,
let ty : expr := (expr.const d.to_name d.univ_levels).mk_app args,
some coe_fn_inst ←
try_core $ to_expr ``(_root_.has_coe_to_fun %%ty) >>= mk_instance | pure none,
some trans_inst@(expr.app (expr.app _ trans_inst_1) trans_inst_2) ←
try_core $ to_expr ``(@_root_.coe_fn_trans %%ty _ _ _) | pure none,
tt ← succeeds $ unify trans_inst coe_fn_inst transparency.reducible | pure none,
set_bool_option `pp.all true,
trans_inst_1 ← pp trans_inst_1,
trans_inst_2 ← pp trans_inst_2,
pure $ format.to_string $
"`has_coe_to_fun` instance is definitionally equal to a transitive instance composed of: " ++
trans_inst_1.group.indent 2 ++
format.line ++ "and" ++
trans_inst_2.group.indent 2
/-- Linter that checks whether `has_coe_to_fun` instances comply with Note [function coercion]. -/
@[linter] meta def linter.has_coe_to_fun : linter :=
{ test := has_coe_to_fun_linter,
auto_decls := tt,
no_errors_found := "has_coe_to_fun is used correctly",
errors_found := "INVALID/MISSING `has_coe_to_fun` instances.
You should add a `has_coe_to_fun` instance for the following types.
See Note [function coercion]." }
|
ccdd84f7aebc324f6d4218f5d987e6547ff797f0 | cc060cf567f81c404a13ee79bf21f2e720fa6db0 | /lean/20170502-segfault.lean | 85e99f9cbaeb9cf9f2919e0efbed202181a84de0 | [
"Apache-2.0"
] | permissive | semorrison/proof | cf0a8c6957153bdb206fd5d5a762a75958a82bca | 5ee398aa239a379a431190edbb6022b1a0aa2c70 | refs/heads/master | 1,610,414,502,842 | 1,518,696,851,000 | 1,518,696,851,000 | 78,375,937 | 2 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 487 | lean | open tactic
meta def simp_at' (h : expr) (extra_lemmas : list expr := []) (cfg : simp_config := {}) : tactic unit :=
do when (expr.is_local_constant h = ff) (fail "tactic simp_at failed, the given expression is not a hypothesis"),
htype ← infer_type h,
S ← simp_lemmas.mk_default,
S ← S.append extra_lemmas,
(new_htype, heq) ← simplify S htype cfg,
guard (new_htype \ne )
assert (expr.local_pp_name h) new_htype,
mk_eq_mp heq h >>= exact,
clear h |
cf1ee34739ac50529c5f3c159712d44524663e20 | 82e44445c70db0f03e30d7be725775f122d72f3e | /src/algebra/opposites.lean | be2e2a9b66e36dc1a44182f54fc106d0cfb0a418 | [
"Apache-2.0"
] | permissive | stjordanis/mathlib | 51e286d19140e3788ef2c470bc7b953e4991f0c9 | 2568d41bca08f5d6bf39d915434c8447e21f42ee | refs/heads/master | 1,631,748,053,501 | 1,627,938,886,000 | 1,627,938,886,000 | 228,728,358 | 0 | 0 | Apache-2.0 | 1,576,630,588,000 | 1,576,630,587,000 | null | UTF-8 | Lean | false | false | 17,366 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import data.opposite
import algebra.field
import algebra.group.commute
import group_theory.group_action.defs
import data.equiv.mul_add
/-!
# Algebraic operations on `αᵒᵖ`
This file records several basic facts about the opposite of an algebraic structure, e.g. the
opposite of a ring is a ring (with multiplication `x * y = yx`). Use is made of the identity
functions `op : α → αᵒᵖ` and `unop : αᵒᵖ → α`.
-/
namespace opposite
universes u
variables (α : Type u)
instance [has_add α] : has_add (opposite α) :=
{ add := λ x y, op (unop x + unop y) }
instance [has_sub α] : has_sub (opposite α) :=
{ sub := λ x y, op (unop x - unop y) }
instance [add_semigroup α] : add_semigroup (opposite α) :=
{ add_assoc := λ x y z, unop_injective $ add_assoc (unop x) (unop y) (unop z),
.. opposite.has_add α }
instance [add_left_cancel_semigroup α] : add_left_cancel_semigroup (opposite α) :=
{ add_left_cancel := λ x y z H, unop_injective $ add_left_cancel $ op_injective H,
.. opposite.add_semigroup α }
instance [add_right_cancel_semigroup α] : add_right_cancel_semigroup (opposite α) :=
{ add_right_cancel := λ x y z H, unop_injective $ add_right_cancel $ op_injective H,
.. opposite.add_semigroup α }
instance [add_comm_semigroup α] : add_comm_semigroup (opposite α) :=
{ add_comm := λ x y, unop_injective $ add_comm (unop x) (unop y),
.. opposite.add_semigroup α }
instance [has_zero α] : has_zero (opposite α) :=
{ zero := op 0 }
instance [nontrivial α] : nontrivial (opposite α) :=
let ⟨x, y, h⟩ := exists_pair_ne α in nontrivial_of_ne (op x) (op y) (op_injective.ne h)
section
local attribute [semireducible] opposite
@[simp] lemma unop_eq_zero_iff {α} [has_zero α] (a : αᵒᵖ) : a.unop = (0 : α) ↔ a = (0 : αᵒᵖ) :=
iff.refl _
@[simp] lemma op_eq_zero_iff {α} [has_zero α] (a : α) : op a = (0 : αᵒᵖ) ↔ a = (0 : α) :=
iff.refl _
end
lemma unop_ne_zero_iff {α} [has_zero α] (a : αᵒᵖ) : a.unop ≠ (0 : α) ↔ a ≠ (0 : αᵒᵖ) :=
not_iff_not.mpr $ unop_eq_zero_iff a
lemma op_ne_zero_iff {α} [has_zero α] (a : α) : op a ≠ (0 : αᵒᵖ) ↔ a ≠ (0 : α) :=
not_iff_not.mpr $ op_eq_zero_iff a
instance [add_zero_class α] : add_zero_class (opposite α) :=
{ zero_add := λ x, unop_injective $ zero_add $ unop x,
add_zero := λ x, unop_injective $ add_zero $ unop x,
.. opposite.has_add α, .. opposite.has_zero α }
instance [add_monoid α] : add_monoid (opposite α) :=
{ .. opposite.add_semigroup α, .. opposite.add_zero_class α }
instance [add_comm_monoid α] : add_comm_monoid (opposite α) :=
{ .. opposite.add_monoid α, .. opposite.add_comm_semigroup α }
instance [has_neg α] : has_neg (opposite α) :=
{ neg := λ x, op $ -(unop x) }
instance [add_group α] : add_group (opposite α) :=
{ add_left_neg := λ x, unop_injective $ add_left_neg $ unop x,
sub_eq_add_neg := λ x y, unop_injective $ sub_eq_add_neg (unop x) (unop y),
.. opposite.add_monoid α, .. opposite.has_neg α, .. opposite.has_sub α }
instance [add_comm_group α] : add_comm_group (opposite α) :=
{ .. opposite.add_group α, .. opposite.add_comm_monoid α }
instance [has_mul α] : has_mul (opposite α) :=
{ mul := λ x y, op (unop y * unop x) }
instance [semigroup α] : semigroup (opposite α) :=
{ mul_assoc := λ x y z, unop_injective $ eq.symm $ mul_assoc (unop z) (unop y) (unop x),
.. opposite.has_mul α }
instance [right_cancel_semigroup α] : left_cancel_semigroup (opposite α) :=
{ mul_left_cancel := λ x y z H, unop_injective $ mul_right_cancel $ op_injective H,
.. opposite.semigroup α }
instance [left_cancel_semigroup α] : right_cancel_semigroup (opposite α) :=
{ mul_right_cancel := λ x y z H, unop_injective $ mul_left_cancel $ op_injective H,
.. opposite.semigroup α }
instance [comm_semigroup α] : comm_semigroup (opposite α) :=
{ mul_comm := λ x y, unop_injective $ mul_comm (unop y) (unop x),
.. opposite.semigroup α }
instance [has_one α] : has_one (opposite α) :=
{ one := op 1 }
section
local attribute [semireducible] opposite
@[simp] lemma unop_eq_one_iff {α} [has_one α] (a : αᵒᵖ) : a.unop = 1 ↔ a = 1 :=
iff.refl _
@[simp] lemma op_eq_one_iff {α} [has_one α] (a : α) : op a = 1 ↔ a = 1 :=
iff.refl _
end
instance [mul_one_class α] : mul_one_class (opposite α) :=
{ one_mul := λ x, unop_injective $ mul_one $ unop x,
mul_one := λ x, unop_injective $ one_mul $ unop x,
.. opposite.has_mul α, .. opposite.has_one α }
instance [monoid α] : monoid (opposite α) :=
{ .. opposite.semigroup α, .. opposite.mul_one_class α }
instance [comm_monoid α] : comm_monoid (opposite α) :=
{ .. opposite.monoid α, .. opposite.comm_semigroup α }
instance [has_inv α] : has_inv (opposite α) :=
{ inv := λ x, op $ (unop x)⁻¹ }
instance [group α] : group (opposite α) :=
{ mul_left_inv := λ x, unop_injective $ mul_inv_self $ unop x,
.. opposite.monoid α, .. opposite.has_inv α }
instance [comm_group α] : comm_group (opposite α) :=
{ .. opposite.group α, .. opposite.comm_monoid α }
instance [distrib α] : distrib (opposite α) :=
{ left_distrib := λ x y z, unop_injective $ add_mul (unop y) (unop z) (unop x),
right_distrib := λ x y z, unop_injective $ mul_add (unop z) (unop x) (unop y),
.. opposite.has_add α, .. opposite.has_mul α }
instance [mul_zero_class α] : mul_zero_class (opposite α) :=
{ zero := 0,
mul := (*),
zero_mul := λ x, unop_injective $ mul_zero $ unop x,
mul_zero := λ x, unop_injective $ zero_mul $ unop x }
instance [mul_zero_one_class α] : mul_zero_one_class (opposite α) :=
{ .. opposite.mul_zero_class α, .. opposite.mul_one_class α }
instance [semigroup_with_zero α] : semigroup_with_zero (opposite α) :=
{ .. opposite.semigroup α, .. opposite.mul_zero_class α }
instance [monoid_with_zero α] : monoid_with_zero (opposite α) :=
{ .. opposite.monoid α, .. opposite.mul_zero_one_class α }
instance [non_unital_non_assoc_semiring α] : non_unital_non_assoc_semiring (opposite α) :=
{ .. opposite.add_comm_monoid α, .. opposite.mul_zero_class α, .. opposite.distrib α }
instance [non_unital_semiring α] : non_unital_semiring (opposite α) :=
{ .. opposite.semigroup_with_zero α, .. opposite.non_unital_non_assoc_semiring α }
instance [non_assoc_semiring α] : non_assoc_semiring (opposite α) :=
{ .. opposite.mul_zero_one_class α, .. opposite.non_unital_non_assoc_semiring α }
instance [semiring α] : semiring (opposite α) :=
{ .. opposite.non_unital_semiring α, .. opposite.non_assoc_semiring α }
instance [ring α] : ring (opposite α) :=
{ .. opposite.add_comm_group α, .. opposite.monoid α, .. opposite.semiring α }
instance [comm_ring α] : comm_ring (opposite α) :=
{ .. opposite.ring α, .. opposite.comm_semigroup α }
instance [has_zero α] [has_mul α] [no_zero_divisors α] : no_zero_divisors (opposite α) :=
{ eq_zero_or_eq_zero_of_mul_eq_zero := λ x y (H : op (_ * _) = op (0:α)),
or.cases_on (eq_zero_or_eq_zero_of_mul_eq_zero $ op_injective H)
(λ hy, or.inr $ unop_injective $ hy) (λ hx, or.inl $ unop_injective $ hx), }
instance [integral_domain α] : integral_domain (opposite α) :=
{ .. opposite.no_zero_divisors α, .. opposite.comm_ring α, .. opposite.nontrivial α }
instance [group_with_zero α] : group_with_zero (opposite α) :=
{ mul_inv_cancel := λ x hx, unop_injective $ inv_mul_cancel $ unop_injective.ne hx,
inv_zero := unop_injective inv_zero,
.. opposite.monoid_with_zero α, .. opposite.nontrivial α, .. opposite.has_inv α }
instance [division_ring α] : division_ring (opposite α) :=
{ .. opposite.group_with_zero α, .. opposite.ring α }
instance [field α] : field (opposite α) :=
{ .. opposite.division_ring α, .. opposite.comm_ring α }
instance (R : Type*) [has_scalar R α] : has_scalar R (opposite α) :=
{ smul := λ c x, op (c • unop x) }
instance (R : Type*) [monoid R] [mul_action R α] : mul_action R (opposite α) :=
{ one_smul := λ x, unop_injective $ one_smul R (unop x),
mul_smul := λ r₁ r₂ x, unop_injective $ mul_smul r₁ r₂ (unop x),
..opposite.has_scalar α R }
instance (R : Type*) [monoid R] [add_monoid α] [distrib_mul_action R α] :
distrib_mul_action R (opposite α) :=
{ smul_add := λ r x₁ x₂, unop_injective $ smul_add r (unop x₁) (unop x₂),
smul_zero := λ r, unop_injective $ smul_zero r,
..opposite.mul_action α R }
/-- Like `monoid.to_mul_action`, but multiplies on the right. -/
instance monoid.to_opposite_mul_action [monoid α] : mul_action (opposite α) α :=
{ smul := λ c x, x * c.unop,
one_smul := mul_one,
mul_smul := λ x y r, (mul_assoc _ _ _).symm }
-- The above instance does not create an unwanted diamond, the two paths to
-- `mul_action (opposite α) (opposite α)` are defeq.
example [monoid α] : monoid.to_mul_action (opposite α) = opposite.mul_action α (opposite α) := rfl
lemma op_smul_eq_mul [monoid α] {a a' : α} : op a • a' = a' * a := rfl
@[simp] lemma op_zero [has_zero α] : op (0 : α) = 0 := rfl
@[simp] lemma unop_zero [has_zero α] : unop (0 : αᵒᵖ) = 0 := rfl
@[simp] lemma op_one [has_one α] : op (1 : α) = 1 := rfl
@[simp] lemma unop_one [has_one α] : unop (1 : αᵒᵖ) = 1 := rfl
variable {α}
@[simp] lemma op_add [has_add α] (x y : α) : op (x + y) = op x + op y := rfl
@[simp] lemma unop_add [has_add α] (x y : αᵒᵖ) : unop (x + y) = unop x + unop y := rfl
@[simp] lemma op_neg [has_neg α] (x : α) : op (-x) = -op x := rfl
@[simp] lemma unop_neg [has_neg α] (x : αᵒᵖ) : unop (-x) = -unop x := rfl
@[simp] lemma op_mul [has_mul α] (x y : α) : op (x * y) = op y * op x := rfl
@[simp] lemma unop_mul [has_mul α] (x y : αᵒᵖ) : unop (x * y) = unop y * unop x := rfl
@[simp] lemma op_inv [has_inv α] (x : α) : op (x⁻¹) = (op x)⁻¹ := rfl
@[simp] lemma unop_inv [has_inv α] (x : αᵒᵖ) : unop (x⁻¹) = (unop x)⁻¹ := rfl
@[simp] lemma op_sub [add_group α] (x y : α) : op (x - y) = op x - op y := rfl
@[simp] lemma unop_sub [add_group α] (x y : αᵒᵖ) : unop (x - y) = unop x - unop y := rfl
@[simp] lemma op_smul {R : Type*} [has_scalar R α] (c : R) (a : α) : op (c • a) = c • op a := rfl
@[simp] lemma unop_smul {R : Type*} [has_scalar R α] (c : R) (a : αᵒᵖ) :
unop (c • a) = c • unop a := rfl
lemma semiconj_by.op [has_mul α] {a x y : α} (h : semiconj_by a x y) :
semiconj_by (op a) (op y) (op x) :=
begin
dunfold semiconj_by,
rw [← op_mul, ← op_mul, h.eq]
end
lemma semiconj_by.unop [has_mul α] {a x y : αᵒᵖ} (h : semiconj_by a x y) :
semiconj_by (unop a) (unop y) (unop x) :=
begin
dunfold semiconj_by,
rw [← unop_mul, ← unop_mul, h.eq]
end
@[simp] lemma semiconj_by_op [has_mul α] {a x y : α} :
semiconj_by (op a) (op y) (op x) ↔ semiconj_by a x y :=
begin
split,
{ intro h,
rw [← unop_op a, ← unop_op x, ← unop_op y],
exact semiconj_by.unop h },
{ intro h,
exact semiconj_by.op h }
end
@[simp] lemma semiconj_by_unop [has_mul α] {a x y : αᵒᵖ} :
semiconj_by (unop a) (unop y) (unop x) ↔ semiconj_by a x y :=
by conv_rhs { rw [← op_unop a, ← op_unop x, ← op_unop y, semiconj_by_op] }
lemma commute.op [has_mul α] {x y : α} (h : commute x y) : commute (op x) (op y) :=
begin
dunfold commute at h ⊢,
exact semiconj_by.op h
end
lemma commute.unop [has_mul α] {x y : αᵒᵖ} (h : commute x y) : commute (unop x) (unop y) :=
begin
dunfold commute at h ⊢,
exact semiconj_by.unop h
end
@[simp] lemma commute_op [has_mul α] {x y : α} :
commute (op x) (op y) ↔ commute x y :=
begin
dunfold commute,
rw semiconj_by_op
end
@[simp] lemma commute_unop [has_mul α] {x y : αᵒᵖ} :
commute (unop x) (unop y) ↔ commute x y :=
begin
dunfold commute,
rw semiconj_by_unop
end
/-- The function `op` is an additive equivalence. -/
def op_add_equiv [has_add α] : α ≃+ αᵒᵖ :=
{ map_add' := λ a b, rfl, .. equiv_to_opposite }
@[simp] lemma coe_op_add_equiv [has_add α] : (op_add_equiv : α → αᵒᵖ) = op := rfl
@[simp] lemma coe_op_add_equiv_symm [has_add α] :
(op_add_equiv.symm : αᵒᵖ → α) = unop := rfl
@[simp] lemma op_add_equiv_to_equiv [has_add α] :
(op_add_equiv : α ≃+ αᵒᵖ).to_equiv = equiv_to_opposite :=
rfl
end opposite
open opposite
/-- Inversion on a group is a `mul_equiv` to the opposite group. When `G` is commutative, there is
`mul_equiv.inv`. -/
@[simps {fully_applied := ff}]
def mul_equiv.inv' (G : Type*) [group G] : G ≃* Gᵒᵖ :=
{ to_fun := opposite.op ∘ has_inv.inv,
inv_fun := has_inv.inv ∘ opposite.unop,
map_mul' := λ x y, unop_injective $ mul_inv_rev x y,
..(equiv.inv G).trans equiv_to_opposite}
/-- A monoid homomorphism `f : R →* S` such that `f x` commutes with `f y` for all `x, y` defines
a monoid homomorphism to `Sᵒᵖ`. -/
@[simps {fully_applied := ff}]
def monoid_hom.to_opposite {R S : Type*} [mul_one_class R] [mul_one_class S] (f : R →* S)
(hf : ∀ x y, commute (f x) (f y)) : R →* Sᵒᵖ :=
{ to_fun := opposite.op ∘ f,
map_one' := congr_arg op f.map_one,
map_mul' := λ x y, by simp [(hf x y).eq] }
/-- A ring homomorphism `f : R →+* S` such that `f x` commutes with `f y` for all `x, y` defines
a ring homomorphism to `Sᵒᵖ`. -/
@[simps {fully_applied := ff}]
def ring_hom.to_opposite {R S : Type*} [semiring R] [semiring S] (f : R →+* S)
(hf : ∀ x y, commute (f x) (f y)) : R →+* Sᵒᵖ :=
{ to_fun := opposite.op ∘ f,
.. ((opposite.op_add_equiv : S ≃+ Sᵒᵖ).to_add_monoid_hom.comp ↑f : R →+ Sᵒᵖ),
.. f.to_monoid_hom.to_opposite hf }
/-- The units of the opposites are equivalent to the opposites of the units. -/
def units.op_equiv {R} [monoid R] : units Rᵒᵖ ≃* (units R)ᵒᵖ :=
{ to_fun := λ u, op ⟨unop u, unop ↑(u⁻¹), op_injective u.4, op_injective u.3⟩,
inv_fun := op_induction $ λ u, ⟨op ↑(u), op ↑(u⁻¹), unop_injective $ u.4, unop_injective u.3⟩,
map_mul' := λ x y, unop_injective $ units.ext $ rfl,
left_inv := λ x, units.ext $ rfl,
right_inv := λ x, unop_injective $ units.ext $ rfl }
@[simp]
lemma units.coe_unop_op_equiv {R} [monoid R] (u : units Rᵒᵖ) :
((units.op_equiv u).unop : R) = unop (u : Rᵒᵖ) :=
rfl
@[simp]
lemma units.coe_op_equiv_symm {R} [monoid R] (u : (units R)ᵒᵖ) :
(units.op_equiv.symm u : Rᵒᵖ) = op (u.unop : R) :=
rfl
/-- A hom `α →* β` can equivalently be viewed as a hom `αᵒᵖ →* βᵒᵖ`. This is the action of the
(fully faithful) `ᵒᵖ`-functor on morphisms. -/
@[simps]
def monoid_hom.op {α β} [mul_one_class α] [mul_one_class β] :
(α →* β) ≃ (αᵒᵖ →* βᵒᵖ) :=
{ to_fun := λ f, { to_fun := op ∘ f ∘ unop,
map_one' := unop_injective f.map_one,
map_mul' := λ x y, unop_injective (f.map_mul y.unop x.unop) },
inv_fun := λ f, { to_fun := unop ∘ f ∘ op,
map_one' := congr_arg unop f.map_one,
map_mul' := λ x y, congr_arg unop (f.map_mul (op y) (op x)) },
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, refl } }
/-- The 'unopposite' of a monoid hom `αᵒᵖ →* βᵒᵖ`. Inverse to `monoid_hom.op`. -/
@[simp] def monoid_hom.unop {α β} [mul_one_class α] [mul_one_class β] :
(αᵒᵖ →* βᵒᵖ) ≃ (α →* β) := monoid_hom.op.symm
/-- A hom `α →+ β` can equivalently be viewed as a hom `αᵒᵖ →+ βᵒᵖ`. This is the action of the
(fully faithful) `ᵒᵖ`-functor on morphisms. -/
@[simps]
def add_monoid_hom.op {α β} [add_zero_class α] [add_zero_class β] :
(α →+ β) ≃ (αᵒᵖ →+ βᵒᵖ) :=
{ to_fun := λ f, { to_fun := op ∘ f ∘ unop,
map_zero' := unop_injective f.map_zero,
map_add' := λ x y, unop_injective (f.map_add x.unop y.unop) },
inv_fun := λ f, { to_fun := unop ∘ f ∘ op,
map_zero' := congr_arg unop f.map_zero,
map_add' := λ x y, congr_arg unop (f.map_add (op x) (op y)) },
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, refl } }
/-- The 'unopposite' of an additive monoid hom `αᵒᵖ →+ βᵒᵖ`. Inverse to `add_monoid_hom.op`. -/
@[simp] def add_monoid_hom.unop {α β} [add_zero_class α] [add_zero_class β] :
(αᵒᵖ →+ βᵒᵖ) ≃ (α →+ β) := add_monoid_hom.op.symm
/-- A ring hom `α →+* β` can equivalently be viewed as a ring hom `αᵒᵖ →+* βᵒᵖ`. This is the action
of the (fully faithful) `ᵒᵖ`-functor on morphisms. -/
@[simps]
def ring_hom.op {α β} [non_assoc_semiring α] [non_assoc_semiring β] :
(α →+* β) ≃ (αᵒᵖ →+* βᵒᵖ) :=
{ to_fun := λ f, { ..f.to_add_monoid_hom.op, ..f.to_monoid_hom.op },
inv_fun := λ f, { ..f.to_add_monoid_hom.unop, ..f.to_monoid_hom.unop },
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, refl } }
/-- The 'unopposite' of a ring hom `αᵒᵖ →+* βᵒᵖ`. Inverse to `ring_hom.op`. -/
@[simp] def ring_hom.unop {α β} [non_assoc_semiring α] [non_assoc_semiring β] :
(αᵒᵖ →+* βᵒᵖ) ≃ (α →+* β) := ring_hom.op.symm
|
e98383cede8cae38e835ce0519d136b2886cab93 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/tactic/lint/frontend.lean | 7d1f54f28445d4524d508ef606018f06cd7698e0 | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 13,858 | lean | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Robert Y. Lewis, Gabriel Ebner
-/
import tactic.lint.basic
/-!
# Linter frontend and commands
This file defines the linter commands which spot common mistakes in the code.
* `#lint`: check all declarations in the current file
* `#lint_mathlib`: check all declarations in mathlib (so excluding core or other projects,
and also excluding the current file)
* `#lint_all`: check all declarations in the environment (the current file and all
imported files)
For a list of default / non-default linters, see the "Linting Commands" user command doc entry.
The command `#list_linters` prints a list of the names of all available linters.
You can append a `*` to any command (e.g. `#lint_mathlib*`) to omit the slow tests (4).
You can append a `-` to any command (e.g. `#lint_mathlib-`) to run a silent lint
that suppresses the output if all checks pass.
A silent lint will fail if any test fails.
You can append a `+` to any command (e.g. `#lint_mathlib+`) to run a verbose lint
that reports the result of each linter, including the successes.
You can append a sequence of linter names to any command to run extra tests, in addition to the
default ones. e.g. `#lint doc_blame_thm` will run all default tests and `doc_blame_thm`.
You can append `only name1 name2 ...` to any command to run a subset of linters, e.g.
`#lint only unused_arguments`
You can add custom linters by defining a term of type `linter` in the `linter` namespace.
A linter defined with the name `linter.my_new_check` can be run with `#lint my_new_check`
or `lint only my_new_check`.
If you add the attribute `@[linter]` to `linter.my_new_check` it will run by default.
Adding the attribute `@[nolint doc_blame unused_arguments]` to a declaration
omits it from only the specified linter checks.
## Tags
sanity check, lint, cleanup, command, tactic
-/
open tactic expr native
setup_tactic_parser
/--
Verbosity for the linter output.
* `low`: only print failing checks, print nothing on success.
* `medium`: only print failing checks, print confirmation on success.
* `high`: print output of every check.
-/
@[derive [decidable_eq, inhabited]]
inductive lint_verbosity | low | medium | high
/-- `get_checks slow extra use_only` produces a list of linters.
`extras` is a list of names that should resolve to declarations with type `linter`.
If `use_only` is true, it only uses the linters in `extra`.
Otherwise, it uses all linters in the environment tagged with `@[linter]`.
If `slow` is false, it only uses the fast default tests. -/
meta def get_checks (slow : bool) (extra : list name) (use_only : bool) :
tactic (list (name × linter)) := do
default ← if use_only then return [] else attribute.get_instances `linter >>= get_linters,
let default := if slow then default else default.filter (λ l, l.2.is_fast),
list.append default <$> get_linters extra
/--
`lint_core all_decls non_auto_decls checks` applies the linters `checks` to the list of
declarations.
If `auto_decls` is false for a linter (default) the linter is applied to `non_auto_decls`.
If `auto_decls` is true, then it is applied to `all_decls`.
The resulting list has one element for each linter, containing the linter as
well as a map from declaration name to warning.
-/
meta def lint_core (all_decls non_auto_decls : list declaration) (checks : list (name × linter)) :
tactic (list (name × linter × rb_map name string)) := do
checks.mmap $ λ ⟨linter_name, linter⟩, do
let test_decls := if linter.auto_decls then all_decls else non_auto_decls,
test_decls ← test_decls.mfilter (λ decl, should_be_linted linter_name decl.to_name),
s ← read,
let results := test_decls.map_async_chunked $ λ decl, prod.mk decl.to_name $
match linter.test decl s with
| result.success w _ := w
| result.exception msg _ _ :=
some $ "LINTER FAILED:\n" ++ msg.elim "(no message)" (λ msg, to_string $ msg ())
end,
let results := results.foldl (λ (results : rb_map name string) warning,
match warning with
| (decl_name, some w) := results.insert decl_name w
| (_, none) := results
end) mk_rb_map,
pure (linter_name, linter, results)
/-- Sorts a map with declaration keys as names by line number. -/
meta def sort_results {α} (e : environment) (results : rb_map name α) : list (name × α) :=
list.reverse $ rb_lmap.values $ rb_lmap.of_list $
results.fold [] $ λ decl linter_warning results,
(((e.decl_pos decl).get_or_else ⟨0,0⟩).line, (decl, linter_warning)) :: results
/-- Formats a linter warning as `#print` command with comment. -/
meta def print_warning (decl_name : name) (warning : string) : format :=
"#print " ++ to_fmt decl_name ++ " /- " ++ warning ++ " -/"
/-- Formats a map of linter warnings using `print_warning`, sorted by line number. -/
meta def print_warnings (env : environment) (results : rb_map name string) : format :=
format.intercalate format.line $ (sort_results env results).map $
λ ⟨decl_name, warning⟩, print_warning decl_name warning
/--
Formats a map of linter warnings grouped by filename with `-- filename` comments.
The first `drop_fn_chars` characters are stripped from the filename.
-/
meta def grouped_by_filename (e : environment) (results : rb_map name string) (drop_fn_chars := 0)
(formatter: rb_map name string → format) : format :=
let results := results.fold (rb_map.mk string (rb_map name string)) $
λ decl_name linter_warning results,
let fn := (e.decl_olean decl_name).get_or_else "" in
results.insert fn (((results.find fn).get_or_else mk_rb_map).insert
decl_name linter_warning) in
let l := results.to_list.reverse.map (λ ⟨fn, results⟩,
("-- " ++ fn.popn drop_fn_chars ++ "\n" ++ formatter results : format)) in
format.intercalate "\n\n" l ++ "\n"
/--
Formats the linter results as Lean code with comments and `#print` commands.
-/
meta def format_linter_results
(env : environment)
(results : list (name × linter × rb_map name string))
(decls non_auto_decls : list declaration)
(group_by_filename : option nat)
(where_desc : string) (slow : bool) (verbose : lint_verbosity) :
format := do
let formatted_results := results.map $ λ ⟨linter_name, linter, results⟩,
let report_str : format := to_fmt "/- The `" ++ to_fmt linter_name ++ "` linter reports: -/\n" in
if ¬ results.empty then
let warnings := match group_by_filename with
| none := print_warnings env results
| some dropped := grouped_by_filename env results dropped (print_warnings env)
end in
report_str ++ "/- " ++ linter.errors_found ++ " -/\n" ++ warnings ++ "\n"
else if verbose = lint_verbosity.high then
"/- OK: " ++ linter.no_errors_found ++ " -/"
else format.nil,
let s := format.intercalate "\n" (formatted_results.filter (λ f, ¬ f.is_nil)),
let s := if verbose = lint_verbosity.low then s else
format!("/- Checking {non_auto_decls.length} declarations (plus " ++
"{decls.length - non_auto_decls.length} automatically generated ones) {where_desc} -/\n\n") ++ s,
let s := if slow then s else s ++ "/- (slow tests skipped) -/\n",
s
/-- The common denominator of `#lint[|mathlib|all]`.
The different commands have different configurations for `l`,
`group_by_filename` and `where_desc`.
If `slow` is false, doesn't do the checks that take a lot of time.
If `verbose` is false, it will suppress messages from passing checks.
By setting `checks` you can customize which checks are performed.
Returns a `name_set` containing the names of all declarations that fail any check in `check`,
and a `format` object describing the failures. -/
meta def lint_aux (decls : list declaration) (group_by_filename : option nat)
(where_desc : string) (slow : bool) (verbose : lint_verbosity) (checks : list (name × linter)) :
tactic (name_set × format) := do
e ← get_env,
let non_auto_decls := decls.filter (λ d, ¬ d.is_auto_or_internal e),
results ← lint_core decls non_auto_decls checks,
let s := format_linter_results e results decls non_auto_decls
group_by_filename where_desc slow verbose,
let ns := name_set.of_list (do (_,_,rs) ← results, rs.keys),
pure (ns, s)
/-- Return the message printed by `#lint` and a `name_set` containing all declarations that fail. -/
meta def lint (slow : bool := tt) (verbose : lint_verbosity := lint_verbosity.medium)
(extra : list name := []) (use_only : bool := ff) : tactic (name_set × format) := do
checks ← get_checks slow extra use_only,
e ← get_env,
let l := e.filter (λ d, e.in_current_file d.to_name),
lint_aux l none "in the current file" slow verbose checks
/-- Returns the declarations considered by the mathlib linter. -/
meta def lint_mathlib_decls : tactic (list declaration) := do
e ← get_env,
ml ← get_mathlib_dir,
pure $ e.filter $ λ d, e.is_prefix_of_file ml d.to_name
/-- Return the message printed by `#lint_mathlib` and a `name_set` containing all declarations
that fail. -/
meta def lint_mathlib (slow : bool := tt) (verbose : lint_verbosity := lint_verbosity.medium)
(extra : list name := []) (use_only : bool := ff) : tactic (name_set × format) := do
checks ← get_checks slow extra use_only,
decls ← lint_mathlib_decls,
mathlib_path_len ← string.length <$> get_mathlib_dir,
lint_aux decls mathlib_path_len "in mathlib (only in imported files)" slow verbose checks
/-- Return the message printed by `#lint_all` and a `name_set` containing all declarations
that fail. -/
meta def lint_all (slow : bool := tt) (verbose : lint_verbosity := lint_verbosity.medium)
(extra : list name := []) (use_only : bool := ff) : tactic (name_set × format) := do
checks ← get_checks slow extra use_only,
e ← get_env,
let l := e.get_decls,
lint_aux l (some 0) "in all imported files (including this one)" slow verbose checks
/-- Parses an optional `only`, followed by a sequence of zero or more identifiers.
Prepends `linter.` to each of these identifiers. -/
private meta def parse_lint_additions : parser (bool × list name) :=
prod.mk <$> only_flag <*> (list.map (name.append `linter) <$> ident_*)
/--
Parses a "-" or "+", returning `lint_verbosity.low` or `lint_verbosity.high` respectively,
or returns `none`.
-/
private meta def parse_verbosity : parser (option lint_verbosity) :=
tk "-" >> return lint_verbosity.low <|>
tk "+" >> return lint_verbosity.high <|>
return none
/-- The common denominator of `lint_cmd`, `lint_mathlib_cmd`, `lint_all_cmd` -/
private meta def lint_cmd_aux
(scope : bool → lint_verbosity → list name → bool → tactic (name_set × format)) :
parser unit :=
do verbosity ← parse_verbosity,
fast_only ← optional (tk "*"),
-- allow either order of *-
verbosity ← if verbosity.is_some then return verbosity else parse_verbosity,
let verbosity := verbosity.get_or_else lint_verbosity.medium,
(use_only, extra) ← parse_lint_additions,
(failed, s) ← scope fast_only.is_none verbosity extra use_only,
when (¬ s.is_nil) $ trace s,
when (verbosity = lint_verbosity.low ∧ ¬ failed.empty) $
fail "Linting did not succeed",
when (verbosity = lint_verbosity.medium ∧ failed.empty) $
trace "/- All linting checks passed! -/"
/-- The command `#lint` at the bottom of a file will warn you about some common mistakes
in that file. Usage: `#lint`, `#lint linter_1 linter_2`, `#lint only linter_1 linter_2`.
`#lint-` will suppress the output if all checks pass.
`#lint+` will enable verbose output.
Use the command `#list_linters` to see all available linters. -/
@[user_command] meta def lint_cmd (_ : parse $ tk "#lint") : parser unit :=
lint_cmd_aux @lint
/-- The command `#lint_mathlib` checks all of mathlib for certain mistakes.
Usage: `#lint_mathlib`, `#lint_mathlib linter_1 linter_2`, `#lint_mathlib only linter_1 linter_2`.
`#lint_mathlib-` will suppress the output if all checks pass.
`lint_mathlib+` will enable verbose output.
Use the command `#list_linters` to see all available linters. -/
@[user_command] meta def lint_mathlib_cmd (_ : parse $ tk "#lint_mathlib") : parser unit :=
lint_cmd_aux @lint_mathlib
/-- The command `#lint_all` checks all imported files for certain mistakes.
Usage: `#lint_all`, `#lint_all linter_1 linter_2`, `#lint_all only linter_1 linter_2`.
`#lint_all-` will suppress the output if all checks pass.
`lint_all+` will enable verbose output.
Use the command `#list_linters` to see all available linters. -/
@[user_command] meta def lint_all_cmd (_ : parse $ tk "#lint_all") : parser unit :=
lint_cmd_aux @lint_all
/-- The command `#list_linters` prints a list of all available linters. -/
@[user_command] meta def list_linters (_ : parse $ tk "#list_linters") : parser unit :=
do env ← get_env,
let ns := env.decl_filter_map $ λ dcl,
if (dcl.to_name.get_prefix = `linter) && (dcl.type = `(linter)) then some dcl.to_name else none,
trace "Available linters:\n linters marked with (*) are in the default lint set\n",
ns.mmap' $ λ n, do
b ← has_attribute' `linter n,
trace $ n.pop_prefix.to_string ++ if b then " (*)" else ""
/--
Invoking the hole command `lint` ("Find common mistakes in current file") will print text that
indicates mistakes made in the file above the command. It is equivalent to copying and pasting the
output of `#lint`. On large files, it may take some time before the output appears.
-/
@[hole_command] meta def lint_hole_cmd : hole_command :=
{ name := "Lint",
descr := "Lint: Find common mistakes in current file.",
action := λ es, do (_, s) ← lint, return [(s.to_string,"")] }
add_tactic_doc
{ name := "Lint",
category := doc_category.hole_cmd,
decl_names := [`lint_hole_cmd],
tags := ["linting"] }
|
c1e754c23c13a14b0dc6b230b2519efe966b3ea1 | 618003631150032a5676f229d13a079ac875ff77 | /src/category_theory/concrete_category/bundled.lean | ee3612cf0e5d2e4fde4900a32eddd169ae6922cc | [
"Apache-2.0"
] | permissive | awainverse/mathlib | 939b68c8486df66cfda64d327ad3d9165248c777 | ea76bd8f3ca0a8bf0a166a06a475b10663dec44a | refs/heads/master | 1,659,592,962,036 | 1,590,987,592,000 | 1,590,987,592,000 | 268,436,019 | 1 | 0 | Apache-2.0 | 1,590,990,500,000 | 1,590,990,500,000 | null | UTF-8 | Lean | false | false | 1,877 | lean | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johannes Hölzl, Reid Barton, Sean Leather
Bundled types.
-/
import tactic.doc_commands
/-!
`bundled c` provides a uniform structure for bundling a type equipped with a type class.
We provide `category` instances for these in `unbundled_hom.lean` (for categories with unbundled
homs, e.g. topological spaces) and in `bundled_hom.lean` (for categories with bundled homs, e.g.
monoids).
-/
universes u v
namespace category_theory
variables {c d : Type u → Type v} {α : Type u}
/-- `bundled` is a type bundled with a type class instance for that type. Only
the type class is exposed as a parameter. -/
structure bundled (c : Type u → Type v) : Type (max (u+1) v) :=
(α : Type u)
(str : c α . tactic.apply_instance)
namespace bundled
/-- A generic function for lifting a type equipped with an instance to a bundled object. -/
-- Usually explicit instances will provide their own version of this, e.g. `Mon.of` and `Top.of`.
def of {c : Type u → Type v} (α : Type u) [str : c α] : bundled c := ⟨α, str⟩
instance : has_coe_to_sort (bundled c) :=
{ S := Type u, coe := bundled.α }
@[simp]
lemma coe_mk (α) (str) : (@bundled.mk c α str : Type u) = α := rfl
/-
`bundled.map` is reducible so that, if we define a category
def Ring : Type (u+1) := induced_category SemiRing (bundled.map @ring.to_semiring)
instance search is able to "see" that a morphism R ⟶ S in Ring is really
a (semi)ring homomorphism from R.α to S.α, and not merely from
`(bundled.map @ring.to_semiring R).α` to `(bundled.map @ring.to_semiring S).α`.
-/
/-- Map over the bundled structure -/
@[reducible] def map (f : Π {α}, c α → d α) (b : bundled c) : bundled d :=
⟨b, f b.str⟩
end bundled
end category_theory
|
cd8586f156719da10044b99f50d47b540029f115 | 94e33a31faa76775069b071adea97e86e218a8ee | /src/analysis/locally_convex/basic.lean | 1932ee782c0666e3036d88ac0ae6f31c013c1e1f | [
"Apache-2.0"
] | permissive | urkud/mathlib | eab80095e1b9f1513bfb7f25b4fa82fa4fd02989 | 6379d39e6b5b279df9715f8011369a301b634e41 | refs/heads/master | 1,658,425,342,662 | 1,658,078,703,000 | 1,658,078,703,000 | 186,910,338 | 0 | 0 | Apache-2.0 | 1,568,512,083,000 | 1,557,958,709,000 | Lean | UTF-8 | Lean | false | false | 13,897 | lean | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Bhavik Mehta, Yaël Dillies
-/
import analysis.normed_space.basic
/-!
# Local convexity
This file defines absorbent and balanced sets.
An absorbent set is one that "surrounds" the origin. The idea is made precise by requiring that any
point belongs to all large enough scalings of the set. This is the vector world analog of a
topological neighborhood of the origin.
A balanced set is one that is everywhere around the origin. This means that `a • s ⊆ s` for all `a`
of norm less than `1`.
## Main declarations
For a module over a normed ring:
* `absorbs`: A set `s` absorbs a set `t` if all large scalings of `s` contain `t`.
* `absorbent`: A set `s` is absorbent if every point eventually belongs to all large scalings of
`s`.
* `balanced`: A set `s` is balanced if `a • s ⊆ s` for all `a` of norm less than `1`.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
absorbent, balanced, locally convex, LCTVS
-/
open set
open_locale pointwise topological_space
variables {𝕜 𝕝 E : Type*} {ι : Sort*} {κ : ι → Sort*}
section semi_normed_ring
variables [semi_normed_ring 𝕜]
section has_smul
variables (𝕜) [has_smul 𝕜 E]
/-- A set `A` absorbs another set `B` if `B` is contained in all scalings of `A` by elements of
sufficiently large norm. -/
def absorbs (A B : set E) := ∃ r, 0 < r ∧ ∀ a : 𝕜, r ≤ ∥a∥ → B ⊆ a • A
variables {𝕜} {s t u v A B : set E}
@[simp] lemma absorbs_empty {s : set E}: absorbs 𝕜 s (∅ : set E) :=
⟨1, one_pos, λ a ha, set.empty_subset _⟩
lemma absorbs.mono (hs : absorbs 𝕜 s u) (hst : s ⊆ t) (hvu : v ⊆ u) : absorbs 𝕜 t v :=
let ⟨r, hr, h⟩ := hs in ⟨r, hr, λ a ha, hvu.trans $ (h _ ha).trans $ smul_set_mono hst⟩
lemma absorbs.mono_left (hs : absorbs 𝕜 s u) (h : s ⊆ t) : absorbs 𝕜 t u := hs.mono h subset.rfl
lemma absorbs.mono_right (hs : absorbs 𝕜 s u) (h : v ⊆ u) : absorbs 𝕜 s v := hs.mono subset.rfl h
lemma absorbs.union (hu : absorbs 𝕜 s u) (hv : absorbs 𝕜 s v) : absorbs 𝕜 s (u ∪ v) :=
begin
obtain ⟨a, ha, hu⟩ := hu,
obtain ⟨b, hb, hv⟩ := hv,
exact ⟨max a b, lt_max_of_lt_left ha,
λ c hc, union_subset (hu _ $ le_of_max_le_left hc) (hv _ $ le_of_max_le_right hc)⟩,
end
@[simp] lemma absorbs_union : absorbs 𝕜 s (u ∪ v) ↔ absorbs 𝕜 s u ∧ absorbs 𝕜 s v :=
⟨λ h, ⟨h.mono_right $ subset_union_left _ _, h.mono_right $ subset_union_right _ _⟩,
λ h, h.1.union h.2⟩
lemma absorbs_Union_finset {ι : Type*} {t : finset ι} {f : ι → set E} :
absorbs 𝕜 s (⋃ i ∈ t, f i) ↔ ∀ i ∈ t, absorbs 𝕜 s (f i) :=
begin
classical,
induction t using finset.induction_on with i t ht hi,
{ simp only [finset.not_mem_empty, set.Union_false, set.Union_empty, absorbs_empty,
is_empty.forall_iff, implies_true_iff] },
rw [finset.set_bUnion_insert, absorbs_union, hi],
split; intro h,
{ refine λ _ hi', (finset.mem_insert.mp hi').elim _ (h.2 _),
exact (λ hi'', by { rw hi'', exact h.1 }) },
exact ⟨h i (finset.mem_insert_self i t), λ i' hi', h i' (finset.mem_insert_of_mem hi')⟩,
end
lemma set.finite.absorbs_Union {ι : Type*} {s : set E} {t : set ι} {f : ι → set E} (hi : t.finite) :
absorbs 𝕜 s (⋃ i ∈ t, f i) ↔ ∀ i ∈ t, absorbs 𝕜 s (f i) :=
begin
lift t to finset ι using hi,
simp only [finset.mem_coe],
exact absorbs_Union_finset,
end
variables (𝕜)
/-- A set is absorbent if it absorbs every singleton. -/
def absorbent (A : set E) := ∀ x, ∃ r, 0 < r ∧ ∀ a : 𝕜, r ≤ ∥a∥ → x ∈ a • A
variables {𝕜}
lemma absorbent.subset (hA : absorbent 𝕜 A) (hAB : A ⊆ B) : absorbent 𝕜 B :=
begin
refine forall_imp (λ x, _) hA,
exact Exists.imp (λ r, and.imp_right $ forall₂_imp $ λ a ha hx, set.smul_set_mono hAB hx),
end
lemma absorbent_iff_forall_absorbs_singleton : absorbent 𝕜 A ↔ ∀ x, absorbs 𝕜 A {x} :=
by simp_rw [absorbs, absorbent, singleton_subset_iff]
lemma absorbent.absorbs (hs : absorbent 𝕜 s) {x : E} : absorbs 𝕜 s {x} :=
absorbent_iff_forall_absorbs_singleton.1 hs _
lemma absorbent_iff_nonneg_lt : absorbent 𝕜 A ↔ ∀ x, ∃ r, 0 ≤ r ∧ ∀ ⦃a : 𝕜⦄, r < ∥a∥ → x ∈ a • A :=
forall_congr $ λ x, ⟨λ ⟨r, hr, hx⟩, ⟨r, hr.le, λ a ha, hx a ha.le⟩, λ ⟨r, hr, hx⟩,
⟨r + 1, add_pos_of_nonneg_of_pos hr zero_lt_one,
λ a ha, hx ((lt_add_of_pos_right r zero_lt_one).trans_le ha)⟩⟩
lemma absorbent.absorbs_finite {s : set E} (hs : absorbent 𝕜 s) {v : set E} (hv : v.finite) :
absorbs 𝕜 s v :=
begin
rw ←set.bUnion_of_singleton v,
exact hv.absorbs_Union.mpr (λ _ _, hs.absorbs),
end
variables (𝕜)
/-- A set `A` is balanced if `a • A` is contained in `A` whenever `a` has norm at most `1`. -/
def balanced (A : set E) := ∀ a : 𝕜, ∥a∥ ≤ 1 → a • A ⊆ A
variables {𝕜}
lemma balanced_iff_smul_mem : balanced 𝕜 s ↔ ∀ ⦃a : 𝕜⦄, ∥a∥ ≤ 1 → ∀ ⦃x : E⦄, x ∈ s → a • x ∈ s :=
forall₂_congr $ λ a ha, smul_set_subset_iff
alias balanced_iff_smul_mem ↔ balanced.smul_mem _
@[simp] lemma balanced_empty : balanced 𝕜 (∅ : set E) :=
λ _ _, by { rw smul_set_empty }
@[simp] lemma balanced_univ : balanced 𝕜 (univ : set E) := λ a ha, subset_univ _
lemma balanced.union (hA : balanced 𝕜 A) (hB : balanced 𝕜 B) : balanced 𝕜 (A ∪ B) :=
λ a ha, smul_set_union.subset.trans $ union_subset_union (hA _ ha) $ hB _ ha
lemma balanced.inter (hA : balanced 𝕜 A) (hB : balanced 𝕜 B) : balanced 𝕜 (A ∩ B) :=
λ a ha, smul_set_inter_subset.trans $ inter_subset_inter (hA _ ha) $ hB _ ha
lemma balanced_Union {f : ι → set E} (h : ∀ i, balanced 𝕜 (f i)) : balanced 𝕜 (⋃ i, f i) :=
λ a ha, (smul_set_Union _ _).subset.trans $ Union_mono $ λ _, h _ _ ha
lemma balanced_Union₂ {f : Π i, κ i → set E} (h : ∀ i j, balanced 𝕜 (f i j)) :
balanced 𝕜 (⋃ i j, f i j) :=
balanced_Union $ λ _, balanced_Union $ h _
lemma balanced_Inter {f : ι → set E} (h : ∀ i, balanced 𝕜 (f i)) : balanced 𝕜 (⋂ i, f i) :=
λ a ha, (smul_set_Inter_subset _ _).trans $ Inter_mono $ λ _, h _ _ ha
lemma balanced_Inter₂ {f : Π i, κ i → set E} (h : ∀ i j, balanced 𝕜 (f i j)) :
balanced 𝕜 (⋂ i j, f i j) :=
balanced_Inter $ λ _, balanced_Inter $ h _
variables [has_smul 𝕝 E] [smul_comm_class 𝕜 𝕝 E]
lemma balanced.smul (a : 𝕝) (hs : balanced 𝕜 s) : balanced 𝕜 (a • s) :=
λ b hb, (smul_comm _ _ _).subset.trans $ smul_set_mono $ hs _ hb
end has_smul
section module
variables [add_comm_group E] [module 𝕜 E] {s s₁ s₂ t t₁ t₂ : set E}
lemma absorbs.neg : absorbs 𝕜 s t → absorbs 𝕜 (-s) (-t) :=
Exists.imp $ λ r, and.imp_right $ forall₂_imp $ λ _ _ h,
(neg_subset_neg.2 h).trans set.smul_set_neg.superset
lemma balanced.neg : balanced 𝕜 s → balanced 𝕜 (-s) :=
forall₂_imp $ λ _ _ h, smul_set_neg.subset.trans $ neg_subset_neg.2 h
lemma absorbs.add : absorbs 𝕜 s₁ t₁ → absorbs 𝕜 s₂ t₂ → absorbs 𝕜 (s₁ + s₂) (t₁ + t₂) :=
λ ⟨r₁, hr₁, h₁⟩ ⟨r₂, hr₂, h₂⟩, ⟨max r₁ r₂, lt_max_of_lt_left hr₁, λ a ha, (add_subset_add
(h₁ _ $ le_of_max_le_left ha) $ h₂ _ $ le_of_max_le_right ha).trans (smul_add _ _ _).superset⟩
lemma balanced.add (hs : balanced 𝕜 s) (ht : balanced 𝕜 t) : balanced 𝕜 (s + t) :=
λ a ha, (smul_add _ _ _).subset.trans $ add_subset_add (hs _ ha) $ ht _ ha
lemma absorbs.sub (h₁ : absorbs 𝕜 s₁ t₁) (h₂ : absorbs 𝕜 s₂ t₂) : absorbs 𝕜 (s₁ - s₂) (t₁ - t₂) :=
by { simp_rw sub_eq_add_neg, exact h₁.add h₂.neg }
lemma balanced.sub (hs : balanced 𝕜 s) (ht : balanced 𝕜 t) : balanced 𝕜 (s - t) :=
by { simp_rw sub_eq_add_neg, exact hs.add ht.neg }
lemma balanced_zero : balanced 𝕜 (0 : set E) := λ a ha, (smul_zero _).subset
end module
end semi_normed_ring
section normed_field
variables [normed_field 𝕜] [normed_ring 𝕝] [normed_space 𝕜 𝕝] [add_comm_group E] [module 𝕜 E]
[smul_with_zero 𝕝 E] [is_scalar_tower 𝕜 𝕝 E] {s t u v A B : set E} {x : E} {a b : 𝕜}
/-- Scalar multiplication (by possibly different types) of a balanced set is monotone. -/
lemma balanced.smul_mono (hs : balanced 𝕝 s) {a : 𝕝} {b : 𝕜} (h : ∥a∥ ≤ ∥b∥) : a • s ⊆ b • s :=
begin
obtain rfl | hb := eq_or_ne b 0,
{ rw norm_zero at h,
rw norm_eq_zero.1 (h.antisymm $ norm_nonneg _),
obtain rfl | h := s.eq_empty_or_nonempty,
{ simp_rw [smul_set_empty] },
{ simp_rw [zero_smul_set h] } },
rintro _ ⟨x, hx, rfl⟩,
refine ⟨b⁻¹ • a • x, _, smul_inv_smul₀ hb _⟩,
rw ←smul_assoc,
refine hs _ _ (smul_mem_smul_set hx),
rw [norm_smul, norm_inv, ←div_eq_inv_mul],
exact div_le_one_of_le h (norm_nonneg _),
end
/-- A balanced set absorbs itself. -/
lemma balanced.absorbs_self (hA : balanced 𝕜 A) : absorbs 𝕜 A A :=
begin
refine ⟨1, zero_lt_one, λ a ha x hx, _⟩,
rw mem_smul_set_iff_inv_smul_mem₀ (norm_pos_iff.1 $ zero_lt_one.trans_le ha),
refine hA a⁻¹ _ (smul_mem_smul_set hx),
rw norm_inv,
exact inv_le_one ha,
end
lemma balanced.subset_smul (hA : balanced 𝕜 A) (ha : 1 ≤ ∥a∥) : A ⊆ a • A :=
begin
refine (subset_set_smul_iff₀ _).2 (hA (a⁻¹) _),
{ rintro rfl,
rw norm_zero at ha,
exact zero_lt_one.not_le ha },
{ rw norm_inv,
exact inv_le_one ha }
end
lemma balanced.smul_eq (hA : balanced 𝕜 A) (ha : ∥a∥ = 1) : a • A = A :=
(hA _ ha.le).antisymm $ hA.subset_smul ha.ge
lemma balanced.mem_smul_iff (hs : balanced 𝕜 s) (h : ∥a∥ = ∥b∥) : a • x ∈ s ↔ b • x ∈ s :=
begin
obtain rfl | hb := eq_or_ne b 0,
{ rw [norm_zero, norm_eq_zero] at h,
rw h },
have ha : a ≠ 0 := norm_ne_zero_iff.1 (ne_of_eq_of_ne h $ norm_ne_zero_iff.2 hb),
split; intro h'; [rw ←inv_mul_cancel_right₀ ha b, rw ←inv_mul_cancel_right₀ hb a];
{ rw [←smul_eq_mul, smul_assoc],
refine hs.smul_mem _ h',
simp [←h, ha] }
end
lemma balanced.neg_mem_iff (hs : balanced 𝕜 s) : -x ∈ s ↔ x ∈ s :=
by convert hs.mem_smul_iff (norm_neg 1); simp only [neg_smul, one_smul]
lemma absorbs.inter (hs : absorbs 𝕜 s u) (ht : absorbs 𝕜 t u) : absorbs 𝕜 (s ∩ t) u :=
begin
obtain ⟨a, ha, hs⟩ := hs,
obtain ⟨b, hb, ht⟩ := ht,
have h : 0 < max a b := lt_max_of_lt_left ha,
refine ⟨max a b, lt_max_of_lt_left ha, λ c hc, _⟩,
rw smul_set_inter₀ (norm_pos_iff.1 $ h.trans_le hc),
exact subset_inter (hs _ $ le_of_max_le_left hc) (ht _ $ le_of_max_le_right hc),
end
@[simp] lemma absorbs_inter : absorbs 𝕜 (s ∩ t) u ↔ absorbs 𝕜 s u ∧ absorbs 𝕜 t u :=
⟨λ h, ⟨h.mono_left $ inter_subset_left _ _, h.mono_left $ inter_subset_right _ _⟩,
λ h, h.1.inter h.2⟩
lemma absorbent_univ : absorbent 𝕜 (univ : set E) :=
begin
refine λ x, ⟨1, zero_lt_one, λ a ha, _⟩,
rw smul_set_univ₀ (norm_pos_iff.1 $ zero_lt_one.trans_le ha),
exact trivial,
end
variables [topological_space E] [has_continuous_smul 𝕜 E]
/-- Every neighbourhood of the origin is absorbent. -/
lemma absorbent_nhds_zero (hA : A ∈ 𝓝 (0 : E)) : absorbent 𝕜 A :=
begin
intro x,
obtain ⟨w, hw₁, hw₂, hw₃⟩ := mem_nhds_iff.mp hA,
have hc : continuous (λ t : 𝕜, t • x) := continuous_id.smul continuous_const,
obtain ⟨r, hr₁, hr₂⟩ := metric.is_open_iff.mp (hw₂.preimage hc) 0
(by rwa [mem_preimage, zero_smul]),
have hr₃ := inv_pos.mpr (half_pos hr₁),
refine ⟨(r / 2)⁻¹, hr₃, λ a ha₁, _⟩,
have ha₂ : 0 < ∥a∥ := hr₃.trans_le ha₁,
refine (mem_smul_set_iff_inv_smul_mem₀ (norm_pos_iff.mp ha₂) _ _).2 (hw₁ $ hr₂ _),
rw [metric.mem_ball, dist_zero_right, norm_inv],
calc ∥a∥⁻¹ ≤ r/2 : (inv_le (half_pos hr₁) ha₂).mp ha₁
... < r : half_lt_self hr₁,
end
/-- The union of `{0}` with the interior of a balanced set is balanced. -/
lemma balanced_zero_union_interior (hA : balanced 𝕜 A) : balanced 𝕜 ((0 : set E) ∪ interior A) :=
begin
intros a ha,
obtain rfl | h := eq_or_ne a 0,
{ rw zero_smul_set,
exacts [subset_union_left _ _, ⟨0, or.inl rfl⟩] },
{ rw [←image_smul, image_union],
apply union_subset_union,
{ rw [image_zero, smul_zero],
refl },
{ calc a • interior A ⊆ interior (a • A) : (is_open_map_smul₀ h).image_interior_subset A
... ⊆ interior A : interior_mono (hA _ ha) } }
end
/-- The interior of a balanced set is balanced if it contains the origin. -/
lemma balanced.interior (hA : balanced 𝕜 A) (h : (0 : E) ∈ interior A) : balanced 𝕜 (interior A) :=
begin
rw ←union_eq_self_of_subset_left (singleton_subset_iff.2 h),
exact balanced_zero_union_interior hA,
end
lemma balanced.closure (hA : balanced 𝕜 A) : balanced 𝕜 (closure A) :=
λ a ha,
(image_closure_subset_closure_image $ continuous_id.const_smul _).trans $ closure_mono $ hA _ ha
end normed_field
section nondiscrete_normed_field
variables [nondiscrete_normed_field 𝕜] [add_comm_group E] [module 𝕜 E] {s : set E}
lemma absorbs_zero_iff : absorbs 𝕜 s 0 ↔ (0 : E) ∈ s :=
begin
refine ⟨_, λ h, ⟨1, zero_lt_one, λ a _, zero_subset.2 $ zero_mem_smul_set h⟩⟩,
rintro ⟨r, hr, h⟩,
obtain ⟨a, ha⟩ := normed_space.exists_lt_norm 𝕜 𝕜 r,
have := h _ ha.le,
rwa [zero_subset, zero_mem_smul_set_iff] at this,
exact norm_ne_zero_iff.1 (hr.trans ha).ne',
end
lemma absorbent.zero_mem (hs : absorbent 𝕜 s) : (0 : E) ∈ s :=
absorbs_zero_iff.1 $ absorbent_iff_forall_absorbs_singleton.1 hs _
end nondiscrete_normed_field
|
17cdf14663823163712014b90c8ee92ac5920c86 | d751a70f46ed26dc0111a87f5bbe83e5c6648904 | /Code/src/inst/exec/topo.lean | d76aa54b4819f04c38337ce8d82210d07105e3e5 | [] | no_license | marcusrossel/bachelors-thesis | 92cb12ae8436c10fbfab9bfe4929a0081e615b37 | d1ec2c2b5c3c6700a506f2e3cc93f1160e44b422 | refs/heads/main | 1,682,873,547,703 | 1,619,795,735,000 | 1,619,795,735,000 | 306,041,494 | 2 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 7,444 | lean | import topo
import inst.exec.propagate
import inst.network.prec
open reactor
open reactor.ports
-- Cf. inst/primitives.lean
variables {υ : Type*} [decidable_eq υ]
namespace inst
namespace network
namespace graph
-- Runs a reaction in a network graph and propagates all of its affected output ports.
noncomputable def run_reaction (η : graph υ) (i : reaction.id) : graph υ :=
(η.run_local i).propagate_ports ((η.run_local i).index_diff η i.rtr role.output).val.to_list
-- Running a reaction is a network graph produces an equivalent network graph.
lemma run_reaction_equiv (η : graph υ) (i : reaction.id) : run_reaction η i ≈ η :=
begin
unfold run_reaction,
transitivity,
exact propagate_ports_equiv _ _,
exact run_local_equiv _ _,
end
-- A specialized version of `no_dep_no_eₒ_dep` for `run_reaction`.
-- This is needed in the proof of `run_reaction_comm`.
lemma run_reaction_no_path_no_eₒ_dep {η : graph υ} {ρ : prec.graph υ} (hw : ρ ⋈ η) {i i' : reaction.id} (hᵢ : ¬(i~ρ~>i')) (hₙ : i.rtr ≠ i'.rtr) :
∀ (p ∈ ((η.run_local i).index_diff η i.rtr role.output).val.to_list) (e : edge), (e ∈ (η.run_local i).eₒ p) → e.dst ∉ ((η.run_local i).deps i' role.input) ∧ (e.src ∉ (η.run_local i).deps i' role.output) :=
begin
intros p hₚ e hₑ,
have hₛ, from index_diff_sub_dₒ η i,
have h, from prec.graph.no_path_no_eₒ_dep hw hᵢ hₙ hₛ p,
rw multiset.mem_to_list at hₚ,
replace h := h hₚ e,
have hq, from run_local_equiv η i,
simp only [(≈)] at hq,
unfold eₒ at hₑ,
rw hq.left at hₑ,
replace h := h hₑ,
simp only [deps, rcn, (hq.right.right i'.rtr).right],
exact h
end
-- If two reactions are independent, then their order of execution doesn't matter.
-- The main part of the proof is the last line, which equates to:
-- (run i)(run i')(prop i)(prop i') -> (run i')(run i)(prop i)(prop i') -> (run i')(run i)(prop i')(prop i) -> (run i')(prop i')(run i)(prop i)
lemma run_reaction_comm {η : graph υ} (hᵤ : η.has_unique_port_ins) {ρ : prec.graph υ} (hw : ρ ⋈ η) {i i' : reaction.id} (hᵢ : ρ.indep i i') :
(η.run_reaction i).run_reaction i' = (η.run_reaction i').run_reaction i :=
begin
by_cases hc : i = i',
rw hc,
{
have hₙ, from prec.graph.indep_rcns_neq_rtrs hw hc hᵢ,
unfold run_reaction,
have hd, from run_reaction_no_path_no_eₒ_dep hw hᵢ.left hₙ,
have hd', from run_reaction_no_path_no_eₒ_dep hw hᵢ.right (ne.symm hₙ),
have hₑ, from eq_rel_to.multiple (propagate_ports_eq_rel_to (by { intros p hₚ e hₑ, exact (hd p hₚ e hₑ).left })) (run_local_eq_rel_to η hₙ),
have hₑ', from eq_rel_to.multiple (propagate_ports_eq_rel_to (by { intros p hₚ e hₑ, exact (hd' p hₚ e hₑ).left })) (run_local_eq_rel_to η (ne.symm hₙ)),
rw [run_local_index_diff_eq_rel_to hₑ, run_local_index_diff_eq_rel_to hₑ'],
have hq, from equiv_trans (run_local_equiv (η.run_local i') i) (run_local_equiv η i'),
unfold has_unique_port_ins at hᵤ,
rw ←hq.left at hᵤ,
rw [propagate_ports_run_local_comm hd, run_local_comm η hₙ, propagate_ports_comm _ _ hᵤ, ←(propagate_ports_run_local_comm hd')]
}
end
-- Runs a given reaction queue on a network graph.
noncomputable def run_topo (η : graph υ) (t : list reaction.id) : graph υ :=
t.foldl run_reaction η
-- Running a reaction queue produces an equivalent network graph.
lemma run_topo_equiv (η : graph υ) (t : list reaction.id) : run_topo η t ≈ η :=
begin
induction t with tₕ tₜ hᵢ generalizing η,
case list.nil { simp [run_topo] },
case list.cons {
unfold run_topo,
transitivity,
exact hᵢ (run_reaction η tₕ),
exact run_reaction_equiv η tₕ
}
end
-- Pulling a fully independent reaction out of a reaction queue and moving it to the front does not change the result of executing the queue.
lemma run_topo_swap {η : graph υ} (hᵤ : η.has_unique_port_ins) {ρ : prec.graph υ} (h_w : ρ ⋈ η) {t : list reaction.id} (hₜ : t.is_topo_over ρ) {i : reaction.id} (h_ti : i ∈ t) (hᵢ : topo.indep i t ρ) :
run_topo η t = run_topo η (i :: (t.erase i)) :=
begin
induction t generalizing i η,
{ exfalso, exact h_ti },
{
unfold run_topo,
repeat { rw list.foldl_cons },
have h_tc, from (topo.cons_is_topo hₜ),
by_cases h_c : i = t_hd,
simp [h_c],
{
have h_e, from run_reaction_equiv η t_hd,
have h_ti', from or.resolve_left (list.eq_or_mem_of_mem_cons h_ti) h_c,
have h_fi', from topo.indep_cons hᵢ,
have hᵤ' : (run_reaction η t_hd).has_unique_port_ins,
from graph.eq_edges_unique_port_ins (graph.equiv_symm h_e).left hᵤ,
have h_wf' : (ρ ⋈ run_reaction η t_hd), from equiv_wf_prec_inv h_e h_w,
have hᵢ', from @t_ih h_tc i (run_reaction η t_hd) hᵤ' h_wf' h_ti' h_fi',
have h_rr : run_topo (run_reaction η t_hd) t_tl = list.foldl run_reaction (run_reaction η t_hd) t_tl, from refl _,
rw [←h_rr, hᵢ'],
unfold run_topo,
rw list.erase_cons_tail _ (ne.symm h_c),
repeat { rw list.foldl_cons },
have h_ind : topo.indep t_hd (t_hd :: t_tl) ρ, from topo.indep_head _ _ hₜ,
unfold topo.indep at hᵢ h_ind,
rw run_reaction_comm hᵤ h_w ⟨(hᵢ t_hd (list.mem_cons_self _ _)), (h_ind i h_ti)⟩
}
}
end
-- Executing different permutations of a reaction queue on a network graph, produces the same results as long as they're topologically ordered.
theorem run_topo_comm {η : graph υ} (hᵤ : η.has_unique_port_ins) {ρ : prec.graph υ} (h_wf : ρ ⋈ η) :
∀ {t t' : list reaction.id} (h_t : t.is_topo_over ρ) (h_t' : t'.is_topo_over ρ) (hₚ : t ~ t'), run_topo η t = run_topo η t' :=
begin
intros t t' h_t h_t' hₚ,
induction t generalizing t' η,
rw list.perm.eq_nil (list.perm.symm hₚ),
{
have h_e, from run_reaction_equiv η t_hd,
have h_pe, from list.cons_perm_iff_perm_erase.mp hₚ,
have h_tc, from (topo.cons_is_topo h_t),
have hte' : (t'.erase t_hd).is_topo_over ρ, from topo.erase_is_topo _ h_t',
have htep' : t_tl ~ (t'.erase t_hd), from h_pe.right,
have hᵤ' : (run_reaction η t_hd).has_unique_port_ins,
from graph.eq_edges_unique_port_ins (graph.equiv_symm h_e).left hᵤ,
have h_wf' : (ρ ⋈ run_reaction η t_hd), from equiv_wf_prec_inv h_e h_wf,
have h_fi : topo.indep t_hd t' ρ, {
have h_fi₁ : topo.indep t_hd (t_hd :: t_tl) ρ, from topo.indep_head _ _ h_t,
exact topo.indep_perm hₚ h_fi₁,
},
have hₘ : t_hd ∈ t', from h_pe.left,
rw (run_topo_swap hᵤ h_wf h_t' hₘ h_fi),
unfold run_topo,
repeat { rw list.foldl_cons },
exact t_ih h_tc hᵤ' h_wf' hte' htep'
}
end
end graph
end network
end inst |
6b56dd9cdbf2779d3e6d04a91d2ab27116ea1e8d | 8d65764a9e5f0923a67fc435eb1a5a1d02fd80e3 | /src/order/filter/basic.lean | 07962d9f04376604eb4fa523200531b347f7a61f | [
"Apache-2.0"
] | permissive | troyjlee/mathlib | e18d4b8026e32062ab9e89bc3b003a5d1cfec3f5 | 45e7eb8447555247246e3fe91c87066506c14875 | refs/heads/master | 1,689,248,035,046 | 1,629,470,528,000 | 1,629,470,528,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 111,154 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import data.set.finite
import order.copy
import order.zorn
import tactic.monotonicity
/-!
# Theory of filters on sets
## Main definitions
* `filter` : filters on a set;
* `at_top`, `at_bot`, `cofinite`, `principal` : specific filters;
* `map`, `comap`, `prod` : operations on filters;
* `tendsto` : limit with respect to filters;
* `eventually` : `f.eventually p` means `{x | p x} ∈ f`;
* `frequently` : `f.frequently p` means `{x | ¬p x} ∉ f`;
* `filter_upwards [h₁, ..., hₙ]` : takes a list of proofs `hᵢ : sᵢ ∈ f`, and replaces a goal `s ∈ f`
with `∀ x, x ∈ s₁ → ... → x ∈ sₙ → x ∈ s`;
* `ne_bot f` : an utility class stating that `f` is a non-trivial filter.
Filters on a type `X` are sets of sets of `X` satisfying three conditions. They are mostly used to
abstract two related kinds of ideas:
* *limits*, including finite or infinite limits of sequences, finite or infinite limits of functions
at a point or at infinity, etc...
* *things happening eventually*, including things happening for large enough `n : ℕ`, or near enough
a point `x`, or for close enough pairs of points, or things happening almost everywhere in the
sense of measure theory. Dually, filters can also express the idea of *things happening often*:
for arbitrarily large `n`, or at a point in any neighborhood of given a point etc...
In this file, we define the type `filter X` of filters on `X`, and endow it with a complete lattice
structure. This structure is lifted from the lattice structure on `set (set X)` using the Galois
insertion which maps a filter to its elements in one direction, and an arbitrary set of sets to
the smallest filter containing it in the other direction.
We also prove `filter` is a monadic functor, with a push-forward operation
`filter.map` and a pull-back operation `filter.comap` that form a Galois connections for the
order on filters.
Finally we describe a product operation `filter X → filter Y → filter (X × Y)`.
The examples of filters appearing in the description of the two motivating ideas are:
* `(at_top : filter ℕ)` : made of sets of `ℕ` containing `{n | n ≥ N}` for some `N`
* `𝓝 x` : made of neighborhoods of `x` in a topological space (defined in topology.basic)
* `𝓤 X` : made of entourages of a uniform space (those space are generalizations of metric spaces
defined in topology.uniform_space.basic)
* `μ.ae` : made of sets whose complement has zero measure with respect to `μ` (defined in
`measure_theory.measure_space`)
The general notion of limit of a map with respect to filters on the source and target types
is `filter.tendsto`. It is defined in terms of the order and the push-forward operation.
The predicate "happening eventually" is `filter.eventually`, and "happening often" is
`filter.frequently`, whose definitions are immediate after `filter` is defined (but they come
rather late in this file in order to immediately relate them to the lattice structure).
For instance, anticipating on topology.basic, the statement: "if a sequence `u` converges to
some `x` and `u n` belongs to a set `M` for `n` large enough then `x` is in the closure of
`M`" is formalized as: `tendsto u at_top (𝓝 x) → (∀ᶠ n in at_top, u n ∈ M) → x ∈ closure M`,
which is a special case of `mem_closure_of_tendsto` from topology.basic.
## Notations
* `∀ᶠ x in f, p x` : `f.eventually p`;
* `∃ᶠ x in f, p x` : `f.frequently p`;
* `f =ᶠ[l] g` : `∀ᶠ x in l, f x = g x`;
* `f ≤ᶠ[l] g` : `∀ᶠ x in l, f x ≤ g x`;
* `f ×ᶠ g` : `filter.prod f g`, localized in `filter`;
* `𝓟 s` : `principal s`, localized in `filter`.
## References
* [N. Bourbaki, *General Topology*][bourbaki1966]
Important note: Bourbaki requires that a filter on `X` cannot contain all sets of `X`, which
we do *not* require. This gives `filter X` better formal properties, in particular a bottom element
`⊥` for its lattice structure, at the cost of including the assumption
`[ne_bot f]` in a number of lemmas and definitions.
-/
open set
universes u v w x y
open_locale classical
/-- A filter `F` on a type `α` is a collection of sets of `α` which contains the whole `α`,
is upwards-closed, and is stable under intersection. We do not forbid this collection to be
all sets of `α`. -/
structure filter (α : Type*) :=
(sets : set (set α))
(univ_sets : set.univ ∈ sets)
(sets_of_superset {x y} : x ∈ sets → x ⊆ y → y ∈ sets)
(inter_sets {x y} : x ∈ sets → y ∈ sets → x ∩ y ∈ sets)
/-- If `F` is a filter on `α`, and `U` a subset of `α` then we can write `U ∈ F` as on paper. -/
instance {α : Type*}: has_mem (set α) (filter α) := ⟨λ U F, U ∈ F.sets⟩
namespace filter
variables {α : Type u} {f g : filter α} {s t : set α}
@[simp] protected lemma mem_mk {t : set (set α)} {h₁ h₂ h₃} : s ∈ mk t h₁ h₂ h₃ ↔ s ∈ t := iff.rfl
@[simp] protected lemma mem_sets : s ∈ f.sets ↔ s ∈ f := iff.rfl
instance inhabited_mem : inhabited {s : set α // s ∈ f} := ⟨⟨univ, f.univ_sets⟩⟩
lemma filter_eq : ∀ {f g : filter α}, f.sets = g.sets → f = g
| ⟨a, _, _, _⟩ ⟨._, _, _, _⟩ rfl := rfl
lemma filter_eq_iff : f = g ↔ f.sets = g.sets :=
⟨congr_arg _, filter_eq⟩
protected lemma ext_iff : f = g ↔ ∀ s, s ∈ f ↔ s ∈ g :=
by simp only [filter_eq_iff, ext_iff, filter.mem_sets]
@[ext]
protected lemma ext : (∀ s, s ∈ f ↔ s ∈ g) → f = g :=
filter.ext_iff.2
@[simp] lemma univ_mem : univ ∈ f :=
f.univ_sets
lemma mem_of_superset : ∀ {x y : set α}, x ∈ f → x ⊆ y → y ∈ f :=
f.sets_of_superset
lemma inter_mem : ∀ {s t}, s ∈ f → t ∈ f → s ∩ t ∈ f :=
f.inter_sets
@[simp] lemma inter_mem_iff {s t} : s ∩ t ∈ f ↔ s ∈ f ∧ t ∈ f :=
⟨λ h, ⟨mem_of_superset h (inter_subset_left s t),
mem_of_superset h (inter_subset_right s t)⟩, and_imp.2 inter_mem⟩
lemma univ_mem' (h : ∀ a, a ∈ s) : s ∈ f :=
mem_of_superset univ_mem (λ x _, h x)
lemma mp_mem (hs : s ∈ f) (h : {x | x ∈ s → x ∈ t} ∈ f) : t ∈ f :=
mem_of_superset (inter_mem hs h) $ λ x ⟨h₁, h₂⟩, h₂ h₁
lemma congr_sets (h : {x | x ∈ s ↔ x ∈ t} ∈ f) : s ∈ f ↔ t ∈ f :=
⟨λ hs, mp_mem hs (mem_of_superset h (λ x, iff.mp)),
λ hs, mp_mem hs (mem_of_superset h (λ x, iff.mpr))⟩
@[simp] lemma bInter_mem {β : Type v} {s : β → set α} {is : set β} (hf : finite is) :
(⋂ i ∈ is, s i) ∈ f ↔ ∀ i ∈ is, s i ∈ f :=
finite.induction_on hf (by simp) (λ i s hi _ hs, by simp [hs])
@[simp] lemma bInter_finset_mem {β : Type v} {s : β → set α} (is : finset β) :
(⋂ i ∈ is, s i) ∈ f ↔ ∀ i ∈ is, s i ∈ f :=
bInter_mem is.finite_to_set
alias bInter_finset_mem ← finset.Inter_mem_sets
attribute [protected] finset.Inter_mem_sets
@[simp] lemma sInter_mem {s : set (set α)} (hfin : finite s) :
⋂₀ s ∈ f ↔ ∀ U ∈ s, U ∈ f :=
by rw [sInter_eq_bInter, bInter_mem hfin]
@[simp] lemma Inter_mem {β : Type v} {s : β → set α} [fintype β] :
(⋂ i, s i) ∈ f ↔ ∀ i, s i ∈ f :=
by simpa using bInter_mem finite_univ
lemma exists_mem_subset_iff : (∃ t ∈ f, t ⊆ s) ↔ s ∈ f :=
⟨λ ⟨t, ht, ts⟩, mem_of_superset ht ts, λ hs, ⟨s, hs, subset.rfl⟩⟩
lemma monotone_mem {f : filter α} : monotone (λ s, s ∈ f) :=
λ s t hst h, mem_of_superset h hst
end filter
namespace tactic.interactive
open tactic interactive
/-- `filter_upwards [h1, ⋯, hn]` replaces a goal of the form `s ∈ f`
and terms `h1 : t1 ∈ f, ⋯, hn : tn ∈ f` with `∀ x, x ∈ t1 → ⋯ → x ∈ tn → x ∈ s`.
`filter_upwards [h1, ⋯, hn] e` is a short form for `{ filter_upwards [h1, ⋯, hn], exact e }`.
-/
meta def filter_upwards
(s : parse types.pexpr_list)
(e' : parse $ optional types.texpr) : tactic unit :=
do
s.reverse.mmap (λ e, eapplyc `filter.mp_mem >> eapply e),
eapplyc `filter.univ_mem',
`[dsimp only [set.mem_set_of_eq]],
match e' with
| some e := interactive.exact e
| none := skip
end
add_tactic_doc
{ name := "filter_upwards",
category := doc_category.tactic,
decl_names := [`tactic.interactive.filter_upwards],
tags := ["goal management", "lemma application"] }
end tactic.interactive
namespace filter
variables {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
section principal
/-- The principal filter of `s` is the collection of all supersets of `s`. -/
def principal (s : set α) : filter α :=
{ sets := {t | s ⊆ t},
univ_sets := subset_univ s,
sets_of_superset := λ x y hx, subset.trans hx,
inter_sets := λ x y, subset_inter }
localized "notation `𝓟` := filter.principal" in filter
instance : inhabited (filter α) :=
⟨𝓟 ∅⟩
@[simp] lemma mem_principal {s t : set α} : s ∈ 𝓟 t ↔ t ⊆ s := iff.rfl
lemma mem_principal_self (s : set α) : s ∈ 𝓟 s := subset.rfl
end principal
open_locale filter
section join
/-- The join of a filter of filters is defined by the relation `s ∈ join f ↔ {t | s ∈ t} ∈ f`. -/
def join (f : filter (filter α)) : filter α :=
{ sets := {s | {t : filter α | s ∈ t} ∈ f},
univ_sets := by simp only [mem_set_of_eq, univ_sets, ← filter.mem_sets, set_of_true],
sets_of_superset := λ x y hx xy,
mem_of_superset hx $ λ f h, mem_of_superset h xy,
inter_sets := λ x y hx hy,
mem_of_superset (inter_mem hx hy) $ λ f ⟨h₁, h₂⟩, inter_mem h₁ h₂ }
@[simp] lemma mem_join {s : set α} {f : filter (filter α)} :
s ∈ join f ↔ {t | s ∈ t} ∈ f := iff.rfl
end join
section lattice
instance : partial_order (filter α) :=
{ le := λ f g, ∀ ⦃U : set α⦄, U ∈ g → U ∈ f,
le_antisymm := λ a b h₁ h₂, filter_eq $ subset.antisymm h₂ h₁,
le_refl := λ a, subset.rfl,
le_trans := λ a b c h₁ h₂, subset.trans h₂ h₁ }
theorem le_def {f g : filter α} : f ≤ g ↔ ∀ x ∈ g, x ∈ f := iff.rfl
/-- `generate_sets g s`: `s` is in the filter closure of `g`. -/
inductive generate_sets (g : set (set α)) : set α → Prop
| basic {s : set α} : s ∈ g → generate_sets s
| univ : generate_sets univ
| superset {s t : set α} : generate_sets s → s ⊆ t → generate_sets t
| inter {s t : set α} : generate_sets s → generate_sets t → generate_sets (s ∩ t)
/-- `generate g` is the smallest filter containing the sets `g`. -/
def generate (g : set (set α)) : filter α :=
{ sets := generate_sets g,
univ_sets := generate_sets.univ,
sets_of_superset := λ x y, generate_sets.superset,
inter_sets := λ s t, generate_sets.inter }
lemma sets_iff_generate {s : set (set α)} {f : filter α} : f ≤ filter.generate s ↔ s ⊆ f.sets :=
iff.intro
(λ h u hu, h $ generate_sets.basic $ hu)
(λ h u hu, hu.rec_on h univ_mem
(λ x y _ hxy hx, mem_of_superset hx hxy)
(λ x y _ _ hx hy, inter_mem hx hy))
lemma mem_generate_iff {s : set $ set α} {U : set α} :
U ∈ generate s ↔ ∃ t ⊆ s, finite t ∧ ⋂₀ t ⊆ U :=
begin
split ; intro h,
{ induction h with V V_in V W V_in hVW hV V W V_in W_in hV hW,
{ use {V},
simp [V_in] },
{ use ∅,
simp [subset.refl, univ] },
{ rcases hV with ⟨t, hts, htfin, hinter⟩,
exact ⟨t, hts, htfin, hinter.trans hVW⟩ },
{ rcases hV with ⟨t, hts, htfin, htinter⟩,
rcases hW with ⟨z, hzs, hzfin, hzinter⟩,
refine ⟨t ∪ z, union_subset hts hzs, htfin.union hzfin, _⟩,
rw sInter_union,
exact inter_subset_inter htinter hzinter } },
{ rcases h with ⟨t, ts, tfin, h⟩,
apply generate_sets.superset _ h,
revert ts,
apply finite.induction_on tfin,
{ intro h,
rw sInter_empty,
exact generate_sets.univ },
{ intros V r hV rfin hinter h,
cases insert_subset.mp h with V_in r_sub,
rw [insert_eq V r, sInter_union],
apply generate_sets.inter _ (hinter r_sub),
rw sInter_singleton,
exact generate_sets.basic V_in } },
end
/-- `mk_of_closure s hs` constructs a filter on `α` whose elements set is exactly
`s : set (set α)`, provided one gives the assumption `hs : (generate s).sets = s`. -/
protected def mk_of_closure (s : set (set α)) (hs : (generate s).sets = s) : filter α :=
{ sets := s,
univ_sets := hs ▸ (univ_mem : univ ∈ generate s),
sets_of_superset := λ x y, hs ▸ (mem_of_superset : x ∈ generate s → x ⊆ y → y ∈ generate s),
inter_sets := λ x y, hs ▸ (inter_mem : x ∈ generate s → y ∈ generate s →
x ∩ y ∈ generate s) }
lemma mk_of_closure_sets {s : set (set α)} {hs : (generate s).sets = s} :
filter.mk_of_closure s hs = generate s :=
filter.ext $ λ u,
show u ∈ (filter.mk_of_closure s hs).sets ↔ u ∈ (generate s).sets, from hs.symm ▸ iff.rfl
/-- Galois insertion from sets of sets into filters. -/
def gi_generate (α : Type*) :
@galois_insertion (set (set α)) (order_dual (filter α)) _ _ filter.generate filter.sets :=
{ gc := λ s f, sets_iff_generate,
le_l_u := λ f u h, generate_sets.basic h,
choice := λ s hs, filter.mk_of_closure s (le_antisymm hs $ sets_iff_generate.1 $ le_rfl),
choice_eq := λ s hs, mk_of_closure_sets }
/-- The infimum of filters is the filter generated by intersections
of elements of the two filters. -/
instance : has_inf (filter α) := ⟨λf g : filter α,
{ sets := {s | ∃ (a ∈ f) (b ∈ g), s = a ∩ b },
univ_sets := ⟨_, univ_mem, _, univ_mem, by simp⟩,
sets_of_superset := begin
rintro x y ⟨a, ha, b, hb, rfl⟩ xy,
refine ⟨a ∪ y, mem_of_superset ha (subset_union_left a y),
b ∪ y, mem_of_superset hb (subset_union_left b y), _⟩,
rw [← inter_union_distrib_right, union_eq_self_of_subset_left xy]
end,
inter_sets := begin
rintro x y ⟨a, ha, b, hb, rfl⟩ ⟨c, hc, d, hd, rfl⟩,
refine ⟨a ∩ c, inter_mem ha hc, b ∩ d, inter_mem hb hd, _⟩,
ac_refl
end }⟩
lemma mem_inf_iff {f g : filter α} {s : set α} :
s ∈ f ⊓ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, s = t₁ ∩ t₂ := iff.rfl
lemma mem_inf_of_left {f g : filter α} {s : set α} (h : s ∈ f) : s ∈ f ⊓ g :=
⟨s, h, univ, univ_mem, (inter_univ s).symm⟩
lemma mem_inf_of_right {f g : filter α} {s : set α} (h : s ∈ g) : s ∈ f ⊓ g :=
⟨univ, univ_mem, s, h, (univ_inter s).symm⟩
lemma inter_mem_inf {α : Type u} {f g : filter α} {s t : set α}
(hs : s ∈ f) (ht : t ∈ g) : s ∩ t ∈ f ⊓ g :=
⟨s, hs, t, ht, rfl⟩
lemma mem_inf_of_inter {f g : filter α} {s t u : set α} (hs : s ∈ f) (ht : t ∈ g) (h : s ∩ t ⊆ u) :
u ∈ f ⊓ g :=
mem_of_superset (inter_mem_inf hs ht) h
lemma mem_inf_iff_superset {f g : filter α} {s : set α} :
s ∈ f ⊓ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ∩ t₂ ⊆ s :=
⟨λ ⟨t₁, h₁, t₂, h₂, eq⟩, ⟨t₁, h₁, t₂, h₂, eq ▸ subset.rfl⟩,
λ ⟨t₁, h₁, t₂, h₂, sub⟩, mem_inf_of_inter h₁ h₂ sub⟩
instance : has_top (filter α) :=
⟨{ sets := {s | ∀ x, x ∈ s},
univ_sets := λ x, mem_univ x,
sets_of_superset := λ x y hx hxy a, hxy (hx a),
inter_sets := λ x y hx hy a, mem_inter (hx _) (hy _) }⟩
lemma mem_top_iff_forall {s : set α} : s ∈ (⊤ : filter α) ↔ (∀ x, x ∈ s) :=
iff.rfl
@[simp] lemma mem_top {s : set α} : s ∈ (⊤ : filter α) ↔ s = univ :=
by rw [mem_top_iff_forall, eq_univ_iff_forall]
section complete_lattice
/- We lift the complete lattice along the Galois connection `generate` / `sets`. Unfortunately,
we want to have different definitional equalities for the lattice operations. So we define them
upfront and change the lattice operations for the complete lattice instance. -/
private def original_complete_lattice : complete_lattice (filter α) :=
@order_dual.complete_lattice _ (gi_generate α).lift_complete_lattice
local attribute [instance] original_complete_lattice
instance : complete_lattice (filter α) := original_complete_lattice.copy
/- le -/ filter.partial_order.le rfl
/- top -/ (filter.has_top).1
(top_unique $ λ s hs, by simp [mem_top.1 hs])
/- bot -/ _ rfl
/- sup -/ _ rfl
/- inf -/ (filter.has_inf).1
begin
ext f g : 2,
exact le_antisymm
(le_inf (λ s, mem_inf_of_left) (λ s, mem_inf_of_right))
(begin
rintro s ⟨a, ha, b, hb, rfl⟩,
exact inter_sets _ (@inf_le_left (filter α) _ _ _ _ ha)
(@inf_le_right (filter α) _ _ _ _ hb)
end)
end
/- Sup -/ (join ∘ 𝓟) (by ext s x; exact (@mem_bInter_iff _ _ s filter.sets x).symm)
/- Inf -/ _ rfl
end complete_lattice
/-- A filter is `ne_bot` if it is not equal to `⊥`, or equivalently the empty set
does not belong to the filter. Bourbaki include this assumption in the definition
of a filter but we prefer to have a `complete_lattice` structure on filter, so
we use a typeclass argument in lemmas instead. -/
class ne_bot (f : filter α) : Prop := (ne' : f ≠ ⊥)
lemma ne_bot_iff {f : filter α} : ne_bot f ↔ f ≠ ⊥ := ⟨λ h, h.1, λ h, ⟨h⟩⟩
lemma ne_bot.ne {f : filter α} (hf : ne_bot f) : f ≠ ⊥ := ne_bot.ne'
@[simp] lemma not_ne_bot {α : Type*} {f : filter α} : ¬ f.ne_bot ↔ f = ⊥ :=
not_iff_comm.1 ne_bot_iff.symm
lemma ne_bot.mono {f g : filter α} (hf : ne_bot f) (hg : f ≤ g) : ne_bot g :=
⟨ne_bot_of_le_ne_bot hf.1 hg⟩
lemma ne_bot_of_le {f g : filter α} [hf : ne_bot f] (hg : f ≤ g) : ne_bot g :=
hf.mono hg
lemma bot_sets_eq : (⊥ : filter α).sets = univ := rfl
lemma sup_sets_eq {f g : filter α} : (f ⊔ g).sets = f.sets ∩ g.sets :=
(gi_generate α).gc.u_inf
lemma Sup_sets_eq {s : set (filter α)} : (Sup s).sets = (⋂ f ∈ s, (f : filter α).sets) :=
(gi_generate α).gc.u_Inf
lemma supr_sets_eq {f : ι → filter α} : (supr f).sets = (⋂ i, (f i).sets) :=
(gi_generate α).gc.u_infi
lemma generate_empty : filter.generate ∅ = (⊤ : filter α) :=
(gi_generate α).gc.l_bot
lemma generate_univ : filter.generate univ = (⊥ : filter α) :=
mk_of_closure_sets.symm
lemma generate_union {s t : set (set α)} :
filter.generate (s ∪ t) = filter.generate s ⊓ filter.generate t :=
(gi_generate α).gc.l_sup
lemma generate_Union {s : ι → set (set α)} :
filter.generate (⋃ i, s i) = (⨅ i, filter.generate (s i)) :=
(gi_generate α).gc.l_supr
@[simp] lemma mem_bot {s : set α} : s ∈ (⊥ : filter α) :=
trivial
@[simp] lemma mem_sup {f g : filter α} {s : set α} :
s ∈ f ⊔ g ↔ s ∈ f ∧ s ∈ g :=
iff.rfl
lemma union_mem_sup {f g : filter α} {s t : set α} (hs : s ∈ f) (ht : t ∈ g) :
s ∪ t ∈ f ⊔ g :=
⟨mem_of_superset hs (subset_union_left s t), mem_of_superset ht (subset_union_right s t)⟩
@[simp] lemma mem_Sup {x : set α} {s : set (filter α)} :
x ∈ Sup s ↔ (∀ f ∈ s, x ∈ (f : filter α)) :=
iff.rfl
@[simp] lemma mem_supr {x : set α} {f : ι → filter α} :
x ∈ supr f ↔ (∀ i, x ∈ f i) :=
by simp only [← filter.mem_sets, supr_sets_eq, iff_self, mem_Inter]
lemma infi_eq_generate (s : ι → filter α) : infi s = generate (⋃ i, (s i).sets) :=
show generate _ = generate _, from congr_arg _ supr_range
lemma mem_infi {ι} {s : ι → filter α} {U : set α} : (U ∈ ⨅ i, s i) ↔
∃ I : set ι, finite I ∧ ∃ V : I → set α, (∀ i, V i ∈ s i) ∧ (⋂ i, V i) ⊆ U :=
begin
rw [infi_eq_generate, mem_generate_iff],
split,
{ rintro ⟨t, tsub, tfin, tinter⟩,
rcases eq_finite_Union_of_finite_subset_Union tfin tsub with ⟨I, Ifin, σ, σfin, σsub, rfl⟩,
rw sInter_Union at tinter,
let V := λ i, ⋂₀ σ i,
have V_in : ∀ i, V i ∈ s i,
{ rintro ⟨i, i_in⟩,
rw sInter_mem (σfin _),
apply σsub },
exact ⟨I, Ifin, V, V_in, tinter⟩ },
{ rintro ⟨I, Ifin, V, V_in, h⟩,
refine ⟨range V, _, _, h⟩,
{ rintro _ ⟨i, rfl⟩,
rw mem_Union,
use [i, V_in i] },
{ haveI : fintype I := finite.fintype Ifin,
exact finite_range _ } },
end
lemma mem_infi' {ι} {s : ι → filter α} {U : set α} : (U ∈ ⨅ i, s i) ↔
∃ I : set ι, finite I ∧ ∃ V : ι → set α, (∀ i ∈ I, V i ∈ s i) ∧ (⋂ i ∈ I, V i) ⊆ U :=
begin
simp only [mem_infi, set_coe.forall', bInter_eq_Inter],
refine ⟨_, λ ⟨I, If, V, hV⟩, ⟨I, If, λ i, V i, hV⟩⟩,
rintro ⟨I, If, V, hV⟩,
lift V to ι → set α using trivial,
exact ⟨I, If, V, hV⟩
end
@[simp] lemma le_principal_iff {s : set α} {f : filter α} : f ≤ 𝓟 s ↔ s ∈ f :=
show (∀ {t}, s ⊆ t → t ∈ f) ↔ s ∈ f,
from ⟨λ h, h (subset.refl s), λ hs t ht, mem_of_superset hs ht⟩
lemma principal_mono {s t : set α} : 𝓟 s ≤ 𝓟 t ↔ s ⊆ t :=
by simp only [le_principal_iff, iff_self, mem_principal]
@[mono] lemma monotone_principal : monotone (𝓟 : set α → filter α) :=
λ _ _, principal_mono.2
@[simp] lemma principal_eq_iff_eq {s t : set α} : 𝓟 s = 𝓟 t ↔ s = t :=
by simp only [le_antisymm_iff, le_principal_iff, mem_principal]; refl
@[simp] lemma join_principal_eq_Sup {s : set (filter α)} : join (𝓟 s) = Sup s := rfl
@[simp] lemma principal_univ : 𝓟 (univ : set α) = ⊤ :=
top_unique $ by simp only [le_principal_iff, mem_top, eq_self_iff_true]
@[simp] lemma principal_empty : 𝓟 (∅ : set α) = ⊥ :=
bot_unique $ λ s _, empty_subset _
/-! ### Lattice equations -/
lemma empty_mem_iff_bot {f : filter α} : ∅ ∈ f ↔ f = ⊥ :=
⟨λ h, bot_unique $ λ s _, mem_of_superset h (empty_subset s),
λ h, h.symm ▸ mem_bot⟩
lemma nonempty_of_mem {f : filter α} [hf : ne_bot f] {s : set α} (hs : s ∈ f) :
s.nonempty :=
s.eq_empty_or_nonempty.elim (λ h, absurd hs (h.symm ▸ mt empty_mem_iff_bot.mp hf.1)) id
lemma ne_bot.nonempty_of_mem {f : filter α} (hf : ne_bot f) {s : set α} (hs : s ∈ f) :
s.nonempty :=
@nonempty_of_mem α f hf s hs
@[simp] lemma empty_not_mem (f : filter α) [ne_bot f] : ¬(∅ ∈ f) :=
λ h, (nonempty_of_mem h).ne_empty rfl
lemma nonempty_of_ne_bot (f : filter α) [ne_bot f] : nonempty α :=
nonempty_of_exists $ nonempty_of_mem (univ_mem : univ ∈ f)
lemma compl_not_mem {f : filter α} {s : set α} [ne_bot f] (h : s ∈ f) : sᶜ ∉ f :=
λ hsc, (nonempty_of_mem (inter_mem h hsc)).ne_empty $ inter_compl_self s
lemma filter_eq_bot_of_is_empty [is_empty α] (f : filter α) : f = ⊥ :=
empty_mem_iff_bot.mp $ univ_mem' is_empty_elim
lemma filter_eq_bot_of_not_nonempty (f : filter α) (ne : ¬ nonempty α) : f = ⊥ :=
empty_mem_iff_bot.mp $ univ_mem' $ λ x, (ne ⟨x⟩).elim
/-- There is exactly one filter on an empty type. --/
-- TODO[gh-6025]: make this globally an instance once safe to do so
local attribute [instance]
protected def unique [is_empty α] : unique (filter α) :=
{ default := ⊥, uniq := filter_eq_bot_of_is_empty }
lemma forall_mem_nonempty_iff_ne_bot {f : filter α} :
(∀ (s : set α), s ∈ f → s.nonempty) ↔ ne_bot f :=
⟨λ h, ⟨λ hf, empty_not_nonempty (h ∅ $ hf.symm ▸ mem_bot)⟩, @nonempty_of_mem _ _⟩
lemma nontrivial_iff_nonempty : nontrivial (filter α) ↔ nonempty α :=
⟨λ ⟨⟨f, g, hfg⟩⟩, by_contra $
λ h, hfg $ by haveI : is_empty α := not_nonempty_iff.1 h; exact subsingleton.elim _ _,
λ ⟨x⟩, ⟨⟨⊤, ⊥, ne_bot.ne $ forall_mem_nonempty_iff_ne_bot.1 $ λ s hs,
by rwa [mem_top.1 hs, ← nonempty_iff_univ_nonempty]⟩⟩⟩
lemma eq_Inf_of_mem_iff_exists_mem {S : set (filter α)} {l : filter α}
(h : ∀ {s}, s ∈ l ↔ ∃ f ∈ S, s ∈ f) : l = Inf S :=
le_antisymm (le_Inf $ λ f hf s hs, h.2 ⟨f, hf, hs⟩)
(λ s hs, let ⟨f, hf, hs⟩ := h.1 hs in (Inf_le hf : Inf S ≤ f) hs)
lemma eq_infi_of_mem_iff_exists_mem {f : ι → filter α} {l : filter α}
(h : ∀ {s}, s ∈ l ↔ ∃ i, s ∈ f i) :
l = infi f :=
eq_Inf_of_mem_iff_exists_mem $ λ s, h.trans exists_range_iff.symm
lemma eq_binfi_of_mem_iff_exists_mem {f : ι → filter α} {p : ι → Prop} {l : filter α}
(h : ∀ {s}, s ∈ l ↔ ∃ i (_ : p i), s ∈ f i) :
l = ⨅ i (_ : p i), f i :=
begin
rw [infi_subtype'],
apply eq_infi_of_mem_iff_exists_mem,
intro s,
exact h.trans ⟨λ ⟨i, pi, si⟩, ⟨⟨i, pi⟩, si⟩, λ ⟨⟨i, pi⟩, si⟩, ⟨i, pi, si⟩⟩
end
lemma infi_sets_eq {f : ι → filter α} (h : directed (≥) f) [ne : nonempty ι] :
(infi f).sets = (⋃ i, (f i).sets) :=
let ⟨i⟩ := ne, u := { filter .
sets := (⋃ i, (f i).sets),
univ_sets := by simp only [mem_Union]; exact ⟨i, univ_mem⟩,
sets_of_superset := by simp only [mem_Union, exists_imp_distrib];
intros x y i hx hxy; exact ⟨i, mem_of_superset hx hxy⟩,
inter_sets :=
begin
simp only [mem_Union, exists_imp_distrib],
intros x y a hx b hy,
rcases h a b with ⟨c, ha, hb⟩,
exact ⟨c, inter_mem (ha hx) (hb hy)⟩
end } in
have u = infi f, from eq_infi_of_mem_iff_exists_mem
(λ s, by simp only [filter.mem_mk, mem_Union, filter.mem_sets]),
congr_arg filter.sets this.symm
lemma mem_infi_of_directed {f : ι → filter α} (h : directed (≥) f) [nonempty ι] (s) :
s ∈ infi f ↔ ∃ i, s ∈ f i :=
by simp only [← filter.mem_sets, infi_sets_eq h, mem_Union]
lemma mem_binfi_of_directed {f : β → filter α} {s : set β}
(h : directed_on (f ⁻¹'o (≥)) s) (ne : s.nonempty) {t : set α} :
t ∈ (⨅ i ∈ s, f i) ↔ ∃ i ∈ s, t ∈ f i :=
by haveI : nonempty {x // x ∈ s} := ne.to_subtype;
erw [infi_subtype', mem_infi_of_directed h.directed_coe, subtype.exists]; refl
lemma binfi_sets_eq {f : β → filter α} {s : set β}
(h : directed_on (f ⁻¹'o (≥)) s) (ne : s.nonempty) :
(⨅ i ∈ s, f i).sets = ⋃ i ∈ s, (f i).sets :=
ext $ λ t, by simp [mem_binfi_of_directed h ne]
lemma infi_sets_eq_finite {ι : Type*} (f : ι → filter α) :
(⨅ i, f i).sets = (⋃ t : finset ι, (⨅ i ∈ t, f i).sets) :=
begin
rw [infi_eq_infi_finset, infi_sets_eq],
exact (directed_of_sup $ λ s₁ s₂ hs, infi_le_infi $ λ i, infi_le_infi_const $ λ h, hs h),
end
lemma infi_sets_eq_finite' (f : ι → filter α) :
(⨅ i, f i).sets = (⋃ t : finset (plift ι), (⨅ i ∈ t, f (plift.down i)).sets) :=
by { rw [← infi_sets_eq_finite, ← equiv.plift.surjective.infi_comp], refl }
lemma mem_infi_finite {ι : Type*} {f : ι → filter α} (s) :
s ∈ infi f ↔ ∃ t : finset ι, s ∈ ⨅ i ∈ t, f i :=
(set.ext_iff.1 (infi_sets_eq_finite f) s).trans mem_Union
lemma mem_infi_finite' {f : ι → filter α} (s) :
s ∈ infi f ↔ ∃ t : finset (plift ι), s ∈ ⨅ i ∈ t, f (plift.down i) :=
(set.ext_iff.1 (infi_sets_eq_finite' f) s).trans mem_Union
@[simp] lemma sup_join {f₁ f₂ : filter (filter α)} : (join f₁ ⊔ join f₂) = join (f₁ ⊔ f₂) :=
filter.ext $ λ x, by simp only [mem_sup, mem_join]
@[simp] lemma supr_join {ι : Sort w} {f : ι → filter (filter α)} :
(⨆ x, join (f x)) = join (⨆ x, f x) :=
filter.ext $ λ x, by simp only [mem_supr, mem_join]
instance : bounded_distrib_lattice (filter α) :=
{ le_sup_inf :=
begin
intros x y z s,
simp only [and_assoc, mem_inf_iff, mem_sup, exists_prop, exists_imp_distrib, and_imp],
rintro hs t₁ ht₁ t₂ ht₂ rfl,
exact ⟨t₁, x.sets_of_superset hs (inter_subset_left t₁ t₂),
ht₁,
t₂,
x.sets_of_superset hs (inter_subset_right t₁ t₂),
ht₂,
rfl⟩
end,
..filter.complete_lattice }
/- the complementary version with ⨆ i, f ⊓ g i does not hold! -/
lemma infi_sup_left {f : filter α} {g : ι → filter α} : (⨅ x, f ⊔ g x) = f ⊔ infi g :=
begin
refine le_antisymm _ (le_infi $ λ i, sup_le_sup_left (infi_le _ _) _),
rintro t ⟨h₁, h₂⟩,
rw [infi_sets_eq_finite'] at h₂,
simp only [mem_Union, (finset.inf_eq_infi _ _).symm] at h₂,
rcases h₂ with ⟨s, hs⟩,
suffices : (⨅ i, f ⊔ g i) ≤ f ⊔ s.inf (λ i, g i.down), { exact this ⟨h₁, hs⟩ },
refine finset.induction_on s _ _,
{ exact le_sup_of_le_right le_top },
{ rintro ⟨i⟩ s his ih,
rw [finset.inf_insert, sup_inf_left],
exact le_inf (infi_le _ _) ih }
end
lemma infi_sup_right {f : filter α} {g : ι → filter α} : (⨅ x, g x ⊔ f) = infi g ⊔ f :=
by simp [sup_comm, ← infi_sup_left]
lemma binfi_sup_right (p : ι → Prop) (f : ι → filter α) (g : filter α) :
(⨅ i (h : p i), (f i ⊔ g)) = (⨅ i (h : p i), f i) ⊔ g :=
by rw [infi_subtype', infi_sup_right, infi_subtype']
lemma binfi_sup_left (p : ι → Prop) (f : ι → filter α) (g : filter α) :
(⨅ i (h : p i), (g ⊔ f i)) = g ⊔ (⨅ i (h : p i), f i) :=
by rw [infi_subtype', infi_sup_left, infi_subtype']
lemma mem_infi_finset {s : finset α} {f : α → filter β} :
∀ t, t ∈ (⨅ a ∈ s, f a) ↔ (∃ p : α → set β, (∀ a ∈ s, p a ∈ f a) ∧ t = ⋂ a ∈ s, p a) :=
show ∀ t, t ∈ (⨅ a ∈ s, f a) ↔ (∃ p : α → set β, (∀ a ∈ s, p a ∈ f a) ∧ t = ⨅ a ∈ s, p a),
begin
simp only [(finset.inf_eq_infi _ _).symm],
refine finset.induction_on s _ _,
{ simp only [finset.not_mem_empty, false_implies_iff, finset.inf_empty, top_le_iff,
imp_true_iff, mem_top, true_and, exists_const],
intros; refl },
{ intros a s has ih t,
simp only [ih, finset.forall_mem_insert, finset.inf_insert, mem_inf_iff, exists_prop,
iff_iff_implies_and_implies, exists_imp_distrib, and_imp, and_assoc] {contextual := tt},
split,
{ rintro t₁ ht₁ t₂ p hp ht₂ rfl,
use function.update p a t₁,
have : ∀ a' ∈ s, function.update p a t₁ a' = p a',
from λ a' ha',
have a' ≠ a, from λ h, has $ h ▸ ha',
function.update_noteq this _ _,
have eq : s.inf (λj, function.update p a t₁ j) = s.inf (λj, p j) :=
finset.inf_congr rfl this,
simp only [this, ht₁, hp, function.update_same, true_and, imp_true_iff, eq]
{contextual := tt},
refl },
rintro p hpa hp rfl,
exact ⟨p a, hpa, s.inf p, ⟨p, hp, rfl⟩, rfl⟩ }
end
/-- If `f : ι → filter α` is directed, `ι` is not empty, and `∀ i, f i ≠ ⊥`, then `infi f ≠ ⊥`.
See also `infi_ne_bot_of_directed` for a version assuming `nonempty α` instead of `nonempty ι`. -/
lemma infi_ne_bot_of_directed' {f : ι → filter α} [nonempty ι]
(hd : directed (≥) f) (hb : ∀ i, ne_bot (f i)) : ne_bot (infi f) :=
⟨begin
intro h,
have he : ∅ ∈ (infi f), from h.symm ▸ (mem_bot : ∅ ∈ (⊥ : filter α)),
obtain ⟨i, hi⟩ : ∃ i, ∅ ∈ f i,
from (mem_infi_of_directed hd ∅).1 he,
exact (hb i).ne (empty_mem_iff_bot.1 hi)
end⟩
/-- If `f : ι → filter α` is directed, `α` is not empty, and `∀ i, f i ≠ ⊥`, then `infi f ≠ ⊥`.
See also `infi_ne_bot_of_directed'` for a version assuming `nonempty ι` instead of `nonempty α`. -/
lemma infi_ne_bot_of_directed {f : ι → filter α}
[hn : nonempty α] (hd : directed (≥) f) (hb : ∀ i, ne_bot (f i)) : ne_bot (infi f) :=
if hι : nonempty ι then @infi_ne_bot_of_directed' _ _ _ hι hd hb else
⟨λ h : infi f = ⊥,
have univ ⊆ (∅ : set α),
begin
rw [←principal_mono, principal_univ, principal_empty, ←h],
exact (le_infi $ λ i, false.elim $ hι ⟨i⟩)
end,
let ⟨x⟩ := hn in this (mem_univ x)⟩
lemma infi_ne_bot_iff_of_directed' {f : ι → filter α} [nonempty ι] (hd : directed (≥) f) :
ne_bot (infi f) ↔ ∀ i, ne_bot (f i) :=
⟨λ H i, H.mono (infi_le _ i), infi_ne_bot_of_directed' hd⟩
lemma infi_ne_bot_iff_of_directed {f : ι → filter α} [nonempty α] (hd : directed (≥) f) :
ne_bot (infi f) ↔ (∀ i, ne_bot (f i)) :=
⟨λ H i, H.mono (infi_le _ i), infi_ne_bot_of_directed hd⟩
lemma mem_infi_of_mem {f : ι → filter α} (i : ι) : ∀ {s}, s ∈ f i → s ∈ ⨅ i, f i :=
show (⨅ i, f i) ≤ f i, from infi_le _ _
@[elab_as_eliminator]
lemma infi_sets_induct {f : ι → filter α} {s : set α} (hs : s ∈ infi f) {p : set α → Prop}
(uni : p univ)
(ins : ∀ {i s₁ s₂}, s₁ ∈ f i → p s₂ → p (s₁ ∩ s₂)) : p s :=
begin
rw [mem_infi_finite'] at hs,
simp only [← finset.inf_eq_infi] at hs,
rcases hs with ⟨is, his⟩,
revert s,
refine finset.induction_on is _ _,
{ intros s hs, rwa [mem_top.1 hs] },
{ rintro ⟨i⟩ js his ih s hs,
rw [finset.inf_insert, mem_inf_iff] at hs,
rcases hs with ⟨s₁, hs₁, s₂, hs₂, rfl⟩,
exact ins hs₁ (ih hs₂) }
end
/-! #### `principal` equations -/
@[simp] lemma inf_principal {s t : set α} : 𝓟 s ⊓ 𝓟 t = 𝓟 (s ∩ t) :=
le_antisymm
(by simp only [le_principal_iff, mem_inf_iff]; exact ⟨s, subset.rfl, t, subset.rfl, rfl⟩)
(by simp [le_inf_iff, inter_subset_left, inter_subset_right])
@[simp] lemma sup_principal {s t : set α} : 𝓟 s ⊔ 𝓟 t = 𝓟 (s ∪ t) :=
filter.ext $ λ u, by simp only [union_subset_iff, mem_sup, mem_principal]
@[simp] lemma supr_principal {ι : Sort w} {s : ι → set α} : (⨆ x, 𝓟 (s x)) = 𝓟 (⋃ i, s i) :=
filter.ext $ λ x, by simp only [mem_supr, mem_principal, Union_subset_iff]
@[simp] lemma principal_eq_bot_iff {s : set α} : 𝓟 s = ⊥ ↔ s = ∅ :=
empty_mem_iff_bot.symm.trans $ mem_principal.trans subset_empty_iff
@[simp] lemma principal_ne_bot_iff {s : set α} : ne_bot (𝓟 s) ↔ s.nonempty :=
ne_bot_iff.trans $ (not_congr principal_eq_bot_iff).trans ne_empty_iff_nonempty
lemma is_compl_principal (s : set α) : is_compl (𝓟 s) (𝓟 sᶜ) :=
⟨by simp only [inf_principal, inter_compl_self, principal_empty, le_refl],
by simp only [sup_principal, union_compl_self, principal_univ, le_refl]⟩
theorem mem_inf_principal {f : filter α} {s t : set α} :
s ∈ f ⊓ 𝓟 t ↔ {x | x ∈ t → x ∈ s} ∈ f :=
begin
simp only [← le_principal_iff, (is_compl_principal s).le_left_iff, disjoint, inf_assoc,
inf_principal, imp_iff_not_or],
rw [← disjoint, ← (is_compl_principal (t ∩ sᶜ)).le_right_iff, compl_inter, compl_compl],
refl
end
lemma inf_principal_eq_bot {f : filter α} {s : set α} : f ⊓ 𝓟 s = ⊥ ↔ sᶜ ∈ f :=
by { rw [← empty_mem_iff_bot, mem_inf_principal], refl }
lemma mem_of_eq_bot {f : filter α} {s : set α} (h : f ⊓ 𝓟 sᶜ = ⊥) : s ∈ f :=
by rwa [inf_principal_eq_bot, compl_compl] at h
lemma diff_mem_inf_principal_compl {f : filter α} {s : set α} (hs : s ∈ f) (t : set α) :
s \ t ∈ f ⊓ 𝓟 tᶜ :=
begin
rw mem_inf_principal,
filter_upwards [hs],
intros a has hat,
exact ⟨has, hat⟩
end
lemma principal_le_iff {s : set α} {f : filter α} :
𝓟 s ≤ f ↔ ∀ V ∈ f, s ⊆ V :=
begin
change (∀ V, V ∈ f → V ∈ _) ↔ _,
simp_rw mem_principal,
end
@[simp] lemma infi_principal_finset {ι : Type w} (s : finset ι) (f : ι → set α) :
(⨅ i ∈ s, 𝓟 (f i)) = 𝓟 (⋂ i ∈ s, f i) :=
begin
induction s using finset.induction_on with i s hi hs,
{ simp },
{ rw [finset.infi_insert, finset.set_bInter_insert, hs, inf_principal] },
end
@[simp] lemma infi_principal_fintype {ι : Type w} [fintype ι] (f : ι → set α) :
(⨅ i, 𝓟 (f i)) = 𝓟 (⋂ i, f i) :=
by simpa using infi_principal_finset finset.univ f
lemma infi_principal_finite {ι : Type w} {s : set ι} (hs : finite s) (f : ι → set α) :
(⨅ i ∈ s, 𝓟 (f i)) = 𝓟 (⋂ i ∈ s, f i) :=
begin
unfreezingI { lift s to finset ι using hs }, -- TODO: why `unfreezingI` is needed?
exact_mod_cast infi_principal_finset s f
end
end lattice
@[mono] lemma join_mono {f₁ f₂ : filter (filter α)} (h : f₁ ≤ f₂) :
join f₁ ≤ join f₂ :=
λ s hs, h hs
/-! ### Eventually -/
/-- `f.eventually p` or `∀ᶠ x in f, p x` mean that `{x | p x} ∈ f`. E.g., `∀ᶠ x in at_top, p x`
means that `p` holds true for sufficiently large `x`. -/
protected def eventually (p : α → Prop) (f : filter α) : Prop := {x | p x} ∈ f
notation `∀ᶠ` binders ` in ` f `, ` r:(scoped p, filter.eventually p f) := r
lemma eventually_iff {f : filter α} {P : α → Prop} : (∀ᶠ x in f, P x) ↔ {x | P x} ∈ f :=
iff.rfl
protected lemma ext' {f₁ f₂ : filter α}
(h : ∀ p : α → Prop, (∀ᶠ x in f₁, p x) ↔ (∀ᶠ x in f₂, p x)) :
f₁ = f₂ :=
filter.ext h
lemma eventually.filter_mono {f₁ f₂ : filter α} (h : f₁ ≤ f₂) {p : α → Prop}
(hp : ∀ᶠ x in f₂, p x) :
∀ᶠ x in f₁, p x :=
h hp
lemma eventually_of_mem {f : filter α} {P : α → Prop} {U : set α} (hU : U ∈ f) (h : ∀ x ∈ U, P x) :
∀ᶠ x in f, P x :=
mem_of_superset hU h
protected lemma eventually.and {p q : α → Prop} {f : filter α} :
f.eventually p → f.eventually q → ∀ᶠ x in f, p x ∧ q x :=
inter_mem
@[simp]
lemma eventually_true (f : filter α) : ∀ᶠ x in f, true := univ_mem
lemma eventually_of_forall {p : α → Prop} {f : filter α} (hp : ∀ x, p x) :
∀ᶠ x in f, p x :=
univ_mem' hp
@[simp] lemma eventually_false_iff_eq_bot {f : filter α} :
(∀ᶠ x in f, false) ↔ f = ⊥ :=
empty_mem_iff_bot
@[simp] lemma eventually_const {f : filter α} [t : ne_bot f] {p : Prop} :
(∀ᶠ x in f, p) ↔ p :=
classical.by_cases (λ h : p, by simp [h]) (λ h, by simpa [h] using t.ne)
lemma eventually_iff_exists_mem {p : α → Prop} {f : filter α} :
(∀ᶠ x in f, p x) ↔ ∃ v ∈ f, ∀ y ∈ v, p y :=
exists_mem_subset_iff.symm
lemma eventually.exists_mem {p : α → Prop} {f : filter α} (hp : ∀ᶠ x in f, p x) :
∃ v ∈ f, ∀ y ∈ v, p y :=
eventually_iff_exists_mem.1 hp
lemma eventually.mp {p q : α → Prop} {f : filter α} (hp : ∀ᶠ x in f, p x)
(hq : ∀ᶠ x in f, p x → q x) :
∀ᶠ x in f, q x :=
mp_mem hp hq
lemma eventually.mono {p q : α → Prop} {f : filter α} (hp : ∀ᶠ x in f, p x)
(hq : ∀ x, p x → q x) :
∀ᶠ x in f, q x :=
hp.mp (eventually_of_forall hq)
@[simp] lemma eventually_and {p q : α → Prop} {f : filter α} :
(∀ᶠ x in f, p x ∧ q x) ↔ (∀ᶠ x in f, p x) ∧ (∀ᶠ x in f, q x) :=
inter_mem_iff
lemma eventually.congr {f : filter α} {p q : α → Prop} (h' : ∀ᶠ x in f, p x)
(h : ∀ᶠ x in f, p x ↔ q x) : ∀ᶠ x in f, q x :=
h'.mp (h.mono $ λ x hx, hx.mp)
lemma eventually_congr {f : filter α} {p q : α → Prop} (h : ∀ᶠ x in f, p x ↔ q x) :
(∀ᶠ x in f, p x) ↔ (∀ᶠ x in f, q x) :=
⟨λ hp, hp.congr h, λ hq, hq.congr $ by simpa only [iff.comm] using h⟩
@[simp] lemma eventually_all {ι} [fintype ι] {l} {p : ι → α → Prop} :
(∀ᶠ x in l, ∀ i, p i x) ↔ ∀ i, ∀ᶠ x in l, p i x :=
by simpa only [filter.eventually, set_of_forall] using Inter_mem
@[simp] lemma eventually_all_finite {ι} {I : set ι} (hI : I.finite) {l} {p : ι → α → Prop} :
(∀ᶠ x in l, ∀ i ∈ I, p i x) ↔ (∀ i ∈ I, ∀ᶠ x in l, p i x) :=
by simpa only [filter.eventually, set_of_forall] using bInter_mem hI
alias eventually_all_finite ← set.finite.eventually_all
attribute [protected] set.finite.eventually_all
@[simp] lemma eventually_all_finset {ι} (I : finset ι) {l} {p : ι → α → Prop} :
(∀ᶠ x in l, ∀ i ∈ I, p i x) ↔ ∀ i ∈ I, ∀ᶠ x in l, p i x :=
I.finite_to_set.eventually_all
alias eventually_all_finset ← finset.eventually_all
attribute [protected] finset.eventually_all
@[simp] lemma eventually_or_distrib_left {f : filter α} {p : Prop} {q : α → Prop} :
(∀ᶠ x in f, p ∨ q x) ↔ (p ∨ ∀ᶠ x in f, q x) :=
classical.by_cases (λ h : p, by simp [h]) (λ h, by simp [h])
@[simp] lemma eventually_or_distrib_right {f : filter α} {p : α → Prop} {q : Prop} :
(∀ᶠ x in f, p x ∨ q) ↔ ((∀ᶠ x in f, p x) ∨ q) :=
by simp only [or_comm _ q, eventually_or_distrib_left]
@[simp] lemma eventually_imp_distrib_left {f : filter α} {p : Prop} {q : α → Prop} :
(∀ᶠ x in f, p → q x) ↔ (p → ∀ᶠ x in f, q x) :=
by simp only [imp_iff_not_or, eventually_or_distrib_left]
@[simp]
lemma eventually_bot {p : α → Prop} : ∀ᶠ x in ⊥, p x := ⟨⟩
@[simp]
lemma eventually_top {p : α → Prop} : (∀ᶠ x in ⊤, p x) ↔ (∀ x, p x) :=
iff.rfl
@[simp] lemma eventually_sup {p : α → Prop} {f g : filter α} :
(∀ᶠ x in f ⊔ g, p x) ↔ (∀ᶠ x in f, p x) ∧ (∀ᶠ x in g, p x) :=
iff.rfl
@[simp]
lemma eventually_Sup {p : α → Prop} {fs : set (filter α)} :
(∀ᶠ x in Sup fs, p x) ↔ (∀ f ∈ fs, ∀ᶠ x in f, p x) :=
iff.rfl
@[simp]
lemma eventually_supr {p : α → Prop} {fs : β → filter α} :
(∀ᶠ x in (⨆ b, fs b), p x) ↔ (∀ b, ∀ᶠ x in fs b, p x) :=
mem_supr
@[simp]
lemma eventually_principal {a : set α} {p : α → Prop} :
(∀ᶠ x in 𝓟 a, p x) ↔ (∀ x ∈ a, p x) :=
iff.rfl
lemma eventually_inf {f g : filter α} {p : α → Prop} :
(∀ᶠ x in f ⊓ g, p x) ↔ ∃ (s ∈ f) (t ∈ g), ∀ x ∈ s ∩ t, p x :=
mem_inf_iff_superset
theorem eventually_inf_principal {f : filter α} {p : α → Prop} {s : set α} :
(∀ᶠ x in f ⊓ 𝓟 s, p x) ↔ ∀ᶠ x in f, x ∈ s → p x :=
mem_inf_principal
/-! ### Frequently -/
/-- `f.frequently p` or `∃ᶠ x in f, p x` mean that `{x | ¬p x} ∉ f`. E.g., `∃ᶠ x in at_top, p x`
means that there exist arbitrarily large `x` for which `p` holds true. -/
protected def frequently (p : α → Prop) (f : filter α) : Prop := ¬∀ᶠ x in f, ¬p x
notation `∃ᶠ` binders ` in ` f `, ` r:(scoped p, filter.frequently p f) := r
lemma eventually.frequently {f : filter α} [ne_bot f] {p : α → Prop} (h : ∀ᶠ x in f, p x) :
∃ᶠ x in f, p x :=
compl_not_mem h
lemma frequently_of_forall {f : filter α} [ne_bot f] {p : α → Prop} (h : ∀ x, p x) :
∃ᶠ x in f, p x :=
eventually.frequently (eventually_of_forall h)
lemma frequently.mp {p q : α → Prop} {f : filter α} (h : ∃ᶠ x in f, p x)
(hpq : ∀ᶠ x in f, p x → q x) :
∃ᶠ x in f, q x :=
mt (λ hq, hq.mp $ hpq.mono $ λ x, mt) h
lemma frequently.filter_mono {p : α → Prop} {f g : filter α} (h : ∃ᶠ x in f, p x) (hle : f ≤ g) :
∃ᶠ x in g, p x :=
mt (λ h', h'.filter_mono hle) h
lemma frequently.mono {p q : α → Prop} {f : filter α} (h : ∃ᶠ x in f, p x)
(hpq : ∀ x, p x → q x) :
∃ᶠ x in f, q x :=
h.mp (eventually_of_forall hpq)
lemma frequently.and_eventually {p q : α → Prop} {f : filter α}
(hp : ∃ᶠ x in f, p x) (hq : ∀ᶠ x in f, q x) :
∃ᶠ x in f, p x ∧ q x :=
begin
refine mt (λ h, hq.mp $ h.mono _) hp,
exact λ x hpq hq hp, hpq ⟨hp, hq⟩
end
lemma frequently.exists {p : α → Prop} {f : filter α} (hp : ∃ᶠ x in f, p x) : ∃ x, p x :=
begin
by_contradiction H,
replace H : ∀ᶠ x in f, ¬ p x, from eventually_of_forall (not_exists.1 H),
exact hp H
end
lemma eventually.exists {p : α → Prop} {f : filter α} [ne_bot f] (hp : ∀ᶠ x in f, p x) :
∃ x, p x :=
hp.frequently.exists
lemma frequently_iff_forall_eventually_exists_and {p : α → Prop} {f : filter α} :
(∃ᶠ x in f, p x) ↔ ∀ {q : α → Prop}, (∀ᶠ x in f, q x) → ∃ x, p x ∧ q x :=
⟨λ hp q hq, (hp.and_eventually hq).exists,
λ H hp, by simpa only [and_not_self, exists_false] using H hp⟩
lemma frequently_iff {f : filter α} {P : α → Prop} :
(∃ᶠ x in f, P x) ↔ ∀ {U}, U ∈ f → ∃ x ∈ U, P x :=
begin
rw frequently_iff_forall_eventually_exists_and,
split ; intro h,
{ intros U U_in,
simpa [exists_prop, and_comm] using h U_in },
{ intros H H',
simpa [and_comm] using h H' },
end
@[simp] lemma not_eventually {p : α → Prop} {f : filter α} :
(¬ ∀ᶠ x in f, p x) ↔ (∃ᶠ x in f, ¬ p x) :=
by simp [filter.frequently]
@[simp] lemma not_frequently {p : α → Prop} {f : filter α} :
(¬ ∃ᶠ x in f, p x) ↔ (∀ᶠ x in f, ¬ p x) :=
by simp only [filter.frequently, not_not]
@[simp] lemma frequently_true_iff_ne_bot (f : filter α) : (∃ᶠ x in f, true) ↔ ne_bot f :=
by simp [filter.frequently, -not_eventually, eventually_false_iff_eq_bot, ne_bot_iff]
@[simp] lemma frequently_false (f : filter α) : ¬ ∃ᶠ x in f, false := by simp
@[simp] lemma frequently_const {f : filter α} [ne_bot f] {p : Prop} :
(∃ᶠ x in f, p) ↔ p :=
classical.by_cases (λ h : p, by simpa [h]) (λ h, by simp [h])
@[simp] lemma frequently_or_distrib {f : filter α} {p q : α → Prop} :
(∃ᶠ x in f, p x ∨ q x) ↔ (∃ᶠ x in f, p x) ∨ (∃ᶠ x in f, q x) :=
by simp only [filter.frequently, ← not_and_distrib, not_or_distrib, eventually_and]
lemma frequently_or_distrib_left {f : filter α} [ne_bot f] {p : Prop} {q : α → Prop} :
(∃ᶠ x in f, p ∨ q x) ↔ (p ∨ ∃ᶠ x in f, q x) :=
by simp
lemma frequently_or_distrib_right {f : filter α} [ne_bot f] {p : α → Prop} {q : Prop} :
(∃ᶠ x in f, p x ∨ q) ↔ (∃ᶠ x in f, p x) ∨ q :=
by simp
@[simp] lemma frequently_imp_distrib {f : filter α} {p q : α → Prop} :
(∃ᶠ x in f, p x → q x) ↔ ((∀ᶠ x in f, p x) → ∃ᶠ x in f, q x) :=
by simp [imp_iff_not_or, not_eventually, frequently_or_distrib]
lemma frequently_imp_distrib_left {f : filter α} [ne_bot f] {p : Prop} {q : α → Prop} :
(∃ᶠ x in f, p → q x) ↔ (p → ∃ᶠ x in f, q x) :=
by simp
lemma frequently_imp_distrib_right {f : filter α} [ne_bot f] {p : α → Prop} {q : Prop} :
(∃ᶠ x in f, p x → q) ↔ ((∀ᶠ x in f, p x) → q) :=
by simp
@[simp] lemma eventually_imp_distrib_right {f : filter α} {p : α → Prop} {q : Prop} :
(∀ᶠ x in f, p x → q) ↔ ((∃ᶠ x in f, p x) → q) :=
by simp only [imp_iff_not_or, eventually_or_distrib_right, not_frequently]
@[simp] lemma frequently_bot {p : α → Prop} : ¬ ∃ᶠ x in ⊥, p x := by simp
@[simp]
lemma frequently_top {p : α → Prop} : (∃ᶠ x in ⊤, p x) ↔ (∃ x, p x) :=
by simp [filter.frequently]
@[simp]
lemma frequently_principal {a : set α} {p : α → Prop} :
(∃ᶠ x in 𝓟 a, p x) ↔ (∃ x ∈ a, p x) :=
by simp [filter.frequently, not_forall]
lemma frequently_sup {p : α → Prop} {f g : filter α} :
(∃ᶠ x in f ⊔ g, p x) ↔ (∃ᶠ x in f, p x) ∨ (∃ᶠ x in g, p x) :=
by simp only [filter.frequently, eventually_sup, not_and_distrib]
@[simp]
lemma frequently_Sup {p : α → Prop} {fs : set (filter α)} :
(∃ᶠ x in Sup fs, p x) ↔ (∃ f ∈ fs, ∃ᶠ x in f, p x) :=
by simp [filter.frequently, -not_eventually, not_forall]
@[simp]
lemma frequently_supr {p : α → Prop} {fs : β → filter α} :
(∃ᶠ x in (⨆ b, fs b), p x) ↔ (∃ b, ∃ᶠ x in fs b, p x) :=
by simp [filter.frequently, -not_eventually, not_forall]
/-!
### Relation “eventually equal”
-/
/-- Two functions `f` and `g` are *eventually equal* along a filter `l` if the set of `x` such that
`f x = g x` belongs to `l`. -/
def eventually_eq (l : filter α) (f g : α → β) : Prop := ∀ᶠ x in l, f x = g x
notation f ` =ᶠ[`:50 l:50 `] `:0 g:50 := eventually_eq l f g
lemma eventually_eq.eventually {l : filter α} {f g : α → β} (h : f =ᶠ[l] g) :
∀ᶠ x in l, f x = g x :=
h
lemma eventually_eq.rw {l : filter α} {f g : α → β} (h : f =ᶠ[l] g) (p : α → β → Prop)
(hf : ∀ᶠ x in l, p x (f x)) :
∀ᶠ x in l, p x (g x) :=
hf.congr $ h.mono $ λ x hx, hx ▸ iff.rfl
lemma eventually_eq_set {s t : set α} {l : filter α} :
s =ᶠ[l] t ↔ ∀ᶠ x in l, x ∈ s ↔ x ∈ t :=
eventually_congr $ eventually_of_forall $ λ x, ⟨eq.to_iff, iff.to_eq⟩
alias eventually_eq_set ↔ filter.eventually_eq.mem_iff filter.eventually.set_eq
lemma eventually_eq.exists_mem {l : filter α} {f g : α → β} (h : f =ᶠ[l] g) :
∃ s ∈ l, eq_on f g s :=
h.exists_mem
lemma eventually_eq_of_mem {l : filter α} {f g : α → β} {s : set α}
(hs : s ∈ l) (h : eq_on f g s) : f =ᶠ[l] g :=
eventually_of_mem hs h
lemma eventually_eq_iff_exists_mem {l : filter α} {f g : α → β} :
(f =ᶠ[l] g) ↔ ∃ s ∈ l, eq_on f g s :=
eventually_iff_exists_mem
lemma eventually_eq.filter_mono {l l' : filter α} {f g : α → β} (h₁ : f =ᶠ[l] g) (h₂ : l' ≤ l) :
f =ᶠ[l'] g :=
h₂ h₁
@[refl] lemma eventually_eq.refl (l : filter α) (f : α → β) :
f =ᶠ[l] f :=
eventually_of_forall $ λ x, rfl
lemma eventually_eq.rfl {l : filter α} {f : α → β} : f =ᶠ[l] f := eventually_eq.refl l f
@[symm] lemma eventually_eq.symm {f g : α → β} {l : filter α} (H : f =ᶠ[l] g) :
g =ᶠ[l] f :=
H.mono $ λ _, eq.symm
@[trans] lemma eventually_eq.trans {f g h : α → β} {l : filter α}
(H₁ : f =ᶠ[l] g) (H₂ : g =ᶠ[l] h) :
f =ᶠ[l] h :=
H₂.rw (λ x y, f x = y) H₁
lemma eventually_eq.prod_mk {l} {f f' : α → β} (hf : f =ᶠ[l] f') {g g' : α → γ} (hg : g =ᶠ[l] g') :
(λ x, (f x, g x)) =ᶠ[l] (λ x, (f' x, g' x)) :=
hf.mp $ hg.mono $ by { intros, simp only * }
lemma eventually_eq.fun_comp {f g : α → β} {l : filter α} (H : f =ᶠ[l] g) (h : β → γ) :
(h ∘ f) =ᶠ[l] (h ∘ g) :=
H.mono $ λ x hx, congr_arg h hx
lemma eventually_eq.comp₂ {δ} {f f' : α → β} {g g' : α → γ} {l} (Hf : f =ᶠ[l] f') (h : β → γ → δ)
(Hg : g =ᶠ[l] g') :
(λ x, h (f x) (g x)) =ᶠ[l] (λ x, h (f' x) (g' x)) :=
(Hf.prod_mk Hg).fun_comp (function.uncurry h)
@[to_additive]
lemma eventually_eq.mul [has_mul β] {f f' g g' : α → β} {l : filter α} (h : f =ᶠ[l] g)
(h' : f' =ᶠ[l] g') :
((λ x, f x * f' x) =ᶠ[l] (λ x, g x * g' x)) :=
h.comp₂ (*) h'
@[to_additive]
lemma eventually_eq.inv [has_inv β] {f g : α → β} {l : filter α} (h : f =ᶠ[l] g) :
((λ x, (f x)⁻¹) =ᶠ[l] (λ x, (g x)⁻¹)) :=
h.fun_comp has_inv.inv
lemma eventually_eq.div [group_with_zero β] {f f' g g' : α → β} {l : filter α} (h : f =ᶠ[l] g)
(h' : f' =ᶠ[l] g') :
((λ x, f x / f' x) =ᶠ[l] (λ x, g x / g' x)) :=
by simpa only [div_eq_mul_inv] using h.mul h'.inv
lemma eventually_eq.div' [group β] {f f' g g' : α → β} {l : filter α} (h : f =ᶠ[l] g)
(h' : f' =ᶠ[l] g') :
((λ x, f x / f' x) =ᶠ[l] (λ x, g x / g' x)) :=
by simpa only [div_eq_mul_inv] using h.mul h'.inv
lemma eventually_eq.sub [add_group β] {f f' g g' : α → β} {l : filter α} (h : f =ᶠ[l] g)
(h' : f' =ᶠ[l] g') :
((λ x, f x - f' x) =ᶠ[l] (λ x, g x - g' x)) :=
by simpa only [sub_eq_add_neg] using h.add h'.neg
lemma eventually_eq.inter {s t s' t' : set α} {l : filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') :
(s ∩ s' : set α) =ᶠ[l] (t ∩ t' : set α) :=
h.comp₂ (∧) h'
lemma eventually_eq.union {s t s' t' : set α} {l : filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') :
(s ∪ s' : set α) =ᶠ[l] (t ∪ t' : set α) :=
h.comp₂ (∨) h'
lemma eventually_eq.compl {s t : set α} {l : filter α} (h : s =ᶠ[l] t) :
(sᶜ : set α) =ᶠ[l] (tᶜ : set α) :=
h.fun_comp not
lemma eventually_eq.diff {s t s' t' : set α} {l : filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') :
(s \ s' : set α) =ᶠ[l] (t \ t' : set α) :=
h.inter h'.compl
lemma eventually_eq_empty {s : set α} {l : filter α} :
s =ᶠ[l] (∅ : set α) ↔ ∀ᶠ x in l, x ∉ s :=
eventually_eq_set.trans $ by simp
lemma inter_eventually_eq_left {s t : set α} {l : filter α} :
(s ∩ t : set α) =ᶠ[l] s ↔ ∀ᶠ x in l, x ∈ s → x ∈ t :=
by simp only [eventually_eq_set, mem_inter_eq, and_iff_left_iff_imp]
lemma inter_eventually_eq_right {s t : set α} {l : filter α} :
(s ∩ t : set α) =ᶠ[l] t ↔ ∀ᶠ x in l, x ∈ t → x ∈ s :=
by rw [inter_comm, inter_eventually_eq_left]
@[simp] lemma eventually_eq_principal {s : set α} {f g : α → β} :
f =ᶠ[𝓟 s] g ↔ eq_on f g s :=
iff.rfl
lemma eventually_eq_inf_principal_iff {F : filter α} {s : set α} {f g : α → β} :
(f =ᶠ[F ⊓ 𝓟 s] g) ↔ ∀ᶠ x in F, x ∈ s → f x = g x :=
eventually_inf_principal
lemma eventually_eq.sub_eq [add_group β] {f g : α → β} {l : filter α} (h : f =ᶠ[l] g) :
f - g =ᶠ[l] 0 :=
by simpa using (eventually_eq.sub (eventually_eq.refl l f) h).symm
lemma eventually_eq_iff_sub [add_group β] {f g : α → β} {l : filter α} :
f =ᶠ[l] g ↔ f - g =ᶠ[l] 0 :=
⟨λ h, h.sub_eq, λ h, by simpa using h.add (eventually_eq.refl l g)⟩
section has_le
variables [has_le β] {l : filter α}
/-- A function `f` is eventually less than or equal to a function `g` at a filter `l`. -/
def eventually_le (l : filter α) (f g : α → β) : Prop := ∀ᶠ x in l, f x ≤ g x
notation f ` ≤ᶠ[`:50 l:50 `] `:0 g:50 := eventually_le l f g
lemma eventually_le.congr {f f' g g' : α → β} (H : f ≤ᶠ[l] g) (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
f' ≤ᶠ[l] g' :=
H.mp $ hg.mp $ hf.mono $ λ x hf hg H, by rwa [hf, hg] at H
lemma eventually_le_congr {f f' g g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
f ≤ᶠ[l] g ↔ f' ≤ᶠ[l] g' :=
⟨λ H, H.congr hf hg, λ H, H.congr hf.symm hg.symm⟩
end has_le
section preorder
variables [preorder β] {l : filter α} {f g h : α → β}
lemma eventually_eq.le (h : f =ᶠ[l] g) : f ≤ᶠ[l] g := h.mono $ λ x, le_of_eq
@[refl] lemma eventually_le.refl (l : filter α) (f : α → β) :
f ≤ᶠ[l] f :=
eventually_eq.rfl.le
lemma eventually_le.rfl : f ≤ᶠ[l] f := eventually_le.refl l f
@[trans] lemma eventually_le.trans (H₁ : f ≤ᶠ[l] g) (H₂ : g ≤ᶠ[l] h) : f ≤ᶠ[l] h :=
H₂.mp $ H₁.mono $ λ x, le_trans
@[trans] lemma eventually_eq.trans_le (H₁ : f =ᶠ[l] g) (H₂ : g ≤ᶠ[l] h) : f ≤ᶠ[l] h :=
H₁.le.trans H₂
@[trans] lemma eventually_le.trans_eq (H₁ : f ≤ᶠ[l] g) (H₂ : g =ᶠ[l] h) : f ≤ᶠ[l] h :=
H₁.trans H₂.le
end preorder
lemma eventually_le.antisymm [partial_order β] {l : filter α} {f g : α → β}
(h₁ : f ≤ᶠ[l] g) (h₂ : g ≤ᶠ[l] f) :
f =ᶠ[l] g :=
h₂.mp $ h₁.mono $ λ x, le_antisymm
lemma eventually_le_antisymm_iff [partial_order β] {l : filter α} {f g : α → β} :
f =ᶠ[l] g ↔ f ≤ᶠ[l] g ∧ g ≤ᶠ[l] f :=
by simp only [eventually_eq, eventually_le, le_antisymm_iff, eventually_and]
lemma eventually_le.le_iff_eq [partial_order β] {l : filter α} {f g : α → β} (h : f ≤ᶠ[l] g) :
g ≤ᶠ[l] f ↔ g =ᶠ[l] f :=
⟨λ h', h'.antisymm h, eventually_eq.le⟩
@[mono] lemma eventually_le.inter {s t s' t' : set α} {l : filter α} (h : s ≤ᶠ[l] t)
(h' : s' ≤ᶠ[l] t') :
(s ∩ s' : set α) ≤ᶠ[l] (t ∩ t' : set α) :=
h'.mp $ h.mono $ λ x, and.imp
@[mono] lemma eventually_le.union {s t s' t' : set α} {l : filter α} (h : s ≤ᶠ[l] t)
(h' : s' ≤ᶠ[l] t') :
(s ∪ s' : set α) ≤ᶠ[l] (t ∪ t' : set α) :=
h'.mp $ h.mono $ λ x, or.imp
@[mono] lemma eventually_le.compl {s t : set α} {l : filter α} (h : s ≤ᶠ[l] t) :
(tᶜ : set α) ≤ᶠ[l] (sᶜ : set α) :=
h.mono $ λ x, mt
@[mono] lemma eventually_le.diff {s t s' t' : set α} {l : filter α} (h : s ≤ᶠ[l] t)
(h' : t' ≤ᶠ[l] s') :
(s \ s' : set α) ≤ᶠ[l] (t \ t' : set α) :=
h.inter h'.compl
lemma join_le {f : filter (filter α)} {l : filter α} (h : ∀ᶠ m in f, m ≤ l) : join f ≤ l :=
λ s hs, h.mono $ λ m hm, hm hs
/-! ### Push-forwards, pull-backs, and the monad structure -/
section map
/-- The forward map of a filter -/
def map (m : α → β) (f : filter α) : filter β :=
{ sets := preimage m ⁻¹' f.sets,
univ_sets := univ_mem,
sets_of_superset := λ s t hs st, mem_of_superset hs $ preimage_mono st,
inter_sets := λ s t hs ht, inter_mem hs ht }
@[simp] lemma map_principal {s : set α} {f : α → β} :
map f (𝓟 s) = 𝓟 (set.image f s) :=
filter.ext $ λ a, image_subset_iff.symm
variables {f : filter α} {m : α → β} {m' : β → γ} {s : set α} {t : set β}
@[simp] lemma eventually_map {P : β → Prop} :
(∀ᶠ b in map m f, P b) ↔ ∀ᶠ a in f, P (m a) :=
iff.rfl
@[simp] lemma frequently_map {P : β → Prop} :
(∃ᶠ b in map m f, P b) ↔ ∃ᶠ a in f, P (m a) :=
iff.rfl
@[simp] lemma mem_map : t ∈ map m f ↔ m ⁻¹' t ∈ f := iff.rfl
lemma mem_map' : t ∈ map m f ↔ {x | m x ∈ t} ∈ f := iff.rfl
lemma image_mem_map (hs : s ∈ f) : m '' s ∈ map m f :=
f.sets_of_superset hs $ subset_preimage_image m s
lemma image_mem_map_iff (hf : function.injective m) : m '' s ∈ map m f ↔ s ∈ f :=
⟨λ h, by rwa [← preimage_image_eq s hf], image_mem_map⟩
lemma range_mem_map : range m ∈ map m f :=
by { rw ←image_univ, exact image_mem_map univ_mem }
lemma mem_map_iff_exists_image : t ∈ map m f ↔ (∃ s ∈ f, m '' s ⊆ t) :=
⟨λ ht, ⟨m ⁻¹' t, ht, image_preimage_subset _ _⟩,
λ ⟨s, hs, ht⟩, mem_of_superset (image_mem_map hs) ht⟩
@[simp] lemma map_id : filter.map id f = f :=
filter_eq $ rfl
@[simp] lemma map_id' : filter.map (λ x, x) f = f := map_id
@[simp] lemma map_compose : filter.map m' ∘ filter.map m = filter.map (m' ∘ m) :=
funext $ λ _, filter_eq $ rfl
@[simp] lemma map_map : filter.map m' (filter.map m f) = filter.map (m' ∘ m) f :=
congr_fun (@@filter.map_compose m m') f
/-- If functions `m₁` and `m₂` are eventually equal at a filter `f`, then
they map this filter to the same filter. -/
lemma map_congr {m₁ m₂ : α → β} {f : filter α} (h : m₁ =ᶠ[f] m₂) :
map m₁ f = map m₂ f :=
filter.ext' $ λ p,
by { simp only [eventually_map], exact eventually_congr (h.mono $ λ x hx, hx ▸ iff.rfl) }
end map
section comap
/-- The inverse map of a filter -/
def comap (m : α → β) (f : filter β) : filter α :=
{ sets := { s | ∃ t ∈ f, m ⁻¹' t ⊆ s },
univ_sets := ⟨univ, univ_mem, by simp only [subset_univ, preimage_univ]⟩,
sets_of_superset := λ a b ⟨a', ha', ma'a⟩ ab, ⟨a', ha', ma'a.trans ab⟩,
inter_sets := λ a b ⟨a', ha₁, ha₂⟩ ⟨b', hb₁, hb₂⟩,
⟨a' ∩ b', inter_mem ha₁ hb₁, inter_subset_inter ha₂ hb₂⟩ }
lemma eventually_comap' {f : filter β} {φ : α → β} {p : β → Prop} (hf : ∀ᶠ b in f, p b) :
∀ᶠ a in comap φ f, p (φ a) :=
⟨_, hf, (λ a h, h)⟩
@[simp] lemma eventually_comap {f : filter β} {φ : α → β} {P : α → Prop} :
(∀ᶠ a in comap φ f, P a) ↔ ∀ᶠ b in f, ∀ a, φ a = b → P a :=
begin
split ; intro h,
{ rcases h with ⟨t, t_in, ht⟩,
apply mem_of_superset t_in,
rintro y y_in _ rfl,
apply ht y_in },
{ exact ⟨_, h, λ _ x_in, x_in _ rfl⟩ }
end
@[simp] lemma frequently_comap {f : filter β} {φ : α → β} {P : α → Prop} :
(∃ᶠ a in comap φ f, P a) ↔ ∃ᶠ b in f, ∃ a, φ a = b ∧ P a :=
begin
classical,
erw [← not_iff_not, not_not, not_not, filter.eventually_comap],
simp only [not_exists, not_and],
end
end comap
/-- The monadic bind operation on filter is defined the usual way in terms of `map` and `join`.
Unfortunately, this `bind` does not result in the expected applicative. See `filter.seq` for the
applicative instance. -/
def bind (f : filter α) (m : α → filter β) : filter β := join (map m f)
/-- The applicative sequentiation operation. This is not induced by the bind operation. -/
def seq (f : filter (α → β)) (g : filter α) : filter β :=
⟨{ s | ∃ u ∈ f, ∃ t ∈ g, (∀ m ∈ u, ∀ x ∈ t, (m : α → β) x ∈ s) },
⟨univ, univ_mem, univ, univ_mem,
by simp only [forall_prop_of_true, mem_univ, forall_true_iff]⟩,
λ s₀ s₁ ⟨t₀, t₁, h₀, h₁, h⟩ hst, ⟨t₀, t₁, h₀, h₁, λ x hx y hy, hst $ h _ hx _ hy⟩,
λ s₀ s₁ ⟨t₀, ht₀, t₁, ht₁, ht⟩ ⟨u₀, hu₀, u₁, hu₁, hu⟩,
⟨t₀ ∩ u₀, inter_mem ht₀ hu₀, t₁ ∩ u₁, inter_mem ht₁ hu₁,
λ x ⟨hx₀, hx₁⟩ x ⟨hy₀, hy₁⟩, ⟨ht _ hx₀ _ hy₀, hu _ hx₁ _ hy₁⟩⟩⟩
/-- `pure x` is the set of sets that contain `x`. It is equal to `𝓟 {x}` but
with this definition we have `s ∈ pure a` defeq `a ∈ s`. -/
instance : has_pure filter :=
⟨λ (α : Type u) x,
{ sets := {s | x ∈ s},
inter_sets := λ s t, and.intro,
sets_of_superset := λ s t hs hst, hst hs,
univ_sets := trivial }⟩
instance : has_bind filter := ⟨@filter.bind⟩
instance : has_seq filter := ⟨@filter.seq⟩
instance : functor filter := { map := @filter.map }
lemma pure_sets (a : α) : (pure a : filter α).sets = {s | a ∈ s} := rfl
@[simp] lemma mem_pure {a : α} {s : set α} : s ∈ (pure a : filter α) ↔ a ∈ s := iff.rfl
@[simp] lemma eventually_pure {a : α} {p : α → Prop} :
(∀ᶠ x in pure a, p x) ↔ p a :=
iff.rfl
@[simp] lemma principal_singleton (a : α) : 𝓟 {a} = pure a :=
filter.ext $ λ s, by simp only [mem_pure, mem_principal, singleton_subset_iff]
@[simp] lemma map_pure (f : α → β) (a : α) : map f (pure a) = pure (f a) :=
rfl
@[simp] lemma join_pure (f : filter α) : join (pure f) = f := filter.ext $ λ s, iff.rfl
@[simp] lemma pure_bind (a : α) (m : α → filter β) :
bind (pure a) m = m a :=
by simp only [has_bind.bind, bind, map_pure, join_pure]
section
-- this section needs to be before applicative, otherwise the wrong instance will be chosen
/-- The monad structure on filters. -/
protected def monad : monad filter := { map := @filter.map }
local attribute [instance] filter.monad
protected lemma is_lawful_monad : is_lawful_monad filter :=
{ id_map := λ α f, filter_eq rfl,
pure_bind := λ α β, pure_bind,
bind_assoc := λ α β γ f m₁ m₂, filter_eq rfl,
bind_pure_comp_eq_map := λ α β f x, filter.ext $ λ s,
by simp only [has_bind.bind, bind, functor.map, mem_map', mem_join, mem_set_of_eq,
function.comp, mem_pure] }
end
instance : applicative filter := { map := @filter.map, seq := @filter.seq }
instance : alternative filter :=
{ failure := λ α, ⊥,
orelse := λ α x y, x ⊔ y }
@[simp] lemma map_def {α β} (m : α → β) (f : filter α) : m <$> f = map m f := rfl
@[simp] lemma bind_def {α β} (f : filter α) (m : α → filter β) : f >>= m = bind f m := rfl
/-! #### `map` and `comap` equations -/
section map
variables {f f₁ f₂ : filter α} {g g₁ g₂ : filter β} {m : α → β} {m' : β → γ} {s : set α} {t : set β}
@[simp] theorem mem_comap : s ∈ comap m g ↔ ∃ t ∈ g, m ⁻¹' t ⊆ s := iff.rfl
theorem preimage_mem_comap (ht : t ∈ g) : m ⁻¹' t ∈ comap m g :=
⟨t, ht, subset.rfl⟩
lemma comap_id : comap id f = f :=
le_antisymm (λ s, preimage_mem_comap) (λ s ⟨t, ht, hst⟩, mem_of_superset ht hst)
lemma comap_const_of_not_mem {x : α} {f : filter α} {V : set α} (hV : V ∈ f) (hx : x ∉ V) :
comap (λ y : α, x) f = ⊥ :=
begin
ext W,
suffices : ∃ t ∈ f, (λ (y : α), x) ⁻¹' t ⊆ W, by simpa,
use [V, hV],
simp [preimage_const_of_not_mem hx],
end
lemma comap_const_of_mem {x : α} {f : filter α} (h : ∀ V ∈ f, x ∈ V) : comap (λ y : α, x) f = ⊤ :=
begin
ext W,
suffices : (∃ (t : set α), t ∈ f ∧ (λ (y : α), x) ⁻¹' t ⊆ W) ↔ W = univ,
by simpa,
split,
{ rintro ⟨V, V_in, hW⟩,
simpa [preimage_const_of_mem (h V V_in), univ_subset_iff] using hW },
{ rintro rfl,
use univ,
simp [univ_mem] },
end
lemma comap_comap {m : γ → β} {n : β → α} : comap m (comap n f) = comap (n ∘ m) f :=
le_antisymm
(λ c ⟨b, hb, (h : preimage (n ∘ m) b ⊆ c)⟩, ⟨preimage n b, preimage_mem_comap hb, h⟩)
(λ c ⟨b, ⟨a, ha, (h₁ : preimage n a ⊆ b)⟩, (h₂ : preimage m b ⊆ c)⟩,
⟨a, ha, show preimage m (preimage n a) ⊆ c, from (preimage_mono h₁).trans h₂⟩)
section comm
variables {δ : Type*}
/-!
The variables in the following lemmas are used as in this diagram:
```
φ
α → β
θ ↓ ↓ ψ
γ → δ
ρ
```
-/
variables {φ : α → β} {θ : α → γ} {ψ : β → δ} {ρ : γ → δ} (H : ψ ∘ φ = ρ ∘ θ)
include H
lemma map_comm (F : filter α) : map ψ (map φ F) = map ρ (map θ F) :=
by rw [filter.map_map, H, ← filter.map_map]
lemma comap_comm (G : filter δ) : comap φ (comap ψ G) = comap θ (comap ρ G) :=
by rw [filter.comap_comap, H, ← filter.comap_comap]
end comm
@[simp] theorem comap_principal {t : set β} : comap m (𝓟 t) = 𝓟 (m ⁻¹' t) :=
filter.ext $ λ s,
⟨λ ⟨u, (hu : t ⊆ u), (b : preimage m u ⊆ s)⟩, (preimage_mono hu).trans b,
λ h, ⟨t, subset.refl t, h⟩⟩
@[simp] theorem comap_pure {b : β} : comap m (pure b) = 𝓟 (m ⁻¹' {b}) :=
by rw [← principal_singleton, comap_principal]
lemma map_le_iff_le_comap : map m f ≤ g ↔ f ≤ comap m g :=
⟨λ h s ⟨t, ht, hts⟩, mem_of_superset (h ht) hts, λ h s ht, h ⟨_, ht, subset.rfl⟩⟩
lemma gc_map_comap (m : α → β) : galois_connection (map m) (comap m) :=
λ f g, map_le_iff_le_comap
@[mono] lemma map_mono : monotone (map m) := (gc_map_comap m).monotone_l
@[mono] lemma comap_mono : monotone (comap m) := (gc_map_comap m).monotone_u
@[simp] lemma map_bot : map m ⊥ = ⊥ := (gc_map_comap m).l_bot
@[simp] lemma map_sup : map m (f₁ ⊔ f₂) = map m f₁ ⊔ map m f₂ := (gc_map_comap m).l_sup
@[simp] lemma map_supr {f : ι → filter α} : map m (⨆ i, f i) = (⨆ i, map m (f i)) :=
(gc_map_comap m).l_supr
@[simp] lemma comap_top : comap m ⊤ = ⊤ := (gc_map_comap m).u_top
@[simp] lemma comap_inf : comap m (g₁ ⊓ g₂) = comap m g₁ ⊓ comap m g₂ := (gc_map_comap m).u_inf
@[simp] lemma comap_infi {f : ι → filter β} : comap m (⨅ i, f i) = (⨅ i, comap m (f i)) :=
(gc_map_comap m).u_infi
lemma le_comap_top (f : α → β) (l : filter α) : l ≤ comap f ⊤ :=
by { rw [comap_top], exact le_top }
lemma map_comap_le : map m (comap m g) ≤ g := (gc_map_comap m).l_u_le _
lemma le_comap_map : f ≤ comap m (map m f) := (gc_map_comap m).le_u_l _
@[simp] lemma comap_bot : comap m ⊥ = ⊥ :=
bot_unique $ λ s _, ⟨∅, by simp only [mem_bot], by simp only [empty_subset, preimage_empty]⟩
lemma comap_supr {ι} {f : ι → filter β} {m : α → β} :
comap m (supr f) = (⨆ i, comap m (f i)) :=
le_antisymm
(λ s hs,
have ∀ i, ∃ t, t ∈ f i ∧ m ⁻¹' t ⊆ s,
by simpa only [mem_comap, exists_prop, mem_supr] using mem_supr.1 hs,
let ⟨t, ht⟩ := classical.axiom_of_choice this in
⟨⋃ i, t i, mem_supr.2 $ λ i, (f i).sets_of_superset (ht i).1 (subset_Union _ _),
begin
rw [preimage_Union, Union_subset_iff],
exact λ i, (ht i).2
end⟩)
(supr_le $ λ i, comap_mono $ le_supr _ _)
lemma comap_Sup {s : set (filter β)} {m : α → β} : comap m (Sup s) = (⨆ f ∈ s, comap m f) :=
by simp only [Sup_eq_supr, comap_supr, eq_self_iff_true]
lemma comap_sup : comap m (g₁ ⊔ g₂) = comap m g₁ ⊔ comap m g₂ :=
by rw [sup_eq_supr, comap_supr, supr_bool_eq, bool.cond_tt, bool.cond_ff]
lemma map_comap (f : filter β) (m : α → β) : (f.comap m).map m = f ⊓ 𝓟 (range m) :=
begin
refine le_antisymm (le_inf map_comap_le $ le_principal_iff.2 range_mem_map) _,
rintro t' ⟨t, ht, sub⟩,
refine mem_inf_principal.2 (mem_of_superset ht _),
rintro _ hxt ⟨x, rfl⟩,
exact sub hxt
end
lemma map_comap_of_mem {f : filter β} {m : α → β} (hf : range m ∈ f) : (f.comap m).map m = f :=
by rw [map_comap, inf_eq_left.2 (le_principal_iff.2 hf)]
lemma comap_le_comap_iff {f g : filter β} {m : α → β} (hf : range m ∈ f) :
comap m f ≤ comap m g ↔ f ≤ g :=
⟨λ h, map_comap_of_mem hf ▸ (map_mono h).trans map_comap_le, λ h, comap_mono h⟩
theorem map_comap_of_surjective {f : α → β} (hf : function.surjective f) (l : filter β) :
map f (comap f l) = l :=
map_comap_of_mem $ by simp only [hf.range_eq, univ_mem]
lemma subtype_coe_map_comap (s : set α) (f : filter α) :
map (coe : s → α) (comap (coe : s → α) f) = f ⊓ 𝓟 s :=
by rw [map_comap, subtype.range_coe]
lemma subtype_coe_map_comap_prod (s : set α) (f : filter (α × α)) :
map (coe : s × s → α × α) (comap (coe : s × s → α × α) f) = f ⊓ 𝓟 (s.prod s) :=
have (coe : s × s → α × α) = (λ x, (x.1, x.2)), by ext ⟨x, y⟩; refl,
by simp [this, map_comap, ← prod_range_range_eq]
lemma image_mem_of_mem_comap {f : filter α} {c : β → α} (h : range c ∈ f) {W : set β}
(W_in : W ∈ comap c f) : c '' W ∈ f :=
begin
rw ← map_comap_of_mem h,
exact image_mem_map W_in
end
lemma image_coe_mem_of_mem_comap {f : filter α} {U : set α} (h : U ∈ f) {W : set U}
(W_in : W ∈ comap (coe : U → α) f) : coe '' W ∈ f :=
image_mem_of_mem_comap (by simp [h]) W_in
lemma comap_map {f : filter α} {m : α → β} (h : function.injective m) :
comap m (map m f) = f :=
le_antisymm
(λ s hs, mem_of_superset (preimage_mem_comap $ image_mem_map hs) $
by simp only [preimage_image_eq s h])
le_comap_map
lemma mem_comap_iff {f : filter β} {m : α → β} (inj : function.injective m)
(large : set.range m ∈ f) {S : set α} : S ∈ comap m f ↔ m '' S ∈ f :=
by rw [← image_mem_map_iff inj, map_comap_of_mem large]
lemma le_of_map_le_map_inj' {f g : filter α} {m : α → β} {s : set α}
(hsf : s ∈ f) (hsg : s ∈ g) (hm : ∀ x ∈ s, ∀ y ∈ s, m x = m y → x = y)
(h : map m f ≤ map m g) : f ≤ g :=
λ t ht, by filter_upwards [hsf, h $ image_mem_map (inter_mem hsg ht)]
λ a has ⟨b, ⟨hbs, hb⟩, h⟩,
hm _ hbs _ has h ▸ hb
lemma le_of_map_le_map_inj_iff {f g : filter α} {m : α → β} {s : set α}
(hsf : s ∈ f) (hsg : s ∈ g) (hm : ∀ x ∈ s, ∀ y ∈ s, m x = m y → x = y) :
map m f ≤ map m g ↔ f ≤ g :=
iff.intro (le_of_map_le_map_inj' hsf hsg hm) (λ h, map_mono h)
lemma eq_of_map_eq_map_inj' {f g : filter α} {m : α → β} {s : set α}
(hsf : s ∈ f) (hsg : s ∈ g) (hm : ∀ x ∈ s, ∀ y ∈ s, m x = m y → x = y)
(h : map m f = map m g) : f = g :=
le_antisymm
(le_of_map_le_map_inj' hsf hsg hm $ le_of_eq h)
(le_of_map_le_map_inj' hsg hsf hm $ le_of_eq h.symm)
lemma map_inj {f g : filter α} {m : α → β} (hm : function.injective m) (h : map m f = map m g) :
f = g :=
have comap m (map m f) = comap m (map m g), by rw h,
by rwa [comap_map hm, comap_map hm] at this
lemma comap_ne_bot_iff {f : filter β} {m : α → β} : ne_bot (comap m f) ↔ ∀ t ∈ f, ∃ a, m a ∈ t :=
begin
rw ← forall_mem_nonempty_iff_ne_bot,
exact ⟨λ h t t_in, h (m ⁻¹' t) ⟨t, t_in, subset.rfl⟩,
λ h s ⟨u, u_in, hu⟩, let ⟨x, hx⟩ := h u u_in in ⟨x, hu hx⟩⟩,
end
lemma comap_ne_bot {f : filter β} {m : α → β} (hm : ∀ t ∈ f, ∃ a, m a ∈ t) : ne_bot (comap m f) :=
comap_ne_bot_iff.mpr hm
lemma comap_ne_bot_iff_frequently {f : filter β} {m : α → β} :
ne_bot (comap m f) ↔ ∃ᶠ y in f, y ∈ range m :=
by simp [comap_ne_bot_iff, frequently_iff, ← exists_and_distrib_left, and.comm]
lemma comap_ne_bot_iff_compl_range {f : filter β} {m : α → β} :
ne_bot (comap m f) ↔ (range m)ᶜ ∉ f :=
comap_ne_bot_iff_frequently
lemma ne_bot.comap_of_range_mem {f : filter β} {m : α → β}
(hf : ne_bot f) (hm : range m ∈ f) : ne_bot (comap m f) :=
comap_ne_bot_iff_frequently.2 $ eventually.frequently hm
lemma comap_inf_principal_ne_bot_of_image_mem {f : filter β} {m : α → β}
(hf : ne_bot f) {s : set α} (hs : m '' s ∈ f) :
ne_bot (comap m f ⊓ 𝓟 s) :=
begin
refine ⟨compl_compl s ▸ mt mem_of_eq_bot _⟩,
rintro ⟨t, ht, hts⟩,
rcases hf.nonempty_of_mem (inter_mem hs ht) with ⟨_, ⟨x, hxs, rfl⟩, hxt⟩,
exact absurd hxs (hts hxt)
end
lemma comap_coe_ne_bot_of_le_principal {s : set γ} {l : filter γ} [h : ne_bot l] (h' : l ≤ 𝓟 s) :
ne_bot (comap (coe : s → γ) l) :=
h.comap_of_range_mem $ (@subtype.range_coe γ s).symm ▸ h' (mem_principal_self s)
lemma ne_bot.comap_of_surj {f : filter β} {m : α → β}
(hf : ne_bot f) (hm : function.surjective m) :
ne_bot (comap m f) :=
hf.comap_of_range_mem $ univ_mem' hm
lemma ne_bot.comap_of_image_mem {f : filter β} {m : α → β} (hf : ne_bot f)
{s : set α} (hs : m '' s ∈ f) :
ne_bot (comap m f) :=
hf.comap_of_range_mem $ mem_of_superset hs (image_subset_range _ _)
@[simp] lemma map_eq_bot_iff : map m f = ⊥ ↔ f = ⊥ :=
⟨by { rw [←empty_mem_iff_bot, ←empty_mem_iff_bot], exact id },
λ h, by simp only [h, eq_self_iff_true, map_bot]⟩
lemma map_ne_bot_iff (f : α → β) {F : filter α} : ne_bot (map f F) ↔ ne_bot F :=
by simp only [ne_bot_iff, ne, map_eq_bot_iff]
lemma ne_bot.map (hf : ne_bot f) (m : α → β) : ne_bot (map m f) :=
(map_ne_bot_iff m).2 hf
instance map_ne_bot [hf : ne_bot f] : ne_bot (f.map m) := hf.map m
lemma sInter_comap_sets (f : α → β) (F : filter β) :
⋂₀ (comap f F).sets = ⋂ U ∈ F, f ⁻¹' U :=
begin
ext x,
suffices : (∀ (A : set α) (B : set β), B ∈ F → f ⁻¹' B ⊆ A → x ∈ A) ↔
∀ (B : set β), B ∈ F → f x ∈ B,
by simp only [mem_sInter, mem_Inter, filter.mem_sets, mem_comap, this, and_imp,
exists_prop, mem_preimage, exists_imp_distrib],
split,
{ intros h U U_in,
simpa only [subset.refl, forall_prop_of_true, mem_preimage] using h (f ⁻¹' U) U U_in },
{ intros h V U U_in f_U_V,
exact f_U_V (h U U_in) },
end
end map
-- this is a generic rule for monotone functions:
lemma map_infi_le {f : ι → filter α} {m : α → β} :
map m (infi f) ≤ (⨅ i, map m (f i)) :=
le_infi $ λ i, map_mono $ infi_le _ _
lemma map_infi_eq {f : ι → filter α} {m : α → β} (hf : directed (≥) f) [nonempty ι] :
map m (infi f) = (⨅ i, map m (f i)) :=
map_infi_le.antisymm
(λ s (hs : preimage m s ∈ infi f),
let ⟨i, hi⟩ := (mem_infi_of_directed hf _).1 hs in
have (⨅ i, map m (f i)) ≤ 𝓟 s, from
infi_le_of_le i $ by { simp only [le_principal_iff, mem_map], assumption },
filter.le_principal_iff.1 this)
lemma map_binfi_eq {ι : Type w} {f : ι → filter α} {m : α → β} {p : ι → Prop}
(h : directed_on (f ⁻¹'o (≥)) {x | p x}) (ne : ∃ i, p i) :
map m (⨅ i (h : p i), f i) = (⨅ i (h : p i), map m (f i)) :=
begin
haveI := nonempty_subtype.2 ne,
simp only [infi_subtype'],
exact map_infi_eq h.directed_coe
end
lemma map_inf_le {f g : filter α} {m : α → β} : map m (f ⊓ g) ≤ map m f ⊓ map m g :=
(@map_mono _ _ m).map_inf_le f g
lemma map_inf' {f g : filter α} {m : α → β} {t : set α} (htf : t ∈ f) (htg : t ∈ g)
(h : ∀ x ∈ t, ∀ y ∈ t, m x = m y → x = y) : map m (f ⊓ g) = map m f ⊓ map m g :=
begin
refine le_antisymm map_inf_le (λ s hs, _),
simp only [mem_inf_iff, exists_prop, mem_map, mem_preimage, mem_inf_iff] at hs,
rcases hs with ⟨t₁, h₁, t₂, h₂, hs : m ⁻¹' s = t₁ ∩ t₂⟩,
have : m '' (t₁ ∩ t) ∩ m '' (t₂ ∩ t) ∈ map m f ⊓ map m g,
{ apply inter_mem_inf ; apply image_mem_map,
exacts [inter_mem h₁ htf, inter_mem h₂ htg] },
apply mem_of_superset this,
{ rw [image_inter_on],
{ refine image_subset_iff.2 _,
rw hs,
exact λ x ⟨⟨h₁, _⟩, h₂, _⟩, ⟨h₁, h₂⟩ },
{ exact λ x ⟨_, hx⟩ y ⟨_, hy⟩, h x hx y hy } }
end
lemma map_inf {f g : filter α} {m : α → β} (h : function.injective m) :
map m (f ⊓ g) = map m f ⊓ map m g :=
map_inf' univ_mem univ_mem (λ x _ y _ hxy, h hxy)
lemma map_eq_comap_of_inverse {f : filter α} {m : α → β} {n : β → α}
(h₁ : m ∘ n = id) (h₂ : n ∘ m = id) : map m f = comap n f :=
le_antisymm
(λ b ⟨a, ha, (h : preimage n a ⊆ b)⟩, f.sets_of_superset ha $
calc a = preimage (n ∘ m) a : by simp only [h₂, preimage_id, eq_self_iff_true]
... ⊆ preimage m b : preimage_mono h)
(λ b (hb : preimage m b ∈ f),
⟨preimage m b, hb, show preimage (m ∘ n) b ⊆ b, by simp only [h₁]; apply subset.refl⟩)
lemma map_swap_eq_comap_swap {f : filter (α × β)} : prod.swap <$> f = comap prod.swap f :=
map_eq_comap_of_inverse prod.swap_swap_eq prod.swap_swap_eq
lemma le_map {f : filter α} {m : α → β} {g : filter β} (h : ∀ s ∈ f, m '' s ∈ g) :
g ≤ f.map m :=
λ s hs, mem_of_superset (h _ hs) $ image_preimage_subset _ _
protected lemma push_pull (f : α → β) (F : filter α) (G : filter β) :
map f (F ⊓ comap f G) = map f F ⊓ G :=
begin
apply le_antisymm,
{ calc map f (F ⊓ comap f G) ≤ map f F ⊓ (map f $ comap f G) : map_inf_le
... ≤ map f F ⊓ G : inf_le_inf_left (map f F) map_comap_le },
{ rintro U ⟨V, V_in, W, ⟨Z, Z_in, hZ⟩, h⟩,
apply mem_inf_of_inter (image_mem_map V_in) Z_in,
calc
f '' V ∩ Z = f '' (V ∩ f ⁻¹' Z) : by rw image_inter_preimage
... ⊆ f '' (V ∩ W) : image_subset _ (inter_subset_inter_right _ ‹_›)
... = f '' (f ⁻¹' U) : by rw h
... ⊆ U : image_preimage_subset f U }
end
protected lemma push_pull' (f : α → β) (F : filter α) (G : filter β) :
map f (comap f G ⊓ F) = G ⊓ map f F :=
by simp only [filter.push_pull, inf_comm]
section applicative
lemma singleton_mem_pure {a : α} : {a} ∈ (pure a : filter α) :=
mem_singleton a
lemma pure_injective : function.injective (pure : α → filter α) :=
λ a b hab, (filter.ext_iff.1 hab {x | a = x}).1 rfl
instance pure_ne_bot {α : Type u} {a : α} : ne_bot (pure a) :=
⟨mt empty_mem_iff_bot.2 $ not_mem_empty a⟩
@[simp] lemma le_pure_iff {f : filter α} {a : α} : f ≤ pure a ↔ {a} ∈ f :=
⟨λ h, h singleton_mem_pure,
λ h s hs, mem_of_superset h $ singleton_subset_iff.2 hs⟩
lemma mem_seq_def {f : filter (α → β)} {g : filter α} {s : set β} :
s ∈ f.seq g ↔ (∃ u ∈ f, ∃ t ∈ g, ∀ x ∈ u, ∀ y ∈ t, (x : α → β) y ∈ s) :=
iff.rfl
lemma mem_seq_iff {f : filter (α → β)} {g : filter α} {s : set β} :
s ∈ f.seq g ↔ (∃ u ∈ f, ∃ t ∈ g, set.seq u t ⊆ s) :=
by simp only [mem_seq_def, seq_subset, exists_prop, iff_self]
lemma mem_map_seq_iff {f : filter α} {g : filter β} {m : α → β → γ} {s : set γ} :
s ∈ (f.map m).seq g ↔ (∃ t u, t ∈ g ∧ u ∈ f ∧ ∀ x ∈ u, ∀ y ∈ t, m x y ∈ s) :=
iff.intro
(λ ⟨t, ht, s, hs, hts⟩, ⟨s, m ⁻¹' t, hs, ht, λ a, hts _⟩)
(λ ⟨t, s, ht, hs, hts⟩, ⟨m '' s, image_mem_map hs, t, ht, λ f ⟨a, has, eq⟩, eq ▸ hts _ has⟩)
lemma seq_mem_seq {f : filter (α → β)} {g : filter α} {s : set (α → β)} {t : set α}
(hs : s ∈ f) (ht : t ∈ g) : s.seq t ∈ f.seq g :=
⟨s, hs, t, ht, λ f hf a ha, ⟨f, hf, a, ha, rfl⟩⟩
lemma le_seq {f : filter (α → β)} {g : filter α} {h : filter β}
(hh : ∀ t ∈ f, ∀ u ∈ g, set.seq t u ∈ h) : h ≤ seq f g :=
λ s ⟨t, ht, u, hu, hs⟩, mem_of_superset (hh _ ht _ hu) $
λ b ⟨m, hm, a, ha, eq⟩, eq ▸ hs _ hm _ ha
@[mono] lemma seq_mono {f₁ f₂ : filter (α → β)} {g₁ g₂ : filter α}
(hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : f₁.seq g₁ ≤ f₂.seq g₂ :=
le_seq $ λ s hs t ht, seq_mem_seq (hf hs) (hg ht)
@[simp] lemma pure_seq_eq_map (g : α → β) (f : filter α) : seq (pure g) f = f.map g :=
begin
refine le_antisymm (le_map $ λ s hs, _) (le_seq $ λ s hs t ht, _),
{ rw ← singleton_seq, apply seq_mem_seq _ hs,
exact singleton_mem_pure },
{ refine sets_of_superset (map g f) (image_mem_map ht) _,
rintro b ⟨a, ha, rfl⟩, exact ⟨g, hs, a, ha, rfl⟩ }
end
@[simp] lemma seq_pure (f : filter (α → β)) (a : α) : seq f (pure a) = map (λ g : α → β, g a) f :=
begin
refine le_antisymm (le_map $ λ s hs, _) (le_seq $ λ s hs t ht, _),
{ rw ← seq_singleton,
exact seq_mem_seq hs singleton_mem_pure },
{ refine sets_of_superset (map (λg:α→β, g a) f) (image_mem_map hs) _,
rintro b ⟨g, hg, rfl⟩, exact ⟨g, hg, a, ht, rfl⟩ }
end
@[simp] lemma seq_assoc (x : filter α) (g : filter (α → β)) (h : filter (β → γ)) :
seq h (seq g x) = seq (seq (map (∘) h) g) x :=
begin
refine le_antisymm (le_seq $ λ s hs t ht, _) (le_seq $ λ s hs t ht, _),
{ rcases mem_seq_iff.1 hs with ⟨u, hu, v, hv, hs⟩,
rcases mem_map_iff_exists_image.1 hu with ⟨w, hw, hu⟩,
refine mem_of_superset _
(set.seq_mono ((set.seq_mono hu subset.rfl).trans hs) subset.rfl),
rw ← set.seq_seq,
exact seq_mem_seq hw (seq_mem_seq hv ht) },
{ rcases mem_seq_iff.1 ht with ⟨u, hu, v, hv, ht⟩,
refine mem_of_superset _ (set.seq_mono subset.rfl ht),
rw set.seq_seq,
exact seq_mem_seq (seq_mem_seq (image_mem_map hs) hu) hv }
end
lemma prod_map_seq_comm (f : filter α) (g : filter β) :
(map prod.mk f).seq g = seq (map (λ b a, (a, b)) g) f :=
begin
refine le_antisymm (le_seq $ λ s hs t ht, _) (le_seq $ λ s hs t ht, _),
{ rcases mem_map_iff_exists_image.1 hs with ⟨u, hu, hs⟩,
refine mem_of_superset _ (set.seq_mono hs subset.rfl),
rw ← set.prod_image_seq_comm,
exact seq_mem_seq (image_mem_map ht) hu },
{ rcases mem_map_iff_exists_image.1 hs with ⟨u, hu, hs⟩,
refine mem_of_superset _ (set.seq_mono hs subset.rfl),
rw set.prod_image_seq_comm,
exact seq_mem_seq (image_mem_map ht) hu }
end
instance : is_lawful_functor (filter : Type u → Type u) :=
{ id_map := λ α f, map_id,
comp_map := λ α β γ f g a, map_map.symm }
instance : is_lawful_applicative (filter : Type u → Type u) :=
{ pure_seq_eq_map := λ α β, pure_seq_eq_map,
map_pure := λ α β, map_pure,
seq_pure := λ α β, seq_pure,
seq_assoc := λ α β γ, seq_assoc }
instance : is_comm_applicative (filter : Type u → Type u) :=
⟨λ α β f g, prod_map_seq_comm f g⟩
lemma {l} seq_eq_filter_seq {α β : Type l} (f : filter (α → β)) (g : filter α) :
f <*> g = seq f g := rfl
end applicative
/-! #### `bind` equations -/
section bind
@[simp] lemma eventually_bind {f : filter α} {m : α → filter β} {p : β → Prop} :
(∀ᶠ y in bind f m, p y) ↔ ∀ᶠ x in f, ∀ᶠ y in m x, p y :=
iff.rfl
@[simp] lemma eventually_eq_bind {f : filter α} {m : α → filter β} {g₁ g₂ : β → γ} :
(g₁ =ᶠ[bind f m] g₂) ↔ ∀ᶠ x in f, g₁ =ᶠ[m x] g₂ :=
iff.rfl
@[simp] lemma eventually_le_bind [has_le γ] {f : filter α} {m : α → filter β} {g₁ g₂ : β → γ} :
(g₁ ≤ᶠ[bind f m] g₂) ↔ ∀ᶠ x in f, g₁ ≤ᶠ[m x] g₂ :=
iff.rfl
lemma mem_bind' {s : set β} {f : filter α} {m : α → filter β} :
s ∈ bind f m ↔ {a | s ∈ m a} ∈ f :=
iff.rfl
@[simp] lemma mem_bind {s : set β} {f : filter α} {m : α → filter β} :
s ∈ bind f m ↔ ∃ t ∈ f, ∀ x ∈ t, s ∈ m x :=
calc s ∈ bind f m ↔ {a | s ∈ m a} ∈ f : iff.rfl
... ↔ (∃ t ∈ f, t ⊆ {a | s ∈ m a}) : exists_mem_subset_iff.symm
... ↔ (∃ t ∈ f, ∀ x ∈ t, s ∈ m x) : iff.rfl
lemma bind_le {f : filter α} {g : α → filter β} {l : filter β} (h : ∀ᶠ x in f, g x ≤ l) :
f.bind g ≤ l :=
join_le $ eventually_map.2 h
@[mono] lemma bind_mono {f₁ f₂ : filter α} {g₁ g₂ : α → filter β} (hf : f₁ ≤ f₂)
(hg : g₁ ≤ᶠ[f₁] g₂) :
bind f₁ g₁ ≤ bind f₂ g₂ :=
begin
refine le_trans (λ s hs, _) (join_mono $ map_mono hf),
simp only [mem_join, mem_bind', mem_map] at hs ⊢,
filter_upwards [hg, hs],
exact λ x hx hs, hx hs
end
lemma bind_inf_principal {f : filter α} {g : α → filter β} {s : set β} :
f.bind (λ x, g x ⊓ 𝓟 s) = (f.bind g) ⊓ 𝓟 s :=
filter.ext $ λ s, by simp only [mem_bind, mem_inf_principal]
lemma sup_bind {f g : filter α} {h : α → filter β} :
bind (f ⊔ g) h = bind f h ⊔ bind g h :=
by simp only [bind, sup_join, map_sup, eq_self_iff_true]
lemma principal_bind {s : set α} {f : α → filter β} :
(bind (𝓟 s) f) = (⨆ x ∈ s, f x) :=
show join (map f (𝓟 s)) = (⨆ x ∈ s, f x),
by simp only [Sup_image, join_principal_eq_Sup, map_principal, eq_self_iff_true]
end bind
section list_traverse
/- This is a separate section in order to open `list`, but mostly because of universe
equality requirements in `traverse` -/
open list
lemma sequence_mono :
∀ (as bs : list (filter α)), forall₂ (≤) as bs → sequence as ≤ sequence bs
| [] [] forall₂.nil := le_rfl
| (a :: as) (b :: bs) (forall₂.cons h hs) := seq_mono (map_mono h) (sequence_mono as bs hs)
variables {α' β' γ' : Type u} {f : β' → filter α'} {s : γ' → set α'}
lemma mem_traverse :
∀ (fs : list β') (us : list γ'),
forall₂ (λ b c, s c ∈ f b) fs us → traverse s us ∈ traverse f fs
| [] [] forall₂.nil := mem_pure.2 $ mem_singleton _
| (f :: fs) (u :: us) (forall₂.cons h hs) := seq_mem_seq (image_mem_map h) (mem_traverse fs us hs)
lemma mem_traverse_iff (fs : list β') (t : set (list α')) :
t ∈ traverse f fs ↔
(∃ us : list (set α'), forall₂ (λ b (s : set α'), s ∈ f b) fs us ∧ sequence us ⊆ t) :=
begin
split,
{ induction fs generalizing t,
case nil { simp only [sequence, mem_pure, imp_self, forall₂_nil_left_iff,
exists_eq_left, set.pure_def, singleton_subset_iff, traverse_nil] },
case cons : b fs ih t {
intro ht,
rcases mem_seq_iff.1 ht with ⟨u, hu, v, hv, ht⟩,
rcases mem_map_iff_exists_image.1 hu with ⟨w, hw, hwu⟩,
rcases ih v hv with ⟨us, hus, hu⟩,
exact ⟨w :: us, forall₂.cons hw hus, (set.seq_mono hwu hu).trans ht⟩ } },
{ rintro ⟨us, hus, hs⟩,
exact mem_of_superset (mem_traverse _ _ hus) hs }
end
end list_traverse
/-! ### Limits -/
/-- `tendsto` is the generic "limit of a function" predicate.
`tendsto f l₁ l₂` asserts that for every `l₂` neighborhood `a`,
the `f`-preimage of `a` is an `l₁` neighborhood. -/
def tendsto (f : α → β) (l₁ : filter α) (l₂ : filter β) := l₁.map f ≤ l₂
lemma tendsto_def {f : α → β} {l₁ : filter α} {l₂ : filter β} :
tendsto f l₁ l₂ ↔ ∀ s ∈ l₂, f ⁻¹' s ∈ l₁ := iff.rfl
lemma tendsto_iff_eventually {f : α → β} {l₁ : filter α} {l₂ : filter β} :
tendsto f l₁ l₂ ↔ ∀ ⦃p : β → Prop⦄, (∀ᶠ y in l₂, p y) → ∀ᶠ x in l₁, p (f x) :=
iff.rfl
lemma tendsto.eventually {f : α → β} {l₁ : filter α} {l₂ : filter β} {p : β → Prop}
(hf : tendsto f l₁ l₂) (h : ∀ᶠ y in l₂, p y) :
∀ᶠ x in l₁, p (f x) :=
hf h
lemma tendsto.frequently {f : α → β} {l₁ : filter α} {l₂ : filter β} {p : β → Prop}
(hf : tendsto f l₁ l₂) (h : ∃ᶠ x in l₁, p (f x)) :
∃ᶠ y in l₂, p y :=
mt hf.eventually h
lemma tendsto.frequently_map {l₁ : filter α} {l₂ : filter β} {p : α → Prop} {q : β → Prop}
(f : α → β) (c : filter.tendsto f l₁ l₂) (w : ∀ x, p x → q (f x)) (h : ∃ᶠ x in l₁, p x) :
∃ᶠ y in l₂, q y :=
c.frequently (h.mono w)
@[simp] lemma tendsto_bot {f : α → β} {l : filter β} : tendsto f ⊥ l := by simp [tendsto]
@[simp] lemma tendsto_top {f : α → β} {l : filter α} : tendsto f l ⊤ := le_top
lemma le_map_of_right_inverse {mab : α → β} {mba : β → α} {f : filter α} {g : filter β}
(h₁ : mab ∘ mba =ᶠ[g] id) (h₂ : tendsto mba g f) :
g ≤ map mab f :=
by { rw [← @map_id _ g, ← map_congr h₁, ← map_map], exact map_mono h₂ }
lemma tendsto_of_not_nonempty {f : α → β} {la : filter α} {lb : filter β} (h : ¬nonempty α) :
tendsto f la lb :=
by simp only [filter_eq_bot_of_not_nonempty la h, tendsto_bot]
lemma eventually_eq_of_left_inv_of_right_inv {f : α → β} {g₁ g₂ : β → α} {fa : filter α}
{fb : filter β} (hleft : ∀ᶠ x in fa, g₁ (f x) = x) (hright : ∀ᶠ y in fb, f (g₂ y) = y)
(htendsto : tendsto g₂ fb fa) :
g₁ =ᶠ[fb] g₂ :=
(htendsto.eventually hleft).mp $ hright.mono $ λ y hr hl, (congr_arg g₁ hr.symm).trans hl
lemma tendsto_iff_comap {f : α → β} {l₁ : filter α} {l₂ : filter β} :
tendsto f l₁ l₂ ↔ l₁ ≤ l₂.comap f :=
map_le_iff_le_comap
alias tendsto_iff_comap ↔ filter.tendsto.le_comap _
lemma tendsto_congr' {f₁ f₂ : α → β} {l₁ : filter α} {l₂ : filter β} (hl : f₁ =ᶠ[l₁] f₂) :
tendsto f₁ l₁ l₂ ↔ tendsto f₂ l₁ l₂ :=
by rw [tendsto, tendsto, map_congr hl]
lemma tendsto.congr' {f₁ f₂ : α → β} {l₁ : filter α} {l₂ : filter β}
(hl : f₁ =ᶠ[l₁] f₂) (h : tendsto f₁ l₁ l₂) : tendsto f₂ l₁ l₂ :=
(tendsto_congr' hl).1 h
theorem tendsto_congr {f₁ f₂ : α → β} {l₁ : filter α} {l₂ : filter β}
(h : ∀ x, f₁ x = f₂ x) : tendsto f₁ l₁ l₂ ↔ tendsto f₂ l₁ l₂ :=
tendsto_congr' (univ_mem' h)
theorem tendsto.congr {f₁ f₂ : α → β} {l₁ : filter α} {l₂ : filter β}
(h : ∀ x, f₁ x = f₂ x) : tendsto f₁ l₁ l₂ → tendsto f₂ l₁ l₂ :=
(tendsto_congr h).1
lemma tendsto_id' {x y : filter α} : x ≤ y → tendsto id x y :=
by simp only [tendsto, map_id, forall_true_iff] {contextual := tt}
lemma tendsto_id {x : filter α} : tendsto id x x := tendsto_id' $ le_refl x
lemma tendsto.comp {f : α → β} {g : β → γ} {x : filter α} {y : filter β} {z : filter γ}
(hg : tendsto g y z) (hf : tendsto f x y) : tendsto (g ∘ f) x z :=
calc map (g ∘ f) x = map g (map f x) : by rw [map_map]
... ≤ map g y : map_mono hf
... ≤ z : hg
lemma tendsto.mono_left {f : α → β} {x y : filter α} {z : filter β}
(hx : tendsto f x z) (h : y ≤ x) : tendsto f y z :=
(map_mono h).trans hx
lemma tendsto.mono_right {f : α → β} {x : filter α} {y z : filter β}
(hy : tendsto f x y) (hz : y ≤ z) : tendsto f x z :=
le_trans hy hz
lemma tendsto.ne_bot {f : α → β} {x : filter α} {y : filter β} (h : tendsto f x y) [hx : ne_bot x] :
ne_bot y :=
(hx.map _).mono h
lemma tendsto_map {f : α → β} {x : filter α} : tendsto f x (map f x) := le_refl (map f x)
lemma tendsto_map' {f : β → γ} {g : α → β} {x : filter α} {y : filter γ}
(h : tendsto (f ∘ g) x y) : tendsto f (map g x) y :=
by rwa [tendsto, map_map]
lemma tendsto_map'_iff {f : β → γ} {g : α → β} {x : filter α} {y : filter γ} :
tendsto f (map g x) y ↔ tendsto (f ∘ g) x y :=
by { rw [tendsto, map_map], refl }
lemma tendsto_comap {f : α → β} {x : filter β} : tendsto f (comap f x) x :=
map_comap_le
lemma tendsto_comap_iff {f : α → β} {g : β → γ} {a : filter α} {c : filter γ} :
tendsto f a (c.comap g) ↔ tendsto (g ∘ f) a c :=
⟨λ h, tendsto_comap.comp h, λ h, map_le_iff_le_comap.mp $ by rwa [map_map]⟩
lemma tendsto_comap'_iff {m : α → β} {f : filter α} {g : filter β} {i : γ → α}
(h : range i ∈ f) : tendsto (m ∘ i) (comap i f) g ↔ tendsto m f g :=
by { rw [tendsto, ← map_compose], simp only [(∘), map_comap_of_mem h, tendsto] }
lemma comap_eq_of_inverse {f : filter α} {g : filter β} {φ : α → β} (ψ : β → α)
(eq : ψ ∘ φ = id) (hφ : tendsto φ f g) (hψ : tendsto ψ g f) : comap φ g = f :=
begin
refine ((comap_mono $ map_le_iff_le_comap.1 hψ).trans _).antisymm (map_le_iff_le_comap.1 hφ),
rw [comap_comap, eq, comap_id],
exact le_rfl
end
lemma map_eq_of_inverse {f : filter α} {g : filter β} {φ : α → β} (ψ : β → α)
(eq : φ ∘ ψ = id) (hφ : tendsto φ f g) (hψ : tendsto ψ g f) : map φ f = g :=
begin
refine le_antisymm hφ (le_trans _ (map_mono hψ)),
rw [map_map, eq, map_id],
exact le_rfl
end
lemma tendsto_inf {f : α → β} {x : filter α} {y₁ y₂ : filter β} :
tendsto f x (y₁ ⊓ y₂) ↔ tendsto f x y₁ ∧ tendsto f x y₂ :=
by simp only [tendsto, le_inf_iff, iff_self]
lemma tendsto_inf_left {f : α → β} {x₁ x₂ : filter α} {y : filter β}
(h : tendsto f x₁ y) : tendsto f (x₁ ⊓ x₂) y :=
le_trans (map_mono inf_le_left) h
lemma tendsto_inf_right {f : α → β} {x₁ x₂ : filter α} {y : filter β}
(h : tendsto f x₂ y) : tendsto f (x₁ ⊓ x₂) y :=
le_trans (map_mono inf_le_right) h
lemma tendsto.inf {f : α → β} {x₁ x₂ : filter α} {y₁ y₂ : filter β}
(h₁ : tendsto f x₁ y₁) (h₂ : tendsto f x₂ y₂) : tendsto f (x₁ ⊓ x₂) (y₁ ⊓ y₂) :=
tendsto_inf.2 ⟨tendsto_inf_left h₁, tendsto_inf_right h₂⟩
@[simp] lemma tendsto_infi {f : α → β} {x : filter α} {y : ι → filter β} :
tendsto f x (⨅ i, y i) ↔ ∀ i, tendsto f x (y i) :=
by simp only [tendsto, iff_self, le_infi_iff]
lemma tendsto_infi' {f : α → β} {x : ι → filter α} {y : filter β} (i : ι) (hi : tendsto f (x i) y) :
tendsto f (⨅ i, x i) y :=
hi.mono_left $ infi_le _ _
lemma tendsto_sup {f : α → β} {x₁ x₂ : filter α} {y : filter β} :
tendsto f (x₁ ⊔ x₂) y ↔ tendsto f x₁ y ∧ tendsto f x₂ y :=
by simp only [tendsto, map_sup, sup_le_iff]
lemma tendsto.sup {f : α → β} {x₁ x₂ : filter α} {y : filter β} :
tendsto f x₁ y → tendsto f x₂ y → tendsto f (x₁ ⊔ x₂) y :=
λ h₁ h₂, tendsto_sup.mpr ⟨ h₁, h₂ ⟩
@[simp] lemma tendsto_principal {f : α → β} {l : filter α} {s : set β} :
tendsto f l (𝓟 s) ↔ ∀ᶠ a in l, f a ∈ s :=
by simp only [tendsto, le_principal_iff, mem_map', filter.eventually]
@[simp] lemma tendsto_principal_principal {f : α → β} {s : set α} {t : set β} :
tendsto f (𝓟 s) (𝓟 t) ↔ ∀ a ∈ s, f a ∈ t :=
by simp only [tendsto_principal, eventually_principal]
@[simp] lemma tendsto_pure {f : α → β} {a : filter α} {b : β} :
tendsto f a (pure b) ↔ ∀ᶠ x in a, f x = b :=
by simp only [tendsto, le_pure_iff, mem_map', mem_singleton_iff, filter.eventually]
lemma tendsto_pure_pure (f : α → β) (a : α) :
tendsto f (pure a) (pure (f a)) :=
tendsto_pure.2 rfl
lemma tendsto_const_pure {a : filter α} {b : β} : tendsto (λ x, b) a (pure b) :=
tendsto_pure.2 $ univ_mem' $ λ _, rfl
lemma pure_le_iff {a : α} {l : filter α} : pure a ≤ l ↔ ∀ s ∈ l, a ∈ s :=
iff.rfl
lemma tendsto_pure_left {f : α → β} {a : α} {l : filter β} :
tendsto f (pure a) l ↔ ∀ s ∈ l, f a ∈ s :=
iff.rfl
@[simp] lemma map_inf_principal_preimage {f : α → β} {s : set β} {l : filter α} :
map f (l ⊓ 𝓟 (f ⁻¹' s)) = map f l ⊓ 𝓟 s :=
filter.ext $ λ t, by simp only [mem_map', mem_inf_principal, mem_set_of_eq, mem_preimage]
/-- If two filters are disjoint, then a function cannot tend to both of them along a non-trivial
filter. -/
lemma tendsto.not_tendsto {f : α → β} {a : filter α} {b₁ b₂ : filter β} (hf : tendsto f a b₁)
[ne_bot a] (hb : disjoint b₁ b₂) :
¬ tendsto f a b₂ :=
λ hf', (tendsto_inf.2 ⟨hf, hf'⟩).ne_bot.ne hb.eq_bot
lemma tendsto.if {l₁ : filter α} {l₂ : filter β} {f g : α → β} {p : α → Prop} [∀ x, decidable (p x)]
(h₀ : tendsto f (l₁ ⊓ 𝓟 {x | p x}) l₂) (h₁ : tendsto g (l₁ ⊓ 𝓟 { x | ¬ p x }) l₂) :
tendsto (λ x, if p x then f x else g x) l₁ l₂ :=
begin
simp only [tendsto_def, mem_inf_principal] at *,
intros s hs,
filter_upwards [h₀ s hs, h₁ s hs],
simp only [mem_preimage], intros x hp₀ hp₁,
split_ifs,
exacts [hp₀ h, hp₁ h]
end
lemma tendsto.piecewise {l₁ : filter α} {l₂ : filter β} {f g : α → β}
{s : set α} [∀ x, decidable (x ∈ s)]
(h₀ : tendsto f (l₁ ⊓ 𝓟 s) l₂) (h₁ : tendsto g (l₁ ⊓ 𝓟 sᶜ) l₂) :
tendsto (piecewise s f g) l₁ l₂ :=
h₀.if h₁
/-! ### Products of filters -/
section prod
variables {s : set α} {t : set β} {f : filter α} {g : filter β}
/- The product filter cannot be defined using the monad structure on filters. For example:
F := do {x ← seq, y ← top, return (x, y)}
hence:
s ∈ F ↔ ∃ n, [n..∞] × univ ⊆ s
G := do {y ← top, x ← seq, return (x, y)}
hence:
s ∈ G ↔ ∀ i:ℕ, ∃ n, [n..∞] × {i} ⊆ s
Now ⋃ i, [i..∞] × {i} is in G but not in F.
As product filter we want to have F as result.
-/
/-- Product of filters. This is the filter generated by cartesian products
of elements of the component filters. -/
protected def prod (f : filter α) (g : filter β) : filter (α × β) :=
f.comap prod.fst ⊓ g.comap prod.snd
localized "infix ` ×ᶠ `:60 := filter.prod" in filter
lemma prod_mem_prod {s : set α} {t : set β} {f : filter α} {g : filter β}
(hs : s ∈ f) (ht : t ∈ g) : set.prod s t ∈ f ×ᶠ g :=
inter_mem_inf (preimage_mem_comap hs) (preimage_mem_comap ht)
lemma mem_prod_iff {s : set (α×β)} {f : filter α} {g : filter β} :
s ∈ f ×ᶠ g ↔ (∃ t₁ ∈ f, ∃ t₂ ∈ g, set.prod t₁ t₂ ⊆ s) :=
begin
simp only [filter.prod],
split,
{ rintro ⟨t₁, ⟨s₁, hs₁, hts₁⟩, t₂, ⟨s₂, hs₂, hts₂⟩, rfl⟩,
exact ⟨s₁, hs₁, s₂, hs₂, λ p ⟨h, h'⟩, ⟨hts₁ h, hts₂ h'⟩⟩ },
{ rintro ⟨t₁, ht₁, t₂, ht₂, h⟩,
exact mem_inf_of_inter (preimage_mem_comap ht₁) (preimage_mem_comap ht₂) h }
end
@[simp] lemma prod_mem_prod_iff {s : set α} {t : set β} {f : filter α} {g : filter β}
[f.ne_bot] [g.ne_bot] :
s.prod t ∈ f ×ᶠ g ↔ s ∈ f ∧ t ∈ g :=
⟨λ h, let ⟨s', hs', t', ht', H⟩ := mem_prod_iff.1 h in (prod_subset_prod_iff.1 H).elim
(λ ⟨hs's, ht't⟩, ⟨mem_of_superset hs' hs's, mem_of_superset ht' ht't⟩)
(λ h, h.elim
(λ hs'e, absurd hs'e (nonempty_of_mem hs').ne_empty)
(λ ht'e, absurd ht'e (nonempty_of_mem ht').ne_empty)),
λ h, prod_mem_prod h.1 h.2⟩
lemma comap_prod (f : α → β × γ) (b : filter β) (c : filter γ) :
comap f (b ×ᶠ c) = (comap (prod.fst ∘ f) b) ⊓ (comap (prod.snd ∘ f) c) :=
by erw [comap_inf, filter.comap_comap, filter.comap_comap]
lemma sup_prod (f₁ f₂ : filter α) (g : filter β) : (f₁ ⊔ f₂) ×ᶠ g = (f₁ ×ᶠ g) ⊔ (f₂ ×ᶠ g) :=
by rw [filter.prod, comap_sup, inf_sup_right, ← filter.prod, ← filter.prod]
lemma prod_sup (f : filter α) (g₁ g₂ : filter β) : f ×ᶠ (g₁ ⊔ g₂) = (f ×ᶠ g₁) ⊔ (f ×ᶠ g₂) :=
by rw [filter.prod, comap_sup, inf_sup_left, ← filter.prod, ← filter.prod]
lemma eventually_prod_iff {p : α × β → Prop} {f : filter α} {g : filter β} :
(∀ᶠ x in f ×ᶠ g, p x) ↔ ∃ (pa : α → Prop) (ha : ∀ᶠ x in f, pa x)
(pb : β → Prop) (hb : ∀ᶠ y in g, pb y), ∀ {x}, pa x → ∀ {y}, pb y → p (x, y) :=
by simpa only [set.prod_subset_iff] using @mem_prod_iff α β p f g
lemma tendsto_fst {f : filter α} {g : filter β} : tendsto prod.fst (f ×ᶠ g) f :=
tendsto_inf_left tendsto_comap
lemma tendsto_snd {f : filter α} {g : filter β} : tendsto prod.snd (f ×ᶠ g) g :=
tendsto_inf_right tendsto_comap
lemma tendsto.prod_mk {f : filter α} {g : filter β} {h : filter γ} {m₁ : α → β} {m₂ : α → γ}
(h₁ : tendsto m₁ f g) (h₂ : tendsto m₂ f h) : tendsto (λ x, (m₁ x, m₂ x)) f (g ×ᶠ h) :=
tendsto_inf.2 ⟨tendsto_comap_iff.2 h₁, tendsto_comap_iff.2 h₂⟩
lemma eventually.prod_inl {la : filter α} {p : α → Prop} (h : ∀ᶠ x in la, p x) (lb : filter β) :
∀ᶠ x in la ×ᶠ lb, p (x : α × β).1 :=
tendsto_fst.eventually h
lemma eventually.prod_inr {lb : filter β} {p : β → Prop} (h : ∀ᶠ x in lb, p x) (la : filter α) :
∀ᶠ x in la ×ᶠ lb, p (x : α × β).2 :=
tendsto_snd.eventually h
lemma eventually.prod_mk {la : filter α} {pa : α → Prop} (ha : ∀ᶠ x in la, pa x)
{lb : filter β} {pb : β → Prop} (hb : ∀ᶠ y in lb, pb y) :
∀ᶠ p in la ×ᶠ lb, pa (p : α × β).1 ∧ pb p.2 :=
(ha.prod_inl lb).and (hb.prod_inr la)
lemma eventually.curry {la : filter α} {lb : filter β} {p : α × β → Prop}
(h : ∀ᶠ x in la ×ᶠ lb, p x) :
∀ᶠ x in la, ∀ᶠ y in lb, p (x, y) :=
begin
rcases eventually_prod_iff.1 h with ⟨pa, ha, pb, hb, h⟩,
exact ha.mono (λ a ha, hb.mono $ λ b hb, h ha hb)
end
lemma prod_infi_left [nonempty ι] {f : ι → filter α} {g : filter β}:
(⨅ i, f i) ×ᶠ g = (⨅ i, (f i) ×ᶠ g) :=
by { rw [filter.prod, comap_infi, infi_inf], simp only [filter.prod, eq_self_iff_true] }
lemma prod_infi_right [nonempty ι] {f : filter α} {g : ι → filter β} :
f ×ᶠ (⨅ i, g i) = (⨅ i, f ×ᶠ (g i)) :=
by { rw [filter.prod, comap_infi, inf_infi], simp only [filter.prod, eq_self_iff_true] }
@[mono] lemma prod_mono {f₁ f₂ : filter α} {g₁ g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) :
f₁ ×ᶠ g₁ ≤ f₂ ×ᶠ g₂ :=
inf_le_inf (comap_mono hf) (comap_mono hg)
lemma prod_comap_comap_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
{f₁ : filter α₁} {f₂ : filter α₂} {m₁ : β₁ → α₁} {m₂ : β₂ → α₂} :
(comap m₁ f₁) ×ᶠ (comap m₂ f₂) = comap (λ p : β₁×β₂, (m₁ p.1, m₂ p.2)) (f₁ ×ᶠ f₂) :=
by simp only [filter.prod, comap_comap, eq_self_iff_true, comap_inf]
lemma prod_comm' : f ×ᶠ g = comap (prod.swap) (g ×ᶠ f) :=
by simp only [filter.prod, comap_comap, (∘), inf_comm, prod.fst_swap,
eq_self_iff_true, prod.snd_swap, comap_inf]
lemma prod_comm : f ×ᶠ g = map (λ p : β×α, (p.2, p.1)) (g ×ᶠ f) :=
by { rw [prod_comm', ← map_swap_eq_comap_swap], refl }
lemma prod_map_map_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
{f₁ : filter α₁} {f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} :
(map m₁ f₁) ×ᶠ (map m₂ f₂) = map (λ p : α₁×α₂, (m₁ p.1, m₂ p.2)) (f₁ ×ᶠ f₂) :=
le_antisymm
(λ s hs,
let ⟨s₁, hs₁, s₂, hs₂, h⟩ := mem_prod_iff.mp hs in
filter.sets_of_superset _ (prod_mem_prod (image_mem_map hs₁) (image_mem_map hs₂)) $
calc set.prod (m₁ '' s₁) (m₂ '' s₂) = (λ p : α₁×α₂, (m₁ p.1, m₂ p.2)) '' set.prod s₁ s₂ :
set.prod_image_image_eq
... ⊆ _ : by rwa [image_subset_iff])
((tendsto.comp le_rfl tendsto_fst).prod_mk (tendsto.comp le_rfl tendsto_snd))
lemma prod_map_map_eq' {α₁ : Type*} {α₂ : Type*} {β₁ : Type*} {β₂ : Type*}
(f : α₁ → α₂) (g : β₁ → β₂) (F : filter α₁) (G : filter β₁) :
(map f F) ×ᶠ (map g G) = map (prod.map f g) (F ×ᶠ G) :=
by { rw filter.prod_map_map_eq, refl }
lemma tendsto.prod_map {δ : Type*} {f : α → γ} {g : β → δ} {a : filter α} {b : filter β}
{c : filter γ} {d : filter δ} (hf : tendsto f a c) (hg : tendsto g b d) :
tendsto (prod.map f g) (a ×ᶠ b) (c ×ᶠ d) :=
begin
erw [tendsto, ← prod_map_map_eq],
exact filter.prod_mono hf hg,
end
lemma map_prod (m : α × β → γ) (f : filter α) (g : filter β) :
map m (f ×ᶠ g) = (f.map (λ a b, m (a, b))).seq g :=
begin
simp [filter.ext_iff, mem_prod_iff, mem_map_seq_iff],
intro s,
split,
exact λ ⟨t, ht, s, hs, h⟩, ⟨s, hs, t, ht, λ x hx y hy, @h ⟨x, y⟩ ⟨hx, hy⟩⟩,
exact λ ⟨s, hs, t, ht, h⟩, ⟨t, ht, s, hs, λ ⟨x, y⟩ ⟨hx, hy⟩, h x hx y hy⟩
end
lemma prod_eq {f : filter α} {g : filter β} : f ×ᶠ g = (f.map prod.mk).seq g :=
have h : _ := map_prod id f g, by rwa [map_id] at h
lemma prod_inf_prod {f₁ f₂ : filter α} {g₁ g₂ : filter β} :
(f₁ ×ᶠ g₁) ⊓ (f₂ ×ᶠ g₂) = (f₁ ⊓ f₂) ×ᶠ (g₁ ⊓ g₂) :=
by simp only [filter.prod, comap_inf, inf_comm, inf_assoc, inf_left_comm]
@[simp] lemma prod_bot {f : filter α} : f ×ᶠ (⊥ : filter β) = ⊥ := by simp [filter.prod]
@[simp] lemma bot_prod {g : filter β} : (⊥ : filter α) ×ᶠ g = ⊥ := by simp [filter.prod]
@[simp] lemma prod_principal_principal {s : set α} {t : set β} :
(𝓟 s) ×ᶠ (𝓟 t) = 𝓟 (set.prod s t) :=
by simp only [filter.prod, comap_principal, principal_eq_iff_eq, comap_principal, inf_principal];
refl
@[simp] lemma pure_prod {a : α} {f : filter β} : pure a ×ᶠ f = map (prod.mk a) f :=
by rw [prod_eq, map_pure, pure_seq_eq_map]
@[simp] lemma prod_pure {f : filter α} {b : β} : f ×ᶠ pure b = map (λ a, (a, b)) f :=
by rw [prod_eq, seq_pure, map_map]
lemma prod_pure_pure {a : α} {b : β} : (pure a) ×ᶠ (pure b) = pure (a, b) :=
by simp
lemma prod_eq_bot {f : filter α} {g : filter β} : f ×ᶠ g = ⊥ ↔ (f = ⊥ ∨ g = ⊥) :=
begin
split,
{ intro h,
rcases mem_prod_iff.1 (empty_mem_iff_bot.2 h) with ⟨s, hs, t, ht, hst⟩,
rw [subset_empty_iff, set.prod_eq_empty_iff] at hst,
cases hst with s_eq t_eq,
{ left, exact empty_mem_iff_bot.1 (s_eq ▸ hs) },
{ right, exact empty_mem_iff_bot.1 (t_eq ▸ ht) } },
{ rintro (rfl | rfl),
exact bot_prod,
exact prod_bot }
end
lemma prod_ne_bot {f : filter α} {g : filter β} : ne_bot (f ×ᶠ g) ↔ (ne_bot f ∧ ne_bot g) :=
by simp only [ne_bot_iff, ne, prod_eq_bot, not_or_distrib]
lemma ne_bot.prod {f : filter α} {g : filter β} (hf : ne_bot f) (hg : ne_bot g) :
ne_bot (f ×ᶠ g) :=
prod_ne_bot.2 ⟨hf, hg⟩
instance prod_ne_bot' {f : filter α} {g : filter β} [hf : ne_bot f] [hg : ne_bot g] :
ne_bot (f ×ᶠ g) :=
hf.prod hg
lemma tendsto_prod_iff {f : α × β → γ} {x : filter α} {y : filter β} {z : filter γ} :
filter.tendsto f (x ×ᶠ y) z ↔
∀ W ∈ z, ∃ U ∈ x, ∃ V ∈ y, ∀ x y, x ∈ U → y ∈ V → f (x, y) ∈ W :=
by simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff, exists_prop, iff_self]
end prod
/-! ### Coproducts of filters -/
section coprod
variables {s : set α} {t : set β} {f : filter α} {g : filter β}
/-- Coproduct of filters. -/
protected def coprod (f : filter α) (g : filter β) : filter (α × β) :=
f.comap prod.fst ⊔ g.comap prod.snd
lemma mem_coprod_iff {s : set (α×β)} {f : filter α} {g : filter β} :
s ∈ f.coprod g ↔ ((∃ t₁ ∈ f, prod.fst ⁻¹' t₁ ⊆ s) ∧ (∃ t₂ ∈ g, prod.snd ⁻¹' t₂ ⊆ s)) :=
by simp [filter.coprod]
@[mono] lemma coprod_mono {f₁ f₂ : filter α} {g₁ g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) :
f₁.coprod g₁ ≤ f₂.coprod g₂ :=
sup_le_sup (comap_mono hf) (comap_mono hg)
lemma principal_coprod_principal (s : set α) (t : set β) :
(𝓟 s).coprod (𝓟 t) = 𝓟 (sᶜ.prod tᶜ)ᶜ :=
begin
rw [filter.coprod, comap_principal, comap_principal, sup_principal],
congr,
ext x,
simp ; tauto,
end
-- this inequality can be strict; see `map_const_principal_coprod_map_id_principal` and
-- `map_prod_map_const_id_principal_coprod_principal` below.
lemma map_prod_map_coprod_le {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
{f₁ : filter α₁} {f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} :
map (prod.map m₁ m₂) (f₁.coprod f₂) ≤ (map m₁ f₁).coprod (map m₂ f₂) :=
begin
intros s,
simp only [mem_map, mem_coprod_iff],
rintro ⟨⟨u₁, hu₁, h₁⟩, u₂, hu₂, h₂⟩,
refine ⟨⟨m₁ ⁻¹' u₁, hu₁, λ _ hx, h₁ _⟩, ⟨m₂ ⁻¹' u₂, hu₂, λ _ hx, h₂ _⟩⟩; convert hx
end
/-- Characterization of the coproduct of the `filter.map`s of two principal filters `𝓟 {a}` and
`𝓟 {i}`, the first under the constant function `λ a, b` and the second under the identity function.
Together with the next lemma, `map_prod_map_const_id_principal_coprod_principal`, this provides an
example showing that the inequality in the lemma `map_prod_map_coprod_le` can be strict. -/
lemma map_const_principal_coprod_map_id_principal {α β ι : Type*} (a : α) (b : β) (i : ι) :
(map (λ _ : α, b) (𝓟 {a})).coprod (map id (𝓟 {i}))
= 𝓟 (({b} : set β).prod (univ : set ι) ∪ (univ : set β).prod {i}) :=
begin
rw [map_principal, map_principal, principal_coprod_principal],
congr,
ext ⟨b', i'⟩,
simp,
tauto,
end
/-- Characterization of the `filter.map` of the coproduct of two principal filters `𝓟 {a}` and
`𝓟 {i}`, under the `prod.map` of two functions, respectively the constant function `λ a, b` and the
identity function. Together with the previous lemma,
`map_const_principal_coprod_map_id_principal`, this provides an example showing that the inequality
in the lemma `map_prod_map_coprod_le` can be strict. -/
lemma map_prod_map_const_id_principal_coprod_principal {α β ι : Type*} (a : α) (b : β) (i : ι) :
map (prod.map (λ _ : α, b) id) ((𝓟 {a}).coprod (𝓟 {i}))
= 𝓟 (({b} : set β).prod (univ : set ι)) :=
begin
rw [principal_coprod_principal, map_principal],
congr,
ext ⟨b', i'⟩,
split,
{ rintro ⟨⟨a'', i''⟩, h₁, h₂, h₃⟩,
simp },
{ rintro ⟨h₁, h₂⟩,
use (a, i'),
simpa using h₁.symm }
end
lemma tendsto.prod_map_coprod {δ : Type*} {f : α → γ} {g : β → δ} {a : filter α} {b : filter β}
{c : filter γ} {d : filter δ} (hf : tendsto f a c) (hg : tendsto g b d) :
tendsto (prod.map f g) (a.coprod b) (c.coprod d) :=
map_prod_map_coprod_le.trans (coprod_mono hf hg)
end coprod
/-! ### `n`-ary coproducts of filters -/
section Coprod
variables {δ : Type*} {κ : δ → Type*} -- {f : Π d, filter (κ d)}
/-- Coproduct of filters. -/
protected def Coprod (f : Π d, filter (κ d)) : filter (Π d, κ d) :=
⨆ d : δ, (f d).comap (λ k, k d)
lemma mem_Coprod_iff {s : set (Π d, κ d)} {f : Π d, filter (κ d)} :
(s ∈ (filter.Coprod f)) ↔ (∀ d : δ, (∃ t₁ ∈ f d, (λ k : (Π d, κ d), k d) ⁻¹' t₁ ⊆ s)) :=
by simp [filter.Coprod]
@[mono] lemma Coprod_mono {f₁ f₂ : Π d, filter (κ d)} (hf : ∀ d, f₁ d ≤ f₂ d) :
filter.Coprod f₁ ≤ filter.Coprod f₂ :=
supr_le_supr $ λ d, comap_mono (hf d)
lemma map_pi_map_Coprod_le {μ : δ → Type*}
{f : Π d, filter (κ d)} {m : Π d, κ d → μ d} :
map (λ (k : Π d, κ d), λ d, m d (k d)) (filter.Coprod f) ≤ filter.Coprod (λ d, map (m d) (f d)) :=
begin
intros s h,
rw [mem_map', mem_Coprod_iff],
intros d,
rw mem_Coprod_iff at h,
obtain ⟨t, H, hH⟩ := h d,
rw mem_map at H,
refine ⟨{x : κ d | m d x ∈ t}, H, _⟩,
intros x hx,
simp only [mem_set_of_eq, preimage_set_of_eq] at hx,
rw mem_set_of_eq,
exact set.mem_of_subset_of_mem hH (mem_preimage.mpr hx),
end
lemma tendsto.pi_map_Coprod {μ : δ → Type*} {f : Π d, filter (κ d)} {m : Π d, κ d → μ d}
{g : Π d, filter (μ d)} (hf : ∀ d, tendsto (m d) (f d) (g d)) :
tendsto (λ (k : Π d, κ d), λ d, m d (k d)) (filter.Coprod f) (filter.Coprod g) :=
map_pi_map_Coprod_le.trans (Coprod_mono hf)
end Coprod
end filter
open_locale filter
lemma set.eq_on.eventually_eq {α β} {s : set α} {f g : α → β} (h : eq_on f g s) :
f =ᶠ[𝓟 s] g :=
h
lemma set.eq_on.eventually_eq_of_mem {α β} {s : set α} {l : filter α} {f g : α → β}
(h : eq_on f g s) (hl : s ∈ l) :
f =ᶠ[l] g :=
h.eventually_eq.filter_mono $ filter.le_principal_iff.2 hl
lemma set.subset.eventually_le {α} {l : filter α} {s t : set α} (h : s ⊆ t) : s ≤ᶠ[l] t :=
filter.eventually_of_forall h
lemma set.maps_to.tendsto {α β} {s : set α} {t : set β} {f : α → β} (h : maps_to f s t) :
filter.tendsto f (𝓟 s) (𝓟 t) :=
filter.tendsto_principal_principal.2 h
|
4c035daf97ede574dfe9ed56c721bfed5f32fedb | dd24e6c3b8dc14dc504f8a906fc04c51e4312e6b | /src/mywork/practice_1.lean | 57b79c07f24fb89650eb94ab1e4862200d676961 | [] | no_license | njeyasingh/CS-2120 | dd781a90dd0645b74e61cee1813483fb7cb4a111 | b3356f665a246f295b3f1e6d61901bfca331810d | refs/heads/main | 1,693,294,711,274 | 1,635,188,659,000 | 1,635,188,659,000 | 399,945,420 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 6,537 | lean | /-
-/mnl6sc@virginia.edu https://github.com/mlarusso/CS-2120.git
npj5kr@virginia.edu https://github.com/njeyasingh/CS-2120.git
fpg2kv@virginia.edu https://github.com/franklin-glance/cs2120f21.git
/-
EQUALITY
-/
/- #1
Suppose that x, y, z, and w are arbitrary objects of some type,
T; and suppose further that we know (have proofs of the facts)
that x = y, y = z, and w = z. Give a very, very short English
proof of the conjecture that z = w. You can use not only the
axioms of equality, but either of the theorems about properties
of equality that we have proven. Hint: There's something about
this question that makes it much easier to answer than it might
at first appear.
-/
/-
Assuming z and w are arbitrary objects of type T, and using
the symmetric axiom of equality, we can say that z = w because w = z
x = y → y = z → w = z → z = w
-/
/- #2
Give a formal statement of the conjecture (proposition) from
#1 by filling in the "hole" in the following definition. The
def is a keyword. The name you're binding to your proposition
is prop_1. The type of the value is Prop (which is the type of
all propositions in Lean).
-/
def prop_1 : Prop :=
∀ (T : Type)
(x y z w : T),
x = y → y = z → w = z → z = w
/- #3 (extra credit)
Give a formal proof of the proposition from #2 by filling in
the hole in this next definition. Hint: Use Lean's versions of
the axioms and basic theorems concerning equality. They are,
again, called eq.refl, eq.subst, eq.symm, eq.trans.
-/
theorem prop_1_proof : prop_1 :=
begin
unfold prop_1,
intros T x y z w xy yz wz,
apply eq.symm wz
end
/-
unfold prop_1
(T : Type) -- first argument
(z w: T) -- second argument
: w = z -- return "type"
:= eq.refl z-/
/-
FOR ALL: ∀.
-/
/- #4
Give a very brief explanation in English of the introduction
rule for ∀. For example, suppose you need to prove (∀ x, P x);
what do you do? (I'm being a little informal in leaving out the
type of X.)
First you need to define some type, we can call it T
Then you state that x is of type T
For all x, x can be applied to proposition P, and P will return True
-/
/- #5
Suppose you have a proof, let's call it pf, of the proposition,
(∀ x, P x), and you need a proof of P t, for some particular t.
Write an expression then uses the elimination rule for ∀ to get
such a proof. Complete the answer by replacing the underscores
in the following expression: (P t).
-/
/-
IMPLIES: →
In the "code" that follows, we define two predicates, each
taking one natural number as an argument. We call them ev and
odd. When applied to any value, n, ev yields the proposition
that n is even (n % 2 = 0), while odd yields the proposition
that n is odd (n % 2 = 1).
-/
def ev (n : ℕ) := n % 2 = 0
def odd (n : ℕ) := n % 2 = 1
/- #6
Write a formal version of the proposition that, for *any*
natural number n, *if* n is even, *then* n + 1 is odd. Give
your answer by filling the hole in the following definition.
Hint: put parenthesis around "n + 1" in your answer.
-/
def successor_of_even_is_odd : Prop :=
∀ (n : ℕ),
ev n → odd(n + 1)
/- #7
Suppose that "its_raining" and "the_streets_are_wet" are
propositions. (We formalize these assumptions as axioms in
what follows. Then give a formal definition of the (larger)
proposition, "if it's raining out then the streets are wet")
by filling in the hole
-/
axioms (raining streets_wet : Prop)
axiom if_raining_then_streets_wet : raining → streets_wet
/- #9
Now suppose that in addition, its_raining is true, and
we have a proof of it, pf_raining. Again, we again give
you this assumption formally as an axiom below. Finish
the formal proof that the streets must be wet. Hint: here
you are asked to use the elimination rule for →.
-/
axiom pf_raining : raining
example : streets_wet :=
if_raining_then_streets_wet pf_raining
/-
AND: ∧
-/
/- #10
In our last class, we proved that "∧ is *commutative*."
That is, for any given *propositions*, P and Q, (P ∧ Q) →
(Q ∧ P). The way we proved it was to *assume* that we're
given such a P, Q, and proof, pq, of (P ∧ Q) -- applying
the introduction rules for ∀ and →). In this context, we
*use* the proof, pq, to derive separate proofs, let's call
them p, a proof of P, and q, a proof of Q. With these in
hand, we then apply the introduction rule for ∧ to put
them back together into a proof of (Q ∧ P). We give you
a formal version of this proof as a reminder, next.
-/
theorem and_commutative : ∀ (P Q : Prop), P ∧ Q → Q ∧ P :=
begin
assume P Q pq,
apply and.intro _ _,
exact (and.elim_right pq),
exact (and.elim_left pq),
end
/-
Your task now is to prove the theorem, "∧ is *associative*."
What this means is that for arbitrary propositions, P, Q, and
R, if (P ∧ (Q ∧ R)) is true, then ((P ∧ Q) ∧ R) is true, *and
vice versa*. You just need to prove it in the first direction.
Hint, if you have a proof, p_qr, of (P ∧ (Q ∧ R)), then the
application of and.elim_left will give you a proof of P, and
and.elim_right will give you a proof of (Q ∧ R).
To help you along, we give you the first part of the proof,
including an example of a new Lean tactic called have, which
allows you to give a name to a new value in the middle of a
proof script.
-/
theorem and_associative :
∀ (P Q R : Prop),
(P ∧ (Q ∧ R)) → ((P ∧ Q) ∧ R) :=
begin
intros P Q R h,
have p : P := and.elim_left h,
apply and.intro,
apply and.intro,
exact (and.elim_left h),
exact ((and.elim_left)) (and.elim_right h),
exact ((and.elim_right) (and.elim_right h)),
end
/- #11
Give an English language proof of the preceding
theorem. Do it by finishing off the following
partial "proof explanation."
Proof. We assume that P, Q, and R are arbitrary
but specific propositions, and that we have a
proof, let's call it p_qr, of (P ∧ (Q ∧ R)) [by
application of ∧ and → introduction.] What now
remains to be proved is ((P ∧ Q) ∧ R). We can
construct a proof of this proposition by applying
and introduction rule to a proof of (P ∧ Q) and a proof of R.
What remains, then, is to obtain these proofs.
But this is easily done by the application of
and elmination rule to p_qr. QED.
-/
/-
Note that Lean includes versions of these
theorems (and many, many, many others) in
its extensive library of formalized maths,
as the following check commands reveal.
Note the difference in naming relative to
the definitions we give in this file.
-/
#check @and.comm
#check @and.assoc
|
3478a1b8c50048851b94feacf3a6ba9554a65e8e | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/algebra/jordan/basic.lean | 2c239fa99c6def23f391971ed89b4639dc26c60b | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 11,207 | lean | /-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import algebra.lie.of_associative
/-!
# Jordan rings
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
Let `A` be a non-unital, non-associative ring. Then `A` is said to be a (commutative, linear) Jordan
ring if the multiplication is commutative and satisfies a weak associativity law known as the
Jordan Identity: for all `a` and `b` in `A`,
```
(a * b) * a^2 = a * (b * a^2)
```
i.e. the operators of multiplication by `a` and `a^2` commute.
A more general concept of a (non-commutative) Jordan ring can also be defined, as a
(non-commutative, non-associative) ring `A` where, for each `a` in `A`, the operators of left and
right multiplication by `a` and `a^2` commute.
Every associative algebra can be equipped with a symmetrized multiplication (characterized by
`sym_alg.sym_mul_sym`) making it into a commutative Jordan algebra (`sym_alg.is_comm_jordan`).
Jordan algebras arising this way are said to be special.
A real Jordan algebra `A` can be introduced by
```lean
variables {A : Type*} [non_unital_non_assoc_ring A] [module ℝ A] [smul_comm_class ℝ A A]
[is_scalar_tower ℝ A A] [is_comm_jordan A]
```
## Main results
- `two_nsmul_lie_lmul_lmul_add_add_eq_zero` : Linearisation of the commutative Jordan axiom
## Implementation notes
We shall primarily be interested in linear Jordan algebras (i.e. over rings of characteristic not
two) leaving quadratic algebras to those better versed in that theory.
The conventional way to linearise the Jordan axiom is to equate coefficients (more formally, assume
that the axiom holds in all field extensions). For simplicity we use brute force algebraic expansion
and substitution instead.
## Motivation
Every Jordan algebra `A` has a triple product defined, for `a` `b` and `c` in `A` by
$$
{a\,b\,c} = (a * b) * c - (a * c) * b + a * (b * c).
$$
Via this triple product Jordan algebras are related to a number of other mathematical structures:
Jordan triples, partial Jordan triples, Jordan pairs and quadratic Jordan algebras. In addition to
their considerable algebraic interest ([mccrimmon2004]) these structures have been shown to have
deep connections to mathematical physics, functional analysis and differential geometry. For more
information about these connections the interested reader is referred to [alfsenshultz2003],
[chu2012], [friedmanscarr2005], [iordanescu2003] and [upmeier1987].
There are also exceptional Jordan algebras which can be shown not to be the symmetrization of any
associative algebra. The 3x3 matrices of octonions is the canonical example.
Non-commutative Jordan algebras have connections to the Vidav-Palmer theorem
[cabreragarciarodriguezpalacios2014].
## References
* [Cabrera García and Rodríguez Palacios, Non-associative normed algebras. Volume 1]
[cabreragarciarodriguezpalacios2014]
* [Hanche-Olsen and Størmer, Jordan Operator Algebras][hancheolsenstormer1984]
* [McCrimmon, A taste of Jordan algebras][mccrimmon2004]
-/
variables (A : Type*)
/-- A (non-commutative) Jordan multiplication. -/
class is_jordan [has_mul A] :=
(lmul_comm_rmul : ∀ a b : A, (a * b) * a = a * (b * a))
(lmul_lmul_comm_lmul : ∀ a b : A, (a * a) * (a * b) = a * ((a * a) * b))
(lmul_lmul_comm_rmul : ∀ a b : A, (a * a) * (b * a) = ((a * a) * b) * a)
(lmul_comm_rmul_rmul : ∀ a b : A, (a * b) * (a * a) = a * (b * (a * a)))
(rmul_comm_rmul_rmul : ∀ a b : A, (b * a) * (a * a) = (b * (a * a)) * a)
/-- A commutative Jordan multipication -/
class is_comm_jordan [has_mul A] :=
(mul_comm : ∀ a b : A, a * b = b * a)
(lmul_comm_rmul_rmul : ∀ a b : A, (a * b) * (a * a) = a * (b * (a * a)))
/-- A (commutative) Jordan multiplication is also a Jordan multipication -/
@[priority 100] -- see Note [lower instance priority]
instance is_comm_jordan.to_is_jordan [has_mul A] [is_comm_jordan A] : is_jordan A :=
{ lmul_comm_rmul := λ a b, by rw [is_comm_jordan.mul_comm, is_comm_jordan.mul_comm a b],
lmul_lmul_comm_lmul := λ a b, by rw [is_comm_jordan.mul_comm (a * a) (a * b),
is_comm_jordan.lmul_comm_rmul_rmul, is_comm_jordan.mul_comm b (a * a)],
lmul_comm_rmul_rmul := is_comm_jordan.lmul_comm_rmul_rmul,
lmul_lmul_comm_rmul := λ a b, by rw [is_comm_jordan.mul_comm (a * a) (b * a),
is_comm_jordan.mul_comm b a, is_comm_jordan.lmul_comm_rmul_rmul, is_comm_jordan.mul_comm,
is_comm_jordan.mul_comm b (a * a)],
rmul_comm_rmul_rmul := λ a b, by rw [is_comm_jordan.mul_comm b a,
is_comm_jordan.lmul_comm_rmul_rmul, is_comm_jordan.mul_comm], }
/-- Semigroup multiplication satisfies the (non-commutative) Jordan axioms-/
@[priority 100] -- see Note [lower instance priority]
instance semigroup.is_jordan [semigroup A] : is_jordan A :=
{ lmul_comm_rmul := λ a b, by rw mul_assoc,
lmul_lmul_comm_lmul := λ a b, by rw [mul_assoc, mul_assoc],
lmul_comm_rmul_rmul := λ a b, by rw [mul_assoc],
lmul_lmul_comm_rmul := λ a b, by rw [←mul_assoc],
rmul_comm_rmul_rmul := λ a b, by rw [← mul_assoc, ← mul_assoc], }
@[priority 100] -- see Note [lower instance priority]
instance comm_semigroup.is_comm_jordan [comm_semigroup A] : is_comm_jordan A :=
{ mul_comm := mul_comm,
lmul_comm_rmul_rmul := λ a b, mul_assoc _ _ _, }
local notation `L` := add_monoid.End.mul_left
local notation `R` := add_monoid.End.mul_right
/-!
The Jordan axioms can be expressed in terms of commuting multiplication operators.
-/
section commute
variables {A} [non_unital_non_assoc_ring A] [is_jordan A]
@[simp] lemma commute_lmul_rmul (a : A) : commute (L a) (R a) :=
add_monoid_hom.ext $ λ b, (is_jordan.lmul_comm_rmul _ _).symm
@[simp] lemma commute_lmul_lmul_sq (a : A) : commute (L a) (L (a * a)) :=
add_monoid_hom.ext $ λ b, (is_jordan.lmul_lmul_comm_lmul _ _).symm
@[simp] lemma commute_lmul_rmul_sq (a : A) : commute (L a) (R (a * a)) :=
add_monoid_hom.ext $ λ b, (is_jordan.lmul_comm_rmul_rmul _ _).symm
@[simp] lemma commute_lmul_sq_rmul (a : A) : commute (L (a * a)) (R a) :=
add_monoid_hom.ext $ λ b, (is_jordan.lmul_lmul_comm_rmul _ _)
@[simp] lemma commute_rmul_rmul_sq (a : A) : commute (R a) (R (a * a)) :=
add_monoid_hom.ext $ λ b, (is_jordan.rmul_comm_rmul_rmul _ _).symm
end commute
variables {A} [non_unital_non_assoc_ring A] [is_comm_jordan A]
/-!
The endomorphisms on an additive monoid `add_monoid.End` form a `ring`, and this may be equipped
with a Lie Bracket via `ring.has_bracket`.
-/
lemma two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add (a b : A) :
2•(⁅L a, L (a * b)⁆ + ⁅L b, L (b * a)⁆) = ⁅L (a * a), L b⁆ + ⁅L (b * b), L a⁆ :=
begin
suffices : 2 • ⁅L a, L (a * b)⁆ + 2 • ⁅L b, L (b * a)⁆ + ⁅L b, L (a * a)⁆ + ⁅L a, L (b * b)⁆ = 0,
{ rwa [← sub_eq_zero, ← sub_sub, sub_eq_add_neg, sub_eq_add_neg, lie_skew, lie_skew, nsmul_add] },
convert (commute_lmul_lmul_sq (a + b)).lie_eq,
simp only [add_mul, mul_add, map_add, lie_add, add_lie, is_comm_jordan.mul_comm b a,
(commute_lmul_lmul_sq a).lie_eq, (commute_lmul_lmul_sq b).lie_eq],
abel,
end
lemma two_nsmul_lie_lmul_lmul_add_add_eq_zero (a b c : A) :
2•(⁅L a, L (b * c)⁆ + ⁅L b, L (c * a)⁆ + ⁅L c, L (a * b)⁆) = 0 :=
begin
symmetry,
calc 0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ :
by rw (commute_lmul_lmul_sq (a + b + c)).lie_eq
... = ⁅L a + L b + L c,
L (a * a) + L (a * b) + L (a * c) + (L (b * a) + L (b * b) + L (b * c))
+ (L (c * a) + L (c * b) + L (c * c))⁆ :
by rw [add_mul, add_mul, mul_add, mul_add, mul_add, mul_add, mul_add, mul_add,
map_add, map_add, map_add, map_add, map_add, map_add, map_add, map_add, map_add, map_add]
... = ⁅L a + L b + L c,
L (a * a) + L (a * b) + L (c * a) + (L (a * b) + L (b * b) + L (b * c))
+ (L (c * a) + L (b * c) + L (c * c))⁆ :
by rw [is_comm_jordan.mul_comm b a, is_comm_jordan.mul_comm c a, is_comm_jordan.mul_comm c b]
... = ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) + 2•L (a * b) + 2•L (c * a)
+ 2•L (b * c) ⁆ :
by {rw [two_smul, two_smul, two_smul],
simp only [lie_add, add_lie, commute_lmul_lmul_sq, zero_add, add_zero], abel}
... = ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2•L (a * b)⁆
+ ⁅L a, 2•L(c * a)⁆ + ⁅L a, 2•L (b * c)⁆
+ (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ + ⁅L b, 2•L (a * b)⁆
+ ⁅L b, 2•L (c * a)⁆ + ⁅L b, 2•L (b * c)⁆)
+ (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ + ⁅L c, 2•L (a * b)⁆
+ ⁅L c, 2•L (c * a)⁆ + ⁅L c, 2•L (b * c)⁆) :
by rw [add_lie, add_lie, lie_add, lie_add, lie_add, lie_add, lie_add, lie_add, lie_add, lie_add,
lie_add, lie_add, lie_add, lie_add, lie_add, lie_add, lie_add]
... = ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2•L (a * b)⁆ + ⁅L a, 2•L (c * a)⁆
+ ⁅L a, 2•L (b * c)⁆
+ (⁅L b, L (a * a)⁆ + ⁅L b, L (c * c)⁆ + ⁅L b, 2•L (a * b)⁆ + ⁅L b, 2•L (c * a)⁆
+ ⁅L b, 2•L (b * c)⁆)
+ (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, 2•L (a * b)⁆ + ⁅L c, 2•L (c * a)⁆
+ ⁅L c, 2•L (b * c)⁆) :
by rw [(commute_lmul_lmul_sq a).lie_eq, (commute_lmul_lmul_sq b).lie_eq,
(commute_lmul_lmul_sq c).lie_eq, zero_add, add_zero, add_zero]
... = ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + 2•⁅L a, L (a * b)⁆ + 2•⁅L a, L (c * a)⁆
+ 2•⁅L a, L (b * c)⁆
+ (⁅L b, L (a * a)⁆ + ⁅L b, L (c * c)⁆ + 2•⁅L b, L (a * b)⁆ + 2•⁅L b, L (c * a)⁆
+ 2•⁅L b, L (b * c)⁆)
+ (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + 2•⁅L c, L (a * b)⁆ + 2•⁅L c, L (c * a)⁆
+ 2•⁅L c, L (b * c)⁆) :
by simp only [lie_nsmul]
... = (⁅L a, L (b * b)⁆+ ⁅L b, L (a * a)⁆ + 2•(⁅L a, L (a * b)⁆ + ⁅L b, L (a * b)⁆))
+ (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2•(⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆))
+ (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2•(⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆))
+ (2•⁅L a, L (b * c)⁆ + 2•⁅L b, L (c * a)⁆ + 2•⁅L c, L (a * b)⁆) : by abel
... = 2•⁅L a, L (b * c)⁆ + 2•⁅L b, L (c * a)⁆ + 2•⁅L c, L (a * b)⁆ :
by begin
rw add_left_eq_self,
nth_rewrite 1 is_comm_jordan.mul_comm a b,
nth_rewrite 0 is_comm_jordan.mul_comm c a,
nth_rewrite 1 is_comm_jordan.mul_comm b c,
rw [two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add,
two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add,
two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add,
← lie_skew (L (a * a)), ← lie_skew (L (b * b)), ← lie_skew (L (c * c)),
← lie_skew (L (a * a)), ← lie_skew (L (b * b)), ← lie_skew (L (c * c))],
abel,
end
... = 2•(⁅L a, L (b * c)⁆ + ⁅L b, L (c * a)⁆ + ⁅L c, L (a * b)⁆) : by rw [nsmul_add, nsmul_add]
end
|
7009c88471ce1e810071e0e1f10d9752f8eea3e1 | 618003631150032a5676f229d13a079ac875ff77 | /src/data/list/forall2.lean | 9fb7038286183377a052cc637b7b56a7ba9952b9 | [
"Apache-2.0"
] | permissive | awainverse/mathlib | 939b68c8486df66cfda64d327ad3d9165248c777 | ea76bd8f3ca0a8bf0a166a06a475b10663dec44a | refs/heads/master | 1,659,592,962,036 | 1,590,987,592,000 | 1,590,987,592,000 | 268,436,019 | 1 | 0 | Apache-2.0 | 1,590,990,500,000 | 1,590,990,500,000 | null | UTF-8 | Lean | false | false | 10,049 | lean | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import data.list.basic
universes u v w z
open nat function
variables {α : Type u} {β : Type v} {γ : Type w} {δ : Type z}
namespace list
/- forall₂ -/
variables {r : α → β → Prop} {p : γ → δ → Prop}
open relator
mk_iff_of_inductive_prop list.forall₂ list.forall₂_iff
@[simp] theorem forall₂_cons {R : α → β → Prop} {a b l₁ l₂} :
forall₂ R (a::l₁) (b::l₂) ↔ R a b ∧ forall₂ R l₁ l₂ :=
⟨λ h, by cases h with h₁ h₂; split; assumption, λ ⟨h₁, h₂⟩, forall₂.cons h₁ h₂⟩
theorem forall₂.imp {R S : α → β → Prop}
(H : ∀ a b, R a b → S a b) {l₁ l₂}
(h : forall₂ R l₁ l₂) : forall₂ S l₁ l₂ :=
by induction h; constructor; solve_by_elim
lemma forall₂.mp {r q s : α → β → Prop} (h : ∀a b, r a b → q a b → s a b) :
∀{l₁ l₂}, forall₂ r l₁ l₂ → forall₂ q l₁ l₂ → forall₂ s l₁ l₂
| [] [] forall₂.nil forall₂.nil := forall₂.nil
| (a::l₁) (b::l₂) (forall₂.cons hr hrs) (forall₂.cons hq hqs) :=
forall₂.cons (h a b hr hq) (forall₂.mp hrs hqs)
lemma forall₂.flip : ∀{a b}, forall₂ (flip r) b a → forall₂ r a b
| _ _ forall₂.nil := forall₂.nil
| (a :: as) (b :: bs) (forall₂.cons h₁ h₂) := forall₂.cons h₁ h₂.flip
lemma forall₂_same {r : α → α → Prop} : ∀{l}, (∀x∈l, r x x) → forall₂ r l l
| [] _ := forall₂.nil
| (a::as) h := forall₂.cons
(h _ (mem_cons_self _ _))
(forall₂_same $ assume a ha, h a $ mem_cons_of_mem _ ha)
lemma forall₂_refl {r} [is_refl α r] (l : list α) : forall₂ r l l :=
forall₂_same $ assume a h, is_refl.refl _
lemma forall₂_eq_eq_eq : forall₂ ((=) : α → α → Prop) = (=) :=
begin
funext a b, apply propext,
split,
{ assume h, induction h, {refl}, simp only [*]; split; refl },
{ assume h, subst h, exact forall₂_refl _ }
end
@[simp, priority 900] lemma forall₂_nil_left_iff {l} : forall₂ r nil l ↔ l = nil :=
⟨λ H, by cases H; refl, by rintro rfl; exact forall₂.nil⟩
@[simp, priority 900] lemma forall₂_nil_right_iff {l} : forall₂ r l nil ↔ l = nil :=
⟨λ H, by cases H; refl, by rintro rfl; exact forall₂.nil⟩
lemma forall₂_cons_left_iff {a l u} : forall₂ r (a::l) u ↔ (∃b u', r a b ∧ forall₂ r l u' ∧ u = b :: u') :=
iff.intro
(assume h, match u, h with (b :: u'), forall₂.cons h₁ h₂ := ⟨b, u', h₁, h₂, rfl⟩ end)
(assume h, match u, h with _, ⟨b, u', h₁, h₂, rfl⟩ := forall₂.cons h₁ h₂ end)
lemma forall₂_cons_right_iff {b l u} :
forall₂ r u (b::l) ↔ (∃a u', r a b ∧ forall₂ r u' l ∧ u = a :: u') :=
iff.intro
(assume h, match u, h with (b :: u'), forall₂.cons h₁ h₂ := ⟨b, u', h₁, h₂, rfl⟩ end)
(assume h, match u, h with _, ⟨b, u', h₁, h₂, rfl⟩ := forall₂.cons h₁ h₂ end)
lemma forall₂_and_left {r : α → β → Prop} {p : α → Prop} :
∀l u, forall₂ (λa b, p a ∧ r a b) l u ↔ (∀a∈l, p a) ∧ forall₂ r l u
| [] u := by simp only [forall₂_nil_left_iff, forall_prop_of_false (not_mem_nil _), imp_true_iff, true_and]
| (a::l) u := by simp only [forall₂_and_left l, forall₂_cons_left_iff, forall_mem_cons,
and_assoc, and_comm, and.left_comm, exists_and_distrib_left.symm]
@[simp] lemma forall₂_map_left_iff {f : γ → α} :
∀{l u}, forall₂ r (map f l) u ↔ forall₂ (λc b, r (f c) b) l u
| [] _ := by simp only [map, forall₂_nil_left_iff]
| (a::l) _ := by simp only [map, forall₂_cons_left_iff, forall₂_map_left_iff]
@[simp] lemma forall₂_map_right_iff {f : γ → β} :
∀{l u}, forall₂ r l (map f u) ↔ forall₂ (λa c, r a (f c)) l u
| _ [] := by simp only [map, forall₂_nil_right_iff]
| _ (b::u) := by simp only [map, forall₂_cons_right_iff, forall₂_map_right_iff]
lemma left_unique_forall₂ (hr : left_unique r) : left_unique (forall₂ r)
| a₀ nil a₁ forall₂.nil forall₂.nil := rfl
| (a₀::l₀) (b::l) (a₁::l₁) (forall₂.cons ha₀ h₀) (forall₂.cons ha₁ h₁) :=
hr ha₀ ha₁ ▸ left_unique_forall₂ h₀ h₁ ▸ rfl
lemma right_unique_forall₂ (hr : right_unique r) : right_unique (forall₂ r)
| nil a₀ a₁ forall₂.nil forall₂.nil := rfl
| (b::l) (a₀::l₀) (a₁::l₁) (forall₂.cons ha₀ h₀) (forall₂.cons ha₁ h₁) :=
hr ha₀ ha₁ ▸ right_unique_forall₂ h₀ h₁ ▸ rfl
lemma bi_unique_forall₂ (hr : bi_unique r) : bi_unique (forall₂ r) :=
⟨assume a b c, left_unique_forall₂ hr.1, assume a b c, right_unique_forall₂ hr.2⟩
theorem forall₂_length_eq {R : α → β → Prop} :
∀ {l₁ l₂}, forall₂ R l₁ l₂ → length l₁ = length l₂
| _ _ forall₂.nil := rfl
| _ _ (forall₂.cons h₁ h₂) := congr_arg succ (forall₂_length_eq h₂)
theorem forall₂_zip {R : α → β → Prop} :
∀ {l₁ l₂}, forall₂ R l₁ l₂ → ∀ {a b}, (a, b) ∈ zip l₁ l₂ → R a b
| _ _ (forall₂.cons h₁ h₂) x y (or.inl rfl) := h₁
| _ _ (forall₂.cons h₁ h₂) x y (or.inr h₃) := forall₂_zip h₂ h₃
theorem forall₂_iff_zip {R : α → β → Prop} {l₁ l₂} : forall₂ R l₁ l₂ ↔
length l₁ = length l₂ ∧ ∀ {a b}, (a, b) ∈ zip l₁ l₂ → R a b :=
⟨λ h, ⟨forall₂_length_eq h, @forall₂_zip _ _ _ _ _ h⟩,
λ h, begin
cases h with h₁ h₂,
induction l₁ with a l₁ IH generalizing l₂,
{ cases length_eq_zero.1 h₁.symm, constructor },
{ cases l₂ with b l₂; injection h₁ with h₁,
exact forall₂.cons (h₂ $ or.inl rfl) (IH h₁ $ λ a b h, h₂ $ or.inr h) }
end⟩
theorem forall₂_take {R : α → β → Prop} :
∀ n {l₁ l₂}, forall₂ R l₁ l₂ → forall₂ R (take n l₁) (take n l₂)
| 0 _ _ _ := by simp only [forall₂.nil, take]
| (n+1) _ _ (forall₂.nil) := by simp only [forall₂.nil, take]
| (n+1) _ _ (forall₂.cons h₁ h₂) := by simp [and.intro h₁ h₂, forall₂_take n]
theorem forall₂_drop {R : α → β → Prop} :
∀ n {l₁ l₂}, forall₂ R l₁ l₂ → forall₂ R (drop n l₁) (drop n l₂)
| 0 _ _ h := by simp only [drop, h]
| (n+1) _ _ (forall₂.nil) := by simp only [forall₂.nil, drop]
| (n+1) _ _ (forall₂.cons h₁ h₂) := by simp [and.intro h₁ h₂, forall₂_drop n]
theorem forall₂_take_append {R : α → β → Prop} (l : list α) (l₁ : list β) (l₂ : list β)
(h : forall₂ R l (l₁ ++ l₂)) : forall₂ R (list.take (length l₁) l) l₁ :=
have h': forall₂ R (take (length l₁) l) (take (length l₁) (l₁ ++ l₂)), from forall₂_take (length l₁) h,
by rwa [take_left] at h'
theorem forall₂_drop_append {R : α → β → Prop} (l : list α) (l₁ : list β) (l₂ : list β)
(h : forall₂ R l (l₁ ++ l₂)) : forall₂ R (list.drop (length l₁) l) l₂ :=
have h': forall₂ R (drop (length l₁) l) (drop (length l₁) (l₁ ++ l₂)), from forall₂_drop (length l₁) h,
by rwa [drop_left] at h'
lemma rel_mem (hr : bi_unique r) : (r ⇒ forall₂ r ⇒ iff) (∈) (∈)
| a b h [] [] forall₂.nil := by simp only [not_mem_nil]
| a b h (a'::as) (b'::bs) (forall₂.cons h₁ h₂) := rel_or (rel_eq hr h h₁) (rel_mem h h₂)
lemma rel_map : ((r ⇒ p) ⇒ forall₂ r ⇒ forall₂ p) map map
| f g h [] [] forall₂.nil := forall₂.nil
| f g h (a::as) (b::bs) (forall₂.cons h₁ h₂) := forall₂.cons (h h₁) (rel_map @h h₂)
lemma rel_append : (forall₂ r ⇒ forall₂ r ⇒ forall₂ r) append append
| [] [] h l₁ l₂ hl := hl
| (a::as) (b::bs) (forall₂.cons h₁ h₂) l₁ l₂ hl := forall₂.cons h₁ (rel_append h₂ hl)
lemma rel_join : (forall₂ (forall₂ r) ⇒ forall₂ r) join join
| [] [] forall₂.nil := forall₂.nil
| (a::as) (b::bs) (forall₂.cons h₁ h₂) := rel_append h₁ (rel_join h₂)
lemma rel_bind : (forall₂ r ⇒ (r ⇒ forall₂ p) ⇒ forall₂ p) list.bind list.bind :=
assume a b h₁ f g h₂, rel_join (rel_map @h₂ h₁)
lemma rel_foldl : ((p ⇒ r ⇒ p) ⇒ p ⇒ forall₂ r ⇒ p) foldl foldl
| f g hfg _ _ h _ _ forall₂.nil := h
| f g hfg x y hxy _ _ (forall₂.cons hab hs) := rel_foldl @hfg (hfg hxy hab) hs
lemma rel_foldr : ((r ⇒ p ⇒ p) ⇒ p ⇒ forall₂ r ⇒ p) foldr foldr
| f g hfg _ _ h _ _ forall₂.nil := h
| f g hfg x y hxy _ _ (forall₂.cons hab hs) := hfg hab (rel_foldr @hfg hxy hs)
lemma rel_filter {p : α → Prop} {q : β → Prop} [decidable_pred p] [decidable_pred q]
(hpq : (r ⇒ (↔)) p q) :
(forall₂ r ⇒ forall₂ r) (filter p) (filter q)
| _ _ forall₂.nil := forall₂.nil
| (a::as) (b::bs) (forall₂.cons h₁ h₂) :=
begin
by_cases p a,
{ have : q b, { rwa [← hpq h₁] },
simp only [filter_cons_of_pos _ h, filter_cons_of_pos _ this, forall₂_cons, h₁, rel_filter h₂, and_true], },
{ have : ¬ q b, { rwa [← hpq h₁] },
simp only [filter_cons_of_neg _ h, filter_cons_of_neg _ this, rel_filter h₂], },
end
theorem filter_map_cons (f : α → option β) (a : α) (l : list α) :
filter_map f (a :: l) = option.cases_on (f a) (filter_map f l) (λb, b :: filter_map f l) :=
begin
generalize eq : f a = b,
cases b,
{ rw filter_map_cons_none _ _ eq },
{ rw filter_map_cons_some _ _ _ eq },
end
lemma rel_filter_map : ((r ⇒ option.rel p) ⇒ forall₂ r ⇒ forall₂ p) filter_map filter_map
| f g hfg _ _ forall₂.nil := forall₂.nil
| f g hfg (a::as) (b::bs) (forall₂.cons h₁ h₂) :=
by rw [filter_map_cons, filter_map_cons];
from match f a, g b, hfg h₁ with
| _, _, option.rel.none := rel_filter_map @hfg h₂
| _, _, option.rel.some h := forall₂.cons h (rel_filter_map @hfg h₂)
end
@[to_additive]
lemma rel_prod [monoid α] [monoid β]
(h : r 1 1) (hf : (r ⇒ r ⇒ r) (*) (*)) : (forall₂ r ⇒ r) prod prod :=
rel_foldl hf h
end list
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