blob_id stringlengths 40 40 | directory_id stringlengths 40 40 | path stringlengths 7 139 | content_id stringlengths 40 40 | detected_licenses listlengths 0 16 | license_type stringclasses 2
values | repo_name stringlengths 7 55 | snapshot_id stringlengths 40 40 | revision_id stringlengths 40 40 | branch_name stringclasses 6
values | visit_date int64 1,471B 1,694B | revision_date int64 1,378B 1,694B | committer_date int64 1,378B 1,694B | github_id float64 1.33M 604M ⌀ | star_events_count int64 0 43.5k | fork_events_count int64 0 1.5k | gha_license_id stringclasses 6
values | gha_event_created_at int64 1,402B 1,695B ⌀ | gha_created_at int64 1,359B 1,637B ⌀ | gha_language stringclasses 19
values | src_encoding stringclasses 2
values | language stringclasses 1
value | is_vendor bool 1
class | is_generated bool 1
class | length_bytes int64 3 6.4M | extension stringclasses 4
values | content stringlengths 3 6.12M |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
b4d1fa092b416c42419d0c1d438f217bcd382f46 | bb31430994044506fa42fd667e2d556327e18dfe | /src/topology/constructions.lean | 565bf50b7a7b704c144f46b222d639b5607b8c72 | [
"Apache-2.0"
] | permissive | sgouezel/mathlib | 0cb4e5335a2ba189fa7af96d83a377f83270e503 | 00638177efd1b2534fc5269363ebf42a7871df9a | refs/heads/master | 1,674,527,483,042 | 1,673,665,568,000 | 1,673,665,568,000 | 119,598,202 | 0 | 0 | null | 1,517,348,647,000 | 1,517,348,646,000 | null | UTF-8 | Lean | false | false | 58,023 | 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, Patrick Massot
-/
import topology.maps
import topology.locally_finite
import order.filter.pi
/-!
# Constructions of new topological spaces from old ones
This file constructs products, sums, subtypes and quotients of topological spaces
and sets up their basic theory, such as criteria for maps into or out of these
constructions to be continuous; descriptions of the open sets, neighborhood filters,
and generators of these constructions; and their behavior with respect to embeddings
and other specific classes of maps.
## Implementation note
The constructed topologies are defined using induced and coinduced topologies
along with the complete lattice structure on topologies. Their universal properties
(for example, a map `X → Y × Z` is continuous if and only if both projections
`X → Y`, `X → Z` are) follow easily using order-theoretic descriptions of
continuity. With more work we can also extract descriptions of the open sets,
neighborhood filters and so on.
## Tags
product, sum, disjoint union, subspace, quotient space
-/
noncomputable theory
open topological_space set filter function
open_locale classical topological_space filter
universes u v
variables {α : Type u} {β : Type v} {γ δ ε ζ : Type*}
section constructions
instance {p : α → Prop} [t : topological_space α] : topological_space (subtype p) :=
induced coe t
instance {r : α → α → Prop} [t : topological_space α] : topological_space (quot r) :=
coinduced (quot.mk r) t
instance {s : setoid α} [t : topological_space α] : topological_space (quotient s) :=
coinduced quotient.mk t
instance [t₁ : topological_space α] [t₂ : topological_space β] : topological_space (α × β) :=
induced prod.fst t₁ ⊓ induced prod.snd t₂
instance [t₁ : topological_space α] [t₂ : topological_space β] : topological_space (α ⊕ β) :=
coinduced sum.inl t₁ ⊔ coinduced sum.inr t₂
instance {β : α → Type v} [t₂ : Πa, topological_space (β a)] : topological_space (sigma β) :=
⨆a, coinduced (sigma.mk a) (t₂ a)
instance Pi.topological_space {β : α → Type v} [t₂ : Πa, topological_space (β a)] :
topological_space (Πa, β a) :=
⨅a, induced (λf, f a) (t₂ a)
instance ulift.topological_space [t : topological_space α] : topological_space (ulift.{v u} α) :=
t.induced ulift.down
/-!
### `additive`, `multiplicative`
The topology on those type synonyms is inherited without change.
-/
section
variables [topological_space α]
open additive multiplicative
instance : topological_space (additive α) := ‹topological_space α›
instance : topological_space (multiplicative α) := ‹topological_space α›
instance [discrete_topology α] : discrete_topology (additive α) := ‹discrete_topology α›
instance [discrete_topology α] : discrete_topology (multiplicative α) := ‹discrete_topology α›
lemma continuous_of_mul : continuous (of_mul : α → additive α) := continuous_id
lemma continuous_to_mul : continuous (to_mul : additive α → α) := continuous_id
lemma continuous_of_add : continuous (of_add : α → multiplicative α) := continuous_id
lemma continuous_to_add : continuous (to_add : multiplicative α → α) := continuous_id
lemma is_open_map_of_mul : is_open_map (of_mul : α → additive α) := is_open_map.id
lemma is_open_map_to_mul : is_open_map (to_mul : additive α → α) := is_open_map.id
lemma is_open_map_of_add : is_open_map (of_add : α → multiplicative α) := is_open_map.id
lemma is_open_map_to_add : is_open_map (to_add : multiplicative α → α) := is_open_map.id
lemma is_closed_map_of_mul : is_closed_map (of_mul : α → additive α) := is_closed_map.id
lemma is_closed_map_to_mul : is_closed_map (to_mul : additive α → α) := is_closed_map.id
lemma is_closed_map_of_add : is_closed_map (of_add : α → multiplicative α) := is_closed_map.id
lemma is_closed_map_to_add : is_closed_map (to_add : multiplicative α → α) := is_closed_map.id
lemma nhds_of_mul (a : α) : 𝓝 (of_mul a) = map of_mul (𝓝 a) := by { unfold nhds, refl, }
lemma nhds_of_add (a : α) : 𝓝 (of_add a) = map of_add (𝓝 a) := by { unfold nhds, refl, }
lemma nhds_to_mul (a : additive α) : 𝓝 (to_mul a) = map to_mul (𝓝 a) := by { unfold nhds, refl, }
lemma nhds_to_add (a : multiplicative α) : 𝓝 (to_add a) = map to_add (𝓝 a) :=
by { unfold nhds, refl, }
end
/-!
### Order dual
The topology on this type synonym is inherited without change.
-/
section
variables [topological_space α]
open order_dual
instance : topological_space αᵒᵈ := ‹topological_space α›
instance [discrete_topology α] : discrete_topology (αᵒᵈ) := ‹discrete_topology α›
lemma continuous_to_dual : continuous (to_dual : α → αᵒᵈ) := continuous_id
lemma continuous_of_dual : continuous (of_dual : αᵒᵈ → α) := continuous_id
lemma is_open_map_to_dual : is_open_map (to_dual : α → αᵒᵈ) := is_open_map.id
lemma is_open_map_of_dual : is_open_map (of_dual : αᵒᵈ → α) := is_open_map.id
lemma is_closed_map_to_dual : is_closed_map (to_dual : α → αᵒᵈ) := is_closed_map.id
lemma is_closed_map_of_dual : is_closed_map (of_dual : αᵒᵈ → α) := is_closed_map.id
lemma nhds_to_dual (a : α) : 𝓝 (to_dual a) = map to_dual (𝓝 a) := by { unfold nhds, refl, }
lemma nhds_of_dual (a : α) : 𝓝 (of_dual a) = map of_dual (𝓝 a) := by { unfold nhds, refl, }
end
lemma quotient.preimage_mem_nhds [topological_space α] [s : setoid α]
{V : set $ quotient s} {a : α} (hs : V ∈ 𝓝 (quotient.mk a)) : quotient.mk ⁻¹' V ∈ 𝓝 a :=
preimage_nhds_coinduced hs
/-- The image of a dense set under `quotient.mk` is a dense set. -/
lemma dense.quotient [setoid α] [topological_space α] {s : set α} (H : dense s) :
dense (quotient.mk '' s) :=
(surjective_quotient_mk α).dense_range.dense_image continuous_coinduced_rng H
/-- The composition of `quotient.mk` and a function with dense range has dense range. -/
lemma dense_range.quotient [setoid α] [topological_space α] {f : β → α} (hf : dense_range f) :
dense_range (quotient.mk ∘ f) :=
(surjective_quotient_mk α).dense_range.comp hf continuous_coinduced_rng
instance {p : α → Prop} [topological_space α] [discrete_topology α] :
discrete_topology (subtype p) :=
⟨bot_unique $ assume s hs,
⟨coe '' s, is_open_discrete _, (set.preimage_image_eq _ subtype.coe_injective)⟩⟩
instance sum.discrete_topology [topological_space α] [topological_space β]
[hα : discrete_topology α] [hβ : discrete_topology β] : discrete_topology (α ⊕ β) :=
⟨by unfold sum.topological_space; simp [hα.eq_bot, hβ.eq_bot]⟩
instance sigma.discrete_topology {β : α → Type v} [Πa, topological_space (β a)]
[h : Πa, discrete_topology (β a)] : discrete_topology (sigma β) :=
⟨by { unfold sigma.topological_space, simp [λ a, (h a).eq_bot] }⟩
section topα
variable [topological_space α]
/-
The 𝓝 filter and the subspace topology.
-/
theorem mem_nhds_subtype (s : set α) (a : {x // x ∈ s}) (t : set {x // x ∈ s}) :
t ∈ 𝓝 a ↔ ∃ u ∈ 𝓝 (a : α), coe ⁻¹' u ⊆ t :=
mem_nhds_induced coe a t
theorem nhds_subtype (s : set α) (a : {x // x ∈ s}) :
𝓝 a = comap coe (𝓝 (a : α)) :=
nhds_induced coe a
lemma nhds_within_subtype_eq_bot_iff {s t : set α} {x : s} :
𝓝[(coe : s → α) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : α) ⊓ 𝓟 s = ⊥ :=
by rw [inf_principal_eq_bot_iff_comap, nhds_within, nhds_within, comap_inf, comap_principal,
nhds_induced]
lemma nhds_ne_subtype_eq_bot_iff {S : set α} {x : S} : 𝓝[{x}ᶜ] x = ⊥ ↔ 𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S = ⊥ :=
by rw [← nhds_within_subtype_eq_bot_iff, preimage_compl, ← image_singleton,
subtype.coe_injective.preimage_image ]
lemma nhds_ne_subtype_ne_bot_iff {S : set α} {x : S} :
(𝓝[{x}ᶜ] x).ne_bot ↔ (𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S).ne_bot :=
by rw [ne_bot_iff, ne_bot_iff, not_iff_not, nhds_ne_subtype_eq_bot_iff]
lemma discrete_topology_subtype_iff {S : set α} :
discrete_topology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ :=
by simp_rw [discrete_topology_iff_nhds_ne, set_coe.forall', nhds_ne_subtype_eq_bot_iff]
end topα
/-- A type synonym equiped with the topology whose open sets are the empty set and the sets with
finite complements. -/
def cofinite_topology (α : Type*) := α
namespace cofinite_topology
/-- The identity equivalence between `α` and `cofinite_topology α`. -/
def of : α ≃ cofinite_topology α := equiv.refl α
instance [inhabited α] : inhabited (cofinite_topology α) :=
{ default := of default }
instance : topological_space (cofinite_topology α) :=
{ is_open := λ s, s.nonempty → set.finite sᶜ,
is_open_univ := by simp,
is_open_inter := λ s t, begin
rintros hs ht ⟨x, hxs, hxt⟩,
rw compl_inter,
exact (hs ⟨x, hxs⟩).union (ht ⟨x, hxt⟩),
end,
is_open_sUnion := begin
rintros s h ⟨x, t, hts, hzt⟩,
rw set.compl_sUnion,
exact set.finite.sInter (mem_image_of_mem _ hts) (h t hts ⟨x, hzt⟩),
end }
lemma is_open_iff {s : set (cofinite_topology α)} :
is_open s ↔ (s.nonempty → (sᶜ).finite) := iff.rfl
lemma is_open_iff' {s : set (cofinite_topology α)} :
is_open s ↔ (s = ∅ ∨ (sᶜ).finite) :=
by simp only [is_open_iff, nonempty_iff_ne_empty, or_iff_not_imp_left]
lemma is_closed_iff {s : set (cofinite_topology α)} :
is_closed s ↔ s = univ ∨ s.finite :=
by simp [← is_open_compl_iff, is_open_iff']
lemma nhds_eq (a : cofinite_topology α) : 𝓝 a = pure a ⊔ cofinite :=
begin
ext U,
rw mem_nhds_iff,
split,
{ rintro ⟨V, hVU, V_op, haV⟩,
exact mem_sup.mpr ⟨hVU haV, mem_of_superset (V_op ⟨_, haV⟩) hVU⟩ },
{ rintros ⟨hU : a ∈ U, hU' : (Uᶜ).finite⟩,
exact ⟨U, subset.rfl, λ h, hU', hU⟩ }
end
lemma mem_nhds_iff {a : cofinite_topology α} {s : set (cofinite_topology α)} :
s ∈ 𝓝 a ↔ a ∈ s ∧ sᶜ.finite :=
by simp [nhds_eq]
end cofinite_topology
end constructions
section prod
variables [topological_space α] [topological_space β] [topological_space γ] [topological_space δ]
[topological_space ε] [topological_space ζ]
@[continuity] lemma continuous_fst : continuous (@prod.fst α β) :=
continuous_inf_dom_left continuous_induced_dom
/-- Postcomposing `f` with `prod.fst` is continuous -/
lemma continuous.fst {f : α → β × γ} (hf : continuous f) : continuous (λ a : α, (f a).1) :=
continuous_fst.comp hf
/-- Precomposing `f` with `prod.fst` is continuous -/
lemma continuous.fst' {f : α → γ} (hf : continuous f) : continuous (λ x : α × β, f x.fst) :=
hf.comp continuous_fst
lemma continuous_at_fst {p : α × β} : continuous_at prod.fst p :=
continuous_fst.continuous_at
/-- Postcomposing `f` with `prod.fst` is continuous at `x` -/
lemma continuous_at.fst {f : α → β × γ} {x : α} (hf : continuous_at f x) :
continuous_at (λ a : α, (f a).1) x :=
continuous_at_fst.comp hf
/-- Precomposing `f` with `prod.fst` is continuous at `(x, y)` -/
lemma continuous_at.fst' {f : α → γ} {x : α} {y : β} (hf : continuous_at f x) :
continuous_at (λ x : α × β, f x.fst) (x, y) :=
continuous_at.comp hf continuous_at_fst
/-- Precomposing `f` with `prod.fst` is continuous at `x : α × β` -/
lemma continuous_at.fst'' {f : α → γ} {x : α × β} (hf : continuous_at f x.fst) :
continuous_at (λ x : α × β, f x.fst) x :=
hf.comp continuous_at_fst
@[continuity] lemma continuous_snd : continuous (@prod.snd α β) :=
continuous_inf_dom_right continuous_induced_dom
/-- Postcomposing `f` with `prod.snd` is continuous -/
lemma continuous.snd {f : α → β × γ} (hf : continuous f) : continuous (λ a : α, (f a).2) :=
continuous_snd.comp hf
/-- Precomposing `f` with `prod.snd` is continuous -/
lemma continuous.snd' {f : β → γ} (hf : continuous f) : continuous (λ x : α × β, f x.snd) :=
hf.comp continuous_snd
lemma continuous_at_snd {p : α × β} : continuous_at prod.snd p :=
continuous_snd.continuous_at
/-- Postcomposing `f` with `prod.snd` is continuous at `x` -/
lemma continuous_at.snd {f : α → β × γ} {x : α} (hf : continuous_at f x) :
continuous_at (λ a : α, (f a).2) x :=
continuous_at_snd.comp hf
/-- Precomposing `f` with `prod.snd` is continuous at `(x, y)` -/
lemma continuous_at.snd' {f : β → γ} {x : α} {y : β} (hf : continuous_at f y) :
continuous_at (λ x : α × β, f x.snd) (x, y) :=
continuous_at.comp hf continuous_at_snd
/-- Precomposing `f` with `prod.snd` is continuous at `x : α × β` -/
lemma continuous_at.snd'' {f : β → γ} {x : α × β} (hf : continuous_at f x.snd) :
continuous_at (λ x : α × β, f x.snd) x :=
hf.comp continuous_at_snd
@[continuity] lemma continuous.prod_mk {f : γ → α} {g : γ → β}
(hf : continuous f) (hg : continuous g) : continuous (λx, (f x, g x)) :=
continuous_inf_rng.2 ⟨continuous_induced_rng.2 hf, continuous_induced_rng.2 hg⟩
@[simp] lemma continuous_prod_mk {f : α → β} {g : α → γ} :
continuous (λ x, (f x, g x)) ↔ continuous f ∧ continuous g :=
⟨λ h, ⟨h.fst, h.snd⟩, λ h, h.1.prod_mk h.2⟩
@[continuity] lemma continuous.prod.mk (a : α) : continuous (λ b : β, (a, b)) :=
continuous_const.prod_mk continuous_id'
@[continuity] lemma continuous.prod.mk_left (b : β) : continuous (λ a : α, (a, b)) :=
continuous_id'.prod_mk continuous_const
lemma continuous.comp₂ {g : α × β → γ} (hg : continuous g) {e : δ → α} (he : continuous e)
{f : δ → β} (hf : continuous f) : continuous (λ x, g (e x, f x)) :=
hg.comp $ he.prod_mk hf
lemma continuous.comp₃ {g : α × β × γ → ε} (hg : continuous g)
{e : δ → α} (he : continuous e) {f : δ → β} (hf : continuous f)
{k : δ → γ} (hk : continuous k) : continuous (λ x, g (e x, f x, k x)) :=
hg.comp₂ he $ hf.prod_mk hk
lemma continuous.comp₄ {g : α × β × γ × ζ → ε} (hg : continuous g)
{e : δ → α} (he : continuous e) {f : δ → β} (hf : continuous f)
{k : δ → γ} (hk : continuous k) {l : δ → ζ} (hl : continuous l) :
continuous (λ x, g (e x, f x, k x, l x)) :=
hg.comp₃ he hf $ hk.prod_mk hl
lemma continuous.prod_map {f : γ → α} {g : δ → β} (hf : continuous f) (hg : continuous g) :
continuous (λ x : γ × δ, (f x.1, g x.2)) :=
hf.fst'.prod_mk hg.snd'
/-- A version of `continuous_inf_dom_left` for binary functions -/
lemma continuous_inf_dom_left₂ {α β γ} {f : α → β → γ}
{ta1 ta2 : topological_space α} {tb1 tb2 : topological_space β} {tc1 : topological_space γ}
(h : by haveI := ta1; haveI := tb1; exact continuous (λ p : α × β, f p.1 p.2)) :
by haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact continuous (λ p : α × β, f p.1 p.2) :=
begin
have ha := @continuous_inf_dom_left _ _ id ta1 ta2 ta1 (@continuous_id _ (id _)),
have hb := @continuous_inf_dom_left _ _ id tb1 tb2 tb1 (@continuous_id _ (id _)),
have h_continuous_id := @continuous.prod_map _ _ _ _ ta1 tb1 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb,
exact @continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id,
end
/-- A version of `continuous_inf_dom_right` for binary functions -/
lemma continuous_inf_dom_right₂ {α β γ} {f : α → β → γ}
{ta1 ta2 : topological_space α} {tb1 tb2 : topological_space β} {tc1 : topological_space γ}
(h : by haveI := ta2; haveI := tb2; exact continuous (λ p : α × β, f p.1 p.2)) :
by haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact continuous (λ p : α × β, f p.1 p.2) :=
begin
have ha := @continuous_inf_dom_right _ _ id ta1 ta2 ta2 (@continuous_id _ (id _)),
have hb := @continuous_inf_dom_right _ _ id tb1 tb2 tb2 (@continuous_id _ (id _)),
have h_continuous_id := @continuous.prod_map _ _ _ _ ta2 tb2 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb,
exact @continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id,
end
/-- A version of `continuous_Inf_dom` for binary functions -/
lemma continuous_Inf_dom₂ {α β γ} {f : α → β → γ}
{tas : set (topological_space α)} {tbs : set (topological_space β)}
{ta : topological_space α} {tb : topological_space β} {tc : topological_space γ}
(ha : ta ∈ tas) (hb : tb ∈ tbs)
(hf : continuous (λ p : α × β, f p.1 p.2)):
by haveI := Inf tas; haveI := Inf tbs; exact @continuous _ _ _ tc (λ p : α × β, f p.1 p.2) :=
begin
let t : topological_space (α × β) := prod.topological_space,
have ha := continuous_Inf_dom ha continuous_id,
have hb := continuous_Inf_dom hb continuous_id,
have h_continuous_id := @continuous.prod_map _ _ _ _ ta tb (Inf tas) (Inf tbs) _ _ ha hb,
exact @continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id,
end
lemma filter.eventually.prod_inl_nhds {p : α → Prop} {a : α} (h : ∀ᶠ x in 𝓝 a, p x) (b : β) :
∀ᶠ x in 𝓝 (a, b), p (x : α × β).1 :=
continuous_at_fst h
lemma filter.eventually.prod_inr_nhds {p : β → Prop} {b : β} (h : ∀ᶠ x in 𝓝 b, p x) (a : α) :
∀ᶠ x in 𝓝 (a, b), p (x : α × β).2 :=
continuous_at_snd h
lemma filter.eventually.prod_mk_nhds {pa : α → Prop} {a} (ha : ∀ᶠ x in 𝓝 a, pa x)
{pb : β → Prop} {b} (hb : ∀ᶠ y in 𝓝 b, pb y) :
∀ᶠ p in 𝓝 (a, b), pa (p : α × β).1 ∧ pb p.2 :=
(ha.prod_inl_nhds b).and (hb.prod_inr_nhds a)
lemma continuous_swap : continuous (prod.swap : α × β → β × α) :=
continuous_snd.prod_mk continuous_fst
lemma continuous_uncurry_left {f : α → β → γ} (a : α)
(h : continuous (uncurry f)) : continuous (f a) :=
show continuous (uncurry f ∘ (λ b, (a, b))), from h.comp (by continuity)
lemma continuous_uncurry_right {f : α → β → γ} (b : β)
(h : continuous (uncurry f)) : continuous (λ a, f a b) :=
show continuous (uncurry f ∘ (λ a, (a, b))), from h.comp (by continuity)
lemma continuous_curry {g : α × β → γ} (a : α)
(h : continuous g) : continuous (curry g a) :=
show continuous (g ∘ (λ b, (a, b))), from h.comp (by continuity)
lemma is_open.prod {s : set α} {t : set β} (hs : is_open s) (ht : is_open t) :
is_open (s ×ˢ t) :=
(hs.preimage continuous_fst).inter (ht.preimage continuous_snd)
lemma nhds_prod_eq {a : α} {b : β} : 𝓝 (a, b) = 𝓝 a ×ᶠ 𝓝 b :=
by rw [filter.prod, prod.topological_space, nhds_inf, nhds_induced, nhds_induced]
/-- If a function `f x y` is such that `y ↦ f x y` is continuous for all `x`, and `x` lives in a
discrete space, then `f` is continuous. -/
lemma continuous_uncurry_of_discrete_topology [discrete_topology α]
{f : α → β → γ} (hf : ∀ a, continuous (f a)) : continuous (uncurry f) :=
begin
apply continuous_iff_continuous_at.2,
rintros ⟨a, x⟩,
change map _ _ ≤ _,
rw [nhds_prod_eq, nhds_discrete, filter.map_pure_prod],
exact (hf a).continuous_at
end
lemma mem_nhds_prod_iff {a : α} {b : β} {s : set (α × β)} :
s ∈ 𝓝 (a, b) ↔ ∃ (u ∈ 𝓝 a) (v ∈ 𝓝 b), u ×ˢ v ⊆ s :=
by rw [nhds_prod_eq, mem_prod_iff]
lemma mem_nhds_prod_iff' {a : α} {b : β} {s : set (α × β)} :
s ∈ 𝓝 (a, b) ↔ ∃ (u : set α) (v : set β), is_open u ∧ a ∈ u ∧ is_open v ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
begin
rw mem_nhds_prod_iff,
split,
{ rintros ⟨u, Hu, v, Hv, h⟩,
rcases mem_nhds_iff.1 Hu with ⟨u', u'u, u'_open, Hu'⟩,
rcases mem_nhds_iff.1 Hv with ⟨v', v'v, v'_open, Hv'⟩,
exact ⟨u', v', u'_open, Hu', v'_open, Hv', (set.prod_mono u'u v'v).trans h⟩ },
{ rintros ⟨u, v, u_open, au, v_open, bv, huv⟩,
exact ⟨u, u_open.mem_nhds au, v, v_open.mem_nhds bv, huv⟩ }
end
lemma _root_.prod.tendsto_iff {α} (seq : α → β × γ) {f : filter α} (x : β × γ) :
tendsto seq f (𝓝 x)
↔ tendsto (λ n, (seq n).fst) f (𝓝 x.fst) ∧ tendsto (λ n, (seq n).snd) f (𝓝 x.snd) :=
by { cases x, rw [nhds_prod_eq, filter.tendsto_prod_iff'], }
lemma filter.has_basis.prod_nhds {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
{sa : ιa → set α} {sb : ιb → set β} {a : α} {b : β} (ha : (𝓝 a).has_basis pa sa)
(hb : (𝓝 b).has_basis pb sb) :
(𝓝 (a, b)).has_basis (λ i : ιa × ιb, pa i.1 ∧ pb i.2) (λ i, sa i.1 ×ˢ sb i.2) :=
by { rw nhds_prod_eq, exact ha.prod hb }
lemma filter.has_basis.prod_nhds' {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
{sa : ιa → set α} {sb : ιb → set β} {ab : α × β} (ha : (𝓝 ab.1).has_basis pa sa)
(hb : (𝓝 ab.2).has_basis pb sb) :
(𝓝 ab).has_basis (λ i : ιa × ιb, pa i.1 ∧ pb i.2) (λ i, sa i.1 ×ˢ sb i.2) :=
by { cases ab, exact ha.prod_nhds hb }
instance [discrete_topology α] [discrete_topology β] : discrete_topology (α × β) :=
⟨eq_of_nhds_eq_nhds $ assume ⟨a, b⟩,
by rw [nhds_prod_eq, nhds_discrete α, nhds_discrete β, nhds_bot, filter.prod_pure_pure]⟩
lemma prod_mem_nhds_iff {s : set α} {t : set β} {a : α} {b : β} :
s ×ˢ t ∈ 𝓝 (a, b) ↔ s ∈ 𝓝 a ∧ t ∈ 𝓝 b :=
by rw [nhds_prod_eq, prod_mem_prod_iff]
lemma prod_mem_nhds {s : set α} {t : set β} {a : α} {b : β}
(ha : s ∈ 𝓝 a) (hb : t ∈ 𝓝 b) : s ×ˢ t ∈ 𝓝 (a, b) :=
prod_mem_nhds_iff.2 ⟨ha, hb⟩
lemma filter.eventually.prod_nhds {p : α → Prop} {q : β → Prop} {a : α} {b : β}
(ha : ∀ᶠ x in 𝓝 a, p x) (hb : ∀ᶠ y in 𝓝 b, q y) :
∀ᶠ z : α × β in 𝓝 (a, b), p z.1 ∧ q z.2 :=
prod_mem_nhds ha hb
lemma nhds_swap (a : α) (b : β) : 𝓝 (a, b) = (𝓝 (b, a)).map prod.swap :=
by rw [nhds_prod_eq, filter.prod_comm, nhds_prod_eq]; refl
lemma filter.tendsto.prod_mk_nhds {γ} {a : α} {b : β} {f : filter γ} {ma : γ → α} {mb : γ → β}
(ha : tendsto ma f (𝓝 a)) (hb : tendsto mb f (𝓝 b)) :
tendsto (λc, (ma c, mb c)) f (𝓝 (a, b)) :=
by rw [nhds_prod_eq]; exact filter.tendsto.prod_mk ha hb
lemma filter.eventually.curry_nhds {p : α × β → Prop} {x : α} {y : β} (h : ∀ᶠ x in 𝓝 (x, y), p x) :
∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') :=
by { rw [nhds_prod_eq] at h, exact h.curry }
lemma continuous_at.prod {f : α → β} {g : α → γ} {x : α}
(hf : continuous_at f x) (hg : continuous_at g x) : continuous_at (λx, (f x, g x)) x :=
hf.prod_mk_nhds hg
lemma continuous_at.prod_map {f : α → γ} {g : β → δ} {p : α × β}
(hf : continuous_at f p.fst) (hg : continuous_at g p.snd) :
continuous_at (λ p : α × β, (f p.1, g p.2)) p :=
hf.fst''.prod hg.snd''
lemma continuous_at.prod_map' {f : α → γ} {g : β → δ} {x : α} {y : β}
(hf : continuous_at f x) (hg : continuous_at g y) :
continuous_at (λ p : α × β, (f p.1, g p.2)) (x, y) :=
hf.fst'.prod hg.snd'
lemma prod_generate_from_generate_from_eq {α β : Type*} {s : set (set α)} {t : set (set β)}
(hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
@prod.topological_space α β (generate_from s) (generate_from t) =
generate_from {g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v} :=
let G := generate_from {g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v} in
le_antisymm
(le_generate_from $ λ g ⟨u, hu, v, hv, g_eq⟩, g_eq.symm ▸
@is_open.prod _ _ (generate_from s) (generate_from t) _ _
(generate_open.basic _ hu) (generate_open.basic _ hv))
(le_inf
(coinduced_le_iff_le_induced.mp $ le_generate_from $ λ u hu,
have (⋃ v ∈ t, u ×ˢ v) = prod.fst ⁻¹' u,
by simp_rw [← prod_Union, ← sUnion_eq_bUnion, ht, prod_univ],
show G.is_open (prod.fst ⁻¹' u),
by { rw [← this], exactI is_open_Union (λ v, is_open_Union $ λ hv,
generate_open.basic _ ⟨_, hu, _, hv, rfl⟩) })
(coinduced_le_iff_le_induced.mp $ le_generate_from $ λ v hv,
have (⋃ u ∈ s, u ×ˢ v) = prod.snd ⁻¹' v,
by simp_rw [← Union_prod_const, ← sUnion_eq_bUnion, hs, univ_prod],
show G.is_open (prod.snd ⁻¹' v),
by { rw [← this], exactI is_open_Union (λ u, is_open_Union $ λ hu,
generate_open.basic _ ⟨_, hu, _, hv, rfl⟩) }))
lemma prod_eq_generate_from :
prod.topological_space =
generate_from {g | ∃(s:set α) (t:set β), is_open s ∧ is_open t ∧ g = s ×ˢ t} :=
le_antisymm
(le_generate_from $ λ g ⟨s, t, hs, ht, g_eq⟩, g_eq.symm ▸ hs.prod ht)
(le_inf
(ball_image_of_ball $ λt ht, generate_open.basic _ ⟨t, univ, by simpa [set.prod_eq] using ht⟩)
(ball_image_of_ball $ λt ht, generate_open.basic _ ⟨univ, t, by simpa [set.prod_eq] using ht⟩))
lemma is_open_prod_iff {s : set (α × β)} : is_open s ↔
(∀a b, (a, b) ∈ s →
∃ (u : set α) (v : set β), is_open u ∧ is_open v ∧ a ∈ u ∧ b ∈ v ∧ u ×ˢ v ⊆ s) :=
begin
rw [is_open_iff_nhds],
simp_rw [le_principal_iff, prod.forall,
((nhds_basis_opens _).prod_nhds (nhds_basis_opens _)).mem_iff, prod.exists, exists_prop],
simp only [and_assoc, and.left_comm]
end
/-- A product of induced topologies is induced by the product map -/
lemma prod_induced_induced {α γ : Type*} (f : α → β) (g : γ → δ) :
@prod.topological_space α γ (induced f ‹_›) (induced g ‹_›) =
induced (λ p, (f p.1, g p.2)) prod.topological_space :=
by simp_rw [prod.topological_space, induced_inf, induced_compose]
lemma continuous_uncurry_of_discrete_topology_left [discrete_topology α]
{f : α → β → γ} (h : ∀ a, continuous (f a)) : continuous (uncurry f) :=
continuous_iff_continuous_at.2 $ λ ⟨a, b⟩,
by simp only [continuous_at, nhds_prod_eq, nhds_discrete α, pure_prod, tendsto_map'_iff, (∘),
uncurry, (h a).tendsto]
/-- Given a neighborhood `s` of `(x, x)`, then `(x, x)` has a square open neighborhood
that is a subset of `s`. -/
lemma exists_nhds_square {s : set (α × α)} {x : α} (hx : s ∈ 𝓝 (x, x)) :
∃ U : set α, is_open U ∧ x ∈ U ∧ U ×ˢ U ⊆ s :=
by simpa [nhds_prod_eq, (nhds_basis_opens x).prod_self.mem_iff, and.assoc, and.left_comm] using hx
/-- `prod.fst` maps neighborhood of `x : α × β` within the section `prod.snd ⁻¹' {x.2}`
to `𝓝 x.1`. -/
lemma map_fst_nhds_within (x : α × β) : map prod.fst (𝓝[prod.snd ⁻¹' {x.2}] x) = 𝓝 x.1 :=
begin
refine le_antisymm (continuous_at_fst.mono_left inf_le_left) (λ s hs, _),
rcases x with ⟨x, y⟩,
rw [mem_map, nhds_within, mem_inf_principal, mem_nhds_prod_iff] at hs,
rcases hs with ⟨u, hu, v, hv, H⟩,
simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H,
exact mem_of_superset hu (λ z hz, H _ hz _ (mem_of_mem_nhds hv) rfl)
end
@[simp] lemma map_fst_nhds (x : α × β) : map prod.fst (𝓝 x) = 𝓝 x.1 :=
le_antisymm continuous_at_fst $ (map_fst_nhds_within x).symm.trans_le (map_mono inf_le_left)
/-- The first projection in a product of topological spaces sends open sets to open sets. -/
lemma is_open_map_fst : is_open_map (@prod.fst α β) :=
is_open_map_iff_nhds_le.2 $ λ x, (map_fst_nhds x).ge
/-- `prod.snd` maps neighborhood of `x : α × β` within the section `prod.fst ⁻¹' {x.1}`
to `𝓝 x.2`. -/
lemma map_snd_nhds_within (x : α × β) : map prod.snd (𝓝[prod.fst ⁻¹' {x.1}] x) = 𝓝 x.2 :=
begin
refine le_antisymm (continuous_at_snd.mono_left inf_le_left) (λ s hs, _),
rcases x with ⟨x, y⟩,
rw [mem_map, nhds_within, mem_inf_principal, mem_nhds_prod_iff] at hs,
rcases hs with ⟨u, hu, v, hv, H⟩,
simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H,
exact mem_of_superset hv (λ z hz, H _ (mem_of_mem_nhds hu) _ hz rfl)
end
@[simp] lemma map_snd_nhds (x : α × β) : map prod.snd (𝓝 x) = 𝓝 x.2 :=
le_antisymm continuous_at_snd $ (map_snd_nhds_within x).symm.trans_le (map_mono inf_le_left)
/-- The second projection in a product of topological spaces sends open sets to open sets. -/
lemma is_open_map_snd : is_open_map (@prod.snd α β) :=
is_open_map_iff_nhds_le.2 $ λ x, (map_snd_nhds x).ge
/-- A product set is open in a product space if and only if each factor is open, or one of them is
empty -/
lemma is_open_prod_iff' {s : set α} {t : set β} :
is_open (s ×ˢ t) ↔ (is_open s ∧ is_open t) ∨ (s = ∅) ∨ (t = ∅) :=
begin
cases (s ×ˢ t).eq_empty_or_nonempty with h h,
{ simp [h, prod_eq_empty_iff.1 h] },
{ have st : s.nonempty ∧ t.nonempty, from prod_nonempty_iff.1 h,
split,
{ assume H : is_open (s ×ˢ t),
refine or.inl ⟨_, _⟩,
show is_open s,
{ rw ← fst_image_prod s st.2,
exact is_open_map_fst _ H },
show is_open t,
{ rw ← snd_image_prod st.1 t,
exact is_open_map_snd _ H } },
{ assume H,
simp only [st.1.ne_empty, st.2.ne_empty, not_false_iff, or_false] at H,
exact H.1.prod H.2 } }
end
lemma closure_prod_eq {s : set α} {t : set β} :
closure (s ×ˢ t) = closure s ×ˢ closure t :=
set.ext $ assume ⟨a, b⟩,
have (𝓝 a ×ᶠ 𝓝 b) ⊓ 𝓟 (s ×ˢ t) = (𝓝 a ⊓ 𝓟 s) ×ᶠ (𝓝 b ⊓ 𝓟 t),
by rw [←prod_inf_prod, prod_principal_principal],
by simp [closure_eq_cluster_pts, cluster_pt, nhds_prod_eq, this]; exact prod_ne_bot
lemma interior_prod_eq (s : set α) (t : set β) :
interior (s ×ˢ t) = interior s ×ˢ interior t :=
set.ext $ λ ⟨a, b⟩, by simp only [mem_interior_iff_mem_nhds, mem_prod, prod_mem_nhds_iff]
lemma frontier_prod_eq (s : set α) (t : set β) :
frontier (s ×ˢ t) = closure s ×ˢ frontier t ∪ frontier s ×ˢ closure t :=
by simp only [frontier, closure_prod_eq, interior_prod_eq, prod_diff_prod]
@[simp] lemma frontier_prod_univ_eq (s : set α) :
frontier (s ×ˢ (univ : set β)) = frontier s ×ˢ univ :=
by simp [frontier_prod_eq]
@[simp] lemma frontier_univ_prod_eq (s : set β) :
frontier ((univ : set α) ×ˢ s) = univ ×ˢ frontier s :=
by simp [frontier_prod_eq]
lemma map_mem_closure₂ {f : α → β → γ} {a : α} {b : β} {s : set α} {t : set β} {u : set γ}
(hf : continuous (uncurry f)) (ha : a ∈ closure s) (hb : b ∈ closure t)
(h : ∀ (a ∈ s) (b ∈ t), f a b ∈ u) :
f a b ∈ closure u :=
have H₁ : (a, b) ∈ closure (s ×ˢ t), by simpa only [closure_prod_eq] using mk_mem_prod ha hb,
have H₂ : maps_to (uncurry f) (s ×ˢ t) u, from forall_prod_set.2 h,
H₂.closure hf H₁
lemma is_closed.prod {s₁ : set α} {s₂ : set β} (h₁ : is_closed s₁) (h₂ : is_closed s₂) :
is_closed (s₁ ×ˢ s₂) :=
closure_eq_iff_is_closed.mp $ by simp only [h₁.closure_eq, h₂.closure_eq, closure_prod_eq]
/-- The product of two dense sets is a dense set. -/
lemma dense.prod {s : set α} {t : set β} (hs : dense s) (ht : dense t) :
dense (s ×ˢ t) :=
λ x, by { rw closure_prod_eq, exact ⟨hs x.1, ht x.2⟩ }
/-- If `f` and `g` are maps with dense range, then `prod.map f g` has dense range. -/
lemma dense_range.prod_map {ι : Type*} {κ : Type*} {f : ι → β} {g : κ → γ}
(hf : dense_range f) (hg : dense_range g) : dense_range (prod.map f g) :=
by simpa only [dense_range, prod_range_range_eq] using hf.prod hg
lemma inducing.prod_mk {f : α → β} {g : γ → δ} (hf : inducing f) (hg : inducing g) :
inducing (λx:α×γ, (f x.1, g x.2)) :=
⟨by rw [prod.topological_space, prod.topological_space, hf.induced, hg.induced,
induced_compose, induced_compose, induced_inf, induced_compose, induced_compose]⟩
lemma embedding.prod_mk {f : α → β} {g : γ → δ} (hf : embedding f) (hg : embedding g) :
embedding (λx:α×γ, (f x.1, g x.2)) :=
{ inj := assume ⟨x₁, x₂⟩ ⟨y₁, y₂⟩, by simp; exact assume h₁ h₂, ⟨hf.inj h₁, hg.inj h₂⟩,
..hf.to_inducing.prod_mk hg.to_inducing }
protected lemma is_open_map.prod {f : α → β} {g : γ → δ} (hf : is_open_map f) (hg : is_open_map g) :
is_open_map (λ p : α × γ, (f p.1, g p.2)) :=
begin
rw [is_open_map_iff_nhds_le],
rintros ⟨a, b⟩,
rw [nhds_prod_eq, nhds_prod_eq, ← filter.prod_map_map_eq],
exact filter.prod_mono (hf.nhds_le a) (hg.nhds_le b)
end
protected lemma open_embedding.prod {f : α → β} {g : γ → δ}
(hf : open_embedding f) (hg : open_embedding g) : open_embedding (λ x : α × γ, (f x.1, g x.2)) :=
open_embedding_of_embedding_open (hf.1.prod_mk hg.1)
(hf.is_open_map.prod hg.is_open_map)
lemma embedding_graph {f : α → β} (hf : continuous f) : embedding (λ x, (x, f x)) :=
embedding_of_embedding_compose (continuous_id.prod_mk hf) continuous_fst embedding_id
end prod
section sum
open sum
variables [topological_space α] [topological_space β] [topological_space γ] [topological_space δ]
@[continuity] lemma continuous_inl : continuous (@inl α β) :=
continuous_sup_rng_left continuous_coinduced_rng
@[continuity] lemma continuous_inr : continuous (@inr α β) :=
continuous_sup_rng_right continuous_coinduced_rng
lemma is_open_sum_iff {s : set (α ⊕ β)} :
is_open s ↔ is_open (inl ⁻¹' s) ∧ is_open (inr ⁻¹' s) :=
iff.rfl
lemma is_open_map_inl : is_open_map (@inl α β) :=
λ u hu, by simpa [is_open_sum_iff, preimage_image_eq u sum.inl_injective]
lemma is_open_map_inr : is_open_map (@inr α β) :=
λ u hu, by simpa [is_open_sum_iff, preimage_image_eq u sum.inr_injective]
lemma open_embedding_inl : open_embedding (@inl α β) :=
open_embedding_of_continuous_injective_open continuous_inl inl_injective is_open_map_inl
lemma open_embedding_inr : open_embedding (@inr α β) :=
open_embedding_of_continuous_injective_open continuous_inr inr_injective is_open_map_inr
lemma embedding_inl : embedding (@inl α β) := open_embedding_inl.1
lemma embedding_inr : embedding (@inr α β) := open_embedding_inr.1
lemma is_open_range_inl : is_open (range (inl : α → α ⊕ β)) := open_embedding_inl.2
lemma is_open_range_inr : is_open (range (inr : β → α ⊕ β)) := open_embedding_inr.2
lemma is_closed_range_inl : is_closed (range (inl : α → α ⊕ β)) :=
by { rw [← is_open_compl_iff, compl_range_inl], exact is_open_range_inr }
lemma is_closed_range_inr : is_closed (range (inr : β → α ⊕ β)) :=
by { rw [← is_open_compl_iff, compl_range_inr], exact is_open_range_inl }
lemma closed_embedding_inl : closed_embedding (inl : α → α ⊕ β) :=
⟨embedding_inl, is_closed_range_inl⟩
lemma closed_embedding_inr : closed_embedding (inr : β → α ⊕ β) :=
⟨embedding_inr, is_closed_range_inr⟩
lemma nhds_inl (x : α) : 𝓝 (inl x : α ⊕ β) = map inl (𝓝 x) :=
(open_embedding_inl.map_nhds_eq _).symm
lemma nhds_inr (x : β) : 𝓝 (inr x : α ⊕ β) = map inr (𝓝 x) :=
(open_embedding_inr.map_nhds_eq _).symm
lemma continuous_sum_elim {f : α → γ} {g : β → γ} :
continuous (sum.elim f g) ↔ continuous f ∧ continuous g :=
by simp only [continuous_sup_dom, continuous_coinduced_dom, sum.elim_comp_inl, sum.elim_comp_inr]
@[continuity] lemma continuous.sum_elim {f : α → γ} {g : β → γ}
(hf : continuous f) (hg : continuous g) : continuous (sum.elim f g) :=
continuous_sum_elim.2 ⟨hf, hg⟩
@[simp] lemma continuous_sum_map {f : α → β} {g : γ → δ} :
continuous (sum.map f g) ↔ continuous f ∧ continuous g :=
continuous_sum_elim.trans $ embedding_inl.continuous_iff.symm.and embedding_inr.continuous_iff.symm
@[continuity] lemma continuous.sum_map {f : α → β} {g : γ → δ} (hf : continuous f)
(hg : continuous g) : continuous (sum.map f g) :=
continuous_sum_map.2 ⟨hf, hg⟩
lemma is_open_map_sum {f : α ⊕ β → γ} :
is_open_map f ↔ is_open_map (λ a, f (inl a)) ∧ is_open_map (λ b, f (inr b)) :=
by simp only [is_open_map_iff_nhds_le, sum.forall, nhds_inl, nhds_inr, filter.map_map]
@[simp] lemma is_open_map_sum_elim {f : α → γ} {g : β → γ} :
is_open_map (sum.elim f g) ↔ is_open_map f ∧ is_open_map g :=
by simp only [is_open_map_sum, elim_inl, elim_inr]
lemma is_open_map.sum_elim {f : α → γ} {g : β → γ} (hf : is_open_map f) (hg : is_open_map g) :
is_open_map (sum.elim f g) :=
is_open_map_sum_elim.2 ⟨hf, hg⟩
end sum
section subtype
variables [topological_space α] [topological_space β] [topological_space γ] {p : α → Prop}
lemma inducing_coe {b : set β} : inducing (coe : b → β) := ⟨rfl⟩
lemma inducing.of_cod_restrict {f : α → β} {b : set β} (hb : ∀ a, f a ∈ b)
(h : inducing (b.cod_restrict f hb)) : inducing f := inducing_coe.comp h
lemma embedding_subtype_coe : embedding (coe : subtype p → α) :=
⟨⟨rfl⟩, subtype.coe_injective⟩
lemma closed_embedding_subtype_coe (h : is_closed {a | p a}) :
closed_embedding (coe : subtype p → α) :=
⟨embedding_subtype_coe, by rwa [subtype.range_coe_subtype]⟩
@[continuity] lemma continuous_subtype_val : continuous (@subtype.val α p) :=
continuous_induced_dom
lemma continuous_subtype_coe : continuous (coe : subtype p → α) :=
continuous_subtype_val
lemma continuous.subtype_coe {f : β → subtype p} (hf : continuous f) :
continuous (λ x, (f x : α)) :=
continuous_subtype_coe.comp hf
lemma is_open.open_embedding_subtype_coe {s : set α} (hs : is_open s) :
open_embedding (coe : s → α) :=
{ induced := rfl,
inj := subtype.coe_injective,
open_range := (subtype.range_coe : range coe = s).symm ▸ hs }
lemma is_open.is_open_map_subtype_coe {s : set α} (hs : is_open s) :
is_open_map (coe : s → α) :=
hs.open_embedding_subtype_coe.is_open_map
lemma is_open_map.restrict {f : α → β} (hf : is_open_map f) {s : set α} (hs : is_open s) :
is_open_map (s.restrict f) :=
hf.comp hs.is_open_map_subtype_coe
lemma is_closed.closed_embedding_subtype_coe {s : set α} (hs : is_closed s) :
closed_embedding (coe : {x // x ∈ s} → α) :=
{ induced := rfl,
inj := subtype.coe_injective,
closed_range := (subtype.range_coe : range coe = s).symm ▸ hs }
@[continuity] lemma continuous.subtype_mk {f : β → α} (h : continuous f)
(hp : ∀x, p (f x)) : continuous (λx, (⟨f x, hp x⟩ : subtype p)) :=
continuous_induced_rng.2 h
lemma continuous.subtype_map {f : α → β} (h : continuous f) {q : β → Prop}
(hpq : ∀ x, p x → q (f x)) : continuous (subtype.map f hpq) :=
(h.comp continuous_subtype_coe).subtype_mk _
lemma continuous_inclusion {s t : set α} (h : s ⊆ t) : continuous (inclusion h) :=
continuous_id.subtype_map h
lemma continuous_at_subtype_coe {p : α → Prop} {a : subtype p} :
continuous_at (coe : subtype p → α) a :=
continuous_iff_continuous_at.mp continuous_subtype_coe _
lemma subtype.dense_iff {s : set α} {t : set s} : dense t ↔ s ⊆ closure (coe '' t) :=
by { rw [inducing_coe.dense_iff, set_coe.forall], refl }
lemma map_nhds_subtype_coe_eq {a : α} (ha : p a) (h : {a | p a} ∈ 𝓝 a) :
map (coe : subtype p → α) (𝓝 ⟨a, ha⟩) = 𝓝 a :=
map_nhds_induced_of_mem $ by simpa only [subtype.coe_mk, subtype.range_coe] using h
lemma nhds_subtype_eq_comap {a : α} {h : p a} :
𝓝 (⟨a, h⟩ : subtype p) = comap coe (𝓝 a) :=
nhds_induced _ _
lemma tendsto_subtype_rng {β : Type*} {p : α → Prop} {b : filter β} {f : β → subtype p} :
∀{a:subtype p}, tendsto f b (𝓝 a) ↔ tendsto (λx, (f x : α)) b (𝓝 (a : α))
| ⟨a, ha⟩ := by rw [nhds_subtype_eq_comap, tendsto_comap_iff, subtype.coe_mk]
lemma continuous_subtype_nhds_cover {ι : Sort*} {f : α → β} {c : ι → α → Prop}
(c_cover : ∀x:α, ∃i, {x | c i x} ∈ 𝓝 x)
(f_cont : ∀i, continuous (λ(x : subtype (c i)), f x)) :
continuous f :=
continuous_iff_continuous_at.mpr $ assume x,
let ⟨i, (c_sets : {x | c i x} ∈ 𝓝 x)⟩ := c_cover x in
let x' : subtype (c i) := ⟨x, mem_of_mem_nhds c_sets⟩ in
calc map f (𝓝 x) = map f (map coe (𝓝 x')) :
congr_arg (map f) (map_nhds_subtype_coe_eq _ $ c_sets).symm
... = map (λx:subtype (c i), f x) (𝓝 x') : rfl
... ≤ 𝓝 (f x) : continuous_iff_continuous_at.mp (f_cont i) x'
lemma continuous_subtype_is_closed_cover {ι : Sort*} {f : α → β} (c : ι → α → Prop)
(h_lf : locally_finite (λi, {x | c i x}))
(h_is_closed : ∀i, is_closed {x | c i x})
(h_cover : ∀x, ∃i, c i x)
(f_cont : ∀i, continuous (λ(x : subtype (c i)), f x)) :
continuous f :=
continuous_iff_is_closed.mpr $
assume s hs,
have ∀i, is_closed ((coe : {x | c i x} → α) '' (f ∘ coe ⁻¹' s)),
from assume i,
(closed_embedding_subtype_coe (h_is_closed _)).is_closed_map _ (hs.preimage (f_cont i)),
have is_closed (⋃i, (coe : {x | c i x} → α) '' (f ∘ coe ⁻¹' s)),
from locally_finite.is_closed_Union
(h_lf.subset $ assume i x ⟨⟨x', hx'⟩, _, heq⟩, heq ▸ hx')
this,
have f ⁻¹' s = (⋃i, (coe : {x | c i x} → α) '' (f ∘ coe ⁻¹' s)),
begin
apply set.ext,
have : ∀ (x : α), f x ∈ s ↔ ∃ (i : ι), c i x ∧ f x ∈ s :=
λ x, ⟨λ hx, let ⟨i, hi⟩ := h_cover x in ⟨i, hi, hx⟩,
λ ⟨i, hi, hx⟩, hx⟩,
simpa [and.comm, @and.left_comm (c _ _), ← exists_and_distrib_right],
end,
by rwa [this]
lemma closure_subtype {x : {a // p a}} {s : set {a // p a}}:
x ∈ closure s ↔ (x : α) ∈ closure ((coe : _ → α) '' s) :=
closure_induced
lemma continuous_at_cod_restrict_iff {f : α → β} {t : set β} (h1 : ∀ x, f x ∈ t) {x : α} :
continuous_at (cod_restrict f t h1) x ↔ continuous_at f x :=
by simp_rw [inducing_coe.continuous_at_iff, function.comp, coe_cod_restrict_apply]
alias continuous_at_cod_restrict_iff ↔ _ continuous_at.cod_restrict
lemma continuous_at.restrict {f : α → β} {s : set α} {t : set β} (h1 : maps_to f s t) {x : s}
(h2 : continuous_at f x) : continuous_at (h1.restrict f s t) x :=
(h2.comp continuous_at_subtype_coe).cod_restrict _
lemma continuous_at.restrict_preimage {f : α → β} {s : set β} {x : f ⁻¹' s}
(h : continuous_at f x) : continuous_at (s.restrict_preimage f) x :=
h.restrict _
@[continuity] lemma continuous.cod_restrict {f : α → β} {s : set β} (hf : continuous f)
(hs : ∀ a, f a ∈ s) : continuous (s.cod_restrict f hs) := hf.subtype_mk hs
lemma inducing.cod_restrict {e : α → β} (he : inducing e) {s : set β} (hs : ∀ x, e x ∈ s) :
inducing (cod_restrict e s hs) :=
inducing_of_inducing_compose (he.continuous.cod_restrict hs) continuous_subtype_coe he
lemma embedding.cod_restrict {e : α → β} (he : embedding e) (s : set β) (hs : ∀ x, e x ∈ s) :
embedding (cod_restrict e s hs) :=
embedding_of_embedding_compose (he.continuous.cod_restrict hs) continuous_subtype_coe he
end subtype
section quotient
variables [topological_space α] [topological_space β] [topological_space γ]
variables {r : α → α → Prop} {s : setoid α}
lemma quotient_map_quot_mk : quotient_map (@quot.mk α r) :=
⟨quot.exists_rep, rfl⟩
@[continuity] lemma continuous_quot_mk : continuous (@quot.mk α r) :=
continuous_coinduced_rng
@[continuity] lemma continuous_quot_lift {f : α → β} (hr : ∀ a b, r a b → f a = f b)
(h : continuous f) : continuous (quot.lift f hr : quot r → β) :=
continuous_coinduced_dom.2 h
lemma quotient_map_quotient_mk : quotient_map (@quotient.mk α s) :=
quotient_map_quot_mk
lemma continuous_quotient_mk : continuous (@quotient.mk α s) :=
continuous_coinduced_rng
lemma continuous.quotient_lift {f : α → β} (h : continuous f) (hs : ∀ a b, a ≈ b → f a = f b) :
continuous (quotient.lift f hs : quotient s → β) :=
continuous_coinduced_dom.2 h
lemma continuous.quotient_lift_on' {f : α → β} (h : continuous f)
(hs : ∀ a b, @setoid.r _ s a b → f a = f b) :
continuous (λ x, quotient.lift_on' x f hs : quotient s → β) :=
h.quotient_lift hs
lemma continuous.quotient_map' {t : setoid β} {f : α → β} (hf : continuous f)
(H : (s.r ⇒ t.r) f f) : continuous (quotient.map' f H) :=
(continuous_quotient_mk.comp hf).quotient_lift _
end quotient
section pi
variables {ι : Type*} {π : ι → Type*} {κ : Type*}
[topological_space α] [∀i, topological_space (π i)] {f : α → Πi:ι, π i}
lemma continuous_pi_iff : continuous f ↔ ∀ i, continuous (λ a, f a i) :=
by simp only [continuous_infi_rng, continuous_induced_rng]
@[continuity] lemma continuous_pi (h : ∀ i, continuous (λ a, f a i)) : continuous f :=
continuous_pi_iff.2 h
@[continuity] lemma continuous_apply (i : ι) : continuous (λp:Πi, π i, p i) :=
continuous_infi_dom continuous_induced_dom
@[continuity]
lemma continuous_apply_apply {ρ : κ → ι → Type*} [∀ j i, topological_space (ρ j i)]
(j : κ) (i : ι) : continuous (λ p : (Π j, Π i, ρ j i), p j i) :=
(continuous_apply i).comp (continuous_apply j)
lemma continuous_at_apply (i : ι) (x : Π i, π i) : continuous_at (λ p : Π i, π i, p i) x :=
(continuous_apply i).continuous_at
lemma filter.tendsto.apply {l : filter β} {f : β → Π i, π i}
{x : Π i, π i} (h : tendsto f l (𝓝 x)) (i : ι) :
tendsto (λ a, f a i) l (𝓝 $ x i) :=
(continuous_at_apply i _).tendsto.comp h
lemma nhds_pi {a : Πi, π i} : 𝓝 a = pi (λ i, 𝓝 (a i)) :=
by simp only [nhds_infi, nhds_induced, filter.pi]
lemma tendsto_pi_nhds {f : β → Πi, π i} {g : Πi, π i} {u : filter β} :
tendsto f u (𝓝 g) ↔ ∀ x, tendsto (λ i, f i x) u (𝓝 (g x)) :=
by rw [nhds_pi, filter.tendsto_pi]
lemma continuous_at_pi {f : α → Π i, π i} {x : α} :
continuous_at f x ↔ ∀ i, continuous_at (λ y, f y i) x :=
tendsto_pi_nhds
lemma filter.tendsto.update [decidable_eq ι]
{l : filter β} {f : β → Π i, π i} {x : Π i, π i} (hf : tendsto f l (𝓝 x)) (i : ι)
{g : β → π i} {xi : π i} (hg : tendsto g l (𝓝 xi)) :
tendsto (λ a, update (f a) i (g a)) l (𝓝 $ update x i xi) :=
tendsto_pi_nhds.2 $ λ j, by { rcases em (j = i) with rfl|hj; simp [*, hf.apply] }
lemma continuous_at.update [decidable_eq ι] {a : α} (hf : continuous_at f a) (i : ι) {g : α → π i}
(hg : continuous_at g a) :
continuous_at (λ a, update (f a) i (g a)) a :=
hf.update i hg
lemma continuous.update [decidable_eq ι] (hf : continuous f) (i : ι) {g : α → π i}
(hg : continuous g) :
continuous (λ a, update (f a) i (g a)) :=
continuous_iff_continuous_at.2 $ λ x, hf.continuous_at.update i hg.continuous_at
/-- `update f i x` is continuous in `(f, x)`. -/
@[continuity] lemma continuous_update [decidable_eq ι] (i : ι) :
continuous (λ f : (Π j, π j) × π i, update f.1 i f.2) :=
continuous_fst.update i continuous_snd
lemma filter.tendsto.fin_insert_nth {n} {π : fin (n + 1) → Type*} [Π i, topological_space (π i)]
(i : fin (n + 1)) {f : β → π i} {l : filter β} {x : π i} (hf : tendsto f l (𝓝 x))
{g : β → Π j : fin n, π (i.succ_above j)} {y : Π j, π (i.succ_above j)} (hg : tendsto g l (𝓝 y)) :
tendsto (λ a, i.insert_nth (f a) (g a)) l (𝓝 $ i.insert_nth x y) :=
tendsto_pi_nhds.2 (λ j, fin.succ_above_cases i (by simpa) (by simpa using tendsto_pi_nhds.1 hg) j)
lemma continuous_at.fin_insert_nth {n} {π : fin (n + 1) → Type*} [Π i, topological_space (π i)]
(i : fin (n + 1)) {f : α → π i} {a : α} (hf : continuous_at f a)
{g : α → Π j : fin n, π (i.succ_above j)} (hg : continuous_at g a) :
continuous_at (λ a, i.insert_nth (f a) (g a)) a :=
hf.fin_insert_nth i hg
lemma continuous.fin_insert_nth {n} {π : fin (n + 1) → Type*} [Π i, topological_space (π i)]
(i : fin (n + 1)) {f : α → π i} (hf : continuous f)
{g : α → Π j : fin n, π (i.succ_above j)} (hg : continuous g) :
continuous (λ a, i.insert_nth (f a) (g a)) :=
continuous_iff_continuous_at.2 $ λ a, hf.continuous_at.fin_insert_nth i hg.continuous_at
lemma is_open_set_pi {i : set ι} {s : Πa, set (π a)} (hi : i.finite) (hs : ∀a∈i, is_open (s a)) :
is_open (pi i s) :=
by rw [pi_def]; exact (is_open_bInter hi $ assume a ha, (hs _ ha).preimage (continuous_apply _))
lemma is_closed_set_pi {i : set ι} {s : Πa, set (π a)} (hs : ∀a∈i, is_closed (s a)) :
is_closed (pi i s) :=
by rw [pi_def];
exact (is_closed_Inter $ λ a, is_closed_Inter $ λ ha, (hs _ ha).preimage (continuous_apply _))
lemma mem_nhds_of_pi_mem_nhds {I : set ι} {s : Π i, set (π i)} (a : Π i, π i) (hs : I.pi s ∈ 𝓝 a)
{i : ι} (hi : i ∈ I) :
s i ∈ 𝓝 (a i) :=
by { rw nhds_pi at hs, exact mem_of_pi_mem_pi hs hi }
lemma set_pi_mem_nhds {i : set ι} {s : Π a, set (π a)}
{x : Π a, π a} (hi : i.finite) (hs : ∀ a ∈ i, s a ∈ 𝓝 (x a)) :
pi i s ∈ 𝓝 x :=
by { rw [pi_def, bInter_mem hi], exact λ a ha, (continuous_apply a).continuous_at (hs a ha) }
lemma set_pi_mem_nhds_iff {I : set ι} (hI : I.finite) {s : Π i, set (π i)} (a : Π i, π i) :
I.pi s ∈ 𝓝 a ↔ ∀ (i : ι), i ∈ I → s i ∈ 𝓝 (a i) :=
by { rw [nhds_pi, pi_mem_pi_iff hI], apply_instance }
lemma interior_pi_set {I : set ι} (hI : I.finite) {s : Π i, set (π i)} :
interior (pi I s) = I.pi (λ i, interior (s i)) :=
by { ext a, simp only [set.mem_pi, mem_interior_iff_mem_nhds, set_pi_mem_nhds_iff hI] }
lemma exists_finset_piecewise_mem_of_mem_nhds [decidable_eq ι]
{s : set (Π a, π a)} {x : Π a, π a} (hs : s ∈ 𝓝 x) (y : Π a, π a) :
∃ I : finset ι, I.piecewise x y ∈ s :=
begin
simp only [nhds_pi, filter.mem_pi'] at hs,
rcases hs with ⟨I, t, htx, hts⟩,
refine ⟨I, hts $ λ i hi, _⟩,
simpa [finset.mem_coe.1 hi] using mem_of_mem_nhds (htx i)
end
lemma pi_eq_generate_from :
Pi.topological_space =
generate_from {g | ∃(s:Πa, set (π a)) (i : finset ι), (∀a∈i, is_open (s a)) ∧ g = pi ↑i s} :=
le_antisymm
(le_generate_from $ assume g ⟨s, i, hi, eq⟩, eq.symm ▸ is_open_set_pi (finset.finite_to_set _) hi)
(le_infi $ assume a s ⟨t, ht, s_eq⟩, generate_open.basic _ $
⟨update (λa, univ) a t, {a}, by simpa using ht, s_eq ▸ by ext f; simp [set.pi]⟩)
lemma pi_generate_from_eq {π : ι → Type*} {g : Πa, set (set (π a))} :
@Pi.topological_space ι π (λa, generate_from (g a)) =
generate_from {t | ∃(s:Πa, set (π a)) (i : finset ι), (∀a∈i, s a ∈ g a) ∧ t = pi ↑i s} :=
let G := {t | ∃(s:Πa, set (π a)) (i : finset ι), (∀a∈i, s a ∈ g a) ∧ t = pi ↑i s} in
begin
rw [pi_eq_generate_from],
refine le_antisymm (generate_from_mono _) (le_generate_from _),
exact assume s ⟨t, i, ht, eq⟩, ⟨t, i, assume a ha, generate_open.basic _ (ht a ha), eq⟩,
{ rintros s ⟨t, i, hi, rfl⟩,
rw [pi_def],
apply is_open_bInter (finset.finite_to_set _),
assume a ha, show ((generate_from G).coinduced (λf:Πa, π a, f a)).is_open (t a),
refine le_generate_from _ _ (hi a ha),
exact assume s hs, generate_open.basic _ ⟨update (λa, univ) a s, {a}, by simp [hs]⟩ }
end
lemma pi_generate_from_eq_finite {π : ι → Type*} {g : Πa, set (set (π a))} [finite ι]
(hg : ∀a, ⋃₀ g a = univ) :
@Pi.topological_space ι π (λa, generate_from (g a)) =
generate_from {t | ∃(s:Πa, set (π a)), (∀a, s a ∈ g a) ∧ t = pi univ s} :=
begin
casesI nonempty_fintype ι,
rw [pi_generate_from_eq],
refine le_antisymm (generate_from_mono _) (le_generate_from _),
exact assume s ⟨t, ht, eq⟩, ⟨t, finset.univ, by simp [ht, eq]⟩,
{ rintros s ⟨t, i, ht, rfl⟩,
apply is_open_iff_forall_mem_open.2 _,
assume f hf,
choose c hc using show ∀a, ∃s, s ∈ g a ∧ f a ∈ s,
{ assume a, have : f a ∈ ⋃₀ g a, { rw [hg], apply mem_univ }, simpa },
refine ⟨pi univ (λa, if a ∈ i then t a else (c : Πa, set (π a)) a), _, _, _⟩,
{ simp [pi_if] },
{ refine generate_open.basic _ ⟨_, assume a, _, rfl⟩,
by_cases a ∈ i; simp [*, set.pi] at * },
{ have : f ∈ pi {a | a ∉ i} c, { simp [*, set.pi] at * },
simpa [pi_if, hf] } }
end
/-- Suppose `π i` is a family of topological spaces indexed by `i : ι`, and `X` is a type
endowed with a family of maps `f i : X → π i` for every `i : ι`, hence inducing a
map `g : X → Π i, π i`. This lemma shows that infimum of the topologies on `X` induced by
the `f i` as `i : ι` varies is simply the topology on `X` induced by `g : X → Π i, π i`
where `Π i, π i` is endowed with the usual product topology. -/
lemma inducing_infi_to_pi {X : Type*} (f : Π i, X → π i) :
@inducing X (Π i, π i) (⨅ i, induced (f i) infer_instance) _ (λ x i, f i x) :=
begin
constructor,
erw induced_infi,
congr' 1,
funext,
erw induced_compose,
end
variables [finite ι] [∀ i, discrete_topology (π i)]
/-- A finite product of discrete spaces is discrete. -/
instance Pi.discrete_topology : discrete_topology (Π i, π i) :=
singletons_open_iff_discrete.mp (λ x,
begin
rw show {x} = ⋂ i, {y : Π i, π i | y i = x i},
{ ext, simp only [funext_iff, set.mem_singleton_iff, set.mem_Inter, set.mem_set_of_eq] },
exact is_open_Inter (λ i, (continuous_apply i).is_open_preimage {x i} (is_open_discrete {x i}))
end)
end pi
section sigma
variables {ι κ : Type*} {σ : ι → Type*} {τ : κ → Type*}
[Π i, topological_space (σ i)] [Π k, topological_space (τ k)] [topological_space α]
@[continuity]
lemma continuous_sigma_mk {i : ι} : continuous (@sigma.mk ι σ i) :=
continuous_supr_rng continuous_coinduced_rng
lemma is_open_sigma_iff {s : set (sigma σ)} : is_open s ↔ ∀ i, is_open (sigma.mk i ⁻¹' s) :=
by simp only [is_open_supr_iff, is_open_coinduced]
lemma is_closed_sigma_iff {s : set (sigma σ)} : is_closed s ↔ ∀ i, is_closed (sigma.mk i ⁻¹' s) :=
by simp only [← is_open_compl_iff, is_open_sigma_iff, preimage_compl]
lemma is_open_map_sigma_mk {i : ι} : is_open_map (@sigma.mk ι σ i) :=
begin
intros s hs,
rw is_open_sigma_iff,
intro j,
rcases eq_or_ne j i with (rfl|hne),
{ rwa set.preimage_image_eq _ sigma_mk_injective },
{ rw [preimage_image_sigma_mk_of_ne hne],
exact is_open_empty }
end
lemma is_open_range_sigma_mk {i : ι} : is_open (set.range (@sigma.mk ι σ i)) :=
is_open_map_sigma_mk.is_open_range
lemma is_closed_map_sigma_mk {i : ι} : is_closed_map (@sigma.mk ι σ i) :=
begin
intros s hs,
rw is_closed_sigma_iff,
intro j,
rcases eq_or_ne j i with (rfl|hne),
{ rwa set.preimage_image_eq _ sigma_mk_injective },
{ rw [preimage_image_sigma_mk_of_ne hne],
exact is_closed_empty }
end
lemma is_closed_range_sigma_mk {i : ι} : is_closed (set.range (@sigma.mk ι σ i)) :=
is_closed_map_sigma_mk.closed_range
lemma open_embedding_sigma_mk {i : ι} : open_embedding (@sigma.mk ι σ i) :=
open_embedding_of_continuous_injective_open
continuous_sigma_mk sigma_mk_injective is_open_map_sigma_mk
lemma closed_embedding_sigma_mk {i : ι} : closed_embedding (@sigma.mk ι σ i) :=
closed_embedding_of_continuous_injective_closed
continuous_sigma_mk sigma_mk_injective is_closed_map_sigma_mk
lemma embedding_sigma_mk {i : ι} : embedding (@sigma.mk ι σ i) :=
closed_embedding_sigma_mk.1
lemma sigma.nhds_mk (i : ι) (x : σ i) : 𝓝 (⟨i, x⟩ : sigma σ) = map (sigma.mk i) (𝓝 x) :=
(open_embedding_sigma_mk.map_nhds_eq x).symm
lemma sigma.nhds_eq (x : sigma σ) : 𝓝 x = map (sigma.mk x.1) (𝓝 x.2) :=
by { cases x, apply sigma.nhds_mk }
lemma comap_sigma_mk_nhds (i : ι) (x : σ i) : comap (sigma.mk i) (𝓝 ⟨i, x⟩) = 𝓝 x :=
(embedding_sigma_mk.to_inducing.nhds_eq_comap _).symm
lemma is_open_sigma_fst_preimage (s : set ι) : is_open (sigma.fst ⁻¹' s : set (Σ a, σ a)) :=
begin
rw [← bUnion_of_singleton s, preimage_Union₂],
simp only [← range_sigma_mk],
exact is_open_bUnion (λ _ _, is_open_range_sigma_mk)
end
/-- A map out of a sum type is continuous iff its restriction to each summand is. -/
@[simp] lemma continuous_sigma_iff {f : sigma σ → α} :
continuous f ↔ ∀ i, continuous (λ a, f ⟨i, a⟩) :=
by simp only [continuous_supr_dom, continuous_coinduced_dom]
/-- A map out of a sum type is continuous if its restriction to each summand is. -/
@[continuity] lemma continuous_sigma {f : sigma σ → α} (hf : ∀ i, continuous (λ a, f ⟨i, a⟩)) :
continuous f :=
continuous_sigma_iff.2 hf
@[simp] lemma continuous_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} :
continuous (sigma.map f₁ f₂) ↔ ∀ i, continuous (f₂ i) :=
continuous_sigma_iff.trans $ by simp only [sigma.map, embedding_sigma_mk.continuous_iff]
@[continuity] lemma continuous.sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)}
(hf : ∀ i, continuous (f₂ i)) :
continuous (sigma.map f₁ f₂) :=
continuous_sigma_map.2 hf
lemma is_open_map_sigma {f : sigma σ → α} : is_open_map f ↔ ∀ i, is_open_map (λ a, f ⟨i, a⟩) :=
by simp only [is_open_map_iff_nhds_le, sigma.forall, sigma.nhds_eq, map_map]
lemma is_open_map_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} :
is_open_map (sigma.map f₁ f₂) ↔ ∀ i, is_open_map (f₂ i) :=
is_open_map_sigma.trans $ forall_congr $
λ i, (@open_embedding_sigma_mk _ _ _ (f₁ i)).is_open_map_iff.symm
lemma inducing_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} (h₁ : injective f₁) :
inducing (sigma.map f₁ f₂) ↔ ∀ i, inducing (f₂ i) :=
by simp only [inducing_iff_nhds, sigma.forall, sigma.nhds_mk, sigma.map, ← map_sigma_mk_comap h₁,
map_inj sigma_mk_injective]
lemma embedding_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} (h : injective f₁) :
embedding (sigma.map f₁ f₂) ↔ ∀ i, embedding (f₂ i) :=
by simp only [embedding_iff, injective.sigma_map, inducing_sigma_map h, forall_and_distrib,
h.sigma_map_iff]
lemma open_embedding_sigma_map {f₁ : ι → κ} {f₂ : Π i, σ i → τ (f₁ i)} (h : injective f₁) :
open_embedding (sigma.map f₁ f₂) ↔ ∀ i, open_embedding (f₂ i) :=
by simp only [open_embedding_iff_embedding_open, is_open_map_sigma_map, embedding_sigma_map h,
forall_and_distrib]
end sigma
section ulift
@[continuity] lemma continuous_ulift_down [topological_space α] :
continuous (ulift.down : ulift.{v u} α → α) :=
continuous_induced_dom
@[continuity] lemma continuous_ulift_up [topological_space α] :
continuous (ulift.up : α → ulift.{v u} α) :=
continuous_induced_rng.2 continuous_id
end ulift
|
47b4834c210a80aab95e76b18abc49dd3e66fdc1 | 05f637fa14ac28031cb1ea92086a0f4eb23ff2b1 | /examples/lean/abelian.lean | cf3d35dc43c2529b2dcd2056cdb41d87e66da189 | [
"Apache-2.0"
] | permissive | codyroux/lean0.1 | 1ce92751d664aacff0529e139083304a7bbc8a71 | 0dc6fb974aa85ed6f305a2f4b10a53a44ee5f0ef | refs/heads/master | 1,610,830,535,062 | 1,402,150,480,000 | 1,402,150,480,000 | 19,588,851 | 2 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 7,424 | lean | import macros
definition associative {A : (Type U)} (f : A → A → A) := ∀ x y z, f (f x y) z = f x (f y z)
definition is_identity {A : (Type U)} (f : A → A → A) (id : A) := ∀ x, f x id = x
definition inverse_ex {A : (Type U)} (f : A → A → A) (id : A) := ∀ x, ∃ y, f x y = id
universe s ≥ 1
definition group := sig A : (Type s), sig mul : A → A → A, sig one : A, (associative mul) # (is_identity mul one) # (inverse_ex mul one)
definition to_group (A : (Type s)) (mul : A → A → A) (one : A) (H1 : associative mul) (H2 : is_identity mul one) (H3 : inverse_ex mul one) : group
:= pair A (pair mul (pair one (pair H1 (pair H2 H3))))
-- The following definitions can be generated automatically.
definition carrier (g : group) := proj1 g
definition G_mul {g : group} : carrier g → carrier g → carrier g
:= proj1 (proj2 g)
infixl 70 * : G_mul
definition one {g : group} : carrier g
:= proj1 (proj2 (proj2 g))
theorem G_assoc {g : group} (x y z : carrier g) : (x * y) * z = x * (y * z)
:= (proj1 (proj2 (proj2 (proj2 g)))) x y z
theorem G_id {g : group} (x : carrier g) : x * one = x
:= (proj1 (proj2 (proj2 (proj2 (proj2 g))))) x
theorem G_inv {g : group} (x : carrier g) : ∃ y, x * y = one
:= (proj2 (proj2 (proj2 (proj2 (proj2 g))))) x
set_opaque group true
set_opaque carrier true
set_opaque G_mul true
set_opaque one true
-- First example: the pairwise product of two groups is a group
definition product (g1 g2 : group) : group
:= let S := carrier g1 # carrier g2,
-- It would be nice to be able to define local notation, and write _*_ instead of f
f := λ x y, pair (proj1 x * proj1 y) (proj2 x * proj2 y),
o := pair one one -- this is actually (pair (@one g1) (@one g2))
in have assoc : associative f,
-- The elaborator failed to infer the type of the pairs.
-- I had to annotate the pairs with their types.
from take x y z : S, -- We don't really need to provide S, but it will make the elaborator to work much harder
-- since * is an overloaded operator, we also have * as notation for Nat::mul in the context.
calc f (f x y) z = (pair ((proj1 x * proj1 y) * proj1 z) ((proj2 x * proj2 y) * proj2 z) : S) : refl (f (f x y) z)
... = (pair (proj1 x * (proj1 y * proj1 z)) ((proj2 x * proj2 y) * proj2 z) : S) : { G_assoc (proj1 x) (proj1 y) (proj1 z) }
... = (pair (proj1 x * (proj1 y * proj1 z)) (proj2 x * (proj2 y * proj2 z)) : S) : { G_assoc (proj2 x) (proj2 y) (proj2 z) }
... = f x (f y z) : refl (f x (f y z)),
have id : is_identity f o,
from take x : S,
calc f x o = (pair (proj1 x * one) (proj2 x * one) : S) : refl (f x o)
... = (pair (proj1 x) (proj2 x * one) : S) : { G_id (proj1 x) }
... = (pair (proj1 x) (proj2 x) : S) : { G_id (proj2 x) }
... = x : pair_proj_eq x,
have inv : inverse_ex f o,
from take x : S,
obtain (y1 : carrier g1) (Hy1 : proj1 x * y1 = one), from G_inv (proj1 x),
obtain (y2 : carrier g2) (Hy2 : proj2 x * y2 = one), from G_inv (proj2 x),
show ∃ y, f x y = o,
from exists_intro (pair y1 y2 : S)
(calc f x (pair y1 y2 : S) = (pair (proj1 x * y1) (proj2 x * y2) : S) : refl (f x (pair y1 y2 : S))
... = (pair one (proj2 x * y2) : S) : { Hy1 }
... = (pair one one : S) : { Hy2 }
... = o : refl o),
to_group S f o assoc id inv
set_opaque product true
-- It would be nice to be able to write x.first and x.second instead of (proj1 x) and (proj2 x)
-- Remark: * is overloaded since Lean loads Nat.lean by default.
-- The type errors related to * are quite cryptic because of that
-- Use 'star' for creating products
infixr 50 ⋆ : product
-- It would be nice to be able to write (p1 p2 : g1 ⋆ g2 ⋆ g3)
check λ (g1 g2 g3 : group) (p1 p2 : carrier (g1 ⋆ g2 ⋆ g3)), p1 * p2 = p2 * p1
theorem group_inhab (g : group) : inhabited (carrier g)
:= inhabited_intro (@one g)
definition inv {g : group} (a : carrier g) : carrier g
:= ε (group_inhab g) (λ x : carrier g, a * x = one)
theorem G_idl {g : group} (x : carrier g) : x * one = x
:= G_id x
theorem G_invl {g : group} (x : carrier g) : x * inv x = one
:= obtain (y : carrier g) (Hy : x * y = one), from G_inv x,
eps_ax (group_inhab g) y Hy
set_opaque inv true
theorem G_inv_aux {g : group} (x : carrier g) : inv x = (inv x * x) * inv x
:= symm (calc (inv x * x) * inv x = inv x * (x * inv x) : G_assoc (inv x) x (inv x)
... = inv x * one : { G_invl x }
... = inv x : G_idl (inv x))
theorem G_invr {g : group} (x : carrier g) : inv x * x = one
:= calc inv x * x = (inv x * x) * one : symm (G_idl (inv x * x))
... = (inv x * x) * (inv x * inv (inv x)) : { symm (G_invl (inv x)) }
... = ((inv x * x) * inv x) * inv (inv x) : symm (G_assoc (inv x * x) (inv x) (inv (inv x)))
... = (inv x * (x * inv x)) * inv (inv x) : { G_assoc (inv x) x (inv x) }
... = (inv x * one) * inv (inv x) : { G_invl x }
... = (inv x) * inv (inv x) : { G_idl (inv x) }
... = one : G_invl (inv x)
theorem G_idr {g : group} (x : carrier g) : one * x = x
:= calc one * x = (x * inv x) * x : { symm (G_invl x) }
... = x * (inv x * x) : G_assoc x (inv x) x
... = x * one : { G_invr x }
... = x : G_idl x
theorem G_inv_inv {g : group} (x : carrier g) : inv (inv x) = x
:= calc inv (inv x) = inv (inv x) * one : symm (G_idl (inv (inv x)))
... = inv (inv x) * (inv x * x) : { symm (G_invr x) }
... = (inv (inv x) * inv x) * x : symm (G_assoc (inv (inv x)) (inv x) x)
... = one * x : { G_invr (inv x) }
... = x : G_idr x
definition commutative {A : (Type U)} (f : A → A → A) := ∀ x y, f x y = f y x
definition abelian_group := sig g : group, commutative (@G_mul g)
definition ab_to_g (ag : abelian_group) : group
:= proj1 ag
-- Coercions currently only work with opaque types
-- We must first define "extract" the information we want, and then
-- mark abelian_group as opaque
definition AG_comm {g : abelian_group} (x y : carrier (ab_to_g g)) : x * y = y * x
:= (proj2 g) x y
set_opaque abelian_group true
set_opaque ab_to_g true
set_opaque AG_comm true
coercion ab_to_g
-- Now, we can use abelian groups where groups are expected.
theorem AG_left_comm {g : abelian_group} (x y z : carrier g) : x * (y * z) = y * (x * z)
:= calc x * (y * z) = (x * y) * z : symm (G_assoc x y z)
... = (y * x) * z : { AG_comm x y }
... = y * (x * z) : G_assoc y x z
|
70ee77e53611a42c12c524125a61e7b7b6112f1d | 80cc5bf14c8ea85ff340d1d747a127dcadeb966f | /src/data/setoid/partition.lean | b5120189338aa89b01aec53180cf7421ff437612 | [
"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 | 8,973 | lean | import data.setoid.basic
import data.set.lattice
/-!
# Equivalence relations: partitions
This file comprises properties of equivalence relations viewed as partitions.
## Tags
setoid, equivalence, iseqv, relation, equivalence relation, partition, equivalence class
-/
namespace setoid
variables {α : Type*}
/-- If x ∈ α is in 2 elements of a set of sets partitioning α, those 2 sets are equal. -/
lemma eq_of_mem_eqv_class {c : set (set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b)
{x b b'} (hc : b ∈ c) (hb : x ∈ b) (hc' : b' ∈ c) (hb' : x ∈ b') :
b = b' :=
(H x).unique2 hc hb hc' hb'
/-- Makes an equivalence relation from a set of sets partitioning α. -/
def mk_classes (c : set (set α)) (H : ∀ a, ∃! b ∈ c, a ∈ b) :
setoid α :=
⟨λ x y, ∀ s ∈ c, x ∈ s → y ∈ s, ⟨λ _ _ _ hx, hx,
λ x y h s hs hy, (H x).elim2 $ λ t ht hx _,
have s = t, from eq_of_mem_eqv_class H hs hy ht (h t ht hx),
this.symm ▸ hx,
λ x y z h1 h2 s hs hx, (H y).elim2 $ λ t ht hy _, (H z).elim2 $ λ t' ht' hz _,
have hst : s = t, from eq_of_mem_eqv_class H hs (h1 _ hs hx) ht hy,
have htt' : t = t', from eq_of_mem_eqv_class H ht (h2 _ ht hy) ht' hz,
(hst.trans htt').symm ▸ hz⟩⟩
/-- Makes the equivalence classes of an equivalence relation. -/
def classes (r : setoid α) : set (set α) :=
{s | ∃ y, s = {x | r.rel x y}}
lemma mem_classes (r : setoid α) (y) : {x | r.rel x y} ∈ r.classes := ⟨y, rfl⟩
/-- Two equivalence relations are equal iff all their equivalence classes are equal. -/
lemma eq_iff_classes_eq {r₁ r₂ : setoid α} :
r₁ = r₂ ↔ ∀ x, {y | r₁.rel x y} = {y | r₂.rel x y} :=
⟨λ h x, h ▸ rfl, λ h, ext' $ λ x, set.ext_iff.1 $ h x⟩
lemma rel_iff_exists_classes (r : setoid α) {x y} :
r.rel x y ↔ ∃ c ∈ r.classes, x ∈ c ∧ y ∈ c :=
⟨λ h, ⟨_, r.mem_classes y, h, r.refl' y⟩,
λ ⟨c, ⟨z, hz⟩, hx, hy⟩, by { subst c, exact r.trans' hx (r.symm' hy) }⟩
/-- Two equivalence relations are equal iff their equivalence classes are equal. -/
lemma classes_inj {r₁ r₂ : setoid α} :
r₁ = r₂ ↔ r₁.classes = r₂.classes :=
⟨λ h, h ▸ rfl, λ h, ext' $ λ a b, by simp only [rel_iff_exists_classes, exists_prop, h] ⟩
/-- The empty set is not an equivalence class. -/
lemma empty_not_mem_classes {r : setoid α} : ∅ ∉ r.classes :=
λ ⟨y, hy⟩, set.not_mem_empty y $ hy.symm ▸ r.refl' y
/-- Equivalence classes partition the type. -/
lemma classes_eqv_classes {r : setoid α} (a) : ∃! b ∈ r.classes, a ∈ b :=
exists_unique.intro2 {x | r.rel x a} (r.mem_classes a) (r.refl' _) $
begin
rintros _ ⟨y, rfl⟩ ha,
ext x,
exact ⟨λ hx, r.trans' hx (r.symm' ha), λ hx, r.trans' hx ha⟩
end
/-- If x ∈ α is in 2 equivalence classes, the equivalence classes are equal. -/
lemma eq_of_mem_classes {r : setoid α} {x b} (hc : b ∈ r.classes)
(hb : x ∈ b) {b'} (hc' : b' ∈ r.classes) (hb' : x ∈ b') : b = b' :=
eq_of_mem_eqv_class classes_eqv_classes hc hb hc' hb'
/-- The elements of a set of sets partitioning α are the equivalence classes of the
equivalence relation defined by the set of sets. -/
lemma eq_eqv_class_of_mem {c : set (set α)}
(H : ∀ a, ∃! b ∈ c, a ∈ b) {s y} (hs : s ∈ c) (hy : y ∈ s) :
s = {x | (mk_classes c H).rel x y} :=
set.ext $ λ x,
⟨λ hs', symm' (mk_classes c H) $ λ b' hb' h', eq_of_mem_eqv_class H hs hy hb' h' ▸ hs',
λ hx, (H x).elim2 $ λ b' hc' hb' h',
(eq_of_mem_eqv_class H hs hy hc' $ hx b' hc' hb').symm ▸ hb'⟩
/-- The equivalence classes of the equivalence relation defined by a set of sets
partitioning α are elements of the set of sets. -/
lemma eqv_class_mem {c : set (set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) {y} :
{x | (mk_classes c H).rel x y} ∈ c :=
(H y).elim2 $ λ b hc hy hb, eq_eqv_class_of_mem H hc hy ▸ hc
/-- Distinct elements of a set of sets partitioning α are disjoint. -/
lemma eqv_classes_disjoint {c : set (set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) :
set.pairwise_disjoint c :=
λ b₁ h₁ b₂ h₂ h, set.disjoint_left.2 $
λ x hx1 hx2, (H x).elim2 $ λ b hc hx hb, h $ eq_of_mem_eqv_class H h₁ hx1 h₂ hx2
/-- A set of disjoint sets covering α partition α (classical). -/
lemma eqv_classes_of_disjoint_union {c : set (set α)}
(hu : set.sUnion c = @set.univ α) (H : set.pairwise_disjoint c) (a) :
∃! b ∈ c, a ∈ b :=
let ⟨b, hc, ha⟩ := set.mem_sUnion.1 $ show a ∈ _, by rw hu; exact set.mem_univ a in
exists_unique.intro2 b hc ha $ λ b' hc' ha', H.elim hc' hc a ha' ha
/-- Makes an equivalence relation from a set of disjoints sets covering α. -/
def setoid_of_disjoint_union {c : set (set α)} (hu : set.sUnion c = @set.univ α)
(H : set.pairwise_disjoint c) : setoid α :=
setoid.mk_classes c $ eqv_classes_of_disjoint_union hu H
/-- The equivalence relation made from the equivalence classes of an equivalence
relation r equals r. -/
theorem mk_classes_classes (r : setoid α) :
mk_classes r.classes classes_eqv_classes = r :=
ext' $ λ x y, ⟨λ h, r.symm' (h {z | r.rel z x} (r.mem_classes x) $ r.refl' x),
λ h b hb hx, eq_of_mem_classes (r.mem_classes x) (r.refl' x) hb hx ▸ r.symm' h⟩
@[simp] theorem sUnion_classes (r : setoid α) : ⋃₀ r.classes = set.univ :=
set.eq_univ_of_forall $ λ x, set.mem_sUnion.2 ⟨{ y | r.rel y x }, ⟨x, rfl⟩, setoid.refl _⟩
section partition
/-- A collection `c : set (set α)` of sets is a partition of `α` into pairwise
disjoint sets if `∅ ∉ c` and each element `a : α` belongs to a unique set `b ∈ c`. -/
def is_partition (c : set (set α)) :=
∅ ∉ c ∧ ∀ a, ∃! b ∈ c, a ∈ b
/-- A partition of `α` does not contain the empty set. -/
lemma nonempty_of_mem_partition {c : set (set α)} (hc : is_partition c) {s} (h : s ∈ c) :
s.nonempty :=
set.ne_empty_iff_nonempty.1 $ λ hs0, hc.1 $ hs0 ▸ h
lemma is_partition_classes (r : setoid α) : is_partition r.classes :=
⟨empty_not_mem_classes, classes_eqv_classes⟩
lemma is_partition.pairwise_disjoint {c : set (set α)} (hc : is_partition c) :
c.pairwise_disjoint :=
eqv_classes_disjoint hc.2
lemma is_partition.sUnion_eq_univ {c : set (set α)} (hc : is_partition c) :
⋃₀ c = set.univ :=
set.eq_univ_of_forall $ λ x, set.mem_sUnion.2 $
let ⟨t, ht⟩ := hc.2 x in ⟨t, by clear_aux_decl; finish⟩
/-- All elements of a partition of α are the equivalence class of some y ∈ α. -/
lemma exists_of_mem_partition {c : set (set α)} (hc : is_partition c) {s} (hs : s ∈ c) :
∃ y, s = {x | (mk_classes c hc.2).rel x y} :=
let ⟨y, hy⟩ := nonempty_of_mem_partition hc hs in
⟨y, eq_eqv_class_of_mem hc.2 hs hy⟩
/-- The equivalence classes of the equivalence relation defined by a partition of α equal
the original partition. -/
theorem classes_mk_classes (c : set (set α)) (hc : is_partition c) :
(mk_classes c hc.2).classes = c :=
set.ext $ λ s,
⟨λ ⟨y, hs⟩, (hc.2 y).elim2 $ λ b hm hb hy,
by rwa (show s = b, from hs.symm ▸ set.ext
(λ x, ⟨λ hx, symm' (mk_classes c hc.2) hx b hm hb,
λ hx b' hc' hx', eq_of_mem_eqv_class hc.2 hm hx hc' hx' ▸ hb⟩)),
exists_of_mem_partition hc⟩
/-- Defining `≤` on partitions as the `≤` defined on their induced equivalence relations. -/
instance partition.le : has_le (subtype (@is_partition α)) :=
⟨λ x y, mk_classes x.1 x.2.2 ≤ mk_classes y.1 y.2.2⟩
/-- Defining a partial order on partitions as the partial order on their induced
equivalence relations. -/
instance partition.partial_order : partial_order (subtype (@is_partition α)) :=
{ le := (≤),
lt := λ x y, x ≤ y ∧ ¬y ≤ x,
le_refl := λ _, @le_refl (setoid α) _ _,
le_trans := λ _ _ _, @le_trans (setoid α) _ _ _ _,
lt_iff_le_not_le := λ _ _, iff.rfl,
le_antisymm := λ x y hx hy, let h := @le_antisymm (setoid α) _ _ _ hx hy in by
rw [subtype.ext_iff_val, ←classes_mk_classes x.1 x.2, ←classes_mk_classes y.1 y.2, h] }
variables (α)
/-- The order-preserving bijection between equivalence relations and partitions of sets. -/
def partition.rel_iso :
setoid α ≃o subtype (@is_partition α) :=
{ to_fun := λ r, ⟨r.classes, empty_not_mem_classes, classes_eqv_classes⟩,
inv_fun := λ x, mk_classes x.1 x.2.2,
left_inv := mk_classes_classes,
right_inv := λ x, by rw [subtype.ext_iff_val, ←classes_mk_classes x.1 x.2],
map_rel_iff' := λ x y, by conv {to_lhs, rw [←mk_classes_classes x, ←mk_classes_classes y]}; refl }
variables {α}
/-- A complete lattice instance for partitions; there is more infrastructure for the
equivalent complete lattice on equivalence relations. -/
instance partition.complete_lattice : complete_lattice (subtype (@is_partition α)) :=
galois_insertion.lift_complete_lattice $ @rel_iso.to_galois_insertion
_ (subtype (@is_partition α)) _ (partial_order.to_preorder _) $ partition.rel_iso α
end partition
end setoid
|
4f3c13795263674d82c52a467484818cc175a761 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/analysis/calculus/parametric_integral.lean | 70202a072b66af4356fc44807621fd5f00bf26e6 | [
"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 | 14,720 | lean | /-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import measure_theory.set_integral
import analysis.calculus.mean_value
/-!
# Derivatives of integrals depending on parameters
A parametric integral is a function with shape `f = λ x : H, ∫ a : α, F x a ∂μ` for some
`F : H → α → E`, where `H` and `E` are normed spaces and `α` is a measured space with measure `μ`.
We already know from `continuous_of_dominated` in `measure_theory.bochner_integral` how to
guarantee that `f` is continuous using the dominated convergence theorem. In this file,
we want to express the derivative of `f` as the integral of the derivative of `F` with respect
to `x`.
## Main results
As explained above, all results express the derivative of a parametric integral as the integral of
a derivative. The variations come from the assumptions and from the different ways of expressing
derivative, especially Fréchet derivatives vs elementary derivative of function of one real
variable.
* `has_fderiv_at_of_dominated_loc_of_lip`: this version assumes
`F x` is ae-measurable for x near `x₀`, `F x₀` is integrable,
`λ x, F x a` has derivative `F' a : H →L[ℝ] E` at `x₀` which is ae-measurable,
`λ x, F x a` is locally Lipschitz near `x₀` for almost every `a`, with a Lipschitz bound which
is integrable with respect to `a`. A subtle point is that the "near x₀" in the last condition
has to be uniform in `a`. This is controlled by a positive number `ε`.
* `has_fderiv_at_of_dominated_of_fderiv_le`: this version assume `λ x, F x a` has derivative
`F' x a` for `x` near `x₀` and `F' x` is bounded by an integrable function independent from
`x` near `x₀`.
`has_deriv_at_of_dominated_loc_of_lip` and `has_deriv_at_of_dominated_loc_of_deriv_le ` are versions
of the above two results that assume `H = ℝ` and use the high-school derivative `deriv` instead of
Fréchet derivative `fderiv`.
-/
noncomputable theory
open topological_space measure_theory filter metric
open_locale topological_space filter
variables {α : Type*} [measurable_space α] {μ : measure α}
{E : Type*} [normed_group E] [normed_space ℝ E]
[complete_space E] [second_countable_topology E]
[measurable_space E] [borel_space E]
{H : Type*} [normed_group H] [normed_space ℝ H]
[second_countable_topology $ H →L[ℝ] E]
/-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a` with
integrable Lipschitz bound (with a ball radius independent of `a`), and `F x` is
ae-measurable for `x` in the same ball. See `has_fderiv_at_of_dominated_loc_of_lip` for a
slightly more general version. -/
lemma has_fderiv_at_of_dominated_loc_of_lip' {F : H → α → E} {F' : α → (H →L[ℝ] E)} {x₀ : H}
{bound : α → ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ x ∈ ball x₀ ε, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs $ bound a) (λ x, F x a) (ball x₀ ε))
(bound_integrable : integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, has_fderiv_at (λ x, F x a) (F' a) x₀) :
integrable F' μ ∧ has_fderiv_at (λ x, ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
begin
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos,
have nneg : ∀ x, 0 ≤ ∥x - x₀∥⁻¹ := λ x, inv_nonneg.mpr (norm_nonneg _) ,
set b : α → ℝ := λ a, abs (bound a),
have b_int : integrable b μ := bound_integrable.norm,
have b_nonneg : ∀ a, 0 ≤ b a := λ a, abs_nonneg _,
have hF_int' : ∀ x ∈ ball x₀ ε, integrable (F x) μ,
{ intros x x_in,
have : ∀ᵐ a ∂μ, ∥F x₀ a - F x a∥ ≤ ε * ∥(bound a : ℝ)∥,
{ apply h_lipsch.mono,
intros a ha,
rw lipschitz_on_with_iff_norm_sub_le at ha,
apply (ha x₀ x₀_in x x_in).trans,
rw [mul_comm, nnreal.coe_nnabs, real.norm_eq_abs],
rw [mem_ball, dist_eq_norm, norm_sub_rev] at x_in,
exact mul_le_mul_of_nonneg_right (le_of_lt x_in) (abs_nonneg _) },
exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int
(integrable.const_mul bound_integrable.norm ε) this },
have hF'_int : integrable F' μ,
{ have : ∀ᵐ a ∂μ, ∥F' a∥ ≤ b a,
{ apply (h_diff.and h_lipsch).mono,
rintros a ⟨ha_diff, ha_lip⟩,
exact ha_diff.le_of_lip (ball_mem_nhds _ ε_pos) ha_lip },
exact b_int.mono' hF'_meas this },
refine ⟨hF'_int, _⟩,
have h_ball: ball x₀ ε ∈ 𝓝 x₀ := ball_mem_nhds x₀ ε_pos,
have : ∀ᶠ x in 𝓝 x₀,
∥x - x₀∥⁻¹ * ∥∫ a, F x a ∂μ - ∫ a, F x₀ a ∂μ - (∫ a, F' a ∂μ) (x - x₀)∥ =
∥∫ a, ∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀)) ∂μ∥,
{ apply mem_sets_of_superset (ball_mem_nhds _ ε_pos),
intros x x_in,
rw [set.mem_set_of_eq, ← norm_smul_of_nonneg (nneg _), integral_smul,
integral_sub, integral_sub, ← continuous_linear_map.integral_apply hF'_int],
exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,
hF'_int.apply_continuous_linear_map _] },
rw [has_fderiv_at_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero,
← show ∫ (a : α), ∥x₀ - x₀∥⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ = 0, by simp],
apply tendsto_integral_filter_of_dominated_convergence,
{ apply is_countably_generated_nhds },
{ filter_upwards [h_ball],
intros x x_in,
apply ae_measurable.const_smul,
exact ((hF_meas _ x_in).sub (hF_meas _ x₀_in)).sub (hF'_meas.apply_continuous_linear_map _) },
{ simp [measurable_const] },
{ apply mem_sets_of_superset h_ball,
intros x hx,
apply (h_diff.and h_lipsch).mono,
rintros a ⟨ha_deriv, ha_bound⟩,
show ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀))∥ ≤ b a + ∥F' a∥,
replace ha_bound : ∥F x a - F x₀ a∥ ≤ b a * ∥x - x₀∥,
{ rw lipschitz_on_with_iff_norm_sub_le at ha_bound,
exact ha_bound _ hx _ x₀_in },
calc ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀))∥
= ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a) - ∥x - x₀∥⁻¹ • F' a (x - x₀)∥ : by rw smul_sub
... ≤ ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a)∥ + ∥∥x - x₀∥⁻¹ • F' a (x - x₀)∥ : norm_sub_le _ _
... = ∥x - x₀∥⁻¹ * ∥F x a - F x₀ a∥ + ∥x - x₀∥⁻¹ * ∥F' a (x - x₀)∥ :
by { rw [norm_smul_of_nonneg, norm_smul_of_nonneg] ; exact nneg _}
... ≤ ∥x - x₀∥⁻¹ * (b a * ∥x - x₀∥) + ∥x - x₀∥⁻¹ * (∥F' a∥ * ∥x - x₀∥) : add_le_add _ _
... ≤ b a + ∥F' a∥ : _,
exact mul_le_mul_of_nonneg_left ha_bound (nneg _),
apply mul_le_mul_of_nonneg_left ((F' a).le_op_norm _) (nneg _),
by_cases h : ∥x - x₀∥ = 0,
{ simpa [h] using add_nonneg (b_nonneg a) (norm_nonneg (F' a)) },
{ field_simp [h] } },
{ exact b_int.add hF'_int.norm },
{ apply h_diff.mono,
intros a ha,
suffices : tendsto (λ x, ∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀))) (𝓝 x₀) (𝓝 0),
by simpa,
rw tendsto_zero_iff_norm_tendsto_zero,
have : (λ x, ∥x - x₀∥⁻¹ * ∥F x a - F x₀ a - F' a (x - x₀)∥) =
λ x, ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀))∥,
{ ext x,
rw norm_smul_of_nonneg (nneg _) },
rwa [has_fderiv_at_iff_tendsto, this] at ha },
end
/-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
(with a ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is ae-measurable
for `x` in a possibly smaller neighborhood of `x₀`. -/
lemma has_fderiv_at_of_dominated_loc_of_lip {F : H → α → E} {F' : α → (H →L[ℝ] E)} {x₀ : H}
{bound : α → ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable F' μ)
(h_lip : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs $ bound a) (λ x, F x a) (ball x₀ ε))
(bound_integrable : integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, has_fderiv_at (λ x, F x a) (F' a) x₀) :
integrable F' μ ∧ has_fderiv_at (λ x, ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
begin
obtain ⟨ε', ε'_pos, h'⟩ : ∃ ε' > 0, ∀ x ∈ ball x₀ ε', ae_measurable (F x) μ,
by simpa using nhds_basis_ball.eventually_iff.mp hF_meas,
set δ := min ε ε',
have δ_pos : 0 < δ := lt_min ε_pos ε'_pos,
replace h' : ∀ x, x ∈ ball x₀ δ → ae_measurable (F x) μ,
{ intros x x_in,
exact h' _ (ball_subset_ball (min_le_right ε ε') x_in) },
replace h_lip : ∀ᵐ (a : α) ∂μ, lipschitz_on_with (real.nnabs $ bound a) (λ x, F x a) (ball x₀ δ),
{ apply h_lip.mono,
intros a lip,
exact lip.mono (ball_subset_ball $ min_le_left ε ε') },
apply has_fderiv_at_of_dominated_loc_of_lip' δ_pos ; assumption
end
/-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is differentiable on a ball around `x₀` for ae `a` with
derivative norm uniformly bounded by an integrable function (the ball radius is independent of `a`),
and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
lemma has_fderiv_at_of_dominated_of_fderiv_le {F : H → α → E} {F' : H → α → (H →L[ℝ] E)} {x₀ : H}
{bound : α → ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable (F' x₀) μ)
(h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ∥F' x a∥ ≤ bound a)
(bound_integrable : integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, has_fderiv_at (λ x, F x a) (F' x a) x) :
has_fderiv_at (λ x, ∫ a, F x a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ :=
begin
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos,
have diff_x₀ : ∀ᵐ a ∂μ, has_fderiv_at (λ x, F x a) (F' x₀ a) x₀ :=
h_diff.mono (λ a ha, ha x₀ x₀_in),
have : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs (bound a)) (λ x, F x a) (ball x₀ ε),
{ apply (h_diff.and h_bound).mono,
rintros a ⟨ha_deriv, ha_bound⟩,
have bound_nonneg : 0 ≤ bound a := (norm_nonneg (F' x₀ a)).trans (ha_bound x₀ x₀_in),
rw show real.nnabs (bound a) = real.to_nnreal (bound a), by simp [bound_nonneg],
apply convex.lipschitz_on_with_of_norm_has_fderiv_within_le _ ha_bound (convex_ball _ _),
intros x x_in,
exact (ha_deriv x x_in).has_fderiv_within_at, },
exact (has_fderiv_at_of_dominated_loc_of_lip ε_pos hF_meas hF_int
hF'_meas this bound_integrable diff_x₀).2
end
/-- Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : ℝ`, assuming
`F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on an interval around `x₀` for ae `a`
(with interval radius independent of `a`) with integrable Lipschitz bound, and `F x` is
ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
lemma has_deriv_at_of_dominated_loc_of_lip {F : ℝ → α → E} {F' : α → E} {x₀ : ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable F' μ) {bound : α → ℝ}
(h_lipsch : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs $ bound a) (λ x, F x a) (ball x₀ ε))
(bound_integrable : integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, has_deriv_at (λ x, F x a) (F' a) x₀) :
(integrable F' μ) ∧ has_deriv_at (λ x, ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
begin
have hm := (continuous_linear_map.smul_rightL ℝ ℝ E 1).continuous.measurable.comp_ae_measurable
hF'_meas,
cases has_fderiv_at_of_dominated_loc_of_lip ε_pos hF_meas hF_int hm h_lipsch bound_integrable
h_diff with hF'_int key,
replace hF'_int : integrable F' μ,
{ rw [← integrable_norm_iff hm] at hF'_int,
simpa only [integrable_norm_iff, hF'_meas, one_mul, continuous_linear_map.norm_id_field',
continuous_linear_map.norm_smul_rightL_apply] using hF'_int},
refine ⟨hF'_int, _⟩,
simp_rw has_deriv_at_iff_has_fderiv_at at h_diff ⊢,
rwa continuous_linear_map.integral_comp_comm _ hF'_int at key,
end
/-- Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : ℝ`, assuming
`F x₀` is integrable, `x ↦ F x a` is differentiable on an interval around `x₀` for ae `a`
(with interval radius independent of `a`) with derivative uniformly bounded by an integrable
function, and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
lemma has_deriv_at_of_dominated_loc_of_deriv_le {F : ℝ → α → E} {F' : ℝ → α → E} {x₀ : ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable (F' x₀) μ)
{bound : α → ℝ}
(h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ∥F' x a∥ ≤ bound a)
(bound_integrable : integrable bound μ)
(h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, has_deriv_at (λ x, F x a) (F' x a) x) :
(integrable (F' x₀) μ) ∧ has_deriv_at (λn, ∫ a, F n a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ :=
begin
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos,
have diff_x₀ : ∀ᵐ a ∂μ, has_deriv_at (λ x, F x a) (F' x₀ a) x₀ :=
h_diff.mono (λ a ha, ha x₀ x₀_in),
have : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs (bound a)) (λ (x : ℝ), F x a) (ball x₀ ε),
{ apply (h_diff.and h_bound).mono,
rintros a ⟨ha_deriv, ha_bound⟩,
have bound_nonneg : 0 ≤ bound a := (norm_nonneg (F' x₀ a)).trans (ha_bound x₀ x₀_in),
rw show real.nnabs (bound a) = real.to_nnreal (bound a), by simp [bound_nonneg],
apply convex.lipschitz_on_with_of_norm_has_deriv_within_le (convex_ball _ _)
(λ x x_in, (ha_deriv x x_in).has_deriv_within_at) ha_bound },
exact has_deriv_at_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this
bound_integrable diff_x₀
end
|
ddd9bb957dd73e867033e7bc38b46cd19c283362 | 98beff2e97d91a54bdcee52f922c4e1866a6c9b9 | /src/pullback_colimit.lean | 1141a1f4d58805fe2d2677c2bec8017e9d21b004 | [] | no_license | b-mehta/topos | c3fc43fb04ba16bae1965ce5c26c6461172e5bc6 | c9032b11789e36038bc841a1e2b486972421b983 | refs/heads/master | 1,629,609,492,867 | 1,609,907,263,000 | 1,609,907,263,000 | 240,943,034 | 43 | 3 | null | 1,598,210,062,000 | 1,581,877,668,000 | Lean | UTF-8 | Lean | false | false | 2,392 | lean | import locally_cartesian_closed
namespace category_theory
open category limits
noncomputable theory
universes v u
variables {C : Type u} [category.{v} C]
variables {J : Type v} [small_category J]
variables {K₁ K₂ : J ⥤ C}
variables (c₁ : cocone K₁) {c₂ : cocone K₂} (t₂ : is_colimit c₂)
variables (τ : K₁ ⟶ K₂) (f : c₁.X ⟶ c₂.X)
variables (all_comm : ∀ (j : J), τ.app j ≫ c₂.ι.app j = c₁.ι.app j ≫ f)
variables (all_pb : Π (j : J), is_limit (pullback_cone.mk _ _ (all_comm j)))
variables [has_finite_limits C] [is_locally_cartesian_closed C] [has_colimits_of_shape J C]
include t₂ all_comm all_pb
def iso_diagrams : K₁ ≅ pullback_diagram f K₂ c₂ (𝟙 c₂.X) :=
begin
apply nat_iso.of_components (λ j, _) _,
{ refine is_limit.cone_points_iso_of_nat_iso (all_pb j) (cone_is_pullback _ _) _,
{ apply nat_iso.of_components _ _,
{ intro j, refl },
{ rintros X Y ⟨f⟩,
dsimp, simp,
cases f_1_1,
dsimp, simp,
dsimp, simp } } },
{ intros X Y g,
dsimp only [pullback_diagram_map, is_limit.cone_points_iso_of_nat_iso],
ext1,
{ rw [assoc, assoc, pullback.lift_fst],
change _ ≫ _ ≫ (pullback_cone.mk _ _ _).fst = _ ≫ (pullback_cone.mk _ _ _).fst ≫ _,
rw is_limit.map_π,
rw is_limit.map_π_assoc,
dsimp,
rw [comp_id, τ.naturality, id_comp] },
{ rw [assoc, assoc, pullback.lift_snd],
change _ ≫ _ ≫ (pullback_cone.mk _ _ _).snd = _ ≫ (pullback_cone.mk _ _ _).snd,
rw is_limit.map_π,
rw is_limit.map_π,
dsimp,
rw [comp_id, comp_id],
rw c₁.w,
} }
end
def pullback_colimit : is_colimit c₁ :=
begin
apply is_colimit.of_iso_colimit ((is_colimit.precompose_hom_equiv (iso_diagrams c₁ t₂ τ f all_comm all_pb) _).inv_fun (pullback_preserves f c₂ t₂ (𝟙 _))),
apply cocones.ext _ _,
{ refine {iso . hom := pullback.snd, inv := pullback.lift f (𝟙 _) (by simp), hom_inv_id' := _},
change _ = 𝟙 _,
apply pullback.hom_ext,
rw [assoc, pullback.lift_fst, ← pullback.condition, comp_id, id_comp],
rw [assoc, pullback.lift_snd, comp_id, id_comp] },
{ intro j,
dsimp [iso_diagrams],
rw [assoc],
rw pullback.lift_snd,
change _ ≫ (pullback_cone.mk _ _ _).snd = _,
rw is_limit.map_π,
dsimp,
apply comp_id }
end
end category_theory |
4ddb225a374945af08cb6f4d49d1da7abd92c286 | 1abd1ed12aa68b375cdef28959f39531c6e95b84 | /src/algebra/group_power/lemmas.lean | c5f41361d61d3c93f102ec5be2d3871a8c809db3 | [
"Apache-2.0"
] | permissive | jumpy4/mathlib | d3829e75173012833e9f15ac16e481e17596de0f | af36f1a35f279f0e5b3c2a77647c6bf2cfd51a13 | refs/heads/master | 1,693,508,842,818 | 1,636,203,271,000 | 1,636,203,271,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 35,935 | lean | /-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import algebra.group_power.basic
import algebra.invertible
import algebra.opposites
import data.list.basic
import data.int.cast
import data.equiv.basic
import data.equiv.mul_add
/-!
# Lemmas about power operations on monoids and groups
This file contains lemmas about `monoid.pow`, `group.pow`, `nsmul`, `zsmul`
which require additional imports besides those available in `algebra.group_power.basic`.
-/
universes u v w x y z u₁ u₂
variables {M : Type u} {N : Type v} {G : Type w} {H : Type x} {A : Type y} {B : Type z}
{R : Type u₁} {S : Type u₂}
/-!
### (Additive) monoid
-/
section monoid
variables [monoid M] [monoid N] [add_monoid A] [add_monoid B]
@[simp] theorem nsmul_one [has_one A] : ∀ n : ℕ, n • (1 : A) = n :=
add_monoid_hom.eq_nat_cast
⟨λ n, n • (1 : A), zero_nsmul _, λ _ _, add_nsmul _ _ _⟩
(one_nsmul _)
@[simp, norm_cast, to_additive]
lemma units.coe_pow (u : units M) (n : ℕ) : ((u ^ n : units M) : M) = u ^ n :=
(units.coe_hom M).map_pow u n
instance invertible_pow (m : M) [invertible m] (n : ℕ) : invertible (m ^ n) :=
{ inv_of := ⅟ m ^ n,
inv_of_mul_self := by rw [← (commute_inv_of m).symm.mul_pow, inv_of_mul_self, one_pow],
mul_inv_of_self := by rw [← (commute_inv_of m).mul_pow, mul_inv_of_self, one_pow] }
lemma inv_of_pow (m : M) [invertible m] (n : ℕ) [invertible (m ^ n)] :
⅟(m ^ n) = ⅟m ^ n :=
@invertible_unique M _ (m ^ n) (m ^ n) rfl ‹_› (invertible_pow m n)
lemma is_unit.pow {m : M} (n : ℕ) : is_unit m → is_unit (m ^ n) :=
λ ⟨u, hu⟩, ⟨u ^ n, by simp *⟩
@[simp] lemma is_unit_pow_succ_iff {m : M} {n : ℕ} :
is_unit (m ^ (n + 1)) ↔ is_unit m :=
begin
refine ⟨_, λ h, h.pow _⟩,
rw [pow_succ, ((commute.refl _).pow_right _).is_unit_mul_iff],
exact and.left
end
lemma is_unit_pos_pow_iff {m : M} :
∀ {n : ℕ} (h : 0 < n), is_unit (m ^ n) ↔ is_unit m
| (n + 1) _ := is_unit_pow_succ_iff
/-- If `x ^ n.succ = 1` then `x` has an inverse, `x^n`. -/
def invertible_of_pow_succ_eq_one (x : M) (n : ℕ) (hx : x ^ n.succ = 1) :
invertible x :=
⟨x ^ n, (pow_succ' x n).symm.trans hx, (pow_succ x n).symm.trans hx⟩
/-- If `x ^ n = 1` then `x` has an inverse, `x^(n - 1)`. -/
def invertible_of_pow_eq_one (x : M) (n : ℕ) (hx : x ^ n = 1) (hn : 0 < n) :
invertible x :=
begin
apply invertible_of_pow_succ_eq_one x (n - 1),
convert hx,
exact tsub_add_cancel_of_le (nat.succ_le_of_lt hn),
end
lemma is_unit_of_pow_eq_one (x : M) (n : ℕ) (hx : x ^ n = 1) (hn : 0 < n) :
is_unit x :=
begin
haveI := invertible_of_pow_eq_one x n hx hn,
exact is_unit_of_invertible x
end
lemma smul_pow [mul_action M N] [is_scalar_tower M N N] [smul_comm_class M N N]
(k : M) (x : N) (p : ℕ) :
(k • x) ^ p = k ^ p • x ^ p :=
begin
induction p with p IH,
{ simp },
{ rw [pow_succ', IH, smul_mul_smul, ←pow_succ', ←pow_succ'] }
end
@[simp] lemma smul_pow' [mul_distrib_mul_action M N] (x : M) (m : N) (n : ℕ) :
x • m ^ n = (x • m) ^ n :=
begin
induction n with n ih,
{ rw [pow_zero, pow_zero], exact smul_one x },
{ rw [pow_succ, pow_succ], exact (smul_mul' x m (m ^ n)).trans (congr_arg _ ih) }
end
end monoid
section group
variables [group G] [group H] [add_group A] [add_group B]
open int
local attribute [ematch] le_of_lt
open nat
theorem zsmul_one [has_one A] (n : ℤ) : n • (1 : A) = n :=
by cases n; simp
@[to_additive add_one_zsmul]
lemma zpow_add_one (a : G) : ∀ n : ℤ, a ^ (n + 1) = a ^ n * a
| (of_nat n) := by simp [← int.coe_nat_succ, pow_succ']
| -[1+0] := by simp [int.neg_succ_of_nat_eq]
| -[1+(n+1)] := by rw [int.neg_succ_of_nat_eq, zpow_neg, neg_add, neg_add_cancel_right, zpow_neg,
← int.coe_nat_succ, zpow_coe_nat, zpow_coe_nat, pow_succ _ (n + 1), mul_inv_rev,
inv_mul_cancel_right]
@[to_additive zsmul_sub_one]
lemma zpow_sub_one (a : G) (n : ℤ) : a ^ (n - 1) = a ^ n * a⁻¹ :=
calc a ^ (n - 1) = a ^ (n - 1) * a * a⁻¹ : (mul_inv_cancel_right _ _).symm
... = a^n * a⁻¹ : by rw [← zpow_add_one, sub_add_cancel]
@[to_additive add_zsmul]
lemma zpow_add (a : G) (m n : ℤ) : a ^ (m + n) = a ^ m * a ^ n :=
begin
induction n using int.induction_on with n ihn n ihn,
case hz : { simp },
{ simp only [← add_assoc, zpow_add_one, ihn, mul_assoc] },
{ rw [zpow_sub_one, ← mul_assoc, ← ihn, ← zpow_sub_one, add_sub_assoc] }
end
@[to_additive add_zsmul_self]
lemma mul_self_zpow (b : G) (m : ℤ) : b*b^m = b^(m+1) :=
by { conv_lhs {congr, rw ← zpow_one b }, rw [← zpow_add, add_comm] }
@[to_additive add_self_zsmul]
lemma mul_zpow_self (b : G) (m : ℤ) : b^m*b = b^(m+1) :=
by { conv_lhs {congr, skip, rw ← zpow_one b }, rw [← zpow_add, add_comm] }
@[to_additive sub_zsmul]
lemma zpow_sub (a : G) (m n : ℤ) : a ^ (m - n) = a ^ m * (a ^ n)⁻¹ :=
by rw [sub_eq_add_neg, zpow_add, zpow_neg]
@[to_additive one_add_zsmul]
theorem zpow_one_add (a : G) (i : ℤ) : a ^ (1 + i) = a * a ^ i :=
by rw [zpow_add, zpow_one]
@[to_additive]
theorem zpow_mul_comm (a : G) (i j : ℤ) : a ^ i * a ^ j = a ^ j * a ^ i :=
by rw [← zpow_add, ← zpow_add, add_comm]
-- note that `mul_zsmul` and `zpow_mul` have the primes swapped since their argument order
-- and therefore the more "natural" choice of lemma is reversed.
@[to_additive mul_zsmul']
theorem zpow_mul (a : G) (m n : ℤ) : a ^ (m * n) = (a ^ m) ^ n :=
int.induction_on n (by simp) (λ n ihn, by simp [mul_add, zpow_add, ihn])
(λ n ihn, by simp only [mul_sub, zpow_sub, ihn, mul_one, zpow_one])
@[to_additive mul_zsmul]
theorem zpow_mul' (a : G) (m n : ℤ) : a ^ (m * n) = (a ^ n) ^ m :=
by rw [mul_comm, zpow_mul]
@[to_additive bit0_zsmul]
theorem zpow_bit0 (a : G) (n : ℤ) : a ^ bit0 n = a ^ n * a ^ n := zpow_add _ _ _
@[to_additive bit1_zsmul]
theorem zpow_bit1 (a : G) (n : ℤ) : a ^ bit1 n = a ^ n * a ^ n * a :=
by rw [bit1, zpow_add, zpow_bit0, zpow_one]
@[simp, norm_cast, to_additive]
lemma units.coe_zpow (u : units G) (n : ℤ) : ((u ^ n : units G) : G) = u ^ n :=
(units.coe_hom G).map_zpow u n
end group
section ordered_add_comm_group
variables [ordered_add_comm_group A]
/-! Lemmas about `zsmul` under ordering, placed here (rather than in `algebra.group_power.order`
with their friends) because they require facts from `data.int.basic`-/
open int
lemma zsmul_pos {a : A} (ha : 0 < a) {k : ℤ} (hk : (0:ℤ) < k) : 0 < k • a :=
begin
lift k to ℕ using int.le_of_lt hk,
rw coe_nat_zsmul,
apply nsmul_pos ha,
exact (coe_nat_pos.mp hk).ne',
end
theorem zsmul_strict_mono_left {a : A} (ha : 0 < a) : strict_mono (λ n : ℤ, n • a) :=
λ n m h,
calc n • a = n • a + 0 : (add_zero _).symm
... < n • a + (m - n) • a : add_lt_add_left (zsmul_pos ha (sub_pos.mpr h)) _
... = m • a : by { rw [← add_zsmul], simp }
theorem zsmul_mono_left {a : A} (ha : 0 ≤ a) : monotone (λ n : ℤ, n • a) :=
λ n m h,
calc n • a = n • a + 0 : (add_zero _).symm
... ≤ n • a + (m - n) • a : add_le_add_left (zsmul_nonneg ha (sub_nonneg.mpr h)) _
... = m • a : by { rw [← add_zsmul], simp }
theorem zsmul_le_zsmul {a : A} {n m : ℤ} (ha : 0 ≤ a) (h : n ≤ m) : n • a ≤ m • a :=
zsmul_mono_left ha h
theorem zsmul_lt_zsmul {a : A} {n m : ℤ} (ha : 0 < a) (h : n < m) : n • a < m • a :=
zsmul_strict_mono_left ha h
theorem zsmul_le_zsmul_iff {a : A} {n m : ℤ} (ha : 0 < a) : n • a ≤ m • a ↔ n ≤ m :=
(zsmul_strict_mono_left ha).le_iff_le
theorem zsmul_lt_zsmul_iff {a : A} {n m : ℤ} (ha : 0 < a) : n • a < m • a ↔ n < m :=
(zsmul_strict_mono_left ha).lt_iff_lt
variables (A)
lemma zsmul_strict_mono_right {n : ℤ} (hn : 0 < n) :
strict_mono ((•) n : A → A) :=
λ a b hab, begin
rw ← sub_pos at hab,
rw [← sub_pos, ← zsmul_sub],
exact zsmul_pos hab hn,
end
lemma zsmul_mono_right {n : ℤ} (hn : 0 ≤ n) :
monotone ((•) n : A → A) :=
λ a b hab, begin
rw ← sub_nonneg at hab,
rw [← sub_nonneg, ← zsmul_sub],
exact zsmul_nonneg hab hn,
end
variables {A}
theorem zsmul_le_zsmul' {n : ℤ} (hn : 0 ≤ n) {a₁ a₂ : A} (h : a₁ ≤ a₂) : n • a₁ ≤ n • a₂ :=
zsmul_mono_right A hn h
theorem zsmul_lt_zsmul' {n : ℤ} (hn : 0 < n) {a₁ a₂ : A} (h : a₁ < a₂) : n • a₁ < n • a₂ :=
zsmul_strict_mono_right A hn h
lemma abs_nsmul {α : Type*} [linear_ordered_add_comm_group α] (n : ℕ) (a : α) :
|n • a| = n • |a| :=
begin
cases le_total a 0 with hneg hpos,
{ rw [abs_of_nonpos hneg, ← abs_neg, ← neg_nsmul, abs_of_nonneg],
exact nsmul_nonneg (neg_nonneg.mpr hneg) n },
{ rw [abs_of_nonneg hpos, abs_of_nonneg],
exact nsmul_nonneg hpos n }
end
lemma abs_zsmul {α : Type*} [linear_ordered_add_comm_group α] (n : ℤ) (a : α) :
|n • a| = |n| • |a| :=
begin
by_cases n0 : 0 ≤ n,
{ lift n to ℕ using n0,
simp only [abs_nsmul, coe_nat_abs, coe_nat_zsmul] },
{ lift (- n) to ℕ using int.le_of_lt (neg_pos.mpr (not_le.mp n0)) with m h,
rw [← abs_neg (n • a), ← neg_zsmul, ← abs_neg n, ← h, coe_nat_zsmul, coe_nat_abs,
coe_nat_zsmul],
exact abs_nsmul m _ },
end
lemma abs_add_eq_add_abs_le {α : Type*} [linear_ordered_add_comm_group α] {a b : α} (hle : a ≤ b) :
|a + b| = |a| + |b| ↔ (0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0) :=
begin
by_cases a0 : 0 ≤ a; by_cases b0 : 0 ≤ b,
{ simp [a0, b0, abs_of_nonneg, add_nonneg a0 b0] },
{ exact (lt_irrefl (0 : α) (a0.trans_lt (hle.trans_lt (not_le.mp b0)))).elim },
any_goals { simp [(not_le.mp a0).le, (not_le.mp b0).le, abs_of_nonpos, add_nonpos, add_comm] },
obtain F := (not_le.mp a0),
have : (|a + b| = -a + b ↔ b ≤ 0) ↔ (|a + b| =
|a| + |b| ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0),
{ simp [a0, b0, abs_of_neg, abs_of_nonneg, F, F.le] },
refine this.mp ⟨λ h, _, λ h, by simp only [le_antisymm h b0, abs_of_neg F, add_zero]⟩,
by_cases ba : a + b ≤ 0,
{ refine le_of_eq (eq_zero_of_neg_eq _),
rwa [abs_of_nonpos ba, neg_add_rev, add_comm, add_right_inj] at h },
{ refine (lt_irrefl (0 : α) _).elim,
rw [abs_of_pos (not_le.mp ba), add_left_inj] at h,
rwa eq_zero_of_neg_eq h.symm at F }
end
lemma abs_add_eq_add_abs_iff {α : Type*} [linear_ordered_add_comm_group α] (a b : α) :
|a + b| = |a| + |b| ↔ (0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0) :=
begin
by_cases ab : a ≤ b,
{ exact abs_add_eq_add_abs_le ab },
{ rw [add_comm a, add_comm (abs _), abs_add_eq_add_abs_le ((not_le.mp ab).le), and.comm,
@and.comm (b ≤ 0 ) _] }
end
end ordered_add_comm_group
section linear_ordered_add_comm_group
variable [linear_ordered_add_comm_group A]
theorem zsmul_le_zsmul_iff' {n : ℤ} (hn : 0 < n) {a₁ a₂ : A} : n • a₁ ≤ n • a₂ ↔ a₁ ≤ a₂ :=
(zsmul_strict_mono_right A hn).le_iff_le
theorem zsmul_lt_zsmul_iff' {n : ℤ} (hn : 0 < n) {a₁ a₂ : A} : n • a₁ < n • a₂ ↔ a₁ < a₂ :=
(zsmul_strict_mono_right A hn).lt_iff_lt
theorem nsmul_le_nsmul_iff {a : A} {n m : ℕ} (ha : 0 < a) : n • a ≤ m • a ↔ n ≤ m :=
begin
refine ⟨λ h, _, nsmul_le_nsmul $ le_of_lt ha⟩,
by_contra H,
exact lt_irrefl _ (lt_of_lt_of_le (nsmul_lt_nsmul ha (not_le.mp H)) h)
end
theorem nsmul_lt_nsmul_iff {a : A} {n m : ℕ} (ha : 0 < a) : n • a < m • a ↔ n < m :=
begin
refine ⟨λ h, _, nsmul_lt_nsmul ha⟩,
by_contra H,
exact lt_irrefl _ (lt_of_le_of_lt (nsmul_le_nsmul (le_of_lt ha) $ not_lt.mp H) h)
end
/-- See also `smul_right_injective`. TODO: provide a `no_zero_smul_divisors` instance. We can't
do that here because importing that definition would create import cycles. -/
lemma zsmul_right_injective {m : ℤ} (hm : m ≠ 0) : function.injective ((•) m : A → A) :=
begin
cases hm.symm.lt_or_lt,
{ exact (zsmul_strict_mono_right A h).injective, },
{ intros a b hab,
refine (zsmul_strict_mono_right A (neg_pos.mpr h)).injective _,
rw [neg_zsmul, neg_zsmul, hab], },
end
lemma zsmul_right_inj {a b : A} {m : ℤ} (hm : m ≠ 0) : m • a = m • b ↔ a = b :=
(zsmul_right_injective hm).eq_iff
/-- Alias of `zsmul_right_inj`, for ease of discovery alongside `zsmul_le_zsmul_iff'` and
`zsmul_lt_zsmul_iff'`. -/
lemma zsmul_eq_zsmul_iff' {a b : A} {m : ℤ} (hm : m ≠ 0) : m • a = m • b ↔ a = b :=
zsmul_right_inj hm
end linear_ordered_add_comm_group
@[simp] lemma with_bot.coe_nsmul [add_monoid A] (a : A) (n : ℕ) :
((n • a : A) : with_bot A) = n • a :=
add_monoid_hom.map_nsmul ⟨(coe : A → with_bot A), with_bot.coe_zero, with_bot.coe_add⟩ a n
theorem nsmul_eq_mul' [semiring R] (a : R) (n : ℕ) : n • a = a * n :=
by induction n with n ih; [rw [zero_nsmul, nat.cast_zero, mul_zero],
rw [succ_nsmul', ih, nat.cast_succ, mul_add, mul_one]]
@[simp] theorem nsmul_eq_mul [semiring R] (n : ℕ) (a : R) : n • a = n * a :=
by rw [nsmul_eq_mul', (n.cast_commute a).eq]
theorem mul_nsmul_left [semiring R] (a b : R) (n : ℕ) : n • (a * b) = a * (n • b) :=
by rw [nsmul_eq_mul', nsmul_eq_mul', mul_assoc]
theorem mul_nsmul_assoc [semiring R] (a b : R) (n : ℕ) : n • (a * b) = n • a * b :=
by rw [nsmul_eq_mul, nsmul_eq_mul, mul_assoc]
@[simp, norm_cast] theorem nat.cast_pow [semiring R] (n m : ℕ) : (↑(n ^ m) : R) = ↑n ^ m :=
begin
induction m with m ih,
{ rw [pow_zero, pow_zero], exact nat.cast_one },
{ rw [pow_succ', pow_succ', nat.cast_mul, ih] }
end
@[simp, norm_cast] theorem int.coe_nat_pow (n m : ℕ) : ((n ^ m : ℕ) : ℤ) = n ^ m :=
by induction m with m ih; [exact int.coe_nat_one, rw [pow_succ', pow_succ', int.coe_nat_mul, ih]]
theorem int.nat_abs_pow (n : ℤ) (k : ℕ) : int.nat_abs (n ^ k) = (int.nat_abs n) ^ k :=
by induction k with k ih; [refl, rw [pow_succ', int.nat_abs_mul, pow_succ', ih]]
-- The next four lemmas allow us to replace multiplication by a numeral with a `zsmul` expression.
-- They are used by the `noncomm_ring` tactic, to normalise expressions before passing to `abel`.
lemma bit0_mul [ring R] {n r : R} : bit0 n * r = (2 : ℤ) • (n * r) :=
by { dsimp [bit0], rw [add_mul, add_zsmul, one_zsmul], }
lemma mul_bit0 [ring R] {n r : R} : r * bit0 n = (2 : ℤ) • (r * n) :=
by { dsimp [bit0], rw [mul_add, add_zsmul, one_zsmul], }
lemma bit1_mul [ring R] {n r : R} : bit1 n * r = (2 : ℤ) • (n * r) + r :=
by { dsimp [bit1], rw [add_mul, bit0_mul, one_mul], }
lemma mul_bit1 [ring R] {n r : R} : r * bit1 n = (2 : ℤ) • (r * n) + r :=
by { dsimp [bit1], rw [mul_add, mul_bit0, mul_one], }
@[simp] theorem zsmul_eq_mul [ring R] (a : R) : ∀ (n : ℤ), n • a = n * a
| (n : ℕ) := by { rw [coe_nat_zsmul, nsmul_eq_mul], refl }
| -[1+ n] := by simp [nat.cast_succ, neg_add_rev, int.cast_neg_succ_of_nat, add_mul]
theorem zsmul_eq_mul' [ring R] (a : R) (n : ℤ) : n • a = a * n :=
by rw [zsmul_eq_mul, (n.cast_commute a).eq]
theorem mul_zsmul_left [ring R] (a b : R) (n : ℤ) : n • (a * b) = a * (n • b) :=
by rw [zsmul_eq_mul', zsmul_eq_mul', mul_assoc]
theorem mul_zsmul_assoc [ring R] (a b : R) (n : ℤ) : n • (a * b) = n • a * b :=
by rw [zsmul_eq_mul, zsmul_eq_mul, mul_assoc]
lemma zsmul_int_int (a b : ℤ) : a • b = a * b := by simp
lemma zsmul_int_one (n : ℤ) : n • 1 = n := by simp
@[simp, norm_cast] theorem int.cast_pow [ring R] (n : ℤ) (m : ℕ) : (↑(n ^ m) : R) = ↑n ^ m :=
begin
induction m with m ih,
{ rw [pow_zero, pow_zero, int.cast_one] },
{ rw [pow_succ, pow_succ, int.cast_mul, ih] }
end
lemma neg_one_pow_eq_pow_mod_two [ring R] {n : ℕ} : (-1 : R) ^ n = (-1) ^ (n % 2) :=
by rw [← nat.mod_add_div n 2, pow_add, pow_mul]; simp [sq]
section ordered_semiring
variables [ordered_semiring R] {a : R}
/-- Bernoulli's inequality. This version works for semirings but requires
additional hypotheses `0 ≤ a * a` and `0 ≤ (1 + a) * (1 + a)`. -/
theorem one_add_mul_le_pow' (Hsq : 0 ≤ a * a) (Hsq' : 0 ≤ (1 + a) * (1 + a))
(H : 0 ≤ 2 + a) :
∀ (n : ℕ), 1 + (n : R) * a ≤ (1 + a) ^ n
| 0 := by simp
| 1 := by simp
| (n+2) :=
have 0 ≤ (n : R) * (a * a * (2 + a)) + a * a,
from add_nonneg (mul_nonneg n.cast_nonneg (mul_nonneg Hsq H)) Hsq,
calc 1 + (↑(n + 2) : R) * a ≤ 1 + ↑(n + 2) * a + (n * (a * a * (2 + a)) + a * a) :
(le_add_iff_nonneg_right _).2 this
... = (1 + a) * (1 + a) * (1 + n * a) :
by { simp [add_mul, mul_add, bit0, mul_assoc, (n.cast_commute (_ : R)).left_comm],
ac_refl }
... ≤ (1 + a) * (1 + a) * (1 + a)^n :
mul_le_mul_of_nonneg_left (one_add_mul_le_pow' n) Hsq'
... = (1 + a)^(n + 2) : by simp only [pow_succ, mul_assoc]
private lemma pow_lt_pow_of_lt_one_aux (h : 0 < a) (ha : a < 1) (i : ℕ) :
∀ k : ℕ, a ^ (i + k + 1) < a ^ i
| 0 :=
begin
rw [←one_mul (a^i), add_zero, pow_succ],
exact mul_lt_mul ha (le_refl _) (pow_pos h _) zero_le_one
end
| (k+1) :=
begin
rw [←one_mul (a^i), pow_succ],
apply mul_lt_mul ha _ _ zero_le_one,
{ apply le_of_lt, apply pow_lt_pow_of_lt_one_aux },
{ show 0 < a ^ (i + (k + 1) + 0), apply pow_pos h }
end
private lemma pow_le_pow_of_le_one_aux (h : 0 ≤ a) (ha : a ≤ 1) (i : ℕ) :
∀ k : ℕ, a ^ (i + k) ≤ a ^ i
| 0 := by simp
| (k+1) := by { rw [←add_assoc, ←one_mul (a^i), pow_succ],
exact mul_le_mul ha (pow_le_pow_of_le_one_aux _) (pow_nonneg h _) zero_le_one }
lemma pow_lt_pow_of_lt_one (h : 0 < a) (ha : a < 1)
{i j : ℕ} (hij : i < j) : a ^ j < a ^ i :=
let ⟨k, hk⟩ := nat.exists_eq_add_of_lt hij in
by rw hk; exact pow_lt_pow_of_lt_one_aux h ha _ _
lemma pow_lt_pow_iff_of_lt_one {n m : ℕ} (hpos : 0 < a) (h : a < 1) :
a ^ m < a ^ n ↔ n < m :=
begin
have : strict_mono (λ (n : order_dual ℕ), a ^ (id n : ℕ)) := λ m n, pow_lt_pow_of_lt_one hpos h,
exact this.lt_iff_lt
end
lemma pow_le_pow_of_le_one (h : 0 ≤ a) (ha : a ≤ 1) {i j : ℕ} (hij : i ≤ j) :
a ^ j ≤ a ^ i :=
let ⟨k, hk⟩ := nat.exists_eq_add_of_le hij in
by rw hk; exact pow_le_pow_of_le_one_aux h ha _ _
lemma pow_le_of_le_one (h₀ : 0 ≤ a) (h₁ : a ≤ 1) {n : ℕ} (hn : n ≠ 0) : a ^ n ≤ a :=
(pow_one a).subst (pow_le_pow_of_le_one h₀ h₁ (nat.pos_of_ne_zero hn))
lemma sq_le (h₀ : 0 ≤ a) (h₁ : a ≤ 1) : a ^ 2 ≤ a := pow_le_of_le_one h₀ h₁ two_ne_zero
end ordered_semiring
section linear_ordered_semiring
variables [linear_ordered_semiring R]
lemma sign_cases_of_C_mul_pow_nonneg {C r : R} (h : ∀ n : ℕ, 0 ≤ C * r ^ n) :
C = 0 ∨ (0 < C ∧ 0 ≤ r) :=
begin
have : 0 ≤ C, by simpa only [pow_zero, mul_one] using h 0,
refine this.eq_or_lt.elim (λ h, or.inl h.symm) (λ hC, or.inr ⟨hC, _⟩),
refine nonneg_of_mul_nonneg_left _ hC,
simpa only [pow_one] using h 1
end
end linear_ordered_semiring
section linear_ordered_ring
variables [linear_ordered_ring R] {a : R} {n : ℕ}
@[simp] lemma abs_pow (a : R) (n : ℕ) : |a ^ n| = |a| ^ n :=
(pow_abs a n).symm
@[simp] theorem pow_bit1_neg_iff : a ^ bit1 n < 0 ↔ a < 0 :=
⟨λ h, not_le.1 $ λ h', not_le.2 h $ pow_nonneg h' _, λ ha, pow_bit1_neg ha n⟩
@[simp] theorem pow_bit1_nonneg_iff : 0 ≤ a ^ bit1 n ↔ 0 ≤ a :=
le_iff_le_iff_lt_iff_lt.2 pow_bit1_neg_iff
@[simp] theorem pow_bit1_nonpos_iff : a ^ bit1 n ≤ 0 ↔ a ≤ 0 :=
by simp only [le_iff_lt_or_eq, pow_bit1_neg_iff, pow_eq_zero_iff (bit1_pos (zero_le n))]
@[simp] theorem pow_bit1_pos_iff : 0 < a ^ bit1 n ↔ 0 < a :=
lt_iff_lt_of_le_iff_le pow_bit1_nonpos_iff
lemma even.pow_nonneg (hn : even n) (a : R) : 0 ≤ a ^ n :=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit0_nonneg a k
lemma even.pow_pos (hn : even n) (ha : a ≠ 0) : 0 < a ^ n :=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit0_pos ha k
lemma odd.pow_nonpos (hn : odd n) (ha : a ≤ 0) : a ^ n ≤ 0:=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit1_nonpos_iff.mpr ha
lemma odd.pow_neg (hn : odd n) (ha : a < 0) : a ^ n < 0:=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit1_neg_iff.mpr ha
lemma odd.pow_nonneg_iff (hn : odd n) : 0 ≤ a ^ n ↔ 0 ≤ a :=
⟨λ h, le_of_not_lt (λ ha, h.not_lt $ hn.pow_neg ha), λ ha, pow_nonneg ha n⟩
lemma odd.pow_nonpos_iff (hn : odd n) : a ^ n ≤ 0 ↔ a ≤ 0 :=
⟨λ h, le_of_not_lt (λ ha, h.not_lt $ pow_pos ha _), hn.pow_nonpos⟩
lemma odd.pow_pos_iff (hn : odd n) : 0 < a ^ n ↔ 0 < a :=
⟨λ h, lt_of_not_ge' (λ ha, h.not_le $ hn.pow_nonpos ha), λ ha, pow_pos ha n⟩
lemma odd.pow_neg_iff (hn : odd n) : a ^ n < 0 ↔ a < 0 :=
⟨λ h, lt_of_not_ge' (λ ha, h.not_le $ pow_nonneg ha _), hn.pow_neg⟩
lemma even.pow_pos_iff (hn : even n) (h₀ : 0 < n) : 0 < a ^ n ↔ a ≠ 0 :=
⟨λ h ha, by { rw [ha, zero_pow h₀] at h, exact lt_irrefl 0 h }, hn.pow_pos⟩
lemma even.pow_abs {p : ℕ} (hp : even p) (a : R) : |a| ^ p = a ^ p :=
begin
rw [←abs_pow, abs_eq_self],
exact hp.pow_nonneg _
end
@[simp] lemma pow_bit0_abs (a : R) (p : ℕ) : |a| ^ bit0 p = a ^ bit0 p := (even_bit0 _).pow_abs _
lemma strict_mono_pow_bit1 (n : ℕ) : strict_mono (λ a : R, a ^ bit1 n) :=
begin
intros a b hab,
cases le_total a 0 with ha ha,
{ cases le_or_lt b 0 with hb hb,
{ rw [← neg_lt_neg_iff, ← neg_pow_bit1, ← neg_pow_bit1],
exact pow_lt_pow_of_lt_left (neg_lt_neg hab) (neg_nonneg.2 hb) (bit1_pos (zero_le n)) },
{ exact (pow_bit1_nonpos_iff.2 ha).trans_lt (pow_bit1_pos_iff.2 hb) } },
{ exact pow_lt_pow_of_lt_left hab ha (bit1_pos (zero_le n)) }
end
lemma odd.strict_mono_pow (hn : odd n) : strict_mono (λ a : R, a ^ n) :=
by cases hn with k hk; simpa only [hk, two_mul] using strict_mono_pow_bit1 _
/-- Bernoulli's inequality for `n : ℕ`, `-2 ≤ a`. -/
theorem one_add_mul_le_pow (H : -2 ≤ a) (n : ℕ) : 1 + (n : R) * a ≤ (1 + a) ^ n :=
one_add_mul_le_pow' (mul_self_nonneg _) (mul_self_nonneg _) (neg_le_iff_add_nonneg'.1 H) _
/-- Bernoulli's inequality reformulated to estimate `a^n`. -/
theorem one_add_mul_sub_le_pow (H : -1 ≤ a) (n : ℕ) : 1 + (n : R) * (a - 1) ≤ a ^ n :=
have -2 ≤ a - 1, by rwa [bit0, neg_add, ← sub_eq_add_neg, sub_le_sub_iff_right],
by simpa only [add_sub_cancel'_right] using one_add_mul_le_pow this n
end linear_ordered_ring
/-- Bernoulli's inequality reformulated to estimate `(n : K)`. -/
theorem nat.cast_le_pow_sub_div_sub {K : Type*} [linear_ordered_field K] {a : K} (H : 1 < a)
(n : ℕ) :
(n : K) ≤ (a ^ n - 1) / (a - 1) :=
(le_div_iff (sub_pos.2 H)).2 $ le_sub_left_of_add_le $
one_add_mul_sub_le_pow ((neg_le_self $ @zero_le_one K _).trans H.le) _
/-- For any `a > 1` and a natural `n` we have `n ≤ a ^ n / (a - 1)`. See also
`nat.cast_le_pow_sub_div_sub` for a stronger inequality with `a ^ n - 1` in the numerator. -/
theorem nat.cast_le_pow_div_sub {K : Type*} [linear_ordered_field K] {a : K} (H : 1 < a) (n : ℕ) :
(n : K) ≤ a ^ n / (a - 1) :=
(n.cast_le_pow_sub_div_sub H).trans $ div_le_div_of_le (sub_nonneg.2 H.le)
(sub_le_self _ zero_le_one)
namespace int
lemma units_sq (u : units ℤ) : u ^ 2 = 1 :=
(sq u).symm ▸ units_mul_self u
alias int.units_sq ← int.units_pow_two
lemma units_pow_eq_pow_mod_two (u : units ℤ) (n : ℕ) : u ^ n = u ^ (n % 2) :=
by conv {to_lhs, rw ← nat.mod_add_div n 2}; rw [pow_add, pow_mul, units_sq, one_pow, mul_one]
@[simp] lemma nat_abs_sq (x : ℤ) : (x.nat_abs ^ 2 : ℤ) = x ^ 2 :=
by rw [sq, int.nat_abs_mul_self', sq]
alias int.nat_abs_sq ← int.nat_abs_pow_two
lemma abs_le_self_sq (a : ℤ) : (int.nat_abs a : ℤ) ≤ a ^ 2 :=
by { rw [← int.nat_abs_sq a, sq], norm_cast, apply nat.le_mul_self }
alias int.abs_le_self_sq ← int.abs_le_self_pow_two
lemma le_self_sq (b : ℤ) : b ≤ b ^ 2 := le_trans (le_nat_abs) (abs_le_self_sq _)
alias int.le_self_sq ← int.le_self_pow_two
lemma pow_right_injective {x : ℤ} (h : 1 < x.nat_abs) : function.injective ((^) x : ℕ → ℤ) :=
begin
suffices : function.injective (nat_abs ∘ ((^) x : ℕ → ℤ)),
{ exact function.injective.of_comp this },
convert nat.pow_right_injective h,
ext n,
rw [function.comp_app, nat_abs_pow]
end
end int
variables (M G A)
/-- Monoid homomorphisms from `multiplicative ℕ` are defined by the image
of `multiplicative.of_add 1`. -/
def powers_hom [monoid M] : M ≃ (multiplicative ℕ →* M) :=
{ to_fun := λ x, ⟨λ n, x ^ n.to_add, by { convert pow_zero x, exact to_add_one },
λ m n, pow_add x m n⟩,
inv_fun := λ f, f (multiplicative.of_add 1),
left_inv := pow_one,
right_inv := λ f, monoid_hom.ext $ λ n, by { simp [← f.map_pow, ← of_add_nsmul] } }
/-- Monoid homomorphisms from `multiplicative ℤ` are defined by the image
of `multiplicative.of_add 1`. -/
def zpowers_hom [group G] : G ≃ (multiplicative ℤ →* G) :=
{ to_fun := λ x, ⟨λ n, x ^ n.to_add, zpow_zero x, λ m n, zpow_add x m n⟩,
inv_fun := λ f, f (multiplicative.of_add 1),
left_inv := zpow_one,
right_inv := λ f, monoid_hom.ext $ λ n, by { simp [← f.map_zpow, ← of_add_zsmul ] } }
/-- Additive homomorphisms from `ℕ` are defined by the image of `1`. -/
def multiples_hom [add_monoid A] : A ≃ (ℕ →+ A) :=
{ to_fun := λ x, ⟨λ n, n • x, zero_nsmul x, λ m n, add_nsmul _ _ _⟩,
inv_fun := λ f, f 1,
left_inv := one_nsmul,
right_inv := λ f, add_monoid_hom.ext_nat $ one_nsmul (f 1) }
/-- Additive homomorphisms from `ℤ` are defined by the image of `1`. -/
def zmultiples_hom [add_group A] : A ≃ (ℤ →+ A) :=
{ to_fun := λ x, ⟨λ n, n • x, zero_zsmul x, λ m n, add_zsmul _ _ _⟩,
inv_fun := λ f, f 1,
left_inv := one_zsmul,
right_inv := λ f, add_monoid_hom.ext_int $ one_zsmul (f 1) }
attribute [to_additive multiples_hom] powers_hom
attribute [to_additive zmultiples_hom] zpowers_hom
variables {M G A}
@[simp] lemma powers_hom_apply [monoid M] (x : M) (n : multiplicative ℕ) :
powers_hom M x n = x ^ n.to_add := rfl
@[simp] lemma powers_hom_symm_apply [monoid M] (f : multiplicative ℕ →* M) :
(powers_hom M).symm f = f (multiplicative.of_add 1) := rfl
@[simp] lemma zpowers_hom_apply [group G] (x : G) (n : multiplicative ℤ) :
zpowers_hom G x n = x ^ n.to_add := rfl
@[simp] lemma zpowers_hom_symm_apply [group G] (f : multiplicative ℤ →* G) :
(zpowers_hom G).symm f = f (multiplicative.of_add 1) := rfl
@[simp] lemma multiples_hom_apply [add_monoid A] (x : A) (n : ℕ) :
multiples_hom A x n = n • x := rfl
attribute [to_additive multiples_hom_apply] powers_hom_apply
@[simp] lemma multiples_hom_symm_apply [add_monoid A] (f : ℕ →+ A) :
(multiples_hom A).symm f = f 1 := rfl
attribute [to_additive multiples_hom_symm_apply] powers_hom_symm_apply
@[simp] lemma zmultiples_hom_apply [add_group A] (x : A) (n : ℤ) :
zmultiples_hom A x n = n • x := rfl
attribute [to_additive zmultiples_hom_apply] zpowers_hom_apply
@[simp] lemma zmultiples_hom_symm_apply [add_group A] (f : ℤ →+ A) :
(zmultiples_hom A).symm f = f 1 := rfl
attribute [to_additive zmultiples_hom_symm_apply] zpowers_hom_symm_apply
-- TODO use to_additive in the rest of this file
lemma monoid_hom.apply_mnat [monoid M] (f : multiplicative ℕ →* M) (n : multiplicative ℕ) :
f n = (f (multiplicative.of_add 1)) ^ n.to_add :=
by rw [← powers_hom_symm_apply, ← powers_hom_apply, equiv.apply_symm_apply]
@[ext] lemma monoid_hom.ext_mnat [monoid M] ⦃f g : multiplicative ℕ →* M⦄
(h : f (multiplicative.of_add 1) = g (multiplicative.of_add 1)) : f = g :=
monoid_hom.ext $ λ n, by rw [f.apply_mnat, g.apply_mnat, h]
lemma monoid_hom.apply_mint [group M] (f : multiplicative ℤ →* M) (n : multiplicative ℤ) :
f n = (f (multiplicative.of_add 1)) ^ n.to_add :=
by rw [← zpowers_hom_symm_apply, ← zpowers_hom_apply, equiv.apply_symm_apply]
/-! `monoid_hom.ext_mint` is defined in `data.int.cast` -/
lemma add_monoid_hom.apply_nat [add_monoid M] (f : ℕ →+ M) (n : ℕ) :
f n = n • (f 1) :=
by rw [← multiples_hom_symm_apply, ← multiples_hom_apply, equiv.apply_symm_apply]
/-! `add_monoid_hom.ext_nat` is defined in `data.nat.cast` -/
lemma add_monoid_hom.apply_int [add_group M] (f : ℤ →+ M) (n : ℤ) :
f n = n • (f 1) :=
by rw [← zmultiples_hom_symm_apply, ← zmultiples_hom_apply, equiv.apply_symm_apply]
/-! `add_monoid_hom.ext_int` is defined in `data.int.cast` -/
variables (M G A)
/-- If `M` is commutative, `powers_hom` is a multiplicative equivalence. -/
def powers_mul_hom [comm_monoid M] : M ≃* (multiplicative ℕ →* M) :=
{ map_mul' := λ a b, monoid_hom.ext $ by simp [mul_pow],
..powers_hom M}
/-- If `M` is commutative, `zpowers_hom` is a multiplicative equivalence. -/
def zpowers_mul_hom [comm_group G] : G ≃* (multiplicative ℤ →* G) :=
{ map_mul' := λ a b, monoid_hom.ext $ by simp [mul_zpow],
..zpowers_hom G}
/-- If `M` is commutative, `multiples_hom` is an additive equivalence. -/
def multiples_add_hom [add_comm_monoid A] : A ≃+ (ℕ →+ A) :=
{ map_add' := λ a b, add_monoid_hom.ext $ by simp [nsmul_add],
..multiples_hom A}
/-- If `M` is commutative, `zmultiples_hom` is an additive equivalence. -/
def zmultiples_add_hom [add_comm_group A] : A ≃+ (ℤ →+ A) :=
{ map_add' := λ a b, add_monoid_hom.ext $ by simp [zsmul_add],
..zmultiples_hom A}
variables {M G A}
@[simp] lemma powers_mul_hom_apply [comm_monoid M] (x : M) (n : multiplicative ℕ) :
powers_mul_hom M x n = x ^ n.to_add := rfl
@[simp] lemma powers_mul_hom_symm_apply [comm_monoid M] (f : multiplicative ℕ →* M) :
(powers_mul_hom M).symm f = f (multiplicative.of_add 1) := rfl
@[simp] lemma zpowers_mul_hom_apply [comm_group G] (x : G) (n : multiplicative ℤ) :
zpowers_mul_hom G x n = x ^ n.to_add := rfl
@[simp] lemma zpowers_mul_hom_symm_apply [comm_group G] (f : multiplicative ℤ →* G) :
(zpowers_mul_hom G).symm f = f (multiplicative.of_add 1) := rfl
@[simp] lemma multiples_add_hom_apply [add_comm_monoid A] (x : A) (n : ℕ) :
multiples_add_hom A x n = n • x := rfl
@[simp] lemma multiples_add_hom_symm_apply [add_comm_monoid A] (f : ℕ →+ A) :
(multiples_add_hom A).symm f = f 1 := rfl
@[simp] lemma zmultiples_add_hom_apply [add_comm_group A] (x : A) (n : ℤ) :
zmultiples_add_hom A x n = n • x := rfl
@[simp] lemma zmultiples_add_hom_symm_apply [add_comm_group A] (f : ℤ →+ A) :
(zmultiples_add_hom A).symm f = f 1 := rfl
/-!
### Commutativity (again)
Facts about `semiconj_by` and `commute` that require `zpow` or `zsmul`, or the fact that integer
multiplication equals semiring multiplication.
-/
namespace semiconj_by
section
variables [semiring R] {a x y : R}
@[simp] lemma cast_nat_mul_right (h : semiconj_by a x y) (n : ℕ) :
semiconj_by a ((n : R) * x) (n * y) :=
semiconj_by.mul_right (nat.commute_cast _ _) h
@[simp] lemma cast_nat_mul_left (h : semiconj_by a x y) (n : ℕ) : semiconj_by ((n : R) * a) x y :=
semiconj_by.mul_left (nat.cast_commute _ _) h
@[simp] lemma cast_nat_mul_cast_nat_mul (h : semiconj_by a x y) (m n : ℕ) :
semiconj_by ((m : R) * a) (n * x) (n * y) :=
(h.cast_nat_mul_left m).cast_nat_mul_right n
end
variables [monoid M] [group G] [ring R]
@[simp, to_additive] lemma units_zpow_right {a : M} {x y : units M} (h : semiconj_by a x y) :
∀ m : ℤ, semiconj_by a (↑(x^m)) (↑(y^m))
| (n : ℕ) := by simp only [zpow_coe_nat, units.coe_pow, h, pow_right]
| -[1+n] := by simp only [zpow_neg_succ_of_nat, units.coe_pow, units_inv_right, h, pow_right]
variables {a b x y x' y' : R}
@[simp] lemma cast_int_mul_right (h : semiconj_by a x y) (m : ℤ) :
semiconj_by a ((m : ℤ) * x) (m * y) :=
semiconj_by.mul_right (int.commute_cast _ _) h
@[simp] lemma cast_int_mul_left (h : semiconj_by a x y) (m : ℤ) : semiconj_by ((m : R) * a) x y :=
semiconj_by.mul_left (int.cast_commute _ _) h
@[simp] lemma cast_int_mul_cast_int_mul (h : semiconj_by a x y) (m n : ℤ) :
semiconj_by ((m : R) * a) (n * x) (n * y) :=
(h.cast_int_mul_left m).cast_int_mul_right n
end semiconj_by
namespace commute
section
variables [semiring R] {a b : R}
@[simp] theorem cast_nat_mul_right (h : commute a b) (n : ℕ) : commute a ((n : R) * b) :=
h.cast_nat_mul_right n
@[simp] theorem cast_nat_mul_left (h : commute a b) (n : ℕ) : commute ((n : R) * a) b :=
h.cast_nat_mul_left n
@[simp] theorem cast_nat_mul_cast_nat_mul (h : commute a b) (m n : ℕ) :
commute ((m : R) * a) (n * b) :=
h.cast_nat_mul_cast_nat_mul m n
@[simp] theorem self_cast_nat_mul (n : ℕ) : commute a (n * a) :=
(commute.refl a).cast_nat_mul_right n
@[simp] theorem cast_nat_mul_self (n : ℕ) : commute ((n : R) * a) a :=
(commute.refl a).cast_nat_mul_left n
@[simp] theorem self_cast_nat_mul_cast_nat_mul (m n : ℕ) : commute ((m : R) * a) (n * a) :=
(commute.refl a).cast_nat_mul_cast_nat_mul m n
end
variables [monoid M] [group G] [ring R]
@[simp, to_additive] lemma units_zpow_right {a : M} {u : units M} (h : commute a u) (m : ℤ) :
commute a (↑(u^m)) :=
h.units_zpow_right m
@[simp, to_additive] lemma units_zpow_left {u : units M} {a : M} (h : commute ↑u a) (m : ℤ) :
commute (↑(u^m)) a :=
(h.symm.units_zpow_right m).symm
variables {a b : R}
@[simp] lemma cast_int_mul_right (h : commute a b) (m : ℤ) : commute a (m * b) :=
h.cast_int_mul_right m
@[simp] lemma cast_int_mul_left (h : commute a b) (m : ℤ) : commute ((m : R) * a) b :=
h.cast_int_mul_left m
lemma cast_int_mul_cast_int_mul (h : commute a b) (m n : ℤ) : commute ((m : R) * a) (n * b) :=
h.cast_int_mul_cast_int_mul m n
variables (a) (m n : ℤ)
@[simp] lemma cast_int_left : commute (m : R) a :=
by { rw [← mul_one (m : R)], exact (one_left a).cast_int_mul_left m }
@[simp] lemma cast_int_right : commute a m :=
by { rw [← mul_one (m : R)], exact (one_right a).cast_int_mul_right m }
@[simp] theorem self_cast_int_mul : commute a (n * a) := (commute.refl a).cast_int_mul_right n
@[simp] theorem cast_int_mul_self : commute ((n : R) * a) a := (commute.refl a).cast_int_mul_left n
theorem self_cast_int_mul_cast_int_mul : commute ((m : R) * a) (n * a) :=
(commute.refl a).cast_int_mul_cast_int_mul m n
end commute
section multiplicative
open multiplicative
@[simp] lemma nat.to_add_pow (a : multiplicative ℕ) (b : ℕ) : to_add (a ^ b) = to_add a * b :=
begin
induction b with b ih,
{ erw [pow_zero, to_add_one, mul_zero] },
{ simp [*, pow_succ, add_comm, nat.mul_succ] }
end
@[simp] lemma nat.of_add_mul (a b : ℕ) : of_add (a * b) = of_add a ^ b :=
(nat.to_add_pow _ _).symm
@[simp] lemma int.to_add_pow (a : multiplicative ℤ) (b : ℕ) : to_add (a ^ b) = to_add a * b :=
by induction b; simp [*, mul_add, pow_succ, add_comm]
@[simp] lemma int.to_add_zpow (a : multiplicative ℤ) (b : ℤ) : to_add (a ^ b) = to_add a * b :=
int.induction_on b (by simp)
(by simp [zpow_add, mul_add] {contextual := tt})
(by simp [zpow_add, mul_add, sub_eq_add_neg, -int.add_neg_one] {contextual := tt})
@[simp] lemma int.of_add_mul (a b : ℤ) : of_add (a * b) = of_add a ^ b :=
(int.to_add_zpow _ _).symm
end multiplicative
namespace units
variables [monoid M]
lemma conj_pow (u : units M) (x : M) (n : ℕ) : (↑u * x * ↑(u⁻¹))^n = u * x^n * ↑(u⁻¹) :=
(divp_eq_iff_mul_eq.2 ((u.mk_semiconj_by x).pow_right n).eq.symm).symm
lemma conj_pow' (u : units M) (x : M) (n : ℕ) : (↑(u⁻¹) * x * u)^n = ↑(u⁻¹) * x^n * u:=
(u⁻¹).conj_pow x n
end units
namespace opposite
/-- Moving to the opposite monoid commutes with taking powers. -/
@[simp] lemma op_pow [monoid M] (x : M) (n : ℕ) : op (x ^ n) = (op x) ^ n := rfl
@[simp] lemma unop_pow [monoid M] (x : Mᵒᵖ) (n : ℕ) : unop (x ^ n) = (unop x) ^ n := rfl
/-- Moving to the opposite group or group_with_zero commutes with taking powers. -/
@[simp] lemma op_zpow [div_inv_monoid M] (x : M) (z : ℤ) : op (x ^ z) = (op x) ^ z := rfl
@[simp] lemma unop_zpow [div_inv_monoid M] (x : Mᵒᵖ) (z : ℤ) : unop (x ^ z) = (unop x) ^ z := rfl
end opposite
|
0913dda796ab69d63e1d37de8b2376ea69ff4a8a | e151e9053bfd6d71740066474fc500a087837323 | /src/hott/eq2.lean | 40dea371bb5e874f56285343efc30ccb074a6f72 | [
"Apache-2.0"
] | permissive | daniel-carranza/hott3 | 15bac2d90589dbb952ef15e74b2837722491963d | 913811e8a1371d3a5751d7d32ff9dec8aa6815d9 | refs/heads/master | 1,610,091,349,670 | 1,596,222,336,000 | 1,596,222,336,000 | 241,957,822 | 0 | 0 | Apache-2.0 | 1,582,222,839,000 | 1,582,222,838,000 | null | UTF-8 | Lean | false | false | 9,479 | 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
Theorems about 2-dimensional paths
-/
import .cubical.square
universes u v w
hott_theory
namespace hott
open function hott.is_equiv hott.equiv
namespace eq
variables {A : Type _} {B : Type _} {C : Type _} {f : A → B} {a a' a₁ a₂ a₃ a₄ : A} {b b' : B}
@[hott] theorem ap_is_constant_eq (p : Πx, f x = b) (q : a = a') :
ap_is_constant f p q =
eq_con_inv_of_con_eq ((eq_of_square (square_of_pathover (apd p q)))⁻¹ ⬝
whisker_left (p a) (ap_constant q b)) :=
begin
induction q, dsimp [ap_constant, ap_is_constant, apd],
hinduction p a, refl
end
@[hott] def ap_inv2 {p q : a = a'} (r : p = q)
: square (ap (ap f) (inverse2 r))
(inverse2 (ap (ap f) r))
(ap_inv f p)
(ap_inv f q) :=
by induction r;exact hrfl
@[hott] def ap_con2 {p₁ q₁ : a₁ = a₂} {p₂ q₂ : a₂ = a₃} (r₁ : p₁ = q₁) (r₂ : p₂ = q₂)
: square (ap (ap f) (r₁ ◾ r₂))
(ap (ap f) r₁ ◾ ap (ap f) r₂)
(ap_con f p₁ p₂)
(ap_con f q₁ q₂) :=
by induction r₂;induction r₁;exact hrfl
@[hott] theorem ap_con_right_inv_sq {A B : Type _} {a1 a2 : A} (f : A → B) (p : a1 = a2) :
square (ap (ap f) (con.right_inv p))
(con.right_inv (ap f p))
(ap_con f p p⁻¹ ⬝ whisker_left _ (ap_inv f p))
idp :=
by induction p;apply hrefl
@[hott] theorem ap_con_left_inv_sq {A B : Type _} {a1 a2 : A} (f : A → B) (p : a1 = a2) :
square (ap (ap f) (con.left_inv p))
(con.left_inv (ap f p))
(ap_con f p⁻¹ p ⬝ whisker_right _ (ap_inv f p))
idp :=
by induction p;apply vrefl
@[hott] def ap02_compose {A B C : Type _} (g : B → C) (f : A → B) {a a' : A}
{p₁ p₂ : a = a'} (q : p₁ = p₂) :
square (ap_compose g f p₁) (ap_compose g f p₂) (ap02 (g ∘ f) q) (ap02 g (ap02 f q)) :=
by induction q; exact vrfl
@[hott] def ap02_id {A : Type _} {a a' : A}
{p₁ p₂ : a = a'} (q : p₁ = p₂) :
square (ap_id p₁) (ap_id p₂) (ap02 id q) q :=
by induction q; exact vrfl
@[hott] theorem ap_ap_is_constant {A B C : Type _} (g : B → C) {f : A → B} {b : B}
(p : Πx, f x = b) {x y : A} (q : x = y) :
square (ap (ap g) (ap_is_constant f p q))
(by exact (ap_is_constant (g ∘ f) (λa, ap g (p a)) q))
(ap_compose g f q)⁻¹
(ap_con _ _ _ ⬝ whisker_left _ (ap_inv _ _)) :=
begin
induction q, dsimp [ap_is_constant], hinduction (p x), apply ids
end
@[hott] theorem ap_ap_compose {A B C D : Type _} (h : C → D) (g : B → C) (f : A → B)
{x y : A} (p : x = y) :
square (ap_compose (h ∘ g) f p)
(ap (ap h) (ap_compose g f p))
(ap_compose h (g ∘ f) p)
(ap_compose h g (ap f p)) :=
by induction p; exact ids
@[hott] def ap_compose_inv {A B C : Type _} (g : B → C) (f : A → B)
{x y : A} (p : x = y) :
square (ap_compose g f p⁻¹)
(inverse2 (ap_compose g f p) ⬝ (ap_inv g (ap f p))⁻¹)
(ap_inv (g ∘ f) p)
(ap (ap g) (ap_inv f p)) :=
by induction p; exact ids
@[hott] def ap_compose_con (g : B → C) (f : A → B) (p : a₁ = a₂) (q : a₂ = a₃) :
square (ap_compose g f (p ⬝ q))
(ap_compose g f p ◾ ap_compose g f q ⬝ (ap_con g (ap f p) (ap f q))⁻¹)
(ap_con (g ∘ f) p q)
(ap (ap g) (ap_con f p q)) :=
by induction q; induction p; exact ids
@[hott] theorem ap_compose_natural {A B C : Type _} (g : B → C) (f : A → B)
{x y : A} {p q : x = y} (r : p = q) :
square (ap (ap (g ∘ f)) r)
(ap (ap g ∘ ap f) r)
(ap_compose g f p)
(ap_compose g f q) :=
natural_square_tr (ap_compose g f) r
@[hott] theorem whisker_right_eq_of_con_inv_eq_idp {p q : a₁ = a₂} (r : p ⬝ q⁻¹ = idp) :
whisker_right q⁻¹ (eq_of_con_inv_eq_idp r) ⬝ con.right_inv q = r :=
begin induction q, hinduction r using eq.rec_symm, reflexivity end
@[hott] theorem ap_eq_of_con_inv_eq_idp (f : A → B) {p q : a₁ = a₂} (r : p ⬝ q⁻¹ = idp)
: ap02 f (eq_of_con_inv_eq_idp r) =
eq_of_con_inv_eq_idp (whisker_left _ (ap_inv _ _)⁻¹ ⬝ (ap_con _ _ _)⁻¹ ⬝ ap02 f r) :=
by induction q; hinduction r using eq.rec_symm; reflexivity
@[hott] theorem eq_of_con_inv_eq_idp_con2 {p p' q q' : a₁ = a₂} (r : p = p') (s : q = q')
(t : p' ⬝ q'⁻¹ = idp)
: eq_of_con_inv_eq_idp (r ◾ inverse2 s ⬝ t) = r ⬝ eq_of_con_inv_eq_idp t ⬝ s⁻¹ :=
by induction s; induction r; induction q; reflexivity
@[hott] def naturality_apd_eq {A : Type _} {B : A → Type _} {a a₂ : A} {f g : Πa, B a}
(H : f ~ g) (p : a = a₂)
: apd f p = concato_eq (eq_concato (H a) (apd g p)) (H a₂)⁻¹ :=
begin
induction p, dsimp,
hgeneralize : H a = p, revert p,
hgeneralize : g a = x, intro p,
induction p,
reflexivity
end
@[hott] theorem con_tr_idp {P : A → Type _} {x y : A} (q : x = y) (u : P x) :
con_tr idp q u = ap (λp, p ▸ u) (idp_con q) :=
by induction q;reflexivity
@[hott] def whisker_left_idp_con_eq_assoc
{A : Type _} {a₁ a₂ a₃ : A} (p : a₁ = a₂) (q : a₂ = a₃)
: whisker_left p (idp_con q)⁻¹ = con.assoc p idp q :=
by induction q; reflexivity
@[hott] def whisker_left_inverse2 {A : Type _} {a : A} {p : a = a} (q : p = idp)
: whisker_left p q⁻² ⬝ q = con.right_inv p :=
by hinduction q using eq.rec_symm; reflexivity
@[hott] def cast_fn_cast_square {A : Type _} {B C : A → Type _} (f : Π⦃a⦄, B a → C a) {a₁ a₂ : A}
(p : a₁ = a₂) (q : a₂ = a₁) (r : p ⬝ q = idp) (b : B a₁) :
cast (ap C q) (f (cast (ap B p) b)) = f b :=
have q⁻¹ = p, from inv_eq_of_idp_eq_con r⁻¹,
begin induction this, induction q, reflexivity end
@[hott] def ap011_ap_square_right {A B C : Type _} (f : A → B → C) {a a' : A} (p : a = a')
{b₁ b₂ b₃ : B} {q₁₂ : b₁ = b₂} {q₂₃ : b₂ = b₃} {q₁₃ : b₁ = b₃} (r : q₁₂ ⬝ q₂₃ = q₁₃) :
square (ap011 f p q₁₂) (ap (λx, f x b₃) p) (ap (f a) q₁₃) (ap (f a') q₂₃) :=
by induction r; induction q₂₃; induction q₁₂; induction p; exact ids
@[hott] def ap011_ap_square_left {A B C : Type _} (f : B → A → C) {a a' : A} (p : a = a')
{b₁ b₂ b₃ : B} {q₁₂ : b₁ = b₂} {q₂₃ : b₂ = b₃} {q₁₃ : b₁ = b₃} (r : q₁₂ ⬝ q₂₃ = q₁₃) :
square (ap011 f q₁₂ p) (ap (f b₃) p) (ap (λx, f x a) q₁₃) (ap (λx, f x a') q₂₃) :=
by induction r; induction q₂₃; induction q₁₂; induction p; exact ids
@[hott] def con2_assoc {A : Type _} {x y z t : A} {p p' : x = y} {q q' : y = z} {r r' : z = t}
(h : p = p') (h' : q = q') (h'' : r = r') :
square ((h ◾ h') ◾ h'') (h ◾ (h' ◾ h'')) (con.assoc p q r) (con.assoc p' q' r') :=
by induction h; induction h'; induction h''; exact hrfl
@[hott] def con_left_inv_idp {A : Type _} {x : A} {p : x = x} (q : p = idp)
: con.left_inv p = q⁻² ◾ q :=
by hinduction q using eq.rec_symm; reflexivity
@[hott] def eckmann_hilton_con2 {A : Type _} {x : A} {p p' q q': idp = idp :> x = x}
(h : p = p') (h' : q = q') : square (h ◾ h') (h' ◾ h) (eckmann_hilton p q) (eckmann_hilton p' q') :=
by induction h; induction h'; exact hrfl
@[hott] def ap_con_fn {A B : Type _} {a a' : A} {b : B} (g h : A → b = b) (p : a = a') :
ap (λa, g a ⬝ h a) p = ap g p ◾ ap h p :=
by induction p; reflexivity
@[hott] def ap_eq_ap011 {A B C X : Type _} (f : A → B → C) (g : X → A) (h : X → B) {x x' : X}
(p : x = x') : ap (λx, f (g x) (h x)) p = ap011 f (ap g p) (ap h p) :=
by induction p; reflexivity
@[hott] def ap_is_weakly_constant {A B : Type _} {f : A → B}
(h : is_weakly_constant f) {a a' : A} (p : a = a') : ap f p = (h a a)⁻¹ ⬝ h a a' :=
by induction p; exact (con.left_inv _)⁻¹
@[hott] def ap_is_constant_idp {A B : Type _} {f : A → B} {b : B} (p : Πa, f a = b) {a : A} (q : a = a)
(r : q = idp) : ap_is_constant f p q = ap02 f r ⬝ (con.right_inv (p a))⁻¹ :=
by hinduction r using eq.rec_symm; exact (idp_con _)⁻¹
@[hott] def con_right_inv_natural {A : Type _} {a a' : A} {p p' : a = a'} (q : p = p') :
con.right_inv p = q ◾ q⁻² ⬝ con.right_inv p' :=
by induction q; induction p; reflexivity
@[hott] def whisker_right_ap {A B : Type _} {a a' : A}{b₁ b₂ b₃ : B} (q : b₂ = b₃) (f : A → b₁ = b₂)
(p : a = a') : whisker_right q (ap f p) = ap (λa, f a ⬝ q) p :=
by induction p; reflexivity
@[hott] def ap02_ap_constant {A B C : Type _} {a a' : A} (f : B → C) (b : B) (p : a = a') :
square (ap_constant p (f b)) (ap02 f (ap_constant p b)) (ap_compose f (λx, b) p) idp :=
by induction p; exact ids
@[hott] def ap_constant_compose {A B C : Type _} {a a' : A} (c : C) (f : A → B) (p : a = a') :
square (ap_constant p c) (ap_constant (ap f p) c) (ap_compose (λx, c) f p) idp :=
by induction p; exact ids
@[hott] def ap02_constant {A B : Type _} {a a' : A} (b : B) {p p' : a = a'}
(q : p = p') : square (ap_constant p b) (ap_constant p' b) (ap02 (λx, b) q) idp :=
by induction q; exact vrfl
end eq
end hott |
ddcb7ff0b64f9f20900d9495fed906b4d3c5695c | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/topology/list.lean | 0b2106f3a7affc80184d92340f12ed5660649401 | [
"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 | 8,722 | lean | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import topology.constructions
import topology.algebra.monoid
/-!
# Topology on lists and vectors
-/
open topological_space set filter
open_locale topological_space filter
variables {α : Type*} {β : Type*} [topological_space α] [topological_space β]
instance : topological_space (list α) :=
topological_space.mk_of_nhds (traverse nhds)
lemma nhds_list (as : list α) : 𝓝 as = traverse 𝓝 as :=
begin
refine nhds_mk_of_nhds _ _ _ _,
{ assume l, induction l,
case list.nil { exact le_rfl },
case list.cons : a l ih
{ suffices : list.cons <$> pure a <*> pure l ≤ list.cons <$> 𝓝 a <*> traverse 𝓝 l,
{ simpa only [] with functor_norm using this },
exact filter.seq_mono (filter.map_mono $ pure_le_nhds a) ih } },
{ assume l s hs,
rcases (mem_traverse_iff _ _).1 hs with ⟨u, hu, hus⟩, clear as hs,
have : ∃v:list (set α), l.forall₂ (λa s, is_open s ∧ a ∈ s) v ∧ sequence v ⊆ s,
{ induction hu generalizing s,
case list.forall₂.nil : hs this
{ existsi [], simpa only [list.forall₂_nil_left_iff, exists_eq_left] },
case list.forall₂.cons : a s as ss ht h ih t hts
{ rcases mem_nhds_iff.1 ht with ⟨u, hut, hu⟩,
rcases ih (subset.refl _) with ⟨v, hv, hvss⟩,
exact ⟨u::v, list.forall₂.cons hu hv,
subset.trans (set.seq_mono (set.image_subset _ hut) hvss) hts⟩ } },
rcases this with ⟨v, hv, hvs⟩,
refine ⟨sequence v, mem_traverse _ _ _, hvs, _⟩,
{ exact hv.imp (assume a s ⟨hs, ha⟩, is_open.mem_nhds hs ha) },
{ assume u hu,
have hu := (list.mem_traverse _ _).1 hu,
have : list.forall₂ (λa s, is_open s ∧ a ∈ s) u v,
{ refine list.forall₂.flip _,
replace hv := hv.flip,
simp only [list.forall₂_and_left, flip] at ⊢ hv,
exact ⟨hv.1, hu.flip⟩ },
refine mem_of_superset _ hvs,
exact mem_traverse _ _ (this.imp $ assume a s ⟨hs, ha⟩, is_open.mem_nhds hs ha) } }
end
@[simp] lemma nhds_nil : 𝓝 ([] : list α) = pure [] :=
by rw [nhds_list, list.traverse_nil _]; apply_instance
lemma nhds_cons (a : α) (l : list α) :
𝓝 (a :: l) = list.cons <$> 𝓝 a <*> 𝓝 l :=
by rw [nhds_list, list.traverse_cons _, ← nhds_list]; apply_instance
lemma list.tendsto_cons {a : α} {l : list α} :
tendsto (λp:α×list α, list.cons p.1 p.2) (𝓝 a ×ᶠ 𝓝 l) (𝓝 (a :: l)) :=
by rw [nhds_cons, tendsto, map_prod]; exact le_rfl
lemma filter.tendsto.cons {α : Type*} {f : α → β} {g : α → list β}
{a : _root_.filter α} {b : β} {l : list β} (hf : tendsto f a (𝓝 b)) (hg : tendsto g a (𝓝 l)) :
tendsto (λa, list.cons (f a) (g a)) a (𝓝 (b :: l)) :=
list.tendsto_cons.comp (tendsto.prod_mk hf hg)
namespace list
lemma tendsto_cons_iff {β : Type*} {f : list α → β} {b : _root_.filter β} {a : α} {l : list α} :
tendsto f (𝓝 (a :: l)) b ↔ tendsto (λp:α×list α, f (p.1 :: p.2)) (𝓝 a ×ᶠ 𝓝 l) b :=
have 𝓝 (a :: l) = (𝓝 a ×ᶠ 𝓝 l).map (λp:α×list α, (p.1 :: p.2)),
begin
simp only
[nhds_cons, filter.prod_eq, (filter.map_def _ _).symm, (filter.seq_eq_filter_seq _ _).symm],
simp [-filter.seq_eq_filter_seq, -filter.map_def, (∘)] with functor_norm,
end,
by rw [this, filter.tendsto_map'_iff]
lemma continuous_cons : continuous (λ x : α × list α, (x.1 :: x.2 : list α)) :=
continuous_iff_continuous_at.mpr $ λ ⟨x, y⟩, continuous_at_fst.cons continuous_at_snd
lemma tendsto_nhds {β : Type*} {f : list α → β} {r : list α → _root_.filter β}
(h_nil : tendsto f (pure []) (r []))
(h_cons : ∀l a, tendsto f (𝓝 l) (r l) →
tendsto (λp:α×list α, f (p.1 :: p.2)) (𝓝 a ×ᶠ 𝓝 l) (r (a::l))) :
∀l, tendsto f (𝓝 l) (r l)
| [] := by rwa [nhds_nil]
| (a::l) := by rw [tendsto_cons_iff]; exact h_cons l a (tendsto_nhds l)
lemma continuous_at_length :
∀(l : list α), continuous_at list.length l :=
begin
simp only [continuous_at, nhds_discrete],
refine tendsto_nhds _ _,
{ exact tendsto_pure_pure _ _ },
{ assume l a ih,
dsimp only [list.length],
refine tendsto.comp (tendsto_pure_pure (λx, x + 1) _) _,
refine tendsto.comp ih tendsto_snd }
end
lemma tendsto_insert_nth' {a : α} : ∀{n : ℕ} {l : list α},
tendsto (λp:α×list α, insert_nth n p.1 p.2) (𝓝 a ×ᶠ 𝓝 l) (𝓝 (insert_nth n a l))
| 0 l := tendsto_cons
| (n+1) [] := by simp
| (n+1) (a'::l) :=
have 𝓝 a ×ᶠ 𝓝 (a' :: l) =
(𝓝 a ×ᶠ (𝓝 a' ×ᶠ 𝓝 l)).map (λp:α×α×list α, (p.1, p.2.1 :: p.2.2)),
begin
simp only [nhds_cons, filter.prod_eq, ← filter.map_def, ← filter.seq_eq_filter_seq],
simp [-filter.seq_eq_filter_seq, -filter.map_def, (∘)] with functor_norm
end,
begin
rw [this, tendsto_map'_iff],
exact (tendsto_fst.comp tendsto_snd).cons
((@tendsto_insert_nth' n l).comp $ tendsto_fst.prod_mk $ tendsto_snd.comp tendsto_snd)
end
lemma tendsto_insert_nth {β} {n : ℕ} {a : α} {l : list α} {f : β → α} {g : β → list α}
{b : _root_.filter β} (hf : tendsto f b (𝓝 a)) (hg : tendsto g b (𝓝 l)) :
tendsto (λb:β, insert_nth n (f b) (g b)) b (𝓝 (insert_nth n a l)) :=
tendsto_insert_nth'.comp (tendsto.prod_mk hf hg)
lemma continuous_insert_nth {n : ℕ} : continuous (λp:α×list α, insert_nth n p.1 p.2) :=
continuous_iff_continuous_at.mpr $
assume ⟨a, l⟩, by rw [continuous_at, nhds_prod_eq]; exact tendsto_insert_nth'
lemma tendsto_remove_nth : ∀{n : ℕ} {l : list α},
tendsto (λl, remove_nth l n) (𝓝 l) (𝓝 (remove_nth l n))
| _ [] := by rw [nhds_nil]; exact tendsto_pure_nhds _ _
| 0 (a::l) := by rw [tendsto_cons_iff]; exact tendsto_snd
| (n+1) (a::l) :=
begin
rw [tendsto_cons_iff],
dsimp [remove_nth],
exact tendsto_fst.cons ((@tendsto_remove_nth n l).comp tendsto_snd)
end
lemma continuous_remove_nth {n : ℕ} : continuous (λl : list α, remove_nth l n) :=
continuous_iff_continuous_at.mpr $ assume a, tendsto_remove_nth
@[to_additive]
lemma tendsto_prod [monoid α] [has_continuous_mul α] {l : list α} :
tendsto list.prod (𝓝 l) (𝓝 l.prod) :=
begin
induction l with x l ih,
{ simp [nhds_nil, mem_of_mem_nhds, tendsto_pure_left] {contextual := tt} },
simp_rw [tendsto_cons_iff, prod_cons],
have := continuous_iff_continuous_at.mp continuous_mul (x, l.prod),
rw [continuous_at, nhds_prod_eq] at this,
exact this.comp (tendsto_id.prod_map ih)
end
@[to_additive]
lemma continuous_prod [monoid α] [has_continuous_mul α] : continuous (prod : list α → α) :=
continuous_iff_continuous_at.mpr $ λ l, tendsto_prod
end list
namespace vector
open list
instance (n : ℕ) : topological_space (vector α n) :=
by unfold vector; apply_instance
lemma tendsto_cons {n : ℕ} {a : α} {l : vector α n}:
tendsto (λp:α×vector α n, p.1 ::ᵥ p.2) (𝓝 a ×ᶠ 𝓝 l) (𝓝 (a ::ᵥ l)) :=
by { simp [tendsto_subtype_rng, ←subtype.val_eq_coe, cons_val],
exact tendsto_fst.cons (tendsto.comp continuous_at_subtype_coe tendsto_snd) }
lemma tendsto_insert_nth
{n : ℕ} {i : fin (n+1)} {a:α} :
∀{l:vector α n}, tendsto (λp:α×vector α n, insert_nth p.1 i p.2)
(𝓝 a ×ᶠ 𝓝 l) (𝓝 (insert_nth a i l))
| ⟨l, hl⟩ :=
begin
rw [insert_nth, tendsto_subtype_rng],
simp [insert_nth_val],
exact list.tendsto_insert_nth tendsto_fst (tendsto.comp continuous_at_subtype_coe tendsto_snd : _)
end
lemma continuous_insert_nth' {n : ℕ} {i : fin (n+1)} :
continuous (λp:α×vector α n, insert_nth p.1 i p.2) :=
continuous_iff_continuous_at.mpr $ assume ⟨a, l⟩,
by rw [continuous_at, nhds_prod_eq]; exact tendsto_insert_nth
lemma continuous_insert_nth {n : ℕ} {i : fin (n+1)}
{f : β → α} {g : β → vector α n} (hf : continuous f) (hg : continuous g) :
continuous (λb, insert_nth (f b) i (g b)) :=
continuous_insert_nth'.comp (hf.prod_mk hg : _)
lemma continuous_at_remove_nth {n : ℕ} {i : fin (n+1)} :
∀{l:vector α (n+1)}, continuous_at (remove_nth i) l
| ⟨l, hl⟩ :=
-- ∀{l:vector α (n+1)}, tendsto (remove_nth i) (𝓝 l) (𝓝 (remove_nth i l))
--| ⟨l, hl⟩ :=
begin
rw [continuous_at, remove_nth, tendsto_subtype_rng],
simp only [← subtype.val_eq_coe, vector.remove_nth_val],
exact tendsto.comp list.tendsto_remove_nth continuous_at_subtype_coe,
end
lemma continuous_remove_nth {n : ℕ} {i : fin (n+1)} :
continuous (remove_nth i : vector α (n+1) → vector α n) :=
continuous_iff_continuous_at.mpr $ assume ⟨a, l⟩, continuous_at_remove_nth
end vector
|
6079b8bb77b46638207e8a49eaeb7c317e8afab9 | a9fe717b93ccfa4b2e64faeb24f96dfefb390240 | /main.lean | e0ce6b5fff22d8b2416f78163dad628f011c3a82 | [] | no_license | skbaek/omega | ab1f4a6daadfc8c855f14c39d9459ab841527141 | 715e384ed14e8eb177a326700066e7c98269e078 | refs/heads/master | 1,588,000,876,352 | 1,552,645,917,000 | 1,552,645,917,000 | 174,442,914 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 113 | lean | import .is_lia .int.main .nat.main
meta def omega : tactic unit :=
monad.cond is_lia_goal int.omega nat.omega
|
f908b66b0286879005a70a20bd54ad126357f46b | e21db629d2e37a833531fdcb0b37ce4d71825408 | /src/parlang/defs.lean | a7631a11836e45907589e4b54544b247453c8a64 | [] | 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 | 8,423 | lean | import data.list.basic -- basic operations on `list`
import data.option.basic -- basic operations on `option`
import data.set.basic
import data.vector
import data.vector2
import logic.function -- function update and inverses
import aux
namespace parlang
variables {n : ℕ} {σ : Type} {ι : Type} {τ : ι → Type} [decidable_eq ι]
/-
We use the following conventions for type variables:
`σ` -- thread internal states
`ι` -- shared memory index
`τ` -- shared memory type map
-/
/-- Kernel of a parallel program.
The general idea is to not have explicit expressions, but use Lean functions to compute values. What
we are explicit shared loads and stores.
Σ is constructor where the second argument may depend on the type of the first (in this case i). Can be constructed using ⟨...⟩
-/
inductive kernel {ι : Type} (σ : Type) (τ : ι → Type) : Type
| load : (σ → (Σi:ι, (τ i → σ))) → kernel
| store : (σ → (Σi:ι, τ i)) → kernel
| compute {} : (σ → σ) → kernel
| seq : kernel → kernel → kernel
| ite : (σ → bool) → kernel → kernel → kernel
| loop : (σ → bool) → kernel → kernel
| sync {} : kernel
infixr ` ;; `:90 := kernel.seq
open kernel
/-- Memory view -/
def memory {ι : Type} (τ : ι → Type) := Π (i : ι), τ i
namespace memory
def get (m : memory τ) (i : ι) : τ i := m i
def update (m : memory τ) (i : ι) (v : τ i) : memory τ := function.update m i v
end memory
/-- Thread state inclusing a shared memory *view*, the list of loads and stores tells what should
differ between differnet threads. -/
structure thread_state {ι : Type} (σ : Type) (τ : ι → Type) : Type :=
(tlocal : σ)
(shared : memory τ)
(loads : set ι := ∅)
(stores : set ι := ∅)
namespace thread_state
def load (f : σ → (Σi:ι, (τ i → σ))) (t : thread_state σ τ) : thread_state σ τ :=
let ⟨i, tr⟩ := f t.tlocal in
{ tlocal := tr (t.shared.get i),
loads := insert i t.loads,
.. t }
def store (f : σ → (Σi:ι, τ i)) (t : thread_state σ τ) : thread_state σ τ :=
let ⟨i, v⟩ := f t.tlocal in
{ shared := t.shared.update i v,
stores := insert i t.stores,
.. t}
def compute (f : σ → σ) (t : thread_state σ τ) : thread_state σ τ :=
{ tlocal := f t.tlocal,
.. t}
def sync (g : memory τ) (t : thread_state σ τ) : thread_state σ τ :=
{ shared := g,
loads := ∅,
stores := ∅,
.. t}
def accesses (t : thread_state σ τ) : set ι := t.stores ∪ t.loads
end thread_state
def no_thread_active (ac : vector bool n) : bool := ¬ac.to_list.any id
def any_thread_active (ac : vector bool n) : bool := ac.to_list.any id
def all_threads_active (ac : vector bool n) : bool := ac.to_list.all id
/-- thread can only be active either in ac₁ or ac₂ -/
def ac_distinct (ac₁ ac₂ : vector bool n) : Prop := ∀ (i : fin n), ac₁.nth i = ff ∨ ac₂.nth i = ff
def ac_ge (ac' : vector bool n) (ac : vector bool n) : Prop := ∀ (t : fin n), ¬ (ac.nth t) → ¬ (ac'.nth t)
instance : has_le (vector bool n) := ⟨ac_ge⟩
/-- shared program state -/
structure state {ι : Type} (n : ℕ) (σ : Type) (τ : ι → Type) : Type :=
(threads : vector (thread_state σ τ) n)
namespace state
def map_threads (f : thread_state σ τ → thread_state σ τ) (s : state n σ τ) : state n σ τ :=
{ threads := s.threads.map f, ..s }
-- we generally don't want to unfold this if possible
-- this would for example happen when you do cases in (exec_state (compute f) ...)
-- TODO: rename this to mat? It would shorten a lot of names
@[irreducible]
def map_active_threads (ac : vector bool n) (f : thread_state σ τ → thread_state σ τ) (s : state n σ τ) : state n σ τ :=
{ threads := (s.threads.map₂ (λ t (a : bool), if a then f t else t) ac), ..s }
def active_threads (ac : vector bool n) (s : state n σ τ) : list (thread_state σ τ) :=
((s.threads.map₂ prod.mk ac).to_list.filter (λ c : (thread_state σ τ × bool), c.2)).map (λ ⟨t, a⟩, t)
-- case 1: no thread changed ι and shadows must be equal at ι
-- case 2: thread t changed ι and all other threads must not access ι
def syncable (s : state n σ τ) (m : memory τ) : Prop :=
∀i:ι,
(∀ tid, i ∉ (s.threads.nth tid).stores ∧ m i = (s.threads.nth tid).shared i) ∨
(∃ tid, i ∈ (s.threads.nth tid).stores ∧ m i = (s.threads.nth tid).shared i ∧
(∀ tid', tid ≠ tid' → i ∉ (s.threads.nth tid').accesses))
def precedes (s u : state n σ τ) : Prop :=
∀ (t : thread_state σ τ × thread_state σ τ), t ∈ (s.threads.map₂ prod.mk u.threads) → t.1.stores ⊆ t.2.stores ∧ t.1.loads ⊆ t.2.loads
end state
/-- If condition *f* evaluates to *tt*, the thread is deactivated -/
@[irreducible]
def deactivate_threads (f : σ → bool) (ac : vector bool n) (s : state n σ τ) : vector bool n :=
ac.map₂ (λ a (ts : thread_state σ τ), (bnot ∘ f) ts.tlocal && a) s.threads
def subkernel (q : kernel σ τ) : kernel σ τ → Prop
| (seq k₁ k₂) := k₁ = q ∨ k₂ = q ∨ subkernel k₁ ∨ subkernel k₂
| (ite c th el) := th = q ∨ el = q ∨ subkernel th ∨ subkernel el
| (loop c body) := body = q ∨ subkernel body
| k := k = q
/-- Execute a kernel on a shared state, i.e. a list of threads -/
inductive exec_state {n : ℕ} : kernel σ τ → vector bool n → state n σ τ → state n σ τ → Prop
| load (f) (s : state n σ τ) (ac : vector bool n) :
exec_state (load f) ac s (s.map_active_threads ac $ thread_state.load f)
| store (f) (s : state n σ τ) (ac : vector bool n) :
exec_state (store f) ac s (s.map_active_threads ac $ thread_state.store f)
| compute (f : σ → σ) (s : state n σ τ) (ac : vector bool n) :
exec_state (compute f) ac s (s.map_active_threads ac $ thread_state.compute f)
| sync_all (s : state n σ τ) (ac : vector bool n) (m : memory τ) (hs : s.syncable m)
(ha : all_threads_active ac) :
exec_state sync ac s (s.map_threads $ thread_state.sync m)
| sync_none (s : state n σ τ) (ac : vector bool n) (h : no_thread_active ac) :
exec_state sync ac s s
| seq (s t u : state n σ τ) (ac : vector bool n) (k₁ k₂ : kernel σ τ) :
exec_state k₁ ac s t → exec_state k₂ ac t u → exec_state (seq k₁ k₂) ac s u
-- in the then-branch we deactivate the threads where the condition is false and similar for else
| ite (s t u : state n σ τ) (ac : vector bool n) (f : σ → bool) (k₁ k₂ : kernel σ τ) :
exec_state k₁ (deactivate_threads (bnot ∘ f) ac s) s t →
exec_state k₂ (deactivate_threads f ac s) t u →
exec_state (ite f k₁ k₂) ac s u
| loop_stop (s : state n σ τ) (ac : vector bool n) (f : σ → bool) (k : kernel σ τ) :
no_thread_active (deactivate_threads (bnot ∘ f) ac s) →
exec_state (loop f k) ac s s
| loop_step (s t u : state n σ τ) (ac : vector bool n) (f : σ → bool) (k : kernel σ τ) :
any_thread_active (deactivate_threads (bnot ∘ f) ac s) →
exec_state k (deactivate_threads (bnot ∘ f) ac s) s t →
exec_state (loop f k) (deactivate_threads (bnot ∘ f) ac s) t u →
exec_state (loop f k) ac s u
def kernel_transform_func (k) (f) (n) (ac) : Prop := ∀ (s u : state n σ τ), exec_state k ac s u ↔ (u = s.map_active_threads ac f)
def contains_sync : kernel σ τ → Prop
| (sync) := true
| (seq k₁ k₂) := contains_sync k₁ ∨ contains_sync k₂
| (load _) := false
| (store _) := false
| (compute _) := false
| (ite c k₁ k₂) := contains_sync k₁ ∨ contains_sync k₂
| (loop c k) := contains_sync k
inductive program {ι : Type} (σ : Type) (τ : ι → Type)
| intro (f : memory τ → ℕ) (k : kernel σ τ) : program
def state_initializer := ℕ → σ
@[reducible]
def init_state (init : ℕ → σ) (f : memory τ → ℕ) (m : memory τ) : state (f m) σ τ :=
{ threads := (vector.range (f m)).map (λ n, { tlocal := init n, shared := m, loads := ∅, stores := ∅ })}
inductive exec_prog : (ℕ → σ) → program σ τ → memory τ → memory τ → Prop
| intro (k : kernel σ τ) (f : memory τ → ℕ) (a b : memory τ) (init : ℕ → σ) (s' : state (f a) σ τ) (hsync : s'.syncable b)
(he : exec_state k (vector.repeat tt (f a)) (init_state init f a) s') :
exec_prog init (program.intro f k) a b
def list_to_kernel_seq (ks : list (kernel σ τ)) : kernel σ τ := ks.foldl (λ k₁ k₂, k₁ ;; k₂) (kernel.compute id)
end parlang |
213b2cd8d96e1e5329ab1bf23d05afbf8e7ee8b9 | 7cef822f3b952965621309e88eadf618da0c8ae9 | /src/data/matrix/basic.lean | 3bf474a617d61859bb90cb3e78e3f6a4b045402d | [
"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 | 12,043 | lean | /-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
Matrices
-/
import algebra.module algebra.pi_instances
import data.fintype
universes u v
def matrix (m n : Type u) [fintype m] [fintype n] (α : Type v) : Type (max u v) :=
m → n → α
namespace matrix
variables {l m n o : Type u} [fintype l] [fintype m] [fintype n] [fintype o]
variables {α : Type v}
section ext
variables {M N : matrix m n α}
theorem ext_iff : (∀ i j, M i j = N i j) ↔ M = N :=
⟨λ h, funext $ λ i, funext $ h i, λ h, by simp [h]⟩
@[ext] theorem ext : (∀ i j, M i j = N i j) → M = N :=
ext_iff.mp
end ext
def transpose (M : matrix m n α) : matrix n m α
| x y := M y x
localized "postfix `ᵀ`:1500 := matrix.transpose" in matrix
def col (w : m → α) : matrix m punit α
| x y := w x
def row (v : n → α) : matrix punit n α
| x y := v y
instance [has_add α] : has_add (matrix m n α) := pi.has_add
instance [add_semigroup α] : add_semigroup (matrix m n α) := pi.add_semigroup
instance [add_comm_semigroup α] : add_comm_semigroup (matrix m n α) := pi.add_comm_semigroup
instance [has_zero α] : has_zero (matrix m n α) := pi.has_zero
instance [add_monoid α] : add_monoid (matrix m n α) := pi.add_monoid
instance [add_comm_monoid α] : add_comm_monoid (matrix m n α) := pi.add_comm_monoid
instance [has_neg α] : has_neg (matrix m n α) := pi.has_neg
instance [add_group α] : add_group (matrix m n α) := pi.add_group
instance [add_comm_group α] : add_comm_group (matrix m n α) := pi.add_comm_group
@[simp] theorem zero_val [has_zero α] (i j) : (0 : matrix m n α) i j = 0 := rfl
@[simp] theorem neg_val [has_neg α] (M : matrix m n α) (i j) : (- M) i j = - M i j := rfl
@[simp] theorem add_val [has_add α] (M N : matrix m n α) (i j) : (M + N) i j = M i j + N i j := rfl
section diagonal
variables [decidable_eq n]
def diagonal [has_zero α] (d : n → α) : matrix n n α := λ i j, if i = j then d i else 0
@[simp] theorem diagonal_val_eq [has_zero α] {d : n → α} (i : n) : (diagonal d) i i = d i :=
by simp [diagonal]
@[simp] theorem diagonal_val_ne [has_zero α] {d : n → α} {i j : n} (h : i ≠ j) :
(diagonal d) i j = 0 := by simp [diagonal, h]
theorem diagonal_val_ne' [has_zero α] {d : n → α} {i j : n} (h : j ≠ i) :
(diagonal d) i j = 0 := diagonal_val_ne h.symm
@[simp] theorem diagonal_zero [has_zero α] : (diagonal (λ _, 0) : matrix n n α) = 0 :=
by simp [diagonal]; refl
section one
variables [has_zero α] [has_one α]
instance : has_one (matrix n n α) := ⟨diagonal (λ _, 1)⟩
@[simp] theorem diagonal_one : (diagonal (λ _, 1) : matrix n n α) = 1 := rfl
theorem one_val {i j} : (1 : matrix n n α) i j = if i = j then 1 else 0 := rfl
@[simp] theorem one_val_eq (i) : (1 : matrix n n α) i i = 1 := diagonal_val_eq i
@[simp] theorem one_val_ne {i j} : i ≠ j → (1 : matrix n n α) i j = 0 :=
diagonal_val_ne
theorem one_val_ne' {i j} : j ≠ i → (1 : matrix n n α) i j = 0 :=
diagonal_val_ne'
end one
end diagonal
@[simp] theorem diagonal_add [decidable_eq n] [add_monoid α] (d₁ d₂ : n → α) :
diagonal d₁ + diagonal d₂ = diagonal (λ i, d₁ i + d₂ i) :=
by ext i j; by_cases i = j; simp [h]
protected def mul [has_mul α] [add_comm_monoid α] (M : matrix l m α) (N : matrix m n α) :
matrix l n α :=
λ i k, finset.univ.sum (λ j, M i j * N j k)
localized "infixl ` ⬝ `:75 := matrix.mul" in matrix
theorem mul_val [has_mul α] [add_comm_monoid α] {M : matrix l m α} {N : matrix m n α} {i k} :
(M ⬝ N) i k = finset.univ.sum (λ j, M i j * N j k) := rfl
local attribute [simp] mul_val
instance [has_mul α] [add_comm_monoid α] : has_mul (matrix n n α) := ⟨matrix.mul⟩
@[simp] theorem mul_eq_mul [has_mul α] [add_comm_monoid α] (M N : matrix n n α) :
M * N = M ⬝ N := rfl
theorem mul_val' [has_mul α] [add_comm_monoid α] {M N : matrix n n α} {i k} :
(M * N) i k = finset.univ.sum (λ j, M i j * N j k) := rfl
section semigroup
variables [semiring α]
protected theorem mul_assoc (L : matrix l m α) (M : matrix m n α) (N : matrix n o α) :
(L ⬝ M) ⬝ N = L ⬝ (M ⬝ N) :=
by classical; funext i k;
simp [finset.mul_sum, finset.sum_mul, mul_assoc];
rw finset.sum_comm
instance : semigroup (matrix n n α) :=
{ mul_assoc := matrix.mul_assoc, ..matrix.has_mul }
end semigroup
@[simp] theorem diagonal_neg [decidable_eq n] [add_group α] (d : n → α) :
-diagonal d = diagonal (λ i, -d i) :=
by ext i j; by_cases i = j; simp [h]
section semiring
variables [semiring α]
@[simp] protected theorem mul_zero (M : matrix m n α) : M ⬝ (0 : matrix n o α) = 0 :=
by ext i j; simp
@[simp] protected theorem zero_mul (M : matrix m n α) : (0 : matrix l m α) ⬝ M = 0 :=
by ext i j; simp
protected theorem mul_add (L : matrix m n α) (M N : matrix n o α) : L ⬝ (M + N) = L ⬝ M + L ⬝ N :=
by ext i j; simp [finset.sum_add_distrib, mul_add]
protected theorem add_mul (L M : matrix l m α) (N : matrix m n α) : (L + M) ⬝ N = L ⬝ N + M ⬝ N :=
by ext i j; simp [finset.sum_add_distrib, add_mul]
@[simp] theorem diagonal_mul [decidable_eq m]
(d : m → α) (M : matrix m n α) (i j) : (diagonal d).mul M i j = d i * M i j :=
by simp; rw finset.sum_eq_single i; simp [diagonal_val_ne'] {contextual := tt}
@[simp] theorem mul_diagonal [decidable_eq n]
(d : n → α) (M : matrix m n α) (i j) : (M ⬝ diagonal d) i j = M i j * d j :=
by simp; rw finset.sum_eq_single j; simp {contextual := tt}
@[simp] protected theorem one_mul [decidable_eq m] (M : matrix m n α) : (1 : matrix m m α) ⬝ M = M :=
by ext i j; rw [← diagonal_one, diagonal_mul, one_mul]
@[simp] protected theorem mul_one [decidable_eq n] (M : matrix m n α) : M ⬝ (1 : matrix n n α) = M :=
by ext i j; rw [← diagonal_one, mul_diagonal, mul_one]
instance [decidable_eq n] : monoid (matrix n n α) :=
{ one_mul := matrix.one_mul,
mul_one := matrix.mul_one,
..matrix.has_one, ..matrix.semigroup }
instance [decidable_eq n] : semiring (matrix n n α) :=
{ mul_zero := matrix.mul_zero,
zero_mul := matrix.zero_mul,
left_distrib := matrix.mul_add,
right_distrib := matrix.add_mul,
..matrix.add_comm_monoid,
..matrix.monoid }
@[simp] theorem diagonal_mul_diagonal' [decidable_eq n] (d₁ d₂ : n → α) :
(diagonal d₁) ⬝ (diagonal d₂) = diagonal (λ i, d₁ i * d₂ i) :=
by ext i j; by_cases i = j; simp [h]
theorem diagonal_mul_diagonal [decidable_eq n] (d₁ d₂ : n → α) :
diagonal d₁ * diagonal d₂ = diagonal (λ i, d₁ i * d₂ i) :=
diagonal_mul_diagonal' _ _
lemma is_add_monoid_hom_mul_left (M : matrix l m α) :
is_add_monoid_hom (λ x : matrix m n α, M ⬝ x) :=
{ to_is_add_hom := ⟨matrix.mul_add _⟩, map_zero := matrix.mul_zero _ }
lemma is_add_monoid_hom_mul_right (M : matrix m n α) :
is_add_monoid_hom (λ x : matrix l m α, x ⬝ M) :=
{ to_is_add_hom := ⟨λ _ _, matrix.add_mul _ _ _⟩, map_zero := matrix.zero_mul _ }
protected lemma sum_mul {β : Type*} (s : finset β) (f : β → matrix l m α)
(M : matrix m n α) : s.sum f ⬝ M = s.sum (λ a, f a ⬝ M) :=
(@finset.sum_hom _ _ _ s f _ _ (λ x, x ⬝ M)
/- This line does not type-check without `id` and `: _`. Lean did not recognize that two different
`add_monoid` instances were def-eq -/
(id (@is_add_monoid_hom_mul_right l _ _ _ _ _ _ _ M) : _)).symm
protected lemma mul_sum {β : Type*} (s : finset β) (f : β → matrix m n α)
(M : matrix l m α) : M ⬝ s.sum f = s.sum (λ a, M ⬝ f a) :=
(@finset.sum_hom _ _ _ s f _ _ (λ x, M ⬝ x)
/- This line does not type-check without `id` and `: _`. Lean did not recognize that two different
`add_monoid` instances were def-eq -/
(id (@is_add_monoid_hom_mul_left _ _ n _ _ _ _ _ M) : _)).symm
end semiring
section ring
variables [ring α]
@[simp] theorem neg_mul (M : matrix m n α) (N : matrix n o α) :
(-M) ⬝ N = -(M ⬝ N) := by ext; simp [matrix.mul]
@[simp] theorem mul_neg (M : matrix m n α) (N : matrix n o α) :
M ⬝ (-N) = -(M ⬝ N) := by ext; simp [matrix.mul]
end ring
instance [decidable_eq n] [ring α] : ring (matrix n n α) :=
{ ..matrix.add_comm_group, ..matrix.semiring }
instance [semiring α] : has_scalar α (matrix m n α) := pi.has_scalar
instance [ring α] : module α (matrix m n α) := pi.module _
@[simp] lemma smul_val [semiring α] (a : α) (A : matrix m n α) (i : m) (j : n) : (a • A) i j = a * A i j := rfl
section comm_ring
variables [comm_ring α]
@[simp] lemma mul_smul (M : matrix m n α) (a : α) (N : matrix n l α) : M ⬝ (a • N) = a • M ⬝ N :=
begin
ext i j,
unfold matrix.mul has_scalar.smul,
rw finset.mul_sum,
congr,
ext,
ac_refl
end
@[simp] lemma smul_mul (M : matrix m n α) (a : α) (N : matrix n l α) : (a • M) ⬝ N = a • M ⬝ N :=
begin
ext i j,
unfold matrix.mul has_scalar.smul,
rw finset.mul_sum,
congr,
ext,
ac_refl
end
end comm_ring
section semiring
variables [semiring α]
def vec_mul_vec (w : m → α) (v : n → α) : matrix m n α
| x y := w x * v y
def mul_vec (M : matrix m n α) (v : n → α) : m → α
| x := finset.univ.sum (λy:n, M x y * v y)
def vec_mul (v : m → α) (M : matrix m n α) : n → α
| y := finset.univ.sum (λx:m, v x * M x y)
instance mul_vec.is_add_monoid_hom_left (v : n → α) :
is_add_monoid_hom (λM:matrix m n α, mul_vec M v) :=
{ map_zero := by ext; simp [mul_vec]; refl,
map_add :=
begin
intros x y,
ext m,
rw pi.add_apply (mul_vec x v) (mul_vec y v) m,
simp [mul_vec, finset.sum_add_distrib, right_distrib]
end }
lemma mul_vec_diagonal [decidable_eq m] (v w : m → α) (x : m) :
mul_vec (diagonal v) w x = v x * w x :=
begin
transitivity,
refine finset.sum_eq_single x _ _,
{ assume b _ ne, simp [diagonal, ne.symm] },
{ simp },
{ rw [diagonal_val_eq] }
end
lemma vec_mul_vec_eq (w : m → α) (v : n → α) :
vec_mul_vec w v = (col w) ⬝ (row v) :=
by simp [matrix.mul]; refl
end semiring
section transpose
open_locale matrix
@[simp] lemma transpose_transpose (M : matrix m n α) :
Mᵀᵀ = M :=
by ext; refl
@[simp] lemma transpose_zero [has_zero α] : (0 : matrix m n α)ᵀ = 0 :=
by ext i j; refl
@[simp] lemma transpose_add [has_add α] (M : matrix m n α) (N : matrix m n α) :
(M + N)ᵀ = Mᵀ + Nᵀ :=
begin
ext i j,
dsimp [transpose],
refl
end
@[simp] lemma transpose_mul [comm_ring α] (M : matrix m n α) (N : matrix n l α) :
(M ⬝ N)ᵀ = Nᵀ ⬝ Mᵀ :=
begin
ext i j,
unfold matrix.mul transpose,
congr,
ext,
ac_refl
end
@[simp] lemma transpose_neg [comm_ring α] (M : matrix m n α) :
(- M)ᵀ = - Mᵀ :=
by ext i j; refl
end transpose
def minor (A : matrix m n α) (row : l → m) (col : o → n) : matrix l o α :=
λ i j, A (row i) (col j)
@[reducible]
def sub_left {m l r : nat} (A : matrix (fin m) (fin (l + r)) α) : matrix (fin m) (fin l) α :=
minor A id (fin.cast_add r)
@[reducible]
def sub_right {m l r : nat} (A : matrix (fin m) (fin (l + r)) α) : matrix (fin m) (fin r) α :=
minor A id (fin.nat_add l)
@[reducible]
def sub_up {d u n : nat} (A : matrix (fin (u + d)) (fin n) α) : matrix (fin u) (fin n) α :=
minor A (fin.cast_add d) id
@[reducible]
def sub_down {d u n : nat} (A : matrix (fin (u + d)) (fin n) α) : matrix (fin d) (fin n) α :=
minor A (fin.nat_add u) id
@[reducible]
def sub_up_right {d u l r : nat} (A: matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin u) (fin r) α :=
sub_up (sub_right A)
@[reducible]
def sub_down_right {d u l r : nat} (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin d) (fin r) α :=
sub_down (sub_right A)
@[reducible]
def sub_up_left {d u l r : nat} (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin u) (fin (l)) α :=
sub_up (sub_left A)
@[reducible]
def sub_down_left {d u l r : nat} (A: matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin d) (fin (l)) α :=
sub_down (sub_left A)
end matrix
|
8e237f5b7bd3b5df7ecc26441c6c1c4a46581650 | 5a6ff5f8d173cbfe51967eb4c96837e3a791fe3d | /mm0-lean/x86/matching.lean | 44b3dc049454362ce86e6390cfb128be5cf189fa | [
"CC0-1.0"
] | permissive | digama0/mm0 | 491ac09146708aa1bb775007bf3dbe339ffc0096 | 98496badaf6464e56ed7b4204e7d54b85667cb01 | refs/heads/master | 1,692,321,030,902 | 1,686,254,458,000 | 1,686,254,458,000 | 172,456,790 | 273 | 38 | CC0-1.0 | 1,689,939,563,000 | 1,551,080,059,000 | Rust | UTF-8 | Lean | false | false | 32,865 | lean | import x86.lemmas data.set.lattice data.list.basic data.pfun data.list.alist
namespace x86
inductive flag_place | CFP | ZFP | SFP | OFP
def flag_place.read (f : flags) : flag_place → bool
| flag_place.CFP := f.CF
| flag_place.ZFP := f.ZF
| flag_place.SFP := f.SF
| flag_place.OFP := f.OF
def flag_place.write (f : flags) (b : bool) : flag_place → flags
| flag_place.CFP := {CF := b, ..f}
| flag_place.ZFP := {ZF := b, ..f}
| flag_place.SFP := {SF := b, ..f}
| flag_place.OFP := {OF := b, ..f}
inductive place
| flag : flag_place → place
| reg : regnum → place
| rip : place
| mem : qword → place
inductive place.read (k : config) : place → ∀ {n}, bitvec n → Prop
| flag {f} : place.read (place.flag f) (bitvec.singleton (f.read k.flags))
| reg {r} : place.read (place.reg r) (k.regs r)
| rip : place.read place.rip k.rip
| mem {a b} : k.mem.read1 perm.R a b → place.read (place.mem a) b
inductive place.write (k : config) : place → ∀ {n}, bitvec n → config → Prop
| flag {f b} : place.write (place.flag f) (bitvec.singleton b) {flags := f.write k.flags b, ..k}
| reg {r n q} : place.write (place.reg r) q (k.write_reg r n q)
| rip {q} : place.write place.rip q {rip := q, ..k}
| mem {a b m'} : k.mem.write1 a b m' → place.write (place.mem a) b {mem := m', ..k}
def place.stable (k k' : config) : place → Prop
| (place.flag f) := f.read k.flags = f.read k'.flags
| (place.reg r) := k.regs r = k'.regs r
| place.rip := k.rip = k'.rip
| (place.mem a) :=
roption.mk (k.mem.valid a) (λ h, (k.mem.mem a h, k.mem.perm a h)) =
roption.mk (k'.mem.valid a) (λ h, (k'.mem.mem a h, k'.mem.perm a h))
theorem place.stable.refl (k p) : place.stable k k p :=
by cases p; exact rfl
theorem place.stable.trans {k₁ k₂ k₃ p}
(h₁ : place.stable k₁ k₂ p) (h₂ : place.stable k₂ k₃ p) : place.stable k₁ k₃ p :=
by cases p; exact eq.trans h₁ h₂
theorem mem.read1.stable {k k' a} (s : place.stable k k' (place.mem a))
{p b} : mem.read1 p k.mem a b → mem.read1 p k'.mem a b :=
@@eq.subst (λ o : roption (byte × perm),
∃ (h : o.1), b = (o.2 h).1 ∧ p ≤ (o.2 h).2) s
theorem place.read.stable {k k'} : ∀ {p}, place.stable k k' p →
∀ {n v}, @place.read k p n v → place.read k' p v :=
begin
rintro p s n v h, induction h,
{ rw show _=_, from s, constructor },
{ rw show _=_, from s, constructor },
{ rw show _=_, from s, constructor },
{ exact place.read.mem (mem.read1.stable s h_a_1) }
end
def stability (k k' : config) : set place := {p | place.stable k k' p}
def split {α} (H h₁ h₂ : set α) : Prop :=
H = h₁ ∪ h₂ ∧ disjoint h₁ h₂
def sProp := config → Prop
def sProp.stable (P : sProp) (D : set place) : Prop :=
∀ k k', D ⊆ stability k k' → P k → P k'
def sProp.reserve (S : set place) : sProp :=
λ k, ∀ p ∈ S, ∃ n v, @place.read k p n v
theorem sProp.reserve.stable {S D : set place} (h : S ⊆ D) :
(sProp.reserve S).stable D :=
λ k k' h' H p hp, let ⟨n, v, hp'⟩ := H p hp in
⟨n, v, place.read.stable (h' (h hp)) hp'⟩
def mProp := config → config → Prop
def mProp.exterior (P : mProp) : set place :=
{p | ∀ k k', P k k' → place.stable k k' p}
def mProp.dom (P : mProp) : set place := (P.exterior)ᶜ
def stable_at (D : set place) : mProp := λ k k', D ⊆ stability k k'
theorem stable_at.refl {D k} : stable_at D k k :=
λ x _, place.stable.refl _ _
theorem stable_at.trans {D k₁ k₂ k₃}
(h₁ : stable_at D k₁ k₂) (h₂ : stable_at D k₂ k₃) : stable_at D k₁ k₃ :=
λ x h, place.stable.trans (h₁ h) (h₂ h)
def mProp.initial (Q : mProp) (P : sProp) : mProp := λ k k', Q k k' ∧ P k
def mProp.final (Q : mProp) (P : sProp) : mProp := λ k k', Q k k' ∧ P k'
def mProp.apply (Q : mProp) (P : sProp) : sProp :=
λ k', ∃ k, P k ∧ Q k k'
def mProp.id : mProp := eq
def mProp.comp (P Q : mProp) : mProp :=
λ k₁ k₃, ∃ k₂, P k₁ k₂ ∧ Q k₂ k₃
def mProp.imp (P Q : mProp) : mProp :=
λ k₂ k₃, ∀ k₁, P k₁ k₂ → Q k₁ k₃
def slift (p : Prop) : sProp := λ _, p
def mlift (p : Prop) : mProp := λ _ _, p
instance lattice.complete_lattice.sProp : complete_lattice sProp :=
pi.complete_lattice
instance lattice.complete_lattice.mProp : complete_lattice mProp :=
pi.complete_lattice
def sProp.with (p : sProp) (q : Prop) : sProp := p ⊓ slift q
def mProp.with (p : mProp) (q : Prop) : mProp := p ⊓ mlift q
def sProp.ex {α} (p : α → sProp) : sProp := λ k, ∃ a, p a k
def mProp.ex {α} (p : α → mProp) : mProp := λ k k', ∃ a, p a k k'
def sProp.all {α} (p : α → sProp) : sProp := λ k, ∀ a, p a k
def mProp.all {α} (p : α → mProp) : mProp := λ k k', ∀ a, p a k k'
def sProp.sn (p : place) {n} (v : bitvec n) : sProp :=
λ k, p.read k v
def mProp.write (p : place) {n} (v' : bitvec n) : mProp :=
λ k k', p.write k v' k'
def mProp.sn (p : place) {n} (v v' : bitvec n) : mProp :=
(mProp.write p v').initial (sProp.sn p v)
def mProp.clob (p : place) {n} (v : bitvec n) : mProp :=
mProp.ex $ λ v', mProp.sn p v v'
def sProp.flags (f : flags) : sProp :=
sProp.all $ λ p : flag_place,
sProp.sn (place.flag p) (bitvec.singleton $ p.read f)
inductive block
| reg : regnum → wsize → block
| mem : qword → qword → block
| const : list byte → block
def block.size : block → qword
| (block.reg _ sz) := (sz.to_nat / 8 : ℕ)
| (block.mem _ sz) := sz
| (block.const l) := l.length
inductive block.places : block → set place
| reg {r sz} : block.places (block.reg r sz) (place.reg r)
| mem {a b sz : qword} :
(bitvec.sub b a).to_nat ≤ sz.to_nat →
block.places (block.mem a sz) (place.mem b)
def block.disj (b1 b2 : block) := disjoint b1.places b2.places
def sProp.mem_block (p : perm) (a : qword) (v : list byte) : sProp :=
λ k, k.mem.read' p a v
def block.read : block → list byte → sProp
| (block.reg r sz) v :=
(sProp.reserve (block.reg r sz).places) ⊓ λ k,
read_full_imm sz (k.regs r) v
| (block.mem a sz) v :=
(sProp.reserve (block.mem a sz).places) ⊓ λ k,
k.mem.read a v ∧ v.length = sz.to_nat
| (block.const l) v := slift $ v = l
def block.writable : block → sProp
| (block.reg r sz) := sProp.reserve (block.reg r sz).places
| (block.mem a sz) :=
(sProp.reserve (block.mem a sz).places) ⊓ λ k,
∃ v, k.mem.read' perm.W a v ∧ v.length = sz.to_nat
| (block.const _) := slift false
def block.write : block → list byte → mProp
| (block.reg r sz) l k k' := ∃ v,
@bits_to_byte (sz.to_nat * 8) sz.to_nat v l ∧
mProp.write (place.reg r) v k k'
| (block.mem a sz) l k k' :=
∃ m', k.mem.write a l m' ∧ k' = {mem := m', ..k}
| (block.const _) _ _ _ := false
theorem block.read.size : ∀ {b v k},
block.read b v k → v.length = b.size.to_nat := sorry
theorem block.write.size : ∀ {b v k k'},
block.write b v k k' → v.length = b.size.to_nat := sorry
theorem block.read_write : ∀ {b v k k'},
block.write b v k k' → block.read b v k' := sorry
theorem block.read.stable {l v k k'}
(h₁ : block.read l v k) (ss : places l ⊆ stability k k') :
block.read l v k' := sorry
def hoare (P : kcfg → Prop) (Q : kcfg → kcfg → Prop) :=
∀ {{k}}, P k → hoare_p (Q k) k
def mHoareIO (P : sProp) (Q : list byte → list byte → mProp) :=
hoare (λ k, P k.k)
(λ k k', ∃ i' o', k.input = i' ++ k'.input ∧ k'.output = k.output ++ o' ∧
(Q i' o') k.k k'.k)
def noIO (Q : mProp) (i o : list byte) : mProp := Q.with (i = [] ∧ o = [])
def mHoare (P : sProp) (Q : mProp) := mHoareIO P (noIO Q)
def locals_ctx := alist (λ _ : ℕ, block)
def locals_ctx.get (Γ : locals_ctx) (n : ℕ) : option block := Γ.lookup n
def labels_ctx := qword × list qword
def labels_ctx.cons (q : qword) (L : labels_ctx) : labels_ctx :=
⟨L.1, q :: L.2⟩
structure asm_ctx :=
(L : list qword)
(Γ : alist (λ _ : ℕ, block))
(exit : qword)
(die : ∃ l, mHoare (sProp.sn place.rip exit ⊓
sProp.mem_block (perm.R + perm.X) exit l) ⊥)
def asm_ctx.read (A : asm_ctx) (n : ℕ) : option block := A.Γ.lookup n
def asm_ctx.push_local (A : asm_ctx) (i : ℕ) (b : block) : asm_ctx :=
⟨A.1, A.2.insert i b, A.3, A.4⟩
def asm_ctx.push_label (A : asm_ctx) (q : qword) : asm_ctx :=
⟨q :: A.1, A.2, A.3, A.4⟩
inductive exit_kind
| straight
| label (n : ℕ)
def exit_kind.result (A : asm_ctx) (pos : qword) : exit_kind → sProp
| exit_kind.straight := sProp.sn place.rip pos
| (exit_kind.label n) := sProp.ex $ λ h, sProp.sn place.rip (A.1.nth_le n h)
def stmt : Type :=
asm_ctx → set place →
∀ rip : qword, list byte → Prop
def stmt.hoareIO (A) (P : sProp) (C : stmt)
(Q : exit_kind → list byte → list byte → mProp) : Prop :=
∀ D rip l, C A D rip l →
mHoareIO
(P ⊓ sProp.sn place.rip rip ⊓
sProp.mem_block (perm.R + perm.X) rip l)
(λ i o, stable_at D ⊓
mProp.ex (λ e, (Q e i o).final (e.result A (rip + l.length))))
def stmt.hoare (A) (P : sProp) (C : stmt) (Q : exit_kind → mProp) : Prop :=
stmt.hoareIO A P C (λ e i o, noIO (Q e) i o)
def hstmt (P : sProp) (Q : mProp) : stmt :=
λ A D rip v,
mHoare (sProp.sn place.rip rip ⊓ sProp.mem_block (perm.R + perm.X) rip v)
((Q ⊓ stable_at D).final (sProp.sn place.rip (rip + v.length)))
instance complete_lattice.stmt : complete_lattice stmt :=
pi.complete_lattice
def stmt.all {α} (s : α → stmt) : stmt :=
λ A D rip v, ∀ a, s a A D rip v
def stmt.ex {α} (s : α → stmt) : stmt :=
λ A D rip v, ∃ a, s a A D rip v
def stmt.with (p : Prop) (s : stmt) : stmt :=
stmt.ex $ λ h : p, s
def stmt.or (s₁ s₂ : stmt) : stmt :=
λ A D rip v, s₁ A D rip v ∨ s₂ A D rip v
def stmt.stabilize (S : set place) (s : stmt) : stmt :=
λ A D, s A (D ∪ S)
def expr (α : Type) := block → stmt
instance complete_lattice.expr {α} : complete_lattice (expr α) :=
pi.complete_lattice
def expr.hoareIO {α} (A) (P : sProp) (E : expr α)
(Q : block → list byte → list byte → mProp) : Prop :=
∀ b, (E b).hoareIO A P $
λ e i o, (Q b i o).with (e = exit_kind.straight)
def expr.hoare {α} (A) (P : sProp) (E : expr α) (Q : block → mProp) : Prop :=
expr.hoareIO A P E (λ ret i o, noIO (Q ret) i o)
def expr.stabilize {α} (S : set place) (s : expr α) : expr α :=
λ b, (s b).stabilize S
class value (α : Type*) :=
(size : ℕ)
(eval : α → list byte → Prop)
(eval_eq : ∀ {{a l}}, eval a l → l.length = size)
def value.evalB {α} [value α] (x : α) (b : block) : sProp :=
λ k, ∃ l, block.read b l k ∧ value.eval x l
def value.read_sized {α} [value α] (a : qword) (x : α) (b : block) : sProp :=
value.evalB x b ⊓ slift (b = block.mem a (value.size α))
class type (α : Type*) :=
(size : ℕ)
(read : α → block → set place → sProp)
(write : α → block → mProp)
(read_eq : ∀ {{a l s k}}, read a l s k → l.size.to_nat = size)
(write_eq : ∀ {{a l k k'}}, write a l k k' → l.size.to_nat = size)
(read_write : ∀ {{a l k k'}}, write a l k k' → ∃ s, read a l s k')
(read_stable : ∀ {{a l s k k'}},
read a l s k → s ⊆ stability k k' → read a l s k')
def type.read' {α} [type α] (a : α) (b : block) : sProp :=
sProp.ex $ type.read a b
instance (α) [value α] : type α :=
⟨value.size α,
λ a b s, sProp.ex $ λ v,
block.read b v ⊓ slift (value.eval a v ∧ s = b.places),
λ a b, mProp.ex $ λ v,
block.write b v ⊓ mlift (value.eval a v),
λ a b s k ⟨v, h₁, h₂, _⟩, by rw [← h₁.size, @value.eval_eq α _ a v h₂],
λ a b k k' ⟨v, h₁, h₂⟩, by rw [← h₁.size, @value.eval_eq α _ a v h₂],
λ a v k k' ⟨s, h₁, h₂⟩, ⟨_, _, block.read_write h₁, h₂, rfl⟩,
λ a l s k k' ⟨v, h₁, h₂, e⟩ ss, ⟨v, h₁.stable (e ▸ ss), h₂, e⟩⟩
def bits.value {n} (m : ℕ) : value (bitvec n) :=
⟨m, bits_to_byte m, λ a v h, h.1⟩
instance unit.value : value unit :=
⟨0, λ _ v, v = [], by rintro _ _ ⟨⟩; refl⟩
instance byte.value : value byte :=
⟨1, λ b v, v = [b], by rintro _ _ ⟨⟩; refl⟩
instance word.value : value word := bits.value 4
instance qword.value : value qword := bits.value 8
class box (α) [type α] := (deref : α)
instance box.type (α) [type α] : type (box α) :=
⟨8, λ x b s k, ∃ a l s',
block.read b l k ∧
qword.to_list_byte a l ∧
type.read x.deref (block.mem a (type.size α)) s' k ∧
s = b.places ∪ s',
λ x b k k', ∃ a l s',
block.write b l k k' ∧
qword.to_list_byte a l ∧
type.read x.deref (block.mem a (type.size α)) s' k',
λ x b s k ⟨a, l, s', h₁, h₂, _⟩,
by rw [← h₁.size, h₂.1],
λ x b k k' ⟨a, l, s', h₁, h₂, _⟩,
by rw [← h₁.size, h₂.1],
λ x b k k' ⟨a, l, s', h₁, h₂, h₃⟩,
⟨_, _, _, _, block.read_write h₁, h₂, h₃, rfl⟩,
λ x b s k k' ⟨a, l, s', h₁, h₂, h₃, e⟩ ss,
let ⟨ss₁, ss₂⟩ := set.union_subset_iff.1 (by rwa e at ss) in
⟨a, l, s', h₁.stable ss₁, h₂, type.read_stable h₃ ss₂, e⟩⟩
def ret (α) (b : block) : expr α :=
λ bl A D a v, bl = b ∧ v = []
def const (α) (l : list byte) : expr α := ret α (block.const l)
def name (α : Type) := ℕ
def var {α} (i : name α) : expr α :=
λ bl A D a v, bl ∈ A.read i ∧ v = []
def hexpr {α} (P : sProp) (Q : block → mProp) : expr α :=
λ ret, hstmt P (Q ret)
def expr.all {α β} (e : α → expr β) : expr β :=
λ ret, stmt.all $ λ a, e a ret
def expr.ex {α β} (e : α → expr β) : expr β :=
λ ret, stmt.ex $ λ a, e a ret
def expr.with {α} (p : Prop) (s : expr α) : expr α :=
expr.ex $ λ h : p, s
def const' {α} [type α] (a : α) : expr α :=
hexpr ⊤ $ λ b, mProp.final mProp.id (type.read' a b)
inductive stmt.seq (s₁ s₂ : stmt) : stmt
| mk {A D rip v₁ v₂} :
s₁ A D rip v₁ →
s₂ A D (rip + v₁.length) v₂ →
stmt.seq A D rip (v₁ ++ v₂)
inductive expr.bindS {α} (e₁ : expr α) (s₂ : block → stmt) : stmt
| mk {b A D rip v₁ v₂} :
e₁ b A D rip v₁ →
s₂ b A D (rip + v₁.length) v₂ →
expr.bindS A D rip (v₁ ++ v₂)
def expr.bind {α β} (e₁ : expr α) (e₂ : block → expr β) : expr β :=
λ b₂, expr.bindS e₁ $ λ b₁, e₂ b₁ b₂
def block.mov (dst src : block) : stmt :=
stmt.with (dst.size = src.size) $
stmt.all $ λ val, hstmt (block.read src val) (block.write dst val)
def expr.set {α} (e₁ e₂ : expr α) : stmt :=
expr.bindS e₁ $ λ dst, expr.bindS e₂ $ λ src, block.mov dst src
inductive label | fail | label (n : ℕ)
inductive label.loc (A : asm_ctx) : label → qword → Prop
| fail : label.loc label.fail A.exit
| label (n h) : label.loc (label.label n) (A.L.nth_le n h)
def stmt.jump_cc (p : flags → bool) (l : label) : stmt :=
λ A D rip v, ∀ tgt, l.loc A tgt →
mHoare (sProp.sn place.rip rip ⊓ sProp.mem_block (perm.R + perm.X) rip v)
(λ k k', stable_at D k k' ∧
mProp.write place.rip (cond (p k.flags) tgt (rip + v.length)) k k')
def stmt.jump : label → stmt := stmt.jump_cc (λ _, tt)
def boolexpr := (flags → bool) → stmt
def boolexpr.hoare (A) (P : sProp) (E : boolexpr) (Q : bool → mProp) : Prop :=
∃ p, stmt.hoare A P (E p)
(λ e k k', e = exit_kind.straight ∧ Q (p k'.flags) k k')
def boolexpr.not (c : boolexpr) : boolexpr :=
λ p, c (bnot ∘ p)
def boolexpr.jump_if (c : boolexpr) (l : label) : stmt :=
stmt.ex $ λ p, (c p).seq $ stmt.jump_cc p l
def stmt.nop : stmt := λ A D rip v, v = []
def if_stmt (c : boolexpr) (s₁ s₂ : stmt) : stmt :=
stmt.ex $ λ p, stmt.seq (c p) $ λ A D rip v,
∃ v₁ v₂ v₃, v = v₁ ++ v₂ ++ v₃ ∧
let q₁ := rip + v₁.length, q₂ := q₁ + v₂.length in
stmt.jump_cc (bnot ∘ p) (label.label 0) (A.push_label q₁) D rip v₁ ∧
s₁.seq (stmt.jump (label.label 0)) (A.push_label q₂) D q₁ v₂ ∧
s₂.seq (stmt.jump (label.label 0)) (A.push_label q₂) D q₂ v₃
def loop (s : stmt) : stmt :=
λ A D rip, s (A.push_label rip) D rip
def block_stmt (s : stmt) : stmt :=
λ A D rip v, s (A.push_label (rip + v.length)) D rip v
def while (c : boolexpr) (s : stmt) : stmt :=
block_stmt $ loop $
(c.not.jump_if (label.label 1)).seq $
s.seq $
stmt.jump (label.label 0)
def decl_block {α} (b : block) (s : name α → stmt) : stmt :=
λ A D rip v, ∃ i, s i (A.push_local i b) D rip v
def decl {α} (sz : qword) (s : name α → stmt) : stmt :=
stmt.ex $ λ b, stmt.with (block.size b = sz) $ decl_block b s
def init {α} (e : expr α) (s : name α → stmt) : stmt :=
e.bindS $ λ b, decl_block b s
def binop_expr {α β γ} [type α] [type β] [type γ]
(f : α → β → γ) (e₁ : expr α) (e₂ : expr β) : expr γ :=
e₁.bind $ λ b₁, e₂.bind $ λ b₂ b, stmt.all $ λ x, stmt.all $ λ y,
hstmt (type.read' x b₁ ⊓ type.read' y b₂) (type.write (f x y) b)
def asn_binop {α β} [type α] [type β] (f : α → β → α) (e₁ : expr α) (e₂ : expr β) : stmt :=
e₁.bindS $ λ b₁, (ret α b₁).set (binop_expr f (ret α b₁) e₂)
def unop_expr {α β} [type α] [type β]
(f : α → β) (e : expr α) : expr β :=
e.bind $ λ b₁ b, stmt.ex $ λ x,
hstmt (type.read' x b₁) (type.write (f x) b)
def asn_unop {α} [type α] (f : α → α) (e : expr α) : stmt :=
e.bindS $ λ b, (ret α b).set (unop_expr f (ret α b))
def for {α} (start : expr α) (test : name α → boolexpr) (incr body : name α → stmt) : stmt :=
init start $ λ i, while (test i) $ (body i).seq (incr i)
def incr {α} [type α] [has_add α] [has_one α] : expr α → stmt :=
asn_unop (+ 1)
def bool_binop {α β} [type α] [type β]
(f : α → β → bool) (e₁ : expr α) (e₂ : expr β) : boolexpr :=
λ p, e₁.bindS $ λ b₁, e₂.bindS $ λ b₂, stmt.ex $ λ x, stmt.ex $ λ y,
hstmt (type.read' x b₁ ⊓ type.read' y b₂)
(mProp.final mProp.id (λ k, p k.flags = f x y))
def ltq (e₁ e₂ : expr qword) : boolexpr :=
bool_binop (λ a b : qword, a.to_nat < b.to_nat) e₁ e₂
def for_seq (sz : qword) (max : expr qword) (body : name qword → stmt) : stmt :=
for (const' (0 : qword)) (λ i, ltq (const' 0) max) (λ i, incr (var i)) body
----------------------------------------
-- Program logic
----------------------------------------
theorem mProp.comp_imp {P Q} : mProp.comp P (mProp.imp P Q) ≤ Q :=
λ k₁ k₃ ⟨k₂, h₁, h₂⟩, h₂ _ h₁
theorem mProp.comp_ex_imp {α P Q} :
mProp.comp (@mProp.ex α P) (mProp.all $ λ a, mProp.imp (P a) Q) ≤ Q :=
λ k₁ k₃ ⟨k₂, ⟨a, h₁⟩, h₂⟩, h₂ a _ h₁
theorem mProp.ex_apply {α Q P} :
(@mProp.ex α Q).apply P = sProp.ex (λ a, (Q a).apply P) :=
funext $ λ k, propext
⟨λ ⟨k', h₁, a, h₂⟩, ⟨a, k', h₁, h₂⟩,
λ ⟨a, k', h₁, h₂⟩, ⟨k', h₁, a, h₂⟩⟩
theorem mProp.ex_comp {α Q Q'} :
(@mProp.ex α Q).comp Q' = mProp.ex (λ a, (Q a).comp Q') :=
funext $ λ k, funext $ λ k', propext
⟨λ ⟨k'', ⟨a, h₁⟩, h₂⟩, ⟨a, k'', h₁, h₂⟩,
λ ⟨a, k'', h₁, h₂⟩, ⟨k'', ⟨a, h₁⟩, h₂⟩⟩
theorem mem.read'_left {m p a v₁ v₂} (ss : v₁ <+: v₂) :
mem.read' m p a v₂ → mem.read' m p a v₁ :=
begin
intro h, induction h generalizing v₁,
{ cases list.eq_nil_of_prefix_nil ss, constructor },
{ cases v₁, constructor,
rcases ss with ⟨_, ⟨⟩⟩,
exact mem.read'.cons h_a (h_ih ⟨_, rfl⟩) }
end
theorem mem.read'_right {m p a v₁ v₂} :
mem.read' m p a (v₁ ++ v₂) → mem.read' m p (a + v₁.length) v₂ :=
begin
generalize e : v₁ ++ v₂ = v,
intro h, induction h generalizing v₁ v₂,
{ cases list.eq_nil_of_suffix_nil ⟨_, e⟩, constructor },
{ cases v₁; cases e,
{ simp, exact mem.read'.cons h_a h_a_1 },
{ simp [-add_comm], rw [← add_assoc, add_right_comm],
exact h_ih rfl } }
end
theorem hoare.zero {P : kcfg → Prop} {Q : kcfg → kcfg → Prop}
(H : ∀ {{k}}, P k → Q k k) : hoare P Q :=
λ k p, hoare_p.zero (H p)
theorem hoare.step {P P' : kcfg → Prop} {Q Q' : kcfg → kcfg → Prop}
(h₁ : ∀ {{k}}, P k → ∃ k', k.step k')
(h₂ : ∀ {{k}}, P k → ∀ {{k'}}, k.step k' → P' k' ∧ ∀ {{k''}}, Q' k' k'' → Q k k'')
(h₃ : hoare P' Q') : hoare P Q :=
λ k p, hoare_p.step (h₁ p) (λ k' s, hoare_p.mono (h₂ p s).2 (h₃ (h₂ p s).1))
theorem hoare.mono_l {P P' : kcfg → Prop} {Q : kcfg → kcfg → Prop}
(H : ∀ {{k}}, P k → P' k) : hoare P' Q → hoare P Q :=
λ H' k h, H' (H h)
theorem hoare.mono_r {P : kcfg → Prop} {Q Q' : kcfg → kcfg → Prop}
(H : ∀ {{k k'}}, P k → Q k k' → Q' k k') : hoare P Q → hoare P Q' :=
λ H' k h, hoare_p.mono (λ k' h', H h (by exact h')) (H' h)
theorem hoare.comp {P : kcfg → Prop} {Q Q' : kcfg → kcfg → Prop}
(H₁ : hoare P Q)
(H₂ : hoare (λ k', ∃ k, Q k k' ∧ P k) Q') :
hoare P (rel.comp Q Q') :=
λ k p, begin
refine (hoare.mono_r _ H₁ p).bind _,
exact (λ k k', Q k k' ∧ P k),
rintro k' ⟨h₁, h₂⟩,
refine (H₂ ⟨_, h₁, h₂⟩).mono (λ k₂ h, ⟨_, h₁, h⟩),
exact λ k k' h₁ h₂, ⟨h₂, h₁⟩
end
theorem mHoareIO.zero {P : sProp} {Q : list byte → list byte → mProp}
(H : ∀ {{k}}, P k → Q [] [] k k) : mHoareIO P Q :=
hoare.zero $ λ k h, ⟨[], [], rfl, (list.append_nil _).symm, H h⟩
theorem mHoareIO.mono_l {P P' : sProp} {Q : list byte → list byte → mProp}
(H : P ≤ P') : mHoareIO P' Q → mHoareIO P Q :=
λ h, hoare.mono_l (λ k h, H _ h) h
theorem mHoareIO.mono_r {P : sProp} {Q Q' : list byte → list byte → mProp}
(H : ∀ {{i o}}, Q i o ≤ Q' i o) : mHoareIO P Q → mHoareIO P Q' :=
λ h, hoare.mono_r (by exact λ k k' _ ⟨i', o', h₁, h₂, h₃⟩,
⟨i', o', h₁, h₂, H _ _ h₃⟩) h
theorem mHoareIO.step {P P' : sProp} {Q Q' : list byte → list byte → mProp}
(H₁ : ∀ ⦃k i o⦄, P k → ∃ k', kcfg.step ⟨i, o, k⟩ k')
(H₂ : ∀ {{k₁ k₂ i o i₁ o₁}}, P k₁ →
kcfg.step ⟨i₁ ++ i, o, k₁⟩ ⟨i, o ++ o₁, k₂⟩ → P' k₂ ∧
∀ {{i₂ o₂ k₃}}, Q' i₂ o₂ k₂ k₃ → Q (i₁ ++ i₂) (o₁ ++ o₂) k₁ k₃)
(H₃ : mHoareIO P' Q') : mHoareIO P Q :=
begin
refine hoare.step (λ ⟨i', o', k⟩ h, H₁ h) _ H₃,
rintro ⟨i₁, o₁, k₁⟩ h ⟨i₂, o₂, k₂⟩ h',
rcases kcfg.step.io_part h' with ⟨⟨i, rfl⟩, ⟨o, rfl⟩⟩,
rcases H₂ h h' with ⟨h₁, h₄⟩,
refine ⟨h₁, _⟩,
rintro k₃ ⟨i', o', e₁, e₂, h₅⟩,
exact ⟨_, _,
(congr_arg ((++) i) e₁).trans (list.append_assoc _ _ _).symm,
e₂.trans (list.append_assoc _ _ _),
h₄ h₅⟩
end
theorem mHoareIO.comp {P : sProp} {Q Q' : list byte → list byte → mProp}
(H₁ : mHoareIO P Q)
(H₂ : mHoareIO (sProp.ex $ λ i, sProp.ex $ λ o, (Q i o).apply P) Q') :
mHoareIO P (λ i o,
mProp.ex $ λ i₁, mProp.ex $ λ i₂,
mProp.ex $ λ o₁, mProp.ex $ λ o₂,
((Q i₁ o₁).comp (Q' i₂ o₂)).with (i = i₁ ++ i₂ ∧ o = o₁ ++ o₂)) :=
λ k p, begin
refine (H₁ p).bind _,
rintro k₁ ⟨i₁, o₁, e₁, e₂, h₁⟩,
refine (H₂ ⟨_, _, _, p, h₁⟩).mono _,
rintro k₂ ⟨i₂, o₂, e₃, e₄, h₂⟩,
refine ⟨_, _, _, _, _, _, _, _, ⟨_, h₁, h₂⟩, rfl, rfl⟩,
rw [e₁, list.append_assoc, e₃],
rw [e₄, ← list.append_assoc, e₂],
end
theorem mHoareIO.ex_l {α} {P : α → sProp} {Q} :
mHoareIO (sProp.ex P) Q ↔ ∀ a, mHoareIO (P a) Q :=
⟨λ H a k h, H ⟨a, h⟩, λ H k ⟨a, h⟩, H a h⟩
theorem mHoare.zero {P : sProp} {Q : mProp}
(H : ∀ {{k}}, P k → Q k k) : mHoare P Q :=
mHoareIO.zero $ λ k h, ⟨H h, rfl, rfl⟩
theorem mHoare.mono_l {P P' : sProp} {Q : mProp}
(H : P ≤ P') : mHoare P' Q → mHoare P Q :=
mHoareIO.mono_l H
theorem mHoare.mono_r {P : sProp} {Q Q' : mProp}
(H : Q ≤ Q') : mHoare P Q → mHoare P Q' :=
mHoareIO.mono_r $ λ k k' i o ⟨h₁, h₂, h₃⟩, ⟨H _ _ h₁, h₂, h₃⟩
theorem mHoare.step {P P' : sProp} {Q Q' : mProp}
(H₁ : ∀ ⦃k⦄, P k → ∃ k', config.step k k')
(H₂ : ∀ {{k₁ k₂}}, P k₁ → config.step k₁ k₂ →
P' k₂ ∧ ∀ {{k₃}}, Q' k₂ k₃ → Q k₁ k₃) :
mHoare P' Q' → mHoare P Q :=
mHoareIO.step
(λ k i o h, let ⟨k', h'⟩ := H₁ h in ⟨⟨i, o, k'⟩, kcfg.step.noio h'⟩)
(λ k₁ k₂ i o i₁ o₁ h h', begin
generalize_hyp ei : i₁ ++ i = i' at h',
generalize_hyp eo : o ++ o₁ = o' at h',
cases h',
{ cases (@list.append_left_inj _ _ [] i).1 ei,
cases (@list.append_right_inj _ _ [] o).1 (by simpa using eo),
exact ⟨(H₂ h h'_a).1,
λ i₂ o₂ k₃ h'', ⟨(H₂ h h'_a).2 h''.1, h''.2⟩⟩ },
{ cases H₁ h with k₂ h',
cases config.step_noIO h' h'_a }
end)
theorem mHoare.comp {P : sProp} {Q Q' : mProp}
(H₁ : mHoare P Q)
(H₂ : mHoare (Q.apply P) Q') :
mHoare P (Q.comp Q') :=
(H₁.comp (H₂.mono_l
(λ k₂ ⟨_, _, k₁, h₁, h₂, rfl, rfl⟩, ⟨_, h₁, h₂⟩))).mono_r $
by rintro _ _ k₁ k₃ ⟨i₁, o₁, i₂, o₂,
⟨k₂, ⟨h₁, rfl, rfl⟩, ⟨h₂, rfl, rfl⟩⟩, rfl, rfl⟩;
exact ⟨⟨_, h₁, h₂⟩, rfl, rfl⟩
theorem mHoare.ex_l {α} {P : α → sProp} {Q} :
mHoare (sProp.ex P) Q ↔ ∀ a, mHoare (P a) Q :=
⟨λ H a k h, H ⟨a, h⟩, λ H k ⟨a, h⟩, H a h⟩
theorem stmt.hoareIO.mono_s {A P Q s₁ s₂}
(h : s₂ ≤ s₁) : stmt.hoareIO A P s₁ Q → stmt.hoareIO A P s₂ Q :=
λ H D rip l h', H _ _ _ (h _ _ _ _ h')
theorem stmt.hoareIO.mono_l {A P₁ P₂ s Q}
(h : P₂ ≤ P₁) : stmt.hoareIO A P₁ s Q → stmt.hoareIO A P₂ s Q :=
λ H D rip l h', (H _ _ _ h').mono_l
(λ k ⟨⟨h₁, h₂⟩, h₃⟩, ⟨⟨h _ h₁, h₂⟩, h₃⟩)
theorem stmt.hoareIO.mono_r {A P s}
{Q₁ Q₂ : exit_kind → list byte → list byte → mProp}
(h : ∀ {{e i o}}, Q₁ e i o ≤ Q₂ e i o) :
stmt.hoareIO A P s Q₁ → stmt.hoareIO A P s Q₂ :=
λ H D rip l h', (H _ _ _ h').mono_r
(λ i o k k', and.imp_right $
Exists.imp $ λ e, and.imp_left $ λ h₁, h _ _ h₁)
theorem stmt.hoareIO.ex_l {α A s} {P : α → sProp} {Q} :
stmt.hoareIO A (sProp.ex P) s Q ↔ ∀ a, stmt.hoareIO A (P a) s Q :=
⟨λ H a, H.mono_l $ λ k h, ⟨a, h⟩,
λ H S rip k h k ⟨⟨⟨a, h₁⟩, h₂⟩, h₃⟩,
H a _ _ _ h ⟨⟨h₁, h₂⟩, h₃⟩⟩
theorem stmt.hoare_iff {A P s Q} :
stmt.hoare A P s Q ↔
∀ {{D rip l}}, s A D rip l →
mHoare
(P ⊓ sProp.sn place.rip rip ⊓
sProp.mem_block (perm.R + perm.X) rip l)
(mProp.ex (λ e, stable_at D ⊓
(Q e).final (e.result A (rip + l.length)))) :=
forall_congr $ λ D, forall_congr $ λ rip, forall_congr $ λ l,
forall_congr $ λ h, iff_of_eq begin
congr, funext i o k k', exact propext
⟨λ ⟨h₁, e, ⟨h₂, ei, eo⟩, h₃⟩, ⟨⟨e, h₁, h₂, h₃⟩, ei, eo⟩,
λ ⟨⟨e, h₁, h₂, h₃⟩, ei, eo⟩, ⟨h₁, e, ⟨h₂, ei, eo⟩, h₃⟩⟩,
end
theorem stmt.hoare.mono_s {A P Q s₁ s₂}
(h : s₂ ≤ s₁) : stmt.hoare A P s₁ Q → stmt.hoare A P s₂ Q :=
λ H D rip l h', H _ _ _ (h _ _ _ _ h')
theorem stmt.hoare.mono_l {A P₁ P₂ s Q}
(h : P₂ ≤ P₁) : stmt.hoare A P₁ s Q → stmt.hoare A P₂ s Q :=
λ H D rip l h', (H _ _ _ h').mono_l
(λ k ⟨⟨h₁, h₂⟩, h₃⟩, ⟨⟨h _ h₁, h₂⟩, h₃⟩)
theorem stmt.hoare.mono_r {A P s} {Q₁ Q₂ : exit_kind → mProp}
(h : ∀ {{e}}, Q₁ e ≤ Q₂ e) :
stmt.hoare A P s Q₁ → stmt.hoare A P s Q₂ :=
stmt.hoareIO.mono_r $ λ e i o k k' ⟨h₁, h₂⟩, ⟨h _ _ h₁, h₂⟩
theorem stmt.hoare.zero {A P} {s : stmt} {p : Prop}
(H : ∀ {{D rip l}}, s A D rip l → p ∧ l = []) :
s.hoare A P (λ e, (mlift p).with (e = exit_kind.straight)) :=
stmt.hoare_iff.2 $ λ D rip l h', begin
rcases H h' with ⟨h, rfl⟩, apply mHoare.zero,
rintro k ⟨⟨h₁, h₂⟩, h₃⟩,
refine ⟨_, stable_at.refl, ⟨h, rfl⟩, _⟩,
simpa using h₂
end
theorem stmt.hoare.ex_l {α A s} {P : α → sProp} {Q} :
stmt.hoare A (sProp.ex P) s Q ↔ ∀ a, stmt.hoare A (P a) s Q :=
⟨λ H a, H.mono_l $ λ k h, ⟨a, h⟩,
λ H S rip k h k ⟨⟨⟨a, h₁⟩, h₂⟩, h₃⟩,
H a _ _ _ h ⟨⟨h₁, h₂⟩, h₃⟩⟩
theorem expr.hoare_iff {α A P E Q} :
@expr.hoare α A P E Q ↔
∀ {{b}}, (E b).hoare A P $
λ e, (Q b).with (e = exit_kind.straight) :=
forall_congr $ λ b, iff_of_eq begin
congr, funext e i o k k', exact propext
⟨λ ⟨⟨h, ei, eo⟩, ee⟩, ⟨⟨h, ee⟩, ei, eo⟩,
λ ⟨⟨h, ee⟩, ei, eo⟩, ⟨⟨h, ei, eo⟩, ee⟩⟩
end
theorem expr.hoare.zero {α A P} {e : expr α} {p : block → Prop}
(H : ∀ {{b D rip l}}, e b A D rip l → p b ∧ l = []) :
e.hoare A P (λ b, mlift (p b)) :=
expr.hoare_iff.2 $ λ b, stmt.hoare.zero (@H b)
theorem expr.hoareIO.mono_e {α A P Q} {e₁ e₂ : expr α}
(h : e₂ ≤ e₁) : expr.hoareIO A P e₁ Q → expr.hoareIO A P e₂ Q :=
λ H b, (H b).mono_s (@h b)
theorem expr.hoareIO.mono_l {α A P₁ P₂ s Q}
(h : P₂ ≤ P₁) : @expr.hoareIO α A P₁ s Q → expr.hoareIO A P₂ s Q :=
λ H b, (H b).mono_l h
theorem expr.hoareIO.mono_r {α A P s}
{Q₁ Q₂ : block → list byte → list byte → mProp}
(h : ∀ {{b i o}}, Q₁ b i o ≤ Q₂ b i o) :
@expr.hoareIO α A P s Q₁ → expr.hoareIO A P s Q₂ :=
λ H b, (H b).mono_r $ λ e i o, inf_le_inf (@h _ _ _) (le_refl _)
theorem expr.hoareIO.ex_l {α β A} {e : expr β} {P : α → sProp} {Q} :
expr.hoareIO A (sProp.ex P) e Q ↔ ∀ a, expr.hoareIO A (P a) e Q :=
iff.trans (forall_congr $ λ b, stmt.hoareIO.ex_l) forall_swap
theorem expr.to_hoare {α A P} {e : expr α} {Q : block → mProp}
(H : ∀ b, stmt.hoare A P (e b) (λ e, (Q b).with (e = exit_kind.straight))) :
expr.hoare A P e Q :=
begin
refine λ b, stmt.hoareIO.mono_r _ (H b),
rintro e _ _ k k' ⟨⟨h, e'⟩, rfl, rfl⟩, exact ⟨⟨h, rfl, rfl⟩, e'⟩
end
theorem expr.hoare.ex_l {α β A} {e : expr β} {P : α → sProp} {Q} :
expr.hoare A (sProp.ex P) e Q ↔ ∀ a, expr.hoare A (P a) e Q :=
expr.hoareIO.ex_l
theorem var.hoare {α} (i : name α) {A} (P : sProp) :
(var i).hoare A P (λ b, mlift (b ∈ A.read i)) :=
expr.hoare.zero $ λ _ _ _ _, id
theorem ret.hoare {α b A} (P : sProp) :
(ret α b).hoare A P (λ b', mlift (b' = b)) :=
expr.hoare.zero $ λ _ _ _ _, id
theorem hstmt.hoare {A} (P : sProp) (Q : mProp) :
(hstmt P Q).hoare A P (λ e, Q.with (e = exit_kind.straight)) :=
stmt.hoare_iff.2 $ λ D rip l h',
(mHoare.mono_l
(by exact λ _, and.imp_left and.right) h').mono_r $
by rintro k k' ⟨⟨h₁, h₂⟩, h₃⟩; exact ⟨_, h₂, ⟨h₁, rfl⟩, h₃⟩
theorem hexpr.hoare {α A} (P : sProp) (Q : block → mProp) :
(@hexpr α P Q).hoare A P Q :=
expr.to_hoare $ λ b, hstmt.hoare _ _
def comp_exit (Q₁ Q₂ : exit_kind → mProp) (e : exit_kind) : mProp :=
mProp.ex $ λ e₁, exit_kind.cases_on e₁ ((Q₁ e₁).comp (Q₂ e)) $ λ _, (Q₁ e₁).with (e₁ = e)
/-
theorem stmt.seq.hoare {A s₁ s₂}
{P₁ P₂ : sProp} {Q₁ Q₂ : exit_kind → mProp}
(H₁ : stmt.hoare A P₁ s₁ Q₁)
(H₂ : stmt.hoare A P₂ s₂ Q₂)
(H : (Q₁ exit_kind.straight).apply P₁ ≤ P₂) :
(stmt.seq s₁ s₂).hoare A P₁ (comp_exit Q₁ Q₂) :=
stmt.hoare_iff.2 begin
rintro D rip l ⟨_, _, _, v₁, v₂, h₁, h₂⟩,
change v₁.append v₂ with v₁ ++ v₂ at a ⊢,
have := stmt.hoare_iff.1 H₁ h₁,
refine (((stmt.hoare_iff.1 H₁ h₁).mono_l _).comp _).mono_r mProp.comp_ex_imp,
{ refine inf_le_inf (le_refl _) _,
exact λ k, mem.read'_left ⟨_, rfl⟩ },
rw [mProp.ex_apply, mHoare.ex_l], rintro (_|⟨n⟩),
{ refine ((stmt.hoare_iff.1 H₂ h₂).mono_l _).mono_r _,
{ rintro k₂ k₃ ⟨e₂, s₂, q₂, r₂⟩ e₁ k₁ ⟨s₁, q₁, r₁⟩,
rw [list.length_append, ← bitvec.add_nat_assoc],
refine ⟨e₂, s₁.trans s₂, ⟨exit_kind.straight, _, _, q₂⟩, r₂⟩,
sorry },
{ } },
{ exact ⟨⟨exit_kind.label n, q₁, _⟩, r₂⟩ },
have:= (((stmt.hoare_iff.1 H₁ h₁).mono_l _).comp
((stmt.hoare_iff.1 H₂ h₂).mono_l _)).mono_r _,
{ rintro k₁ k₃ ⟨k₂, ⟨s₁, e₁, q₁, r₁⟩, ⟨s₂, e₂, q₂, r₂⟩⟩,
cases e₁,
{ refine ⟨s₁.trans s₂, e₂, ⟨exit_kind.straight, _, q₁, q₂⟩, _⟩,
rwa [list.length_append, ← bitvec.add_nat_assoc] },
{ refine ⟨s₁.trans s₂, _, ⟨exit_kind.label e₁, _, rfl⟩, _⟩,
} },
end
-/
----------------------------------------
-- Assembly
----------------------------------------
-- theorem var.asm {α} {i : name α} {A D rip bl}
-- (h : bl ∈ locals_ctx.get Γ i) : var i bl A D rip [] :=
-- ⟨h, rfl⟩
theorem ret.asm {α A D rip bl} : ret α bl bl A D rip [] :=
⟨rfl, rfl⟩
def stmt.asm (s : stmt) (D : set place) := Π A rip, ∃ v, s A D rip v
end x86 |
032b0e15f9c629e10b5ae24bb286b4394e3f4068 | 22e97a5d648fc451e25a06c668dc03ac7ed7bc25 | /src/data/equiv/functor.lean | 71e568780ecfeb511d35e6472a386a8067e6918e | [
"Apache-2.0"
] | permissive | keeferrowan/mathlib | f2818da875dbc7780830d09bd4c526b0764a4e50 | aad2dfc40e8e6a7e258287a7c1580318e865817e | refs/heads/master | 1,661,736,426,952 | 1,590,438,032,000 | 1,590,438,032,000 | 266,892,663 | 0 | 0 | Apache-2.0 | 1,590,445,835,000 | 1,590,445,835,000 | null | UTF-8 | Lean | false | false | 2,646 | lean | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Simon Hudon, Scott Morrison
-/
import data.equiv.basic
import control.bifunctor
/-!
# Functor and bifunctors can be applied to `equiv`s.
We define
```lean
def functor.map_equiv (f : Type u → Type v) [functor f] [is_lawful_functor f] :
α ≃ β → f α ≃ f β
```
and
```lean
def bifunctor.map_equiv (F : Type u → Type v → Type w) [bifunctor F] [is_lawful_bifunctor F] :
α ≃ β → α' ≃ β' → F α α' ≃ F β β'
```
-/
universes u v w
variables {α β : Type u}
open equiv
namespace functor
variables (f : Type u → Type v) [functor f] [is_lawful_functor f]
/-- Apply a functor to an `equiv`. -/
def map_equiv (h : α ≃ β) : f α ≃ f β :=
{ to_fun := map h,
inv_fun := map h.symm,
left_inv := λ x,
by { rw map_map, convert is_lawful_functor.id_map x, ext a, apply symm_apply_apply },
right_inv := λ x,
by { rw map_map, convert is_lawful_functor.id_map x, ext a, apply apply_symm_apply } }
@[simp]
lemma map_equiv_apply (h : α ≃ β) (x : f α) :
(map_equiv f h : f α ≃ f β) x = map h x := rfl
@[simp]
lemma map_equiv_symm_apply (h : α ≃ β) (y : f β) :
(map_equiv f h : f α ≃ f β).symm y = map h.symm y := rfl
@[simp]
lemma map_equiv_refl : map_equiv f (equiv.refl α) = equiv.refl (f α) :=
begin
ext x,
simp only [map_equiv_apply, refl_apply],
exact is_lawful_functor.id_map x,
end
end functor
namespace bifunctor
variables {α' β' : Type v} (F : Type u → Type v → Type w) [bifunctor F] [is_lawful_bifunctor F]
/-- Apply a bifunctor to a pair of `equiv`s. -/
def map_equiv (h : α ≃ β) (h' : α' ≃ β') : F α α' ≃ F β β' :=
{ to_fun := bimap h h',
inv_fun := bimap h.symm h'.symm,
left_inv := λ x,
by { rw bimap_bimap, convert is_lawful_bifunctor.id_bimap x; { ext a, apply symm_apply_apply } },
right_inv := λ x,
by { rw bimap_bimap, convert is_lawful_bifunctor.id_bimap x; { ext a, apply apply_symm_apply } } }
@[simp]
lemma map_equiv_apply (h : α ≃ β) (h' : α' ≃ β') (x : F α α') :
(map_equiv F h h' : F α α' ≃ F β β') x = bimap h h' x := rfl
@[simp]
lemma map_equiv_symm_apply (h : α ≃ β) (h' : α' ≃ β') (y : F β β') :
(map_equiv F h h' : F α α' ≃ F β β').symm y = bimap h.symm h'.symm y := rfl
@[simp]
lemma map_equiv_refl_refl : map_equiv F (equiv.refl α) (equiv.refl α') = equiv.refl (F α α') :=
begin
ext x,
simp only [map_equiv_apply, refl_apply],
exact is_lawful_bifunctor.id_bimap x,
end
end bifunctor
|
47c806f99eabae9e469437f87372ec4a2628d021 | 92bfaf170880e47d55bf51d5a782fffd76db2f5f | /melting_point/ens.lean | 87d6bf7b1094c865351af84da99b04730d320e4b | [] | no_license | forked-from-1kasper/melting_point | d33403e1985d876a2c7c06859962cc0c37570189 | e5ea4a0917de086b7e5b122e8d5aa90d2761d147 | refs/heads/master | 1,624,785,375,577 | 1,618,305,367,000 | 1,618,305,367,000 | 222,729,018 | 2 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,872 | lean | noncomputable theory
open classical
axiom cls : Type
axiom V : set cls
def ens := subtype V
namespace cls
axiom contains : ens → cls → Prop
instance : has_mem ens cls := ⟨contains⟩
axiom comp : (ens → Prop) → cls
@[simp] axiom compβrule : Π (φ : ens → Prop) (x : ens), (x ∈ comp φ) = φ x
axiom ext : ∀ {α β : cls}, (∀ x, x ∈ α ↔ x ∈ β) → α = β
def empty := comp (λ _, false)
def univ := comp (λ _, true)
def insert (x : ens) (α : cls) :=
comp (λ y, x = y ∨ y ∈ α)
instance : has_insert ens cls := ⟨insert⟩
def sep (φ : ens → Prop) (α : cls) :=
comp (λ x, x ∈ α ∧ φ x)
instance : has_sep ens cls := ⟨sep⟩
def union (α β : cls) := comp {x | x ∈ α ∨ x ∈ β}
def diff (α β : cls) := {x ∈ α | x ∉ β}
def inter (α β : cls) := {x ∈ α | x ∈ β}
def subset (α β : cls) := ∀ x, x ∈ α → x ∈ β
def compl (α : cls) := comp {x | x ∉ α}
instance : has_emptyc cls := ⟨empty⟩
instance : has_union cls := ⟨union⟩
instance : has_inter cls := ⟨inter⟩
instance : has_subset cls := ⟨subset⟩
instance : has_neg cls := ⟨compl⟩
instance : has_sdiff cls := ⟨diff⟩
instance : has_singleton ens cls := ⟨λ x, insert x ∅⟩
def powerset (α : cls) : cls := comp {β | β.val ⊆ α}
prefix `𝒫`:100 := powerset
end cls
namespace ens
axiom empty : ∅ ∈ V
axiom singleton : ∀ x, {x} ∈ V
axiom union : ∀ α β, α ∈ V → β ∈ V → α ∪ β ∈ V
axiom inter : ∀ α β, α ∈ V → β ∈ V → α ∩ β ∈ V
axiom diff : ∀ α β, α ∈ V → β ∈ V → α \ β ∈ V
axiom powerset : ∀ (α : cls), α ∈ V → 𝒫 α ∈ V
axiom sep : ∀ (φ : ens → Prop) (α : cls), α ∈ V → { x ∈ α | φ x } ∈ V
end ens |
49c57296774976861cc861a895e3897912705b6b | f1dc39e1c68f71465c8bf79910c4664d03824751 | /library/init/data/int/order.lean | 28436c94cb2a15e3b0b743ae67cf5b804265d04a | [
"Apache-2.0"
] | permissive | kckennylau/lean-2 | 6504f45da07bc98b098d726b74130103be25885c | c9a9368bc0fd600d832bd56c5cb2124b8a523ef9 | refs/heads/master | 1,659,140,308,864 | 1,589,361,166,000 | 1,589,361,166,000 | 263,748,786 | 0 | 0 | null | 1,589,405,915,000 | 1,589,405,915,000 | null | UTF-8 | Lean | false | false | 11,818 | lean | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
The order relation on the integers.
-/
prelude
import init.data.int.basic init.data.ordering.basic
local attribute [simp] sub_eq_add_neg
namespace int
private def nonneg (a : ℤ) : Prop := int.cases_on a (assume n, true) (assume n, false)
protected def le (a b : ℤ) : Prop := nonneg (b - a)
instance : has_le int := ⟨int.le⟩
protected def lt (a b : ℤ) : Prop := (a + 1) ≤ b
instance : has_lt int := ⟨int.lt⟩
private def decidable_nonneg (a : ℤ) : decidable (nonneg a) :=
int.cases_on a (assume a, decidable.true) (assume a, decidable.false)
instance decidable_le (a b : ℤ) : decidable (a ≤ b) := decidable_nonneg _
instance decidable_lt (a b : ℤ) : decidable (a < b) := decidable_nonneg _
lemma lt_iff_add_one_le (a b : ℤ) : a < b ↔ a + 1 ≤ b := iff.refl _
private lemma nonneg.elim {a : ℤ} : nonneg a → ∃ n : ℕ, a = n :=
int.cases_on a (assume n H, exists.intro n rfl) (assume n', false.elim)
private lemma nonneg_or_nonneg_neg (a : ℤ) : nonneg a ∨ nonneg (-a) :=
int.cases_on a (assume n, or.inl trivial) (assume n, or.inr trivial)
lemma le.intro_sub {a b : ℤ} {n : ℕ} (h : b - a = n) : a ≤ b :=
show nonneg (b - a), by rw h; trivial
lemma le.intro {a b : ℤ} {n : ℕ} (h : a + n = b) : a ≤ b :=
le.intro_sub (by rw [← h, add_comm]; simp)
lemma le.dest_sub {a b : ℤ} (h : a ≤ b) : ∃ n : ℕ, b - a = n := nonneg.elim h
lemma le.dest {a b : ℤ} (h : a ≤ b) : ∃ n : ℕ, a + n = b :=
match (le.dest_sub h) with
| ⟨n, h₁⟩ := exists.intro n begin rw [← h₁, add_comm], simp end
end
lemma le.elim {a b : ℤ} (h : a ≤ b) {P : Prop} (h' : ∀ n : ℕ, a + ↑n = b → P) : P :=
exists.elim (le.dest h) h'
protected lemma le_total (a b : ℤ) : a ≤ b ∨ b ≤ a :=
or.imp_right
(assume H : nonneg (-(b - a)),
have -(b - a) = a - b, by simp,
show nonneg (a - b), from this ▸ H)
(nonneg_or_nonneg_neg (b - a))
lemma coe_nat_le_coe_nat_of_le {m n : ℕ} (h : m ≤ n) : (↑m : ℤ) ≤ ↑n :=
match nat.le.dest h with
| ⟨k, (hk : m + k = n)⟩ := le.intro (begin rw [← hk], reflexivity end)
end
lemma le_of_coe_nat_le_coe_nat {m n : ℕ} (h : (↑m : ℤ) ≤ ↑n) : m ≤ n :=
le.elim h (assume k, assume hk : ↑m + ↑k = ↑n,
have m + k = n, from int.coe_nat_inj ((int.coe_nat_add m k).trans hk),
nat.le.intro this)
lemma coe_nat_le_coe_nat_iff (m n : ℕ) : (↑m : ℤ) ≤ ↑n ↔ m ≤ n :=
iff.intro le_of_coe_nat_le_coe_nat coe_nat_le_coe_nat_of_le
lemma coe_zero_le (n : ℕ) : 0 ≤ (↑n : ℤ) :=
coe_nat_le_coe_nat_of_le n.zero_le
lemma eq_coe_of_zero_le {a : ℤ} (h : 0 ≤ a) : ∃ n : ℕ, a = n :=
by have t := le.dest_sub h; simp at t; exact t
lemma eq_succ_of_zero_lt {a : ℤ} (h : 0 < a) : ∃ n : ℕ, a = n.succ :=
let ⟨n, (h : ↑(1+n) = a)⟩ := le.dest h in
⟨n, by rw add_comm at h; exact h.symm⟩
lemma lt_add_succ (a : ℤ) (n : ℕ) : a < a + ↑(nat.succ n) :=
le.intro (show a + 1 + n = a + nat.succ n, begin simp [int.coe_nat_eq, add_comm, add_left_comm], reflexivity end)
lemma lt.intro {a b : ℤ} {n : ℕ} (h : a + nat.succ n = b) : a < b :=
h ▸ lt_add_succ a n
lemma lt.dest {a b : ℤ} (h : a < b) : ∃ n : ℕ, a + ↑(nat.succ n) = b :=
le.elim h (assume n, assume hn : a + 1 + n = b,
exists.intro n begin rw [← hn, add_assoc, add_comm (1 : int)], reflexivity end)
lemma lt.elim {a b : ℤ} (h : a < b) {P : Prop} (h' : ∀ n : ℕ, a + ↑(nat.succ n) = b → P) : P :=
exists.elim (lt.dest h) h'
lemma coe_nat_lt_coe_nat_iff (n m : ℕ) : (↑n : ℤ) < ↑m ↔ n < m :=
begin rw [lt_iff_add_one_le, ← int.coe_nat_succ, coe_nat_le_coe_nat_iff], reflexivity end
lemma lt_of_coe_nat_lt_coe_nat {m n : ℕ} (h : (↑m : ℤ) < ↑n) : m < n :=
(coe_nat_lt_coe_nat_iff _ _).mp h
lemma coe_nat_lt_coe_nat_of_lt {m n : ℕ} (h : m < n) : (↑m : ℤ) < ↑n :=
(coe_nat_lt_coe_nat_iff _ _).mpr h
/- show that the integers form an ordered additive group -/
protected lemma le_refl (a : ℤ) : a ≤ a :=
le.intro (add_zero a)
protected lemma le_trans {a b c : ℤ} (h₁ : a ≤ b) (h₂ : b ≤ c) : a ≤ c :=
le.elim h₁ (assume n, assume hn : a + n = b,
le.elim h₂ (assume m, assume hm : b + m = c,
begin apply le.intro, rw [← hm, ← hn, add_assoc], reflexivity end))
protected lemma le_antisymm {a b : ℤ} (h₁ : a ≤ b) (h₂ : b ≤ a) : a = b :=
le.elim h₁ (assume n, assume hn : a + n = b,
le.elim h₂ (assume m, assume hm : b + m = a,
have a + ↑(n + m) = a + 0, by rw [int.coe_nat_add, ← add_assoc, hn, hm, add_zero a],
have (↑(n + m) : ℤ) = 0, from add_left_cancel this,
have n + m = 0, from int.coe_nat_inj this,
have n = 0, from nat.eq_zero_of_add_eq_zero_right this,
show a = b, begin rw [← hn, this, int.coe_nat_zero, add_zero a] end))
protected lemma lt_irrefl (a : ℤ) : ¬ a < a :=
assume : a < a,
lt.elim this (assume n, assume hn : a + nat.succ n = a,
have a + nat.succ n = a + 0, by rw [hn, add_zero],
have nat.succ n = 0, from int.coe_nat_inj (add_left_cancel this),
show false, from nat.succ_ne_zero _ this)
protected lemma ne_of_lt {a b : ℤ} (h : a < b) : a ≠ b :=
(assume : a = b, absurd (begin rewrite this at h, exact h end) (int.lt_irrefl b))
lemma le_of_lt {a b : ℤ} (h : a < b) : a ≤ b :=
lt.elim h (assume n, assume hn : a + nat.succ n = b, le.intro hn)
protected lemma lt_iff_le_and_ne (a b : ℤ) : a < b ↔ (a ≤ b ∧ a ≠ b) :=
iff.intro
(assume h, ⟨le_of_lt h, int.ne_of_lt h⟩)
(assume ⟨aleb, aneb⟩,
le.elim aleb (assume n, assume hn : a + n = b,
have n ≠ 0,
from (assume : n = 0, aneb begin rw [← hn, this, int.coe_nat_zero, add_zero] end),
have n = nat.succ (nat.pred n),
from eq.symm (nat.succ_pred_eq_of_pos (nat.pos_of_ne_zero this)),
lt.intro (begin rewrite this at hn, exact hn end)))
lemma lt_succ (a : ℤ) : a < a + 1 :=
int.le_refl (a + 1)
protected lemma add_le_add_left {a b : ℤ} (h : a ≤ b) (c : ℤ) : c + a ≤ c + b :=
le.elim h (assume n, assume hn : a + n = b,
le.intro (show c + a + n = c + b, begin rw [add_assoc, hn] end))
protected lemma add_lt_add_left {a b : ℤ} (h : a < b) (c : ℤ) : c + a < c + b :=
iff.mpr (int.lt_iff_le_and_ne _ _)
(and.intro
(int.add_le_add_left (le_of_lt h) _)
(assume heq, int.lt_irrefl b begin rw add_left_cancel heq at h, exact h end))
protected lemma mul_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b :=
le.elim ha (assume n, assume hn,
le.elim hb (assume m, assume hm,
le.intro (show 0 + ↑n * ↑m = a * b, begin rw [← hn, ← hm], simp [zero_add] end)))
protected lemma mul_pos {a b : ℤ} (ha : 0 < a) (hb : 0 < b) : 0 < a * b :=
lt.elim ha (assume n, assume hn,
lt.elim hb (assume m, assume hm,
lt.intro (show 0 + ↑(nat.succ (nat.succ n * m + n)) = a * b,
begin rw [← hn, ← hm], simp [int.coe_nat_zero],
rw [← int.coe_nat_mul], simp [nat.mul_succ, nat.succ_add] end)))
protected lemma zero_lt_one : (0 : ℤ) < 1 := trivial
protected lemma lt_iff_le_not_le {a b : ℤ} : a < b ↔ (a ≤ b ∧ ¬ b ≤ a) :=
begin
simp [int.lt_iff_le_and_ne], split; intro h,
{ cases h with hab hn, split,
{ assumption },
{ intro hba, simp [int.le_antisymm hab hba] at *, contradiction } },
{ cases h with hab hn, split,
{ assumption },
{ intro h, simp [*] at * } }
end
instance : decidable_linear_ordered_comm_ring int :=
{ le := int.le,
le_refl := int.le_refl,
le_trans := @int.le_trans,
le_antisymm := @int.le_antisymm,
lt := int.lt,
lt_iff_le_not_le := @int.lt_iff_le_not_le,
add_le_add_left := @int.add_le_add_left,
zero_ne_one := int.zero_ne_one,
mul_pos := @int.mul_pos,
le_total := int.le_total,
zero_lt_one := int.zero_lt_one,
decidable_eq := int.decidable_eq,
decidable_le := int.decidable_le,
decidable_lt := int.decidable_lt,
..int.comm_ring }
instance : decidable_linear_ordered_add_comm_group int :=
by apply_instance
lemma eq_nat_abs_of_zero_le {a : ℤ} (h : 0 ≤ a) : a = nat_abs a :=
let ⟨n, e⟩ := eq_coe_of_zero_le h in by rw e; refl
lemma le_nat_abs {a : ℤ} : a ≤ nat_abs a :=
or.elim (le_total 0 a)
(λh, by rw eq_nat_abs_of_zero_le h; refl)
(λh, le_trans h (coe_zero_le _))
lemma neg_succ_lt_zero (n : ℕ) : -[1+ n] < 0 :=
lt_of_not_ge $ λ h, let ⟨m, h⟩ := eq_coe_of_zero_le h in by contradiction
lemma eq_neg_succ_of_lt_zero : ∀ {a : ℤ}, a < 0 → ∃ n : ℕ, a = -[1+ n]
| (n : ℕ) h := absurd h (not_lt_of_ge (coe_zero_le _))
| -[1+ n] h := ⟨n, rfl⟩
/- more facts specific to int -/
theorem of_nat_nonneg (n : ℕ) : 0 ≤ of_nat n := trivial
theorem coe_succ_pos (n : nat) : (nat.succ n : ℤ) > 0 :=
coe_nat_lt_coe_nat_of_lt (nat.succ_pos _)
theorem exists_eq_neg_of_nat {a : ℤ} (H : a ≤ 0) : ∃n : ℕ, a = -n :=
let ⟨n, h⟩ := eq_coe_of_zero_le (neg_nonneg_of_nonpos H) in
⟨n, eq_neg_of_eq_neg h.symm⟩
theorem nat_abs_of_nonneg {a : ℤ} (H : a ≥ 0) : (nat_abs a : ℤ) = a :=
match a, eq_coe_of_zero_le H with ._, ⟨n, rfl⟩ := rfl end
theorem of_nat_nat_abs_of_nonpos {a : ℤ} (H : a ≤ 0) : (nat_abs a : ℤ) = -a :=
by rw [← nat_abs_neg, nat_abs_of_nonneg (neg_nonneg_of_nonpos H)]
theorem abs_eq_nat_abs : ∀ a : ℤ, abs a = nat_abs a
| (n : ℕ) := abs_of_nonneg $ coe_zero_le _
| -[1+ n] := abs_of_nonpos $ le_of_lt $ neg_succ_lt_zero _
theorem nat_abs_abs (a : ℤ) : nat_abs (abs a) = nat_abs a :=
by rw [abs_eq_nat_abs]; refl
theorem lt_of_add_one_le {a b : ℤ} (H : a + 1 ≤ b) : a < b := H
theorem add_one_le_of_lt {a b : ℤ} (H : a < b) : a + 1 ≤ b := H
theorem lt_add_one_of_le {a b : ℤ} (H : a ≤ b) : a < b + 1 :=
add_le_add_right H 1
theorem le_of_lt_add_one {a b : ℤ} (H : a < b + 1) : a ≤ b :=
le_of_add_le_add_right H
theorem sub_one_le_of_lt {a b : ℤ} (H : a ≤ b) : a - 1 < b :=
sub_right_lt_of_lt_add $ lt_add_one_of_le H
theorem lt_of_sub_one_le {a b : ℤ} (H : a - 1 < b) : a ≤ b :=
le_of_lt_add_one $ lt_add_of_sub_right_lt H
theorem le_sub_one_of_lt {a b : ℤ} (H : a < b) : a ≤ b - 1 :=
le_sub_right_of_add_le H
theorem lt_of_le_sub_one {a b : ℤ} (H : a ≤ b - 1) : a < b :=
add_le_of_le_sub_right H
theorem sign_of_succ (n : nat) : sign (nat.succ n) = 1 := rfl
theorem sign_eq_one_of_pos {a : ℤ} (h : 0 < a) : sign a = 1 :=
match a, eq_succ_of_zero_lt h with ._, ⟨n, rfl⟩ := rfl end
theorem sign_eq_neg_one_of_neg {a : ℤ} (h : a < 0) : sign a = -1 :=
match a, eq_neg_succ_of_lt_zero h with ._, ⟨n, rfl⟩ := rfl end
lemma eq_zero_of_sign_eq_zero : Π {a : ℤ}, sign a = 0 → a = 0
| 0 _ := rfl
theorem pos_of_sign_eq_one : ∀ {a : ℤ}, sign a = 1 → 0 < a
| (n+1:ℕ) _ := coe_nat_lt_coe_nat_of_lt (nat.succ_pos _)
theorem neg_of_sign_eq_neg_one : ∀ {a : ℤ}, sign a = -1 → a < 0
| (n+1:ℕ) h := match h with end
| 0 h := match h with end
| -[1+ n] _ := neg_succ_lt_zero _
theorem sign_eq_one_iff_pos (a : ℤ) : sign a = 1 ↔ 0 < a :=
⟨pos_of_sign_eq_one, sign_eq_one_of_pos⟩
theorem sign_eq_neg_one_iff_neg (a : ℤ) : sign a = -1 ↔ a < 0 :=
⟨neg_of_sign_eq_neg_one, sign_eq_neg_one_of_neg⟩
theorem sign_eq_zero_iff_zero (a : ℤ) : sign a = 0 ↔ a = 0 :=
⟨eq_zero_of_sign_eq_zero, λ h, by rw [h, sign_zero]⟩
theorem sign_mul_abs (a : ℤ) : sign a * abs a = a :=
by rw [abs_eq_nat_abs, sign_mul_nat_abs]
theorem eq_one_of_mul_eq_self_left {a b : ℤ} (Hpos : a ≠ 0) (H : b * a = a) : b = 1 :=
eq_of_mul_eq_mul_right Hpos (by rw [one_mul, H])
theorem eq_one_of_mul_eq_self_right {a b : ℤ} (Hpos : b ≠ 0) (H : b * a = b) : a = 1 :=
eq_of_mul_eq_mul_left Hpos (by rw [mul_one, H])
end int
|
8af7e8155ad6775e1e4b7b128759a15401a8a8d6 | a3db153d66921f9d5a199b60b52ab3c21cf0e023 | /src/basic.lean | a8acd8a01546962b79968b15fb89e4723eac1b98 | [
"MIT"
] | permissive | metalogical/sia-lean | 3bd87a76b3ee6d910cbf182eb2a0e6d9eca3c056 | f8e354dd2ff6c09c4e001c1f80f6112c62da8592 | refs/heads/master | 1,629,775,811,523 | 1,512,684,874,000 | 1,512,684,874,000 | 106,763,566 | 4 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 4,807 | lean | import .sia
namespace sia
section
parameters {R : Type} [sia R]
variables {a b c : R}
@[reducible] private def Delta := Delta R
@[reducible] private def DeltaT := subtype Delta
-- Useful properties of Delta
lemma neg_delta : forall d: DeltaT, Delta (-d.val) :=
assume d, calc -d.val * -d.val
= d.val * d.val : by rw [neg_mul_neg]
... = 0 : d.property
instance : has_neg DeltaT := { neg := fun d : DeltaT, { val := -d.val, property := neg_delta d } }
lemma mul_delta : forall d: DeltaT, forall a: R, Delta (d.val * a) :=
assume d,
assume a,
have d.val * d.val = 0, from d.property,
show (d.val * a) * (d.val * a) = 0, from calc
(d.val * a) * (d.val * a)
= d.val * d.val * a * a : by simp
... = 0 * a * a : by rw this
... = 0 : by simp [zero_mul]
instance : has_mul DeltaT := { mul := fun d e : DeltaT, { val := d.val * e.val, property := mul_delta d e.val } }
-- Other theorems
theorem microaffinity : forall f: R -> R, forall x: R, exists! a: R, forall d: DeltaT, f (x + d.val) = f x + a * d.val :=
assume f: R -> R,
assume x: R,
let g (d: DeltaT) : R := f (x + d.val) in
have nice: f x = g 0, from eq.symm (eq.subst (add_group.add_zero x) (eq.refl _)),
by {rewrite nice, apply kock_lawvere}
theorem delta_near_zero : forall d: DeltaT, 0 <= d.val /\ d.val <= 0 :=
assume d,
have left: not (d.val < 0), from
assume bad,
have 0 < -d.val, by {rw <-neg_zero, apply st_ordered_field.lt_neg_flip bad},
have 0 < d.val * d.val, by {rw <-neg_mul_neg, rw <-mul_zero, apply st_ordered_field.lt_mul_pos_left this, assumption},
absurd d.property (ne.symm (st_order.lt_ne this)),
have right: not (0 < d.val), from
assume bad,
have 0 < d.val * d.val, by {rw <-mul_zero, apply st_ordered_field.lt_mul_pos_left bad bad},
absurd d.property (ne.symm (st_order.lt_ne this)),
and.intro left right
section -- Theorem 1.1
theorem delta_in_zero_interval : set.subset Delta [[(0: R) ... 0]] :=
assume a,
assume a_in_Delta,
delta_near_zero { val := a, property := a_in_Delta }
@[reducible]
def degenerate (S : set R) : Prop := forall x y : subtype S, x.val = y.val
theorem delta_nondegenerate : not (degenerate Delta) :=
assume deg,
have d_eq_zero: forall y: DeltaT, (0: R) = y.val, from deg 0,
let f := fun r: R, r in
have bad: exists! a: R, forall d: DeltaT, 0 + d.val = 0 + a * d.val, from microaffinity f 0,
have pf_zero: forall d: DeltaT, 0 + d.val = 0 + 0 * d.val, from
assume d,
calc
0 + d.val = d.val + 0 * d.val : by simp
... = 0 + 0 * d.val : by rw d_eq_zero d,
have pf_one: forall d: DeltaT, 0 + d.val = 0 + 1 * d.val, by simp,
have (0: R) = 1, from unique_of_exists_unique bad pf_zero pf_one,
absurd this (st_order.lt_ne (st_ordered_field.lt_zero_one R))
theorem delta_indistinguishable_zero : forall d: DeltaT, not (not (d.val = 0)) :=
assume d,
assume bad: d.val != 0,
have d.val < 0 \/ 0 < d.val, from st_order.ne_lt bad,
have left: not (d.val < 0), from and.elim_left (delta_near_zero d),
have right: not (0 < d.val), from and.elim_right (delta_near_zero d),
or.elim this left right
theorem not_lem_eq_delta_zero : not (forall d: DeltaT, d.val = 0 \/ d.val != 0) :=
assume bad,
have forall d: DeltaT, d.val = 0, from
assume d,
or.resolve_right (bad d) (delta_indistinguishable_zero d),
have forall x y: DeltaT, x.val = y.val, from
assume x y,
eq.trans (this x) (eq.symm (this y)),
delta_nondegenerate this
theorem microcancellation : forall {a b: R}, (forall d: DeltaT, a * d.val = b * d.val) -> a = b :=
assume a b,
assume ea_eq_eb : forall d: DeltaT, a * d.val = b * d.val,
let f (d : DeltaT) : R := a * d.val in
begin
apply (unique_of_exists_unique (kock_lawvere f)),
show forall d, f d = a * 0 + a * d.val,
simp,
show forall d, f d = a * 0 + b * d.val,
simp,
show forall d, f d = b * d.val,
intro d,
apply eq.trans (ea_eq_eb d),
reflexivity,
end
end
end
end sia
|
ab60e74c445f7b7754a669df6b190a2b75d002a8 | 491068d2ad28831e7dade8d6dff871c3e49d9431 | /tests/lean/sec_param_pp.lean | 9e1a41360f2c64b7f2e215525f29c68d6c2be599 | [
"Apache-2.0"
] | permissive | davidmueller13/lean | 65a3ed141b4088cd0a268e4de80eb6778b21a0e9 | c626e2e3c6f3771e07c32e82ee5b9e030de5b050 | refs/heads/master | 1,611,278,313,401 | 1,444,021,177,000 | 1,444,021,177,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 271 | lean | section
parameters {A : Type} (a : A)
variable f : A → A → A
definition id : A := a
check id
definition pr (b : A) : A := f a b
check pr f id
set_option pp.universes true
check pr f id
definition pr2 (B : Type) (b : B) : A := a
check pr2 num 10
end
|
85d10ed2d34d48114f47cd1a1472c2a51ff5dc52 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/analysis/normed_space/star/exponential.lean | f7a147543e7816cd6f8673ccaf5210846ab74a3b | [
"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 | 2,435 | lean | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import analysis.normed_space.star.basic
import algebra.star.module
import analysis.special_functions.exponential
/-! # The exponential map from selfadjoint to unitary
In this file, we establish various propreties related to the map `λ a, exp ℂ A (I • a)` between the
subtypes `self_adjoint A` and `unitary A`.
## TODO
* Show that any exponential unitary is path-connected in `unitary A` to `1 : unitary A`.
* Prove any unitary whose distance to `1 : unitary A` is less than `1` can be expressed as an
exponential unitary.
* A unitary is in the path component of `1` if and only if it is a finite product of exponential
unitaries.
-/
section star
variables {A : Type*}
[normed_ring A] [normed_algebra ℂ A] [star_ring A] [has_continuous_star A] [complete_space A]
[star_module ℂ A]
open complex
lemma self_adjoint.exp_i_smul_unitary {a : A} (ha : a ∈ self_adjoint A) :
exp ℂ (I • a) ∈ unitary A :=
begin
rw [unitary.mem_iff, star_exp],
simp only [star_smul, is_R_or_C.star_def, self_adjoint.mem_iff.mp ha, conj_I, neg_smul],
rw ←@exp_add_of_commute ℂ A _ _ _ _ _ _ ((commute.refl (I • a)).neg_left),
rw ←@exp_add_of_commute ℂ A _ _ _ _ _ _ ((commute.refl (I • a)).neg_right),
simpa only [add_right_neg, add_left_neg, and_self] using (exp_zero : exp ℂ (0 : A) = 1),
end
/-- The map from the selfadjoint real subspace to the unitary group. This map only makes sense
over ℂ. -/
@[simps]
noncomputable def self_adjoint.exp_unitary (a : self_adjoint A) : unitary A :=
⟨exp ℂ (I • a), self_adjoint.exp_i_smul_unitary (a.property)⟩
open self_adjoint
lemma commute.exp_unitary_add {a b : self_adjoint A} (h : commute (a : A) (b : A)) :
exp_unitary (a + b) = exp_unitary a * exp_unitary b :=
begin
ext,
have hcomm : commute (I • (a : A)) (I • (b : A)),
calc _ = _ : by simp only [h.eq, algebra.smul_mul_assoc, algebra.mul_smul_comm],
simpa only [exp_unitary_coe, add_subgroup.coe_add, smul_add] using exp_add_of_commute hcomm,
end
lemma commute.exp_unitary {a b : self_adjoint A} (h : commute (a : A) (b : A)) :
commute (exp_unitary a) (exp_unitary b) :=
calc (exp_unitary a) * (exp_unitary b) = (exp_unitary b) * (exp_unitary a)
: by rw [←h.exp_unitary_add, ←h.symm.exp_unitary_add, add_comm]
end star
|
bcbc559719939a644c4a7aa348f5b0a50e0cf6a0 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/linear_algebra/matrix/charpoly/eigs.lean | 684f1bb8e94be0b0de12b3f373db24af911bdf19 | [
"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 | 3,739 | lean | /-
Copyright (c) 2023 Mohanad Ahmed. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mohanad Ahmed
-/
import data.polynomial.basic
import field_theory.is_alg_closed.basic
/-!
# Eigenvalues are characteristic polynomial roots.
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
In fields we show that:
* `matrix.det_eq_prod_roots_charpoly_of_splits`: the determinant (in the field of the matrix)
is the product of the roots of the characteristic polynomial if the polynomial splits in the field
of the matrix.
* `matrix.trace_eq_sum_roots_charpoly_of_splits`: the trace is the sum of the roots of the
characteristic polynomial if the polynomial splits in the field of the matrix.
In an algebraically closed field we show that:
* `matrix.det_eq_prod_roots_charpoly`: the determinant is the product of the roots of the
characteristic polynomial.
* `matrix.trace_eq_sum_roots_charpoly`: the trace is the sum of the roots of the
characteristic polynomial.
Note that over other fields such as `ℝ`, these results can be used by using
`A.map (algebra_map ℝ ℂ)` as the matrix, and then applying `ring_hom.map_det`.
The two lemmas `matrix.det_eq_prod_roots_charpoly` and `matrix.trace_eq_sum_roots_charpoly` are more
commonly stated as trace is the sum of eigenvalues and determinant is the product of eigenvalues.
Mathlib has already defined eigenvalues in `linear_algebra.eigenspace` as the roots of the minimal
polynomial of a linear endomorphism. These do not have correct multiplicity and cannot be used in
the theorems above. Hence we express these theorems in terms of the roots of the characteristic
polynomial directly.
## TODO
The proofs of `det_eq_prod_roots_charpoly_of_splits` and
`trace_eq_sum_roots_charpoly_of_splits` closely resemble
`norm_gen_eq_prod_roots` and `trace_gen_eq_sum_roots` respectively, but the
dependencies are not general enough to unify them. We should refactor
`polynomial.prod_roots_eq_coeff_zero_of_monic_of_split` and
`polynomial.sum_roots_eq_next_coeff_of_monic_of_split` to assume splitting over an arbitrary map.
-/
variables {n : Type*} [fintype n] [decidable_eq n]
variables {R : Type*} [field R]
variables {A : matrix n n R}
open matrix polynomial
open_locale matrix big_operators
namespace matrix
lemma det_eq_prod_roots_charpoly_of_splits (hAps : A.charpoly.splits (ring_hom.id R)) :
A.det = (matrix.charpoly A).roots.prod :=
begin
rw [det_eq_sign_charpoly_coeff, ← (charpoly_nat_degree_eq_dim A),
polynomial.prod_roots_eq_coeff_zero_of_monic_of_split A.charpoly_monic (hAps),
← mul_assoc, ← pow_two, pow_right_comm, neg_one_sq, one_pow, one_mul],
end
lemma trace_eq_sum_roots_charpoly_of_splits (hAps : A.charpoly.splits (ring_hom.id R)) :
A.trace = (matrix.charpoly A).roots.sum :=
begin
casesI is_empty_or_nonempty n,
{ rw [matrix.trace, fintype.sum_empty, matrix.charpoly,
det_eq_one_of_card_eq_zero (fintype.card_eq_zero_iff.2 h), polynomial.roots_one,
multiset.empty_eq_zero, multiset.sum_zero], },
{ rw [trace_eq_neg_charpoly_coeff, neg_eq_iff_eq_neg,
← polynomial.sum_roots_eq_next_coeff_of_monic_of_split A.charpoly_monic (hAps),
next_coeff, charpoly_nat_degree_eq_dim,
if_neg (fintype.card_ne_zero : fintype.card n ≠ 0)], },
end
variables (A)
lemma det_eq_prod_roots_charpoly [is_alg_closed R] : A.det = (matrix.charpoly A).roots.prod :=
det_eq_prod_roots_charpoly_of_splits (is_alg_closed.splits A.charpoly)
lemma trace_eq_sum_roots_charpoly [is_alg_closed R] : A.trace = (matrix.charpoly A).roots.sum :=
trace_eq_sum_roots_charpoly_of_splits (is_alg_closed.splits A.charpoly)
end matrix
|
eaa9ff6c56825c520600007d3533a169963ac93f | bbecf0f1968d1fba4124103e4f6b55251d08e9c4 | /src/ring_theory/polynomial/rational_root.lean | 79dd1115324a8cc43b6069dfd4e775441e623fc3 | [
"Apache-2.0"
] | permissive | waynemunro/mathlib | e3fd4ff49f4cb43d4a8ded59d17be407bc5ee552 | 065a70810b5480d584033f7bbf8e0409480c2118 | refs/heads/master | 1,693,417,182,397 | 1,634,644,781,000 | 1,634,644,781,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 5,041 | lean | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import ring_theory.integrally_closed
import ring_theory.polynomial.scale_roots
/-!
# Rational root theorem and integral root theorem
This file contains the rational root theorem and integral root theorem.
The rational root theorem for a unique factorization domain `A`
with localization `S`, states that the roots of `p : polynomial A` in `A`'s
field of fractions are of the form `x / y` with `x y : A`, `x ∣ p.coeff 0` and
`y ∣ p.leading_coeff`.
The corollary is the integral root theorem `is_integer_of_is_root_of_monic`:
if `p` is monic, its roots must be integers.
Finally, we use this to show unique factorization domains are integrally closed.
## References
* https://en.wikipedia.org/wiki/Rational_root_theorem
-/
section scale_roots
variables {A K R S : Type*} [comm_ring A] [field K] [comm_ring R] [comm_ring S]
variables {M : submonoid A} [algebra A S] [is_localization M S] [algebra A K] [is_fraction_ring A K]
open finsupp is_fraction_ring is_localization polynomial
lemma scale_roots_aeval_eq_zero_of_aeval_mk'_eq_zero {p : polynomial A} {r : A} {s : M}
(hr : aeval (mk' S r s) p = 0) :
aeval (algebra_map A S r) (scale_roots p s) = 0 :=
begin
convert scale_roots_eval₂_eq_zero (algebra_map A S) hr,
rw [aeval_def, mk'_spec' _ r s]
end
variables [integral_domain A]
lemma num_is_root_scale_roots_of_aeval_eq_zero
[unique_factorization_monoid A] {p : polynomial A} {x : K} (hr : aeval x p = 0) :
is_root (scale_roots p (denom A x)) (num A x) :=
begin
apply is_root_of_eval₂_map_eq_zero (is_fraction_ring.injective A K),
refine scale_roots_aeval_eq_zero_of_aeval_mk'_eq_zero _,
rw mk'_num_denom,
exact hr
end
end scale_roots
section rational_root_theorem
variables {A K : Type*} [comm_ring A] [integral_domain A] [unique_factorization_monoid A] [field K]
variables [algebra A K] [is_fraction_ring A K]
open is_fraction_ring is_localization polynomial unique_factorization_monoid
/-- Rational root theorem part 1:
if `r : f.codomain` is a root of a polynomial over the ufd `A`,
then the numerator of `r` divides the constant coefficient -/
theorem num_dvd_of_is_root {p : polynomial A} {r : K} (hr : aeval r p = 0) :
num A r ∣ p.coeff 0 :=
begin
suffices : num A r ∣ (scale_roots p (denom A r)).coeff 0,
{ simp only [coeff_scale_roots, nat.sub_zero] at this,
haveI := classical.prop_decidable,
by_cases hr : num A r = 0,
{ obtain ⟨u, hu⟩ := (is_unit_denom_of_num_eq_zero hr).pow p.nat_degree,
rw ←hu at this,
exact units.dvd_mul_right.mp this },
{ refine dvd_of_dvd_mul_left_of_no_prime_factors hr _ this,
intros q dvd_num dvd_denom_pow hq,
apply hq.not_unit,
exact num_denom_reduced A r dvd_num (hq.dvd_of_dvd_pow dvd_denom_pow) } },
convert dvd_term_of_is_root_of_dvd_terms 0 (num_is_root_scale_roots_of_aeval_eq_zero hr) _,
{ rw [pow_zero, mul_one] },
intros j hj,
apply dvd_mul_of_dvd_right,
convert pow_dvd_pow (num A r) (nat.succ_le_of_lt (bot_lt_iff_ne_bot.mpr hj)),
exact (pow_one _).symm
end
/-- Rational root theorem part 2:
if `r : f.codomain` is a root of a polynomial over the ufd `A`,
then the denominator of `r` divides the leading coefficient -/
theorem denom_dvd_of_is_root {p : polynomial A} {r : K} (hr : aeval r p = 0) :
(denom A r : A) ∣ p.leading_coeff :=
begin
suffices : (denom A r : A) ∣ p.leading_coeff * num A r ^ p.nat_degree,
{ refine dvd_of_dvd_mul_left_of_no_prime_factors
(mem_non_zero_divisors_iff_ne_zero.mp (denom A r).2) _ this,
intros q dvd_denom dvd_num_pow hq,
apply hq.not_unit,
exact num_denom_reduced A r (hq.dvd_of_dvd_pow dvd_num_pow) dvd_denom },
rw ←coeff_scale_roots_nat_degree,
apply dvd_term_of_is_root_of_dvd_terms _ (num_is_root_scale_roots_of_aeval_eq_zero hr),
intros j hj,
by_cases h : j < p.nat_degree,
{ rw coeff_scale_roots,
refine (dvd_mul_of_dvd_right _ _).mul_right _,
convert pow_dvd_pow _ (nat.succ_le_iff.mpr (lt_sub_iff_left.mpr _)),
{ exact (pow_one _).symm },
simpa using h },
rw [←nat_degree_scale_roots p (denom A r)] at *,
rw [coeff_eq_zero_of_nat_degree_lt (lt_of_le_of_ne (le_of_not_gt h) hj.symm), zero_mul],
exact dvd_zero _
end
/-- Integral root theorem:
if `r : f.codomain` is a root of a monic polynomial over the ufd `A`,
then `r` is an integer -/
theorem is_integer_of_is_root_of_monic {p : polynomial A} (hp : monic p) {r : K}
(hr : aeval r p = 0) : is_integer A r :=
is_integer_of_is_unit_denom (is_unit_of_dvd_one _ (hp ▸ denom_dvd_of_is_root hr))
namespace unique_factorization_monoid
lemma integer_of_integral {x : K} :
is_integral A x → is_integer A x :=
λ ⟨p, hp, hx⟩, is_integer_of_is_root_of_monic hp hx
@[priority 100] -- See library note [lower instance priority]
instance : is_integrally_closed A :=
⟨λ x, integer_of_integral⟩
end unique_factorization_monoid
end rational_root_theorem
|
a0787e2fcf7cf2db063aa66a400d9c17c2c56d7b | 92b50235facfbc08dfe7f334827d47281471333b | /tests/lean/run/tactic26.lean | 70e229eba1a8d6e44113640de0b34e5ca8270f09 | [
"Apache-2.0"
] | permissive | htzh/lean | 24f6ed7510ab637379ec31af406d12584d31792c | d70c79f4e30aafecdfc4a60b5d3512199200ab6e | refs/heads/master | 1,607,677,731,270 | 1,437,089,952,000 | 1,437,089,952,000 | 37,078,816 | 0 | 0 | null | 1,433,780,956,000 | 1,433,780,955,000 | null | UTF-8 | Lean | false | false | 838 | lean | import logic data.num
open tactic inhabited
namespace foo
inductive sum (A : Type) (B : Type) : Type :=
| inl : A → sum A B
| inr : B → sum A B
theorem inl_inhabited {A : Type} (B : Type) (H : inhabited A) : inhabited (sum A B)
:= inhabited.destruct H (λ a, inhabited.mk (sum.inl B a))
theorem inr_inhabited (A : Type) {B : Type} (H : inhabited B) : inhabited (sum A B)
:= inhabited.destruct H (λ b, inhabited.mk (sum.inr A b))
notation `(` h `|` r:(foldl `|` (e r, tactic.or_else r e) h) `)` := r
infixl `;`:15 := tactic.and_then
definition my_tac := fixpoint (λ t, ( apply @inl_inhabited; t
| apply @inr_inhabited; t
| apply @num.is_inhabited
))
tactic_hint my_tac
theorem T : inhabited (sum false num)
end foo
|
9f36d675c6b45b2028217b458f0714a66ae52e3c | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /src/linear_algebra/affine_space.lean | 7c2556a36ce4cd4bf1a76d6e29de9530d33b4101 | [
"Apache-2.0"
] | permissive | dupuisf/mathlib | 62de4ec6544bf3b79086afd27b6529acfaf2c1bb | 8582b06b0a5d06c33ee07d0bdf7c646cae22cf36 | refs/heads/master | 1,669,494,854,016 | 1,595,692,409,000 | 1,595,692,409,000 | 272,046,630 | 0 | 0 | Apache-2.0 | 1,592,066,143,000 | 1,592,066,142,000 | null | UTF-8 | Lean | false | false | 62,528 | lean | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Joseph Myers.
-/
import algebra.add_torsor
import data.indicator_function
import linear_algebra.basis
noncomputable theory
open_locale big_operators
open_locale classical
/-!
# Affine spaces
This file defines affine spaces (over modules) and subspaces, affine
maps, affine combinations of points, and the affine span of a set of
points.
## Implementation notes
This file is very minimal and many things are surely omitted. Most
results can be deduced from corresponding results for modules or
vector spaces. The variables `k` and `V` are explicit rather than
implicit arguments to lemmas because otherwise the elaborator
sometimes has problems inferring appropriate types and type class
instances. Definitions of affine spaces vary as to whether a space
with no points is permitted; here, we require a nonempty type of
points (via the definition of torsors requiring a nonempty type).
## References
* https://en.wikipedia.org/wiki/Affine_space
* https://en.wikipedia.org/wiki/Principal_homogeneous_space
-/
open add_action
open add_torsor
/-- `affine_space` is an abbreviation for `add_torsor` in the case
where the group is a vector space, or more generally a module, but we
omit the type classes `[ring k]` and `[module k V]` in the type
synonym itself to simplify type class search.. -/
@[nolint unused_arguments]
abbreviation affine_space (k : Type*) (V : Type*) (P : Type*) [add_comm_group V] :=
add_torsor V P
namespace affine_space
variables (k : Type*) (V : Type*) {P : Type*} [ring k] [add_comm_group V] [module k V]
variables [affine_space k V P]
/-- The submodule spanning the differences of a (possibly empty) set
of points. -/
def vector_span (s : set P) : submodule k V := submodule.span k (vsub_set V s)
/-- The definition of `vector_span`, for rewriting. -/
lemma vector_span_def (s : set P) : vector_span k V s = submodule.span k (vsub_set V s) :=
rfl
variables (P)
/-- The `vector_span` of the empty set is `⊥`. -/
@[simp] lemma vector_span_empty : vector_span k V (∅ : set P) = ⊥ :=
by rw [vector_span_def, vsub_set_empty, submodule.span_empty]
variables {P}
/-- The `vsub_set` lies within the `vector_span`. -/
lemma vsub_set_subset_vector_span (s : set P) : vsub_set V s ⊆ vector_span k V s :=
submodule.subset_span
/-- Each pairwise difference is in the `vector_span`. -/
lemma vsub_mem_vector_span {s : set P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
p1 -ᵥ p2 ∈ vector_span k V s :=
begin
rw ←submodule.mem_coe,
exact set.mem_of_mem_of_subset (vsub_mem_vsub_set V hp1 hp2)
(vsub_set_subset_vector_span k V s)
end
/-- The points in the affine span of a (possibly empty) set of
points. Use `affine_span` instead to get an `affine_subspace k V P`. -/
def span_points (s : set P) : set P :=
{p | ∃ p1 ∈ s, ∃ v ∈ (vector_span k V s), p = v +ᵥ p1}
/-- A point in a set is in its affine span. -/
lemma mem_span_points (p : P) (s : set P) : p ∈ s → p ∈ span_points k V s
| hp := ⟨p, hp, 0, submodule.zero_mem _, (zero_vadd V p).symm⟩
/-- A set is contained in its `span_points`. -/
lemma subset_span_points (s : set P) : s ⊆ span_points k V s :=
λ p, mem_span_points k V p s
/-- The `span_points` of a set is nonempty if and only if that set
is. -/
@[simp] lemma span_points_nonempty (s : set P) :
(span_points k V s).nonempty ↔ s.nonempty :=
begin
split,
{ contrapose,
rw [set.not_nonempty_iff_eq_empty, set.not_nonempty_iff_eq_empty],
intro h,
simp [h, span_points] },
{ exact λ ⟨p, hp⟩, ⟨p, mem_span_points k V p s hp⟩ }
end
/-- Adding a point in the affine span and a vector in the spanning
submodule produces a point in the affine span. -/
lemma vadd_mem_span_points_of_mem_span_points_of_mem_vector_span {s : set P} {p : P} {v : V}
(hp : p ∈ span_points k V s) (hv : v ∈ vector_span k V s) : v +ᵥ p ∈ span_points k V s :=
begin
rcases hp with ⟨p2, ⟨hp2, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩,
rw [hv2p, vadd_assoc],
use [p2, hp2, v + v2, (vector_span k V s).add_mem hv hv2, rfl]
end
/-- Subtracting two points in the affine span produces a vector in the
spanning submodule. -/
lemma vsub_mem_vector_span_of_mem_span_points_of_mem_span_points {s : set P} {p1 p2 : P}
(hp1 : p1 ∈ span_points k V s) (hp2 : p2 ∈ span_points k V s) :
p1 -ᵥ p2 ∈ vector_span k V s :=
begin
rcases hp1 with ⟨p1a, ⟨hp1a, ⟨v1, ⟨hv1, hv1p⟩⟩⟩⟩,
rcases hp2 with ⟨p2a, ⟨hp2a, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩,
rw [hv1p, hv2p, vsub_vadd_eq_vsub_sub V (v1 +ᵥ p1a), vadd_vsub_assoc, add_comm, add_sub_assoc],
have hv1v2 : v1 - v2 ∈ vector_span k V s,
{ apply (vector_span k V s).add_mem hv1,
rw ←neg_one_smul k v2,
exact (vector_span k V s).smul_mem (-1 : k) hv2 },
refine (vector_span k V s).add_mem _ hv1v2,
exact vsub_mem_vector_span k V hp1a hp2a
end
end affine_space
open affine_space
namespace finset
variables {k : Type*} (V : Type*) {P : Type*} [ring k] [add_comm_group V] [module k V]
variables [S : affine_space k V P]
include S
variables {ι : Type*} (s : finset ι)
/-- A weighted sum of the results of subtracting a base point from the
given points, as a linear map on the weights. The main cases of
interest are where the sum of the weights is 0, in which case the sum
is independent of the choice of base point, and where the sum of the
weights is 1, in which case the sum added to the base point is
independent of the choice of base point. -/
def weighted_vsub_of_point (p : ι → P) (b : P) : (ι → k) →ₗ[k] V :=
∑ i in s, (linear_map.proj i : (ι → k) →ₗ[k] k).smul_right (p i -ᵥ b)
@[simp] lemma weighted_vsub_of_point_apply (w : ι → k) (p : ι → P) (b : P) :
s.weighted_vsub_of_point V p b w = ∑ i in s, w i • (p i -ᵥ b) :=
by simp [weighted_vsub_of_point, linear_map.sum_apply]
/-- The weighted sum is independent of the base point when the sum of
the weights is 0. -/
lemma weighted_vsub_of_point_eq_of_sum_eq_zero (w : ι → k) (p : ι → P) (h : ∑ i in s, w i = 0)
(b₁ b₂ : P) : s.weighted_vsub_of_point V p b₁ w = s.weighted_vsub_of_point V p b₂ w :=
begin
apply eq_of_sub_eq_zero,
rw [weighted_vsub_of_point_apply, weighted_vsub_of_point_apply, ←finset.sum_sub_distrib],
conv_lhs {
congr,
skip,
funext,
rw [←smul_sub, vsub_sub_vsub_cancel_left]
},
rw [←finset.sum_smul, h, zero_smul]
end
/-- The weighted sum, added to the base point, is independent of the
base point when the sum of the weights is 1. -/
lemma weighted_vsub_of_point_vadd_eq_of_sum_eq_one (w : ι → k) (p : ι → P) (h : ∑ i in s, w i = 1)
(b₁ b₂ : P) :
s.weighted_vsub_of_point V p b₁ w +ᵥ b₁ = s.weighted_vsub_of_point V p b₂ w +ᵥ b₂ :=
begin
erw [weighted_vsub_of_point_apply, weighted_vsub_of_point_apply, ←vsub_eq_zero_iff_eq V,
vadd_vsub_assoc, vsub_vadd_eq_vsub_sub, ←add_sub_assoc, add_comm, add_sub_assoc,
←finset.sum_sub_distrib],
conv_lhs {
congr,
skip,
congr,
skip,
funext,
rw [←smul_sub, vsub_sub_vsub_cancel_left]
},
rw [←finset.sum_smul, h, one_smul, vsub_add_vsub_cancel, vsub_self]
end
/-- The weighted sum is unaffected by removing the base point, if
present, from the set of points. -/
@[simp] lemma weighted_vsub_of_point_erase (w : ι → k) (p : ι → P) (i : ι) :
(s.erase i).weighted_vsub_of_point V p (p i) w = s.weighted_vsub_of_point V p (p i) w :=
begin
rw [weighted_vsub_of_point_apply, weighted_vsub_of_point_apply],
apply finset.sum_erase,
rw [vsub_self, smul_zero]
end
/-- The weighted sum is unaffected by adding the base point, whether
or not present, to the set of points. -/
@[simp] lemma weighted_vsub_of_point_insert (w : ι → k) (p : ι → P) (i : ι) :
(insert i s).weighted_vsub_of_point V p (p i) w = s.weighted_vsub_of_point V p (p i) w :=
begin
rw [weighted_vsub_of_point_apply, weighted_vsub_of_point_apply],
apply finset.sum_insert_zero,
rw [vsub_self, smul_zero]
end
/-- The weighted sum is unaffected by changing the weights to the
corresponding indicator function and adding points to the set. -/
lemma weighted_vsub_of_point_indicator_subset (w : ι → k) (p : ι → P) (b : P) {s₁ s₂ : finset ι}
(h : s₁ ⊆ s₂) :
s₁.weighted_vsub_of_point V p b w = s₂.weighted_vsub_of_point V p b (set.indicator ↑s₁ w) :=
begin
rw [weighted_vsub_of_point_apply, weighted_vsub_of_point_apply],
exact set.sum_indicator_subset_of_eq_zero w (λ i wi, wi • (p i -ᵥ b : V)) h (λ i, zero_smul k _)
end
/-- A weighted sum of the results of subtracting a default base point
from the given points, as a linear map on the weights. This is
intended to be used when the sum of the weights is 0; that condition
is specified as a hypothesis on those lemmas that require it. -/
def weighted_vsub (p : ι → P) : (ι → k) →ₗ[k] V :=
s.weighted_vsub_of_point V p (classical.choice S.nonempty)
/-- Applying `weighted_vsub` with given weights. This is for the case
where a result involving a default base point is OK (for example, when
that base point will cancel out later); a more typical use case for
`weighted_vsub` would involve selecting a preferred base point with
`weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero` and then
using `weighted_vsub_of_point_apply`. -/
lemma weighted_vsub_apply (w : ι → k) (p : ι → P) :
s.weighted_vsub V p w = ∑ i in s, w i • (p i -ᵥ (classical.choice S.nonempty)) :=
by simp [weighted_vsub, linear_map.sum_apply]
/-- `weighted_vsub` gives the sum of the results of subtracting any
base point, when the sum of the weights is 0. -/
lemma weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero (w : ι → k) (p : ι → P)
(h : ∑ i in s, w i = 0) (b : P) : s.weighted_vsub V p w = s.weighted_vsub_of_point V p b w :=
s.weighted_vsub_of_point_eq_of_sum_eq_zero V w p h _ _
/-- The `weighted_vsub` for an empty set is 0. -/
@[simp] lemma weighted_vsub_empty (w : ι → k) (p : ι → P) :
(∅ : finset ι).weighted_vsub V p w = 0 :=
by simp [weighted_vsub_apply]
/-- The weighted sum is unaffected by changing the weights to the
corresponding indicator function and adding points to the set. -/
lemma weighted_vsub_indicator_subset (w : ι → k) (p : ι → P) {s₁ s₂ : finset ι} (h : s₁ ⊆ s₂) :
s₁.weighted_vsub V p w = s₂.weighted_vsub V p (set.indicator ↑s₁ w) :=
weighted_vsub_of_point_indicator_subset _ _ _ _ h
/-- A weighted sum of the results of subtracting a default base point
from the given points, added to that base point. This is intended to
be used when the sum of the weights is 1, in which case it is an
affine combination (barycenter) of the points with the given weights;
that condition is specified as a hypothesis on those lemmas that
require it. -/
def affine_combination (w : ι → k) (p : ι → P) : P :=
s.weighted_vsub_of_point V p (classical.choice S.nonempty) w +ᵥ (classical.choice S.nonempty)
/-- Applying `affine_combination` with given weights. This is for the
case where a result involving a default base point is OK (for example,
when that base point will cancel out later); a more typical use case
for `affine_combination` would involve selecting a preferred base
point with
`affine_combination_eq_weighted_vsub_of_point_vadd_of_sum_eq_one` and
then using `weighted_vsub_of_point_apply`. -/
lemma affine_combination_apply (w : ι → k) (p : ι → P) :
s.affine_combination V w p =
s.weighted_vsub_of_point V p (classical.choice S.nonempty) w +ᵥ (classical.choice S.nonempty) :=
rfl
/-- `affine_combination` gives the sum with any base point, when the
sum of the weights is 1. -/
lemma affine_combination_eq_weighted_vsub_of_point_vadd_of_sum_eq_one (w : ι → k) (p : ι → P)
(h : ∑ i in s, w i = 1) (b : P) :
s.affine_combination V w p = s.weighted_vsub_of_point V p b w +ᵥ b :=
s.weighted_vsub_of_point_vadd_eq_of_sum_eq_one V w p h _ _
/-- Adding a `weighted_vsub` to an `affine_combination`. -/
lemma weighted_vsub_vadd_affine_combination (w₁ w₂ : ι → k) (p : ι → P) :
s.weighted_vsub V p w₁ +ᵥ s.affine_combination V w₂ p = s.affine_combination V (w₁ + w₂) p :=
begin
erw vadd_assoc,
congr,
exact (linear_map.map_add _ _ _).symm
end
/-- Subtracting two `affine_combination`s. -/
lemma affine_combination_vsub (w₁ w₂ : ι → k) (p : ι → P) :
s.affine_combination V w₁ p -ᵥ s.affine_combination V w₂ p = s.weighted_vsub V p (w₁ - w₂) :=
begin
erw vadd_vsub_vadd_cancel_right,
exact (linear_map.map_sub _ _ _).symm
end
/-- An `affine_combination` equals a point if that point is in the set
and has weight 1 and the other points in the set have weight 0. -/
@[simp] lemma affine_combination_of_eq_one_of_eq_zero (w : ι → k) (p : ι → P) {i : ι}
(his : i ∈ s) (hwi : w i = 1) (hw0 : ∀ i2 ∈ s, i2 ≠ i → w i2 = 0) :
s.affine_combination V w p = p i :=
begin
have h1 : ∑ i in s, w i = 1 := hwi ▸ finset.sum_eq_single i hw0 (λ h, false.elim (h his)),
rw [s.affine_combination_eq_weighted_vsub_of_point_vadd_of_sum_eq_one V w p h1 (p i),
weighted_vsub_of_point_apply],
convert zero_vadd V (p i),
convert finset.sum_eq_zero _,
intros i2 hi2,
by_cases h : i2 = i,
{ simp [h] },
{ simp [hw0 i2 hi2 h] }
end
/-- An affine combination is unaffected by changing the weights to the
corresponding indicator function and adding points to the set. -/
lemma affine_combination_indicator_subset (w : ι → k) (p : ι → P) {s₁ s₂ : finset ι}
(h : s₁ ⊆ s₂) :
s₁.affine_combination V w p = s₂.affine_combination V (set.indicator ↑s₁ w) p :=
by rw [affine_combination_apply, affine_combination_apply,
weighted_vsub_of_point_indicator_subset _ _ _ _ h]
end finset
section affine_independent
variables (k : Type*) (V : Type*) {P : Type*} [ring k] [add_comm_group V] [module k V]
variables [affine_space k V P] {ι : Type*}
/-- An indexed family is said to be affinely independent if no
nontrivial weighted subtractions (where the sum of weights is 0) are
0. -/
def affine_independent (p : ι → P) : Prop :=
∀ (s : finset ι) (w : ι → k), ∑ i in s, w i = 0 → s.weighted_vsub V p w = 0 → ∀ i ∈ s, w i = 0
/-- A family with at most one point is affinely independent. -/
lemma affine_independent_of_subsingleton [subsingleton ι] (p : ι → P) :
affine_independent k V p :=
λ s w h hs i hi, fintype.eq_of_subsingleton_of_sum_eq h i hi
/-- A family is affinely independent if and only if the differences
from a base point in that family are linearly independent. -/
lemma affine_independent_iff_linear_independent_vsub (p : ι → P) (i1 : ι) :
affine_independent k V p ↔ linear_independent k (λ i : {x // x ≠ i1}, (p i -ᵥ p i1 : V)) :=
begin
split,
{ intro h,
rw linear_independent_iff',
intros s g hg i hi,
set f : ι → k := λ x, if hx : x = i1 then -∑ y in s, g y else g ⟨x, hx⟩ with hfdef,
let s2 : finset ι := insert i1 (s.map (function.embedding.subtype _)),
have hfg : ∀ x : {x // x ≠ i1}, g x = f x,
{ intro x,
rw hfdef,
dsimp only [],
erw [dif_neg x.property, subtype.coe_eta] },
rw hfg,
have hf : ∑ ι in s2, f ι = 0,
{ rw [finset.sum_insert (finset.not_mem_map_subtype_of_not_property s (not_not.2 rfl)),
finset.sum_subtype_map_embedding (λ x hx, (hfg x).symm)],
rw hfdef,
dsimp only [],
rw dif_pos rfl,
exact neg_add_self _ },
have hs2 : s2.weighted_vsub V p f = 0,
{ set f2 : ι → V := λ x, f x • (p x -ᵥ p i1) with hf2def,
set g2 : {x // x ≠ i1} → V := λ x, g x • (p x -ᵥ p i1) with hg2def,
have hf2g2 : ∀ x : {x // x ≠ i1}, f2 x = g2 x,
{ simp_rw [hf2def, hg2def, hfg],
exact λ x, rfl },
rw [finset.weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero V s2 f p hf (p i1),
finset.weighted_vsub_of_point_insert, finset.weighted_vsub_of_point_apply,
finset.sum_subtype_map_embedding (λ x hx, hf2g2 x)],
exact hg },
exact h s2 f hf hs2 i (finset.mem_insert_of_mem (finset.mem_map.2 ⟨i, hi, rfl⟩)) },
{ intro h,
rw linear_independent_iff' at h,
intros s w hw hs i hi,
rw [finset.weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero V s w p hw (p i1),
←s.weighted_vsub_of_point_erase V w p i1, finset.weighted_vsub_of_point_apply] at hs,
let f : ι → V := λ i, w i • (p i -ᵥ p i1),
have hs2 : ∑ i in (s.erase i1).subtype (λ i, i ≠ i1), f i = 0,
{ rw [←hs],
convert finset.sum_subtype_of_mem f (λ x, finset.ne_of_mem_erase) },
have h2 := h ((s.erase i1).subtype (λ i, i ≠ i1)) (λ x, w x) hs2,
simp_rw [finset.mem_subtype] at h2,
have h2b : ∀ i ∈ s, i ≠ i1 → w i = 0 :=
λ i his hi, h2 ⟨i, hi⟩ (finset.mem_erase_of_ne_of_mem hi his),
exact finset.eq_zero_of_sum_eq_zero hw h2b i hi }
end
end affine_independent
/-- An `affine_subspace k V P` is a subset of an `affine_space k V P`
that, if not empty, has an affine space structure induced by a
corresponding subspace of the `module k V`. -/
structure affine_subspace (k : Type*) (V : Type*) (P : Type*) [ring k] [add_comm_group V]
[module k V] [affine_space k V P] :=
(carrier : set P)
(smul_vsub_vadd_mem : ∀ (c : k) (p1 p2 p3 ∈ carrier), c • (p1 -ᵥ p2 : V) +ᵥ p3 ∈ carrier)
namespace affine_subspace
variables (k : Type*) (V : Type*) (P : Type*) [ring k] [add_comm_group V] [module k V]
[affine_space k V P]
instance : has_coe (affine_subspace k V P) (set P) := ⟨carrier⟩
instance : has_mem P (affine_subspace k V P) := ⟨λ p s, p ∈ (s : set P)⟩
/-- A point is in an affine subspace coerced to a set if and only if
it is in that affine subspace. -/
@[simp] lemma mem_coe (p : P) (s : affine_subspace k V P) :
p ∈ (s : set P) ↔ p ∈ s :=
iff.rfl
variables {k V P}
/-- The direction of an affine subspace is the submodule spanned by
the pairwise differences of points. (Except in the case of an empty
affine subspace, where the direction is the zero submodule, every
vector in the direction is the difference of two points in the affine
subspace.) -/
def direction (s : affine_subspace k V P) : submodule k V := vector_span k V (s : set P)
/-- The direction equals the `vector_span`. -/
lemma direction_eq_vector_span (s : affine_subspace k V P) :
s.direction = vector_span k V (s : set P) :=
rfl
/-- Alternative definition of the direction when the affine subspace
is nonempty. This is defined so that the order on submodules (as used
in the definition of `submodule.span`) can be used in the proof of
`coe_direction_eq_vsub_set`, and is not intended to be used beyond
that proof. -/
def direction_of_nonempty {s : affine_subspace k V P} (h : (s : set P).nonempty) :
submodule k V :=
{ carrier := vsub_set V (s : set P),
zero_mem' := begin
cases h with p hp,
exact (vsub_self V p) ▸ vsub_mem_vsub_set V hp hp
end,
add_mem' := begin
intros a b ha hb,
rcases ha with ⟨p1, hp1, p2, hp2, ha⟩,
rcases hb with ⟨p3, hp3, p4, hp4, hb⟩,
rw [ha, hb, ←vadd_vsub_assoc],
refine vsub_mem_vsub_set V _ hp4,
convert s.smul_vsub_vadd_mem 1 p1 p2 p3 hp1 hp2 hp3,
rw one_smul
end,
smul_mem' := begin
intros c v hv,
rcases hv with ⟨p1, hp1, p2, hp2, hv⟩,
rw [hv, ←vadd_vsub V (c • (p1 -ᵥ p2)) p2],
refine vsub_mem_vsub_set V _ hp2,
exact s.smul_vsub_vadd_mem c p1 p2 p2 hp1 hp2 hp2
end }
/-- `direction_of_nonempty` gives the same submodule as
`direction`. -/
lemma direction_of_nonempty_eq_direction {s : affine_subspace k V P} (h : (s : set P).nonempty) :
direction_of_nonempty h = s.direction :=
le_antisymm (vsub_set_subset_vector_span k V s) (submodule.span_le.2 set.subset.rfl)
/-- The set of vectors in the direction of a nonempty affine subspace
is given by `vsub_set`. -/
lemma coe_direction_eq_vsub_set {s : affine_subspace k V P} (h : (s : set P).nonempty) :
(s.direction : set V) = vsub_set V (s : set P) :=
direction_of_nonempty_eq_direction h ▸ rfl
/-- A vector is in the direction of a nonempty affine subspace if and
only if it is the subtraction of two vectors in the subspace. -/
lemma mem_direction_iff_eq_vsub {s : affine_subspace k V P} (h : (s : set P).nonempty) (v : V) :
v ∈ s.direction ↔ ∃ p1 ∈ s, ∃ p2 ∈ s, v = p1 -ᵥ p2 :=
begin
rw [←submodule.mem_coe, coe_direction_eq_vsub_set h],
exact iff.rfl
end
/-- Adding a vector in the direction to a point in the subspace
produces a point in the subspace. -/
lemma vadd_mem_of_mem_direction {s : affine_subspace k V P} {v : V} (hv : v ∈ s.direction) {p : P}
(hp : p ∈ s) : v +ᵥ p ∈ s :=
begin
rw mem_direction_iff_eq_vsub ⟨p, hp⟩ at hv,
rcases hv with ⟨p1, hp1, p2, hp2, hv⟩,
rw hv,
convert s.smul_vsub_vadd_mem 1 p1 p2 p hp1 hp2 hp,
rw one_smul
end
/-- Subtracting two points in the subspace produces a vector in the
direction. -/
lemma vsub_mem_direction {s : affine_subspace k V P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
(p1 -ᵥ p2) ∈ s.direction :=
vsub_mem_vector_span k V hp1 hp2
/-- Given a point in an affine subspace, the set of vectors in its
direction equals the set of vectors subtracting that point on the
right. -/
lemma coe_direction_eq_vsub_set_right {s : affine_subspace k V P} {p : P} (hp : p ∈ s) :
(s.direction : set V) = {v | ∃ p2 ∈ s, v = p2 -ᵥ p} :=
begin
rw coe_direction_eq_vsub_set ⟨p, hp⟩,
refine le_antisymm _ _,
{ rintros v ⟨p1, hp1, p2, hp2, hv⟩,
exact ⟨v +ᵥ p,
vadd_mem_of_mem_direction (hv.symm ▸ vsub_mem_direction hp1 hp2) hp,
(vadd_vsub _ _ _).symm⟩ },
{ rintros v ⟨p2, hp2, hv⟩,
exact ⟨p2, hp2, p, hp, hv⟩ }
end
/-- Given a point in an affine subspace, the set of vectors in its
direction equals the set of vectors subtracting that point on the
left. -/
lemma coe_direction_eq_vsub_set_left {s : affine_subspace k V P} {p : P} (hp : p ∈ s) :
(s.direction : set V) = {v | ∃ p2 ∈ s, v = p -ᵥ p2} :=
begin
ext v,
rw [submodule.mem_coe, ←submodule.neg_mem_iff, ←submodule.mem_coe,
coe_direction_eq_vsub_set_right hp, set.mem_set_of_eq, set.mem_set_of_eq],
conv_lhs { congr, funext, rw [←neg_vsub_eq_vsub_rev, neg_inj] }
end
/-- Given a point in an affine subspace, a vector is in its direction
if and only if it results from subtracting that point on the right. -/
lemma mem_direction_iff_eq_vsub_right {s : affine_subspace k V P} {p : P} (hp : p ∈ s) (v : V) :
v ∈ s.direction ↔ ∃ p2 ∈ s, v = p2 -ᵥ p :=
begin
rw [←submodule.mem_coe, coe_direction_eq_vsub_set_right hp],
exact iff.rfl
end
/-- Given a point in an affine subspace, a vector is in its direction
if and only if it results from subtracting that point on the left. -/
lemma mem_direction_iff_eq_vsub_left {s : affine_subspace k V P} {p : P} (hp : p ∈ s) (v : V) :
v ∈ s.direction ↔ ∃ p2 ∈ s, v = p -ᵥ p2 :=
begin
rw [←submodule.mem_coe, coe_direction_eq_vsub_set_left hp],
exact iff.rfl
end
/-- Given a point in an affine subspace, a result of subtracting that
point on the right is in the direction if and only if the other point
is in the subspace. -/
lemma vsub_right_mem_direction_iff_mem {s : affine_subspace k V P} {p : P} (hp : p ∈ s) (p2 : P) :
p2 -ᵥ p ∈ s.direction ↔ p2 ∈ s :=
begin
rw mem_direction_iff_eq_vsub_right hp,
simp
end
/-- Given a point in an affine subspace, a result of subtracting that
point on the left is in the direction if and only if the other point
is in the subspace. -/
lemma vsub_left_mem_direction_iff_mem {s : affine_subspace k V P} {p : P} (hp : p ∈ s) (p2 : P) :
p -ᵥ p2 ∈ s.direction ↔ p2 ∈ s :=
begin
rw mem_direction_iff_eq_vsub_left hp,
simp
end
/-- Two affine subspaces are equal if they have the same points. -/
@[ext] lemma ext {s1 s2 : affine_subspace k V P} (h : (s1 : set P) = s2) : s1 = s2 :=
begin
cases s1,
cases s2,
congr,
exact h
end
/-- Two affine subspaces with the same direction and nonempty
intersection are equal. -/
lemma ext_of_direction_eq {s1 s2 : affine_subspace k V P} (hd : s1.direction = s2.direction)
(hn : ((s1 : set P) ∩ s2).nonempty) : s1 = s2 :=
begin
ext p,
have hq1 := set.mem_of_mem_inter_left hn.some_mem,
have hq2 := set.mem_of_mem_inter_right hn.some_mem,
split,
{ intro hp,
rw ←vsub_vadd V p hn.some,
refine vadd_mem_of_mem_direction _ hq2,
rw ←hd,
exact vsub_mem_direction hp hq1 },
{ intro hp,
rw ←vsub_vadd V p hn.some,
refine vadd_mem_of_mem_direction _ hq1,
rw hd,
exact vsub_mem_direction hp hq2 }
end
/-- Construct an affine subspace from a point and a direction. -/
def mk' (p : P) (direction : submodule k V) : affine_subspace k V P :=
{ carrier := {q | ∃ v ∈ direction, q = v +ᵥ p},
smul_vsub_vadd_mem := λ c p1 p2 p3 hp1 hp2 hp3, begin
rcases hp1 with ⟨v1, hv1, hp1⟩,
rcases hp2 with ⟨v2, hv2, hp2⟩,
rcases hp3 with ⟨v3, hv3, hp3⟩,
use [c • (v1 - v2) + v3,
direction.add_mem (direction.smul_mem c (direction.sub_mem hv1 hv2)) hv3],
simp [hp1, hp2, hp3, vadd_assoc]
end }
/-- An affine space constructed from a point and a direction contains
that point. -/
lemma self_mem_mk' (p : P) (direction : submodule k V) :
p ∈ mk' p direction :=
⟨0, ⟨direction.zero_mem, (add_action.zero_vadd _ _).symm⟩⟩
/-- An affine space constructed from a point and a direction contains
the result of adding a vector in that direction to that point. -/
lemma vadd_mem_mk' {v : V} (p : P) {direction : submodule k V} (hv : v ∈ direction) :
v +ᵥ p ∈ mk' p direction :=
⟨v, hv, rfl⟩
/-- An affine space constructed from a point and a direction is
nonempty. -/
lemma mk'_nonempty (p : P) (direction : submodule k V) : (mk' p direction : set P).nonempty :=
⟨p, self_mem_mk' p direction⟩
/-- The direction of an affine space constructed from a point and a
direction. -/
@[simp] lemma direction_mk' (p : P) (direction : submodule k V) :
(mk' p direction).direction = direction :=
begin
ext v,
rw mem_direction_iff_eq_vsub (mk'_nonempty _ _),
split,
{ rintros ⟨p1, ⟨v1, hv1, hp1⟩, p2, ⟨v2, hv2, hp2⟩, hv⟩,
rw [hv, hp1, hp2, vadd_vsub_vadd_cancel_right],
exact direction.sub_mem hv1 hv2 },
{ exact λ hv, ⟨v +ᵥ p, vadd_mem_mk' _ hv, p,
self_mem_mk' _ _, (vadd_vsub _ _ _).symm⟩ }
end
/-- Constructing an affine subspace from a point in a subspace and
that subspace's direction yields the original subspace. -/
@[simp] lemma mk'_eq {s : affine_subspace k V P} {p : P} (hp : p ∈ s) : mk' p s.direction = s :=
ext_of_direction_eq (direction_mk' p s.direction)
⟨p, set.mem_inter (self_mem_mk' _ _) hp⟩
/-- If an affine subspace contains a set of points, it contains the
`span_points` of that set. -/
lemma span_points_subset_coe_of_subset_coe {s : set P} {s1 : affine_subspace k V P} (h : s ⊆ s1) :
span_points k V s ⊆ s1 :=
begin
rintros p ⟨p1, hp1, v, hv, hp⟩,
rw hp,
have hp1s1 : p1 ∈ (s1 : set P) := set.mem_of_mem_of_subset hp1 h,
refine vadd_mem_of_mem_direction _ hp1s1,
have hs : vector_span k V s ≤ s1.direction := submodule.span_mono (vsub_set_mono V h),
rw submodule.le_def at hs,
rw ←submodule.mem_coe,
exact set.mem_of_mem_of_subset hv hs
end
end affine_subspace
section affine_span
variables (k : Type*) (V : Type*) {P : Type*} [ring k] [add_comm_group V] [module k V]
[affine_space k V P]
/-- The affine span of a set of points is the smallest affine subspace
containing those points. (Actually defined here in terms of spans in
modules.) -/
def affine_span (s : set P) : affine_subspace k V P :=
{ carrier := span_points k V s,
smul_vsub_vadd_mem := λ c p1 p2 p3 hp1 hp2 hp3,
vadd_mem_span_points_of_mem_span_points_of_mem_vector_span k V hp3
((vector_span k V s).smul_mem c
(vsub_mem_vector_span_of_mem_span_points_of_mem_span_points k V hp1 hp2)) }
/-- The affine span, converted to a set, is `span_points`. -/
@[simp] lemma coe_affine_span (s : set P) :
(affine_span k V s : set P) = span_points k V s :=
rfl
/-- The direction of the affine span is the `vector_span`. -/
lemma direction_affine_span (s : set P) : (affine_span k V s).direction = vector_span k V s :=
begin
apply le_antisymm,
{ refine submodule.span_le.2 _,
rintros v ⟨p1, ⟨p2, hp2, v1, hv1, hp1⟩, p3, ⟨p4, hp4, v2, hv2, hp3⟩, hv⟩,
rw [hv, hp1, hp3, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, submodule.mem_coe],
exact (vector_span k V s).sub_mem ((vector_span k V s).add_mem hv1
(vsub_mem_vector_span k V hp2 hp4)) hv2 },
{ exact submodule.span_mono (vsub_set_mono V (subset_span_points k V s)) }
end
/-- A point in a set is in its affine span. -/
lemma mem_affine_span {p : P} {s : set P} (hp : p ∈ s) : p ∈ affine_span k V s :=
mem_span_points k V p s hp
end affine_span
namespace affine_subspace
variables {k : Type*} {V : Type*} {P : Type*} [ring k] [add_comm_group V] [module k V]
[S : affine_space k V P]
include S
instance : complete_lattice (affine_subspace k V P) :=
{ sup := λ s1 s2, affine_span k V (s1 ∪ s2),
le_sup_left := λ s1 s2, set.subset.trans (set.subset_union_left s1 s2)
(subset_span_points k V _),
le_sup_right := λ s1 s2, set.subset.trans (set.subset_union_right s1 s2)
(subset_span_points k V _),
sup_le := λ s1 s2 s3 hs1 hs2, span_points_subset_coe_of_subset_coe (set.union_subset hs1 hs2),
inf := λ s1 s2, mk (s1 ∩ s2)
(λ c p1 p2 p3 hp1 hp2 hp3,
⟨s1.smul_vsub_vadd_mem c p1 p2 p3 hp1.1 hp2.1 hp3.1,
s2.smul_vsub_vadd_mem c p1 p2 p3 hp1.2 hp2.2 hp3.2⟩),
inf_le_left := λ _ _, set.inter_subset_left _ _,
inf_le_right := λ _ _, set.inter_subset_right _ _,
le_inf := λ _ _ _, set.subset_inter,
top := { carrier := set.univ,
smul_vsub_vadd_mem := λ _ _ _ _ _ _ _, set.mem_univ _ },
le_top := λ _ _ _, set.mem_univ _,
bot := { carrier := ∅,
smul_vsub_vadd_mem := λ _ _ _ _, false.elim },
bot_le := λ _ _, false.elim,
Sup := λ s, affine_span k V (⋃ s' ∈ s, (s' : set P)),
Inf := λ s, mk (⋂ s' ∈ s, (s' : set P))
(λ c p1 p2 p3 hp1 hp2 hp3, set.mem_bInter_iff.2 $ λ s2 hs2,
s2.smul_vsub_vadd_mem c p1 p2 p3 (set.mem_bInter_iff.1 hp1 s2 hs2)
(set.mem_bInter_iff.1 hp2 s2 hs2)
(set.mem_bInter_iff.1 hp3 s2 hs2)),
le_Sup := λ _ _ h, set.subset.trans (set.subset_bUnion_of_mem h) (subset_span_points k V _),
Sup_le := λ _ _ h, span_points_subset_coe_of_subset_coe (set.bUnion_subset h),
Inf_le := λ _ _, set.bInter_subset_of_mem,
le_Inf := λ _ _, set.subset_bInter,
.. partial_order.lift (coe : affine_subspace k V P → set P) (λ _ _, ext) }
instance : inhabited (affine_subspace k V P) := ⟨⊤⟩
/-- The `≤` order on subspaces is the same as that on the corresponding
sets. -/
lemma le_def (s1 s2 : affine_subspace k V P) : s1 ≤ s2 ↔ (s1 : set P) ⊆ s2 :=
iff.rfl
/-- One subspace is less than or equal to another if and only if all
its points are in the second subspace. -/
lemma le_def' (s1 s2 : affine_subspace k V P) : s1 ≤ s2 ↔ ∀ p ∈ s1, p ∈ s2 :=
iff.rfl
/-- The `<` order on subspaces is the same as that on the corresponding
sets. -/
lemma lt_def (s1 s2 : affine_subspace k V P) : s1 < s2 ↔ (s1 : set P) ⊂ s2 :=
iff.rfl
/-- One subspace is not less than or equal to another if and only if
it has a point not in the second subspace. -/
lemma not_le_iff_exists (s1 s2 : affine_subspace k V P) : ¬ s1 ≤ s2 ↔ ∃ p ∈ s1, p ∉ s2 :=
set.not_subset
/-- If a subspace is less than another, there is a point only in the
second. -/
lemma exists_of_lt {s1 s2 : affine_subspace k V P} (h : s1 < s2) : ∃ p ∈ s2, p ∉ s1 :=
set.exists_of_ssubset h
/-- A subspace is less than another if and only if it is less than or
equal to the second subspace and there is a point only in the
second. -/
lemma lt_iff_le_and_exists (s1 s2 : affine_subspace k V P) : s1 < s2 ↔ s1 ≤ s2 ∧ ∃ p ∈ s2, p ∉ s1 :=
by rw [lt_iff_le_not_le, not_le_iff_exists]
variables (k V)
/-- The affine span is the `Inf` of subspaces containing the given
points. -/
lemma affine_span_eq_Inf (s : set P) : affine_span k V s = Inf {s' | s ⊆ s'} :=
le_antisymm (span_points_subset_coe_of_subset_coe (set.subset_bInter (λ _ h, h)))
(Inf_le (subset_span_points k V _))
variables (P)
/-- The Galois insertion formed by `affine_span` and coercion back to
a set. -/
protected def gi : galois_insertion (affine_span k V) (coe : affine_subspace k V P → set P) :=
{ choice := λ s _, affine_span k V s,
gc := λ s1 s2, ⟨λ h, set.subset.trans (subset_span_points k V s1) h,
span_points_subset_coe_of_subset_coe⟩,
le_l_u := λ _, subset_span_points k V _,
choice_eq := λ _ _, rfl }
/-- The span of the empty set is `⊥`. -/
@[simp] lemma span_empty : affine_span k V (∅ : set P) = ⊥ :=
(affine_subspace.gi k V P).gc.l_bot
/-- The span of `univ` is `⊤`. -/
@[simp] lemma span_univ : affine_span k V (set.univ : set P) = ⊤ :=
eq_top_iff.2 $ subset_span_points k V _
variables {P}
/-- The span of a union of sets is the sup of their spans. -/
lemma span_union (s t : set P) : affine_span k V (s ∪ t) = affine_span k V s ⊔ affine_span k V t :=
(affine_subspace.gi k V P).gc.l_sup
/-- The span of a union of an indexed family of sets is the sup of
their spans. -/
lemma span_Union {ι : Type*} (s : ι → set P) :
affine_span k V (⋃ i, s i) = ⨆ i, affine_span k V (s i) :=
(affine_subspace.gi k V P).gc.l_supr
variables (P)
/-- `⊤`, coerced to a set, is the whole set of points. -/
@[simp] lemma top_coe : ((⊤ : affine_subspace k V P) : set P) = set.univ :=
rfl
variables {P}
/-- All points are in `⊤`. -/
lemma mem_top (p : P) : p ∈ (⊤ : affine_subspace k V P) :=
set.mem_univ p
variables (P)
/-- The direction of `⊤` is the whole module as a submodule. -/
@[simp] lemma direction_top : (⊤ : affine_subspace k V P).direction = ⊤ :=
begin
cases S.nonempty with p,
ext v,
refine ⟨imp_intro submodule.mem_top, λ hv, _⟩,
have hpv : (v +ᵥ p -ᵥ p : V) ∈ (⊤ : affine_subspace k V P).direction :=
vsub_mem_direction (mem_top k V _) (mem_top k V _),
rwa vadd_vsub at hpv
end
/-- `⊥`, coerced to a set, is the empty set. -/
@[simp] lemma bot_coe : ((⊥ : affine_subspace k V P) : set P) = ∅ :=
rfl
variables {P}
/-- No points are in `⊥`. -/
lemma not_mem_bot (p : P) : p ∉ (⊥ : affine_subspace k V P) :=
set.not_mem_empty p
variables (P)
/-- The direction of `⊥` is the submodule `⊥`. -/
@[simp] lemma direction_bot : (⊥ : affine_subspace k V P).direction = ⊥ :=
by rw [direction_eq_vector_span, bot_coe, vector_span_def, vsub_set_empty, submodule.span_empty]
variables {k V P}
/-- The inf of two affine subspaces, coerced to a set, is the
intersection of the two sets of points. -/
@[simp] lemma inf_coe (s1 s2 : affine_subspace k V P) : ((s1 ⊓ s2) : set P) = s1 ∩ s2 :=
rfl
/-- A point is in the inf of two affine subspaces if and only if it is
in both of them. -/
lemma mem_inf_iff (p : P) (s1 s2 : affine_subspace k V P) : p ∈ s1 ⊓ s2 ↔ p ∈ s1 ∧ p ∈ s2 :=
iff.rfl
/-- The direction of the inf of two affine subspaces is less than or
equal to the inf of their directions. -/
lemma direction_inf (s1 s2 : affine_subspace k V P) :
(s1 ⊓ s2).direction ≤ s1.direction ⊓ s2.direction :=
begin
repeat { rw [direction_eq_vector_span, vector_span_def] },
exact le_inf
(Inf_le_Inf (λ p hp, set.subset.trans (vsub_set_mono V (set.inter_subset_left _ _)) hp))
(Inf_le_Inf (λ p hp, set.subset.trans (vsub_set_mono V (set.inter_subset_right _ _)) hp))
end
/-- If one affine subspace is less than or equal to another, the same
applies to their directions. -/
lemma direction_le {s1 s2 : affine_subspace k V P} (h : s1 ≤ s2) : s1.direction ≤ s2.direction :=
begin
repeat { rw [direction_eq_vector_span, vector_span_def] },
exact submodule.span_mono (vsub_set_mono _ h)
end
/-- If one nonempty affine subspace is less than another, the same
applies to their directions -/
lemma direction_lt_of_nonempty {s1 s2 : affine_subspace k V P} (h : s1 < s2)
(hn : (s1 : set P).nonempty) : s1.direction < s2.direction :=
begin
cases hn with p hp,
rw lt_iff_le_and_exists at h,
rcases h with ⟨hle, p2, hp2, hp2s1⟩,
rw submodule.lt_iff_le_and_exists,
use [direction_le hle, p2 -ᵥ p, vsub_mem_direction hp2 (hle hp)],
intro hm,
rw vsub_right_mem_direction_iff_mem hp p2 at hm,
exact hp2s1 hm
end
/-- The sup of the directions of two affine subspaces is less than or
equal to the direction of their sup. -/
lemma sup_direction_le (s1 s2 : affine_subspace k V P) :
s1.direction ⊔ s2.direction ≤ (s1 ⊔ s2).direction :=
begin
repeat { rw [direction_eq_vector_span, vector_span_def] },
exact sup_le
(Inf_le_Inf (λ p hp, set.subset.trans (vsub_set_mono V (le_sup_left : s1 ≤ s1 ⊔ s2)) hp))
(Inf_le_Inf (λ p hp, set.subset.trans (vsub_set_mono V (le_sup_right : s2 ≤ s1 ⊔ s2)) hp))
end
/-- The sup of the directions of two nonempty affine subspaces with
empty intersection is less than the direction of their sup. -/
lemma sup_direction_lt_of_nonempty_of_inter_empty {s1 s2 : affine_subspace k V P}
(h1 : (s1 : set P).nonempty) (h2 : (s2 : set P).nonempty) (he : (s1 ∩ s2 : set P) = ∅) :
s1.direction ⊔ s2.direction < (s1 ⊔ s2).direction :=
begin
cases h1 with p1 hp1,
cases h2 with p2 hp2,
rw submodule.lt_iff_le_and_exists,
use [sup_direction_le s1 s2, p2 -ᵥ p1,
vsub_mem_direction ((le_sup_right : s2 ≤ s1 ⊔ s2) hp2) ((le_sup_left : s1 ≤ s1 ⊔ s2) hp1)],
intro h,
rw submodule.mem_sup at h,
rcases h with ⟨v1, hv1, v2, hv2, hv1v2⟩,
rw [←sub_eq_zero, sub_eq_add_neg, neg_vsub_eq_vsub_rev, add_comm v1, add_assoc,
←vadd_vsub_assoc, ←neg_neg v2, add_comm, ←sub_eq_add_neg, ←vsub_vadd_eq_vsub_sub,
vsub_eq_zero_iff_eq] at hv1v2,
refine set.nonempty.ne_empty _ he,
use [v1 +ᵥ p1, vadd_mem_of_mem_direction hv1 hp1],
rw hv1v2,
exact vadd_mem_of_mem_direction (submodule.neg_mem _ hv2) hp2
end
/-- If the directions of two nonempty affine subspaces span the whole
module, they have nonempty intersection. -/
lemma inter_nonempty_of_nonempty_of_sup_direction_eq_top {s1 s2 : affine_subspace k V P}
(h1 : (s1 : set P).nonempty) (h2 : (s2 : set P).nonempty)
(hd : s1.direction ⊔ s2.direction = ⊤) : ((s1 : set P) ∩ s2).nonempty :=
begin
by_contradiction h,
rw set.not_nonempty_iff_eq_empty at h,
have hlt := sup_direction_lt_of_nonempty_of_inter_empty h1 h2 h,
rw hd at hlt,
exact not_top_lt hlt
end
/-- If the directions of two nonempty affine subspaces are complements
of each other, they intersect in exactly one point. -/
lemma inter_eq_singleton_of_nonempty_of_is_compl {s1 s2 : affine_subspace k V P}
(h1 : (s1 : set P).nonempty) (h2 : (s2 : set P).nonempty)
(hd : is_compl s1.direction s2.direction) : ∃ p, (s1 : set P) ∩ s2 = {p} :=
begin
cases inter_nonempty_of_nonempty_of_sup_direction_eq_top h1 h2 hd.sup_eq_top with p hp,
use p,
ext q,
rw set.mem_singleton_iff,
split,
{ rintros ⟨hq1, hq2⟩,
have hqp : q -ᵥ p ∈ s1.direction ⊓ s2.direction :=
⟨vsub_mem_direction hq1 hp.1, vsub_mem_direction hq2 hp.2⟩,
rwa [hd.inf_eq_bot, submodule.mem_bot, vsub_eq_zero_iff_eq] at hqp },
{ exact λ h, h.symm ▸ hp }
end
end affine_subspace
namespace affine_space
variables (k : Type*) (V : Type*) {P : Type*} [ring k] [add_comm_group V] [module k V]
[affine_space k V P]
variables {ι : Type*}
/-- The `vector_span` is the span of the pairwise subtractions with a
given point on the left. -/
lemma vector_span_eq_span_vsub_set_left {s : set P} {p : P} (hp : p ∈ s) :
vector_span k V s = submodule.span k {v | ∃ p2 ∈ s, v = p -ᵥ p2} :=
begin
rw vector_span_def,
refine le_antisymm _ (submodule.span_mono _),
{ rw submodule.span_le,
rintros v ⟨p1, hp1, p2, hp2, hv⟩,
rw ←vsub_sub_vsub_cancel_left V p1 p2 p at hv,
rw [hv, submodule.mem_coe, submodule.mem_span],
exact λ m hm, submodule.sub_mem _ (hm ⟨p2, hp2, rfl⟩) (hm ⟨p1, hp1, rfl⟩) },
{ rintros v ⟨p2, hp2, hv⟩,
exact ⟨p, hp, p2, hp2, hv⟩ }
end
/-- The `vector_span` is the span of the pairwise subtractions with a
given point on the right. -/
lemma vector_span_eq_span_vsub_set_right {s : set P} {p : P} (hp : p ∈ s) :
vector_span k V s = submodule.span k {v | ∃ p2 ∈ s, v = p2 -ᵥ p} :=
begin
rw vector_span_def,
refine le_antisymm _ (submodule.span_mono _),
{ rw submodule.span_le,
rintros v ⟨p1, hp1, p2, hp2, hv⟩,
rw ←vsub_sub_vsub_cancel_right V p1 p2 p at hv,
rw [hv, submodule.mem_coe, submodule.mem_span],
exact λ m hm, submodule.sub_mem _ (hm ⟨p1, hp1, rfl⟩) (hm ⟨p2, hp2, rfl⟩) },
{ rintros v ⟨p2, hp2, hv⟩,
exact ⟨p2, hp2, p, hp, hv⟩ }
end
/-- The `vector_span` of an indexed family is the span of the pairwise
subtractions with a given point on the left. -/
lemma vector_span_range_eq_span_range_vsub_left (p : ι → P) (i0 : ι) :
vector_span k V (set.range p) = submodule.span k (set.range (λ (i : ι), p i0 -ᵥ p i)) :=
begin
simp_rw [vector_span_eq_span_vsub_set_left k V (set.mem_range_self i0), set.exists_range_iff],
conv_lhs { congr, congr, funext, conv { congr, funext, rw eq_comm } },
refl
end
/-- The `vector_span` of an indexed family is the span of the pairwise
subtractions with a given point on the right. -/
lemma vector_span_range_eq_span_range_vsub_right (p : ι → P) (i0 : ι) :
vector_span k V (set.range p) = submodule.span k (set.range (λ (i : ι), p i -ᵥ p i0)) :=
begin
simp_rw [vector_span_eq_span_vsub_set_right k V (set.mem_range_self i0), set.exists_range_iff],
conv_lhs { congr, congr, funext, conv { congr, funext, rw eq_comm } },
refl
end
/-- The affine span of a set is nonempty if and only if that set
is. -/
lemma affine_span_nonempty (s : set P) :
(affine_span k V s : set P).nonempty ↔ s.nonempty :=
span_points_nonempty k V s
variables {k}
/-- A `weighted_vsub` with sum of weights 0 is in the `vector_span` of
an indexed family. -/
lemma weighted_vsub_mem_vector_span {s : finset ι} {w : ι → k}
(h : ∑ i in s, w i = 0) (p : ι → P) :
s.weighted_vsub V p w ∈ vector_span k V (set.range p) :=
begin
by_cases hn : nonempty ι,
{ cases hn with i0,
rw [vector_span_range_eq_span_range_vsub_right k V p i0, ←set.image_univ,
finsupp.mem_span_iff_total,
finset.weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero V s w p h (p i0),
finset.weighted_vsub_of_point_apply],
let w' := set.indicator ↑s w,
have hwx : ∀ i, w' i ≠ 0 → i ∈ s := λ i, set.mem_of_indicator_ne_zero,
use [finsupp.on_finset s w' hwx, set.subset_univ _],
rw [finsupp.total_apply, finsupp.on_finset_sum hwx],
{ apply finset.sum_congr rfl,
intros i hi,
simp [w', set.indicator_apply, if_pos hi] },
{ exact λ _, zero_smul k _ } },
{ simp [finset.eq_empty_of_not_nonempty hn s] }
end
/-- An `affine_combination` with sum of weights 1 is in the
`affine_span` of an indexed family, if the underlying ring is
nontrivial. -/
lemma affine_combination_mem_affine_span [nontrivial k] {s : finset ι} {w : ι → k}
(h : ∑ i in s, w i = 1) (p : ι → P) :
s.affine_combination V w p ∈ affine_span k V (set.range p) :=
begin
have hnz : ∑ i in s, w i ≠ 0 := h.symm ▸ one_ne_zero,
have hn : s.nonempty := finset.nonempty_of_sum_ne_zero hnz,
cases hn with i1 hi1,
let w1 : ι → k := function.update (function.const ι 0) i1 1,
have hw1 : ∑ i in s, w1 i = 1,
{ rw [finset.sum_update_of_mem hi1, finset.sum_const_zero, add_zero] },
have hw1s : s.affine_combination V w1 p = p i1 :=
s.affine_combination_of_eq_one_of_eq_zero V w1 p hi1 (function.update_same _ _ _)
(λ _ _ hne, function.update_noteq hne _ _),
have hv : s.affine_combination V w p -ᵥ p i1 ∈ (affine_span k V (set.range p)).direction,
{ rw [direction_affine_span, ←hw1s, finset.affine_combination_vsub],
apply weighted_vsub_mem_vector_span,
simp [pi.sub_apply, h, hw1] },
rw ←vsub_vadd V (s.affine_combination V w p) (p i1),
exact affine_subspace.vadd_mem_of_mem_direction hv (mem_affine_span k V (set.mem_range_self _))
end
variables (k) {V}
/-- A vector is in the `vector_span` of an indexed family if and only
if it is a `weighted_vsub` with sum of weights 0. -/
lemma mem_vector_span_iff_eq_weighted_vsub {v : V} {p : ι → P} :
v ∈ vector_span k V (set.range p) ↔
∃ (s : finset ι) (w : ι → k) (h : ∑ i in s, w i = 0), v = s.weighted_vsub V p w :=
begin
split,
{ by_cases hn : nonempty ι,
{ cases hn with i0,
rw [vector_span_range_eq_span_range_vsub_right k V p i0, ←set.image_univ,
finsupp.mem_span_iff_total],
rintros ⟨l, hl, hv⟩,
use insert i0 l.support,
set w := (l : ι → k) -
function.update (function.const ι 0 : ι → k) i0 (∑ i in l.support, l i) with hwdef,
use w,
have hw : ∑ i in insert i0 l.support, w i = 0,
{ rw hwdef,
simp_rw [pi.sub_apply, finset.sum_sub_distrib,
finset.sum_update_of_mem (finset.mem_insert_self _ _), finset.sum_const_zero,
finset.sum_insert_of_eq_zero_if_not_mem finsupp.not_mem_support_iff.1,
add_zero, sub_self] },
use hw,
have hz : w i0 • (p i0 -ᵥ p i0 : V) = 0 := (vsub_self V (p i0)).symm ▸ smul_zero _,
change (λ i, w i • (p i -ᵥ p i0 : V)) i0 = 0 at hz,
rw [finset.weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero V _ w p hw (p i0),
finset.weighted_vsub_of_point_apply, ←hv, finsupp.total_apply,
finset.sum_insert_zero hz],
change ∑ i in l.support, l i • _ = _,
congr,
ext i,
by_cases h : i = i0,
{ simp [h] },
{ simp [hwdef, h] } },
{ rw [set.range_eq_empty.2 hn, vector_span_empty, submodule.mem_bot],
intro hv,
use [∅],
simp [hv] } },
{ rintros ⟨s, w, hw, rfl⟩,
exact weighted_vsub_mem_vector_span V hw p }
end
variables {k}
/-- A point in the `affine_span` of an indexed family is an
`affine_combination` with sum of weights 1. -/
lemma eq_affine_combination_of_mem_affine_span {p1 : P} {p : ι → P}
(h : p1 ∈ affine_span k V (set.range p)) :
∃ (s : finset ι) (w : ι → k) (hw : ∑ i in s, w i = 1), p1 = s.affine_combination V w p :=
begin
have hn : ((affine_span k V (set.range p)) : set P).nonempty := ⟨p1, h⟩,
rw [affine_span_nonempty, set.range_nonempty_iff_nonempty] at hn,
cases hn with i0,
have h0 : p i0 ∈ affine_span k V (set.range p) := mem_affine_span k V (set.mem_range_self i0),
have hd : p1 -ᵥ p i0 ∈ (affine_span k V (set.range p)).direction :=
affine_subspace.vsub_mem_direction h h0,
rw [direction_affine_span, mem_vector_span_iff_eq_weighted_vsub] at hd,
rcases hd with ⟨s, w, h, hs⟩,
let s' := insert i0 s,
let w' := set.indicator ↑s w,
have h' : ∑ i in s', w' i = 0,
{ rw [←h, set.sum_indicator_subset _ (finset.subset_insert i0 s)] },
have hs' : s'.weighted_vsub V p w' = p1 -ᵥ p i0,
{ rw hs,
exact (finset.weighted_vsub_indicator_subset _ _ _ (finset.subset_insert i0 s)).symm },
let w0 : ι → k := function.update (function.const ι 0) i0 1,
have hw0 : ∑ i in s', w0 i = 1,
{ rw [finset.sum_update_of_mem (finset.mem_insert_self _ _), finset.sum_const_zero, add_zero] },
have hw0s : s'.affine_combination V w0 p = p i0 :=
s'.affine_combination_of_eq_one_of_eq_zero V w0 p
(finset.mem_insert_self _ _)
(function.update_same _ _ _)
(λ _ _ hne, function.update_noteq hne _ _),
use [s', w0 + w'],
split,
{ simp [pi.add_apply, finset.sum_add_distrib, hw0, h'] },
{ rw [add_comm, ←finset.weighted_vsub_vadd_affine_combination, hw0s, hs', vsub_vadd] }
end
variables (k V)
/-- A point is in the `affine_span` of an indexed family if and only
if it is an `affine_combination` with sum of weights 1, provided the
underlying ring is nontrivial. -/
lemma mem_affine_span_iff_eq_affine_combination [nontrivial k] {p1 : P} {p : ι → P} :
p1 ∈ affine_span k V (set.range p) ↔
∃ (s : finset ι) (w : ι → k) (hw : ∑ i in s, w i = 1), p1 = s.affine_combination V w p :=
begin
split,
{ exact eq_affine_combination_of_mem_affine_span },
{ rintros ⟨s, w, hw, rfl⟩,
exact affine_combination_mem_affine_span V hw p }
end
end affine_space
/-- An `affine_map k V1 P1 V2 P2` is a map from `P1` to `P2` that
induces a corresponding linear map from `V1` to `V2`. -/
structure affine_map (k : Type*) (V1 : Type*) (P1 : Type*) (V2 : Type*) (P2 : Type*)
[ring k]
[add_comm_group V1] [module k V1] [affine_space k V1 P1]
[add_comm_group V2] [module k V2] [affine_space k V2 P2] :=
(to_fun : P1 → P2)
(linear : linear_map k V1 V2)
(map_vadd' : ∀ (p : P1) (v : V1), to_fun (v +ᵥ p) = linear v +ᵥ to_fun p)
namespace affine_map
variables {k : Type*} {V1 : Type*} {P1 : Type*} {V2 : Type*} {P2 : Type*}
{V3 : Type*} {P3 : Type*} {V4 : Type*} {P4 : Type*} [ring k]
[add_comm_group V1] [module k V1] [affine_space k V1 P1]
[add_comm_group V2] [module k V2] [affine_space k V2 P2]
[add_comm_group V3] [module k V3] [affine_space k V3 P3]
[add_comm_group V4] [module k V4] [affine_space k V4 P4]
instance: has_coe_to_fun (affine_map k V1 P1 V2 P2) := ⟨_, to_fun⟩
/-- Constructing an affine map and coercing back to a function
produces the same map. -/
@[simp] lemma coe_mk (f : P1 → P2) (linear add) :
((mk f linear add : affine_map k V1 P1 V2 P2) : P1 → P2) = f := rfl
/-- `to_fun` is the same as the result of coercing to a function. -/
@[simp] lemma to_fun_eq_coe (f : affine_map k V1 P1 V2 P2) : f.to_fun = ⇑f := rfl
/-- An affine map on the result of adding a vector to a point produces
the same result as the linear map applied to that vector, added to the
affine map applied to that point. -/
@[simp] lemma map_vadd (f : affine_map k V1 P1 V2 P2) (p : P1) (v : V1) :
f (v +ᵥ p) = f.linear v +ᵥ f p := f.map_vadd' p v
/-- The linear map on the result of subtracting two points is the
result of subtracting the result of the affine map on those two
points. -/
@[simp] lemma linear_map_vsub (f : affine_map k V1 P1 V2 P2) (p1 p2 : P1) :
f.linear (p1 -ᵥ p2) = f p1 -ᵥ f p2 :=
by conv_rhs { rw [←vsub_vadd V1 p1 p2, map_vadd, vadd_vsub] }
/-- Two affine maps are equal if they coerce to the same function. -/
@[ext] lemma ext {f g : affine_map k V1 P1 V2 P2} (h : ∀ p, f p = g p) : f = g :=
begin
rcases f with ⟨f, f_linear, f_add⟩,
rcases g with ⟨g, g_linear, g_add⟩,
have : f = g := funext h,
subst g,
congr',
ext v,
cases (add_torsor.nonempty V1 : nonempty P1) with p,
apply vadd_right_cancel (f p),
erw [← f_add, ← g_add]
end
lemma ext_iff {f g : affine_map k V1 P1 V2 P2} : f = g ↔ ∀ p, f p = g p := ⟨λ h p, h ▸ rfl, ext⟩
variables (k V1 P1 V2)
/-- Constant function as an `affine_map`. -/
def const (p : P2) : affine_map k V1 P1 V2 P2 :=
{ to_fun := function.const P1 p,
linear := 0,
map_vadd' := λ p v, by simp }
@[simp] lemma coe_const (p : P2) : ⇑(const k V1 P1 V2 p) = function.const P1 p := rfl
@[simp] lemma const_linear (p : P2) : (const k V1 P1 V2 p).linear = 0 := rfl
variables {k V1 P1 V2}
instance nonempty : nonempty (affine_map k V1 P1 V2 P2) :=
⟨const k V1 P1 V2 (classical.choice $ add_torsor.nonempty V2)⟩
/-- Construct an affine map by verifying the relation between the map and its linear part at one
base point. Namely, this function takes a map `f : P₁ → P₂`, a linear map `f' : V₁ →ₗ[k] V₂`, and
a point `p` such that for any other point `p'` we have `f p' = f' (p' -ᵥ p) +ᵥ f p`. -/
def mk' (f : P1 → P2) (f' : V1 →ₗ[k] V2) (p : P1) (h : ∀ p' : P1, f p' = f' (p' -ᵥ p) +ᵥ f p) :
affine_map k V1 P1 V2 P2 :=
{ to_fun := f,
linear := f',
map_vadd' := λ p' v, by rw [h, h p', vadd_vsub_assoc, f'.map_add, add_action.vadd_assoc] }
@[simp] lemma coe_mk' (f : P1 → P2) (f' : V1 →ₗ[k] V2) (p h) : ⇑(mk' f f' p h) = f := rfl
@[simp] lemma mk'_linear (f : P1 → P2) (f' : V1 →ₗ[k] V2) (p h) : (mk' f f' p h).linear = f' := rfl
/-- The set of affine maps to a vector space is an additive commutative group. -/
instance : add_comm_group (affine_map k V1 P1 V2 V2) :=
{ zero := ⟨0, 0, λ p v, (add_action.zero_vadd _ _).symm⟩,
add := λ f g, ⟨f + g, f.linear + g.linear, λ p v, by simp [add_add_add_comm]⟩,
neg := λ f, ⟨-f, -f.linear, λ p v, by simp [add_comm]⟩,
add_assoc := λ f₁ f₂ f₃, ext $ λ p, add_assoc _ _ _,
zero_add := λ f, ext $ λ p, zero_add (f p),
add_zero := λ f, ext $ λ p, add_zero (f p),
add_comm := λ f g, ext $ λ p, add_comm (f p) (g p),
add_left_neg := λ f, ext $ λ p, add_left_neg (f p) }
@[simp, norm_cast] lemma coe_zero : ⇑(0 : affine_map k V1 P1 V2 V2) = 0 := rfl
@[simp] lemma zero_linear : (0 : affine_map k V1 P1 V2 V2).linear = 0 := rfl
@[simp, norm_cast] lemma coe_add (f g : affine_map k V1 P1 V2 V2) : ⇑(f + g) = f + g := rfl
@[simp]
lemma add_linear (f g : affine_map k V1 P1 V2 V2) : (f + g).linear = f.linear + g.linear := rfl
/-- The space of affine maps from `P1` to `P2` is an affine space over the space of affine spaces
from `P1` to the vector `V2` corresponding to `P2`. -/
instance : affine_space k (affine_map k V1 P1 V2 V2) (affine_map k V1 P1 V2 P2) :=
{ vadd := λ f g, ⟨λ p, f p +ᵥ g p, f.linear + g.linear, λ p v,
by simp [add_action.vadd_assoc, add_right_comm]⟩,
zero_vadd' := λ f, ext $ λ p, add_action.zero_vadd _ (f p),
vadd_assoc' := λ f₁ f₂ f₃, ext $ λ p, add_action.vadd_assoc V2 (f₁ p) (f₂ p) (f₃ p),
vsub := λ f g, ⟨λ p, f p -ᵥ g p, f.linear - g.linear, λ p v,
by simp [vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_sub, sub_add_eq_add_sub]⟩,
vsub_vadd' := λ f g, ext $ λ p, vsub_vadd V2 (f p) (g p),
vadd_vsub' := λ f g, ext $ λ p, vadd_vsub V2 (f p) (g p) }
@[simp] lemma vadd_apply (f : affine_map k V1 P1 V2 V2) (g : affine_map k V1 P1 V2 P2) (p : P1) :
(f +ᵥ g) p = f p +ᵥ g p :=
rfl
@[simp] lemma vsub_apply (f g : affine_map k V1 P1 V2 P2) (p : P1) :
(f -ᵥ g : affine_map k V1 P1 V2 V2) p = f p -ᵥ g p :=
rfl
variables (k V1 P1)
/-- Identity map as an affine map. -/
def id : affine_map k V1 P1 V1 P1 :=
{ to_fun := id,
linear := linear_map.id,
map_vadd' := λ p v, rfl }
/-- The identity affine map acts as the identity. -/
@[simp] lemma coe_id : ⇑(id k V1 P1) = _root_.id := rfl
@[simp] lemma id_linear : (id k V1 P1).linear = linear_map.id := rfl
variable {P1}
/-- The identity affine map acts as the identity. -/
lemma id_apply (p : P1) : id k V1 P1 p = p := rfl
variables {k V1 P1 V2}
instance : inhabited (affine_map k V1 P1 V1 P1) := ⟨id k V1 P1⟩
/-- Composition of affine maps. -/
def comp (f : affine_map k V2 P2 V3 P3) (g : affine_map k V1 P1 V2 P2) :
affine_map k V1 P1 V3 P3 :=
{ to_fun := f ∘ g,
linear := f.linear.comp g.linear,
map_vadd' := begin
intros p v,
rw [function.comp_app, g.map_vadd, f.map_vadd],
refl
end }
/-- Composition of affine maps acts as applying the two functions. -/
@[simp] lemma coe_comp (f : affine_map k V2 P2 V3 P3) (g : affine_map k V1 P1 V2 P2) :
⇑(f.comp g) = f ∘ g := rfl
/-- Composition of affine maps acts as applying the two functions. -/
lemma comp_apply (f : affine_map k V2 P2 V3 P3) (g : affine_map k V1 P1 V2 P2) (p : P1) :
f.comp g p = f (g p) := rfl
@[simp] lemma comp_id (f : affine_map k V1 P1 V2 P2) : f.comp (id k V1 P1) = f := ext $ λ p, rfl
@[simp] lemma id_comp (f : affine_map k V1 P1 V2 P2) : (id k V2 P2).comp f = f := ext $ λ p, rfl
lemma comp_assoc (f₃₄ : affine_map k V3 P3 V4 P4) (f₂₃ : affine_map k V2 P2 V3 P3)
(f₁₂ : affine_map k V1 P1 V2 P2) :
(f₃₄.comp f₂₃).comp f₁₂ = f₃₄.comp (f₂₃.comp f₁₂) :=
rfl
instance : monoid (affine_map k V1 P1 V1 P1) :=
{ one := id k V1 P1,
mul := comp,
one_mul := id_comp,
mul_one := comp_id,
mul_assoc := comp_assoc }
@[simp] lemma coe_mul (f g : affine_map k V1 P1 V1 P1) : ⇑(f * g) = f ∘ g := rfl
@[simp] lemma coe_one : ⇑(1 : affine_map k V1 P1 V1 P1) = _root_.id := rfl
/-- The affine map from `k` to `P1` sending `0` to `p` and `1` to `v +ᵥ p`. -/
def line_map (p : P1) (v : V1) : affine_map k k k V1 P1 :=
{ to_fun := λ c, c • v +ᵥ p,
linear := linear_map.id.smul_right v,
map_vadd' := λ a b, by simp [add_smul, add_action.vadd_assoc] }
lemma line_map_apply (p : P1) (v : V1) (c : k) : line_map p v c = c • v +ᵥ p := rfl
@[simp] lemma line_map_linear (p : P1) (v : V1) :
(line_map p v : affine_map k k k V1 P1).linear = linear_map.id.smul_right v :=
rfl
@[simp] lemma line_map_zero (p : P1) : line_map p (0:V1) = const k k k V1 p :=
by { ext c, simp [line_map_apply] }
@[simp] lemma line_map_apply_zero (p : P1) (v : V1) : line_map p v (0:k) = p :=
by simp [line_map_apply]
@[simp] lemma affine_apply_line_map (f : affine_map k V1 P1 V2 P2) (p : P1) (v : V1) (c : k) :
f (line_map p v c) = line_map (f p) (f.linear v) c :=
by simp [line_map_apply]
@[simp] lemma affine_comp_line_map (f : affine_map k V1 P1 V2 P2) (p : P1) (v : V1) :
f.comp (line_map p v) = line_map (f p) (f.linear v) :=
ext $ f.affine_apply_line_map p v
lemma line_map_vadd_neg (p : P1) (v : V1) :
line_map (v +ᵥ p) (-v) = (line_map p v).comp (line_map (1:k) (-1:k)) :=
by { rw [affine_comp_line_map], simp [line_map_apply] }
end affine_map
namespace affine_map
variables {k : Type*} {V1 : Type*} {P1 : Type*} {V2 : Type*} [comm_ring k]
[add_comm_group V1] [module k V1] [affine_space k V1 P1] [add_comm_group V2] [module k V2]
/-- If `k` is a commutative ring, then the set of affine maps with codomain in a `k`-module
is a `k`-module. -/
instance : module k (affine_map k V1 P1 V2 V2) :=
{ smul := λ c f, ⟨c • f, c • f.linear, λ p v, by simp [smul_add]⟩,
one_smul := λ f, ext $ λ p, one_smul _ _,
mul_smul := λ c₁ c₂ f, ext $ λ p, mul_smul _ _ _,
smul_add := λ c f g, ext $ λ p, smul_add _ _ _,
smul_zero := λ c, ext $ λ p, smul_zero _,
add_smul := λ c₁ c₂ f, ext $ λ p, add_smul _ _ _,
zero_smul := λ f, ext $ λ p, zero_smul _ _ }
@[simp] lemma coe_smul (c : k) (f : affine_map k V1 P1 V2 V2) : ⇑(c • f) = c • f := rfl
variable (V1)
/-- `homothety V c r` is the homothety about `c` with scale factor `r`. -/
def homothety (c : P1) (r : k) : affine_map k V1 P1 V1 P1 :=
r • (id k V1 P1 -ᵥ const k V1 P1 V1 c : affine_map k V1 P1 V1 V1) +ᵥ const k V1 P1 V1 c
lemma homothety_def (c : P1) (r : k) :
homothety V1 c r = r • (id k V1 P1 -ᵥ const k V1 P1 V1 c : affine_map k V1 P1 V1 V1) +ᵥ
const k V1 P1 V1 c :=
rfl
lemma homothety_apply (c : P1) (r : k) (p : P1) :
homothety V1 c r p = r • (p -ᵥ c : V1) +ᵥ c := rfl
@[simp] lemma homothety_one (c : P1) : homothety V1 c (1:k) = id k V1 P1 :=
by { ext p, simp [homothety_apply] }
lemma homothety_mul (c : P1) (r₁ r₂ : k) :
homothety V1 c (r₁ * r₂) = (homothety V1 c r₁).comp (homothety V1 c r₂) :=
by { ext p, simp [homothety_apply, mul_smul] }
@[simp] lemma homothety_zero (c : P1) : homothety V1 c (0:k) = const k V1 P1 V1 c :=
by { ext p, simp [homothety_apply] }
@[simp] lemma homothety_add (c : P1) (r₁ r₂ : k) :
homothety V1 c (r₁ + r₂) =
r₁ • (id k V1 P1 -ᵥ const k V1 P1 V1 c : affine_map k V1 P1 V1 V1) +ᵥ homothety V1 c r₂ :=
by simp only [homothety_def, add_smul, add_action.vadd_assoc]
/-- `homothety` as a multiplicative monoid homomorphism. -/
def homothety_hom (c : P1) : k →* affine_map k V1 P1 V1 P1 :=
⟨homothety V1 c, homothety_one V1 c, homothety_mul V1 c⟩
@[simp] lemma coe_homothety_hom (c : P1) : ⇑(homothety_hom V1 c : k →* _) = homothety V1 c := rfl
/-- `homothety` as an affine map. -/
def homothety_affine (c : P1) :
affine_map k k k (affine_map k V1 P1 V1 V1) (affine_map k V1 P1 V1 P1) :=
⟨homothety V1 c, (linear_map.lsmul k _).flip (id k V1 P1 -ᵥ const k V1 P1 V1 c),
function.swap (homothety_add V1 c)⟩
@[simp] lemma coe_homothety_affine (c : P1) :
⇑(homothety_affine V1 c : affine_map k k k _ _) = homothety V1 c :=
rfl
end affine_map
namespace affine_map
variables {k : Type*} (V : Type*) (P : Type*) [comm_ring k] [add_comm_group V] [module k V]
variables [affine_space k V P] {ι : Type*} (s : finset ι)
-- TODO: define `affine_map.proj`, `affine_map.fst`, `affine_map.snd`
/-- A weighted sum, as an affine map on the points involved. -/
def weighted_vsub_of_point (w : ι → k) : affine_map k ((ι → V) × V) ((ι → P) × P) V V :=
{ to_fun := λ p, s.weighted_vsub_of_point _ p.fst p.snd w,
linear := ∑ i in s,
w i • ((linear_map.proj i).comp (linear_map.fst _ _ _) - linear_map.snd _ _ _),
map_vadd' := begin
rintros ⟨p, b⟩ ⟨v, b'⟩,
simp [linear_map.sum_apply, finset.weighted_vsub_of_point, vsub_vadd_eq_vsub_sub,
vadd_vsub_assoc, add_sub, ← sub_add_eq_add_sub, smul_add, finset.sum_add_distrib]
end }
end affine_map
namespace linear_map
variables {k : Type*} {V₁ : Type*} {V₂ : Type*} [ring k] [add_comm_group V₁] [module k V₁]
[add_comm_group V₂] [module k V₂] (f : V₁ →ₗ[k] V₂)
/-- Reinterpret a linear map as an affine map. -/
def to_affine_map : affine_map k V₁ V₁ V₂ V₂ :=
{ to_fun := f,
linear := f,
map_vadd' := λ p v, f.map_add v p }
@[simp] lemma coe_to_affine_map : ⇑f.to_affine_map = f := rfl
@[simp] lemma to_affine_map_linear : f.to_affine_map.linear = f := rfl
end linear_map
|
151680e55315bb0f2447a39d9975fbaf9560266c | 761d983a78bc025071bac14a3facced881cf5e53 | /new_affine/real_affine_space.lean | 77b07603aab2f8fc0f2b70591d82c07f04f1c095 | [] | no_license | rohanrajnair/affine_lib | bcf22ff892cf74ccb36a95bc9b7fff8e0adb2d0d | 83076864245ac547b9d615bc6a23804b1b4a8f70 | refs/heads/master | 1,673,320,928,343 | 1,603,036,653,000 | 1,603,036,653,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,459 | lean | import linear_algebra.affine_space.basic
import ..new_affine.affine_coordinate_space
import data.real.basic
namespace real_affine
abbreviation real_vec := aff_vec ℝ
abbreviation real_pt := aff_pt ℝ
abbreviation r3_vec := real_vec 3
abbreviation r3_pt := real_pt 3
universes u v w
structure aff_struct :=
(X : Type u)
(K : Type v)
(V : Type w)
(fld : ring K)
(grp : add_comm_group V)
(vec : module K V)
(aff : affine_space V X)
structure vec_struct :=
(K : Type u)
(V : Type v)
(fld : field K)
(grp : add_comm_group V)
(vec : vector_space K V)
inductive Algebra
| aff_space (a : aff_struct)
| vec_space (v : vec_struct)
| nat_monoid -- placeholder, commutative monoid with monus operator
def real_aff_pt_n := λn : ℕ, aff_pt ℝ n
def real_scalar := ℝ
def real_aff_vec_n := λn : ℕ, aff_vec ℝ n
def prf_ring := real.ring
def prf_real_add_comm_grp_n := λn : ℕ, aff_comm_group ℝ n
def prf_vec_module_n := λn : ℕ, aff_module ℝ n
def prf_aff_crd_sp := λn : ℕ, aff_coord_is ℝ n
--⟨aff_pt ℝ n, ℝ, aff_vec ℝ n, real.ring, aff_comm_group ℝ n, aff_module ℝ n, aff_coord_is ℝ n⟩
def to_affine : ℕ → aff_struct := λ n,
⟨real_aff_pt_n n, ℝ , (real_aff_vec_n n), real.ring, prf_real_add_comm_grp_n n,prf_vec_module_n n,prf_aff_crd_sp n⟩
noncomputable def to_vector : ℕ → vec_struct := λ n,
⟨ℝ, aff_vec ℝ n, real.field, aff_comm_group ℝ n, aff_module ℝ n⟩
end real_affine |
1295b8cc000f3dab2d808cc982ffce16260c8884 | aa101d73b1a3173c7ec56de02b96baa8ca64c42e | /src/my_exercises/07_first_negations.lean | edc2bae977561a16cadada2b70eeb85bfdbf2a1d | [
"Apache-2.0"
] | permissive | gihanmarasingha/tutorials | b554d4d53866c493c4341dc13e914b01444e95a6 | 56617114ef0f9f7b808476faffd11e22e4380918 | refs/heads/master | 1,671,141,758,153 | 1,599,173,318,000 | 1,599,173,318,000 | 282,405,870 | 0 | 0 | Apache-2.0 | 1,595,666,751,000 | 1,595,666,750,000 | null | UTF-8 | Lean | false | false | 9,929 | lean | import tuto_lib
import data.int.parity
/-
Negations, proof by contradiction and contraposition.
This file introduces the logical rules and tactics related to negation:
exfalso, by_contradiction, contrapose, by_cases and push_neg.
There is a special statement denoted by `false` which, by definition,
has no proof.
So `false` implies everything. Indeed `false → P` means any proof of
`false` could be turned into a proof of P.
This fact is known by its latin name
"ex falso quod libet" (from false follows whatever you want).
Hence Lean's tactic to invoke this is called `exfalso`.
-/
example : false → 0 = 1 :=
begin
intro h,
exfalso,
exact h,
end
/-
The preceding example suggests that this definition of `false` isn't very useful.
But actually it allows us to define the negation of a statement P as
"P implies false" that we can read as "if P were true, we would get
a contradiction". Lean denotes this by `¬ P`.
One can prove that (¬ P) ↔ (P ↔ false). But in practice we directly
use the definition of `¬ P`.
-/
example {x : ℝ} : ¬ x < x :=
begin
intro hyp,
rw lt_iff_le_and_ne at hyp,
cases hyp with hyp_inf hyp_non,
clear hyp_inf, -- we won't use that one, so let's discard it
change x = x → false at hyp_non, -- Lean doesn't need this psychological line
apply hyp_non,
refl,
end
open int
-- 0045
example (n : ℤ) (h_pair : even n) (h_non_pair : ¬ even n) : 0 = 1 :=
begin
exfalso,
exact h_non_pair h_pair,
end
-- 0046
/- I give two solutions. The second is longer and misses the obvious idea of trying proof by negation to derive ¬P . -/
example (P Q : Prop) (h₁ : P ∨ Q) (h₂ : ¬ (P ∧ Q)) : ¬ P ↔ Q :=
begin
split, {
intro hnp,
cases h₁ with hp hq,
exfalso, exact hnp hp,
exact hq,
}, {
intros hq hp,
exact h₂ ⟨hp,hq⟩,
}
end
example (P Q : Prop) (h₁ : P ∨ Q) (h₂ : ¬ (P ∧ Q)) : ¬ P ↔ Q :=
begin
split, {
intro hnp,
cases h₁ with hp hq, {
exfalso,
exact hnp hp,
}, {
exact hq,
}
}, {
intro hq,
cases h₁ with hp hq, {
exfalso,
exact h₂ ⟨hp, hq⟩,
}, {
intro hp,
exact h₂ ⟨hp, hq⟩,
}
}
end
/-
The definition of negation easily implies that, for every statement P,
P → ¬ ¬ P
The excluded middle axiom, which asserts P ∨ ¬ P allows us to
prove the converse implication.
Together those two implications form the principle of double negation elimination.
not_not {P : Prop} : (¬ ¬ P) ↔ P
The implication `¬ ¬ P → P` is the basis for proofs by contradiction:
in order to prove P, it suffices to prove ¬¬ P, ie `¬ P → false`.
Of course there is no need to keep explaining all this. The tactic
`by_contradiction Hyp` will transform any goal P into `false` and
add Hyp : ¬ P to the local context.
Let's return to a proof from the 5th file: uniqueness of limits for a sequence.
This cannot be proved without using some version of the excluded middle
axiom. We used it secretely in
eq_of_abs_sub_le_all (x y : ℝ) : (∀ ε > 0, |x - y| ≤ ε) → x = y
(we'll prove a variation on this lemma below).
-/
example (u : ℕ → ℝ) (l l' : ℝ) : seq_limit u l → seq_limit u l' → l = l' :=
begin
intros hl hl',
by_contradiction H,
change l ≠ l' at H, -- Lean does not need this line
have ineg : |l-l'| > 0,
exact abs_pos_of_ne_zero (sub_ne_zero_of_ne H),
cases hl ( |l-l'|/4 ) (by linarith) with N hN,
cases hl' ( |l-l'|/4 ) (by linarith) with N' hN',
let N₀ := max N N', -- this is a new tactic, whose effect should be clear
specialize hN N₀ (le_max_left _ _),
specialize hN' N₀ (le_max_right _ _),
have clef : |l-l'| < |l-l'|,
calc
|l - l'| = |(l-u N₀) + (u N₀ -l')| : by ring
... ≤ |l - u N₀| + |u N₀ - l'| : by apply abs_add
... = |u N₀ - l| + |u N₀ - l'| : by rw abs_sub
... < |l-l'| : by linarith,
linarith, -- linarith can also find simple numerical contradictions
end
/-
Another incarnation of the excluded middle axiom is the principle of
contraposition: in order to prove P ⇒ Q, it suffices to prove
non Q ⇒ non P.
-/
-- Using a proof by contradiction, let's prove the contraposition principle
-- 0047
example (P Q : Prop) (h : ¬ Q → ¬ P) : P → Q :=
begin
intro hnqp,
by_contradiction hq,
exact h hq hnqp,
end
/-
Again Lean doesn't need to be explain this principle. We can use the
`contrapose` tactic.
-/
example (P Q : Prop) (h : ¬ Q → ¬ P) : P → Q :=
begin
contrapose,
exact h,
end
/-
In the next exercise, we'll use
odd n : ∃ k, n = 2*k + 1
not_even_iff_odd : ¬ even n ↔ odd n,
-/
-- 0048
/- Again two solutions. The second improves on the first by using rcases with rfl. -/
example (n : ℤ) : even (n^2) ↔ even n :=
begin
split, {
contrapose,
simp [not_even_iff_odd],
intro nodd,
cases nodd with k hk,
rw hk,
use (2*k^2 + 2*k),
ring,
}, {
intro neven,
cases neven with k hk,
rw hk,
use (2*k^2),
ring,
}
end
example (n : ℤ) : even (n^2) ↔ even n :=
begin
split, {
contrapose,
simp [not_even_iff_odd],
rintro ⟨k, rfl⟩,
use (2*k^2 + 2*k),
ring,
}, {
rintro ⟨k, rfl⟩,
use (2*k^2),
ring,
}
end
/-
As a last step on our law of the excluded middle tour, let's notice that, especially
in pure logic exercises, it can sometimes be useful to use the
excluded middle axiom in its original form:
classical.em : ∀ P, P ∨ ¬ P
Instead of applying this lemma and then using the `cases` tactic, we
have the shortcut
by_cases h : P,
combining both steps to create two proof branches: one assuming
h : P, and the other assuming h : ¬ P
For instance, let's prove a reformulation of this implication relation,
which is sometimes used as a definition in other logical foundations,
especially those based on truth tables (hence very strongly using
excluded middle from the very beginning).
-/
variables (P Q : Prop)
example : (P → Q) ↔ (¬ P ∨ Q) :=
begin
split,
{ intro h,
by_cases hP : P,
{ right,
exact h hP },
{ left,
exact hP } },
{ intros h hP,
cases h with hnP hQ,
{ exfalso,
exact hnP hP },
{ exact hQ } },
end
-- 0049
example : ¬ (P ∧ Q) ↔ ¬ P ∨ ¬ Q :=
begin
split, {
intro hnpq,
by_cases hp : P, {
right,
intro hq,
exact hnpq ⟨hp, hq⟩,
}, {
exact or.inl hp,
}
}, {
rintros hnpq ⟨hp, hq⟩,
cases hnpq with hnp hnq, {
exact hnp hp,
}, {
exact hnq hq,
}
}
end
/-
It is crucial to understand negation of quantifiers.
Let's do it by hand for a little while.
In the first exercise, only the definition of negation is needed.
-/
-- 0050
/- Two proofs. The second is simpler: it uses rintros, an anoymous constructor for the existential quantifier, and uses 'application' rather than specialize. -/
example (n : ℤ) : ¬ (∃ k, n = 2*k) ↔ ∀ k, n ≠ 2*k :=
begin
split, {
intros hnex k hnk,
apply hnex,
use k,
exact hnk,
}, {
intros hallk hexk,
cases hexk with k hk,
specialize hallk k,
exact hallk hk,
}
end
example (n : ℤ) : ¬ (∃ k, n = 2*k) ↔ ∀ k, n ≠ 2*k :=
begin
split, {
intros hnex k hnk,
-- apply hnex,
--exact ⟨k, hnk⟩,
exact hnex ⟨k, hnk⟩,
}, {
rintros hallk ⟨k,hnk⟩,
exact hallk k hnk,
}
end
/-
Contrary to negation of the existential quantifier, negation of the
universal quantifier requires excluded middle for the first implication.
In order to prove this, we can use either
* a double proof by contradiction
* a contraposition, not_not : (¬ ¬ P) ↔ P) and a proof by contradiction.
-/
def even_fun (f : ℝ → ℝ) := ∀ x, f (-x) = f x
-- 0051
/- Two solutions. The second is shorter. -/
example (f : ℝ → ℝ) : ¬ even_fun f ↔ ∃ x, f (-x) ≠ f x :=
begin
split, {
intro hnef,
by_contradiction H,
unfold even_fun at hnef,
apply hnef,
intro x,
by_contradiction K,
apply H,
use x,
}, {
intros hex hnef,
cases hex with x hef,
apply hef,
exact hnef x,
}
end
example (f : ℝ → ℝ) : ¬ even_fun f ↔ ∃ x, f (-x) ≠ f x :=
begin
split, {
intro hnef,
by_contradiction H,
apply hnef,
intro x,
by_contradiction K,
apply H,
use x,
}, {
rintros ⟨x,hnef⟩ hef,
exact hnef (hef x),
}
end
/-
Of course we can't keep repeating the above proofs, especially the second one.
So we use the `push_neg` tactic.
-/
example : ¬ even_fun (λ x, 2*x) :=
begin
unfold even_fun, -- Here unfolding is important because push_neg won't do it.
push_neg,
use 42,
linarith,
end
-- 0052
example (f : ℝ → ℝ) : ¬ even_fun f ↔ ∃ x, f (-x) ≠ f x :=
begin
unfold even_fun,
push_neg,
end
def bounded_above (f : ℝ → ℝ) := ∃ M, ∀ x, f x ≤ M
example : ¬ bounded_above (λ x, x) :=
begin
unfold bounded_above,
push_neg,
intro M,
use M + 1,
linarith,
end
-- Let's contrapose
-- 0053
/- The following is shorter than that in the model solution. -/
example (x : ℝ) : (∀ ε > 0, x ≤ ε) → x ≤ 0 :=
begin
contrapose,
intros hxneg hε,
specialize hε (x/2) (by linarith),
linarith,
end
/-
The "contrapose, push_neg" combo is so common that we can abreviate it to
`contrapose!`
Let's use this trick, together with:
eq_or_lt_of_le : a ≤ b → a = b ∨ a < b
-/
-- 0054
example (f : ℝ → ℝ) : (∀ x y, x < y → f x < f y) ↔ (∀ x y, (x ≤ y ↔ f x ≤ f y)) :=
begin
split, {
intros h x y,
split , {
intro hxy,
cases (eq_or_lt_of_le hxy) with h₁ h₂, {
rw h₁,
}, {
specialize h x y h₂,
linarith,
}
}, {
contrapose!,
exact h y x,
}
}, {
intros h x y,
contrapose!,
exact (h y x).mpr,
}
end
|
b62949f9f8010990d3beafd43690e18481ebef41 | 0845ae2ca02071debcfd4ac24be871236c01784f | /tests/compiler/t4.lean | 10785ed045bc0e07db0bd6ba20dd2fa2e19939ba | [
"Apache-2.0"
] | permissive | GaloisInc/lean4 | 74c267eb0e900bfaa23df8de86039483ecbd60b7 | 228ddd5fdcd98dd4e9c009f425284e86917938aa | refs/heads/master | 1,643,131,356,301 | 1,562,715,572,000 | 1,562,715,572,000 | 192,390,898 | 0 | 0 | null | 1,560,792,750,000 | 1,560,792,749,000 | null | UTF-8 | Lean | false | false | 3,362 | lean | /- Benchmark for new code generator -/
inductive Expr
| Val : Int → Expr
| Var : String → Expr
| Add : Expr → Expr → Expr
| Mul : Expr → Expr → Expr
| Pow : Expr → Expr → Expr
| Ln : Expr → Expr
open Expr
def Expr.toString : Expr → String
| (Val n) := toString n
| (Var x) := x
| (Add f g) := "(" ++ Expr.toString f ++ " + " ++ Expr.toString g ++ ")"
| (Mul f g) := "(" ++ Expr.toString f ++ " * " ++ Expr.toString g ++ ")"
| (Pow f g) := "(" ++ Expr.toString f ++ " ^ " ++ Expr.toString g ++ ")"
| (Ln f) := "ln(" ++ Expr.toString f ++ ")"
instance : HasToString Expr :=
⟨Expr.toString⟩
partial def pown : Int → Int → Int
| a 0 := 1
| a 1 := a
| a n :=
let b := pown a (n / 2) in
b * b * (if n % 2 = 0 then 1 else a)
partial def addAux : Expr → Expr → Expr
| (Val n) (Val m) := Val (n + m)
| (Val 0) f := f
| f (Val 0) := f
| f (Val n) := addAux (Val n) f
| (Val n) (Add (Val m) f) := addAux (Val (n+m)) f
| f (Add (Val n) g) := addAux (Val n) (addAux f g)
| (Add f g) h := addAux f (addAux g h)
| f g := Add f g
def add (a b : Expr) : Expr :=
-- dbgTrace (">> add (" ++ toString a ++ ", " ++ toString b ++ ")") $ fun _ =>
addAux a b
-- set_option trace.compiler.borrowed_inference true
partial def mulAux : Expr → Expr → Expr
| (Val n) (Val m) := Val (n*m)
| (Val 0) _ := Val 0
| _ (Val 0) := Val 0
| (Val 1) f := f
| f (Val 1) := f
| f (Val n) := mulAux (Val n) f
| (Val n) (Mul (Val m) f) := mulAux (Val (n*m)) f
| f (Mul (Val n) g) := mulAux (Val n) (mulAux f g)
| (Mul f g) h := mulAux f (mulAux g h)
| f g := Mul f g
def mul (a b : Expr) : Expr :=
-- dbgTrace (">> mul (" ++ toString a ++ ", " ++ toString b ++ ")") $ fun _ =>
mulAux a b
def pow : Expr → Expr → Expr
| (Val m) (Val n) := Val (pown m n)
| _ (Val 0) := Val 1
| f (Val 1) := f
| (Val 0) _ := Val 0
| f g := Pow f g
def ln : Expr → Expr
| (Val 1) := Val 0
| f := Ln f
def d (x : String) : Expr → Expr
| (Val _) := Val 0
| (Var y) := if x = y then Val 1 else Val 0
| (Add f g) := add (d f) (d g)
| (Mul f g) :=
-- dbgTrace (">> d (" ++ toString f ++ ", " ++ toString g ++ ")") $ fun _ =>
add (mul f (d g)) (mul g (d f))
| (Pow f g) := mul (pow f g) (add (mul (mul g (d f)) (pow f (Val (-1)))) (mul (ln f) (d g)))
| (Ln f) := mul (d f) (pow f (Val (-1)))
def count : Expr → Nat
| (Val _) := 1
| (Var _) := 1
| (Add f g) := count f + count g
| (Mul f g) := count f + count g
| (Pow f g) := count f + count g
| (Ln f) := count f
def nestAux (s : Nat) (f : Nat → Expr → IO Expr) : Nat → Expr → IO Expr
| 0 x := pure x
| m@(n+1) x := f (s - m) x >>= nestAux n
def nest (f : Nat → Expr → IO Expr) (n : Nat) (e : Expr) : IO Expr :=
nestAux n f n e
def deriv (i : Nat) (f : Expr) : IO Expr :=
do
let d := d "x" f;
IO.println (toString (i+1) ++ " count: " ++ (toString $ count d));
IO.println (toString d);
pure d
def main (xs : List String) : IO UInt32 :=
do let x := Var "x";
let f := add x (mul x (mul x (add x x)));
IO.println f;
nest deriv 3 f;
pure 0
-- setOption profiler True
-- #eval main []
|
e0e7f50ed379be3672aa0ad8df6659c1c90c8fd8 | 80cc5bf14c8ea85ff340d1d747a127dcadeb966f | /src/group_theory/submonoid/membership.lean | b08568d2c533f08eaf7f4f0a876f2429e15bccef | [
"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 | 9,534 | lean | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import group_theory.submonoid.operations
import algebra.big_operators.basic
import algebra.free_monoid
/-!
# Submonoids: membership criteria
In this file we prove various facts about membership in a submonoid:
* `list_prod_mem`, `multiset_prod_mem`, `prod_mem`: if each element of a collection belongs
to a multiplicative submonoid, then so does their product;
* `list_sum_mem`, `multiset_sum_mem`, `sum_mem`: if each element of a collection belongs
to an additive submonoid, then so does their sum;
* `pow_mem`, `nsmul_mem`: if `x ∈ S` where `S` is a multiplicative (resp., additive) submonoid and
`n` is a natural number, then `x^n` (resp., `n •ℕ x`) belongs to `S`;
* `mem_supr_of_directed`, `coe_supr_of_directed`, `mem_Sup_of_directed_on`,
`coe_Sup_of_directed_on`: the supremum of a directed collection of submonoid is their union.
* `sup_eq_range`, `mem_sup`: supremum of two submonoids `S`, `T` of a commutative monoid is the set
of products;
* `closure_singleton_eq`, `mem_closure_singleton`: the multiplicative (resp., additive) closure
of `{x}` consists of powers (resp., natural multiples) of `x`.
## Tags
submonoid, submonoids
-/
open_locale big_operators
variables {M : Type*} [monoid M] {s : set M}
variables {A : Type*} [add_monoid A] {t : set A}
namespace submonoid
variables (S : submonoid M)
/-- Product of a list of elements in a submonoid is in the submonoid. -/
@[to_additive "Sum of a list of elements in an `add_submonoid` is in the `add_submonoid`."]
lemma list_prod_mem : ∀ {l : list M}, (∀x ∈ l, x ∈ S) → l.prod ∈ S
| [] h := S.one_mem
| (a::l) h :=
suffices a * l.prod ∈ S, by rwa [list.prod_cons],
have a ∈ S ∧ (∀ x ∈ l, x ∈ S), from list.forall_mem_cons.1 h,
S.mul_mem this.1 (list_prod_mem this.2)
/-- Product of a multiset of elements in a submonoid of a `comm_monoid` is in the submonoid. -/
@[to_additive "Sum of a multiset of elements in an `add_submonoid` of an `add_comm_monoid` is
in the `add_submonoid`."]
lemma multiset_prod_mem {M} [comm_monoid M] (S : submonoid M) (m : multiset M) :
(∀a ∈ m, a ∈ S) → m.prod ∈ S :=
begin
refine quotient.induction_on m (assume l hl, _),
rw [multiset.quot_mk_to_coe, multiset.coe_prod],
exact S.list_prod_mem hl
end
/-- Product of elements of a submonoid of a `comm_monoid` indexed by a `finset` is in the
submonoid. -/
@[to_additive "Sum of elements in an `add_submonoid` of an `add_comm_monoid` indexed by a `finset`
is in the `add_submonoid`."]
lemma prod_mem {M : Type*} [comm_monoid M] (S : submonoid M)
{ι : Type*} {t : finset ι} {f : ι → M} (h : ∀c ∈ t, f c ∈ S) :
∏ c in t, f c ∈ S :=
S.multiset_prod_mem (t.1.map f) $ λ x hx, let ⟨i, hi, hix⟩ := multiset.mem_map.1 hx in hix ▸ h i hi
lemma pow_mem {x : M} (hx : x ∈ S) : ∀ n:ℕ, x^n ∈ S
| 0 := S.one_mem
| (n+1) := S.mul_mem hx (pow_mem n)
open set
@[to_additive]
lemma mem_supr_of_directed {ι} [hι : nonempty ι] {S : ι → submonoid M} (hS : directed (≤) S)
{x : M} :
x ∈ (⨆ i, S i) ↔ ∃ i, x ∈ S i :=
begin
refine ⟨_, λ ⟨i, hi⟩, (le_def.1 $ le_supr S i) hi⟩,
suffices : x ∈ closure (⋃ i, (S i : set M)) → ∃ i, x ∈ S i,
by simpa only [closure_Union, closure_eq (S _)] using this,
refine (λ hx, closure_induction hx (λ _, mem_Union.1) _ _),
{ exact hι.elim (λ i, ⟨i, (S i).one_mem⟩) },
{ rintros x y ⟨i, hi⟩ ⟨j, hj⟩,
rcases hS i j with ⟨k, hki, hkj⟩,
exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩ }
end
@[to_additive]
lemma coe_supr_of_directed {ι} [nonempty ι] {S : ι → submonoid M} (hS : directed (≤) S) :
((⨆ i, S i : submonoid M) : set M) = ⋃ i, ↑(S i) :=
set.ext $ λ x, by simp [mem_supr_of_directed hS]
@[to_additive]
lemma mem_Sup_of_directed_on {S : set (submonoid M)} (Sne : S.nonempty)
(hS : directed_on (≤) S) {x : M} :
x ∈ Sup S ↔ ∃ s ∈ S, x ∈ s :=
begin
haveI : nonempty S := Sne.to_subtype,
simp only [Sup_eq_supr', mem_supr_of_directed hS.directed_coe, set_coe.exists, subtype.coe_mk]
end
@[to_additive]
lemma coe_Sup_of_directed_on {S : set (submonoid M)} (Sne : S.nonempty) (hS : directed_on (≤) S) :
(↑(Sup S) : set M) = ⋃ s ∈ S, ↑s :=
set.ext $ λ x, by simp [mem_Sup_of_directed_on Sne hS]
@[to_additive]
lemma mem_sup_left {S T : submonoid M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T :=
show S ≤ S ⊔ T, from le_sup_left
@[to_additive]
lemma mem_sup_right {S T : submonoid M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T :=
show T ≤ S ⊔ T, from le_sup_right
@[to_additive]
lemma mem_supr_of_mem {ι : Type*} {S : ι → submonoid M} (i : ι) :
∀ {x : M}, x ∈ S i → x ∈ supr S :=
show S i ≤ supr S, from le_supr _ _
@[to_additive]
lemma mem_Sup_of_mem {S : set (submonoid M)} {s : submonoid M}
(hs : s ∈ S) : ∀ {x : M}, x ∈ s → x ∈ Sup S :=
show s ≤ Sup S, from le_Sup hs
end submonoid
namespace free_monoid
variables {α : Type*}
open submonoid
@[to_additive]
theorem closure_range_of : closure (set.range $ @of α) = ⊤ :=
eq_top_iff.2 $ λ x hx, free_monoid.rec_on x (one_mem _) $ λ x xs hxs,
mul_mem _ (subset_closure $ set.mem_range_self _) hxs
end free_monoid
namespace submonoid
variables {N : Type*} [monoid N]
open monoid_hom
lemma closure_singleton_eq (x : M) : closure ({x} : set M) = (powers_hom M x).mrange :=
closure_eq_of_le (set.singleton_subset_iff.2 ⟨multiplicative.of_add 1, trivial, pow_one x⟩) $
λ x ⟨n, _, hn⟩, hn ▸ pow_mem _ (subset_closure $ set.mem_singleton _) _
/-- The submonoid generated by an element of a monoid equals the set of natural number powers of
the element. -/
lemma mem_closure_singleton {x y : M} : y ∈ closure ({x} : set M) ↔ ∃ n:ℕ, x^n=y :=
by rw [closure_singleton_eq, mem_mrange]; refl
lemma mem_closure_singleton_self {y : M} : y ∈ closure ({y} : set M) :=
mem_closure_singleton.2 ⟨1, pow_one y⟩
@[to_additive]
lemma closure_eq_mrange (s : set M) : closure s = (free_monoid.lift (coe : s → M)).mrange :=
by rw [mrange, ← free_monoid.closure_range_of, map_mclosure, ← set.range_comp,
free_monoid.lift_comp_of, subtype.range_coe]
@[to_additive]
lemma exists_list_of_mem_closure {s : set M} {x : M} (hx : x ∈ closure s) :
∃ (l : list M) (hl : ∀ y ∈ l, y ∈ s), l.prod = x :=
begin
rw [closure_eq_mrange, mem_mrange] at hx,
rcases hx with ⟨l, hx⟩,
exact ⟨list.map coe l, λ y hy, let ⟨z, hz, hy⟩ := list.mem_map.1 hy in hy ▸ z.2, hx⟩
end
/-- The submonoid generated by an element. -/
def powers (n : N) : submonoid N :=
submonoid.copy (powers_hom N n).mrange (set.range ((^) n : ℕ → N)) $
set.ext (λ n, exists_congr $ λ i, by simp; refl)
@[simp] lemma mem_powers (n : N) : n ∈ powers n := ⟨1, pow_one _⟩
lemma powers_eq_closure (n : N) : powers n = closure {n} :=
by { ext, exact mem_closure_singleton.symm }
lemma powers_subset {n : N} {P : submonoid N} (h : n ∈ P) : powers n ≤ P :=
λ x hx, match x, hx with _, ⟨i, rfl⟩ := P.pow_mem h i end
end submonoid
namespace submonoid
variables {N : Type*} [comm_monoid N]
open monoid_hom
@[to_additive]
lemma sup_eq_range (s t : submonoid N) : s ⊔ t = (s.subtype.coprod t.subtype).mrange :=
by rw [mrange, ← mrange_inl_sup_mrange_inr, map_sup, map_mrange, coprod_comp_inl,
map_mrange, coprod_comp_inr, range_subtype, range_subtype]
@[to_additive]
lemma mem_sup {s t : submonoid N} {x : N} :
x ∈ s ⊔ t ↔ ∃ (y ∈ s) (z ∈ t), y * z = x :=
by simp only [sup_eq_range, mem_mrange, coprod_apply, prod.exists, submonoid.exists,
coe_subtype, subtype.coe_mk]
end submonoid
namespace add_submonoid
open set
lemma nsmul_mem (S : add_submonoid A) {x : A} (hx : x ∈ S) :
∀ n : ℕ, n •ℕ x ∈ S
| 0 := S.zero_mem
| (n+1) := S.add_mem hx (nsmul_mem n)
lemma closure_singleton_eq (x : A) : closure ({x} : set A) = (multiples_hom A x).mrange :=
closure_eq_of_le (set.singleton_subset_iff.2 ⟨1, trivial, one_nsmul x⟩) $
λ x ⟨n, _, hn⟩, hn ▸ nsmul_mem _ (subset_closure $ set.mem_singleton _) _
/-- The `add_submonoid` generated by an element of an `add_monoid` equals the set of
natural number multiples of the element. -/
lemma mem_closure_singleton {x y : A} :
y ∈ closure ({x} : set A) ↔ ∃ n:ℕ, n •ℕ x = y :=
by rw [closure_singleton_eq, add_monoid_hom.mem_mrange]; refl
/-- The additive submonoid generated by an element. -/
def multiples (x : A) : add_submonoid A :=
add_submonoid.copy (multiples_hom A x).mrange (set.range (λ i, nsmul i x : ℕ → A)) $
set.ext (λ n, exists_congr $ λ i, by simp; refl)
@[simp] lemma mem_multiples (x : A) : x ∈ multiples x := ⟨1, one_nsmul _⟩
lemma multiples_eq_closure (x : A) : multiples x = closure {x} :=
by { ext, exact mem_closure_singleton.symm }
lemma multiples_subset {x : A} {P : add_submonoid A} (h : x ∈ P) : multiples x ≤ P :=
λ x hx, match x, hx with _, ⟨i, rfl⟩ := P.nsmul_mem h i end
attribute [to_additive add_submonoid.multiples] submonoid.powers
attribute [to_additive add_submonoid.mem_multiples] submonoid.mem_powers
attribute [to_additive add_submonoid.multiples_eq_closure] submonoid.powers_eq_closure
attribute [to_additive add_submonoid.multiples_subset] submonoid.powers_subset
end add_submonoid
|
92f7b762a7272238748d056f94676f5ca74bc99a | 64874bd1010548c7f5a6e3e8902efa63baaff785 | /tests/lean/run/ho.lean | 8daa9d9e385d7ec96d8272cb139974d904c31bce | [
"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 | 207 | lean | import data.nat
inductive star : Type₁ :=
z : star,
s : (nat → star) → star
check @star.rec
check @star.cases_on
example (f : nat → star) : ¬ star.z = star.s f :=
assume H, star.no_confusion H
|
743d1e510a6792a0cab37a154397b39555bd3477 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/logic/unique.lean | 9f1b2ae34b3b4099a3b98483cd0da0209ddde272 | [
"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 | 5,099 | 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 tactic.basic
import logic.is_empty
/-!
# Types with a unique term
In this file we define a typeclass `unique`,
which expresses that a type has a unique term.
In other words, a type that is `inhabited` and a `subsingleton`.
## Main declaration
* `unique`: a typeclass that expresses that a type has a unique term.
## Main statements
* `unique.mk'`: an inhabited subsingleton type is `unique`. This can not be an instance because it
would lead to loops in typeclass inference.
* `function.surjective.unique`: if the domain of a surjective function is `unique`, then its
codomain is `unique` as well.
* `function.injective.subsingleton`: if the codomain of an injective function is `subsingleton`,
then its domain is `subsingleton` as well.
* `function.injective.unique`: if the codomain of an injective function is `subsingleton` and its
domain is `inhabited`, then its domain is `unique`.
## Implementation details
The typeclass `unique α` is implemented as a type,
rather than a `Prop`-valued predicate,
for good definitional properties of the default term.
-/
universes u v w
variables {α : Sort u} {β : Sort v} {γ : Sort w}
/-- `unique α` expresses that `α` is a type with a unique term `default α`.
This is implemented as a type, rather than a `Prop`-valued predicate,
for good definitional properties of the default term. -/
@[ext]
structure unique (α : Sort u) extends inhabited α :=
(uniq : ∀ a:α, a = default)
attribute [class] unique
instance punit.unique : unique punit.{u} :=
{ default := punit.star,
uniq := λ x, punit_eq x _ }
/-- Every provable proposition is unique, as all proofs are equal. -/
def unique_prop {p : Prop} (h : p) : unique p :=
{ default := h, uniq := λ x, rfl }
instance : unique true := unique_prop trivial
lemma fin.eq_zero : ∀ n : fin 1, n = 0
| ⟨n, hn⟩ := fin.eq_of_veq (nat.eq_zero_of_le_zero (nat.le_of_lt_succ hn))
instance {n : ℕ} : inhabited (fin n.succ) := ⟨0⟩
instance inhabited_fin_one_add (n : ℕ) : inhabited (fin (1 + n)) := ⟨⟨0, nat.zero_lt_one_add n⟩⟩
@[simp] lemma fin.default_eq_zero (n : ℕ) : default (fin n.succ) = 0 := rfl
instance fin.unique : unique (fin 1) :=
{ uniq := fin.eq_zero, .. fin.inhabited }
namespace unique
open function
section
variables [unique α]
@[priority 100] -- see Note [lower instance priority]
instance : inhabited α := to_inhabited ‹unique α›
lemma eq_default (a : α) : a = default α := uniq _ a
lemma default_eq (a : α) : default α = a := (uniq _ a).symm
@[priority 100] -- see Note [lower instance priority]
instance : subsingleton α := subsingleton_of_forall_eq _ eq_default
lemma forall_iff {p : α → Prop} : (∀ a, p a) ↔ p (default α) :=
⟨λ h, h _, λ h x, by rwa [unique.eq_default x]⟩
lemma exists_iff {p : α → Prop} : Exists p ↔ p (default α) :=
⟨λ ⟨a, ha⟩, eq_default a ▸ ha, exists.intro (default α)⟩
end
@[ext] protected lemma subsingleton_unique' : ∀ (h₁ h₂ : unique α), h₁ = h₂
| ⟨⟨x⟩, h⟩ ⟨⟨y⟩, _⟩ := by congr; rw [h x, h y]
instance subsingleton_unique : subsingleton (unique α) :=
⟨unique.subsingleton_unique'⟩
/-- Construct `unique` from `inhabited` and `subsingleton`. Making this an instance would create
a loop in the class inheritance graph. -/
def mk' (α : Sort u) [h₁ : inhabited α] [subsingleton α] : unique α :=
{ uniq := λ x, subsingleton.elim _ _, .. h₁ }
end unique
@[simp] lemma pi.default_def {β : Π a : α, Sort v} [Π a, inhabited (β a)] :
default (Π a, β a) = λ a, default (β a) :=
rfl
lemma pi.default_apply {β : Π a : α, Sort v} [Π a, inhabited (β a)] (a : α) :
default (Π a, β a) a = default (β a) :=
rfl
instance pi.unique {β : Π a : α, Sort v} [Π a, unique (β a)] : unique (Π a, β a) :=
{ uniq := λ f, funext $ λ x, unique.eq_default _,
.. pi.inhabited α }
/-- There is a unique function on an empty domain. -/
instance pi.unique_of_is_empty [is_empty α] (β : Π a : α, Sort v) :
unique (Π a, β a) :=
{ default := is_empty_elim,
uniq := λ f, funext is_empty_elim }
namespace function
variable {f : α → β}
/-- If the domain of a surjective function is a singleton,
then the codomain is a singleton as well. -/
protected def surjective.unique (hf : surjective f) [unique α] : unique β :=
{ default := f (default _),
uniq := λ b, let ⟨a, ha⟩ := hf b in ha ▸ congr_arg f (unique.eq_default _) }
/-- If the codomain of an injective function is a subsingleton, then the domain
is a subsingleton as well. -/
protected lemma injective.subsingleton (hf : injective f) [subsingleton β] :
subsingleton α :=
⟨λ x y, hf $ subsingleton.elim _ _⟩
/-- If `α` is inhabited and admits an injective map to a subsingleton type, then `α` is `unique`. -/
protected def injective.unique [inhabited α] [subsingleton β] (hf : injective f) : unique α :=
@unique.mk' _ _ hf.subsingleton
end function
|
b5afe14a3c52e26bedf56d345da38614f5aca631 | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/run/kcomp.lean | d86373cfffe6c0d5faab9e230810548690521e69 | [
"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 | 973 | lean | set_option pp.notation false
constant A : Type
constants a b : A
constant P : A → Type
constant H₁ : a = a
constant H₂ : P a
constant H₃ : a = b
constant f {A : Type} (a : A) : a = a
#reduce (eq.rec H₂ (@f A a) : P a)
#reduce (eq.rec H₂ H₁ : P a)
#reduce (eq.rec H₂ H₃ : P b)
#reduce (eq.rec H₂ (@eq.refl A a) : P a)
-- eval λ (A : Type) (a b : A) (H₁ : a = a) (P : A → Prop) (H₂ : P a) (H₃ : a = a) (c : A), eq.rec (eq.rec H₂ H₁) H₃
#check @eq.rec A a P H₂ a
#check λ H : a = a, H₂
inductive to_type {B : Type} : B → Type
| mk : Π (b : B), to_type b
noncomputable definition tst1 : to_type (λ H : a = a, H₂) := to_type.mk (@eq.rec A a P H₂ a)
#check to_type.mk(λ H : a = a, H₂)
#check to_type.mk(@eq.rec A a P H₂ a)
#check to_type.mk(λ H : a = a, H₂) = to_type.mk(@eq.rec A a P H₂ a)
-- #check to_type.mk(eq.rec H₂ H₁) = to_type.mk(H₂)
-- #check to_type.mk(eq.rec H₂ (f a)) = to_type.mk(H₂)
|
132a73f3a82d0fb3db9d82c30028bfeee5c3bab1 | 4efff1f47634ff19e2f786deadd394270a59ecd2 | /src/algebraic_geometry/locally_ringed_space.lean | a48931df75a9cd01a395314078e8d6e6b3fc81a6 | [
"Apache-2.0"
] | permissive | agjftucker/mathlib | d634cd0d5256b6325e3c55bb7fb2403548371707 | 87fe50de17b00af533f72a102d0adefe4a2285e8 | refs/heads/master | 1,625,378,131,941 | 1,599,166,526,000 | 1,599,166,526,000 | 160,748,509 | 0 | 0 | Apache-2.0 | 1,544,141,789,000 | 1,544,141,789,000 | null | UTF-8 | Lean | false | false | 4,881 | 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 algebraic_geometry.sheafed_space
import algebra.category.CommRing.limits
import algebra.category.CommRing.colimits
import algebraic_geometry.stalks
import ring_theory.ideal.basic
/-!
# The category of locally ringed spaces
We define (bundled) locally ringed spaces
(as `SheafedSpace CommRing` along with the fact that the stalks are local rings),
and morphisms between these (morphisms in `SheafedSpace` with `is_local_ring_hom` on the stalk maps).
## Future work
* Define the restriction along an open embedding
-/
universes v u
open category_theory
open Top
open topological_space
open opposite
open category_theory.category category_theory.functor
namespace algebraic_geometry
/-- A `LocallyRingedSpace` is a topological space equipped with a sheaf of commutative rings
such that all the stalks are local rings.
A morphism of locally ringed spaces is a morphism of ringed spaces
such that the morphims induced on stalks are local ring homomorphisms. -/
@[nolint has_inhabited_instance]
structure LocallyRingedSpace extends SheafedSpace CommRing :=
(local_ring : ∀ x, local_ring (presheaf.stalk x))
attribute [instance] LocallyRingedSpace.local_ring
namespace LocallyRingedSpace
variables (X : LocallyRingedSpace)
/-- The underlying topological space of a locally ringed space. -/
def to_Top : Top := X.1.carrier
instance : has_coe_to_sort LocallyRingedSpace :=
{ S := Type u,
coe := λ X : LocallyRingedSpace, (X.to_Top : Type u), }
-- PROJECT: how about a typeclass "has_structure_sheaf" to mediate the 𝒪 notation, rather
-- than defining it over and over for PresheafedSpace, LRS, Scheme, etc.
/-- The structure sheaf of a locally ringed space. -/
def 𝒪 : sheaf CommRing X.to_Top := X.to_SheafedSpace.sheaf
/-- A morphism of locally ringed spaces is a morphism of ringed spaces
such that the morphims induced on stalks are local ring homomorphisms. -/
def hom (X Y : LocallyRingedSpace) : Type* :=
{ f : X.to_SheafedSpace ⟶ Y.to_SheafedSpace // ∀ x, is_local_ring_hom (PresheafedSpace.stalk_map f x) }
instance : has_hom LocallyRingedSpace := ⟨hom⟩
@[ext] lemma hom_ext {X Y : LocallyRingedSpace} (f g : hom X Y) (w : f.1 = g.1) : f = g :=
subtype.eq w
/--
The stalk of a locally ringed space, just as a `CommRing`.
-/
-- TODO perhaps we should make a bundled `LocalRing` and return one here?
-- TODO define `sheaf.stalk` so we can write `X.𝒪.stalk` here?
def stalk (X : LocallyRingedSpace) (x : X) : CommRing := X.presheaf.stalk x
/--
A morphism of locally ringed spaces `f : X ⟶ Y` induces
a local ring homomorphism from `Y.stalk (f x)` to `X.stalk x` for any `x : X`.
-/
def stalk_map {X Y : LocallyRingedSpace} (f : X ⟶ Y) (x : X) :
Y.stalk (f.1.1 x) ⟶ X.stalk x :=
PresheafedSpace.stalk_map f.1 x
instance {X Y : LocallyRingedSpace} (f : X ⟶ Y) (x : X) :
is_local_ring_hom (stalk_map f x) := f.2 x
/-- The identity morphism on a locally ringed space. -/
@[simps]
def id (X : LocallyRingedSpace) : hom X X :=
⟨𝟙 _, λ x, by { erw PresheafedSpace.stalk_map.id, apply is_local_ring_hom_id, }⟩
instance (X : LocallyRingedSpace) : inhabited (hom X X) := ⟨id X⟩
/-- Composition of morphisms of locally ringed spaces. -/
@[simps]
def comp {X Y Z : LocallyRingedSpace} (f : hom X Y) (g : hom Y Z) : hom X Z :=
⟨f.val ≫ g.val, λ x,
begin
erw PresheafedSpace.stalk_map.comp,
exact @is_local_ring_hom_comp _ _ _ _ _ _ _ _ (f.2 _) (g.2 _),
end⟩
/-- The category of locally ringed spaces. -/
instance : category LocallyRingedSpace :=
{ hom := hom,
id := id,
comp := λ X Y Z f g, comp f g,
comp_id' := by { intros, ext1, simp, },
id_comp' := by { intros, ext1, simp, },
assoc' := by { intros, ext1, simp, }, }.
/-- The forgetful functor from `LocallyRingedSpace` to `SheafedSpace CommRing`. -/
def forget_to_SheafedSpace : LocallyRingedSpace ⥤ SheafedSpace CommRing :=
{ obj := λ X, X.to_SheafedSpace,
map := λ X Y f, f.1, }
instance : faithful forget_to_SheafedSpace := {}
-- PROJECT: once we have `PresheafedSpace.restrict_stalk_iso`
-- (that restriction doesn't change stalks) we can uncomment this.
/-
def restrict {U : Top} (X : LocallyRingedSpace)
(f : U ⟶ X.to_Top) (h : open_embedding f) : LocallyRingedSpace :=
{ local_ring :=
begin
intro x,
dsimp at *,
-- We show that the stalk of the restriction is isomorphic to the original stalk,
have := X.to_SheafedSpace.to_PresheafedSpace.restrict_stalk_iso f h x,
-- and then transfer `local_ring` across the ring equivalence.
apply (this.CommRing_iso_to_ring_equiv).local_ring, -- import data.equiv.transfer_instance
apply X.local_ring,
end,
.. X.to_SheafedSpace.restrict _ f h }
-/
end LocallyRingedSpace
end algebraic_geometry
|
f88783f25dd67fc8f7d12d874067d704bea786a9 | dc253be9829b840f15d96d986e0c13520b085033 | /algebra/left_module.hlean | 8bbb2246fc56678325ab6e1bc472d1517d19da68 | [
"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 | 26,663 | hlean | /-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Floris van Doorn
Modules prod vector spaces over a ring.
(We use "left_module," which is more precise, because "module" is a keyword.)
-/
import algebra.field ..move_to_lib .exactness algebra.group_power
open is_trunc pointed function sigma eq algebra prod is_equiv equiv group
structure has_scalar [class] (F V : Type) :=
(smul : F → V → V)
infixl ` • `:73 := has_scalar.smul
/- modules over a ring -/
namespace left_module
structure left_module (R M : Type) [ringR : ring R] extends has_scalar R M, ab_group M renaming
mul → add mul_assoc → add_assoc one → zero one_mul → zero_add mul_one → add_zero inv → neg
mul_left_inv → add_left_inv mul_comm → add_comm :=
(smul_left_distrib : Π (r : R) (x y : M), smul r (add x y) = (add (smul r x) (smul r y)))
(smul_right_distrib : Π (r s : R) (x : M), smul (ring.add _ r s) x = (add (smul r x) (smul s x)))
(mul_smul : Π r s x, smul (mul r s) x = smul r (smul s x))
(one_smul : Π x, smul one x = x)
/- we make it a class now (and not as part of the structure) to avoid
left_module.to_ab_group to be an instance -/
attribute left_module [class]
definition add_ab_group_of_left_module [reducible] [trans_instance] (R M : Type) [K : ring R]
[H : left_module R M] : add_ab_group M :=
@left_module.to_ab_group R M K H
definition has_scalar_of_left_module [reducible] [trans_instance] (R M : Type) [K : ring R]
[H : left_module R M] : has_scalar R M :=
@left_module.to_has_scalar R M K H
section left_module
variables {R M : Type}
variable [ringR : ring R]
variable [moduleRM : left_module R M]
include ringR moduleRM
-- Note: the anonymous include does not work in the propositions below.
proposition smul_left_distrib (a : R) (u v : M) : a • (u + v) = a • u + a • v :=
!left_module.smul_left_distrib
proposition smul_right_distrib (a b : R) (u : M) : (a + b) • u = a • u + b • u :=
!left_module.smul_right_distrib
proposition mul_smul (a : R) (b : R) (u : M) : (a * b) • u = a • (b • u) :=
!left_module.mul_smul
proposition one_smul (u : M) : (1 : R) • u = u := !left_module.one_smul
proposition zero_smul (u : M) : (0 : R) • u = 0 :=
have (0 : R) • u + 0 • u = 0 • u + 0, by rewrite [-smul_right_distrib, *add_zero],
!add.left_cancel this
proposition smul_zero (a : R) : a • (0 : M) = 0 :=
have a • (0:M) + a • 0 = a • 0 + 0, by rewrite [-smul_left_distrib, *add_zero],
!add.left_cancel this
proposition neg_smul (a : R) (u : M) : (-a) • u = - (a • u) :=
eq_neg_of_add_eq_zero (by rewrite [-smul_right_distrib, add.left_inv, zero_smul])
proposition neg_one_smul (u : M) : -(1 : R) • u = -u :=
by rewrite [neg_smul, one_smul]
proposition smul_neg (a : R) (u : M) : a • (-u) = -(a • u) :=
by rewrite [-neg_one_smul, -mul_smul, mul_neg_one_eq_neg, neg_smul]
proposition smul_sub_left_distrib (a : R) (u v : M) : a • (u - v) = a • u - a • v :=
by rewrite [sub_eq_add_neg, smul_left_distrib, smul_neg]
proposition sub_smul_right_distrib (a b : R) (v : M) : (a - b) • v = a • v - b • v :=
by rewrite [sub_eq_add_neg, smul_right_distrib, neg_smul]
end left_module
/- vector spaces -/
structure vector_space [class] (F V : Type) [fieldF : field F]
extends left_module F V
/- homomorphisms -/
definition is_smul_hom [class] (R : Type) {M₁ M₂ : Type} [has_scalar R M₁] [has_scalar R M₂]
(f : M₁ → M₂) : Type :=
∀ r : R, ∀ a : M₁, f (r • a) = r • f a
definition is_prop_is_smul_hom [instance] (R : Type) {M₁ M₂ : Type} [is_set M₂]
[has_scalar R M₁] [has_scalar R M₂] (f : M₁ → M₂) : is_prop (is_smul_hom R f) :=
begin unfold is_smul_hom, apply _ end
definition respect_smul (R : Type) {M₁ M₂ : Type} [has_scalar R M₁] [has_scalar R M₂]
(f : M₁ → M₂) [H : is_smul_hom R f] :
∀ r : R, ∀ a : M₁, f (r • a) = r • f a :=
H
definition is_module_hom [class] (R : Type) {M₁ M₂ : Type}
[has_scalar R M₁] [has_scalar R M₂] [add_group M₁] [add_group M₂]
(f : M₁ → M₂) :=
is_add_hom f × is_smul_hom R f
definition is_add_hom_of_is_module_hom [instance] (R : Type) {M₁ M₂ : Type}
[has_scalar R M₁] [has_scalar R M₂] [add_group M₁] [add_group M₂]
(f : M₁ → M₂) [H : is_module_hom R f] : is_add_hom f :=
prod.pr1 H
definition is_smul_hom_of_is_module_hom [instance] {R : Type} {M₁ M₂ : Type}
[has_scalar R M₁] [has_scalar R M₂] [add_group M₁] [add_group M₂]
(f : M₁ → M₂) [H : is_module_hom R f] : is_smul_hom R f :=
prod.pr2 H
-- Why do we have to give the instance explicitly?
definition is_prop_is_module_hom [instance] (R : Type) {M₁ M₂ : Type}
[has_scalar R M₁] [has_scalar R M₂] [add_group M₁] [add_group M₂]
(f : M₁ → M₂) : is_prop (is_module_hom R f) :=
have h₁ : is_prop (is_add_hom f), from is_prop_is_add_hom f,
begin unfold is_module_hom, apply _ end
section module_hom
variables {R : Type} {M₁ M₂ M₃ : Type}
variables [has_scalar R M₁] [has_scalar R M₂] [has_scalar R M₃]
variables [add_group M₁] [add_group M₂] [add_group M₃]
variables (g : M₂ → M₃) (f : M₁ → M₂) [is_module_hom R g] [is_module_hom R f]
proposition is_module_hom_id : is_module_hom R (@id M₁) :=
pair (λ a₁ a₂, rfl) (λ r a, rfl)
proposition is_module_hom_comp : is_module_hom R (g ∘ f) :=
pair
(take a₁ a₂, begin esimp, rewrite [respect_add f, respect_add g] end)
(take r a, by esimp; rewrite [respect_smul R f, respect_smul R g])
proposition respect_smul_add_smul (a b : R) (u v : M₁) : f (a • u + b • v) = a • f u + b • f v :=
by rewrite [respect_add f, +respect_smul R f]
end module_hom
section hom_constant
variables {R : Type} {M₁ M₂ : Type}
variables [ring R] [has_scalar R M₁] [add_group M₁] [left_module R M₂]
proposition is_module_hom_constant : is_module_hom R (const M₁ (0 : M₂)) :=
(λm₁ m₂, !add_zero⁻¹, λr m, (smul_zero r)⁻¹ᵖ)
end hom_constant
structure LeftModule (R : Ring) :=
(carrier : Type) (struct : left_module R carrier)
attribute LeftModule.struct [instance]
section
local attribute LeftModule.carrier [coercion]
definition AddAbGroup_of_LeftModule [coercion] {R : Ring} (M : LeftModule R) : AddAbGroup :=
AddAbGroup.mk M (LeftModule.struct M)
end
definition LeftModule.struct2 [instance] {R : Ring} (M : LeftModule R) : left_module R M :=
LeftModule.struct M
-- definition LeftModule.struct3 [instance] {R : Ring} (M : LeftModule R) :
-- left_module R (AddAbGroup_of_LeftModule M) :=
-- _
definition pointed_LeftModule_carrier [instance] {R : Ring} (M : LeftModule R) :
pointed (LeftModule.carrier M) :=
pointed.mk zero
definition pSet_of_LeftModule {R : Ring} (M : LeftModule R) : Set* :=
pSet.mk' (LeftModule.carrier M)
definition left_module_AddAbGroup_of_LeftModule [instance] {R : Ring} (M : LeftModule R) :
left_module R (AddAbGroup_of_LeftModule M) :=
LeftModule.struct M
definition left_module_AddAbGroup_of_LeftModule2 [instance] {R : Ring} (M : LeftModule R) :
left_module R (Group_of_AbGroup (AddAbGroup_of_LeftModule M)) :=
LeftModule.struct M
definition left_module_of_ab_group {G : Type} [gG : add_ab_group G] {R : Type} [ring R]
(smul : R → G → G)
(h1 : Π (r : R) (x y : G), smul r (x + y) = (smul r x + smul r y))
(h2 : Π (r s : R) (x : G), smul (r + s) x = (smul r x + smul s x))
(h3 : Π r s x, smul (r * s) x = smul r (smul s x))
(h4 : Π x, smul 1 x = x) : left_module R G :=
left_module.mk smul _ add add.assoc 0 zero_add add_zero neg add.left_inv add.comm h1 h2 h3 h4
definition LeftModule_of_AddAbGroup {R : Ring} (G : AddAbGroup) (smul : R → G → G)
(h1 h2 h3 h4) : LeftModule R :=
LeftModule.mk G (left_module_of_ab_group smul h1 h2 h3 h4)
open unit
definition trivial_LeftModule [constructor] (R : Ring) : LeftModule R :=
LeftModule_of_AddAbGroup trivial_ab_group (λr u, star)
(λr u₁ u₂, idp) (λr₁ r₂ u, idp) (λr₁ r₂ u, idp) unit.eta
section
variables {R : Ring} {M M₁ M₂ M₃ : LeftModule R}
definition smul_homomorphism [constructor] (M : LeftModule R) (r : R) : M →a M :=
add_homomorphism.mk (λg, r • g) (smul_left_distrib r)
proposition to_smul_left_distrib (a : R) (u v : M) : a • (u + v) = a • u + a • v :=
!smul_left_distrib
proposition to_smul_right_distrib (a b : R) (u : M) : (a + b) • u = a • u + b • u :=
!smul_right_distrib
proposition to_mul_smul (a : R) (b : R) (u : M) : (a * b) • u = a • (b • u) :=
!mul_smul
proposition to_one_smul (u : M) : (1 : R) • u = u := !one_smul
structure homomorphism (M₁ M₂ : LeftModule R) : Type :=
(fn : LeftModule.carrier M₁ → LeftModule.carrier M₂)
(p : is_module_hom R fn)
infix ` →lm `:55 := homomorphism
definition homomorphism_fn [unfold 4] [coercion] := @homomorphism.fn
definition is_module_hom_of_homomorphism [unfold 4] [instance] [priority 900]
{M₁ M₂ : LeftModule R} (φ : M₁ →lm M₂) : is_module_hom R φ :=
homomorphism.p φ
section
variable (φ : M₁ →lm M₂)
definition to_respect_add (x y : M₁) : φ (x + y) = φ x + φ y :=
respect_add φ x y
definition to_respect_smul (a : R) (x : M₁) : φ (a • x) = a • (φ x) :=
respect_smul R φ a x
definition to_respect_sub (x y : M₁) : φ (x - y) = φ x - φ y :=
respect_sub φ x y
definition is_embedding_of_homomorphism /- φ -/ (H : Π{x}, φ x = 0 → x = 0) : is_embedding φ :=
is_embedding_of_is_add_hom φ @H
variables (M₁ M₂)
definition is_set_homomorphism [instance] : is_set (M₁ →lm M₂) :=
begin
have H : M₁ →lm M₂ ≃ Σ(f : LeftModule.carrier M₁ → LeftModule.carrier M₂),
is_module_hom (Ring.carrier R) f,
begin
fapply equiv.MK,
{ intro φ, induction φ, constructor, exact p},
{ intro v, induction v with f H, constructor, exact H},
{ intro v, induction v, reflexivity},
{ intro φ, induction φ, reflexivity}
end,
have ∀ f : LeftModule.carrier M₁ → LeftModule.carrier M₂,
is_set (is_module_hom (Ring.carrier R) f), from _,
exact is_trunc_equiv_closed_rev _ H _
end
variables {M₁ M₂}
definition pmap_of_homomorphism [constructor] /- φ -/ :
pSet_of_LeftModule M₁ →* pSet_of_LeftModule M₂ :=
have H : φ 0 = 0, from respect_zero φ,
pmap.mk φ begin esimp, exact H end
definition homomorphism_change_fun [constructor]
(φ : M₁ →lm M₂) (f : M₁ → M₂) (p : φ ~ f) : M₁ →lm M₂ :=
homomorphism.mk f
(prod.mk
(λx₁ x₂, (p (x₁ + x₂))⁻¹ ⬝ to_respect_add φ x₁ x₂ ⬝ ap011 _ (p x₁) (p x₂))
(λ a x, (p (a • x))⁻¹ ⬝ to_respect_smul φ a x ⬝ ap01 _ (p x)))
definition homomorphism_eq (φ₁ φ₂ : M₁ →lm M₂) (p : φ₁ ~ φ₂) : φ₁ = φ₂ :=
begin
induction φ₁ with φ₁ q₁, induction φ₂ with φ₂ q₂, esimp at p, induction p,
exact ap (homomorphism.mk φ₁) !is_prop.elim
end
end
section
definition homomorphism.mk' [constructor] (φ : M₁ → M₂)
(p : Π(g₁ g₂ : M₁), φ (g₁ + g₂) = φ g₁ + φ g₂)
(q : Π(r : R) x, φ (r • x) = r • φ x) : M₁ →lm M₂ :=
homomorphism.mk φ (p, q)
definition to_respect_zero (φ : M₁ →lm M₂) : φ 0 = 0 :=
respect_zero φ
definition homomorphism_compose [reducible] [constructor] (f' : M₂ →lm M₃) (f : M₁ →lm M₂) :
M₁ →lm M₃ :=
homomorphism.mk (f' ∘ f) !is_module_hom_comp
variable (M)
definition homomorphism_id [reducible] [constructor] [refl] : M →lm M :=
homomorphism.mk (@id M) !is_module_hom_id
variable {M}
abbreviation lmid [constructor] := homomorphism_id M
infixr ` ∘lm `:75 := homomorphism_compose
definition lm_constant [constructor] (M₁ M₂ : LeftModule R) : M₁ →lm M₂ :=
homomorphism.mk (const M₁ 0) !is_module_hom_constant
definition trivial_image_of_is_contr {R} {M₁ M₂ : LeftModule R} {φ : M₁ →lm M₂} (H : is_contr M₁)
⦃m : M₂⦄ (hm : image φ m) : m = 0 :=
begin
induction hm with m' p, induction p,
exact ap φ (@eq_of_is_contr _ H _ _) ⬝ to_respect_zero φ
end
structure isomorphism (M₁ M₂ : LeftModule R) :=
(to_hom : M₁ →lm M₂)
(is_equiv_to_hom : is_equiv to_hom)
infix ` ≃lm `:25 := isomorphism
attribute isomorphism.to_hom [coercion]
attribute isomorphism.is_equiv_to_hom [instance]
attribute isomorphism._trans_of_to_hom [unfold 4]
definition equiv_of_isomorphism [constructor] (φ : M₁ ≃lm M₂) : M₁ ≃ M₂ :=
equiv.mk φ !isomorphism.is_equiv_to_hom
section
local attribute pSet_of_LeftModule [coercion]
definition pequiv_of_isomorphism [constructor] (φ : M₁ ≃lm M₂) : M₁ ≃* M₂ :=
pequiv_of_equiv (equiv_of_isomorphism φ) (to_respect_zero φ)
end
definition isomorphism_of_equiv [constructor] (φ : M₁ ≃ M₂)
(p : Π(g₁ g₂ : M₁), φ (g₁ + g₂) = φ g₁ + φ g₂)
(q : Πr x, φ (r • x) = r • φ x) : M₁ ≃lm M₂ :=
isomorphism.mk (homomorphism.mk φ (p, q)) !to_is_equiv
definition isomorphism_of_eq [constructor] {M₁ M₂ : LeftModule R} (p : M₁ = M₂ :> LeftModule R)
: M₁ ≃lm M₂ :=
isomorphism_of_equiv (equiv_of_eq (ap LeftModule.carrier p))
begin intros, induction p, reflexivity end
begin intros, induction p, reflexivity end
-- definition pequiv_of_isomorphism_of_eq {M₁ M₂ : LeftModule R} (p : M₁ = M₂ :> LeftModule R) :
-- pequiv_of_isomorphism (isomorphism_of_eq p) = pequiv_of_eq (ap pType_of_LeftModule p) :=
-- begin
-- induction p,
-- apply pequiv_eq,
-- fapply pmap_eq,
-- { intro g, reflexivity},
-- { apply is_prop.elim}
-- end
definition to_lminv [constructor] (φ : M₁ ≃lm M₂) : M₂ →lm M₁ :=
homomorphism.mk φ⁻¹
abstract begin
split,
intro g₁ g₂, apply inj' φ,
rewrite [respect_add φ, +right_inv φ],
intro r x, apply inj' φ,
rewrite [to_respect_smul φ, +right_inv φ],
end end
variable (M)
definition isomorphism.refl [refl] [constructor] : M ≃lm M :=
isomorphism.mk lmid !is_equiv_id
variable {M}
definition isomorphism.rfl [refl] [constructor] : M ≃lm M := isomorphism.refl M
definition isomorphism.symm [symm] [constructor] (φ : M₁ ≃lm M₂) : M₂ ≃lm M₁ :=
isomorphism.mk (to_lminv φ) !is_equiv_inv
definition isomorphism.trans [trans] [constructor] (φ : M₁ ≃lm M₂) (ψ : M₂ ≃lm M₃) : M₁ ≃lm M₃ :=
isomorphism.mk (ψ ∘lm φ) (is_equiv_compose ψ φ _ _)
definition isomorphism.eq_trans [trans] [constructor]
{M₁ M₂ : LeftModule R} {M₃ : LeftModule R} (φ : M₁ = M₂) (ψ : M₂ ≃lm M₃) : M₁ ≃lm M₃ :=
proof isomorphism.trans (isomorphism_of_eq φ) ψ qed
definition isomorphism.trans_eq [trans] [constructor]
{M₁ : LeftModule R} {M₂ M₃ : LeftModule R} (φ : M₁ ≃lm M₂) (ψ : M₂ = M₃) : M₁ ≃lm M₃ :=
isomorphism.trans φ (isomorphism_of_eq ψ)
postfix `⁻¹ˡᵐ`:(max + 1) := isomorphism.symm
infixl ` ⬝lm `:75 := isomorphism.trans
infixl ` ⬝lmp `:75 := isomorphism.trans_eq
infixl ` ⬝plm `:75 := isomorphism.eq_trans
definition homomorphism_of_eq [constructor] {M₁ M₂ : LeftModule R} (p : M₁ = M₂ :> LeftModule R)
: M₁ →lm M₂ :=
isomorphism_of_eq p
definition group_homomorphism_of_lm_homomorphism [constructor] {M₁ M₂ : LeftModule R}
(φ : M₁ →lm M₂) : M₁ →a M₂ :=
add_homomorphism.mk φ (to_respect_add φ)
definition lm_homomorphism_of_group_homomorphism [constructor] {M₁ M₂ : LeftModule R}
(φ : M₁ →a M₂) (h : Π(r : R) g, φ (r • g) = r • φ g) : M₁ →lm M₂ :=
homomorphism.mk' φ (group.to_respect_add φ) h
definition group_isomorphism_of_lm_isomorphism [constructor] {M₁ M₂ : LeftModule R}
(φ : M₁ ≃lm M₂) : AddGroup_of_AddAbGroup M₁ ≃g AddGroup_of_AddAbGroup M₂ :=
group.isomorphism.mk (group_homomorphism_of_lm_homomorphism φ) (isomorphism.is_equiv_to_hom φ)
definition lm_isomorphism_of_group_isomorphism [constructor] {M₁ M₂ : LeftModule R}
(φ : AddGroup_of_AddAbGroup M₁ ≃g AddGroup_of_AddAbGroup M₂)
(h : Π(r : R) g, φ (r • g) = r • φ g) : M₁ ≃lm M₂ :=
isomorphism.mk (lm_homomorphism_of_group_homomorphism φ h) (group.isomorphism.is_equiv_to_hom φ)
definition trivial_homomorphism [constructor] (M₁ M₂ : LeftModule R) : M₁ →lm M₂ :=
lm_homomorphism_of_group_homomorphism (group.trivial_add_homomorphism M₁ M₂)
(λs m, (smul_zero s)⁻¹)
section
local attribute pSet_of_LeftModule [coercion]
definition is_exact_mod (f : M₁ →lm M₂) (f' : M₂ →lm M₃) : Type :=
@is_exact M₁ M₂ M₃ (homomorphism_fn f) (homomorphism_fn f')
definition is_exact_mod.mk {f : M₁ →lm M₂} {f' : M₂ →lm M₃}
(h₁ : Πm, f' (f m) = 0) (h₂ : Πm, f' m = 0 → image f m) : is_exact_mod f f' :=
is_exact.mk h₁ h₂
structure short_exact_mod (A B C : LeftModule R) :=
(f : A →lm B)
(g : B →lm C)
(h : @is_short_exact A B C f g)
structure five_exact_mod (A B C D E : LeftModule R) :=
(f₁ : A →lm B)
(f₂ : B →lm C)
(f₃ : C →lm D)
(f₄ : D →lm E)
(h₁ : @is_exact A B C f₁ f₂)
(h₂ : @is_exact B C D f₂ f₃)
(h₃ : @is_exact C D E f₃ f₄)
local abbreviation g_of_lm := @group_homomorphism_of_lm_homomorphism
definition short_exact_mod_of_is_exact {X A B C Y : LeftModule R}
(k : X →lm A) (f : A →lm B) (g : B →lm C) (l : C →lm Y)
(hX : is_contr X) (hY : is_contr Y)
(kf : is_exact_mod k f) (fg : is_exact_mod f g) (gl : is_exact_mod g l) :
short_exact_mod A B C :=
short_exact_mod.mk f g
(is_short_exact_of_is_exact (g_of_lm k) (g_of_lm f) (g_of_lm g) (g_of_lm l) hX hY kf fg gl)
definition short_exact_mod_isomorphism {A B A' B' C C' : LeftModule R}
(eA : A ≃lm A') (eB : B ≃lm B') (eC : C ≃lm C')
(H : short_exact_mod A' B' C') : short_exact_mod A B C :=
short_exact_mod.mk (eB⁻¹ˡᵐ ∘lm short_exact_mod.f H ∘lm eA) (eC⁻¹ˡᵐ ∘lm short_exact_mod.g H ∘lm eB)
(is_short_exact_equiv _ _
(equiv_of_isomorphism eA) (equiv_of_isomorphism eB) (pequiv_of_isomorphism eC)
(λa, to_right_inv (equiv_of_isomorphism eB) _) (λb, to_right_inv (equiv_of_isomorphism eC) _)
(short_exact_mod.h H))
definition is_contr_middle_of_short_exact_mod {A B C : LeftModule R} (H : short_exact_mod A B C)
(HA : is_contr A) (HC : is_contr C) : is_contr B :=
is_contr_middle_of_is_exact (is_exact_of_is_short_exact (short_exact_mod.h H))
definition is_contr_right_of_short_exact_mod {A B C : LeftModule R} (H : short_exact_mod A B C)
(HB : is_contr B) : is_contr C :=
is_contr_right_of_is_short_exact (short_exact_mod.h H) _ _
definition is_contr_left_of_short_exact_mod {A B C : LeftModule R} (H : short_exact_mod A B C)
(HB : is_contr B) : is_contr A :=
is_contr_left_of_is_short_exact (short_exact_mod.h H) _ pt
definition isomorphism_of_is_contr_left {A B C : LeftModule R} (H : short_exact_mod A B C)
(HA : is_contr A) : B ≃lm C :=
isomorphism.mk (short_exact_mod.g H)
begin
apply @is_equiv_right_of_is_short_exact _ _ _
(group_homomorphism_of_lm_homomorphism (short_exact_mod.f H))
(group_homomorphism_of_lm_homomorphism (short_exact_mod.g H)),
rexact short_exact_mod.h H, exact HA,
end
definition isomorphism_of_is_contr_right {A B C : LeftModule R} (H : short_exact_mod A B C)
(HC : is_contr C) : A ≃lm B :=
isomorphism.mk (short_exact_mod.f H)
begin
apply @is_equiv_left_of_is_short_exact _ _ _
(group_homomorphism_of_lm_homomorphism (short_exact_mod.f H))
(group_homomorphism_of_lm_homomorphism (short_exact_mod.g H)),
rexact short_exact_mod.h H, exact HC,
end
end
end
end
/- we say that an left module D is built from the sequence E if
D is a "twisted sum" of the E, and E has only finitely many nontrivial values -/
open nat
structure is_built_from.{u v w} {R : Ring} (D : LeftModule.{u v} R)
(E : ℕ → LeftModule.{u w} R) : Type.{max u (v+1) w} :=
(part : ℕ → LeftModule.{u v} R)
(ses : Πn, short_exact_mod (E n) (part n) (part (n+1)))
(e0 : part 0 ≃lm D)
(n₀ : ℕ)
(HD' : Π(s : ℕ), n₀ ≤ s → is_contr (part s))
open is_built_from
universe variables u v w
variables {R : Ring.{u}} {D D' : LeftModule.{u v} R} {E E' : ℕ → LeftModule.{u w} R}
definition is_built_from_shift (H : is_built_from D E) :
is_built_from (part H 1) (λn, E (n+1)) :=
is_built_from.mk (λn, part H (n+1)) (λn, ses H (n+1)) isomorphism.rfl (pred (n₀ H))
(λs Hle, HD' H _ (le_succ_of_pred_le Hle))
definition is_built_from_isomorphism (e : D ≃lm D') (f : Πn, E n ≃lm E' n)
(H : is_built_from D E) : is_built_from D' E' :=
⦃is_built_from, H, e0 := e0 H ⬝lm e,
ses := λn, short_exact_mod_isomorphism (f n)⁻¹ˡᵐ isomorphism.rfl isomorphism.rfl (ses H n)⦄
definition is_built_from_isomorphism_left (e : D ≃lm D') (H : is_built_from D E) :
is_built_from D' E :=
⦃is_built_from, H, e0 := e0 H ⬝lm e⦄
definition isomorphism_of_is_contr_submodule (H : is_built_from D E) (n : ℕ)
(HE : is_contr (E n)) : part H n ≃lm part H (n+1) :=
isomorphism_of_is_contr_left (ses H n) HE
definition isomorphism_of_is_contr_submodules_range (H : is_built_from D E) {n k : ℕ}
(Hnk : n ≤ k) (HE : Πl, n ≤ l → l < k → is_contr (E l)) : part H n ≃lm part H k :=
begin
induction Hnk with k Hnk IH,
{ reflexivity },
{ refine IH (λl Hnl Hlk, HE l Hnl (lt.step Hlk)) ⬝lm
isomorphism_of_is_contr_submodule H k (HE k Hnk !self_lt_succ), }
end
definition is_contr_quotients_of_is_contr_total (H : is_built_from D E) (HD : is_contr D) (n : ℕ) :
is_contr (part H n) :=
begin
induction n with n IH,
{ exact is_trunc_equiv_closed_rev -2 (equiv_of_isomorphism (e0 H)) _ },
{ exact is_contr_right_of_short_exact_mod (ses H n) IH }
end
definition is_contr_quotients_of_is_contr_submodules (H : is_built_from D E) {n : ℕ}
(HE : Πk, n ≤ k → is_contr (E k)) : is_contr (part H n) :=
begin
refine is_contr_equiv_closed_rev _ (HD' H (max n (n₀ H)) !le_max_right),
apply equiv_of_isomorphism,
apply isomorphism_of_is_contr_submodules_range H !le_max_left,
intros l Hnl Hl', exact HE l Hnl
end
/- alternate direct proof -/
-- nat.rec_down (λk, is_contr (part H (n + k))) _ (HD' H _ !nat.le_add_left)
-- (λk H2, is_contr_middle_of_short_exact_mod (ses H (n + k)) (HE (n + k) !nat.le_add_right)
-- proof H2 qed)
definition isomorphism_of_is_contr_submodules (H : is_built_from D E) {n : ℕ}
(HE : Πk, n < k → is_contr (E k)) : E n ≃lm part H n :=
isomorphism_of_is_contr_right (ses H n) (is_contr_quotients_of_is_contr_submodules H HE)
-- definition is_contr_quotients_of_is_contr_submodules1 (H : is_built_from D E) (n : ℕ)
-- (HE : Πk, n ≤ k → is_contr (E k)) : is_contr (part H n) :=
-- nat.rec_down (λk, is_contr (part H (n + k))) _ (HD' H _ !nat.le_add_left)
-- (λk H2, is_contr_middle_of_short_exact_mod (ses H (n + k)) (HE (n + k) !nat.le_add_right)
-- proof H2 qed)
definition isomorphism_zero_of_is_built_from (H : is_built_from D E) (p : n₀ H = 1) : E 0 ≃lm D :=
isomorphism_of_is_contr_right (ses H 0) (HD' H 1 (le_of_eq p)) ⬝lm e0 H
definition isomorphism_total_of_is_contr_submodules (H : is_built_from D E) {n : ℕ}
(HE : Πk, k < n → is_contr (E k)) : D ≃lm part H n :=
(e0 H)⁻¹ˡᵐ ⬝lm isomorphism_of_is_contr_submodules_range H !zero_le (λk H0k Hkn, HE k Hkn)
definition isomorphism_of_is_contr_submodules_but_one (H : is_built_from D E) {n : ℕ}
(HE : Πk, k ≠ n → is_contr (E k)) : D ≃lm E n :=
(e0 H)⁻¹ˡᵐ ⬝lm
isomorphism_of_is_contr_submodules_range H !zero_le (λk H0k Hkn, HE k (ne_of_lt Hkn)) ⬝lm
(isomorphism_of_is_contr_submodules H (λk Hk, HE k (ne.symm (ne_of_lt Hk))))⁻¹ˡᵐ
definition short_exact_mod_of_is_contr_submodules (H : is_built_from D E) {n m : ℕ} (Hnm : n < m)
(HE : Πk, k ≠ n → k ≠ m → is_contr (E k)) : short_exact_mod (E n) D (E m) :=
begin
refine short_exact_mod_isomorphism isomorphism.rfl _ _ (ses H n),
{ exact isomorphism_total_of_is_contr_submodules H (λk Hk, HE k (ne_of_lt Hk)
(ne_of_lt (lt.trans Hk Hnm))) },
{ exact isomorphism_of_is_contr_submodules H
(λk Hk, HE k (ne.symm (ne_of_lt (lt.trans Hnm Hk))) (ne.symm (ne_of_lt Hk))) ⬝lm
(isomorphism_of_is_contr_submodules_range H Hnm
(λk Hnk Hkm, HE k (ne.symm (ne_of_lt Hnk)) (ne_of_lt Hkm)))⁻¹ˡᵐ }
end
definition is_contr_submodules (H : is_built_from D E) (HD : is_contr D) (n : ℕ) :
is_contr (E n) :=
begin
apply is_contr_left_of_short_exact_mod (ses H n),
exact is_contr_quotients_of_is_contr_total H HD n
end
definition is_contr_total (H : is_built_from D E) (HE : Πn, is_contr (E n)) : is_contr D :=
have is_contr (part H 0), from is_contr_quotients_of_is_contr_submodules H (λn H, HE n),
is_contr_equiv_closed (equiv_of_isomorphism (e0 H)) _
section int
open int
definition left_module_int_of_ab_group [constructor] (A : Type) [add_ab_group A] : left_module rℤ A :=
left_module_of_ab_group imul imul_add add_imul mul_imul one_imul
definition LeftModule_int_of_AbGroup [constructor] (A : AddAbGroup) : LeftModule rℤ :=
LeftModule.mk A (left_module_int_of_ab_group A)
definition lm_hom_int.mk [constructor] {A B : AbGroup} (φ : A →g B) :
LeftModule_int_of_AbGroup A →lm LeftModule_int_of_AbGroup B :=
lm_homomorphism_of_group_homomorphism φ (to_respect_imul φ)
definition lm_iso_int.mk [constructor] {A B : AbGroup} (φ : A ≃g B) :
LeftModule_int_of_AbGroup A ≃lm LeftModule_int_of_AbGroup B :=
isomorphism.mk (lm_hom_int.mk φ) (group.isomorphism.is_equiv_to_hom φ)
definition group_isomorphism_of_lm_isomorphism_int [constructor] {A B : AbGroup}
(φ : LeftModule_int_of_AbGroup A ≃lm LeftModule_int_of_AbGroup B) : A ≃g B :=
group_isomorphism_of_lm_isomorphism φ
end int
end left_module
|
7fac0d900dcb7ccb9ef6a8a2426661e6339e22e7 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/run/choiceMacroRules.lean | 78284b5035012b468265fb2cb6f7a30aef212c28 | [
"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 | 457 | lean | syntax:65 (name := myAdd1) term "+++" term:65 : term
syntax:65 (name := myAdd2) term "+++" term:65 : term
macro_rules (kind := myAdd1)
| `($a +++ $b) => `(Nat.add $a $b)
macro_rules (kind := myAdd2)
| `($a +++ $b) => `(Append.append $a $b)
#check (1:Nat) +++ 3
theorem tst1 : ((1:Nat) +++ 3) = 1 + 3 :=
rfl
#check fun (x : Nat) => if x +++ 3 = x then x else x + 1
#check [1, 2] +++ [3, 4]
theorem tst2 : ([1, 2] +++ [3, 4]) = [1, 2] ++ [3, 4] :=
rfl
|
0ea898a6c3fba554f3f321ecdfb3afc76b38f119 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/Lean3Lib/init/meta/smt/congruence_closure.lean | a8bb307d9de915402e751c414f267915fc99a2d8 | [] | 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 | 566 | lean | /-
Copyright (c) 2016 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.meta.interactive_base
import Mathlib.Lean3Lib.init.meta.tactic
import Mathlib.Lean3Lib.init.meta.set_get_option_tactics
universes l
namespace Mathlib
/- If tt, congruence closure will treat implicit instance arguments as constants. -/
structure cc_config
where
ignore_instances : Bool
ac : Bool
ho_fns : Option (List name)
em : Bool
|
d2d8f0b39ac2bdcbcaaa5e02e6db6222ab14bad2 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/data/equiv/encodable/basic.lean | d6f36f012ed9fe55b91658e61262bd730ba0c823 | [] | 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 | 15,532 | lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Mario Carneiro
Type class for encodable Types.
Note that every encodable Type is countable.
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.equiv.nat
import Mathlib.order.rel_iso
import Mathlib.order.directed
import Mathlib.PostPort
universes u_1 l u_2 u_3
namespace Mathlib
/-- An encodable type is a "constructively countable" type. This is where
we have an explicit injection `encode : α → nat` and a partial inverse
`decode : nat → option α`. This makes the range of `encode` decidable,
although it is not decidable if `α` is finite or not. -/
class encodable (α : Type u_1)
where
encode : α → ℕ
decode : ℕ → Option α
encodek : ∀ (a : α), decode (encode a) = some a
namespace encodable
theorem encode_injective {α : Type u_1} [encodable α] : function.injective encode := sorry
/- This is not set as an instance because this is usually not the best way
to infer decidability. -/
def decidable_eq_of_encodable (α : Type u_1) [encodable α] : DecidableEq α :=
sorry
def of_left_injection {α : Type u_1} {β : Type u_2} [encodable α] (f : β → α) (finv : α → Option β) (linv : ∀ (b : β), finv (f b) = some b) : encodable β :=
mk (fun (b : β) => encode (f b)) (fun (n : ℕ) => option.bind (decode α n) finv) sorry
def of_left_inverse {α : Type u_1} {β : Type u_2} [encodable α] (f : β → α) (finv : α → β) (linv : ∀ (b : β), finv (f b) = b) : encodable β :=
of_left_injection f (some ∘ finv) sorry
/-- If `α` is encodable and `β ≃ α`, then so is `β` -/
def of_equiv {β : Type u_2} (α : Type u_1) [encodable α] (e : β ≃ α) : encodable β :=
of_left_inverse (⇑e) (⇑(equiv.symm e)) (equiv.left_inv e)
@[simp] theorem encode_of_equiv {α : Type u_1} {β : Type u_2} [encodable α] (e : β ≃ α) (b : β) : encode b = encode (coe_fn e b) :=
rfl
@[simp] theorem decode_of_equiv {α : Type u_1} {β : Type u_2} [encodable α] (e : β ≃ α) (n : ℕ) : decode β n = option.map (⇑(equiv.symm e)) (decode α n) :=
rfl
protected instance nat : encodable ℕ :=
mk id some sorry
@[simp] theorem encode_nat (n : ℕ) : encode n = n :=
rfl
@[simp] theorem decode_nat (n : ℕ) : decode ℕ n = some n :=
rfl
protected instance empty : encodable empty :=
mk (fun (a : empty) => empty.rec (fun (a : empty) => ℕ) a) (fun (n : ℕ) => none) sorry
protected instance unit : encodable PUnit :=
mk (fun (_x : PUnit) => 0) (fun (n : ℕ) => nat.cases_on n (some PUnit.unit) fun (_x : ℕ) => none) sorry
@[simp] theorem encode_star : encode PUnit.unit = 0 :=
rfl
@[simp] theorem decode_unit_zero : decode PUnit 0 = some PUnit.unit :=
rfl
@[simp] theorem decode_unit_succ (n : ℕ) : decode PUnit (Nat.succ n) = none :=
rfl
protected instance option {α : Type u_1} [h : encodable α] : encodable (Option α) :=
mk (fun (o : Option α) => option.cases_on o 0 fun (a : α) => Nat.succ (encode a))
(fun (n : ℕ) => nat.cases_on n (some none) fun (m : ℕ) => option.map some (decode α m)) sorry
@[simp] theorem encode_none {α : Type u_1} [encodable α] : encode none = 0 :=
rfl
@[simp] theorem encode_some {α : Type u_1} [encodable α] (a : α) : encode (some a) = Nat.succ (encode a) :=
rfl
@[simp] theorem decode_option_zero {α : Type u_1} [encodable α] : decode (Option α) 0 = some none :=
rfl
@[simp] theorem decode_option_succ {α : Type u_1} [encodable α] (n : ℕ) : decode (Option α) (Nat.succ n) = option.map some (decode α n) :=
rfl
def decode2 (α : Type u_1) [encodable α] (n : ℕ) : Option α :=
option.bind (decode α n) (option.guard fun (a : α) => encode a = n)
theorem mem_decode2' {α : Type u_1} [encodable α] {n : ℕ} {a : α} : a ∈ decode2 α n ↔ a ∈ decode α n ∧ encode a = n := sorry
theorem mem_decode2 {α : Type u_1} [encodable α] {n : ℕ} {a : α} : a ∈ decode2 α n ↔ encode a = n :=
iff.trans mem_decode2' (and_iff_right_of_imp fun (e : encode a = n) => e ▸ encodek a)
theorem decode2_is_partial_inv {α : Type u_1} [encodable α] : function.is_partial_inv encode (decode2 α) :=
fun (a : α) (n : ℕ) => mem_decode2
theorem decode2_inj {α : Type u_1} [encodable α] {n : ℕ} {a₁ : α} {a₂ : α} (h₁ : a₁ ∈ decode2 α n) (h₂ : a₂ ∈ decode2 α n) : a₁ = a₂ :=
encode_injective (Eq.trans (iff.mp mem_decode2 h₁) (Eq.symm (iff.mp mem_decode2 h₂)))
theorem encodek2 {α : Type u_1} [encodable α] (a : α) : decode2 α (encode a) = some a :=
iff.mpr mem_decode2 rfl
def decidable_range_encode (α : Type u_1) [encodable α] : decidable_pred (set.range encode) :=
fun (x : ℕ) => decidable_of_iff ↥(option.is_some (decode2 α x)) sorry
def equiv_range_encode (α : Type u_1) [encodable α] : α ≃ ↥(set.range encode) :=
equiv.mk (fun (a : α) => { val := encode a, property := sorry }) (fun (n : ↥(set.range encode)) => option.get sorry)
sorry sorry
def encode_sum {α : Type u_1} {β : Type u_2} [encodable α] [encodable β] : α ⊕ β → ℕ :=
sorry
def decode_sum {α : Type u_1} {β : Type u_2} [encodable α] [encodable β] (n : ℕ) : Option (α ⊕ β) :=
sorry
protected instance sum {α : Type u_1} {β : Type u_2} [encodable α] [encodable β] : encodable (α ⊕ β) :=
mk encode_sum decode_sum sorry
@[simp] theorem encode_inl {α : Type u_1} {β : Type u_2} [encodable α] [encodable β] (a : α) : encode (sum.inl a) = bit0 (encode a) :=
rfl
@[simp] theorem encode_inr {α : Type u_1} {β : Type u_2} [encodable α] [encodable β] (b : β) : encode (sum.inr b) = bit1 (encode b) :=
rfl
@[simp] theorem decode_sum_val {α : Type u_1} {β : Type u_2} [encodable α] [encodable β] (n : ℕ) : decode (α ⊕ β) n = decode_sum n :=
rfl
protected instance bool : encodable Bool :=
of_equiv (Unit ⊕ Unit) equiv.bool_equiv_punit_sum_punit
@[simp] theorem encode_tt : encode tt = 1 :=
rfl
@[simp] theorem encode_ff : encode false = 0 :=
rfl
@[simp] theorem decode_zero : decode Bool 0 = some false :=
rfl
@[simp] theorem decode_one : decode Bool 1 = some tt :=
rfl
theorem decode_ge_two (n : ℕ) (h : bit0 1 ≤ n) : decode Bool n = none := sorry
def encode_sigma {α : Type u_1} {γ : α → Type u_3} [encodable α] [(a : α) → encodable (γ a)] : sigma γ → ℕ :=
sorry
def decode_sigma {α : Type u_1} {γ : α → Type u_3} [encodable α] [(a : α) → encodable (γ a)] (n : ℕ) : Option (sigma γ) :=
sorry
protected instance sigma {α : Type u_1} {γ : α → Type u_3} [encodable α] [(a : α) → encodable (γ a)] : encodable (sigma γ) :=
mk encode_sigma decode_sigma sorry
@[simp] theorem decode_sigma_val {α : Type u_1} {γ : α → Type u_3} [encodable α] [(a : α) → encodable (γ a)] (n : ℕ) : decode (sigma γ) n =
option.bind (decode α (prod.fst (nat.unpair n)))
fun (a : α) => option.map (sigma.mk a) (decode (γ a) (prod.snd (nat.unpair n))) := sorry
@[simp] theorem encode_sigma_val {α : Type u_1} {γ : α → Type u_3} [encodable α] [(a : α) → encodable (γ a)] (a : α) (b : γ a) : encode (sigma.mk a b) = nat.mkpair (encode a) (encode b) :=
rfl
protected instance prod {α : Type u_1} {β : Type u_2} [encodable α] [encodable β] : encodable (α × β) :=
of_equiv (sigma fun (_x : α) => β) (equiv.symm (equiv.sigma_equiv_prod α β))
@[simp] theorem decode_prod_val {α : Type u_1} {β : Type u_2} [encodable α] [encodable β] (n : ℕ) : decode (α × β) n =
option.bind (decode α (prod.fst (nat.unpair n)))
fun (a : α) => option.map (Prod.mk a) (decode β (prod.snd (nat.unpair n))) := sorry
@[simp] theorem encode_prod_val {α : Type u_1} {β : Type u_2} [encodable α] [encodable β] (a : α) (b : β) : encode (a, b) = nat.mkpair (encode a) (encode b) :=
rfl
def encode_subtype {α : Type u_1} {P : α → Prop} [encA : encodable α] : (Subtype fun (a : α) => P a) → ℕ :=
sorry
def decode_subtype {α : Type u_1} {P : α → Prop} [encA : encodable α] [decP : decidable_pred P] (v : ℕ) : Option (Subtype fun (a : α) => P a) :=
option.bind (decode α v)
fun (a : α) => dite (P a) (fun (h : P a) => some { val := a, property := h }) fun (h : ¬P a) => none
protected instance subtype {α : Type u_1} {P : α → Prop} [encA : encodable α] [decP : decidable_pred P] : encodable (Subtype fun (a : α) => P a) :=
mk encode_subtype decode_subtype sorry
theorem subtype.encode_eq {α : Type u_1} {P : α → Prop} [encA : encodable α] [decP : decidable_pred P] (a : Subtype P) : encode a = encode (subtype.val a) :=
subtype.cases_on a fun (a_val : α) (a_property : P a_val) => Eq.refl (encode { val := a_val, property := a_property })
protected instance fin (n : ℕ) : encodable (fin n) :=
of_equiv (Subtype fun (m : ℕ) => m < n) (equiv.fin_equiv_subtype n)
protected instance int : encodable ℤ :=
of_equiv ℕ equiv.int_equiv_nat
protected instance ulift {α : Type u_1} [encodable α] : encodable (ulift α) :=
of_equiv α equiv.ulift
protected instance plift {α : Type u_1} [encodable α] : encodable (plift α) :=
of_equiv α equiv.plift
def of_inj {α : Type u_1} {β : Type u_2} [encodable β] (f : α → β) (hf : function.injective f) : encodable α :=
of_left_injection f (function.partial_inv f) sorry
end encodable
/--
`ulower α : Type 0` is an equivalent type in the lowest universe, given `encodable α`.
-/
def ulower (α : Type u_1) [encodable α] :=
↥(set.range encodable.encode)
namespace ulower
/--
The equivalence between the encodable type `α` and `ulower α : Type 0`.
-/
def equiv (α : Type u_1) [encodable α] : α ≃ ulower α :=
encodable.equiv_range_encode α
/--
Lowers an `a : α` into `ulower α`.
-/
def down {α : Type u_1} [encodable α] (a : α) : ulower α :=
coe_fn (equiv α) a
protected instance inhabited {α : Type u_1} [encodable α] [Inhabited α] : Inhabited (ulower α) :=
{ default := down Inhabited.default }
/--
Lifts an `a : ulower α` into `α`.
-/
def up {α : Type u_1} [encodable α] (a : ulower α) : α :=
coe_fn (equiv.symm (equiv α)) a
@[simp] theorem down_up {α : Type u_1} [encodable α] {a : ulower α} : down (up a) = a :=
equiv.right_inv (equiv α) a
@[simp] theorem up_down {α : Type u_1} [encodable α] {a : α} : up (down a) = a :=
equiv.left_inv (equiv α) a
@[simp] theorem up_eq_up {α : Type u_1} [encodable α] {a : ulower α} {b : ulower α} : up a = up b ↔ a = b :=
equiv.apply_eq_iff_eq (equiv.symm (equiv α))
@[simp] theorem down_eq_down {α : Type u_1} [encodable α] {a : α} {b : α} : down a = down b ↔ a = b :=
equiv.apply_eq_iff_eq (equiv α)
protected theorem ext {α : Type u_1} [encodable α] {a : ulower α} {b : ulower α} : up a = up b → a = b :=
iff.mp up_eq_up
end ulower
/-
Choice function for encodable types and decidable predicates.
We provide the following API
choose {α : Type*} {p : α → Prop} [c : encodable α] [d : decidable_pred p] : (∃ x, p x) → α :=
choose_spec {α : Type*} {p : α → Prop} [c : encodable α] [d : decidable_pred p] (ex : ∃ x, p x) :
p (choose ex) :=
-/
namespace encodable
def choose_x {α : Type u_1} {p : α → Prop} [encodable α] [decidable_pred p] (h : ∃ (x : α), p x) : Subtype fun (x : α) => p x :=
(fun (this : ∃ (n : ℕ), good p (decode α n)) => sorry) sorry
def choose {α : Type u_1} {p : α → Prop} [encodable α] [decidable_pred p] (h : ∃ (x : α), p x) : α :=
subtype.val (choose_x h)
theorem choose_spec {α : Type u_1} {p : α → Prop} [encodable α] [decidable_pred p] (h : ∃ (x : α), p x) : p (choose h) :=
subtype.property (choose_x h)
theorem axiom_of_choice {α : Type u_1} {β : α → Type u_2} {R : (x : α) → β x → Prop} [(a : α) → encodable (β a)] [(x : α) → (y : β x) → Decidable (R x y)] (H : ∀ (x : α), ∃ (y : β x), R x y) : ∃ (f : (a : α) → β a), ∀ (x : α), R x (f x) :=
Exists.intro (fun (x : α) => choose (H x)) fun (x : α) => choose_spec (H x)
theorem skolem {α : Type u_1} {β : α → Type u_2} {P : (x : α) → β x → Prop} [c : (a : α) → encodable (β a)] [d : (x : α) → (y : β x) → Decidable (P x y)] : (∀ (x : α), ∃ (y : β x), P x y) ↔ ∃ (f : (a : α) → β a), ∀ (x : α), P x (f x) := sorry
/-
There is a total ordering on the elements of an encodable type, induced by the map to ℕ.
-/
/-- The `encode` function, viewed as an embedding. -/
def encode' (α : Type u_1) [encodable α] : α ↪ ℕ :=
function.embedding.mk encode encode_injective
protected instance order.preimage.is_trans {α : Type u_1} [encodable α] : is_trans α (⇑(encode' α) ⁻¹'o LessEq) :=
rel_embedding.is_trans (rel_embedding.preimage (encode' α) LessEq)
protected instance order.preimage.is_antisymm {α : Type u_1} [encodable α] : is_antisymm α (⇑(encode' α) ⁻¹'o LessEq) :=
rel_embedding.is_antisymm (rel_embedding.preimage (encode' α) LessEq)
protected instance order.preimage.is_total {α : Type u_1} [encodable α] : is_total α (⇑(encode' α) ⁻¹'o LessEq) :=
rel_embedding.is_total (rel_embedding.preimage (encode' α) LessEq)
end encodable
namespace directed
/-- Given a `directed r` function `f : α → β` defined on an encodable inhabited type,
construct a noncomputable sequence such that `r (f (x n)) (f (x (n + 1)))`
and `r (f a) (f (x (encode a + 1))`. -/
protected def sequence {α : Type u_1} {β : Type u_2} [encodable α] [Inhabited α] {r : β → β → Prop} (f : α → β) (hf : directed r f) : ℕ → α :=
sorry
theorem sequence_mono_nat {α : Type u_1} {β : Type u_2} [encodable α] [Inhabited α] {r : β → β → Prop} {f : α → β} (hf : directed r f) (n : ℕ) : r (f (directed.sequence f hf n)) (f (directed.sequence f hf (n + 1))) := sorry
theorem rel_sequence {α : Type u_1} {β : Type u_2} [encodable α] [Inhabited α] {r : β → β → Prop} {f : α → β} (hf : directed r f) (a : α) : r (f a) (f (directed.sequence f hf (encodable.encode a + 1))) := sorry
theorem sequence_mono {α : Type u_1} {β : Type u_2} [encodable α] [Inhabited α] [preorder β] {f : α → β} (hf : directed LessEq f) : monotone (f ∘ directed.sequence f hf) :=
monotone_of_monotone_nat (sequence_mono_nat hf)
theorem le_sequence {α : Type u_1} {β : Type u_2} [encodable α] [Inhabited α] [preorder β] {f : α → β} (hf : directed LessEq f) (a : α) : f a ≤ f (directed.sequence f hf (encodable.encode a + 1)) :=
rel_sequence hf a
end directed
/-- Representative of an equivalence class. This is a computable version of `quot.out` for a setoid
on an encodable type. -/
def quotient.rep {α : Type u_1} {s : setoid α} [DecidableRel has_equiv.equiv] [encodable α] (q : quotient s) : α :=
encodable.choose (quotient.exists_rep q)
theorem quotient.rep_spec {α : Type u_1} {s : setoid α} [DecidableRel has_equiv.equiv] [encodable α] (q : quotient s) : quotient.mk (quotient.rep q) = q :=
encodable.choose_spec (quotient.exists_rep q)
/-- The quotient of an encodable space by a decidable equivalence relation is encodable. -/
def encodable_quotient {α : Type u_1} {s : setoid α} [DecidableRel has_equiv.equiv] [encodable α] : encodable (quotient s) :=
encodable.mk (fun (q : quotient s) => encodable.encode (quotient.rep q))
(fun (n : ℕ) => quotient.mk <$> encodable.decode α n) sorry
|
afd79873ef223bfdf0364be37341394c483300fb | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/order/filter/cofinite.lean | 2ef81659503488bec1645b1e862b01400ca2453e | [
"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 | 2,811 | 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, Yury Kudryashov
-/
import order.filter.at_top_bot
/-!
# The cofinite filter
In this file we define
`cofinite`: the filter of sets with finite complement
and prove its basic properties. In particular, we prove that for `ℕ` it is equal to `at_top`.
## TODO
Define filters for other cardinalities of the complement.
-/
open set
open_locale classical
variables {α : Type*}
namespace filter
/-- The cofinite filter is the filter of subsets whose complements are finite. -/
def cofinite : filter α :=
{ sets := {s | finite sᶜ},
univ_sets := by simp only [compl_univ, finite_empty, mem_set_of_eq],
sets_of_superset := assume s t (hs : finite sᶜ) (st: s ⊆ t),
hs.subset $ compl_subset_compl.2 st,
inter_sets := assume s t (hs : finite sᶜ) (ht : finite (tᶜ)),
by simp only [compl_inter, finite.union, ht, hs, mem_set_of_eq] }
@[simp] lemma mem_cofinite {s : set α} : s ∈ (@cofinite α) ↔ finite sᶜ := iff.rfl
instance cofinite_ne_bot [infinite α] : ne_bot (@cofinite α) :=
mt empty_in_sets_eq_bot.mpr $ by { simp only [mem_cofinite, compl_empty], exact infinite_univ }
lemma frequently_cofinite_iff_infinite {p : α → Prop} :
(∃ᶠ x in cofinite, p x) ↔ set.infinite {x | p x} :=
by simp only [filter.frequently, filter.eventually, mem_cofinite, compl_set_of, not_not,
set.infinite]
end filter
open filter
lemma set.finite.compl_mem_cofinite {s : set α} (hs : s.finite) : sᶜ ∈ (@cofinite α) :=
mem_cofinite.2 $ (compl_compl s).symm ▸ hs
lemma set.finite.eventually_cofinite_nmem {s : set α} (hs : s.finite) : ∀ᶠ x in cofinite, x ∉ s :=
hs.compl_mem_cofinite
lemma finset.eventually_cofinite_nmem (s : finset α) : ∀ᶠ x in cofinite, x ∉ s :=
s.finite_to_set.eventually_cofinite_nmem
lemma set.infinite_iff_frequently_cofinite {s : set α} :
set.infinite s ↔ (∃ᶠ x in cofinite, x ∈ s) :=
frequently_cofinite_iff_infinite.symm
lemma filter.eventually_cofinite_ne (x : α) : ∀ᶠ a in cofinite, a ≠ x :=
(set.finite_singleton x).eventually_cofinite_nmem
/-- For natural numbers the filters `cofinite` and `at_top` coincide. -/
lemma nat.cofinite_eq_at_top : @cofinite ℕ = at_top :=
begin
ext s,
simp only [mem_cofinite, mem_at_top_sets],
split,
{ assume hs,
use (hs.to_finset.sup id) + 1,
assume b hb,
by_contradiction hbs,
have := hs.to_finset.subset_range_sup_succ (finite.mem_to_finset.2 hbs),
exact not_lt_of_le hb (finset.mem_range.1 this) },
{ rintros ⟨N, hN⟩,
apply (finite_lt_nat N).subset,
assume n hn,
change n < N,
exact lt_of_not_ge (λ hn', hn $ hN n hn') }
end
|
4a7d90c4539c14a04cad81a291fb5fb1c2b5bb0b | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/Lean3Lib/init/meta/derive.lean | 42879132647bbc1ac0fea25945da471ab69fefb2 | [] | 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 | 796 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sebastian Ullrich
Attribute that can automatically derive typeclass instances.
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.meta.attribute
import Mathlib.Lean3Lib.init.meta.interactive_base
import Mathlib.Lean3Lib.init.meta.mk_has_reflect_instance
import Mathlib.Lean3Lib.init.meta.mk_has_sizeof_instance
import Mathlib.Lean3Lib.init.meta.mk_inhabited_instance
namespace Mathlib
/-- A handler that may or may not be able to implement the typeclass `cls` for `decl`.
It should return `tt` if it was able to derive `cls` and `ff` if it does not know
how to derive `cls`, in which case lower-priority handlers will be tried next. -/
|
26420dd10cd6d28cab3f1df62b8af79883b9dc4f | 556aeb81a103e9e0ac4e1fe0ce1bc6e6161c3c5e | /src/starkware/cairo/common/cairo_secp/elliptic_curves.lean | a921869b62dea1107ad21e1c4c7844d5fcc0227a | [] | permissive | starkware-libs/formal-proofs | d6b731604461bf99e6ba820e68acca62a21709e8 | f5fa4ba6a471357fd171175183203d0b437f6527 | refs/heads/master | 1,691,085,444,753 | 1,690,507,386,000 | 1,690,507,386,000 | 410,476,629 | 32 | 9 | Apache-2.0 | 1,690,506,773,000 | 1,632,639,790,000 | Lean | UTF-8 | Lean | false | false | 4,896 | lean | import data.zmod.algebra
import tactic
class ec_field (F : Type*) extends field F :=
(deceqF : decidable_eq F)
(two_ne_zero : (2 : F) ≠ 0)
instance ec_field.decidable_eq (F : Type*) [h : ec_field F] : decidable_eq F :=
h.deceqF
def on_ec {F : Type*} [field F] (p : F × F) := p.2^2 = p.1^3 + 7
lemma eq_or_eq_neg_of_eq_of_on_ec {F : Type*} [field F] {x1 y1 x2 y2 : F}
(hp : on_ec (x1, y1)) (hq : on_ec (x2, y2)) (heq : x1 = x2 ) :
y1 = y2 ∨ y1 = -y2 :=
begin
have : y1^2 = y2^2,
{ apply hp.trans, symmetry, apply hq.trans, dsimp, rw heq },
apply eq_or_eq_neg_of_sq_eq_sq _ _ this
end
lemma eq_of_on_ec {F : Type*} [field F] {x1 y1 x2 y2 : F}
(hp : on_ec (x1, y1)) (hq : on_ec (x2, y2)) (heq : x1 = x2 ) (hne : y1 ≠ -y2) :
y1 = y2 :=
or.resolve_right (eq_or_eq_neg_of_eq_of_on_ec hp hq heq) hne
@[ext] structure AffineECPoint (F : Type*) [field F] :=
(x y : F) (h : on_ec (x, y))
inductive ECPoint (F : Type*) [field F]
| ZeroPoint : ECPoint
| AffinePoint : AffineECPoint F → ECPoint
variables {F : Type*} [ec_field F]
def ec_double_slope (p : F × F) := 3 * p.1^2 / (2 * p.2)
def ec_double (p : F × F) :=
let slope := ec_double_slope p,
new_x := slope^2 - 2 * p.1 in
(new_x, (p.1 - new_x) * slope - p.2)
theorem on_ec_ec_double {p : F × F} (h : on_ec p) (h' : p.2 ≠ 0) :
on_ec (ec_double p) :=
begin
cases p with x y,
have pow_three : ∀ z : F, z^3 = z * z * z, { intro z, simp [pow_succ, mul_assoc] },
simp [on_ec, ec_double, ec_double_slope] at *,
apply eq_of_sub_eq_zero,
field_simp [ec_field.two_ne_zero, h, h', pow_two, pow_three], ring_nf,
simp [h], ring_nf,
end
def ec_add_slope (p1 p2 : F × F) := (p2.2 - p1.2) / (p2.1 - p1.1)
def ec_add (p1 p2 : F × F) : F × F :=
let slope := ec_add_slope p1 p2,
new_x := slope^2 - (p1.1 + p2.1) in
(new_x, (p1.1 - new_x) * slope - p1.2)
theorem on_ec_ec_add {p1 p2 : F × F} (h1 : on_ec p1) (h2 : on_ec p2) (h' : p1.1 ≠ p2.1) :
on_ec (ec_add p1 p2) :=
begin
cases p1 with x1 y1,
cases p2 with x2 y2,
have pow_three : ∀ z : F, z^3 = z * z^2, { intro z, simp [pow_succ, mul_assoc], },
have pow_four : ∀ z : F, z^4 = z^2 * z^2, { intro z, simp [pow_succ, mul_assoc], },
simp [on_ec, ec_add, ec_add_slope] at *,
have : x2 - x1 ≠ 0, { contrapose! h', symmetry, apply eq_of_sub_eq_zero h' },
apply eq_of_sub_eq_zero,
field_simp [ec_field.two_ne_zero, pow_two, pow_three, this], ring_nf,
simp [h1, h2, pow_three, pow_four], ring_nf,
simp [h1, h2], ring_nf
end
namespace ECPoint
protected def add : ECPoint F → ECPoint F → ECPoint F
| ZeroPoint b := b
| a ZeroPoint := a
| (AffinePoint a) (AffinePoint b) :=
if axbx: a.x = b.x then
if ayby: a.y = -b.y then
-- a = -b
ZeroPoint
else
have a.y = b.y, from eq_of_on_ec a.h b.h axbx ayby,
have a.y ≠ 0, by { contrapose! ayby, rw [←this, ayby, neg_zero] },
let p := ec_double (a.x, a.y) in
AffinePoint ⟨p.1, p.2, on_ec_ec_double a.h this⟩
else
let p := ec_add (a.x, a.y) (b.x, b.y) in
AffinePoint ⟨p.1, p.2, on_ec_ec_add a.h b.h axbx⟩
protected def neg : ECPoint F → ECPoint F
| ZeroPoint := ZeroPoint
| (AffinePoint a) := AffinePoint ⟨a.x, -a.y, by { simp [on_ec], exact a.h} ⟩
protected theorem add_left_neg (a : ECPoint F) : a.neg.add a = ZeroPoint :=
by { cases a; simp [ECPoint.neg, ECPoint.add] }
protected theorem add_comm (a b: ECPoint F) : a.add b = b.add a :=
begin
cases a; cases b; simp [ECPoint.add],
by_cases h : a.x = b.x,
{ rw [dif_pos h, dif_pos h.symm], simp [h],
by_cases h': a.y = -b.y,
{ rw [dif_pos h', dif_pos (eq_neg_of_eq_neg h')] },
rw [dif_neg h', dif_neg (λ h'', h' (eq_neg_of_eq_neg h''))],
simp [eq_of_on_ec a.h b.h h h'] },
rw [dif_neg h, dif_neg (ne.symm h)],
have : ec_add_slope (a.x, a.y) (b.x, b.y) = ec_add_slope (b.x, b.y) (a.x, a.y),
{ simp [ec_add_slope], rw [←neg_sub b.y a.y, ←neg_sub b.x a.x, neg_div_neg_eq] },
congr' 1, ext; dsimp [ec_add]; simp [add_comm],
{ simp [this] },
have h1 : a.x - b.x ≠ 0, { rwa [ne, sub_eq_zero] },
have h2 : b.x - a.x ≠ 0, { rw [ne, sub_eq_zero], exact ne.symm h },
field_simp [ec_add_slope, h1, h2, pow_two],
ring_nf
end
end ECPoint
instance : add_comm_group (ECPoint F) :=
{ add := ECPoint.add,
neg := ECPoint.neg,
zero := ECPoint.ZeroPoint,
add_assoc := sorry,
zero_add := by { intro a, cases a; simp [ECPoint.add] },
add_zero := by { intro a, cases a; simp [ECPoint.add] },
add_left_neg := ECPoint.add_left_neg,
add_comm := ECPoint.add_comm }
theorem ECPoint.add_def (x y : ECPoint F) : x + y = ECPoint.add x y := rfl
theorem ECPoint.neg_def (x : ECPoint F) : - x = ECPoint.neg x := rfl
theorem ECPoint.zero_def : (0 : ECPoint F) = ECPoint.ZeroPoint := rfl
|
758dd71c802dc58266d94ad0b839dd621f4cc679 | bb31430994044506fa42fd667e2d556327e18dfe | /src/measure_theory/group/add_circle.lean | 93388f293ccf4c6c97449de3ff92bd9a62929a3a | [
"Apache-2.0"
] | permissive | sgouezel/mathlib | 0cb4e5335a2ba189fa7af96d83a377f83270e503 | 00638177efd1b2534fc5269363ebf42a7871df9a | refs/heads/master | 1,674,527,483,042 | 1,673,665,568,000 | 1,673,665,568,000 | 119,598,202 | 0 | 0 | null | 1,517,348,647,000 | 1,517,348,646,000 | null | UTF-8 | Lean | false | false | 5,356 | lean | /-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import measure_theory.integral.periodic
import data.zmod.quotient
/-!
# Measure-theoretic results about the additive circle
The file is a place to collect measure-theoretic results about the additive circle.
## Main definitions:
* `add_circle.closed_ball_ae_eq_ball`: open and closed balls in the additive circle are almost
equal
* `add_circle.is_add_fundamental_domain_of_ae_ball`: a ball is a fundamental domain for rational
angle rotation in the additive circle
-/
open set function filter measure_theory measure_theory.measure metric
open_locale measure_theory pointwise big_operators topological_space ennreal
namespace add_circle
variables {T : ℝ} [hT : fact (0 < T)]
include hT
lemma closed_ball_ae_eq_ball {x : add_circle T} {ε : ℝ} :
closed_ball x ε =ᵐ[volume] ball x ε :=
begin
cases le_or_lt ε 0 with hε hε,
{ rw [ball_eq_empty.mpr hε, ae_eq_empty, volume_closed_ball,
min_eq_right (by linarith [hT.out] : 2 * ε ≤ T), ennreal.of_real_eq_zero],
exact mul_nonpos_of_nonneg_of_nonpos zero_le_two hε, },
{ suffices : volume (closed_ball x ε) ≤ volume (ball x ε),
{ exact (ae_eq_of_subset_of_measure_ge ball_subset_closed_ball this measurable_set_ball
(measure_ne_top _ _)).symm, },
have : tendsto (λ δ, volume (closed_ball x δ)) (𝓝[<] ε) (𝓝 $ volume (closed_ball x ε)),
{ simp_rw volume_closed_ball,
refine ennreal.tendsto_of_real (tendsto.min tendsto_const_nhds $ tendsto.const_mul _ _),
convert (@monotone_id ℝ _).tendsto_nhds_within_Iio ε,
simp, },
refine le_of_tendsto this (mem_nhds_within_Iio_iff_exists_Ioo_subset.mpr ⟨0, hε, λ r hr, _⟩),
exact measure_mono (closed_ball_subset_ball hr.2), },
end
/-- Let `G` be the subgroup of `add_circle T` generated by a point `u` of finite order `n : ℕ`. Then
any set `I` that is almost equal to a ball of radius `T / 2n` is a fundamental domain for the action
of `G` on `add_circle T` by left addition. -/
lemma is_add_fundamental_domain_of_ae_ball (I : set $ add_circle T)
(u x : add_circle T) (hu : is_of_fin_add_order u)
(hI : I =ᵐ[volume] ball x (T / (2 * add_order_of u))) :
is_add_fundamental_domain (add_subgroup.zmultiples u) I :=
begin
set G := add_subgroup.zmultiples u,
set n := add_order_of u,
set B := ball x (T / (2 * n)),
have hn : 1 ≤ (n : ℝ), { norm_cast, linarith [add_order_of_pos' hu], },
refine is_add_fundamental_domain.mk_of_measure_univ_le _ _ _ _,
{ -- `null_measurable_set I volume`
exact (measurable_set_ball.null_measurable_set.congr hI.symm), },
{ -- `∀ (g : G), g ≠ 0 → ae_disjoint volume (g +ᵥ I) I`
rintros ⟨g, hg⟩ hg',
replace hg' : g ≠ 0, by simpa only [ne.def, add_subgroup.mk_eq_zero_iff] using hg',
change ae_disjoint volume (g +ᵥ I) I,
refine ae_disjoint.congr (disjoint.ae_disjoint _)
((quasi_measure_preserving_add_left volume (-g)).vadd_ae_eq_of_ae_eq g hI) hI,
have hBg : g +ᵥ B = ball (g + x) (T / (2 * n)),
{ rw [add_comm g x, ← singleton_add_ball _ x g, add_ball, thickening_singleton], },
rw hBg,
apply ball_disjoint_ball,
rw [dist_eq_norm, add_sub_cancel, div_mul_eq_div_div, ← add_div, ← add_div, add_self_div_two,
div_le_iff' (by positivity : 0 < (n : ℝ)), ← nsmul_eq_mul],
refine (le_add_order_smul_norm_of_is_of_fin_add_order (hu.of_mem_zmultiples hg) hg').trans
(nsmul_le_nsmul (norm_nonneg g) _),
exact nat.le_of_dvd (add_order_of_pos_iff.mpr hu) (add_order_of_dvd_of_mem_zmultiples hg), },
{ -- `∀ (g : G), quasi_measure_preserving (has_vadd.vadd g) volume volume`
exact (λ g, quasi_measure_preserving_add_left volume g), },
{ -- `volume univ ≤ ∑' (g : G), volume (g +ᵥ I)`
replace hI : I =ᵐ[volume] closed_ball x (T / (2 * ↑n)) :=
hI.trans closed_ball_ae_eq_ball.symm,
haveI : fintype G := @fintype.of_finite _ hu.finite_zmultiples,
have hG_card : (finset.univ : finset G).card = n,
{ show _ = add_order_of u, rw [add_order_eq_card_zmultiples', nat.card_eq_fintype_card], refl },
simp_rw [measure_vadd],
rw [add_circle.measure_univ, tsum_fintype, finset.sum_const, measure_congr hI,
volume_closed_ball, ← ennreal.of_real_nsmul, mul_div, mul_div_mul_comm,
div_self (@two_ne_zero ℝ _ _ _ _), one_mul, min_eq_right (div_le_self hT.out.le hn), hG_card,
nsmul_eq_mul, mul_div_cancel' T (lt_of_lt_of_le zero_lt_one hn).ne.symm],
exact le_refl _, },
end
lemma volume_of_add_preimage_eq (s I : set $ add_circle T) (u x : add_circle T)
(hu : is_of_fin_add_order u) (hs : (u +ᵥ s : set $ add_circle T) =ᵐ[volume] s)
(hI : I =ᵐ[volume] ball x (T / (2 * add_order_of u))) :
volume s = add_order_of u • volume (s ∩ I) :=
begin
let G := add_subgroup.zmultiples u,
haveI : fintype G := @fintype.of_finite _ hu.finite_zmultiples,
have hsG : ∀ (g : G), (g +ᵥ s : set $ add_circle T) =ᵐ[volume] s,
{ rintros ⟨y, hy⟩, exact (vadd_ae_eq_self_of_mem_zmultiples hs hy : _), },
rw [(is_add_fundamental_domain_of_ae_ball I u x hu hI).measure_eq_card_smul_of_vadd_ae_eq_self
s hsG, add_order_eq_card_zmultiples' u, nat.card_eq_fintype_card],
end
end add_circle
|
347edc671c675554108802ac4d1e8ad5a5841078 | b7f22e51856f4989b970961f794f1c435f9b8f78 | /tests/lean/hott/580.hlean | e48e83b3a3d5d4ccb62558f9cdeae2e79b6604ba | [
"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 | 359 | hlean | section
parameters (A : Type) (a b z : A) (add : A → A → A) (le : A → A → Type₀)
local notation 0 := z
local infix + := add
local infix ≤ := le
definition add_zero (x : A) : x + 0 = x := sorry
set_option pp.universes true
definition le_add_of_nonneg_right (H : a + 0 ≤ a + b) : a ≤ a + b :=
begin
rewrite add_zero at H,
exact H
end
end
|
18705930326df3884fb90b5ad21c35f421da8ae4 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/analysis/convex/extrema.lean | 7cdb67adfffcf57ce208f83026c44b068eb815fc | [
"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 | 5,441 | lean | /-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import analysis.calculus.local_extr
import topology.algebra.affine
/-!
# Minima and maxima of convex functions
We show that if a function `f : E → β` is convex, then a local minimum is also
a global minimum, and likewise for concave functions.
-/
variables {E β: Type*} [add_comm_group E] [topological_space E]
[module ℝ E] [topological_add_group E] [topological_vector_space ℝ E]
[linear_ordered_add_comm_group β] [semimodule ℝ β] [ordered_semimodule ℝ β]
{s : set E}
open set filter
open_locale classical
/--
Helper lemma for the more general case: `is_min_on.of_is_local_min_on_of_convex_on`.
-/
lemma is_min_on.of_is_local_min_on_of_convex_on_Icc {f : ℝ → β} {a b : ℝ}
(a_lt_b : a < b) (h_local_min : is_local_min_on f (Icc a b) a) (h_conv : convex_on (Icc a b) f) :
∀ x ∈ Icc a b, f a ≤ f x :=
begin
by_contradiction H_cont,
push_neg at H_cont,
rcases H_cont with ⟨x, ⟨h_ax, h_xb⟩, fx_lt_fa⟩,
obtain ⟨z, hz, ge_on_nhd⟩ : ∃ z > a, ∀ y ∈ (Icc a z), f y ≥ f a,
{ rcases eventually_iff_exists_mem.mp h_local_min with ⟨U, U_in_nhds_within, fy_ge_fa⟩,
rw [nhds_within_Icc_eq_nhds_within_Ici a_lt_b, mem_nhds_within_Ici_iff_exists_Icc_subset]
at U_in_nhds_within,
rcases U_in_nhds_within with ⟨ε, ε_in_Ioi, Ioc_in_U⟩,
exact ⟨ε, mem_Ioi.mp ε_in_Ioi, λ y y_in_Ioc, fy_ge_fa y $ Ioc_in_U y_in_Ioc⟩ },
have a_lt_x : a < x := lt_of_le_of_ne h_ax (λ H, by subst H; exact lt_irrefl (f a) fx_lt_fa),
have lt_on_nhd : ∀ y ∈ Ioc a x, f y < f a,
{ intros y y_in_Ioc,
rcases (convex.mem_Ioc a_lt_x).mp y_in_Ioc with ⟨ya, yx, ya_pos, yx_pos, yax, y_combo⟩,
calc
f y = f (ya * a + yx * x) : by rw [y_combo]
... ≤ ya • f a + yx • f x
: h_conv.2 (left_mem_Icc.mpr (le_of_lt a_lt_b)) ⟨h_ax, h_xb⟩ (ya_pos)
(le_of_lt yx_pos) yax
... < ya • f a + yx • f a : add_lt_add_left (smul_lt_smul_of_pos fx_lt_fa yx_pos) (ya • f a)
... = f a : by rw [←add_smul, yax, one_smul] },
by_cases h_xz : x ≤ z,
{ exact not_lt_of_ge (ge_on_nhd x (show x ∈ Icc a z, by exact ⟨h_ax, h_xz⟩)) fx_lt_fa, },
{ have h₁ : z ∈ Ioc a x := ⟨hz, le_of_not_ge h_xz⟩,
have h₂ : z ∈ Icc a z := ⟨le_of_lt hz, le_refl z⟩,
exact not_lt_of_ge (ge_on_nhd z h₂) (lt_on_nhd z h₁) }
end
/--
A local minimum of a convex function is a global minimum, restricted to a set `s`.
-/
lemma is_min_on.of_is_local_min_on_of_convex_on {f : E → β} {a : E}
(a_in_s : a ∈ s) (h_localmin: is_local_min_on f s a) (h_conv : convex_on s f) :
∀ x ∈ s, f a ≤ f x :=
begin
by_contradiction H_cont,
push_neg at H_cont,
rcases H_cont with ⟨x, ⟨x_in_s, fx_lt_fa⟩⟩,
let g : ℝ →ᵃ[ℝ] E := affine_map.line_map a x,
have hg0 : g 0 = a := affine_map.line_map_apply_zero a x,
have hg1 : g 1 = x := affine_map.line_map_apply_one a x,
have fg_local_min_on : is_local_min_on (f ∘ g) (g ⁻¹' s) 0,
{ rw ←hg0 at h_localmin,
refine is_local_min_on.comp_continuous_on h_localmin subset.rfl
(continuous.continuous_on (affine_map.line_map_continuous)) _,
simp [mem_preimage, hg0, a_in_s] },
have fg_min_on : ∀ x ∈ (Icc 0 1 : set ℝ), (f ∘ g) 0 ≤ (f ∘ g) x,
{ have Icc_in_s' : Icc 0 1 ⊆ (g ⁻¹' s),
{ have h0 : (0 : ℝ) ∈ (g ⁻¹' s) := by simp [mem_preimage, a_in_s],
have h1 : (1 : ℝ) ∈ (g ⁻¹' s) := by simp [mem_preimage, hg1, x_in_s],
rw ←segment_eq_Icc (show (0 : ℝ) ≤ 1, by linarith),
exact (convex.affine_preimage g h_conv.1).segment_subset
(by simp [mem_preimage, hg0, a_in_s]) (by simp [mem_preimage, hg1, x_in_s]) },
have fg_local_min_on' : is_local_min_on (f ∘ g) (Icc 0 1) 0 :=
is_local_min_on.on_subset fg_local_min_on Icc_in_s',
refine is_min_on.of_is_local_min_on_of_convex_on_Icc (by linarith) fg_local_min_on' _,
exact (convex_on.comp_affine_map g h_conv).subset Icc_in_s' (convex_Icc 0 1) },
have gx_lt_ga : (f ∘ g) 1 < (f ∘ g) 0 := by simp [hg1, fx_lt_fa, hg0],
exact not_lt_of_ge (fg_min_on 1 (mem_Icc.mpr ⟨zero_le_one, le_refl 1⟩)) gx_lt_ga,
end
/-- A local maximum of a concave function is a global maximum, restricted to a set `s`. -/
lemma is_max_on.of_is_local_max_on_of_concave_on {f : E → β} {a : E}
(a_in_s : a ∈ s) (h_localmax: is_local_max_on f s a) (h_conc : concave_on s f) :
∀ x ∈ s, f x ≤ f a :=
@is_min_on.of_is_local_min_on_of_convex_on _ (order_dual β) _ _ _ _ _ _ _ _ s f a a_in_s h_localmax h_conc
/-- A local minimum of a convex function is a global minimum. -/
lemma is_min_on.of_is_local_min_of_convex_univ {f : E → β} {a : E}
(h_local_min : is_local_min f a) (h_conv : convex_on univ f) : ∀ x, f a ≤ f x :=
λ x, (is_min_on.of_is_local_min_on_of_convex_on (mem_univ a)
(is_local_min.on h_local_min univ) h_conv) x (mem_univ x)
/-- A local maximum of a concave function is a global maximum. -/
lemma is_max_on.of_is_local_max_of_convex_univ {f : E → β} {a : E}
(h_local_max : is_local_max f a) (h_conc : concave_on univ f) : ∀ x, f x ≤ f a :=
@is_min_on.of_is_local_min_of_convex_univ _ (order_dual β) _ _ _ _ _ _ _ _ f a h_local_max h_conc
|
6188e999fc2d585b5a5817bed90872228571b10d | 7fc0ec5c526f706107537a6e2e34ac4cdf88d232 | /src/linear_algebra/monoid_star_algebra.lean | 3e805bf0eee8498cedca52a6af61ce394e2ff2e4 | [
"Apache-2.0"
] | permissive | fpvandoorn/group-representations | 7440a81f2ac9e0d2defa44dc1643c3167f7a2d99 | bd9d72311749187d3bd4f542d5eab83e8341856c | refs/heads/master | 1,684,128,141,684 | 1,623,338,135,000 | 1,623,338,135,000 | 256,117,709 | 0 | 2 | null | null | null | null | UTF-8 | Lean | false | false | 26,917 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury G. Kudryashov, Scott Morrison
-/
import data.finsupp
import ring_theory.algebra
import linear_algebra.star_algebra
import topology.algebra.infinite_sum
import data.monoid_algebra
/-!
# Monoid star algebras
This is formally parallel to monoid_star_algebra, but instead of taking finite formal
combinations of the generators, we take L^1 combinations.
We take the measure to be the sum over all elements. This makes the convolution product
definable by restricting the sum ∑_x,y f(x) g(y) to x*y = a.
Ideally we would take the definitions from monoid_algebra, interpreted formally, and
check the summability conditions. This requires generalizing finsupp to summable or some
stronger norm condition (we could also use ℓ_2).
NOTE: we are trying to use summable as defined in topology.algebra.infinite_sum but this is
NOT the same as ℓ_1 whose definition involves the sum of the norm of the elements.
So we would rather use l1_space but using the discreteness to define the convolution.
## Implementation note
Unfortunately because additive and multiplicative structures both appear in both cases,
it doesn't appear to be possible to make much use of `to_additive`, and we just settle for
saying everything twice.
Similarly, I attempted to just define `add_monoid_star_algebra k G := monoid_star_algebra k (multiplicative G)`,
but the definitional equality `multiplicative G = G` leaks through everywhere, and
seems impossible to use.
-/
noncomputable theory
open_locale classical
open finset finsupp
universes u₁ u₂ u₃
variables (k : Type u₁) (G : Type u₂)
section
variables [add_comm_monoid k] [topological_space k] [monoid G]
/--
The monoid algebra over a semiring `k` generated by the monoid `G`.
It is the type of finite formal `k`-linear combinations of terms of `G`,
endowed with the convolution product.
-/
-- this is not right but we want a similar class of summable functions.
@[derive [inhabited, add_comm_monoid]]
def monoid_star_algebra : Type (max u₁ u₂) := G →₀ k
end
namespace monoid_star_algebra
variables {k G}
local attribute [reducible] monoid_star_algebra
section
variables [add_comm_monoid k] [topological_space k] [monoid G]
/-- The product of `f g : monoid_star_algebra k G` is the summable function
whose value at `a` is the sum of `f x * g y` over all pairs `x, y`
such that `x * y = a`. (Think of the group ring of a group.) -/
def mul_def1 (f g : monoid_star_algebra k G) := (f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, single (a₁ * a₂) (b₁ * b₂))
#check mul_def1
lemma mul_convergence (f g : monoid_star_algebra k G) (hf : summable f) (hg : summable g) : summable (mul_def1 f g) :=
begin unfold summable at *, unfold mul_def1, rcases hf, rcases hg,
end
instance : has_mul (monoid_star_algebra k G) :=
⟨λf g, f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, single (a₁ * a₂) (b₁ * b₂)⟩
lemma mul_def {f g : monoid_star_algebra k G} :
f * g = (f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, single (a₁ * a₂) (b₁ * b₂)) :=
rfl
lemma mul_apply (f g : monoid_star_algebra k G) (x : G) :
(f * g) x = (f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, if a₁ * a₂ = x then b₁ * b₂ else 0) :=
begin
rw [mul_def],
simp only [finsupp.sum_apply, single_apply],
end
lemma mul_apply_antidiagonal (f g : monoid_star_algebra k G) (x : G) (s : finset (G × G))
(hs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x) :
(f * g) x = s.sum (λ p, f p.1 * g p.2) :=
let F : G × G → k := λ p, if p.1 * p.2 = x then f p.1 * g p.2 else 0 in
calc (f * g) x = (f.support.sum $ λ a₁, g.support.sum $ λ a₂, F (a₁, a₂)) :
mul_apply f g x
... = (f.support.product g.support).sum F : finset.sum_product.symm
... = ((f.support.product g.support).filter (λ p : G × G, p.1 * p.2 = x)).sum (λ p, f p.1 * g p.2) :
(finset.sum_filter _ _).symm
... = (s.filter (λ p : G × G, p.1 ∈ f.support ∧ p.2 ∈ g.support)).sum (λ p, f p.1 * g p.2) :
sum_congr (by { ext, simp [hs, and_comm] }) (λ _ _, rfl)
... = s.sum (λ p, f p.1 * g p.2) : sum_subset (filter_subset _) $ λ p hps hp,
begin
simp only [mem_filter, mem_support_iff, not_and, not_not] at hp ⊢,
by_cases h1 : f p.1 = 0,
{ rw [h1, zero_mul] },
{ rw [hp hps h1, mul_zero] }
end
end
section
variables [semiring k] [monoid G]
lemma support_mul (a b : monoid_star_algebra k G) :
(a * b).support ⊆ a.support.bind (λa₁, b.support.bind $ λa₂, {a₁ * a₂}) :=
subset.trans support_sum $ bind_mono $ assume a₁ _,
subset.trans support_sum $ bind_mono $ assume a₂ _, support_single_subset
/-- The unit of the multiplication is `single 1 1`, i.e. the function
that is `1` at `1` and zero elsewhere. -/
instance : has_one (monoid_star_algebra k G) :=
⟨single 1 1⟩
lemma one_def : (1 : monoid_star_algebra k G) = single 1 1 :=
rfl
-- TODO: the simplifier unfolds 0 in the instance proof!
protected lemma zero_mul (f : monoid_star_algebra k G) : 0 * f = 0 :=
by simp only [mul_def, sum_zero_index]
protected lemma mul_zero (f : monoid_star_algebra k G) : f * 0 = 0 :=
by simp only [mul_def, sum_zero_index, sum_zero]
private lemma left_distrib (a b c : monoid_star_algebra k G) : a * (b + c) = a * b + a * c :=
by simp only [mul_def, sum_add_index, mul_add, mul_zero, single_zero, single_add,
eq_self_iff_true, forall_true_iff, forall_3_true_iff, sum_add]
private lemma right_distrib (a b c : monoid_star_algebra k G) : (a + b) * c = a * c + b * c :=
by simp only [mul_def, sum_add_index, add_mul, mul_zero, zero_mul, single_zero,
single_add, eq_self_iff_true, forall_true_iff, forall_3_true_iff, sum_zero, sum_add]
instance : semiring (monoid_star_algebra k G) :=
{ one := 1,
mul := (*),
one_mul := assume f, by simp only [mul_def, one_def, sum_single_index, zero_mul,
single_zero, sum_zero, zero_add, one_mul, sum_single],
mul_one := assume f, by simp only [mul_def, one_def, sum_single_index, mul_zero,
single_zero, sum_zero, add_zero, mul_one, sum_single],
zero_mul := monoid_star_algebra.zero_mul,
mul_zero := monoid_star_algebra.mul_zero,
mul_assoc := assume f g h, by simp only [mul_def, sum_sum_index, sum_zero_index, sum_add_index,
sum_single_index, single_zero, single_add, eq_self_iff_true, forall_true_iff, forall_3_true_iff,
add_mul, mul_add, add_assoc, mul_assoc, zero_mul, mul_zero, sum_zero, sum_add],
left_distrib := left_distrib,
right_distrib := right_distrib,
.. finsupp.add_comm_monoid }
@[simp] lemma single_mul_single {a₁ a₂ : G} {b₁ b₂ : k} :
(single a₁ b₁ : monoid_star_algebra k G) * single a₂ b₂ = single (a₁ * a₂) (b₁ * b₂) :=
(sum_single_index (by simp only [zero_mul, single_zero, sum_zero])).trans
(sum_single_index (by rw [mul_zero, single_zero]))
@[simp] lemma single_pow {a : G} {b : k} :
∀ n : ℕ, (single a b : monoid_star_algebra k G)^n = single (a^n) (b ^ n)
| 0 := rfl
| (n+1) := by simp only [pow_succ, single_pow n, single_mul_single]
section
variables (k G)
/-- Embedding of a monoid into its monoid algebra. -/
def of : G →* monoid_star_algebra k G :=
{ to_fun := λ a, single a 1,
map_one' := rfl,
map_mul' := λ a b, by rw [single_mul_single, one_mul] }
end
@[simp] lemma of_apply (a : G) : of k G a = single a 1 := rfl
lemma mul_single_apply_aux (f : monoid_star_algebra k G) {r : k}
{x y z : G} (H : ∀ a, a * x = z ↔ a = y) :
(f * single x r) z = f y * r :=
have A : ∀ a₁ b₁, (single x r).sum (λ a₂ b₂, ite (a₁ * a₂ = z) (b₁ * b₂) 0) =
ite (a₁ * x = z) (b₁ * r) 0,
from λ a₁ b₁, sum_single_index $ by simp,
calc (f * single x r) z = sum f (λ a b, if (a = y) then (b * r) else 0) :
-- different `decidable` instances make it not trivial
by { simp only [mul_apply, A, H], congr, funext, split_ifs; refl }
... = if y ∈ f.support then f y * r else 0 : f.support.sum_ite_eq' _ _
... = f y * r : by split_ifs with h; simp at h; simp [h]
lemma mul_single_one_apply (f : monoid_star_algebra k G) (r : k) (x : G) :
(f * single 1 r) x = f x * r :=
f.mul_single_apply_aux $ λ a, by rw [mul_one]
lemma single_mul_apply_aux (f : monoid_star_algebra k G) {r : k} {x y z : G}
(H : ∀ a, x * a = y ↔ a = z) :
(single x r * f) y = r * f z :=
have f.sum (λ a b, ite (x * a = y) (0 * b) 0) = 0, by simp,
calc (single x r * f) y = sum f (λ a b, ite (x * a = y) (r * b) 0) :
(mul_apply _ _ _).trans $ sum_single_index this
... = f.sum (λ a b, ite (a = z) (r * b) 0) :
by { simp only [H], congr, ext; split_ifs; refl }
... = if z ∈ f.support then (r * f z) else 0 : f.support.sum_ite_eq' _ _
... = _ : by split_ifs with h; simp at h; simp [h]
lemma single_one_mul_apply (f : monoid_star_algebra k G) (r : k) (x : G) :
(single 1 r * f) x = r * f x :=
f.single_mul_apply_aux $ λ a, by rw [one_mul]
end
instance [comm_semiring k] [comm_monoid G] : comm_semiring (monoid_star_algebra k G) :=
{ mul_comm := assume f g,
begin
simp only [mul_def, finsupp.sum, mul_comm],
rw [finset.sum_comm],
simp only [mul_comm]
end,
.. monoid_star_algebra.semiring }
instance [ring k] : has_neg (monoid_star_algebra k G) :=
by apply_instance
instance [ring k] [monoid G] : ring (monoid_star_algebra k G) :=
{ neg := has_neg.neg,
add_left_neg := add_left_neg,
.. monoid_star_algebra.semiring }
instance [comm_ring k] [comm_monoid G] : comm_ring (monoid_star_algebra k G) :=
{ mul_comm := mul_comm, .. monoid_star_algebra.ring}
instance [semiring k] : has_scalar k (monoid_star_algebra k G) :=
finsupp.has_scalar
instance [semiring k] : semimodule k (monoid_star_algebra k G) :=
finsupp.semimodule G k
instance [ring k] : module k (monoid_star_algebra k G) :=
by {delta monoid_star_algebra, apply_instance }
lemma single_one_comm [comm_semiring k] [monoid G] (r : k) (f : monoid_star_algebra k G) :
single 1 r * f = f * single 1 r :=
by { ext, rw [single_one_mul_apply, mul_single_one_apply, mul_comm] }
instance [comm_semiring k] [monoid G] : algebra k (monoid_star_algebra k G) :=
{ to_fun := single 1,
map_one' := rfl,
map_mul' := λ x y, by rw [single_mul_single, one_mul],
map_zero' := single_zero,
map_add' := λ x y, single_add,
smul_def' := λ r a, ext (λ _, smul_apply.trans (single_one_mul_apply _ _ _).symm),
commutes' := λ r f, ext $ λ _, by rw [single_one_mul_apply, mul_single_one_apply, mul_comm] }
@[simp] lemma coe_algebra_map [comm_semiring k] [monoid G] :
(algebra_map k (monoid_star_algebra k G) : k → monoid_star_algebra k G) = single 1 :=
rfl
lemma single_eq_algebra_map_mul_of [comm_semiring k] [monoid G] (a : G) (b : k) :
single a b = (algebra_map k (monoid_star_algebra k G) : k → monoid_star_algebra k G) b * of k G a :=
by simp
instance [group G] [semiring k] :
distrib_mul_action G (monoid_star_algebra k G) :=
finsupp.comap_distrib_mul_action_self
section lift
variables (k G) [comm_semiring k] [monoid G] (R : Type u₃) [semiring R] [algebra k R]
/-- Any monoid homomorphism `G →* R` can be lifted to an algebra homomorphism
`monoid_star_algebra k G →ₐ[k] R`. -/
def lift : (G →* R) ≃ (monoid_star_algebra k G →ₐ[k] R) :=
{ inv_fun := λ f, (f : monoid_star_algebra k G →* R).comp (of k G),
to_fun := λ F, { to_fun := λ f, f.sum (λ a b, b • F a),
map_one' := by { rw [one_def, sum_single_index, one_smul, F.map_one], apply zero_smul },
map_mul' :=
begin
intros f g,
rw [mul_def, finsupp.sum_mul, finsupp.sum_sum_index];
try { intros, simp only [zero_smul, add_smul], done },
refine finset.sum_congr rfl (λ a ha, _), simp only [],
rw [finsupp.mul_sum, finsupp.sum_sum_index];
try { intros, simp only [zero_smul, add_smul], done },
refine finset.sum_congr rfl (λ a' ha', _), simp only [],
rw [sum_single_index, F.map_mul, algebra.mul_smul_comm, algebra.smul_mul_assoc,
smul_smul, mul_comm],
apply zero_smul
end,
map_zero' := sum_zero_index,
map_add' := λ f g, by rw [sum_add_index]; intros; simp only [zero_smul, add_smul],
commutes' := λ r, by rw [coe_algebra_map, sum_single_index, F.map_one, algebra.smul_def,
mul_one]; apply zero_smul },
left_inv := λ f, begin ext x, simp [sum_single_index] end,
right_inv := λ F,
begin
ext f,
conv_rhs { rw ← f.sum_single },
simp [← F.map_smul, finsupp.sum, ← F.map_sum]
end }
variables {k G R}
lemma lift_apply (F : G →* R) (f : monoid_star_algebra k G) :
lift k G R F f = f.sum (λ a b, b • F a) := rfl
@[simp] lemma lift_symm_apply (F : monoid_star_algebra k G →ₐ[k] R) (x : G) :
(lift k G R).symm F x = F (single x 1) := rfl
lemma lift_of (F : G →* R) (x) :
lift k G R F (of k G x) = F x :=
by rw [of_apply, ← lift_symm_apply, equiv.symm_apply_apply]
@[simp] lemma lift_single (F : G →* R) (a b) :
lift k G R F (single a b) = b • F a :=
by rw [single_eq_algebra_map_mul_of, ← algebra.smul_def, alg_hom.map_smul, lift_of]
lemma lift_unique' (F : monoid_star_algebra k G →ₐ[k] R) :
F = lift k G R ((F : monoid_star_algebra k G →* R).comp (of k G)) :=
((lift k G R).apply_symm_apply F).symm
/-- Decomposition of a `k`-algebra homomorphism from `monoid_star_algebra k G` by
its values on `F (single a 1)`. -/
lemma lift_unique (F : monoid_star_algebra k G →ₐ[k] R) (f : monoid_star_algebra k G) :
F f = f.sum (λ a b, b • F (single a 1)) :=
by conv_lhs { rw lift_unique' F, simp [lift_apply] }
/-- A `k`-algebra homomorphism from `monoid_star_algebra k G` is uniquely defined by its
values on the functions `single a 1`. -/
lemma alg_hom_ext ⦃φ₁ φ₂ : monoid_star_algebra k G →ₐ[k] R⦄
(h : ∀ x, φ₁ (single x 1) = φ₂ (single x 1)) : φ₁ = φ₂ :=
(lift k G R).symm.injective $ monoid_hom.ext h
end lift
section
variables (k)
/-- When `V` is a `k[G]`-module, multiplication by a group element `g` is a `k`-linear map. -/
def group_smul.linear_map [group G] [comm_ring k]
(V : Type u₃) [add_comm_group V] [module (monoid_star_algebra k G) V] (g : G) :
(module.restrict_scalars k (monoid_star_algebra k G) V) →ₗ[k]
(module.restrict_scalars k (monoid_star_algebra k G) V) :=
{ to_fun := λ v, (single g (1 : k) • v : V),
map_add' := λ x y, smul_add (single g (1 : k)) x y,
map_smul' := λ c x,
by simp only [module.restrict_scalars_smul_def, coe_algebra_map, ←mul_smul, single_one_comm], }.
@[simp]
lemma group_smul.linear_map_apply [group G] [comm_ring k]
(V : Type u₃) [add_comm_group V] [module (monoid_star_algebra k G) V] (g : G) (v : V) :
(group_smul.linear_map k V g) v = (single g (1 : k) • v : V) :=
rfl
section
variables {k}
variables [group G] [comm_ring k]
{V : Type u₃} {gV : add_comm_group V} {mV : module (monoid_star_algebra k G) V}
{W : Type u₃} {gW : add_comm_group W} {mW : module (monoid_star_algebra k G) W}
(f : (module.restrict_scalars k (monoid_star_algebra k G) V) →ₗ[k]
(module.restrict_scalars k (monoid_star_algebra k G) W))
(h : ∀ (g : G) (v : V), f (single g (1 : k) • v : V) = (single g (1 : k) • (f v) : W))
include h
/-- Build a `k[G]`-linear map from a `k`-linear map and evidence that it is `G`-equivariant. -/
def equivariant_of_linear_of_comm : V →ₗ[monoid_star_algebra k G] W :=
{ to_fun := f,
map_add' := λ v v', by simp,
map_smul' := λ c v,
begin
apply finsupp.induction c,
{ simp, },
{ intros g r c' nm nz w,
rw [add_smul, linear_map.map_add, w, add_smul, add_left_inj,
single_eq_algebra_map_mul_of, ←smul_smul, ←smul_smul],
erw [f.map_smul, h g v],
refl, }
end, }
@[simp]
lemma equivariant_of_linear_of_comm_apply (v : V) : (equivariant_of_linear_of_comm f h) v = f v :=
rfl
end
end
universe ui
variable {ι : Type ui}
lemma prod_single [comm_semiring k] [comm_monoid G]
{s : finset ι} {a : ι → G} {b : ι → k} :
s.prod (λi, single (a i) (b i)) = single (s.prod a) (s.prod b) :=
finset.induction_on s rfl $ λ a s has ih, by rw [prod_insert has, ih,
single_mul_single, prod_insert has, prod_insert has]
section -- We now prove some additional statements that hold for group algebras.
variables [semiring k] [group G]
@[simp]
lemma mul_single_apply (f : monoid_star_algebra k G) (r : k) (x y : G) :
(f * single x r) y = f (y * x⁻¹) * r :=
f.mul_single_apply_aux $ λ a, eq_mul_inv_iff_mul_eq.symm
@[simp]
lemma single_mul_apply (r : k) (x : G) (f : monoid_star_algebra k G) (y : G) :
(single x r * f) y = r * f (x⁻¹ * y) :=
f.single_mul_apply_aux $ λ z, eq_inv_mul_iff_mul_eq.symm
lemma mul_apply_left (f g : monoid_star_algebra k G) (x : G) :
(f * g) x = (f.sum $ λ a b, b * (g (a⁻¹ * x))) :=
calc (f * g) x = sum f (λ a b, (single a (f a) * g) x) :
by rw [← finsupp.sum_apply, ← finsupp.sum_mul, f.sum_single]
... = _ : by simp only [single_mul_apply, finsupp.sum]
-- If we'd assumed `comm_semiring`, we could deduce this from `mul_apply_left`.
lemma mul_apply_right (f g : monoid_star_algebra k G) (x : G) :
(f * g) x = (g.sum $ λa b, (f (x * a⁻¹)) * b) :=
calc (f * g) x = sum g (λ a b, (f * single a (g a)) x) :
by rw [← finsupp.sum_apply, ← finsupp.mul_sum, g.sum_single]
... = _ : by simp only [mul_single_apply, finsupp.sum]
end
end monoid_star_algebra
section
variables [semiring k]
/--
The monoid algebra over a semiring `k` generated by the additive monoid `G`.
It is the type of finite formal `k`-linear combinations of terms of `G`,
endowed with the convolution product.
-/
@[derive [inhabited, add_comm_monoid]]
def add_monoid_star_algebra := G →₀ k
end
namespace add_monoid_star_algebra
variables {k G}
local attribute [reducible] add_monoid_star_algebra
section
variables [semiring k] [add_monoid G]
/-- The product of `f g : add_monoid_star_algebra k G` is the finitely supported function
whose value at `a` is the sum of `f x * g y` over all pairs `x, y`
such that `x + y = a`. (Think of the product of multivariate
polynomials where `α` is the additive monoid of monomial exponents.) -/
instance : has_mul (add_monoid_star_algebra k G) :=
⟨λf g, f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, single (a₁ + a₂) (b₁ * b₂)⟩
lemma mul_def {f g : add_monoid_star_algebra k G} :
f * g = (f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, single (a₁ + a₂) (b₁ * b₂)) :=
rfl
lemma mul_apply (f g : add_monoid_star_algebra k G) (x : G) :
(f * g) x = (f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, if a₁ + a₂ = x then b₁ * b₂ else 0) :=
begin
rw [mul_def],
simp only [finsupp.sum_apply, single_apply],
end
lemma support_mul (a b : add_monoid_star_algebra k G) :
(a * b).support ⊆ a.support.bind (λa₁, b.support.bind $ λa₂, {a₁ + a₂}) :=
subset.trans support_sum $ bind_mono $ assume a₁ _,
subset.trans support_sum $ bind_mono $ assume a₂ _, support_single_subset
/-- The unit of the multiplication is `single 1 1`, i.e. the function
that is `1` at `0` and zero elsewhere. -/
instance : has_one (add_monoid_star_algebra k G) :=
⟨single 0 1⟩
lemma one_def : (1 : add_monoid_star_algebra k G) = single 0 1 :=
rfl
-- TODO: the simplifier unfolds 0 in the instance proof!
protected lemma zero_mul (f : add_monoid_star_algebra k G) : 0 * f = 0 :=
by simp only [mul_def, sum_zero_index]
protected lemma mul_zero (f : add_monoid_star_algebra k G) : f * 0 = 0 :=
by simp only [mul_def, sum_zero_index, sum_zero]
private lemma left_distrib (a b c : add_monoid_star_algebra k G) : a * (b + c) = a * b + a * c :=
by simp only [mul_def, sum_add_index, mul_add, mul_zero, single_zero, single_add,
eq_self_iff_true, forall_true_iff, forall_3_true_iff, sum_add]
private lemma right_distrib (a b c : add_monoid_star_algebra k G) : (a + b) * c = a * c + b * c :=
by simp only [mul_def, sum_add_index, add_mul, mul_zero, zero_mul, single_zero,
single_add, eq_self_iff_true, forall_true_iff, forall_3_true_iff, sum_zero, sum_add]
instance : semiring (add_monoid_star_algebra k G) :=
{ one := 1,
mul := (*),
one_mul := assume f, by simp only [mul_def, one_def, sum_single_index, zero_mul,
single_zero, sum_zero, zero_add, one_mul, sum_single],
mul_one := assume f, by simp only [mul_def, one_def, sum_single_index, mul_zero,
single_zero, sum_zero, add_zero, mul_one, sum_single],
zero_mul := add_monoid_star_algebra.zero_mul,
mul_zero := add_monoid_star_algebra.mul_zero,
mul_assoc := assume f g h, by simp only [mul_def, sum_sum_index, sum_zero_index, sum_add_index,
sum_single_index, single_zero, single_add, eq_self_iff_true, forall_true_iff, forall_3_true_iff,
add_mul, mul_add, add_assoc, mul_assoc, zero_mul, mul_zero, sum_zero, sum_add],
left_distrib := left_distrib,
right_distrib := right_distrib,
.. finsupp.add_comm_monoid }
lemma single_mul_single {a₁ a₂ : G} {b₁ b₂ : k} :
(single a₁ b₁ : add_monoid_star_algebra k G) * single a₂ b₂ = single (a₁ + a₂) (b₁ * b₂) :=
(sum_single_index (by simp only [zero_mul, single_zero, sum_zero])).trans
(sum_single_index (by rw [mul_zero, single_zero]))
section
variables (k G)
/-- Embedding of a monoid into its monoid algebra. -/
def of : multiplicative G →* add_monoid_star_algebra k G :=
{ to_fun := λ a, single a 1,
map_one' := rfl,
map_mul' := λ a b, by { rw [single_mul_single, one_mul], refl } }
end
@[simp] lemma of_apply (a : G) : of k G a = single a 1 := rfl
lemma mul_single_apply_aux (f : add_monoid_star_algebra k G) (r : k)
(x y z : G) (H : ∀ a, a + x = z ↔ a = y) :
(f * single x r) z = f y * r :=
have A : ∀ a₁ b₁, (single x r).sum (λ a₂ b₂, ite (a₁ + a₂ = z) (b₁ * b₂) 0) =
ite (a₁ + x = z) (b₁ * r) 0,
from λ a₁ b₁, sum_single_index $ by simp,
calc (f * single x r) z = sum f (λ a b, if (a = y) then (b * r) else 0) :
-- different `decidable` instances make it not trivial
by { simp only [mul_apply, A, H], congr, funext, split_ifs; refl }
... = if y ∈ f.support then f y * r else 0 : f.support.sum_ite_eq' _ _
... = f y * r : by split_ifs with h; simp at h; simp [h]
lemma mul_single_zero_apply (f : add_monoid_star_algebra k G) (r : k) (x : G) :
(f * single 0 r) x = f x * r :=
f.mul_single_apply_aux r _ _ _ $ λ a, by rw [add_zero]
lemma single_mul_apply_aux (f : add_monoid_star_algebra k G) (r : k) (x y z : G)
(H : ∀ a, x + a = y ↔ a = z) :
(single x r * f) y = r * f z :=
have f.sum (λ a b, ite (x + a = y) (0 * b) 0) = 0, by simp,
calc (single x r * f) y = sum f (λ a b, ite (x + a = y) (r * b) 0) :
(mul_apply _ _ _).trans $ sum_single_index this
... = f.sum (λ a b, ite (a = z) (r * b) 0) :
by { simp only [H], congr, ext; split_ifs; refl }
... = if z ∈ f.support then (r * f z) else 0 : f.support.sum_ite_eq' _ _
... = _ : by split_ifs with h; simp at h; simp [h]
lemma single_zero_mul_apply (f : add_monoid_star_algebra k G) (r : k) (x : G) :
(single 0 r * f) x = r * f x :=
f.single_mul_apply_aux r _ _ _ $ λ a, by rw [zero_add]
end
instance [comm_semiring k] [add_comm_monoid G] : comm_semiring (add_monoid_star_algebra k G) :=
{ mul_comm := assume f g,
begin
simp only [mul_def, finsupp.sum, mul_comm],
rw [finset.sum_comm],
simp only [add_comm]
end,
.. add_monoid_star_algebra.semiring }
instance [ring k] : has_neg (add_monoid_star_algebra k G) :=
by apply_instance
instance [ring k] [add_monoid G] : ring (add_monoid_star_algebra k G) :=
{ neg := has_neg.neg,
add_left_neg := add_left_neg,
.. add_monoid_star_algebra.semiring }
instance [comm_ring k] [add_comm_monoid G] : comm_ring (add_monoid_star_algebra k G) :=
{ mul_comm := mul_comm, .. add_monoid_star_algebra.ring}
instance [semiring k] : has_scalar k (add_monoid_star_algebra k G) :=
finsupp.has_scalar
instance [semiring k] : semimodule k (add_monoid_star_algebra k G) :=
finsupp.semimodule G k
instance [ring k] : module k (add_monoid_star_algebra k G) :=
by { delta add_monoid_star_algebra, apply_instance }
instance [comm_semiring k] [add_monoid G] : algebra k (add_monoid_star_algebra k G) :=
{ to_fun := single 0,
map_one' := rfl,
map_mul' := λ x y, by rw [single_mul_single, zero_add],
map_zero' := single_zero,
map_add' := λ x y, single_add,
smul_def' := λ r a, by { ext x, exact smul_apply.trans (single_zero_mul_apply _ _ _).symm },
commutes' := λ r f, show single 0 r * f = f * single 0 r,
by ext; rw [single_zero_mul_apply, mul_single_zero_apply, mul_comm] }
@[simp] lemma coe_algebra_map [comm_semiring k] [add_monoid G] :
(algebra_map k (add_monoid_star_algebra k G) : k → add_monoid_star_algebra k G) = single 0 :=
rfl
/-- Any monoid homomorphism `multiplicative G →* R` can be lifted to an algebra homomorphism
`add_monoid_star_algebra k G →ₐ[k] R`. -/
def lift [comm_semiring k] [add_monoid G] {R : Type u₃} [semiring R] [algebra k R] :
(multiplicative G →* R) ≃ (add_monoid_star_algebra k G →ₐ[k] R) :=
{ inv_fun := λ f, ((f : add_monoid_star_algebra k G →+* R) : add_monoid_star_algebra k G →* R).comp (of k G),
to_fun := λ F, { to_fun := λ f, f.sum (λ a b, b • F a),
map_one' := by { rw [one_def, sum_single_index, one_smul], erw [F.map_one], apply zero_smul },
map_mul' :=
begin
intros f g,
rw [mul_def, finsupp.sum_mul, finsupp.sum_sum_index];
try { intros, simp only [zero_smul, add_smul], done },
refine finset.sum_congr rfl (λ a ha, _), simp only [],
rw [finsupp.mul_sum, finsupp.sum_sum_index];
try { intros, simp only [zero_smul, add_smul], done },
refine finset.sum_congr rfl (λ a' ha', _), simp only [],
rw [sum_single_index],
erw [F.map_mul],
rw [algebra.mul_smul_comm, algebra.smul_mul_assoc, smul_smul, mul_comm],
apply zero_smul
end,
map_zero' := sum_zero_index,
map_add' := λ f g, by rw [sum_add_index]; intros; simp only [zero_smul, add_smul],
commutes' := λ r,
begin
rw [coe_algebra_map, sum_single_index],
erw [F.map_one],
rw [algebra.smul_def, mul_one],
apply zero_smul
end, },
left_inv := λ f, begin ext x, simp [sum_single_index] end,
right_inv := λ F,
begin
ext f,
conv_rhs { rw ← f.sum_single },
simp [← F.map_smul, finsupp.sum, ← F.map_sum]
end }
-- It is hard to state the equivalent of `distrib_mul_action G (monoid_star_algebra k G)`
-- because we've never discussed actions of additive groups.
universe ui
variable {ι : Type ui}
lemma prod_single [comm_semiring k] [add_comm_monoid G]
{s : finset ι} {a : ι → G} {b : ι → k} :
s.prod (λi, single (a i) (b i)) = single (s.sum a) (s.prod b) :=
finset.induction_on s rfl $ λ a s has ih, by rw [prod_insert has, ih,
single_mul_single, sum_insert has, prod_insert has]
end add_monoid_star_algebra
|
fdfb678691bba56b25a79be677772b9b37e4c6ae | 618003631150032a5676f229d13a079ac875ff77 | /src/linear_algebra/lagrange.lean | 058eed1e48129eec201e10bc3f369aec1a7d7dd2 | [
"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 | 6,946 | lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Kenny Lau.
-/
import ring_theory.polynomial algebra.big_operators
/-!
# Lagrange interpolation
## Main definitions
* `lagrange.basis s x` where `s : finset F` and `x : F`: the Lagrange basis polynomial
that evaluates to `1` at `x` and `0` at other elements of `s`.
* `lagrange.interpolate s f` where `s : finset F` and `f : s → F`: the Lagrange interpolant
that evaluates to `f x` at `x` for `x ∈ s`.
-/
noncomputable theory
open_locale big_operators classical
universe u
namespace lagrange
variables {F : Type u} [decidable_eq F] [field F] (s : finset F)
variables {F' : Type u} [field F'] (s' : finset F')
open polynomial
/-- Lagrange basis polynomials that evaluate to 1 at `x` and 0 at other elements of `s`. -/
def basis (x : F) : polynomial F :=
∏ y in s.erase x, C (x - y)⁻¹ * (X - C y)
@[simp] theorem basis_empty (x : F) : basis ∅ x = 1 :=
rfl
@[simp] theorem eval_basis_self (x : F) : (basis s x).eval x = 1 :=
begin
rw [basis, ← (s.erase x).prod_hom (eval x), finset.prod_eq_one],
intros y hy, simp_rw [eval_mul, eval_sub, eval_C, eval_X],
exact inv_mul_cancel (sub_ne_zero_of_ne (finset.ne_of_mem_erase hy).symm)
end
@[simp] theorem eval_basis_ne (x y : F) (h1 : y ∈ s) (h2 : y ≠ x) : (basis s x).eval y = 0 :=
begin
rw [basis, ← (s.erase x).prod_hom (eval y), finset.prod_eq_zero (finset.mem_erase.2 ⟨h2, h1⟩)],
simp_rw [eval_mul, eval_sub, eval_C, eval_X, sub_self, mul_zero]
end
theorem eval_basis (x y : F) (h : y ∈ s) : (basis s x).eval y = if y = x then 1 else 0 :=
by { split_ifs with H, { subst H, apply eval_basis_self }, { exact eval_basis_ne s x y h H } }
@[simp] theorem nat_degree_basis (x : F) (hx : x ∈ s) : (basis s x).nat_degree = s.card - 1 :=
begin
unfold basis, generalize hsx : s.erase x = sx,
have : x ∉ sx := hsx ▸ finset.not_mem_erase x s,
rw [← finset.insert_erase hx, hsx, finset.card_insert_of_not_mem this, nat.add_sub_cancel],
clear hx hsx s, revert this, apply sx.induction_on,
{ intros hx, rw [finset.prod_empty, nat_degree_one], refl },
{ intros y s hys ih hx, rw [finset.mem_insert, not_or_distrib] at hx,
have h1 : C (x - y)⁻¹ ≠ C 0 := λ h, hx.1 (eq_of_sub_eq_zero $ inv_eq_zero.1 $ C_inj.1 h),
have h2 : X ^ 1 - C y ≠ 0 := by convert X_pow_sub_C_ne_zero zero_lt_one y,
rw C_0 at h1, rw pow_one at h2,
rw [finset.prod_insert hys, nat_degree_mul_eq (mul_ne_zero h1 h2), ih hx.2,
finset.card_insert_of_not_mem hys, nat_degree_mul_eq h1 h2,
nat_degree_C, zero_add, nat_degree, degree_X_sub_C, add_comm], refl,
rw [ne, finset.prod_eq_zero_iff], rintro ⟨z, hzs, hz⟩,
rw mul_eq_zero at hz, cases hz with hz hz,
{ rw [← C_0, C_inj, inv_eq_zero, sub_eq_zero] at hz, exact hx.2 (hz.symm ▸ hzs) },
{ rw ← pow_one (X : polynomial F) at hz, exact X_pow_sub_C_ne_zero zero_lt_one _ hz } }
end
variables (f : (↑s : set F) → F)
/-- Lagrange interpolation: given a finset `s` and a function `f : s → F`,
`interpolate s f` is the unique polynomial of degree `< s.card`
that takes value `f x` on all `x` in `s`. -/
def interpolate : polynomial F :=
∑ x in s.attach, C (f x) * basis s x
@[simp] theorem interpolate_empty (f) : interpolate (∅ : finset F) f = 0 :=
rfl
@[simp] theorem eval_interpolate (x) (H : x ∈ s) : eval x (interpolate s f) = f ⟨x, H⟩ :=
begin
rw [interpolate, ← finset.sum_hom _ (eval x), finset.sum_eq_single (⟨x, H⟩ : { x // x ∈ s })],
{ rw [eval_mul, eval_C, subtype.coe_mk, eval_basis_self, mul_one] },
{ rintros ⟨y, hy⟩ _ hyx, rw [eval_mul, subtype.coe_mk, eval_basis_ne s y x H, mul_zero],
{ rintros rfl, exact hyx rfl } },
{ intro h, exact absurd (finset.mem_attach _ _) h }
end
theorem degree_interpolate_lt : (interpolate s f).degree < s.card :=
if H : s = ∅ then by { subst H, rw [interpolate_empty, degree_zero], exact with_bot.bot_lt_coe _ }
else lt_of_le_of_lt (degree_sum_le _ _) $ (finset.sup_lt_iff $ with_bot.bot_lt_coe s.card).2 $ λ b _,
calc (C (f b) * basis s b).degree
≤ (C (f b)).degree + (basis s b).degree : degree_mul_le _ _
... ≤ 0 + (basis s b).degree : add_le_add_right' degree_C_le
... = (basis s b).degree : zero_add _
... ≤ (basis s b).nat_degree : degree_le_nat_degree
... = (s.card - 1 : ℕ) : by { rw nat_degree_basis s b b.2 }
... < s.card : with_bot.coe_lt_coe.2 (nat.pred_lt $ mt finset.card_eq_zero.1 H)
/-- Linear version of `interpolate`. -/
def linterpolate : ((↑s : set F) → F) →ₗ[F] polynomial F :=
{ to_fun := interpolate s,
add := λ f g, by { simp_rw [interpolate, ← finset.sum_add_distrib, ← add_mul, ← C_add], refl },
smul := λ c f, by { simp_rw [interpolate, finset.smul_sum, C_mul', smul_smul], refl } }
@[simp] lemma interpolate_add (f g) : interpolate s (f + g) = interpolate s f + interpolate s g :=
(linterpolate s).map_add f g
@[simp] lemma interpolate_zero : interpolate s 0 = 0 :=
(linterpolate s).map_zero
@[simp] lemma interpolate_neg (f) : interpolate s (-f) = -interpolate s f :=
(linterpolate s).map_neg f
@[simp] lemma interpolate_sub (f g) : interpolate s (f - g) = interpolate s f - interpolate s g :=
(linterpolate s).map_sub f g
@[simp] lemma interpolate_smul (c : F) (f) : interpolate s (c • f) = c • interpolate s f :=
(linterpolate s).map_smul c f
theorem eq_zero_of_eval_eq_zero {f : polynomial F'} (hf1 : f.degree < s'.card)
(hf2 : ∀ x ∈ s', f.eval x = 0) : f = 0 :=
by_contradiction $ λ hf3, not_le_of_lt hf1 $
calc (s'.card : with_bot ℕ)
≤ f.roots.card : with_bot.coe_le_coe.2 $ finset.card_le_of_subset $ λ x hx,
(mem_roots hf3).2 $ hf2 x hx
... ≤ f.degree : card_roots hf3
theorem eq_of_eval_eq {f g : polynomial F'} (hf : f.degree < s'.card) (hg : g.degree < s'.card)
(hfg : ∀ x ∈ s', f.eval x = g.eval x) : f = g :=
eq_of_sub_eq_zero $ eq_zero_of_eval_eq_zero s'
(lt_of_le_of_lt (degree_sub_le f g) $ max_lt hf hg)
(λ x hx, by rw [eval_sub, hfg x hx, sub_self])
theorem eq_interpolate (f : polynomial F) (hf : f.degree < s.card) :
interpolate s (λ x, f.eval x) = f :=
eq_of_eval_eq s (degree_interpolate_lt s _) hf $ λ x hx, eval_interpolate s _ x hx
/-- Lagrange interpolation induces isomorphism between functions from `s` and polynomials
of degree less than `s.card`. -/
def fun_equiv_degree_lt : degree_lt F s.card ≃ₗ[F] ((↑s : set F) → F) :=
{ to_fun := λ f x, f.1.eval x,
add := λ f g, funext $ λ x, eval_add,
smul := λ c f, funext $ λ x, by { rw [pi.smul_apply, smul_eq_mul, ← @eval_C F c _ x,
← eval_mul, eval_C, C_mul'], refl },
inv_fun := λ f, ⟨interpolate s f, mem_degree_lt.2 $ degree_interpolate_lt s f⟩,
left_inv := λ f, subtype.eq $ eq_interpolate s f $ mem_degree_lt.1 f.2,
right_inv := λ f, funext $ λ ⟨x, hx⟩, eval_interpolate s f x hx }
end lagrange
|
30e076b3aad20bcdbcc76ecdbd937317b4ea3629 | 3984ab8555ab1e1084e22ef652544acdfc231f27 | /src/Regret Minimization in Games with Incomplete Information - template.lean | 2be70c4492c764265b7cddb76e92ae06a0c6b191 | [] | no_license | mrakgr/CFR-in-Lean | a35c7a478795cc794cc0caff3199cf28c8ee5448 | 720a3260297bcc158e08833d38964450dcaad2eb | refs/heads/master | 1,598,515,917,940 | 1,572,612,355,000 | 1,572,612,355,000 | 217,296,108 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 4,499 | lean | import data.rat
-- `constant`s are axiomatic definitions. They are inherently noncomputable.
constant Infoset : Type
constant Policy : Type
def maximize {α : Type} : α × list α → (α → ℚ) → ℚ | (x,xs) f := xs.foldl (fun s a, max s (f a)) (f x)
def list.fsum {α : Type} : (α → ℚ) → list α → ℚ | f l := l.foldl (fun s a, s + f a) 0
constant Infoset.fsum : (Infoset → ℚ) → Infoset → ℚ
-- Compared to the paper, here I am moving the indexing so it obeys the C notation.
def nat.fsum (f : ℕ → ℚ) : ℕ → ℚ | 0 := 0 | (n+1) := n.fsum + f n
constant σ : ℕ → Policy
constant σ_all : (ℕ → Policy) × list (ℕ → Policy)
constant Policy.at : Policy → Infoset → Policy → Policy
constant Player : Type
constant u : Player → Policy → Infoset → ℚ
constant D : Infoset → Infoset
constant Succ : Player → Infoset → Infoset
constant SuccA : Player → Infoset → Policy → Infoset
constant π : Player → Policy → Infoset → ℚ
constant π_is_probability : ∀ i σ I, 0 <= π i σ I ∧ π i σ I <= 1
constant neg : Player → Player
def R.template (update_policy : Policy → Infoset → Policy → Policy)
(i : Player) (T : ℕ) (I : Infoset) : ℚ :=
let u := u i in
1 / T *
maximize σ_all (fun σ',
T.fsum (fun t,
let (σ, σ') := (σ t, σ' t) in
let σ' := update_policy σ' I σ in
π (neg i) σ I * (u σ' I - u σ I)
)
)
def R.full := R.template (fun σ' I σ, σ.at (D I) σ')
def R.full.pos (i : Player) (T : ℕ) (I : Infoset) := max (R.full i T I) 0
def R.imm := R.template (fun σ' I σ, σ.at I σ')
def R.imm.pos (i : Player) (T : ℕ) (I : Infoset) := max (R.imm i T I) 0
constant A : Infoset → Policy × list Policy
constant succ : Player → Policy → Infoset → Policy → Infoset → rat
constant succ_is_probability : ∀ i σ I a I', 0 <= succ i σ I a I' ∧ succ i σ I a I' <= 1
def R.full.expanded (i : Player) (T : ℕ) (I : Infoset) : ℚ :=
let u := u i in
1 / T * (maximize (A I) $ fun a, maximize σ_all $ fun σ', T.fsum $ fun t,
let (σ, σ') := (σ t, σ' t) in
π (neg i) σ I * (u (σ.at I a) I - u σ I + (Succ i I).fsum (fun I', succ i σ I a I' * (u (σ.at (D I) σ') I' - u σ I')))
)
constant Rfull_eq_expanded (i : Player) (T : ℕ) (I : Infoset) : R.full i T I = R.full.expanded i T I
lemma five (i : Player) (T : ℕ) (I : Infoset)
: R.full i T I <= R.imm i T I + (Succ i I).fsum (fun I', R.full.pos i T I) :=
let u := u i in
calc
R.full i T I = R.full.expanded i T I : Rfull_eq_expanded i T I
... <=
1 / T * (maximize (A I) $ fun a, maximize σ_all $ fun σ', T.fsum $ fun t,
let (σ, σ') := (σ t, σ' t) in
π (neg i) σ I * (u (σ.at I a) I - u σ I)
)
+ 1 / T * (maximize (A I) $ fun a, maximize σ_all $ fun σ', T.fsum $ fun t,
let (σ, σ') := (σ t, σ' t) in
π (neg i) σ I * (SuccA i I a).fsum (fun I', succ i σ I a I' * (u (σ.at (D I) σ') I' - u σ I'))
) : sorry
... =
R.imm i T I + 1 / T * (maximize (A I) $ fun a, maximize σ_all $ fun σ', T.fsum $ fun t,
let (σ, σ') := (σ t, σ' t) in
(SuccA i I a).fsum (fun I', π (neg i) σ I * succ i σ I a I' * (u (σ.at (D I') σ') I' - u σ I'))
) : sorry
... =
R.imm i T I + (maximize (A I) $ fun a, (SuccA i I a).fsum $ fun I',
1 / T * (maximize (A I) $ fun a, maximize σ_all $ fun σ', T.fsum $ fun t,
let (σ, σ') := (σ t, σ' t) in
π (neg i) σ I * succ i σ I a I' * (u (σ.at (D I') σ') I' - u σ I')
)
) : sorry
... = R.imm i T I + (maximize (A I) $ fun a, (SuccA i I a).fsum $ fun I', R.full i T I') : sorry
... <= R.imm i T I + (Succ i I).fsum (fun I', R.full.pos i T I) : sorry
lemma six (i : Player) (T : ℕ) (I : Infoset) : R.full i T I <= (D I).fsum (fun I', R.full.pos i T I') :=
calc
R.full i T I <= R.imm i T I + ((Succ i I).fsum $ fun I', (Succ i I').fsum $ fun I'', R.full.pos i T I'') : sorry
... <= R.imm.pos i T I + ((Succ i I).fsum $ fun I', (Succ i I').fsum $ fun I'', R.full.pos i T I'') : sorry
... <= (D I).fsum (fun I', R.full.pos i T I') : sorry |
760253e13690a8813f0f791f88bbe1110eb087d1 | 4d7079ae603c07560e99d1ce35876f769cbb3e24 | /src/stability.lean | a62b20c50252222263994123785cf31be16ebe9f | [] | no_license | chasenorman/Formalized-Voting | 72493a08aa09d98d0fb589731b842e2d08c991d0 | de04e630b83525b042db166670ba97f9952b5691 | refs/heads/main | 1,687,282,160,284 | 1,627,155,031,000 | 1,627,155,031,000 | 370,472,025 | 13 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,064 | lean | import main
import split_cycle
open_locale classical
variables {V X : Type}
def minus_candidate (P : Prof V X) (b : X) : Prof V {x : X // x ≠ b} := λ v x y, P v x.val y.val
def strong_stability [fintype V] (F : VSCC) (P : Prof V X) : Prop :=
∀ (a b : X) (e : a ≠ b), 0 ≤ margin P a b → (⟨a, e⟩ : {x : X // x ≠ b}) ∈ F V {x : X // x ≠ b} (minus_candidate P b) → a ∈ F V X P
lemma margin_eq_margin_minus_candidate [fintype V] {P : Prof V X} (b : X) {x y : X} (h : x ≠ b) (e : y ≠ b) :
margin (minus_candidate P b) ⟨x, h⟩ ⟨y, e⟩ = margin P x y :=
begin
obviously,
end
theorem split_cycle_strong_stability [fintype V] (P : Prof V X) : strong_stability split_cycle P :=
begin
unfold strong_stability,
intros a b e m w y,
unfold split_cycle at *, unfold max_el_VSCC at *,
simp at *,
unfold split_cycle_VCCR, unfold split_cycle_CCR,
simp at *,
use _inst_1,
--intro p,
by_cases y ≠ b,
{specialize w y, specialize w h,
unfold split_cycle_VCCR at w, unfold split_cycle_CCR at w,
simp at w,
cases w with i2 w,
intro m2,
unfold margin_pos at m2,
rw ←(margin_eq_margin_minus_candidate b h e) at m2,
change margin_pos (minus_candidate P b) ⟨y, h⟩ ⟨a, e⟩ at m2,
unfold margin_pos at w,
rw margin_eq_margin i2 _inst_1 at w,
specialize w m2,
cases w with c w,
cases w with y_mem w,
cases w with a_mem w,
use (c.map subtype.val),
simp,
use h, exact y_mem,
use e, exact a_mem,
rw ←margin_eq_margin_minus_candidate b h e,
unfold cycle at *,
cases w with n w,
split,
work_on_goal 1
{by_contradiction, push_neg at h, exact n (list.map_eq_nil.mp h),},
rw list.last_map subtype.val n,
refine list.chain_map_of_chain subtype.val _ w,
obviously,
rw margin_eq_margin i2 _inst_1 at ᾰ,
obviously,},
push_neg at h,
rw h,
intro m2,
rw margin_antisymmetric at m,
exfalso,
simp at m,
have o : ∀ (n : ℤ), n ≤ 0 → ¬ 0 < n := by omega,
exact (o (margin P b a) m) m2,
end
|
09a4479104d2ec349f18d13e3748b59ea75701ff | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/tactic/monotonicity/interactive.lean | be3ca23b63c6c8fa47dfaa29f547a6af8b52ce9e | [
"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 | 23,516 | lean | /-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import tactic.monotonicity.basic
import control.traversable
import control.traversable.derive
import data.dlist
variables {a b c p : Prop}
namespace tactic.interactive
open lean lean.parser interactive
open interactive.types
open tactic
local postfix `?`:9001 := optional
local postfix *:9001 := many
meta inductive mono_function (elab : bool := tt)
| non_assoc : expr elab → list (expr elab) → list (expr elab) → mono_function
| assoc : expr elab → option (expr elab) → option (expr elab) → mono_function
| assoc_comm : expr elab → expr elab → mono_function
meta instance : decidable_eq mono_function :=
by mk_dec_eq_instance
meta def mono_function.to_tactic_format : mono_function → tactic format
| (mono_function.non_assoc fn xs ys) := do
fn' ← pp fn,
xs' ← mmap pp xs,
ys' ← mmap pp ys,
return format!"{fn'} {xs'} _ {ys'}"
| (mono_function.assoc fn xs ys) := do
fn' ← pp fn,
xs' ← pp xs,
ys' ← pp ys,
return format!"{fn'} {xs'} _ {ys'}"
| (mono_function.assoc_comm fn xs) := do
fn' ← pp fn,
xs' ← pp xs,
return format!"{fn'} _ {xs'}"
meta instance has_to_tactic_format_mono_function : has_to_tactic_format mono_function :=
{ to_tactic_format := mono_function.to_tactic_format }
@[derive traversable]
meta structure ac_mono_ctx' (rel : Type) :=
(to_rel : rel)
(function : mono_function)
(left right rel_def : expr)
@[reducible]
meta def ac_mono_ctx := ac_mono_ctx' (option (expr → expr → expr))
@[reducible]
meta def ac_mono_ctx_ne := ac_mono_ctx' (expr → expr → expr)
meta def ac_mono_ctx.to_tactic_format (ctx : ac_mono_ctx) : tactic format :=
do fn ← pp ctx.function,
l ← pp ctx.left,
r ← pp ctx.right,
rel ← pp ctx.rel_def,
return format!"{{ function := {fn}\n, left := {l}\n, right := {r}\n, rel_def := {rel} }"
meta instance has_to_tactic_format_mono_ctx : has_to_tactic_format ac_mono_ctx :=
{ to_tactic_format := ac_mono_ctx.to_tactic_format }
meta def as_goal (e : expr) (tac : tactic unit) : tactic unit :=
do gs ← get_goals,
set_goals [e],
tac,
set_goals gs
open list (hiding map) functor dlist
section config
parameter opt : mono_cfg
parameter asms : list expr
meta def unify_with_instance (e : expr) : tactic unit :=
as_goal e $
apply_instance
<|>
apply_opt_param
<|>
apply_auto_param
<|>
tactic.solve_by_elim { lemmas := some asms }
<|>
reflexivity
<|>
applyc ``id
<|>
return ()
private meta def match_rule_head (p : expr)
: list expr → expr → expr → tactic expr
| vs e t :=
(unify t p >> mmap' unify_with_instance vs >> instantiate_mvars e)
<|>
do (expr.pi _ _ d b) ← return t | failed,
v ← mk_meta_var d,
match_rule_head (v::vs) (expr.app e v) (b.instantiate_var v)
meta def pi_head : expr → tactic expr
| (expr.pi n _ t b) :=
do v ← mk_meta_var t,
pi_head (b.instantiate_var v)
| e := return e
meta def delete_expr (e : expr)
: list expr → tactic (option (list expr))
| [] := return none
| (x :: xs) :=
(compare opt e x >> return (some xs))
<|>
(map (cons x) <$> delete_expr xs)
meta def match_ac'
: list expr → list expr → tactic (list expr × list expr × list expr)
| es (x :: xs) := do
es' ← delete_expr x es,
match es' with
| (some es') := do
(c,l,r) ← match_ac' es' xs, return (x::c,l,r)
| none := do
(c,l,r) ← match_ac' es xs, return (c,l,x::r)
end
| es [] := do
return ([],es,[])
meta def match_ac (l : list expr) (r : list expr)
: tactic (list expr × list expr × list expr) :=
do (s',l',r') ← match_ac' l r,
s' ← mmap instantiate_mvars s',
l' ← mmap instantiate_mvars l',
r' ← mmap instantiate_mvars r',
return (s',l',r')
meta def match_prefix
: list expr → list expr → tactic (list expr × list expr × list expr)
| (x :: xs) (y :: ys) :=
(do compare opt x y,
prod.map ((::) x) id <$> match_prefix xs ys)
<|> return ([],x :: xs,y :: ys)
| xs ys := return ([],xs,ys)
/--
`(prefix,left,right,suffix) ← match_assoc unif l r` finds the
longest prefix and suffix common to `l` and `r` and
returns them along with the differences -/
meta def match_assoc (l : list expr) (r : list expr)
: tactic (list expr × list expr × list expr × list expr) :=
do (pre,l₁,r₁) ← match_prefix l r,
(suf,l₂,r₂) ← match_prefix (reverse l₁) (reverse r₁),
return (pre,reverse l₂,reverse r₂,reverse suf)
meta def check_ac : expr → tactic (bool × bool × option (expr × expr × expr) × expr)
| (expr.app (expr.app f x) y) :=
do t ← infer_type x,
a ← try_core $ to_expr ``(is_associative %%t %%f) >>= mk_instance,
c ← try_core $ to_expr ``(is_commutative %%t %%f) >>= mk_instance,
i ← try_core (do
v ← mk_meta_var t,
l_inst_p ← to_expr ``(is_left_id %%t %%f %%v),
r_inst_p ← to_expr ``(is_right_id %%t %%f %%v),
l_v ← mk_meta_var l_inst_p,
r_v ← mk_meta_var r_inst_p ,
l_id ← mk_mapp `is_left_id.left_id [some t,f,v,some l_v],
mk_instance l_inst_p >>= unify l_v,
r_id ← mk_mapp `is_right_id.right_id [none,f,v,some r_v],
mk_instance r_inst_p >>= unify r_v,
v' ← instantiate_mvars v,
return (l_id,r_id,v')),
return (a.is_some,c.is_some,i,f)
| _ := return (ff,ff,none,expr.var 1)
meta def parse_assoc_chain' (f : expr) : expr → tactic (dlist expr)
| e :=
(do (expr.app (expr.app f' x) y) ← return e,
is_def_eq f f',
(++) <$> parse_assoc_chain' x <*> parse_assoc_chain' y)
<|> return (singleton e)
meta def parse_assoc_chain (f : expr) : expr → tactic (list expr) :=
map dlist.to_list ∘ parse_assoc_chain' f
meta def fold_assoc (op : expr) : option (expr × expr × expr) → list expr → option (expr × list expr)
| _ (x::xs) := some (foldl (expr.app ∘ expr.app op) x xs, [])
| none [] := none
| (some (l_id,r_id,x₀)) [] := some (x₀,[l_id,r_id])
meta def fold_assoc1 (op : expr) : list expr → option expr
| (x::xs) := some $ foldl (expr.app ∘ expr.app op) x xs
| [] := none
meta def same_function_aux
: list expr → list expr → expr → expr → tactic (expr × list expr × list expr)
| xs₀ xs₁ (expr.app f₀ a₀) (expr.app f₁ a₁) :=
same_function_aux (a₀ :: xs₀) (a₁ :: xs₁) f₀ f₁
| xs₀ xs₁ e₀ e₁ := is_def_eq e₀ e₁ >> return (e₀,xs₀,xs₁)
meta def same_function : expr → expr → tactic (expr × list expr × list expr) :=
same_function_aux [] []
meta def parse_ac_mono_function (l r : expr)
: tactic (expr × expr × list expr × mono_function) :=
do (full_f,ls,rs) ← same_function l r,
(a,c,i,f) ← check_ac l,
if a
then if c
then do
(s,ls,rs) ← monad.join (match_ac
<$> parse_assoc_chain f l
<*> parse_assoc_chain f r),
(l',l_id) ← fold_assoc f i ls,
(r',r_id) ← fold_assoc f i rs,
s' ← fold_assoc1 f s,
return (l',r',l_id ++ r_id,mono_function.assoc_comm f s')
else do -- a ∧ ¬ c
(pre,ls,rs,suff) ← monad.join (match_assoc
<$> parse_assoc_chain f l
<*> parse_assoc_chain f r),
(l',l_id) ← fold_assoc f i ls,
(r',r_id) ← fold_assoc f i rs,
let pre' := fold_assoc1 f pre,
let suff' := fold_assoc1 f suff,
return (l',r',l_id ++ r_id,mono_function.assoc f pre' suff')
else do -- ¬ a
(xs₀,x₀,x₁,xs₁) ← find_one_difference opt ls rs,
return (x₀,x₁,[],mono_function.non_assoc full_f xs₀ xs₁)
meta def parse_ac_mono_function' (l r : pexpr) :=
do l' ← to_expr l,
r' ← to_expr r,
parse_ac_mono_function l' r'
meta def ac_monotonicity_goal : expr → tactic (expr × expr × list expr × ac_mono_ctx)
| `(%%e₀ → %%e₁) :=
do (l,r,id_rs,f) ← parse_ac_mono_function e₀ e₁,
t₀ ← infer_type e₀,
t₁ ← infer_type e₁,
rel_def ← to_expr ``(λ x₀ x₁, (x₀ : %%t₀) → (x₁ : %%t₁)),
return (e₀, e₁, id_rs,
{ function := f
, left := l, right := r
, to_rel := some $ expr.pi `x binder_info.default
, rel_def := rel_def })
| `(%%e₀ = %%e₁) :=
do (l,r,id_rs,f) ← parse_ac_mono_function e₀ e₁,
t₀ ← infer_type e₀,
t₁ ← infer_type e₁,
rel_def ← to_expr ``(λ x₀ x₁, (x₀ : %%t₀) = (x₁ : %%t₁)),
return (e₀, e₁, id_rs,
{ function := f
, left := l, right := r
, to_rel := none
, rel_def := rel_def })
| (expr.app (expr.app rel e₀) e₁) :=
do (l,r,id_rs,f) ← parse_ac_mono_function e₀ e₁,
return (e₀, e₁, id_rs,
{ function := f
, left := l, right := r
, to_rel := expr.app ∘ expr.app rel
, rel_def := rel })
| _ := fail "invalid monotonicity goal"
meta def bin_op_left (f : expr) : option expr → expr → expr
| none e := e
| (some e₀) e₁ := f.mk_app [e₀,e₁]
meta def bin_op (f a b : expr) : expr :=
f.mk_app [a,b]
meta def bin_op_right (f : expr) : expr → option expr → expr
| e none := e
| e₀ (some e₁) := f.mk_app [e₀,e₁]
meta def mk_fun_app : mono_function → expr → expr
| (mono_function.non_assoc f x y) z := f.mk_app (x ++ z :: y)
| (mono_function.assoc f x y) z := bin_op_left f x (bin_op_right f z y)
| (mono_function.assoc_comm f x) z := f.mk_app [z,x]
meta inductive mono_law
/- `assoc (l₀,r₀) (r₁,l₁)` gives first how to find rules to prove
x+(y₀+z) R x+(y₁+z);
if that fails, helps prove (x+y₀)+z R (x+y₁)+z -/
| assoc : expr × expr → expr × expr → mono_law
/- `congr r` gives the rule to prove `x = y → f x = f y` -/
| congr : expr → mono_law
| other : expr → mono_law
meta def mono_law.to_tactic_format : mono_law → tactic format
| (mono_law.other e) := do e ← pp e, return format!"other {e}"
| (mono_law.congr r) := do e ← pp r, return format!"congr {e}"
| (mono_law.assoc (x₀,x₁) (y₀,y₁)) :=
do x₀ ← pp x₀,
x₁ ← pp x₁,
y₀ ← pp y₀,
y₁ ← pp y₁,
return format!"assoc {x₀}; {x₁} | {y₀}; {y₁}"
meta instance has_to_tactic_format_mono_law : has_to_tactic_format mono_law :=
{ to_tactic_format := mono_law.to_tactic_format }
meta def mk_rel (ctx : ac_mono_ctx_ne) (f : expr → expr) : expr :=
ctx.to_rel (f ctx.left) (f ctx.right)
meta def mk_congr_args (fn : expr) (xs₀ xs₁ : list expr) (l r : expr) : tactic expr :=
do p ← mk_app `eq [fn.mk_app $ xs₀ ++ l :: xs₁,fn.mk_app $ xs₀ ++ r :: xs₁],
prod.snd <$> solve_aux p
(do iterate_exactly (xs₁.length) (applyc `congr_fun),
applyc `congr_arg)
meta def mk_congr_law (ctx : ac_mono_ctx) : tactic expr :=
match ctx.function with
| (mono_function.assoc f x₀ x₁) :=
if (x₀ <|> x₁).is_some
then mk_congr_args f x₀.to_monad x₁.to_monad ctx.left ctx.right
else failed
| (mono_function.assoc_comm f x₀) := mk_congr_args f [x₀] [] ctx.left ctx.right
| (mono_function.non_assoc f x₀ x₁) := mk_congr_args f x₀ x₁ ctx.left ctx.right
end
meta def mk_pattern (ctx : ac_mono_ctx) : tactic mono_law :=
match (sequence ctx : option (ac_mono_ctx' _)) with
| (some ctx) :=
match ctx.function with
| (mono_function.assoc f (some x) (some y)) :=
return $ mono_law.assoc
( mk_rel ctx (λ i, bin_op f x (bin_op f i y))
, mk_rel ctx (λ i, bin_op f i y))
( mk_rel ctx (λ i, bin_op f (bin_op f x i) y)
, mk_rel ctx (λ i, bin_op f x i))
| (mono_function.assoc f (some x) none) :=
return $ mono_law.other $
mk_rel ctx (λ e, mk_fun_app ctx.function e)
| (mono_function.assoc f none (some y)) :=
return $ mono_law.other $
mk_rel ctx (λ e, mk_fun_app ctx.function e)
| (mono_function.assoc f none none) :=
none
| _ :=
return $ mono_law.other $
mk_rel ctx (λ e, mk_fun_app ctx.function e)
end
| none := mono_law.congr <$> mk_congr_law ctx
end
meta def match_rule (pat : expr) (r : name) : tactic expr :=
do r' ← mk_const r,
t ← infer_type r',
t ← expr.dsimp t { fail_if_unchanged := ff } tt [] [
simp_arg_type.expr ``(monotone), simp_arg_type.expr ``(strict_mono)],
match_rule_head pat [] r' t
meta def find_lemma (pat : expr) : list name → tactic (list expr)
| [] := return []
| (r :: rs) :=
do (cons <$> match_rule pat r <|> pure id) <*> find_lemma rs
meta def match_chaining_rules (ls : list name) (x₀ x₁ : expr) : tactic (list expr) :=
do x' ← to_expr ``(%%x₁ → %%x₀),
r₀ ← find_lemma x' ls,
r₁ ← find_lemma x₁ ls,
return (expr.app <$> r₀ <*> r₁)
meta def find_rule (ls : list name) : mono_law → tactic (list expr)
| (mono_law.assoc (x₀,x₁) (y₀,y₁)) :=
(match_chaining_rules ls x₀ x₁)
<|> (match_chaining_rules ls y₀ y₁)
| (mono_law.congr r) := return [r]
| (mono_law.other p) := find_lemma p ls
universes u v
def apply_rel {α : Sort u} (R : α → α → Sort v) {x y : α}
(x' y' : α)
(h : R x y)
(hx : x = x')
(hy : y = y')
: R x' y' :=
by { rw [← hx,← hy], apply h }
meta def ac_refine (e : expr) : tactic unit :=
refine ``(eq.mp _ %%e) ; ac_refl
meta def one_line (e : expr) : tactic format :=
do lbl ← pp e,
asm ← infer_type e >>= pp,
return format!"\t{asm}\n"
meta def side_conditions (e : expr) : tactic format :=
do let vs := e.list_meta_vars,
ts ← mmap one_line vs.tail,
let r := e.get_app_fn.const_name,
return format!"{r}:\n{format.join ts}"
open monad
/-- tactic-facing function, similar to `interactive.tactic.generalize` with the
exception that meta variables -/
private meta def monotonicity.generalize' (h : name) (v : expr) (x : name) : tactic (expr × expr) :=
do tgt ← target,
t ← infer_type v,
tgt' ← do {
⟨tgt', _⟩ ← solve_aux tgt (tactic.generalize v x >> target),
to_expr ``(λ y : %%t, Π x, y = x → %%(tgt'.binding_body.lift_vars 0 1))
} <|> to_expr ``(λ y : %%t, Π x, %%v = x → %%tgt),
t ← head_beta (tgt' v) >>= assert h,
swap,
r ← mk_eq_refl v,
solve1 $ tactic.exact (t v r),
prod.mk <$> tactic.intro x <*> tactic.intro h
private meta def hide_meta_vars (tac : list expr → tactic unit) : tactic unit :=
focus1 $
do tgt ← target >>= instantiate_mvars,
tactic.change tgt,
ctx ← local_context,
let vs := tgt.list_meta_vars,
vs' ← mmap (λ v,
do h ← get_unused_name `h,
x ← get_unused_name `x,
prod.snd <$> monotonicity.generalize' h v x) vs,
tac ctx;
vs'.mmap' (try ∘ tactic.subst)
meta def hide_meta_vars' (tac : itactic) : itactic :=
hide_meta_vars $ λ _, tac
end config
meta def solve_mvar (v : expr) (tac : tactic unit) : tactic unit :=
do gs ← get_goals,
set_goals [v],
target >>= instantiate_mvars >>= tactic.change,
tac, done,
set_goals $ gs
def list.minimum_on {α β} [linear_order β] (f : α → β) : list α → list α
| [] := []
| (x :: xs) := prod.snd $ xs.foldl (λ ⟨k,a⟩ b,
let k' := f b in
if k < k' then (k,a)
else if k' < k then (k', [b])
else (k,b :: a)) (f x, [x])
open format mono_selection
meta def best_match {β} (xs : list expr) (tac : expr → tactic β) : tactic unit :=
do t ← target,
xs ← xs.mmap (λ x,
try_core $ prod.mk x <$> solve_aux t (tac x >> get_goals)),
let xs := xs.filter_map id,
let r := list.minimum_on (list.length ∘ prod.fst ∘ prod.snd) xs,
match r with
| [(_,gs,pr)] := tactic.exact pr >> set_goals gs
| [] := fail "no good match found"
| _ :=
do lmms ← r.mmap (λ ⟨l,gs,_⟩,
do ts ← gs.mmap infer_type,
msg ← ts.mmap pp,
pure $ foldl compose "\n\n" (list.intersperse "\n" $ to_fmt l.get_app_fn.const_name :: msg)),
let msg := foldl compose "" lmms,
fail format!"ambiguous match: {msg}\n\nTip: try asserting a side condition to distinguish between the lemmas"
end
meta def mono_aux (dir : parse side) :
tactic unit :=
do t ← target >>= instantiate_mvars,
ns ← get_monotonicity_lemmas t dir,
asms ← local_context,
rs ← find_lemma asms t ns,
focus1 $ () <$ best_match rs (λ law, tactic.refine $ to_pexpr law)
/--
- `mono` applies a monotonicity rule.
- `mono*` applies monotonicity rules repetitively.
- `mono with x ≤ y` or `mono with [0 ≤ x,0 ≤ y]` creates an assertion for the listed
propositions. Those help to select the right monotonicity rule.
- `mono left` or `mono right` is useful when proving strict orderings:
for `x + y < w + z` could be broken down into either
- left: `x ≤ w` and `y < z` or
- right: `x < w` and `y ≤ z`
- `mono using [rule1,rule2]` calls `simp [rule1,rule2]` before applying mono.
- The general syntax is `mono '*'? ('with' hyp | 'with' [hyp1,hyp2])? ('using' [hyp1,hyp2])? mono_cfg?
To use it, first import `tactic.monotonicity`.
Here is an example of mono:
```lean
example (x y z k : ℤ)
(h : 3 ≤ (4 : ℤ))
(h' : z ≤ y) :
(k + 3 + x) - y ≤ (k + 4 + x) - z :=
begin
mono, -- unfold `(-)`, apply add_le_add
{ -- ⊢ k + 3 + x ≤ k + 4 + x
mono, -- apply add_le_add, refl
-- ⊢ k + 3 ≤ k + 4
mono },
{ -- ⊢ -y ≤ -z
mono /- apply neg_le_neg -/ }
end
```
More succinctly, we can prove the same goal as:
```lean
example (x y z k : ℤ)
(h : 3 ≤ (4 : ℤ))
(h' : z ≤ y) :
(k + 3 + x) - y ≤ (k + 4 + x) - z :=
by mono*
```
-/
meta def mono (many : parse (tk "*")?)
(dir : parse side)
(hyps : parse $ tk "with" *> pexpr_list_or_texpr <|> pure [])
(simp_rules : parse $ tk "using" *> simp_arg_list <|> pure []) :
tactic unit :=
do hyps ← hyps.mmap (λ p, to_expr p >>= mk_meta_var),
hyps.mmap' (λ pr, do h ← get_unused_name `h, note h none pr),
when (¬ simp_rules.empty) (simp_core { } failed tt simp_rules [] (loc.ns [none]) >> skip),
if many.is_some
then repeat $ mono_aux dir
else mono_aux dir,
gs ← get_goals,
set_goals $ hyps ++ gs
add_tactic_doc
{ name := "mono",
category := doc_category.tactic,
decl_names := [`tactic.interactive.mono],
tags := ["monotonicity"] }
/--
transforms a goal of the form `f x ≼ f y` into `x ≤ y` using lemmas
marked as `monotonic`.
Special care is taken when `f` is the repeated application of an
associative operator and if the operator is commutative
-/
meta def ac_mono_aux (cfg : mono_cfg := { mono_cfg . }) :
tactic unit :=
hide_meta_vars $ λ asms,
do try `[simp only [sub_eq_add_neg]],
tgt ← target >>= instantiate_mvars,
(l,r,id_rs,g) ← ac_monotonicity_goal cfg tgt
<|> fail "monotonic context not found",
ns ← get_monotonicity_lemmas tgt both,
p ← mk_pattern g,
rules ← find_rule asms ns p <|> fail "no applicable rules found",
when (rules = []) (fail "no applicable rules found"),
err ← format.join <$> mmap side_conditions rules,
focus1 $ best_match rules (λ rule, do
t₀ ← mk_meta_var `(Prop),
v₀ ← mk_meta_var t₀,
t₁ ← mk_meta_var `(Prop),
v₁ ← mk_meta_var t₁,
tactic.refine $ ``(apply_rel %%(g.rel_def) %%l %%r %%rule %%v₀ %%v₁),
solve_mvar v₀ (try (any_of id_rs rewrite_target) >>
( done <|>
refl <|>
ac_refl <|>
`[simp only [is_associative.assoc]]) ),
solve_mvar v₁ (try (any_of id_rs rewrite_target) >>
( done <|>
refl <|>
ac_refl <|>
`[simp only [is_associative.assoc]]) ),
n ← num_goals,
iterate_exactly (n-1) (try $ solve1 $ apply_instance <|>
tactic.solve_by_elim { lemmas := some asms }))
open sum nat
/-- (repeat_until_or_at_most n t u): repeat tactic `t` at most n times or until u succeeds -/
meta def repeat_until_or_at_most : nat → tactic unit → tactic unit → tactic unit
| 0 t _ := fail "too many applications"
| (succ n) t u := u <|> (t >> repeat_until_or_at_most n t u)
meta def repeat_until : tactic unit → tactic unit → tactic unit :=
repeat_until_or_at_most 100000
@[derive _root_.has_reflect, derive _root_.inhabited]
inductive rep_arity : Type
| one | exactly (n : ℕ) | many
meta def repeat_or_not : rep_arity → tactic unit → option (tactic unit) → tactic unit
| rep_arity.one tac none := tac
| rep_arity.many tac none := repeat tac
| (rep_arity.exactly n) tac none := iterate_exactly' n tac
| rep_arity.one tac (some until) := tac >> until
| rep_arity.many tac (some until) := repeat_until tac until
| (rep_arity.exactly n) tac (some until) := iterate_exactly n tac >> until
meta def assert_or_rule : lean.parser (pexpr ⊕ pexpr) :=
(tk ":=" *> inl <$> texpr <|> (tk ":" *> inr <$> texpr))
meta def arity : lean.parser rep_arity :=
rep_arity.many <$ tk "*" <|>
rep_arity.exactly <$> (tk "^" *> small_nat) <|>
pure rep_arity.one
/--
`ac_mono` reduces the `f x ⊑ f y`, for some relation `⊑` and a
monotonic function `f` to `x ≺ y`.
`ac_mono*` unwraps monotonic functions until it can't.
`ac_mono^k`, for some literal number `k` applies monotonicity `k`
times.
`ac_mono h`, with `h` a hypothesis, unwraps monotonic functions and
uses `h` to solve the remaining goal. Can be combined with `*` or `^k`:
`ac_mono* h`
`ac_mono : p` asserts `p` and uses it to discharge the goal result
unwrapping a series of monotonic functions. Can be combined with * or
^k: `ac_mono* : p`
In the case where `f` is an associative or commutative operator,
`ac_mono` will consider any possible permutation of its arguments and
use the one the minimizes the difference between the left-hand side
and the right-hand side.
To use it, first import `tactic.monotonicity`.
`ac_mono` can be used as follows:
```lean
example (x y z k m n : ℕ)
(h₀ : z ≥ 0)
(h₁ : x ≤ y) :
(m + x + n) * z + k ≤ z * (y + n + m) + k :=
begin
ac_mono,
-- ⊢ (m + x + n) * z ≤ z * (y + n + m)
ac_mono,
-- ⊢ m + x + n ≤ y + n + m
ac_mono,
end
```
As with `mono*`, `ac_mono*` solves the goal in one go and so does
`ac_mono* h₁`. The latter syntax becomes especially interesting in the
following example:
```lean
example (x y z k m n : ℕ)
(h₀ : z ≥ 0)
(h₁ : m + x + n ≤ y + n + m) :
(m + x + n) * z + k ≤ z * (y + n + m) + k :=
by ac_mono* h₁.
```
By giving `ac_mono` the assumption `h₁`, we are asking `ac_refl` to
stop earlier than it would normally would.
-/
meta def ac_mono (rep : parse arity) :
parse assert_or_rule? →
opt_param mono_cfg { mono_cfg . } →
tactic unit
| none opt := focus1 $ repeat_or_not rep (ac_mono_aux opt) none
| (some (inl h)) opt :=
do focus1 $ repeat_or_not rep (ac_mono_aux opt) (some $ done <|> to_expr h >>= ac_refine)
| (some (inr t)) opt :=
do h ← i_to_expr t >>= assert `h,
tactic.swap,
focus1 $ repeat_or_not rep (ac_mono_aux opt) (some $ done <|> ac_refine h)
/-
TODO(Simon): with `ac_mono h` and `ac_mono : p` split the remaining
gaol if the provided rule does not solve it completely.
-/
add_tactic_doc
{ name := "ac_mono",
category := doc_category.tactic,
decl_names := [`tactic.interactive.ac_mono],
tags := ["monotonicity"] }
attribute [mono] and.imp or.imp
end tactic.interactive
|
210b981341eeb015949b351dd2b878051c66f91a | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/category_theory/lifting_properties.lean | 4ebc2ffa9b26495f65ceb71111ec71e4084a5af3 | [
"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 | 5,342 | lean | /-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import category_theory.arrow
import category_theory.limits.preserves.shapes.terminal
/-!
# Lifting properties
This file defines the lifting property of two arrows in a category and shows basic properties of
this notion.
We also construct the subcategory consisting of those morphisms which have the right lifting
property with respect to arrows in a given diagram.
## Main results
- `has_lifting_property`: the definition of the lifting property
- `iso_has_right_lifting_property`: any isomorphism satisfies the right lifting property (rlp)
- `id_has_right_lifting_property`: any identity has the rlp
- `right_lifting_property_initial_iff`: spells out the rlp with respect to a map whose source is an
initial object
- `right_lifting_subcat`: given a set of arrows `F : D → arrow C`, we construct the subcategory
of those morphisms `p` in `C` that satisfy the rlp w.r.t. `F i`, for any element `i` of `D`.
## Tags
lifting property
-/
open category_theory.limits
namespace category_theory
universes v u v₁
variables {C : Type u} [category.{v} C]
variables {D : Type v₁}
variables {X Y Z : C}
/-- The lifting property of a morphism `i` with respect to a morphism `p`.
This can be interpreted as the right lifting property of `i` with respect to `p`,
or the left lifting property of `p` with respect to `i`. -/
class has_lifting_property (i p : arrow C) : Prop :=
(sq_has_lift : ∀ (sq : i ⟶ p), arrow.has_lift sq)
@[priority 100] -- see Note [lower instance priority]
instance has_lifting_property' {i p : arrow C} [has_lifting_property i p] (sq : i ⟶ p) :
arrow.has_lift sq :=
has_lifting_property.sq_has_lift sq
/-- Any isomorphism has the right lifting property with respect to any map.
A → X
↓i ↓p≅
B → Y
-/
lemma iso_has_right_lifting_property (i : arrow C) (p : X ≅ Y) :
has_lifting_property i (arrow.mk p.hom) :=
⟨λ sq, ⟨⟨{ lift := sq.right ≫ p.inv, }⟩⟩⟩ -- the lift is obtained by p⁻¹ ∘ (B → Y)
/-- Any identity has the right lifting property with respect to any map. -/
lemma id_has_right_lifting_property (i : arrow C) : has_lifting_property i (arrow.mk (𝟙 X)) :=
iso_has_right_lifting_property i (iso.refl _)
/-- An equivalent characterization for right lifting with respect to a map `i` whose source is
initial.
∅ → X
↓ ↓
B → Y has a lifting iff there is a map B → X making the right part commute.
-/
lemma right_lifting_property_initial_iff (i p : arrow C) (h : is_initial i.left) :
has_lifting_property i p ↔ ∀ {e : i.right ⟶ p.right}, ∃ l : i.right ⟶ p.left, l ≫ p.hom = e :=
begin
fsplit,
{ introsI hlift e,
have comm : (is_initial.to h p.left) ≫ p.hom = i.hom ≫ e :=
is_initial.hom_ext h _ _,
use arrow.lift (arrow.hom_mk comm),
simp },
{ refine λ hlift, ⟨λ sq, _⟩,
obtain ⟨l, hl⟩ : ∃ (l : i.right ⟶ p.left), l ≫ p.hom = sq.right := hlift,
exact arrow.has_lift.mk ⟨l, is_initial.hom_ext h _ _⟩, }
end
/-- The condition of having the rlp with respect to a morphism `i` is stable under composition. -/
lemma has_right_lifting_property_comp {i : arrow C} {f : X ⟶ Y} {g : Y ⟶ Z}
(hf : has_lifting_property i (arrow.mk f))
(hg : has_lifting_property i (arrow.mk g)) :
has_lifting_property i (arrow.mk (f ≫ g)) :=
{ sq_has_lift := λ sq1,
-- construct a square i ⟶ f
let sq2 : i ⟶ (arrow.mk f) := ⟨sq1.left, arrow.lift (arrow.square_to_snd sq1)⟩ in
-- show that the lift of this square is a lift of i with respect to g ∘ f
⟨⟨⟨(arrow.lift sq2 : _ ⟶ _), by simp⟩⟩⟩ }
/-- The objects of the subcategory `right_lifting_subcategory` are the ones in the
underlying category. -/
def right_lifting_subcat (R : Type u) := R
instance right_lifting_subcat.inhabited (R : Type u) [inhabited R] :
inhabited (right_lifting_subcat R) :=
{ default := (default R : R) }
/-- The objects of the subcategory `right_lifting_subcategory` are the ones in the
underlying category. -/
def right_lifting_subcat.X {R : Type u} (x : right_lifting_subcat R) : R := x
lemma id_has_right_lifting_property' {F : D → arrow C} (X : C) :
∀ i : D, has_lifting_property (F i) (arrow.mk (𝟙 X)) :=
λ i, id_has_right_lifting_property (F i)
lemma has_right_lifting_property_comp'
{F : D → arrow C} {f : X ⟶ Y} (hf : ∀ i : D, has_lifting_property (F i) (arrow.mk f))
{g : Y ⟶ Z} (hg : ∀ i : D, has_lifting_property (F i) (arrow.mk g)) :
∀ i : D, has_lifting_property (F i) (arrow.mk (f ≫ g)) :=
λ i, has_right_lifting_property_comp (hf i) (hg i)
/-- Given a set of arrows in C, indexed by `F : D → arrow C`,
we construct the (non-full) subcategory of `C`
spanned by those morphisms that have the right lifting property relative to all maps
of the form `F i`, where `i` is any element in `D`. -/
def right_lifting_subcategory (F : D → arrow C) : category (right_lifting_subcat C) :=
{ hom := λ X Y, { p : X ⟶ Y // ∀ {i : D}, has_lifting_property (F i) (arrow.mk p) },
id := λ X, ⟨𝟙 X, id_has_right_lifting_property' X⟩,
comp := λ X Y Z f g, ⟨f.val ≫ g.val, has_right_lifting_property_comp' f.property g.property⟩ }
end category_theory
|
1f145690f27f4dfe37881398061e4df846193786 | 853df553b1d6ca524e3f0a79aedd32dde5d27ec3 | /src/order/order_iso.lean | a9e7fad94bc65905a3be27b8ad0c3e33f6bb633d | [
"Apache-2.0"
] | permissive | DanielFabian/mathlib | efc3a50b5dde303c59eeb6353ef4c35a345d7112 | f520d07eba0c852e96fe26da71d85bf6d40fcc2a | refs/heads/master | 1,668,739,922,971 | 1,595,201,756,000 | 1,595,201,756,000 | 279,469,476 | 0 | 0 | null | 1,594,696,604,000 | 1,594,696,604,000 | null | UTF-8 | Lean | false | false | 14,861 | lean | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import logic.embedding
import logic.function.iterate
import order.rel_classes
open function
universes u v w
variables {α : Type*} {β : Type*} {γ : Type*}
{r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop}
/-- An increasing function is injective -/
lemma injective_of_increasing (r : α → α → Prop) (s : β → β → Prop) [is_trichotomous α r]
[is_irrefl β s] (f : α → β) (hf : ∀{x y}, r x y → s (f x) (f y)) : injective f :=
begin
intros x y hxy,
rcases trichotomous_of r x y with h | h | h,
have := hf h, rw hxy at this, exfalso, exact irrefl_of s (f y) this,
exact h,
have := hf h, rw hxy at this, exfalso, exact irrefl_of s (f y) this
end
/-- An order embedding with respect to a given pair of orders `r` and `s`
is an embedding `f : α ↪ β` such that `r a b ↔ s (f a) (f b)`. -/
structure order_embedding {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends α ↪ β :=
(ord' : ∀ {a b}, r a b ↔ s (to_embedding a) (to_embedding b))
infix ` ≼o `:25 := order_embedding
/-- the induced order on a subtype is an embedding under the natural inclusion. -/
definition subtype.order_embedding {X : Type*} (r : X → X → Prop) (p : X → Prop) :
((subtype.val : subtype p → X) ⁻¹'o r) ≼o r :=
⟨⟨subtype.val,subtype.val_injective⟩,by intros;refl⟩
theorem preimage_equivalence {α β} (f : α → β) {s : β → β → Prop}
(hs : equivalence s) : equivalence (f ⁻¹'o s) :=
⟨λ a, hs.1 _, λ a b h, hs.2.1 h, λ a b c h₁ h₂, hs.2.2 h₁ h₂⟩
namespace order_embedding
instance : has_coe_to_fun (r ≼o s) := ⟨λ _, α → β, λ o, o.to_embedding⟩
theorem injective (f : r ≼o s) : injective f := f.inj'
theorem ord (f : r ≼o s) : ∀ {a b}, r a b ↔ s (f a) (f b) := f.ord'
@[simp] theorem coe_fn_mk (f : α ↪ β) (o) :
(@order_embedding.mk _ _ r s f o : α → β) = f := rfl
@[simp] theorem coe_fn_to_embedding (f : r ≼o s) : (f.to_embedding : α → β) = f := rfl
/-- The map `coe_fn : (r ≼o s) → (r → s)` is injective. We can't use `function.injective`
here but mimic its signature by using `⦃e₁ e₂⦄`. -/
theorem coe_fn_inj : ∀ ⦃e₁ e₂ : r ≼o s⦄, (e₁ : α → β) = e₂ → e₁ = e₂
| ⟨⟨f₁, h₁⟩, o₁⟩ ⟨⟨f₂, h₂⟩, o₂⟩ h := by { congr, exact h }
@[refl] protected def refl (r : α → α → Prop) : r ≼o r :=
⟨embedding.refl _, λ a b, iff.rfl⟩
@[trans] protected def trans (f : r ≼o s) (g : s ≼o t) : r ≼o t :=
⟨f.1.trans g.1, λ a b, by rw [f.2, g.2]; simp⟩
@[simp] theorem refl_apply (x : α) : order_embedding.refl r x = x := rfl
@[simp] theorem trans_apply (f : r ≼o s) (g : s ≼o t) (a : α) : (f.trans g) a = g (f a) := rfl
/-- An order embedding is also an order embedding between dual orders. -/
def rsymm (f : r ≼o s) : swap r ≼o swap s :=
⟨f.to_embedding, λ a b, f.ord⟩
/-- If `f` is injective, then it is an order embedding from the
preimage order of `s` to `s`. -/
def preimage (f : α ↪ β) (s : β → β → Prop) : f ⁻¹'o s ≼o s := ⟨f, λ a b, iff.rfl⟩
theorem eq_preimage (f : r ≼o s) : r = f ⁻¹'o s :=
by { ext a b, exact f.ord }
protected theorem is_irrefl : ∀ (f : r ≼o s) [is_irrefl β s], is_irrefl α r
| ⟨f, o⟩ ⟨H⟩ := ⟨λ a h, H _ (o.1 h)⟩
protected theorem is_refl : ∀ (f : r ≼o s) [is_refl β s], is_refl α r
| ⟨f, o⟩ ⟨H⟩ := ⟨λ a, o.2 (H _)⟩
protected theorem is_symm : ∀ (f : r ≼o s) [is_symm β s], is_symm α r
| ⟨f, o⟩ ⟨H⟩ := ⟨λ a b h, o.2 (H _ _ (o.1 h))⟩
protected theorem is_asymm : ∀ (f : r ≼o s) [is_asymm β s], is_asymm α r
| ⟨f, o⟩ ⟨H⟩ := ⟨λ a b h₁ h₂, H _ _ (o.1 h₁) (o.1 h₂)⟩
protected theorem is_antisymm : ∀ (f : r ≼o s) [is_antisymm β s], is_antisymm α r
| ⟨f, o⟩ ⟨H⟩ := ⟨λ a b h₁ h₂, f.inj' (H _ _ (o.1 h₁) (o.1 h₂))⟩
protected theorem is_trans : ∀ (f : r ≼o s) [is_trans β s], is_trans α r
| ⟨f, o⟩ ⟨H⟩ := ⟨λ a b c h₁ h₂, o.2 (H _ _ _ (o.1 h₁) (o.1 h₂))⟩
protected theorem is_total : ∀ (f : r ≼o s) [is_total β s], is_total α r
| ⟨f, o⟩ ⟨H⟩ := ⟨λ a b, (or_congr o o).2 (H _ _)⟩
protected theorem is_preorder : ∀ (f : r ≼o s) [is_preorder β s], is_preorder α r
| f H := by exactI {..f.is_refl, ..f.is_trans}
protected theorem is_partial_order : ∀ (f : r ≼o s) [is_partial_order β s], is_partial_order α r
| f H := by exactI {..f.is_preorder, ..f.is_antisymm}
protected theorem is_linear_order : ∀ (f : r ≼o s) [is_linear_order β s], is_linear_order α r
| f H := by exactI {..f.is_partial_order, ..f.is_total}
protected theorem is_strict_order : ∀ (f : r ≼o s) [is_strict_order β s], is_strict_order α r
| f H := by exactI {..f.is_irrefl, ..f.is_trans}
protected theorem is_trichotomous : ∀ (f : r ≼o s) [is_trichotomous β s], is_trichotomous α r
| ⟨f, o⟩ ⟨H⟩ := ⟨λ a b, (or_congr o (or_congr f.inj'.eq_iff.symm o)).2 (H _ _)⟩
protected theorem is_strict_total_order' : ∀ (f : r ≼o s) [is_strict_total_order' β s], is_strict_total_order' α r
| f H := by exactI {..f.is_trichotomous, ..f.is_strict_order}
protected theorem acc (f : r ≼o s) (a : α) : acc s (f a) → acc r a :=
begin
generalize h : f a = b, intro ac,
induction ac with _ H IH generalizing a, subst h,
exact ⟨_, λ a' h, IH (f a') (f.ord.1 h) _ rfl⟩
end
protected theorem well_founded : ∀ (f : r ≼o s) (h : well_founded s), well_founded r
| f ⟨H⟩ := ⟨λ a, f.acc _ (H _)⟩
protected theorem is_well_order : ∀ (f : r ≼o s) [is_well_order β s], is_well_order α r
| f H := by exactI {wf := f.well_founded H.wf, ..f.is_strict_total_order'}
/-- It suffices to prove `f` is monotone between strict orders
to show it is an order embedding. -/
def of_monotone [is_trichotomous α r] [is_asymm β s] (f : α → β) (H : ∀ a b, r a b → s (f a) (f b)) : r ≼o s :=
begin
haveI := @is_asymm.is_irrefl β s _,
refine ⟨⟨f, λ a b e, _⟩, λ a b, ⟨H _ _, λ h, _⟩⟩,
{ refine ((@trichotomous _ r _ a b).resolve_left _).resolve_right _;
exact λ h, @irrefl _ s _ _ (by simpa [e] using H _ _ h) },
{ refine (@trichotomous _ r _ a b).resolve_right (or.rec (λ e, _) (λ h', _)),
{ subst e, exact irrefl _ h },
{ exact asymm (H _ _ h') h } }
end
@[simp] theorem of_monotone_coe [is_trichotomous α r] [is_asymm β s] (f : α → β) (H) :
(@of_monotone _ _ r s _ _ f H : α → β) = f := rfl
-- If le is preserved by an order embedding of preorders, then lt is too
def lt_embedding_of_le_embedding [preorder α] [preorder β]
(f : (has_le.le : α → α → Prop) ≼o (has_le.le : β → β → Prop)) :
(has_lt.lt : α → α → Prop) ≼o (has_lt.lt : β → β → Prop) :=
{ ord' := by intros; simp [lt_iff_le_not_le,f.ord], .. f }
end order_embedding
/-- The inclusion map `fin n → ℕ` is an order embedding. -/
def fin.val.order_embedding (n) : @order_embedding (fin n) ℕ (<) (<) :=
⟨⟨fin.val, @fin.eq_of_veq _⟩, λ a b, iff.rfl⟩
/-- The inclusion map `fin m → fin n` is an order embedding. -/
def fin_fin.order_embedding {m n} (h : m ≤ n) : @order_embedding (fin m) (fin n) (<) (<) :=
⟨⟨λ ⟨x, h'⟩, ⟨x, lt_of_lt_of_le h' h⟩,
λ ⟨a, _⟩ ⟨b, _⟩ h, by congr; injection h⟩,
by intros; cases a; cases b; refl⟩
instance fin.lt.is_well_order (n) : is_well_order (fin n) (<) :=
(fin.val.order_embedding _).is_well_order
/-- An order isomorphism is an equivalence that is also an order embedding. -/
structure order_iso {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends α ≃ β :=
(ord' : ∀ {a b}, r a b ↔ s (to_equiv a) (to_equiv b))
infix ` ≃o `:25 := order_iso
namespace order_iso
/-- Convert an `order_iso` to an `order_embedding`. This function is also available as a coercion
but often it is easier to write `f.to_order_embedding` than to write explicitly `r` and `s`
in the target type. -/
def to_order_embedding (f : r ≃o s) : r ≼o s :=
⟨f.to_equiv.to_embedding, f.ord'⟩
instance : has_coe (r ≃o s) (r ≼o s) := ⟨to_order_embedding⟩
-- see Note [function coercion]
instance : has_coe_to_fun (r ≃o s) := ⟨λ _, α → β, λ f, f⟩
@[simp] lemma to_order_embedding_eq_coe (f : r ≃o s) : f.to_order_embedding = f := rfl
@[simp] lemma coe_coe_fn (f : r ≃o s) : ((f : r ≼o s) : α → β) = f := rfl
theorem ord (f : r ≃o s) : ∀ {a b}, r a b ↔ s (f a) (f b) := f.ord'
lemma ord'' {r : α → α → Prop} {s : β → β → Prop} (f : r ≃o s) {x y : α} :
r x y ↔ s ((↑f : r ≼o s) x) ((↑f : r ≼o s) y) := f.ord
@[simp] theorem coe_fn_mk (f : α ≃ β) (o) :
(@order_iso.mk _ _ r s f o : α → β) = f := rfl
@[simp] theorem coe_fn_to_equiv (f : r ≃o s) : (f.to_equiv : α → β) = f := rfl
theorem to_equiv_injective : injective (to_equiv : (r ≃o s) → α ≃ β)
| ⟨e₁, o₁⟩ ⟨e₂, o₂⟩ h := by { congr, exact h }
/-- The map `coe_fn : (r ≃o s) → (r → s)` is injective. We can't use `function.injective`
here but mimic its signature by using `⦃e₁ e₂⦄`. -/
theorem coe_fn_injective ⦃e₁ e₂ : r ≃o s⦄ (h : (e₁ : α → β) = e₂) : e₁ = e₂ :=
to_equiv_injective $ equiv.coe_fn_injective h
@[ext] theorem ext {e₁ e₂ : r ≃o s} (h : ∀ x, e₁ x = e₂ x) : e₁ = e₂ :=
coe_fn_injective $ funext h
/-- Identity map is an order isomorphism. -/
@[refl] protected def refl (r : α → α → Prop) : r ≃o r :=
⟨equiv.refl _, λ a b, iff.rfl⟩
/-- Inverse map of an order isomorphism is an order isomorphism. -/
@[symm] protected def symm (f : r ≃o s) : s ≃o r :=
⟨f.to_equiv.symm, λ a b, by cases f with f o; rw o; simp⟩
/-- Composition of two order isomorphisms is an order isomorphism. -/
@[trans] protected def trans (f₁ : r ≃o s) (f₂ : s ≃o t) : r ≃o t :=
⟨f₁.to_equiv.trans f₂.to_equiv, λ a b, f₁.ord.trans f₂.ord⟩
/-- An order isomorphism is also an order isomorphism between dual orders. -/
def rsymm (f : r ≃o s) : (swap r) ≃o (swap s) :=
⟨f.to_equiv, λ _ _, f.ord⟩
@[simp] theorem coe_fn_symm_mk (f o) : ((@order_iso.mk _ _ r s f o).symm : β → α) = f.symm :=
rfl
@[simp] theorem refl_apply (x : α) : order_iso.refl r x = x := rfl
@[simp] theorem trans_apply (f : r ≃o s) (g : s ≃o t) (a : α) : (f.trans g) a = g (f a) :=
rfl
@[simp] theorem apply_symm_apply (e : r ≃o s) (x : β) : e (e.symm x) = x :=
e.to_equiv.apply_symm_apply x
@[simp] theorem symm_apply_apply (e : r ≃o s) (x : α) : e.symm (e x) = x :=
e.to_equiv.symm_apply_apply x
theorem rel_symm_apply (e : r ≃o s) {x y} : r x (e.symm y) ↔ s (e x) y :=
by rw [e.ord, e.apply_symm_apply]
theorem symm_apply_rel (e : r ≃o s) {x y} : r (e.symm x) y ↔ s x (e y) :=
by rw [e.ord, e.apply_symm_apply]
protected lemma bijective (e : r ≃o s) : bijective e := e.to_equiv.bijective
protected lemma injective (e : r ≃o s) : injective e := e.to_equiv.injective
protected lemma surjective (e : r ≃o s) : surjective e := e.to_equiv.surjective
/-- Any equivalence lifts to an order isomorphism between `s` and its preimage. -/
protected def preimage (f : α ≃ β) (s : β → β → Prop) : f ⁻¹'o s ≃o s := ⟨f, λ a b, iff.rfl⟩
/-- A surjective order embedding is an order isomorphism. -/
noncomputable def of_surjective (f : r ≼o s) (H : surjective f) : r ≃o s :=
⟨equiv.of_bijective f ⟨f.injective, H⟩, by simp [f.ord']⟩
@[simp] theorem of_surjective_coe (f : r ≼o s) (H) : (of_surjective f H : α → β) = f :=
rfl
def sum_lex_congr {α₁ α₂ β₁ β₂ r₁ r₂ s₁ s₂}
(e₁ : @order_iso α₁ α₂ r₁ r₂) (e₂ : @order_iso β₁ β₂ s₁ s₂) :
sum.lex r₁ s₁ ≃o sum.lex r₂ s₂ :=
⟨equiv.sum_congr e₁.to_equiv e₂.to_equiv, λ a b,
by cases e₁ with f hf; cases e₂ with g hg;
cases a; cases b; simp [hf, hg]⟩
def prod_lex_congr {α₁ α₂ β₁ β₂ r₁ r₂ s₁ s₂}
(e₁ : @order_iso α₁ α₂ r₁ r₂) (e₂ : @order_iso β₁ β₂ s₁ s₂) :
prod.lex r₁ s₁ ≃o prod.lex r₂ s₂ :=
⟨equiv.prod_congr e₁.to_equiv e₂.to_equiv, λ a b, begin
cases e₁ with f hf; cases e₂ with g hg,
cases a with a₁ a₂; cases b with b₁ b₂,
suffices : prod.lex r₁ s₁ (a₁, a₂) (b₁, b₂) ↔
prod.lex r₂ s₂ (f a₁, g a₂) (f b₁, g b₂), {simpa [hf, hg]},
split,
{ intro h, cases h with _ _ _ _ h _ _ _ h,
{ left, exact hf.1 h },
{ right, exact hg.1 h } },
{ generalize e : f b₁ = fb₁,
intro h, cases h with _ _ _ _ h _ _ _ h,
{ subst e, left, exact hf.2 h },
{ have := f.injective e, subst b₁,
right, exact hg.2 h } }
end⟩
instance : group (r ≃o r) :=
{ one := order_iso.refl r,
mul := λ f₁ f₂, f₂.trans f₁,
inv := order_iso.symm,
mul_assoc := λ f₁ f₂ f₃, rfl,
one_mul := λ f, ext $ λ _, rfl,
mul_one := λ f, ext $ λ _, rfl,
mul_left_inv := λ f, ext f.symm_apply_apply }
@[simp] lemma coe_one : ⇑(1 : r ≃o r) = id := rfl
@[simp] lemma coe_mul (e₁ e₂ : r ≃o r) : ⇑(e₁ * e₂) = e₁ ∘ e₂ := rfl
lemma mul_apply (e₁ e₂ : r ≃o r) (x : α) : (e₁ * e₂) x = e₁ (e₂ x) := rfl
@[simp] lemma inv_apply_self (e : r ≃o r) (x) : e⁻¹ (e x) = x := e.symm_apply_apply x
@[simp] lemma apply_inv_self (e : r ≃o r) (x) : e (e⁻¹ x) = x := e.apply_symm_apply x
end order_iso
/-- A subset `p : set α` embeds into `α` -/
def set_coe_embedding {α : Type*} (p : set α) : p ↪ α := ⟨subtype.val, @subtype.eq _ _⟩
/-- `subrel r p` is the inherited relation on a subset. -/
def subrel (r : α → α → Prop) (p : set α) : p → p → Prop :=
@subtype.val _ p ⁻¹'o r
@[simp] theorem subrel_val (r : α → α → Prop) (p : set α)
{a b} : subrel r p a b ↔ r a.1 b.1 := iff.rfl
namespace subrel
protected def order_embedding (r : α → α → Prop) (p : set α) :
subrel r p ≼o r := ⟨set_coe_embedding _, λ a b, iff.rfl⟩
@[simp] theorem order_embedding_apply (r : α → α → Prop) (p a) :
subrel.order_embedding r p a = a.1 := rfl
instance (r : α → α → Prop) [is_well_order α r]
(p : set α) : is_well_order p (subrel r p) :=
order_embedding.is_well_order (subrel.order_embedding r p)
end subrel
/-- Restrict the codomain of an order embedding -/
def order_embedding.cod_restrict (p : set β) (f : r ≼o s) (H : ∀ a, f a ∈ p) : r ≼o subrel s p :=
⟨f.to_embedding.cod_restrict p H, f.ord'⟩
@[simp] theorem order_embedding.cod_restrict_apply (p) (f : r ≼o s) (H a) :
order_embedding.cod_restrict p f H a = ⟨f a, H a⟩ := rfl
|
066478548bb5a4447898c7ecdce2a7e4e2045842 | 957a80ea22c5abb4f4670b250d55534d9db99108 | /tests/lean/run/do_match_else.lean | bf16e9dfd427b46c46d2c287b0c8737ebe5b5ccf | [
"Apache-2.0"
] | permissive | GaloisInc/lean | aa1e64d604051e602fcf4610061314b9a37ab8cd | f1ec117a24459b59c6ff9e56a1d09d9e9e60a6c0 | refs/heads/master | 1,592,202,909,807 | 1,504,624,387,000 | 1,504,624,387,000 | 75,319,626 | 2 | 1 | Apache-2.0 | 1,539,290,164,000 | 1,480,616,104,000 | C++ | UTF-8 | Lean | false | false | 566 | lean | open tactic
set_option pp.all true
meta def app2 (f a b : expr) :=
expr.app (expr.app f a) b
example (a b c x y : nat) (H : nat.add (nat.add x y) y = 0) : true :=
by do
a ← get_local `a, b ← get_local `b, c ← get_local `c,
H ← get_local `H >>= infer_type,
(lhs, rhs) ← match_eq H,
nat_add : expr ← mk_const `nat.add,
p : pattern ← mk_pattern [] [a, b] (app2 nat_add a b) [] [app2 nat_add b a, a, b],
trace (pattern.moutput p),
[v₁, v₂, v₃] ← match_pattern p lhs | failed,
trace v₁, trace v₂, trace v₃,
constructor
|
78b7fe649367aadf641fed41a4e03894bfd29269 | fecda8e6b848337561d6467a1e30cf23176d6ad0 | /src/data/padics/ring_homs.lean | 06ec8c3472d65d00c93443d0accd24d4f7fae22a | [
"Apache-2.0"
] | permissive | spolu/mathlib | bacf18c3d2a561d00ecdc9413187729dd1f705ed | 480c92cdfe1cf3c2d083abded87e82162e8814f4 | refs/heads/master | 1,671,684,094,325 | 1,600,736,045,000 | 1,600,736,045,000 | 297,564,749 | 1 | 0 | null | 1,600,758,368,000 | 1,600,758,367,000 | null | UTF-8 | Lean | false | false | 24,775 | lean | /-
Copyright (c) 2020 Johan Commelin and Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin and Robert Y. Lewis
-/
import data.padics.padic_integers
/-!
# Relating `ℤ_[p]` to `zmod (p ^ n)`
In this file we establish connections between the `p`-adic integers $\mathbb{Z}_p$
and the integers modulo powers of `p`, $\mathbb{Z}/p^n\mathbb{Z}$.
## Main declarations
We show that $\mathbb{Z}_p$ has a ring hom to $\mathbb{Z}/p^n\mathbb{Z}$ for each `n`.
The case for `n = 1` is handled separately, since it is used in the general construction
and we may want to use it without the `^1` getting in the way.
* `padic_int.to_zmod`: ring hom to `zmod p`
* `padic_int.to_zmod_pow`: ring hom to `zmod (p^n)`
* `padic_int.ker_to_zmod` / `padic_int.ker_to_zmod_pow`: the kernels of these maps are the ideals
generated by `p^n`
We also establish the universal property of $\mathbb{Z}_p$ as a projective limit.
Given a family of compatible ring homs $f_k : R \to \mathbb{Z}/p^n\mathbb{Z}$,
there is a unique limit $R \to \mathbb{Z}_p$.
* `padic_int.lift`: the limit function
* `padic_int.lift_spec` / `padic_int.lift_unique`: the universal property
## Implementation notes
The ring hom constructions go through an auxiliary constructor `padic_int.to_zmod_hom`,
which removes some boilerplate code.
-/
noncomputable theory
open_locale classical
namespace padic_int
open nat local_ring padic
variables {p : ℕ} [hp_prime : fact (p.prime)]
include hp_prime
section ring_homs
/-! ### Ring homomorphisms to `zmod p` and `zmod (p ^ n)` -/
variables (p) (r : ℚ)
omit hp_prime
/--
`mod_part p r` is an integer that satisfies
`∥(r - mod_part p r : ℚ_[p])∥ < 1` when `∥(r : ℚ_[p])∥ ≤ 1`,
see `padic_int.norm_sub_mod_part`.
It is the unique non-negative integer that is `< p` with this property.
(Note that this definition assumes `r : ℚ`.
See `padic_int.zmod_repr` for a version that takes values in `ℕ`
and works for arbitrary `x : ℤ_[p]`.) -/
def mod_part : ℤ :=
(r.num * gcd_a r.denom p) % p
include hp_prime
variable {p}
lemma mod_part_lt_p : mod_part p r < p :=
begin
convert int.mod_lt _ _,
{ simp },
{ exact_mod_cast hp_prime.ne_zero }
end
lemma mod_part_nonneg : 0 ≤ mod_part p r :=
int.mod_nonneg _ $ by exact_mod_cast hp_prime.ne_zero
lemma is_unit_denom (r : ℚ) (h : ∥(r : ℚ_[p])∥ ≤ 1) : is_unit (r.denom : ℤ_[p]) :=
begin
rw is_unit_iff,
apply le_antisymm (r.denom : ℤ_[p]).2,
rw [← not_lt, val_eq_coe, coe_coe],
intro norm_denom_lt,
have hr : ∥(r * r.denom : ℚ_[p])∥ = ∥(r.num : ℚ_[p])∥,
{ rw_mod_cast @rat.mul_denom_eq_num r, refl, },
rw padic_norm_e.mul at hr,
have key : ∥(r.num : ℚ_[p])∥ < 1,
{ calc _ = _ : hr.symm
... < 1 * 1 : mul_lt_mul' h norm_denom_lt (norm_nonneg _) zero_lt_one
... = 1 : mul_one 1 },
have : ↑p ∣ r.num ∧ (p : ℤ) ∣ r.denom,
{ simp only [← norm_int_lt_one_iff_dvd, ← padic_norm_e_of_padic_int],
norm_cast, exact ⟨key, norm_denom_lt⟩ },
apply hp_prime.not_dvd_one,
rwa [← r.cop.gcd_eq_one, nat.dvd_gcd_iff, ← int.coe_nat_dvd_left, ← int.coe_nat_dvd],
end
lemma norm_sub_mod_part_aux (r : ℚ) (h : ∥(r : ℚ_[p])∥ ≤ 1) :
↑p ∣ r.num - r.num * r.denom.gcd_a p % p * ↑(r.denom) :=
begin
rw ← zmod.int_coe_zmod_eq_zero_iff_dvd,
simp only [int.cast_coe_nat, zmod.cast_mod_nat p, int.cast_mul, int.cast_sub],
have := congr_arg (coe : ℤ → zmod p) (gcd_eq_gcd_ab r.denom p),
simp only [int.cast_coe_nat, add_zero, int.cast_add, zmod.cast_self, int.cast_mul, zero_mul]
at this,
push_cast,
rw [mul_right_comm, mul_assoc, ←this],
suffices rdcp : r.denom.coprime p,
{ rw rdcp.gcd_eq_one, simp only [mul_one, cast_one, sub_self], },
apply coprime.symm,
apply (coprime_or_dvd_of_prime ‹_› _).resolve_right,
rw [← int.coe_nat_dvd, ← norm_int_lt_one_iff_dvd, not_lt],
apply ge_of_eq,
rw ← is_unit_iff,
exact is_unit_denom r h,
end
lemma norm_sub_mod_part (h : ∥(r : ℚ_[p])∥ ≤ 1) : ∥(⟨r,h⟩ - mod_part p r : ℤ_[p])∥ < 1 :=
begin
let n := mod_part p r,
by_cases aux : (⟨r,h⟩ - n : ℤ_[p]) = 0,
{ rw [aux, norm_zero], exact zero_lt_one, },
rw [norm_lt_one_iff_dvd, ← (is_unit_denom r h).dvd_mul_right],
suffices : ↑p ∣ r.num - n * r.denom,
{ convert (int.cast_ring_hom ℤ_[p]).map_dvd this,
simp only [sub_mul, int.cast_coe_nat, ring_hom.eq_int_cast, int.cast_mul,
sub_left_inj, int.cast_sub],
apply subtype.coe_injective,
simp only [coe_mul, subtype.coe_mk, coe_coe],
rw_mod_cast @rat.mul_denom_eq_num r, refl },
exact norm_sub_mod_part_aux r h
end
lemma exists_mem_range_of_norm_rat_le_one (h : ∥(r : ℚ_[p])∥ ≤ 1) :
∃ n : ℤ, 0 ≤ n ∧ n < p ∧ ∥(⟨r,h⟩ - n : ℤ_[p])∥ < 1 :=
⟨mod_part p r, mod_part_nonneg _, mod_part_lt_p _, norm_sub_mod_part _ h⟩
lemma zmod_congr_of_sub_mem_span_aux (n : ℕ) (x : ℤ_[p]) (a b : ℤ)
(ha : x - a ∈ (ideal.span {p ^ n} : ideal ℤ_[p]))
(hb : x - b ∈ (ideal.span {p ^ n} : ideal ℤ_[p])) :
(a : zmod (p ^ n)) = b :=
begin
rw [ideal.mem_span_singleton] at ha hb,
rw [← sub_eq_zero, ← int.cast_sub,
zmod.int_coe_zmod_eq_zero_iff_dvd, int.coe_nat_pow],
rw [← dvd_neg, neg_sub] at ha,
have := dvd_add ha hb,
rwa [sub_eq_add_neg, sub_eq_add_neg, add_assoc, neg_add_cancel_left,
← sub_eq_add_neg, ← int.cast_sub, pow_p_dvd_int_iff] at this,
end
lemma zmod_congr_of_sub_mem_span (n : ℕ) (x : ℤ_[p]) (a b : ℕ)
(ha : x - a ∈ (ideal.span {p ^ n} : ideal ℤ_[p]))
(hb : x - b ∈ (ideal.span {p ^ n} : ideal ℤ_[p])) :
(a : zmod (p ^ n)) = b :=
zmod_congr_of_sub_mem_span_aux n x a b ha hb
lemma zmod_congr_of_sub_mem_max_ideal (x : ℤ_[p]) (m n : ℕ)
(hm : x - m ∈ maximal_ideal ℤ_[p]) (hn : x - n ∈ maximal_ideal ℤ_[p]) :
(m : zmod p) = n :=
begin
rw maximal_ideal_eq_span_p at hm hn,
have := zmod_congr_of_sub_mem_span_aux 1 x m n,
simp only [_root_.pow_one] at this,
specialize this hm hn,
apply_fun zmod.cast_hom (show p ∣ p ^ 1, by rw nat.pow_one) (zmod p) at this,
simpa only [ring_hom.map_int_cast],
end
variable (x : ℤ_[p])
lemma exists_mem_range : ∃ n : ℕ, n < p ∧ (x - n ∈ maximal_ideal ℤ_[p]) :=
begin
simp only [maximal_ideal_eq_span_p, ideal.mem_span_singleton, ← norm_lt_one_iff_dvd],
obtain ⟨r, hr⟩ := rat_dense (x : ℚ_[p]) zero_lt_one,
have H : ∥(r : ℚ_[p])∥ ≤ 1,
{ rw norm_sub_rev at hr,
calc _ = ∥(r : ℚ_[p]) - x + x∥ : by ring
... ≤ _ : padic_norm_e.nonarchimedean _ _
... ≤ _ : max_le (le_of_lt hr) x.2 },
obtain ⟨n, hzn, hnp, hn⟩ := exists_mem_range_of_norm_rat_le_one r H,
lift n to ℕ using hzn,
use n,
split, {exact_mod_cast hnp},
simp only [norm_def, coe_sub, subtype.coe_mk, coe_coe] at hn ⊢,
rw show (x - n : ℚ_[p]) = (x - r) + (r - n), by ring,
apply lt_of_le_of_lt (padic_norm_e.nonarchimedean _ _),
apply max_lt hr,
simpa using hn
end
/--
`zmod_repr x` is the unique natural number smaller than `p`
satisfying `∥(x - zmod_repr x : ℤ_[p])∥ < 1`.
-/
def zmod_repr : ℕ :=
classical.some (exists_mem_range x)
lemma zmod_repr_spec : zmod_repr x < p ∧ (x - zmod_repr x ∈ maximal_ideal ℤ_[p]) :=
classical.some_spec (exists_mem_range x)
lemma zmod_repr_lt_p : zmod_repr x < p := (zmod_repr_spec _).1
lemma sub_zmod_repr_mem : (x - zmod_repr x ∈ maximal_ideal ℤ_[p]) := (zmod_repr_spec _).2
/--
`to_zmod_hom` is an auxiliary constructor for creating ring homs from `ℤ_[p]` to `zmod v`.
-/
def to_zmod_hom (v : ℕ) (f : ℤ_[p] → ℕ) (f_spec : ∀ x, x - f x ∈ (ideal.span {v} : ideal ℤ_[p]))
(f_congr : ∀ (x : ℤ_[p]) (a b : ℕ),
x - a ∈ (ideal.span {v} : ideal ℤ_[p]) → x - b ∈ (ideal.span {v} : ideal ℤ_[p]) →
(a : zmod v) = b) :
ℤ_[p] →+* zmod v :=
{ to_fun := λ x, f x,
map_zero' :=
begin
rw [f_congr (0 : ℤ_[p]) _ 0, cast_zero],
{ exact f_spec _ },
{ simp only [sub_zero, cast_zero, submodule.zero_mem], }
end,
map_one' :=
begin
rw [f_congr (1 : ℤ_[p]) _ 1, cast_one],
{ exact f_spec _ },
{ simp only [sub_self, cast_one, submodule.zero_mem], }
end,
map_add' :=
begin
intros x y,
rw [f_congr (x + y) _ (f x + f y), cast_add],
{ exact f_spec _ },
{ convert ideal.add_mem _ (f_spec x) (f_spec y),
rw cast_add,
ring, }
end,
map_mul' :=
begin
intros x y,
rw [f_congr (x * y) _ (f x * f y), cast_mul],
{ exact f_spec _ },
{ let I : ideal ℤ_[p] := ideal.span {v},
have A : x * (y - f y) ∈ I := I.mul_mem_left (f_spec _),
have B : (x - f x) * (f y) ∈ I := I.mul_mem_right (f_spec _),
convert I.add_mem A B,
rw cast_mul,
ring, }
end, }
/--
`to_zmod` is a ring hom from `ℤ_[p]` to `zmod p`,
with the equality `to_zmod x = (zmod_repr x : zmod p)`.
-/
def to_zmod : ℤ_[p] →+* zmod p :=
to_zmod_hom p zmod_repr
(by { rw ←maximal_ideal_eq_span_p, exact sub_zmod_repr_mem })
(by { rw ←maximal_ideal_eq_span_p, exact zmod_congr_of_sub_mem_max_ideal } )
/--
`z - (to_zmod z : ℤ_[p])` is contained in the maximal ideal of `ℤ_[p]`, for every `z : ℤ_[p]`.
The coercion from `zmod p` to `ℤ_[p]` is `zmod.has_coe_t`,
which coerces `zmod p` into artibrary rings.
This is unfortunate, but a consequence of the fact that we allow `zmod p`
to coerce to rings of arbitrary characteristic, instead of only rings of characteristic `p`.
This coercion is only a ring homomorphism if it coerces into a ring whose characteristic divides
`p`. While this is not the case here we can still make use of the coercion.
-/
lemma to_zmod_spec (z : ℤ_[p]) : z - (to_zmod z : ℤ_[p]) ∈ maximal_ideal ℤ_[p] :=
begin
convert sub_zmod_repr_mem z using 2,
dsimp [to_zmod, to_zmod_hom],
unfreezingI { rcases (exists_eq_add_of_lt (hp_prime.pos)) with ⟨p', rfl⟩ },
change ↑(zmod.val _) = _,
simp only [zmod.val_cast_nat, add_zero, add_def, cast_inj, zero_add],
apply mod_eq_of_lt,
simpa only [zero_add] using zmod_repr_lt_p z,
end
lemma ker_to_zmod : (to_zmod : ℤ_[p] →+* zmod p).ker = maximal_ideal ℤ_[p] :=
begin
ext x,
rw ring_hom.mem_ker,
split,
{ intro h,
simpa only [h, zmod.cast_zero, sub_zero] using to_zmod_spec x, },
{ intro h,
rw ← sub_zero x at h,
dsimp [to_zmod, to_zmod_hom],
convert zmod_congr_of_sub_mem_max_ideal x _ 0 _ h,
apply sub_zmod_repr_mem, }
end
/-- `appr n x` gives a value `v : ℕ` such that `x` and `↑v : ℤ_p` are congruent mod `p^n`.
See `appr_spec`. -/
noncomputable def appr : ℤ_[p] → ℕ → ℕ
| x 0 := 0
| x (n+1) :=
let y := x - appr x n in
if hy : y = 0 then
appr x n
else
let u := unit_coeff hy in
appr x n + p ^ n * (to_zmod ((u : ℤ_[p]) * (p ^ (y.valuation - n).nat_abs))).val
lemma appr_lt (x : ℤ_[p]) (n : ℕ) : x.appr n < p ^ n :=
begin
induction n with n ih generalizing x,
{ simp only [appr, succ_pos', nat.pow_zero], },
simp only [appr, ring_hom.map_nat_cast, zmod.cast_self, ring_hom.map_pow, int.nat_abs,
ring_hom.map_mul],
have hp : p ^ n < p ^ (n + 1),
{ apply pow_lt_pow hp_prime.one_lt (lt_add_one n) },
split_ifs with h,
{ apply lt_trans (ih _) hp, },
{ calc _ < p ^ n + p ^ n * (p - 1) : _
... = p ^ (n + 1) : _,
{ apply add_lt_add_of_lt_of_le (ih _),
apply nat.mul_le_mul_left,
apply le_pred_of_lt,
apply zmod.val_lt },
{ rw [nat.mul_sub_left_distrib, mul_one, ← nat.pow_succ],
apply nat.add_sub_cancel' (le_of_lt hp) } }
end
lemma appr_mono (x : ℤ_[p]) : monotone x.appr :=
begin
apply monotone_of_monotone_nat,
intro n,
dsimp [appr],
split_ifs, { refl, },
apply nat.le_add_right,
end
lemma dvd_appr_sub_appr (x : ℤ_[p]) (m n : ℕ) (h : m ≤ n) :
p ^ m ∣ x.appr n - x.appr m :=
begin
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le h, clear h,
induction k with k ih,
{ simp only [add_zero, nat.sub_self, dvd_zero], },
rw [nat.succ_eq_add_one, ← add_assoc],
dsimp [appr],
split_ifs with h,
{ exact ih },
rw [add_comm, nat.add_sub_assoc (appr_mono _ (nat.le_add_right m k))],
apply dvd_add _ ih,
apply dvd_mul_of_dvd_left,
apply nat.pow_dvd_pow _ (nat.le_add_right m k),
end
lemma appr_spec (n : ℕ) : ∀ (x : ℤ_[p]), x - appr x n ∈ (ideal.span {p^n} : ideal ℤ_[p]) :=
begin
simp only [ideal.mem_span_singleton],
induction n with n ih,
{ simp only [is_unit_one, is_unit.dvd, pow_zero, forall_true_iff], },
{ intro x,
dsimp only [appr],
split_ifs with h,
{ rw h, apply dvd_zero },
{ push_cast, rw sub_add_eq_sub_sub,
obtain ⟨c, hc⟩ := ih x,
simp only [ring_hom.map_nat_cast, zmod.cast_self, ring_hom.map_pow, ring_hom.map_mul,
zmod.nat_cast_val],
have hc' : c ≠ 0,
{ rintro rfl, simp only [mul_zero] at hc, contradiction },
conv_rhs { congr, simp only [hc], },
rw show (x - ↑(appr x n)).valuation = (↑p ^ n * c).valuation,
{ rw hc },
rw [valuation_p_pow_mul _ _ hc', add_sub_cancel', pow_succ', ← mul_sub],
apply mul_dvd_mul_left,
by_cases hc0 : c.valuation.nat_abs = 0,
{ simp only [hc0, mul_one, pow_zero],
rw [mul_comm, unit_coeff_spec h] at hc,
suffices : c = unit_coeff h,
{ rw [← this, ← ideal.mem_span_singleton, ← maximal_ideal_eq_span_p],
apply to_zmod_spec },
obtain ⟨c, rfl⟩ : is_unit c, -- TODO: write a can_lift instance for units
{ rw int.nat_abs_eq_zero at hc0,
rw [is_unit_iff, norm_eq_pow_val hc', hc0, neg_zero, fpow_zero], },
rw discrete_valuation_ring.unit_mul_pow_congr_unit _ _ _ _ _ hc,
exact irreducible_p },
{ rw [_root_.zero_pow (nat.pos_of_ne_zero hc0)],
simp only [sub_zero, zmod.cast_zero, mul_zero],
rw unit_coeff_spec hc',
apply dvd_mul_of_dvd_right,
apply dvd_pow (dvd_refl _),
exact hc0 } } }
end
attribute [irreducible] appr
/-- A ring hom from `ℤ_[p]` to `zmod (p^n)`, with underlying function `padic_int.appr n`. -/
def to_zmod_pow (n : ℕ) : ℤ_[p] →+* zmod (p ^ n) :=
to_zmod_hom (p^n) (λ x, appr x n)
(by { intros, convert appr_spec n _ using 1, simp })
(by { intros x a b ha hb,
apply zmod_congr_of_sub_mem_span n x a b,
{ simpa using ha },
{ simpa using hb } })
lemma ker_to_zmod_pow (n : ℕ) : (to_zmod_pow n : ℤ_[p] →+* zmod (p ^ n)).ker = ideal.span {p ^ n} :=
begin
ext x,
rw ring_hom.mem_ker,
split,
{ intro h,
suffices : x.appr n = 0,
{ convert appr_spec n x, simp only [this, sub_zero, cast_zero], },
dsimp [to_zmod_pow, to_zmod_hom] at h,
rw zmod.nat_coe_zmod_eq_zero_iff_dvd at h,
apply eq_zero_of_dvd_of_lt h (appr_lt _ _), },
{ intro h,
rw ← sub_zero x at h,
dsimp [to_zmod_pow, to_zmod_hom],
rw [zmod_congr_of_sub_mem_span n x _ 0 _ h, cast_zero],
apply appr_spec, }
end
@[simp] lemma zmod_cast_comp_to_zmod_pow (m n : ℕ) (h : m ≤ n) :
(zmod.cast_hom (nat.pow_dvd_pow p h) (zmod (p ^ m))).comp (to_zmod_pow n) = to_zmod_pow m :=
begin
apply zmod.ring_hom_eq_of_ker_eq,
ext x,
rw [ring_hom.mem_ker, ring_hom.mem_ker],
simp only [function.comp_app, zmod.cast_hom_apply, ring_hom.coe_comp],
simp only [to_zmod_pow, to_zmod_hom, ring_hom.coe_mk],
rw [zmod.cast_nat_cast (nat.pow_dvd_pow p h),
zmod_congr_of_sub_mem_span m (x.appr n) (x.appr n) (x.appr m)],
{ rw [sub_self], apply ideal.zero_mem _, },
{ rw ideal.mem_span_singleton,
rcases dvd_appr_sub_appr x m n h with ⟨c, hc⟩,
use c,
rw [← nat.cast_sub (appr_mono _ h), hc, nat.cast_mul, nat.cast_pow], },
{ apply_instance }
end
@[simp] lemma cast_to_zmod_pow (m n : ℕ) (h : m ≤ n) (x : ℤ_[p]) :
↑(to_zmod_pow n x) = to_zmod_pow m x :=
by { rw ← zmod_cast_comp_to_zmod_pow _ _ h, refl }
lemma dense_range_nat_cast :
dense_range (nat.cast : ℕ → ℤ_[p]) :=
begin
intro x,
rw metric.mem_closure_range_iff,
intros ε hε,
obtain ⟨n, hn⟩ := exists_pow_neg_lt p hε,
use (x.appr n),
rw dist_eq_norm,
apply lt_of_le_of_lt _ hn,
rw norm_le_pow_iff_mem_span_pow,
apply appr_spec,
end
lemma dense_range_int_cast :
dense_range (int.cast : ℤ → ℤ_[p]) :=
begin
intro x,
apply dense_range_nat_cast.induction_on x,
{ exact is_closed_closure, },
{ intro a,
change (a.cast : ℤ_[p]) with (a : ℤ).cast,
apply subset_closure,
exact set.mem_range_self _ }
end
end ring_homs
section lift
/-! ### Universal property as projective limit -/
open cau_seq padic_seq
variables {R : Type*} [comm_ring R] (f : Π k : ℕ, R →+* zmod (p^k))
(f_compat : ∀ k1 k2 (hk : k1 ≤ k2), (zmod.cast_hom (nat.pow_dvd_pow p hk) _).comp (f k2) = f k1)
omit hp_prime
/--
Given a family of ring homs `f : Π n : ℕ, R →+* zmod (p ^ n)`,
`nth_hom f r` is an integer-valued sequence
whose `n`th value is the unique integer `k` such that `0 ≤ k < p ^ n`
and `f n r = (k : zmod (p ^ n))`.
-/
def nth_hom (r : R) : ℕ → ℤ :=
λ n, (f n r : zmod (p^n)).val
@[simp] lemma nth_hom_zero : nth_hom f 0 = 0 :=
by simp [nth_hom]; refl
variable {f}
include hp_prime
include f_compat
lemma pow_dvd_nth_hom_sub (r : R) (i j : ℕ) (h : i ≤ j) :
↑p ^ i ∣ nth_hom f r j - nth_hom f r i :=
begin
specialize f_compat (i) (j) h,
rw [← int.coe_nat_pow, ← zmod.int_coe_zmod_eq_zero_iff_dvd,
int.cast_sub],
dsimp [nth_hom],
rw [← f_compat, ring_hom.comp_apply],
have : fact (p ^ (i) > 0) := nat.pow_pos (nat.prime.pos ‹_›) _,
have : fact (p ^ (j) > 0) := nat.pow_pos (nat.prime.pos ‹_›) _,
unfreezingI { simp only [zmod.cast_id, zmod.cast_hom_apply, sub_self, zmod.nat_cast_val], },
end
lemma is_cau_seq_nth_hom (r : R): is_cau_seq (padic_norm p) (λ n, nth_hom f r n) :=
begin
intros ε hε,
obtain ⟨k, hk⟩ : ∃ k : ℕ, (p ^ - (↑(k : ℕ) : ℤ) : ℚ) < ε := exists_pow_neg_lt_rat p hε,
use k,
intros j hj,
refine lt_of_le_of_lt _ hk,
norm_cast,
rw ← padic_norm.dvd_iff_norm_le,
exact_mod_cast pow_dvd_nth_hom_sub f_compat r k j hj
end
/--
`nth_hom_seq f_compat r` bundles `padic_int.nth_hom f r`
as a Cauchy sequence of rationals with respect to the `p`-adic norm.
The `n`th value of the sequence is `((f n r).val : ℚ)`.
-/
def nth_hom_seq (r : R) : padic_seq p := ⟨λ n, nth_hom f r n, is_cau_seq_nth_hom f_compat r⟩
lemma nth_hom_seq_one : nth_hom_seq f_compat 1 ≈ 1 :=
begin
intros ε hε,
change _ < _ at hε,
use 1,
intros j hj,
haveI : fact (1 < p^j) := nat.one_lt_pow _ _ (by linarith) (nat.prime.one_lt ‹_›),
simp [nth_hom_seq, nth_hom, zmod.val_one, hε],
end
lemma nth_hom_seq_add (r s : R) :
nth_hom_seq f_compat (r + s) ≈ nth_hom_seq f_compat r + nth_hom_seq f_compat s :=
begin
intros ε hε,
obtain ⟨n, hn⟩ := exists_pow_neg_lt_rat p hε,
use n,
intros j hj,
dsimp [nth_hom_seq],
apply lt_of_le_of_lt _ hn,
rw [← int.cast_add, ← int.cast_sub, ← padic_norm.dvd_iff_norm_le,
← zmod.int_coe_zmod_eq_zero_iff_dvd],
dsimp [nth_hom],
have : fact (p ^ n > 0) := nat.pow_pos (nat.prime.pos ‹_›) _,
have : fact (p ^ j > 0) := nat.pow_pos (nat.prime.pos ‹_›) _,
unfreezingI
{ simp only [int.cast_coe_nat, int.cast_add, ring_hom.map_add, int.cast_sub, zmod.nat_cast_val] },
rw [zmod.cast_add (show p ^ n ∣ p ^ j, from _), sub_self],
{ apply_instance },
{ apply nat.pow_dvd_pow, linarith only [hj] },
end
lemma nth_hom_seq_mul (r s : R) :
nth_hom_seq f_compat (r * s) ≈ nth_hom_seq f_compat r * nth_hom_seq f_compat s :=
begin
intros ε hε,
obtain ⟨n, hn⟩ := exists_pow_neg_lt_rat p hε,
use n,
intros j hj,
dsimp [nth_hom_seq],
apply lt_of_le_of_lt _ hn,
rw [← int.cast_mul, ← int.cast_sub, ← padic_norm.dvd_iff_norm_le,
← zmod.int_coe_zmod_eq_zero_iff_dvd],
dsimp [nth_hom],
have : fact (p ^ n > 0) := nat.pow_pos (nat.prime.pos ‹_›) _,
have : fact (p ^ j > 0) := nat.pow_pos (nat.prime.pos ‹_›) _,
unfreezingI
{ simp only [int.cast_coe_nat, int.cast_mul, int.cast_sub, ring_hom.map_mul, zmod.nat_cast_val] },
rw [zmod.cast_mul (show p ^ n ∣ p ^ j, from _), sub_self],
{ apply_instance },
{ apply nat.pow_dvd_pow, linarith only [hj] },
end
/--
`lim_nth_hom f_compat r` is the limit of a sequence `f` of compatible ring homs `R →+* zmod (p^k)`.
This is itself a ring hom: see `padic_int.lift`.
-/
def lim_nth_hom (r : R) : ℤ_[p] :=
of_int_seq (nth_hom f r) (is_cau_seq_nth_hom f_compat r)
lemma lim_nth_hom_spec (r : R) :
∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ n ≥ N, ∥lim_nth_hom f_compat r - nth_hom f r n∥ < ε :=
begin
intros ε hε,
obtain ⟨ε', hε'0, hε'⟩ : ∃ v : ℚ, (0 : ℝ) < v ∧ ↑v < ε := exists_rat_btwn hε,
norm_cast at hε'0,
obtain ⟨N, hN⟩ := padic_norm_e.defn (nth_hom_seq f_compat r) hε'0,
use N,
intros n hn,
apply lt.trans _ hε',
change ↑(padic_norm_e _) < _,
norm_cast,
convert hN _ hn,
simp [nth_hom, lim_nth_hom, nth_hom_seq, of_int_seq],
end
lemma lim_nth_hom_zero : lim_nth_hom f_compat 0 = 0 :=
by simp [lim_nth_hom]; refl
lemma lim_nth_hom_one : lim_nth_hom f_compat 1 = 1 :=
subtype.ext $ quot.sound $ nth_hom_seq_one _
lemma lim_nth_hom_add (r s : R) :
lim_nth_hom f_compat (r + s) = lim_nth_hom f_compat r + lim_nth_hom f_compat s :=
subtype.ext $ quot.sound $ nth_hom_seq_add _ _ _
lemma lim_nth_hom_mul (r s : R) :
lim_nth_hom f_compat (r * s) = lim_nth_hom f_compat r * lim_nth_hom f_compat s :=
subtype.ext $ quot.sound $ nth_hom_seq_mul _ _ _
-- TODO: generalize this to arbitrary complete discrete valuation rings
/--
`lift f_compat` is the limit of a sequence `f` of compatible ring homs `R →+* zmod (p^k)`,
with the equality `lift f_compat r = padic_int.lim_nth_hom f_compat r`.
-/
def lift : R →+* ℤ_[p] :=
{ to_fun := lim_nth_hom f_compat,
map_one' := lim_nth_hom_one f_compat,
map_mul' := lim_nth_hom_mul f_compat,
map_zero' := lim_nth_hom_zero f_compat,
map_add' := lim_nth_hom_add f_compat }
omit f_compat
lemma lift_sub_val_mem_span (r : R) (n : ℕ) :
(lift f_compat r - (f n r).val) ∈ (ideal.span {↑p ^ n} : ideal ℤ_[p]) :=
begin
obtain ⟨k, hk⟩ := lim_nth_hom_spec f_compat r _
(show (0 : ℝ) < p ^ (-n : ℤ), from nat.fpow_pos_of_pos hp_prime.pos _),
have := le_of_lt (hk (max n k) (le_max_right _ _)),
rw norm_le_pow_iff_mem_span_pow at this,
dsimp [lift],
rw sub_eq_sub_add_sub (lim_nth_hom f_compat r) _ ↑(nth_hom f r (max n k)),
apply ideal.add_mem _ _ this,
rw [ideal.mem_span_singleton],
simpa only [ring_hom.eq_int_cast, ring_hom.map_pow, int.cast_sub] using
(int.cast_ring_hom ℤ_[p]).map_dvd (pow_dvd_nth_hom_sub f_compat r n (max n k) (le_max_left _ _)),
end
/--
One part of the universal property of `ℤ_[p]` as a projective limit.
See also `padic_int.lift_unique`.
-/
lemma lift_spec (n : ℕ) : (to_zmod_pow n).comp (lift f_compat) = f n :=
begin
ext r,
haveI : fact (0 < p ^ n) := nat.pow_pos (nat.prime.pos ‹_›) n,
rw [ring_hom.comp_apply, ← zmod.cast_val (f n r), ← (to_zmod_pow n).map_nat_cast,
← sub_eq_zero, ← ring_hom.map_sub, ← ring_hom.mem_ker, ker_to_zmod_pow],
apply lift_sub_val_mem_span,
end
/--
One part of the universal property of `ℤ_[p]` as a projective limit.
See also `padic_int.lift_spec`.
-/
lemma lift_unique (g : R →+* ℤ_[p]) (hg : ∀ n, (to_zmod_pow n).comp g = f n) :
lift f_compat = g :=
begin
ext1 r,
apply eq_of_forall_dist_le,
intros ε hε,
obtain ⟨n, hn⟩ := exists_pow_neg_lt p hε,
apply le_trans _ (le_of_lt hn),
rw [dist_eq_norm, norm_le_pow_iff_mem_span_pow, ← ker_to_zmod_pow, ring_hom.mem_ker,
ring_hom.map_sub, ← ring_hom.comp_apply, ← ring_hom.comp_apply, lift_spec, hg, sub_self],
end
@[simp] lemma lift_self (z : ℤ_[p]) : @lift p _ ℤ_[p] _ to_zmod_pow
zmod_cast_comp_to_zmod_pow z = z :=
begin
show _ = ring_hom.id _ z,
rw @lift_unique p _ ℤ_[p] _ _ zmod_cast_comp_to_zmod_pow (ring_hom.id ℤ_[p]),
intro, rw ring_hom.comp_id,
end
end lift
lemma ext_of_to_zmod_pow {x y : ℤ_[p]} :
(∀ n, to_zmod_pow n x = to_zmod_pow n y) ↔ x = y :=
begin
split,
{ intro h,
rw [← lift_self x, ← lift_self y],
simp [lift, lim_nth_hom, nth_hom, h] },
{ rintro rfl _, refl }
end
lemma to_zmod_pow_eq_iff_ext {R : Type*} [comm_ring R] {g g' : R →+* ℤ_[p]} :
(∀ n, (to_zmod_pow n).comp g = (to_zmod_pow n).comp g') ↔ g = g' :=
begin
split,
{ intro hg,
ext x : 1,
apply ext_of_to_zmod_pow.mp,
intro n,
show (to_zmod_pow n).comp g x = (to_zmod_pow n).comp g' x,
rw hg n },
{ rintro rfl _, refl }
end
end padic_int
|
2ed9e4243f17914fea1b72d9826c6b62e0eb5702 | 7490bf5d40d31857a58062614642bb5a41c36154 | /dm_prod_test.lean | d62723a352d108169aa15e44dcdf8a252c29e188 | [] | no_license | reesegrayallen/Lean-Discrete-Mathematics | 9f1d6fe1c814cc9264ce868a67adcf5a82566e22 | 00c875284613ea12e0a729f519738aab8599456b | refs/heads/main | 1,674,181,372,629 | 1,606,801,004,000 | 1,606,801,004,000 | 317,387,970 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,497 | lean | /-
HOMEWORK 3
Reese Allen (rga2uz)
CS2102 - Sullivan
-/
import .dm_prod
-- defining ordered pairs to use in testing
def p1 := dm_prod.mk 5 2
def p2 := dm_prod.mk 3 tt
def p3 := dm_prod.mk ff tt
#check p1 -- dm_prod ℕ ℕ
#check p2 -- dm_prod ℕ bool
#check p3 -- dm_prod bool bool
-- additional ordered pairs
def p4 := dm_prod.mk "yes" "no"
def p5 := dm_prod.mk (3, 4) ff
def p6 := dm_prod.mk "seven" 7
-- test cases for fst
#reduce fst p1
#reduce fst p2
#reduce fst p3
#eval fst' p4
#reduce fst' p5
#eval fst' p6
#reduce fst'' p1
#reduce fst'' p3
#reduce fst'' p5
-- test cases for snd
#reduce snd p1
#reduce snd p2
#reduce snd p3
#eval snd' p4
#reduce snd' p5
#eval snd' p6
#reduce snd'' p1
#reduce snd'' p3
#reduce snd'' p5
-- test cases for set_fst
#reduce set_fst p1 ff
#reduce set_fst p2 (1, 100)
#reduce set_fst p3 1
#reduce set_fst' p1 7
#reduce set_fst' p2 (1, 100)
#reduce set_fst' p3 tt
#reduce set_fst'' p1 (100, 1)
#reduce set_fst'' p2 ff
#reduce set_fst'' p3 4
-- test cases for set_snd
#reduce set_snd p1 ff
#reduce set_snd p2 (1, 100)
#reduce set_snd p3 1
#reduce set_snd' p1 7
#reduce set_snd' p2 (1, 100, 1000)
#reduce set_snd' p3 (tt, 5)
#reduce set_snd'' p5 (ff, tt)
#reduce set_snd'' p3 ff
#reduce set_snd'' p5 9
-- test cases for swap
#reduce swap p1
#reduce swap p2
#reduce swap p3
#reduce swap' p3
#reduce swap' p5
#reduce swap' p1
#reduce swap'' p1
#reduce swap'' p2
#reduce swap'' p5
|
57e06b0984e00e05764e7a4e500f9a3cc2ef6c79 | 92b50235facfbc08dfe7f334827d47281471333b | /hott/init/num.hlean | 5e67d8aa70eb65feabcf6386ed900ceb7350073c | [
"Apache-2.0"
] | permissive | htzh/lean | 24f6ed7510ab637379ec31af406d12584d31792c | d70c79f4e30aafecdfc4a60b5d3512199200ab6e | refs/heads/master | 1,607,677,731,270 | 1,437,089,952,000 | 1,437,089,952,000 | 37,078,816 | 0 | 0 | null | 1,433,780,956,000 | 1,433,780,955,000 | null | UTF-8 | Lean | false | false | 3,360 | 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.logic init.bool
open bool
definition pos_num.is_inhabited [instance] : inhabited pos_num :=
inhabited.mk pos_num.one
namespace pos_num
definition is_one (a : pos_num) : bool :=
pos_num.rec_on a tt (λn r, ff) (λn r, ff)
definition pred (a : pos_num) : pos_num :=
pos_num.rec_on a one (λn r, bit0 n) (λn r, cond (is_one n) one (bit1 r))
definition size (a : pos_num) : pos_num :=
pos_num.rec_on a one (λn r, succ r) (λn r, succ r)
definition add (a b : pos_num) : pos_num :=
pos_num.rec_on a
succ
(λn f b, pos_num.rec_on b
(succ (bit1 n))
(λm r, succ (bit1 (f m)))
(λm r, bit1 (f m)))
(λn f b, pos_num.rec_on b
(bit1 n)
(λm r, bit1 (f m))
(λm r, bit0 (f m)))
b
notation a + b := add a b
definition mul (a b : pos_num) : pos_num :=
pos_num.rec_on a
b
(λn r, bit0 r + b)
(λn r, bit0 r)
notation a * b := mul a b
definition lt (a b : pos_num) : bool :=
pos_num.rec_on a
(λ b, pos_num.cases_on b
ff
(λm, tt)
(λm, tt))
(λn f b, pos_num.cases_on b
ff
(λm, f m)
(λm, f m))
(λn f b, pos_num.cases_on b
ff
(λm, f (succ m))
(λm, f m))
b
definition le (a b : pos_num) : bool :=
lt a (succ b)
definition equal (a b : pos_num) : bool :=
le a b && le b a
end pos_num
definition num.is_inhabited [instance] : inhabited num :=
inhabited.mk num.zero
namespace num
open pos_num
definition pred (a : num) : num :=
num.rec_on a zero (λp, cond (is_one p) zero (pos (pred p)))
definition size (a : num) : num :=
num.rec_on a (pos one) (λp, pos (size p))
definition add (a b : num) : num :=
num.rec_on a b (λpa, num.rec_on b (pos pa) (λpb, pos (pos_num.add pa pb)))
definition mul (a b : num) : num :=
num.rec_on a zero (λpa, num.rec_on b zero (λpb, pos (pos_num.mul pa pb)))
notation a + b := add a b
notation a * b := mul a b
definition le (a b : num) : bool :=
num.rec_on a tt (λpa, num.rec_on b ff (λpb, pos_num.le pa pb))
private definition psub (a b : pos_num) : num :=
pos_num.rec_on a
(λb, zero)
(λn f b,
cond (pos_num.le (bit1 n) b)
zero
(pos_num.cases_on b
(pos (bit0 n))
(λm, 2 * f m)
(λm, 2 * f m + 1)))
(λn f b,
cond (pos_num.le (bit0 n) b)
zero
(pos_num.cases_on b
(pos (pos_num.pred (bit0 n)))
(λm, pred (2 * f m))
(λm, 2 * f m)))
b
definition sub (a b : num) : num :=
num.rec_on a zero (λpa, num.rec_on b a (λpb, psub pa pb))
notation a ≤ b := le a b
notation a - b := sub a b
end num
-- the coercion from num to nat is defined here,
-- so that it can already be used in init.trunc and init.tactic
namespace nat
definition add (a b : nat) : nat :=
nat.rec_on b a (λ b₁ r, succ r)
notation a + b := add a b
definition of_num [coercion] (n : num) : nat :=
num.rec zero
(λ n, pos_num.rec (succ zero) (λ n r, r + r + (succ zero)) (λ n r, r + r) n) n
end nat
attribute nat.of_num [reducible] [constructor] -- of_num is also reducible if namespace "nat" is not opened
|
90e7a627b191e18545d9372491725e8f897615a4 | 35677d2df3f081738fa6b08138e03ee36bc33cad | /src/data/list/chain.lean | 29a547f48b65223b98a5a9e301caca543d09954e | [
"Apache-2.0"
] | permissive | gebner/mathlib | eab0150cc4f79ec45d2016a8c21750244a2e7ff0 | cc6a6edc397c55118df62831e23bfbd6e6c6b4ab | refs/heads/master | 1,625,574,853,976 | 1,586,712,827,000 | 1,586,712,827,000 | 99,101,412 | 1 | 0 | Apache-2.0 | 1,586,716,389,000 | 1,501,667,958,000 | Lean | UTF-8 | Lean | false | false | 7,553 | lean | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import data.list.pairwise
universes u v
open nat
variables {α : Type u} {β : Type v}
namespace list
/- chain relation (conjunction of R a b ∧ R b c ∧ R c d ...) -/
run_cmd tactic.mk_iff_of_inductive_prop `list.chain `list.chain_iff
variable {R : α → α → Prop}
theorem rel_of_chain_cons {a b : α} {l : list α}
(p : chain R a (b::l)) : R a b :=
(chain_cons.1 p).1
theorem chain_of_chain_cons {a b : α} {l : list α}
(p : chain R a (b::l)) : chain R b l :=
(chain_cons.1 p).2
theorem chain.imp' {S : α → α → Prop}
(HRS : ∀ ⦃a b⦄, R a b → S a b) {a b : α} (Hab : ∀ ⦃c⦄, R a c → S b c)
{l : list α} (p : chain R a l) : chain S b l :=
by induction p with _ a c l r p IH generalizing b; constructor;
[exact Hab r, exact IH (@HRS _)]
theorem chain.imp {S : α → α → Prop}
(H : ∀ a b, R a b → S a b) {a : α} {l : list α} (p : chain R a l) : chain S a l :=
p.imp' H (H a)
theorem chain.iff {S : α → α → Prop}
(H : ∀ a b, R a b ↔ S a b) {a : α} {l : list α} : chain R a l ↔ chain S a l :=
⟨chain.imp (λ a b, (H a b).1), chain.imp (λ a b, (H a b).2)⟩
theorem chain.iff_mem {a : α} {l : list α} :
chain R a l ↔ chain (λ x y, x ∈ a :: l ∧ y ∈ l ∧ R x y) a l :=
⟨λ p, by induction p with _ a b l r p IH; constructor;
[exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩,
exact IH.imp (λ a b ⟨am, bm, h⟩,
⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩)],
chain.imp (λ a b h, h.2.2)⟩
theorem chain_singleton {a b : α} : chain R a [b] ↔ R a b :=
by simp only [chain_cons, chain.nil, and_true]
theorem chain_split {a b : α} {l₁ l₂ : list α} : chain R a (l₁++b::l₂) ↔
chain R a (l₁++[b]) ∧ chain R b l₂ :=
by induction l₁ with x l₁ IH generalizing a;
simp only [*, nil_append, cons_append, chain.nil, chain_cons, and_true, and_assoc]
theorem chain_map (f : β → α) {b : β} {l : list β} :
chain R (f b) (map f l) ↔ chain (λ a b : β, R (f a) (f b)) b l :=
by induction l generalizing b; simp only [map, chain.nil, chain_cons, *]
theorem chain_of_chain_map {S : β → β → Prop} (f : α → β)
(H : ∀ a b : α, S (f a) (f b) → R a b) {a : α} {l : list α}
(p : chain S (f a) (map f l)) : chain R a l :=
((chain_map f).1 p).imp H
theorem chain_map_of_chain {S : β → β → Prop} (f : α → β)
(H : ∀ a b : α, R a b → S (f a) (f b)) {a : α} {l : list α}
(p : chain R a l) : chain S (f a) (map f l) :=
(chain_map f).2 $ p.imp H
theorem chain_of_pairwise {a : α} {l : list α} (p : pairwise R (a::l)) : chain R a l :=
begin
cases pairwise_cons.1 p with r p', clear p,
induction p' with b l r' p IH generalizing a, {exact chain.nil},
simp only [chain_cons, forall_mem_cons] at r,
exact chain_cons.2 ⟨r.1, IH r'⟩
end
theorem chain_iff_pairwise (tr : transitive R) {a : α} {l : list α} :
chain R a l ↔ pairwise R (a::l) :=
⟨λ c, begin
induction c with b b c l r p IH, {exact pairwise_singleton _ _},
apply IH.cons _, simp only [mem_cons_iff, forall_mem_cons', r, true_and],
show ∀ x ∈ l, R b x, from λ x m, (tr r (rel_of_pairwise_cons IH m)),
end, chain_of_pairwise⟩
theorem chain_iff_nth_le {R} : ∀ {a : α} {l : list α},
chain R a l ↔ (∀ h : 0 < length l, R a (nth_le l 0 h)) ∧ (∀ i (h : i < length l - 1),
R (nth_le l i (lt_of_lt_pred h)) (nth_le l (i+1) (lt_pred_iff.mp h)))
| a [] := by simp
| a (b :: t) :=
begin
rw [chain_cons, chain_iff_nth_le],
split,
{ rintros ⟨R, ⟨h0, h⟩⟩,
split,
{ intro w, exact R, },
{ intros i w,
cases i,
{ apply h0, },
{ convert h i _ using 1,
simp only [succ_eq_add_one, add_succ_sub_one, add_zero, length, add_lt_add_iff_right] at w,
exact lt_pred_iff.mpr w, } } },
{ rintros ⟨h0, h⟩, split,
{ apply h0, simp, },
{ split,
{ apply h 0, },
{ intros i w, convert h (i+1) _,
exact lt_pred_iff.mp w, } } },
end
theorem chain'.imp {S : α → α → Prop}
(H : ∀ a b, R a b → S a b) {l : list α} (p : chain' R l) : chain' S l :=
by cases l; [trivial, exact p.imp H]
theorem chain'.iff {S : α → α → Prop}
(H : ∀ a b, R a b ↔ S a b) {l : list α} : chain' R l ↔ chain' S l :=
⟨chain'.imp (λ a b, (H a b).1), chain'.imp (λ a b, (H a b).2)⟩
theorem chain'.iff_mem : ∀ {l : list α}, chain' R l ↔ chain' (λ x y, x ∈ l ∧ y ∈ l ∧ R x y) l
| [] := iff.rfl
| (x::l) :=
⟨λ h, (chain.iff_mem.1 h).imp $ λ a b ⟨h₁, h₂, h₃⟩, ⟨h₁, or.inr h₂, h₃⟩,
chain'.imp $ λ a b h, h.2.2⟩
@[simp] theorem chain'_nil : chain' R [] := trivial
@[simp] theorem chain'_singleton (a : α) : chain' R [a] := chain.nil
theorem chain'_split {a : α} : ∀ {l₁ l₂ : list α}, chain' R (l₁++a::l₂) ↔
chain' R (l₁++[a]) ∧ chain' R (a::l₂)
| [] l₂ := (and_iff_right (chain'_singleton a)).symm
| (b::l₁) l₂ := chain_split
theorem chain'_map (f : β → α) {l : list β} :
chain' R (map f l) ↔ chain' (λ a b : β, R (f a) (f b)) l :=
by cases l; [refl, exact chain_map _]
theorem chain'_of_chain'_map {S : β → β → Prop} (f : α → β)
(H : ∀ a b : α, S (f a) (f b) → R a b) {l : list α}
(p : chain' S (map f l)) : chain' R l :=
((chain'_map f).1 p).imp H
theorem chain'_map_of_chain' {S : β → β → Prop} (f : α → β)
(H : ∀ a b : α, R a b → S (f a) (f b)) {l : list α}
(p : chain' R l) : chain' S (map f l) :=
(chain'_map f).2 $ p.imp H
theorem pairwise.chain' : ∀ {l : list α}, pairwise R l → chain' R l
| [] _ := trivial
| (a::l) h := chain_of_pairwise h
theorem chain'_iff_pairwise (tr : transitive R) : ∀ {l : list α},
chain' R l ↔ pairwise R l
| [] := (iff_true_intro pairwise.nil).symm
| (a::l) := chain_iff_pairwise tr
@[simp] theorem chain'_cons {x y l} : chain' R (x :: y :: l) ↔ R x y ∧ chain' R (y :: l) :=
chain_cons
theorem chain'.cons {x y l} (h₁ : R x y) (h₂ : chain' R (y :: l)) :
chain' R (x :: y :: l) :=
chain'_cons.2 ⟨h₁, h₂⟩
theorem chain'.tail : ∀ {l} (h : chain' R l), chain' R l.tail
| [] _ := trivial
| [x] _ := trivial
| (x :: y :: l) h := (chain'_cons.mp h).right
theorem chain'_pair {x y} : chain' R [x, y] ↔ R x y :=
by simp only [chain'_singleton, chain'_cons, and_true]
theorem chain'_reverse : ∀ {l}, chain' R (reverse l) ↔ chain' (flip R) l
| [] := iff.rfl
| [a] := by simp only [chain'_singleton, reverse_singleton]
| (a :: b :: l) := by rw [chain'_cons, reverse_cons, reverse_cons, append_assoc, cons_append,
nil_append, chain'_split, ← reverse_cons, @chain'_reverse (b :: l), and_comm, chain'_pair, flip]
theorem chain'_iff_nth_le {R} : ∀ {l : list α},
chain' R l ↔ ∀ i (h : i < length l - 1),
R (nth_le l i (lt_of_lt_pred h)) (nth_le l (i+1) (lt_pred_iff.mp h))
| [] := by simp
| (a :: nil) := by simp
| (a :: b :: t) :=
begin
rw [chain'_cons, chain'_iff_nth_le],
split,
{ rintros ⟨R, h⟩ i w,
cases i,
{ exact R, },
{ convert h i _ using 1,
simp only [succ_eq_add_one, add_succ_sub_one, add_zero, length, add_lt_add_iff_right] at w,
simpa using w, },
},
{ rintros h, split,
{ apply h 0, simp, },
{ intros i w, convert h (i+1) _,
simp only [add_zero, length, add_succ_sub_one] at w,
simpa using w, }
},
end
end list
|
ba34740881e78e7631a3efcf036b41fbd65ff48c | 8be24982c807641260370bd09243eac768750811 | /src/test.lean | 3b8aaa3c79bb1c61472cd9783a808017288f4cc5 | [] | no_license | jjaassoonn/transcendental | 8008813253af3aa80b5a5c56551317e7ab4246ab | 99bc6ea6089f04ed90a0f55f0533ebb7f47b22ff | refs/heads/master | 1,607,869,957,944 | 1,599,659,687,000 | 1,599,659,687,000 | 234,444,826 | 9 | 1 | null | 1,598,218,307,000 | 1,579,224,232,000 | HTML | UTF-8 | Lean | false | false | 8,676 | lean | import measure_theory.set_integral
import measure_theory.bochner_integration
import measure_theory.interval_integral
import measure_theory.lebesgue_measure
import analysis.special_functions.exp_log
import small_things
import tactic
noncomputable theory
open_locale classical
open_locale big_operators
notation α`[X]` := polynomial α
-- F(x) = ∫ t in a .. x, f t
-- F'(x) = f(x) for x in [a,b]
private lemma ftc1 (f : ℝ -> ℝ) {hf : measurable f} {hf2 : continuous f} (a b : ℝ) (h : a ≤ b) {hf3 : measure_theory.integrable_on f (set.Icc a b)} (x0 : ℝ) (hx0 : x0 ∈ set.Icc a b) :
has_deriv_at (λ (b : ℝ), ∫ (x : ℝ) in a..b, f x) (f x0) x0 :=
begin
apply interval_integral.integral_has_deriv_at_of_tendsto_ae,
simp only [], exact hf,
unfold interval_integrable,
split,
apply measure_theory.integrable_on.mono hf3,
intros y hy,
simp only [set.mem_Ioc, set.mem_Icc] at ⊢ hy hx0,
split, linarith, linarith,
apply le_refl _,
apply measure_theory.integrable_on.mono hf3,
intros y hy,
simp only [set.mem_Ioc, set.mem_Icc] at ⊢ hy hx0,
split, linarith, linarith,
apply le_refl _,
apply filter.tendsto_inf_left,
exact continuous.tendsto hf2 x0,
end
/-- # Assumption-/
/-Theorem
fundamental theorem of calculus and integration by part is assumed. I am waiting for them to arrive in `mathlib` and I will update this part and prove relatvent additional assumptions.
-/
theorem ftc' (F f: ℝ -> ℝ) {hF : differentiable ℝ F}
{F_deriv : deriv F = f} {hf : measurable f} {hf1 : continuous f}
(a b : ℝ) (h : b ≥ a) : (∫ x in a..b, f x) = F b - F a :=
begin
by_cases hab : (a = b),
rw hab, simp only [interval_integral.integral_same, sub_self],
set G := (λ x, (∫ t in a..x, f t)) with hG,
have prop := ftc1 f a b h,
rw <-hG at prop,
have hG1 : differentiable_on ℝ G (set.Icc a b),
{
intros x hx,
have prop := ftc1 f a b h x hx,
refine differentiable_at.differentiable_within_at _,
rw hG,
exact has_deriv_at.differentiable_at prop,
exact hf, exact hf1,
apply continuous.integrable_on_compact _ hf1,
exact real.locally_finite_volume, exact compact_Icc,
},
have H : (set.Icc a b).indicator (deriv G) = (set.Icc a b).indicator f,
{
apply set.indicator_congr,
intros x0 hx0,
replace prop := prop x0 hx0,
exact has_deriv_at.deriv prop,
},
have H2 : (set.Icc a b).indicator (deriv G) = (set.Icc a b).indicator (deriv F),
{
exact eq.trans H (congr_arg (set.Icc a b).indicator (eq.symm F_deriv)),
},
replace H2 : ∀ y ∈ set.Icc a b, (deriv (F - G)) y = 0,
{
intros y hy,
change deriv (λ t, F t - G t) y = 0,
rw deriv_sub, rw sub_eq_zero,
have eq1 : (set.Icc a b).indicator (deriv F) y = deriv F y,
exact if_pos hy, rw <-eq1,
have eq2 : (set.Icc a b).indicator (deriv G) y = deriv G y,
exact if_pos hy, rw <-eq2, exact congr_fun H2.symm y,
dsimp only [],
exact hF y,
dsimp only [],
exact has_deriv_at.differentiable_at (prop y hy),
},
have key : ∀ y ∈ set.Ioc a b, (F - G) y = (F - G) a,
{
intros y hy,
have ineq : a < y, simp only [set.mem_Ioc] at hy, exact hy.1,
have key := exists_deriv_eq_slope (F - G) ineq _ _,
rcases key with ⟨c, hc, hc2⟩,
have hc' : c ∈ set.Icc a b,
simp only [set.mem_Icc, set.mem_Ioc, set.mem_Ioo] at hy ⊢ hc,
split, linarith, linarith,
rw H2 c hc' at hc2,
replace hc2 := eq.symm hc2,
rw div_eq_zero_iff at hc2,
cases hc2, exact sub_eq_zero.mp hc2,
simp only [set.mem_Icc, set.mem_Ioc, set.mem_Ioo] at hy ⊢ hc, linarith,
apply continuous_on.sub,
simp only [], apply continuous.continuous_on,
apply differentiable.continuous hF,
have hG1' : continuous_on G (set.Icc a b),
apply differentiable_on.continuous_on hG1,
simp only [], apply continuous_on.mono hG1',
apply set.Icc_subset_Icc, exact le_refl a, exact hy.2,
apply differentiable_on.sub,
simp only [], exact differentiable.differentiable_on hF,
simp only [], apply differentiable_on.mono hG1,
intros z hz,
simp only [set.mem_Icc, set.mem_Ioc, set.mem_Ioo] at *,
split, linarith, linarith,
},
have G_a : G a = 0,
{
rw hG, simp only [interval_integral.integral_same],
},
have G_b : G b = ∫ x in a .. b, f x,
{
rw hG,
},
rw <-G_b,
have eq : G b = G b - 0, rw sub_zero, rw eq, rw <-G_a,
rw sub_eq_sub_iff_sub_eq_sub,
suffices : F b - G b = F a - G a, linarith,
replace key := key b _,
simp only [pi.sub_apply] at key ⊢, exact key,
simp only [set.mem_Icc, set.mem_Ioc, set.mem_Ioo] at *,
split, exact lt_of_le_of_ne h hab, exact le_refl b,
exact hf, exact hf1,
apply measure_theory.integrable_on.mono_set (continuous.integrable_on_compact (@compact_Icc a b) hf1),
exact set.subset.rfl,
exact real.locally_finite_volume,
end
theorem ftc (f: ℝ -> ℝ) {hf : differentiable ℝ f} {hf2 : measurable (deriv f)} {hf3 : continuous (deriv f)} (a b : ℝ) (h : b ≥ a) : (∫ x in a..b, (deriv f) x) = f b - f a :=
begin
refine ftc' f (deriv f) a b h,
simp only [], exact hf,
refl, exact hf2, exact hf3,
end
theorem integrate_by_part (f g : ℝ -> ℝ)
{hf : differentiable ℝ f} {hf2 : measurable (deriv f)}
{hf3 : measurable f} {hf4 : continuous (deriv f)}
{hg : differentiable ℝ g} {hg2 : measurable (deriv g)}
{hg3 : measurable g} {hg4 : continuous (deriv g)}
(a b : ℝ) (h : b ≥ a) :
(∫ x in a..b, (f x)*(deriv g x)) = (f b) * (g b) - (f a) * (g a) - (∫ x in a..b, (deriv f x) * (g x)) :=
begin
have eq1 := ftc (f * g) a b h,
have eq2 : (∫ (x : ℝ) in a..b, deriv (f * g) x) = (∫ (x : ℝ) in a..b, (deriv f x) * g x + f x * (deriv g x)),
{
rw interval_integral.integral_of_le h,
rw interval_integral.integral_of_le h,
apply congr_arg, ext y,
apply deriv_mul,
simp only [],
exact hf y,
simp only [],
exact hg y,
},
rw eq2 at eq1,
rw interval_integral.integral_add at eq1,
simp only [pi.mul_apply] at eq1 ⊢,
rw <-eq1, simp only [add_sub_cancel'],
apply measurable.mul,
simp only [], exact hf2,
simp only [], exact hg3,
rw interval_integrable,
have H1 : continuous (λ x, (deriv f x) * g x),
{
apply continuous.mul _ _, exact normed_ring_top_monoid,
exact hf4, apply @differentiable.continuous ℝ _ ℝ _ _ ℝ _ _ _,
exact hg,
},
split,
apply measure_theory.integrable_on.mono_set (continuous.integrable_on_compact (@compact_Icc a b) H1),
exact set.Ioc_subset_Icc_self,
exact real.locally_finite_volume,
apply measure_theory.integrable_on.mono_set (continuous.integrable_on_compact (@compact_Icc a b) H1),
rw set.Ioc_eq_empty, exact (set.Icc a b).empty_subset, exact h,
exact real.locally_finite_volume,
apply measurable.mul, exact hf3, exact hg2,
rw interval_integrable,
have H2 : continuous (λ x, f x * deriv g x),
{
apply continuous.mul _ _, exact normed_ring_top_monoid,
apply @differentiable.continuous ℝ _ ℝ _ _ ℝ _ _ _,
exact hf, exact hg4,
},
split,
apply measure_theory.integrable_on.mono_set (continuous.integrable_on_compact (@compact_Icc a b) H2),
exact set.Ioc_subset_Icc_self,
exact real.locally_finite_volume,
apply measure_theory.integrable_on.mono_set (continuous.integrable_on_compact (@compact_Icc a b) H2),
rw set.Ioc_eq_empty, exact (set.Icc a b).empty_subset, exact h,
exact real.locally_finite_volume,
apply differentiable.mul hf hg,
apply continuous.measurable,
{
have eq1 : deriv (f * g) = (deriv f) * g + f * (deriv g),
ext, simp only [pi.add_apply, pi.mul_apply],
change (deriv (λ x, (f x * g x))) x = deriv f x * g x + f x * deriv g x,
rw deriv_mul,
simp only [], exact hf x,
simp only [], exact hg x,
rw eq1, apply continuous.add _ _,
exact normed_top_monoid, apply continuous.mul _ _,
exact normed_ring_top_monoid, exact hf4,
apply differentiable.continuous hg,
apply continuous.mul _ _, exact normed_ring_top_monoid,
apply differentiable.continuous hf, exact hg4,
},
{
have eq1 : deriv (f * g) = (deriv f) * g + f * (deriv g),
ext, simp only [pi.add_apply, pi.mul_apply],
change (deriv (λ x, (f x * g x))) x = deriv f x * g x + f x * deriv g x,
rw deriv_mul,
simp only [], exact hf x,
simp only [], exact hg x,
rw eq1, apply continuous.add _ _,
exact normed_top_monoid, apply continuous.mul _ _,
exact normed_ring_top_monoid, exact hf4,
apply differentiable.continuous hg,
apply continuous.mul _ _, exact normed_ring_top_monoid,
apply differentiable.continuous hf, exact hg4,
},
end
|
908f21c9b01bbd37f15d4a03d7b8dc0dd250cafc | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /tests/lean/mismatch.lean | 9c18bf3502620466d96e4c629ac5370e3af3f74a | [
"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 | 82 | lean | -- definition id {A : Type} {a : A} := a
definition o : num := 1
check @id nat o
|
5e6c75c64cd3f43719ef477916221b4923a2ac0e | 958488bc7f3c2044206e0358e56d7690b6ae696c | /lean/hhg/nat.lean | 92358b313ddb59ae6786d334d8dee6a632d89de4 | [] | no_license | possientis/Prog | a08eec1c1b121c2fd6c70a8ae89e2fbef952adb4 | d4b3debc37610a88e0dac3ac5914903604fd1d1f | refs/heads/master | 1,692,263,717,723 | 1,691,757,179,000 | 1,691,757,179,000 | 40,361,602 | 3 | 0 | null | 1,679,896,438,000 | 1,438,953,859,000 | Coq | UTF-8 | Lean | false | false | 406 | lean | inductive nat2 : Type
| zero : nat2
| succ : nat2 -> nat2
--#print nat2
inductive aexp : Type
| num : nat -> aexp -- TODO: need Z
| var : string -> aexp
| add : aexp -> aexp -> aexp
| sub : aexp -> aexp -> aexp
| mul : aexp -> aexp -> aexp
| div : aexp -> aexp -> aexp
lemma add_zero2 : ∀ (n : ℕ), 0 + n = n :=
begin
intros n,
induction n with n IH,
{reflexivity},
{simp}
end
|
a40b1a06efdda94c432601682b65650cd60d9f3a | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/data/real/pi/bounds.lean | 48790c51e6d406b7a8e5c22549d5d37c7fc9e965 | [
"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 | 7,798 | lean | /-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Mario Carneiro
-/
import analysis.special_functions.trigonometric.basic
/-!
# Pi
This file contains lemmas which establish bounds on `real.pi`.
Notably, these include `pi_gt_sqrt_two_add_series` and `pi_lt_sqrt_two_add_series`,
which bound `π` using series;
numerical bounds on `π` such as `pi_gt_314`and `pi_lt_315` (more precise versions are given, too).
See also `data.real.pi.leibniz` and `data.real.pi.wallis` for infinite formulas for `π`.
-/
open_locale real
namespace real
lemma pi_gt_sqrt_two_add_series (n : ℕ) : 2 ^ (n+1) * sqrt (2 - sqrt_two_add_series 0 n) < π :=
begin
have : sqrt (2 - sqrt_two_add_series 0 n) / 2 * 2 ^ (n+2) < π,
{ rw [← lt_div_iff, ←sin_pi_over_two_pow_succ], apply sin_lt, apply div_pos pi_pos,
all_goals { apply pow_pos, norm_num } },
apply lt_of_le_of_lt (le_of_eq _) this,
rw [pow_succ _ (n+1), ←mul_assoc, div_mul_cancel, mul_comm], norm_num
end
lemma pi_lt_sqrt_two_add_series (n : ℕ) :
π < 2 ^ (n+1) * sqrt (2 - sqrt_two_add_series 0 n) + 1 / 4 ^ n :=
begin
have : π < (sqrt (2 - sqrt_two_add_series 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n+2),
{ rw [← div_lt_iff, ← sin_pi_over_two_pow_succ],
refine lt_of_lt_of_le (lt_add_of_sub_right_lt (sin_gt_sub_cube _ _)) _,
{ apply div_pos pi_pos, apply pow_pos, norm_num },
{ rw div_le_iff',
{ refine le_trans pi_le_four _,
simp only [show ((4 : ℝ) = 2 ^ 2), by norm_num, mul_one],
apply pow_le_pow, norm_num, apply le_add_of_nonneg_left, apply nat.zero_le },
{ apply pow_pos, norm_num } },
apply add_le_add_left, rw div_le_div_right,
rw [le_div_iff, ←mul_pow],
refine le_trans _ (le_of_eq (one_pow 3)), apply pow_le_pow_of_le_left,
{ apply le_of_lt, apply mul_pos, apply div_pos pi_pos, apply pow_pos, norm_num, apply pow_pos,
norm_num },
rw ← le_div_iff,
refine le_trans ((div_le_div_right _).mpr pi_le_four) _, apply pow_pos, norm_num,
rw [pow_succ, pow_succ, ←mul_assoc, ←div_div_eq_div_mul],
convert le_rfl,
all_goals { repeat {apply pow_pos}, norm_num }},
apply lt_of_lt_of_le this (le_of_eq _), rw [add_mul], congr' 1,
{ rw [pow_succ _ (n+1), ←mul_assoc, div_mul_cancel, mul_comm], norm_num },
rw [pow_succ, ←pow_mul, mul_comm n 2, pow_mul, show (2 : ℝ) ^ 2 = 4, by norm_num, pow_succ,
pow_succ, ←mul_assoc (2 : ℝ), show (2 : ℝ) * 2 = 4, by norm_num, ←mul_assoc, div_mul_cancel,
mul_comm ((2 : ℝ) ^ n), ←div_div_eq_div_mul, div_mul_cancel],
apply pow_ne_zero, norm_num, norm_num
end
/-- From an upper bound on `sqrt_two_add_series 0 n = 2 cos (π / 2 ^ (n+1))` of the form
`sqrt_two_add_series 0 n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2)`, one can deduce the lower bound `a < π`
thanks to basic trigonometric inequalities as expressed in `pi_gt_sqrt_two_add_series`. -/
theorem pi_lower_bound_start (n : ℕ) {a}
(h : sqrt_two_add_series ((0:ℕ) / (1:ℕ)) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2) : a < π :=
begin
refine lt_of_le_of_lt _ (pi_gt_sqrt_two_add_series n), rw [mul_comm],
refine (div_le_iff (pow_pos (by norm_num) _ : (0 : ℝ) < _)).mp (le_sqrt_of_sq_le _),
rwa [le_sub, show (0:ℝ) = (0:ℕ)/(1:ℕ), by rw [nat.cast_zero, zero_div]],
end
lemma sqrt_two_add_series_step_up (c d : ℕ) {a b n : ℕ} {z : ℝ}
(hz : sqrt_two_add_series (c/d) n ≤ z) (hb : 0 < b) (hd : 0 < d)
(h : (2 * b + a) * d ^ 2 ≤ c ^ 2 * b) : sqrt_two_add_series (a/b) (n+1) ≤ z :=
begin
refine le_trans _ hz, rw sqrt_two_add_series_succ, apply sqrt_two_add_series_monotone_left,
have hb' : 0 < (b:ℝ) := nat.cast_pos.2 hb,
have hd' : 0 < (d:ℝ) := nat.cast_pos.2 hd,
rw [sqrt_le_left (div_nonneg c.cast_nonneg d.cast_nonneg), div_pow,
add_div_eq_mul_add_div _ _ (ne_of_gt hb'), div_le_div_iff hb' (pow_pos hd' _)],
exact_mod_cast h
end
/-- Create a proof of `a < π` for a fixed rational number `a`, given a witness, which is a
sequence of rational numbers `sqrt 2 < r 1 < r 2 < ... < r n < 2` satisfying the property that
`sqrt (2 + r i) ≤ r(i+1)`, where `r 0 = 0` and `sqrt (2 - r n) ≥ a/2^(n+1)`. -/
meta def pi_lower_bound (l : list ℚ) : tactic unit :=
do let n := l.length,
tactic.apply `(@pi_lower_bound_start %%(reflect n)),
l.mmap' (λ r, do
let a := r.num.to_nat, let b := r.denom,
(() <$ tactic.apply `(@sqrt_two_add_series_step_up %%(reflect a) %%(reflect b)));
[tactic.skip, `[norm_num1], `[norm_num1], `[norm_num1]]),
`[simp only [sqrt_two_add_series, nat.cast_bit0, nat.cast_bit1, nat.cast_one, nat.cast_zero]],
`[norm_num1]
/-- From a lower bound on `sqrt_two_add_series 0 n = 2 cos (π / 2 ^ (n+1))` of the form
`2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrt_two_add_series 0 n`, one can deduce the upper bound
`π < a` thanks to basic trigonometric formulas as expressed in `pi_lt_sqrt_two_add_series`. -/
theorem pi_upper_bound_start (n : ℕ) {a}
(h : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrt_two_add_series ((0:ℕ) / (1:ℕ)) n)
(h₂ : 1 / 4 ^ n ≤ a) : π < a :=
begin
refine lt_of_lt_of_le (pi_lt_sqrt_two_add_series n) _,
rw [← le_sub_iff_add_le, ← le_div_iff', sqrt_le_left, sub_le],
{ rwa [nat.cast_zero, zero_div] at h },
{ exact div_nonneg (sub_nonneg.2 h₂) (pow_nonneg (le_of_lt zero_lt_two) _) },
{ exact pow_pos zero_lt_two _ }
end
lemma sqrt_two_add_series_step_down (a b : ℕ) {c d n : ℕ} {z : ℝ}
(hz : z ≤ sqrt_two_add_series (a/b) n) (hb : 0 < b) (hd : 0 < d)
(h : a ^ 2 * d ≤ (2 * d + c) * b ^ 2) : z ≤ sqrt_two_add_series (c/d) (n+1) :=
begin
apply le_trans hz, rw sqrt_two_add_series_succ, apply sqrt_two_add_series_monotone_left,
apply le_sqrt_of_sq_le,
have hb' : 0 < (b:ℝ) := nat.cast_pos.2 hb,
have hd' : 0 < (d:ℝ) := nat.cast_pos.2 hd,
rw [div_pow, add_div_eq_mul_add_div _ _ (ne_of_gt hd'), div_le_div_iff (pow_pos hb' _) hd'],
exact_mod_cast h
end
/-- Create a proof of `π < a` for a fixed rational number `a`, given a witness, which is a
sequence of rational numbers `sqrt 2 < r 1 < r 2 < ... < r n < 2` satisfying the property that
`sqrt (2 + r i) ≥ r(i+1)`, where `r 0 = 0` and `sqrt (2 - r n) ≥ (a - 1/4^n) / 2^(n+1)`. -/
meta def pi_upper_bound (l : list ℚ) : tactic unit :=
do let n := l.length,
(() <$ tactic.apply `(@pi_upper_bound_start %%(reflect n))); [pure (), `[norm_num1]],
l.mmap' (λ r, do
let a := r.num.to_nat, let b := r.denom,
(() <$ tactic.apply `(@sqrt_two_add_series_step_down %%(reflect a) %%(reflect b)));
[pure (), `[norm_num1], `[norm_num1], `[norm_num1]]),
`[simp only [sqrt_two_add_series, nat.cast_bit0, nat.cast_bit1, nat.cast_one, nat.cast_zero]],
`[norm_num]
lemma pi_gt_three : 3 < π := by pi_lower_bound [23/16]
lemma pi_gt_314 : 3.14 < π := by pi_lower_bound [99/70, 874/473, 1940/989, 1447/727]
lemma pi_lt_315 : π < 3.15 := by pi_upper_bound [140/99, 279/151, 51/26, 412/207]
lemma pi_gt_31415 : 3.1415 < π := by pi_lower_bound [
11482/8119, 5401/2923, 2348/1197, 11367/5711, 25705/12868, 23235/11621]
lemma pi_lt_31416 : π < 3.1416 := by pi_upper_bound [
4756/3363, 101211/54775, 505534/257719, 83289/41846,
411278/205887, 438142/219137, 451504/225769, 265603/132804, 849938/424971]
lemma pi_gt_3141592 : 3.141592 < π := by pi_lower_bound [
11482/8119, 7792/4217, 54055/27557, 949247/476920, 3310126/1657059,
2635492/1318143, 1580265/790192, 1221775/610899, 3612247/1806132, 849943/424972]
lemma pi_lt_3141593 : π < 3.141593 := by pi_upper_bound [
27720/19601, 56935/30813, 49359/25163, 258754/130003, 113599/56868, 1101994/551163,
8671537/4336095, 3877807/1938940, 52483813/26242030, 56946167/28473117, 23798415/11899211]
end real
|
f0ccf372a3348af24fb2084aa64b696735bb24de | ac89c256db07448984849346288e0eeffe8b20d0 | /src/Init/Prelude.lean | 6861c0cfc0b658f6a830ecb73bcf0e5449d96c31 | [
"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 | 75,439 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
universe u v w
@[inline] def id {α : Sort u} (a : α) : α := a
/- `idRhs` is an auxiliary declaration used to implement "smart unfolding". It is used as a marker. -/
@[macroInline, reducible] def idRhs (α : Sort u) (a : α) : α := a
abbrev Function.comp {α : Sort u} {β : Sort v} {δ : Sort w} (f : β → δ) (g : α → β) : α → δ :=
fun x => f (g x)
abbrev Function.const {α : Sort u} (β : Sort v) (a : α) : β → α :=
fun x => a
set_option checkBinderAnnotations false in
@[reducible] def inferInstance {α : Sort u} [i : α] : α := i
set_option checkBinderAnnotations false in
@[reducible] def inferInstanceAs (α : Sort u) [i : α] : α := i
set_option bootstrap.inductiveCheckResultingUniverse false in
inductive PUnit : Sort u where
| unit : PUnit
/-- An abbreviation for `PUnit.{0}`, its most common instantiation.
This Type should be preferred over `PUnit` where possible to avoid
unnecessary universe parameters. -/
abbrev Unit : Type := PUnit
@[matchPattern] abbrev Unit.unit : Unit := PUnit.unit
/-- Auxiliary unsafe constant used by the Compiler when erasing proofs from code. -/
unsafe axiom lcProof {α : Prop} : α
/-- Auxiliary unsafe constant used by the Compiler to mark unreachable code. -/
unsafe axiom lcUnreachable {α : Sort u} : α
inductive True : Prop where
| intro : True
inductive False : Prop
inductive Empty : Type
set_option bootstrap.inductiveCheckResultingUniverse false in
inductive PEmpty : Sort u where
def Not (a : Prop) : Prop := a → False
@[macroInline] def False.elim {C : Sort u} (h : False) : C :=
False.rec (fun _ => C) h
@[macroInline] def absurd {a : Prop} {b : Sort v} (h₁ : a) (h₂ : Not a) : b :=
False.elim (h₂ h₁)
inductive Eq {α : Sort u} (a : α) : α → Prop where
| refl {} : Eq a a
@[simp] abbrev Eq.ndrec.{u1, u2} {α : Sort u2} {a : α} {motive : α → Sort u1} (m : motive a) {b : α} (h : Eq a b) : motive b :=
Eq.rec (motive := fun α _ => motive α) m h
@[matchPattern] def rfl {α : Sort u} {a : α} : Eq a a := Eq.refl a
@[simp] theorem id_eq (a : α) : Eq (id a) a := rfl
theorem Eq.subst {α : Sort u} {motive : α → Prop} {a b : α} (h₁ : Eq a b) (h₂ : motive a) : motive b :=
Eq.ndrec h₂ h₁
theorem Eq.symm {α : Sort u} {a b : α} (h : Eq a b) : Eq b a :=
h ▸ rfl
theorem Eq.trans {α : Sort u} {a b c : α} (h₁ : Eq a b) (h₂ : Eq b c) : Eq a c :=
h₂ ▸ h₁
@[macroInline] def cast {α β : Sort u} (h : Eq α β) (a : α) : β :=
Eq.rec (motive := fun α _ => α) a h
theorem congrArg {α : Sort u} {β : Sort v} {a₁ a₂ : α} (f : α → β) (h : Eq a₁ a₂) : Eq (f a₁) (f a₂) :=
h ▸ rfl
theorem congr {α : Sort u} {β : Sort v} {f₁ f₂ : α → β} {a₁ a₂ : α} (h₁ : Eq f₁ f₂) (h₂ : Eq a₁ a₂) : Eq (f₁ a₁) (f₂ a₂) :=
h₁ ▸ h₂ ▸ rfl
theorem congrFun {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x} (h : Eq f g) (a : α) : Eq (f a) (g a) :=
h ▸ rfl
/-
Initialize the Quotient Module, which effectively adds the following definitions:
constant Quot {α : Sort u} (r : α → α → Prop) : Sort u
constant Quot.mk {α : Sort u} (r : α → α → Prop) (a : α) : Quot r
constant Quot.lift {α : Sort u} {r : α → α → Prop} {β : Sort v} (f : α → β) :
(∀ a b : α, r a b → Eq (f a) (f b)) → Quot r → β
constant Quot.ind {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop} :
(∀ a : α, β (Quot.mk r a)) → ∀ q : Quot r, β q
-/
init_quot
inductive HEq {α : Sort u} (a : α) : {β : Sort u} → β → Prop where
| refl {} : HEq a a
@[matchPattern] protected def HEq.rfl {α : Sort u} {a : α} : HEq a a :=
HEq.refl a
theorem eq_of_heq {α : Sort u} {a a' : α} (h : HEq a a') : Eq a a' :=
have : (α β : Sort u) → (a : α) → (b : β) → HEq a b → (h : Eq α β) → Eq (cast h a) b :=
fun α β a b h₁ =>
HEq.rec (motive := fun {β} (b : β) (h : HEq a b) => (h₂ : Eq α β) → Eq (cast h₂ a) b)
(fun (h₂ : Eq α α) => rfl)
h₁
this α α a a' h rfl
structure Prod (α : Type u) (β : Type v) where
fst : α
snd : β
attribute [unbox] Prod
/-- Similar to `Prod`, but `α` and `β` can be propositions.
We use this Type internally to automatically generate the brecOn recursor. -/
structure PProd (α : Sort u) (β : Sort v) where
fst : α
snd : β
/-- Similar to `Prod`, but `α` and `β` are in the same universe. -/
structure MProd (α β : Type u) where
fst : α
snd : β
structure And (a b : Prop) : Prop where
intro :: (left : a) (right : b)
inductive Or (a b : Prop) : Prop where
| inl (h : a) : Or a b
| inr (h : b) : Or a b
inductive Bool : Type where
| false : Bool
| true : Bool
export Bool (false true)
/- Remark: Subtype must take a Sort instead of Type because of the axiom strongIndefiniteDescription. -/
structure Subtype {α : Sort u} (p : α → Prop) where
val : α
property : p val
/-- Gadget for optional parameter support. -/
@[reducible] def optParam (α : Sort u) (default : α) : Sort u := α
/-- Gadget for marking output parameters in type classes. -/
@[reducible] def outParam (α : Sort u) : Sort u := α
/-- Auxiliary Declaration used to implement the notation (a : α) -/
@[reducible] def typedExpr (α : Sort u) (a : α) : α := a
/-- Auxiliary Declaration used to implement the named patterns `x@p` -/
@[reducible] def namedPattern {α : Sort u} (x a : α) : α := a
/- Auxiliary axiom used to implement `sorry`. -/
@[extern "lean_sorry", neverExtract]
axiom sorryAx (α : Sort u) (synthetic := true) : α
theorem eq_false_of_ne_true : {b : Bool} → Not (Eq b true) → Eq b false
| true, h => False.elim (h rfl)
| false, h => rfl
theorem eq_true_of_ne_false : {b : Bool} → Not (Eq b false) → Eq b true
| true, h => rfl
| false, h => False.elim (h rfl)
theorem ne_false_of_eq_true : {b : Bool} → Eq b true → Not (Eq b false)
| true, _ => fun h => Bool.noConfusion h
| false, h => Bool.noConfusion h
theorem ne_true_of_eq_false : {b : Bool} → Eq b false → Not (Eq b true)
| true, h => Bool.noConfusion h
| false, _ => fun h => Bool.noConfusion h
class Inhabited (α : Sort u) where
mk {} :: (default : α)
constant arbitrary [Inhabited α] : α :=
Inhabited.default
instance : Inhabited (Sort u) where
default := PUnit
instance (α : Sort u) {β : Sort v} [Inhabited β] : Inhabited (α → β) where
default := fun _ => arbitrary
instance (α : Sort u) {β : α → Sort v} [(a : α) → Inhabited (β a)] : Inhabited ((a : α) → β a) where
default := fun _ => arbitrary
deriving instance Inhabited for Bool
/-- Universe lifting operation from Sort to Type -/
structure PLift (α : Sort u) : Type u where
up :: (down : α)
/- Bijection between α and PLift α -/
theorem PLift.up_down {α : Sort u} : ∀ (b : PLift α), Eq (up (down b)) b
| up a => rfl
theorem PLift.down_up {α : Sort u} (a : α) : Eq (down (up a)) a :=
rfl
/- Pointed types -/
structure PointedType where
(type : Type u)
(val : type)
instance : Inhabited PointedType.{u} where
default := { type := PUnit.{u+1}, val := ⟨⟩ }
/-- Universe lifting operation -/
structure ULift.{r, s} (α : Type s) : Type (max s r) where
up :: (down : α)
/- Bijection between α and ULift.{v} α -/
theorem ULift.up_down {α : Type u} : ∀ (b : ULift.{v} α), Eq (up (down b)) b
| up a => rfl
theorem ULift.down_up {α : Type u} (a : α) : Eq (down (up.{v} a)) a :=
rfl
class inductive Decidable (p : Prop) where
| isFalse (h : Not p) : Decidable p
| isTrue (h : p) : Decidable p
@[inlineIfReduce, nospecialize] def Decidable.decide (p : Prop) [h : Decidable p] : Bool :=
Decidable.casesOn (motive := fun _ => Bool) h (fun _ => false) (fun _ => true)
export Decidable (isTrue isFalse decide)
abbrev DecidablePred {α : Sort u} (r : α → Prop) :=
(a : α) → Decidable (r a)
abbrev DecidableRel {α : Sort u} (r : α → α → Prop) :=
(a b : α) → Decidable (r a b)
abbrev DecidableEq (α : Sort u) :=
(a b : α) → Decidable (Eq a b)
def decEq {α : Sort u} [s : DecidableEq α] (a b : α) : Decidable (Eq a b) :=
s a b
theorem decide_eq_true : [s : Decidable p] → p → Eq (decide p) true
| isTrue _, _ => rfl
| isFalse h₁, h₂ => absurd h₂ h₁
theorem decide_eq_false : [s : Decidable p] → Not p → Eq (decide p) false
| isTrue h₁, h₂ => absurd h₁ h₂
| isFalse h, _ => rfl
theorem of_decide_eq_true [s : Decidable p] : Eq (decide p) true → p := fun h =>
match (generalizing := false) s with
| isTrue h₁ => h₁
| isFalse h₁ => absurd h (ne_true_of_eq_false (decide_eq_false h₁))
theorem of_decide_eq_false [s : Decidable p] : Eq (decide p) false → Not p := fun h =>
match (generalizing := false) s with
| isTrue h₁ => absurd h (ne_false_of_eq_true (decide_eq_true h₁))
| isFalse h₁ => h₁
@[inline] instance : DecidableEq Bool :=
fun a b => match a, b with
| false, false => isTrue rfl
| false, true => isFalse (fun h => Bool.noConfusion h)
| true, false => isFalse (fun h => Bool.noConfusion h)
| true, true => isTrue rfl
class BEq (α : Type u) where
beq : α → α → Bool
open BEq (beq)
instance [DecidableEq α] : BEq α where
beq a b := decide (Eq a b)
-- We use "dependent" if-then-else to be able to communicate the if-then-else condition
-- to the branches
@[macroInline] def dite {α : Sort u} (c : Prop) [h : Decidable c] (t : c → α) (e : Not c → α) : α :=
Decidable.casesOn (motive := fun _ => α) h e t
/- if-then-else -/
@[macroInline] def ite {α : Sort u} (c : Prop) [h : Decidable c] (t e : α) : α :=
Decidable.casesOn (motive := fun _ => α) h (fun _ => e) (fun _ => t)
@[macroInline] instance {p q} [dp : Decidable p] [dq : Decidable q] : Decidable (And p q) :=
match dp with
| isTrue hp =>
match dq with
| isTrue hq => isTrue ⟨hp, hq⟩
| isFalse hq => isFalse (fun h => hq (And.right h))
| isFalse hp =>
isFalse (fun h => hp (And.left h))
@[macroInline] instance [dp : Decidable p] [dq : Decidable q] : Decidable (Or p q) :=
match dp with
| isTrue hp => isTrue (Or.inl hp)
| isFalse hp =>
match dq with
| isTrue hq => isTrue (Or.inr hq)
| isFalse hq =>
isFalse fun h => match h with
| Or.inl h => hp h
| Or.inr h => hq h
instance [dp : Decidable p] : Decidable (Not p) :=
match dp with
| isTrue hp => isFalse (absurd hp)
| isFalse hp => isTrue hp
/- Boolean operators -/
@[macroInline] def cond {α : Type u} (c : Bool) (x y : α) : α :=
match c with
| true => x
| false => y
@[macroInline] def or (x y : Bool) : Bool :=
match x with
| true => true
| false => y
@[macroInline] def and (x y : Bool) : Bool :=
match x with
| false => false
| true => y
@[inline] def not : Bool → Bool
| true => false
| false => true
inductive Nat where
| zero : Nat
| succ (n : Nat) : Nat
instance : Inhabited Nat where
default := Nat.zero
/- For numeric literals notation -/
class OfNat (α : Type u) (n : Nat) where
ofNat : α
@[defaultInstance 100] /- low prio -/
instance (n : Nat) : OfNat Nat n where
ofNat := n
class LE (α : Type u) where le : α → α → Prop
class LT (α : Type u) where lt : α → α → Prop
@[reducible] def GE.ge {α : Type u} [LE α] (a b : α) : Prop := LE.le b a
@[reducible] def GT.gt {α : Type u} [LT α] (a b : α) : Prop := LT.lt b a
@[inline] def max [LT α] [DecidableRel (@LT.lt α _)] (a b : α) : α :=
ite (LT.lt b a) a b
@[inline] def min [LE α] [DecidableRel (@LE.le α _)] (a b : α) : α :=
ite (LE.le a b) a b
class HAdd (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hAdd : α → β → γ
class HSub (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hSub : α → β → γ
class HMul (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hMul : α → β → γ
class HDiv (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hDiv : α → β → γ
class HMod (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hMod : α → β → γ
class HPow (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hPow : α → β → γ
class HAppend (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hAppend : α → β → γ
class HOrElse (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hOrElse : α → β → γ
class HAndThen (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hAndThen : α → β → γ
class HAnd (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hAnd : α → β → γ
class HXor (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hXor : α → β → γ
class HOr (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hOr : α → β → γ
class HShiftLeft (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hShiftLeft : α → β → γ
class HShiftRight (α : Type u) (β : Type v) (γ : outParam (Type w)) where
hShiftRight : α → β → γ
class Add (α : Type u) where
add : α → α → α
class Sub (α : Type u) where
sub : α → α → α
class Mul (α : Type u) where
mul : α → α → α
class Neg (α : Type u) where
neg : α → α
class Div (α : Type u) where
div : α → α → α
class Mod (α : Type u) where
mod : α → α → α
class Pow (α : Type u) (β : Type v) where
pow : α → β → α
class Append (α : Type u) where
append : α → α → α
class OrElse (α : Type u) where
orElse : α → α → α
class AndThen (α : Type u) where
andThen : α → α → α
class AndOp (α : Type u) where
and : α → α → α
class Xor (α : Type u) where
xor : α → α → α
class OrOp (α : Type u) where
or : α → α → α
class Complement (α : Type u) where
complement : α → α
class ShiftLeft (α : Type u) where
shiftLeft : α → α → α
class ShiftRight (α : Type u) where
shiftRight : α → α → α
@[defaultInstance]
instance [Add α] : HAdd α α α where
hAdd a b := Add.add a b
@[defaultInstance]
instance [Sub α] : HSub α α α where
hSub a b := Sub.sub a b
@[defaultInstance]
instance [Mul α] : HMul α α α where
hMul a b := Mul.mul a b
@[defaultInstance]
instance [Div α] : HDiv α α α where
hDiv a b := Div.div a b
@[defaultInstance]
instance [Mod α] : HMod α α α where
hMod a b := Mod.mod a b
@[defaultInstance]
instance [Pow α β] : HPow α β α where
hPow a b := Pow.pow a b
@[defaultInstance]
instance [Append α] : HAppend α α α where
hAppend a b := Append.append a b
@[defaultInstance]
instance [OrElse α] : HOrElse α α α where
hOrElse a b := OrElse.orElse a b
@[defaultInstance]
instance [AndThen α] : HAndThen α α α where
hAndThen a b := AndThen.andThen a b
@[defaultInstance]
instance [AndOp α] : HAnd α α α where
hAnd a b := AndOp.and a b
@[defaultInstance]
instance [Xor α] : HXor α α α where
hXor a b := Xor.xor a b
@[defaultInstance]
instance [OrOp α] : HOr α α α where
hOr a b := OrOp.or a b
@[defaultInstance]
instance [ShiftLeft α] : HShiftLeft α α α where
hShiftLeft a b := ShiftLeft.shiftLeft a b
@[defaultInstance]
instance [ShiftRight α] : HShiftRight α α α where
hShiftRight a b := ShiftRight.shiftRight a b
open HAdd (hAdd)
open HMul (hMul)
open HPow (hPow)
open HAppend (hAppend)
set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_add"]
protected def Nat.add : (@& Nat) → (@& Nat) → Nat
| a, Nat.zero => a
| a, Nat.succ b => Nat.succ (Nat.add a b)
instance : Add Nat where
add := Nat.add
/- We mark the following definitions as pattern to make sure they can be used in recursive equations,
and reduced by the equation Compiler. -/
attribute [matchPattern] Nat.add Add.add HAdd.hAdd Neg.neg
set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_mul"]
protected def Nat.mul : (@& Nat) → (@& Nat) → Nat
| a, 0 => 0
| a, Nat.succ b => Nat.add (Nat.mul a b) a
instance : Mul Nat where
mul := Nat.mul
set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_pow"]
protected def Nat.pow (m : @& Nat) : (@& Nat) → Nat
| 0 => 1
| succ n => Nat.mul (Nat.pow m n) m
instance : Pow Nat Nat where
pow := Nat.pow
set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_dec_eq"]
def Nat.beq : (@& Nat) → (@& Nat) → Bool
| zero, zero => true
| zero, succ m => false
| succ n, zero => false
| succ n, succ m => beq n m
theorem Nat.eq_of_beq_eq_true : {n m : Nat} → Eq (beq n m) true → Eq n m
| zero, zero, h => rfl
| zero, succ m, h => Bool.noConfusion h
| succ n, zero, h => Bool.noConfusion h
| succ n, succ m, h =>
have : Eq (beq n m) true := h
have : Eq n m := eq_of_beq_eq_true this
this ▸ rfl
theorem Nat.ne_of_beq_eq_false : {n m : Nat} → Eq (beq n m) false → Not (Eq n m)
| zero, zero, h₁, h₂ => Bool.noConfusion h₁
| zero, succ m, h₁, h₂ => Nat.noConfusion h₂
| succ n, zero, h₁, h₂ => Nat.noConfusion h₂
| succ n, succ m, h₁, h₂ =>
have : Eq (beq n m) false := h₁
Nat.noConfusion h₂ (fun h₂ => absurd h₂ (ne_of_beq_eq_false this))
@[extern "lean_nat_dec_eq"]
protected def Nat.decEq (n m : @& Nat) : Decidable (Eq n m) :=
match h:beq n m with
| true => isTrue (eq_of_beq_eq_true h)
| false => isFalse (ne_of_beq_eq_false h)
@[inline] instance : DecidableEq Nat := Nat.decEq
set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_dec_le"]
def Nat.ble : @& Nat → @& Nat → Bool
| zero, zero => true
| zero, succ m => true
| succ n, zero => false
| succ n, succ m => ble n m
protected def Nat.le (n m : Nat) : Prop :=
Eq (ble n m) true
instance : LE Nat where
le := Nat.le
protected def Nat.lt (n m : Nat) : Prop :=
Nat.le (succ n) m
instance : LT Nat where
lt := Nat.lt
theorem Nat.not_succ_le_zero : ∀ (n : Nat), LE.le (succ n) 0 → False
| 0, h => nomatch h
| succ n, h => nomatch h
theorem Nat.not_lt_zero (n : Nat) : Not (LT.lt n 0) :=
not_succ_le_zero n
@[extern "lean_nat_dec_le"]
instance Nat.decLe (n m : @& Nat) : Decidable (LE.le n m) :=
decEq (Nat.ble n m) true
@[extern "lean_nat_dec_lt"]
instance Nat.decLt (n m : @& Nat) : Decidable (LT.lt n m) :=
decLe (succ n) m
theorem Nat.zero_le : (n : Nat) → LE.le 0 n
| zero => rfl
| succ n => rfl
theorem Nat.succ_le_succ {n m : Nat} (h : LE.le n m) : LE.le (succ n) (succ m) :=
h
theorem Nat.zero_lt_succ (n : Nat) : LT.lt 0 (succ n) :=
succ_le_succ (zero_le n)
theorem Nat.le_step : {n m : Nat} → LE.le n m → LE.le n (succ m)
| zero, zero, h => rfl
| zero, succ n, h => rfl
| succ n, zero, h => Bool.noConfusion h
| succ n, succ m, h =>
have : LE.le n m := h
have : LE.le n (succ m) := le_step this
succ_le_succ this
protected theorem Nat.le_trans : {n m k : Nat} → LE.le n m → LE.le m k → LE.le n k
| zero, m, k, h₁, h₂ => zero_le _
| succ n, zero, k, h₁, h₂ => Bool.noConfusion h₁
| succ n, succ m, zero, h₁, h₂ => Bool.noConfusion h₂
| succ n, succ m, succ k, h₁, h₂ =>
have h₁' : LE.le n m := h₁
have h₂' : LE.le m k := h₂
show LE.le n k from
Nat.le_trans h₁' h₂'
protected theorem Nat.lt_trans {n m k : Nat} (h₁ : LT.lt n m) : LT.lt m k → LT.lt n k :=
Nat.le_trans (le_step h₁)
theorem Nat.le_succ : (n : Nat) → LE.le n (succ n)
| zero => rfl
| succ n => le_succ n
theorem Nat.le_succ_of_le {n m : Nat} (h : LE.le n m) : LE.le n (succ m) :=
Nat.le_trans h (le_succ m)
protected theorem Nat.eq_or_lt_of_le : {n m: Nat} → LE.le n m → Or (Eq n m) (LT.lt n m)
| zero, zero, h => Or.inl rfl
| zero, succ n, h => Or.inr (zero_le n)
| succ n, zero, h => Bool.noConfusion h
| succ n, succ m, h =>
have : LE.le n m := h
match Nat.eq_or_lt_of_le this with
| Or.inl h => Or.inl (h ▸ rfl)
| Or.inr h => Or.inr (succ_le_succ h)
protected def Nat.le_refl : (n : Nat) → LE.le n n
| zero => rfl
| succ n => Nat.le_refl n
protected theorem Nat.lt_or_ge (n m : Nat) : Or (LT.lt n m) (GE.ge n m) :=
match m with
| zero => Or.inr (zero_le n)
| succ m =>
match Nat.lt_or_ge n m with
| Or.inl h => Or.inl (le_succ_of_le h)
| Or.inr h =>
match Nat.eq_or_lt_of_le h with
| Or.inl h1 => Or.inl (h1 ▸ Nat.le_refl _)
| Or.inr h1 => Or.inr h1
protected theorem Nat.le_antisymm : {n m : Nat} → LE.le n m → LE.le m n → Eq n m
| zero, zero, h₁, h₂ => rfl
| succ n, zero, h₁, h₂ => Bool.noConfusion h₁
| zero, succ m, h₁, h₂ => Bool.noConfusion h₂
| succ n, succ m, h₁, h₂ =>
have h₁' : LE.le n m := h₁
have h₂' : LE.le m n := h₂
(Nat.le_antisymm h₁' h₂') ▸ rfl
protected theorem Nat.lt_of_le_of_ne {n m : Nat} (h₁ : LE.le n m) (h₂ : Not (Eq n m)) : LT.lt n m :=
match Nat.lt_or_ge n m with
| Or.inl h₃ => h₃
| Or.inr h₃ => absurd (Nat.le_antisymm h₁ h₃) h₂
set_option bootstrap.genMatcherCode false in
@[extern c inline "lean_nat_sub(#1, lean_box(1))"]
def Nat.pred : (@& Nat) → Nat
| 0 => 0
| succ a => a
set_option bootstrap.genMatcherCode false in
@[extern "lean_nat_sub"]
protected def Nat.sub : (@& Nat) → (@& Nat) → Nat
| a, 0 => a
| a, succ b => pred (Nat.sub a b)
instance : Sub Nat where
sub := Nat.sub
theorem Nat.pred_le_pred : {n m : Nat} → LE.le n m → LE.le (pred n) (pred m)
| zero, zero, h => rfl
| zero, succ n, h => zero_le n
| succ n, zero, h => Bool.noConfusion h
| succ n, succ m, h => h
theorem Nat.le_of_succ_le_succ {n m : Nat} : LE.le (succ n) (succ m) → LE.le n m :=
pred_le_pred
theorem Nat.le_of_lt_succ {m n : Nat} : LT.lt m (succ n) → LE.le m n :=
le_of_succ_le_succ
@[extern "lean_system_platform_nbits"] constant System.Platform.getNumBits : Unit → Subtype fun (n : Nat) => Or (Eq n 32) (Eq n 64) :=
fun _ => ⟨64, Or.inr rfl⟩ -- inhabitant
def System.Platform.numBits : Nat :=
(getNumBits ()).val
theorem System.Platform.numBits_eq : Or (Eq numBits 32) (Eq numBits 64) :=
(getNumBits ()).property
structure Fin (n : Nat) where
val : Nat
isLt : LT.lt val n
theorem Fin.eq_of_val_eq {n} : ∀ {i j : Fin n}, Eq i.val j.val → Eq i j
| ⟨v, h⟩, ⟨_, _⟩, rfl => rfl
theorem Fin.val_eq_of_eq {n} {i j : Fin n} (h : Eq i j) : Eq i.val j.val :=
h ▸ rfl
theorem Fin.ne_of_val_ne {n} {i j : Fin n} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
fun h' => absurd (val_eq_of_eq h') h
instance (n : Nat) : DecidableEq (Fin n) :=
fun i j =>
match decEq i.val j.val with
| isTrue h => isTrue (Fin.eq_of_val_eq h)
| isFalse h => isFalse (Fin.ne_of_val_ne h)
instance {n} : LT (Fin n) where
lt a b := LT.lt a.val b.val
instance {n} : LE (Fin n) where
le a b := LE.le a.val b.val
instance Fin.decLt {n} (a b : Fin n) : Decidable (LT.lt a b) := Nat.decLt ..
instance Fin.decLe {n} (a b : Fin n) : Decidable (LE.le a b) := Nat.decLe ..
def UInt8.size : Nat := 256
structure UInt8 where
val : Fin UInt8.size
attribute [extern "lean_uint8_of_nat_mk"] UInt8.mk
attribute [extern "lean_uint8_to_nat"] UInt8.val
@[extern "lean_uint8_of_nat"]
def UInt8.ofNatCore (n : @& Nat) (h : LT.lt n UInt8.size) : UInt8 := {
val := { val := n, isLt := h }
}
set_option bootstrap.genMatcherCode false in
@[extern c inline "#1 == #2"]
def UInt8.decEq (a b : UInt8) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt8.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt8 := UInt8.decEq
instance : Inhabited UInt8 where
default := UInt8.ofNatCore 0 (by decide)
def UInt16.size : Nat := 65536
structure UInt16 where
val : Fin UInt16.size
attribute [extern "lean_uint16_of_nat_mk"] UInt16.mk
attribute [extern "lean_uint16_to_nat"] UInt16.val
@[extern "lean_uint16_of_nat"]
def UInt16.ofNatCore (n : @& Nat) (h : LT.lt n UInt16.size) : UInt16 := {
val := { val := n, isLt := h }
}
set_option bootstrap.genMatcherCode false in
@[extern c inline "#1 == #2"]
def UInt16.decEq (a b : UInt16) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt16.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt16 := UInt16.decEq
instance : Inhabited UInt16 where
default := UInt16.ofNatCore 0 (by decide)
def UInt32.size : Nat := 4294967296
structure UInt32 where
val : Fin UInt32.size
attribute [extern "lean_uint32_of_nat_mk"] UInt32.mk
attribute [extern "lean_uint32_to_nat"] UInt32.val
@[extern "lean_uint32_of_nat"]
def UInt32.ofNatCore (n : @& Nat) (h : LT.lt n UInt32.size) : UInt32 := {
val := { val := n, isLt := h }
}
@[extern "lean_uint32_to_nat"]
def UInt32.toNat (n : UInt32) : Nat := n.val.val
set_option bootstrap.genMatcherCode false in
@[extern c inline "#1 == #2"]
def UInt32.decEq (a b : UInt32) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt32.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt32 := UInt32.decEq
instance : Inhabited UInt32 where
default := UInt32.ofNatCore 0 (by decide)
instance : LT UInt32 where
lt a b := LT.lt a.val b.val
instance : LE UInt32 where
le a b := LE.le a.val b.val
set_option bootstrap.genMatcherCode false in
@[extern c inline "#1 < #2"]
def UInt32.decLt (a b : UInt32) : Decidable (LT.lt a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ => inferInstanceAs (Decidable (LT.lt n m))
set_option bootstrap.genMatcherCode false in
@[extern c inline "#1 <= #2"]
def UInt32.decLe (a b : UInt32) : Decidable (LE.le a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ => inferInstanceAs (Decidable (LE.le n m))
instance (a b : UInt32) : Decidable (LT.lt a b) := UInt32.decLt a b
instance (a b : UInt32) : Decidable (LE.le a b) := UInt32.decLe a b
def UInt64.size : Nat := 18446744073709551616
structure UInt64 where
val : Fin UInt64.size
attribute [extern "lean_uint64_of_nat_mk"] UInt64.mk
attribute [extern "lean_uint64_to_nat"] UInt64.val
@[extern "lean_uint64_of_nat"]
def UInt64.ofNatCore (n : @& Nat) (h : LT.lt n UInt64.size) : UInt64 := {
val := { val := n, isLt := h }
}
set_option bootstrap.genMatcherCode false in
@[extern c inline "#1 == #2"]
def UInt64.decEq (a b : UInt64) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt64.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt64 := UInt64.decEq
instance : Inhabited UInt64 where
default := UInt64.ofNatCore 0 (by decide)
def USize.size : Nat := hPow 2 System.Platform.numBits
theorem usize_size_eq : Or (Eq USize.size 4294967296) (Eq USize.size 18446744073709551616) :=
show Or (Eq (hPow 2 System.Platform.numBits) 4294967296) (Eq (hPow 2 System.Platform.numBits) 18446744073709551616) from
match System.Platform.numBits, System.Platform.numBits_eq with
| _, Or.inl rfl => Or.inl (by decide)
| _, Or.inr rfl => Or.inr (by decide)
structure USize where
val : Fin USize.size
attribute [extern "lean_usize_of_nat_mk"] USize.mk
attribute [extern "lean_usize_to_nat"] USize.val
@[extern "lean_usize_of_nat"]
def USize.ofNatCore (n : @& Nat) (h : LT.lt n USize.size) : USize := {
val := { val := n, isLt := h }
}
set_option bootstrap.genMatcherCode false in
@[extern c inline "#1 == #2"]
def USize.decEq (a b : USize) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h =>isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => USize.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq USize := USize.decEq
instance : Inhabited USize where
default := USize.ofNatCore 0 (match USize.size, usize_size_eq with
| _, Or.inl rfl => by decide
| _, Or.inr rfl => by decide)
@[extern "lean_usize_of_nat"]
def USize.ofNat32 (n : @& Nat) (h : LT.lt n 4294967296) : USize := {
val := {
val := n
isLt := match USize.size, usize_size_eq with
| _, Or.inl rfl => h
| _, Or.inr rfl => Nat.lt_trans h (by decide)
}
}
abbrev Nat.isValidChar (n : Nat) : Prop :=
Or (LT.lt n 0xd800) (And (LT.lt 0xdfff n) (LT.lt n 0x110000))
abbrev UInt32.isValidChar (n : UInt32) : Prop :=
n.toNat.isValidChar
/-- The `Char` Type represents an unicode scalar value.
See http://www.unicode.org/glossary/#unicode_scalar_value). -/
structure Char where
val : UInt32
valid : val.isValidChar
private theorem isValidChar_UInt32 {n : Nat} (h : n.isValidChar) : LT.lt n UInt32.size :=
match h with
| Or.inl h => Nat.lt_trans h (by decide)
| Or.inr ⟨_, h⟩ => Nat.lt_trans h (by decide)
@[extern "lean_uint32_of_nat"]
private def Char.ofNatAux (n : @& Nat) (h : n.isValidChar) : Char :=
{ val := ⟨{ val := n, isLt := isValidChar_UInt32 h }⟩, valid := h }
@[noinline, matchPattern]
def Char.ofNat (n : Nat) : Char :=
dite (n.isValidChar)
(fun h => Char.ofNatAux n h)
(fun _ => { val := ⟨{ val := 0, isLt := by decide }⟩, valid := Or.inl (by decide) })
theorem Char.eq_of_val_eq : ∀ {c d : Char}, Eq c.val d.val → Eq c d
| ⟨v, h⟩, ⟨_, _⟩, rfl => rfl
theorem Char.val_eq_of_eq : ∀ {c d : Char}, Eq c d → Eq c.val d.val
| _, _, rfl => rfl
theorem Char.ne_of_val_ne {c d : Char} (h : Not (Eq c.val d.val)) : Not (Eq c d) :=
fun h' => absurd (val_eq_of_eq h') h
theorem Char.val_ne_of_ne {c d : Char} (h : Not (Eq c d)) : Not (Eq c.val d.val) :=
fun h' => absurd (eq_of_val_eq h') h
instance : DecidableEq Char :=
fun c d =>
match decEq c.val d.val with
| isTrue h => isTrue (Char.eq_of_val_eq h)
| isFalse h => isFalse (Char.ne_of_val_ne h)
def Char.utf8Size (c : Char) : UInt32 :=
let v := c.val
ite (LE.le v (UInt32.ofNatCore 0x7F (by decide)))
(UInt32.ofNatCore 1 (by decide))
(ite (LE.le v (UInt32.ofNatCore 0x7FF (by decide)))
(UInt32.ofNatCore 2 (by decide))
(ite (LE.le v (UInt32.ofNatCore 0xFFFF (by decide)))
(UInt32.ofNatCore 3 (by decide))
(UInt32.ofNatCore 4 (by decide))))
inductive Option (α : Type u) where
| none : Option α
| some (val : α) : Option α
attribute [unbox] Option
export Option (none some)
instance {α} : Inhabited (Option α) where
default := none
@[macroInline] def Option.getD : Option α → α → α
| some x, _ => x
| none, e => e
inductive List (α : Type u) where
| nil : List α
| cons (head : α) (tail : List α) : List α
instance {α} : Inhabited (List α) where
default := List.nil
protected def List.hasDecEq {α: Type u} [DecidableEq α] : (a b : List α) → Decidable (Eq a b)
| nil, nil => isTrue rfl
| cons a as, nil => isFalse (fun h => List.noConfusion h)
| nil, cons b bs => isFalse (fun h => List.noConfusion h)
| cons a as, cons b bs =>
match decEq a b with
| isTrue hab =>
match List.hasDecEq as bs with
| isTrue habs => isTrue (hab ▸ habs ▸ rfl)
| isFalse nabs => isFalse (fun h => List.noConfusion h (fun _ habs => absurd habs nabs))
| isFalse nab => isFalse (fun h => List.noConfusion h (fun hab _ => absurd hab nab))
instance {α : Type u} [DecidableEq α] : DecidableEq (List α) := List.hasDecEq
@[specialize]
def List.foldl {α β} (f : α → β → α) : (init : α) → List β → α
| a, nil => a
| a, cons b l => foldl f (f a b) l
def List.set : List α → Nat → α → List α
| cons a as, 0, b => cons b as
| cons a as, Nat.succ n, b => cons a (set as n b)
| nil, _, _ => nil
def List.lengthAux {α : Type u} : List α → Nat → Nat
| nil, n => n
| cons a as, n => lengthAux as (Nat.succ n)
def List.length {α : Type u} (as : List α) : Nat :=
lengthAux as 0
@[simp] theorem List.length_cons {α} (a : α) (as : List α) : Eq (cons a as).length as.length.succ :=
let rec aux (a : α) (as : List α) : (n : Nat) → Eq ((cons a as).lengthAux n) (as.lengthAux n).succ :=
match as with
| nil => fun _ => rfl
| cons a as => fun n => aux a as n.succ
aux a as 0
def List.concat {α : Type u} : List α → α → List α
| nil, b => cons b nil
| cons a as, b => cons a (concat as b)
def List.get {α : Type u} : (as : List α) → (i : Nat) → LT.lt i as.length → α
| nil, i, h => absurd h (Nat.not_lt_zero _)
| cons a as, 0, h => a
| cons a as, Nat.succ i, h =>
have : LT.lt i.succ as.length.succ := length_cons .. ▸ h
get as i (Nat.le_of_succ_le_succ this)
structure String where
data : List Char
attribute [extern "lean_string_mk"] String.mk
attribute [extern "lean_string_data"] String.data
@[extern "lean_string_dec_eq"]
def String.decEq (s₁ s₂ : @& String) : Decidable (Eq s₁ s₂) :=
match s₁, s₂ with
| ⟨s₁⟩, ⟨s₂⟩ =>
dite (Eq s₁ s₂) (fun h => isTrue (congrArg _ h)) (fun h => isFalse (fun h' => String.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq String := String.decEq
/-- A byte position in a `String`. Internally, `String`s are UTF-8 encoded.
Codepoint positions (counting the Unicode codepoints rather than bytes)
are represented by plain `Nat`s instead.
Indexing a `String` by a byte position is constant-time, while codepoint
positions need to be translated internally to byte positions in linear-time. -/
abbrev String.Pos := Nat
structure Substring where
str : String
startPos : String.Pos
stopPos : String.Pos
@[inline] def Substring.bsize : Substring → Nat
| ⟨_, b, e⟩ => e.sub b
def String.csize (c : Char) : Nat :=
c.utf8Size.toNat
private def String.utf8ByteSizeAux : List Char → Nat → Nat
| List.nil, r => r
| List.cons c cs, r => utf8ByteSizeAux cs (hAdd r (csize c))
@[extern "lean_string_utf8_byte_size"]
def String.utf8ByteSize : (@& String) → Nat
| ⟨s⟩ => utf8ByteSizeAux s 0
@[inline] def String.bsize (s : String) : Nat :=
utf8ByteSize s
@[inline] def String.toSubstring (s : String) : Substring := {
str := s
startPos := 0
stopPos := s.bsize
}
@[extern c inline "#3"]
unsafe def unsafeCast {α : Type u} {β : Type v} (a : α) : β :=
cast lcProof (PUnit.{v})
@[neverExtract, extern "lean_panic_fn"]
constant panic {α : Type u} [Inhabited α] (msg : String) : α
/-
The Compiler has special support for arrays.
They are implemented using dynamic arrays: https://en.wikipedia.org/wiki/Dynamic_array
-/
structure Array (α : Type u) where
data : List α
attribute [extern "lean_array_data"] Array.data
attribute [extern "lean_array_mk"] Array.mk
/- The parameter `c` is the initial capacity -/
@[extern "lean_mk_empty_array_with_capacity"]
def Array.mkEmpty {α : Type u} (c : @& Nat) : Array α := {
data := List.nil
}
def Array.empty {α : Type u} : Array α :=
mkEmpty 0
@[reducible, extern "lean_array_get_size"]
def Array.size {α : Type u} (a : @& Array α) : Nat :=
a.data.length
@[extern "lean_array_fget"]
def Array.get {α : Type u} (a : @& Array α) (i : @& Fin a.size) : α :=
a.data.get i.val i.isLt
@[inline] def Array.getD (a : Array α) (i : Nat) (v₀ : α) : α :=
dite (LT.lt i a.size) (fun h => a.get ⟨i, h⟩) (fun _ => v₀)
/- "Comfortable" version of `fget`. It performs a bound check at runtime. -/
@[extern "lean_array_get"]
def Array.get! {α : Type u} [Inhabited α] (a : @& Array α) (i : @& Nat) : α :=
Array.getD a i arbitrary
def Array.getOp {α : Type u} [Inhabited α] (self : Array α) (idx : Nat) : α :=
self.get! idx
@[extern "lean_array_push"]
def Array.push {α : Type u} (a : Array α) (v : α) : Array α := {
data := List.concat a.data v
}
@[extern "lean_array_fset"]
def Array.set (a : Array α) (i : @& Fin a.size) (v : α) : Array α := {
data := a.data.set i.val v
}
@[inline] def Array.setD (a : Array α) (i : Nat) (v : α) : Array α :=
dite (LT.lt i a.size) (fun h => a.set ⟨i, h⟩ v) (fun _ => a)
@[extern "lean_array_set"]
def Array.set! (a : Array α) (i : @& Nat) (v : α) : Array α :=
Array.setD a i v
-- Slower `Array.append` used in quotations.
protected def Array.appendCore {α : Type u} (as : Array α) (bs : Array α) : Array α :=
let rec loop (i : Nat) (j : Nat) (as : Array α) : Array α :=
dite (LT.lt j bs.size)
(fun hlt =>
match i with
| 0 => as
| Nat.succ i' => loop i' (hAdd j 1) (as.push (bs.get ⟨j, hlt⟩)))
(fun _ => as)
loop bs.size 0 as
@[inlineIfReduce]
def List.toArrayAux : List α → Array α → Array α
| nil, r => r
| cons a as, r => toArrayAux as (r.push a)
@[inlineIfReduce]
def List.redLength : List α → Nat
| nil => 0
| cons _ as => as.redLength.succ
@[inline, matchPattern, export lean_list_to_array]
def List.toArray (as : List α) : Array α :=
as.toArrayAux (Array.mkEmpty as.redLength)
class Bind (m : Type u → Type v) where
bind : {α β : Type u} → m α → (α → m β) → m β
export Bind (bind)
class Pure (f : Type u → Type v) where
pure {α : Type u} : α → f α
export Pure (pure)
class Functor (f : Type u → Type v) : Type (max (u+1) v) where
map : {α β : Type u} → (α → β) → f α → f β
mapConst : {α β : Type u} → α → f β → f α := Function.comp map (Function.const _)
class Seq (f : Type u → Type v) : Type (max (u+1) v) where
seq : {α β : Type u} → f (α → β) → f α → f β
class SeqLeft (f : Type u → Type v) : Type (max (u+1) v) where
seqLeft : {α β : Type u} → f α → f β → f α
class SeqRight (f : Type u → Type v) : Type (max (u+1) v) where
seqRight : {α β : Type u} → f α → f β → f β
class Applicative (f : Type u → Type v) extends Functor f, Pure f, Seq f, SeqLeft f, SeqRight f where
map := fun x y => Seq.seq (pure x) y
seqLeft := fun a b => Seq.seq (Functor.map (Function.const _) a) b
seqRight := fun a b => Seq.seq (Functor.map (Function.const _ id) a) b
class Monad (m : Type u → Type v) extends Applicative m, Bind m : Type (max (u+1) v) where
map f x := bind x (Function.comp pure f)
seq f x := bind f fun y => Functor.map y x
seqLeft x y := bind x fun a => bind y (fun _ => pure a)
seqRight x y := bind x fun _ => y
instance {α : Type u} {m : Type u → Type v} [Monad m] : Inhabited (α → m α) where
default := pure
instance {α : Type u} {m : Type u → Type v} [Monad m] [Inhabited α] : Inhabited (m α) where
default := pure arbitrary
-- A fusion of Haskell's `sequence` and `map`
def Array.sequenceMap {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : α → m β) : m (Array β) :=
let rec loop (i : Nat) (j : Nat) (bs : Array β) : m (Array β) :=
dite (LT.lt j as.size)
(fun hlt =>
match i with
| 0 => pure bs
| Nat.succ i' => Bind.bind (f (as.get ⟨j, hlt⟩)) fun b => loop i' (hAdd j 1) (bs.push b))
(fun _ => bs)
loop as.size 0 Array.empty
/-- A Function for lifting a computation from an inner Monad to an outer Monad.
Like [MonadTrans](https://hackage.haskell.org/package/transformers-0.5.5.0/docs/Control-Monad-Trans-Class.html),
but `n` does not have to be a monad transformer.
Alternatively, an implementation of [MonadLayer](https://hackage.haskell.org/package/layers-0.1/docs/Control-Monad-Layer.html#t:MonadLayer) without `layerInvmap` (so far). -/
class MonadLift (m : Type u → Type v) (n : Type u → Type w) where
monadLift : {α : Type u} → m α → n α
/-- The reflexive-transitive closure of `MonadLift`.
`monadLift` is used to transitively lift monadic computations such as `StateT.get` or `StateT.put s`.
Corresponds to [MonadLift](https://hackage.haskell.org/package/layers-0.1/docs/Control-Monad-Layer.html#t:MonadLift). -/
class MonadLiftT (m : Type u → Type v) (n : Type u → Type w) where
monadLift : {α : Type u} → m α → n α
export MonadLiftT (monadLift)
abbrev liftM := @monadLift
instance (m n o) [MonadLift n o] [MonadLiftT m n] : MonadLiftT m o where
monadLift x := MonadLift.monadLift (m := n) (monadLift x)
instance (m) : MonadLiftT m m where
monadLift x := x
/-- A functor in the category of monads. Can be used to lift monad-transforming functions.
Based on pipes' [MFunctor](https://hackage.haskell.org/package/pipes-2.4.0/docs/Control-MFunctor.html),
but not restricted to monad transformers.
Alternatively, an implementation of [MonadTransFunctor](http://duairc.netsoc.ie/layers-docs/Control-Monad-Layer.html#t:MonadTransFunctor). -/
class MonadFunctor (m : Type u → Type v) (n : Type u → Type w) where
monadMap {α : Type u} : ({β : Type u} → m β → m β) → n α → n α
/-- The reflexive-transitive closure of `MonadFunctor`.
`monadMap` is used to transitively lift Monad morphisms -/
class MonadFunctorT (m : Type u → Type v) (n : Type u → Type w) where
monadMap {α : Type u} : ({β : Type u} → m β → m β) → n α → n α
export MonadFunctorT (monadMap)
instance (m n o) [MonadFunctor n o] [MonadFunctorT m n] : MonadFunctorT m o where
monadMap f := MonadFunctor.monadMap (m := n) (monadMap (m := m) f)
instance monadFunctorRefl (m) : MonadFunctorT m m where
monadMap f := f
inductive Except (ε : Type u) (α : Type v) where
| error : ε → Except ε α
| ok : α → Except ε α
attribute [unbox] Except
instance {ε : Type u} {α : Type v} [Inhabited ε] : Inhabited (Except ε α) where
default := Except.error arbitrary
/-- An implementation of [MonadError](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Except.html#t:MonadError) -/
class MonadExceptOf (ε : Type u) (m : Type v → Type w) where
throw {α : Type v} : ε → m α
tryCatch {α : Type v} : m α → (ε → m α) → m α
abbrev throwThe (ε : Type u) {m : Type v → Type w} [MonadExceptOf ε m] {α : Type v} (e : ε) : m α :=
MonadExceptOf.throw e
abbrev tryCatchThe (ε : Type u) {m : Type v → Type w} [MonadExceptOf ε m] {α : Type v} (x : m α) (handle : ε → m α) : m α :=
MonadExceptOf.tryCatch x handle
/-- Similar to `MonadExceptOf`, but `ε` is an outParam for convenience -/
class MonadExcept (ε : outParam (Type u)) (m : Type v → Type w) where
throw {α : Type v} : ε → m α
tryCatch {α : Type v} : m α → (ε → m α) → m α
export MonadExcept (throw tryCatch)
instance (ε : outParam (Type u)) (m : Type v → Type w) [MonadExceptOf ε m] : MonadExcept ε m where
throw := throwThe ε
tryCatch := tryCatchThe ε
namespace MonadExcept
variable {ε : Type u} {m : Type v → Type w}
@[inline] protected def orelse [MonadExcept ε m] {α : Type v} (t₁ t₂ : m α) : m α :=
tryCatch t₁ fun _ => t₂
instance [MonadExcept ε m] {α : Type v} : OrElse (m α) where
orElse := MonadExcept.orelse
end MonadExcept
/-- An implementation of [ReaderT](https://hackage.haskell.org/package/transformers-0.5.5.0/docs/Control-Monad-Trans-Reader.html#t:ReaderT) -/
def ReaderT (ρ : Type u) (m : Type u → Type v) (α : Type u) : Type (max u v) :=
ρ → m α
instance (ρ : Type u) (m : Type u → Type v) (α : Type u) [Inhabited (m α)] : Inhabited (ReaderT ρ m α) where
default := fun _ => arbitrary
@[inline] def ReaderT.run {ρ : Type u} {m : Type u → Type v} {α : Type u} (x : ReaderT ρ m α) (r : ρ) : m α :=
x r
namespace ReaderT
section
variable {ρ : Type u} {m : Type u → Type v} {α : Type u}
instance : MonadLift m (ReaderT ρ m) where
monadLift x := fun _ => x
instance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ReaderT ρ m) where
throw e := liftM (m := m) (throw e)
tryCatch := fun x c r => tryCatchThe ε (x r) (fun e => (c e) r)
end
section
variable {ρ : Type u} {m : Type u → Type v} [Monad m] {α β : Type u}
@[inline] protected def read : ReaderT ρ m ρ :=
pure
@[inline] protected def pure (a : α) : ReaderT ρ m α :=
fun r => pure a
@[inline] protected def bind (x : ReaderT ρ m α) (f : α → ReaderT ρ m β) : ReaderT ρ m β :=
fun r => bind (x r) fun a => f a r
@[inline] protected def map (f : α → β) (x : ReaderT ρ m α) : ReaderT ρ m β :=
fun r => Functor.map f (x r)
instance : Monad (ReaderT ρ m) where
pure := ReaderT.pure
bind := ReaderT.bind
map := ReaderT.map
instance (ρ m) [Monad m] : MonadFunctor m (ReaderT ρ m) where
monadMap f x := fun ctx => f (x ctx)
@[inline] protected def adapt {ρ' : Type u} [Monad m] {α : Type u} (f : ρ' → ρ) : ReaderT ρ m α → ReaderT ρ' m α :=
fun x r => x (f r)
end
end ReaderT
/-- An implementation of [MonadReader](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Reader-Class.html#t:MonadReader).
It does not contain `local` because this Function cannot be lifted using `monadLift`.
Instead, the `MonadReaderAdapter` class provides the more general `adaptReader` Function.
Note: This class can be seen as a simplification of the more "principled" definition
```
class MonadReader (ρ : outParam (Type u)) (n : Type u → Type u) where
lift {α : Type u} : ({m : Type u → Type u} → [Monad m] → ReaderT ρ m α) → n α
```
-/
class MonadReaderOf (ρ : Type u) (m : Type u → Type v) where
read : m ρ
@[inline] def readThe (ρ : Type u) {m : Type u → Type v} [MonadReaderOf ρ m] : m ρ :=
MonadReaderOf.read
/-- Similar to `MonadReaderOf`, but `ρ` is an outParam for convenience -/
class MonadReader (ρ : outParam (Type u)) (m : Type u → Type v) where
read : m ρ
export MonadReader (read)
instance (ρ : Type u) (m : Type u → Type v) [MonadReaderOf ρ m] : MonadReader ρ m where
read := readThe ρ
instance {ρ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadLift m n] [MonadReaderOf ρ m] : MonadReaderOf ρ n where
read := liftM (m := m) read
instance {ρ : Type u} {m : Type u → Type v} [Monad m] : MonadReaderOf ρ (ReaderT ρ m) where
read := ReaderT.read
class MonadWithReaderOf (ρ : Type u) (m : Type u → Type v) where
withReader {α : Type u} : (ρ → ρ) → m α → m α
@[inline] def withTheReader (ρ : Type u) {m : Type u → Type v} [MonadWithReaderOf ρ m] {α : Type u} (f : ρ → ρ) (x : m α) : m α :=
MonadWithReaderOf.withReader f x
class MonadWithReader (ρ : outParam (Type u)) (m : Type u → Type v) where
withReader {α : Type u} : (ρ → ρ) → m α → m α
export MonadWithReader (withReader)
instance (ρ : Type u) (m : Type u → Type v) [MonadWithReaderOf ρ m] : MonadWithReader ρ m where
withReader := withTheReader ρ
instance {ρ : Type u} {m : Type u → Type v} {n : Type u → Type v} [MonadFunctor m n] [MonadWithReaderOf ρ m] : MonadWithReaderOf ρ n where
withReader f := monadMap (m := m) (withTheReader ρ f)
instance {ρ : Type u} {m : Type u → Type v} [Monad m] : MonadWithReaderOf ρ (ReaderT ρ m) where
withReader f x := fun ctx => x (f ctx)
/-- An implementation of [MonadState](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-State-Class.html).
In contrast to the Haskell implementation, we use overlapping instances to derive instances
automatically from `monadLift`. -/
class MonadStateOf (σ : Type u) (m : Type u → Type v) where
/- Obtain the top-most State of a Monad stack. -/
get : m σ
/- Set the top-most State of a Monad stack. -/
set : σ → m PUnit
/- Map the top-most State of a Monad stack.
Note: `modifyGet f` may be preferable to `do s <- get; let (a, s) := f s; put s; pure a`
because the latter does not use the State linearly (without sufficient inlining). -/
modifyGet {α : Type u} : (σ → Prod α σ) → m α
export MonadStateOf (set)
abbrev getThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] : m σ :=
MonadStateOf.get
@[inline] abbrev modifyThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → σ) : m PUnit :=
MonadStateOf.modifyGet fun s => (PUnit.unit, f s)
@[inline] abbrev modifyGetThe {α : Type u} (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → Prod α σ) : m α :=
MonadStateOf.modifyGet f
/-- Similar to `MonadStateOf`, but `σ` is an outParam for convenience -/
class MonadState (σ : outParam (Type u)) (m : Type u → Type v) where
get : m σ
set : σ → m PUnit
modifyGet {α : Type u} : (σ → Prod α σ) → m α
export MonadState (get modifyGet)
instance (σ : Type u) (m : Type u → Type v) [MonadStateOf σ m] : MonadState σ m where
set := MonadStateOf.set
get := getThe σ
modifyGet f := MonadStateOf.modifyGet f
@[inline] def modify {σ : Type u} {m : Type u → Type v} [MonadState σ m] (f : σ → σ) : m PUnit :=
modifyGet fun s => (PUnit.unit, f s)
@[inline] def getModify {σ : Type u} {m : Type u → Type v} [MonadState σ m] [Monad m] (f : σ → σ) : m σ :=
modifyGet fun s => (s, f s)
-- NOTE: The Ordering of the following two instances determines that the top-most `StateT` Monad layer
-- will be picked first
instance {σ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadLift m n] [MonadStateOf σ m] : MonadStateOf σ n where
get := liftM (m := m) MonadStateOf.get
set s := liftM (m := m) (MonadStateOf.set s)
modifyGet f := monadLift (m := m) (MonadState.modifyGet f)
namespace EStateM
inductive Result (ε σ α : Type u) where
| ok : α → σ → Result ε σ α
| error : ε → σ → Result ε σ α
variable {ε σ α : Type u}
instance [Inhabited ε] [Inhabited σ] : Inhabited (Result ε σ α) where
default := Result.error arbitrary arbitrary
end EStateM
open EStateM (Result) in
def EStateM (ε σ α : Type u) := σ → Result ε σ α
namespace EStateM
variable {ε σ α β : Type u}
instance [Inhabited ε] : Inhabited (EStateM ε σ α) where
default := fun s => Result.error arbitrary s
@[inline] protected def pure (a : α) : EStateM ε σ α := fun s =>
Result.ok a s
@[inline] protected def set (s : σ) : EStateM ε σ PUnit := fun _ =>
Result.ok ⟨⟩ s
@[inline] protected def get : EStateM ε σ σ := fun s =>
Result.ok s s
@[inline] protected def modifyGet (f : σ → Prod α σ) : EStateM ε σ α := fun s =>
match f s with
| (a, s) => Result.ok a s
@[inline] protected def throw (e : ε) : EStateM ε σ α := fun s =>
Result.error e s
/-- Auxiliary instance for saving/restoring the "backtrackable" part of the state. -/
class Backtrackable (δ : outParam (Type u)) (σ : Type u) where
save : σ → δ
restore : σ → δ → σ
@[inline] protected def tryCatch {δ} [Backtrackable δ σ] {α} (x : EStateM ε σ α) (handle : ε → EStateM ε σ α) : EStateM ε σ α := fun s =>
let d := Backtrackable.save s
match x s with
| Result.error e s => handle e (Backtrackable.restore s d)
| ok => ok
@[inline] protected def orElse {δ} [Backtrackable δ σ] (x₁ x₂ : EStateM ε σ α) : EStateM ε σ α := fun s =>
let d := Backtrackable.save s;
match x₁ s with
| Result.error _ s => x₂ (Backtrackable.restore s d)
| ok => ok
@[inline] def adaptExcept {ε' : Type u} (f : ε → ε') (x : EStateM ε σ α) : EStateM ε' σ α := fun s =>
match x s with
| Result.error e s => Result.error (f e) s
| Result.ok a s => Result.ok a s
@[inline] protected def bind (x : EStateM ε σ α) (f : α → EStateM ε σ β) : EStateM ε σ β := fun s =>
match x s with
| Result.ok a s => f a s
| Result.error e s => Result.error e s
@[inline] protected def map (f : α → β) (x : EStateM ε σ α) : EStateM ε σ β := fun s =>
match x s with
| Result.ok a s => Result.ok (f a) s
| Result.error e s => Result.error e s
@[inline] protected def seqRight (x : EStateM ε σ α) (y : EStateM ε σ β) : EStateM ε σ β := fun s =>
match x s with
| Result.ok _ s => y s
| Result.error e s => Result.error e s
instance : Monad (EStateM ε σ) where
bind := EStateM.bind
pure := EStateM.pure
map := EStateM.map
seqRight := EStateM.seqRight
instance {δ} [Backtrackable δ σ] : OrElse (EStateM ε σ α) where
orElse := EStateM.orElse
instance : MonadStateOf σ (EStateM ε σ) where
set := EStateM.set
get := EStateM.get
modifyGet := EStateM.modifyGet
instance {δ} [Backtrackable δ σ] : MonadExceptOf ε (EStateM ε σ) where
throw := EStateM.throw
tryCatch := EStateM.tryCatch
@[inline] def run (x : EStateM ε σ α) (s : σ) : Result ε σ α :=
x s
@[inline] def run' (x : EStateM ε σ α) (s : σ) : Option α :=
match run x s with
| Result.ok v _ => some v
| Result.error .. => none
@[inline] def dummySave : σ → PUnit := fun _ => ⟨⟩
@[inline] def dummyRestore : σ → PUnit → σ := fun s _ => s
/- Dummy default instance -/
instance nonBacktrackable : Backtrackable PUnit σ where
save := dummySave
restore := dummyRestore
end EStateM
class Hashable (α : Sort u) where
hash : α → UInt64
export Hashable (hash)
@[extern c inline "(size_t)#1"]
constant UInt64.toUSize (u : UInt64) : USize
@[extern c inline "(uint64_t)#1"]
constant USize.toUInt64 (u : USize) : UInt64
@[extern "lean_uint64_mix_hash"]
constant mixHash (u₁ u₂ : UInt64) : UInt64
@[extern "lean_string_hash"]
protected constant String.hash (s : @& String) : UInt64
instance : Hashable String where
hash := String.hash
namespace Lean
/- Hierarchical names -/
inductive Name where
| anonymous : Name
| str : Name → String → UInt64 → Name
| num : Name → Nat → UInt64 → Name
instance : Inhabited Name where
default := Name.anonymous
protected def Name.hash : Name → UInt64
| Name.anonymous => UInt64.ofNatCore 1723 (by decide)
| Name.str p s h => h
| Name.num p v h => h
instance : Hashable Name where
hash := Name.hash
namespace Name
@[export lean_name_mk_string]
def mkStr (p : Name) (s : String) : Name :=
Name.str p s (mixHash (hash p) (hash s))
@[export lean_name_mk_numeral]
def mkNum (p : Name) (v : Nat) : Name :=
Name.num p v (mixHash (hash p) (dite (LT.lt v UInt64.size) (fun h => UInt64.ofNatCore v h) (fun _ => UInt64.ofNatCore 17 (by decide))))
def mkSimple (s : String) : Name :=
mkStr Name.anonymous s
@[extern "lean_name_eq"]
protected def beq : (@& Name) → (@& Name) → Bool
| anonymous, anonymous => true
| str p₁ s₁ _, str p₂ s₂ _ => and (BEq.beq s₁ s₂) (Name.beq p₁ p₂)
| num p₁ n₁ _, num p₂ n₂ _ => and (BEq.beq n₁ n₂) (Name.beq p₁ p₂)
| _, _ => false
instance : BEq Name where
beq := Name.beq
protected def append : Name → Name → Name
| n, anonymous => n
| n, str p s _ => Name.mkStr (Name.append n p) s
| n, num p d _ => Name.mkNum (Name.append n p) d
instance : Append Name where
append := Name.append
end Name
/- Syntax -/
/-- Source information of tokens. -/
inductive SourceInfo where
/-
Token from original input with whitespace and position information.
`leading` will be inferred after parsing by `Syntax.updateLeading`. During parsing,
it is not at all clear what the preceding token was, especially with backtracking. -/
| original (leading : Substring) (pos : String.Pos) (trailing : Substring) (endPos : String.Pos)
/-
Synthesized token (e.g. from a quotation) annotated with a span from the original source.
In the delaborator, we "misuse" this constructor to store synthetic positions identifying
subterms. -/
| synthetic (pos : String.Pos) (endPos : String.Pos)
/- Synthesized token without position information. -/
| protected none
instance : Inhabited SourceInfo := ⟨SourceInfo.none⟩
namespace SourceInfo
def getPos? (info : SourceInfo) (originalOnly := false) : Option String.Pos :=
match info, originalOnly with
| original (pos := pos) .., _ => some pos
| synthetic (pos := pos) .., false => some pos
| _, _ => none
end SourceInfo
abbrev SyntaxNodeKind := Name
/- Syntax AST -/
inductive Syntax where
| missing : Syntax
| node (kind : SyntaxNodeKind) (args : Array Syntax) : Syntax
| atom (info : SourceInfo) (val : String) : Syntax
| ident (info : SourceInfo) (rawVal : Substring) (val : Name) (preresolved : List (Prod Name (List String))) : Syntax
instance : Inhabited Syntax where
default := Syntax.missing
/- Builtin kinds -/
def choiceKind : SyntaxNodeKind := `choice
def nullKind : SyntaxNodeKind := `null
def groupKind : SyntaxNodeKind := `group
def identKind : SyntaxNodeKind := `ident
def strLitKind : SyntaxNodeKind := `strLit
def charLitKind : SyntaxNodeKind := `charLit
def numLitKind : SyntaxNodeKind := `numLit
def scientificLitKind : SyntaxNodeKind := `scientificLit
def nameLitKind : SyntaxNodeKind := `nameLit
def fieldIdxKind : SyntaxNodeKind := `fieldIdx
def interpolatedStrLitKind : SyntaxNodeKind := `interpolatedStrLitKind
def interpolatedStrKind : SyntaxNodeKind := `interpolatedStrKind
namespace Syntax
def getKind (stx : Syntax) : SyntaxNodeKind :=
match stx with
| Syntax.node k args => k
-- We use these "pseudo kinds" for antiquotation kinds.
-- For example, an antiquotation `$id:ident` (using Lean.Parser.Term.ident)
-- is compiled to ``if stx.isOfKind `ident ...``
| Syntax.missing => `missing
| Syntax.atom _ v => Name.mkSimple v
| Syntax.ident .. => identKind
def setKind (stx : Syntax) (k : SyntaxNodeKind) : Syntax :=
match stx with
| Syntax.node _ args => Syntax.node k args
| _ => stx
def isOfKind (stx : Syntax) (k : SyntaxNodeKind) : Bool :=
beq stx.getKind k
def getArg (stx : Syntax) (i : Nat) : Syntax :=
match stx with
| Syntax.node _ args => args.getD i Syntax.missing
| _ => Syntax.missing
-- Add `stx[i]` as sugar for `stx.getArg i`
@[inline] def getOp (self : Syntax) (idx : Nat) : Syntax :=
self.getArg idx
def getArgs (stx : Syntax) : Array Syntax :=
match stx with
| Syntax.node _ args => args
| _ => Array.empty
def getNumArgs (stx : Syntax) : Nat :=
match stx with
| Syntax.node _ args => args.size
| _ => 0
def isMissing : Syntax → Bool
| Syntax.missing => true
| _ => false
def isNodeOf (stx : Syntax) (k : SyntaxNodeKind) (n : Nat) : Bool :=
and (stx.isOfKind k) (beq stx.getNumArgs n)
def isIdent : Syntax → Bool
| ident _ _ _ _ => true
| _ => false
def getId : Syntax → Name
| ident _ _ val _ => val
| _ => Name.anonymous
def matchesNull (stx : Syntax) (n : Nat) : Bool :=
isNodeOf stx nullKind n
def matchesIdent (stx : Syntax) (id : Name) : Bool :=
and stx.isIdent (beq stx.getId id)
def setArgs (stx : Syntax) (args : Array Syntax) : Syntax :=
match stx with
| node k _ => node k args
| stx => stx
def setArg (stx : Syntax) (i : Nat) (arg : Syntax) : Syntax :=
match stx with
| node k args => node k (args.setD i arg)
| stx => stx
/-- Retrieve the left-most leaf's info in the Syntax tree. -/
partial def getHeadInfo? : Syntax → Option SourceInfo
| atom info _ => some info
| ident info .. => some info
| node _ args =>
let rec loop (i : Nat) : Option SourceInfo :=
match decide (LT.lt i args.size) with
| true => match getHeadInfo? (args.get! i) with
| some info => some info
| none => loop (hAdd i 1)
| false => none
loop 0
| _ => none
/-- Retrieve the left-most leaf's info in the Syntax tree, or `none` if there is no token. -/
partial def getHeadInfo (stx : Syntax) : SourceInfo :=
match stx.getHeadInfo? with
| some info => info
| none => SourceInfo.none
def getPos? (stx : Syntax) (originalOnly := false) : Option String.Pos :=
stx.getHeadInfo.getPos? originalOnly
partial def getTailPos? (stx : Syntax) (originalOnly := false) : Option String.Pos :=
match stx, originalOnly with
| atom (SourceInfo.original (endPos := pos) ..) .., _ => some pos
| atom (SourceInfo.synthetic (endPos := pos) ..) _, false => some pos
| ident (SourceInfo.original (endPos := pos) ..) .., _ => some pos
| ident (SourceInfo.synthetic (endPos := pos) ..) .., false => some pos
| node _ args, _ =>
let rec loop (i : Nat) : Option String.Pos :=
match decide (LT.lt i args.size) with
| true => match getTailPos? (args.get! ((args.size.sub i).sub 1)) originalOnly with
| some info => some info
| none => loop (hAdd i 1)
| false => none
loop 0
| _, _ => none
/--
An array of syntax elements interspersed with separators. Can be coerced to/from `Array Syntax` to automatically
remove/insert the separators. -/
structure SepArray (sep : String) where
elemsAndSeps : Array Syntax
end Syntax
def SourceInfo.fromRef (ref : Syntax) : SourceInfo :=
match ref.getPos?, ref.getTailPos? with
| some pos, some tailPos => SourceInfo.synthetic pos tailPos
| _, _ => SourceInfo.none
def mkAtom (val : String) : Syntax :=
Syntax.atom SourceInfo.none val
def mkAtomFrom (src : Syntax) (val : String) : Syntax :=
Syntax.atom (SourceInfo.fromRef src) val
/- Parser descriptions -/
inductive ParserDescr where
| const (name : Name)
| unary (name : Name) (p : ParserDescr)
| binary (name : Name) (p₁ p₂ : ParserDescr)
| node (kind : SyntaxNodeKind) (prec : Nat) (p : ParserDescr)
| trailingNode (kind : SyntaxNodeKind) (prec lhsPrec : Nat) (p : ParserDescr)
| symbol (val : String)
| nonReservedSymbol (val : String) (includeIdent : Bool)
| cat (catName : Name) (rbp : Nat)
| parser (declName : Name)
| nodeWithAntiquot (name : String) (kind : SyntaxNodeKind) (p : ParserDescr)
| sepBy (p : ParserDescr) (sep : String) (psep : ParserDescr) (allowTrailingSep : Bool := false)
| sepBy1 (p : ParserDescr) (sep : String) (psep : ParserDescr) (allowTrailingSep : Bool := false)
instance : Inhabited ParserDescr where
default := ParserDescr.symbol ""
abbrev TrailingParserDescr := ParserDescr
/-
Runtime support for making quotation terms auto-hygienic, by mangling identifiers
introduced by them with a "macro scope" supplied by the context. Details to appear in a
paper soon.
-/
abbrev MacroScope := Nat
/-- Macro scope used internally. It is not available for our frontend. -/
def reservedMacroScope := 0
/-- First macro scope available for our frontend -/
def firstFrontendMacroScope := hAdd reservedMacroScope 1
class MonadRef (m : Type → Type) where
getRef : m Syntax
withRef {α} : Syntax → m α → m α
export MonadRef (getRef)
instance (m n : Type → Type) [MonadLift m n] [MonadFunctor m n] [MonadRef m] : MonadRef n where
getRef := liftM (getRef : m _)
withRef ref x := monadMap (m := m) (MonadRef.withRef ref) x
def replaceRef (ref : Syntax) (oldRef : Syntax) : Syntax :=
match ref.getPos? with
| some _ => ref
| _ => oldRef
@[inline] def withRef {m : Type → Type} [Monad m] [MonadRef m] {α} (ref : Syntax) (x : m α) : m α :=
bind getRef fun oldRef =>
let ref := replaceRef ref oldRef
MonadRef.withRef ref x
/-- A monad that supports syntax quotations. Syntax quotations (in term
position) are monadic values that when executed retrieve the current "macro
scope" from the monad and apply it to every identifier they introduce
(independent of whether this identifier turns out to be a reference to an
existing declaration, or an actually fresh binding during further
elaboration). We also apply the position of the result of `getRef` to each
introduced symbol, which results in better error positions than not applying
any position. -/
class MonadQuotation (m : Type → Type) extends MonadRef m where
-- Get the fresh scope of the current macro invocation
getCurrMacroScope : m MacroScope
getMainModule : m Name
/- Execute action in a new macro invocation context. This transformer should be
used at all places that morally qualify as the beginning of a "macro call",
e.g. `elabCommand` and `elabTerm` in the case of the elaborator. However, it
can also be used internally inside a "macro" if identifiers introduced by
e.g. different recursive calls should be independent and not collide. While
returning an intermediate syntax tree that will recursively be expanded by
the elaborator can be used for the same effect, doing direct recursion inside
the macro guarded by this transformer is often easier because one is not
restricted to passing a single syntax tree. Modelling this helper as a
transformer and not just a monadic action ensures that the current macro
scope before the recursive call is restored after it, as expected. -/
withFreshMacroScope {α : Type} : m α → m α
export MonadQuotation (getCurrMacroScope getMainModule withFreshMacroScope)
def MonadRef.mkInfoFromRefPos [Monad m] [MonadRef m] : m SourceInfo := do
SourceInfo.fromRef (← getRef)
instance {m n : Type → Type} [MonadFunctor m n] [MonadLift m n] [MonadQuotation m] : MonadQuotation n where
getCurrMacroScope := liftM (m := m) getCurrMacroScope
getMainModule := liftM (m := m) getMainModule
withFreshMacroScope := monadMap (m := m) withFreshMacroScope
/-
We represent a name with macro scopes as
```
<actual name>._@.(<module_name>.<scopes>)*.<module_name>._hyg.<scopes>
```
Example: suppose the module name is `Init.Data.List.Basic`, and name is `foo.bla`, and macroscopes [2, 5]
```
foo.bla._@.Init.Data.List.Basic._hyg.2.5
```
We may have to combine scopes from different files/modules.
The main modules being processed is always the right most one.
This situation may happen when we execute a macro generated in
an imported file in the current file.
```
foo.bla._@.Init.Data.List.Basic.2.1.Init.Lean.Expr_hyg.4
```
The delimiter `_hyg` is used just to improve the `hasMacroScopes` performance.
-/
def Name.hasMacroScopes : Name → Bool
| str _ s _ => beq s "_hyg"
| num p _ _ => hasMacroScopes p
| _ => false
private def eraseMacroScopesAux : Name → Name
| Name.str p s _ => match beq s "_@" with
| true => p
| false => eraseMacroScopesAux p
| Name.num p _ _ => eraseMacroScopesAux p
| Name.anonymous => Name.anonymous
@[export lean_erase_macro_scopes]
def Name.eraseMacroScopes (n : Name) : Name :=
match n.hasMacroScopes with
| true => eraseMacroScopesAux n
| false => n
private def simpMacroScopesAux : Name → Name
| Name.num p i _ => Name.mkNum (simpMacroScopesAux p) i
| n => eraseMacroScopesAux n
/- Helper function we use to create binder names that do not need to be unique. -/
@[export lean_simp_macro_scopes]
def Name.simpMacroScopes (n : Name) : Name :=
match n.hasMacroScopes with
| true => simpMacroScopesAux n
| false => n
structure MacroScopesView where
name : Name
imported : Name
mainModule : Name
scopes : List MacroScope
instance : Inhabited MacroScopesView where
default := ⟨arbitrary, arbitrary, arbitrary, arbitrary⟩
def MacroScopesView.review (view : MacroScopesView) : Name :=
match view.scopes with
| List.nil => view.name
| List.cons _ _ =>
let base := (Name.mkStr (hAppend (hAppend (Name.mkStr view.name "_@") view.imported) view.mainModule) "_hyg")
view.scopes.foldl Name.mkNum base
private def assembleParts : List Name → Name → Name
| List.nil, acc => acc
| List.cons (Name.str _ s _) ps, acc => assembleParts ps (Name.mkStr acc s)
| List.cons (Name.num _ n _) ps, acc => assembleParts ps (Name.mkNum acc n)
| _, acc => panic "Error: unreachable @ assembleParts"
private def extractImported (scps : List MacroScope) (mainModule : Name) : Name → List Name → MacroScopesView
| n@(Name.str p str _), parts =>
match beq str "_@" with
| true => { name := p, mainModule := mainModule, imported := assembleParts parts Name.anonymous, scopes := scps }
| false => extractImported scps mainModule p (List.cons n parts)
| n@(Name.num p str _), parts => extractImported scps mainModule p (List.cons n parts)
| _, _ => panic "Error: unreachable @ extractImported"
private def extractMainModule (scps : List MacroScope) : Name → List Name → MacroScopesView
| n@(Name.str p str _), parts =>
match beq str "_@" with
| true => { name := p, mainModule := assembleParts parts Name.anonymous, imported := Name.anonymous, scopes := scps }
| false => extractMainModule scps p (List.cons n parts)
| n@(Name.num p num _), acc => extractImported scps (assembleParts acc Name.anonymous) n List.nil
| _, _ => panic "Error: unreachable @ extractMainModule"
private def extractMacroScopesAux : Name → List MacroScope → MacroScopesView
| Name.num p scp _, acc => extractMacroScopesAux p (List.cons scp acc)
| Name.str p str _, acc => extractMainModule acc p List.nil -- str must be "_hyg"
| _, _ => panic "Error: unreachable @ extractMacroScopesAux"
/--
Revert all `addMacroScope` calls. `v = extractMacroScopes n → n = v.review`.
This operation is useful for analyzing/transforming the original identifiers, then adding back
the scopes (via `MacroScopesView.review`). -/
def extractMacroScopes (n : Name) : MacroScopesView :=
match n.hasMacroScopes with
| true => extractMacroScopesAux n List.nil
| false => { name := n, scopes := List.nil, imported := Name.anonymous, mainModule := Name.anonymous }
def addMacroScope (mainModule : Name) (n : Name) (scp : MacroScope) : Name :=
match n.hasMacroScopes with
| true =>
let view := extractMacroScopes n
match beq view.mainModule mainModule with
| true => Name.mkNum n scp
| false =>
{ view with
imported := view.scopes.foldl Name.mkNum (hAppend view.imported view.mainModule)
mainModule := mainModule
scopes := List.cons scp List.nil
}.review
| false =>
Name.mkNum (Name.mkStr (hAppend (Name.mkStr n "_@") mainModule) "_hyg") scp
@[inline] def MonadQuotation.addMacroScope {m : Type → Type} [MonadQuotation m] [Monad m] (n : Name) : m Name :=
bind getMainModule fun mainModule =>
bind getCurrMacroScope fun scp =>
pure (Lean.addMacroScope mainModule n scp)
def defaultMaxRecDepth := 512
def maxRecDepthErrorMessage : String :=
"maximum recursion depth has been reached (use `set_option maxRecDepth <num>` to increase limit)"
namespace Macro
/- References -/
private constant MethodsRefPointed : PointedType.{0}
private def MethodsRef : Type := MethodsRefPointed.type
structure Context where
methods : MethodsRef
mainModule : Name
currMacroScope : MacroScope
currRecDepth : Nat := 0
maxRecDepth : Nat := defaultMaxRecDepth
ref : Syntax
inductive Exception where
| error : Syntax → String → Exception
| unsupportedSyntax : Exception
structure State where
macroScope : MacroScope
traceMsgs : List (Prod Name String) := List.nil
deriving Inhabited
end Macro
abbrev MacroM := ReaderT Macro.Context (EStateM Macro.Exception Macro.State)
abbrev Macro := Syntax → MacroM Syntax
namespace Macro
instance : MonadRef MacroM where
getRef := bind read fun ctx => pure ctx.ref
withRef := fun ref x => withReader (fun ctx => { ctx with ref := ref }) x
def addMacroScope (n : Name) : MacroM Name :=
bind read fun ctx =>
pure (Lean.addMacroScope ctx.mainModule n ctx.currMacroScope)
def throwUnsupported {α} : MacroM α :=
throw Exception.unsupportedSyntax
def throwError {α} (msg : String) : MacroM α :=
bind getRef fun ref =>
throw (Exception.error ref msg)
def throwErrorAt {α} (ref : Syntax) (msg : String) : MacroM α :=
withRef ref (throwError msg)
@[inline] protected def withFreshMacroScope {α} (x : MacroM α) : MacroM α :=
bind (modifyGet (fun s => (s.macroScope, { s with macroScope := hAdd s.macroScope 1 }))) fun fresh =>
withReader (fun ctx => { ctx with currMacroScope := fresh }) x
@[inline] def withIncRecDepth {α} (ref : Syntax) (x : MacroM α) : MacroM α :=
bind read fun ctx =>
match beq ctx.currRecDepth ctx.maxRecDepth with
| true => throw (Exception.error ref maxRecDepthErrorMessage)
| false => withReader (fun ctx => { ctx with currRecDepth := hAdd ctx.currRecDepth 1 }) x
instance : MonadQuotation MacroM where
getCurrMacroScope ctx := pure ctx.currMacroScope
getMainModule ctx := pure ctx.mainModule
withFreshMacroScope := Macro.withFreshMacroScope
structure Methods where
expandMacro? : Syntax → MacroM (Option Syntax)
getCurrNamespace : MacroM Name
hasDecl : Name → MacroM Bool
resolveNamespace? : Name → MacroM (Option Name)
resolveGlobalName : Name → MacroM (List (Prod Name (List String)))
deriving Inhabited
unsafe def mkMethodsImp (methods : Methods) : MethodsRef :=
unsafeCast methods
@[implementedBy mkMethodsImp]
constant mkMethods (methods : Methods) : MethodsRef := MethodsRefPointed.val
instance : Inhabited MethodsRef where
default := mkMethods arbitrary
unsafe def getMethodsImp : MacroM Methods :=
bind read fun ctx => pure (unsafeCast (ctx.methods))
@[implementedBy getMethodsImp] constant getMethods : MacroM Methods
/-- `expandMacro? stx` return `some stxNew` if `stx` is a macro, and `stxNew` is its expansion. -/
def expandMacro? (stx : Syntax) : MacroM (Option Syntax) := do
(← getMethods).expandMacro? stx
/-- Return `true` if the environment contains a declaration with name `declName` -/
def hasDecl (declName : Name) : MacroM Bool := do
(← getMethods).hasDecl declName
def getCurrNamespace : MacroM Name := do
(← getMethods).getCurrNamespace
def resolveNamespace? (n : Name) : MacroM (Option Name) := do
(← getMethods).resolveNamespace? n
def resolveGlobalName (n : Name) : MacroM (List (Prod Name (List String))) := do
(← getMethods).resolveGlobalName n
def trace (clsName : Name) (msg : String) : MacroM Unit := do
modify fun s => { s with traceMsgs := List.cons (Prod.mk clsName msg) s.traceMsgs }
end Macro
export Macro (expandMacro?)
namespace PrettyPrinter
abbrev UnexpandM := EStateM Unit Unit
/--
Function that tries to reverse macro expansions as a post-processing step of delaboration.
While less general than an arbitrary delaborator, it can be declared without importing `Lean`.
Used by the `[appUnexpander]` attribute. -/
-- a `kindUnexpander` could reasonably be added later
abbrev Unexpander := Syntax → UnexpandM Syntax
-- unexpanders should not need to introduce new names
instance : MonadQuotation UnexpandM where
getRef := pure Syntax.missing
withRef := fun _ => id
getCurrMacroScope := pure 0
getMainModule := pure `_fakeMod
withFreshMacroScope := id
end PrettyPrinter
end Lean
|
cae54bee826ad5fac2d63a24feabe8188e188c77 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/analysis/calculus/fderiv.lean | 2dcaaae1573f0b4bfcfb22ea7a17390f26d00d63 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 139,309 | lean | /-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import analysis.asymptotics.asymptotic_equivalent
import analysis.calculus.tangent_cone
import analysis.normed_space.bounded_linear_maps
import analysis.normed_space.units
/-!
# The Fréchet derivative
Let `E` and `F` be normed spaces, `f : E → F`, and `f' : E →L[𝕜] F` a
continuous 𝕜-linear map, where `𝕜` is a non-discrete normed field. Then
`has_fderiv_within_at f f' s x`
says that `f` has derivative `f'` at `x`, where the domain of interest
is restricted to `s`. We also have
`has_fderiv_at f f' x := has_fderiv_within_at f f' x univ`
Finally,
`has_strict_fderiv_at f f' x`
means that `f : E → F` has derivative `f' : E →L[𝕜] F` in the sense of strict differentiability,
i.e., `f y - f z - f'(y - z) = o(y - z)` as `y, z → x`. This notion is used in the inverse
function theorem, and is defined here only to avoid proving theorems like
`is_bounded_bilinear_map.has_fderiv_at` twice: first for `has_fderiv_at`, then for
`has_strict_fderiv_at`.
## Main results
In addition to the definition and basic properties of the derivative, this file contains the
usual formulas (and existence assertions) for the derivative of
* constants
* the identity
* bounded linear maps
* bounded bilinear maps
* sum of two functions
* sum of finitely many functions
* multiplication of a function by a scalar constant
* negative of a function
* subtraction of two functions
* multiplication of a function by a scalar function
* multiplication of two scalar functions
* composition of functions (the chain rule)
* inverse function (assuming that it exists; the inverse function theorem is in `inverse.lean`)
For most binary operations we also define `const_op` and `op_const` theorems for the cases when
the first or second argument is a constant. This makes writing chains of `has_deriv_at`'s easier,
and they more frequently lead to the desired result.
One can also interpret the derivative of a function `f : 𝕜 → E` as an element of `E` (by identifying
a linear function from `𝕜` to `E` with its value at `1`). Results on the Fréchet derivative are
translated to this more elementary point of view on the derivative in the file `deriv.lean`. The
derivative of polynomials is handled there, as it is naturally one-dimensional.
The simplifier is set up to prove automatically that some functions are differentiable, or
differentiable at a point (but not differentiable on a set or within a set at a point, as checking
automatically that the good domains are mapped one to the other when using composition is not
something the simplifier can easily do). This means that one can write
`example (x : ℝ) : differentiable ℝ (λ x, sin (exp (3 + x^2)) - 5 * cos x) := by simp`.
If there are divisions, one needs to supply to the simplifier proofs that the denominators do
not vanish, as in
```lean
example (x : ℝ) (h : 1 + sin x ≠ 0) : differentiable_at ℝ (λ x, exp x / (1 + sin x)) x :=
by simp [h]
```
Of course, these examples only work once `exp`, `cos` and `sin` have been shown to be
differentiable, in `analysis.special_functions.trigonometric`.
The simplifier is not set up to compute the Fréchet derivative of maps (as these are in general
complicated multidimensional linear maps), but it will compute one-dimensional derivatives,
see `deriv.lean`.
## Implementation details
The derivative is defined in terms of the `is_o` relation, but also
characterized in terms of the `tendsto` relation.
We also introduce predicates `differentiable_within_at 𝕜 f s x` (where `𝕜` is the base field,
`f` the function to be differentiated, `x` the point at which the derivative is asserted to exist,
and `s` the set along which the derivative is defined), as well as `differentiable_at 𝕜 f x`,
`differentiable_on 𝕜 f s` and `differentiable 𝕜 f` to express the existence of a derivative.
To be able to compute with derivatives, we write `fderiv_within 𝕜 f s x` and `fderiv 𝕜 f x`
for some choice of a derivative if it exists, and the zero function otherwise. This choice only
behaves well along sets for which the derivative is unique, i.e., those for which the tangent
directions span a dense subset of the whole space. The predicates `unique_diff_within_at s x` and
`unique_diff_on s`, defined in `tangent_cone.lean` express this property. We prove that indeed
they imply the uniqueness of the derivative. This is satisfied for open subsets, and in particular
for `univ`. This uniqueness only holds when the field is non-discrete, which we request at the very
beginning: otherwise, a derivative can be defined, but it has no interesting properties whatsoever.
To make sure that the simplifier can prove automatically that functions are differentiable, we tag
many lemmas with the `simp` attribute, for instance those saying that the sum of differentiable
functions is differentiable, as well as their product, their cartesian product, and so on. A notable
exception is the chain rule: we do not mark as a simp lemma the fact that, if `f` and `g` are
differentiable, then their composition also is: `simp` would always be able to match this lemma,
by taking `f` or `g` to be the identity. Instead, for every reasonable function (say, `exp`),
we add a lemma that if `f` is differentiable then so is `(λ x, exp (f x))`. This means adding
some boilerplate lemmas, but these can also be useful in their own right.
Tests for this ability of the simplifier (with more examples) are provided in
`tests/differentiable.lean`.
## Tags
derivative, differentiable, Fréchet, calculus
-/
open filter asymptotics continuous_linear_map set metric
open_locale topological_space classical nnreal filter asymptotics ennreal
noncomputable theory
section
variables {𝕜 : Type*} [nontrivially_normed_field 𝕜]
variables {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E]
variables {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F]
variables {G : Type*} [normed_add_comm_group G] [normed_space 𝕜 G]
variables {G' : Type*} [normed_add_comm_group G'] [normed_space 𝕜 G']
/-- A function `f` has the continuous linear map `f'` as derivative along the filter `L` if
`f x' = f x + f' (x' - x) + o (x' - x)` when `x'` converges along the filter `L`. This definition
is designed to be specialized for `L = 𝓝 x` (in `has_fderiv_at`), giving rise to the usual notion
of Fréchet derivative, and for `L = 𝓝[s] x` (in `has_fderiv_within_at`), giving rise to
the notion of Fréchet derivative along the set `s`. -/
def has_fderiv_at_filter (f : E → F) (f' : E →L[𝕜] F) (x : E) (L : filter E) :=
(λ x', f x' - f x - f' (x' - x)) =o[L] (λ x', x' - x)
/-- A function `f` has the continuous linear map `f'` as derivative at `x` within a set `s` if
`f x' = f x + f' (x' - x) + o (x' - x)` when `x'` tends to `x` inside `s`. -/
def has_fderiv_within_at (f : E → F) (f' : E →L[𝕜] F) (s : set E) (x : E) :=
has_fderiv_at_filter f f' x (𝓝[s] x)
/-- A function `f` has the continuous linear map `f'` as derivative at `x` if
`f x' = f x + f' (x' - x) + o (x' - x)` when `x'` tends to `x`. -/
def has_fderiv_at (f : E → F) (f' : E →L[𝕜] F) (x : E) :=
has_fderiv_at_filter f f' x (𝓝 x)
/-- A function `f` has derivative `f'` at `a` in the sense of *strict differentiability*
if `f x - f y - f' (x - y) = o(x - y)` as `x, y → a`. This form of differentiability is required,
e.g., by the inverse function theorem. Any `C^1` function on a vector space over `ℝ` is strictly
differentiable but this definition works, e.g., for vector spaces over `p`-adic numbers. -/
def has_strict_fderiv_at (f : E → F) (f' : E →L[𝕜] F) (x : E) :=
(λ p : E × E, f p.1 - f p.2 - f' (p.1 - p.2)) =o[𝓝 (x, x)] (λ p : E × E, p.1 - p.2)
variables (𝕜)
/-- A function `f` is differentiable at a point `x` within a set `s` if it admits a derivative
there (possibly non-unique). -/
def differentiable_within_at (f : E → F) (s : set E) (x : E) :=
∃f' : E →L[𝕜] F, has_fderiv_within_at f f' s x
/-- A function `f` is differentiable at a point `x` if it admits a derivative there (possibly
non-unique). -/
def differentiable_at (f : E → F) (x : E) :=
∃f' : E →L[𝕜] F, has_fderiv_at f f' x
/-- If `f` has a derivative at `x` within `s`, then `fderiv_within 𝕜 f s x` is such a derivative.
Otherwise, it is set to `0`. -/
def fderiv_within (f : E → F) (s : set E) (x : E) : E →L[𝕜] F :=
if h : ∃f', has_fderiv_within_at f f' s x then classical.some h else 0
/-- If `f` has a derivative at `x`, then `fderiv 𝕜 f x` is such a derivative. Otherwise, it is
set to `0`. -/
def fderiv (f : E → F) (x : E) : E →L[𝕜] F :=
if h : ∃f', has_fderiv_at f f' x then classical.some h else 0
/-- `differentiable_on 𝕜 f s` means that `f` is differentiable within `s` at any point of `s`. -/
def differentiable_on (f : E → F) (s : set E) :=
∀x ∈ s, differentiable_within_at 𝕜 f s x
/-- `differentiable 𝕜 f` means that `f` is differentiable at any point. -/
def differentiable (f : E → F) :=
∀x, differentiable_at 𝕜 f x
variables {𝕜}
variables {f f₀ f₁ g : E → F}
variables {f' f₀' f₁' g' : E →L[𝕜] F}
variables (e : E →L[𝕜] F)
variables {x : E}
variables {s t : set E}
variables {L L₁ L₂ : filter E}
lemma fderiv_within_zero_of_not_differentiable_within_at
(h : ¬ differentiable_within_at 𝕜 f s x) : fderiv_within 𝕜 f s x = 0 :=
have ¬ ∃ f', has_fderiv_within_at f f' s x, from h,
by simp [fderiv_within, this]
lemma fderiv_zero_of_not_differentiable_at (h : ¬ differentiable_at 𝕜 f x) : fderiv 𝕜 f x = 0 :=
have ¬ ∃ f', has_fderiv_at f f' x, from h,
by simp [fderiv, this]
section derivative_uniqueness
/- In this section, we discuss the uniqueness of the derivative.
We prove that the definitions `unique_diff_within_at` and `unique_diff_on` indeed imply the
uniqueness of the derivative. -/
/-- If a function f has a derivative f' at x, a rescaled version of f around x converges to f',
i.e., `n (f (x + (1/n) v) - f x)` converges to `f' v`. More generally, if `c n` tends to infinity
and `c n * d n` tends to `v`, then `c n * (f (x + d n) - f x)` tends to `f' v`. This lemma expresses
this fact, for functions having a derivative within a set. Its specific formulation is useful for
tangent cone related discussions. -/
theorem has_fderiv_within_at.lim (h : has_fderiv_within_at f f' s x) {α : Type*} (l : filter α)
{c : α → 𝕜} {d : α → E} {v : E} (dtop : ∀ᶠ n in l, x + d n ∈ s)
(clim : tendsto (λ n, ‖c n‖) l at_top)
(cdlim : tendsto (λ n, c n • d n) l (𝓝 v)) :
tendsto (λn, c n • (f (x + d n) - f x)) l (𝓝 (f' v)) :=
begin
have tendsto_arg : tendsto (λ n, x + d n) l (𝓝[s] x),
{ conv in (𝓝[s] x) { rw ← add_zero x },
rw [nhds_within, tendsto_inf],
split,
{ apply tendsto_const_nhds.add (tangent_cone_at.lim_zero l clim cdlim) },
{ rwa tendsto_principal } },
have : (λ y, f y - f x - f' (y - x)) =o[𝓝[s] x] (λ y, y - x) := h,
have : (λ n, f (x + d n) - f x - f' ((x + d n) - x)) =o[l] (λ n, (x + d n) - x) :=
this.comp_tendsto tendsto_arg,
have : (λ n, f (x + d n) - f x - f' (d n)) =o[l] d := by simpa only [add_sub_cancel'],
have : (λ n, c n • (f (x + d n) - f x - f' (d n))) =o[l] (λ n, c n • d n) :=
(is_O_refl c l).smul_is_o this,
have : (λ n, c n • (f (x + d n) - f x - f' (d n))) =o[l] (λ n, (1:ℝ)) :=
this.trans_is_O (cdlim.is_O_one ℝ),
have L1 : tendsto (λn, c n • (f (x + d n) - f x - f' (d n))) l (𝓝 0) :=
(is_o_one_iff ℝ).1 this,
have L2 : tendsto (λn, f' (c n • d n)) l (𝓝 (f' v)) :=
tendsto.comp f'.cont.continuous_at cdlim,
have L3 : tendsto (λn, (c n • (f (x + d n) - f x - f' (d n)) + f' (c n • d n)))
l (𝓝 (0 + f' v)) :=
L1.add L2,
have : (λn, (c n • (f (x + d n) - f x - f' (d n)) + f' (c n • d n)))
= (λn, c n • (f (x + d n) - f x)),
by { ext n, simp [smul_add, smul_sub] },
rwa [this, zero_add] at L3
end
/-- If `f'` and `f₁'` are two derivatives of `f` within `s` at `x`, then they are equal on the
tangent cone to `s` at `x` -/
theorem has_fderiv_within_at.unique_on (hf : has_fderiv_within_at f f' s x)
(hg : has_fderiv_within_at f f₁' s x) :
eq_on f' f₁' (tangent_cone_at 𝕜 s x) :=
λ y ⟨c, d, dtop, clim, cdlim⟩,
tendsto_nhds_unique (hf.lim at_top dtop clim cdlim) (hg.lim at_top dtop clim cdlim)
/-- `unique_diff_within_at` achieves its goal: it implies the uniqueness of the derivative. -/
theorem unique_diff_within_at.eq (H : unique_diff_within_at 𝕜 s x)
(hf : has_fderiv_within_at f f' s x) (hg : has_fderiv_within_at f f₁' s x) : f' = f₁' :=
continuous_linear_map.ext_on H.1 (hf.unique_on hg)
theorem unique_diff_on.eq (H : unique_diff_on 𝕜 s) (hx : x ∈ s)
(h : has_fderiv_within_at f f' s x) (h₁ : has_fderiv_within_at f f₁' s x) : f' = f₁' :=
(H x hx).eq h h₁
end derivative_uniqueness
section fderiv_properties
/-! ### Basic properties of the derivative -/
theorem has_fderiv_at_filter_iff_tendsto :
has_fderiv_at_filter f f' x L ↔
tendsto (λ x', ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0) :=
have h : ∀ x', ‖x' - x‖ = 0 → ‖f x' - f x - f' (x' - x)‖ = 0, from λ x' hx',
by { rw [sub_eq_zero.1 (norm_eq_zero.1 hx')], simp },
begin
unfold has_fderiv_at_filter,
rw [←is_o_norm_left, ←is_o_norm_right, is_o_iff_tendsto h],
exact tendsto_congr (λ _, div_eq_inv_mul _ _),
end
theorem has_fderiv_within_at_iff_tendsto : has_fderiv_within_at f f' s x ↔
tendsto (λ x', ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) (𝓝[s] x) (𝓝 0) :=
has_fderiv_at_filter_iff_tendsto
theorem has_fderiv_at_iff_tendsto : has_fderiv_at f f' x ↔
tendsto (λ x', ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) (𝓝 x) (𝓝 0) :=
has_fderiv_at_filter_iff_tendsto
theorem has_fderiv_at_iff_is_o_nhds_zero : has_fderiv_at f f' x ↔
(λ h : E, f (x + h) - f x - f' h) =o[𝓝 0] (λh, h) :=
begin
rw [has_fderiv_at, has_fderiv_at_filter, ← map_add_left_nhds_zero x, is_o_map],
simp [(∘)]
end
/-- Converse to the mean value inequality: if `f` is differentiable at `x₀` and `C`-lipschitz
on a neighborhood of `x₀` then it its derivative at `x₀` has norm bounded by `C`. This version
only assumes that `‖f x - f x₀‖ ≤ C * ‖x - x₀‖` in a neighborhood of `x`. -/
lemma has_fderiv_at.le_of_lip' {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : has_fderiv_at f f' x₀)
{C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) : ‖f'‖ ≤ C :=
begin
refine le_of_forall_pos_le_add (λ ε ε0, op_norm_le_of_nhds_zero _ _),
exact add_nonneg hC₀ ε0.le,
rw [← map_add_left_nhds_zero x₀, eventually_map] at hlip,
filter_upwards [is_o_iff.1 (has_fderiv_at_iff_is_o_nhds_zero.1 hf) ε0, hlip] with y hy hyC,
rw add_sub_cancel' at hyC,
calc ‖f' y‖ ≤ ‖f (x₀ + y) - f x₀‖ + ‖f (x₀ + y) - f x₀ - f' y‖ : norm_le_insert _ _
... ≤ C * ‖y‖ + ε * ‖y‖ : add_le_add hyC hy
... = (C + ε) * ‖y‖ : (add_mul _ _ _).symm
end
/-- Converse to the mean value inequality: if `f` is differentiable at `x₀` and `C`-lipschitz
on a neighborhood of `x₀` then it its derivative at `x₀` has norm bounded by `C`. -/
lemma has_fderiv_at.le_of_lip {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : has_fderiv_at f f' x₀)
{s : set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : lipschitz_on_with C f s) : ‖f'‖ ≤ C :=
begin
refine hf.le_of_lip' C.coe_nonneg _,
filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs),
end
theorem has_fderiv_at_filter.mono (h : has_fderiv_at_filter f f' x L₂) (hst : L₁ ≤ L₂) :
has_fderiv_at_filter f f' x L₁ :=
h.mono hst
theorem has_fderiv_within_at.mono_of_mem (h : has_fderiv_within_at f f' t x) (hst : t ∈ 𝓝[s] x) :
has_fderiv_within_at f f' s x :=
h.mono $ nhds_within_le_iff.mpr hst
theorem has_fderiv_within_at.mono (h : has_fderiv_within_at f f' t x) (hst : s ⊆ t) :
has_fderiv_within_at f f' s x :=
h.mono $ nhds_within_mono _ hst
theorem has_fderiv_at.has_fderiv_at_filter (h : has_fderiv_at f f' x) (hL : L ≤ 𝓝 x) :
has_fderiv_at_filter f f' x L :=
h.mono hL
theorem has_fderiv_at.has_fderiv_within_at
(h : has_fderiv_at f f' x) : has_fderiv_within_at f f' s x :=
h.has_fderiv_at_filter inf_le_left
lemma has_fderiv_within_at.differentiable_within_at (h : has_fderiv_within_at f f' s x) :
differentiable_within_at 𝕜 f s x :=
⟨f', h⟩
lemma has_fderiv_at.differentiable_at (h : has_fderiv_at f f' x) : differentiable_at 𝕜 f x :=
⟨f', h⟩
@[simp] lemma has_fderiv_within_at_univ :
has_fderiv_within_at f f' univ x ↔ has_fderiv_at f f' x :=
by { simp only [has_fderiv_within_at, nhds_within_univ], refl }
alias has_fderiv_within_at_univ ↔ has_fderiv_within_at.has_fderiv_at_of_univ _
lemma has_fderiv_within_at_insert {y : E} {g' : E →L[𝕜] F} :
has_fderiv_within_at g g' (insert y s) x ↔ has_fderiv_within_at g g' s x :=
begin
rcases eq_or_ne x y with rfl|h,
{ simp_rw [has_fderiv_within_at, has_fderiv_at_filter],
apply asymptotics.is_o_insert,
simp only [sub_self, g'.map_zero] },
refine ⟨λ h, h.mono $ subset_insert y s, λ hg, hg.mono_of_mem _⟩,
simp_rw [nhds_within_insert_of_ne h, self_mem_nhds_within]
end
alias has_fderiv_within_at_insert ↔ has_fderiv_within_at.of_insert has_fderiv_within_at.insert'
lemma has_fderiv_within_at.insert {g' : E →L[𝕜] F} (h : has_fderiv_within_at g g' s x) :
has_fderiv_within_at g g' (insert x s) x :=
h.insert'
lemma has_strict_fderiv_at.is_O_sub (hf : has_strict_fderiv_at f f' x) :
(λ p : E × E, f p.1 - f p.2) =O[𝓝 (x, x)] (λ p : E × E, p.1 - p.2) :=
hf.is_O.congr_of_sub.2 (f'.is_O_comp _ _)
lemma has_fderiv_at_filter.is_O_sub (h : has_fderiv_at_filter f f' x L) :
(λ x', f x' - f x) =O[L] (λ x', x' - x) :=
h.is_O.congr_of_sub.2 (f'.is_O_sub _ _)
protected lemma has_strict_fderiv_at.has_fderiv_at (hf : has_strict_fderiv_at f f' x) :
has_fderiv_at f f' x :=
begin
rw [has_fderiv_at, has_fderiv_at_filter, is_o_iff],
exact (λ c hc, tendsto_id.prod_mk_nhds tendsto_const_nhds (is_o_iff.1 hf hc))
end
protected lemma has_strict_fderiv_at.differentiable_at (hf : has_strict_fderiv_at f f' x) :
differentiable_at 𝕜 f x :=
hf.has_fderiv_at.differentiable_at
/-- If `f` is strictly differentiable at `x` with derivative `f'` and `K > ‖f'‖₊`, then `f` is
`K`-Lipschitz in a neighborhood of `x`. -/
lemma has_strict_fderiv_at.exists_lipschitz_on_with_of_nnnorm_lt (hf : has_strict_fderiv_at f f' x)
(K : ℝ≥0) (hK : ‖f'‖₊ < K) : ∃ s ∈ 𝓝 x, lipschitz_on_with K f s :=
begin
have := hf.add_is_O_with (f'.is_O_with_comp _ _) hK,
simp only [sub_add_cancel, is_O_with] at this,
rcases exists_nhds_square this with ⟨U, Uo, xU, hU⟩,
exact ⟨U, Uo.mem_nhds xU, lipschitz_on_with_iff_norm_sub_le.2 $
λ x hx y hy, hU (mk_mem_prod hx hy)⟩
end
/-- If `f` is strictly differentiable at `x` with derivative `f'`, then `f` is Lipschitz in a
neighborhood of `x`. See also `has_strict_fderiv_at.exists_lipschitz_on_with_of_nnnorm_lt` for a
more precise statement. -/
lemma has_strict_fderiv_at.exists_lipschitz_on_with (hf : has_strict_fderiv_at f f' x) :
∃ K (s ∈ 𝓝 x), lipschitz_on_with K f s :=
(exists_gt _).imp hf.exists_lipschitz_on_with_of_nnnorm_lt
/-- Directional derivative agrees with `has_fderiv`. -/
lemma has_fderiv_at.lim (hf : has_fderiv_at f f' x) (v : E) {α : Type*} {c : α → 𝕜}
{l : filter α} (hc : tendsto (λ n, ‖c n‖) l at_top) :
tendsto (λ n, (c n) • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v)) :=
begin
refine (has_fderiv_within_at_univ.2 hf).lim _ univ_mem hc _,
assume U hU,
refine (eventually_ne_of_tendsto_norm_at_top hc (0:𝕜)).mono (λ y hy, _),
convert mem_of_mem_nhds hU,
dsimp only,
rw [← mul_smul, mul_inv_cancel hy, one_smul]
end
theorem has_fderiv_at.unique
(h₀ : has_fderiv_at f f₀' x) (h₁ : has_fderiv_at f f₁' x) : f₀' = f₁' :=
begin
rw ← has_fderiv_within_at_univ at h₀ h₁,
exact unique_diff_within_at_univ.eq h₀ h₁
end
lemma has_fderiv_within_at_inter' (h : t ∈ 𝓝[s] x) :
has_fderiv_within_at f f' (s ∩ t) x ↔ has_fderiv_within_at f f' s x :=
by simp [has_fderiv_within_at, nhds_within_restrict'' s h]
lemma has_fderiv_within_at_inter (h : t ∈ 𝓝 x) :
has_fderiv_within_at f f' (s ∩ t) x ↔ has_fderiv_within_at f f' s x :=
by simp [has_fderiv_within_at, nhds_within_restrict' s h]
lemma has_fderiv_within_at.union (hs : has_fderiv_within_at f f' s x)
(ht : has_fderiv_within_at f f' t x) :
has_fderiv_within_at f f' (s ∪ t) x :=
begin
simp only [has_fderiv_within_at, nhds_within_union],
exact hs.sup ht,
end
lemma has_fderiv_within_at.nhds_within (h : has_fderiv_within_at f f' s x)
(ht : s ∈ 𝓝[t] x) : has_fderiv_within_at f f' t x :=
(has_fderiv_within_at_inter' ht).1 (h.mono (inter_subset_right _ _))
lemma has_fderiv_within_at.has_fderiv_at (h : has_fderiv_within_at f f' s x) (hs : s ∈ 𝓝 x) :
has_fderiv_at f f' x :=
by rwa [← univ_inter s, has_fderiv_within_at_inter hs, has_fderiv_within_at_univ] at h
lemma differentiable_within_at.differentiable_at
(h : differentiable_within_at 𝕜 f s x) (hs : s ∈ 𝓝 x) : differentiable_at 𝕜 f x :=
h.imp (λ f' hf', hf'.has_fderiv_at hs)
lemma differentiable_within_at.has_fderiv_within_at (h : differentiable_within_at 𝕜 f s x) :
has_fderiv_within_at f (fderiv_within 𝕜 f s x) s x :=
begin
dunfold fderiv_within,
dunfold differentiable_within_at at h,
rw dif_pos h,
exact classical.some_spec h
end
lemma differentiable_at.has_fderiv_at (h : differentiable_at 𝕜 f x) :
has_fderiv_at f (fderiv 𝕜 f x) x :=
begin
dunfold fderiv,
dunfold differentiable_at at h,
rw dif_pos h,
exact classical.some_spec h
end
lemma differentiable_on.has_fderiv_at (h : differentiable_on 𝕜 f s) (hs : s ∈ 𝓝 x) :
has_fderiv_at f (fderiv 𝕜 f x) x :=
((h x (mem_of_mem_nhds hs)).differentiable_at hs).has_fderiv_at
lemma differentiable_on.differentiable_at (h : differentiable_on 𝕜 f s) (hs : s ∈ 𝓝 x) :
differentiable_at 𝕜 f x :=
(h.has_fderiv_at hs).differentiable_at
lemma differentiable_on.eventually_differentiable_at (h : differentiable_on 𝕜 f s) (hs : s ∈ 𝓝 x) :
∀ᶠ y in 𝓝 x, differentiable_at 𝕜 f y :=
(eventually_eventually_nhds.2 hs).mono $ λ y, h.differentiable_at
lemma has_fderiv_at.fderiv (h : has_fderiv_at f f' x) : fderiv 𝕜 f x = f' :=
by { ext, rw h.unique h.differentiable_at.has_fderiv_at }
lemma fderiv_eq {f' : E → E →L[𝕜] F} (h : ∀ x, has_fderiv_at f (f' x) x) : fderiv 𝕜 f = f' :=
funext $ λ x, (h x).fderiv
/-- Converse to the mean value inequality: if `f` is differentiable at `x₀` and `C`-lipschitz
on a neighborhood of `x₀` then it its derivative at `x₀` has norm bounded by `C`.
Version using `fderiv`. -/
lemma fderiv_at.le_of_lip {f : E → F} {x₀ : E} (hf : differentiable_at 𝕜 f x₀)
{s : set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : lipschitz_on_with C f s) : ‖fderiv 𝕜 f x₀‖ ≤ C :=
hf.has_fderiv_at.le_of_lip hs hlip
lemma has_fderiv_within_at.fderiv_within
(h : has_fderiv_within_at f f' s x) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 f s x = f' :=
(hxs.eq h h.differentiable_within_at.has_fderiv_within_at).symm
/-- If `x` is not in the closure of `s`, then `f` has any derivative at `x` within `s`,
as this statement is empty. -/
lemma has_fderiv_within_at_of_not_mem_closure (h : x ∉ closure s) :
has_fderiv_within_at f f' s x :=
begin
simp only [mem_closure_iff_nhds_within_ne_bot, ne_bot_iff, ne.def, not_not] at h,
simp [has_fderiv_within_at, has_fderiv_at_filter, h, is_o, is_O_with],
end
lemma differentiable_within_at.mono (h : differentiable_within_at 𝕜 f t x) (st : s ⊆ t) :
differentiable_within_at 𝕜 f s x :=
begin
rcases h with ⟨f', hf'⟩,
exact ⟨f', hf'.mono st⟩
end
lemma differentiable_within_at.mono_of_mem (h : differentiable_within_at 𝕜 f s x) {t : set E}
(hst : s ∈ nhds_within x t) :
differentiable_within_at 𝕜 f t x :=
(h.has_fderiv_within_at.mono_of_mem hst).differentiable_within_at
lemma differentiable_within_at_univ :
differentiable_within_at 𝕜 f univ x ↔ differentiable_at 𝕜 f x :=
by simp only [differentiable_within_at, has_fderiv_within_at_univ, differentiable_at]
lemma differentiable_within_at_inter (ht : t ∈ 𝓝 x) :
differentiable_within_at 𝕜 f (s ∩ t) x ↔ differentiable_within_at 𝕜 f s x :=
by simp only [differentiable_within_at, has_fderiv_within_at, has_fderiv_at_filter,
nhds_within_restrict' s ht]
lemma differentiable_within_at_inter' (ht : t ∈ 𝓝[s] x) :
differentiable_within_at 𝕜 f (s ∩ t) x ↔ differentiable_within_at 𝕜 f s x :=
by simp only [differentiable_within_at, has_fderiv_within_at, has_fderiv_at_filter,
nhds_within_restrict'' s ht]
lemma differentiable_within_at.antimono (h : differentiable_within_at 𝕜 f s x) (hst : s ⊆ t)
(hx : s ∈ 𝓝[t] x) :
differentiable_within_at 𝕜 f t x :=
by rwa [← differentiable_within_at_inter' hx, inter_eq_self_of_subset_right hst]
lemma has_fderiv_within_at.antimono (h : has_fderiv_within_at f f' s x) (hst : s ⊆ t)
(hs : unique_diff_within_at 𝕜 s x) (hx : s ∈ 𝓝[t] x) :
has_fderiv_within_at f f' t x :=
begin
have h' : has_fderiv_within_at f _ t x :=
(h.differentiable_within_at.antimono hst hx).has_fderiv_within_at,
rwa hs.eq h (h'.mono hst),
end
lemma differentiable_at.differentiable_within_at
(h : differentiable_at 𝕜 f x) : differentiable_within_at 𝕜 f s x :=
(differentiable_within_at_univ.2 h).mono (subset_univ _)
lemma differentiable.differentiable_at (h : differentiable 𝕜 f) :
differentiable_at 𝕜 f x :=
h x
lemma differentiable_at.fderiv_within
(h : differentiable_at 𝕜 f x) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 f s x = fderiv 𝕜 f x :=
h.has_fderiv_at.has_fderiv_within_at.fderiv_within hxs
lemma differentiable_on.mono (h : differentiable_on 𝕜 f t) (st : s ⊆ t) :
differentiable_on 𝕜 f s :=
λ x hx, (h x (st hx)).mono st
lemma differentiable_on_univ :
differentiable_on 𝕜 f univ ↔ differentiable 𝕜 f :=
by simp only [differentiable_on, differentiable, differentiable_within_at_univ, mem_univ,
forall_true_left]
lemma differentiable.differentiable_on (h : differentiable 𝕜 f) : differentiable_on 𝕜 f s :=
(differentiable_on_univ.2 h).mono (subset_univ _)
lemma differentiable_on_of_locally_differentiable_on
(h : ∀x∈s, ∃u, is_open u ∧ x ∈ u ∧ differentiable_on 𝕜 f (s ∩ u)) : differentiable_on 𝕜 f s :=
begin
assume x xs,
rcases h x xs with ⟨t, t_open, xt, ht⟩,
exact (differentiable_within_at_inter (is_open.mem_nhds t_open xt)).1 (ht x ⟨xs, xt⟩)
end
lemma fderiv_within_subset (st : s ⊆ t) (ht : unique_diff_within_at 𝕜 s x)
(h : differentiable_within_at 𝕜 f t x) :
fderiv_within 𝕜 f s x = fderiv_within 𝕜 f t x :=
((differentiable_within_at.has_fderiv_within_at h).mono st).fderiv_within ht
lemma fderiv_within_subset' (st : s ⊆ t) (ht : unique_diff_within_at 𝕜 s x) (hx : s ∈ 𝓝[t] x)
(h : differentiable_within_at 𝕜 f s x) :
fderiv_within 𝕜 f s x = fderiv_within 𝕜 f t x :=
fderiv_within_subset st ht (h.antimono st hx)
@[simp] lemma fderiv_within_univ : fderiv_within 𝕜 f univ = fderiv 𝕜 f :=
begin
ext x : 1,
by_cases h : differentiable_at 𝕜 f x,
{ apply has_fderiv_within_at.fderiv_within _ unique_diff_within_at_univ,
rw has_fderiv_within_at_univ,
apply h.has_fderiv_at },
{ have : ¬ differentiable_within_at 𝕜 f univ x,
{ rwa differentiable_within_at_univ },
rw [fderiv_zero_of_not_differentiable_at h,
fderiv_within_zero_of_not_differentiable_within_at this] }
end
lemma fderiv_within_inter (ht : t ∈ 𝓝 x) (hs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 f (s ∩ t) x = fderiv_within 𝕜 f s x :=
begin
by_cases h : differentiable_within_at 𝕜 f (s ∩ t) x,
{ apply fderiv_within_subset (inter_subset_left _ _) _ ((differentiable_within_at_inter ht).1 h),
apply hs.inter ht },
{ have : ¬ differentiable_within_at 𝕜 f s x,
{ rwa ←differentiable_within_at_inter ht },
rw [fderiv_within_zero_of_not_differentiable_within_at h,
fderiv_within_zero_of_not_differentiable_within_at this] }
end
lemma fderiv_within_of_mem_nhds (h : s ∈ 𝓝 x) :
fderiv_within 𝕜 f s x = fderiv 𝕜 f x :=
begin
have : s = univ ∩ s, by simp only [univ_inter],
rw [this, ← fderiv_within_univ],
exact fderiv_within_inter h (unique_diff_on_univ _ (mem_univ _))
end
lemma fderiv_within_of_open (hs : is_open s) (hx : x ∈ s) :
fderiv_within 𝕜 f s x = fderiv 𝕜 f x :=
fderiv_within_of_mem_nhds (is_open.mem_nhds hs hx)
lemma fderiv_within_eq_fderiv (hs : unique_diff_within_at 𝕜 s x) (h : differentiable_at 𝕜 f x) :
fderiv_within 𝕜 f s x = fderiv 𝕜 f x :=
begin
rw ← fderiv_within_univ,
exact fderiv_within_subset (subset_univ _) hs h.differentiable_within_at
end
lemma fderiv_mem_iff {f : E → F} {s : set (E →L[𝕜] F)} {x : E} :
fderiv 𝕜 f x ∈ s ↔ (differentiable_at 𝕜 f x ∧ fderiv 𝕜 f x ∈ s) ∨
(¬differentiable_at 𝕜 f x ∧ (0 : E →L[𝕜] F) ∈ s) :=
by by_cases hx : differentiable_at 𝕜 f x; simp [fderiv_zero_of_not_differentiable_at, *]
lemma fderiv_within_mem_iff {f : E → F} {t : set E} {s : set (E →L[𝕜] F)} {x : E} :
fderiv_within 𝕜 f t x ∈ s ↔ (differentiable_within_at 𝕜 f t x ∧ fderiv_within 𝕜 f t x ∈ s) ∨
(¬differentiable_within_at 𝕜 f t x ∧ (0 : E →L[𝕜] F) ∈ s) :=
by by_cases hx : differentiable_within_at 𝕜 f t x;
simp [fderiv_within_zero_of_not_differentiable_within_at, *]
lemma asymptotics.is_O.has_fderiv_within_at {s : set E} {x₀ : E} {n : ℕ}
(h : f =O[𝓝[s] x₀] λ x, ‖x - x₀‖^n) (hx₀ : x₀ ∈ s) (hn : 1 < n) :
has_fderiv_within_at f (0 : E →L[𝕜] F) s x₀ :=
by simp_rw [has_fderiv_within_at, has_fderiv_at_filter,
h.eq_zero_of_norm_pow_within hx₀ $ zero_lt_one.trans hn, zero_apply, sub_zero,
h.trans_is_o ((is_o_pow_sub_sub x₀ hn).mono nhds_within_le_nhds)]
lemma asymptotics.is_O.has_fderiv_at {x₀ : E} {n : ℕ}
(h : f =O[𝓝 x₀] λ x, ‖x - x₀‖^n) (hn : 1 < n) :
has_fderiv_at f (0 : E →L[𝕜] F) x₀ :=
begin
rw [← nhds_within_univ] at h,
exact (h.has_fderiv_within_at (mem_univ _) hn).has_fderiv_at_of_univ
end
lemma has_fderiv_within_at.is_O {f : E → F} {s : set E} {x₀ : E} {f' : E →L[𝕜] F}
(h : has_fderiv_within_at f f' s x₀) :
(λ x, f x - f x₀) =O[𝓝[s] x₀] λ x, x - x₀ :=
by simpa only [sub_add_cancel] using h.is_O.add (is_O_sub f' (𝓝[s] x₀) x₀)
lemma has_fderiv_at.is_O {f : E → F} {x₀ : E} {f' : E →L[𝕜] F} (h : has_fderiv_at f f' x₀) :
(λ x, f x - f x₀) =O[𝓝 x₀] λ x, x - x₀ :=
by simpa only [sub_add_cancel] using h.is_O.add (is_O_sub f' (𝓝 x₀) x₀)
end fderiv_properties
section continuous
/-! ### Deducing continuity from differentiability -/
theorem has_fderiv_at_filter.tendsto_nhds
(hL : L ≤ 𝓝 x) (h : has_fderiv_at_filter f f' x L) :
tendsto f L (𝓝 (f x)) :=
begin
have : tendsto (λ x', f x' - f x) L (𝓝 0),
{ refine h.is_O_sub.trans_tendsto (tendsto.mono_left _ hL),
rw ← sub_self x, exact tendsto_id.sub tendsto_const_nhds },
have := tendsto.add this tendsto_const_nhds,
rw zero_add (f x) at this,
exact this.congr (by simp only [sub_add_cancel, eq_self_iff_true, forall_const])
end
theorem has_fderiv_within_at.continuous_within_at
(h : has_fderiv_within_at f f' s x) : continuous_within_at f s x :=
has_fderiv_at_filter.tendsto_nhds inf_le_left h
theorem has_fderiv_at.continuous_at (h : has_fderiv_at f f' x) :
continuous_at f x :=
has_fderiv_at_filter.tendsto_nhds le_rfl h
lemma differentiable_within_at.continuous_within_at (h : differentiable_within_at 𝕜 f s x) :
continuous_within_at f s x :=
let ⟨f', hf'⟩ := h in hf'.continuous_within_at
lemma differentiable_at.continuous_at (h : differentiable_at 𝕜 f x) : continuous_at f x :=
let ⟨f', hf'⟩ := h in hf'.continuous_at
lemma differentiable_on.continuous_on (h : differentiable_on 𝕜 f s) : continuous_on f s :=
λx hx, (h x hx).continuous_within_at
lemma differentiable.continuous (h : differentiable 𝕜 f) : continuous f :=
continuous_iff_continuous_at.2 $ λx, (h x).continuous_at
protected lemma has_strict_fderiv_at.continuous_at (hf : has_strict_fderiv_at f f' x) :
continuous_at f x :=
hf.has_fderiv_at.continuous_at
lemma has_strict_fderiv_at.is_O_sub_rev {f' : E ≃L[𝕜] F}
(hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) x) :
(λ p : E × E, p.1 - p.2) =O[𝓝 (x, x)](λ p : E × E, f p.1 - f p.2) :=
((f'.is_O_comp_rev _ _).trans (hf.trans_is_O (f'.is_O_comp_rev _ _)).right_is_O_add).congr
(λ _, rfl) (λ _, sub_add_cancel _ _)
lemma has_fderiv_at_filter.is_O_sub_rev (hf : has_fderiv_at_filter f f' x L) {C}
(hf' : antilipschitz_with C f') :
(λ x', x' - x) =O[L] (λ x', f x' - f x) :=
have (λ x', x' - x) =O[L] (λ x', f' (x' - x)),
from is_O_iff.2 ⟨C, eventually_of_forall $ λ x',
add_monoid_hom_class.bound_of_antilipschitz f' hf' _⟩,
(this.trans (hf.trans_is_O this).right_is_O_add).congr (λ _, rfl) (λ _, sub_add_cancel _ _)
end continuous
section congr
/-! ### congr properties of the derivative -/
theorem filter.eventually_eq.has_strict_fderiv_at_iff
(h : f₀ =ᶠ[𝓝 x] f₁) (h' : ∀ y, f₀' y = f₁' y) :
has_strict_fderiv_at f₀ f₀' x ↔ has_strict_fderiv_at f₁ f₁' x :=
begin
refine is_o_congr ((h.prod_mk_nhds h).mono _) (eventually_of_forall $ λ _, rfl),
rintros p ⟨hp₁, hp₂⟩,
simp only [*]
end
theorem has_strict_fderiv_at.congr_of_eventually_eq (h : has_strict_fderiv_at f f' x)
(h₁ : f =ᶠ[𝓝 x] f₁) : has_strict_fderiv_at f₁ f' x :=
(h₁.has_strict_fderiv_at_iff (λ _, rfl)).1 h
theorem filter.eventually_eq.has_fderiv_at_filter_iff
(h₀ : f₀ =ᶠ[L] f₁) (hx : f₀ x = f₁ x) (h₁ : ∀ x, f₀' x = f₁' x) :
has_fderiv_at_filter f₀ f₀' x L ↔ has_fderiv_at_filter f₁ f₁' x L :=
is_o_congr (h₀.mono $ λ y hy, by simp only [hy, h₁, hx]) (eventually_of_forall $ λ _, rfl)
lemma has_fderiv_at_filter.congr_of_eventually_eq (h : has_fderiv_at_filter f f' x L)
(hL : f₁ =ᶠ[L] f) (hx : f₁ x = f x) : has_fderiv_at_filter f₁ f' x L :=
(hL.has_fderiv_at_filter_iff hx $ λ _, rfl).2 h
theorem filter.eventually_eq.has_fderiv_at_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
has_fderiv_at f₀ f' x ↔ has_fderiv_at f₁ f' x :=
h.has_fderiv_at_filter_iff h.eq_of_nhds (λ _, rfl)
theorem filter.eventually_eq.differentiable_at_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
differentiable_at 𝕜 f₀ x ↔ differentiable_at 𝕜 f₁ x :=
exists_congr $ λ f', h.has_fderiv_at_iff
theorem filter.eventually_eq.has_fderiv_within_at_iff (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x) :
has_fderiv_within_at f₀ f' s x ↔ has_fderiv_within_at f₁ f' s x :=
h.has_fderiv_at_filter_iff hx (λ _, rfl)
theorem filter.eventually_eq.has_fderiv_within_at_iff_of_mem (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s) :
has_fderiv_within_at f₀ f' s x ↔ has_fderiv_within_at f₁ f' s x :=
h.has_fderiv_within_at_iff (h.eq_of_nhds_within hx)
theorem filter.eventually_eq.differentiable_within_at_iff (h : f₀ =ᶠ[𝓝[s] x] f₁)
(hx : f₀ x = f₁ x) :
differentiable_within_at 𝕜 f₀ s x ↔ differentiable_within_at 𝕜 f₁ s x :=
exists_congr $ λ f', h.has_fderiv_within_at_iff hx
theorem filter.eventually_eq.differentiable_within_at_iff_of_mem (h : f₀ =ᶠ[𝓝[s] x] f₁)
(hx : x ∈ s) :
differentiable_within_at 𝕜 f₀ s x ↔ differentiable_within_at 𝕜 f₁ s x :=
h.differentiable_within_at_iff (h.eq_of_nhds_within hx)
lemma has_fderiv_within_at.congr_mono (h : has_fderiv_within_at f f' s x) (ht : ∀x ∈ t, f₁ x = f x)
(hx : f₁ x = f x) (h₁ : t ⊆ s) : has_fderiv_within_at f₁ f' t x :=
has_fderiv_at_filter.congr_of_eventually_eq (h.mono h₁) (filter.mem_inf_of_right ht) hx
lemma has_fderiv_within_at.congr (h : has_fderiv_within_at f f' s x) (hs : ∀x ∈ s, f₁ x = f x)
(hx : f₁ x = f x) : has_fderiv_within_at f₁ f' s x :=
h.congr_mono hs hx (subset.refl _)
lemma has_fderiv_within_at.congr' (h : has_fderiv_within_at f f' s x) (hs : ∀x ∈ s, f₁ x = f x)
(hx : x ∈ s) : has_fderiv_within_at f₁ f' s x :=
h.congr hs (hs x hx)
lemma has_fderiv_within_at.congr_of_eventually_eq (h : has_fderiv_within_at f f' s x)
(h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : has_fderiv_within_at f₁ f' s x :=
has_fderiv_at_filter.congr_of_eventually_eq h h₁ hx
lemma has_fderiv_at.congr_of_eventually_eq (h : has_fderiv_at f f' x)
(h₁ : f₁ =ᶠ[𝓝 x] f) : has_fderiv_at f₁ f' x :=
has_fderiv_at_filter.congr_of_eventually_eq h h₁ (mem_of_mem_nhds h₁ : _)
lemma differentiable_within_at.congr_mono (h : differentiable_within_at 𝕜 f s x)
(ht : ∀x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (h₁ : t ⊆ s) : differentiable_within_at 𝕜 f₁ t x :=
(has_fderiv_within_at.congr_mono h.has_fderiv_within_at ht hx h₁).differentiable_within_at
lemma differentiable_within_at.congr (h : differentiable_within_at 𝕜 f s x)
(ht : ∀x ∈ s, f₁ x = f x) (hx : f₁ x = f x) : differentiable_within_at 𝕜 f₁ s x :=
differentiable_within_at.congr_mono h ht hx (subset.refl _)
lemma differentiable_within_at.congr_of_eventually_eq
(h : differentiable_within_at 𝕜 f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f)
(hx : f₁ x = f x) : differentiable_within_at 𝕜 f₁ s x :=
(h.has_fderiv_within_at.congr_of_eventually_eq h₁ hx).differentiable_within_at
lemma differentiable_on.congr_mono (h : differentiable_on 𝕜 f s) (h' : ∀x ∈ t, f₁ x = f x)
(h₁ : t ⊆ s) : differentiable_on 𝕜 f₁ t :=
λ x hx, (h x (h₁ hx)).congr_mono h' (h' x hx) h₁
lemma differentiable_on.congr (h : differentiable_on 𝕜 f s) (h' : ∀x ∈ s, f₁ x = f x) :
differentiable_on 𝕜 f₁ s :=
λ x hx, (h x hx).congr h' (h' x hx)
lemma differentiable_on_congr (h' : ∀x ∈ s, f₁ x = f x) :
differentiable_on 𝕜 f₁ s ↔ differentiable_on 𝕜 f s :=
⟨λ h, differentiable_on.congr h (λy hy, (h' y hy).symm),
λ h, differentiable_on.congr h h'⟩
lemma differentiable_at.congr_of_eventually_eq (h : differentiable_at 𝕜 f x) (hL : f₁ =ᶠ[𝓝 x] f) :
differentiable_at 𝕜 f₁ x :=
hL.differentiable_at_iff.2 h
lemma differentiable_within_at.fderiv_within_congr_mono (h : differentiable_within_at 𝕜 f s x)
(hs : ∀x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (hxt : unique_diff_within_at 𝕜 t x) (h₁ : t ⊆ s) :
fderiv_within 𝕜 f₁ t x = fderiv_within 𝕜 f s x :=
(has_fderiv_within_at.congr_mono h.has_fderiv_within_at hs hx h₁).fderiv_within hxt
lemma filter.eventually_eq.fderiv_within_eq (hs : unique_diff_within_at 𝕜 s x)
(hL : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
fderiv_within 𝕜 f₁ s x = fderiv_within 𝕜 f s x :=
if h : differentiable_within_at 𝕜 f s x
then has_fderiv_within_at.fderiv_within (h.has_fderiv_within_at.congr_of_eventually_eq hL hx) hs
else
have h' : ¬ differentiable_within_at 𝕜 f₁ s x,
from mt (λ h, h.congr_of_eventually_eq (hL.mono $ λ x, eq.symm) hx.symm) h,
by rw [fderiv_within_zero_of_not_differentiable_within_at h,
fderiv_within_zero_of_not_differentiable_within_at h']
lemma filter.eventually_eq.fderiv_within_eq_nhds (hs : unique_diff_within_at 𝕜 s x)
(hL : f₁ =ᶠ[𝓝 x] f) :
fderiv_within 𝕜 f₁ s x = fderiv_within 𝕜 f s x :=
(show f₁ =ᶠ[𝓝[s] x] f, from nhds_within_le_nhds hL).fderiv_within_eq hs (mem_of_mem_nhds hL : _)
lemma fderiv_within_congr (hs : unique_diff_within_at 𝕜 s x)
(hL : ∀y∈s, f₁ y = f y) (hx : f₁ x = f x) :
fderiv_within 𝕜 f₁ s x = fderiv_within 𝕜 f s x :=
begin
apply filter.eventually_eq.fderiv_within_eq hs _ hx,
apply mem_of_superset self_mem_nhds_within,
exact hL
end
lemma filter.eventually_eq.fderiv_eq (hL : f₁ =ᶠ[𝓝 x] f) :
fderiv 𝕜 f₁ x = fderiv 𝕜 f x :=
begin
have A : f₁ x = f x := hL.eq_of_nhds,
rw [← fderiv_within_univ, ← fderiv_within_univ],
rw ← nhds_within_univ at hL,
exact hL.fderiv_within_eq unique_diff_within_at_univ A
end
protected lemma filter.eventually_eq.fderiv (h : f₁ =ᶠ[𝓝 x] f) :
fderiv 𝕜 f₁ =ᶠ[𝓝 x] fderiv 𝕜 f :=
h.eventually_eq_nhds.mono $ λ x h, h.fderiv_eq
end congr
section id
/-! ### Derivative of the identity -/
theorem has_strict_fderiv_at_id (x : E) :
has_strict_fderiv_at id (id 𝕜 E) x :=
(is_o_zero _ _).congr_left $ by simp
theorem has_fderiv_at_filter_id (x : E) (L : filter E) :
has_fderiv_at_filter id (id 𝕜 E) x L :=
(is_o_zero _ _).congr_left $ by simp
theorem has_fderiv_within_at_id (x : E) (s : set E) :
has_fderiv_within_at id (id 𝕜 E) s x :=
has_fderiv_at_filter_id _ _
theorem has_fderiv_at_id (x : E) : has_fderiv_at id (id 𝕜 E) x :=
has_fderiv_at_filter_id _ _
@[simp] lemma differentiable_at_id : differentiable_at 𝕜 id x :=
(has_fderiv_at_id x).differentiable_at
@[simp] lemma differentiable_at_id' : differentiable_at 𝕜 (λ x, x) x :=
(has_fderiv_at_id x).differentiable_at
lemma differentiable_within_at_id : differentiable_within_at 𝕜 id s x :=
differentiable_at_id.differentiable_within_at
@[simp] lemma differentiable_id : differentiable 𝕜 (id : E → E) :=
λx, differentiable_at_id
@[simp] lemma differentiable_id' : differentiable 𝕜 (λ (x : E), x) :=
λx, differentiable_at_id
lemma differentiable_on_id : differentiable_on 𝕜 id s :=
differentiable_id.differentiable_on
lemma fderiv_id : fderiv 𝕜 id x = id 𝕜 E :=
has_fderiv_at.fderiv (has_fderiv_at_id x)
@[simp] lemma fderiv_id' : fderiv 𝕜 (λ (x : E), x) x = continuous_linear_map.id 𝕜 E :=
fderiv_id
lemma fderiv_within_id (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 id s x = id 𝕜 E :=
begin
rw differentiable_at.fderiv_within (differentiable_at_id) hxs,
exact fderiv_id
end
lemma fderiv_within_id' (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λ (x : E), x) s x = continuous_linear_map.id 𝕜 E :=
fderiv_within_id hxs
end id
section const
/-! ### derivative of a constant function -/
theorem has_strict_fderiv_at_const (c : F) (x : E) :
has_strict_fderiv_at (λ _, c) (0 : E →L[𝕜] F) x :=
(is_o_zero _ _).congr_left $ λ _, by simp only [zero_apply, sub_self]
theorem has_fderiv_at_filter_const (c : F) (x : E) (L : filter E) :
has_fderiv_at_filter (λ x, c) (0 : E →L[𝕜] F) x L :=
(is_o_zero _ _).congr_left $ λ _, by simp only [zero_apply, sub_self]
theorem has_fderiv_within_at_const (c : F) (x : E) (s : set E) :
has_fderiv_within_at (λ x, c) (0 : E →L[𝕜] F) s x :=
has_fderiv_at_filter_const _ _ _
theorem has_fderiv_at_const (c : F) (x : E) :
has_fderiv_at (λ x, c) (0 : E →L[𝕜] F) x :=
has_fderiv_at_filter_const _ _ _
@[simp] lemma differentiable_at_const (c : F) : differentiable_at 𝕜 (λx, c) x :=
⟨0, has_fderiv_at_const c x⟩
lemma differentiable_within_at_const (c : F) : differentiable_within_at 𝕜 (λx, c) s x :=
differentiable_at.differentiable_within_at (differentiable_at_const _)
lemma fderiv_const_apply (c : F) : fderiv 𝕜 (λy, c) x = 0 :=
has_fderiv_at.fderiv (has_fderiv_at_const c x)
@[simp] lemma fderiv_const (c : F) : fderiv 𝕜 (λ (y : E), c) = 0 :=
by { ext m, rw fderiv_const_apply, refl }
lemma fderiv_within_const_apply (c : F) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λy, c) s x = 0 :=
begin
rw differentiable_at.fderiv_within (differentiable_at_const _) hxs,
exact fderiv_const_apply _
end
@[simp] lemma differentiable_const (c : F) : differentiable 𝕜 (λx : E, c) :=
λx, differentiable_at_const _
lemma differentiable_on_const (c : F) : differentiable_on 𝕜 (λx, c) s :=
(differentiable_const _).differentiable_on
lemma has_fderiv_within_at_singleton (f : E → F) (x : E) :
has_fderiv_within_at f (0 : E →L[𝕜] F) {x} x :=
by simp only [has_fderiv_within_at, nhds_within_singleton, has_fderiv_at_filter, is_o_pure,
continuous_linear_map.zero_apply, sub_self]
lemma has_fderiv_at_of_subsingleton [h : subsingleton E] (f : E → F) (x : E) :
has_fderiv_at f (0 : E →L[𝕜] F) x :=
begin
rw [← has_fderiv_within_at_univ, subsingleton_univ.eq_singleton_of_mem (mem_univ x)],
exact has_fderiv_within_at_singleton f x
end
lemma differentiable_on_empty : differentiable_on 𝕜 f ∅ := λ x, false.elim
lemma differentiable_on_singleton : differentiable_on 𝕜 f {x} :=
forall_eq.2 (has_fderiv_within_at_singleton f x).differentiable_within_at
lemma set.subsingleton.differentiable_on (hs : s.subsingleton) : differentiable_on 𝕜 f s :=
hs.induction_on differentiable_on_empty (λ x, differentiable_on_singleton)
lemma has_fderiv_at_zero_of_eventually_const
(c : F) (hf : f =ᶠ[𝓝 x] (λ y, c)) :
has_fderiv_at f (0 : E →L[𝕜] F) x :=
(has_fderiv_at_const _ _).congr_of_eventually_eq hf
end const
section continuous_linear_map
/-!
### Continuous linear maps
There are currently two variants of these in mathlib, the bundled version
(named `continuous_linear_map`, and denoted `E →L[𝕜] F`), and the unbundled version (with a
predicate `is_bounded_linear_map`). We give statements for both versions. -/
protected theorem continuous_linear_map.has_strict_fderiv_at {x : E} :
has_strict_fderiv_at e e x :=
(is_o_zero _ _).congr_left $ λ x, by simp only [e.map_sub, sub_self]
protected lemma continuous_linear_map.has_fderiv_at_filter :
has_fderiv_at_filter e e x L :=
(is_o_zero _ _).congr_left $ λ x, by simp only [e.map_sub, sub_self]
protected lemma continuous_linear_map.has_fderiv_within_at : has_fderiv_within_at e e s x :=
e.has_fderiv_at_filter
protected lemma continuous_linear_map.has_fderiv_at : has_fderiv_at e e x :=
e.has_fderiv_at_filter
@[simp] protected lemma continuous_linear_map.differentiable_at : differentiable_at 𝕜 e x :=
e.has_fderiv_at.differentiable_at
protected lemma continuous_linear_map.differentiable_within_at : differentiable_within_at 𝕜 e s x :=
e.differentiable_at.differentiable_within_at
@[simp] protected lemma continuous_linear_map.fderiv : fderiv 𝕜 e x = e :=
e.has_fderiv_at.fderiv
protected lemma continuous_linear_map.fderiv_within (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 e s x = e :=
begin
rw differentiable_at.fderiv_within e.differentiable_at hxs,
exact e.fderiv
end
@[simp] protected lemma continuous_linear_map.differentiable : differentiable 𝕜 e :=
λx, e.differentiable_at
protected lemma continuous_linear_map.differentiable_on : differentiable_on 𝕜 e s :=
e.differentiable.differentiable_on
lemma is_bounded_linear_map.has_fderiv_at_filter (h : is_bounded_linear_map 𝕜 f) :
has_fderiv_at_filter f h.to_continuous_linear_map x L :=
h.to_continuous_linear_map.has_fderiv_at_filter
lemma is_bounded_linear_map.has_fderiv_within_at (h : is_bounded_linear_map 𝕜 f) :
has_fderiv_within_at f h.to_continuous_linear_map s x :=
h.has_fderiv_at_filter
lemma is_bounded_linear_map.has_fderiv_at (h : is_bounded_linear_map 𝕜 f) :
has_fderiv_at f h.to_continuous_linear_map x :=
h.has_fderiv_at_filter
lemma is_bounded_linear_map.differentiable_at (h : is_bounded_linear_map 𝕜 f) :
differentiable_at 𝕜 f x :=
h.has_fderiv_at.differentiable_at
lemma is_bounded_linear_map.differentiable_within_at (h : is_bounded_linear_map 𝕜 f) :
differentiable_within_at 𝕜 f s x :=
h.differentiable_at.differentiable_within_at
lemma is_bounded_linear_map.fderiv (h : is_bounded_linear_map 𝕜 f) :
fderiv 𝕜 f x = h.to_continuous_linear_map :=
has_fderiv_at.fderiv (h.has_fderiv_at)
lemma is_bounded_linear_map.fderiv_within (h : is_bounded_linear_map 𝕜 f)
(hxs : unique_diff_within_at 𝕜 s x) : fderiv_within 𝕜 f s x = h.to_continuous_linear_map :=
begin
rw differentiable_at.fderiv_within h.differentiable_at hxs,
exact h.fderiv
end
lemma is_bounded_linear_map.differentiable (h : is_bounded_linear_map 𝕜 f) :
differentiable 𝕜 f :=
λx, h.differentiable_at
lemma is_bounded_linear_map.differentiable_on (h : is_bounded_linear_map 𝕜 f) :
differentiable_on 𝕜 f s :=
h.differentiable.differentiable_on
end continuous_linear_map
section composition
/-!
### Derivative of the composition of two functions
For composition lemmas, we put x explicit to help the elaborator, as otherwise Lean tends to
get confused since there are too many possibilities for composition -/
variable (x)
theorem has_fderiv_at_filter.comp {g : F → G} {g' : F →L[𝕜] G} {L' : filter F}
(hg : has_fderiv_at_filter g g' (f x) L')
(hf : has_fderiv_at_filter f f' x L) (hL : tendsto f L L') :
has_fderiv_at_filter (g ∘ f) (g'.comp f') x L :=
let eq₁ := (g'.is_O_comp _ _).trans_is_o hf in
let eq₂ := (hg.comp_tendsto hL).trans_is_O hf.is_O_sub in
by { refine eq₂.triangle (eq₁.congr_left (λ x', _)), simp }
/- A readable version of the previous theorem,
a general form of the chain rule. -/
example {g : F → G} {g' : F →L[𝕜] G}
(hg : has_fderiv_at_filter g g' (f x) (L.map f))
(hf : has_fderiv_at_filter f f' x L) :
has_fderiv_at_filter (g ∘ f) (g'.comp f') x L :=
begin
unfold has_fderiv_at_filter at hg,
have := calc (λ x', g (f x') - g (f x) - g' (f x' - f x)) =o[L] (λ x', f x' - f x) :
hg.comp_tendsto le_rfl
... =O[L] (λ x', x' - x) : hf.is_O_sub,
refine this.triangle _,
calc (λ x' : E, g' (f x' - f x) - g'.comp f' (x' - x))
=ᶠ[L] λ x', g' (f x' - f x - f' (x' - x)) : eventually_of_forall (λ x', by simp)
... =O[L] λ x', f x' - f x - f' (x' - x) : g'.is_O_comp _ _
... =o[L] λ x', x' - x : hf
end
theorem has_fderiv_within_at.comp {g : F → G} {g' : F →L[𝕜] G} {t : set F}
(hg : has_fderiv_within_at g g' t (f x)) (hf : has_fderiv_within_at f f' s x)
(hst : maps_to f s t) :
has_fderiv_within_at (g ∘ f) (g'.comp f') s x :=
hg.comp x hf $ hf.continuous_within_at.tendsto_nhds_within hst
theorem has_fderiv_at.comp_has_fderiv_within_at {g : F → G} {g' : F →L[𝕜] G}
(hg : has_fderiv_at g g' (f x)) (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (g ∘ f) (g'.comp f') s x :=
hg.comp x hf hf.continuous_within_at
theorem has_fderiv_within_at.comp_of_mem {g : F → G} {g' : F →L[𝕜] G} {t : set F}
(hg : has_fderiv_within_at g g' t (f x)) (hf : has_fderiv_within_at f f' s x)
(hst : tendsto f (𝓝[s] x) (𝓝[t] f x)) :
has_fderiv_within_at (g ∘ f) (g'.comp f') s x :=
has_fderiv_at_filter.comp x hg hf hst
/-- The chain rule. -/
theorem has_fderiv_at.comp {g : F → G} {g' : F →L[𝕜] G}
(hg : has_fderiv_at g g' (f x)) (hf : has_fderiv_at f f' x) :
has_fderiv_at (g ∘ f) (g'.comp f') x :=
hg.comp x hf hf.continuous_at
lemma differentiable_within_at.comp {g : F → G} {t : set F}
(hg : differentiable_within_at 𝕜 g t (f x)) (hf : differentiable_within_at 𝕜 f s x)
(h : maps_to f s t) : differentiable_within_at 𝕜 (g ∘ f) s x :=
(hg.has_fderiv_within_at.comp x hf.has_fderiv_within_at h).differentiable_within_at
lemma differentiable_within_at.comp' {g : F → G} {t : set F}
(hg : differentiable_within_at 𝕜 g t (f x)) (hf : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (g ∘ f) (s ∩ f⁻¹' t) x :=
hg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
lemma differentiable_at.comp {g : F → G}
(hg : differentiable_at 𝕜 g (f x)) (hf : differentiable_at 𝕜 f x) :
differentiable_at 𝕜 (g ∘ f) x :=
(hg.has_fderiv_at.comp x hf.has_fderiv_at).differentiable_at
lemma differentiable_at.comp_differentiable_within_at {g : F → G}
(hg : differentiable_at 𝕜 g (f x)) (hf : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (g ∘ f) s x :=
hg.differentiable_within_at.comp x hf (maps_to_univ _ _)
lemma fderiv_within.comp {g : F → G} {t : set F}
(hg : differentiable_within_at 𝕜 g t (f x)) (hf : differentiable_within_at 𝕜 f s x)
(h : maps_to f s t) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (g ∘ f) s x = (fderiv_within 𝕜 g t (f x)).comp (fderiv_within 𝕜 f s x) :=
(hg.has_fderiv_within_at.comp x (hf.has_fderiv_within_at) h).fderiv_within hxs
/-- Ternary version of `fderiv_within.comp`, with equality assumptions of basepoints added, in
order to apply more easily as a rewrite from right-to-left. -/
lemma fderiv_within.comp₃ {g' : G → G'} {g : F → G} {t : set F} {u : set G} {y : F} {y' : G}
(hg' : differentiable_within_at 𝕜 g' u y') (hg : differentiable_within_at 𝕜 g t y)
(hf : differentiable_within_at 𝕜 f s x)
(h2g : maps_to g t u) (h2f : maps_to f s t)
(h3g : g y = y') (h3f : f x = y) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (g' ∘ g ∘ f) s x = (fderiv_within 𝕜 g' u y').comp
((fderiv_within 𝕜 g t y).comp (fderiv_within 𝕜 f s x)) :=
begin
substs h3g h3f,
exact (hg'.has_fderiv_within_at.comp x
(hg.has_fderiv_within_at.comp x (hf.has_fderiv_within_at) h2f) $ h2g.comp h2f).fderiv_within hxs
end
lemma fderiv.comp {g : F → G}
(hg : differentiable_at 𝕜 g (f x)) (hf : differentiable_at 𝕜 f x) :
fderiv 𝕜 (g ∘ f) x = (fderiv 𝕜 g (f x)).comp (fderiv 𝕜 f x) :=
(hg.has_fderiv_at.comp x hf.has_fderiv_at).fderiv
lemma fderiv.comp_fderiv_within {g : F → G}
(hg : differentiable_at 𝕜 g (f x)) (hf : differentiable_within_at 𝕜 f s x)
(hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (g ∘ f) s x = (fderiv 𝕜 g (f x)).comp (fderiv_within 𝕜 f s x) :=
(hg.has_fderiv_at.comp_has_fderiv_within_at x hf.has_fderiv_within_at).fderiv_within hxs
lemma differentiable_on.comp {g : F → G} {t : set F}
(hg : differentiable_on 𝕜 g t) (hf : differentiable_on 𝕜 f s) (st : maps_to f s t) :
differentiable_on 𝕜 (g ∘ f) s :=
λx hx, differentiable_within_at.comp x (hg (f x) (st hx)) (hf x hx) st
lemma differentiable.comp {g : F → G} (hg : differentiable 𝕜 g) (hf : differentiable 𝕜 f) :
differentiable 𝕜 (g ∘ f) :=
λx, differentiable_at.comp x (hg (f x)) (hf x)
lemma differentiable.comp_differentiable_on {g : F → G} (hg : differentiable 𝕜 g)
(hf : differentiable_on 𝕜 f s) :
differentiable_on 𝕜 (g ∘ f) s :=
hg.differentiable_on.comp hf (maps_to_univ _ _)
/-- The chain rule for derivatives in the sense of strict differentiability. -/
protected lemma has_strict_fderiv_at.comp {g : F → G} {g' : F →L[𝕜] G}
(hg : has_strict_fderiv_at g g' (f x)) (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, g (f x)) (g'.comp f') x :=
((hg.comp_tendsto (hf.continuous_at.prod_map' hf.continuous_at)).trans_is_O hf.is_O_sub).triangle $
by simpa only [g'.map_sub, f'.coe_comp'] using (g'.is_O_comp _ _).trans_is_o hf
protected lemma differentiable.iterate {f : E → E} (hf : differentiable 𝕜 f) (n : ℕ) :
differentiable 𝕜 (f^[n]) :=
nat.rec_on n differentiable_id (λ n ihn, ihn.comp hf)
protected lemma differentiable_on.iterate {f : E → E} (hf : differentiable_on 𝕜 f s)
(hs : maps_to f s s) (n : ℕ) :
differentiable_on 𝕜 (f^[n]) s :=
nat.rec_on n differentiable_on_id (λ n ihn, ihn.comp hf hs)
variable {x}
protected lemma has_fderiv_at_filter.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : has_fderiv_at_filter f f' x L) (hL : tendsto f L L) (hx : f x = x) (n : ℕ) :
has_fderiv_at_filter (f^[n]) (f'^n) x L :=
begin
induction n with n ihn,
{ exact has_fderiv_at_filter_id x L },
{ rw [function.iterate_succ, pow_succ'],
rw ← hx at ihn,
exact ihn.comp x hf hL }
end
protected lemma has_fderiv_at.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : has_fderiv_at f f' x) (hx : f x = x) (n : ℕ) :
has_fderiv_at (f^[n]) (f'^n) x :=
begin
refine hf.iterate _ hx n,
convert hf.continuous_at,
exact hx.symm
end
protected lemma has_fderiv_within_at.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : has_fderiv_within_at f f' s x) (hx : f x = x) (hs : maps_to f s s) (n : ℕ) :
has_fderiv_within_at (f^[n]) (f'^n) s x :=
begin
refine hf.iterate _ hx n,
convert tendsto_inf.2 ⟨hf.continuous_within_at, _⟩,
exacts [hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]
end
protected lemma has_strict_fderiv_at.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : has_strict_fderiv_at f f' x) (hx : f x = x) (n : ℕ) :
has_strict_fderiv_at (f^[n]) (f'^n) x :=
begin
induction n with n ihn,
{ exact has_strict_fderiv_at_id x },
{ rw [function.iterate_succ, pow_succ'],
rw ← hx at ihn,
exact ihn.comp x hf }
end
protected lemma differentiable_at.iterate {f : E → E} (hf : differentiable_at 𝕜 f x)
(hx : f x = x) (n : ℕ) :
differentiable_at 𝕜 (f^[n]) x :=
(hf.has_fderiv_at.iterate hx n).differentiable_at
protected lemma differentiable_within_at.iterate {f : E → E} (hf : differentiable_within_at 𝕜 f s x)
(hx : f x = x) (hs : maps_to f s s) (n : ℕ) :
differentiable_within_at 𝕜 (f^[n]) s x :=
(hf.has_fderiv_within_at.iterate hx hs n).differentiable_within_at
end composition
section cartesian_product
/-! ### Derivative of the cartesian product of two functions -/
section prod
variables {f₂ : E → G} {f₂' : E →L[𝕜] G}
protected lemma has_strict_fderiv_at.prod
(hf₁ : has_strict_fderiv_at f₁ f₁' x) (hf₂ : has_strict_fderiv_at f₂ f₂' x) :
has_strict_fderiv_at (λx, (f₁ x, f₂ x)) (f₁'.prod f₂') x :=
hf₁.prod_left hf₂
lemma has_fderiv_at_filter.prod
(hf₁ : has_fderiv_at_filter f₁ f₁' x L) (hf₂ : has_fderiv_at_filter f₂ f₂' x L) :
has_fderiv_at_filter (λx, (f₁ x, f₂ x)) (f₁'.prod f₂') x L :=
hf₁.prod_left hf₂
lemma has_fderiv_within_at.prod
(hf₁ : has_fderiv_within_at f₁ f₁' s x) (hf₂ : has_fderiv_within_at f₂ f₂' s x) :
has_fderiv_within_at (λx, (f₁ x, f₂ x)) (f₁'.prod f₂') s x :=
hf₁.prod hf₂
lemma has_fderiv_at.prod (hf₁ : has_fderiv_at f₁ f₁' x) (hf₂ : has_fderiv_at f₂ f₂' x) :
has_fderiv_at (λx, (f₁ x, f₂ x)) (f₁'.prod f₂') x :=
hf₁.prod hf₂
lemma has_fderiv_at_prod_mk_left (e₀ : E) (f₀ : F) :
has_fderiv_at (λ e : E, (e, f₀)) (inl 𝕜 E F) e₀ :=
(has_fderiv_at_id e₀).prod (has_fderiv_at_const f₀ e₀)
lemma has_fderiv_at_prod_mk_right (e₀ : E) (f₀ : F) :
has_fderiv_at (λ f : F, (e₀, f)) (inr 𝕜 E F) f₀ :=
(has_fderiv_at_const e₀ f₀).prod (has_fderiv_at_id f₀)
lemma differentiable_within_at.prod
(hf₁ : differentiable_within_at 𝕜 f₁ s x) (hf₂ : differentiable_within_at 𝕜 f₂ s x) :
differentiable_within_at 𝕜 (λx:E, (f₁ x, f₂ x)) s x :=
(hf₁.has_fderiv_within_at.prod hf₂.has_fderiv_within_at).differentiable_within_at
@[simp]
lemma differentiable_at.prod (hf₁ : differentiable_at 𝕜 f₁ x) (hf₂ : differentiable_at 𝕜 f₂ x) :
differentiable_at 𝕜 (λx:E, (f₁ x, f₂ x)) x :=
(hf₁.has_fderiv_at.prod hf₂.has_fderiv_at).differentiable_at
lemma differentiable_on.prod (hf₁ : differentiable_on 𝕜 f₁ s) (hf₂ : differentiable_on 𝕜 f₂ s) :
differentiable_on 𝕜 (λx:E, (f₁ x, f₂ x)) s :=
λx hx, differentiable_within_at.prod (hf₁ x hx) (hf₂ x hx)
@[simp]
lemma differentiable.prod (hf₁ : differentiable 𝕜 f₁) (hf₂ : differentiable 𝕜 f₂) :
differentiable 𝕜 (λx:E, (f₁ x, f₂ x)) :=
λ x, differentiable_at.prod (hf₁ x) (hf₂ x)
lemma differentiable_at.fderiv_prod
(hf₁ : differentiable_at 𝕜 f₁ x) (hf₂ : differentiable_at 𝕜 f₂ x) :
fderiv 𝕜 (λx:E, (f₁ x, f₂ x)) x = (fderiv 𝕜 f₁ x).prod (fderiv 𝕜 f₂ x) :=
(hf₁.has_fderiv_at.prod hf₂.has_fderiv_at).fderiv
lemma differentiable_at.fderiv_within_prod
(hf₁ : differentiable_within_at 𝕜 f₁ s x) (hf₂ : differentiable_within_at 𝕜 f₂ s x)
(hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λx:E, (f₁ x, f₂ x)) s x =
(fderiv_within 𝕜 f₁ s x).prod (fderiv_within 𝕜 f₂ s x) :=
(hf₁.has_fderiv_within_at.prod hf₂.has_fderiv_within_at).fderiv_within hxs
end prod
section fst
variables {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} {p : E × F}
lemma has_strict_fderiv_at_fst : has_strict_fderiv_at (@prod.fst E F) (fst 𝕜 E F) p :=
(fst 𝕜 E F).has_strict_fderiv_at
protected lemma has_strict_fderiv_at.fst (h : has_strict_fderiv_at f₂ f₂' x) :
has_strict_fderiv_at (λ x, (f₂ x).1) ((fst 𝕜 F G).comp f₂') x :=
has_strict_fderiv_at_fst.comp x h
lemma has_fderiv_at_filter_fst {L : filter (E × F)} :
has_fderiv_at_filter (@prod.fst E F) (fst 𝕜 E F) p L :=
(fst 𝕜 E F).has_fderiv_at_filter
protected lemma has_fderiv_at_filter.fst (h : has_fderiv_at_filter f₂ f₂' x L) :
has_fderiv_at_filter (λ x, (f₂ x).1) ((fst 𝕜 F G).comp f₂') x L :=
has_fderiv_at_filter_fst.comp x h tendsto_map
lemma has_fderiv_at_fst : has_fderiv_at (@prod.fst E F) (fst 𝕜 E F) p :=
has_fderiv_at_filter_fst
protected lemma has_fderiv_at.fst (h : has_fderiv_at f₂ f₂' x) :
has_fderiv_at (λ x, (f₂ x).1) ((fst 𝕜 F G).comp f₂') x :=
h.fst
lemma has_fderiv_within_at_fst {s : set (E × F)} :
has_fderiv_within_at (@prod.fst E F) (fst 𝕜 E F) s p :=
has_fderiv_at_filter_fst
protected lemma has_fderiv_within_at.fst (h : has_fderiv_within_at f₂ f₂' s x) :
has_fderiv_within_at (λ x, (f₂ x).1) ((fst 𝕜 F G).comp f₂') s x :=
h.fst
lemma differentiable_at_fst : differentiable_at 𝕜 prod.fst p :=
has_fderiv_at_fst.differentiable_at
@[simp] protected lemma differentiable_at.fst (h : differentiable_at 𝕜 f₂ x) :
differentiable_at 𝕜 (λ x, (f₂ x).1) x :=
differentiable_at_fst.comp x h
lemma differentiable_fst : differentiable 𝕜 (prod.fst : E × F → E) :=
λ x, differentiable_at_fst
@[simp] protected lemma differentiable.fst (h : differentiable 𝕜 f₂) :
differentiable 𝕜 (λ x, (f₂ x).1) :=
differentiable_fst.comp h
lemma differentiable_within_at_fst {s : set (E × F)} : differentiable_within_at 𝕜 prod.fst s p :=
differentiable_at_fst.differentiable_within_at
protected lemma differentiable_within_at.fst (h : differentiable_within_at 𝕜 f₂ s x) :
differentiable_within_at 𝕜 (λ x, (f₂ x).1) s x :=
differentiable_at_fst.comp_differentiable_within_at x h
lemma differentiable_on_fst {s : set (E × F)} : differentiable_on 𝕜 prod.fst s :=
differentiable_fst.differentiable_on
protected lemma differentiable_on.fst (h : differentiable_on 𝕜 f₂ s) :
differentiable_on 𝕜 (λ x, (f₂ x).1) s :=
differentiable_fst.comp_differentiable_on h
lemma fderiv_fst : fderiv 𝕜 prod.fst p = fst 𝕜 E F := has_fderiv_at_fst.fderiv
lemma fderiv.fst (h : differentiable_at 𝕜 f₂ x) :
fderiv 𝕜 (λ x, (f₂ x).1) x = (fst 𝕜 F G).comp (fderiv 𝕜 f₂ x) :=
h.has_fderiv_at.fst.fderiv
lemma fderiv_within_fst {s : set (E × F)} (hs : unique_diff_within_at 𝕜 s p) :
fderiv_within 𝕜 prod.fst s p = fst 𝕜 E F :=
has_fderiv_within_at_fst.fderiv_within hs
lemma fderiv_within.fst (hs : unique_diff_within_at 𝕜 s x) (h : differentiable_within_at 𝕜 f₂ s x) :
fderiv_within 𝕜 (λ x, (f₂ x).1) s x = (fst 𝕜 F G).comp (fderiv_within 𝕜 f₂ s x) :=
h.has_fderiv_within_at.fst.fderiv_within hs
end fst
section snd
variables {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} {p : E × F}
lemma has_strict_fderiv_at_snd : has_strict_fderiv_at (@prod.snd E F) (snd 𝕜 E F) p :=
(snd 𝕜 E F).has_strict_fderiv_at
protected lemma has_strict_fderiv_at.snd (h : has_strict_fderiv_at f₂ f₂' x) :
has_strict_fderiv_at (λ x, (f₂ x).2) ((snd 𝕜 F G).comp f₂') x :=
has_strict_fderiv_at_snd.comp x h
lemma has_fderiv_at_filter_snd {L : filter (E × F)} :
has_fderiv_at_filter (@prod.snd E F) (snd 𝕜 E F) p L :=
(snd 𝕜 E F).has_fderiv_at_filter
protected lemma has_fderiv_at_filter.snd (h : has_fderiv_at_filter f₂ f₂' x L) :
has_fderiv_at_filter (λ x, (f₂ x).2) ((snd 𝕜 F G).comp f₂') x L :=
has_fderiv_at_filter_snd.comp x h tendsto_map
lemma has_fderiv_at_snd : has_fderiv_at (@prod.snd E F) (snd 𝕜 E F) p :=
has_fderiv_at_filter_snd
protected lemma has_fderiv_at.snd (h : has_fderiv_at f₂ f₂' x) :
has_fderiv_at (λ x, (f₂ x).2) ((snd 𝕜 F G).comp f₂') x :=
h.snd
lemma has_fderiv_within_at_snd {s : set (E × F)} :
has_fderiv_within_at (@prod.snd E F) (snd 𝕜 E F) s p :=
has_fderiv_at_filter_snd
protected lemma has_fderiv_within_at.snd (h : has_fderiv_within_at f₂ f₂' s x) :
has_fderiv_within_at (λ x, (f₂ x).2) ((snd 𝕜 F G).comp f₂') s x :=
h.snd
lemma differentiable_at_snd : differentiable_at 𝕜 prod.snd p :=
has_fderiv_at_snd.differentiable_at
@[simp] protected lemma differentiable_at.snd (h : differentiable_at 𝕜 f₂ x) :
differentiable_at 𝕜 (λ x, (f₂ x).2) x :=
differentiable_at_snd.comp x h
lemma differentiable_snd : differentiable 𝕜 (prod.snd : E × F → F) :=
λ x, differentiable_at_snd
@[simp] protected lemma differentiable.snd (h : differentiable 𝕜 f₂) :
differentiable 𝕜 (λ x, (f₂ x).2) :=
differentiable_snd.comp h
lemma differentiable_within_at_snd {s : set (E × F)} : differentiable_within_at 𝕜 prod.snd s p :=
differentiable_at_snd.differentiable_within_at
protected lemma differentiable_within_at.snd (h : differentiable_within_at 𝕜 f₂ s x) :
differentiable_within_at 𝕜 (λ x, (f₂ x).2) s x :=
differentiable_at_snd.comp_differentiable_within_at x h
lemma differentiable_on_snd {s : set (E × F)} : differentiable_on 𝕜 prod.snd s :=
differentiable_snd.differentiable_on
protected lemma differentiable_on.snd (h : differentiable_on 𝕜 f₂ s) :
differentiable_on 𝕜 (λ x, (f₂ x).2) s :=
differentiable_snd.comp_differentiable_on h
lemma fderiv_snd : fderiv 𝕜 prod.snd p = snd 𝕜 E F := has_fderiv_at_snd.fderiv
lemma fderiv.snd (h : differentiable_at 𝕜 f₂ x) :
fderiv 𝕜 (λ x, (f₂ x).2) x = (snd 𝕜 F G).comp (fderiv 𝕜 f₂ x) :=
h.has_fderiv_at.snd.fderiv
lemma fderiv_within_snd {s : set (E × F)} (hs : unique_diff_within_at 𝕜 s p) :
fderiv_within 𝕜 prod.snd s p = snd 𝕜 E F :=
has_fderiv_within_at_snd.fderiv_within hs
lemma fderiv_within.snd (hs : unique_diff_within_at 𝕜 s x) (h : differentiable_within_at 𝕜 f₂ s x) :
fderiv_within 𝕜 (λ x, (f₂ x).2) s x = (snd 𝕜 F G).comp (fderiv_within 𝕜 f₂ s x) :=
h.has_fderiv_within_at.snd.fderiv_within hs
end snd
section prod_map
variables {f₂ : G → G'} {f₂' : G →L[𝕜] G'} {y : G} (p : E × G)
protected theorem has_strict_fderiv_at.prod_map (hf : has_strict_fderiv_at f f' p.1)
(hf₂ : has_strict_fderiv_at f₂ f₂' p.2) :
has_strict_fderiv_at (prod.map f f₂) (f'.prod_map f₂') p :=
(hf.comp p has_strict_fderiv_at_fst).prod (hf₂.comp p has_strict_fderiv_at_snd)
protected theorem has_fderiv_at.prod_map (hf : has_fderiv_at f f' p.1)
(hf₂ : has_fderiv_at f₂ f₂' p.2) :
has_fderiv_at (prod.map f f₂) (f'.prod_map f₂') p :=
(hf.comp p has_fderiv_at_fst).prod (hf₂.comp p has_fderiv_at_snd)
@[simp] protected theorem differentiable_at.prod_map (hf : differentiable_at 𝕜 f p.1)
(hf₂ : differentiable_at 𝕜 f₂ p.2) :
differentiable_at 𝕜 (λ p : E × G, (f p.1, f₂ p.2)) p :=
(hf.comp p differentiable_at_fst).prod (hf₂.comp p differentiable_at_snd)
end prod_map
end cartesian_product
section const_smul
variables {R : Type*} [semiring R] [module R F] [smul_comm_class 𝕜 R F]
[has_continuous_const_smul R F]
/-! ### Derivative of a function multiplied by a constant -/
theorem has_strict_fderiv_at.const_smul (h : has_strict_fderiv_at f f' x) (c : R) :
has_strict_fderiv_at (λ x, c • f x) (c • f') x :=
(c • (1 : F →L[𝕜] F)).has_strict_fderiv_at.comp x h
theorem has_fderiv_at_filter.const_smul (h : has_fderiv_at_filter f f' x L) (c : R) :
has_fderiv_at_filter (λ x, c • f x) (c • f') x L :=
(c • (1 : F →L[𝕜] F)).has_fderiv_at_filter.comp x h tendsto_map
theorem has_fderiv_within_at.const_smul (h : has_fderiv_within_at f f' s x) (c : R) :
has_fderiv_within_at (λ x, c • f x) (c • f') s x :=
h.const_smul c
theorem has_fderiv_at.const_smul (h : has_fderiv_at f f' x) (c : R) :
has_fderiv_at (λ x, c • f x) (c • f') x :=
h.const_smul c
lemma differentiable_within_at.const_smul (h : differentiable_within_at 𝕜 f s x) (c : R) :
differentiable_within_at 𝕜 (λy, c • f y) s x :=
(h.has_fderiv_within_at.const_smul c).differentiable_within_at
lemma differentiable_at.const_smul (h : differentiable_at 𝕜 f x) (c : R) :
differentiable_at 𝕜 (λy, c • f y) x :=
(h.has_fderiv_at.const_smul c).differentiable_at
lemma differentiable_on.const_smul (h : differentiable_on 𝕜 f s) (c : R) :
differentiable_on 𝕜 (λy, c • f y) s :=
λx hx, (h x hx).const_smul c
lemma differentiable.const_smul (h : differentiable 𝕜 f) (c : R) :
differentiable 𝕜 (λy, c • f y) :=
λx, (h x).const_smul c
lemma fderiv_within_const_smul (hxs : unique_diff_within_at 𝕜 s x)
(h : differentiable_within_at 𝕜 f s x) (c : R) :
fderiv_within 𝕜 (λy, c • f y) s x = c • fderiv_within 𝕜 f s x :=
(h.has_fderiv_within_at.const_smul c).fderiv_within hxs
lemma fderiv_const_smul (h : differentiable_at 𝕜 f x) (c : R) :
fderiv 𝕜 (λy, c • f y) x = c • fderiv 𝕜 f x :=
(h.has_fderiv_at.const_smul c).fderiv
end const_smul
section add
/-! ### Derivative of the sum of two functions -/
theorem has_strict_fderiv_at.add (hf : has_strict_fderiv_at f f' x)
(hg : has_strict_fderiv_at g g' x) :
has_strict_fderiv_at (λ y, f y + g y) (f' + g') x :=
(hf.add hg).congr_left $ λ y,
by { simp only [linear_map.sub_apply, linear_map.add_apply, map_sub, map_add, add_apply], abel }
theorem has_fderiv_at_filter.add
(hf : has_fderiv_at_filter f f' x L) (hg : has_fderiv_at_filter g g' x L) :
has_fderiv_at_filter (λ y, f y + g y) (f' + g') x L :=
(hf.add hg).congr_left $ λ _,
by { simp only [linear_map.sub_apply, linear_map.add_apply, map_sub, map_add, add_apply], abel }
theorem has_fderiv_within_at.add
(hf : has_fderiv_within_at f f' s x) (hg : has_fderiv_within_at g g' s x) :
has_fderiv_within_at (λ y, f y + g y) (f' + g') s x :=
hf.add hg
theorem has_fderiv_at.add
(hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) :
has_fderiv_at (λ x, f x + g x) (f' + g') x :=
hf.add hg
lemma differentiable_within_at.add
(hf : differentiable_within_at 𝕜 f s x) (hg : differentiable_within_at 𝕜 g s x) :
differentiable_within_at 𝕜 (λ y, f y + g y) s x :=
(hf.has_fderiv_within_at.add hg.has_fderiv_within_at).differentiable_within_at
@[simp] lemma differentiable_at.add
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
differentiable_at 𝕜 (λ y, f y + g y) x :=
(hf.has_fderiv_at.add hg.has_fderiv_at).differentiable_at
lemma differentiable_on.add
(hf : differentiable_on 𝕜 f s) (hg : differentiable_on 𝕜 g s) :
differentiable_on 𝕜 (λy, f y + g y) s :=
λx hx, (hf x hx).add (hg x hx)
@[simp] lemma differentiable.add
(hf : differentiable 𝕜 f) (hg : differentiable 𝕜 g) :
differentiable 𝕜 (λy, f y + g y) :=
λx, (hf x).add (hg x)
lemma fderiv_within_add (hxs : unique_diff_within_at 𝕜 s x)
(hf : differentiable_within_at 𝕜 f s x) (hg : differentiable_within_at 𝕜 g s x) :
fderiv_within 𝕜 (λy, f y + g y) s x = fderiv_within 𝕜 f s x + fderiv_within 𝕜 g s x :=
(hf.has_fderiv_within_at.add hg.has_fderiv_within_at).fderiv_within hxs
lemma fderiv_add
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
fderiv 𝕜 (λy, f y + g y) x = fderiv 𝕜 f x + fderiv 𝕜 g x :=
(hf.has_fderiv_at.add hg.has_fderiv_at).fderiv
theorem has_strict_fderiv_at.add_const (hf : has_strict_fderiv_at f f' x) (c : F) :
has_strict_fderiv_at (λ y, f y + c) f' x :=
add_zero f' ▸ hf.add (has_strict_fderiv_at_const _ _)
theorem has_fderiv_at_filter.add_const
(hf : has_fderiv_at_filter f f' x L) (c : F) :
has_fderiv_at_filter (λ y, f y + c) f' x L :=
add_zero f' ▸ hf.add (has_fderiv_at_filter_const _ _ _)
theorem has_fderiv_within_at.add_const
(hf : has_fderiv_within_at f f' s x) (c : F) :
has_fderiv_within_at (λ y, f y + c) f' s x :=
hf.add_const c
theorem has_fderiv_at.add_const (hf : has_fderiv_at f f' x) (c : F):
has_fderiv_at (λ x, f x + c) f' x :=
hf.add_const c
lemma differentiable_within_at.add_const
(hf : differentiable_within_at 𝕜 f s x) (c : F) :
differentiable_within_at 𝕜 (λ y, f y + c) s x :=
(hf.has_fderiv_within_at.add_const c).differentiable_within_at
@[simp] lemma differentiable_within_at_add_const_iff (c : F) :
differentiable_within_at 𝕜 (λ y, f y + c) s x ↔ differentiable_within_at 𝕜 f s x :=
⟨λ h, by simpa using h.add_const (-c), λ h, h.add_const c⟩
lemma differentiable_at.add_const
(hf : differentiable_at 𝕜 f x) (c : F) :
differentiable_at 𝕜 (λ y, f y + c) x :=
(hf.has_fderiv_at.add_const c).differentiable_at
@[simp] lemma differentiable_at_add_const_iff (c : F) :
differentiable_at 𝕜 (λ y, f y + c) x ↔ differentiable_at 𝕜 f x :=
⟨λ h, by simpa using h.add_const (-c), λ h, h.add_const c⟩
lemma differentiable_on.add_const
(hf : differentiable_on 𝕜 f s) (c : F) :
differentiable_on 𝕜 (λy, f y + c) s :=
λx hx, (hf x hx).add_const c
@[simp] lemma differentiable_on_add_const_iff (c : F) :
differentiable_on 𝕜 (λ y, f y + c) s ↔ differentiable_on 𝕜 f s :=
⟨λ h, by simpa using h.add_const (-c), λ h, h.add_const c⟩
lemma differentiable.add_const
(hf : differentiable 𝕜 f) (c : F) :
differentiable 𝕜 (λy, f y + c) :=
λx, (hf x).add_const c
@[simp] lemma differentiable_add_const_iff (c : F) :
differentiable 𝕜 (λ y, f y + c) ↔ differentiable 𝕜 f :=
⟨λ h, by simpa using h.add_const (-c), λ h, h.add_const c⟩
lemma fderiv_within_add_const (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
fderiv_within 𝕜 (λy, f y + c) s x = fderiv_within 𝕜 f s x :=
if hf : differentiable_within_at 𝕜 f s x
then (hf.has_fderiv_within_at.add_const c).fderiv_within hxs
else by { rw [fderiv_within_zero_of_not_differentiable_within_at hf,
fderiv_within_zero_of_not_differentiable_within_at], simpa }
lemma fderiv_add_const (c : F) : fderiv 𝕜 (λy, f y + c) x = fderiv 𝕜 f x :=
by simp only [← fderiv_within_univ, fderiv_within_add_const unique_diff_within_at_univ]
theorem has_strict_fderiv_at.const_add (hf : has_strict_fderiv_at f f' x) (c : F) :
has_strict_fderiv_at (λ y, c + f y) f' x :=
zero_add f' ▸ (has_strict_fderiv_at_const _ _).add hf
theorem has_fderiv_at_filter.const_add
(hf : has_fderiv_at_filter f f' x L) (c : F) :
has_fderiv_at_filter (λ y, c + f y) f' x L :=
zero_add f' ▸ (has_fderiv_at_filter_const _ _ _).add hf
theorem has_fderiv_within_at.const_add
(hf : has_fderiv_within_at f f' s x) (c : F) :
has_fderiv_within_at (λ y, c + f y) f' s x :=
hf.const_add c
theorem has_fderiv_at.const_add
(hf : has_fderiv_at f f' x) (c : F):
has_fderiv_at (λ x, c + f x) f' x :=
hf.const_add c
lemma differentiable_within_at.const_add
(hf : differentiable_within_at 𝕜 f s x) (c : F) :
differentiable_within_at 𝕜 (λ y, c + f y) s x :=
(hf.has_fderiv_within_at.const_add c).differentiable_within_at
@[simp] lemma differentiable_within_at_const_add_iff (c : F) :
differentiable_within_at 𝕜 (λ y, c + f y) s x ↔ differentiable_within_at 𝕜 f s x :=
⟨λ h, by simpa using h.const_add (-c), λ h, h.const_add c⟩
lemma differentiable_at.const_add
(hf : differentiable_at 𝕜 f x) (c : F) :
differentiable_at 𝕜 (λ y, c + f y) x :=
(hf.has_fderiv_at.const_add c).differentiable_at
@[simp] lemma differentiable_at_const_add_iff (c : F) :
differentiable_at 𝕜 (λ y, c + f y) x ↔ differentiable_at 𝕜 f x :=
⟨λ h, by simpa using h.const_add (-c), λ h, h.const_add c⟩
lemma differentiable_on.const_add (hf : differentiable_on 𝕜 f s) (c : F) :
differentiable_on 𝕜 (λy, c + f y) s :=
λx hx, (hf x hx).const_add c
@[simp] lemma differentiable_on_const_add_iff (c : F) :
differentiable_on 𝕜 (λ y, c + f y) s ↔ differentiable_on 𝕜 f s :=
⟨λ h, by simpa using h.const_add (-c), λ h, h.const_add c⟩
lemma differentiable.const_add (hf : differentiable 𝕜 f) (c : F) :
differentiable 𝕜 (λy, c + f y) :=
λx, (hf x).const_add c
@[simp] lemma differentiable_const_add_iff (c : F) :
differentiable 𝕜 (λ y, c + f y) ↔ differentiable 𝕜 f :=
⟨λ h, by simpa using h.const_add (-c), λ h, h.const_add c⟩
lemma fderiv_within_const_add (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
fderiv_within 𝕜 (λy, c + f y) s x = fderiv_within 𝕜 f s x :=
by simpa only [add_comm] using fderiv_within_add_const hxs c
lemma fderiv_const_add (c : F) : fderiv 𝕜 (λy, c + f y) x = fderiv 𝕜 f x :=
by simp only [add_comm c, fderiv_add_const]
end add
section sum
/-! ### Derivative of a finite sum of functions -/
open_locale big_operators
variables {ι : Type*} {u : finset ι} {A : ι → (E → F)} {A' : ι → (E →L[𝕜] F)}
theorem has_strict_fderiv_at.sum (h : ∀ i ∈ u, has_strict_fderiv_at (A i) (A' i) x) :
has_strict_fderiv_at (λ y, ∑ i in u, A i y) (∑ i in u, A' i) x :=
begin
dsimp [has_strict_fderiv_at] at *,
convert is_o.sum h,
simp [finset.sum_sub_distrib, continuous_linear_map.sum_apply]
end
theorem has_fderiv_at_filter.sum (h : ∀ i ∈ u, has_fderiv_at_filter (A i) (A' i) x L) :
has_fderiv_at_filter (λ y, ∑ i in u, A i y) (∑ i in u, A' i) x L :=
begin
dsimp [has_fderiv_at_filter] at *,
convert is_o.sum h,
simp [continuous_linear_map.sum_apply]
end
theorem has_fderiv_within_at.sum (h : ∀ i ∈ u, has_fderiv_within_at (A i) (A' i) s x) :
has_fderiv_within_at (λ y, ∑ i in u, A i y) (∑ i in u, A' i) s x :=
has_fderiv_at_filter.sum h
theorem has_fderiv_at.sum (h : ∀ i ∈ u, has_fderiv_at (A i) (A' i) x) :
has_fderiv_at (λ y, ∑ i in u, A i y) (∑ i in u, A' i) x :=
has_fderiv_at_filter.sum h
theorem differentiable_within_at.sum (h : ∀ i ∈ u, differentiable_within_at 𝕜 (A i) s x) :
differentiable_within_at 𝕜 (λ y, ∑ i in u, A i y) s x :=
has_fderiv_within_at.differentiable_within_at $ has_fderiv_within_at.sum $
λ i hi, (h i hi).has_fderiv_within_at
@[simp] theorem differentiable_at.sum (h : ∀ i ∈ u, differentiable_at 𝕜 (A i) x) :
differentiable_at 𝕜 (λ y, ∑ i in u, A i y) x :=
has_fderiv_at.differentiable_at $ has_fderiv_at.sum $ λ i hi, (h i hi).has_fderiv_at
theorem differentiable_on.sum (h : ∀ i ∈ u, differentiable_on 𝕜 (A i) s) :
differentiable_on 𝕜 (λ y, ∑ i in u, A i y) s :=
λ x hx, differentiable_within_at.sum $ λ i hi, h i hi x hx
@[simp] theorem differentiable.sum (h : ∀ i ∈ u, differentiable 𝕜 (A i)) :
differentiable 𝕜 (λ y, ∑ i in u, A i y) :=
λ x, differentiable_at.sum $ λ i hi, h i hi x
theorem fderiv_within_sum (hxs : unique_diff_within_at 𝕜 s x)
(h : ∀ i ∈ u, differentiable_within_at 𝕜 (A i) s x) :
fderiv_within 𝕜 (λ y, ∑ i in u, A i y) s x = (∑ i in u, fderiv_within 𝕜 (A i) s x) :=
(has_fderiv_within_at.sum (λ i hi, (h i hi).has_fderiv_within_at)).fderiv_within hxs
theorem fderiv_sum (h : ∀ i ∈ u, differentiable_at 𝕜 (A i) x) :
fderiv 𝕜 (λ y, ∑ i in u, A i y) x = (∑ i in u, fderiv 𝕜 (A i) x) :=
(has_fderiv_at.sum (λ i hi, (h i hi).has_fderiv_at)).fderiv
end sum
section pi
/-!
### Derivatives of functions `f : E → Π i, F' i`
In this section we formulate `has_*fderiv*_pi` theorems as `iff`s, and provide two versions of each
theorem:
* the version without `'` deals with `φ : Π i, E → F' i` and `φ' : Π i, E →L[𝕜] F' i`
and is designed to deduce differentiability of `λ x i, φ i x` from differentiability
of each `φ i`;
* the version with `'` deals with `Φ : E → Π i, F' i` and `Φ' : E →L[𝕜] Π i, F' i`
and is designed to deduce differentiability of the components `λ x, Φ x i` from
differentiability of `Φ`.
-/
variables {ι : Type*} [fintype ι] {F' : ι → Type*} [Π i, normed_add_comm_group (F' i)]
[Π i, normed_space 𝕜 (F' i)] {φ : Π i, E → F' i} {φ' : Π i, E →L[𝕜] F' i}
{Φ : E → Π i, F' i} {Φ' : E →L[𝕜] Π i, F' i}
@[simp] lemma has_strict_fderiv_at_pi' :
has_strict_fderiv_at Φ Φ' x ↔
∀ i, has_strict_fderiv_at (λ x, Φ x i) ((proj i).comp Φ') x :=
begin
simp only [has_strict_fderiv_at, continuous_linear_map.coe_pi],
exact is_o_pi
end
@[simp] lemma has_strict_fderiv_at_pi :
has_strict_fderiv_at (λ x i, φ i x) (continuous_linear_map.pi φ') x ↔
∀ i, has_strict_fderiv_at (φ i) (φ' i) x :=
has_strict_fderiv_at_pi'
@[simp] lemma has_fderiv_at_filter_pi' :
has_fderiv_at_filter Φ Φ' x L ↔
∀ i, has_fderiv_at_filter (λ x, Φ x i) ((proj i).comp Φ') x L :=
begin
simp only [has_fderiv_at_filter, continuous_linear_map.coe_pi],
exact is_o_pi
end
lemma has_fderiv_at_filter_pi :
has_fderiv_at_filter (λ x i, φ i x) (continuous_linear_map.pi φ') x L ↔
∀ i, has_fderiv_at_filter (φ i) (φ' i) x L :=
has_fderiv_at_filter_pi'
@[simp] lemma has_fderiv_at_pi' :
has_fderiv_at Φ Φ' x ↔
∀ i, has_fderiv_at (λ x, Φ x i) ((proj i).comp Φ') x :=
has_fderiv_at_filter_pi'
lemma has_fderiv_at_pi :
has_fderiv_at (λ x i, φ i x) (continuous_linear_map.pi φ') x ↔
∀ i, has_fderiv_at (φ i) (φ' i) x :=
has_fderiv_at_filter_pi
@[simp] lemma has_fderiv_within_at_pi' :
has_fderiv_within_at Φ Φ' s x ↔
∀ i, has_fderiv_within_at (λ x, Φ x i) ((proj i).comp Φ') s x :=
has_fderiv_at_filter_pi'
lemma has_fderiv_within_at_pi :
has_fderiv_within_at (λ x i, φ i x) (continuous_linear_map.pi φ') s x ↔
∀ i, has_fderiv_within_at (φ i) (φ' i) s x :=
has_fderiv_at_filter_pi
@[simp] lemma differentiable_within_at_pi :
differentiable_within_at 𝕜 Φ s x ↔
∀ i, differentiable_within_at 𝕜 (λ x, Φ x i) s x :=
⟨λ h i, (has_fderiv_within_at_pi'.1 h.has_fderiv_within_at i).differentiable_within_at,
λ h, (has_fderiv_within_at_pi.2 (λ i, (h i).has_fderiv_within_at)).differentiable_within_at⟩
@[simp] lemma differentiable_at_pi :
differentiable_at 𝕜 Φ x ↔ ∀ i, differentiable_at 𝕜 (λ x, Φ x i) x :=
⟨λ h i, (has_fderiv_at_pi'.1 h.has_fderiv_at i).differentiable_at,
λ h, (has_fderiv_at_pi.2 (λ i, (h i).has_fderiv_at)).differentiable_at⟩
lemma differentiable_on_pi :
differentiable_on 𝕜 Φ s ↔ ∀ i, differentiable_on 𝕜 (λ x, Φ x i) s :=
⟨λ h i x hx, differentiable_within_at_pi.1 (h x hx) i,
λ h x hx, differentiable_within_at_pi.2 (λ i, h i x hx)⟩
lemma differentiable_pi :
differentiable 𝕜 Φ ↔ ∀ i, differentiable 𝕜 (λ x, Φ x i) :=
⟨λ h i x, differentiable_at_pi.1 (h x) i, λ h x, differentiable_at_pi.2 (λ i, h i x)⟩
-- TODO: find out which version (`φ` or `Φ`) works better with `rw`/`simp`
lemma fderiv_within_pi (h : ∀ i, differentiable_within_at 𝕜 (φ i) s x)
(hs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λ x i, φ i x) s x = pi (λ i, fderiv_within 𝕜 (φ i) s x) :=
(has_fderiv_within_at_pi.2 (λ i, (h i).has_fderiv_within_at)).fderiv_within hs
lemma fderiv_pi (h : ∀ i, differentiable_at 𝕜 (φ i) x) :
fderiv 𝕜 (λ x i, φ i x) x = pi (λ i, fderiv 𝕜 (φ i) x) :=
(has_fderiv_at_pi.2 (λ i, (h i).has_fderiv_at)).fderiv
end pi
section neg
/-! ### Derivative of the negative of a function -/
theorem has_strict_fderiv_at.neg (h : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, -f x) (-f') x :=
(-1 : F →L[𝕜] F).has_strict_fderiv_at.comp x h
theorem has_fderiv_at_filter.neg (h : has_fderiv_at_filter f f' x L) :
has_fderiv_at_filter (λ x, -f x) (-f') x L :=
(-1 : F →L[𝕜] F).has_fderiv_at_filter.comp x h tendsto_map
theorem has_fderiv_within_at.neg (h : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, -f x) (-f') s x :=
h.neg
theorem has_fderiv_at.neg (h : has_fderiv_at f f' x) :
has_fderiv_at (λ x, -f x) (-f') x :=
h.neg
lemma differentiable_within_at.neg (h : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (λy, -f y) s x :=
h.has_fderiv_within_at.neg.differentiable_within_at
@[simp] lemma differentiable_within_at_neg_iff :
differentiable_within_at 𝕜 (λy, -f y) s x ↔ differentiable_within_at 𝕜 f s x :=
⟨λ h, by simpa only [neg_neg] using h.neg, λ h, h.neg⟩
lemma differentiable_at.neg (h : differentiable_at 𝕜 f x) :
differentiable_at 𝕜 (λy, -f y) x :=
h.has_fderiv_at.neg.differentiable_at
@[simp] lemma differentiable_at_neg_iff :
differentiable_at 𝕜 (λy, -f y) x ↔ differentiable_at 𝕜 f x :=
⟨λ h, by simpa only [neg_neg] using h.neg, λ h, h.neg⟩
lemma differentiable_on.neg (h : differentiable_on 𝕜 f s) :
differentiable_on 𝕜 (λy, -f y) s :=
λx hx, (h x hx).neg
@[simp] lemma differentiable_on_neg_iff :
differentiable_on 𝕜 (λy, -f y) s ↔ differentiable_on 𝕜 f s :=
⟨λ h, by simpa only [neg_neg] using h.neg, λ h, h.neg⟩
lemma differentiable.neg (h : differentiable 𝕜 f) :
differentiable 𝕜 (λy, -f y) :=
λx, (h x).neg
@[simp] lemma differentiable_neg_iff : differentiable 𝕜 (λy, -f y) ↔ differentiable 𝕜 f :=
⟨λ h, by simpa only [neg_neg] using h.neg, λ h, h.neg⟩
lemma fderiv_within_neg (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λy, -f y) s x = - fderiv_within 𝕜 f s x :=
if h : differentiable_within_at 𝕜 f s x
then h.has_fderiv_within_at.neg.fderiv_within hxs
else by { rw [fderiv_within_zero_of_not_differentiable_within_at h,
fderiv_within_zero_of_not_differentiable_within_at, neg_zero], simpa }
@[simp] lemma fderiv_neg : fderiv 𝕜 (λy, -f y) x = - fderiv 𝕜 f x :=
by simp only [← fderiv_within_univ, fderiv_within_neg unique_diff_within_at_univ]
end neg
section sub
/-! ### Derivative of the difference of two functions -/
theorem has_strict_fderiv_at.sub
(hf : has_strict_fderiv_at f f' x) (hg : has_strict_fderiv_at g g' x) :
has_strict_fderiv_at (λ x, f x - g x) (f' - g') x :=
by simpa only [sub_eq_add_neg] using hf.add hg.neg
theorem has_fderiv_at_filter.sub
(hf : has_fderiv_at_filter f f' x L) (hg : has_fderiv_at_filter g g' x L) :
has_fderiv_at_filter (λ x, f x - g x) (f' - g') x L :=
by simpa only [sub_eq_add_neg] using hf.add hg.neg
theorem has_fderiv_within_at.sub
(hf : has_fderiv_within_at f f' s x) (hg : has_fderiv_within_at g g' s x) :
has_fderiv_within_at (λ x, f x - g x) (f' - g') s x :=
hf.sub hg
theorem has_fderiv_at.sub
(hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) :
has_fderiv_at (λ x, f x - g x) (f' - g') x :=
hf.sub hg
lemma differentiable_within_at.sub
(hf : differentiable_within_at 𝕜 f s x) (hg : differentiable_within_at 𝕜 g s x) :
differentiable_within_at 𝕜 (λ y, f y - g y) s x :=
(hf.has_fderiv_within_at.sub hg.has_fderiv_within_at).differentiable_within_at
@[simp] lemma differentiable_at.sub
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
differentiable_at 𝕜 (λ y, f y - g y) x :=
(hf.has_fderiv_at.sub hg.has_fderiv_at).differentiable_at
lemma differentiable_on.sub
(hf : differentiable_on 𝕜 f s) (hg : differentiable_on 𝕜 g s) :
differentiable_on 𝕜 (λy, f y - g y) s :=
λx hx, (hf x hx).sub (hg x hx)
@[simp] lemma differentiable.sub
(hf : differentiable 𝕜 f) (hg : differentiable 𝕜 g) :
differentiable 𝕜 (λy, f y - g y) :=
λx, (hf x).sub (hg x)
lemma fderiv_within_sub (hxs : unique_diff_within_at 𝕜 s x)
(hf : differentiable_within_at 𝕜 f s x) (hg : differentiable_within_at 𝕜 g s x) :
fderiv_within 𝕜 (λy, f y - g y) s x = fderiv_within 𝕜 f s x - fderiv_within 𝕜 g s x :=
(hf.has_fderiv_within_at.sub hg.has_fderiv_within_at).fderiv_within hxs
lemma fderiv_sub
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
fderiv 𝕜 (λy, f y - g y) x = fderiv 𝕜 f x - fderiv 𝕜 g x :=
(hf.has_fderiv_at.sub hg.has_fderiv_at).fderiv
theorem has_strict_fderiv_at.sub_const
(hf : has_strict_fderiv_at f f' x) (c : F) :
has_strict_fderiv_at (λ x, f x - c) f' x :=
by simpa only [sub_eq_add_neg] using hf.add_const (-c)
theorem has_fderiv_at_filter.sub_const
(hf : has_fderiv_at_filter f f' x L) (c : F) :
has_fderiv_at_filter (λ x, f x - c) f' x L :=
by simpa only [sub_eq_add_neg] using hf.add_const (-c)
theorem has_fderiv_within_at.sub_const
(hf : has_fderiv_within_at f f' s x) (c : F) :
has_fderiv_within_at (λ x, f x - c) f' s x :=
hf.sub_const c
theorem has_fderiv_at.sub_const
(hf : has_fderiv_at f f' x) (c : F) :
has_fderiv_at (λ x, f x - c) f' x :=
hf.sub_const c
lemma differentiable_within_at.sub_const
(hf : differentiable_within_at 𝕜 f s x) (c : F) :
differentiable_within_at 𝕜 (λ y, f y - c) s x :=
(hf.has_fderiv_within_at.sub_const c).differentiable_within_at
@[simp] lemma differentiable_within_at_sub_const_iff (c : F) :
differentiable_within_at 𝕜 (λ y, f y - c) s x ↔ differentiable_within_at 𝕜 f s x :=
by simp only [sub_eq_add_neg, differentiable_within_at_add_const_iff]
lemma differentiable_at.sub_const (hf : differentiable_at 𝕜 f x) (c : F) :
differentiable_at 𝕜 (λ y, f y - c) x :=
(hf.has_fderiv_at.sub_const c).differentiable_at
@[simp] lemma differentiable_at_sub_const_iff (c : F) :
differentiable_at 𝕜 (λ y, f y - c) x ↔ differentiable_at 𝕜 f x :=
by simp only [sub_eq_add_neg, differentiable_at_add_const_iff]
lemma differentiable_on.sub_const (hf : differentiable_on 𝕜 f s) (c : F) :
differentiable_on 𝕜 (λy, f y - c) s :=
λx hx, (hf x hx).sub_const c
@[simp] lemma differentiable_on_sub_const_iff (c : F) :
differentiable_on 𝕜 (λ y, f y - c) s ↔ differentiable_on 𝕜 f s :=
by simp only [sub_eq_add_neg, differentiable_on_add_const_iff]
lemma differentiable.sub_const (hf : differentiable 𝕜 f) (c : F) :
differentiable 𝕜 (λy, f y - c) :=
λx, (hf x).sub_const c
@[simp] lemma differentiable_sub_const_iff (c : F) :
differentiable 𝕜 (λ y, f y - c) ↔ differentiable 𝕜 f :=
by simp only [sub_eq_add_neg, differentiable_add_const_iff]
lemma fderiv_within_sub_const (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
fderiv_within 𝕜 (λy, f y - c) s x = fderiv_within 𝕜 f s x :=
by simp only [sub_eq_add_neg, fderiv_within_add_const hxs]
lemma fderiv_sub_const (c : F) : fderiv 𝕜 (λy, f y - c) x = fderiv 𝕜 f x :=
by simp only [sub_eq_add_neg, fderiv_add_const]
theorem has_strict_fderiv_at.const_sub
(hf : has_strict_fderiv_at f f' x) (c : F) :
has_strict_fderiv_at (λ x, c - f x) (-f') x :=
by simpa only [sub_eq_add_neg] using hf.neg.const_add c
theorem has_fderiv_at_filter.const_sub
(hf : has_fderiv_at_filter f f' x L) (c : F) :
has_fderiv_at_filter (λ x, c - f x) (-f') x L :=
by simpa only [sub_eq_add_neg] using hf.neg.const_add c
theorem has_fderiv_within_at.const_sub
(hf : has_fderiv_within_at f f' s x) (c : F) :
has_fderiv_within_at (λ x, c - f x) (-f') s x :=
hf.const_sub c
theorem has_fderiv_at.const_sub
(hf : has_fderiv_at f f' x) (c : F) :
has_fderiv_at (λ x, c - f x) (-f') x :=
hf.const_sub c
lemma differentiable_within_at.const_sub
(hf : differentiable_within_at 𝕜 f s x) (c : F) :
differentiable_within_at 𝕜 (λ y, c - f y) s x :=
(hf.has_fderiv_within_at.const_sub c).differentiable_within_at
@[simp] lemma differentiable_within_at_const_sub_iff (c : F) :
differentiable_within_at 𝕜 (λ y, c - f y) s x ↔ differentiable_within_at 𝕜 f s x :=
by simp [sub_eq_add_neg]
lemma differentiable_at.const_sub
(hf : differentiable_at 𝕜 f x) (c : F) :
differentiable_at 𝕜 (λ y, c - f y) x :=
(hf.has_fderiv_at.const_sub c).differentiable_at
@[simp] lemma differentiable_at_const_sub_iff (c : F) :
differentiable_at 𝕜 (λ y, c - f y) x ↔ differentiable_at 𝕜 f x :=
by simp [sub_eq_add_neg]
lemma differentiable_on.const_sub (hf : differentiable_on 𝕜 f s) (c : F) :
differentiable_on 𝕜 (λy, c - f y) s :=
λx hx, (hf x hx).const_sub c
@[simp] lemma differentiable_on_const_sub_iff (c : F) :
differentiable_on 𝕜 (λ y, c - f y) s ↔ differentiable_on 𝕜 f s :=
by simp [sub_eq_add_neg]
lemma differentiable.const_sub (hf : differentiable 𝕜 f) (c : F) :
differentiable 𝕜 (λy, c - f y) :=
λx, (hf x).const_sub c
@[simp] lemma differentiable_const_sub_iff (c : F) :
differentiable 𝕜 (λ y, c - f y) ↔ differentiable 𝕜 f :=
by simp [sub_eq_add_neg]
lemma fderiv_within_const_sub (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
fderiv_within 𝕜 (λy, c - f y) s x = -fderiv_within 𝕜 f s x :=
by simp only [sub_eq_add_neg, fderiv_within_const_add, fderiv_within_neg, hxs]
lemma fderiv_const_sub (c : F) : fderiv 𝕜 (λy, c - f y) x = -fderiv 𝕜 f x :=
by simp only [← fderiv_within_univ, fderiv_within_const_sub unique_diff_within_at_univ]
end sub
section bilinear_map
/-! ### Derivative of a bounded bilinear map -/
variables {b : E × F → G} {u : set (E × F) }
open normed_field
lemma is_bounded_bilinear_map.has_strict_fderiv_at (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
has_strict_fderiv_at b (h.deriv p) p :=
begin
rw has_strict_fderiv_at,
set T := (E × F) × (E × F),
have : (λ q : T, b (q.1 - q.2)) =o[𝓝 (p, p)] (λ q : T, ‖q.1 - q.2‖ * 1),
{ refine (h.is_O'.comp_tendsto le_top).trans_is_o _,
simp only [(∘)],
refine (is_O_refl (λ q : T, ‖q.1 - q.2‖) _).mul_is_o (is_o.norm_left $ (is_o_one_iff _).2 _),
rw [← sub_self p],
exact continuous_at_fst.sub continuous_at_snd },
simp only [mul_one, is_o_norm_right] at this,
refine (is_o.congr_of_sub _).1 this, clear this,
convert_to (λ q : T, h.deriv (p - q.2) (q.1 - q.2)) =o[𝓝 (p, p)] (λ q : T, q.1 - q.2),
{ ext ⟨⟨x₁, y₁⟩, ⟨x₂, y₂⟩⟩, rcases p with ⟨x, y⟩,
simp only [is_bounded_bilinear_map_deriv_coe, prod.mk_sub_mk, h.map_sub_left, h.map_sub_right],
abel },
have : (λ q : T, p - q.2) =o[𝓝 (p, p)] (λ q, (1:ℝ)),
from (is_o_one_iff _).2 (sub_self p ▸ tendsto_const_nhds.sub continuous_at_snd),
apply is_bounded_bilinear_map_apply.is_O_comp.trans_is_o,
refine is_o.trans_is_O _ (is_O_const_mul_self 1 _ _).of_norm_right,
refine is_o.mul_is_O _ (is_O_refl _ _),
exact (((h.is_bounded_linear_map_deriv.is_O_id ⊤).comp_tendsto le_top : _).trans_is_o
this).norm_left
end
lemma is_bounded_bilinear_map.has_fderiv_at (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
has_fderiv_at b (h.deriv p) p :=
(h.has_strict_fderiv_at p).has_fderiv_at
lemma is_bounded_bilinear_map.has_fderiv_within_at (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
has_fderiv_within_at b (h.deriv p) u p :=
(h.has_fderiv_at p).has_fderiv_within_at
lemma is_bounded_bilinear_map.differentiable_at (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
differentiable_at 𝕜 b p :=
(h.has_fderiv_at p).differentiable_at
lemma is_bounded_bilinear_map.differentiable_within_at (h : is_bounded_bilinear_map 𝕜 b)
(p : E × F) :
differentiable_within_at 𝕜 b u p :=
(h.differentiable_at p).differentiable_within_at
lemma is_bounded_bilinear_map.fderiv (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
fderiv 𝕜 b p = h.deriv p :=
has_fderiv_at.fderiv (h.has_fderiv_at p)
lemma is_bounded_bilinear_map.fderiv_within (h : is_bounded_bilinear_map 𝕜 b) (p : E × F)
(hxs : unique_diff_within_at 𝕜 u p) : fderiv_within 𝕜 b u p = h.deriv p :=
begin
rw differentiable_at.fderiv_within (h.differentiable_at p) hxs,
exact h.fderiv p
end
lemma is_bounded_bilinear_map.differentiable (h : is_bounded_bilinear_map 𝕜 b) :
differentiable 𝕜 b :=
λx, h.differentiable_at x
lemma is_bounded_bilinear_map.differentiable_on (h : is_bounded_bilinear_map 𝕜 b) :
differentiable_on 𝕜 b u :=
h.differentiable.differentiable_on
end bilinear_map
section clm_comp_apply
/-! ### Derivative of the pointwise composition/application of continuous linear maps -/
variables {H : Type*} [normed_add_comm_group H] [normed_space 𝕜 H] {c : E → G →L[𝕜] H}
{c' : E →L[𝕜] G →L[𝕜] H} {d : E → F →L[𝕜] G} {d' : E →L[𝕜] F →L[𝕜] G} {u : E → G}
{u' : E →L[𝕜] G}
lemma has_strict_fderiv_at.clm_comp (hc : has_strict_fderiv_at c c' x)
(hd : has_strict_fderiv_at d d' x) : has_strict_fderiv_at (λ y, (c y).comp (d y))
((compL 𝕜 F G H (c x)).comp d' + ((compL 𝕜 F G H).flip (d x)).comp c') x :=
(is_bounded_bilinear_map_comp.has_strict_fderiv_at (c x, d x)).comp x $ hc.prod hd
lemma has_fderiv_within_at.clm_comp (hc : has_fderiv_within_at c c' s x)
(hd : has_fderiv_within_at d d' s x) : has_fderiv_within_at (λ y, (c y).comp (d y))
((compL 𝕜 F G H (c x)).comp d' + ((compL 𝕜 F G H).flip (d x)).comp c') s x :=
(is_bounded_bilinear_map_comp.has_fderiv_at (c x, d x)).comp_has_fderiv_within_at x $ hc.prod hd
lemma has_fderiv_at.clm_comp (hc : has_fderiv_at c c' x)
(hd : has_fderiv_at d d' x) : has_fderiv_at (λ y, (c y).comp (d y))
((compL 𝕜 F G H (c x)).comp d' + ((compL 𝕜 F G H).flip (d x)).comp c') x :=
(is_bounded_bilinear_map_comp.has_fderiv_at (c x, d x)).comp x $ hc.prod hd
lemma differentiable_within_at.clm_comp
(hc : differentiable_within_at 𝕜 c s x) (hd : differentiable_within_at 𝕜 d s x) :
differentiable_within_at 𝕜 (λ y, (c y).comp (d y)) s x :=
(hc.has_fderiv_within_at.clm_comp hd.has_fderiv_within_at).differentiable_within_at
lemma differentiable_at.clm_comp (hc : differentiable_at 𝕜 c x)
(hd : differentiable_at 𝕜 d x) : differentiable_at 𝕜 (λ y, (c y).comp (d y)) x :=
(hc.has_fderiv_at.clm_comp hd.has_fderiv_at).differentiable_at
lemma differentiable_on.clm_comp (hc : differentiable_on 𝕜 c s) (hd : differentiable_on 𝕜 d s) :
differentiable_on 𝕜 (λ y, (c y).comp (d y)) s :=
λx hx, (hc x hx).clm_comp (hd x hx)
lemma differentiable.clm_comp (hc : differentiable 𝕜 c) (hd : differentiable 𝕜 d) :
differentiable 𝕜 (λ y, (c y).comp (d y)) :=
λx, (hc x).clm_comp (hd x)
lemma fderiv_within_clm_comp (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hd : differentiable_within_at 𝕜 d s x) :
fderiv_within 𝕜 (λ y, (c y).comp (d y)) s x =
(compL 𝕜 F G H (c x)).comp (fderiv_within 𝕜 d s x) +
((compL 𝕜 F G H).flip (d x)).comp (fderiv_within 𝕜 c s x) :=
(hc.has_fderiv_within_at.clm_comp hd.has_fderiv_within_at).fderiv_within hxs
lemma fderiv_clm_comp (hc : differentiable_at 𝕜 c x) (hd : differentiable_at 𝕜 d x) :
fderiv 𝕜 (λ y, (c y).comp (d y)) x =
(compL 𝕜 F G H (c x)).comp (fderiv 𝕜 d x) +
((compL 𝕜 F G H).flip (d x)).comp (fderiv 𝕜 c x) :=
(hc.has_fderiv_at.clm_comp hd.has_fderiv_at).fderiv
lemma has_strict_fderiv_at.clm_apply (hc : has_strict_fderiv_at c c' x)
(hu : has_strict_fderiv_at u u' x) :
has_strict_fderiv_at (λ y, (c y) (u y)) ((c x).comp u' + c'.flip (u x)) x :=
(is_bounded_bilinear_map_apply.has_strict_fderiv_at (c x, u x)).comp x (hc.prod hu)
lemma has_fderiv_within_at.clm_apply (hc : has_fderiv_within_at c c' s x)
(hu : has_fderiv_within_at u u' s x) :
has_fderiv_within_at (λ y, (c y) (u y)) ((c x).comp u' + c'.flip (u x)) s x :=
(is_bounded_bilinear_map_apply.has_fderiv_at (c x, u x)).comp_has_fderiv_within_at x (hc.prod hu)
lemma has_fderiv_at.clm_apply (hc : has_fderiv_at c c' x) (hu : has_fderiv_at u u' x) :
has_fderiv_at (λ y, (c y) (u y)) ((c x).comp u' + c'.flip (u x)) x :=
(is_bounded_bilinear_map_apply.has_fderiv_at (c x, u x)).comp x (hc.prod hu)
lemma differentiable_within_at.clm_apply
(hc : differentiable_within_at 𝕜 c s x) (hu : differentiable_within_at 𝕜 u s x) :
differentiable_within_at 𝕜 (λ y, (c y) (u y)) s x :=
(hc.has_fderiv_within_at.clm_apply hu.has_fderiv_within_at).differentiable_within_at
lemma differentiable_at.clm_apply (hc : differentiable_at 𝕜 c x)
(hu : differentiable_at 𝕜 u x) : differentiable_at 𝕜 (λ y, (c y) (u y)) x :=
(hc.has_fderiv_at.clm_apply hu.has_fderiv_at).differentiable_at
lemma differentiable_on.clm_apply (hc : differentiable_on 𝕜 c s) (hu : differentiable_on 𝕜 u s) :
differentiable_on 𝕜 (λ y, (c y) (u y)) s :=
λx hx, (hc x hx).clm_apply (hu x hx)
lemma differentiable.clm_apply (hc : differentiable 𝕜 c) (hu : differentiable 𝕜 u) :
differentiable 𝕜 (λ y, (c y) (u y)) :=
λx, (hc x).clm_apply (hu x)
lemma fderiv_within_clm_apply (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hu : differentiable_within_at 𝕜 u s x) :
fderiv_within 𝕜 (λ y, (c y) (u y)) s x =
((c x).comp (fderiv_within 𝕜 u s x) + (fderiv_within 𝕜 c s x).flip (u x)) :=
(hc.has_fderiv_within_at.clm_apply hu.has_fderiv_within_at).fderiv_within hxs
lemma fderiv_clm_apply (hc : differentiable_at 𝕜 c x) (hu : differentiable_at 𝕜 u x) :
fderiv 𝕜 (λ y, (c y) (u y)) x = ((c x).comp (fderiv 𝕜 u x) + (fderiv 𝕜 c x).flip (u x)) :=
(hc.has_fderiv_at.clm_apply hu.has_fderiv_at).fderiv
end clm_comp_apply
section smul
/-! ### Derivative of the product of a scalar-valued function and a vector-valued function
If `c` is a differentiable scalar-valued function and `f` is a differentiable vector-valued
function, then `λ x, c x • f x` is differentiable as well. Lemmas in this section works for
function `c` taking values in the base field, as well as in a normed algebra over the base
field: e.g., they work for `c : E → ℂ` and `f : E → F` provided that `F` is a complex
normed vector space.
-/
variables {𝕜' : Type*} [nontrivially_normed_field 𝕜'] [normed_algebra 𝕜 𝕜']
[normed_space 𝕜' F] [is_scalar_tower 𝕜 𝕜' F]
variables {c : E → 𝕜'} {c' : E →L[𝕜] 𝕜'}
theorem has_strict_fderiv_at.smul (hc : has_strict_fderiv_at c c' x)
(hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ y, c y • f y) (c x • f' + c'.smul_right (f x)) x :=
(is_bounded_bilinear_map_smul.has_strict_fderiv_at (c x, f x)).comp x $
hc.prod hf
theorem has_fderiv_within_at.smul
(hc : has_fderiv_within_at c c' s x) (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ y, c y • f y) (c x • f' + c'.smul_right (f x)) s x :=
(is_bounded_bilinear_map_smul.has_fderiv_at (c x, f x)).comp_has_fderiv_within_at x $
hc.prod hf
theorem has_fderiv_at.smul (hc : has_fderiv_at c c' x) (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ y, c y • f y) (c x • f' + c'.smul_right (f x)) x :=
(is_bounded_bilinear_map_smul.has_fderiv_at (c x, f x)).comp x $
hc.prod hf
lemma differentiable_within_at.smul
(hc : differentiable_within_at 𝕜 c s x) (hf : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (λ y, c y • f y) s x :=
(hc.has_fderiv_within_at.smul hf.has_fderiv_within_at).differentiable_within_at
@[simp] lemma differentiable_at.smul (hc : differentiable_at 𝕜 c x) (hf : differentiable_at 𝕜 f x) :
differentiable_at 𝕜 (λ y, c y • f y) x :=
(hc.has_fderiv_at.smul hf.has_fderiv_at).differentiable_at
lemma differentiable_on.smul (hc : differentiable_on 𝕜 c s) (hf : differentiable_on 𝕜 f s) :
differentiable_on 𝕜 (λ y, c y • f y) s :=
λx hx, (hc x hx).smul (hf x hx)
@[simp] lemma differentiable.smul (hc : differentiable 𝕜 c) (hf : differentiable 𝕜 f) :
differentiable 𝕜 (λ y, c y • f y) :=
λx, (hc x).smul (hf x)
lemma fderiv_within_smul (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hf : differentiable_within_at 𝕜 f s x) :
fderiv_within 𝕜 (λ y, c y • f y) s x =
c x • fderiv_within 𝕜 f s x + (fderiv_within 𝕜 c s x).smul_right (f x) :=
(hc.has_fderiv_within_at.smul hf.has_fderiv_within_at).fderiv_within hxs
lemma fderiv_smul (hc : differentiable_at 𝕜 c x) (hf : differentiable_at 𝕜 f x) :
fderiv 𝕜 (λ y, c y • f y) x =
c x • fderiv 𝕜 f x + (fderiv 𝕜 c x).smul_right (f x) :=
(hc.has_fderiv_at.smul hf.has_fderiv_at).fderiv
theorem has_strict_fderiv_at.smul_const (hc : has_strict_fderiv_at c c' x) (f : F) :
has_strict_fderiv_at (λ y, c y • f) (c'.smul_right f) x :=
by simpa only [smul_zero, zero_add] using hc.smul (has_strict_fderiv_at_const f x)
theorem has_fderiv_within_at.smul_const (hc : has_fderiv_within_at c c' s x) (f : F) :
has_fderiv_within_at (λ y, c y • f) (c'.smul_right f) s x :=
by simpa only [smul_zero, zero_add] using hc.smul (has_fderiv_within_at_const f x s)
theorem has_fderiv_at.smul_const (hc : has_fderiv_at c c' x) (f : F) :
has_fderiv_at (λ y, c y • f) (c'.smul_right f) x :=
by simpa only [smul_zero, zero_add] using hc.smul (has_fderiv_at_const f x)
lemma differentiable_within_at.smul_const
(hc : differentiable_within_at 𝕜 c s x) (f : F) :
differentiable_within_at 𝕜 (λ y, c y • f) s x :=
(hc.has_fderiv_within_at.smul_const f).differentiable_within_at
lemma differentiable_at.smul_const (hc : differentiable_at 𝕜 c x) (f : F) :
differentiable_at 𝕜 (λ y, c y • f) x :=
(hc.has_fderiv_at.smul_const f).differentiable_at
lemma differentiable_on.smul_const (hc : differentiable_on 𝕜 c s) (f : F) :
differentiable_on 𝕜 (λ y, c y • f) s :=
λx hx, (hc x hx).smul_const f
lemma differentiable.smul_const (hc : differentiable 𝕜 c) (f : F) :
differentiable 𝕜 (λ y, c y • f) :=
λx, (hc x).smul_const f
lemma fderiv_within_smul_const (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (f : F) :
fderiv_within 𝕜 (λ y, c y • f) s x =
(fderiv_within 𝕜 c s x).smul_right f :=
(hc.has_fderiv_within_at.smul_const f).fderiv_within hxs
lemma fderiv_smul_const (hc : differentiable_at 𝕜 c x) (f : F) :
fderiv 𝕜 (λ y, c y • f) x = (fderiv 𝕜 c x).smul_right f :=
(hc.has_fderiv_at.smul_const f).fderiv
end smul
section mul
/-! ### Derivative of the product of two functions -/
variables {𝔸 𝔸' : Type*} [normed_ring 𝔸] [normed_comm_ring 𝔸'] [normed_algebra 𝕜 𝔸]
[normed_algebra 𝕜 𝔸'] {a b : E → 𝔸} {a' b' : E →L[𝕜] 𝔸} {c d : E → 𝔸'} {c' d' : E →L[𝕜] 𝔸'}
theorem has_strict_fderiv_at.mul' {x : E} (ha : has_strict_fderiv_at a a' x)
(hb : has_strict_fderiv_at b b' x) :
has_strict_fderiv_at (λ y, a y * b y) (a x • b' + a'.smul_right (b x)) x :=
((continuous_linear_map.mul 𝕜 𝔸).is_bounded_bilinear_map.has_strict_fderiv_at (a x, b x)).comp x
(ha.prod hb)
theorem has_strict_fderiv_at.mul
(hc : has_strict_fderiv_at c c' x) (hd : has_strict_fderiv_at d d' x) :
has_strict_fderiv_at (λ y, c y * d y) (c x • d' + d x • c') x :=
by { convert hc.mul' hd, ext z, apply mul_comm }
theorem has_fderiv_within_at.mul'
(ha : has_fderiv_within_at a a' s x) (hb : has_fderiv_within_at b b' s x) :
has_fderiv_within_at (λ y, a y * b y) (a x • b' + a'.smul_right (b x)) s x :=
((continuous_linear_map.mul 𝕜 𝔸).is_bounded_bilinear_map.has_fderiv_at
(a x, b x)).comp_has_fderiv_within_at x (ha.prod hb)
theorem has_fderiv_within_at.mul
(hc : has_fderiv_within_at c c' s x) (hd : has_fderiv_within_at d d' s x) :
has_fderiv_within_at (λ y, c y * d y) (c x • d' + d x • c') s x :=
by { convert hc.mul' hd, ext z, apply mul_comm }
theorem has_fderiv_at.mul'
(ha : has_fderiv_at a a' x) (hb : has_fderiv_at b b' x) :
has_fderiv_at (λ y, a y * b y) (a x • b' + a'.smul_right (b x)) x :=
((continuous_linear_map.mul 𝕜 𝔸).is_bounded_bilinear_map.has_fderiv_at (a x, b x)).comp x
(ha.prod hb)
theorem has_fderiv_at.mul (hc : has_fderiv_at c c' x) (hd : has_fderiv_at d d' x) :
has_fderiv_at (λ y, c y * d y) (c x • d' + d x • c') x :=
by { convert hc.mul' hd, ext z, apply mul_comm }
lemma differentiable_within_at.mul
(ha : differentiable_within_at 𝕜 a s x) (hb : differentiable_within_at 𝕜 b s x) :
differentiable_within_at 𝕜 (λ y, a y * b y) s x :=
(ha.has_fderiv_within_at.mul' hb.has_fderiv_within_at).differentiable_within_at
@[simp] lemma differentiable_at.mul (ha : differentiable_at 𝕜 a x) (hb : differentiable_at 𝕜 b x) :
differentiable_at 𝕜 (λ y, a y * b y) x :=
(ha.has_fderiv_at.mul' hb.has_fderiv_at).differentiable_at
lemma differentiable_on.mul (ha : differentiable_on 𝕜 a s) (hb : differentiable_on 𝕜 b s) :
differentiable_on 𝕜 (λ y, a y * b y) s :=
λx hx, (ha x hx).mul (hb x hx)
@[simp] lemma differentiable.mul (ha : differentiable 𝕜 a) (hb : differentiable 𝕜 b) :
differentiable 𝕜 (λ y, a y * b y) :=
λx, (ha x).mul (hb x)
lemma differentiable_within_at.pow (ha : differentiable_within_at 𝕜 a s x) :
∀ n : ℕ, differentiable_within_at 𝕜 (λ x, a x ^ n) s x
| 0 := by simp only [pow_zero, differentiable_within_at_const]
| (n + 1) := by simp only [pow_succ, differentiable_within_at.pow n, ha.mul]
@[simp] lemma differentiable_at.pow (ha : differentiable_at 𝕜 a x) (n : ℕ) :
differentiable_at 𝕜 (λ x, a x ^ n) x :=
differentiable_within_at_univ.mp $ ha.differentiable_within_at.pow n
lemma differentiable_on.pow (ha : differentiable_on 𝕜 a s) (n : ℕ) :
differentiable_on 𝕜 (λ x, a x ^ n) s :=
λ x h, (ha x h).pow n
@[simp] lemma differentiable.pow (ha : differentiable 𝕜 a) (n : ℕ) :
differentiable 𝕜 (λ x, a x ^ n) :=
λx, (ha x).pow n
lemma fderiv_within_mul' (hxs : unique_diff_within_at 𝕜 s x)
(ha : differentiable_within_at 𝕜 a s x) (hb : differentiable_within_at 𝕜 b s x) :
fderiv_within 𝕜 (λ y, a y * b y) s x =
a x • fderiv_within 𝕜 b s x + (fderiv_within 𝕜 a s x).smul_right (b x) :=
(ha.has_fderiv_within_at.mul' hb.has_fderiv_within_at).fderiv_within hxs
lemma fderiv_within_mul (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hd : differentiable_within_at 𝕜 d s x) :
fderiv_within 𝕜 (λ y, c y * d y) s x =
c x • fderiv_within 𝕜 d s x + d x • fderiv_within 𝕜 c s x :=
(hc.has_fderiv_within_at.mul hd.has_fderiv_within_at).fderiv_within hxs
lemma fderiv_mul' (ha : differentiable_at 𝕜 a x) (hb : differentiable_at 𝕜 b x) :
fderiv 𝕜 (λ y, a y * b y) x =
a x • fderiv 𝕜 b x + (fderiv 𝕜 a x).smul_right (b x) :=
(ha.has_fderiv_at.mul' hb.has_fderiv_at).fderiv
lemma fderiv_mul (hc : differentiable_at 𝕜 c x) (hd : differentiable_at 𝕜 d x) :
fderiv 𝕜 (λ y, c y * d y) x =
c x • fderiv 𝕜 d x + d x • fderiv 𝕜 c x :=
(hc.has_fderiv_at.mul hd.has_fderiv_at).fderiv
theorem has_strict_fderiv_at.mul_const' (ha : has_strict_fderiv_at a a' x) (b : 𝔸) :
has_strict_fderiv_at (λ y, a y * b) (a'.smul_right b) x :=
(((continuous_linear_map.mul 𝕜 𝔸).flip b).has_strict_fderiv_at).comp x ha
theorem has_strict_fderiv_at.mul_const (hc : has_strict_fderiv_at c c' x) (d : 𝔸') :
has_strict_fderiv_at (λ y, c y * d) (d • c') x :=
by { convert hc.mul_const' d, ext z, apply mul_comm }
theorem has_fderiv_within_at.mul_const' (ha : has_fderiv_within_at a a' s x) (b : 𝔸) :
has_fderiv_within_at (λ y, a y * b) (a'.smul_right b) s x :=
(((continuous_linear_map.mul 𝕜 𝔸).flip b).has_fderiv_at).comp_has_fderiv_within_at x ha
theorem has_fderiv_within_at.mul_const (hc : has_fderiv_within_at c c' s x) (d : 𝔸') :
has_fderiv_within_at (λ y, c y * d) (d • c') s x :=
by { convert hc.mul_const' d, ext z, apply mul_comm }
theorem has_fderiv_at.mul_const' (ha : has_fderiv_at a a' x) (b : 𝔸) :
has_fderiv_at (λ y, a y * b) (a'.smul_right b) x :=
(((continuous_linear_map.mul 𝕜 𝔸).flip b).has_fderiv_at).comp x ha
theorem has_fderiv_at.mul_const (hc : has_fderiv_at c c' x) (d : 𝔸') :
has_fderiv_at (λ y, c y * d) (d • c') x :=
by { convert hc.mul_const' d, ext z, apply mul_comm }
lemma differentiable_within_at.mul_const
(ha : differentiable_within_at 𝕜 a s x) (b : 𝔸) :
differentiable_within_at 𝕜 (λ y, a y * b) s x :=
(ha.has_fderiv_within_at.mul_const' b).differentiable_within_at
lemma differentiable_at.mul_const (ha : differentiable_at 𝕜 a x) (b : 𝔸) :
differentiable_at 𝕜 (λ y, a y * b) x :=
(ha.has_fderiv_at.mul_const' b).differentiable_at
lemma differentiable_on.mul_const (ha : differentiable_on 𝕜 a s) (b : 𝔸) :
differentiable_on 𝕜 (λ y, a y * b) s :=
λx hx, (ha x hx).mul_const b
lemma differentiable.mul_const (ha : differentiable 𝕜 a) (b : 𝔸) :
differentiable 𝕜 (λ y, a y * b) :=
λx, (ha x).mul_const b
lemma fderiv_within_mul_const' (hxs : unique_diff_within_at 𝕜 s x)
(ha : differentiable_within_at 𝕜 a s x) (b : 𝔸) :
fderiv_within 𝕜 (λ y, a y * b) s x = (fderiv_within 𝕜 a s x).smul_right b :=
(ha.has_fderiv_within_at.mul_const' b).fderiv_within hxs
lemma fderiv_within_mul_const (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (d : 𝔸') :
fderiv_within 𝕜 (λ y, c y * d) s x = d • fderiv_within 𝕜 c s x :=
(hc.has_fderiv_within_at.mul_const d).fderiv_within hxs
lemma fderiv_mul_const' (ha : differentiable_at 𝕜 a x) (b : 𝔸) :
fderiv 𝕜 (λ y, a y * b) x = (fderiv 𝕜 a x).smul_right b :=
(ha.has_fderiv_at.mul_const' b).fderiv
lemma fderiv_mul_const (hc : differentiable_at 𝕜 c x) (d : 𝔸') :
fderiv 𝕜 (λ y, c y * d) x = d • fderiv 𝕜 c x :=
(hc.has_fderiv_at.mul_const d).fderiv
theorem has_strict_fderiv_at.const_mul (ha : has_strict_fderiv_at a a' x) (b : 𝔸) :
has_strict_fderiv_at (λ y, b * a y) (b • a') x :=
(((continuous_linear_map.mul 𝕜 𝔸) b).has_strict_fderiv_at).comp x ha
theorem has_fderiv_within_at.const_mul
(ha : has_fderiv_within_at a a' s x) (b : 𝔸) :
has_fderiv_within_at (λ y, b * a y) (b • a') s x :=
(((continuous_linear_map.mul 𝕜 𝔸) b).has_fderiv_at).comp_has_fderiv_within_at x ha
theorem has_fderiv_at.const_mul (ha : has_fderiv_at a a' x) (b : 𝔸) :
has_fderiv_at (λ y, b * a y) (b • a') x :=
(((continuous_linear_map.mul 𝕜 𝔸) b).has_fderiv_at).comp x ha
lemma differentiable_within_at.const_mul
(ha : differentiable_within_at 𝕜 a s x) (b : 𝔸) :
differentiable_within_at 𝕜 (λ y, b * a y) s x :=
(ha.has_fderiv_within_at.const_mul b).differentiable_within_at
lemma differentiable_at.const_mul (ha : differentiable_at 𝕜 a x) (b : 𝔸) :
differentiable_at 𝕜 (λ y, b * a y) x :=
(ha.has_fderiv_at.const_mul b).differentiable_at
lemma differentiable_on.const_mul (ha : differentiable_on 𝕜 a s) (b : 𝔸) :
differentiable_on 𝕜 (λ y, b * a y) s :=
λx hx, (ha x hx).const_mul b
lemma differentiable.const_mul (ha : differentiable 𝕜 a) (b : 𝔸) :
differentiable 𝕜 (λ y, b * a y) :=
λx, (ha x).const_mul b
lemma fderiv_within_const_mul (hxs : unique_diff_within_at 𝕜 s x)
(ha : differentiable_within_at 𝕜 a s x) (b : 𝔸) :
fderiv_within 𝕜 (λ y, b * a y) s x = b • fderiv_within 𝕜 a s x :=
(ha.has_fderiv_within_at.const_mul b).fderiv_within hxs
lemma fderiv_const_mul (ha : differentiable_at 𝕜 a x) (b : 𝔸) :
fderiv 𝕜 (λ y, b * a y) x = b • fderiv 𝕜 a x :=
(ha.has_fderiv_at.const_mul b).fderiv
end mul
section algebra_inverse
variables {R : Type*} [normed_ring R] [normed_algebra 𝕜 R] [complete_space R]
open normed_ring continuous_linear_map ring
/-- At an invertible element `x` of a normed algebra `R`, the Fréchet derivative of the inversion
operation is the linear map `λ t, - x⁻¹ * t * x⁻¹`. -/
lemma has_fderiv_at_ring_inverse (x : Rˣ) :
has_fderiv_at ring.inverse (-mul_left_right 𝕜 R ↑x⁻¹ ↑x⁻¹) x :=
begin
have h_is_o : (λ (t : R), inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =o[𝓝 0] (λ (t : R), t),
{ refine (inverse_add_norm_diff_second_order x).trans_is_o ((is_o_norm_norm).mp _),
simp only [norm_pow, norm_norm],
have h12 : 1 < 2 := by norm_num,
convert (asymptotics.is_o_pow_pow h12).comp_tendsto tendsto_norm_zero,
ext, simp },
have h_lim : tendsto (λ (y:R), y - x) (𝓝 x) (𝓝 0),
{ refine tendsto_zero_iff_norm_tendsto_zero.mpr _,
exact tendsto_iff_norm_tendsto_zero.mp tendsto_id },
simp only [has_fderiv_at, has_fderiv_at_filter],
convert h_is_o.comp_tendsto h_lim,
ext y,
simp only [coe_comp', function.comp_app, mul_left_right_apply, neg_apply, inverse_unit x,
units.inv_mul, add_sub_cancel'_right, mul_sub, sub_mul, one_mul, sub_neg_eq_add]
end
lemma differentiable_at_inverse (x : Rˣ) : differentiable_at 𝕜 (@ring.inverse R _) x :=
(has_fderiv_at_ring_inverse x).differentiable_at
lemma fderiv_inverse (x : Rˣ) :
fderiv 𝕜 (@ring.inverse R _) x = - mul_left_right 𝕜 R ↑x⁻¹ ↑x⁻¹ :=
(has_fderiv_at_ring_inverse x).fderiv
end algebra_inverse
namespace continuous_linear_equiv
/-! ### Differentiability of linear equivs, and invariance of differentiability -/
variable (iso : E ≃L[𝕜] F)
protected lemma has_strict_fderiv_at :
has_strict_fderiv_at iso (iso : E →L[𝕜] F) x :=
iso.to_continuous_linear_map.has_strict_fderiv_at
protected lemma has_fderiv_within_at :
has_fderiv_within_at iso (iso : E →L[𝕜] F) s x :=
iso.to_continuous_linear_map.has_fderiv_within_at
protected lemma has_fderiv_at : has_fderiv_at iso (iso : E →L[𝕜] F) x :=
iso.to_continuous_linear_map.has_fderiv_at_filter
protected lemma differentiable_at : differentiable_at 𝕜 iso x :=
iso.has_fderiv_at.differentiable_at
protected lemma differentiable_within_at :
differentiable_within_at 𝕜 iso s x :=
iso.differentiable_at.differentiable_within_at
protected lemma fderiv : fderiv 𝕜 iso x = iso :=
iso.has_fderiv_at.fderiv
protected lemma fderiv_within (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 iso s x = iso :=
iso.to_continuous_linear_map.fderiv_within hxs
protected lemma differentiable : differentiable 𝕜 iso :=
λx, iso.differentiable_at
protected lemma differentiable_on : differentiable_on 𝕜 iso s :=
iso.differentiable.differentiable_on
lemma comp_differentiable_within_at_iff {f : G → E} {s : set G} {x : G} :
differentiable_within_at 𝕜 (iso ∘ f) s x ↔ differentiable_within_at 𝕜 f s x :=
begin
refine ⟨λ H, _, λ H, iso.differentiable.differentiable_at.comp_differentiable_within_at x H⟩,
have : differentiable_within_at 𝕜 (iso.symm ∘ (iso ∘ f)) s x :=
iso.symm.differentiable.differentiable_at.comp_differentiable_within_at x H,
rwa [← function.comp.assoc iso.symm iso f, iso.symm_comp_self] at this,
end
lemma comp_differentiable_at_iff {f : G → E} {x : G} :
differentiable_at 𝕜 (iso ∘ f) x ↔ differentiable_at 𝕜 f x :=
by rw [← differentiable_within_at_univ, ← differentiable_within_at_univ,
iso.comp_differentiable_within_at_iff]
lemma comp_differentiable_on_iff {f : G → E} {s : set G} :
differentiable_on 𝕜 (iso ∘ f) s ↔ differentiable_on 𝕜 f s :=
begin
rw [differentiable_on, differentiable_on],
simp only [iso.comp_differentiable_within_at_iff],
end
lemma comp_differentiable_iff {f : G → E} :
differentiable 𝕜 (iso ∘ f) ↔ differentiable 𝕜 f :=
begin
rw [← differentiable_on_univ, ← differentiable_on_univ],
exact iso.comp_differentiable_on_iff
end
lemma comp_has_fderiv_within_at_iff
{f : G → E} {s : set G} {x : G} {f' : G →L[𝕜] E} :
has_fderiv_within_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') s x ↔ has_fderiv_within_at f f' s x :=
begin
refine ⟨λ H, _, λ H, iso.has_fderiv_at.comp_has_fderiv_within_at x H⟩,
have A : f = iso.symm ∘ (iso ∘ f), by { rw [← function.comp.assoc, iso.symm_comp_self], refl },
have B : f' = (iso.symm : F →L[𝕜] E).comp ((iso : E →L[𝕜] F).comp f'),
by rw [← continuous_linear_map.comp_assoc, iso.coe_symm_comp_coe,
continuous_linear_map.id_comp],
rw [A, B],
exact iso.symm.has_fderiv_at.comp_has_fderiv_within_at x H
end
lemma comp_has_strict_fderiv_at_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
has_strict_fderiv_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ has_strict_fderiv_at f f' x :=
begin
refine ⟨λ H, _, λ H, iso.has_strict_fderiv_at.comp x H⟩,
convert iso.symm.has_strict_fderiv_at.comp x H; ext z; apply (iso.symm_apply_apply _).symm
end
lemma comp_has_fderiv_at_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
has_fderiv_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ has_fderiv_at f f' x :=
by simp_rw [← has_fderiv_within_at_univ, iso.comp_has_fderiv_within_at_iff]
lemma comp_has_fderiv_within_at_iff'
{f : G → E} {s : set G} {x : G} {f' : G →L[𝕜] F} :
has_fderiv_within_at (iso ∘ f) f' s x ↔
has_fderiv_within_at f ((iso.symm : F →L[𝕜] E).comp f') s x :=
by rw [← iso.comp_has_fderiv_within_at_iff, ← continuous_linear_map.comp_assoc,
iso.coe_comp_coe_symm, continuous_linear_map.id_comp]
lemma comp_has_fderiv_at_iff' {f : G → E} {x : G} {f' : G →L[𝕜] F} :
has_fderiv_at (iso ∘ f) f' x ↔ has_fderiv_at f ((iso.symm : F →L[𝕜] E).comp f') x :=
by simp_rw [← has_fderiv_within_at_univ, iso.comp_has_fderiv_within_at_iff']
lemma comp_fderiv_within {f : G → E} {s : set G} {x : G}
(hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (iso ∘ f) s x = (iso : E →L[𝕜] F).comp (fderiv_within 𝕜 f s x) :=
begin
by_cases h : differentiable_within_at 𝕜 f s x,
{ rw [fderiv.comp_fderiv_within x iso.differentiable_at h hxs, iso.fderiv] },
{ have : ¬differentiable_within_at 𝕜 (iso ∘ f) s x,
from mt iso.comp_differentiable_within_at_iff.1 h,
rw [fderiv_within_zero_of_not_differentiable_within_at h,
fderiv_within_zero_of_not_differentiable_within_at this,
continuous_linear_map.comp_zero] }
end
lemma comp_fderiv {f : G → E} {x : G} :
fderiv 𝕜 (iso ∘ f) x = (iso : E →L[𝕜] F).comp (fderiv 𝕜 f x) :=
begin
rw [← fderiv_within_univ, ← fderiv_within_univ],
exact iso.comp_fderiv_within unique_diff_within_at_univ,
end
end continuous_linear_equiv
namespace linear_isometry_equiv
/-! ### Differentiability of linear isometry equivs, and invariance of differentiability -/
variable (iso : E ≃ₗᵢ[𝕜] F)
protected lemma has_strict_fderiv_at : has_strict_fderiv_at iso (iso : E →L[𝕜] F) x :=
(iso : E ≃L[𝕜] F).has_strict_fderiv_at
protected lemma has_fderiv_within_at : has_fderiv_within_at iso (iso : E →L[𝕜] F) s x :=
(iso : E ≃L[𝕜] F).has_fderiv_within_at
protected lemma has_fderiv_at : has_fderiv_at iso (iso : E →L[𝕜] F) x :=
(iso : E ≃L[𝕜] F).has_fderiv_at
protected lemma differentiable_at : differentiable_at 𝕜 iso x :=
iso.has_fderiv_at.differentiable_at
protected lemma differentiable_within_at :
differentiable_within_at 𝕜 iso s x :=
iso.differentiable_at.differentiable_within_at
protected lemma fderiv : fderiv 𝕜 iso x = iso := iso.has_fderiv_at.fderiv
protected lemma fderiv_within (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 iso s x = iso :=
(iso : E ≃L[𝕜] F).fderiv_within hxs
protected lemma differentiable : differentiable 𝕜 iso :=
λx, iso.differentiable_at
protected lemma differentiable_on : differentiable_on 𝕜 iso s :=
iso.differentiable.differentiable_on
lemma comp_differentiable_within_at_iff {f : G → E} {s : set G} {x : G} :
differentiable_within_at 𝕜 (iso ∘ f) s x ↔ differentiable_within_at 𝕜 f s x :=
(iso : E ≃L[𝕜] F).comp_differentiable_within_at_iff
lemma comp_differentiable_at_iff {f : G → E} {x : G} :
differentiable_at 𝕜 (iso ∘ f) x ↔ differentiable_at 𝕜 f x :=
(iso : E ≃L[𝕜] F).comp_differentiable_at_iff
lemma comp_differentiable_on_iff {f : G → E} {s : set G} :
differentiable_on 𝕜 (iso ∘ f) s ↔ differentiable_on 𝕜 f s :=
(iso : E ≃L[𝕜] F).comp_differentiable_on_iff
lemma comp_differentiable_iff {f : G → E} :
differentiable 𝕜 (iso ∘ f) ↔ differentiable 𝕜 f :=
(iso : E ≃L[𝕜] F).comp_differentiable_iff
lemma comp_has_fderiv_within_at_iff
{f : G → E} {s : set G} {x : G} {f' : G →L[𝕜] E} :
has_fderiv_within_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') s x ↔ has_fderiv_within_at f f' s x :=
(iso : E ≃L[𝕜] F).comp_has_fderiv_within_at_iff
lemma comp_has_strict_fderiv_at_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
has_strict_fderiv_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ has_strict_fderiv_at f f' x :=
(iso : E ≃L[𝕜] F).comp_has_strict_fderiv_at_iff
lemma comp_has_fderiv_at_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
has_fderiv_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ has_fderiv_at f f' x :=
(iso : E ≃L[𝕜] F).comp_has_fderiv_at_iff
lemma comp_has_fderiv_within_at_iff'
{f : G → E} {s : set G} {x : G} {f' : G →L[𝕜] F} :
has_fderiv_within_at (iso ∘ f) f' s x ↔
has_fderiv_within_at f ((iso.symm : F →L[𝕜] E).comp f') s x :=
(iso : E ≃L[𝕜] F).comp_has_fderiv_within_at_iff'
lemma comp_has_fderiv_at_iff' {f : G → E} {x : G} {f' : G →L[𝕜] F} :
has_fderiv_at (iso ∘ f) f' x ↔ has_fderiv_at f ((iso.symm : F →L[𝕜] E).comp f') x :=
(iso : E ≃L[𝕜] F).comp_has_fderiv_at_iff'
lemma comp_fderiv_within {f : G → E} {s : set G} {x : G}
(hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (iso ∘ f) s x = (iso : E →L[𝕜] F).comp (fderiv_within 𝕜 f s x) :=
(iso : E ≃L[𝕜] F).comp_fderiv_within hxs
lemma comp_fderiv {f : G → E} {x : G} :
fderiv 𝕜 (iso ∘ f) x = (iso : E →L[𝕜] F).comp (fderiv 𝕜 f x) :=
(iso : E ≃L[𝕜] F).comp_fderiv
end linear_isometry_equiv
/-- If `f (g y) = y` for `y` in some neighborhood of `a`, `g` is continuous at `a`, and `f` has an
invertible derivative `f'` at `g a` in the strict sense, then `g` has the derivative `f'⁻¹` at `a`
in the strict sense.
This is one of the easy parts of the inverse function theorem: it assumes that we already have an
inverse function. -/
theorem has_strict_fderiv_at.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
(hg : continuous_at g a) (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) (g a))
(hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) :
has_strict_fderiv_at g (f'.symm : F →L[𝕜] E) a :=
begin
replace hg := hg.prod_map' hg,
replace hfg := hfg.prod_mk_nhds hfg,
have : (λ p : F × F, g p.1 - g p.2 - f'.symm (p.1 - p.2)) =O[𝓝 (a, a)]
(λ p : F × F, f' (g p.1 - g p.2) - (p.1 - p.2)),
{ refine ((f'.symm : F →L[𝕜] E).is_O_comp _ _).congr (λ x, _) (λ _, rfl),
simp },
refine this.trans_is_o _, clear this,
refine ((hf.comp_tendsto hg).symm.congr' (hfg.mono _)
(eventually_of_forall $ λ _, rfl)).trans_is_O _,
{ rintros p ⟨hp1, hp2⟩,
simp [hp1, hp2] },
{ refine (hf.is_O_sub_rev.comp_tendsto hg).congr'
(eventually_of_forall $ λ _, rfl) (hfg.mono _),
rintros p ⟨hp1, hp2⟩,
simp only [(∘), hp1, hp2] }
end
/-- If `f (g y) = y` for `y` in some neighborhood of `a`, `g` is continuous at `a`, and `f` has an
invertible derivative `f'` at `g a`, then `g` has the derivative `f'⁻¹` at `a`.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. -/
theorem has_fderiv_at.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
(hg : continuous_at g a) (hf : has_fderiv_at f (f' : E →L[𝕜] F) (g a))
(hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) :
has_fderiv_at g (f'.symm : F →L[𝕜] E) a :=
begin
have : (λ x : F, g x - g a - f'.symm (x - a)) =O[𝓝 a] (λ x : F, f' (g x - g a) - (x - a)),
{ refine ((f'.symm : F →L[𝕜] E).is_O_comp _ _).congr (λ x, _) (λ _, rfl),
simp },
refine this.trans_is_o _, clear this,
refine ((hf.comp_tendsto hg).symm.congr' (hfg.mono _)
(eventually_of_forall $ λ _, rfl)).trans_is_O _,
{ rintros p hp,
simp [hp, hfg.self_of_nhds] },
{ refine ((hf.is_O_sub_rev f'.antilipschitz).comp_tendsto hg).congr'
(eventually_of_forall $ λ _, rfl) (hfg.mono _),
rintros p hp,
simp only [(∘), hp, hfg.self_of_nhds] }
end
/-- If `f` is a local homeomorphism defined on a neighbourhood of `f.symm a`, and `f` has an
invertible derivative `f'` in the sense of strict differentiability at `f.symm a`, then `f.symm` has
the derivative `f'⁻¹` at `a`.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. -/
lemma local_homeomorph.has_strict_fderiv_at_symm (f : local_homeomorph E F) {f' : E ≃L[𝕜] F} {a : F}
(ha : a ∈ f.target) (htff' : has_strict_fderiv_at f (f' : E →L[𝕜] F) (f.symm a)) :
has_strict_fderiv_at f.symm (f'.symm : F →L[𝕜] E) a :=
htff'.of_local_left_inverse (f.symm.continuous_at ha) (f.eventually_right_inverse ha)
/-- If `f` is a local homeomorphism defined on a neighbourhood of `f.symm a`, and `f` has an
invertible derivative `f'` at `f.symm a`, then `f.symm` has the derivative `f'⁻¹` at `a`.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. -/
lemma local_homeomorph.has_fderiv_at_symm (f : local_homeomorph E F) {f' : E ≃L[𝕜] F} {a : F}
(ha : a ∈ f.target) (htff' : has_fderiv_at f (f' : E →L[𝕜] F) (f.symm a)) :
has_fderiv_at f.symm (f'.symm : F →L[𝕜] E) a :=
htff'.of_local_left_inverse (f.symm.continuous_at ha) (f.eventually_right_inverse ha)
lemma has_fderiv_within_at.eventually_ne (h : has_fderiv_within_at f f' s x)
(hf' : ∃ C, ∀ z, ‖z‖ ≤ C * ‖f' z‖) :
∀ᶠ z in 𝓝[s \ {x}] x, f z ≠ f x :=
begin
rw [nhds_within, diff_eq, ← inf_principal, ← inf_assoc, eventually_inf_principal],
have A : (λ z, z - x) =O[𝓝[s] x] (λ z, f' (z - x)) :=
(is_O_iff.2 $ hf'.imp $ λ C hC, eventually_of_forall $ λ z, hC _),
have : (λ z, f z - f x) ~[𝓝[s] x] (λ z, f' (z - x)) := h.trans_is_O A,
simpa [not_imp_not, sub_eq_zero] using (A.trans this.is_O_symm).eq_zero_imp
end
lemma has_fderiv_at.eventually_ne (h : has_fderiv_at f f' x) (hf' : ∃ C, ∀ z, ‖z‖ ≤ C * ‖f' z‖) :
∀ᶠ z in 𝓝[≠] x, f z ≠ f x :=
by simpa only [compl_eq_univ_diff] using (has_fderiv_within_at_univ.2 h).eventually_ne hf'
end
section
/-
In the special case of a normed space over the reals,
we can use scalar multiplication in the `tendsto` characterization
of the Fréchet derivative.
-/
variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
variables {F : Type*} [normed_add_comm_group F] [normed_space ℝ F]
variables {f : E → F} {f' : E →L[ℝ] F} {x : E}
theorem has_fderiv_at_filter_real_equiv {L : filter E} :
tendsto (λ x' : E, ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0) ↔
tendsto (λ x' : E, ‖x' - x‖⁻¹ • (f x' - f x - f' (x' - x))) L (𝓝 0) :=
begin
symmetry,
rw [tendsto_iff_norm_tendsto_zero], refine tendsto_congr (λ x', _),
have : ‖x' - x‖⁻¹ ≥ 0, from inv_nonneg.mpr (norm_nonneg _),
simp [norm_smul, abs_of_nonneg this]
end
lemma has_fderiv_at.lim_real (hf : has_fderiv_at f f' x) (v : E) :
tendsto (λ (c:ℝ), c • (f (x + c⁻¹ • v) - f x)) at_top (𝓝 (f' v)) :=
begin
apply hf.lim v,
rw tendsto_at_top_at_top,
exact λ b, ⟨b, λ a ha, le_trans ha (le_abs_self _)⟩
end
end
section tangent_cone
variables {𝕜 : Type*} [nontrivially_normed_field 𝕜]
{E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E]
{F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F]
{f : E → F} {s : set E} {f' : E →L[𝕜] F}
/-- The image of a tangent cone under the differential of a map is included in the tangent cone to
the image. -/
lemma has_fderiv_within_at.maps_to_tangent_cone {x : E} (h : has_fderiv_within_at f f' s x) :
maps_to f' (tangent_cone_at 𝕜 s x) (tangent_cone_at 𝕜 (f '' s) (f x)) :=
begin
rintros v ⟨c, d, dtop, clim, cdlim⟩,
refine ⟨c, (λn, f (x + d n) - f x), mem_of_superset dtop _, clim,
h.lim at_top dtop clim cdlim⟩,
simp [-mem_image, mem_image_of_mem] {contextual := tt}
end
/-- If a set has the unique differentiability property at a point x, then the image of this set
under a map with onto derivative has also the unique differentiability property at the image point.
-/
lemma has_fderiv_within_at.unique_diff_within_at {x : E} (h : has_fderiv_within_at f f' s x)
(hs : unique_diff_within_at 𝕜 s x) (h' : dense_range f') :
unique_diff_within_at 𝕜 (f '' s) (f x) :=
begin
refine ⟨h'.dense_of_maps_to f'.continuous hs.1 _,
h.continuous_within_at.mem_closure_image hs.2⟩,
show submodule.span 𝕜 (tangent_cone_at 𝕜 s x) ≤
(submodule.span 𝕜 (tangent_cone_at 𝕜 (f '' s) (f x))).comap f',
rw [submodule.span_le],
exact h.maps_to_tangent_cone.mono (subset.refl _) submodule.subset_span
end
lemma unique_diff_on.image {f' : E → E →L[𝕜] F} (hs : unique_diff_on 𝕜 s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (hd : ∀ x ∈ s, dense_range (f' x)) :
unique_diff_on 𝕜 (f '' s) :=
ball_image_iff.2 $ λ x hx, (hf' x hx).unique_diff_within_at (hs x hx) (hd x hx)
lemma has_fderiv_within_at.unique_diff_within_at_of_continuous_linear_equiv
{x : E} (e' : E ≃L[𝕜] F) (h : has_fderiv_within_at f (e' : E →L[𝕜] F) s x)
(hs : unique_diff_within_at 𝕜 s x) :
unique_diff_within_at 𝕜 (f '' s) (f x) :=
h.unique_diff_within_at hs e'.surjective.dense_range
lemma continuous_linear_equiv.unique_diff_on_image (e : E ≃L[𝕜] F) (h : unique_diff_on 𝕜 s) :
unique_diff_on 𝕜 (e '' s) :=
h.image (λ x _, e.has_fderiv_within_at) (λ x hx, e.surjective.dense_range)
@[simp] lemma continuous_linear_equiv.unique_diff_on_image_iff (e : E ≃L[𝕜] F) :
unique_diff_on 𝕜 (e '' s) ↔ unique_diff_on 𝕜 s :=
⟨λ h, e.symm_image_image s ▸ e.symm.unique_diff_on_image h, e.unique_diff_on_image⟩
@[simp] lemma continuous_linear_equiv.unique_diff_on_preimage_iff (e : F ≃L[𝕜] E) :
unique_diff_on 𝕜 (e ⁻¹' s) ↔ unique_diff_on 𝕜 s :=
by rw [← e.image_symm_eq_preimage, e.symm.unique_diff_on_image_iff]
end tangent_cone
section restrict_scalars
/-!
### Restricting from `ℂ` to `ℝ`, or generally from `𝕜'` to `𝕜`
If a function is differentiable over `ℂ`, then it is differentiable over `ℝ`. In this paragraph,
we give variants of this statement, in the general situation where `ℂ` and `ℝ` are replaced
respectively by `𝕜'` and `𝕜` where `𝕜'` is a normed algebra over `𝕜`.
-/
variables (𝕜 : Type*) [nontrivially_normed_field 𝕜]
variables {𝕜' : Type*} [nontrivially_normed_field 𝕜'] [normed_algebra 𝕜 𝕜']
variables {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] [normed_space 𝕜' E]
variables [is_scalar_tower 𝕜 𝕜' E]
variables {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] [normed_space 𝕜' F]
variables [is_scalar_tower 𝕜 𝕜' F]
variables {f : E → F} {f' : E →L[𝕜'] F} {s : set E} {x : E}
lemma has_strict_fderiv_at.restrict_scalars (h : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at f (f'.restrict_scalars 𝕜) x := h
lemma has_fderiv_at_filter.restrict_scalars {L} (h : has_fderiv_at_filter f f' x L) :
has_fderiv_at_filter f (f'.restrict_scalars 𝕜) x L := h
lemma has_fderiv_at.restrict_scalars (h : has_fderiv_at f f' x) :
has_fderiv_at f (f'.restrict_scalars 𝕜) x := h
lemma has_fderiv_within_at.restrict_scalars (h : has_fderiv_within_at f f' s x) :
has_fderiv_within_at f (f'.restrict_scalars 𝕜) s x := h
lemma differentiable_at.restrict_scalars (h : differentiable_at 𝕜' f x) :
differentiable_at 𝕜 f x :=
(h.has_fderiv_at.restrict_scalars 𝕜).differentiable_at
lemma differentiable_within_at.restrict_scalars (h : differentiable_within_at 𝕜' f s x) :
differentiable_within_at 𝕜 f s x :=
(h.has_fderiv_within_at.restrict_scalars 𝕜).differentiable_within_at
lemma differentiable_on.restrict_scalars (h : differentiable_on 𝕜' f s) :
differentiable_on 𝕜 f s :=
λx hx, (h x hx).restrict_scalars 𝕜
lemma differentiable.restrict_scalars (h : differentiable 𝕜' f) :
differentiable 𝕜 f :=
λx, (h x).restrict_scalars 𝕜
lemma has_fderiv_within_at_of_restrict_scalars
{g' : E →L[𝕜] F} (h : has_fderiv_within_at f g' s x)
(H : f'.restrict_scalars 𝕜 = g') : has_fderiv_within_at f f' s x :=
by { rw ← H at h, exact h }
lemma has_fderiv_at_of_restrict_scalars {g' : E →L[𝕜] F} (h : has_fderiv_at f g' x)
(H : f'.restrict_scalars 𝕜 = g') : has_fderiv_at f f' x :=
by { rw ← H at h, exact h }
lemma differentiable_at.fderiv_restrict_scalars (h : differentiable_at 𝕜' f x) :
fderiv 𝕜 f x = (fderiv 𝕜' f x).restrict_scalars 𝕜 :=
(h.has_fderiv_at.restrict_scalars 𝕜).fderiv
lemma differentiable_within_at_iff_restrict_scalars
(hf : differentiable_within_at 𝕜 f s x) (hs : unique_diff_within_at 𝕜 s x) :
differentiable_within_at 𝕜' f s x ↔
∃ (g' : E →L[𝕜'] F), g'.restrict_scalars 𝕜 = fderiv_within 𝕜 f s x :=
begin
split,
{ rintros ⟨g', hg'⟩,
exact ⟨g', hs.eq (hg'.restrict_scalars 𝕜) hf.has_fderiv_within_at⟩, },
{ rintros ⟨f', hf'⟩,
exact ⟨f', has_fderiv_within_at_of_restrict_scalars 𝕜 hf.has_fderiv_within_at hf'⟩, },
end
lemma differentiable_at_iff_restrict_scalars (hf : differentiable_at 𝕜 f x) :
differentiable_at 𝕜' f x ↔ ∃ (g' : E →L[𝕜'] F), g'.restrict_scalars 𝕜 = fderiv 𝕜 f x :=
begin
rw [← differentiable_within_at_univ, ← fderiv_within_univ],
exact differentiable_within_at_iff_restrict_scalars 𝕜
hf.differentiable_within_at unique_diff_within_at_univ,
end
end restrict_scalars
/-! ### Support of derivatives -/
section support
open function
variables (𝕜 : Type*) {E F : Type*} [nontrivially_normed_field 𝕜] [normed_add_comm_group E]
[normed_space 𝕜 E] [normed_add_comm_group F] [normed_space 𝕜 F] {f : E → F}
lemma support_fderiv_subset : support (fderiv 𝕜 f) ⊆ tsupport f :=
begin
intros x,
rw [← not_imp_not],
intro h2x,
rw [not_mem_tsupport_iff_eventually_eq] at h2x,
exact nmem_support.mpr (h2x.fderiv_eq.trans $ fderiv_const_apply 0),
end
lemma has_compact_support.fderiv (hf : has_compact_support f) : has_compact_support (fderiv 𝕜 f) :=
hf.mono' $ support_fderiv_subset 𝕜
end support
|
3a8e52a11977b91b7e3bbc2386ec7347ef95ec84 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/order/jordan_holder.lean | ab622f412faaef6114b3ba70a42564f7575cd01f | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 30,653 | lean | /-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import order.lattice
import data.list.sort
import logic.equiv.fin
import logic.equiv.functor
import data.fintype.card
/-!
# Jordan-Hölder Theorem
This file proves the Jordan Hölder theorem for a `jordan_holder_lattice`, a class also defined in
this file. Examples of `jordan_holder_lattice` include `subgroup G` if `G` is a group, and
`submodule R M` if `M` is an `R`-module. Using this approach the theorem need not be proved
seperately for both groups and modules, the proof in this file can be applied to both.
## Main definitions
The main definitions in this file are `jordan_holder_lattice` and `composition_series`,
and the relation `equivalent` on `composition_series`
A `jordan_holder_lattice` is the class for which the Jordan Hölder theorem is proved. A
Jordan Hölder lattice is a lattice equipped with a notion of maximality, `is_maximal`, and a notion
of isomorphism of pairs `iso`. In the example of subgroups of a group, `is_maximal H K` means that
`H` is a maximal normal subgroup of `K`, and `iso (H₁, K₁) (H₂, K₂)` means that the quotient
`H₁ / K₁` is isomorphic to the quotient `H₂ / K₂`. `iso` must be symmetric and transitive and must
satisfy the second isomorphism theorem `iso (H, H ⊔ K) (H ⊓ K, K)`.
A `composition_series X` is a finite nonempty series of elements of the lattice `X` such that
each element is maximal inside the next. The length of a `composition_series X` is
one less than the number of elements in the series. Note that there is no stipulation
that a series start from the bottom of the lattice and finish at the top.
For a composition series `s`, `s.top` is the largest element of the series,
and `s.bot` is the least element.
Two `composition_series X`, `s₁` and `s₂` are equivalent if there is a bijection
`e : fin s₁.length ≃ fin s₂.length` such that for any `i`,
`iso (s₁ i, s₁ i.succ) (s₂ (e i), s₂ (e i.succ))`
## Main theorems
The main theorem is `composition_series.jordan_holder`, which says that if two composition
series have the same least element and the same largest element,
then they are `equivalent`.
## TODO
Provide instances of `jordan_holder_lattice` for both submodules and subgroups, and potentially
for modular lattices.
It is not entirely clear how this should be done. Possibly there should be no global instances
of `jordan_holder_lattice`, and the instances should only be defined locally in order to prove
the Jordan-Hölder theorem for modules/groups and the API should be transferred because many of the
theorems in this file will have stronger versions for modules. There will also need to be an API for
mapping composition series across homomorphisms. It is also probably possible to
provide an instance of `jordan_holder_lattice` for any `modular_lattice`, and in this case the
Jordan-Hölder theorem will say that there is a well defined notion of length of a modular lattice.
However an instance of `jordan_holder_lattice` for a modular lattice will not be able to contain
the correct notion of isomorphism for modules, so a separate instance for modules will still be
required and this will clash with the instance for modular lattices, and so at least one of these
instances should not be a global instance.
-/
universe u
open set
/--
A `jordan_holder_lattice` is the class for which the Jordan Hölder theorem is proved. A
Jordan Hölder lattice is a lattice equipped with a notion of maximality, `is_maximal`, and a notion
of isomorphism of pairs `iso`. In the example of subgroups of a group, `is_maximal H K` means that
`H` is a maximal normal subgroup of `K`, and `iso (H₁, K₁) (H₂, K₂)` means that the quotient
`H₁ / K₁` is isomorphic to the quotient `H₂ / K₂`. `iso` must be symmetric and transitive and must
satisfy the second isomorphism theorem `iso (H, H ⊔ K) (H ⊓ K, K)`.
Examples include `subgroup G` if `G` is a group, and `submodule R M` if `M` is an `R`-module.
-/
class jordan_holder_lattice (X : Type u) [lattice X] :=
(is_maximal : X → X → Prop)
(lt_of_is_maximal : ∀ {x y}, is_maximal x y → x < y)
(sup_eq_of_is_maximal : ∀ {x y z}, is_maximal x z → is_maximal y z →
x ≠ y → x ⊔ y = z)
(is_maximal_inf_left_of_is_maximal_sup : ∀ {x y}, is_maximal x (x ⊔ y) → is_maximal y (x ⊔ y) →
is_maximal (x ⊓ y) x)
(iso : (X × X) → (X × X) → Prop)
(iso_symm : ∀ {x y}, iso x y → iso y x)
(iso_trans : ∀ {x y z}, iso x y → iso y z → iso x z)
(second_iso : ∀ {x y}, is_maximal x (x ⊔ y) → iso (x, x ⊔ y) (x ⊓ y, y))
namespace jordan_holder_lattice
variables {X : Type u} [lattice X] [jordan_holder_lattice X]
lemma is_maximal_inf_right_of_is_maximal_sup {x y : X}
(hxz : is_maximal x (x ⊔ y)) (hyz : is_maximal y (x ⊔ y)) :
is_maximal (x ⊓ y) y :=
begin
rw [inf_comm],
rw [sup_comm] at hxz hyz,
exact is_maximal_inf_left_of_is_maximal_sup hyz hxz
end
lemma is_maximal_of_eq_inf (x b : X) {a y : X}
(ha : x ⊓ y = a) (hxy : x ≠ y) (hxb : is_maximal x b) (hyb : is_maximal y b) :
is_maximal a y :=
begin
have hb : x ⊔ y = b,
from sup_eq_of_is_maximal hxb hyb hxy,
substs a b,
exact is_maximal_inf_right_of_is_maximal_sup hxb hyb
end
lemma second_iso_of_eq {x y a b : X} (hm : is_maximal x a) (ha : x ⊔ y = a) (hb : x ⊓ y = b) :
iso (x, a) (b, y) :=
by substs a b; exact second_iso hm
lemma is_maximal.iso_refl {x y : X} (h : is_maximal x y) : iso (x, y) (x, y) :=
second_iso_of_eq h
(sup_eq_right.2 (le_of_lt (lt_of_is_maximal h)))
(inf_eq_left.2 (le_of_lt (lt_of_is_maximal h)))
end jordan_holder_lattice
open jordan_holder_lattice
attribute [symm] iso_symm
attribute [trans] iso_trans
/--
A `composition_series X` is a finite nonempty series of elements of a
`jordan_holder_lattice` such that each element is maximal inside the next. The length of a
`composition_series X` is one less than the number of elements in the series.
Note that there is no stipulation that a series start from the bottom of the lattice and finish at
the top. For a composition series `s`, `s.top` is the largest element of the series,
and `s.bot` is the least element.
-/
structure composition_series (X : Type u) [lattice X] [jordan_holder_lattice X] : Type u :=
(length : ℕ)
(series : fin (length + 1) → X)
(step' : ∀ i : fin length, is_maximal (series i.cast_succ) (series i.succ))
namespace composition_series
variables {X : Type u} [lattice X] [jordan_holder_lattice X]
instance : has_coe_to_fun (composition_series X) (λ x, fin (x.length + 1) → X) :=
{ coe := composition_series.series }
instance [inhabited X] : inhabited (composition_series X) :=
⟨{ length := 0,
series := default,
step' := λ x, x.elim0 }⟩
variables {X}
lemma step (s : composition_series X) : ∀ i : fin s.length,
is_maximal (s i.cast_succ) (s i.succ) := s.step'
@[simp] lemma coe_fn_mk (length : ℕ) (series step) :
(@composition_series.mk X _ _ length series step : fin length.succ → X) = series := rfl
theorem lt_succ (s : composition_series X) (i : fin s.length) :
s i.cast_succ < s i.succ :=
lt_of_is_maximal (s.step _)
protected theorem strict_mono (s : composition_series X) : strict_mono s :=
fin.strict_mono_iff_lt_succ.2 s.lt_succ
protected theorem injective (s : composition_series X) : function.injective s :=
s.strict_mono.injective
@[simp] protected theorem inj (s : composition_series X) {i j : fin s.length.succ} :
s i = s j ↔ i = j :=
s.injective.eq_iff
instance : has_mem X (composition_series X) :=
⟨λ x s, x ∈ set.range s⟩
lemma mem_def {x : X} {s : composition_series X} : x ∈ s ↔ x ∈ set.range s := iff.rfl
lemma total {s : composition_series X} {x y : X} (hx : x ∈ s) (hy : y ∈ s) : x ≤ y ∨ y ≤ x :=
begin
rcases set.mem_range.1 hx with ⟨i, rfl⟩,
rcases set.mem_range.1 hy with ⟨j, rfl⟩,
rw [s.strict_mono.le_iff_le, s.strict_mono.le_iff_le],
exact le_total i j
end
/-- The ordered `list X` of elements of a `composition_series X`. -/
def to_list (s : composition_series X) : list X := list.of_fn s
/-- Two `composition_series` are equal if they are the same length and
have the same `i`th element for every `i` -/
lemma ext_fun {s₁ s₂ : composition_series X}
(hl : s₁.length = s₂.length)
(h : ∀ i, s₁ i = s₂ (fin.cast (congr_arg nat.succ hl) i)) :
s₁ = s₂ :=
begin
cases s₁, cases s₂,
dsimp at *,
subst hl,
simpa [function.funext_iff] using h
end
@[simp] lemma length_to_list (s : composition_series X) : s.to_list.length = s.length + 1 :=
by rw [to_list, list.length_of_fn]
lemma to_list_ne_nil (s : composition_series X) : s.to_list ≠ [] :=
by rw [← list.length_pos_iff_ne_nil, length_to_list]; exact nat.succ_pos _
lemma to_list_injective : function.injective (@composition_series.to_list X _ _) :=
λ s₁ s₂ (h : list.of_fn s₁ = list.of_fn s₂),
have h₁ : s₁.length = s₂.length,
from nat.succ_injective
((list.length_of_fn s₁).symm.trans $
(congr_arg list.length h).trans $
list.length_of_fn s₂),
have h₂ : ∀ i : fin s₁.length.succ, (s₁ i) = s₂ (fin.cast (congr_arg nat.succ h₁) i),
begin
assume i,
rw [← list.nth_le_of_fn s₁ i, ← list.nth_le_of_fn s₂],
simp [h]
end,
begin
cases s₁, cases s₂,
dsimp at *,
subst h₁,
simp only [heq_iff_eq, eq_self_iff_true, true_and],
simp only [fin.cast_refl] at h₂,
exact funext h₂
end
lemma chain'_to_list (s : composition_series X) :
list.chain' is_maximal s.to_list :=
list.chain'_iff_nth_le.2
begin
assume i hi,
simp only [to_list, list.nth_le_of_fn'],
rw [length_to_list] at hi,
exact s.step ⟨i, hi⟩
end
lemma to_list_sorted (s : composition_series X) : s.to_list.sorted (<) :=
list.pairwise_iff_nth_le.2 (λ i j hi hij,
begin
dsimp [to_list],
rw [list.nth_le_of_fn', list.nth_le_of_fn'],
exact s.strict_mono hij
end)
lemma to_list_nodup (s : composition_series X) : s.to_list.nodup :=
s.to_list_sorted.nodup
@[simp] lemma mem_to_list {s : composition_series X} {x : X} : x ∈ s.to_list ↔ x ∈ s :=
by rw [to_list, list.mem_of_fn, mem_def]
/-- Make a `composition_series X` from the ordered list of its elements. -/
def of_list (l : list X) (hl : l ≠ []) (hc : list.chain' is_maximal l) :
composition_series X :=
{ length := l.length - 1,
series := λ i, l.nth_le i begin
conv_rhs { rw ← tsub_add_cancel_of_le (nat.succ_le_of_lt (list.length_pos_of_ne_nil hl)) },
exact i.2
end,
step' := λ ⟨i, hi⟩, list.chain'_iff_nth_le.1 hc i hi }
lemma length_of_list (l : list X) (hl : l ≠ []) (hc : list.chain' is_maximal l) :
(of_list l hl hc).length = l.length - 1 := rfl
lemma of_list_to_list (s : composition_series X) :
of_list s.to_list s.to_list_ne_nil s.chain'_to_list = s :=
begin
refine ext_fun _ _,
{ rw [length_of_list, length_to_list, nat.succ_sub_one] },
{ rintros ⟨i, hi⟩,
dsimp [of_list, to_list],
rw [list.nth_le_of_fn'] }
end
@[simp] lemma of_list_to_list' (s : composition_series X) :
of_list s.to_list s.to_list_ne_nil s.chain'_to_list = s :=
of_list_to_list s
@[simp] lemma to_list_of_list (l : list X) (hl : l ≠ []) (hc : list.chain' is_maximal l) :
to_list (of_list l hl hc) = l :=
begin
refine list.ext_le _ _,
{ rw [length_to_list, length_of_list,
tsub_add_cancel_of_le (nat.succ_le_of_lt $ list.length_pos_of_ne_nil hl)] },
{ assume i hi hi',
dsimp [of_list, to_list],
rw [list.nth_le_of_fn'],
refl }
end
/-- Two `composition_series` are equal if they have the same elements. See also `ext_fun`. -/
@[ext] lemma ext {s₁ s₂ : composition_series X} (h : ∀ x, x ∈ s₁ ↔ x ∈ s₂) : s₁ = s₂ :=
to_list_injective $ list.eq_of_perm_of_sorted
(by classical; exact list.perm_of_nodup_nodup_to_finset_eq
s₁.to_list_nodup
s₂.to_list_nodup
(finset.ext $ by simp *))
s₁.to_list_sorted s₂.to_list_sorted
/-- The largest element of a `composition_series` -/
def top (s : composition_series X) : X := s (fin.last _)
lemma top_mem (s : composition_series X) : s.top ∈ s :=
mem_def.2 (set.mem_range.2 ⟨fin.last _, rfl⟩)
@[simp] lemma le_top {s : composition_series X} (i : fin (s.length + 1)) : s i ≤ s.top :=
s.strict_mono.monotone (fin.le_last _)
lemma le_top_of_mem {s : composition_series X} {x : X} (hx : x ∈ s) : x ≤ s.top :=
let ⟨i, hi⟩ := set.mem_range.2 hx in hi ▸ le_top _
/-- The smallest element of a `composition_series` -/
def bot (s : composition_series X) : X := s 0
lemma bot_mem (s : composition_series X) : s.bot ∈ s :=
mem_def.2 (set.mem_range.2 ⟨0, rfl⟩)
@[simp] lemma bot_le {s : composition_series X} (i : fin (s.length + 1)) : s.bot ≤ s i :=
s.strict_mono.monotone (fin.zero_le _)
lemma bot_le_of_mem {s : composition_series X} {x : X} (hx : x ∈ s) : s.bot ≤ x :=
let ⟨i, hi⟩ := set.mem_range.2 hx in hi ▸ bot_le _
lemma length_pos_of_mem_ne {s : composition_series X}
{x y : X} (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y) :
0 < s.length :=
let ⟨i, hi⟩ := hx, ⟨j, hj⟩ := hy in
have hij : i ≠ j, from mt s.inj.2 $ λ h, hxy (hi ▸ hj ▸ h),
hij.lt_or_lt.elim
(λ hij, (lt_of_le_of_lt (zero_le i)
(lt_of_lt_of_le hij (nat.le_of_lt_succ j.2))))
(λ hji, (lt_of_le_of_lt (zero_le j)
(lt_of_lt_of_le hji (nat.le_of_lt_succ i.2))))
lemma forall_mem_eq_of_length_eq_zero {s : composition_series X}
(hs : s.length = 0) {x y} (hx : x ∈ s) (hy : y ∈ s) : x = y :=
by_contradiction (λ hxy, pos_iff_ne_zero.1 (length_pos_of_mem_ne hx hy hxy) hs)
/-- Remove the largest element from a `composition_series`. If the series `s`
has length zero, then `s.erase_top = s` -/
@[simps] def erase_top (s : composition_series X) : composition_series X :=
{ length := s.length - 1,
series := λ i, s ⟨i, lt_of_lt_of_le i.2 (nat.succ_le_succ tsub_le_self)⟩,
step' := λ i, begin
have := s.step ⟨i, lt_of_lt_of_le i.2 tsub_le_self⟩,
cases i,
exact this
end }
lemma top_erase_top (s : composition_series X) :
s.erase_top.top = s ⟨s.length - 1, lt_of_le_of_lt tsub_le_self (nat.lt_succ_self _)⟩ :=
show s _ = s _, from congr_arg s
begin
ext,
simp only [erase_top_length, fin.coe_last, fin.coe_cast_succ, fin.coe_of_nat_eq_mod,
fin.coe_mk, coe_coe]
end
lemma erase_top_top_le (s : composition_series X) : s.erase_top.top ≤ s.top :=
by simp [erase_top, top, s.strict_mono.le_iff_le, fin.le_iff_coe_le_coe, tsub_le_self]
@[simp] lemma bot_erase_top (s : composition_series X) : s.erase_top.bot = s.bot := rfl
lemma mem_erase_top_of_ne_of_mem {s : composition_series X} {x : X}
(hx : x ≠ s.top) (hxs : x ∈ s) : x ∈ s.erase_top :=
begin
rcases hxs with ⟨i, rfl⟩,
have hi : (i : ℕ) < (s.length - 1).succ,
{ conv_rhs { rw [← nat.succ_sub (length_pos_of_mem_ne ⟨i, rfl⟩ s.top_mem hx),
nat.succ_sub_one] },
exact lt_of_le_of_ne
(nat.le_of_lt_succ i.2)
(by simpa [top, s.inj, fin.ext_iff] using hx) },
refine ⟨i.cast_succ, _⟩,
simp [fin.ext_iff, nat.mod_eq_of_lt hi]
end
lemma mem_erase_top {s : composition_series X} {x : X}
(h : 0 < s.length) : x ∈ s.erase_top ↔ x ≠ s.top ∧ x ∈ s :=
begin
simp only [mem_def],
dsimp only [erase_top, coe_fn_mk],
split,
{ rintros ⟨i, rfl⟩,
have hi : (i : ℕ) < s.length,
{ conv_rhs { rw [← nat.succ_sub_one s.length, nat.succ_sub h] },
exact i.2 },
simp [top, fin.ext_iff, (ne_of_lt hi)] },
{ intro h,
exact mem_erase_top_of_ne_of_mem h.1 h.2 }
end
lemma lt_top_of_mem_erase_top {s : composition_series X} {x : X} (h : 0 < s.length)
(hx : x ∈ s.erase_top) : x < s.top :=
lt_of_le_of_ne
(le_top_of_mem ((mem_erase_top h).1 hx).2)
((mem_erase_top h).1 hx).1
lemma is_maximal_erase_top_top {s : composition_series X} (h : 0 < s.length) :
is_maximal s.erase_top.top s.top :=
have s.length - 1 + 1 = s.length,
by conv_rhs { rw [← nat.succ_sub_one s.length] }; rw nat.succ_sub h,
begin
rw [top_erase_top, top],
convert s.step ⟨s.length - 1, nat.sub_lt h zero_lt_one⟩;
ext; simp [this]
end
lemma append_cast_add_aux
{s₁ s₂ : composition_series X}
(i : fin s₁.length) :
fin.append (nat.add_succ _ _).symm (s₁ ∘ fin.cast_succ) s₂
(fin.cast_add s₂.length i).cast_succ = s₁ i.cast_succ :=
by { cases i, simp [fin.append, *] }
lemma append_succ_cast_add_aux
{s₁ s₂ : composition_series X}
(i : fin s₁.length)
(h : s₁ (fin.last _) = s₂ 0) :
fin.append (nat.add_succ _ _).symm (s₁ ∘ fin.cast_succ) s₂
(fin.cast_add s₂.length i).succ = s₁ i.succ :=
begin
cases i with i hi,
simp only [fin.append, hi, fin.succ_mk, function.comp_app, fin.cast_succ_mk,
fin.coe_mk, fin.cast_add_mk],
split_ifs,
{ refl },
{ have : i + 1 = s₁.length, from le_antisymm hi (le_of_not_gt h_1),
calc s₂ ⟨i + 1 - s₁.length, by simp [this]⟩
= s₂ 0 : congr_arg s₂ (by simp [fin.ext_iff, this])
... = s₁ (fin.last _) : h.symm
... = _ : congr_arg s₁ (by simp [fin.ext_iff, this]) }
end
lemma append_nat_add_aux
{s₁ s₂ : composition_series X}
(i : fin s₂.length) :
fin.append (nat.add_succ _ _).symm (s₁ ∘ fin.cast_succ) s₂
(fin.nat_add s₁.length i).cast_succ = s₂ i.cast_succ :=
begin
cases i,
simp only [fin.append, nat.not_lt_zero, fin.nat_add_mk, add_lt_iff_neg_left,
add_tsub_cancel_left, dif_neg, fin.cast_succ_mk, not_false_iff, fin.coe_mk]
end
lemma append_succ_nat_add_aux
{s₁ s₂ : composition_series X}
(i : fin s₂.length) :
fin.append (nat.add_succ _ _).symm (s₁ ∘ fin.cast_succ) s₂
(fin.nat_add s₁.length i).succ = s₂ i.succ :=
begin
cases i with i hi,
simp only [fin.append, add_assoc, nat.not_lt_zero, fin.nat_add_mk, add_lt_iff_neg_left,
add_tsub_cancel_left, fin.succ_mk, dif_neg, not_false_iff, fin.coe_mk]
end
/-- Append two composition series `s₁` and `s₂` such that
the least element of `s₁` is the maximum element of `s₂`. -/
@[simps length] def append (s₁ s₂ : composition_series X)
(h : s₁.top = s₂.bot) : composition_series X :=
{ length := s₁.length + s₂.length,
series := fin.append (nat.add_succ _ _).symm (s₁ ∘ fin.cast_succ) s₂,
step' := λ i, begin
refine fin.add_cases _ _ i,
{ intro i,
rw [append_succ_cast_add_aux _ h, append_cast_add_aux],
exact s₁.step i },
{ intro i,
rw [append_nat_add_aux, append_succ_nat_add_aux],
exact s₂.step i }
end }
@[simp] lemma append_cast_add {s₁ s₂ : composition_series X}
(h : s₁.top = s₂.bot) (i : fin s₁.length) :
append s₁ s₂ h (fin.cast_add s₂.length i).cast_succ = s₁ i.cast_succ :=
append_cast_add_aux i
@[simp] lemma append_succ_cast_add {s₁ s₂ : composition_series X}
(h : s₁.top = s₂.bot) (i : fin s₁.length) :
append s₁ s₂ h (fin.cast_add s₂.length i).succ = s₁ i.succ :=
append_succ_cast_add_aux i h
@[simp] lemma append_nat_add {s₁ s₂ : composition_series X}
(h : s₁.top = s₂.bot) (i : fin s₂.length) :
append s₁ s₂ h (fin.nat_add s₁.length i).cast_succ = s₂ i.cast_succ :=
append_nat_add_aux i
@[simp] lemma append_succ_nat_add {s₁ s₂ : composition_series X}
(h : s₁.top = s₂.bot) (i : fin s₂.length) :
append s₁ s₂ h (fin.nat_add s₁.length i).succ = s₂ i.succ :=
append_succ_nat_add_aux i
/-- Add an element to the top of a `composition_series` -/
@[simps length] def snoc (s : composition_series X) (x : X)
(hsat : is_maximal s.top x) : composition_series X :=
{ length := s.length + 1,
series := fin.snoc s x,
step' := λ i, begin
refine fin.last_cases _ _ i,
{ rwa [fin.snoc_cast_succ, fin.succ_last, fin.snoc_last, ← top] },
{ intro i,
rw [fin.snoc_cast_succ, ← fin.cast_succ_fin_succ, fin.snoc_cast_succ],
exact s.step _ }
end }
@[simp] lemma top_snoc (s : composition_series X) (x : X)
(hsat : is_maximal s.top x) : (snoc s x hsat).top = x :=
fin.snoc_last _ _
@[simp] lemma snoc_last (s : composition_series X) (x : X) (hsat : is_maximal s.top x) :
snoc s x hsat (fin.last (s.length + 1)) = x :=
fin.snoc_last _ _
@[simp] lemma snoc_cast_succ (s : composition_series X) (x : X) (hsat : is_maximal s.top x)
(i : fin (s.length + 1)) : snoc s x hsat (i.cast_succ) = s i :=
fin.snoc_cast_succ _ _ _
@[simp] lemma bot_snoc (s : composition_series X) (x : X) (hsat : is_maximal s.top x) :
(snoc s x hsat).bot = s.bot :=
by rw [bot, bot, ← fin.cast_succ_zero, snoc_cast_succ]
lemma mem_snoc {s : composition_series X} {x y: X}
{hsat : is_maximal s.top x} : y ∈ snoc s x hsat ↔ y ∈ s ∨ y = x :=
begin
simp only [snoc, mem_def],
split,
{ rintros ⟨i, rfl⟩,
refine fin.last_cases _ (λ i, _) i,
{ right, simp },
{ left, simp } },
{ intro h,
rcases h with ⟨i, rfl⟩ | rfl,
{ use i.cast_succ, simp },
{ use (fin.last _), simp } }
end
lemma eq_snoc_erase_top {s : composition_series X} (h : 0 < s.length) :
s = snoc (erase_top s) s.top (is_maximal_erase_top_top h) :=
begin
ext x,
simp [mem_snoc, mem_erase_top h],
by_cases h : x = s.top; simp [*, s.top_mem]
end
@[simp] lemma snoc_erase_top_top {s : composition_series X}
(h : is_maximal s.erase_top.top s.top) : s.erase_top.snoc s.top h = s :=
have h : 0 < s.length,
from nat.pos_of_ne_zero begin
assume hs,
refine ne_of_gt (lt_of_is_maximal h) _,
simp [top, fin.ext_iff, hs]
end,
(eq_snoc_erase_top h).symm
/-- Two `composition_series X`, `s₁` and `s₂` are equivalent if there is a bijection
`e : fin s₁.length ≃ fin s₂.length` such that for any `i`,
`iso (s₁ i) (s₁ i.succ) (s₂ (e i), s₂ (e i.succ))` -/
def equivalent (s₁ s₂ : composition_series X) : Prop :=
∃ f : fin s₁.length ≃ fin s₂.length,
∀ i : fin s₁.length,
iso (s₁ i.cast_succ, s₁ i.succ)
(s₂ (f i).cast_succ, s₂ (f i).succ)
namespace equivalent
@[refl] lemma refl (s : composition_series X) : equivalent s s :=
⟨equiv.refl _, λ _, (s.step _).iso_refl⟩
@[symm] lemma symm {s₁ s₂ : composition_series X} (h : equivalent s₁ s₂) :
equivalent s₂ s₁ :=
⟨h.some.symm, λ i, iso_symm (by simpa using h.some_spec (h.some.symm i))⟩
@[trans] lemma trans {s₁ s₂ s₃ : composition_series X}
(h₁ : equivalent s₁ s₂)
(h₂ : equivalent s₂ s₃) :
equivalent s₁ s₃ :=
⟨h₁.some.trans h₂.some, λ i, iso_trans (h₁.some_spec i) (h₂.some_spec (h₁.some i))⟩
lemma append
{s₁ s₂ t₁ t₂ : composition_series X}
(hs : s₁.top = s₂.bot)
(ht : t₁.top = t₂.bot)
(h₁ : equivalent s₁ t₁)
(h₂ : equivalent s₂ t₂) :
equivalent (append s₁ s₂ hs) (append t₁ t₂ ht) :=
let e : fin (s₁.length + s₂.length) ≃ fin (t₁.length + t₂.length) :=
calc fin (s₁.length + s₂.length) ≃ fin s₁.length ⊕ fin s₂.length : fin_sum_fin_equiv.symm
... ≃ fin t₁.length ⊕ fin t₂.length : equiv.sum_congr h₁.some h₂.some
... ≃ fin (t₁.length + t₂.length) : fin_sum_fin_equiv in
⟨e, begin
assume i,
refine fin.add_cases _ _ i,
{ assume i,
simpa [top, bot] using h₁.some_spec i },
{ assume i,
simpa [top, bot] using h₂.some_spec i }
end⟩
protected lemma snoc
{s₁ s₂ : composition_series X}
{x₁ x₂ : X}
{hsat₁ : is_maximal s₁.top x₁}
{hsat₂ : is_maximal s₂.top x₂}
(hequiv : equivalent s₁ s₂)
(htop : iso (s₁.top, x₁) (s₂.top, x₂)) :
equivalent (s₁.snoc x₁ hsat₁) (s₂.snoc x₂ hsat₂) :=
let e : fin s₁.length.succ ≃ fin s₂.length.succ :=
calc fin (s₁.length + 1) ≃ option (fin s₁.length) : fin_succ_equiv_last
... ≃ option (fin s₂.length) : functor.map_equiv option hequiv.some
... ≃ fin (s₂.length + 1) : fin_succ_equiv_last.symm in
⟨e, λ i, begin
refine fin.last_cases _ _ i,
{ simpa [top] using htop },
{ assume i,
simpa [fin.succ_cast_succ] using hequiv.some_spec i }
end⟩
lemma length_eq {s₁ s₂ : composition_series X} (h : equivalent s₁ s₂) : s₁.length = s₂.length :=
by simpa using fintype.card_congr h.some
lemma snoc_snoc_swap
{s : composition_series X}
{x₁ x₂ y₁ y₂ : X}
{hsat₁ : is_maximal s.top x₁}
{hsat₂ : is_maximal s.top x₂}
{hsaty₁ : is_maximal (snoc s x₁ hsat₁).top y₁}
{hsaty₂ : is_maximal (snoc s x₂ hsat₂).top y₂}
(hr₁ : iso (s.top, x₁) (x₂, y₂))
(hr₂ : iso (x₁, y₁) (s.top, x₂)) :
equivalent
(snoc (snoc s x₁ hsat₁) y₁ hsaty₁)
(snoc (snoc s x₂ hsat₂) y₂ hsaty₂) :=
let e : fin (s.length + 1 + 1) ≃ fin (s.length + 1 + 1) :=
equiv.swap (fin.last _) (fin.cast_succ (fin.last _)) in
have h1 : ∀ {i : fin s.length},
i.cast_succ.cast_succ ≠ (fin.last _).cast_succ,
from λ _, ne_of_lt (by simp [fin.cast_succ_lt_last]),
have h2 : ∀ {i : fin s.length},
i.cast_succ.cast_succ ≠ (fin.last _),
from λ _, ne_of_lt (by simp [fin.cast_succ_lt_last]),
⟨e, begin
intro i,
dsimp only [e],
refine fin.last_cases _ (λ i, _) i,
{ erw [equiv.swap_apply_left, snoc_cast_succ, snoc_last, fin.succ_last, snoc_last,
snoc_cast_succ, snoc_cast_succ, fin.succ_cast_succ, snoc_cast_succ,
fin.succ_last, snoc_last],
exact hr₂ },
{ refine fin.last_cases _ (λ i, _) i,
{ erw [equiv.swap_apply_right, snoc_cast_succ, snoc_cast_succ,
snoc_cast_succ, fin.succ_cast_succ, snoc_cast_succ,
fin.succ_last, snoc_last, snoc_last, fin.succ_last, snoc_last],
exact hr₁ },
{ erw [equiv.swap_apply_of_ne_of_ne h2 h1, snoc_cast_succ, snoc_cast_succ,
snoc_cast_succ, snoc_cast_succ, fin.succ_cast_succ, snoc_cast_succ,
fin.succ_cast_succ, snoc_cast_succ, snoc_cast_succ, snoc_cast_succ],
exact (s.step i).iso_refl } }
end⟩
end equivalent
lemma length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero
{s₁ s₂ : composition_series X}
(hb : s₁.bot = s₂.bot) (ht : s₁.top = s₂.top)
(hs₁ : s₁.length = 0) : s₂.length = 0 :=
begin
have : s₁.bot = s₁.top,
from congr_arg s₁ (fin.ext (by simp [hs₁])),
have : (fin.last s₂.length) = (0 : fin s₂.length.succ),
from s₂.injective (hb.symm.trans (this.trans ht)).symm,
simpa [fin.ext_iff]
end
lemma length_pos_of_bot_eq_bot_of_top_eq_top_of_length_pos
{s₁ s₂ : composition_series X}
(hb : s₁.bot = s₂.bot) (ht : s₁.top = s₂.top) :
0 < s₁.length → 0 < s₂.length :=
not_imp_not.1 begin
simp only [pos_iff_ne_zero, ne.def, not_iff_not, not_not],
exact length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero hb.symm ht.symm
end
lemma eq_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero
{s₁ s₂ : composition_series X}
(hb : s₁.bot = s₂.bot) (ht : s₁.top = s₂.top)
(hs₁0 : s₁.length = 0) :
s₁ = s₂ :=
have ∀ x, x ∈ s₁ ↔ x = s₁.top,
from λ x, ⟨λ hx, forall_mem_eq_of_length_eq_zero hs₁0 hx s₁.top_mem, λ hx, hx.symm ▸ s₁.top_mem⟩,
have ∀ x, x ∈ s₂ ↔ x = s₂.top,
from λ x, ⟨λ hx, forall_mem_eq_of_length_eq_zero
(length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero hb ht hs₁0)
hx s₂.top_mem, λ hx, hx.symm ▸ s₂.top_mem⟩,
by { ext, simp * }
/-- Given a `composition_series`, `s`, and an element `x`
such that `x` is maximal inside `s.top` there is a series, `t`,
such that `t.top = x`, `t.bot = s.bot`
and `snoc t s.top _` is equivalent to `s`. -/
lemma exists_top_eq_snoc_equivalant (s : composition_series X) (x : X)
(hm : is_maximal x s.top) (hb : s.bot ≤ x) :
∃ t : composition_series X, t.bot = s.bot ∧ t.length + 1 = s.length ∧
∃ htx : t.top = x, equivalent s (snoc t s.top (htx.symm ▸ hm)) :=
begin
induction hn : s.length with n ih generalizing s x,
{ exact (ne_of_gt (lt_of_le_of_lt hb (lt_of_is_maximal hm))
(forall_mem_eq_of_length_eq_zero hn s.top_mem s.bot_mem)).elim },
{ have h0s : 0 < s.length, from hn.symm ▸ nat.succ_pos _,
by_cases hetx : s.erase_top.top = x,
{ use s.erase_top,
simp [← hetx, hn] },
{ have imxs : is_maximal (x ⊓ s.erase_top.top) s.erase_top.top,
from is_maximal_of_eq_inf x s.top rfl (ne.symm hetx) hm
(is_maximal_erase_top_top h0s),
have := ih _ _ imxs (le_inf (by simpa) (le_top_of_mem s.erase_top.bot_mem)) (by simp [hn]),
rcases this with ⟨t, htb, htl, htt, hteqv⟩,
have hmtx : is_maximal t.top x,
from is_maximal_of_eq_inf s.erase_top.top s.top
(by rw [inf_comm, htt]) hetx
(is_maximal_erase_top_top h0s) hm,
use snoc t x hmtx,
refine ⟨by simp [htb], by simp [htl], by simp, _⟩,
have : s.equivalent ((snoc t s.erase_top.top (htt.symm ▸ imxs)).snoc s.top
(by simpa using is_maximal_erase_top_top h0s)),
{ conv_lhs { rw eq_snoc_erase_top h0s },
exact equivalent.snoc hteqv
(by simpa using (is_maximal_erase_top_top h0s).iso_refl) },
refine this.trans _,
refine equivalent.snoc_snoc_swap _ _,
{ exact iso_symm (second_iso_of_eq hm
(sup_eq_of_is_maximal hm
(is_maximal_erase_top_top h0s)
(ne.symm hetx))
htt.symm) },
{ exact second_iso_of_eq (is_maximal_erase_top_top h0s)
(sup_eq_of_is_maximal
(is_maximal_erase_top_top h0s)
hm hetx)
(by rw [inf_comm, htt]) } } }
end
/-- The **Jordan-Hölder** theorem, stated for any `jordan_holder_lattice`.
If two composition series start and finish at the same place, they are equivalent. -/
theorem jordan_holder (s₁ s₂ : composition_series X)
(hb : s₁.bot = s₂.bot) (ht : s₁.top = s₂.top) :
equivalent s₁ s₂ :=
begin
induction hle : s₁.length with n ih generalizing s₁ s₂,
{ rw [eq_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero hb ht hle] },
{ have h0s₂ : 0 < s₂.length,
from length_pos_of_bot_eq_bot_of_top_eq_top_of_length_pos hb ht (hle.symm ▸ nat.succ_pos _),
rcases exists_top_eq_snoc_equivalant s₁ s₂.erase_top.top
(ht.symm ▸ is_maximal_erase_top_top h0s₂)
(hb.symm ▸ s₂.bot_erase_top ▸ bot_le_of_mem (top_mem _)) with ⟨t, htb, htl, htt, hteq⟩,
have := ih t s₂.erase_top (by simp [htb, ← hb]) htt (nat.succ_inj'.1 (htl.trans hle)),
refine hteq.trans _,
conv_rhs { rw [eq_snoc_erase_top h0s₂] },
simp only [ht],
exact equivalent.snoc this
(by simp [htt, (is_maximal_erase_top_top h0s₂).iso_refl]) }
end
end composition_series
|
2f378d28d91c0f586758992f2c274b5304f9e42b | 874a8d2247ab9a4516052498f80da2e32d0e3a48 | /eulerv2.lean | fd2c36280d83c1aa7df22c230b71719cb780af40 | [] | no_license | AlexKontorovich/Spring2020Math492 | 378b36c643ee029f5ab91c1677889baa591f5e85 | 659108c5d864ff5c75b9b3b13b847aa5cff4348a | refs/heads/master | 1,610,780,595,457 | 1,588,174,859,000 | 1,588,174,859,000 | 243,017,788 | 0 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 6,148 | lean | import analysis.normed_space.basic
import topology.instances.ennreal
import analysis.normed_space.basic
import topology.instances.ennreal
import algebra.archimedean algebra.geom_sum
import data.nat.choose data.complex.basic
import tactic.linarith
import analysis.calculus.deriv
import data.complex.exponential
open finset
open cau_seq
namespace complex
noncomputable theory
lemma is_cau_abs_cos (z : ℂ) : is_cau_seq _root_.abs
(λ n, (range n).sum (λ m, abs (
((-1) ^ m) * z ^ (2 * m ) / nat.fact (2 * m )))) :=
begin
sorry,
end
lemma is_cau_abs_sin (z : ℂ) : is_cau_seq _root_.abs
(λ n, (range n).sum (λ m, abs (
((-1) ^ m) * z ^ (2 * m + 1) / nat.fact (2 * m + 1)))) :=
begin
sorry,
/-
let ⟨n, hn⟩ := exists_nat_gt (abs z) in
have hn0 : (0 : ℝ) < n, from lt_of_le_of_lt (abs_nonneg _) hn,
series_ratio_test n (complex.abs z / n)
(div_nonneg_of_nonneg_of_pos (complex.abs_nonneg _) hn0)
(by rwa [div_lt_iff hn0, one_mul])
(λ m hm,
by rw [abs_abs, abs_abs, nat.fact_succ, pow_succ,
mul_comm m.succ, nat.cast_mul, ← div_div_eq_div_mul, mul_div_assoc,
mul_div_right_comm, abs_mul, abs_div, abs_cast_nat];
exact mul_le_mul_of_nonneg_right
(div_le_div_of_le_left (abs_nonneg _) hn0
(nat.cast_le.2 (le_trans hm (nat.le_succ _)))) (abs_nonneg _))
-/
end
lemma is_cau_sin (z : ℂ) :
is_cau_seq abs (λ n, (range n).sum (λ m,
((-1) ^ m) * z ^ (2 * m + 1) / nat.fact (2 * m + 1)))
:=
begin
exact is_cau_series_of_abv_cau (is_cau_abs_sin z),
end
lemma is_cau_cos (z : ℂ) :
is_cau_seq abs (λ n, (range n).sum (λ m,
((-1) ^ m) * z ^ (2 * m ) / nat.fact (2 * m)))
:=
begin
exact is_cau_series_of_abv_cau (is_cau_abs_cos z),
end
def sin' (z : ℂ) : cau_seq ℂ complex.abs :=
⟨λ n, (range n).sum
(λ m, ((-1) ^ m) * z ^ (2 * m + 1) / nat.fact (2 * m + 1)),
is_cau_sin z⟩
def sin1 (z : ℂ) : ℂ := lim (sin' z)
def cos' (z : ℂ) : cau_seq ℂ complex.abs :=
⟨λ n, (range n).sum
(λ m, (-1) ^ m * z ^ (2 * m) / nat.fact (2 * m)),
is_cau_cos z⟩
def cos1 (z : ℂ) : ℂ := lim (cos' z)
theorem euler : ∀ x, exp (x * I) = cos1 x + sin1 x * I
:=
begin
intros,
have partials: ∀ n:ℕ , (exp' (x*I)).1 (2*n+1) =
(cos' x).1 (n+1) + ((sin' x).1 n) * I,
{
intros,
rw exp',
simp,
rw cos',
simp,
rw sin',
simp,
induction n with n0 hn,
{ -- case n0=0
simp,
-- simp,
},
{ -- induction on n0
rw sum_range_succ _ _, -- takes out last term in cos
have lastSin :
sum (range (nat.succ n0)) (λ (x_1 : ℕ), (-1) ^ x_1 * x ^ (1 + 2 * x_1) / ↑(nat.fact (1 + 2 * x_1)))
=
(-1) ^ n0 * x ^ (1 + 2 * n0) / ↑(nat.fact (1 + 2 * n0))
+
sum (range (n0)) (λ (x_1 : ℕ), (-1) ^ x_1 * x ^ (1 + 2 * x_1) / ↑(nat.fact (1 + 2 * x_1)))
,
{
rw sum_range_succ _ _,
},
have sinFactorial :
sum (range (nat.succ n0)) (λ (x_1 : ℕ), (-1) ^ x_1 * x ^ (2 * x_1 + 1) / ((2 * ↑x_1 + 1) * ↑(nat.fact (2 * x_1))))
=
sum (range (nat.succ n0)) (λ (x_1 : ℕ), (-1) ^ x_1 * x ^ (1 + 2 * x_1) / ↑(nat.fact (1 + 2 * x_1))),
{
rw add_comm,
rw nat.fact_succ,
by library_search,
},
rw sinFactorial,
rw lastSin,
have twoFromExp:
sum (range (nat.succ (2 * nat.succ n0))) (λ (x_1 : ℕ),
(x * I) ^ x_1 / ↑(nat.fact x_1))
=
(x * I) ^ ( 2 * nat.succ n0) / ↑(nat.fact ( 2 * nat.succ n0))+
sum (range ( 2 * nat.succ n0)) (λ (x_1 : ℕ),
(x * I) ^ x_1 / ↑(nat.fact x_1))
,
{
rw sum_range_succ _ ( 2 * nat.succ n0),
},
have twoFromExpv1 :
(x * I) ^ ( 2 * nat.succ n0) / ↑(nat.fact ( 2 * nat.succ n0))+sum (range ( 2 * nat.succ n0)) (λ (x_1 : ℕ), (x * I) ^ x_1 / ↑(nat.fact x_1))
=
sum (range (nat.succ (2 * nat.succ n0))) (λ (x_1 : ℕ),
(x * I) ^ x_1 / ↑(nat.fact x_1)),
{
by exact eq.symm twoFromExp,
},
have twoNP1 : 1+ (2 * nat.succ n0) =nat.succ (2 * nat.succ n0)
,
{
exact add_comm 1 (2 * nat.succ n0),
},
have twoNP1v1 : nat.succ ( 2 * nat.succ n0) = 1 + (2 * nat.succ n0),
{
by exact eq.symm twoNP1,
},
--rw twoNP1v1,
--rw twoFromExpv1,
have oneFromExp:
sum (range (2 * nat.succ n0)) (λ (x_1 : ℕ), (x * I) ^ x_1 / ↑(nat.fact x_1))
=
(x * I) ^ (1+2 * n0) / ↑(nat.fact (1+2 * n0))
+
sum (range (1+ 2 * n0)) (λ (x_1 : ℕ), (x * I) ^ x_1 / ↑(nat.fact x_1))
,
{
have stupid : 2 * nat.succ n0 = nat.succ(1+2*n0),
{
have RRS : nat.succ(1+2*n0) = 1+(1+2*n0) ,
{
ring,
},
rw RRS,
have realSt : 1+(1+2*n0) = 1+1+2*n0,
{
simp,
},
rw realSt,
exact nat.add_comm (nat.mul 2 n0) 2,
},
rw stupid,
rw sum_range_succ _ (1 + 2 * n0),
},
rw oneFromExp,
--rw hn,
simp,
rw nat.succ_eq_add_one,
ring,
sorry,
-- need (x*I)^(2 n0 )= x^(2n0) (-1)^n0 etc, commutativity
},
},
have partialExp : lim (exp' (x * I)) = exp (x*I),
{
sorry,
},
rw exp,
rw partialExp,
-- Now: need to take limits on both sides, they're same,
-- but need to convince Lean that limit of exp' (2n+1) is same as
-- limit of exp' (n) which is what exp is defined to be
end
end complex
|
eec770fd2b1058cf796e0500472edc8381d8ea34 | 6b10c15e653d49d146378acda9f3692e9b5b1950 | /examples/logic/unnamed_293.lean | f9224b471228dc0411abcd8dc9e22fcbc5ead492 | [] | no_license | gebner/mathematics_in_lean | 3cf7f18767208ea6c3307ec3a67c7ac266d8514d | 6d1462bba46d66a9b948fc1aef2714fd265cde0b | refs/heads/master | 1,655,301,945,565 | 1,588,697,505,000 | 1,588,697,505,000 | 261,523,603 | 0 | 0 | null | 1,588,695,611,000 | 1,588,695,610,000 | null | UTF-8 | Lean | false | false | 114 | lean | variables A B : Prop
-- BEGIN
example : A ∧ B → B ∧ A :=
begin
intro h,
exact ⟨h.2, h.1⟩
end
-- END |
7a46146b954a8d7f968259d86662aa94bcf56a90 | 798dd332c1ad790518589a09bc82459fb12e5156 | /tests/linarith.lean | a06c5a25555051a7a9edc180c680c34b064495a7 | [
"Apache-2.0"
] | permissive | tobiasgrosser/mathlib | b040b7eb42d5942206149371cf92c61404de3c31 | 120635628368ec261e031cefc6d30e0304088b03 | refs/heads/master | 1,644,803,442,937 | 1,536,663,752,000 | 1,536,663,907,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,176 | lean | import tactic.linarith
example (h1 : (1 : ℚ) < 1) : false :=
by linarith
example (e b c a v0 v1 : ℚ) (h1 : v0 = 5*a) (h2 : v1 = 3*b) (h3 : v0 + v1 + c = 10) :
v0 + 5 + (v1 - 3) + (c - 2) = 10 :=
by linarith
example (ε : ℚ) (h1 : ε > 0) : ε / 2 + ε / 3 + ε / 7 < ε :=
by linarith
example (x y z : ℚ) (h1 : 2*x < 3*y) (h2 : -4*x + z/2 < 0)
(h3 : 12*y - z < 0) : false :=
by linarith
example (ε : ℚ) (h1 : ε > 0) : ε / 2 < ε :=
by linarith
example (ε : ℚ) (h1 : ε > 0) : ε / 3 + ε / 3 + ε / 3 = ε :=
by linarith
example (a b c : ℚ) (h2 : b + 2 > 3 + b) : false :=
by linarith {discharger := `[ring SOP]}
example (a b c : ℚ) (h2 : b + 2 > 3 + b) : false :=
by linarith
example (a b c : ℚ) (x y : ℤ) (h1 : x ≤ 3*y) (h2 : b + 2 > 3 + b) : false :=
by linarith {restrict_type := ℚ}
example (g v V c h : ℚ) (h1 : h = 0) (h2 : v = V) (h3 : V > 0) (h4 : g > 0)
(h5 : 0 ≤ c) (h6 : c < 1) :
v ≤ V :=
by linarith
example (x y z : ℚ) (h1 : 2*x + ((-3)*y) < 0) (h2 : (-4)*x + 2*z < 0)
(h3 : 12*y + (-4)* z < 0) (h4 : nat.prime 7) : false :=
by linarith
example (x y z : ℚ) (h1 : 2*1*x + (3)*(y*(-1)) < 0) (h2 : (-2)*x*2 < -(z + z))
(h3 : 12*y + (-4)* z < 0) (h4 : nat.prime 7) : false :=
by linarith
example (x y z : ℤ) (h1 : 2*x < 3*y) (h2 : -4*x + 2*z < 0)
(h3 : 12*y - 4* z < 0) : false :=
by linarith
example (x y z : ℤ) (h1 : 2*x < 3*y) (h2 : -4*x + 2*z < 0) (h3 : x*y < 5)
(h3 : 12*y - 4* z < 0) : false :=
by linarith
example (w x y z : ℤ) (h1 : 4*x + (-3)*y + 6*w ≤ 0) (h2 : (-1)*x < 0)
(h3 : y < 0) (h4 : w ≥ 0) (h5 : nat.prime x.nat_abs) : false :=
by linarith
example (a b c : ℚ) (h1 : a > 0) (h2 : b > 5) (h3 : c < -10)
(h4 : a + b - c < 3) : false :=
by linarith
example (a b c : ℚ) (h2 : b > 0) (h3 : b < 0) : false :=
by linarith
example (a b c : ℚ) (h2 : (2 : ℚ) > 3) : a + b - c ≥ 3 :=
by linarith
example (x : ℚ) (hx : x > 0) (h : x.num < 0) : false :=
by linarith using [rat.num_pos_iff_pos.mpr hx]
example (x y z : ℚ) (hx : x ≤ 3*y) (h2 : y ≤ 2*z) (h3 : x ≥ 6*z) : x = 3*y :=
by linarith |
7834355a28c527bd51626636457cb57a195501f6 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/topology/category/Top/basic.lean | f212483491efe5c7391ea2e9f5aeddf7983fef38 | [
"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 | 3,669 | 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 category_theory.concrete_category.bundled_hom
import category_theory.elementwise
import topology.continuous_function.basic
/-!
# Category instance for topological spaces
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
We introduce the bundled category `Top` of topological spaces together with the functors `discrete`
and `trivial` from the category of types to `Top` which equip a type with the corresponding
discrete, resp. trivial, topology. For a proof that these functors are left, resp. right adjoint
to the forgetful functor, see `topology.category.Top.adjunctions`.
-/
open category_theory
open topological_space
universe u
/-- The category of topological spaces and continuous maps. -/
def Top : Type (u+1) := bundled topological_space
namespace Top
instance bundled_hom : bundled_hom @continuous_map :=
⟨@continuous_map.to_fun, @continuous_map.id, @continuous_map.comp, @continuous_map.coe_injective⟩
attribute [derive [large_category, concrete_category]] Top
instance : has_coe_to_sort Top Type* := bundled.has_coe_to_sort
instance topological_space_unbundled (x : Top) : topological_space x := x.str
@[simp] lemma id_app (X : Top.{u}) (x : X) :
(𝟙 X : X → X) x = x := rfl
@[simp] lemma comp_app {X Y Z : Top.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g : X → Z) x = g (f x) := rfl
/-- Construct a bundled `Top` from the underlying type and the typeclass. -/
def of (X : Type u) [topological_space X] : Top := ⟨X⟩
instance (X : Top) : topological_space X := X.str
@[simp] lemma coe_of (X : Type u) [topological_space X] : (of X : Type u) = X := rfl
instance : inhabited Top := ⟨Top.of empty⟩
/-- The discrete topology on any type. -/
def discrete : Type u ⥤ Top.{u} :=
{ obj := λ X, ⟨X, ⊥⟩,
map := λ X Y f, { to_fun := f, continuous_to_fun := continuous_bot } }
instance {X : Type u} : discrete_topology (discrete.obj X) := ⟨rfl⟩
/-- The trivial topology on any type. -/
def trivial : Type u ⥤ Top.{u} :=
{ obj := λ X, ⟨X, ⊤⟩,
map := λ X Y f, { to_fun := f, continuous_to_fun := continuous_top } }
/-- Any homeomorphisms induces an isomorphism in `Top`. -/
@[simps] def iso_of_homeo {X Y : Top.{u}} (f : X ≃ₜ Y) : X ≅ Y :=
{ hom := ⟨f⟩,
inv := ⟨f.symm⟩ }
/-- Any isomorphism in `Top` induces a homeomorphism. -/
@[simps] def homeo_of_iso {X Y : Top.{u}} (f : X ≅ Y) : X ≃ₜ Y :=
{ to_fun := f.hom,
inv_fun := f.inv,
left_inv := λ x, by simp,
right_inv := λ x, by simp,
continuous_to_fun := f.hom.continuous,
continuous_inv_fun := f.inv.continuous }
@[simp] lemma of_iso_of_homeo {X Y : Top.{u}} (f : X ≃ₜ Y) : homeo_of_iso (iso_of_homeo f) = f :=
by { ext, refl }
@[simp] lemma of_homeo_of_iso {X Y : Top.{u}} (f : X ≅ Y) : iso_of_homeo (homeo_of_iso f) = f :=
by { ext, refl }
@[simp]
lemma open_embedding_iff_comp_is_iso {X Y Z : Top} (f : X ⟶ Y) (g : Y ⟶ Z) [is_iso g] :
open_embedding (f ≫ g) ↔ open_embedding f :=
(Top.homeo_of_iso (as_iso g)).open_embedding.of_comp_iff f
@[simp]
lemma open_embedding_iff_is_iso_comp {X Y Z : Top} (f : X ⟶ Y) (g : Y ⟶ Z) [is_iso f] :
open_embedding (f ≫ g) ↔ open_embedding g :=
begin
split,
{ intro h,
convert h.comp (Top.homeo_of_iso (as_iso f).symm).open_embedding,
exact congr_arg _ (is_iso.inv_hom_id_assoc f g).symm },
{ exact λ h, h.comp (Top.homeo_of_iso (as_iso f)).open_embedding }
end
end Top
|
691e5f72f2b9898c38cab9ed9af9cf6afe78355e | b70447c014d9e71cf619ebc9f539b262c19c2e0b | /hott/eq2.hlean | e3d239c094de2c7453272dba56bf5e8771f678cd | [
"Apache-2.0"
] | permissive | ia0/lean2 | c20d8da69657f94b1d161f9590a4c635f8dc87f3 | d86284da630acb78fa5dc3b0b106153c50ffccd0 | refs/heads/master | 1,611,399,322,751 | 1,495,751,007,000 | 1,495,751,007,000 | 93,104,167 | 0 | 0 | null | 1,496,355,488,000 | 1,496,355,487,000 | null | UTF-8 | Lean | false | false | 5,612 | hlean | /-
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
Theorems about 2-dimensional paths
-/
import .cubical.square
open function
namespace eq
variables {A B C : Type} {f : A → B} {a a' a₁ a₂ a₃ a₄ : A} {b b' : B}
theorem ap_is_constant_eq (p : Πx, f x = b) (q : a = a') :
ap_is_constant p q =
eq_con_inv_of_con_eq ((eq_of_square (square_of_pathover (apd p q)))⁻¹ ⬝
whisker_left (p a) (ap_constant q b)) :=
begin
induction q, esimp, generalize (p a), intro p, cases p, apply idpath idp
end
definition ap_inv2 {p q : a = a'} (r : p = q)
: square (ap (ap f) (inverse2 r))
(inverse2 (ap (ap f) r))
(ap_inv f p)
(ap_inv f q) :=
by induction r;exact hrfl
definition ap_con2 {p₁ q₁ : a₁ = a₂} {p₂ q₂ : a₂ = a₃} (r₁ : p₁ = q₁) (r₂ : p₂ = q₂)
: square (ap (ap f) (r₁ ◾ r₂))
(ap (ap f) r₁ ◾ ap (ap f) r₂)
(ap_con f p₁ p₂)
(ap_con f q₁ q₂) :=
by induction r₂;induction r₁;exact hrfl
theorem ap_con_right_inv_sq {A B : Type} {a1 a2 : A} (f : A → B) (p : a1 = a2) :
square (ap (ap f) (con.right_inv p))
(con.right_inv (ap f p))
(ap_con f p p⁻¹ ⬝ whisker_left _ (ap_inv f p))
idp :=
by cases p;apply hrefl
theorem ap_con_left_inv_sq {A B : Type} {a1 a2 : A} (f : A → B) (p : a1 = a2) :
square (ap (ap f) (con.left_inv p))
(con.left_inv (ap f p))
(ap_con f p⁻¹ p ⬝ whisker_right _ (ap_inv f p))
idp :=
by cases p;apply vrefl
definition ap02_compose {A B C : Type} (g : B → C) (f : A → B) {a a' : A}
{p₁ p₂ : a = a'} (q : p₁ = p₂) :
square (ap_compose g f p₁) (ap_compose g f p₂) (ap02 (g ∘ f) q) (ap02 g (ap02 f q)) :=
by induction q; exact vrfl
definition ap02_id {A : Type} {a a' : A}
{p₁ p₂ : a = a'} (q : p₁ = p₂) :
square (ap_id p₁) (ap_id p₂) (ap02 id q) q :=
by induction q; exact vrfl
theorem ap_ap_is_constant {A B C : Type} (g : B → C) {f : A → B} {b : B}
(p : Πx, f x = b) {x y : A} (q : x = y) :
square (ap (ap g) (ap_is_constant p q))
(ap_is_constant (λa, ap g (p a)) q)
(ap_compose g f q)⁻¹
(!ap_con ⬝ whisker_left _ !ap_inv) :=
begin
induction q, esimp, generalize (p x), intro p, cases p, apply ids
-- induction q, rewrite [↑ap_compose,ap_inv], apply hinverse, apply ap_con_right_inv_sq,
end
theorem ap_ap_compose {A B C D : Type} (h : C → D) (g : B → C) (f : A → B)
{x y : A} (p : x = y) :
square (ap_compose (h ∘ g) f p)
(ap (ap h) (ap_compose g f p))
(ap_compose h (g ∘ f) p)
(ap_compose h g (ap f p)) :=
by induction p;exact ids
theorem ap_compose_inv {A B C : Type} (g : B → C) (f : A → B)
{x y : A} (p : x = y) :
square (ap_compose g f p⁻¹)
(inverse2 (ap_compose g f p) ⬝ (ap_inv g (ap f p))⁻¹)
(ap_inv (g ∘ f) p)
(ap (ap g) (ap_inv f p)) :=
by induction p;exact ids
theorem ap_compose_con (g : B → C) (f : A → B) (p : a₁ = a₂) (q : a₂ = a₃) :
square (ap_compose g f (p ⬝ q))
(ap_compose g f p ◾ ap_compose g f q ⬝ (ap_con g (ap f p) (ap f q))⁻¹)
(ap_con (g ∘ f) p q)
(ap (ap g) (ap_con f p q)) :=
by induction q;induction p;exact ids
theorem ap_compose_natural {A B C : Type} (g : B → C) (f : A → B)
{x y : A} {p q : x = y} (r : p = q) :
square (ap (ap (g ∘ f)) r)
(ap (ap g ∘ ap f) r)
(ap_compose g f p)
(ap_compose g f q) :=
natural_square_tr (ap_compose g f) r
theorem whisker_right_eq_of_con_inv_eq_idp {p q : a₁ = a₂} (r : p ⬝ q⁻¹ = idp) :
whisker_right q⁻¹ (eq_of_con_inv_eq_idp r) ⬝ con.right_inv q = r :=
by induction q; esimp at r; cases r; reflexivity
theorem ap_eq_of_con_inv_eq_idp (f : A → B) {p q : a₁ = a₂} (r : p ⬝ q⁻¹ = idp)
: ap02 f (eq_of_con_inv_eq_idp r) =
eq_of_con_inv_eq_idp (whisker_left _ !ap_inv⁻¹ ⬝ !ap_con⁻¹ ⬝ ap02 f r)
:=
by induction q;esimp at *;cases r;reflexivity
theorem eq_of_con_inv_eq_idp_con2 {p p' q q' : a₁ = a₂} (r : p = p') (s : q = q')
(t : p' ⬝ q'⁻¹ = idp)
: eq_of_con_inv_eq_idp (r ◾ inverse2 s ⬝ t) = r ⬝ eq_of_con_inv_eq_idp t ⬝ s⁻¹ :=
by induction s;induction r;induction q;reflexivity
definition naturality_apd_eq {A : Type} {B : A → Type} {a a₂ : A} {f g : Πa, B a}
(H : f ~ g) (p : a = a₂)
: apd f p = concato_eq (eq_concato (H a) (apd g p)) (H a₂)⁻¹ :=
begin
induction p, esimp,
generalizes [H a, g a], intro ga Ha, induction Ha,
reflexivity
end
theorem con_tr_idp {P : A → Type} {x y : A} (q : x = y) (u : P x) :
con_tr idp q u = ap (λp, p ▸ u) (idp_con q) :=
by induction q;reflexivity
definition eq_transport_Fl_idp_left {A B : Type} {a : A} {b : B} (f : A → B) (q : f a = b)
: eq_transport_Fl idp q = !idp_con⁻¹ :=
by induction q; reflexivity
definition whisker_left_idp_con_eq_assoc
{A : Type} {a₁ a₂ a₃ : A} (p : a₁ = a₂) (q : a₂ = a₃)
: whisker_left p (idp_con q)⁻¹ = con.assoc p idp q :=
by induction q; reflexivity
definition whisker_left_inverse2 {A : Type} {a : A} {p : a = a} (q : p = idp)
: whisker_left p q⁻² ⬝ q = con.right_inv p :=
by cases q; reflexivity
end eq
|
5509c4f76cab4d7133bfa748fd3c19c6c4933401 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/analysis/convex/contractible.lean | 88bf959d51e582c6625f67c583b722181d8aebd2 | [
"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 | 1,738 | lean | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import analysis.convex.star
import topology.homotopy.contractible
/-!
# A convex set is contractible
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
In this file we prove that a (star) convex set in a real topological vector space is a contractible
topological space.
-/
variables {E : Type*} [add_comm_group E] [module ℝ E] [topological_space E]
[has_continuous_add E] [has_continuous_smul ℝ E] {s : set E} {x : E}
/-- A non-empty star convex set is a contractible space. -/
protected lemma star_convex.contractible_space (h : star_convex ℝ x s) (hne : s.nonempty) :
contractible_space s :=
begin
refine (contractible_iff_id_nullhomotopic _).2 ⟨⟨x, h.mem hne⟩,
⟨⟨⟨λ p, ⟨p.1.1 • x + (1 - p.1.1) • p.2, _⟩, _⟩, λ x, _, λ x, _⟩⟩⟩,
{ exact h p.2.2 p.1.2.1 (sub_nonneg.2 p.1.2.2) (add_sub_cancel'_right _ _) },
{ exact ((continuous_subtype_val.fst'.smul continuous_const).add
((continuous_const.sub continuous_subtype_val.fst').smul
continuous_subtype_val.snd')).subtype_mk _ },
{ ext1, simp },
{ ext1, simp }
end
/-- A non-empty convex set is a contractible space. -/
protected lemma convex.contractible_space (hs : convex ℝ s) (hne : s.nonempty) :
contractible_space s :=
let ⟨x, hx⟩ := hne in (hs.star_convex hx).contractible_space hne
@[priority 100] instance real_topological_vector_space.contractible_space : contractible_space E :=
(homeomorph.set.univ E).contractible_space_iff.mp $ convex_univ.contractible_space set.univ_nonempty
|
120cad99360d3210091aa0af0d6cefb9eba1aeab | 94e33a31faa76775069b071adea97e86e218a8ee | /src/algebra/group/units.lean | 978fa33f275847ae66273e5aaf37cb188c4c0703 | [
"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 | 18,221 | lean | /-
Copyright (c) 2017 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johannes Hölzl, Chris Hughes, Jens Wagemaker
-/
import algebra.group.basic
import logic.nontrivial
/-!
# Units (i.e., invertible elements) of a monoid
An element of a `monoid` is a unit if it has a two-sided inverse.
## Main declarations
* `units M`: the group of units (i.e., invertible elements) of a monoid.
* `is_unit x`: a predicate asserting that `x` is a unit (i.e., invertible element) of a monoid.
For both declarations, there is an additive counterpart: `add_units` and `is_add_unit`.
## Notation
We provide `Mˣ` as notation for `units M`,
resembling the notation $R^{\times}$ for the units of a ring, which is common in mathematics.
-/
open function
universe u
variable {α : Type u}
/-- Units of a `monoid`, bundled version. Notation: `αˣ`.
An element of a `monoid` is a unit if it has a two-sided inverse.
This version bundles the inverse element so that it can be computed.
For a predicate see `is_unit`. -/
structure units (α : Type u) [monoid α] :=
(val : α)
(inv : α)
(val_inv : val * inv = 1)
(inv_val : inv * val = 1)
postfix `ˣ`:1025 := units
-- We don't provide notation for the additive version, because its use is somewhat rare.
/-- Units of an `add_monoid`, bundled version.
An element of an `add_monoid` is a unit if it has a two-sided additive inverse.
This version bundles the inverse element so that it can be computed.
For a predicate see `is_add_unit`. -/
structure add_units (α : Type u) [add_monoid α] :=
(val : α)
(neg : α)
(val_neg : val + neg = 0)
(neg_val : neg + val = 0)
attribute [to_additive] units
section has_elem
@[to_additive] lemma unique_has_one {α : Type*} [unique α] [has_one α] :
default = (1 : α) :=
unique.default_eq 1
end has_elem
namespace units
variables [monoid α]
@[to_additive] instance : has_coe αˣ α := ⟨val⟩
@[to_additive] instance : has_inv αˣ := ⟨λ u, ⟨u.2, u.1, u.4, u.3⟩⟩
/-- See Note [custom simps projection] -/
@[to_additive /-" See Note [custom simps projection] "-/]
def simps.coe (u : αˣ) : α := u
/-- See Note [custom simps projection] -/
@[to_additive /-" See Note [custom simps projection] "-/]
def simps.coe_inv (u : αˣ) : α := ↑(u⁻¹)
initialize_simps_projections units (val → coe as_prefix, inv → coe_inv as_prefix)
initialize_simps_projections add_units (val → coe as_prefix, neg → coe_neg as_prefix)
@[simp, to_additive] lemma coe_mk (a : α) (b h₁ h₂) : ↑(units.mk a b h₁ h₂) = a := rfl
@[ext, to_additive] theorem ext :
function.injective (coe : αˣ → α)
| ⟨v, i₁, vi₁, iv₁⟩ ⟨v', i₂, vi₂, iv₂⟩ e :=
by change v = v' at e; subst v'; congr;
simpa only [iv₂, vi₁, one_mul, mul_one] using mul_assoc i₂ v i₁
@[norm_cast, to_additive] theorem eq_iff {a b : αˣ} :
(a : α) = b ↔ a = b := ext.eq_iff
@[to_additive] theorem ext_iff {a b : αˣ} :
a = b ↔ (a : α) = b := eq_iff.symm
@[to_additive] instance [decidable_eq α] : decidable_eq αˣ :=
λ a b, decidable_of_iff' _ ext_iff
@[simp, to_additive] theorem mk_coe (u : αˣ) (y h₁ h₂) :
mk (u : α) y h₁ h₂ = u :=
ext rfl
/-- Copy a unit, adjusting definition equalities. -/
@[to_additive /-"Copy an `add_unit`, adjusting definitional equalities."-/, simps]
def copy (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = ↑(u⁻¹)) : αˣ :=
{ val := val, inv := inv,
inv_val := hv.symm ▸ hi.symm ▸ u.inv_val, val_inv := hv.symm ▸ hi.symm ▸ u.val_inv }
@[to_additive]
lemma copy_eq (u : αˣ) (val hv inv hi) :
u.copy val hv inv hi = u :=
ext hv
@[to_additive] instance : mul_one_class αˣ :=
{ mul := λ u₁ u₂, ⟨u₁.val * u₂.val, u₂.inv * u₁.inv,
by rw [mul_assoc, ←mul_assoc u₂.val, val_inv, one_mul, val_inv],
by rw [mul_assoc, ←mul_assoc u₁.inv, inv_val, one_mul, inv_val]⟩,
one := ⟨1, 1, one_mul 1, one_mul 1⟩,
one_mul := λ u, ext $ one_mul u,
mul_one := λ u, ext $ mul_one u }
/-- Units of a monoid form a group. -/
@[to_additive "Additive units of an additive monoid form an additive group."]
instance : group αˣ :=
{ mul := (*),
one := 1,
mul_assoc := λ u₁ u₂ u₃, ext $ mul_assoc u₁ u₂ u₃,
inv := has_inv.inv,
mul_left_inv := λ u, ext u.inv_val,
..units.mul_one_class }
@[to_additive] instance {α} [comm_monoid α] : comm_group αˣ :=
{ mul_comm := λ u₁ u₂, ext $ mul_comm _ _, ..units.group }
@[to_additive] instance : inhabited αˣ := ⟨1⟩
@[to_additive] instance [has_repr α] : has_repr αˣ := ⟨repr ∘ val⟩
variables (a b c : αˣ) {u : αˣ}
@[simp, norm_cast, to_additive] lemma coe_mul : (↑(a * b) : α) = a * b := rfl
@[simp, norm_cast, to_additive] lemma coe_one : ((1 : αˣ) : α) = 1 := rfl
@[simp, norm_cast, to_additive] lemma coe_eq_one {a : αˣ} : (a : α) = 1 ↔ a = 1 :=
by rw [←units.coe_one, eq_iff]
@[simp, to_additive] lemma inv_mk (x y : α) (h₁ h₂) : (mk x y h₁ h₂)⁻¹ = mk y x h₂ h₁ := rfl
@[simp, to_additive] lemma val_eq_coe : a.val = (↑a : α) := rfl
@[simp, to_additive] lemma inv_eq_coe_inv : a.inv = ((a⁻¹ : αˣ) : α) := rfl
@[simp, to_additive] lemma inv_mul : (↑a⁻¹ * a : α) = 1 := inv_val _
@[simp, to_additive] lemma mul_inv : (a * ↑a⁻¹ : α) = 1 := val_inv _
@[to_additive] lemma inv_mul_of_eq {a : α} (h : ↑u = a) : ↑u⁻¹ * a = 1 := by rw [←h, u.inv_mul]
@[to_additive] lemma mul_inv_of_eq {a : α} (h : ↑u = a) : a * ↑u⁻¹ = 1 := by rw [←h, u.mul_inv]
@[simp, to_additive] lemma mul_inv_cancel_left (a : αˣ) (b : α) : (a:α) * (↑a⁻¹ * b) = b :=
by rw [← mul_assoc, mul_inv, one_mul]
@[simp, to_additive] lemma inv_mul_cancel_left (a : αˣ) (b : α) : (↑a⁻¹:α) * (a * b) = b :=
by rw [← mul_assoc, inv_mul, one_mul]
@[simp, to_additive] lemma mul_inv_cancel_right (a : α) (b : αˣ) : a * b * ↑b⁻¹ = a :=
by rw [mul_assoc, mul_inv, mul_one]
@[simp, to_additive] lemma inv_mul_cancel_right (a : α) (b : αˣ) : a * ↑b⁻¹ * b = a :=
by rw [mul_assoc, inv_mul, mul_one]
@[simp, to_additive] theorem mul_right_inj (a : αˣ) {b c : α} : (a:α) * b = a * c ↔ b = c :=
⟨λ h, by simpa only [inv_mul_cancel_left] using congr_arg ((*) ↑(a⁻¹ : αˣ)) h, congr_arg _⟩
@[simp, to_additive] theorem mul_left_inj (a : αˣ) {b c : α} : b * a = c * a ↔ b = c :=
⟨λ h, by simpa only [mul_inv_cancel_right] using congr_arg (* ↑(a⁻¹ : αˣ)) h, congr_arg _⟩
@[to_additive] theorem eq_mul_inv_iff_mul_eq {a b : α} : a = b * ↑c⁻¹ ↔ a * c = b :=
⟨λ h, by rw [h, inv_mul_cancel_right], λ h, by rw [← h, mul_inv_cancel_right]⟩
@[to_additive] theorem eq_inv_mul_iff_mul_eq {a c : α} : a = ↑b⁻¹ * c ↔ ↑b * a = c :=
⟨λ h, by rw [h, mul_inv_cancel_left], λ h, by rw [← h, inv_mul_cancel_left]⟩
@[to_additive] theorem inv_mul_eq_iff_eq_mul {b c : α} : ↑a⁻¹ * b = c ↔ b = a * c :=
⟨λ h, by rw [← h, mul_inv_cancel_left], λ h, by rw [h, inv_mul_cancel_left]⟩
@[to_additive] theorem mul_inv_eq_iff_eq_mul {a c : α} : a * ↑b⁻¹ = c ↔ a = c * b :=
⟨λ h, by rw [← h, inv_mul_cancel_right], λ h, by rw [h, mul_inv_cancel_right]⟩
@[to_additive] protected lemma inv_eq_of_mul_eq_one_left {a : α} (h : a * u = 1) : ↑u⁻¹ = a :=
calc ↑u⁻¹ = 1 * ↑u⁻¹ : by rw one_mul
... = a : by rw [←h, mul_inv_cancel_right]
@[to_additive] protected lemma inv_eq_of_mul_eq_one_right {a : α} (h : ↑u * a = 1) : ↑u⁻¹ = a :=
calc ↑u⁻¹ = ↑u⁻¹ * 1 : by rw mul_one
... = a : by rw [←h, inv_mul_cancel_left]
@[to_additive] protected lemma eq_inv_of_mul_eq_one_left {a : α} (h : ↑u * a = 1) : a = ↑u⁻¹ :=
(units.inv_eq_of_mul_eq_one_right h).symm
@[to_additive] protected lemma eq_inv_of_mul_eq_one_right {a : α} (h : a * u = 1) : a = ↑u⁻¹ :=
(units.inv_eq_of_mul_eq_one_left h).symm
@[simp, to_additive] lemma mul_inv_eq_one {a : α} : a * ↑u⁻¹ = 1 ↔ a = u :=
⟨inv_inv u ▸ units.eq_inv_of_mul_eq_one_right, λ h, mul_inv_of_eq h.symm⟩
@[simp, to_additive] lemma inv_mul_eq_one {a : α} : ↑u⁻¹ * a = 1 ↔ ↑u = a :=
⟨inv_inv u ▸ units.inv_eq_of_mul_eq_one_right, inv_mul_of_eq⟩
@[to_additive] lemma mul_eq_one_iff_eq_inv {a : α} : a * u = 1 ↔ a = ↑u⁻¹ :=
by rw [←mul_inv_eq_one, inv_inv]
@[to_additive] lemma mul_eq_one_iff_inv_eq {a : α} : ↑u * a = 1 ↔ ↑u⁻¹ = a :=
by rw [←inv_mul_eq_one, inv_inv]
lemma inv_unique {u₁ u₂ : αˣ} (h : (↑u₁ : α) = ↑u₂) : (↑u₁⁻¹ : α) = ↑u₂⁻¹ :=
units.inv_eq_of_mul_eq_one_right $ by rw [h, u₂.mul_inv]
end units
/-- For `a, b` in a `comm_monoid` such that `a * b = 1`, makes a unit out of `a`. -/
@[to_additive "For `a, b` in an `add_comm_monoid` such that `a + b = 0`, makes an add_unit
out of `a`."]
def units.mk_of_mul_eq_one [comm_monoid α] (a b : α) (hab : a * b = 1) :
αˣ :=
⟨a, b, hab, (mul_comm b a).trans hab⟩
@[simp, to_additive] lemma units.coe_mk_of_mul_eq_one [comm_monoid α] {a b : α} (h : a * b = 1) :
(units.mk_of_mul_eq_one a b h : α) = a := rfl
section monoid
variables [monoid α] {a b c : α}
/-- Partial division. It is defined when the
second argument is invertible, and unlike the division operator
in `division_ring` it is not totalized at zero. -/
def divp (a : α) (u) : α := a * (u⁻¹ : αˣ)
infix ` /ₚ `:70 := divp
@[simp] theorem divp_self (u : αˣ) : (u : α) /ₚ u = 1 := units.mul_inv _
@[simp] theorem divp_one (a : α) : a /ₚ 1 = a := mul_one _
theorem divp_assoc (a b : α) (u : αˣ) : a * b /ₚ u = a * (b /ₚ u) :=
mul_assoc _ _ _
@[simp] theorem divp_inv (u : αˣ) : a /ₚ u⁻¹ = a * u := rfl
@[simp] theorem divp_mul_cancel (a : α) (u : αˣ) : a /ₚ u * u = a :=
(mul_assoc _ _ _).trans $ by rw [units.inv_mul, mul_one]
@[simp] theorem mul_divp_cancel (a : α) (u : αˣ) : (a * u) /ₚ u = a :=
(mul_assoc _ _ _).trans $ by rw [units.mul_inv, mul_one]
@[simp] theorem divp_left_inj (u : αˣ) {a b : α} : a /ₚ u = b /ₚ u ↔ a = b :=
units.mul_left_inj _
theorem divp_divp_eq_divp_mul (x : α) (u₁ u₂ : αˣ) : (x /ₚ u₁) /ₚ u₂ = x /ₚ (u₂ * u₁) :=
by simp only [divp, mul_inv_rev, units.coe_mul, mul_assoc]
theorem divp_eq_iff_mul_eq {x : α} {u : αˣ} {y : α} : x /ₚ u = y ↔ y * u = x :=
u.mul_left_inj.symm.trans $ by rw [divp_mul_cancel]; exact ⟨eq.symm, eq.symm⟩
theorem divp_eq_one_iff_eq {a : α} {u : αˣ} : a /ₚ u = 1 ↔ a = u :=
(units.mul_left_inj u).symm.trans $ by rw [divp_mul_cancel, one_mul]
@[simp] theorem one_divp (u : αˣ) : 1 /ₚ u = ↑u⁻¹ :=
one_mul _
end monoid
section comm_monoid
variables [comm_monoid α]
theorem divp_eq_divp_iff {x y : α} {ux uy : αˣ} :
x /ₚ ux = y /ₚ uy ↔ x * uy = y * ux :=
by rw [divp_eq_iff_mul_eq, mul_comm, ← divp_assoc, divp_eq_iff_mul_eq, mul_comm y ux]
theorem divp_mul_divp (x y : α) (ux uy : αˣ) :
(x /ₚ ux) * (y /ₚ uy) = (x * y) /ₚ (ux * uy) :=
by rw [← divp_divp_eq_divp_mul, divp_assoc, mul_comm x, divp_assoc, mul_comm]
end comm_monoid
/-!
# `is_unit` predicate
In this file we define the `is_unit` predicate on a `monoid`, and
prove a few basic properties. For the bundled version see `units`. See
also `prime`, `associated`, and `irreducible` in `algebra/associated`.
-/
section is_unit
variables {M : Type*} {N : Type*}
/-- An element `a : M` of a monoid is a unit if it has a two-sided inverse.
The actual definition says that `a` is equal to some `u : Mˣ`, where
`Mˣ` is a bundled version of `is_unit`. -/
@[to_additive "An element `a : M` of an add_monoid is an `add_unit` if it has
a two-sided additive inverse. The actual definition says that `a` is equal to some
`u : add_units M`, where `add_units M` is a bundled version of `is_add_unit`."]
def is_unit [monoid M] (a : M) : Prop := ∃ u : Mˣ, (u : M) = a
@[nontriviality, to_additive]
lemma is_unit_of_subsingleton [monoid M] [subsingleton M] (a : M) : is_unit a :=
⟨⟨a, a, subsingleton.elim _ _, subsingleton.elim _ _⟩, rfl⟩
attribute [nontriviality] is_add_unit_of_subsingleton
@[to_additive] instance [monoid M] : can_lift M Mˣ :=
{ coe := coe,
cond := is_unit,
prf := λ _, id }
@[to_additive] instance [monoid M] [subsingleton M] : unique Mˣ :=
{ default := 1,
uniq := λ a, units.coe_eq_one.mp $ subsingleton.elim (a : M) 1 }
@[simp, to_additive is_add_unit_add_unit]
protected lemma units.is_unit [monoid M] (u : Mˣ) : is_unit (u : M) := ⟨u, rfl⟩
@[simp, to_additive]
theorem is_unit_one [monoid M] : is_unit (1:M) := ⟨1, rfl⟩
@[to_additive] theorem is_unit_of_mul_eq_one [comm_monoid M]
(a b : M) (h : a * b = 1) : is_unit a :=
⟨units.mk_of_mul_eq_one a b h, rfl⟩
@[to_additive is_add_unit.exists_neg] theorem is_unit.exists_right_inv [monoid M]
{a : M} (h : is_unit a) : ∃ b, a * b = 1 :=
by { rcases h with ⟨⟨a, b, hab, _⟩, rfl⟩, exact ⟨b, hab⟩ }
@[to_additive is_add_unit.exists_neg'] theorem is_unit.exists_left_inv [monoid M]
{a : M} (h : is_unit a) : ∃ b, b * a = 1 :=
by { rcases h with ⟨⟨a, b, _, hba⟩, rfl⟩, exact ⟨b, hba⟩ }
@[to_additive] theorem is_unit_iff_exists_inv [comm_monoid M]
{a : M} : is_unit a ↔ ∃ b, a * b = 1 :=
⟨λ h, h.exists_right_inv,
λ ⟨b, hab⟩, is_unit_of_mul_eq_one _ b hab⟩
@[to_additive] theorem is_unit_iff_exists_inv' [comm_monoid M]
{a : M} : is_unit a ↔ ∃ b, b * a = 1 :=
by simp [is_unit_iff_exists_inv, mul_comm]
@[to_additive]
lemma is_unit.mul [monoid M] {x y : M} : is_unit x → is_unit y → is_unit (x * y) :=
by { rintros ⟨x, rfl⟩ ⟨y, rfl⟩, exact ⟨x * y, units.coe_mul _ _⟩ }
/-- Multiplication by a `u : Mˣ` on the right doesn't affect `is_unit`. -/
@[simp, to_additive "Addition of a `u : add_units M` on the right doesn't
affect `is_add_unit`."]
theorem units.is_unit_mul_units [monoid M] (a : M) (u : Mˣ) :
is_unit (a * u) ↔ is_unit a :=
iff.intro
(assume ⟨v, hv⟩,
have is_unit (a * ↑u * ↑u⁻¹), by existsi v * u⁻¹; rw [←hv, units.coe_mul],
by rwa [mul_assoc, units.mul_inv, mul_one] at this)
(λ v, v.mul u.is_unit)
/-- Multiplication by a `u : Mˣ` on the left doesn't affect `is_unit`. -/
@[simp, to_additive "Addition of a `u : add_units M` on the left doesn't affect `is_add_unit`."]
theorem units.is_unit_units_mul {M : Type*} [monoid M] (u : Mˣ) (a : M) :
is_unit (↑u * a) ↔ is_unit a :=
iff.intro
(assume ⟨v, hv⟩,
have is_unit (↑u⁻¹ * (↑u * a)), by existsi u⁻¹ * v; rw [←hv, units.coe_mul],
by rwa [←mul_assoc, units.inv_mul, one_mul] at this)
u.is_unit.mul
@[to_additive]
theorem is_unit_of_mul_is_unit_left [comm_monoid M] {x y : M}
(hu : is_unit (x * y)) : is_unit x :=
let ⟨z, hz⟩ := is_unit_iff_exists_inv.1 hu in
is_unit_iff_exists_inv.2 ⟨y * z, by rwa ← mul_assoc⟩
@[to_additive] theorem is_unit_of_mul_is_unit_right [comm_monoid M] {x y : M}
(hu : is_unit (x * y)) : is_unit y :=
@is_unit_of_mul_is_unit_left _ _ y x $ by rwa mul_comm
@[simp, to_additive]
lemma is_unit.mul_iff [comm_monoid M] {x y : M} : is_unit (x * y) ↔ is_unit x ∧ is_unit y :=
⟨λ h, ⟨is_unit_of_mul_is_unit_left h, is_unit_of_mul_is_unit_right h⟩,
λ h, is_unit.mul h.1 h.2⟩
/-- The element of the group of units, corresponding to an element of a monoid which is a unit. When
`α` is a `division_monoid`, use `is_unit.unit'` instead. -/
@[to_additive "The element of the additive group of additive units, corresponding to an element of
an additive monoid which is an additive unit. When `α` is a `subtraction_monoid`, use
`is_add_unit.add_unit'` instead."]
noncomputable def is_unit.unit [monoid M] {a : M} (h : is_unit a) : Mˣ :=
(classical.some h).copy a (classical.some_spec h).symm _ rfl
@[simp, to_additive]
lemma is_unit.unit_of_coe_units [monoid M] {a : Mˣ} (h : is_unit (a : M)) : h.unit = a :=
units.ext $ rfl
@[simp, to_additive]
lemma is_unit.unit_spec [monoid M] {a : M} (h : is_unit a) : ↑h.unit = a :=
rfl
@[simp, to_additive]
lemma is_unit.coe_inv_mul [monoid M] {a : M} (h : is_unit a) :
↑(h.unit)⁻¹ * a = 1 :=
units.mul_inv _
@[simp, to_additive]
lemma is_unit.mul_coe_inv [monoid M] {a : M} (h : is_unit a) :
a * ↑(h.unit)⁻¹ = 1 :=
begin
convert units.mul_inv _,
simp [h.unit_spec]
end
/-- `is_unit x` is decidable if we can decide if `x` comes from `Mˣ`. -/
instance [monoid M] (x : M) [h : decidable (∃ u : Mˣ, ↑u = x)] : decidable (is_unit x) := h
section monoid
variables [monoid M] {a b c : M}
@[to_additive] lemma is_unit.mul_left_inj (h : is_unit a) : b * a = c * a ↔ b = c :=
let ⟨u, hu⟩ := h in hu ▸ u.mul_left_inj
@[to_additive] lemma is_unit.mul_right_inj (h : is_unit a) : a * b = a * c ↔ b = c :=
let ⟨u, hu⟩ := h in hu ▸ u.mul_right_inj
@[to_additive] protected lemma is_unit.mul_left_cancel (h : is_unit a) : a * b = a * c → b = c :=
h.mul_right_inj.1
@[to_additive] protected lemma is_unit.mul_right_cancel (h : is_unit b) : a * b = c * b → a = c :=
h.mul_left_inj.1
@[to_additive] protected lemma is_unit.mul_right_injective (h : is_unit a) : injective ((*) a) :=
λ _ _, h.mul_left_cancel
@[to_additive] protected lemma is_unit.mul_left_injective (h : is_unit b) : injective (* b) :=
λ _ _, h.mul_right_cancel
end monoid
end is_unit
section noncomputable_defs
variables {M : Type*}
/-- Constructs a `group` structure on a `monoid` consisting only of units. -/
noncomputable def group_of_is_unit [hM : monoid M] (h : ∀ (a : M), is_unit a) : group M :=
{ inv := λ a, ↑((h a).unit)⁻¹,
mul_left_inv := λ a, by
{ change ↑((h a).unit)⁻¹ * a = 1,
rw [units.inv_mul_eq_iff_eq_mul, (h a).unit_spec, mul_one] },
.. hM }
/-- Constructs a `comm_group` structure on a `comm_monoid` consisting only of units. -/
noncomputable def comm_group_of_is_unit [hM : comm_monoid M] (h : ∀ (a : M), is_unit a) :
comm_group M :=
{ inv := λ a, ↑((h a).unit)⁻¹,
mul_left_inv := λ a, by
{ change ↑((h a).unit)⁻¹ * a = 1,
rw [units.inv_mul_eq_iff_eq_mul, (h a).unit_spec, mul_one] },
.. hM }
end noncomputable_defs
|
8477fcb3f06e12c844223c7af0f4f26d08df0bdb | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/tactic/congr.lean | e0d6f2672b6a8b9ce8ead01978c804b9a398c966 | [
"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 | 8,641 | 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 tactic.lint
import tactic.ext
/-!
# Congruence and related tactics
This file contains the tactic `congr'`, which is an extension of `congr`, and various tactics
using `congr'` internally.
`congr'` has some advantages over `congr`:
* It turns `↔` to equalities, before trying another congr lemma
* You can write `congr' n` to give the maximal depth of recursive applications. This is useful if
`congr` breaks down the goal to aggressively, and the resulting goals are false.
* You can write `congr' with ...` to do `congr', ext ...` in a single tactic.
Other tactics in this file:
* `rcongr`: repeatedly apply `congr'` and `ext.`
* `convert`: like `exact`, but produces an equality goal if the type doesn't match.
* `convert_to`: changes the goal, if you prove an equality between the old goal and the new goal.
* `ac_change`: like `convert_to`, but uses `ac_refl` to discharge the goals.
-/
open tactic
setup_tactic_parser
namespace tactic
/-- Apply the constant `iff_of_eq` to the goal. -/
meta def apply_iff_congr_core : tactic unit :=
applyc ``iff_of_eq
/-- The main part of the body for the loop in `congr'`. This will try to replace a goal `f x = f y`
with `x = y`. Also has support for `==` and `↔`. -/
meta def congr_core' : tactic unit :=
do tgt ← target,
apply_eq_congr_core tgt
<|> apply_heq_congr_core
<|> apply_iff_congr_core
<|> fail "congr tactic failed"
/-- The main function in `convert_to`. Changes the goal to `r` and a proof obligation that the goal
is equal to `r`. -/
meta def convert_to_core (r : pexpr) : tactic unit :=
do tgt ← target,
h ← to_expr ``(_ : %%tgt = %%r),
rewrite_target h,
swap
/-- Attempts to prove the goal by proof irrelevance, but avoids unifying universe metavariables
to do so. -/
meta def by_proof_irrel : tactic unit :=
do tgt ← target,
@expr.const tt n [level.zero] ← pure tgt.get_app_fn,
if n = ``eq then `[apply proof_irrel] else
if n = ``heq then `[apply proof_irrel_heq] else
failed
/--
Same as the `congr` tactic, but takes an optional argument which gives
the depth of recursive applications.
* This is useful when `congr` is too aggressive in breaking down the goal.
* For example, given `⊢ f (g (x + y)) = f (g (y + x))`, `congr'` produces the goals `⊢ x = y`
and `⊢ y = x`, while `congr' 2` produces the intended `⊢ x + y = y + x`.
* If, at any point, a subgoal matches a hypothesis then the subgoal will be closed.
-/
meta def congr' : option ℕ → tactic unit
| o := focus1 $
assumption <|> reflexivity transparency.none <|> by_proof_irrel <|>
(guard (o ≠ some 0) >> congr_core' >>
all_goals' (try (congr' (nat.pred <$> o)))) <|>
reflexivity
namespace interactive
/--
Same as the `congr` tactic, but takes an optional argument which gives
the depth of recursive applications.
* This is useful when `congr` is too aggressive in breaking down the goal.
* For example, given `⊢ f (g (x + y)) = f (g (y + x))`, `congr'` produces the goals `⊢ x = y`
and `⊢ y = x`, while `congr' 2` produces the intended `⊢ x + y = y + x`.
* If, at any point, a subgoal matches a hypothesis then the subgoal will be closed.
* You can use `congr' with p (: n)?` to call `ext p (: n)?` to all subgoals generated by `congr'`.
For example, if the goal is `⊢ f '' s = g '' s` then `congr' with x` generates the goal
`x : α ⊢ f x = g x`.
-/
meta def congr' (n : parse (with_desc "n" small_nat)?) :
parse (tk "with" *> prod.mk <$> rcases_patt_parse_hi* <*> (tk ":" *> small_nat)?)? →
tactic unit
| none := tactic.congr' n
| (some ⟨p, m⟩) := focus1 (tactic.congr' n >> all_goals' (tactic.ext p m $> ()))
/--
Repeatedly and apply `congr'` and `ext`, using the the given patterns as arguments for `ext`.
There are two ways this tactic stops:
* `congr'` fails (makes no progress), after having already applied `ext`.
* `congr'` canceled out the last usage of `ext`. In this case, the state is reverted to before
the `congr'` was applied.
For example, when the goal is
```lean
⊢ (λ x, f x + 3) '' s = (λ x, g x + 3) '' s
```
then `rcongr x` produces the goal
```lean
x : α ⊢ f x = g x
```
This gives the same result as `congr', ext x, congr'`.
In contrast, `congr'` would produce
```lean
⊢ (λ x, f x + 3) = (λ x, g x + 3)
```
and `congr' with x` (or `congr', ext x`) would produce
```lean
x : α ⊢ f x + 3 = g x + 3
```
-/
meta def rcongr : parse rcases_patt_parse_hi* → tactic unit
| ps := do
t ← target,
qs ← try_core (tactic.ext ps none),
some () ← try_core (tactic.congr' none >>
(done <|> do s ← target, guard $ ¬ s =ₐ t)) | skip,
done <|> rcongr (qs.lhoare ps)
add_tactic_doc
{ name := "congr'",
category := doc_category.tactic,
decl_names := [`tactic.interactive.congr', `tactic.interactive.congr, `tactic.interactive.rcongr],
tags := ["congruence"],
inherit_description_from := `tactic.interactive.congr' }
/--
The `exact e` and `refine e` tactics require a term `e` whose type is
definitionally equal to the goal. `convert e` is similar to `refine e`,
but the type of `e` is not required to exactly match the
goal. Instead, new goals are created for differences between the type
of `e` and the goal. For example, in the proof state
```lean
n : ℕ,
e : prime (2 * n + 1)
⊢ prime (n + n + 1)
```
the tactic `convert e` will change the goal to
```lean
⊢ n + n = 2 * n
```
In this example, the new goal can be solved using `ring`.
The `convert` tactic applies congruence lemmas eagerly before reducing,
therefore it can fail in cases where `exact` succeeds:
```lean
def p (n : ℕ) := true
example (h : p 0) : p 1 := by exact h -- succeeds
example (h : p 0) : p 1 := by convert h -- fails, with leftover goal `1 = 0`
```
If `x y : t`, and an instance `subsingleton t` is in scope, then any goals of the form
`x = y` are solved automatically.
The syntax `convert ← e` will reverse the direction of the new goals
(producing `⊢ 2 * n = n + n` in this example).
Internally, `convert e` works by creating a new goal asserting that
the goal equals the type of `e`, then simplifying it using
`congr'`. The syntax `convert e using n` can be used to control the
depth of matching (like `congr' n`). In the example, `convert e using
1` would produce a new goal `⊢ n + n + 1 = 2 * n + 1`.
-/
meta def convert (sym : parse (with_desc "←" (tk "<-")?)) (r : parse texpr)
(n : parse (tk "using" *> small_nat)?) : tactic unit :=
do tgt ← target,
u ← infer_type tgt,
r ← i_to_expr ``(%%r : (_ : %%u)),
src ← infer_type r,
src ← simp_lemmas.mk.dsimplify [] src {fail_if_unchanged := ff},
v ← to_expr (if sym.is_some then ``(%%src = %%tgt) else ``(%%tgt = %%src)) tt ff >>= mk_meta_var,
(if sym.is_some then mk_eq_mp v r else mk_eq_mpr v r) >>= tactic.exact,
gs ← get_goals,
set_goals [v],
try (tactic.congr' n),
gs' ← get_goals,
set_goals $ gs' ++ gs
add_tactic_doc
{ name := "convert",
category := doc_category.tactic,
decl_names := [`tactic.interactive.convert],
tags := ["congruence"] }
/--
`convert_to g using n` attempts to change the current goal to `g`, but unlike `change`,
it will generate equality proof obligations using `congr' n` to resolve discrepancies.
`convert_to g` defaults to using `congr' 1`.
`convert_to` is similar to `convert`, but `convert_to` takes a type (the desired subgoal) while
`convert` takes a proof term.
That is, `convert_to g using n` is equivalent to `convert (_ : g) using n`.
-/
meta def convert_to (r : parse texpr) (n : parse (tk "using" *> small_nat)?) : tactic unit :=
match n with
| none := convert_to_core r >> `[congr' 1]
| (some 0) := convert_to_core r
| (some o) := convert_to_core r >> tactic.congr' o
end
/--
`ac_change g using n` is `convert_to g using n` followed by `ac_refl`. It is useful for
rearranging/reassociating e.g. sums:
```lean
example (a b c d e f g N : ℕ) : (a + b) + (c + d) + (e + f) + g ≤ N :=
begin
ac_change a + d + e + f + c + g + b ≤ _,
-- ⊢ a + d + e + f + c + g + b ≤ N
end
```
-/
meta def ac_change (r : parse texpr) (n : parse (tk "using" *> small_nat)?) : tactic unit :=
convert_to r n; try ac_refl
add_tactic_doc
{ name := "convert_to",
category := doc_category.tactic,
decl_names := [`tactic.interactive.convert_to, `tactic.interactive.ac_change],
tags := ["congruence"],
inherit_description_from := `tactic.interactive.convert_to }
end interactive
end tactic
|
7ae8d05607b0a754c10ae3710bef78f4decbf3f0 | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/analysis/inner_product_space/spectrum.lean | ad3cca6d234c95513359217f851137ba3d4e36b0 | [
"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 | 11,374 | lean | /-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import analysis.inner_product_space.rayleigh
import analysis.inner_product_space.pi_L2
/-! # Spectral theory of self-adjoint operators
This file covers the spectral theory of self-adjoint operators on an inner product space.
The first part of the file covers general properties, true without any condition on boundedness or
compactness of the operator or finite-dimensionality of the underlying space, notably:
* `is_self_adjoint.conj_eigenvalue_eq_self`: the eigenvalues are real
* `is_self_adjoint.orthogonal_family_eigenspaces`: the eigenspaces are orthogonal
* `is_self_adjoint.orthogonal_supr_eigenspaces`: the restriction of the operator to the mutual
orthogonal complement of the eigenspaces has, itself, no eigenvectors
The second part of the file covers properties of self-adjoint operators in finite dimension.
Letting `T` be a self-adjoint operator on a finite-dimensional inner product space `T`,
* The definition `is_self_adjoint.diagonalization` provides a linear isometry equivalence `E` to
the direct sum of the eigenspaces of `T`. The theorem
`is_self_adjoint.diagonalization_apply_self_apply` states that, when `T` is transferred via this
equivalence to an operator on the direct sum, it acts diagonally.
* The definition `is_self_adjoint.eigenvector_basis` provides an orthonormal basis for `E`
consisting of eigenvectors of `T`, with `is_self_adjoint.eigenvalues` giving the corresponding
list of eigenvalues, as real numbers. The definition `is_self_adjoint.diagonalization_basis`
gives the associated linear isometry equivalence from `E` to Euclidean space, and the theorem
`is_self_adjoint.diagonalization_basis_apply_self_apply` states that, when `T` is transferred via
this equivalence to an operator on Euclidean space, it acts diagonally.
These are forms of the *diagonalization theorem* for self-adjoint operators on finite-dimensional
inner product spaces.
## TODO
Spectral theory for compact self-adjoint operators, bounded self-adjoint operators.
## Tags
self-adjoint operator, spectral theorem, diagonalization theorem
-/
variables {𝕜 : Type*} [is_R_or_C 𝕜] [dec_𝕜 : decidable_eq 𝕜]
variables {E : Type*} [inner_product_space 𝕜 E]
local notation `⟪`x`, `y`⟫` := @inner 𝕜 E _ x y
open_locale big_operators complex_conjugate
open module.End
namespace inner_product_space
namespace is_self_adjoint
variables {T : E →ₗ[𝕜] E} (hT : is_self_adjoint T)
include hT
/-- A self-adjoint operator preserves orthogonal complements of its eigenspaces. -/
lemma invariant_orthogonal_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
T v ∈ (eigenspace T μ)ᗮ :=
begin
intros w hw,
have : T w = (μ:𝕜) • w := by rwa mem_eigenspace_iff at hw,
simp [← hT w, this, inner_smul_left, hv w hw]
end
/-- The eigenvalues of a self-adjoint operator are real. -/
lemma conj_eigenvalue_eq_self {μ : 𝕜} (hμ : has_eigenvalue T μ) : conj μ = μ :=
begin
obtain ⟨v, hv₁, hv₂⟩ := hμ.exists_has_eigenvector,
rw mem_eigenspace_iff at hv₁,
simpa [hv₂, inner_smul_left, inner_smul_right, hv₁] using hT v v
end
/-- The eigenspaces of a self-adjoint operator are mutually orthogonal. -/
lemma orthogonal_family_eigenspaces :
@orthogonal_family 𝕜 _ _ _ _ (λ μ, eigenspace T μ) _ (λ μ, (eigenspace T μ).subtypeₗᵢ) :=
begin
rintros μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩,
by_cases hv' : v = 0,
{ simp [hv'] },
have H := hT.conj_eigenvalue_eq_self (has_eigenvalue_of_has_eigenvector ⟨hv, hv'⟩),
rw mem_eigenspace_iff at hv hw,
refine or.resolve_left _ hμν.symm,
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
end
lemma orthogonal_family_eigenspaces' :
@orthogonal_family 𝕜 _ _ _ _ (λ μ : eigenvalues T, eigenspace T μ) _
(λ μ, (eigenspace T μ).subtypeₗᵢ) :=
hT.orthogonal_family_eigenspaces.comp subtype.coe_injective
/-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
product space is an invariant subspace of the operator. -/
lemma orthogonal_supr_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
T v ∈ (⨆ μ, eigenspace T μ)ᗮ :=
begin
rw ← submodule.infi_orthogonal at ⊢ hv,
exact T.infi_invariant hT.invariant_orthogonal_eigenspace v hv
end
/-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
product space has no eigenvalues. -/
lemma orthogonal_supr_eigenspaces (μ : 𝕜) :
eigenspace (T.restrict hT.orthogonal_supr_eigenspaces_invariant) μ = ⊥ :=
begin
set p : submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ,
refine eigenspace_restrict_eq_bot hT.orthogonal_supr_eigenspaces_invariant _,
have H₂ : p ≤ (eigenspace T μ)ᗮ := submodule.orthogonal_le (le_supr _ _),
exact (eigenspace T μ).orthogonal_disjoint.mono_right H₂
end
/-! ### Finite-dimensional theory -/
variables [finite_dimensional 𝕜 E]
/-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on a
finite-dimensional inner product space is trivial. -/
lemma orthogonal_supr_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)ᗮ = ⊥ :=
begin
have hT' : is_self_adjoint _ := hT.restrict_invariant hT.orthogonal_supr_eigenspaces_invariant,
-- a self-adjoint operator on a nontrivial inner product space has an eigenvalue
haveI := hT'.subsingleton_of_no_eigenvalue_finite_dimensional hT.orthogonal_supr_eigenspaces,
exact submodule.eq_bot_of_subsingleton _,
end
lemma orthogonal_supr_eigenspaces_eq_bot' : (⨆ μ : eigenvalues T, eigenspace T μ)ᗮ = ⊥ :=
show (⨆ μ : {μ // (eigenspace T μ) ≠ ⊥}, eigenspace T μ)ᗮ = ⊥,
by rw [supr_ne_bot_subtype, hT.orthogonal_supr_eigenspaces_eq_bot]
include dec_𝕜
/-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` give
an internal direct sum decomposition of `E`. -/
lemma direct_sum_submodule_is_internal :
direct_sum.submodule_is_internal (λ μ : eigenvalues T, eigenspace T μ) :=
hT.orthogonal_family_eigenspaces'.submodule_is_internal_iff.mpr
hT.orthogonal_supr_eigenspaces_eq_bot'
section version1
/-- Isometry from an inner product space `E` to the direct sum of the eigenspaces of some
self-adjoint operator `T` on `E`. -/
noncomputable def diagonalization : E ≃ₗᵢ[𝕜] pi_Lp 2 (λ μ : eigenvalues T, eigenspace T μ) :=
hT.direct_sum_submodule_is_internal.isometry_L2_of_orthogonal_family
hT.orthogonal_family_eigenspaces'
@[simp] lemma diagonalization_symm_apply (w : pi_Lp 2 (λ μ : eigenvalues T, eigenspace T μ)) :
hT.diagonalization.symm w = ∑ μ, w μ :=
hT.direct_sum_submodule_is_internal.isometry_L2_of_orthogonal_family_symm_apply
hT.orthogonal_family_eigenspaces' w
/-- *Diagonalization theorem*, *spectral theorem*; version 1: A self-adjoint operator `T` on a
finite-dimensional inner product space `E` acts diagonally on the decomposition of `E` into the
direct sum of the eigenspaces of `T`. -/
lemma diagonalization_apply_self_apply (v : E) (μ : eigenvalues T) :
hT.diagonalization (T v) μ = (μ : 𝕜) • hT.diagonalization v μ :=
begin
suffices : ∀ w : pi_Lp 2 (λ μ : eigenvalues T, eigenspace T μ),
(T (hT.diagonalization.symm w)) = hT.diagonalization.symm (λ μ, (μ : 𝕜) • w μ),
{ simpa [linear_isometry_equiv.symm_apply_apply, -is_self_adjoint.diagonalization_symm_apply]
using congr_arg (λ w, hT.diagonalization w μ) (this (hT.diagonalization v)) },
intros w,
have hwT : ∀ μ : eigenvalues T, T (w μ) = (μ : 𝕜) • w μ,
{ intros μ,
simpa [mem_eigenspace_iff] using (w μ).prop },
simp [hwT],
end
end version1
section version2
variables {n : ℕ} (hn : finite_dimensional.finrank 𝕜 E = n)
/-- A choice of orthonormal basis of eigenvectors for self-adjoint operator `T` on a
finite-dimensional inner product space `E`.
TODO Postcompose with a permutation so that these eigenvectors are listed in increasing order of
eigenvalue. -/
noncomputable def eigenvector_basis : basis (fin n) 𝕜 E :=
hT.direct_sum_submodule_is_internal.subordinate_orthonormal_basis hn
lemma eigenvector_basis_orthonormal : orthonormal 𝕜 (hT.eigenvector_basis hn) :=
hT.direct_sum_submodule_is_internal.subordinate_orthonormal_basis_orthonormal hn
hT.orthogonal_family_eigenspaces'
/-- The sequence of real eigenvalues associated to the standard orthonormal basis of eigenvectors
for a self-adjoint operator `T` on `E`.
TODO Postcompose with a permutation so that these eigenvalues are listed in increasing order. -/
noncomputable def eigenvalues (i : fin n) : ℝ :=
@is_R_or_C.re 𝕜 _ $ hT.direct_sum_submodule_is_internal.subordinate_orthonormal_basis_index hn i
lemma has_eigenvector_eigenvector_basis (i : fin n) :
has_eigenvector T (hT.eigenvalues hn i) (hT.eigenvector_basis hn i) :=
begin
let v : E := hT.eigenvector_basis hn i,
let μ : 𝕜 := hT.direct_sum_submodule_is_internal.subordinate_orthonormal_basis_index hn i,
change has_eigenvector T (is_R_or_C.re μ) v,
have key : has_eigenvector T μ v,
{ have H₁ : v ∈ eigenspace T μ,
{ exact hT.direct_sum_submodule_is_internal.subordinate_orthonormal_basis_subordinate hn i },
have H₂ : v ≠ 0 := (hT.eigenvector_basis_orthonormal hn).ne_zero i,
exact ⟨H₁, H₂⟩ },
have re_μ : ↑(is_R_or_C.re μ) = μ,
{ rw ← is_R_or_C.eq_conj_iff_re,
exact hT.conj_eigenvalue_eq_self (has_eigenvalue_of_has_eigenvector key) },
simpa [re_μ] using key,
end
attribute [irreducible] eigenvector_basis eigenvalues
@[simp] lemma apply_eigenvector_basis (i : fin n) :
T (hT.eigenvector_basis hn i) = (hT.eigenvalues hn i : 𝕜) • hT.eigenvector_basis hn i :=
mem_eigenspace_iff.mp (hT.has_eigenvector_eigenvector_basis hn i).1
/-- An isometry from an inner product space `E` to Euclidean space, induced by a choice of
orthonormal basis of eigenvectors for a self-adjoint operator `T` on `E`. -/
noncomputable def diagonalization_basis : E ≃ₗᵢ[𝕜] euclidean_space 𝕜 (fin n) :=
(hT.eigenvector_basis hn).isometry_euclidean_of_orthonormal (hT.eigenvector_basis_orthonormal hn)
@[simp] lemma diagonalization_basis_symm_apply (w : euclidean_space 𝕜 (fin n)) :
(hT.diagonalization_basis hn).symm w = ∑ i, w i • hT.eigenvector_basis hn i :=
by simp [diagonalization_basis]
/-- *Diagonalization theorem*, *spectral theorem*; version 2: A self-adjoint operator `T` on a
finite-dimensional inner product space `E` acts diagonally on the identification of `E` with
Euclidean space induced by an orthonormal basis of eigenvectors of `T`. -/
lemma diagonalization_basis_apply_self_apply (v : E) (i : fin n) :
hT.diagonalization_basis hn (T v) i = hT.eigenvalues hn i * hT.diagonalization_basis hn v i :=
begin
suffices : ∀ w : euclidean_space 𝕜 (fin n),
T ((hT.diagonalization_basis hn).symm w)
= (hT.diagonalization_basis hn).symm (λ i, hT.eigenvalues hn i * w i),
{ simpa [-diagonalization_basis_symm_apply] using
congr_arg (λ v, hT.diagonalization_basis hn v i) (this (hT.diagonalization_basis hn v)) },
intros w,
simp [mul_comm, mul_smul],
end
end version2
end is_self_adjoint
end inner_product_space
|
6c35ea0890a7d78b3db24ee87056b1ddfa24e984 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/data/mv_polynomial/cardinal.lean | ff7a3335a6beef8c1807558c073cec036d86237f | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 2,046 | lean | /-
Copyright (c) 2021 Chris Hughes, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Junyan Xu
-/
import data.finsupp.fintype
import data.mv_polynomial.equiv
import set_theory.cardinal.ordinal
/-!
# Cardinality of Multivariate Polynomial Ring
The main result in this file is `mv_polynomial.cardinal_mk_le_max`, which says that
the cardinality of `mv_polynomial σ R` is bounded above by the maximum of `#R`, `#σ`
and `ℵ₀`.
-/
universes u v
open cardinal
open_locale cardinal
namespace mv_polynomial
section two_universes
variables {σ : Type u} {R : Type v} [comm_semiring R]
@[simp] lemma cardinal_mk_eq_max_lift [nonempty σ] [nontrivial R] :
#(mv_polynomial σ R) = max (max (cardinal.lift.{u} $ #R) $ cardinal.lift.{v} $ #σ) ℵ₀ :=
(mk_finsupp_lift_of_infinite _ R).trans $
by rw [mk_finsupp_nat, max_assoc, lift_max, lift_aleph_0, max_comm]
@[simp] lemma cardinal_mk_eq_lift [is_empty σ] : #(mv_polynomial σ R) = cardinal.lift.{u} (#R) :=
((is_empty_ring_equiv R σ).to_equiv.trans equiv.ulift.{u}.symm).cardinal_eq
lemma cardinal_lift_mk_le_max {σ : Type u} {R : Type v} [comm_semiring R] :
#(mv_polynomial σ R) ≤ max (max (cardinal.lift.{u} $ #R) $ cardinal.lift.{v} $ #σ) ℵ₀ :=
begin
casesI subsingleton_or_nontrivial R,
{ exact (mk_eq_one _).trans_le (le_max_of_le_right one_le_aleph_0) },
casesI is_empty_or_nonempty σ,
{ exact cardinal_mk_eq_lift.trans_le (le_max_of_le_left $ le_max_left _ _) },
{ exact cardinal_mk_eq_max_lift.le },
end
end two_universes
variables {σ R : Type u} [comm_semiring R]
lemma cardinal_mk_eq_max [nonempty σ] [nontrivial R] :
#(mv_polynomial σ R) = max (max (#R) (#σ)) ℵ₀ := by simp
/-- The cardinality of the multivariate polynomial ring, `mv_polynomial σ R` is at most the maximum
of `#R`, `#σ` and `ℵ₀` -/
lemma cardinal_mk_le_max : #(mv_polynomial σ R) ≤ max (max (#R) (#σ)) ℵ₀ :=
cardinal_lift_mk_le_max.trans $ by rw [lift_id, lift_id]
end mv_polynomial
|
77ee3542c9c5d9df289f0ae63184ed2ccc72eb4a | 9be442d9ec2fcf442516ed6e9e1660aa9071b7bd | /tests/lean/353.lean | bc3ffd78ef9d6edf4d8a93917cfbae50ac4fb8c0 | [
"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 | EdAyers/lean4 | 57ac632d6b0789cb91fab2170e8c9e40441221bd | 37ba0df5841bde51dbc2329da81ac23d4f6a4de4 | refs/heads/master | 1,676,463,245,298 | 1,660,619,433,000 | 1,660,619,433,000 | 183,433,437 | 1 | 0 | Apache-2.0 | 1,657,612,672,000 | 1,556,196,574,000 | Lean | UTF-8 | Lean | false | false | 478 | lean | class ArrSort (α : Sort u1) where
Arr : α → α → Sort u2
class Arr (α : Sort u1) (γ : Sort u2) where
Arr : α → α → γ
infix:70 " ~> " => Arr.Arr
@[defaultInstance]
instance inst1 {α : Sort _} [ArrSort α] : Arr α (Sort _) := { Arr := ArrSort.Arr }
instance inst2 : ArrSort Prop := { Arr := λ a b => a → b }
set_option pp.all true
set_option trace.Meta.debug true
structure Function' where
map : ∀ {a₁ a₂ : Bool}, (a₁ ~> a₂) ~> (a₁ ~> a₂)
|
d0f13cafa569de37ad7ac19948273c4ca5245adc | e0f9ba56b7fedc16ef8697f6caeef5898b435143 | /src/analysis/calculus/inverse.lean | 1e91b9213fe3dc837e35f36f741a69b715a7604f | [
"Apache-2.0"
] | permissive | anrddh/mathlib | 6a374da53c7e3a35cb0298b0cd67824efef362b4 | a4266a01d2dcb10de19369307c986d038c7bb6a6 | refs/heads/master | 1,656,710,827,909 | 1,589,560,456,000 | 1,589,560,456,000 | 264,271,800 | 0 | 0 | Apache-2.0 | 1,589,568,062,000 | 1,589,568,061,000 | null | UTF-8 | Lean | false | false | 21,386 | lean | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov.
-/
import analysis.calculus.deriv
import topology.local_homeomorph
import topology.metric_space.contracting
/-!
# Inverse function theorem
In this file we prove the inverse function theorem. It says that if a map `f : E → F`
has an invertible strict derivative `f'` at `a`, then it is locally invertible,
and the inverse function has derivative `f' ⁻¹`.
We define `has_strict_deriv_at.to_local_homeomorph` that repacks a function `f`
with a `hf : has_strict_fderiv_at f f' a`, `f' : E ≃L[𝕜] F`, into a `local_homeomorph`.
The `to_fun` of this `local_homeomorph` is `defeq` to `f`, so one can apply theorems
about `local_homeomorph` to `hf.to_local_homeomorph f`, and get statements about `f`.
Then we define `has_strict_fderiv_at.local_inverse` to be the `inv_fun` of this `local_homeomorph`,
and prove two versions of the inverse function theorem:
* `has_strict_fderiv_at.to_local_inverse`: if `f` has an invertible derivative `f'` at `a` in the
strict sense (`hf`), then `hf.local_inverse f f' a` has derivative `f'.symm` at `f a` in the
strict sense;
* `has_strict_fderiv_at.to_local_left_inverse`: if `f` has an invertible derivative `f'` at `a` in
the strict sense and `g` is locally left inverse to `f` near `a`, then `g` has derivative
`f'.symm` at `f a` in the strict sense.
In the one-dimensional case we reformulate these theorems in terms of `has_strict_deriv_at` and
`f'⁻¹`. Some other versions of the theorem assuming that we already know the inverse function are
formulated in `fderiv.lean` and `deriv.lean`
## Notations
In the section about `approximates_linear_on` we introduce some `local notation` to make formulas
shorter:
* by `N` we denote `∥f'⁻¹∥`;
* by `g` we denote the auxiliary contracting map `x ↦ x + f'.symm (y - f x)` used to prove that
`{x | f x = y}` is nonempty.
## Tags
derivative, strictly differentiable, inverse function
-/
open function set filter metric
open_locale topological_space classical nnreal
noncomputable theory
variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜]
variables {E : Type*} [normed_group E] [normed_space 𝕜 E]
variables {F : Type*} [normed_group F] [normed_space 𝕜 F]
variables {G : Type*} [normed_group G] [normed_space 𝕜 G]
variables {G' : Type*} [normed_group G'] [normed_space 𝕜 G']
open asymptotics filter metric set
open continuous_linear_map (id)
/-!
### Non-linear maps approximating close to affine maps
In this section we study a map `f` such that `∥f x - f y - f' (x - y)∥ ≤ c * ∥x - y∥` on an open set
`s`, where `f' : E ≃L[𝕜] F` is a continuous linear equivalence and `c < ∥f'⁻¹∥`. Maps of this type
behave like `f a + f' (x - a)` near each `a ∈ s`.
If `E` is a complete space, we prove that the image `f '' s` is open, and `f` is a homeomorphism
between `s` and `f '' s`. More precisely, we define `approximates_linear_on.to_local_homeomorph` to
be a `local_homeomorph` with `to_fun = f`, `source = s`, and `target = f '' s`.
Maps of this type naturally appear in the proof of the inverse function theorem (see next section),
and `approximates_linear_on.to_local_homeomorph` will imply that the locally inverse function
exists.
We define this auxiliary notion to split the proof of the inverse function theorem into small
lemmas. This approach makes it possible
- to prove a lower estimate on the size of the domain of the inverse function;
- to reuse parts of the proofs in the case if a function is not strictly differentiable. E.g., for a
function `f : E × F → G` with estimates on `f x y₁ - f x y₂` but not on `f x₁ y - f x₂ y`.
-/
/-- We say that `f` approximates a continuous linear map `f'` on `s` with constant `c`,
if `∥f x - f y - f' (x - y)∥ ≤ c * ∥x - y∥` whenever `x, y ∈ s`.
This predicate is defined to facilitate the splitting of the inverse function theorem into small
lemmas. Some of these lemmas can be useful, e.g., to prove that the inverse function is defined
on a specific set. -/
def approximates_linear_on (f : E → F) (f' : E →L[𝕜] F) (s : set E) (c : ℝ≥0) : Prop :=
∀ (x ∈ s) (y ∈ s), ∥f x - f y - f' (x - y)∥ ≤ c * ∥x - y∥
namespace approximates_linear_on
variables [cs : complete_space E] {f : E → F}
/-! First we prove some properties of a function that `approximates_linear_on` a (not necessarily
invertible) continuous linear map. -/
section
variables {f' : E →L[𝕜] F} {s t : set E} {c c' : ℝ≥0}
theorem mono_num (hc : c ≤ c') (hf : approximates_linear_on f f' s c) :
approximates_linear_on f f' s c' :=
λ x hx y hy, le_trans (hf x hx y hy) (mul_le_mul_of_nonneg_right hc $ norm_nonneg _)
theorem mono_set (hst : s ⊆ t) (hf : approximates_linear_on f f' t c) :
approximates_linear_on f f' s c :=
λ x hx y hy, hf x (hst hx) y (hst hy)
lemma lipschitz_sub (hf : approximates_linear_on f f' s c) :
lipschitz_with c (λ x : s, f x - f' x) :=
begin
refine lipschitz_with.of_dist_le_mul (λ x y, _),
rw [dist_eq_norm, subtype.dist_eq, dist_eq_norm],
convert hf x x.2 y y.2 using 2,
rw [f'.map_sub], abel
end
protected lemma lipschitz (hf : approximates_linear_on f f' s c) :
lipschitz_with (nnnorm f' + c) (s.restrict f) :=
by simpa only [restrict_apply, add_sub_cancel'_right]
using (f'.lipschitz.restrict s).add hf.lipschitz_sub
protected lemma continuous (hf : approximates_linear_on f f' s c) :
continuous (s.restrict f) :=
hf.lipschitz.continuous
protected lemma continuous_on (hf : approximates_linear_on f f' s c) :
continuous_on f s :=
continuous_on_iff_continuous_restrict.2 hf.continuous
end
/-!
From now on we assume that `f` approximates an invertible continuous linear map `f : E ≃L[𝕜] F`.
We also assume that either `E = {0}`, or `c < ∥f'⁻¹∥⁻¹`. We use `N` as an abbreviation for `∥f'⁻¹∥`.
-/
variables {f' : E ≃L[𝕜] F} {s : set E} {c : ℝ≥0}
local notation `N` := nnnorm (f'.symm : F →L[𝕜] E)
protected lemma antilipschitz (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
antilipschitz_with (N⁻¹ - c)⁻¹ (s.restrict f) :=
begin
cases hc with hE hc,
{ haveI : subsingleton s := ⟨λ x y, subtype.eq $ @subsingleton.elim _ hE _ _⟩,
exact antilipschitz_with.of_subsingleton },
convert (f'.antilipschitz.restrict s).add_lipschitz_with hf.lipschitz_sub hc,
simp [restrict]
end
protected lemma injective (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
injective (s.restrict f) :=
(hf.antilipschitz hc).injective
protected lemma inj_on (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
inj_on f s :=
inj_on_iff_injective.2 $ hf.injective hc
/-- A map approximating a linear equivalence on a set defines a local equivalence on this set.
Should not be used outside of this file, because it is superseded by `to_local_homeomorph` below.
This is a first step towards the inverse function. -/
def to_local_equiv (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
local_equiv E F :=
by haveI : nonempty E := ⟨0⟩; exact (hf.inj_on hc).to_local_equiv _ _
/-- The inverse function is continuous on `f '' s`. Use properties of `local_homeomorph` instead. -/
lemma inverse_continuous_on (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
continuous_on (hf.to_local_equiv hc).symm (f '' s) :=
begin
apply continuous_on_iff_continuous_restrict.2,
refine ((hf.antilipschitz hc).to_right_inv_on' _ (hf.to_local_equiv hc).right_inv').continuous,
exact (λ x hx, (hf.to_local_equiv hc).map_target hx)
end
/-!
Now we prove that `f '' s` is an open set. This follows from the fact that the restriction of `f`
on `s` is an open map. More precisely, we show that the image of a closed ball $$\bar B(a, ε) ⊆ s$$
under `f` includes the closed ball $$\bar B\left(f(a), \frac{ε}{∥{f'}⁻¹∥⁻¹-c}\right)$$.
In order to do this, we introduce an auxiliary map $$g_y(x) = x + {f'}⁻¹ (y - f x)$$. Provided that
$$∥y - f a∥ ≤ \frac{ε}{∥{f'}⁻¹∥⁻¹-c}$$, we prove that $$g_y$$ contracts in $$\bar B(a, ε)$$ and `f`
sends the fixed point of $$g_y$$ to `y`.
-/
section
variables (f f')
/-- Iterations of this map converge to `f⁻¹ y`. The formula is very similar to the one
used in Newton's method, but we use the same `f'.symm` for all `y` instead of evaluating
the derivative at each point along the orbit. -/
def inverse_approx_map (y : F) (x : E) : E := x + f'.symm (y - f x)
end
section inverse_approx_map
variables (y : F) {x x' : E} {ε : ℝ}
local notation `g` := inverse_approx_map f f' y
lemma inverse_approx_map_sub (x x' : E) : g x - g x' = (x - x') - f'.symm (f x - f x') :=
by { simp only [inverse_approx_map, f'.symm.map_sub], abel }
lemma inverse_approx_map_dist_self (x : E) :
dist (g x) x = dist (f'.symm $ f x) (f'.symm y) :=
by simp only [inverse_approx_map, dist_eq_norm, f'.symm.map_sub, add_sub_cancel', norm_sub_rev]
lemma inverse_approx_map_dist_self_le (x : E) :
dist (g x) x ≤ N * dist (f x) y :=
by { rw inverse_approx_map_dist_self, exact f'.symm.lipschitz.dist_le_mul (f x) y }
lemma inverse_approx_map_fixed_iff {x : E} :
g x = x ↔ f x = y :=
by rw [← dist_eq_zero, inverse_approx_map_dist_self, dist_eq_zero, f'.symm.injective.eq_iff]
lemma inverse_approx_map_contracts_on (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
{x x'} (hx : x ∈ s) (hx' : x' ∈ s) :
dist (g x) (g x') ≤ N * c * dist x x' :=
begin
rw [dist_eq_norm, dist_eq_norm, inverse_approx_map_sub, norm_sub_rev],
suffices : ∥f'.symm (f x - f x' - f' (x - x'))∥ ≤ N * (c * ∥x - x'∥),
by simpa only [f'.symm.map_sub, f'.symm_apply_apply, mul_assoc] using this,
exact (f'.symm : F →L[𝕜] E).le_op_norm_of_le (hf x hx x' hx')
end
variable {y}
lemma inverse_approx_map_maps_to (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) {b : E} (hb : b ∈ s) (hε : closed_ball b ε ⊆ s)
(hy : y ∈ closed_ball (f b) ((N⁻¹ - c) * ε)) :
maps_to g (closed_ball b ε) (closed_ball b ε) :=
begin
cases hc with hE hc,
{ exactI λ x hx, mem_preimage.2 (subsingleton.elim x (g x) ▸ hx) },
assume x hx,
simp only [subset_def, mem_closed_ball, mem_preimage] at hx hy ⊢,
rw [dist_comm] at hy,
calc dist (inverse_approx_map f f' y x) b ≤
dist (inverse_approx_map f f' y x) (inverse_approx_map f f' y b) +
dist (inverse_approx_map f f' y b) b : dist_triangle _ _ _
... ≤ N * c * dist x b + N * dist (f b) y :
add_le_add (hf.inverse_approx_map_contracts_on y (hε hx) hb)
(inverse_approx_map_dist_self_le _ _)
... ≤ N * c * ε + N * ((N⁻¹ - c) * ε) :
add_le_add (mul_le_mul_of_nonneg_left hx (mul_nonneg (nnreal.coe_nonneg _) c.coe_nonneg))
(mul_le_mul_of_nonneg_left hy (nnreal.coe_nonneg _))
... = N * (c + (N⁻¹ - c)) * ε : by simp only [mul_add, add_mul, mul_assoc]
... = ε : by { rw [add_sub_cancel'_right, mul_inv_cancel, one_mul],
exact ne_of_gt (inv_pos.1 $ lt_of_le_of_lt c.coe_nonneg hc) }
end
end inverse_approx_map
include cs
variable {ε : ℝ}
theorem surj_on_closed_ball (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) {b : E} (ε0 : 0 ≤ ε) (hε : closed_ball b ε ⊆ s) :
surj_on f (closed_ball b ε) (closed_ball (f b) ((N⁻¹ - c) * ε)) :=
begin
cases hc with hE hc,
{ resetI,
haveI hF : subsingleton F := f'.symm.to_linear_equiv.to_equiv.subsingleton,
intros y hy,
exact ⟨b, mem_closed_ball_self ε0, subsingleton.elim _ _⟩ },
intros y hy,
have : contracting_with (N * c) ((hf.inverse_approx_map_maps_to (or.inr hc)
(hε $ mem_closed_ball_self ε0) hε hy).restrict _ _ _),
{ split,
{ rwa [mul_comm, ← nnreal.lt_inv_iff_mul_lt],
exact ne_of_gt (inv_pos.1 $ lt_of_le_of_lt c.coe_nonneg hc) },
{ exact lipschitz_with.of_dist_le_mul (λ x x', hf.inverse_approx_map_contracts_on
y (hε x.mem) (hε x'.mem)) } },
refine ⟨this.efixed_point' _ _ _ b (mem_closed_ball_self ε0) (edist_lt_top _ _), _, _⟩,
{ exact is_complete_of_is_closed is_closed_ball },
{ apply contracting_with.efixed_point_mem' },
{ exact (inverse_approx_map_fixed_iff y).1 (this.efixed_point_is_fixed' _ _ _ _) }
end
section
variables (f s)
/-- Given a function `f` that approximates a linear equivalence on an open set `s`,
returns a local homeomorph with `to_fun = f` and `source = s`. -/
def to_local_homeomorph (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) : local_homeomorph E F :=
{ to_local_equiv := hf.to_local_equiv hc,
open_source := hs,
open_target :=
begin
cases hc with hE hc,
{ resetI,
haveI hF : subsingleton F := f'.to_linear_equiv.to_equiv.symm.subsingleton,
apply is_open_discrete },
change is_open (f '' s),
simp only [is_open_iff_mem_nhds, nhds_basis_closed_ball.mem_iff, ball_image_iff] at hs ⊢,
intros x hx,
rcases hs x hx with ⟨ε, ε0, hε⟩,
refine ⟨(N⁻¹ - c) * ε, mul_pos (sub_pos.2 hc) ε0, _⟩,
exact (hf.surj_on_closed_ball (or.inr hc) (le_of_lt ε0) hε).mono hε (subset.refl _)
end,
continuous_to_fun := hf.continuous_on,
continuous_inv_fun := hf.inverse_continuous_on hc }
end
@[simp] lemma to_local_homeomorph_coe (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) :
(hf.to_local_homeomorph f s hc hs : E → F) = f := rfl
@[simp] lemma to_local_homeomorph_source (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) :
(hf.to_local_homeomorph f s hc hs).source = s := rfl
@[simp] lemma to_local_homeomorph_target (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) :
(hf.to_local_homeomorph f s hc hs).target = f '' s := rfl
lemma closed_ball_subset_target (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) {b : E} (ε0 : 0 ≤ ε) (hε : closed_ball b ε ⊆ s) :
closed_ball (f b) ((N⁻¹ - c) * ε) ⊆ (hf.to_local_homeomorph f s hc hs).target :=
(hf.surj_on_closed_ball hc ε0 hε).mono hε (subset.refl _)
end approximates_linear_on
/-!
### Inverse function theorem
Now we prove the inverse function theorem. Let `f : E → F` be a map defined on a complete vector
space `E`. Assume that `f` has an invertible derivative `f' : E ≃L[𝕜] F` at `a : E` in the strict
sense. Then `f` approximates `f'` in the sense of `approximates_linear_on` on an open neighborhood
of `a`, and we can apply `approximates_linear_on.to_local_homeomorph` to construct the inverse
function. -/
namespace has_strict_fderiv_at
/-- If `f` has derivative `f'` at `a` in the strict sense and `c > 0`, then `f` approximates `f'`
with constant `c` on some neighborhood of `a`. -/
lemma approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
(hf : has_strict_fderiv_at f f' a) {c : ℝ≥0} (hc : subsingleton E ∨ 0 < c) :
∃ s ∈ 𝓝 a, approximates_linear_on f f' s c :=
begin
cases hc with hE hc,
{ refine ⟨univ, mem_nhds_sets is_open_univ trivial, λ x hx y hy, _⟩,
simp [@subsingleton.elim E hE x y] },
have := hf.def hc,
rw [nhds_prod_eq, filter.eventually, mem_prod_same_iff] at this,
rcases this with ⟨s, has, hs⟩,
exact ⟨s, has, λ x hx y hy, hs (mk_mem_prod hx hy)⟩
end
variables [cs : complete_space E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E}
lemma approximates_deriv_on_open_nhds (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
∃ (s : set E) (hs : a ∈ s ∧ is_open s),
approximates_linear_on f (f' : E →L[𝕜] F) s ((nnnorm (f'.symm : F →L[𝕜] E))⁻¹ / 2) :=
begin
refine ((nhds_basis_opens a).exists_iff _).1 _,
exact (λ s t, approximates_linear_on.mono_set),
exact (hf.approximates_deriv_on_nhds $ f'.subsingleton_or_nnnorm_symm_pos.imp id $
λ hf', nnreal.half_pos $ nnreal.inv_pos.2 $ hf')
end
include cs
variable (f)
/-- Given a function with an invertible strict derivative at `a`, returns a `local_homeomorph`
with `to_fun = f` and `a ∈ source`. This is a part of the inverse function theorem.
The other part `has_strict_fderiv_at.to_local_inverse` states that the inverse function
of this `local_homeomorph` has derivative `f'.symm`. -/
def to_local_homeomorph (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) : local_homeomorph E F :=
approximates_linear_on.to_local_homeomorph f
(classical.some hf.approximates_deriv_on_open_nhds)
(classical.some_spec hf.approximates_deriv_on_open_nhds).snd
(f'.subsingleton_or_nnnorm_symm_pos.imp id $ λ hf', nnreal.half_lt_self $ ne_of_gt $
nnreal.inv_pos.2 $ hf')
(classical.some_spec hf.approximates_deriv_on_open_nhds).fst.2
variable {f}
@[simp] lemma to_local_homeomorph_coe (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
(hf.to_local_homeomorph f : E → F) = f := rfl
lemma mem_to_local_homeomorph_source (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
a ∈ (hf.to_local_homeomorph f).source :=
(classical.some_spec hf.approximates_deriv_on_open_nhds).fst.1
lemma image_mem_to_local_homeomorph_target (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
f a ∈ (hf.to_local_homeomorph f).target :=
(hf.to_local_homeomorph f).map_source hf.mem_to_local_homeomorph_source
variables (f f' a)
/-- Given a function `f` with an invertible derivative, returns a function that is locally inverse
to `f`. -/
def local_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) : F → E :=
(hf.to_local_homeomorph f).symm
variables {f f' a}
lemma eventually_left_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
∀ᶠ x in 𝓝 a, hf.local_inverse f f' a (f x) = x :=
(hf.to_local_homeomorph f).eventually_left_inverse hf.mem_to_local_homeomorph_source
lemma local_inverse_apply_image (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
hf.local_inverse f f' a (f a) = a :=
hf.eventually_left_inverse.self_of_nhds
lemma eventually_right_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
∀ᶠ y in 𝓝 (f a), f (hf.local_inverse f f' a y) = y :=
(hf.to_local_homeomorph f).eventually_right_inverse' hf.mem_to_local_homeomorph_source
lemma local_inverse_continuous_at (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
continuous_at (hf.local_inverse f f' a) (f a) :=
(hf.to_local_homeomorph f).continuous_at_symm hf.image_mem_to_local_homeomorph_target
/-- If `f` has an invertible derivative `f'` at `a` in the sense of strict differentiability `(hf)`,
then the inverse function `hf.local_inverse f` has derivative `f'.symm` at `f a`. -/
theorem to_local_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
has_strict_fderiv_at (hf.local_inverse f f' a) (f'.symm : F →L[𝕜] E) (f a) :=
begin
have : has_strict_fderiv_at f (f' : E →L[𝕜] F) (hf.local_inverse f f' a (f a)),
{ rwa hf.local_inverse_apply_image },
exact this.of_local_left_inverse hf.local_inverse_continuous_at hf.eventually_right_inverse
end
/-- If `f : E → F` has an invertible derivative `f'` at `a` in the sense of strict differentiability
and `g (f x) = x` in a neighborhood of `a`, then `g` has derivative `f'.symm` at `f a`.
For a version assuming `f (g y) = y` and continuity of `g` at `f a` but not `[complete_space E]`
see `of_local_left_inverse`. -/
theorem to_local_left_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) {g : F → E}
(hg : ∀ᶠ x in 𝓝 a, g (f x) = x) :
has_strict_fderiv_at g (f'.symm : F →L[𝕜] E) (f a) :=
begin
apply hf.to_local_inverse.congr_of_mem_sets,
have := ((hf.to_local_homeomorph f).tendsto_symm
hf.mem_to_local_homeomorph_source).eventually hg,
refine this.mp (hf.eventually_right_inverse.mono $ λ y hy hy', _),
exact hy'.symm.trans (congr_arg g hy)
end
end has_strict_fderiv_at
/-!
### Inverse function theorem, 1D case
In this case we prove a version of the inverse function theorem for maps `f : 𝕜 → 𝕜`.
We use `continuous_linear_equiv.units_equiv_aut` to translate `has_strict_deriv_at f f' a` and
`f' ≠ 0` into `has_strict_fderiv_at f (_ : 𝕜 ≃L[𝕜] 𝕜) a`.
-/
namespace has_strict_deriv_at
variables [cs : complete_space 𝕜] {f : 𝕜 → 𝕜} {f' a : 𝕜} (hf : has_strict_deriv_at f f' a)
(hf' : f' ≠ 0)
include cs
variables (f f' a)
/-- A function that is inverse to `f` near `a`. -/
@[reducible] def local_inverse : 𝕜 → 𝕜 :=
(hf.has_strict_fderiv_at_equiv hf').local_inverse _ _ _
variables {f f' a}
theorem to_local_inverse : has_strict_deriv_at (hf.local_inverse f f' a hf') f'⁻¹ (f a) :=
(hf.has_strict_fderiv_at_equiv hf').to_local_inverse
theorem to_local_left_inverse {g : 𝕜 → 𝕜} (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) :
has_strict_deriv_at g f'⁻¹ (f a) :=
(hf.has_strict_fderiv_at_equiv hf').to_local_left_inverse hg
end has_strict_deriv_at
|
b52ba6d808a8e77a350e3dab9cc10176bd43fc12 | fcf3ffa92a3847189ca669cb18b34ef6b2ec2859 | /src/world6/level3.lean | bd72573157ccb010d7c7ba9bcf9b06fc6c8dc152 | [
"Apache-2.0"
] | permissive | nomoid/lean-proofs | 4a80a97888699dee42b092b7b959b22d9aa0c066 | b9f03a24623d1a1d111d6c2bbf53c617e2596d6a | refs/heads/master | 1,674,955,317,080 | 1,607,475,706,000 | 1,607,475,706,000 | 314,104,281 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 198 | lean | lemma maze (P Q R S T U: Prop)
(p : P)
(h : P → Q)
(i : Q → R)
(j : Q → T)
(k : S → T)
(l : T → U)
: U :=
begin
have q := h(p),
have t := j(q),
have u := l(t),
exact u,
end |
c4e3cf5cd26eef1abb3506a734e0353d4b8a84e2 | 8b9f17008684d796c8022dab552e42f0cb6fb347 | /hott/hit/circle.hlean | 694aa8208e47e69ea0d5c555f0101fb5fcd4b9db | [
"Apache-2.0"
] | permissive | chubbymaggie/lean | 0d06ae25f9dd396306fb02190e89422ea94afd7b | d2c7b5c31928c98f545b16420d37842c43b4ae9a | refs/heads/master | 1,611,313,622,901 | 1,430,266,839,000 | 1,430,267,083,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,698 | hlean | /-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: hit.circle
Authors: Floris van Doorn
Declaration of the circle
-/
import .sphere
open eq suspension bool sphere_index equiv
definition circle [reducible] := suspension bool --redefine this as sphere 1
namespace circle
definition base1 : circle := !north
definition base2 : circle := !south
definition seg1 : base1 = base2 := merid tt
definition seg2 : base2 = base1 := (merid ff)⁻¹
definition base : circle := base1
definition loop : base = base := seg1 ⬝ seg2
definition rec2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) (x : circle) : P x :=
begin
fapply (suspension.rec_on x),
{ exact Pb1},
{ exact Pb2},
{ intro b, cases b,
apply tr_eq_of_eq_inv_tr, exact Ps2⁻¹,
exact Ps1},
end
definition rec2_on [reducible] {P : circle → Type} (x : circle) (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) : P x :=
circle.rec2 Pb1 Pb2 Ps1 Ps2 x
definition rec2_seg1 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1)
: apD (rec2 Pb1 Pb2 Ps1 Ps2) seg1 = sorry ⬝ Ps1 ⬝ sorry :=
sorry
definition rec2_seg2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1)
: apD (rec2 Pb1 Pb2 Ps1 Ps2) seg2 = sorry ⬝ Ps2 ⬝ sorry :=
sorry
definition elim2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1) (x : circle) : P :=
rec2 Pb1 Pb2 (!tr_constant ⬝ Ps1) (!tr_constant ⬝ Ps2) x
definition elim2_on [reducible] {P : Type} (x : circle) (Pb1 Pb2 : P)
(Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1) : P :=
elim2 Pb1 Pb2 Ps1 Ps2 x
definition elim2_seg1 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1)
: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg1 = sorry ⬝ Ps1 ⬝ sorry :=
sorry
definition elim2_seg2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1)
: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg2 = sorry ⬝ Ps2 ⬝ sorry :=
sorry
protected definition rec {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase)
(x : circle) : P x :=
begin
fapply (rec2_on x),
{ exact Pbase},
{ exact (transport P seg1 Pbase)},
{ apply idp},
{ apply eq_tr_of_inv_tr_eq, rewrite -tr_con, apply Ploop},
end
protected definition rec_on [reducible] {P : circle → Type} (x : circle) (Pbase : P base)
(Ploop : loop ▹ Pbase = Pbase) : P x :=
rec Pbase Ploop x
definition rec_loop {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase) :
ap (rec Pbase Ploop) loop = sorry ⬝ Ploop ⬝ sorry :=
sorry
protected definition elim {P : Type} (Pbase : P) (Ploop : Pbase = Pbase)
(x : circle) : P :=
rec Pbase (tr_constant loop Pbase ⬝ Ploop) x
protected definition elim_on [reducible] {P : Type} (x : circle) (Pbase : P)
(Ploop : Pbase = Pbase) : P :=
elim Pbase Ploop x
definition elim_loop {P : Type} (Pbase : P) (Ploop : Pbase = Pbase) :
ap (elim Pbase Ploop) loop = sorry ⬝ Ploop ⬝ sorry :=
sorry
protected definition elim_type (Pbase : Type) (Ploop : Pbase ≃ Pbase)
(x : circle) : Type :=
elim Pbase (ua Ploop) x
protected definition elim_type_on [reducible] (x : circle) (Pbase : Type)
(Ploop : Pbase ≃ Pbase) : Type :=
elim_type Pbase Ploop x
definition elim_type_loop (Pbase : Type) (Ploop : Pbase ≃ Pbase) :
transport (elim_type Pbase Ploop) loop = sorry /-Ploop-/ :=
sorry
end circle
|
fef3eb862505514dbab555d46b55c4dc26283c1e | 626e312b5c1cb2d88fca108f5933076012633192 | /src/algebra/ordered_group.lean | f9c9372de4a1e90c3f9f465ddd7c5885a82b0d7b | [
"Apache-2.0"
] | permissive | Bioye97/mathlib | 9db2f9ee54418d29dd06996279ba9dc874fd6beb | 782a20a27ee83b523f801ff34efb1a9557085019 | refs/heads/master | 1,690,305,956,488 | 1,631,067,774,000 | 1,631,067,774,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 45,594 | lean | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import algebra.ordered_monoid
import order.rel_iso
import order.order_dual
/-!
# Ordered groups
This file develops the basics of ordered groups.
## Implementation details
Unfortunately, the number of `'` appended to lemmas in this file
may differ between the multiplicative and the additive version of a lemma.
The reason is that we did not want to change existing names in the library.
-/
set_option old_structure_cmd true
open function
universe u
variable {α : Type u}
@[to_additive]
instance group.covariant_class_le.to_contravariant_class_le
[group α] [has_le α] [covariant_class α α (*) (≤)] : contravariant_class α α (*) (≤) :=
group.covconv
@[to_additive]
instance group.swap.covariant_class_le.to_contravariant_class_le [group α] [has_le α]
[covariant_class α α (swap (*)) (≤)] : contravariant_class α α (swap (*)) (≤) :=
{ elim := λ a b c bc, calc b = b * a * a⁻¹ : eq_mul_inv_of_mul_eq rfl
... ≤ c * a * a⁻¹ : mul_le_mul_right' bc a⁻¹
... = c : mul_inv_eq_of_eq_mul rfl }
@[to_additive]
instance group.covariant_class_lt.to_contravariant_class_lt
[group α] [has_lt α] [covariant_class α α (*) (<)] : contravariant_class α α (*) (<) :=
{ elim := λ a b c bc, calc b = a⁻¹ * (a * b) : eq_inv_mul_of_mul_eq rfl
... < a⁻¹ * (a * c) : mul_lt_mul_left' bc a⁻¹
... = c : inv_mul_cancel_left a c }
@[to_additive]
instance group.swap.covariant_class_lt.to_contravariant_class_lt [group α] [has_lt α]
[covariant_class α α (swap (*)) (<)] : contravariant_class α α (swap (*)) (<) :=
{ elim := λ a b c bc, calc b = b * a * a⁻¹ : eq_mul_inv_of_mul_eq rfl
... < c * a * a⁻¹ : mul_lt_mul_right' bc a⁻¹
... = c : mul_inv_eq_of_eq_mul rfl }
/-- An ordered additive commutative group is an additive commutative group
with a partial order in which addition is strictly monotone. -/
@[protect_proj, ancestor add_comm_group partial_order]
class ordered_add_comm_group (α : Type u) extends add_comm_group α, partial_order α :=
(add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b)
/-- An ordered commutative group is an commutative group
with a partial order in which multiplication is strictly monotone. -/
@[protect_proj, ancestor comm_group partial_order]
class ordered_comm_group (α : Type u) extends comm_group α, partial_order α :=
(mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b)
attribute [to_additive] ordered_comm_group
@[to_additive]
instance ordered_comm_group.to_covariant_class_left_le (α : Type u) [ordered_comm_group α] :
covariant_class α α (*) (≤) :=
{ elim := λ a b c bc, ordered_comm_group.mul_le_mul_left b c bc a }
/--The units of an ordered commutative monoid form an ordered commutative group. -/
@[to_additive]
instance units.ordered_comm_group [ordered_comm_monoid α] : ordered_comm_group (units α) :=
{ mul_le_mul_left := λ a b h c, (mul_le_mul_left' (h : (a : α) ≤ b) _ : (c : α) * a ≤ c * b),
.. units.partial_order,
.. units.comm_group }
@[priority 100, to_additive] -- see Note [lower instance priority]
instance ordered_comm_group.to_ordered_cancel_comm_monoid (α : Type u)
[s : ordered_comm_group α] :
ordered_cancel_comm_monoid α :=
{ mul_left_cancel := λ a b c, (mul_right_inj a).mp,
le_of_mul_le_mul_left := λ a b c, (mul_le_mul_iff_left a).mp,
..s }
@[priority 100, to_additive]
instance ordered_comm_group.has_exists_mul_of_le (α : Type u)
[ordered_comm_group α] :
has_exists_mul_of_le α :=
⟨λ a b hab, ⟨b * a⁻¹, (mul_inv_cancel_comm_assoc a b).symm⟩⟩
@[to_additive] instance [h : has_inv α] : has_inv (order_dual α) := h
@[to_additive] instance [h : has_div α] : has_div (order_dual α) := h
@[to_additive] instance [h : div_inv_monoid α] : div_inv_monoid (order_dual α) := h
@[to_additive] instance [h : group α] : group (order_dual α) := h
@[to_additive] instance [h : comm_group α] : comm_group (order_dual α) := h
@[to_additive] instance [ordered_comm_group α] : ordered_comm_group (order_dual α) :=
{ .. order_dual.ordered_comm_monoid, .. order_dual.group }
section group
variables [group α]
section typeclasses_left_le
variables [has_le α] [covariant_class α α (*) (≤)] {a b c d : α}
/-- Uses `left` co(ntra)variant. -/
@[simp, to_additive left.neg_nonpos_iff]
lemma left.inv_le_one_iff :
a⁻¹ ≤ 1 ↔ 1 ≤ a :=
by { rw [← mul_le_mul_iff_left a], simp }
/-- Uses `left` co(ntra)variant. -/
@[simp, to_additive left.nonneg_neg_iff]
lemma left.one_le_inv_iff :
1 ≤ a⁻¹ ↔ a ≤ 1 :=
by { rw [← mul_le_mul_iff_left a], simp }
@[simp, to_additive]
lemma le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c :=
by { rw ← mul_le_mul_iff_left a, simp }
@[simp, to_additive]
lemma inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c :=
by rw [← mul_le_mul_iff_left b, mul_inv_cancel_left]
@[to_additive neg_le_iff_add_nonneg']
lemma inv_le_iff_one_le_mul' : a⁻¹ ≤ b ↔ 1 ≤ a * b :=
(mul_le_mul_iff_left a).symm.trans $ by rw mul_inv_self
@[to_additive]
lemma le_inv_iff_mul_le_one_left : a ≤ b⁻¹ ↔ b * a ≤ 1 :=
(mul_le_mul_iff_left b).symm.trans $ by rw mul_inv_self
@[to_additive]
lemma le_inv_mul_iff_le : 1 ≤ b⁻¹ * a ↔ b ≤ a :=
by rw [← mul_le_mul_iff_left b, mul_one, mul_inv_cancel_left]
@[to_additive]
lemma inv_mul_le_one_iff : a⁻¹ * b ≤ 1 ↔ b ≤ a :=
trans (inv_mul_le_iff_le_mul) $ by rw mul_one
end typeclasses_left_le
section typeclasses_left_lt
variables [has_lt α] [covariant_class α α (*) (<)] {a b c : α}
/-- Uses `left` co(ntra)variant. -/
@[simp, to_additive left.neg_pos_iff]
lemma left.one_lt_inv_iff :
1 < a⁻¹ ↔ a < 1 :=
by rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
/-- Uses `left` co(ntra)variant. -/
@[simp, to_additive left.neg_neg_iff]
lemma left.inv_lt_one_iff :
a⁻¹ < 1 ↔ 1 < a :=
by rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
@[simp, to_additive]
lemma lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c :=
by { rw [← mul_lt_mul_iff_left a], simp }
@[simp, to_additive]
lemma inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c :=
by rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left]
@[to_additive]
lemma inv_lt_iff_one_lt_mul' : a⁻¹ < b ↔ 1 < a * b :=
(mul_lt_mul_iff_left a).symm.trans $ by rw mul_inv_self
@[to_additive]
lemma lt_inv_iff_mul_lt_one' : a < b⁻¹ ↔ b * a < 1 :=
(mul_lt_mul_iff_left b).symm.trans $ by rw mul_inv_self
@[to_additive]
lemma lt_inv_mul_iff_lt : 1 < b⁻¹ * a ↔ b < a :=
by rw [← mul_lt_mul_iff_left b, mul_one, mul_inv_cancel_left]
@[to_additive]
lemma inv_mul_lt_one_iff : a⁻¹ * b < 1 ↔ b < a :=
trans (inv_mul_lt_iff_lt_mul) $ by rw mul_one
end typeclasses_left_lt
section typeclasses_right_le
variables [has_le α] [covariant_class α α (swap (*)) (≤)] {a b c : α}
/-- Uses `right` co(ntra)variant. -/
@[simp, to_additive right.neg_nonpos_iff]
lemma right.inv_le_one_iff :
a⁻¹ ≤ 1 ↔ 1 ≤ a :=
by { rw [← mul_le_mul_iff_right a], simp }
/-- Uses `right` co(ntra)variant. -/
@[simp, to_additive right.nonneg_neg_iff]
lemma right.one_le_inv_iff :
1 ≤ a⁻¹ ↔ a ≤ 1 :=
by { rw [← mul_le_mul_iff_right a], simp }
@[to_additive neg_le_iff_add_nonneg]
lemma inv_le_iff_one_le_mul : a⁻¹ ≤ b ↔ 1 ≤ b * a :=
(mul_le_mul_iff_right a).symm.trans $ by rw inv_mul_self
@[to_additive]
lemma le_inv_iff_mul_le_one_right : a ≤ b⁻¹ ↔ a * b ≤ 1 :=
(mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_self
@[simp, to_additive]
lemma mul_inv_le_iff_le_mul : a * b⁻¹ ≤ c ↔ a ≤ c * b :=
(mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_cancel_right
@[simp, to_additive]
lemma le_mul_inv_iff_mul_le : c ≤ a * b⁻¹ ↔ c * b ≤ a :=
(mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_cancel_right
@[simp, to_additive]
lemma mul_inv_le_one_iff_le : a * b⁻¹ ≤ 1 ↔ a ≤ b :=
mul_inv_le_iff_le_mul.trans $ by rw one_mul
@[to_additive]
lemma le_mul_inv_iff_le : 1 ≤ a * b⁻¹ ↔ b ≤ a :=
by rw [← mul_le_mul_iff_right b, one_mul, inv_mul_cancel_right]
@[to_additive]
lemma mul_inv_le_one_iff : b * a⁻¹ ≤ 1 ↔ b ≤ a :=
trans (mul_inv_le_iff_le_mul) $ by rw one_mul
end typeclasses_right_le
section typeclasses_right_lt
variables [has_lt α] [covariant_class α α (swap (*)) (<)] {a b c : α}
/-- Uses `right` co(ntra)variant. -/
@[simp, to_additive right.neg_neg_iff]
lemma right.inv_lt_one_iff :
a⁻¹ < 1 ↔ 1 < a :=
by rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
/-- Uses `right` co(ntra)variant. -/
@[simp, to_additive right.neg_pos_iff]
lemma right.one_lt_inv_iff :
1 < a⁻¹ ↔ a < 1 :=
by rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
@[to_additive]
lemma inv_lt_iff_one_lt_mul : a⁻¹ < b ↔ 1 < b * a :=
(mul_lt_mul_iff_right a).symm.trans $ by rw inv_mul_self
@[to_additive]
lemma lt_inv_iff_mul_lt_one : a < b⁻¹ ↔ a * b < 1 :=
(mul_lt_mul_iff_right b).symm.trans $ by rw inv_mul_self
@[simp, to_additive]
lemma mul_inv_lt_iff_lt_mul : a * b⁻¹ < c ↔ a < c * b :=
by rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right]
@[simp, to_additive]
lemma lt_mul_inv_iff_mul_lt : c < a * b⁻¹ ↔ c * b < a :=
(mul_lt_mul_iff_right b).symm.trans $ by rw inv_mul_cancel_right
@[simp, to_additive]
lemma inv_mul_lt_one_iff_lt : a * b⁻¹ < 1 ↔ a < b :=
by rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right, one_mul]
@[to_additive]
lemma lt_mul_inv_iff_lt : 1 < a * b⁻¹ ↔ b < a :=
by rw [← mul_lt_mul_iff_right b, one_mul, inv_mul_cancel_right]
@[to_additive]
lemma mul_inv_lt_one_iff : b * a⁻¹ < 1 ↔ b < a :=
trans (mul_inv_lt_iff_lt_mul) $ by rw one_mul
end typeclasses_right_lt
section typeclasses_left_right_le
variables [has_le α] [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (≤)]
{a b c d : α}
@[simp, to_additive]
lemma inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a :=
by { rw [← mul_le_mul_iff_left a, ← mul_le_mul_iff_right b], simp }
alias neg_le_neg_iff ↔ le_of_neg_le_neg _
section
variable (α)
/-- `x ↦ x⁻¹` as an order-reversing equivalence. -/
@[to_additive "`x ↦ -x` as an order-reversing equivalence.", simps]
def order_iso.inv : α ≃o order_dual α :=
{ to_equiv := (equiv.inv α).trans order_dual.to_dual,
map_rel_iff' := λ a b, @inv_le_inv_iff α _ _ _ _ _ _ }
end
@[to_additive neg_le]
lemma inv_le' : a⁻¹ ≤ b ↔ b⁻¹ ≤ a :=
(order_iso.inv α).symm_apply_le
alias inv_le' ↔ inv_le_of_inv_le' _
attribute [to_additive neg_le_of_neg_le] inv_le_of_inv_le'
@[to_additive le_neg]
lemma le_inv' : a ≤ b⁻¹ ↔ b ≤ a⁻¹ :=
(order_iso.inv α).le_symm_apply
@[to_additive]
lemma mul_inv_le_inv_mul_iff : a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ c * b :=
by rw [← mul_le_mul_iff_left d, ← mul_le_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
@[simp, to_additive] lemma div_le_self_iff (a : α) {b : α} : a / b ≤ a ↔ 1 ≤ b :=
by simp [div_eq_mul_inv]
alias sub_le_self_iff ↔ _ sub_le_self
end typeclasses_left_right_le
section typeclasses_left_right_lt
variables [has_lt α] [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (<)]
{a b c d : α}
@[simp, to_additive]
lemma inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a :=
by { rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b], simp }
@[to_additive neg_lt]
lemma inv_lt' : a⁻¹ < b ↔ b⁻¹ < a :=
by rw [← inv_lt_inv_iff, inv_inv]
@[to_additive lt_neg]
lemma lt_inv' : a < b⁻¹ ↔ b < a⁻¹ :=
by rw [← inv_lt_inv_iff, inv_inv]
alias lt_inv' ↔ lt_inv_of_lt_inv _
attribute [to_additive] lt_inv_of_lt_inv
alias inv_lt' ↔ inv_lt_of_inv_lt' _
attribute [to_additive neg_lt_of_neg_lt] inv_lt_of_inv_lt'
@[to_additive]
lemma mul_inv_lt_inv_mul_iff : a * b⁻¹ < d⁻¹ * c ↔ d * a < c * b :=
by rw [← mul_lt_mul_iff_left d, ← mul_lt_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
@[simp, to_additive] lemma div_lt_self_iff (a : α) {b : α} : a / b < a ↔ 1 < b :=
by simp [div_eq_mul_inv]
alias sub_lt_self_iff ↔ _ sub_lt_self
end typeclasses_left_right_lt
section pre_order
variable [preorder α]
section left_le
variables [covariant_class α α (*) (≤)] {a : α}
@[to_additive]
lemma left.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a :=
le_trans (left.inv_le_one_iff.mpr h) h
alias left.neg_le_self ← neg_le_self
@[to_additive]
lemma left.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ :=
le_trans h (left.one_le_inv_iff.mpr h)
end left_le
section left_lt
variables [covariant_class α α (*) (<)] {a : α}
@[to_additive]
lemma left.inv_lt_self (h : 1 < a) : a⁻¹ < a :=
(left.inv_lt_one_iff.mpr h).trans h
alias left.neg_lt_self ← neg_lt_self
@[to_additive]
lemma left.self_lt_inv (h : a < 1) : a < a⁻¹ :=
lt_trans h (left.one_lt_inv_iff.mpr h)
end left_lt
section right_le
variables [covariant_class α α (swap (*)) (≤)] {a : α}
@[to_additive]
lemma right.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a :=
le_trans (right.inv_le_one_iff.mpr h) h
@[to_additive]
lemma right.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ :=
le_trans h (right.one_le_inv_iff.mpr h)
end right_le
section right_lt
variables [covariant_class α α (swap (*)) (<)] {a : α}
@[to_additive]
lemma right.inv_lt_self (h : 1 < a) : a⁻¹ < a :=
(right.inv_lt_one_iff.mpr h).trans h
@[to_additive]
lemma right.self_lt_inv (h : a < 1) : a < a⁻¹ :=
lt_trans h (right.one_lt_inv_iff.mpr h)
end right_lt
end pre_order
end group
section comm_group
variables [comm_group α]
section has_le
variables [has_le α] [covariant_class α α (*) (≤)] {a b c d : α}
@[to_additive]
lemma inv_mul_le_iff_le_mul' : c⁻¹ * a ≤ b ↔ a ≤ b * c :=
by rw [inv_mul_le_iff_le_mul, mul_comm]
@[simp, to_additive]
lemma mul_inv_le_iff_le_mul' : a * b⁻¹ ≤ c ↔ a ≤ b * c :=
by rw [← inv_mul_le_iff_le_mul, mul_comm]
@[to_additive add_neg_le_add_neg_iff]
lemma mul_inv_le_mul_inv_iff' : a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b :=
by rw [mul_comm c, mul_inv_le_inv_mul_iff, mul_comm]
end has_le
section has_lt
variables [has_lt α] [covariant_class α α (*) (<)] {a b c d : α}
@[to_additive]
lemma inv_mul_lt_iff_lt_mul' : c⁻¹ * a < b ↔ a < b * c :=
by rw [inv_mul_lt_iff_lt_mul, mul_comm]
@[simp, to_additive]
lemma mul_inv_lt_iff_le_mul' : a * b⁻¹ < c ↔ a < b * c :=
by rw [← inv_mul_lt_iff_lt_mul, mul_comm]
@[to_additive add_neg_lt_add_neg_iff]
lemma mul_inv_lt_mul_inv_iff' : a * b⁻¹ < c * d⁻¹ ↔ a * d < c * b :=
by rw [mul_comm c, mul_inv_lt_inv_mul_iff, mul_comm]
end has_lt
end comm_group
alias le_inv' ↔ le_inv_of_le_inv _
attribute [to_additive] le_inv_of_le_inv
alias left.inv_le_one_iff ↔ one_le_of_inv_le_one _
attribute [to_additive] one_le_of_inv_le_one
alias left.one_le_inv_iff ↔ le_one_of_one_le_inv _
attribute [to_additive nonpos_of_neg_nonneg] le_one_of_one_le_inv
alias inv_lt_inv_iff ↔ lt_of_inv_lt_inv _
attribute [to_additive] lt_of_inv_lt_inv
alias left.inv_lt_one_iff ↔ one_lt_of_inv_lt_one _
attribute [to_additive] one_lt_of_inv_lt_one
alias left.inv_lt_one_iff ← inv_lt_one_iff_one_lt
attribute [to_additive] inv_lt_one_iff_one_lt
alias left.inv_lt_one_iff ← inv_lt_one'
attribute [to_additive neg_lt_zero] inv_lt_one'
alias left.one_lt_inv_iff ↔ inv_of_one_lt_inv _
attribute [to_additive neg_of_neg_pos] inv_of_one_lt_inv
alias left.one_lt_inv_iff ↔ _ one_lt_inv_of_inv
attribute [to_additive neg_pos_of_neg] one_lt_inv_of_inv
alias le_inv_mul_iff_mul_le ↔ mul_le_of_le_inv_mul _
attribute [to_additive] mul_le_of_le_inv_mul
alias le_inv_mul_iff_mul_le ↔ _ le_inv_mul_of_mul_le
attribute [to_additive] le_inv_mul_of_mul_le
alias inv_mul_le_iff_le_mul ↔ _ inv_mul_le_of_le_mul
attribute [to_additive] inv_mul_le_iff_le_mul
alias lt_inv_mul_iff_mul_lt ↔ mul_lt_of_lt_inv_mul _
attribute [to_additive] mul_lt_of_lt_inv_mul
alias lt_inv_mul_iff_mul_lt ↔ _ lt_inv_mul_of_mul_lt
attribute [to_additive] lt_inv_mul_of_mul_lt
alias inv_mul_lt_iff_lt_mul ↔ lt_mul_of_inv_mul_lt inv_mul_lt_of_lt_mul
attribute [to_additive] lt_mul_of_inv_mul_lt
attribute [to_additive] inv_mul_lt_of_lt_mul
alias lt_mul_of_inv_mul_lt ← lt_mul_of_inv_mul_lt_left
attribute [to_additive] lt_mul_of_inv_mul_lt_left
alias left.inv_le_one_iff ← inv_le_one'
attribute [to_additive neg_nonpos] inv_le_one'
alias left.one_le_inv_iff ← one_le_inv'
attribute [to_additive neg_nonneg] one_le_inv'
alias left.one_lt_inv_iff ← one_lt_inv'
attribute [to_additive neg_pos] one_lt_inv'
alias mul_lt_mul_left' ← ordered_comm_group.mul_lt_mul_left'
attribute [to_additive ordered_add_comm_group.add_lt_add_left] ordered_comm_group.mul_lt_mul_left'
alias le_of_mul_le_mul_left' ← ordered_comm_group.le_of_mul_le_mul_left
attribute [to_additive ordered_add_comm_group.le_of_add_le_add_left]
ordered_comm_group.le_of_mul_le_mul_left
alias lt_of_mul_lt_mul_left' ← ordered_comm_group.lt_of_mul_lt_mul_left
attribute [to_additive ordered_add_comm_group.lt_of_add_lt_add_left]
ordered_comm_group.lt_of_mul_lt_mul_left
/-- Pullback an `ordered_comm_group` under an injective map.
See note [reducible non-instances]. -/
@[reducible, to_additive function.injective.ordered_add_comm_group
"Pullback an `ordered_add_comm_group` under an injective map."]
def function.injective.ordered_comm_group [ordered_comm_group α] {β : Type*}
[has_one β] [has_mul β] [has_inv β] [has_div β]
(f : β → α) (hf : function.injective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y)
(inv : ∀ x, f (x⁻¹) = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) :
ordered_comm_group β :=
{ ..partial_order.lift f hf,
..hf.ordered_comm_monoid f one mul,
..hf.comm_group f one mul inv div }
/- Most of the lemmas that are primed in this section appear in ordered_field. -/
/- I (DT) did not try to minimise the assumptions. -/
section group
variables [group α] [has_le α]
section right
variables [covariant_class α α (swap (*)) (≤)] {a b c d : α}
@[simp, to_additive]
lemma div_le_div_iff_right (c : α) : a / c ≤ b / c ↔ a ≤ b :=
by simpa only [div_eq_mul_inv] using mul_le_mul_iff_right _
@[to_additive sub_le_sub_right]
lemma div_le_div_right' (h : a ≤ b) (c : α) : a / c ≤ b / c :=
(div_le_div_iff_right c).2 h
@[simp, to_additive sub_nonneg]
lemma one_le_div' : 1 ≤ a / b ↔ b ≤ a :=
by rw [← mul_le_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
alias sub_nonneg ↔ le_of_sub_nonneg sub_nonneg_of_le
@[simp, to_additive sub_nonpos]
lemma div_le_one' : a / b ≤ 1 ↔ a ≤ b :=
by rw [← mul_le_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
alias sub_nonpos ↔ le_of_sub_nonpos sub_nonpos_of_le
@[to_additive]
lemma le_div_iff_mul_le : a ≤ c / b ↔ a * b ≤ c :=
by rw [← mul_le_mul_iff_right b, div_eq_mul_inv, inv_mul_cancel_right]
alias le_sub_iff_add_le ↔ add_le_of_le_sub_right le_sub_right_of_add_le
@[to_additive]
lemma div_le_iff_le_mul : a / c ≤ b ↔ a ≤ b * c :=
by rw [← mul_le_mul_iff_right c, div_eq_mul_inv, inv_mul_cancel_right]
/-- `equiv.mul_right` as an `order_iso`. -/
@[to_additive "`equiv.add_right` as an `order_iso`.", simps to_equiv apply {simp_rhs := tt}]
def order_iso.mul_right (a : α) : α ≃o α :=
{ map_rel_iff' := λ _ _, mul_le_mul_iff_right a, to_equiv := equiv.mul_right a }
@[simp, to_additive] lemma order_iso.mul_right_symm (a : α) :
(order_iso.mul_right a).symm = order_iso.mul_right a⁻¹ :=
by { ext x, refl }
end right
section left
variables [covariant_class α α (*) (≤)]
/-- `equiv.mul_left` as an `order_iso`. -/
@[to_additive "`equiv.add_left` as an `order_iso`.", simps to_equiv apply {simp_rhs := tt}]
def order_iso.mul_left (a : α) : α ≃o α :=
{ map_rel_iff' := λ _ _, mul_le_mul_iff_left a, to_equiv := equiv.mul_left a }
@[simp, to_additive] lemma order_iso.mul_left_symm (a : α) :
(order_iso.mul_left a).symm = order_iso.mul_left a⁻¹ :=
by { ext x, refl }
variables [covariant_class α α (swap (*)) (≤)] {a b c : α}
@[simp, to_additive]
lemma div_le_div_iff_left (a : α) : a / b ≤ a / c ↔ c ≤ b :=
by rw [div_eq_mul_inv, div_eq_mul_inv, ← mul_le_mul_iff_left a⁻¹, inv_mul_cancel_left,
inv_mul_cancel_left, inv_le_inv_iff]
@[to_additive sub_le_sub_left]
lemma div_le_div_left' (h : a ≤ b) (c : α) : c / b ≤ c / a :=
(div_le_div_iff_left c).2 h
end left
end group
section comm_group
variables [comm_group α]
section has_le
variables [has_le α] [covariant_class α α (*) (≤)] {a b c d : α}
@[to_additive sub_le_sub_iff]
lemma div_le_div_iff' : a / b ≤ c / d ↔ a * d ≤ c * b :=
by simpa only [div_eq_mul_inv] using mul_inv_le_mul_inv_iff'
@[to_additive]
lemma le_div_iff_mul_le' : b ≤ c / a ↔ a * b ≤ c :=
by rw [le_div_iff_mul_le, mul_comm]
alias le_sub_iff_add_le' ↔ add_le_of_le_sub_left le_sub_left_of_add_le
@[to_additive]
lemma div_le_iff_le_mul' : a / b ≤ c ↔ a ≤ b * c :=
by rw [div_le_iff_le_mul, mul_comm]
alias sub_le_iff_le_add' ↔ le_add_of_sub_left_le sub_left_le_of_le_add
@[simp, to_additive]
lemma inv_le_div_iff_le_mul : b⁻¹ ≤ a / c ↔ c ≤ a * b :=
le_div_iff_mul_le.trans inv_mul_le_iff_le_mul'
@[to_additive]
lemma inv_le_div_iff_le_mul' : a⁻¹ ≤ b / c ↔ c ≤ a * b :=
by rw [inv_le_div_iff_le_mul, mul_comm]
@[to_additive sub_le]
lemma div_le'' : a / b ≤ c ↔ a / c ≤ b :=
div_le_iff_le_mul'.trans div_le_iff_le_mul.symm
@[to_additive le_sub]
lemma le_div'' : a ≤ b / c ↔ c ≤ b / a :=
le_div_iff_mul_le'.trans le_div_iff_mul_le.symm
end has_le
section preorder
variables [preorder α] [covariant_class α α (*) (≤)] {a b c d : α}
@[to_additive sub_le_sub]
lemma div_le_div'' (hab : a ≤ b) (hcd : c ≤ d) :
a / d ≤ b / c :=
begin
rw [div_eq_mul_inv, div_eq_mul_inv, mul_comm b, mul_inv_le_inv_mul_iff, mul_comm],
exact mul_le_mul' hab hcd
end
end preorder
end comm_group
/- Most of the lemmas that are primed in this section appear in ordered_field. -/
/- I (DT) did not try to minimise the assumptions. -/
section group
variables [group α] [has_lt α]
section right
variables [covariant_class α α (swap (*)) (<)] {a b c d : α}
@[simp, to_additive]
lemma div_lt_div_iff_right (c : α) : a / c < b / c ↔ a < b :=
by simpa only [div_eq_mul_inv] using mul_lt_mul_iff_right _
@[to_additive sub_lt_sub_right]
lemma div_lt_div_right' (h : a < b) (c : α) : a / c < b / c :=
(div_lt_div_iff_right c).2 h
@[simp, to_additive sub_pos]
lemma one_lt_div' : 1 < a / b ↔ b < a :=
by rw [← mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
alias sub_pos ↔ lt_of_sub_pos sub_pos_of_lt
@[simp, to_additive sub_neg]
lemma div_lt_one' : a / b < 1 ↔ a < b :=
by rw [← mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
alias sub_neg ↔ lt_of_sub_neg sub_neg_of_lt
alias sub_neg ← sub_lt_zero
@[to_additive]
lemma lt_div_iff_mul_lt : a < c / b ↔ a * b < c :=
by rw [← mul_lt_mul_iff_right b, div_eq_mul_inv, inv_mul_cancel_right]
alias lt_sub_iff_add_lt ↔ add_lt_of_lt_sub_right lt_sub_right_of_add_lt
@[to_additive]
lemma div_lt_iff_lt_mul : a / c < b ↔ a < b * c :=
by rw [← mul_lt_mul_iff_right c, div_eq_mul_inv, inv_mul_cancel_right]
alias sub_lt_iff_lt_add ↔ lt_add_of_sub_right_lt sub_right_lt_of_lt_add
end right
section left
variables [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (<)] {a b c : α}
@[simp, to_additive]
lemma div_lt_div_iff_left (a : α) : a / b < a / c ↔ c < b :=
by rw [div_eq_mul_inv, div_eq_mul_inv, ← mul_lt_mul_iff_left a⁻¹, inv_mul_cancel_left,
inv_mul_cancel_left, inv_lt_inv_iff]
@[simp, to_additive]
lemma inv_lt_div_iff_lt_mul : a⁻¹ < b / c ↔ c < a * b :=
by rw [div_eq_mul_inv, lt_mul_inv_iff_mul_lt, inv_mul_lt_iff_lt_mul]
@[to_additive sub_lt_sub_left]
lemma div_lt_div_left' (h : a < b) (c : α) : c / b < c / a :=
(div_lt_div_iff_left c).2 h
end left
end group
section comm_group
variables [comm_group α]
section has_lt
variables [has_lt α] [covariant_class α α (*) (<)] {a b c d : α}
@[to_additive sub_lt_sub_iff]
lemma div_lt_div_iff' : a / b < c / d ↔ a * d < c * b :=
by simpa only [div_eq_mul_inv] using mul_inv_lt_mul_inv_iff'
@[to_additive]
lemma lt_div_iff_mul_lt' : b < c / a ↔ a * b < c :=
by rw [lt_div_iff_mul_lt, mul_comm]
alias lt_sub_iff_add_lt' ↔ add_lt_of_lt_sub_left lt_sub_left_of_add_lt
@[to_additive]
lemma div_lt_iff_lt_mul' : a / b < c ↔ a < b * c :=
by rw [div_lt_iff_lt_mul, mul_comm]
alias sub_lt_iff_lt_add' ↔ lt_add_of_sub_left_lt sub_left_lt_of_lt_add
@[to_additive]
lemma inv_lt_div_iff_lt_mul' : b⁻¹ < a / c ↔ c < a * b :=
lt_div_iff_mul_lt.trans inv_mul_lt_iff_lt_mul'
@[to_additive sub_lt]
lemma div_lt'' : a / b < c ↔ a / c < b :=
div_lt_iff_lt_mul'.trans div_lt_iff_lt_mul.symm
@[to_additive lt_sub]
lemma lt_div'' : a < b / c ↔ c < b / a :=
lt_div_iff_mul_lt'.trans lt_div_iff_mul_lt.symm
end has_lt
section preorder
variables [preorder α] [covariant_class α α (*) (<)] {a b c d : α}
@[to_additive sub_lt_sub]
lemma div_lt_div'' (hab : a < b) (hcd : c < d) :
a / d < b / c :=
begin
rw [div_eq_mul_inv, div_eq_mul_inv, mul_comm b, mul_inv_lt_inv_mul_iff, mul_comm],
exact mul_lt_mul_of_lt_of_lt hab hcd
end
end preorder
end comm_group
section linear_order
variables [group α] [linear_order α] [covariant_class α α (*) (≤)]
section variable_names
variables {a b c : α}
@[to_additive]
lemma le_of_forall_one_lt_lt_mul (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b :=
le_of_not_lt (λ h₁, lt_irrefl a (by simpa using (h _ (lt_inv_mul_iff_lt.mpr h₁))))
@[to_additive]
lemma le_iff_forall_one_lt_lt_mul : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε :=
⟨λ h ε, lt_mul_of_le_of_one_lt h, le_of_forall_one_lt_lt_mul⟩
/- I (DT) introduced this lemma to prove (the additive version `sub_le_sub_flip` of)
`div_le_div_flip` below. Now I wonder what is the point of either of these lemmas... -/
@[to_additive]
lemma div_le_inv_mul_iff [covariant_class α α (swap (*)) (≤)] :
a / b ≤ a⁻¹ * b ↔ a ≤ b :=
begin
rw [div_eq_mul_inv, mul_inv_le_inv_mul_iff],
exact ⟨λ h, not_lt.mp (λ k, not_lt.mpr h (mul_lt_mul''' k k)), λ h, mul_le_mul' h h⟩,
end
/- What is the point of this lemma? See comment about `div_le_inv_mul_iff` above. -/
@[simp, to_additive]
lemma div_le_div_flip {α : Type*} [comm_group α] [linear_order α] [covariant_class α α (*) (≤)]
{a b : α}:
a / b ≤ b / a ↔ a ≤ b :=
begin
rw [div_eq_mul_inv b, mul_comm],
exact div_le_inv_mul_iff,
end
@[simp, to_additive] lemma max_one_div_max_inv_one_eq_self (a : α) :
max a 1 / max a⁻¹ 1 = a :=
by { rcases le_total a 1 with h|h; simp [h] }
alias max_zero_sub_max_neg_zero_eq_self ← max_zero_sub_eq_self
end variable_names
section densely_ordered
variables [densely_ordered α] {a b c : α}
@[to_additive]
lemma le_of_forall_one_lt_le_mul (h : ∀ ε : α, 1 < ε → a ≤ b * ε) : a ≤ b :=
le_of_forall_le_of_dense $ λ c hc,
calc a ≤ b * (b⁻¹ * c) : h _ (lt_inv_mul_iff_lt.mpr hc)
... = c : mul_inv_cancel_left b c
@[to_additive]
lemma le_of_forall_lt_one_mul_le (h : ∀ ε < 1, a * ε ≤ b) : a ≤ b :=
@le_of_forall_one_lt_le_mul (order_dual α) _ _ _ _ _ _ h
@[to_additive]
lemma le_of_forall_one_lt_div_le (h : ∀ ε : α, 1 < ε → a / ε ≤ b) : a ≤ b :=
le_of_forall_lt_one_mul_le $ λ ε ε1,
by simpa only [div_eq_mul_inv, inv_inv] using h ε⁻¹ (left.one_lt_inv_iff.2 ε1)
@[to_additive]
lemma le_iff_forall_one_lt_le_mul : a ≤ b ↔ ∀ ε, 1 < ε → a ≤ b * ε :=
⟨λ h ε ε_pos, le_mul_of_le_of_one_le h ε_pos.le, le_of_forall_one_lt_le_mul⟩
@[to_additive]
lemma le_iff_forall_lt_one_mul_le : a ≤ b ↔ ∀ ε < 1, a * ε ≤ b :=
@le_iff_forall_one_lt_le_mul (order_dual α) _ _ _ _ _ _
end densely_ordered
end linear_order
/-!
### Linearly ordered commutative groups
-/
/-- A linearly ordered additive commutative group is an
additive commutative group with a linear order in which
addition is monotone. -/
@[protect_proj, ancestor ordered_add_comm_group linear_order]
class linear_ordered_add_comm_group (α : Type u) extends ordered_add_comm_group α, linear_order α
/-- A linearly ordered commutative monoid with an additively absorbing `⊤` element.
Instances should include number systems with an infinite element adjoined.` -/
@[protect_proj, ancestor linear_ordered_add_comm_monoid_with_top sub_neg_monoid nontrivial]
class linear_ordered_add_comm_group_with_top (α : Type*)
extends linear_ordered_add_comm_monoid_with_top α, sub_neg_monoid α, nontrivial α :=
(neg_top : - (⊤ : α) = ⊤)
(add_neg_cancel : ∀ a:α, a ≠ ⊤ → a + (- a) = 0)
/-- A linearly ordered commutative group is a
commutative group with a linear order in which
multiplication is monotone. -/
@[protect_proj, ancestor ordered_comm_group linear_order, to_additive]
class linear_ordered_comm_group (α : Type u) extends ordered_comm_group α, linear_order α
@[to_additive] instance [linear_ordered_comm_group α] :
linear_ordered_comm_group (order_dual α) :=
{ .. order_dual.ordered_comm_group, .. order_dual.linear_order α }
section linear_ordered_comm_group
variables [linear_ordered_comm_group α] {a b c : α}
@[priority 100, to_additive] -- see Note [lower instance priority]
instance linear_ordered_comm_group.to_linear_ordered_cancel_comm_monoid :
linear_ordered_cancel_comm_monoid α :=
{ le_of_mul_le_mul_left := λ x y z, le_of_mul_le_mul_left',
mul_left_cancel := λ x y z, mul_left_cancel,
..‹linear_ordered_comm_group α› }
/-- Pullback a `linear_ordered_comm_group` under an injective map.
See note [reducible non-instances]. -/
@[reducible, to_additive function.injective.linear_ordered_add_comm_group
"Pullback a `linear_ordered_add_comm_group` under an injective map."]
def function.injective.linear_ordered_comm_group {β : Type*}
[has_one β] [has_mul β] [has_inv β] [has_div β]
(f : β → α) (hf : function.injective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y)
(inv : ∀ x, f (x⁻¹) = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) :
linear_ordered_comm_group β :=
{ ..linear_order.lift f hf,
..hf.ordered_comm_group f one mul inv div }
@[to_additive linear_ordered_add_comm_group.add_lt_add_left]
lemma linear_ordered_comm_group.mul_lt_mul_left'
(a b : α) (h : a < b) (c : α) : c * a < c * b :=
mul_lt_mul_left' h c
@[to_additive min_neg_neg]
lemma min_inv_inv' (a b : α) : min (a⁻¹) (b⁻¹) = (max a b)⁻¹ :=
eq.symm $ @monotone.map_max α (order_dual α) _ _ has_inv.inv a b $ λ a b, inv_le_inv_iff.mpr
@[to_additive max_neg_neg]
lemma max_inv_inv' (a b : α) : max (a⁻¹) (b⁻¹) = (min a b)⁻¹ :=
eq.symm $ @monotone.map_min α (order_dual α) _ _ has_inv.inv a b $ λ a b, inv_le_inv_iff.mpr
@[to_additive min_sub_sub_right]
lemma min_div_div_right' (a b c : α) : min (a / c) (b / c) = min a b / c :=
by simpa only [div_eq_mul_inv] using min_mul_mul_right a b (c⁻¹)
@[to_additive max_sub_sub_right]
lemma max_div_div_right' (a b c : α) : max (a / c) (b / c) = max a b / c :=
by simpa only [div_eq_mul_inv] using max_mul_mul_right a b (c⁻¹)
@[to_additive min_sub_sub_left]
lemma min_div_div_left' (a b c : α) : min (a / b) (a / c) = a / max b c :=
by simp only [div_eq_mul_inv, min_mul_mul_left, min_inv_inv']
@[to_additive max_sub_sub_left]
lemma max_div_div_left' (a b c : α) : max (a / b) (a / c) = a / min b c :=
by simp only [div_eq_mul_inv, max_mul_mul_left, max_inv_inv']
@[to_additive eq_zero_of_neg_eq]
lemma eq_one_of_inv_eq' (h : a⁻¹ = a) : a = 1 :=
match lt_trichotomy a 1 with
| or.inl h₁ :=
have 1 < a, from h ▸ one_lt_inv_of_inv h₁,
absurd h₁ this.asymm
| or.inr (or.inl h₁) := h₁
| or.inr (or.inr h₁) :=
have a < 1, from h ▸ inv_lt_one'.mpr h₁,
absurd h₁ this.asymm
end
@[to_additive exists_zero_lt]
lemma exists_one_lt' [nontrivial α] : ∃ (a:α), 1 < a :=
begin
obtain ⟨y, hy⟩ := decidable.exists_ne (1 : α),
cases hy.lt_or_lt,
{ exact ⟨y⁻¹, one_lt_inv'.mpr h⟩ },
{ exact ⟨y, h⟩ }
end
@[priority 100, to_additive] -- see Note [lower instance priority]
instance linear_ordered_comm_group.to_no_top_order [nontrivial α] :
no_top_order α :=
⟨ begin
obtain ⟨y, hy⟩ : ∃ (a:α), 1 < a := exists_one_lt',
exact λ a, ⟨a * y, lt_mul_of_one_lt_right' a hy⟩
end ⟩
@[priority 100, to_additive] -- see Note [lower instance priority]
instance linear_ordered_comm_group.to_no_bot_order [nontrivial α] : no_bot_order α :=
⟨ begin
obtain ⟨y, hy⟩ : ∃ (a:α), 1 < a := exists_one_lt',
exact λ a, ⟨a / y, (div_lt_self_iff a).mpr hy⟩
end ⟩
end linear_ordered_comm_group
section covariant_add_le
section has_neg
variables [has_neg α] [linear_order α] {a b: α}
/-- `mabs a` is the multiplicative absolute value of `a`. -/
@[to_additive abs
"`abs a` is the additive absolute value of `a`."
]
def mabs {α : Type*} [has_inv α] [lattice α] (a : α) : α := a ⊔ (a⁻¹)
lemma abs_eq_max_neg {α : Type*} [has_neg α] [linear_order α] (a : α) : abs a = max a (-a) :=
begin
exact rfl,
end
lemma abs_choice (x : α) : abs x = x ∨ abs x = -x := max_choice _ _
lemma abs_le' : abs a ≤ b ↔ a ≤ b ∧ -a ≤ b := max_le_iff
lemma le_abs : a ≤ abs b ↔ a ≤ b ∨ a ≤ -b := le_max_iff
lemma le_abs_self (a : α) : a ≤ abs a := le_max_left _ _
lemma neg_le_abs_self (a : α) : -a ≤ abs a := le_max_right _ _
lemma lt_abs : a < abs b ↔ a < b ∨ a < -b := lt_max_iff
theorem abs_le_abs (h₀ : a ≤ b) (h₁ : -a ≤ b) : abs a ≤ abs b :=
(abs_le'.2 ⟨h₀, h₁⟩).trans (le_abs_self b)
lemma abs_by_cases (P : α → Prop) {a : α} (h1 : P a) (h2 : P (-a)) : P (abs a) :=
sup_ind _ _ h1 h2
end has_neg
section add_group
variables [add_group α] [linear_order α]
@[simp] lemma abs_neg (a : α) : abs (-a) = abs a :=
begin
rw [abs_eq_max_neg, max_comm, neg_neg, abs_eq_max_neg]
end
lemma eq_or_eq_neg_of_abs_eq {a b : α} (h : abs a = b) : a = b ∨ a = -b :=
by simpa only [← h, eq_comm, eq_neg_iff_eq_neg] using abs_choice a
lemma abs_eq_abs {a b : α} : abs a = abs b ↔ a = b ∨ a = -b :=
begin
refine ⟨λ h, _, λ h, _⟩,
{ obtain rfl | rfl := eq_or_eq_neg_of_abs_eq h;
simpa only [neg_eq_iff_neg_eq, neg_inj, or.comm, @eq_comm _ (-b)] using abs_choice b },
{ cases h; simp only [h, abs_neg] },
end
lemma abs_sub_comm (a b : α) : abs (a - b) = abs (b - a) :=
calc abs (a - b) = abs (- (b - a)) : congr_arg _ (neg_sub b a).symm
... = abs (b - a) : abs_neg (b - a)
variables [covariant_class α α (+) (≤)] {a b c : α}
lemma abs_of_nonneg (h : 0 ≤ a) : abs a = a :=
max_eq_left $ (neg_nonpos.2 h).trans h
lemma abs_of_pos (h : 0 < a) : abs a = a :=
abs_of_nonneg h.le
lemma abs_of_nonpos (h : a ≤ 0) : abs a = -a :=
max_eq_right $ h.trans (neg_nonneg.2 h)
lemma abs_of_neg (h : a < 0) : abs a = -a :=
abs_of_nonpos h.le
@[simp] lemma abs_zero : abs 0 = (0:α) :=
abs_of_nonneg le_rfl
@[simp] lemma abs_pos : 0 < abs a ↔ a ≠ 0 :=
begin
rcases lt_trichotomy a 0 with (ha|rfl|ha),
{ simp [abs_of_neg ha, neg_pos, ha.ne, ha] },
{ simp },
{ simp [abs_of_pos ha, ha, ha.ne.symm] }
end
lemma abs_pos_of_pos (h : 0 < a) : 0 < abs a := abs_pos.2 h.ne.symm
lemma abs_pos_of_neg (h : a < 0) : 0 < abs a := abs_pos.2 h.ne
lemma neg_abs_le_self (a : α) : -abs a ≤ a :=
begin
cases le_total 0 a with h h,
{ calc -abs a = - a : congr_arg (has_neg.neg) (abs_of_nonneg h)
... ≤ 0 : neg_nonpos.mpr h
... ≤ a : h },
{ calc -abs a = - - a : congr_arg (has_neg.neg) (abs_of_nonpos h)
... ≤ a : (neg_neg a).le }
end
lemma abs_nonneg (a : α) : 0 ≤ abs a :=
(le_total 0 a).elim (λ h, h.trans (le_abs_self a)) (λ h, (neg_nonneg.2 h).trans $ neg_le_abs_self a)
@[simp] lemma abs_abs (a : α) : abs (abs a) = abs a :=
abs_of_nonneg $ abs_nonneg a
@[simp] lemma abs_eq_zero : abs a = 0 ↔ a = 0 :=
decidable.not_iff_not.1 $ ne_comm.trans $ (abs_nonneg a).lt_iff_ne.symm.trans abs_pos
@[simp] lemma abs_nonpos_iff {a : α} : abs a ≤ 0 ↔ a = 0 :=
(abs_nonneg a).le_iff_eq.trans abs_eq_zero
variable [covariant_class α α (swap (+)) (≤)]
lemma abs_lt : abs a < b ↔ - b < a ∧ a < b :=
max_lt_iff.trans $ and.comm.trans $ by rw [neg_lt]
lemma neg_lt_of_abs_lt (h : abs a < b) : -b < a := (abs_lt.mp h).1
lemma lt_of_abs_lt (h : abs a < b) : a < b := (abs_lt.mp h).2
lemma max_sub_min_eq_abs' (a b : α) : max a b - min a b = abs (a - b) :=
begin
cases le_total a b with ab ba,
{ rw [max_eq_right ab, min_eq_left ab, abs_of_nonpos, neg_sub], rwa sub_nonpos },
{ rw [max_eq_left ba, min_eq_right ba, abs_of_nonneg], rwa sub_nonneg }
end
lemma max_sub_min_eq_abs (a b : α) : max a b - min a b = abs (b - a) :=
by { rw abs_sub_comm, exact max_sub_min_eq_abs' _ _ }
end add_group
section add_comm_group
variables [add_comm_group α] [linear_order α] [covariant_class α α (+) (≤)] {a b c d : α}
lemma abs_le : abs a ≤ b ↔ - b ≤ a ∧ a ≤ b :=
by rw [abs_le', and.comm, neg_le]
lemma neg_le_of_abs_le (h : abs a ≤ b) : -b ≤ a := (abs_le.mp h).1
lemma le_of_abs_le (h : abs a ≤ b) : a ≤ b := (abs_le.mp h).2
/--
The **triangle inequality** in `linear_ordered_add_comm_group`s.
-/
lemma abs_add (a b : α) : abs (a + b) ≤ abs a + abs b :=
abs_le.2 ⟨(neg_add (abs a) (abs b)).symm ▸
add_le_add (neg_le.2 $ neg_le_abs_self _) (neg_le.2 $ neg_le_abs_self _),
add_le_add (le_abs_self _) (le_abs_self _)⟩
theorem abs_sub (a b : α) :
abs (a - b) ≤ abs a + abs b :=
by { rw [sub_eq_add_neg, ←abs_neg b], exact abs_add a _ }
lemma abs_sub_le_iff : abs (a - b) ≤ c ↔ a - b ≤ c ∧ b - a ≤ c :=
by rw [abs_le, neg_le_sub_iff_le_add, sub_le_iff_le_add', and_comm, sub_le_iff_le_add']
lemma abs_sub_lt_iff : abs (a - b) < c ↔ a - b < c ∧ b - a < c :=
by rw [abs_lt, neg_lt_sub_iff_lt_add', sub_lt_iff_lt_add', and_comm, sub_lt_iff_lt_add']
lemma sub_le_of_abs_sub_le_left (h : abs (a - b) ≤ c) : b - c ≤ a :=
sub_le.1 $ (abs_sub_le_iff.1 h).2
lemma sub_le_of_abs_sub_le_right (h : abs (a - b) ≤ c) : a - c ≤ b :=
sub_le_of_abs_sub_le_left (abs_sub_comm a b ▸ h)
lemma sub_lt_of_abs_sub_lt_left (h : abs (a - b) < c) : b - c < a :=
sub_lt.1 $ (abs_sub_lt_iff.1 h).2
lemma sub_lt_of_abs_sub_lt_right (h : abs (a - b) < c) : a - c < b :=
sub_lt_of_abs_sub_lt_left (abs_sub_comm a b ▸ h)
lemma abs_sub_abs_le_abs_sub (a b : α) : abs a - abs b ≤ abs (a - b) :=
sub_le_iff_le_add.2 $
calc abs a = abs (a - b + b) : by rw [sub_add_cancel]
... ≤ abs (a - b) + abs b : abs_add _ _
lemma abs_abs_sub_abs_le_abs_sub (a b : α) : abs (abs a - abs b) ≤ abs (a - b) :=
abs_sub_le_iff.2 ⟨abs_sub_abs_le_abs_sub _ _, by rw abs_sub_comm; apply abs_sub_abs_le_abs_sub⟩
lemma abs_eq (hb : 0 ≤ b) : abs a = b ↔ a = b ∨ a = -b :=
begin
refine ⟨eq_or_eq_neg_of_abs_eq, _⟩,
rintro (rfl|rfl); simp only [abs_neg, abs_of_nonneg hb]
end
lemma abs_le_max_abs_abs (hab : a ≤ b) (hbc : b ≤ c) : abs b ≤ max (abs a) (abs c) :=
abs_le'.2
⟨by simp [hbc.trans (le_abs_self c)],
by simp [(neg_le_neg_iff.mpr hab).trans (neg_le_abs_self a)]⟩
lemma eq_of_abs_sub_eq_zero {a b : α} (h : abs (a - b) = 0) : a = b :=
sub_eq_zero.1 $ abs_eq_zero.1 h
lemma abs_sub_le (a b c : α) : abs (a - c) ≤ abs (a - b) + abs (b - c) :=
calc
abs (a - c) = abs (a - b + (b - c)) : by rw [sub_add_sub_cancel]
... ≤ abs (a - b) + abs (b - c) : abs_add _ _
lemma abs_add_three (a b c : α) : abs (a + b + c) ≤ abs a + abs b + abs c :=
(abs_add _ _).trans (add_le_add_right (abs_add _ _) _)
lemma dist_bdd_within_interval {a b lb ub : α} (hal : lb ≤ a) (hau : a ≤ ub)
(hbl : lb ≤ b) (hbu : b ≤ ub) : abs (a - b) ≤ ub - lb :=
abs_sub_le_iff.2 ⟨sub_le_sub hau hbl, sub_le_sub hbu hal⟩
lemma eq_of_abs_sub_nonpos (h : abs (a - b) ≤ 0) : a = b :=
eq_of_abs_sub_eq_zero (le_antisymm h (abs_nonneg (a - b)))
lemma abs_max_sub_max_le_abs (a b c : α) : abs (max a c - max b c) ≤ abs (a - b) :=
begin
simp_rw [abs_le, le_sub_iff_add_le, sub_le_iff_le_add, ← max_add_add_left],
split; apply max_le_max; simp only [← le_sub_iff_add_le, ← sub_le_iff_le_add, sub_self, neg_le,
neg_le_abs_self, neg_zero, abs_nonneg, le_abs_self]
end
end add_comm_group
end covariant_add_le
section linear_ordered_add_comm_group
variable [linear_ordered_add_comm_group α]
instance with_top.linear_ordered_add_comm_group_with_top :
linear_ordered_add_comm_group_with_top (with_top α) :=
{ neg := option.map (λ a : α, -a),
neg_top := @option.map_none _ _ (λ a : α, -a),
add_neg_cancel := begin
rintro (a | a) ha,
{ exact (ha rfl).elim },
{ exact with_top.coe_add.symm.trans (with_top.coe_eq_coe.2 (add_neg_self a)) }
end,
.. with_top.linear_ordered_add_comm_monoid_with_top,
.. option.nontrivial }
end linear_ordered_add_comm_group
/-- This is not so much a new structure as a construction mechanism
for ordered groups, by specifying only the "positive cone" of the group. -/
class nonneg_add_comm_group (α : Type*) extends add_comm_group α :=
(nonneg : α → Prop)
(pos : α → Prop := λ a, nonneg a ∧ ¬ nonneg (neg a))
(pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬ nonneg (-a) . order_laws_tac)
(zero_nonneg : nonneg 0)
(add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b))
(nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0)
namespace nonneg_add_comm_group
variable [s : nonneg_add_comm_group α]
include s
@[reducible, priority 100] -- see Note [lower instance priority]
instance to_ordered_add_comm_group : ordered_add_comm_group α :=
{ le := λ a b, nonneg (b - a),
lt := λ a b, pos (b - a),
lt_iff_le_not_le := λ a b, by simp; rw [pos_iff]; simp,
le_refl := λ a, by simp [zero_nonneg],
le_trans := λ a b c nab nbc, by simp [-sub_eq_add_neg];
rw ← sub_add_sub_cancel; exact add_nonneg nbc nab,
le_antisymm := λ a b nab nba, eq_of_sub_eq_zero $
nonneg_antisymm nba (by rw neg_sub; exact nab),
add_le_add_left := λ a b nab c, by simpa [(≤), preorder.le] using nab,
..s }
theorem nonneg_def {a : α} : nonneg a ↔ 0 ≤ a :=
show _ ↔ nonneg _, by simp
theorem pos_def {a : α} : pos a ↔ 0 < a :=
show _ ↔ pos _, by simp
theorem not_zero_pos : ¬ pos (0 : α) :=
mt pos_def.1 (lt_irrefl _)
theorem zero_lt_iff_nonneg_nonneg {a : α} :
0 < a ↔ nonneg a ∧ ¬ nonneg (-a) :=
pos_def.symm.trans (pos_iff _)
theorem nonneg_total_iff :
(∀ a : α, nonneg a ∨ nonneg (-a)) ↔
(∀ a b : α, a ≤ b ∨ b ≤ a) :=
⟨λ h a b, by have := h (b - a); rwa [neg_sub] at this,
λ h a, by rw [nonneg_def, nonneg_def, neg_nonneg]; apply h⟩
/--
A `nonneg_add_comm_group` is a `linear_ordered_add_comm_group`
if `nonneg` is total and decidable.
-/
def to_linear_ordered_add_comm_group
[decidable_pred (@nonneg α _)]
(nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a))
: linear_ordered_add_comm_group α :=
{ le := (≤),
lt := (<),
le_total := nonneg_total_iff.1 nonneg_total,
decidable_le := by apply_instance,
decidable_lt := by apply_instance,
..@nonneg_add_comm_group.to_ordered_add_comm_group _ s }
end nonneg_add_comm_group
namespace prod
variables {G H : Type*}
@[to_additive]
instance [ordered_comm_group G] [ordered_comm_group H] :
ordered_comm_group (G × H) :=
{ .. prod.comm_group, .. prod.partial_order G H, .. prod.ordered_cancel_comm_monoid }
end prod
section type_tags
instance [ordered_add_comm_group α] : ordered_comm_group (multiplicative α) :=
{ ..multiplicative.comm_group,
..multiplicative.ordered_comm_monoid }
instance [ordered_comm_group α] : ordered_add_comm_group (additive α) :=
{ ..additive.add_comm_group,
..additive.ordered_add_comm_monoid }
instance [linear_ordered_add_comm_group α] : linear_ordered_comm_group (multiplicative α) :=
{ ..multiplicative.linear_order,
..multiplicative.ordered_comm_group }
instance [linear_ordered_comm_group α] : linear_ordered_add_comm_group (additive α) :=
{ ..additive.linear_order,
..additive.ordered_add_comm_group }
end type_tags
section norm_num_lemmas
/- The following lemmas are stated so that the `norm_num` tactic can use them with the
expected signatures. -/
variables [ordered_comm_group α] {a b : α}
@[to_additive neg_le_neg]
lemma inv_le_inv' : a ≤ b → b⁻¹ ≤ a⁻¹ :=
inv_le_inv_iff.mpr
@[to_additive neg_lt_neg]
lemma inv_lt_inv' : a < b → b⁻¹ < a⁻¹ :=
inv_lt_inv_iff.mpr
/- The additive version is also a `linarith` lemma. -/
@[to_additive]
theorem inv_lt_one_of_one_lt : 1 < a → a⁻¹ < 1 :=
inv_lt_one_iff_one_lt.mpr
/- The additive version is also a `linarith` lemma. -/
@[to_additive]
lemma inv_le_one_of_one_le : 1 ≤ a → a⁻¹ ≤ 1 :=
inv_le_one'.mpr
@[to_additive neg_nonneg_of_nonpos]
lemma one_le_inv_of_le_one : a ≤ 1 → 1 ≤ a⁻¹ :=
one_le_inv'.mpr
end norm_num_lemmas
|
e16d7a72412a4f5e3f15776cce9bd7b609cd3dde | d1a52c3f208fa42c41df8278c3d280f075eb020c | /tests/leanpkg/cyclic/Cyclic.lean | 5af4f224d95231cffe6adfe49d145aef68b8d0ea | [
"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 | cipher1024/lean4 | 6e1f98bb58e7a92b28f5364eb38a14c8d0aae393 | 69114d3b50806264ef35b57394391c3e738a9822 | refs/heads/master | 1,642,227,983,603 | 1,642,011,696,000 | 1,642,011,696,000 | 228,607,691 | 0 | 0 | Apache-2.0 | 1,576,584,269,000 | 1,576,584,268,000 | null | UTF-8 | Lean | false | false | 16 | lean | import Cyclic.A
|
4ea430c3bd0b47d5a84784a10edd9583e86c760e | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/linear_algebra/affine_space/affine_subspace.lean | 0aed92209fba84a643a6ba2cabc7cc8b0db19a02 | [
"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 | 54,912 | lean | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import data.set.intervals.unordered_interval
import linear_algebra.affine_space.affine_equiv
/-!
# Affine spaces
This file defines affine subspaces (over modules) and the affine span of a set of points.
## Main definitions
* `affine_subspace k P` is the type of affine subspaces. Unlike
affine spaces, affine subspaces are allowed to be empty, and lemmas
that do not apply to empty affine subspaces have `nonempty`
hypotheses. There is a `complete_lattice` structure on affine
subspaces.
* `affine_subspace.direction` gives the `submodule` spanned by the
pairwise differences of points in an `affine_subspace`. There are
various lemmas relating to the set of vectors in the `direction`,
and relating the lattice structure on affine subspaces to that on
their directions.
* `affine_span` gives the affine subspace spanned by a set of points,
with `vector_span` giving its direction. `affine_span` is defined
in terms of `span_points`, which gives an explicit description of
the points contained in the affine span; `span_points` itself should
generally only be used when that description is required, with
`affine_span` being the main definition for other purposes. Two
other descriptions of the affine span are proved equivalent: it is
the `Inf` of affine subspaces containing the points, and (if
`[nontrivial k]`) it contains exactly those points that are affine
combinations of points in the given set.
## Implementation notes
`out_param` is used in the definiton of `add_torsor V P` to make `V` an implicit argument (deduced
from `P`) in most cases; `include V` is needed in many cases for `V`, and type classes using it, to
be added as implicit arguments to individual lemmas. As for modules, `k` is an explicit argument
rather than implied by `P` or `V`.
This file only provides purely algebraic definitions and results.
Those depending on analysis or topology are defined elsewhere; see
`analysis.normed_space.add_torsor` and `topology.algebra.affine`.
## References
* https://en.wikipedia.org/wiki/Affine_space
* https://en.wikipedia.org/wiki/Principal_homogeneous_space
-/
noncomputable theory
open_locale big_operators classical affine
open set
section
variables (k : Type*) {V : Type*} {P : Type*} [ring k] [add_comm_group V] [module k V]
variables [affine_space V P]
include V
/-- The submodule spanning the differences of a (possibly empty) set
of points. -/
def vector_span (s : set P) : submodule k V := submodule.span k (s -ᵥ s)
/-- The definition of `vector_span`, for rewriting. -/
lemma vector_span_def (s : set P) : vector_span k s = submodule.span k (s -ᵥ s) :=
rfl
/-- `vector_span` is monotone. -/
lemma vector_span_mono {s₁ s₂ : set P} (h : s₁ ⊆ s₂) : vector_span k s₁ ≤ vector_span k s₂ :=
submodule.span_mono (vsub_self_mono h)
variables (P)
/-- The `vector_span` of the empty set is `⊥`. -/
@[simp] lemma vector_span_empty : vector_span k (∅ : set P) = (⊥ : submodule k V) :=
by rw [vector_span_def, vsub_empty, submodule.span_empty]
variables {P}
/-- The `vector_span` of a single point is `⊥`. -/
@[simp] lemma vector_span_singleton (p : P) : vector_span k ({p} : set P) = ⊥ :=
by simp [vector_span_def]
/-- The `s -ᵥ s` lies within the `vector_span k s`. -/
lemma vsub_set_subset_vector_span (s : set P) : s -ᵥ s ⊆ ↑(vector_span k s) :=
submodule.subset_span
/-- Each pairwise difference is in the `vector_span`. -/
lemma vsub_mem_vector_span {s : set P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
p1 -ᵥ p2 ∈ vector_span k s :=
vsub_set_subset_vector_span k s (vsub_mem_vsub hp1 hp2)
/-- The points in the affine span of a (possibly empty) set of
points. Use `affine_span` instead to get an `affine_subspace k P`. -/
def span_points (s : set P) : set P :=
{p | ∃ p1 ∈ s, ∃ v ∈ (vector_span k s), p = v +ᵥ p1}
/-- A point in a set is in its affine span. -/
lemma mem_span_points (p : P) (s : set P) : p ∈ s → p ∈ span_points k s
| hp := ⟨p, hp, 0, submodule.zero_mem _, (zero_vadd V p).symm⟩
/-- A set is contained in its `span_points`. -/
lemma subset_span_points (s : set P) : s ⊆ span_points k s :=
λ p, mem_span_points k p s
/-- The `span_points` of a set is nonempty if and only if that set
is. -/
@[simp] lemma span_points_nonempty (s : set P) :
(span_points k s).nonempty ↔ s.nonempty :=
begin
split,
{ contrapose,
rw [set.not_nonempty_iff_eq_empty, set.not_nonempty_iff_eq_empty],
intro h,
simp [h, span_points] },
{ exact λ h, h.mono (subset_span_points _ _) }
end
/-- Adding a point in the affine span and a vector in the spanning
submodule produces a point in the affine span. -/
lemma vadd_mem_span_points_of_mem_span_points_of_mem_vector_span {s : set P} {p : P} {v : V}
(hp : p ∈ span_points k s) (hv : v ∈ vector_span k s) : v +ᵥ p ∈ span_points k s :=
begin
rcases hp with ⟨p2, ⟨hp2, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩,
rw [hv2p, vadd_vadd],
use [p2, hp2, v + v2, (vector_span k s).add_mem hv hv2, rfl]
end
/-- Subtracting two points in the affine span produces a vector in the
spanning submodule. -/
lemma vsub_mem_vector_span_of_mem_span_points_of_mem_span_points {s : set P} {p1 p2 : P}
(hp1 : p1 ∈ span_points k s) (hp2 : p2 ∈ span_points k s) :
p1 -ᵥ p2 ∈ vector_span k s :=
begin
rcases hp1 with ⟨p1a, ⟨hp1a, ⟨v1, ⟨hv1, hv1p⟩⟩⟩⟩,
rcases hp2 with ⟨p2a, ⟨hp2a, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩,
rw [hv1p, hv2p, vsub_vadd_eq_vsub_sub (v1 +ᵥ p1a), vadd_vsub_assoc, add_comm, add_sub_assoc],
have hv1v2 : v1 - v2 ∈ vector_span k s,
{ rw sub_eq_add_neg,
apply (vector_span k s).add_mem hv1,
rw ←neg_one_smul k v2,
exact (vector_span k s).smul_mem (-1 : k) hv2 },
refine (vector_span k s).add_mem _ hv1v2,
exact vsub_mem_vector_span k hp1a hp2a
end
end
/-- An `affine_subspace k P` is a subset of an `affine_space V P`
that, if not empty, has an affine space structure induced by a
corresponding subspace of the `module k V`. -/
structure affine_subspace (k : Type*) {V : Type*} (P : Type*) [ring k] [add_comm_group V]
[module k V] [affine_space V P] :=
(carrier : set P)
(smul_vsub_vadd_mem : ∀ (c : k) {p1 p2 p3 : P}, p1 ∈ carrier → p2 ∈ carrier → p3 ∈ carrier →
c • (p1 -ᵥ p2 : V) +ᵥ p3 ∈ carrier)
namespace submodule
variables {k V : Type*} [ring k] [add_comm_group V] [module k V]
/-- Reinterpret `p : submodule k V` as an `affine_subspace k V`. -/
def to_affine_subspace (p : submodule k V) : affine_subspace k V :=
{ carrier := p,
smul_vsub_vadd_mem := λ c p₁ p₂ p₃ h₁ h₂ h₃, p.add_mem (p.smul_mem _ (p.sub_mem h₁ h₂)) h₃ }
end submodule
namespace affine_subspace
variables (k : Type*) {V : Type*} (P : Type*) [ring k] [add_comm_group V] [module k V]
[affine_space V P]
include V
-- TODO Refactor to use `instance : set_like (affine_subspace k P) P :=` instead
instance : has_coe (affine_subspace k P) (set P) := ⟨carrier⟩
instance : has_mem P (affine_subspace k P) := ⟨λ p s, p ∈ (s : set P)⟩
/-- A point is in an affine subspace coerced to a set if and only if
it is in that affine subspace. -/
@[simp] lemma mem_coe (p : P) (s : affine_subspace k P) :
p ∈ (s : set P) ↔ p ∈ s :=
iff.rfl
variables {k P}
/-- The direction of an affine subspace is the submodule spanned by
the pairwise differences of points. (Except in the case of an empty
affine subspace, where the direction is the zero submodule, every
vector in the direction is the difference of two points in the affine
subspace.) -/
def direction (s : affine_subspace k P) : submodule k V := vector_span k (s : set P)
/-- The direction equals the `vector_span`. -/
lemma direction_eq_vector_span (s : affine_subspace k P) :
s.direction = vector_span k (s : set P) :=
rfl
/-- Alternative definition of the direction when the affine subspace
is nonempty. This is defined so that the order on submodules (as used
in the definition of `submodule.span`) can be used in the proof of
`coe_direction_eq_vsub_set`, and is not intended to be used beyond
that proof. -/
def direction_of_nonempty {s : affine_subspace k P} (h : (s : set P).nonempty) :
submodule k V :=
{ carrier := (s : set P) -ᵥ s,
zero_mem' := begin
cases h with p hp,
exact (vsub_self p) ▸ vsub_mem_vsub hp hp
end,
add_mem' := begin
intros a b ha hb,
rcases ha with ⟨p1, p2, hp1, hp2, rfl⟩,
rcases hb with ⟨p3, p4, hp3, hp4, rfl⟩,
rw [←vadd_vsub_assoc],
refine vsub_mem_vsub _ hp4,
convert s.smul_vsub_vadd_mem 1 hp1 hp2 hp3,
rw one_smul
end,
smul_mem' := begin
intros c v hv,
rcases hv with ⟨p1, p2, hp1, hp2, rfl⟩,
rw [←vadd_vsub (c • (p1 -ᵥ p2)) p2],
refine vsub_mem_vsub _ hp2,
exact s.smul_vsub_vadd_mem c hp1 hp2 hp2
end }
/-- `direction_of_nonempty` gives the same submodule as
`direction`. -/
lemma direction_of_nonempty_eq_direction {s : affine_subspace k P} (h : (s : set P).nonempty) :
direction_of_nonempty h = s.direction :=
le_antisymm (vsub_set_subset_vector_span k s) (submodule.span_le.2 set.subset.rfl)
/-- The set of vectors in the direction of a nonempty affine subspace
is given by `vsub_set`. -/
lemma coe_direction_eq_vsub_set {s : affine_subspace k P} (h : (s : set P).nonempty) :
(s.direction : set V) = (s : set P) -ᵥ s :=
direction_of_nonempty_eq_direction h ▸ rfl
/-- A vector is in the direction of a nonempty affine subspace if and
only if it is the subtraction of two vectors in the subspace. -/
lemma mem_direction_iff_eq_vsub {s : affine_subspace k P} (h : (s : set P).nonempty) (v : V) :
v ∈ s.direction ↔ ∃ p1 ∈ s, ∃ p2 ∈ s, v = p1 -ᵥ p2 :=
begin
rw [←set_like.mem_coe, coe_direction_eq_vsub_set h],
exact ⟨λ ⟨p1, p2, hp1, hp2, hv⟩, ⟨p1, hp1, p2, hp2, hv.symm⟩,
λ ⟨p1, hp1, p2, hp2, hv⟩, ⟨p1, p2, hp1, hp2, hv.symm⟩⟩
end
/-- Adding a vector in the direction to a point in the subspace
produces a point in the subspace. -/
lemma vadd_mem_of_mem_direction {s : affine_subspace k P} {v : V} (hv : v ∈ s.direction) {p : P}
(hp : p ∈ s) : v +ᵥ p ∈ s :=
begin
rw mem_direction_iff_eq_vsub ⟨p, hp⟩ at hv,
rcases hv with ⟨p1, hp1, p2, hp2, hv⟩,
rw hv,
convert s.smul_vsub_vadd_mem 1 hp1 hp2 hp,
rw one_smul
end
/-- Subtracting two points in the subspace produces a vector in the
direction. -/
lemma vsub_mem_direction {s : affine_subspace k P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
(p1 -ᵥ p2) ∈ s.direction :=
vsub_mem_vector_span k hp1 hp2
/-- Adding a vector to a point in a subspace produces a point in the
subspace if and only if the vector is in the direction. -/
lemma vadd_mem_iff_mem_direction {s : affine_subspace k P} (v : V) {p : P} (hp : p ∈ s) :
v +ᵥ p ∈ s ↔ v ∈ s.direction :=
⟨λ h, by simpa using vsub_mem_direction h hp, λ h, vadd_mem_of_mem_direction h hp⟩
/-- Given a point in an affine subspace, the set of vectors in its
direction equals the set of vectors subtracting that point on the
right. -/
lemma coe_direction_eq_vsub_set_right {s : affine_subspace k P} {p : P} (hp : p ∈ s) :
(s.direction : set V) = (-ᵥ p) '' s :=
begin
rw coe_direction_eq_vsub_set ⟨p, hp⟩,
refine le_antisymm _ _,
{ rintros v ⟨p1, p2, hp1, hp2, rfl⟩,
exact ⟨p1 -ᵥ p2 +ᵥ p,
vadd_mem_of_mem_direction (vsub_mem_direction hp1 hp2) hp,
(vadd_vsub _ _)⟩ },
{ rintros v ⟨p2, hp2, rfl⟩,
exact ⟨p2, p, hp2, hp, rfl⟩ }
end
/-- Given a point in an affine subspace, the set of vectors in its
direction equals the set of vectors subtracting that point on the
left. -/
lemma coe_direction_eq_vsub_set_left {s : affine_subspace k P} {p : P} (hp : p ∈ s) :
(s.direction : set V) = (-ᵥ) p '' s :=
begin
ext v,
rw [set_like.mem_coe, ←submodule.neg_mem_iff, ←set_like.mem_coe,
coe_direction_eq_vsub_set_right hp, set.mem_image_iff_bex, set.mem_image_iff_bex],
conv_lhs { congr, funext, rw [←neg_vsub_eq_vsub_rev, neg_inj] }
end
/-- Given a point in an affine subspace, a vector is in its direction
if and only if it results from subtracting that point on the right. -/
lemma mem_direction_iff_eq_vsub_right {s : affine_subspace k P} {p : P} (hp : p ∈ s) (v : V) :
v ∈ s.direction ↔ ∃ p2 ∈ s, v = p2 -ᵥ p :=
begin
rw [←set_like.mem_coe, coe_direction_eq_vsub_set_right hp],
exact ⟨λ ⟨p2, hp2, hv⟩, ⟨p2, hp2, hv.symm⟩, λ ⟨p2, hp2, hv⟩, ⟨p2, hp2, hv.symm⟩⟩
end
/-- Given a point in an affine subspace, a vector is in its direction
if and only if it results from subtracting that point on the left. -/
lemma mem_direction_iff_eq_vsub_left {s : affine_subspace k P} {p : P} (hp : p ∈ s) (v : V) :
v ∈ s.direction ↔ ∃ p2 ∈ s, v = p -ᵥ p2 :=
begin
rw [←set_like.mem_coe, coe_direction_eq_vsub_set_left hp],
exact ⟨λ ⟨p2, hp2, hv⟩, ⟨p2, hp2, hv.symm⟩, λ ⟨p2, hp2, hv⟩, ⟨p2, hp2, hv.symm⟩⟩
end
/-- Given a point in an affine subspace, a result of subtracting that
point on the right is in the direction if and only if the other point
is in the subspace. -/
lemma vsub_right_mem_direction_iff_mem {s : affine_subspace k P} {p : P} (hp : p ∈ s) (p2 : P) :
p2 -ᵥ p ∈ s.direction ↔ p2 ∈ s :=
begin
rw mem_direction_iff_eq_vsub_right hp,
simp
end
/-- Given a point in an affine subspace, a result of subtracting that
point on the left is in the direction if and only if the other point
is in the subspace. -/
lemma vsub_left_mem_direction_iff_mem {s : affine_subspace k P} {p : P} (hp : p ∈ s) (p2 : P) :
p -ᵥ p2 ∈ s.direction ↔ p2 ∈ s :=
begin
rw mem_direction_iff_eq_vsub_left hp,
simp
end
/-- Two affine subspaces are equal if they have the same points. -/
@[ext] lemma coe_injective : function.injective (coe : affine_subspace k P → set P) :=
λ s1 s2 h, begin
cases s1,
cases s2,
congr,
exact h
end
@[simp] lemma ext_iff (s₁ s₂ : affine_subspace k P) :
(s₁ : set P) = s₂ ↔ s₁ = s₂ :=
⟨λ h, coe_injective h, by tidy⟩
/-- Two affine subspaces with the same direction and nonempty
intersection are equal. -/
lemma ext_of_direction_eq {s1 s2 : affine_subspace k P} (hd : s1.direction = s2.direction)
(hn : ((s1 : set P) ∩ s2).nonempty) : s1 = s2 :=
begin
ext p,
have hq1 := set.mem_of_mem_inter_left hn.some_mem,
have hq2 := set.mem_of_mem_inter_right hn.some_mem,
split,
{ intro hp,
rw ←vsub_vadd p hn.some,
refine vadd_mem_of_mem_direction _ hq2,
rw ←hd,
exact vsub_mem_direction hp hq1 },
{ intro hp,
rw ←vsub_vadd p hn.some,
refine vadd_mem_of_mem_direction _ hq1,
rw hd,
exact vsub_mem_direction hp hq2 }
end
instance to_add_torsor (s : affine_subspace k P) [nonempty s] : add_torsor s.direction s :=
{ vadd := λ a b, ⟨(a:V) +ᵥ (b:P), vadd_mem_of_mem_direction a.2 b.2⟩,
zero_vadd := by simp,
add_vadd := λ a b c, by { ext, apply add_vadd },
vsub := λ a b, ⟨(a:P) -ᵥ (b:P), (vsub_left_mem_direction_iff_mem a.2 _).mpr b.2 ⟩,
nonempty := by apply_instance,
vsub_vadd' := λ a b, by { ext, apply add_torsor.vsub_vadd' },
vadd_vsub' := λ a b, by { ext, apply add_torsor.vadd_vsub' } }
@[simp, norm_cast] lemma coe_vsub (s : affine_subspace k P) [nonempty s] (a b : s) :
↑(a -ᵥ b) = (a:P) -ᵥ (b:P) :=
rfl
@[simp, norm_cast] lemma coe_vadd (s : affine_subspace k P) [nonempty s] (a : s.direction) (b : s) :
↑(a +ᵥ b) = (a:V) +ᵥ (b:P) :=
rfl
/-- Two affine subspaces with nonempty intersection are equal if and
only if their directions are equal. -/
lemma eq_iff_direction_eq_of_mem {s₁ s₂ : affine_subspace k P} {p : P} (h₁ : p ∈ s₁)
(h₂ : p ∈ s₂) : s₁ = s₂ ↔ s₁.direction = s₂.direction :=
⟨λ h, h ▸ rfl, λ h, ext_of_direction_eq h ⟨p, h₁, h₂⟩⟩
/-- Construct an affine subspace from a point and a direction. -/
def mk' (p : P) (direction : submodule k V) : affine_subspace k P :=
{ carrier := {q | ∃ v ∈ direction, q = v +ᵥ p},
smul_vsub_vadd_mem := λ c p1 p2 p3 hp1 hp2 hp3, begin
rcases hp1 with ⟨v1, hv1, hp1⟩,
rcases hp2 with ⟨v2, hv2, hp2⟩,
rcases hp3 with ⟨v3, hv3, hp3⟩,
use [c • (v1 - v2) + v3,
direction.add_mem (direction.smul_mem c (direction.sub_mem hv1 hv2)) hv3],
simp [hp1, hp2, hp3, vadd_vadd]
end }
/-- An affine subspace constructed from a point and a direction contains
that point. -/
lemma self_mem_mk' (p : P) (direction : submodule k V) :
p ∈ mk' p direction :=
⟨0, ⟨direction.zero_mem, (zero_vadd _ _).symm⟩⟩
/-- An affine subspace constructed from a point and a direction contains
the result of adding a vector in that direction to that point. -/
lemma vadd_mem_mk' {v : V} (p : P) {direction : submodule k V} (hv : v ∈ direction) :
v +ᵥ p ∈ mk' p direction :=
⟨v, hv, rfl⟩
/-- An affine subspace constructed from a point and a direction is
nonempty. -/
lemma mk'_nonempty (p : P) (direction : submodule k V) : (mk' p direction : set P).nonempty :=
⟨p, self_mem_mk' p direction⟩
/-- The direction of an affine subspace constructed from a point and a
direction. -/
@[simp] lemma direction_mk' (p : P) (direction : submodule k V) :
(mk' p direction).direction = direction :=
begin
ext v,
rw mem_direction_iff_eq_vsub (mk'_nonempty _ _),
split,
{ rintros ⟨p1, ⟨v1, hv1, hp1⟩, p2, ⟨v2, hv2, hp2⟩, hv⟩,
rw [hv, hp1, hp2, vadd_vsub_vadd_cancel_right],
exact direction.sub_mem hv1 hv2 },
{ exact λ hv, ⟨v +ᵥ p, vadd_mem_mk' _ hv, p,
self_mem_mk' _ _, (vadd_vsub _ _).symm⟩ }
end
/-- Constructing an affine subspace from a point in a subspace and
that subspace's direction yields the original subspace. -/
@[simp] lemma mk'_eq {s : affine_subspace k P} {p : P} (hp : p ∈ s) : mk' p s.direction = s :=
ext_of_direction_eq (direction_mk' p s.direction)
⟨p, set.mem_inter (self_mem_mk' _ _) hp⟩
/-- If an affine subspace contains a set of points, it contains the
`span_points` of that set. -/
lemma span_points_subset_coe_of_subset_coe {s : set P} {s1 : affine_subspace k P} (h : s ⊆ s1) :
span_points k s ⊆ s1 :=
begin
rintros p ⟨p1, hp1, v, hv, hp⟩,
rw hp,
have hp1s1 : p1 ∈ (s1 : set P) := set.mem_of_mem_of_subset hp1 h,
refine vadd_mem_of_mem_direction _ hp1s1,
have hs : vector_span k s ≤ s1.direction := vector_span_mono k h,
rw set_like.le_def at hs,
rw ←set_like.mem_coe,
exact set.mem_of_mem_of_subset hv hs
end
end affine_subspace
lemma affine_map.line_map_mem
{k V P : Type*} [ring k] [add_comm_group V] [module k V] [add_torsor V P]
{Q : affine_subspace k P} {p₀ p₁ : P} (c : k) (h₀ : p₀ ∈ Q) (h₁ : p₁ ∈ Q) :
affine_map.line_map p₀ p₁ c ∈ Q :=
begin
rw affine_map.line_map_apply,
exact Q.smul_vsub_vadd_mem c h₁ h₀ h₀,
end
section affine_span
variables (k : Type*) {V : Type*} {P : Type*} [ring k] [add_comm_group V] [module k V]
[affine_space V P]
include V
/-- The affine span of a set of points is the smallest affine subspace
containing those points. (Actually defined here in terms of spans in
modules.) -/
def affine_span (s : set P) : affine_subspace k P :=
{ carrier := span_points k s,
smul_vsub_vadd_mem := λ c p1 p2 p3 hp1 hp2 hp3,
vadd_mem_span_points_of_mem_span_points_of_mem_vector_span k hp3
((vector_span k s).smul_mem c
(vsub_mem_vector_span_of_mem_span_points_of_mem_span_points k hp1 hp2)) }
/-- The affine span, converted to a set, is `span_points`. -/
@[simp] lemma coe_affine_span (s : set P) :
(affine_span k s : set P) = span_points k s :=
rfl
/-- A set is contained in its affine span. -/
lemma subset_affine_span (s : set P) : s ⊆ affine_span k s :=
subset_span_points k s
/-- The direction of the affine span is the `vector_span`. -/
lemma direction_affine_span (s : set P) : (affine_span k s).direction = vector_span k s :=
begin
apply le_antisymm,
{ refine submodule.span_le.2 _,
rintros v ⟨p1, p3, ⟨p2, hp2, v1, hv1, hp1⟩, ⟨p4, hp4, v2, hv2, hp3⟩, rfl⟩,
rw [hp1, hp3, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, set_like.mem_coe],
exact (vector_span k s).sub_mem ((vector_span k s).add_mem hv1
(vsub_mem_vector_span k hp2 hp4)) hv2 },
{ exact vector_span_mono k (subset_span_points k s) }
end
/-- A point in a set is in its affine span. -/
lemma mem_affine_span {p : P} {s : set P} (hp : p ∈ s) : p ∈ affine_span k s :=
mem_span_points k p s hp
end affine_span
namespace affine_subspace
variables {k : Type*} {V : Type*} {P : Type*} [ring k] [add_comm_group V] [module k V]
[S : affine_space V P]
include S
instance : complete_lattice (affine_subspace k P) :=
{ sup := λ s1 s2, affine_span k (s1 ∪ s2),
le_sup_left := λ s1 s2, set.subset.trans (set.subset_union_left s1 s2)
(subset_span_points k _),
le_sup_right := λ s1 s2, set.subset.trans (set.subset_union_right s1 s2)
(subset_span_points k _),
sup_le := λ s1 s2 s3 hs1 hs2, span_points_subset_coe_of_subset_coe (set.union_subset hs1 hs2),
inf := λ s1 s2, mk (s1 ∩ s2)
(λ c p1 p2 p3 hp1 hp2 hp3,
⟨s1.smul_vsub_vadd_mem c hp1.1 hp2.1 hp3.1,
s2.smul_vsub_vadd_mem c hp1.2 hp2.2 hp3.2⟩),
inf_le_left := λ _ _, set.inter_subset_left _ _,
inf_le_right := λ _ _, set.inter_subset_right _ _,
le_inf := λ _ _ _, set.subset_inter,
top := { carrier := set.univ,
smul_vsub_vadd_mem := λ _ _ _ _ _ _ _, set.mem_univ _ },
le_top := λ _ _ _, set.mem_univ _,
bot := { carrier := ∅,
smul_vsub_vadd_mem := λ _ _ _ _, false.elim },
bot_le := λ _ _, false.elim,
Sup := λ s, affine_span k (⋃ s' ∈ s, (s' : set P)),
Inf := λ s, mk (⋂ s' ∈ s, (s' : set P))
(λ c p1 p2 p3 hp1 hp2 hp3, set.mem_Inter₂.2 $ λ s2 hs2, begin
rw set.mem_Inter₂ at *,
exact s2.smul_vsub_vadd_mem c (hp1 s2 hs2) (hp2 s2 hs2) (hp3 s2 hs2)
end),
le_Sup := λ _ _ h, set.subset.trans (set.subset_bUnion_of_mem h) (subset_span_points k _),
Sup_le := λ _ _ h, span_points_subset_coe_of_subset_coe (set.Union₂_subset h),
Inf_le := λ _ _, set.bInter_subset_of_mem,
le_Inf := λ _ _, set.subset_Inter₂,
.. partial_order.lift (coe : affine_subspace k P → set P) coe_injective }
instance : inhabited (affine_subspace k P) := ⟨⊤⟩
/-- The `≤` order on subspaces is the same as that on the corresponding
sets. -/
lemma le_def (s1 s2 : affine_subspace k P) : s1 ≤ s2 ↔ (s1 : set P) ⊆ s2 :=
iff.rfl
/-- One subspace is less than or equal to another if and only if all
its points are in the second subspace. -/
lemma le_def' (s1 s2 : affine_subspace k P) : s1 ≤ s2 ↔ ∀ p ∈ s1, p ∈ s2 :=
iff.rfl
/-- The `<` order on subspaces is the same as that on the corresponding
sets. -/
lemma lt_def (s1 s2 : affine_subspace k P) : s1 < s2 ↔ (s1 : set P) ⊂ s2 :=
iff.rfl
/-- One subspace is not less than or equal to another if and only if
it has a point not in the second subspace. -/
lemma not_le_iff_exists (s1 s2 : affine_subspace k P) : ¬ s1 ≤ s2 ↔ ∃ p ∈ s1, p ∉ s2 :=
set.not_subset
/-- If a subspace is less than another, there is a point only in the
second. -/
lemma exists_of_lt {s1 s2 : affine_subspace k P} (h : s1 < s2) : ∃ p ∈ s2, p ∉ s1 :=
set.exists_of_ssubset h
/-- A subspace is less than another if and only if it is less than or
equal to the second subspace and there is a point only in the
second. -/
lemma lt_iff_le_and_exists (s1 s2 : affine_subspace k P) : s1 < s2 ↔ s1 ≤ s2 ∧ ∃ p ∈ s2, p ∉ s1 :=
by rw [lt_iff_le_not_le, not_le_iff_exists]
/-- If an affine subspace is nonempty and contained in another with
the same direction, they are equal. -/
lemma eq_of_direction_eq_of_nonempty_of_le {s₁ s₂ : affine_subspace k P}
(hd : s₁.direction = s₂.direction) (hn : (s₁ : set P).nonempty) (hle : s₁ ≤ s₂) :
s₁ = s₂ :=
let ⟨p, hp⟩ := hn in ext_of_direction_eq hd ⟨p, hp, hle hp⟩
variables (k V)
/-- The affine span is the `Inf` of subspaces containing the given
points. -/
lemma affine_span_eq_Inf (s : set P) : affine_span k s = Inf {s' | s ⊆ s'} :=
le_antisymm (span_points_subset_coe_of_subset_coe $ set.subset_Inter₂ $ λ _, id)
(Inf_le (subset_span_points k _))
variables (P)
/-- The Galois insertion formed by `affine_span` and coercion back to
a set. -/
protected def gi : galois_insertion (affine_span k) (coe : affine_subspace k P → set P) :=
{ choice := λ s _, affine_span k s,
gc := λ s1 s2, ⟨λ h, set.subset.trans (subset_span_points k s1) h,
span_points_subset_coe_of_subset_coe⟩,
le_l_u := λ _, subset_span_points k _,
choice_eq := λ _ _, rfl }
/-- The span of the empty set is `⊥`. -/
@[simp] lemma span_empty : affine_span k (∅ : set P) = ⊥ :=
(affine_subspace.gi k V P).gc.l_bot
/-- The span of `univ` is `⊤`. -/
@[simp] lemma span_univ : affine_span k (set.univ : set P) = ⊤ :=
eq_top_iff.2 $ subset_span_points k _
variables {k V P}
lemma _root_.affine_span_le {s : set P} {Q : affine_subspace k P} :
affine_span k s ≤ Q ↔ s ⊆ (Q : set P) :=
(affine_subspace.gi k V P).gc _ _
variables (k V) {P}
/-- The affine span of a single point, coerced to a set, contains just
that point. -/
@[simp] lemma coe_affine_span_singleton (p : P) : (affine_span k ({p} : set P) : set P) = {p} :=
begin
ext x,
rw [mem_coe, ←vsub_right_mem_direction_iff_mem (mem_affine_span k (set.mem_singleton p)) _,
direction_affine_span],
simp
end
/-- A point is in the affine span of a single point if and only if
they are equal. -/
@[simp] lemma mem_affine_span_singleton (p1 p2 : P) :
p1 ∈ affine_span k ({p2} : set P) ↔ p1 = p2 :=
by simp [←mem_coe]
/-- The span of a union of sets is the sup of their spans. -/
lemma span_union (s t : set P) : affine_span k (s ∪ t) = affine_span k s ⊔ affine_span k t :=
(affine_subspace.gi k V P).gc.l_sup
/-- The span of a union of an indexed family of sets is the sup of
their spans. -/
lemma span_Union {ι : Type*} (s : ι → set P) :
affine_span k (⋃ i, s i) = ⨆ i, affine_span k (s i) :=
(affine_subspace.gi k V P).gc.l_supr
variables (P)
/-- `⊤`, coerced to a set, is the whole set of points. -/
@[simp] lemma top_coe : ((⊤ : affine_subspace k P) : set P) = set.univ :=
rfl
variables {P}
/-- All points are in `⊤`. -/
lemma mem_top (p : P) : p ∈ (⊤ : affine_subspace k P) :=
set.mem_univ p
variables (P)
/-- The direction of `⊤` is the whole module as a submodule. -/
@[simp] lemma direction_top : (⊤ : affine_subspace k P).direction = ⊤ :=
begin
cases S.nonempty with p,
ext v,
refine ⟨imp_intro submodule.mem_top, λ hv, _⟩,
have hpv : (v +ᵥ p -ᵥ p : V) ∈ (⊤ : affine_subspace k P).direction :=
vsub_mem_direction (mem_top k V _) (mem_top k V _),
rwa vadd_vsub at hpv
end
/-- `⊥`, coerced to a set, is the empty set. -/
@[simp] lemma bot_coe : ((⊥ : affine_subspace k P) : set P) = ∅ :=
rfl
lemma bot_ne_top : (⊥ : affine_subspace k P) ≠ ⊤ :=
begin
intros contra,
rw [← ext_iff, bot_coe, top_coe] at contra,
exact set.empty_ne_univ contra,
end
instance : nontrivial (affine_subspace k P) := ⟨⟨⊥, ⊤, bot_ne_top k V P⟩⟩
lemma nonempty_of_affine_span_eq_top {s : set P} (h : affine_span k s = ⊤) : s.nonempty :=
begin
rw ← set.ne_empty_iff_nonempty,
rintros rfl,
rw affine_subspace.span_empty at h,
exact bot_ne_top k V P h,
end
/-- If the affine span of a set is `⊤`, then the vector span of the same set is the `⊤`. -/
lemma vector_span_eq_top_of_affine_span_eq_top {s : set P} (h : affine_span k s = ⊤) :
vector_span k s = ⊤ :=
by rw [← direction_affine_span, h, direction_top]
/-- For a nonempty set, the affine span is `⊤` iff its vector span is `⊤`. -/
lemma affine_span_eq_top_iff_vector_span_eq_top_of_nonempty {s : set P} (hs : s.nonempty) :
affine_span k s = ⊤ ↔ vector_span k s = ⊤ :=
begin
refine ⟨vector_span_eq_top_of_affine_span_eq_top k V P, _⟩,
intros h,
suffices : nonempty (affine_span k s),
{ obtain ⟨p, hp : p ∈ affine_span k s⟩ := this,
rw [eq_iff_direction_eq_of_mem hp (mem_top k V p), direction_affine_span, h, direction_top] },
obtain ⟨x, hx⟩ := hs,
exact ⟨⟨x, mem_affine_span k hx⟩⟩,
end
/-- For a non-trivial space, the affine span of a set is `⊤` iff its vector span is `⊤`. -/
lemma affine_span_eq_top_iff_vector_span_eq_top_of_nontrivial {s : set P} [nontrivial P] :
affine_span k s = ⊤ ↔ vector_span k s = ⊤ :=
begin
cases s.eq_empty_or_nonempty with hs hs,
{ simp [hs, subsingleton_iff_bot_eq_top, add_torsor.subsingleton_iff V P, not_subsingleton], },
{ rw affine_span_eq_top_iff_vector_span_eq_top_of_nonempty k V P hs, },
end
lemma card_pos_of_affine_span_eq_top {ι : Type*} [fintype ι] {p : ι → P}
(h : affine_span k (range p) = ⊤) :
0 < fintype.card ι :=
begin
obtain ⟨-, ⟨i, -⟩⟩ := nonempty_of_affine_span_eq_top k V P h,
exact fintype.card_pos_iff.mpr ⟨i⟩,
end
variables {P}
/-- No points are in `⊥`. -/
lemma not_mem_bot (p : P) : p ∉ (⊥ : affine_subspace k P) :=
set.not_mem_empty p
variables (P)
/-- The direction of `⊥` is the submodule `⊥`. -/
@[simp] lemma direction_bot : (⊥ : affine_subspace k P).direction = ⊥ :=
by rw [direction_eq_vector_span, bot_coe, vector_span_def, vsub_empty, submodule.span_empty]
variables {k V P}
lemma subsingleton_of_subsingleton_span_eq_top {s : set P} (h₁ : s.subsingleton)
(h₂ : affine_span k s = ⊤) : subsingleton P :=
begin
obtain ⟨p, hp⟩ := affine_subspace.nonempty_of_affine_span_eq_top k V P h₂,
have : s = {p}, { exact subset.antisymm (λ q hq, h₁ hq hp) (by simp [hp]), },
rw [this, ← affine_subspace.ext_iff, affine_subspace.coe_affine_span_singleton,
affine_subspace.top_coe, eq_comm, ← subsingleton_iff_singleton (mem_univ _)] at h₂,
exact subsingleton_of_univ_subsingleton h₂,
end
lemma eq_univ_of_subsingleton_span_eq_top {s : set P} (h₁ : s.subsingleton)
(h₂ : affine_span k s = ⊤) : s = (univ : set P) :=
begin
obtain ⟨p, hp⟩ := affine_subspace.nonempty_of_affine_span_eq_top k V P h₂,
have : s = {p}, { exact subset.antisymm (λ q hq, h₁ hq hp) (by simp [hp]), },
rw [this, eq_comm, ← subsingleton_iff_singleton (mem_univ p), subsingleton_univ_iff],
exact subsingleton_of_subsingleton_span_eq_top h₁ h₂,
end
/-- A nonempty affine subspace is `⊤` if and only if its direction is
`⊤`. -/
@[simp] lemma direction_eq_top_iff_of_nonempty {s : affine_subspace k P}
(h : (s : set P).nonempty) : s.direction = ⊤ ↔ s = ⊤ :=
begin
split,
{ intro hd,
rw ←direction_top k V P at hd,
refine ext_of_direction_eq hd _,
simp [h] },
{ rintro rfl,
simp }
end
/-- The inf of two affine subspaces, coerced to a set, is the
intersection of the two sets of points. -/
@[simp] lemma inf_coe (s1 s2 : affine_subspace k P) : ((s1 ⊓ s2) : set P) = s1 ∩ s2 :=
rfl
/-- A point is in the inf of two affine subspaces if and only if it is
in both of them. -/
lemma mem_inf_iff (p : P) (s1 s2 : affine_subspace k P) : p ∈ s1 ⊓ s2 ↔ p ∈ s1 ∧ p ∈ s2 :=
iff.rfl
/-- The direction of the inf of two affine subspaces is less than or
equal to the inf of their directions. -/
lemma direction_inf (s1 s2 : affine_subspace k P) :
(s1 ⊓ s2).direction ≤ s1.direction ⊓ s2.direction :=
begin
repeat { rw [direction_eq_vector_span, vector_span_def] },
exact le_inf
(Inf_le_Inf (λ p hp, trans (vsub_self_mono (inter_subset_left _ _)) hp))
(Inf_le_Inf (λ p hp, trans (vsub_self_mono (inter_subset_right _ _)) hp))
end
/-- If two affine subspaces have a point in common, the direction of
their inf equals the inf of their directions. -/
lemma direction_inf_of_mem {s₁ s₂ : affine_subspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂) :
(s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction :=
begin
ext v,
rw [submodule.mem_inf, ←vadd_mem_iff_mem_direction v h₁, ←vadd_mem_iff_mem_direction v h₂,
←vadd_mem_iff_mem_direction v ((mem_inf_iff p s₁ s₂).2 ⟨h₁, h₂⟩), mem_inf_iff]
end
/-- If two affine subspaces have a point in their inf, the direction
of their inf equals the inf of their directions. -/
lemma direction_inf_of_mem_inf {s₁ s₂ : affine_subspace k P} {p : P} (h : p ∈ s₁ ⊓ s₂) :
(s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction :=
direction_inf_of_mem ((mem_inf_iff p s₁ s₂).1 h).1 ((mem_inf_iff p s₁ s₂).1 h).2
/-- If one affine subspace is less than or equal to another, the same
applies to their directions. -/
lemma direction_le {s1 s2 : affine_subspace k P} (h : s1 ≤ s2) : s1.direction ≤ s2.direction :=
begin
repeat { rw [direction_eq_vector_span, vector_span_def] },
exact vector_span_mono k h
end
/-- If one nonempty affine subspace is less than another, the same
applies to their directions -/
lemma direction_lt_of_nonempty {s1 s2 : affine_subspace k P} (h : s1 < s2)
(hn : (s1 : set P).nonempty) : s1.direction < s2.direction :=
begin
cases hn with p hp,
rw lt_iff_le_and_exists at h,
rcases h with ⟨hle, p2, hp2, hp2s1⟩,
rw set_like.lt_iff_le_and_exists,
use [direction_le hle, p2 -ᵥ p, vsub_mem_direction hp2 (hle hp)],
intro hm,
rw vsub_right_mem_direction_iff_mem hp p2 at hm,
exact hp2s1 hm
end
/-- The sup of the directions of two affine subspaces is less than or
equal to the direction of their sup. -/
lemma sup_direction_le (s1 s2 : affine_subspace k P) :
s1.direction ⊔ s2.direction ≤ (s1 ⊔ s2).direction :=
begin
repeat { rw [direction_eq_vector_span, vector_span_def] },
exact sup_le
(Inf_le_Inf (λ p hp, set.subset.trans (vsub_self_mono (le_sup_left : s1 ≤ s1 ⊔ s2)) hp))
(Inf_le_Inf (λ p hp, set.subset.trans (vsub_self_mono (le_sup_right : s2 ≤ s1 ⊔ s2)) hp))
end
/-- The sup of the directions of two nonempty affine subspaces with
empty intersection is less than the direction of their sup. -/
lemma sup_direction_lt_of_nonempty_of_inter_empty {s1 s2 : affine_subspace k P}
(h1 : (s1 : set P).nonempty) (h2 : (s2 : set P).nonempty) (he : (s1 ∩ s2 : set P) = ∅) :
s1.direction ⊔ s2.direction < (s1 ⊔ s2).direction :=
begin
cases h1 with p1 hp1,
cases h2 with p2 hp2,
rw set_like.lt_iff_le_and_exists,
use [sup_direction_le s1 s2, p2 -ᵥ p1,
vsub_mem_direction ((le_sup_right : s2 ≤ s1 ⊔ s2) hp2) ((le_sup_left : s1 ≤ s1 ⊔ s2) hp1)],
intro h,
rw submodule.mem_sup at h,
rcases h with ⟨v1, hv1, v2, hv2, hv1v2⟩,
rw [←sub_eq_zero, sub_eq_add_neg, neg_vsub_eq_vsub_rev, add_comm v1, add_assoc,
←vadd_vsub_assoc, ←neg_neg v2, add_comm, ←sub_eq_add_neg, ←vsub_vadd_eq_vsub_sub,
vsub_eq_zero_iff_eq] at hv1v2,
refine set.nonempty.ne_empty _ he,
use [v1 +ᵥ p1, vadd_mem_of_mem_direction hv1 hp1],
rw hv1v2,
exact vadd_mem_of_mem_direction (submodule.neg_mem _ hv2) hp2
end
/-- If the directions of two nonempty affine subspaces span the whole
module, they have nonempty intersection. -/
lemma inter_nonempty_of_nonempty_of_sup_direction_eq_top {s1 s2 : affine_subspace k P}
(h1 : (s1 : set P).nonempty) (h2 : (s2 : set P).nonempty)
(hd : s1.direction ⊔ s2.direction = ⊤) : ((s1 : set P) ∩ s2).nonempty :=
begin
by_contradiction h,
rw set.not_nonempty_iff_eq_empty at h,
have hlt := sup_direction_lt_of_nonempty_of_inter_empty h1 h2 h,
rw hd at hlt,
exact not_top_lt hlt
end
/-- If the directions of two nonempty affine subspaces are complements
of each other, they intersect in exactly one point. -/
lemma inter_eq_singleton_of_nonempty_of_is_compl {s1 s2 : affine_subspace k P}
(h1 : (s1 : set P).nonempty) (h2 : (s2 : set P).nonempty)
(hd : is_compl s1.direction s2.direction) : ∃ p, (s1 : set P) ∩ s2 = {p} :=
begin
cases inter_nonempty_of_nonempty_of_sup_direction_eq_top h1 h2 hd.sup_eq_top with p hp,
use p,
ext q,
rw set.mem_singleton_iff,
split,
{ rintros ⟨hq1, hq2⟩,
have hqp : q -ᵥ p ∈ s1.direction ⊓ s2.direction :=
⟨vsub_mem_direction hq1 hp.1, vsub_mem_direction hq2 hp.2⟩,
rwa [hd.inf_eq_bot, submodule.mem_bot, vsub_eq_zero_iff_eq] at hqp },
{ exact λ h, h.symm ▸ hp }
end
/-- Coercing a subspace to a set then taking the affine span produces
the original subspace. -/
@[simp] lemma affine_span_coe (s : affine_subspace k P) : affine_span k (s : set P) = s :=
begin
refine le_antisymm _ (subset_span_points _ _),
rintros p ⟨p1, hp1, v, hv, rfl⟩,
exact vadd_mem_of_mem_direction hv hp1
end
end affine_subspace
section affine_space'
variables (k : Type*) {V : Type*} {P : Type*} [ring k] [add_comm_group V] [module k V]
[affine_space V P]
variables {ι : Type*}
include V
open affine_subspace set
/-- The `vector_span` is the span of the pairwise subtractions with a
given point on the left. -/
lemma vector_span_eq_span_vsub_set_left {s : set P} {p : P} (hp : p ∈ s) :
vector_span k s = submodule.span k ((-ᵥ) p '' s) :=
begin
rw vector_span_def,
refine le_antisymm _ (submodule.span_mono _),
{ rw submodule.span_le,
rintros v ⟨p1, p2, hp1, hp2, hv⟩,
rw ←vsub_sub_vsub_cancel_left p1 p2 p at hv,
rw [←hv, set_like.mem_coe, submodule.mem_span],
exact λ m hm, submodule.sub_mem _ (hm ⟨p2, hp2, rfl⟩) (hm ⟨p1, hp1, rfl⟩) },
{ rintros v ⟨p2, hp2, hv⟩,
exact ⟨p, p2, hp, hp2, hv⟩ }
end
/-- The `vector_span` is the span of the pairwise subtractions with a
given point on the right. -/
lemma vector_span_eq_span_vsub_set_right {s : set P} {p : P} (hp : p ∈ s) :
vector_span k s = submodule.span k ((-ᵥ p) '' s) :=
begin
rw vector_span_def,
refine le_antisymm _ (submodule.span_mono _),
{ rw submodule.span_le,
rintros v ⟨p1, p2, hp1, hp2, hv⟩,
rw ←vsub_sub_vsub_cancel_right p1 p2 p at hv,
rw [←hv, set_like.mem_coe, submodule.mem_span],
exact λ m hm, submodule.sub_mem _ (hm ⟨p1, hp1, rfl⟩) (hm ⟨p2, hp2, rfl⟩) },
{ rintros v ⟨p2, hp2, hv⟩,
exact ⟨p2, p, hp2, hp, hv⟩ }
end
/-- The `vector_span` is the span of the pairwise subtractions with a
given point on the left, excluding the subtraction of that point from
itself. -/
lemma vector_span_eq_span_vsub_set_left_ne {s : set P} {p : P} (hp : p ∈ s) :
vector_span k s = submodule.span k ((-ᵥ) p '' (s \ {p})) :=
begin
conv_lhs { rw [vector_span_eq_span_vsub_set_left k hp, ←set.insert_eq_of_mem hp,
←set.insert_diff_singleton, set.image_insert_eq] },
simp [submodule.span_insert_eq_span]
end
/-- The `vector_span` is the span of the pairwise subtractions with a
given point on the right, excluding the subtraction of that point from
itself. -/
lemma vector_span_eq_span_vsub_set_right_ne {s : set P} {p : P} (hp : p ∈ s) :
vector_span k s = submodule.span k ((-ᵥ p) '' (s \ {p})) :=
begin
conv_lhs { rw [vector_span_eq_span_vsub_set_right k hp, ←set.insert_eq_of_mem hp,
←set.insert_diff_singleton, set.image_insert_eq] },
simp [submodule.span_insert_eq_span]
end
/-- The `vector_span` is the span of the pairwise subtractions with a
given point on the right, excluding the subtraction of that point from
itself. -/
lemma vector_span_eq_span_vsub_finset_right_ne {s : finset P} {p : P} (hp : p ∈ s) :
vector_span k (s : set P) = submodule.span k ((s.erase p).image (-ᵥ p)) :=
by simp [vector_span_eq_span_vsub_set_right_ne _ (finset.mem_coe.mpr hp)]
/-- The `vector_span` of the image of a function is the span of the
pairwise subtractions with a given point on the left, excluding the
subtraction of that point from itself. -/
lemma vector_span_image_eq_span_vsub_set_left_ne (p : ι → P) {s : set ι} {i : ι} (hi : i ∈ s) :
vector_span k (p '' s) = submodule.span k ((-ᵥ) (p i) '' (p '' (s \ {i}))) :=
begin
conv_lhs { rw [vector_span_eq_span_vsub_set_left k (set.mem_image_of_mem p hi),
←set.insert_eq_of_mem hi, ←set.insert_diff_singleton, set.image_insert_eq,
set.image_insert_eq] },
simp [submodule.span_insert_eq_span]
end
/-- The `vector_span` of the image of a function is the span of the
pairwise subtractions with a given point on the right, excluding the
subtraction of that point from itself. -/
lemma vector_span_image_eq_span_vsub_set_right_ne (p : ι → P) {s : set ι} {i : ι} (hi : i ∈ s) :
vector_span k (p '' s) = submodule.span k ((-ᵥ (p i)) '' (p '' (s \ {i}))) :=
begin
conv_lhs { rw [vector_span_eq_span_vsub_set_right k (set.mem_image_of_mem p hi),
←set.insert_eq_of_mem hi, ←set.insert_diff_singleton, set.image_insert_eq,
set.image_insert_eq] },
simp [submodule.span_insert_eq_span]
end
/-- The `vector_span` of an indexed family is the span of the pairwise
subtractions with a given point on the left. -/
lemma vector_span_range_eq_span_range_vsub_left (p : ι → P) (i0 : ι) :
vector_span k (set.range p) = submodule.span k (set.range (λ (i : ι), p i0 -ᵥ p i)) :=
by rw [vector_span_eq_span_vsub_set_left k (set.mem_range_self i0), ←set.range_comp]
/-- The `vector_span` of an indexed family is the span of the pairwise
subtractions with a given point on the right. -/
lemma vector_span_range_eq_span_range_vsub_right (p : ι → P) (i0 : ι) :
vector_span k (set.range p) = submodule.span k (set.range (λ (i : ι), p i -ᵥ p i0)) :=
by rw [vector_span_eq_span_vsub_set_right k (set.mem_range_self i0), ←set.range_comp]
/-- The `vector_span` of an indexed family is the span of the pairwise
subtractions with a given point on the left, excluding the subtraction
of that point from itself. -/
lemma vector_span_range_eq_span_range_vsub_left_ne (p : ι → P) (i₀ : ι) :
vector_span k (set.range p) = submodule.span k (set.range (λ (i : {x // x ≠ i₀}), p i₀ -ᵥ p i)) :=
begin
rw [←set.image_univ, vector_span_image_eq_span_vsub_set_left_ne k _ (set.mem_univ i₀)],
congr' with v,
simp only [set.mem_range, set.mem_image, set.mem_diff, set.mem_singleton_iff, subtype.exists,
subtype.coe_mk],
split,
{ rintros ⟨x, ⟨i₁, ⟨⟨hi₁u, hi₁⟩, rfl⟩⟩, hv⟩,
exact ⟨i₁, hi₁, hv⟩ },
{ exact λ ⟨i₁, hi₁, hv⟩, ⟨p i₁, ⟨i₁, ⟨set.mem_univ _, hi₁⟩, rfl⟩, hv⟩ }
end
/-- The `vector_span` of an indexed family is the span of the pairwise
subtractions with a given point on the right, excluding the subtraction
of that point from itself. -/
lemma vector_span_range_eq_span_range_vsub_right_ne (p : ι → P) (i₀ : ι) :
vector_span k (set.range p) = submodule.span k (set.range (λ (i : {x // x ≠ i₀}), p i -ᵥ p i₀)) :=
begin
rw [←set.image_univ, vector_span_image_eq_span_vsub_set_right_ne k _ (set.mem_univ i₀)],
congr' with v,
simp only [set.mem_range, set.mem_image, set.mem_diff, set.mem_singleton_iff, subtype.exists,
subtype.coe_mk],
split,
{ rintros ⟨x, ⟨i₁, ⟨⟨hi₁u, hi₁⟩, rfl⟩⟩, hv⟩,
exact ⟨i₁, hi₁, hv⟩ },
{ exact λ ⟨i₁, hi₁, hv⟩, ⟨p i₁, ⟨i₁, ⟨set.mem_univ _, hi₁⟩, rfl⟩, hv⟩ }
end
/-- The affine span of a set is nonempty if and only if that set
is. -/
lemma affine_span_nonempty (s : set P) :
(affine_span k s : set P).nonempty ↔ s.nonempty :=
span_points_nonempty k s
/-- The affine span of a nonempty set is nonempty. -/
instance {s : set P} [nonempty s] : nonempty (affine_span k s) :=
((affine_span_nonempty k s).mpr (nonempty_subtype.mp ‹_›)).to_subtype
variables {k}
/-- Suppose a set of vectors spans `V`. Then a point `p`, together
with those vectors added to `p`, spans `P`. -/
lemma affine_span_singleton_union_vadd_eq_top_of_span_eq_top {s : set V} (p : P)
(h : submodule.span k (set.range (coe : s → V)) = ⊤) :
affine_span k ({p} ∪ (λ v, v +ᵥ p) '' s) = ⊤ :=
begin
convert ext_of_direction_eq _
⟨p,
mem_affine_span k (set.mem_union_left _ (set.mem_singleton _)),
mem_top k V p⟩,
rw [direction_affine_span, direction_top,
vector_span_eq_span_vsub_set_right k
((set.mem_union_left _ (set.mem_singleton _)) : p ∈ _), eq_top_iff, ←h],
apply submodule.span_mono,
rintros v ⟨v', rfl⟩,
use (v' : V) +ᵥ p,
simp
end
variables (k)
/-- `affine_span` is monotone. -/
@[mono]
lemma affine_span_mono {s₁ s₂ : set P} (h : s₁ ⊆ s₂) : affine_span k s₁ ≤ affine_span k s₂ :=
span_points_subset_coe_of_subset_coe (set.subset.trans h (subset_affine_span k _))
/-- Taking the affine span of a set, adding a point and taking the
span again produces the same results as adding the point to the set
and taking the span. -/
lemma affine_span_insert_affine_span (p : P) (ps : set P) :
affine_span k (insert p (affine_span k ps : set P)) = affine_span k (insert p ps) :=
by rw [set.insert_eq, set.insert_eq, span_union, span_union, affine_span_coe]
/-- If a point is in the affine span of a set, adding it to that set
does not change the affine span. -/
lemma affine_span_insert_eq_affine_span {p : P} {ps : set P} (h : p ∈ affine_span k ps) :
affine_span k (insert p ps) = affine_span k ps :=
begin
rw ←mem_coe at h,
rw [←affine_span_insert_affine_span, set.insert_eq_of_mem h, affine_span_coe]
end
end affine_space'
namespace affine_subspace
variables {k : Type*} {V : Type*} {P : Type*} [ring k] [add_comm_group V] [module k V]
[affine_space V P]
include V
/-- The direction of the sup of two nonempty affine subspaces is the
sup of the two directions and of any one difference between points in
the two subspaces. -/
lemma direction_sup {s1 s2 : affine_subspace k P} {p1 p2 : P} (hp1 : p1 ∈ s1) (hp2 : p2 ∈ s2) :
(s1 ⊔ s2).direction = s1.direction ⊔ s2.direction ⊔ k ∙ (p2 -ᵥ p1) :=
begin
refine le_antisymm _ _,
{ change (affine_span k ((s1 : set P) ∪ s2)).direction ≤ _,
rw ←mem_coe at hp1,
rw [direction_affine_span, vector_span_eq_span_vsub_set_right k (set.mem_union_left _ hp1),
submodule.span_le],
rintros v ⟨p3, hp3, rfl⟩,
cases hp3,
{ rw [sup_assoc, sup_comm, set_like.mem_coe, submodule.mem_sup],
use [0, submodule.zero_mem _, p3 -ᵥ p1, vsub_mem_direction hp3 hp1],
rw zero_add },
{ rw [sup_assoc, set_like.mem_coe, submodule.mem_sup],
use [0, submodule.zero_mem _, p3 -ᵥ p1],
rw [and_comm, zero_add],
use rfl,
rw [←vsub_add_vsub_cancel p3 p2 p1, submodule.mem_sup],
use [p3 -ᵥ p2, vsub_mem_direction hp3 hp2, p2 -ᵥ p1,
submodule.mem_span_singleton_self _] } },
{ refine sup_le (sup_direction_le _ _) _,
rw [direction_eq_vector_span, vector_span_def],
exact Inf_le_Inf (λ p hp, set.subset.trans
(set.singleton_subset_iff.2
(vsub_mem_vsub (mem_span_points k p2 _ (set.mem_union_right _ hp2))
(mem_span_points k p1 _ (set.mem_union_left _ hp1))))
hp) }
end
/-- The direction of the span of the result of adding a point to a
nonempty affine subspace is the sup of the direction of that subspace
and of any one difference between that point and a point in the
subspace. -/
lemma direction_affine_span_insert {s : affine_subspace k P} {p1 p2 : P} (hp1 : p1 ∈ s) :
(affine_span k (insert p2 (s : set P))).direction = submodule.span k {p2 -ᵥ p1} ⊔ s.direction :=
begin
rw [sup_comm, ←set.union_singleton, ←coe_affine_span_singleton k V p2],
change (s ⊔ affine_span k {p2}).direction = _,
rw [direction_sup hp1 (mem_affine_span k (set.mem_singleton _)), direction_affine_span],
simp
end
/-- Given a point `p1` in an affine subspace `s`, and a point `p2`, a
point `p` is in the span of `s` with `p2` added if and only if it is a
multiple of `p2 -ᵥ p1` added to a point in `s`. -/
lemma mem_affine_span_insert_iff {s : affine_subspace k P} {p1 : P} (hp1 : p1 ∈ s) (p2 p : P) :
p ∈ affine_span k (insert p2 (s : set P)) ↔
∃ (r : k) (p0 : P) (hp0 : p0 ∈ s), p = r • (p2 -ᵥ p1 : V) +ᵥ p0 :=
begin
rw ←mem_coe at hp1,
rw [←vsub_right_mem_direction_iff_mem (mem_affine_span k (set.mem_insert_of_mem _ hp1)),
direction_affine_span_insert hp1, submodule.mem_sup],
split,
{ rintros ⟨v1, hv1, v2, hv2, hp⟩,
rw submodule.mem_span_singleton at hv1,
rcases hv1 with ⟨r, rfl⟩,
use [r, v2 +ᵥ p1, vadd_mem_of_mem_direction hv2 hp1],
symmetry' at hp,
rw [←sub_eq_zero, ←vsub_vadd_eq_vsub_sub, vsub_eq_zero_iff_eq] at hp,
rw [hp, vadd_vadd] },
{ rintros ⟨r, p3, hp3, rfl⟩,
use [r • (p2 -ᵥ p1), submodule.mem_span_singleton.2 ⟨r, rfl⟩, p3 -ᵥ p1,
vsub_mem_direction hp3 hp1],
rw [vadd_vsub_assoc, add_comm] }
end
end affine_subspace
section map_comap
variables {k V₁ P₁ V₂ P₂ V₃ P₃ : Type*} [ring k]
variables [add_comm_group V₁] [module k V₁] [add_torsor V₁ P₁]
variables [add_comm_group V₂] [module k V₂] [add_torsor V₂ P₂]
variables [add_comm_group V₃] [module k V₃] [add_torsor V₃ P₃]
include V₁ V₂
section
variables (f : P₁ →ᵃ[k] P₂)
@[simp] lemma affine_map.vector_span_image_eq_submodule_map {s : set P₁} :
submodule.map f.linear (vector_span k s) = vector_span k (f '' s) :=
by simp [f.image_vsub_image, vector_span_def]
namespace affine_subspace
/-- The image of an affine subspace under an affine map as an affine subspace. -/
def map (s : affine_subspace k P₁) : affine_subspace k P₂ :=
{ carrier := f '' s,
smul_vsub_vadd_mem :=
begin
rintros t - - - ⟨p₁, h₁, rfl⟩ ⟨p₂, h₂, rfl⟩ ⟨p₃, h₃, rfl⟩,
use t • (p₁ -ᵥ p₂) +ᵥ p₃,
suffices : t • (p₁ -ᵥ p₂) +ᵥ p₃ ∈ s, { by simp [this], },
exact s.smul_vsub_vadd_mem t h₁ h₂ h₃,
end }
@[simp] lemma coe_map (s : affine_subspace k P₁) : (s.map f : set P₂) = f '' s := rfl
@[simp] lemma mem_map {f : P₁ →ᵃ[k] P₂} {x : P₂} {s : affine_subspace k P₁} :
x ∈ s.map f ↔ ∃ y ∈ s, f y = x := mem_image_iff_bex
@[simp] lemma map_bot : (⊥ : affine_subspace k P₁).map f = ⊥ :=
coe_injective $ image_empty f
omit V₂
@[simp] lemma map_id (s : affine_subspace k P₁) : s.map (affine_map.id k P₁) = s :=
coe_injective $ image_id _
include V₂ V₃
lemma map_map (s : affine_subspace k P₁) (f : P₁ →ᵃ[k] P₂) (g : P₂ →ᵃ[k] P₃) :
(s.map f).map g = s.map (g.comp f) :=
coe_injective $ image_image _ _ _
omit V₃
@[simp] lemma map_direction (s : affine_subspace k P₁) :
(s.map f).direction = s.direction.map f.linear :=
by simp [direction_eq_vector_span]
lemma map_span (s : set P₁) :
(affine_span k s).map f = affine_span k (f '' s) :=
begin
rcases s.eq_empty_or_nonempty with rfl | ⟨p, hp⟩, { simp, },
apply ext_of_direction_eq,
{ simp [direction_affine_span], },
{ exact ⟨f p, mem_image_of_mem f (subset_affine_span k _ hp),
subset_affine_span k _ (mem_image_of_mem f hp)⟩, },
end
end affine_subspace
namespace affine_map
@[simp] lemma map_top_of_surjective (hf : function.surjective f) : affine_subspace.map f ⊤ = ⊤ :=
begin
rw ← affine_subspace.ext_iff,
exact image_univ_of_surjective hf,
end
lemma span_eq_top_of_surjective {s : set P₁}
(hf : function.surjective f) (h : affine_span k s = ⊤) :
affine_span k (f '' s) = ⊤ :=
by rw [← affine_subspace.map_span, h, map_top_of_surjective f hf]
end affine_map
lemma affine_equiv.span_eq_top_iff {s : set P₁} (e : P₁ ≃ᵃ[k] P₂) :
affine_span k s = ⊤ ↔ affine_span k (e '' s) = ⊤ :=
begin
refine ⟨(e : P₁ →ᵃ[k] P₂).span_eq_top_of_surjective e.surjective, _⟩,
intros h,
have : s = e.symm '' (e '' s), { simp [← image_comp], },
rw this,
exact (e.symm : P₂ →ᵃ[k] P₁).span_eq_top_of_surjective e.symm.surjective h,
end
end
namespace affine_subspace
/-- The preimage of an affine subspace under an affine map as an affine subspace. -/
def comap (f : P₁ →ᵃ[k] P₂) (s : affine_subspace k P₂) : affine_subspace k P₁ :=
{ carrier := f ⁻¹' s,
smul_vsub_vadd_mem := λ t p₁ p₂ p₃ (hp₁ : f p₁ ∈ s) (hp₂ : f p₂ ∈ s) (hp₃ : f p₃ ∈ s),
show f _ ∈ s, begin
rw [affine_map.map_vadd, linear_map.map_smul, affine_map.linear_map_vsub],
apply s.smul_vsub_vadd_mem _ hp₁ hp₂ hp₃,
end }
@[simp] lemma coe_comap (f : P₁ →ᵃ[k] P₂) (s : affine_subspace k P₂) :
(s.comap f : set P₁) = f ⁻¹' ↑s := rfl
@[simp] lemma mem_comap {f : P₁ →ᵃ[k] P₂} {x : P₁} {s : affine_subspace k P₂} :
x ∈ s.comap f ↔ f x ∈ s := iff.rfl
lemma comap_mono {f : P₁ →ᵃ[k] P₂} {s t : affine_subspace k P₂} : s ≤ t → s.comap f ≤ t.comap f :=
preimage_mono
@[simp] lemma comap_top {f : P₁ →ᵃ[k] P₂} : (⊤ : affine_subspace k P₂).comap f = ⊤ :=
by { rw ← ext_iff, exact preimage_univ, }
omit V₂
@[simp] lemma comap_id (s : affine_subspace k P₁) : s.comap (affine_map.id k P₁) = s :=
coe_injective rfl
include V₂ V₃
lemma comap_comap (s : affine_subspace k P₃) (f : P₁ →ᵃ[k] P₂) (g : P₂ →ᵃ[k] P₃) :
(s.comap g).comap f = s.comap (g.comp f) :=
coe_injective rfl
omit V₃
-- lemmas about map and comap derived from the galois connection
lemma map_le_iff_le_comap {f : P₁ →ᵃ[k] P₂} {s : affine_subspace k P₁} {t : affine_subspace k P₂} :
s.map f ≤ t ↔ s ≤ t.comap f :=
image_subset_iff
lemma gc_map_comap (f : P₁ →ᵃ[k] P₂) : galois_connection (map f) (comap f) :=
λ _ _, map_le_iff_le_comap
lemma map_comap_le (f : P₁ →ᵃ[k] P₂) (s : affine_subspace k P₂) : (s.comap f).map f ≤ s :=
(gc_map_comap f).l_u_le _
lemma le_comap_map (f : P₁ →ᵃ[k] P₂) (s : affine_subspace k P₁) : s ≤ (s.map f).comap f :=
(gc_map_comap f).le_u_l _
lemma map_sup (s t : affine_subspace k P₁) (f : P₁ →ᵃ[k] P₂) : (s ⊔ t).map f = s.map f ⊔ t.map f :=
(gc_map_comap f).l_sup
lemma map_supr {ι : Sort*} (f : P₁ →ᵃ[k] P₂) (s : ι → affine_subspace k P₁) :
(supr s).map f = ⨆ i, (s i).map f :=
(gc_map_comap f).l_supr
lemma comap_inf (s t : affine_subspace k P₂) (f : P₁ →ᵃ[k] P₂) :
(s ⊓ t).comap f = s.comap f ⊓ t.comap f :=
(gc_map_comap f).u_inf
lemma comap_supr {ι : Sort*} (f : P₁ →ᵃ[k] P₂) (s : ι → affine_subspace k P₂) :
(infi s).comap f = ⨅ i, (s i).comap f :=
(gc_map_comap f).u_infi
end affine_subspace
end map_comap
|
88a13bb13f66e8116510f50e69f54454eb6b36e0 | 0005a183d2afe3c7d097776749c13513cae91b66 | /src/bigop.lean | c475a669bb708cf757e4d28aaff312d7a3951794 | [] | no_license | PatrickMassot/bigop | 9f85014752aa632c270bafbd96400512839220eb | af53ad615b619e3c05fe119f98f939ff010f7e6d | refs/heads/master | 1,584,009,287,023 | 1,562,778,117,000 | 1,562,778,117,000 | 130,576,309 | 1 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 12,452 | lean | import tactic.ext
import tactic.linarith
import pending_lemmas
open list
variables {R : Type*} {I : Type*} (op : R → R → R) (nil: R)
(r : list I) (P : I → Prop) [decidable_pred P] (F : I → R)
local infix ` ◆ `:70 := op -- type using \di
/- Starting from `F : I → R`, `r : list I`, a composition law `op` on `R`,
a element `nil` in R, and a decidable predicate `P` on `I`,
`apply_bigop op nil r P F` is the big "product", for operation `op`,
of all `F i` for `i` in `r` if `P i`. All parenthesis are closed after
inserting `nil` at the very end, like in `(a op (b op (c op nil)))`
(using infix notation for op) -/
def apply_bigop := foldr (λ i, op (F i)) nil (filter P r)
-- alternate definition : foldr (λ i x, if P i then op (F i) x else x) nil r
/- We now define a notation with many variations depending on the list,
predicate, operation -/
/- variable in filtered list -/
notation `big[`:0 op`/`:0 nil`]_(`:0 binder `∈` r `|` P:(scoped p, p)`)` F:(scoped f, f) :=
apply_bigop op nil r P F
notation `Σ_(`:0 binder `∈` r `|` P:(scoped p, p) `)` F:(scoped f, f) :=
apply_bigop (+) 0 r P F
notation `Π_(`:0 binder `∈` r `|` P:(scoped p, p) `)` F:(scoped f, f) :=
apply_bigop (*) 1 r P F
/- variable in unfiltered list -/
notation `big[`:0 op `/`:0 nil `]_(`:0 binder `∈` r `)` F:(scoped f, f) :=
apply_bigop op nil r (λ i, true) F
notation `Σ_(`:0 binder `∈` r `)` F:(scoped f, f) :=
apply_bigop (+) 0 r (λ i, true) F
notation `Π_(`:0 binder `∈` r `)` F:(scoped f, f) :=
apply_bigop (*) 1 r (λ i, true) F
/- variable is integers from a to b, b exclued, filtered -/
notation `big[`op`/`:0 nil`]_(`:0 binder`=`a`..`b `|` P:(scoped p, p)`)` F:(scoped f, f) :=
apply_bigop op nil (int.range a b) P F
notation `Σ_(`:0 binder`=`a`..`b `|` P:(scoped p, p)`)` F:(scoped f, f) :=
apply_bigop (+) 0 (int.range a b) P F
notation `Π_(`:0 binder`=`a`..` b `|` P:(scoped p, p)`)` F:(scoped f, f) :=
apply_bigop (*) 1 (int.range a b) P F
/- variable is integers from a to b, b exclued, unfiltered -/
notation `big[`:0 op `/`:0 nil `]_(`:0 binder `=` a `..` b `)` F:(scoped f, f) :=
apply_bigop op nil (int.range a b) (λ i, true) F
notation `Σ_(`:0 binder `=` a `..` b `)` F:(scoped f, f) :=
apply_bigop (+) 0 (int.range a b) (λ i, true) F
notation `Π_(`:0 binder `=` a `..` b `)` F:(scoped f, f) :=
apply_bigop (*) 1 (int.range a b) (λ i, true) F
local notation `?(F` h`)` := if P h then F h else nil
/- First lemmas, without assuming anything on `op` and `nil` -/
@[simp]
lemma big.nil : (big[(◆)/nil]_(i ∈ [] | (P i)) (F i)) = nil :=
by simp [apply_bigop]
lemma big.filter_mem [decidable_pred (λ i, i ∈ r)] :
(big[(◆)/nil]_(i ∈ r) (F i)) = (big[(◆)/nil]_(i ∈ r | i ∈ r) (F i)) :=
by simp [apply_bigop]
lemma big_cons_true {h} (t) (Ph : P h) :
(big[(◆)/nil]_(i ∈ h::t | (P i)) (F i)) = F h ◆ (big[(◆)/nil]_(i ∈ t | (P i)) (F i)):=
by simp [apply_bigop, Ph]
lemma big_cons_false {h} (t) (Ph : ¬ P h) :
(big[(◆)/nil]_(i ∈ h::t | (P i)) (F i)) = (big[(◆)/nil]_(i ∈ t | (P i)) (F i)) :=
by simp [apply_bigop, Ph]
lemma big_rec (K : R → Prop) (Knil : K nil) (Kop : ∀ i x, P i → K x → K ((F i) ◆ x)) :
K (big[(◆)/nil]_(i ∈ r | (P i)) (F i)) :=
begin
induction r with h t IH,
{ simp[big.nil, Knil] },
{ by_cases H : P h,
{ simp [big_cons_true, H, Kop _ _ H IH] },
{ simp [big_cons_false, H, IH] } }
end
lemma big_ind (K : R → Prop) (Knil : K nil) (Kop : ∀ x y, K x → K y → K (x ◆ y))
(K_F : ∀ i, P i → K (F i)) :
K (big[(◆)/nil]_(i ∈ r | (P i)) (F i)) :=
begin
apply big_rec,
exact Knil,
intros i x P_i K_x,
apply Kop ; tauto
end
lemma big_append_eq_of_not_mem [decidable_eq I] {j} (H : j ∉ r) :
(big[(◆)/nil]_(i ∈ j::r | ((P i) ∧ i ≠ j)) (F i)) = big[(◆)/nil]_(i ∈ r | (P i)) (F i) :=
begin
unfold apply_bigop,
congr' 1,
rw list.filter_cons_of_neg,
apply list.filter_congr,
{ intros x x_in,
simp [show x ≠ j, by { intro H', rw ←H' at H, contradiction }] },
{ finish }
end
-- A version of extensionality where we assume same (◆)/nil and same list
lemma big.ext (P') [decidable_pred P'] (F' : I → R)
(HP : ∀ i ∈ r, P i ↔ P' i) (HF : ∀ i ∈ r, F i = F' i) :
(big[(◆)/nil]_(i ∈ r | (P i)) (F i)) = (big[(◆)/nil]_(i ∈ r | (P' i)) (F' i)) :=
begin
unfold apply_bigop,
rw list.filter_congr HP,
apply list.foldr_ext,
intros _ i_r _,
simp[HF, mem_filter.1 i_r]
end
lemma big.map {J : Type*} (f : I → J) (P : J → Prop) [decidable_pred P] (F : J → R) :
(big[(◆)/nil]_(j ∈ map f r | (P j)) (F j)) = (big[(◆)/nil]_(i ∈ r | (P (f i))) (F (f i))) :=
by simp[apply_bigop, filter_map_comm, foldr_map]
--set_option pp.all true
lemma big.empty_range (P : ℤ → Prop) [decidable_pred P] (F : ℤ → R) (a b : ℤ)
(H : b ≤ a) : (big[(◆)/nil]_(i=a..b | (P i)) (F i)) = nil :=
begin
convert big.nil op nil P F,
exact (int.range_eq_nil _ _).2 H
end
lemma big.shift (P : ℤ → Prop) [decidable_pred P] (F : ℤ → R) (a b k : ℤ) :
(big[(◆)/nil]_(i=a..b | (P i)) (F i)) = (big[(◆)/nil]_(i=(a+k)..(b+k) | (P (i-k))) (F (i-k))) :=
begin
rw [int.range_shift, big.map],
congr ; ext ; ring
end
/- Now we go towards assuming (R, op, nil) is a monoid -/
/- Also need to make sure old hierarchy talks to new one.
Associativity seems ok but we need: -/
instance add_monoid_is_left_id (α : Type*) [add_monoid α] : is_left_id α (+) 0 := ⟨by simp⟩
instance add_monoid_is_right_id (α : Type*) [add_monoid α] : is_right_id α (+) 0 := ⟨by simp⟩
instance monoid_is_left_id (α : Type*) [monoid α] : is_left_id α (*) 1 := ⟨by simp⟩
instance monoid_is_right_id (α : Type*) [monoid α] : is_right_id α (*) 1 := ⟨by simp⟩
section nil_left_id
/- Assuming only that nil is left neutral for op -/
variable [is_left_id R op nil]
open is_left_id
lemma big.cons {h} (t) :
(big[(◆)/nil]_(i ∈ h::t | (P i)) (F i)) = ?(F h) ◆ (big[(◆)/nil]_(i ∈ t | (P i)) (F i)):=
begin
by_cases H : P h,
{ simp [H, big_cons_true] },
{ simp [H, big_cons_false, left_id op] }
end
end nil_left_id
section nil_right_id
/- Assuming only that nil is left neutral for op -/
variable [is_right_id R op nil]
open is_right_id
lemma big.one_term (i₀ : I) : (big[(◆)/nil]_(i ∈ [i₀]) F i) = F i₀ :=
by simp [apply_bigop, right_id op]
lemma big.one_term' (i₀ : I) :
(big[(◆)/nil]_(i ∈ [i₀] | P i) F i) = if P i₀ then F i₀ else nil :=
by by_cases H : P i₀; simp [H, apply_bigop, right_id op]
end nil_right_id
section left_monoid
/- Assume that (R, op) is almost a monoid with neutral element nil -/
variables [is_left_id R op nil] [is_associative R op]
open is_left_id is_associative
lemma big.append (r₁ r₂ : list I) :
(big[(◆)/nil]_(i ∈ r₁++r₂ | (P i)) (F i)) =
(big[(◆)/nil]_(i ∈ r₁ | (P i)) (F i)) ◆ (big[(◆)/nil]_(i ∈ r₂ | (P i)) (F i)) :=
begin
let Op := λ l, big[(◆)/nil]_(i ∈ l | (P i)) (F i),
induction r₁ with h t IH,
{ exact (eq.symm $ calc
Op [] ◆ Op r₂ = nil ◆ (big[(◆)/nil]_(i ∈ r₂ | P i)F i) : by {dsimp [Op], rw big.nil}
... = _ : left_id _ _ )},
{ have : ?(F h) ◆ Op t = Op (h :: t) :=
eq.symm (big.cons _ _ _ _ _),
exact calc
Op (h :: t ++ r₂)
= Op (h :: (t ++ r₂)) : rfl
... = ?(F h) ◆ Op (t ++ r₂) : big.cons _ _ _ _ _
... = ?(F h) ◆ (Op t ◆ Op r₂) : by simp [Op, IH]
... = (?(F h) ◆ Op t) ◆ Op r₂ : eq.symm $ assoc _ _ _ _
... = Op (h::t) ◆ Op r₂ : by rw this }
end
end left_monoid
section monoid
variables [is_left_id R op nil] [is_right_id R op nil] [is_associative R op]
open is_left_id is_right_id is_associative
lemma big.concat (i) :
(big[(◆)/nil]_(j ∈ concat r i | (P j)) (F j)) =
(big[(◆)/nil]_(j ∈ r | (P j)) (F j)) ◆ (if P i then F i else nil) :=
by simp [big.append,big.one_term']
lemma big.concat_true (i) :
(big[(◆)/nil]_(i ∈ concat r i) (F i)) =
(big[(◆)/nil]_(i ∈ r) (F i)) ◆ F i :=
by apply big.concat
lemma big.concat_range_true (F : ℤ → R) {a b : ℤ} (h : a < b) :
(big[(◆)/nil]_(i =a..b) (F i)) =
(big[(◆)/nil]_(i =a..b-1) (F i)) ◆ F (b-1) :=
by rw int_range_eq_concat h ; apply big.concat_true
lemma big.commute_through {a : R} (H : ∀ i, P i → a ◆ F i = F i ◆ a) :
a ◆ (big[(◆)/nil]_(i ∈ r | (P i)) F i) = (big[(◆)/nil]_(i ∈ r | (P i)) (F i)) ◆ a :=
begin
let K := λ x, (a ◆ x = x ◆ a),
change K (big[(◆)/nil]_(i ∈ r | (P i)) F i),
apply big_ind ; dsimp [K],
{ simp [left_id op, right_id op] },
{ intros x y xop yop,
rw [assoc op, ←assoc op, xop, ← yop, assoc op]},
{ exact H }
end
lemma big.reverse_of_commute (H : ∀ i j, P i → P j → F i ◆ F j = F j ◆ F i) :
(big[(◆)/nil]_(i ∈ r | (P i)) F i) = (big[(◆)/nil]_(i ∈ reverse r | (P i)) (F i)) :=
begin
induction r with h t IH,
{ simp },
{ rw [big.cons, reverse_cons, ←concat_eq_append, big.concat],
by_cases Ph : P h,
{ simp only [Ph],
rw IH,
apply big.commute_through,
simp * {contextual := true } },
{ simp [Ph, left_id op, right_id op, IH, H] { contextual := true } } }
end
lemma big.reverse_range_of_commute (P : ℤ → Prop) [decidable_pred P] (F : ℤ → R) (a b : ℤ) (H : ∀ i j, P i → P j → F i ◆ F j = F j ◆ F i) :
(big[(◆)/nil]_(i=a..b | (P i)) F i) = (big[(◆)/nil]_(i=a..b | (P (a+b-i-1))) (F (a+b-i-1))) :=
by rw [big.reverse_of_commute _ _ _ _ _ H, reverse_int_range_map_int_range, big.map]
lemma big.gather_of_commute (F G : I → R) [decidable_eq I]
(H : ∀ n n' (h : n < r.length) (h' : n' < r.length),
n ≠ n' → F (r.nth_le n h) ◆ G (r.nth_le n' h') = G (r.nth_le n' h') ◆ F (r.nth_le n h)) :
(big[(◆)/nil]_(i ∈ r) F i) ◆ (big[(◆)/nil]_(i ∈ r) G i) =
big[(◆)/nil]_(i ∈ r) F i ◆ G i :=
begin
induction r with a t IH,
{ simp,
rw left_id op },
{ have key : ∀ i ∈ t, G a ◆ F i = F i ◆ G a,
{ intros i i_in,
rcases nth_le_of_mem i_in with ⟨n, h, Hn⟩,
specialize H (n+1) 0 (nat.succ_lt_succ h) (nat.succ_pos _),
simp at H,
rw ←Hn,
exact H.symm },
simp only [big_cons_true] ,
conv_lhs {
rw assoc op,
congr, skip,
rw ←assoc op,
congr,
rw big.filter_mem,
rw ←big.commute_through _ _ _ _ _ key,
},
rw [assoc op, assoc op, ←big.filter_mem],
congr,
apply IH,
introv neq,
replace neq : n + 1 ≠ n' + 1 := λ hneq, neq (nat.succ_inj hneq),
specialize H (n+1) (n'+1) (nat.succ_lt_succ h) (nat.succ_lt_succ h') neq,
rwa nth_le_cons at H },
end
lemma big.gather_of_commute_int (F G : ℤ → R) (a b) (H : ∀ i j, i ≠ j → F i ◆ G j = G j ◆ F i) :
(big[(◆)/nil]_(i = a..b) F i) ◆ (big[(◆)/nil]_(i = a..b) G i) = big[(◆)/nil]_(i = a..b) F i ◆ G i :=
begin
apply big.gather_of_commute,
introv neq,
apply H,
simp [int.range, neq],
end
lemma apply_ite {nil : R} {φ : R → R} (Hnil : φ nil = nil) (h : I) :
φ (ite (P h) (F h) nil) = ite (P h) (φ (F h)) nil :=
calc
φ (ite (P h) (F h) nil) = ite (P h) (φ $ F h) (φ nil) : by { by_cases H : P h; simp[H] }
... = ite (P h) (φ $ F h) nil : by rw Hnil
lemma big.mph {φ : R → R}
(Hop : ∀ a b : R, φ (a ◆ b) = φ a ◆ φ b) (Hnil : φ nil = nil) :
φ (big[(◆)/nil]_(i ∈ r | (P i)) F i) = big[(◆)/nil]_(i ∈ r | (P i)) φ (F i) :=
begin
induction r with h t IH,
{ simp [big.nil, Hnil] },
{ rw [big.cons, Hop, IH, apply_ite _ _ Hnil, ←(big.cons op nil P _ t)] }
end
lemma big.anti_mph {φ : R → R}
(Hop : ∀ a b : R, φ (a ◆ b) = φ b ◆ φ a) (Hnil : φ nil = nil) :
φ (big[(◆)/nil]_(i ∈ r | (P i)) F i) = big[(◆)/nil]_(i ∈ r.reverse | (P i)) φ (F i) :=
begin
induction r with h t IH,
{ simp [big.nil, Hnil] },
{ rw [big.cons, Hop, apply_ite _ _ Hnil, reverse_cons', concat_eq_append, big.append, IH, big.one_term'] }
end
lemma big.range_anti_mph {φ : R → R} (P : ℤ → Prop) [decidable_pred P] (F : ℤ → R) (a b : ℤ)
(Hop : ∀ a b : R, φ (a ◆ b) = φ b ◆ φ a) (Hnil : φ nil = nil) :
φ (big[(◆)/nil]_(i=a..b | (P i)) F i) = big[(◆)/nil]_(i=a..b | (P (a+b-i-1))) φ (F (a+b-i-1)) :=
by rw [big.anti_mph _ _ _ _ _ Hop Hnil, reverse_int_range_map_int_range, big.map]
end monoid
|
9db95b32ef91c107b59b33e9936d3a2e23e3420e | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/run/format.lean | 4d02e0d2958a443c2e5b17cc3c2bc8f355a4bc8b | [
"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 | 289 | lean | import system.io
open list
#eval pp ([["aaa", "bbb", "ccc", "dddd", "eeeeee", "ffffff"], ["aaa", "bbb", "ccc", "dddd", "eeeeee", "ffffff"],
["aaa", "bbb", "ccc", "dddd", "eeeeee", "ffffff"], ["aaa", "bbb", "ccc", "dddd", "eeeeee", "ffffff"]],
[(10:nat), 20, 30])
|
675511e42ea0a099c109a4616b1815bfc44b989a | 491068d2ad28831e7dade8d6dff871c3e49d9431 | /library/init/wf.lean | 34b62313e1f4106803055326f6d18b51292d716f | [
"Apache-2.0"
] | permissive | davidmueller13/lean | 65a3ed141b4088cd0a268e4de80eb6778b21a0e9 | c626e2e3c6f3771e07c32e82ee5b9e030de5b050 | refs/heads/master | 1,611,278,313,401 | 1,444,021,177,000 | 1,444,021,177,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 4,384 | lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
prelude
import init.relation init.tactic
inductive acc {A : Type} (R : A → A → Prop) : A → Prop :=
intro : ∀x, (∀ y, R y x → acc R y) → acc R x
namespace acc
variables {A : Type} {R : A → A → Prop}
definition inv {x y : A} (H₁ : acc R x) (H₂ : R y x) : acc R y :=
acc.rec_on H₁ (λ x₁ ac₁ iH H₂, ac₁ y H₂) H₂
-- dependent elimination for acc
protected definition drec [recursor]
{C : Π (a : A), acc R a → Type}
(h₁ : Π (x : A) (acx : Π (y : A), R y x → acc R y),
(Π (y : A) (ryx : R y x), C y (acx y ryx)) → C x (acc.intro x acx))
{a : A} (h₂ : acc R a) : C a h₂ :=
begin
refine acc.rec _ h₂ h₂,
intro x acx ih h₂,
exact h₁ x acx (λ y ryx, ih y ryx (acx y ryx))
end
end acc
inductive well_founded [class] {A : Type} (R : A → A → Prop) : Prop :=
intro : (∀ a, acc R a) → well_founded R
namespace well_founded
definition apply [coercion] {A : Type} {R : A → A → Prop} (wf : well_founded R) : ∀a, acc R a :=
take a, well_founded.rec_on wf (λp, p) a
section
parameters {A : Type} {R : A → A → Prop}
local infix `≺`:50 := R
hypothesis [Hwf : well_founded R]
theorem recursion {C : A → Type} (a : A) (H : Πx, (Πy, y ≺ x → C y) → C x) : C a :=
acc.rec_on (Hwf a) (λ x₁ ac₁ iH, H x₁ iH)
theorem induction {C : A → Prop} (a : A) (H : ∀x, (∀y, y ≺ x → C y) → C x) : C a :=
recursion a H
variable {C : A → Type}
variable F : Πx, (Πy, y ≺ x → C y) → C x
definition fix_F (x : A) (a : acc R x) : C x :=
acc.rec_on a (λ x₁ ac₁ iH, F x₁ iH)
theorem fix_F_eq (x : A) (r : acc R x) :
fix_F F x r = F x (λ (y : A) (p : y ≺ x), fix_F F y (acc.inv r p)) :=
begin
induction r using acc.drec,
reflexivity -- proof is trivial due to proof irrelevance
end
end
variables {A : Type} {C : A → Type} {R : A → A → Prop}
-- Well-founded fixpoint
definition fix [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) : C x :=
fix_F F x (Hwf x)
-- Well-founded fixpoint satisfies fixpoint equation
theorem fix_eq [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) :
fix F x = F x (λy h, fix F y) :=
fix_F_eq F x (Hwf x)
end well_founded
open well_founded
-- Empty relation is well-founded
definition empty.wf {A : Type} : well_founded empty_relation :=
well_founded.intro (λ (a : A),
acc.intro a (λ (b : A) (lt : false), false.rec _ lt))
-- Subrelation of a well-founded relation is well-founded
namespace subrelation
section
parameters {A : Type} {R Q : A → A → Prop}
parameters (H₁ : subrelation Q R)
parameters (H₂ : well_founded R)
definition accessible {a : A} (ac : acc R a) : acc Q a :=
using H₁,
begin
induction ac with x ax ih, constructor,
exact λ (y : A) (lt : Q y x), ih y (H₁ lt)
end
definition wf : well_founded Q :=
well_founded.intro (λ a, accessible (H₂ a))
end
end subrelation
-- The inverse image of a well-founded relation is well-founded
namespace inv_image
section
parameters {A B : Type} {R : B → B → Prop}
parameters (f : A → B)
parameters (H : well_founded R)
private definition acc_aux {b : B} (ac : acc R b) : ∀ x, f x = b → acc (inv_image R f) x :=
begin
induction ac with x acx ih,
intro z e, constructor,
intro y lt, subst x,
exact ih (f y) lt y rfl
end
definition accessible {a : A} (ac : acc R (f a)) : acc (inv_image R f) a :=
acc_aux ac a rfl
definition wf : well_founded (inv_image R f) :=
well_founded.intro (λ a, accessible (H (f a)))
end
end inv_image
-- The transitive closure of a well-founded relation is well-founded
namespace tc
section
parameters {A : Type} {R : A → A → Prop}
local notation `R⁺` := tc R
definition accessible {z} (ac: acc R z) : acc R⁺ z :=
begin
induction ac with x acx ih,
constructor, intro y lt,
induction lt with a b rab a b c rab rbc ih₁ ih₂,
{exact ih a rab},
{exact acc.inv (ih₂ acx ih) rab}
end
definition wf (H : well_founded R) : well_founded R⁺ :=
well_founded.intro (λ a, accessible (H a))
end
end tc
|
183c71e764a138c2c2cf2e19fbc6365b1a8ddd7b | ea7e4e10c8e02ca90fe014eb18bf5a44ce3a3502 | /src/affine_algebraic_set.lean | dfc65bbf831affb609e4f01acb3bb96b82413a18 | [
"Apache-2.0"
] | permissive | roro47/M4P33 | d4f49cc1a45954e03b42eaca559c9ab8024b1a38 | 34a93f45035add9dc909203f7957ce3822c38cdd | refs/heads/master | 1,607,563,697,384 | 1,578,860,560,000 | 1,578,860,560,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 4,029 | lean | /-
Copyright (c) 2020 Kevin Buzzard
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, and whoever else wants to join in.
-/
import for_mathlib.algebraically_closed
import data.mv_polynomial
/-!
# Affine algebraic sets
This file defines affine algebraic subsets of affine n-space and proves basic properties
about them.
## Important definitions
* `affine_algebraic_set k` -- the type of affine algebraic sets over the field `k`.
## Notation
None as yet -- do we need 𝔸ⁿ for affine n-space?
## Implementation notes
Much, but not all, of this file assumes that `k` is algebraically closed.
Remark: analysis/complex/polynomial.lean contains proof that ℂ is alg closed.
## References
Martin Orr's lecture notes!
## Tags
algebraic geometry, algebraic variety
-/
-- let k be a field
variables {k : Type*} [discrete_field k]
-- and let n be a natural number
variable {n : ℕ}
-- In Lean, the multivariable polynomial ring k[X₁, X₂, ..., Xₙ] is
-- denoted `mv_polynomial (fin n) k`. We could use better notation.
-- The set kⁿ is denoted `fin n → k` (which means maps from {0,1,2,...,(n-1)} to k).
-- We now make some definitions which we'll need in the course.
namespace mv_polynomial -- means "multivariable polynomial"
/-- The set of zeros in kⁿ of a function f ∈ k[X₁, X₂, ..., Xₙ] -/
def zeros (f : mv_polynomial (fin n) k) : set (fin n → k) :=
{x | f.eval x = 0}
/-- x is in the zeros of f iff f(x) = 0 -/
@[simp] lemma mem_zeros (f : mv_polynomial (fin n) k) (x : fin n → k) :
x ∈ f.zeros ↔ f.eval x = 0 := iff.rfl
/-- The zeros of f * g are the union of the zeros of f and of g -/
lemma zeros_mul (f g : mv_polynomial (fin n) k) :
zeros (f * g) = zeros f ∪ zeros g :=
begin
-- two sets are equal if they have the same elements
ext,
-- and now it's not hard to prove using `mem_zeros` and other
-- equalities known to Lean's simplifier.
simp,
end
open mv_polynomial
/-- An affine algebraic subset of kⁿ is the common zeros of a set of polynomials -/
structure affine_algebraic_set (k : Type*) [discrete_field k] (n : ℕ) :=
-- a subset of the set of maps {0,1,2,...,n-1} → k (called "carrier")
(carrier : set (fin n → k))
-- ...such that there's a set of polynomials such that the carrier is equal to the
-- intersection of the zeros of the polynomials in the set.
(is_algebraic : ∃ S : set (mv_polynomial (fin n) k), carrier = ⋂ f ∈ S, zeros f) -- ...such that
namespace affine_algebraic_set
-- Now some basic facts about affine algebrai subsets.
set_option trace.simplify.rewrite false
set_option trace.simplify.rewrite true
/-- Two affine algebraic subsets with the same carrier are equal! -/
lemma ext (V W : affine_algebraic_set k n) : V.carrier = W.carrier → V = W :=
begin
intro h,
cases V,
cases W,
simpa,
end
/-- We can talk about elements of affine algebraic subsets of kⁿ -/
instance : has_mem (fin n → k) (affine_algebraic_set k n) :=
⟨λ x V, x ∈ V.carrier⟩
/-- The union of two algebraic subsets of kⁿ is an algebraic subset-/
def union (V W : affine_algebraic_set k n) : affine_algebraic_set k n :=
{ carrier := V.carrier ∪ W.carrier,
is_algebraic :=
begin
-- say S is the set that defines V
cases V.is_algebraic with S hS,
-- and T is the set that defines W
cases W.is_algebraic with T hT,
use {u | ∃ (s ∈ S) (t ∈ T), u = s * t},
-- To prove that V ∪ W is defined by this set, we prove both inclusions
apply set.subset.antisymm,
{ -- say x ∈ V ∪ W,
intros x hx,
-- it's either in V or W.
cases hx with hxV hxW,
{ -- Say it's in V
rw set.mem_Inter,
-- we have to show it's a zero of all the s * t
intro u,
suffices : ∀ s ∈ S, ∀ t ∈ T, u = s * t → x ∈ zeros u,
simpa using this,
rintro s hs t ht rfl,
sorry,
},
{
sorry
}
},
{
sorry
}
end
}
end affine_algebraic_set |
2c6d581590389ac9b7e0582683febfe9898b6a45 | 82b86ba2ae0d5aed0f01f49c46db5afec0eb2bd7 | /src/Init/Prelude.lean | a6e7a6fad85bf945f209f7e55d46a9dfbd59784e | [
"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 | 61,186 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
universes u v w
@[inline] def id {α : Sort u} (a : α) : α := a
abbrev Function.comp {α : Sort u} {β : Sort v} {δ : Sort w} (f : β → δ) (g : α → β) : α → δ :=
fun x => f (g x)
abbrev Function.const {α : Sort u} (β : Sort v) (a : α) : β → α :=
fun x => a
@[reducible] def inferInstance {α : Type u} [i : α] : α := i
@[reducible] def inferInstanceAs (α : Type u) [i : α] : α := i
set_option bootstrap.inductiveCheckResultingUniverse false in
inductive PUnit : Sort u
| unit : PUnit
/-- An abbreviation for `PUnit.{0}`, its most common instantiation.
This Type should be preferred over `PUnit` where possible to avoid
unnecessary universe parameters. -/
abbrev Unit : Type := PUnit
@[matchPattern] abbrev Unit.unit : Unit := PUnit.unit
/-- Auxiliary unsafe constant used by the Compiler when erasing proofs from code. -/
unsafe axiom lcProof {α : Prop} : α
/-- Auxiliary unsafe constant used by the Compiler to mark unreachable code. -/
unsafe axiom lcUnreachable {α : Sort u} : α
inductive True : Prop
| intro : True
inductive False : Prop
inductive Empty : Type
def Not (a : Prop) : Prop := a → False
@[macroInline] def False.elim {C : Sort u} (h : False) : C :=
False.rec (fun _ => C) h
@[macroInline] def absurd {a : Prop} {b : Sort v} (h₁ : a) (h₂ : Not a) : b :=
False.elim (h₂ h₁)
inductive Eq {α : Sort u} (a : α) : α → Prop
| refl {} : Eq a a
abbrev Eq.ndrec.{u1, u2} {α : Sort u2} {a : α} {motive : α → Sort u1} (m : motive a) {b : α} (h : Eq a b) : motive b :=
Eq.rec (motive := fun α _ => motive α) m h
@[matchPattern] def rfl {α : Sort u} {a : α} : Eq a a := Eq.refl a
theorem Eq.subst {α : Sort u} {motive : α → Prop} {a b : α} (h₁ : Eq a b) (h₂ : motive a) : motive b :=
Eq.ndrec h₂ h₁
theorem Eq.symm {α : Sort u} {a b : α} (h : Eq a b) : Eq b a :=
h ▸ rfl
@[macroInline] def cast {α β : Sort u} (h : Eq α β) (a : α) : β :=
Eq.rec (motive := fun α _ => α) a h
theorem congrArg {α : Sort u} {β : Sort v} {a₁ a₂ : α} (f : α → β) (h : Eq a₁ a₂) : Eq (f a₁) (f a₂) :=
h ▸ rfl
/-
Initialize the Quotient Module, which effectively adds the following definitions:
constant Quot {α : Sort u} (r : α → α → Prop) : Sort u
constant Quot.mk {α : Sort u} (r : α → α → Prop) (a : α) : Quot r
constant Quot.lift {α : Sort u} {r : α → α → Prop} {β : Sort v} (f : α → β) :
(∀ a b : α, r a b → Eq (f a) (f b)) → Quot r → β
constant Quot.ind {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop} :
(∀ a : α, β (Quot.mk r a)) → ∀ q : Quot r, β q
-/
init_quot
inductive HEq {α : Sort u} (a : α) : {β : Sort u} → β → Prop
| refl {} : HEq a a
@[matchPattern] def HEq.rfl {α : Sort u} {a : α} : HEq a a :=
HEq.refl a
theorem eqOfHEq {α : Sort u} {a a' : α} (h : HEq a a') : Eq a a' :=
have (α β : Sort u) → (a : α) → (b : β) → HEq a b → (h : Eq α β) → Eq (cast h a) b from
fun α β a b h₁ =>
HEq.rec (motive := fun {β} (b : β) (h : HEq a b) => (h₂ : Eq α β) → Eq (cast h₂ a) b)
(fun (h₂ : Eq α α) => rfl)
h₁
this α α a a' h rfl
structure Prod (α : Type u) (β : Type v) :=
(fst : α) (snd : β)
attribute [unbox] Prod
/-- Similar to `Prod`, but `α` and `β` can be propositions.
We use this Type internally to automatically generate the brecOn recursor. -/
structure PProd (α : Sort u) (β : Sort v) :=
(fst : α) (snd : β)
/-- Similar to `Prod`, but `α` and `β` are in the same universe. -/
structure MProd (α β : Type u) :=
(fst : α) (snd : β)
structure And (a b : Prop) : Prop :=
intro :: (left : a) (right : b)
inductive Or (a b : Prop) : Prop
| inl (h : a) : Or a b
| inr (h : b) : Or a b
inductive Bool : Type
| false : Bool
| true : Bool
export Bool (false true)
/- Remark: Subtype must take a Sort instead of Type because of the axiom strongIndefiniteDescription. -/
structure Subtype {α : Sort u} (p : α → Prop) :=
(val : α) (property : p val)
/-- Gadget for optional parameter support. -/
@[reducible] def optParam (α : Sort u) (default : α) : Sort u := α
/-- Gadget for marking output parameters in type classes. -/
@[reducible] def outParam (α : Sort u) : Sort u := α
/-- Auxiliary Declaration used to implement the notation (a : α) -/
@[reducible] def typedExpr (α : Sort u) (a : α) : α := a
/-- Auxiliary Declaration used to implement the named patterns `x@p` -/
@[reducible] def namedPattern {α : Sort u} (x a : α) : α := a
/- Auxiliary axiom used to implement `sorry`. -/
axiom sorryAx (α : Sort u) (synthetic := true) : α
theorem eqFalseOfNeTrue : {b : Bool} → Not (Eq b true) → Eq b false
| true, h => False.elim (h rfl)
| false, h => rfl
theorem eqTrueOfNeFalse : {b : Bool} → Not (Eq b false) → Eq b true
| true, h => rfl
| false, h => False.elim (h rfl)
theorem neFalseOfEqTrue : {b : Bool} → Eq b true → Not (Eq b false)
| true, _ => fun h => Bool.noConfusion h
| false, h => Bool.noConfusion h
theorem neTrueOfEqFalse : {b : Bool} → Eq b false → Not (Eq b true)
| true, h => Bool.noConfusion h
| false, _ => fun h => Bool.noConfusion h
class Inhabited (α : Sort u) :=
mk {} :: (default : α)
constant arbitrary (α : Sort u) [s : Inhabited α] : α :=
@Inhabited.default α s
instance (α : Sort u) {β : Sort v} [Inhabited β] : Inhabited (α → β) := {
default := fun _ => arbitrary β
}
instance (α : Sort u) {β : α → Sort v} [(a : α) → Inhabited (β a)] : Inhabited ((a : α) → β a) := {
default := fun a => arbitrary (β a)
}
/-- Universe lifting operation from Sort to Type -/
structure PLift (α : Sort u) : Type u :=
up :: (down : α)
/- Bijection between α and PLift α -/
theorem PLift.upDown {α : Sort u} : ∀ (b : PLift α), Eq (up (down b)) b
| up a => rfl
theorem PLift.downUp {α : Sort u} (a : α) : Eq (down (up a)) a :=
rfl
/- Pointed types -/
structure PointedType :=
(type : Type u)
(val : type)
instance : Inhabited PointedType.{u} := {
default := { type := PUnit.{u+1}, val := ⟨⟩ }
}
/-- Universe lifting operation -/
structure ULift.{r, s} (α : Type s) : Type (max s r) :=
up :: (down : α)
/- Bijection between α and ULift.{v} α -/
theorem ULift.upDown {α : Type u} : ∀ (b : ULift.{v} α), Eq (up (down b)) b
| up a => rfl
theorem ULift.downUp {α : Type u} (a : α) : Eq (down (up.{v} a)) a :=
rfl
class inductive Decidable (p : Prop)
| isFalse (h : Not p) : Decidable p
| isTrue (h : p) : Decidable p
@[inlineIfReduce, nospecialize] def Decidable.decide (p : Prop) [h : Decidable p] : Bool :=
Decidable.casesOn (motive := fun _ => Bool) h (fun _ => false) (fun _ => true)
export Decidable (isTrue isFalse decide)
abbrev DecidablePred {α : Sort u} (r : α → Prop) :=
(a : α) → Decidable (r a)
abbrev DecidableRel {α : Sort u} (r : α → α → Prop) :=
(a b : α) → Decidable (r a b)
abbrev DecidableEq (α : Sort u) :=
(a b : α) → Decidable (Eq a b)
def decEq {α : Sort u} [s : DecidableEq α] (a b : α) : Decidable (Eq a b) :=
s a b
theorem decideEqTrue : {p : Prop} → [s : Decidable p] → p → Eq (decide p) true
| _, isTrue _, _ => rfl
| _, isFalse h₁, h₂ => absurd h₂ h₁
theorem decideEqFalse : {p : Prop} → [s : Decidable p] → Not p → Eq (decide p) false
| _, isTrue h₁, h₂ => absurd h₁ h₂
| _, isFalse h, _ => rfl
theorem ofDecideEqTrue {p : Prop} [s : Decidable p] : Eq (decide p) true → p := fun h =>
match s with
| isTrue h₁ => h₁
| isFalse h₁ => absurd h (neTrueOfEqFalse (decideEqFalse h₁))
theorem ofDecideEqFalse {p : Prop} [s : Decidable p] : Eq (decide p) false → Not p := fun h =>
match s with
| isTrue h₁ => absurd h (neFalseOfEqTrue (decideEqTrue h₁))
| isFalse h₁ => h₁
@[inline] instance : DecidableEq Bool :=
fun a b => match a, b with
| false, false => isTrue rfl
| false, true => isFalse (fun h => Bool.noConfusion h)
| true, false => isFalse (fun h => Bool.noConfusion h)
| true, true => isTrue rfl
class BEq (α : Type u) := (beq : α → α → Bool)
open BEq (beq)
instance {α : Type u} [DecidableEq α] : BEq α :=
⟨fun a b => decide (Eq a b)⟩
-- We use "dependent" if-then-else to be able to communicate the if-then-else condition
-- to the branches
@[macroInline] def dite {α : Sort u} (c : Prop) [h : Decidable c] (t : c → α) (e : Not c → α) : α :=
Decidable.casesOn (motive := fun _ => α) h e t
/- if-then-else -/
@[macroInline] def ite {α : Sort u} (c : Prop) [h : Decidable c] (t e : α) : α :=
Decidable.casesOn (motive := fun _ => α) h (fun _ => e) (fun _ => t)
@[macroInline] instance {p q} [dp : Decidable p] [dq : Decidable q] : Decidable (And p q) :=
match dp with
| isTrue hp =>
match dq with
| isTrue hq => isTrue ⟨hp, hq⟩
| isFalse hq => isFalse (fun h => hq (And.right h))
| isFalse hp =>
isFalse (fun h => hp (And.left h))
@[macroInline] instance {p q} [dp : Decidable p] [dq : Decidable q] : Decidable (Or p q) :=
match dp with
| isTrue hp => isTrue (Or.inl hp)
| isFalse hp =>
match dq with
| isTrue hq => isTrue (Or.inr hq)
| isFalse hq =>
isFalse fun h => match h with
| Or.inl h => hp h
| Or.inr h => hq h
instance {p} [dp : Decidable p] : Decidable (Not p) :=
match dp with
| isTrue hp => isFalse (absurd hp)
| isFalse hp => isTrue hp
/- Boolean operators -/
@[macroInline] def cond {a : Type u} : Bool → a → a → a
| true, x, y => x
| false, x, y => y
@[macroInline] def or : Bool → Bool → Bool
| true, _ => true
| false, b => b
@[macroInline] def and : Bool → Bool → Bool
| false, _ => false
| true, b => b
@[macroInline] def not : Bool → Bool
| true => false
| false => true
inductive Nat
| zero : Nat
| succ (n : Nat) : Nat
/- For numeric literals notation -/
class OfNat (α : Type u) :=
(ofNat : Nat → α)
export OfNat (ofNat)
instance : OfNat Nat := ⟨id⟩
instance : Inhabited Nat := {
default := 0
}
class HasLessEq (α : Type u) := (LessEq : α → α → Prop)
class HasLess (α : Type u) := (Less : α → α → Prop)
export HasLess (Less)
export HasLessEq (LessEq)
class Add (α : Type u) := (add : α → α → α)
class Mul (α : Type u) := (mul : α → α → α)
class Neg (α : Type u) := (neg : α → α)
class Sub (α : Type u) := (sub : α → α → α)
class Div (α : Type u) := (div : α → α → α)
class Mod (α : Type u) := (mod : α → α → α)
class ModN (α : Type u) := (modn : α → Nat → α)
class Pow (α : Type u) (β : Type v) := (pow : α → β → α)
class Append (α : Type u) := (append : α → α → α)
class OrElse (α : Type u) := (orElse : α → α → α)
class AndThen (α : Type u) := (andThen : α → α → α)
open Add (add)
open Mul (mul)
open Pow (pow)
open Append (append)
@[reducible] def GreaterEq {α : Type u} [HasLessEq α] (a b : α) : Prop := LessEq b a
@[reducible] def Greater {α : Type u} [HasLess α] (a b : α) : Prop := Less b a
set_option bootstrap.gen_matcher_code false in
@[extern "lean_nat_add"]
protected def Nat.add : (@& Nat) → (@& Nat) → Nat
| a, Nat.zero => a
| a, Nat.succ b => Nat.succ (Nat.add a b)
instance : Add Nat := {
add := Nat.add
}
/- We mark the following definitions as pattern to make sure they can be used in recursive equations,
and reduced by the equation Compiler. -/
attribute [matchPattern] Nat.add Add.add Neg.neg
set_option bootstrap.gen_matcher_code false in
@[extern "lean_nat_mul"]
protected def Nat.mul : (@& Nat) → (@& Nat) → Nat
| a, 0 => 0
| a, Nat.succ b => Nat.add (Nat.mul a b) a
instance : Mul Nat := {
mul := Nat.mul
}
set_option bootstrap.gen_matcher_code false in
@[extern "lean_nat_pow"]
protected def Nat.pow (m : @& Nat) : (@& Nat) → Nat
| 0 => 1
| succ n => Nat.mul (Nat.pow m n) m
instance : Pow Nat Nat := {
pow := Nat.pow
}
set_option bootstrap.gen_matcher_code false in
@[extern "lean_nat_dec_eq"]
def Nat.beq : Nat → Nat → Bool
| zero, zero => true
| zero, succ m => false
| succ n, zero => false
| succ n, succ m => beq n m
theorem Nat.eqOfBeqEqTrue : {n m : Nat} → Eq (beq n m) true → Eq n m
| zero, zero, h => rfl
| zero, succ m, h => Bool.noConfusion h
| succ n, zero, h => Bool.noConfusion h
| succ n, succ m, h =>
have Eq (beq n m) true from h
have Eq n m from eqOfBeqEqTrue this
this ▸ rfl
theorem Nat.neOfBeqEqFalse : {n m : Nat} → Eq (beq n m) false → Not (Eq n m)
| zero, zero, h₁, h₂ => Bool.noConfusion h₁
| zero, succ m, h₁, h₂ => Nat.noConfusion h₂
| succ n, zero, h₁, h₂ => Nat.noConfusion h₂
| succ n, succ m, h₁, h₂ =>
have Eq (beq n m) false from h₁
Nat.noConfusion h₂ (fun h₂ => absurd h₂ (neOfBeqEqFalse this))
@[extern "lean_nat_dec_eq"]
protected def Nat.decEq (n m : @& Nat) : Decidable (Eq n m) :=
match h:beq n m with
| true => isTrue (eqOfBeqEqTrue h)
| false => isFalse (neOfBeqEqFalse h)
@[inline] instance : DecidableEq Nat := Nat.decEq
set_option bootstrap.gen_matcher_code false in
@[extern "lean_nat_dec_le"]
def Nat.ble : Nat → Nat → Bool
| zero, zero => true
| zero, succ m => true
| succ n, zero => false
| succ n, succ m => ble n m
protected def Nat.le (n m : Nat) : Prop :=
Eq (ble n m) true
instance : HasLessEq Nat := ⟨Nat.le⟩
protected def Nat.lt (n m : Nat) : Prop :=
Nat.le (succ n) m
instance : HasLess Nat := ⟨Nat.lt⟩
theorem Nat.notSuccLeZero : ∀ (n : Nat), LessEq (succ n) 0 → False
| 0, h => nomatch h
| succ n, h => nomatch h
theorem Nat.notLtZero (n : Nat) : Not (Less n 0) :=
notSuccLeZero n
@[extern "lean_nat_dec_le"]
instance Nat.decLe (n m : @& Nat) : Decidable (LessEq n m) :=
decEq (Nat.ble n m) true
@[extern "lean_nat_dec_lt"]
instance Nat.decLt (n m : @& Nat) : Decidable (Less n m) :=
decLe (succ n) m
theorem Nat.zeroLe : (n : Nat) → LessEq 0 n
| zero => rfl
| succ n => rfl
theorem Nat.succLeSucc {n m : Nat} (h : LessEq n m) : LessEq (succ n) (succ m) :=
h
theorem Nat.zeroLtSucc (n : Nat) : Less 0 (succ n) :=
succLeSucc (zeroLe n)
theorem Nat.leStep : {n m : Nat} → LessEq n m → LessEq n (succ m)
| zero, zero, h => rfl
| zero, succ n, h => rfl
| succ n, zero, h => Bool.noConfusion h
| succ n, succ m, h =>
have LessEq n m from h
have LessEq n (succ m) from leStep this
succLeSucc this
protected theorem Nat.leTrans : {n m k : Nat} → LessEq n m → LessEq m k → LessEq n k
| zero, m, k, h₁, h₂ => zeroLe _
| succ n, zero, k, h₁, h₂ => Bool.noConfusion h₁
| succ n, succ m, zero, h₁, h₂ => Bool.noConfusion h₂
| succ n, succ m, succ k, h₁, h₂ =>
have h₁' : LessEq n m from h₁
have h₂' : LessEq m k from h₂
Nat.leTrans h₁' h₂'
protected theorem Nat.ltTrans {n m k : Nat} (h₁ : Less n m) : Less m k → Less n k :=
Nat.leTrans (leStep h₁)
theorem Nat.leSucc : (n : Nat) → LessEq n (succ n)
| zero => rfl
| succ n => leSucc n
theorem Nat.leSuccOfLe {n m : Nat} (h : LessEq n m) : LessEq n (succ m) :=
Nat.leTrans h (leSucc m)
protected theorem Nat.eqOrLtOfLe : {n m: Nat} → LessEq n m → Or (Eq n m) (Less n m)
| zero, zero, h => Or.inl rfl
| zero, succ n, h => Or.inr (zeroLe n)
| succ n, zero, h => Bool.noConfusion h
| succ n, succ m, h =>
have LessEq n m from h
match Nat.eqOrLtOfLe this with
| Or.inl h => Or.inl (h ▸ rfl)
| Or.inr h => Or.inr (succLeSucc h)
protected def Nat.leRefl : (n : Nat) → LessEq n n
| zero => rfl
| succ n => Nat.leRefl n
protected theorem Nat.ltOrGe (n m : Nat) : Or (Less n m) (GreaterEq n m) :=
match m with
| zero => Or.inr (zeroLe n)
| succ m =>
match Nat.ltOrGe n m with
| Or.inl h => Or.inl (leSuccOfLe h)
| Or.inr h =>
match Nat.eqOrLtOfLe h with
| Or.inl h1 => Or.inl (h1 ▸ Nat.leRefl _)
| Or.inr h1 => Or.inr h1
protected theorem Nat.leAntisymm : {n m : Nat} → LessEq n m → LessEq m n → Eq n m
| zero, zero, h₁, h₂ => rfl
| succ n, zero, h₁, h₂ => Bool.noConfusion h₁
| zero, succ m, h₁, h₂ => Bool.noConfusion h₂
| succ n, succ m, h₁, h₂ =>
have h₁' : LessEq n m from h₁
have h₂' : LessEq m n from h₂
(Nat.leAntisymm h₁' h₂') ▸ rfl
protected theorem Nat.ltOfLeOfNe {n m : Nat} (h₁ : LessEq n m) (h₂ : Not (Eq n m)) : Less n m :=
match Nat.ltOrGe n m with
| Or.inl h₃ => h₃
| Or.inr h₃ => absurd (Nat.leAntisymm h₁ h₃) h₂
set_option bootstrap.gen_matcher_code false in
@[extern c inline "lean_nat_sub(#1, lean_box(1))"]
def Nat.pred : Nat → Nat
| 0 => 0
| succ a => a
theorem Nat.predLePred : {n m : Nat} → LessEq n m → LessEq (pred n) (pred m)
| zero, zero, h => rfl
| zero, succ n, h => zeroLe n
| succ n, zero, h => Bool.noConfusion h
| succ n, succ m, h => h
theorem Nat.leOfSuccLeSucc {n m : Nat} : LessEq (succ n) (succ m) → LessEq n m :=
predLePred
theorem Nat.leOfLtSucc {m n : Nat} : Less m (succ n) → LessEq m n :=
leOfSuccLeSucc
@[extern "lean_system_platform_nbits"] constant System.Platform.getNumBits : Unit → { n : Nat // Or (Eq n 32) (Eq n 64) } :=
fun _ => ⟨64, Or.inr rfl⟩ -- inhabitant
def System.Platform.numBits : Nat :=
(getNumBits ()).val
theorem System.Platform.numBitsEq : Or (Eq numBits 32) (Eq numBits 64) :=
(getNumBits ()).property
structure Fin (n : Nat) :=
(val : Nat)
(isLt : Less val n)
theorem Fin.eqOfVeq {n} : ∀ {i j : Fin n}, Eq i.val j.val → Eq i j
| ⟨v, h⟩, ⟨_, _⟩, rfl => rfl
theorem Fin.veqOfEq {n} {i j : Fin n} (h : Eq i j) : Eq i.val j.val :=
h ▸ rfl
theorem Fin.neOfVne {n} {i j : Fin n} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
fun h' => absurd (veqOfEq h') h
instance (n : Nat) : DecidableEq (Fin n) :=
fun i j =>
match decEq i.val j.val with
| isTrue h => isTrue (Fin.eqOfVeq h)
| isFalse h => isFalse (Fin.neOfVne h)
protected def Fin.lt {n} (a b : Fin n) : Prop := Less a.val b.val
protected def Fin.le {n} (a b : Fin n) : Prop := LessEq a.val b.val
instance {n} : HasLess (Fin n) := ⟨Fin.lt⟩
instance {n} : HasLessEq (Fin n) := ⟨Fin.le⟩
instance Fin.decLt {n} (a b : Fin n) : Decidable (Less a b) := Nat.decLt ..
instance Fin.decLe {n} (a b : Fin n) : Decidable (LessEq a b) := Nat.decLe ..
def uint8Sz : Nat := 256
structure UInt8 :=
(val : Fin uint8Sz)
set_option bootstrap.gen_matcher_code false in
@[extern c inline "#1 == #2"]
def UInt8.decEq (a b : UInt8) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt8.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt8 := UInt8.decEq
instance : Inhabited UInt8 := {
default := { val := { val := 0, isLt := decide! } }
}
def uint16Sz : Nat := 65536
structure UInt16 :=
(val : Fin uint16Sz)
set_option bootstrap.gen_matcher_code false in
@[extern c inline "#1 == #2"]
def UInt16.decEq (a b : UInt16) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt16.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt16 := UInt16.decEq
instance : Inhabited UInt16 := {
default := { val := { val := 0, isLt := decide! } }
}
def uint32Sz : Nat := 4294967296
structure UInt32 :=
(val : Fin uint32Sz)
@[extern "lean_uint32_to_nat"]
def UInt32.toNat (n : UInt32) : Nat := n.val.val
set_option bootstrap.gen_matcher_code false in
@[extern c inline "#1 == #2"]
def UInt32.decEq (a b : UInt32) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt32.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt32 := UInt32.decEq
instance : Inhabited UInt32 := {
default := { val := { val := 0, isLt := decide! } }
}
def UInt32.lt (a b : UInt32) : Prop := Less a.val b.val
def UInt32.le (a b : UInt32) : Prop := LessEq a.val b.val
instance : HasLess UInt32 := ⟨UInt32.lt⟩
instance : HasLessEq UInt32 := ⟨UInt32.le⟩
set_option bootstrap.gen_matcher_code false in
@[extern c inline "#1 < #2"]
def UInt32.decLt (a b : UInt32) : Decidable (Less a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ => inferInstanceAs (Decidable (Less n m))
set_option bootstrap.gen_matcher_code false in
@[extern c inline "#1 <= #2"]
def UInt32.decLe (a b : UInt32) : Decidable (LessEq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ => inferInstanceAs (Decidable (LessEq n m))
instance (a b : UInt32) : Decidable (Less a b) := UInt32.decLt a b
instance (a b : UInt32) : Decidable (LessEq a b) := UInt32.decLe a b
@[extern "lean_uint32_of_nat"]
def UInt32.ofNatCore (n : @& Nat) (h : Less n uint32Sz) : UInt32 := {
val := { val := n, isLt := h }
}
def uint64Sz : Nat := 18446744073709551616
structure UInt64 :=
(val : Fin uint64Sz)
set_option bootstrap.gen_matcher_code false in
@[extern c inline "#1 == #2"]
def UInt64.decEq (a b : UInt64) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt64.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt64 := UInt64.decEq
instance : Inhabited UInt64 := {
default := { val := { val := 0, isLt := decide! } }
}
def usizeSz : Nat := pow 2 System.Platform.numBits
theorem usizeSzEq : Or (Eq usizeSz 4294967296) (Eq usizeSz 18446744073709551616) :=
show Or (Eq (pow 2 System.Platform.numBits) 4294967296) (Eq (pow 2 System.Platform.numBits) 18446744073709551616) from
match System.Platform.numBits, System.Platform.numBitsEq with
| _, Or.inl rfl => Or.inl (decide! : (Eq (pow 2 32) (4294967296:Nat)))
| _, Or.inr rfl => Or.inr (decide! : (Eq (pow 2 64) (18446744073709551616:Nat)))
structure USize :=
(val : Fin usizeSz)
set_option bootstrap.gen_matcher_code false in
@[extern c inline "#1 == #2"]
def USize.decEq (a b : USize) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h =>isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => USize.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq USize := USize.decEq
instance : Inhabited USize := {
default := { val := { val := 0, isLt := match usizeSz, usizeSzEq with | _, Or.inl rfl => decide! | _, Or.inr rfl => decide! } }
}
@[extern "lean_usize_of_nat"]
def USize.ofNat32 (n : @& Nat) (h : Less n 4294967296) : USize := {
val := {
val := n,
isLt := match usizeSz, usizeSzEq with
| _, Or.inl rfl => h
| _, Or.inr rfl => Nat.ltTrans h (decide! : Less 4294967296 18446744073709551616)
}
}
@[extern "lean_usize_of_nat"]
def USize.ofNatCore (n : @& Nat) (h : Less n usizeSz) : USize := {
val := { val := n, isLt := h }
}
abbrev Nat.isValidChar (n : Nat) : Prop :=
Or (Less n 0xd800) (And (Less 0xdfff n) (Less n 0x110000))
abbrev UInt32.isValidChar (n : UInt32) : Prop :=
n.toNat.isValidChar
/-- The `Char` Type represents an unicode scalar value.
See http://www.unicode.org/glossary/#unicode_scalar_value). -/
structure Char :=
(val : UInt32)
(valid : val.isValidChar)
private theorem validCharIsUInt32 {n : Nat} (h : n.isValidChar) : Less n uint32Sz :=
match h with
| Or.inl h => Nat.ltTrans h (decide! : Less 55296 uint32Sz)
| Or.inr ⟨_, h⟩ => Nat.ltTrans h (decide! : Less 1114112 uint32Sz)
@[extern "lean_uint32_of_nat"]
private def Char.ofNatAux (n : Nat) (h : n.isValidChar) : Char :=
{ val := ⟨{ val := n, isLt := validCharIsUInt32 h }⟩, valid := h }
@[noinline, matchPattern]
def Char.ofNat (n : Nat) : Char :=
dite (n.isValidChar)
(fun h => Char.ofNatAux n h)
(fun _ => { val := ⟨{ val := 0, isLt := decide! }⟩, valid := Or.inl decide! })
theorem Char.eqOfVeq : ∀ {c d : Char}, Eq c.val d.val → Eq c d
| ⟨v, h⟩, ⟨_, _⟩, rfl => rfl
theorem Char.veqOfEq : ∀ {c d : Char}, Eq c d → Eq c.val d.val
| _, _, rfl => rfl
theorem Char.neOfVne {c d : Char} (h : Not (Eq c.val d.val)) : Not (Eq c d) :=
fun h' => absurd (veqOfEq h') h
theorem Char.vneOfNe {c d : Char} (h : Not (Eq c d)) : Not (Eq c.val d.val) :=
fun h' => absurd (eqOfVeq h') h
instance : DecidableEq Char :=
fun c d =>
match decEq c.val d.val with
| isTrue h => isTrue (Char.eqOfVeq h)
| isFalse h => isFalse (Char.neOfVne h)
def Char.utf8Size (c : Char) : UInt32 :=
let v := c.val
ite (LessEq v (UInt32.ofNatCore 0x7F decide!))
(UInt32.ofNatCore 1 decide!)
(ite (LessEq v (UInt32.ofNatCore 0x7FF decide!))
(UInt32.ofNatCore 2 decide!)
(ite (LessEq v (UInt32.ofNatCore 0xFFFF decide!))
(UInt32.ofNatCore 3 decide!)
(UInt32.ofNatCore 4 decide!)))
inductive Option (α : Type u)
| none : Option α
| some (val : α) : Option α
attribute [unbox] Option
export Option (none some)
instance {α} : Inhabited (Option α) := {
default := none
}
inductive List (α : Type u)
| nil : List α
| cons (head : α) (tail : List α) : List α
instance {α} : Inhabited (List α) := {
default := List.nil
}
protected def List.hasDecEq {α: Type u} [DecidableEq α] : (a b : List α) → Decidable (Eq a b)
| nil, nil => isTrue rfl
| cons a as, nil => isFalse (fun h => List.noConfusion h)
| nil, cons b bs => isFalse (fun h => List.noConfusion h)
| cons a as, cons b bs =>
match decEq a b with
| isTrue hab =>
match List.hasDecEq as bs with
| isTrue habs => isTrue (hab ▸ habs ▸ rfl)
| isFalse nabs => isFalse (fun h => List.noConfusion h (fun _ habs => absurd habs nabs))
| isFalse nab => isFalse (fun h => List.noConfusion h (fun hab _ => absurd hab nab))
instance {α : Type u} [DecidableEq α] : DecidableEq (List α) := List.hasDecEq
@[specialize]
def List.foldl {α β} (f : α → β → α) : (init : α) → List β → α
| a, nil => a
| a, cons b l => foldl f (f a b) l
structure String :=
(data : List Char)
attribute [extern "lean_string_mk"] String.mk
attribute [extern "lean_string_data"] String.data
@[extern "lean_string_dec_eq"]
def String.decEq (s₁ s₂ : @& String) : Decidable (Eq s₁ s₂) :=
match s₁, s₂ with
| ⟨s₁⟩, ⟨s₂⟩ =>
dite (Eq s₁ s₂) (fun h => isTrue (congrArg _ h)) (fun h => isFalse (fun h' => String.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq String := String.decEq
/-- A byte position in a `String`. Internally, `String`s are UTF-8 encoded.
Codepoint positions (counting the Unicode codepoints rather than bytes)
are represented by plain `Nat`s instead.
Indexing a `String` by a byte position is constant-time, while codepoint
positions need to be translated internally to byte positions in linear-time. -/
abbrev String.Pos := Nat
structure Substring :=
(str : String)
(startPos : String.Pos)
(stopPos : String.Pos)
def String.csize (c : Char) : Nat :=
c.utf8Size.toNat
private def String.utf8ByteSizeAux : List Char → Nat → Nat
| List.nil, r => r
| List.cons c cs, r => utf8ByteSizeAux cs (add r (csize c))
@[extern "lean_string_utf8_byte_size"]
def String.utf8ByteSize : (@& String) → Nat
| ⟨s⟩ => utf8ByteSizeAux s 0
@[inline] def String.bsize (s : String) : Nat :=
utf8ByteSize s
@[inline] def String.toSubstring (s : String) : Substring := {
str := s,
startPos := 0,
stopPos := s.bsize
}
@[extern c inline "#3"]
unsafe def unsafeCast {α : Type u} {β : Type v} (a : α) : β :=
cast lcProof (PUnit.{v})
@[neverExtract, extern "lean_panic_fn"]
constant panic {α : Type u} [Inhabited α] (msg : String) : α
/-
The Compiler has special support for arrays.
They are implemented using dynamic arrays: https://en.wikipedia.org/wiki/Dynamic_array
-/
structure Array (α : Type u) :=
(sz : Nat)
(data : Fin sz → α)
attribute [extern "lean_array_mk"] Array.mk
attribute [extern "lean_array_data"] Array.data
attribute [extern "lean_array_sz"] Array.sz
/- The parameter `c` is the initial capacity -/
@[extern "lean_mk_empty_array_with_capacity"]
def Array.mkEmpty {α : Type u} (c : @& Nat) : Array α := {
sz := 0,
data := fun ⟨x, h⟩ => absurd h (Nat.notLtZero x)
}
def Array.empty {α : Type u} : Array α :=
mkEmpty 0
@[reducible, extern "lean_array_get_size"]
def Array.size {α : Type u} (a : @& Array α) : Nat :=
a.sz
@[extern "lean_array_fget"]
def Array.get {α : Type u} (a : @& Array α) (i : @& Fin a.size) : α :=
a.data i
/- "Comfortable" version of `fget`. It performs a bound check at runtime. -/
@[extern "lean_array_get"]
def Array.get! {α : Type u} [Inhabited α] (a : @& Array α) (i : @& Nat) : α :=
dite (Less i a.size) (fun h => a.get ⟨i, h⟩) (fun _ => arbitrary α)
@[extern "lean_array_push"]
def Array.push {α : Type u} (a : Array α) (v : α) : Array α := {
sz := Nat.succ a.sz,
data := fun ⟨j, h₁⟩ => dite (Eq j a.sz) (fun _ => v) (fun h₂ => a.data ⟨j, Nat.ltOfLeOfNe (Nat.leOfLtSucc h₁) h₂⟩)
}
class Bind (m : Type u → Type v) :=
(bind : {α β : Type u} → m α → (α → m β) → m β)
export Bind (bind)
class Pure (f : Type u → Type v) :=
(pure {α : Type u} : α → f α)
export Pure (pure)
class Functor (f : Type u → Type v) : Type (max (u+1) v) :=
(map : {α β : Type u} → (α → β) → f α → f β)
(mapConst : {α β : Type u} → α → f β → f α := Function.comp map (Function.const _))
class Seq (f : Type u → Type v) : Type (max (u+1) v) :=
(seq : {α β : Type u} → f (α → β) → f α → f β)
class SeqLeft (f : Type u → Type v) : Type (max (u+1) v) :=
(seqLeft : {α : Type u} → f α → f PUnit → f α)
class SeqRight (f : Type u → Type v) : Type (max (u+1) v) :=
(seqRight : {β : Type u} → f PUnit → f β → f β)
class Applicative (f : Type u → Type v) extends Functor f, Pure f, Seq f, SeqLeft f, SeqRight f :=
(map := fun x y => Seq.seq (pure x) y)
(seqLeft := fun a b => Seq.seq (Functor.map (Function.const _) a) b)
(seqRight := fun a b => Seq.seq (Functor.map (Function.const _ id) a) b)
class Monad (m : Type u → Type v) extends Applicative m, Bind m : Type (max (u+1) v) :=
(map := fun f x => bind x (Function.comp pure f))
(seq := fun f x => bind f (fun y => Functor.map y x))
(seqLeft := fun x y => bind x (fun a => bind y (fun _ => pure a)))
(seqRight := fun x y => bind x (fun _ => y))
instance {α : Type u} {m : Type u → Type v} [Monad m] : Inhabited (α → m α) := ⟨pure⟩
instance {α : Type u} {m : Type u → Type v} [Monad m] [Inhabited α] : Inhabited (m α) := ⟨pure $ arbitrary _⟩
/-- A Function for lifting a computation from an inner Monad to an outer Monad.
Like [MonadTrans](https://hackage.haskell.org/package/transformers-0.5.5.0/docs/Control-Monad-Trans-Class.html),
but `n` does not have to be a monad transformer.
Alternatively, an implementation of [MonadLayer](https://hackage.haskell.org/package/layers-0.1/docs/Control-Monad-Layer.html#t:MonadLayer) without `layerInvmap` (so far). -/
class MonadLift (m : Type u → Type v) (n : Type u → Type w) :=
(monadLift : {α : Type u} → m α → n α)
/-- The reflexive-transitive closure of `MonadLift`.
`monadLift` is used to transitively lift monadic computations such as `StateT.get` or `StateT.put s`.
Corresponds to [MonadLift](https://hackage.haskell.org/package/layers-0.1/docs/Control-Monad-Layer.html#t:MonadLift). -/
class MonadLiftT (m : Type u → Type v) (n : Type u → Type w) :=
(monadLift : {α : Type u} → m α → n α)
export MonadLiftT (monadLift)
abbrev liftM := @monadLift
instance (m n o) [MonadLiftT m n] [MonadLift n o] : MonadLiftT m o := {
monadLift := fun x => MonadLift.monadLift (m := n) (monadLift x)
}
instance (m) : MonadLiftT m m := {
monadLift := fun x => x
}
/-- A functor in the category of monads. Can be used to lift monad-transforming functions.
Based on pipes' [MFunctor](https://hackage.haskell.org/package/pipes-2.4.0/docs/Control-MFunctor.html),
but not restricted to monad transformers.
Alternatively, an implementation of [MonadTransFunctor](http://duairc.netsoc.ie/layers-docs/Control-Monad-Layer.html#t:MonadTransFunctor). -/
class MonadFunctor (m : Type u → Type v) (n : Type u → Type w) :=
(monadMap {α : Type u} : (∀ {β}, m β → m β) → n α → n α)
/-- The reflexive-transitive closure of `MonadFunctor`.
`monadMap` is used to transitively lift Monad morphisms -/
class MonadFunctorT (m : Type u → Type v) (n : Type u → Type w) :=
(monadMap {α : Type u} : (∀ {β}, m β → m β) → n α → n α)
export MonadFunctorT (monadMap)
instance (m n o) [MonadFunctorT m n] [MonadFunctor n o] : MonadFunctorT m o := {
monadMap := fun f => MonadFunctor.monadMap (m := n) (monadMap (m := m) f)
}
instance monadFunctorRefl (m) : MonadFunctorT m m := {
monadMap := fun f => f
}
inductive Except (ε : Type u) (α : Type v)
| error : ε → Except ε α
| ok : α → Except ε α
attribute [unbox] Except
instance {ε : Type u} {α : Type v} [Inhabited ε] : Inhabited (Except ε α) :=
⟨Except.error (arbitrary ε)⟩
/-- An implementation of [MonadError](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Except.html#t:MonadError) -/
class MonadExceptOf (ε : Type u) (m : Type v → Type w) :=
(throw {α : Type v} : ε → m α)
(tryCatch {α : Type v} : m α → (ε → m α) → m α)
abbrev throwThe (ε : Type u) {m : Type v → Type w} [MonadExceptOf ε m] {α : Type v} (e : ε) : m α :=
MonadExceptOf.throw e
abbrev tryCatchThe (ε : Type u) {m : Type v → Type w} [MonadExceptOf ε m] {α : Type v} (x : m α) (handle : ε → m α) : m α :=
MonadExceptOf.tryCatch x handle
/-- Similar to `MonadExceptOf`, but `ε` is an outParam for convenience -/
class MonadExcept (ε : outParam (Type u)) (m : Type v → Type w) :=
(throw {α : Type v} : ε → m α)
(tryCatch {α : Type v} : m α → (ε → m α) → m α)
export MonadExcept (throw tryCatch)
instance (ε : outParam (Type u)) (m : Type v → Type w) [MonadExceptOf ε m] : MonadExcept ε m := {
throw := throwThe ε,
tryCatch := tryCatchThe ε
}
namespace MonadExcept
variables {ε : Type u} {m : Type v → Type w}
@[inline] protected def orelse [MonadExcept ε m] {α : Type v} (t₁ t₂ : m α) : m α :=
tryCatch t₁ fun _ => t₂
instance [MonadExcept ε m] {α : Type v} : OrElse (m α) := ⟨MonadExcept.orelse⟩
end MonadExcept
/-- An implementation of [ReaderT](https://hackage.haskell.org/package/transformers-0.5.5.0/docs/Control-Monad-Trans-Reader.html#t:ReaderT) -/
def ReaderT (ρ : Type u) (m : Type u → Type v) (α : Type u) : Type (max u v) :=
ρ → m α
instance (ρ : Type u) (m : Type u → Type v) (α : Type u) [Inhabited (m α)] : Inhabited (ReaderT ρ m α) :=
⟨fun _ => arbitrary _⟩
@[inline] def ReaderT.run {ρ : Type u} {m : Type u → Type v} {α : Type u} (x : ReaderT ρ m α) (r : ρ) : m α :=
x r
@[reducible] def Reader (ρ : Type u) := ReaderT ρ id
namespace ReaderT
section
variables {ρ : Type u} {m : Type u → Type v} {α : Type u}
@[inline] protected def lift (a : m α) : ReaderT ρ m α :=
fun r => a
instance : MonadLift m (ReaderT ρ m) := ⟨ReaderT.lift⟩
instance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ReaderT ρ m) := {
throw := Function.comp ReaderT.lift (throwThe ε),
tryCatch := fun x c r => tryCatchThe ε (x r) (fun e => (c e) r)
}
end
section
variables {ρ : Type u} {m : Type u → Type v} [Monad m] {α β : Type u}
@[inline] protected def read : ReaderT ρ m ρ :=
pure
@[inline] protected def pure (a : α) : ReaderT ρ m α :=
fun r => pure a
@[inline] protected def bind (x : ReaderT ρ m α) (f : α → ReaderT ρ m β) : ReaderT ρ m β :=
fun r => bind (x r) fun a => f a r
@[inline] protected def map (f : α → β) (x : ReaderT ρ m α) : ReaderT ρ m β :=
fun r => Functor.map f (x r)
instance : Monad (ReaderT ρ m) := {
pure := ReaderT.pure,
bind := ReaderT.bind,
map := ReaderT.map
}
instance (ρ m) [Monad m] : MonadFunctor m (ReaderT ρ m) := ⟨fun f x r => f (x r)⟩
@[inline] protected def adapt {ρ' : Type u} [Monad m] {α : Type u} (f : ρ' → ρ) : ReaderT ρ m α → ReaderT ρ' m α :=
fun x r => x (f r)
end
end ReaderT
/-- An implementation of [MonadReader](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Reader-Class.html#t:MonadReader).
It does not contain `local` because this Function cannot be lifted using `monadLift`.
Instead, the `MonadReaderAdapter` class provides the more general `adaptReader` Function.
Note: This class can be seen as a simplification of the more "principled" definition
```
class MonadReader (ρ : outParam (Type u)) (n : Type u → Type u) :=
(lift {α : Type u} : (∀ {m : Type u → Type u} [Monad m], ReaderT ρ m α) → n α)
```
-/
class MonadReaderOf (ρ : Type u) (m : Type u → Type v) :=
(read : m ρ)
@[inline] def readThe (ρ : Type u) {m : Type u → Type v} [MonadReaderOf ρ m] : m ρ :=
MonadReaderOf.read
/-- Similar to `MonadReaderOf`, but `ρ` is an outParam for convenience -/
class MonadReader (ρ : outParam (Type u)) (m : Type u → Type v) :=
(read : m ρ)
export MonadReader (read)
instance (ρ : Type u) (m : Type u → Type v) [MonadReaderOf ρ m] : MonadReader ρ m :=
⟨readThe ρ⟩
instance {ρ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadReaderOf ρ m] [MonadLift m n] : MonadReaderOf ρ n :=
⟨monadLift (MonadReader.read : m ρ)⟩
instance {ρ : Type u} {m : Type u → Type v} [Monad m] : MonadReaderOf ρ (ReaderT ρ m) :=
⟨ReaderT.read⟩
class MonadWithReaderOf (ρ : Type u) (m : Type u → Type v) :=
(withReader {α : Type u} : (ρ → ρ) → m α → m α)
@[inline] def withTheReader (ρ : Type u) {m : Type u → Type v} [MonadWithReaderOf ρ m] {α : Type u} (f : ρ → ρ) (x : m α) : m α :=
MonadWithReaderOf.withReader f x
class MonadWithReader (ρ : outParam (Type u)) (m : Type u → Type v) :=
(withReader {α : Type u} : (ρ → ρ) → m α → m α)
export MonadWithReader (withReader)
instance (ρ : Type u) (m : Type u → Type v) [MonadWithReaderOf ρ m] : MonadWithReader ρ m := ⟨withTheReader ρ⟩
instance {ρ : Type u} {m : Type u → Type v} {n : Type u → Type v} [MonadWithReaderOf ρ m] [MonadFunctor m n] : MonadWithReaderOf ρ n :=
⟨fun f => monadMap (m := m) (withTheReader ρ f)⟩
instance {ρ : Type u} {m : Type u → Type v} [Monad m] : MonadWithReaderOf ρ (ReaderT ρ m) :=
⟨fun f x ctx => x (f ctx)⟩
/-- An implementation of [MonadState](https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-State-Class.html).
In contrast to the Haskell implementation, we use overlapping instances to derive instances
automatically from `monadLift`. -/
class MonadStateOf (σ : Type u) (m : Type u → Type v) :=
/- Obtain the top-most State of a Monad stack. -/
(get : m σ)
/- Set the top-most State of a Monad stack. -/
(set : σ → m PUnit)
/- Map the top-most State of a Monad stack.
Note: `modifyGet f` may be preferable to `do s <- get; let (a, s) := f s; put s; pure a`
because the latter does not use the State linearly (without sufficient inlining). -/
(modifyGet {α : Type u} : (σ → Prod α σ) → m α)
export MonadStateOf (set)
abbrev getThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] : m σ :=
MonadStateOf.get
@[inline] abbrev modifyThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → σ) : m PUnit :=
MonadStateOf.modifyGet fun s => (PUnit.unit, f s)
@[inline] abbrev modifyGetThe {α : Type u} (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → Prod α σ) : m α :=
MonadStateOf.modifyGet f
/-- Similar to `MonadStateOf`, but `σ` is an outParam for convenience -/
class MonadState (σ : outParam (Type u)) (m : Type u → Type v) :=
(get : m σ)
(set : σ → m PUnit)
(modifyGet {α : Type u} : (σ → Prod α σ) → m α)
export MonadState (get modifyGet)
instance (σ : Type u) (m : Type u → Type v) [MonadStateOf σ m] : MonadState σ m := {
set := MonadStateOf.set,
get := getThe σ,
modifyGet := fun f => MonadStateOf.modifyGet f
}
@[inline] def modify {σ : Type u} {m : Type u → Type v} [MonadState σ m] (f : σ → σ) : m PUnit :=
modifyGet fun s => (PUnit.unit, f s)
@[inline] def getModify {σ : Type u} {m : Type u → Type v} [MonadState σ m] [Monad m] (f : σ → σ) : m σ :=
modifyGet fun s => (s, f s)
-- NOTE: The Ordering of the following two instances determines that the top-most `StateT` Monad layer
-- will be picked first
instance {σ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadStateOf σ m] [MonadLift m n] : MonadStateOf σ n := {
get := liftM (m := m) MonadStateOf.get,
set := fun s => liftM (m := m) (MonadStateOf.set s),
modifyGet := fun f => monadLift (m := m) (MonadState.modifyGet f)
}
namespace EStateM
inductive Result (ε σ α : Type u)
| ok : α → σ → Result ε σ α
| error : ε → σ → Result ε σ α
variables {ε σ α : Type u}
instance [Inhabited ε] [Inhabited σ] : Inhabited (Result ε σ α) := ⟨Result.error (arbitrary _) (arbitrary _)⟩
end EStateM
open EStateM (Result) in
def EStateM (ε σ α : Type u) := σ → Result ε σ α
namespace EStateM
variables {ε σ α β : Type u}
instance [Inhabited ε] : Inhabited (EStateM ε σ α) := ⟨fun s =>
Result.error (arbitrary ε) s⟩
@[inline] protected def pure (a : α) : EStateM ε σ α := fun s =>
Result.ok a s
@[inline] protected def set (s : σ) : EStateM ε σ PUnit := fun _ =>
Result.ok ⟨⟩ s
@[inline] protected def get : EStateM ε σ σ := fun s =>
Result.ok s s
@[inline] protected def modifyGet (f : σ → Prod α σ) : EStateM ε σ α := fun s =>
match f s with
| (a, s) => Result.ok a s
@[inline] protected def throw (e : ε) : EStateM ε σ α := fun s =>
Result.error e s
/-- Auxiliary instance for saving/restoring the "backtrackable" part of the state. -/
class Backtrackable (δ : outParam (Type u)) (σ : Type u) :=
(save : σ → δ)
(restore : σ → δ → σ)
@[inline] protected def tryCatch {δ} [Backtrackable δ σ] {α} (x : EStateM ε σ α) (handle : ε → EStateM ε σ α) : EStateM ε σ α := fun s =>
let d := Backtrackable.save s
match x s with
| Result.error e s => handle e (Backtrackable.restore s d)
| ok => ok
@[inline] protected def orElse {δ} [Backtrackable δ σ] (x₁ x₂ : EStateM ε σ α) : EStateM ε σ α := fun s =>
let d := Backtrackable.save s;
match x₁ s with
| Result.error _ s => x₂ (Backtrackable.restore s d)
| ok => ok
@[inline] def adaptExcept {ε' : Type u} (f : ε → ε') (x : EStateM ε σ α) : EStateM ε' σ α := fun s =>
match x s with
| Result.error e s => Result.error (f e) s
| Result.ok a s => Result.ok a s
@[inline] protected def bind (x : EStateM ε σ α) (f : α → EStateM ε σ β) : EStateM ε σ β := fun s =>
match x s with
| Result.ok a s => f a s
| Result.error e s => Result.error e s
@[inline] protected def map (f : α → β) (x : EStateM ε σ α) : EStateM ε σ β := fun s =>
match x s with
| Result.ok a s => Result.ok (f a) s
| Result.error e s => Result.error e s
@[inline] protected def seqRight (x : EStateM ε σ PUnit) (y : EStateM ε σ β) : EStateM ε σ β := fun s =>
match x s with
| Result.ok _ s => y s
| Result.error e s => Result.error e s
instance : Monad (EStateM ε σ) := {
bind := EStateM.bind,
pure := EStateM.pure,
map := EStateM.map,
seqRight := EStateM.seqRight
}
instance {δ} [Backtrackable δ σ] : OrElse (EStateM ε σ α) := {
orElse := EStateM.orElse
}
instance : MonadStateOf σ (EStateM ε σ) := {
set := EStateM.set,
get := EStateM.get,
modifyGet := EStateM.modifyGet
}
instance {δ} [Backtrackable δ σ] : MonadExceptOf ε (EStateM ε σ) := {
throw := EStateM.throw,
tryCatch := EStateM.tryCatch
}
@[inline] def run (x : EStateM ε σ α) (s : σ) : Result ε σ α :=
x s
@[inline] def run' (x : EStateM ε σ α) (s : σ) : Option α :=
match run x s with
| Result.ok v _ => some v
| Result.error _ _ => none
@[inline] def dummySave : σ → PUnit := fun _ => ⟨⟩
@[inline] def dummyRestore : σ → PUnit → σ := fun s _ => s
/- Dummy default instance -/
instance nonBacktrackable : Backtrackable PUnit σ := {
save := dummySave,
restore := dummyRestore
}
end EStateM
class Hashable (α : Type u) :=
(hash : α → USize)
export Hashable (hash)
@[extern "lean_usize_mix_hash"]
constant mixHash (u₁ u₂ : USize) : USize
@[extern "lean_string_hash"]
protected constant String.hash (s : @& String) : USize
instance : Hashable String := ⟨String.hash⟩
namespace Lean
/- Hierarchical names -/
inductive Name
| anonymous : Name
| str : Name → String → USize → Name
| num : Name → Nat → USize → Name
instance : Inhabited Name := ⟨Name.anonymous⟩
protected def Name.hash : Name → USize
| Name.anonymous => USize.ofNat32 1723 decide!
| Name.str p s h => h
| Name.num p v h => h
instance : Hashable Name := ⟨Name.hash⟩
namespace Name
@[export lean_name_mk_string]
def mkStr (p : Name) (s : String) : Name :=
Name.str p s (mixHash (hash p) (hash s))
@[export lean_name_mk_numeral]
def mkNum (p : Name) (v : Nat) : Name :=
Name.num p v (mixHash (hash p) (dite (Less v usizeSz) (fun h => USize.ofNatCore v h) (fun _ => USize.ofNat32 17 decide!)))
def mkSimple (s : String) : Name :=
mkStr Name.anonymous s
@[extern "lean_name_eq"]
protected def beq : (@& Name) → (@& Name) → Bool
| anonymous, anonymous => true
| str p₁ s₁ _, str p₂ s₂ _ => and (BEq.beq s₁ s₂) (Name.beq p₁ p₂)
| num p₁ n₁ _, num p₂ n₂ _ => and (BEq.beq n₁ n₂) (Name.beq p₁ p₂)
| _, _ => false
instance : BEq Name := ⟨Name.beq⟩
protected def append : Name → Name → Name
| n, anonymous => n
| n, str p s _ => Name.mkStr (Name.append n p) s
| n, num p d _ => Name.mkNum (Name.append n p) d
instance : Append Name := ⟨Name.append⟩
end Name
/- Syntax -/
/--
Source information of syntax atoms. All information is generally set for unquoted syntax and unset for syntax in
syntax quotations, but syntax transformations might want to invalidate only one side to make the pretty printer
reformat it. In the special case of the delaborator, we also use purely synthetic position information without
whitespace information. -/
structure SourceInfo :=
/- Will be inferred after parsing by `Syntax.updateLeading`. During parsing,
it is not at all clear what the preceding token was, especially with backtracking. -/
(leading : Option Substring := none)
(pos : Option String.Pos := none)
(trailing : Option Substring := none)
instance : Inhabited SourceInfo := ⟨{}⟩
abbrev SyntaxNodeKind := Name
/- Syntax AST -/
inductive Syntax
| missing : Syntax
| node (kind : SyntaxNodeKind) (args : Array Syntax) : Syntax
| atom (info : SourceInfo) (val : String) : Syntax
| ident (info : SourceInfo) (rawVal : Substring) (val : Name) (preresolved : List (Prod Name (List String))) : Syntax
instance : Inhabited Syntax := ⟨Syntax.missing⟩
/- Builtin kinds -/
def choiceKind : SyntaxNodeKind := `choice
def nullKind : SyntaxNodeKind := `null
def identKind : SyntaxNodeKind := `ident
def strLitKind : SyntaxNodeKind := `strLit
def charLitKind : SyntaxNodeKind := `charLit
def numLitKind : SyntaxNodeKind := `numLit
def nameLitKind : SyntaxNodeKind := `nameLit
def fieldIdxKind : SyntaxNodeKind := `fieldIdx
def interpolatedStrLitKind : SyntaxNodeKind := `interpolatedStrLitKind
def interpolatedStrKind : SyntaxNodeKind := `interpolatedStrKind
namespace Syntax
def getKind (stx : Syntax) : SyntaxNodeKind :=
match stx with
| Syntax.node k args => k
-- We use these "pseudo kinds" for antiquotation kinds.
-- For example, an antiquotation `$id:ident` (using Lean.Parser.Term.ident)
-- is compiled to ``if stx.isOfKind `ident ...``
| Syntax.missing => `missing
| Syntax.atom _ v => Name.mkSimple v
| Syntax.ident _ _ _ _ => identKind
def setKind (stx : Syntax) (k : SyntaxNodeKind) : Syntax :=
match stx with
| Syntax.node _ args => Syntax.node k args
| _ => stx
def isOfKind (stx : Syntax) (k : SyntaxNodeKind) : Bool :=
beq stx.getKind k
def getArg (stx : Syntax) (i : Nat) : Syntax :=
match stx with
| Syntax.node _ args => args.get! i
| _ => Syntax.missing -- panic! "Syntax.getArg: not a node"
-- Add `stx[i]` as sugar for `stx.getArg i`
@[inline] def getOp (self : Syntax) (idx : Nat) : Syntax :=
self.getArg idx
def getArgs (stx : Syntax) : Array Syntax :=
match stx with
| Syntax.node _ args => args
| _ => Array.empty
end Syntax
inductive ParserDescr
| andthen : ParserDescr → ParserDescr → ParserDescr
| orelse : ParserDescr → ParserDescr → ParserDescr
| optional : ParserDescr → ParserDescr
| lookahead : ParserDescr → ParserDescr
| «try» : ParserDescr → ParserDescr
| many : ParserDescr → ParserDescr
| many1 : ParserDescr → ParserDescr
| sepBy : ParserDescr → ParserDescr → Bool → ParserDescr
| sepBy1 : ParserDescr → ParserDescr → Bool → ParserDescr
| node : Name → Nat → ParserDescr → ParserDescr
| trailingNode : Name → Nat → ParserDescr → ParserDescr
| symbol : String → ParserDescr
| nonReservedSymbol : String → Bool → ParserDescr
| noWs : ParserDescr
| numLit : ParserDescr
| strLit : ParserDescr
| charLit : ParserDescr
| nameLit : ParserDescr
| interpolatedStr : ParserDescr → ParserDescr -- interpolated string
| ident : ParserDescr
| cat : Name → Nat → ParserDescr
| parser : Name → ParserDescr
| notFollowedBy : ParserDescr → ParserDescr
| withPosition : ParserDescr → ParserDescr
| checkCol : Bool → ParserDescr
instance : Inhabited ParserDescr := ⟨ParserDescr.symbol ""⟩
abbrev TrailingParserDescr := ParserDescr
/-
Runtime support for making quotation terms auto-hygienic, by mangling identifiers
introduced by them with a "macro scope" supplied by the context. Details to appear in a
paper soon.
-/
abbrev MacroScope := Nat
/-- Macro scope used internally. It is not available for our frontend. -/
def reservedMacroScope := 0
/-- First macro scope available for our frontend -/
def firstFrontendMacroScope := add reservedMacroScope 1
/-- A monad that supports syntax quotations. Syntax quotations (in term
position) are monadic values that when executed retrieve the current "macro
scope" from the monad and apply it to every identifier they introduce
(independent of whether this identifier turns out to be a reference to an
existing declaration, or an actually fresh binding during further
elaboration). -/
class MonadQuotation (m : Type → Type) :=
-- Get the fresh scope of the current macro invocation
(getCurrMacroScope : m MacroScope)
(getMainModule : m Name)
/- Execute action in a new macro invocation context. This transformer should be
used at all places that morally qualify as the beginning of a "macro call",
e.g. `elabCommand` and `elabTerm` in the case of the elaborator. However, it
can also be used internally inside a "macro" if identifiers introduced by
e.g. different recursive calls should be independent and not collide. While
returning an intermediate syntax tree that will recursively be expanded by
the elaborator can be used for the same effect, doing direct recursion inside
the macro guarded by this transformer is often easier because one is not
restricted to passing a single syntax tree. Modelling this helper as a
transformer and not just a monadic action ensures that the current macro
scope before the recursive call is restored after it, as expected. -/
(withFreshMacroScope {α : Type} : m α → m α)
export MonadQuotation (getCurrMacroScope getMainModule withFreshMacroScope)
instance {m n : Type → Type} [MonadQuotation m] [MonadLift m n] [MonadFunctorT m n] : MonadQuotation n := {
getCurrMacroScope := liftM (m := m) getCurrMacroScope,
getMainModule := liftM (m := m) getMainModule,
withFreshMacroScope := monadMap (m := m) withFreshMacroScope
}
/-
We represent a name with macro scopes as
```
<actual name>._@.(<module_name>.<scopes>)*.<module_name>._hyg.<scopes>
```
Example: suppose the module name is `Init.Data.List.Basic`, and name is `foo.bla`, and macroscopes [2, 5]
```
foo.bla._@.Init.Data.List.Basic._hyg.2.5
```
We may have to combine scopes from different files/modules.
The main modules being processed is always the right most one.
This situation may happen when we execute a macro generated in
an imported file in the current file.
```
foo.bla._@.Init.Data.List.Basic.2.1.Init.Lean.Expr_hyg.4
```
The delimiter `_hyg` is used just to improve the `hasMacroScopes` performance.
-/
def Name.hasMacroScopes : Name → Bool
| str _ s _ => beq s "_hyg"
| num p _ _ => hasMacroScopes p
| _ => false
private def eraseMacroScopesAux : Name → Name
| Name.str p s _ => match beq s "_@" with
| true => p
| false => eraseMacroScopesAux p
| Name.num p _ _ => eraseMacroScopesAux p
| Name.anonymous => Name.anonymous
@[export lean_erase_macro_scopes]
def Name.eraseMacroScopes (n : Name) : Name :=
match n.hasMacroScopes with
| true => eraseMacroScopesAux n
| false => n
private def simpMacroScopesAux : Name → Name
| Name.num p i _ => Name.mkNum (simpMacroScopesAux p) i
| n => eraseMacroScopesAux n
/- Helper function we use to create binder names that do not need to be unique. -/
@[export lean_simp_macro_scopes]
def Name.simpMacroScopes (n : Name) : Name :=
match n.hasMacroScopes with
| true => simpMacroScopesAux n
| false => n
structure MacroScopesView :=
(name : Name)
(imported : Name)
(mainModule : Name)
(scopes : List MacroScope)
instance : Inhabited MacroScopesView := ⟨⟨arbitrary _, arbitrary _, arbitrary _, arbitrary _⟩⟩
def MacroScopesView.review (view : MacroScopesView) : Name :=
match view.scopes with
| List.nil => view.name
| List.cons _ _ =>
let base := (Name.mkStr (append (append (Name.mkStr view.name "_@") view.imported) view.mainModule) "_hyg")
view.scopes.foldl Name.mkNum base
private def assembleParts : List Name → Name → Name
| List.nil, acc => acc
| List.cons (Name.str _ s _) ps, acc => assembleParts ps (Name.mkStr acc s)
| List.cons (Name.num _ n _) ps, acc => assembleParts ps (Name.mkNum acc n)
| _, acc => panic "unreachable @ assembleParts"
private def extractImported (scps : List MacroScope) (mainModule : Name) : Name → List Name → MacroScopesView
| n@(Name.str p str _), parts =>
match beq str "_@" with
| true => { name := p, mainModule := mainModule, imported := assembleParts parts Name.anonymous, scopes := scps }
| false => extractImported scps mainModule p (List.cons n parts)
| n@(Name.num p str _), parts => extractImported scps mainModule p (List.cons n parts)
| _, _ => panic "unreachable @ extractImported"
private def extractMainModule (scps : List MacroScope) : Name → List Name → MacroScopesView
| n@(Name.str p str _), parts =>
match beq str "_@" with
| true => { name := p, mainModule := assembleParts parts Name.anonymous, imported := Name.anonymous, scopes := scps }
| false => extractMainModule scps p (List.cons n parts)
| n@(Name.num p num _), acc => extractImported scps (assembleParts acc Name.anonymous) n List.nil
| _, _ => panic "unreachable @ extractMainModule"
private def extractMacroScopesAux : Name → List MacroScope → MacroScopesView
| Name.num p scp _, acc => extractMacroScopesAux p (List.cons scp acc)
| Name.str p str _, acc => extractMainModule acc p List.nil -- str must be "_hyg"
| _, _ => panic "unreachable @ extractMacroScopesAux"
/--
Revert all `addMacroScope` calls. `v = extractMacroScopes n → n = v.review`.
This operation is useful for analyzing/transforming the original identifiers, then adding back
the scopes (via `MacroScopesView.review`). -/
def extractMacroScopes (n : Name) : MacroScopesView :=
match n.hasMacroScopes with
| true => extractMacroScopesAux n List.nil
| false => { name := n, scopes := List.nil, imported := Name.anonymous, mainModule := Name.anonymous }
def addMacroScope (mainModule : Name) (n : Name) (scp : MacroScope) : Name :=
match n.hasMacroScopes with
| true =>
let view := extractMacroScopes n
match beq view.mainModule mainModule with
| true => Name.mkNum n scp
| false =>
{ view with
imported := view.scopes.foldl Name.mkNum (append view.imported view.mainModule),
mainModule := mainModule,
scopes := List.cons scp List.nil
}.review
| false =>
Name.mkNum (Name.mkStr (append (Name.mkStr n "_@") mainModule) "_hyg") scp
@[inline] def MonadQuotation.addMacroScope {m : Type → Type} [MonadQuotation m] [Monad m] (n : Name) : m Name :=
bind getMainModule fun mainModule =>
bind getCurrMacroScope fun scp =>
pure (Lean.addMacroScope mainModule n scp)
def defaultMaxRecDepth := 512
def maxRecDepthErrorMessage : String :=
"maximum recursion depth has been reached (use `set_option maxRecDepth <num>` to increase limit)"
namespace Macro
/- References -/
constant MacroEnvPointed : PointedType.{0}
def MacroEnv : Type := MacroEnvPointed.type
instance : Inhabited MacroEnv := ⟨MacroEnvPointed.val⟩
structure Context :=
(macroEnv : MacroEnv)
(mainModule : Name)
(currMacroScope : MacroScope)
(currRecDepth : Nat := 0)
(maxRecDepth : Nat := defaultMaxRecDepth)
inductive Exception
| error : Syntax → String → Exception
| unsupportedSyntax : Exception
end Macro
abbrev MacroM := ReaderT Macro.Context (EStateM Macro.Exception MacroScope)
abbrev Macro := Syntax → MacroM Syntax
namespace Macro
def addMacroScope (n : Name) : MacroM Name :=
bind read fun ctx =>
pure (Lean.addMacroScope ctx.mainModule n ctx.currMacroScope)
def throwUnsupported {α} : MacroM α :=
throw Exception.unsupportedSyntax
def throwError {α} (ref : Syntax) (msg : String) : MacroM α :=
throw (Exception.error ref msg)
@[inline] protected def withFreshMacroScope {α} (x : MacroM α) : MacroM α :=
bind (modifyGet (fun s => (s, add s 1))) fun fresh =>
withReader (fun ctx => { ctx with currMacroScope := fresh }) x
@[inline] def withIncRecDepth {α} (ref : Syntax) (x : MacroM α) : MacroM α :=
bind read fun ctx =>
match beq ctx.currRecDepth ctx.maxRecDepth with
| true => throw (Exception.error ref maxRecDepthErrorMessage)
| false => withReader (fun ctx => { ctx with currRecDepth := add ctx.currRecDepth 1 }) x
instance : MonadQuotation MacroM := {
getCurrMacroScope := fun ctx => pure ctx.currMacroScope,
getMainModule := fun ctx => pure ctx.mainModule,
withFreshMacroScope := Macro.withFreshMacroScope
}
unsafe def mkMacroEnvImp (expandMacro? : Syntax → MacroM (Option Syntax)) : MacroEnv :=
unsafeCast expandMacro?
@[implementedBy mkMacroEnvImp]
constant mkMacroEnv (expandMacro? : Syntax → MacroM (Option Syntax)) : MacroEnv
def expandMacroNotAvailable? (stx : Syntax) : MacroM (Option Syntax) :=
throwError stx "expandMacro has not been set"
def mkMacroEnvSimple : MacroEnv :=
mkMacroEnv expandMacroNotAvailable?
unsafe def expandMacro?Imp (stx : Syntax) : MacroM (Option Syntax) :=
bind read fun ctx =>
let f : Syntax → MacroM (Option Syntax) := unsafeCast (ctx.macroEnv)
f stx
/-- `expandMacro? stx` return `some stxNew` if `stx` is a macro, and `stxNew` is its expansion. -/
@[implementedBy expandMacro?Imp] constant expandMacro? : Syntax → MacroM (Option Syntax)
end Macro
export Macro (expandMacro?)
end Lean
|
b10b9c18d6791b74f528bd4be68a1882d9b384cd | aa5a655c05e5359a70646b7154e7cac59f0b4132 | /src/Lean/Meta/WHNF.lean | 157b0544482540798c9babc3be548d722e7f3d1a | [
"Apache-2.0"
] | permissive | lambdaxymox/lean4 | ae943c960a42247e06eff25c35338268d07454cb | 278d47c77270664ef29715faab467feac8a0f446 | refs/heads/master | 1,677,891,867,340 | 1,612,500,005,000 | 1,612,500,005,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 22,042 | 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
-/
import Lean.ToExpr
import Lean.AuxRecursor
import Lean.ProjFns
import Lean.Meta.Basic
import Lean.Meta.LevelDefEq
import Lean.Meta.GetConst
import Lean.Meta.Match.MatcherInfo
namespace Lean.Meta
/- ===========================
Smart unfolding support
=========================== -/
def smartUnfoldingSuffix := "_sunfold"
@[inline] def mkSmartUnfoldingNameFor (n : Name) : Name :=
Name.mkStr n smartUnfoldingSuffix
register_builtin_option smartUnfolding : Bool := {
defValue := true
descr := "when computing weak head normal form, use auxiliary definition created for functions defined by structural recursion"
}
/- ===========================
Helper methods
=========================== -/
def isAuxDef (constName : Name) : MetaM Bool := do
let env ← getEnv
return isAuxRecursor env constName || isNoConfusion env constName
@[inline] private def matchConstAux {α} (e : Expr) (failK : Unit → MetaM α) (k : ConstantInfo → List Level → MetaM α) : MetaM α :=
match e with
| Expr.const name lvls _ => do
let (some cinfo) ← getConst? name | failK ()
k cinfo lvls
| _ => failK ()
/- ===========================
Helper functions for reducing recursors
=========================== -/
private def getFirstCtor (d : Name) : MetaM (Option Name) := do
let some (ConstantInfo.inductInfo { ctors := ctor::_, ..}) ← getConstNoEx? d | pure none
return some ctor
private def mkNullaryCtor (type : Expr) (nparams : Nat) : MetaM (Option Expr) :=
match type.getAppFn with
| Expr.const d lvls _ => do
let (some ctor) ← getFirstCtor d | pure none
return mkAppN (mkConst ctor lvls) (type.getAppArgs.shrink nparams)
| _ =>
return none
def toCtorIfLit : Expr → Expr
| Expr.lit (Literal.natVal v) _ =>
if v == 0 then mkConst `Nat.zero
else mkApp (mkConst `Nat.succ) (mkNatLit (v-1))
| Expr.lit (Literal.strVal v) _ =>
mkApp (mkConst `String.mk) (toExpr v.toList)
| e => e
private def getRecRuleFor (recVal : RecursorVal) (major : Expr) : Option RecursorRule :=
match major.getAppFn with
| Expr.const fn _ _ => recVal.rules.find? fun r => r.ctor == fn
| _ => none
private def toCtorWhenK (recVal : RecursorVal) (major : Expr) : MetaM (Option Expr) := do
let majorType ← inferType major
let majorType ← whnf majorType
let majorTypeI := majorType.getAppFn
if !majorTypeI.isConstOf recVal.getInduct then
return none
else if majorType.hasExprMVar && majorType.getAppArgs[recVal.numParams:].any Expr.hasExprMVar then
return none
else do
let (some newCtorApp) ← mkNullaryCtor majorType recVal.numParams | pure none
let newType ← inferType newCtorApp
if (← isDefEq majorType newType) then
return newCtorApp
else
return none
/-- Auxiliary function for reducing recursor applications. -/
private def reduceRec {α} (recVal : RecursorVal) (recLvls : List Level) (recArgs : Array Expr) (failK : Unit → MetaM α) (successK : Expr → MetaM α) : MetaM α :=
let majorIdx := recVal.getMajorIdx
if h : majorIdx < recArgs.size then do
let major := recArgs.get ⟨majorIdx, h⟩
let mut major ← whnf major
if recVal.k then
let newMajor ← toCtorWhenK recVal major
major := newMajor.getD major
let major := toCtorIfLit major
match getRecRuleFor recVal major with
| some rule =>
let majorArgs := major.getAppArgs
if recLvls.length != recVal.levelParams.length then
failK ()
else
let rhs := rule.rhs.instantiateLevelParams recVal.levelParams recLvls
-- Apply parameters, motives and minor premises from recursor application.
let rhs := mkAppRange rhs 0 (recVal.numParams+recVal.numMotives+recVal.numMinors) recArgs
/- The number of parameters in the constructor is not necessarily
equal to the number of parameters in the recursor when we have
nested inductive types. -/
let nparams := majorArgs.size - rule.nfields
let rhs := mkAppRange rhs nparams majorArgs.size majorArgs
let rhs := mkAppRange rhs (majorIdx + 1) recArgs.size recArgs
successK rhs
| none => failK ()
else
failK ()
/- ===========================
Helper functions for reducing Quot.lift and Quot.ind
=========================== -/
/-- Auxiliary function for reducing `Quot.lift` and `Quot.ind` applications. -/
private def reduceQuotRec {α} (recVal : QuotVal) (recLvls : List Level) (recArgs : Array Expr) (failK : Unit → MetaM α) (successK : Expr → MetaM α) : MetaM α :=
let process (majorPos argPos : Nat) : MetaM α :=
if h : majorPos < recArgs.size then do
let major := recArgs.get ⟨majorPos, h⟩
let major ← whnf major
match major with
| Expr.app (Expr.app (Expr.app (Expr.const majorFn _ _) _ _) _ _) majorArg _ => do
let some (ConstantInfo.quotInfo { kind := QuotKind.ctor, .. }) ← getConstNoEx? majorFn | failK ()
let f := recArgs[argPos]
let r := mkApp f majorArg
let recArity := majorPos + 1
successK $ mkAppRange r recArity recArgs.size recArgs
| _ => failK ()
else
failK ()
match recVal.kind with
| QuotKind.lift => process 5 3
| QuotKind.ind => process 4 3
| _ => failK ()
/- ===========================
Helper function for extracting "stuck term"
=========================== -/
mutual
private partial def isRecStuck? (recVal : RecursorVal) (recLvls : List Level) (recArgs : Array Expr) : MetaM (Option MVarId) :=
if recVal.k then
-- TODO: improve this case
return none
else do
let majorIdx := recVal.getMajorIdx
if h : majorIdx < recArgs.size then do
let major := recArgs.get ⟨majorIdx, h⟩
let major ← whnf major
getStuckMVar? major
else
return none
private partial def isQuotRecStuck? (recVal : QuotVal) (recLvls : List Level) (recArgs : Array Expr) : MetaM (Option MVarId) :=
let process? (majorPos : Nat) : MetaM (Option MVarId) :=
if h : majorPos < recArgs.size then do
let major := recArgs.get ⟨majorPos, h⟩
let major ← whnf major
getStuckMVar? major
else
return none
match recVal.kind with
| QuotKind.lift => process? 5
| QuotKind.ind => process? 4
| _ => return none
/-- Return `some (Expr.mvar mvarId)` if metavariable `mvarId` is blocking reduction. -/
partial def getStuckMVar? : Expr → MetaM (Option MVarId)
| Expr.mdata _ e _ => getStuckMVar? e
| Expr.proj _ _ e _ => do getStuckMVar? (← whnf e)
| e@(Expr.mvar ..) => do
let e ← instantiateMVars e
match e with
| Expr.mvar mvarId _ => pure (some mvarId)
| _ => getStuckMVar? e
| e@(Expr.app f _ _) =>
let f := f.getAppFn
match f with
| Expr.mvar mvarId _ => return some mvarId
| Expr.const fName fLvls _ => do
let cinfo? ← getConstNoEx? fName
match cinfo? with
| some $ ConstantInfo.recInfo recVal => isRecStuck? recVal fLvls e.getAppArgs
| some $ ConstantInfo.quotInfo recVal => isQuotRecStuck? recVal fLvls e.getAppArgs
| _ => return none
| _ => return none
| _ => return none
end
/- ===========================
Weak Head Normal Form auxiliary combinators
=========================== -/
/-- Auxiliary combinator for handling easy WHNF cases. It takes a function for handling the "hard" cases as an argument -/
@[specialize] private partial def whnfEasyCases (e : Expr) (k : Expr → MetaM Expr) : MetaM Expr := do
match e with
| Expr.forallE .. => return e
| Expr.lam .. => return e
| Expr.sort .. => return e
| Expr.lit .. => return e
| Expr.bvar .. => unreachable!
| Expr.letE .. => k e
| Expr.const .. => k e
| Expr.app .. => k e
| Expr.proj .. => k e
| Expr.mdata _ e _ => whnfEasyCases e k
| Expr.fvar fvarId _ =>
let decl ← getLocalDecl fvarId
match decl with
| LocalDecl.cdecl .. => return e
| LocalDecl.ldecl (value := v) (nonDep := nonDep) .. =>
let cfg ← getConfig
if nonDep && !cfg.zetaNonDep then
return e
else
when cfg.trackZeta do
modify fun s => { s with zetaFVarIds := s.zetaFVarIds.insert fvarId }
whnfEasyCases v k
| Expr.mvar mvarId _ =>
match (← getExprMVarAssignment? mvarId) with
| some v => whnfEasyCases v k
| none => return e
/-- Return true iff term is of the form `idRhs ...` -/
private def isIdRhsApp (e : Expr) : Bool :=
e.isAppOf `idRhs
/-- (@idRhs T f a_1 ... a_n) ==> (f a_1 ... a_n) -/
private def extractIdRhs (e : Expr) : Expr :=
if !isIdRhsApp e then e
else
let args := e.getAppArgs
if args.size < 2 then e
else mkAppRange args[1] 2 args.size args
@[specialize] private def deltaDefinition {α} (c : ConstantInfo) (lvls : List Level)
(failK : Unit → α) (successK : Expr → α) : α :=
if c.lparams.length != lvls.length then failK ()
else
let val := c.instantiateValueLevelParams lvls
successK (extractIdRhs val)
@[specialize] private def deltaBetaDefinition {α} (c : ConstantInfo) (lvls : List Level) (revArgs : Array Expr)
(failK : Unit → α) (successK : Expr → α) : α :=
if c.lparams.length != lvls.length then
failK ()
else
let val := c.instantiateValueLevelParams lvls
let val := val.betaRev revArgs
successK (extractIdRhs val)
inductive ReduceMatcherResult where
| reduced (val : Expr)
| stuck (val : Expr)
| notMatcher
| partialApp
def reduceMatcher? (e : Expr) : MetaM ReduceMatcherResult := do
match e.getAppFn with
| Expr.const declName declLevels _ =>
let some info ← getMatcherInfo? declName
| return ReduceMatcherResult.notMatcher
let args := e.getAppArgs
let prefixSz := info.numParams + 1 + info.numDiscrs
if args.size < prefixSz + info.numAlts then
return ReduceMatcherResult.partialApp
else
let constInfo ← getConstInfo declName
let f := constInfo.instantiateValueLevelParams declLevels
let auxApp := mkAppN f args[0:prefixSz]
let auxAppType ← inferType auxApp
forallBoundedTelescope auxAppType info.numAlts fun hs _ => do
let auxApp := mkAppN auxApp hs
let auxApp ← whnf auxApp
let auxAppFn := auxApp.getAppFn
let mut i := prefixSz
for h in hs do
if auxAppFn == h then
let result := mkAppN args[i] auxApp.getAppArgs
let result := mkAppN result args[prefixSz + info.numAlts:args.size]
return ReduceMatcherResult.reduced result.headBeta
i := i + 1
return ReduceMatcherResult.stuck auxApp
| _ => pure ReduceMatcherResult.notMatcher
/- Given an expression `e`, compute its WHNF and if the result is a constructor, return field #i. -/
def project? (e : Expr) (i : Nat) : MetaM (Option Expr) := do
let e ← whnf e
matchConstCtor e.getAppFn (fun _ => pure none) fun ctorVal _ =>
let numArgs := e.getAppNumArgs
let idx := ctorVal.numParams + i
if idx < numArgs then
return some (e.getArg! idx)
else
return none
def reduceProj? (e : Expr) : MetaM (Option Expr) := do
match e with
| Expr.proj _ i c _ => project? c i
| _ => return none
/--
Apply beta-reduction, zeta-reduction (i.e., unfold let local-decls), iota-reduction,
expand let-expressions, expand assigned meta-variables. -/
partial def whnfCore (e : Expr) : MetaM Expr :=
whnfEasyCases e fun e => do
trace[Meta.whnf]! e
match e with
| Expr.const .. => pure e
| Expr.letE _ _ v b _ => whnfCore $ b.instantiate1 v
| Expr.app f .. =>
let f := f.getAppFn
let f' ← whnfCore f
if f'.isLambda then
let revArgs := e.getAppRevArgs
whnfCore <| f'.betaRev revArgs
else match (← reduceMatcher? e) with
| ReduceMatcherResult.reduced eNew => whnfCore eNew
| ReduceMatcherResult.partialApp => pure e
| ReduceMatcherResult.stuck _ => pure e
| ReduceMatcherResult.notMatcher =>
let done : Unit → MetaM Expr := fun _ =>
if f == f' then pure e else pure $ e.updateFn f'
matchConstAux f' done fun cinfo lvls =>
match cinfo with
| ConstantInfo.recInfo rec => reduceRec rec lvls e.getAppArgs done whnfCore
| ConstantInfo.quotInfo rec => reduceQuotRec rec lvls e.getAppArgs done whnfCore
| c@(ConstantInfo.defnInfo _) => do
if (← isAuxDef c.name) then
deltaBetaDefinition c lvls e.getAppRevArgs done whnfCore
else
done ()
| _ => done ()
| Expr.proj .. => match (← reduceProj? e) with
| some e => whnfCore e
| none => return e
| _ => unreachable!
mutual
/-- Reduce `e` until `idRhs` application is exposed or it gets stuck.
This is a helper method for implementing smart unfolding. -/
private partial def whnfUntilIdRhs (e : Expr) : MetaM Expr := do
let e ← whnfCore e
match (← getStuckMVar? e) with
| some mvarId =>
/- Try to "unstuck" by resolving pending TC problems -/
if (← Meta.synthPending mvarId) then
whnfUntilIdRhs e
else
return e -- failed because metavariable is blocking reduction
| _ =>
if isIdRhsApp e then
return e -- done
else
match (← unfoldDefinition? e) with
| some e => whnfUntilIdRhs e
| none => pure e -- failed because of symbolic argument
/--
Auxiliary method for unfolding a class projection when transparency is set to `TransparencyMode.instances`.
Recall that that class instance projections are not marked with `[reducible]` because we want them to be
in "reducible canonical form".
-/
private partial def unfoldProjInst (e : Expr) : MetaM (Option Expr) := do
if (← getTransparency) != TransparencyMode.instances then
return none
else
match e.getAppFn with
| Expr.const declName .. =>
match (← getProjectionFnInfo? declName) with
| some { fromClass := true, .. } =>
match (← withDefault <| unfoldDefinition? e) with
| none => return none
| some e =>
match (← reduceProj? e.getAppFn) with
| none => return none
| some r => return mkAppN r e.getAppArgs
| _ => return none
| _ => return none
/-- Unfold definition using "smart unfolding" if possible. -/
partial def unfoldDefinition? (e : Expr) : MetaM (Option Expr) :=
match e with
| Expr.app f _ _ =>
matchConstAux f.getAppFn (fun _ => unfoldProjInst e) fun fInfo fLvls => do
if fInfo.lparams.length != fLvls.length then
return none
else
let unfoldDefault (_ : Unit) : MetaM (Option Expr) :=
if fInfo.hasValue then
deltaBetaDefinition fInfo fLvls e.getAppRevArgs (fun _ => pure none) (fun e => pure (some e))
else
return none
if smartUnfolding.get (← getOptions) then
let fAuxInfo? ← getConstNoEx? (mkSmartUnfoldingNameFor fInfo.name)
match fAuxInfo? with
| some fAuxInfo@(ConstantInfo.defnInfo _) =>
deltaBetaDefinition fAuxInfo fLvls e.getAppRevArgs (fun _ => pure none) fun e₁ => do
let e₂ ← whnfUntilIdRhs e₁
if isIdRhsApp e₂ then
return some (extractIdRhs e₂)
else
return none
| _ => unfoldDefault ()
else
unfoldDefault ()
| Expr.const name lvls _ => do
let (some (cinfo@(ConstantInfo.defnInfo _))) ← getConstNoEx? name | pure none
deltaDefinition cinfo lvls
(fun _ => pure none)
(fun e => pure (some e))
| _ => return none
end
def unfoldDefinition (e : Expr) : MetaM Expr := do
let some e ← unfoldDefinition? e | throwError! "failed to unfold definition{indentExpr e}"
return e
@[specialize] partial def whnfHeadPred (e : Expr) (pred : Expr → MetaM Bool) : MetaM Expr :=
whnfEasyCases e fun e => do
let e ← whnfCore e
if (← pred e) then
match (← unfoldDefinition? e) with
| some e => whnfHeadPred e pred
| none => return e
else
return e
def whnfUntil (e : Expr) (declName : Name) : MetaM (Option Expr) := do
let e ← whnfHeadPred e (fun e => return !e.isAppOf declName)
if e.isAppOf declName then
return e
else
return none
/-- Try to reduce matcher/recursor/quot applications. We say they are all "morally" recursor applications. -/
def reduceRecMatcher? (e : Expr) : MetaM (Option Expr) := do
if !e.isApp then
return none
else match (← reduceMatcher? e) with
| ReduceMatcherResult.reduced e => return e
| _ => matchConstAux e.getAppFn (fun _ => pure none) fun cinfo lvls => do
match cinfo with
| ConstantInfo.recInfo «rec» => reduceRec «rec» lvls e.getAppArgs (fun _ => pure none) (fun e => pure (some e))
| ConstantInfo.quotInfo «rec» => reduceQuotRec «rec» lvls e.getAppArgs (fun _ => pure none) (fun e => pure (some e))
| c@(ConstantInfo.defnInfo _) =>
if (← isAuxDef c.name) then
deltaBetaDefinition c lvls e.getAppRevArgs (fun _ => pure none) (fun e => pure (some e))
else
return none
| _ => return none
unsafe def reduceBoolNativeUnsafe (constName : Name) : MetaM Bool := evalConstCheck Bool `Bool constName
unsafe def reduceNatNativeUnsafe (constName : Name) : MetaM Nat := evalConstCheck Nat `Nat constName
@[implementedBy reduceBoolNativeUnsafe] constant reduceBoolNative (constName : Name) : MetaM Bool
@[implementedBy reduceNatNativeUnsafe] constant reduceNatNative (constName : Name) : MetaM Nat
def reduceNative? (e : Expr) : MetaM (Option Expr) :=
match e with
| Expr.app (Expr.const fName _ _) (Expr.const argName _ _) _ =>
if fName == `Lean.reduceBool then do
return toExpr (← reduceBoolNative argName)
else if fName == `Lean.reduceNat then do
return toExpr (← reduceNatNative argName)
else
return none
| _ =>
return none
@[inline] def withNatValue {α} (a : Expr) (k : Nat → MetaM (Option α)) : MetaM (Option α) := do
let a ← whnf a
match a with
| Expr.const `Nat.zero _ _ => k 0
| Expr.lit (Literal.natVal v) _ => k v
| _ => return none
def reduceUnaryNatOp (f : Nat → Nat) (a : Expr) : MetaM (Option Expr) :=
withNatValue a fun a =>
return mkNatLit <| f a
def reduceBinNatOp (f : Nat → Nat → Nat) (a b : Expr) : MetaM (Option Expr) :=
withNatValue a fun a =>
withNatValue b fun b => do
trace[Meta.isDefEq.whnf.reduceBinOp]! "{a} op {b}"
return mkNatLit <| f a b
def reduceBinNatPred (f : Nat → Nat → Bool) (a b : Expr) : MetaM (Option Expr) := do
withNatValue a fun a =>
withNatValue b fun b =>
return toExpr <| f a b
def reduceNat? (e : Expr) : MetaM (Option Expr) :=
if e.hasFVar || e.hasMVar then
return none
else match e with
| Expr.app (Expr.const fn _ _) a _ =>
if fn == `Nat.succ then
reduceUnaryNatOp Nat.succ a
else
return none
| Expr.app (Expr.app (Expr.const fn _ _) a1 _) a2 _ =>
if fn == `Nat.add then reduceBinNatOp Nat.add a1 a2
else if fn == `Nat.sub then reduceBinNatOp Nat.sub a1 a2
else if fn == `Nat.mul then reduceBinNatOp Nat.mul a1 a2
else if fn == `Nat.div then reduceBinNatOp Nat.div a1 a2
else if fn == `Nat.mod then reduceBinNatOp Nat.mod a1 a2
else if fn == `Nat.beq then reduceBinNatPred Nat.beq a1 a2
else if fn == `Nat.ble then reduceBinNatPred Nat.ble a1 a2
else return none
| _ =>
return none
@[inline] private def useWHNFCache (e : Expr) : MetaM Bool := do
-- We cache only closed terms without expr metavars.
-- Potential refinement: cache if `e` is not stuck at a metavariable
if e.hasFVar || e.hasExprMVar then
return false
else
match (← getConfig).transparency with
| TransparencyMode.default => true
| TransparencyMode.all => true
| _ => false
@[inline] private def cached? (useCache : Bool) (e : Expr) : MetaM (Option Expr) := do
if useCache then
match (← getConfig).transparency with
| TransparencyMode.default => return (← get).cache.whnfDefault.find? e
| TransparencyMode.all => return (← get).cache.whnfAll.find? e
| _ => unreachable!
else
return none
private def cache (useCache : Bool) (e r : Expr) : MetaM Expr := do
if useCache then
match (← getConfig).transparency with
| TransparencyMode.default => modify fun s => { s with cache.whnfDefault := s.cache.whnfDefault.insert e r }
| TransparencyMode.all => modify fun s => { s with cache.whnfAll := s.cache.whnfAll.insert e r }
| _ => unreachable!
return r
partial def whnfImp (e : Expr) : MetaM Expr :=
whnfEasyCases e fun e => do
checkMaxHeartbeats "whnf"
let useCache ← useWHNFCache e
match (← cached? useCache e) with
| some e' => pure e'
| none =>
let e' ← whnfCore e
match (← reduceNat? e') with
| some v => cache useCache e v
| none =>
match (← reduceNative? e') with
| some v => cache useCache e v
| none =>
match (← unfoldDefinition? e') with
| some e => whnfImp e
| none => cache useCache e e'
@[builtinInit] def setWHNFRef : IO Unit :=
whnfRef.set whnfImp
builtin_initialize
registerTraceClass `Meta.whnf
end Lean.Meta
|
4b6b86409d9fb4ca988b45f0204ce15fe898f67c | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /test/equiv_rw.lean | 701c17160f95533a226d6caded093aaf722c8a9b | [
"Apache-2.0"
] | permissive | dupuisf/mathlib | 62de4ec6544bf3b79086afd27b6529acfaf2c1bb | 8582b06b0a5d06c33ee07d0bdf7c646cae22cf36 | refs/heads/master | 1,669,494,854,016 | 1,595,692,409,000 | 1,595,692,409,000 | 272,046,630 | 0 | 0 | Apache-2.0 | 1,592,066,143,000 | 1,592,066,142,000 | null | UTF-8 | Lean | false | false | 9,075 | 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 tactic.equiv_rw
import control.equiv_functor.instances -- these make equiv_rw more powerful!
-- Uncomment this line to observe the steps of constructing appropriate equivalences.
-- set_option trace.equiv_rw_type true
import tactic.equiv_rw
-- This fails if we use `occurs` rather than `kdepends_on` in `equiv_rw_type`.
instance : equiv_functor set :=
{ map := λ α β e s, by { equiv_rw e.symm, assumption, } }
-- Rewriting a hypothesis along an equivalence.
example {α β : Type} (e : α ≃ β)
(f : α → ℕ) (h : ∀ b : β, f (e.symm b) = 0) (i : α) : f i = 0 :=
begin
equiv_rw e at i,
apply h,
end
-- Check that dependent hypotheses are reverted and reintroduced.
example {α β : Type} (e : α ≃ β) (Z : α → Type) (f : Π a, Z a → ℕ)
(h : ∀ (b : β) (x : Z (e.symm b)), f (e.symm b) x = 0)
(i : α) (x : Z i) : f i x = 0 :=
begin
equiv_rw e at i,
guard_hyp i := β,
guard_target f (e.symm i) x = 0,
guard_hyp x := Z ((e.symm) i),
exact h i x,
end
-- Rewriting the goal along an equivalence.
example {α β : Type} (e : α ≃ β) (b : β) : α :=
begin
equiv_rw e,
exact b,
end
-- Fail if the equivalence can't be used.
example {α β γ : Type} (e : β ≃ γ) (a : α) : α :=
begin
success_if_fail { equiv_rw e at a },
success_if_fail { equiv_rw e },
exact a,
end
-- Verify that `equiv_rw` will rewrite under `equiv_functor` instances.
example {α β : Type} (u : unique α) (e : α ≃ β) : β :=
begin
equiv_rw e at u,
apply inhabited.default,
end
example {α β : Type} (p : equiv.perm α) (e : α ≃ β) : equiv.perm β :=
begin
equiv_rw e at p,
exact p,
end
-- We can rewrite the goal under functors.
example {α β : Type} (e : α ≃ β) (b : β) : option α :=
begin
equiv_rw e,
exact some b,
end
-- We can rewrite hypotheses under functors.
example {α β : Type} (e : α ≃ β) (b : option α) : option β :=
begin
equiv_rw e at b,
exact b,
end
-- We can rewrite hypotheses under compositions of functors.
example {α β : Type} (e : α ≃ β) (b : list (list α)) : list β :=
begin
equiv_rw e at b,
exact b.join,
end
-- Check that we can rewrite in the target position of function types.
example {α β γ : Type} (e : α ≃ β) (f : γ → β) : γ → α :=
begin
equiv_rw e,
exact f,
end
-- Check that we can rewrite in the source position of function types.
example {α β γ : Type} (e : α ≃ β) (f : β → γ) : α → γ :=
begin
equiv_rw e,
exact f,
end
-- Rewriting under multiple functors.
example {α β : Type} (e : α ≃ β) (b : β) : list (option α) :=
begin
equiv_rw e,
exact [none, some b],
end
-- Rewriting under multiple functors, including functions.
example {α β γ : Type} (e : α ≃ β) (b : β) : γ → list (option α) :=
begin
equiv_rw e,
exact (λ g, [none, some b]),
end
-- Rewriting in multiple positions.
example {α β : Type*} [has_add β] (e : α ≃ β) : α → α :=
begin
have : (α → α) ≃ _, {
apply equiv.arrow_congr,
apply e,
apply e,
},
equiv_rw e,
exact (@id β),
end
-- Rewriting in multiple positions.
example {α β : Type} [has_add β] (e : α ≃ β) : β → α → α :=
begin
equiv_rw e,
exact (+),
end
-- Rewriting in multiple positions.
example {α β : Type} [has_add β] (e : α ≃ β) : α → α → α :=
begin
equiv_rw e,
exact (+),
end
example {α β γ : Type} (e : α ≃ β) (s : β ⊕ γ) : α ⊕ γ :=
begin
equiv_rw e,
exact s,
end
example {α β γ : Type} (e : α ≃ β) (s : β ⊕ γ) : (α ⊕ γ) × (γ ⊕ α) :=
begin
equiv_rw e,
exact (s, s.swap),
end
example {α β γ : Type} (e : α ≃ β) (s : α ⊕ γ) : (β ⊕ γ) × (γ ⊕ β) :=
begin
equiv_rw e at s,
exact (s, s.swap),
end
example {α β γ : Type} (e : α ≃ β) (s : (α ⊕ γ) × β) : (β ⊕ γ) :=
begin
success_if_fail { equiv_rw e at s {max_depth := 4} },
equiv_rw e at s,
exact s.1,
end
-- Test generating the actual equivalence using `equiv_rw_type`.
example {α β : Type} (e : α ≃ β) (b : β) : α × (ℕ ⊕ ℕ) :=
begin
have e' : α × (ℕ ⊕ ℕ) ≃ _ := by equiv_rw_type e,
apply e'.inv_fun,
exact (b, sum.inl 0)
end
example {α β : Type} (e : α ≃ β) (P : α → Prop) (h : { a // P a }) : β :=
begin
equiv_rw e at h,
exact h.val,
end
example {α β : Type} (e : α ≃ β) (P : α → Prop) (h : ∀ a, P a) (b : β) : { a // P a } :=
begin
equiv_rw e,
use b,
apply h,
end
example {α β : Type} (e : α ≃ β) (P : α → Prop) (h : ∀ a : α, P a) (b : β) : P (e.symm b) :=
begin
equiv_rw e.symm at b,
exact h b,
end
example {α β : Type} (e : α ≃ β) (P : α → Sort*) (h : Π a : α, P a) (b : β) : P (e.symm b) :=
begin
-- this is a bit perverse, as `equiv_rw e.symm at b` is more natural,
-- but this tests rewriting in the argument of a dependent function
equiv_rw e at h,
exact h _,
end
-- a poor example, rewriting in the base of a dependent pair
example {α β : Type} (P : α → Type) (h : Σ a, P a) (e : α ≃ β) : β :=
begin
equiv_rw e at h,
exact h.1
end
-- rewriting in the argument of a dependent function can't be done in one step
example {α β γ : Type} (e : α ≃ β) (P : α → Type*) (h : Π a : α, (P a) × (option α)) (b : β) :
option β :=
begin
equiv_rw e at h,
have t := h b,
equiv_rw e at t,
exact t.2,
end
-- Demonstrate using `equiv_rw` to build new instances of `equiv_functor`
-- Observe that the next three declarations could easily be implemented by a tactic.
-- This has been automated in the `transport` branch,
-- so we won't attempt to write a deriver handler until we join with that.
def semigroup.map {α β : Type} (e : α ≃ β) : semigroup α → semigroup β :=
begin
intro S,
refine_struct { .. },
-- transport data fields using `equiv_rw`
{ have mul := S.mul,
equiv_rw e at mul,
-- This `equiv_rw` performs the following steps:
-- have e' := (equiv.arrow_congr' e (equiv.arrow_congr' e e)),
-- have h := (e'.symm_apply_apply mul).symm,
-- revert h,
-- generalize : (e' mul) = mul',
-- intro h,
-- clear_dependent mul,
-- rename mul' mul,
exact mul,
},
-- transport axioms by simplifying, and applying the original axiom
{ intros, dsimp, simp, apply S.mul_assoc, }
end
example {α β : Type} (e : α ≃ β) (S : semigroup α) :
(semigroup.map e S).mul =
(equiv.arrow_congr' e (equiv.arrow_congr' e e)) has_mul.mul :=
rfl
example {α β : Type} (e : α ≃ β) (S : semigroup α) (x y : β) :
begin
haveI := semigroup.map e S,
exact x * y = e (e.symm x * e.symm y)
end :=
rfl
attribute [ext] semigroup
lemma semigroup.id_map (α : Type) : semigroup.map (equiv.refl α) = id :=
by { ext, refl, }
lemma semigroup.map_map {α β γ : Type} (e : α ≃ β) (f : β ≃ γ) :
semigroup.map (e.trans f) = (semigroup.map f) ∘ (semigroup.map e) :=
by { ext, dsimp [semigroup.map], simp, }
-- TODO (after joining the `transport` branch) create a derive handler for this
instance : equiv_functor semigroup :=
{ map := λ α β e, semigroup.map e,
map_refl' := semigroup.id_map,
map_trans' := λ α β γ e f, semigroup.map_map e f, }
-- Verify that we can now use `equiv_rw` under `semigroup`:
example {α : Type} [I : semigroup α] {β : Type} (e : α ≃ β) : semigroup β :=
begin
equiv_rw e at I,
exact I,
end
-- Now we do `monoid`, to try out a structure with constants.
-- The constructions and proofs here are written as uniformly as possible.
-- This example is the blueprint for the `transport` tactic.
mk_simp_attribute transport_simps "simps useful inside `transport`"
attribute [transport_simps]
eq_rec_constant
eq_mpr_rfl
equiv.to_fun_as_coe
equiv.arrow_congr'_apply
equiv.symm_apply_apply
equiv.apply_eq_iff_eq_symm_apply
def monoid.map {α β : Type} (e : α ≃ β) (S : monoid α) : monoid β :=
begin
refine_struct { .. },
{ have mul := S.mul, equiv_rw e at mul, exact mul, },
{ try { unfold_projs },
simp only with transport_simps,
have mul_assoc := S.mul_assoc,
equiv_rw e at mul_assoc,
solve_by_elim, },
{ have one := S.one, equiv_rw e at one, exact one, },
{ try { unfold_projs },
simp only with transport_simps,
have one_mul := S.one_mul,
equiv_rw e at one_mul,
solve_by_elim, },
{ try { unfold_projs },
simp only with transport_simps,
have mul_one := S.mul_one,
equiv_rw e at mul_one,
solve_by_elim, },
end
example {α β : Type} (e : α ≃ β) (S : monoid α) :
(monoid.map e S).mul =
(equiv.arrow_congr' e (equiv.arrow_congr' e e)) has_mul.mul :=
rfl
example {α β : Type} (e : α ≃ β) (S : monoid α) (x y : β) :
begin
haveI := monoid.map e S,
exact x * y = e (e.symm x * e.symm y)
end :=
rfl
example {α β : Type} (e : α ≃ β) (S : monoid α) :
begin
haveI := monoid.map e S,
exact (1 : β) = e (1 : α)
end :=
rfl
|
2316ba162a5aa34eb292ff5451b3158df8195223 | 05f637fa14ac28031cb1ea92086a0f4eb23ff2b1 | /tests/lean/alias1.lean | 8b07b25a10486528bd3e3424c995eb99e5cf7587 | [
"Apache-2.0"
] | permissive | codyroux/lean0.1 | 1ce92751d664aacff0529e139083304a7bbc8a71 | 0dc6fb974aa85ed6f305a2f4b10a53a44ee5f0ef | refs/heads/master | 1,610,830,535,062 | 1,402,150,480,000 | 1,402,150,480,000 | 19,588,851 | 2 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 54 | lean | alias BB : Bool.
variable x : BB.
print environment 1. |
7006ea84e3db611098a7db851f91157171672680 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /src/Init/Prelude.lean | 4e2c2b9f8eec0e38e0acc83e2743efcdeaa5081d | [
"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 | 177,361 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
prelude -- Don't import Init, because we're in Init itself
set_option linter.missingDocs true -- keep it documented
/-!
# Init.Prelude
This is the first file in the lean import hierarchy. It is responsible for setting
up basic definitions, most of which lean already has "built in knowledge" about,
so it is important that they be set up in exactly this way. (For example, lean will
use `PUnit` in the desugaring of `do` notation, or in the pattern match compiler.)
-/
universe u v w
/--
The identity function. `id` takes an implicit argument `α : Sort u`
(a type in any universe), and an argument `a : α`, and returns `a`.
Although this may look like a useless function, one application of the identity
function is to explicitly put a type on an expression. If `e` has type `T`,
and `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean
knows that this expression has type `T'` rather than `T`. This can make a
difference for typeclass inference, since `T` and `T'` may have different
typeclass instances on them. `show T' from e` is sugar for an `@id T' e`
expression.
-/
@[inline] def id {α : Sort u} (a : α) : α := a
/--
Function composition is the act of pipelining the result of one function, to the input of another, creating an entirely new function.
Example:
```
#eval Function.comp List.reverse (List.drop 2) [3, 2, 4, 1]
-- [1, 4]
```
You can use the notation `f ∘ g` as shorthand for `Function.comp f g`.
```
#eval (List.reverse ∘ List.drop 2) [3, 2, 4, 1]
-- [1, 4]
```
A simpler way of thinking about it, is that `List.reverse ∘ List.drop 2`
is equivalent to `fun xs => List.reverse (List.drop 2 xs)`,
the benefit is that the meaning of composition is obvious,
and the representation is compact.
-/
@[inline] def Function.comp {α : Sort u} {β : Sort v} {δ : Sort w} (f : β → δ) (g : α → β) : α → δ :=
fun x => f (g x)
/--
The constant function. If `a : α`, then `Function.const β a : β → α` is the
"constant function with value `a`", that is, `Function.const β a b = a`.
```
example (b : Bool) : Function.const Bool 10 b = 10 :=
rfl
#check Function.const Bool 10
-- Bool → Nat
```
-/
@[inline] def Function.const {α : Sort u} (β : Sort v) (a : α) : β → α :=
fun _ => a
set_option checkBinderAnnotations false in
/--
`inferInstance` synthesizes a value of any target type by typeclass
inference. This function has the same type signature as the identity
function, but the square brackets on the `[i : α]` argument means that it will
attempt to construct this argument by typeclass inference. (This will fail if
`α` is not a `class`.) Example:
```
#check (inferInstance : Inhabited Nat) -- Inhabited Nat
def foo : Inhabited (Nat × Nat) :=
inferInstance
example : foo.default = (default, default) :=
rfl
```
-/
abbrev inferInstance {α : Sort u} [i : α] : α := i
set_option checkBinderAnnotations false in
/-- `inferInstanceAs α` synthesizes a value of any target type by typeclass
inference. This is just like `inferInstance` except that `α` is given
explicitly instead of being inferred from the target type. It is especially
useful when the target type is some `α'` which is definitionally equal to `α`,
but the instance we are looking for is only registered for `α` (because
typeclass search does not unfold most definitions, but definitional equality
does.) Example:
```
#check inferInstanceAs (Inhabited Nat) -- Inhabited Nat
```
-/
abbrev inferInstanceAs (α : Sort u) [i : α] : α := i
set_option bootstrap.inductiveCheckResultingUniverse false in
/--
The unit type, the canonical type with one element, named `unit` or `()`.
This is the universe-polymorphic version of `Unit`; it is preferred to use
`Unit` instead where applicable.
For more information about universe levels: [Types as objects](https://leanprover.github.io/theorem_proving_in_lean4/dependent_type_theory.html#types-as-objects)
-/
inductive PUnit : Sort u where
/-- `PUnit.unit : PUnit` is the canonical element of the unit type. -/
| unit : PUnit
/--
The unit type, the canonical type with one element, named `unit` or `()`.
In other words, it describes only a single value, which consists of said constructor applied
to no arguments whatsoever.
The `Unit` type is similar to `void` in languages derived from C.
`Unit` is actually defined as `PUnit.{0}` where `PUnit` is the universe
polymorphic version. The `Unit` should be preferred over `PUnit` where possible to avoid
unnecessary universe parameters.
In functional programming, `Unit` is the return type of things that "return
nothing", since a type with one element conveys no additional information.
When programming with monads, the type `m Unit` represents an action that has
some side effects but does not return a value, while `m α` would be an action
that has side effects and returns a value of type `α`.
-/
abbrev Unit : Type := PUnit
/--
`Unit.unit : Unit` is the canonical element of the unit type.
It can also be written as `()`.
-/
@[match_pattern] abbrev Unit.unit : Unit := PUnit.unit
/-- Marker for information that has been erased by the code generator. -/
unsafe axiom lcErased : Type
/--
Auxiliary unsafe constant used by the Compiler when erasing proofs from code.
It may look strange to have an axiom that says "every proposition is true",
since this is obviously unsound, but the `unsafe` marker ensures that the
kernel will not let this through into regular proofs. The lower levels of the
code generator don't need proofs in terms, so this is used to stub the proofs
out.
-/
unsafe axiom lcProof {α : Prop} : α
/--
Auxiliary unsafe constant used by the Compiler when erasing casts.
-/
unsafe axiom lcCast {α : Sort u} {β : Sort v} (a : α) : β
/--
Auxiliary unsafe constant used by the Compiler to mark unreachable code.
Like `lcProof`, this is an `unsafe axiom`, which means that even though it is
not sound, the kernel will not let us use it for regular proofs.
Executing this expression to actually synthesize a value of type `α` causes
**immediate undefined behavior**, and the compiler does take advantage of this
to optimize the code assuming that it is not called. If it is not optimized out,
it is likely to appear as a print message saying "unreachable code", but this
behavior is not guaranteed or stable in any way.
-/
unsafe axiom lcUnreachable {α : Sort u} : α
/--
`True` is a proposition and has only an introduction rule, `True.intro : True`.
In other words, `True` is simply true, and has a canonical proof, `True.intro`
For more information: [Propositional Logic](https://leanprover.github.io/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
-/
inductive True : Prop where
/-- `True` is true, and `True.intro` (or more commonly, `trivial`)
is the proof. -/
| intro : True
/--
`False` is the empty proposition. Thus, it has no introduction rules.
It represents a contradiction. `False` elimination rule, `False.rec`,
expresses the fact that anything follows from a contradiction.
This rule is sometimes called ex falso (short for ex falso sequitur quodlibet),
or the principle of explosion.
For more information: [Propositional Logic](https://leanprover.github.io/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
-/
inductive False : Prop
/--
The empty type. It has no constructors. The `Empty.rec`
eliminator expresses the fact that anything follows from the empty type.
-/
inductive Empty : Type
set_option bootstrap.inductiveCheckResultingUniverse false in
/--
The universe-polymorphic empty type. Prefer `Empty` or `False` where
possible.
-/
inductive PEmpty : Sort u where
/--
`Not p`, or `¬p`, is the negation of `p`. It is defined to be `p → False`,
so if your goal is `¬p` you can use `intro h` to turn the goal into
`h : p ⊢ False`, and if you have `hn : ¬p` and `h : p` then `hn h : False`
and `(hn h).elim` will prove anything.
For more information: [Propositional Logic](https://leanprover.github.io/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
-/
def Not (a : Prop) : Prop := a → False
/--
`False.elim : False → C` says that from `False`, any desired proposition
`C` holds. Also known as ex falso quodlibet (EFQ) or the principle of explosion.
The target type is actually `C : Sort u` which means it works for both
propositions and types. When executed, this acts like an "unreachable"
instruction: it is **undefined behavior** to run, but it will probably print
"unreachable code". (You would need to construct a proof of false to run it
anyway, which you can only do using `sorry` or unsound axioms.)
-/
@[macro_inline] def False.elim {C : Sort u} (h : False) : C :=
h.rec
/--
Anything follows from two contradictory hypotheses. Example:
```
example (hp : p) (hnp : ¬p) : q := absurd hp hnp
```
For more information: [Propositional Logic](https://leanprover.github.io/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
-/
@[macro_inline] def absurd {a : Prop} {b : Sort v} (h₁ : a) (h₂ : Not a) : b :=
(h₂ h₁).rec
/--
The equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://leanprover.github.io/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
-/
inductive Eq : α → α → Prop where
/-- `Eq.refl a : a = a` is reflexivity, the unique constructor of the
equality type. See also `rfl`, which is usually used instead. -/
| refl (a : α) : Eq a a
/-- Non-dependent recursor for the equality type. -/
@[simp] abbrev Eq.ndrec.{u1, u2} {α : Sort u2} {a : α} {motive : α → Sort u1} (m : motive a) {b : α} (h : Eq a b) : motive b :=
h.rec m
/--
`rfl : a = a` is the unique constructor of the equality type. This is the
same as `Eq.refl` except that it takes `a` implicitly instead of explicitly.
This is a more powerful theorem than it may appear at first, because although
the statement of the theorem is `a = a`, lean will allow anything that is
definitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in
lean by `rfl`, because both sides are the same up to definitional equality.
-/
@[match_pattern] def rfl {α : Sort u} {a : α} : Eq a a := Eq.refl a
/-- `id x = x`, as a `@[simp]` lemma. -/
@[simp] theorem id_eq (a : α) : Eq (id a) a := rfl
/--
The substitution principle for equality. If `a = b ` and `P a` holds,
then `P b` also holds. We conventionally use the name `motive` for `P` here,
so that you can specify it explicitly using e.g.
`Eq.subst (motive := fun x => x < 5)` if it is not otherwise inferred correctly.
This theorem is the underlying mechanism behind the `rw` tactic, which is
essentially a fancy algorithm for finding good `motive` arguments to usefully
apply this theorem to replace occurrences of `a` with `b` in the goal or
hypotheses.
For more information: [Equality](https://leanprover.github.io/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
-/
theorem Eq.subst {α : Sort u} {motive : α → Prop} {a b : α} (h₁ : Eq a b) (h₂ : motive a) : motive b :=
Eq.ndrec h₂ h₁
/--
Equality is symmetric: if `a = b` then `b = a`.
Because this is in the `Eq` namespace, if you have a variable `h : a = b`,
`h.symm` can be used as shorthand for `Eq.symm h` as a proof of `b = a`.
For more information: [Equality](https://leanprover.github.io/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
-/
theorem Eq.symm {α : Sort u} {a b : α} (h : Eq a b) : Eq b a :=
h ▸ rfl
/--
Equality is transitive: if `a = b` and `b = c` then `a = c`.
Because this is in the `Eq` namespace, if you have variables or expressions
`h₁ : a = b` and `h₂ : b = c`, you can use `h₁.trans h₂ : a = c` as shorthand
for `Eq.trans h₁ h₂`.
For more information: [Equality](https://leanprover.github.io/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
-/
theorem Eq.trans {α : Sort u} {a b c : α} (h₁ : Eq a b) (h₂ : Eq b c) : Eq a c :=
h₂ ▸ h₁
/--
Cast across a type equality. If `h : α = β` is an equality of types, and
`a : α`, then `a : β` will usually not typecheck directly, but this function
will allow you to work around this and embed `a` in type `β` as `cast h a : β`.
It is best to avoid this function if you can, because it is more complicated
to reason about terms containing casts, but if the types don't match up
definitionally sometimes there isn't anything better you can do.
For more information: [Equality](https://leanprover.github.io/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
-/
@[macro_inline] def cast {α β : Sort u} (h : Eq α β) (a : α) : β :=
h.rec a
/--
Congruence in the function argument: if `a₁ = a₂` then `f a₁ = f a₂` for
any (nondependent) function `f`. This is more powerful than it might look at first, because
you can also use a lambda expression for `f` to prove that
`<something containing a₁> = <something containing a₂>`. This function is used
internally by tactics like `congr` and `simp` to apply equalities inside
subterms.
For more information: [Equality](https://leanprover.github.io/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
-/
theorem congrArg {α : Sort u} {β : Sort v} {a₁ a₂ : α} (f : α → β) (h : Eq a₁ a₂) : Eq (f a₁) (f a₂) :=
h ▸ rfl
/--
Congruence in both function and argument. If `f₁ = f₂` and `a₁ = a₂` then
`f₁ a₁ = f₂ a₂`. This only works for nondependent functions; the theorem
statement is more complex in the dependent case.
For more information: [Equality](https://leanprover.github.io/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
-/
theorem congr {α : Sort u} {β : Sort v} {f₁ f₂ : α → β} {a₁ a₂ : α} (h₁ : Eq f₁ f₂) (h₂ : Eq a₁ a₂) : Eq (f₁ a₁) (f₂ a₂) :=
h₁ ▸ h₂ ▸ rfl
/-- Congruence in the function part of an application: If `f = g` then `f a = g a`. -/
theorem congrFun {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x} (h : Eq f g) (a : α) : Eq (f a) (g a) :=
h ▸ rfl
/-!
Initialize the Quotient Module, which effectively adds the following definitions:
```
opaque Quot {α : Sort u} (r : α → α → Prop) : Sort u
opaque Quot.mk {α : Sort u} (r : α → α → Prop) (a : α) : Quot r
opaque Quot.lift {α : Sort u} {r : α → α → Prop} {β : Sort v} (f : α → β) :
(∀ a b : α, r a b → Eq (f a) (f b)) → Quot r → β
opaque Quot.ind {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop} :
(∀ a : α, β (Quot.mk r a)) → ∀ q : Quot r, β q
```
-/
init_quot
/--
Let `α` be any type, and let `r` be an equivalence relation on `α`.
It is mathematically common to form the "quotient" `α / r`, that is, the type of
elements of `α` "modulo" `r`. Set theoretically, one can view `α / r` as the set
of equivalence classes of `α` modulo `r`. If `f : α → β` is any function that
respects the equivalence relation in the sense that for every `x y : α`,
`r x y` implies `f x = f y`, then f "lifts" to a function `f' : α / r → β`
defined on each equivalence class `⟦x⟧` by `f' ⟦x⟧ = f x`.
Lean extends the Calculus of Constructions with additional constants that
perform exactly these constructions, and installs this last equation as a
definitional reduction rule.
Given a type `α` and any binary relation `r` on `α`, `Quot r` is a type. Note
that `r` is not required to be an equivalence relation. `Quot` is the basic
building block used to construct later the type `Quotient`.
-/
add_decl_doc Quot
/--
Given a type `α` and any binary relation `r` on `α`, `Quot.mk` maps `α` to `Quot r`.
So that if `r : α → α → Prop` and `a : α`, then `Quot.mk r a` is an element of `Quot r`.
See `Quot`.
-/
add_decl_doc Quot.mk
/--
Given a type `α` and any binary relation `r` on `α`,
`Quot.ind` says that every element of `Quot r` is of the form `Quot.mk r a`.
See `Quot` and `Quot.lift`.
-/
add_decl_doc Quot.ind
/--
Given a type `α`, any binary relation `r` on `α`, a function `f : α → β`, and a proof `h`
that `f` respects the relation `r`, then `Quot.lift f h` is the corresponding function on `Quot r`.
The idea is that for each element `a` in `α`, the function `Quot.lift f h` maps `Quot.mk r a`
(the `r`-class containing `a`) to `f a`, wherein `h` shows that this function is well defined.
In fact, the computation principle is declared as a reduction rule.
-/
add_decl_doc Quot.lift
/--
Unsafe auxiliary constant used by the compiler to erase `Quot.lift`.
-/
unsafe axiom Quot.lcInv {α : Sort u} {r : α → α → Prop} (q : Quot r) : α
/--
Heterogeneous equality. `HEq a b` asserts that `a` and `b` have the same
type, and casting `a` across the equality yields `b`, and vice versa.
You should avoid using this type if you can. Heterogeneous equality does not
have all the same properties as `Eq`, because the assumption that the types of
`a` and `b` are equal is often too weak to prove theorems of interest. One
important non-theorem is the analogue of `congr`: If `HEq f g` and `HEq x y`
and `f x` and `g y` are well typed it does not follow that `HEq (f x) (g y)`.
(This does follow if you have `f = g` instead.) However if `a` and `b` have
the same type then `a = b` and `HEq a b` are equivalent.
-/
inductive HEq : {α : Sort u} → α → {β : Sort u} → β → Prop where
/-- Reflexivity of heterogeneous equality. -/
| refl (a : α) : HEq a a
/-- A version of `HEq.refl` with an implicit argument. -/
@[match_pattern] protected def HEq.rfl {α : Sort u} {a : α} : HEq a a :=
HEq.refl a
theorem eq_of_heq {α : Sort u} {a a' : α} (h : HEq a a') : Eq a a' :=
have : (α β : Sort u) → (a : α) → (b : β) → HEq a b → (h : Eq α β) → Eq (cast h a) b :=
fun _ _ _ _ h₁ =>
h₁.rec (fun _ => rfl)
this α α a a' h rfl
/--
Product type (aka pair). You can use `α × β` as notation for `Prod α β`.
Given `a : α` and `b : β`, `Prod.mk a b : Prod α β`. You can use `(a, b)`
as notation for `Prod.mk a b`. Moreover, `(a, b, c)` is notation for
`Prod.mk a (Prod.mk b c)`.
Given `p : Prod α β`, `p.1 : α` and `p.2 : β`. They are short for `Prod.fst p`
and `Prod.snd p` respectively. You can also write `p.fst` and `p.snd`.
For more information: [Constructors with Arguments](https://leanprover.github.io/theorem_proving_in_lean4/inductive_types.html?highlight=Prod#constructors-with-arguments)
-/
structure Prod (α : Type u) (β : Type v) where
/-- The first projection out of a pair. if `p : α × β` then `p.1 : α`. -/
fst : α
/-- The second projection out of a pair. if `p : α × β` then `p.2 : β`. -/
snd : β
attribute [unbox] Prod
/--
Similar to `Prod`, but `α` and `β` can be propositions.
We use this type internally to automatically generate the `brecOn` recursor.
-/
structure PProd (α : Sort u) (β : Sort v) where
/-- The first projection out of a pair. if `p : PProd α β` then `p.1 : α`. -/
fst : α
/-- The second projection out of a pair. if `p : PProd α β` then `p.2 : β`. -/
snd : β
/--
Similar to `Prod`, but `α` and `β` are in the same universe.
We say `MProd` is the universe monomorphic product type.
-/
structure MProd (α β : Type u) where
/-- The first projection out of a pair. if `p : MProd α β` then `p.1 : α`. -/
fst : α
/-- The second projection out of a pair. if `p : MProd α β` then `p.2 : β`. -/
snd : β
/--
`And a b`, or `a ∧ b`, is the conjunction of propositions. It can be
constructed and destructed like a pair: if `ha : a` and `hb : b` then
`⟨ha, hb⟩ : a ∧ b`, and if `h : a ∧ b` then `h.left : a` and `h.right : b`.
-/
structure And (a b : Prop) : Prop where
/-- `And.intro : a → b → a ∧ b` is the constructor for the And operation. -/
intro ::
/-- Extract the left conjunct from a conjunction. `h : a ∧ b` then
`h.left`, also notated as `h.1`, is a proof of `a`. -/
left : a
/-- Extract the right conjunct from a conjunction. `h : a ∧ b` then
`h.right`, also notated as `h.2`, is a proof of `b`. -/
right : b
/--
`Or a b`, or `a ∨ b`, is the disjunction of propositions. There are two
constructors for `Or`, called `Or.inl : a → a ∨ b` and `Or.inr : b → a ∨ b`,
and you can use `match` or `cases` to destruct an `Or` assumption into the
two cases.
-/
inductive Or (a b : Prop) : Prop where
/-- `Or.inl` is "left injection" into an `Or`. If `h : a` then `Or.inl h : a ∨ b`. -/
| inl (h : a) : Or a b
/-- `Or.inr` is "right injection" into an `Or`. If `h : b` then `Or.inr h : a ∨ b`. -/
| inr (h : b) : Or a b
/-- Alias for `Or.inl`. -/
theorem Or.intro_left (b : Prop) (h : a) : Or a b :=
Or.inl h
/-- Alias for `Or.inr`. -/
theorem Or.intro_right (a : Prop) (h : b) : Or a b :=
Or.inr h
/--
Proof by cases on an `Or`. If `a ∨ b`, and both `a` and `b` imply
proposition `c`, then `c` is true.
-/
theorem Or.elim {c : Prop} (h : Or a b) (left : a → c) (right : b → c) : c :=
match h with
| Or.inl h => left h
| Or.inr h => right h
/--
`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
-/
inductive Bool : Type where
/-- The boolean value `false`, not to be confused with the proposition `False`. -/
| false : Bool
/-- The boolean value `true`, not to be confused with the proposition `True`. -/
| true : Bool
export Bool (false true)
/--
`Subtype p`, usually written as `{x : α // p x}`, is a type which
represents all the elements `x : α` for which `p x` is true. It is structurally
a pair-like type, so if you have `x : α` and `h : p x` then
`⟨x, h⟩ : {x // p x}`. An element `s : {x // p x}` will coerce to `α` but
you can also make it explicit using `s.1` or `s.val`.
-/
structure Subtype {α : Sort u} (p : α → Prop) where
/-- If `s : {x // p x}` then `s.val : α` is the underlying element in the base
type. You can also write this as `s.1`, or simply as `s` when the type is
known from context. -/
val : α
/-- If `s : {x // p x}` then `s.2` or `s.property` is the assertion that
`p s.1`, that is, that `s` is in fact an element for which `p` holds. -/
property : p val
set_option linter.unusedVariables.funArgs false in
/--
Gadget for optional parameter support.
A binder like `(x : α := default)` in a declaration is syntax sugar for
`x : optParam α default`, and triggers the elaborator to attempt to use
`default` to supply the argument if it is not supplied.
-/
@[reducible] def optParam (α : Sort u) (default : α) : Sort u := α
/--
Gadget for marking output parameters in type classes.
For example, the `Membership` class is defined as:
```
class Membership (α : outParam (Type u)) (γ : Type v)
```
This means that whenever a typeclass goal of the form `Membership ?α ?γ` comes
up, lean will wait to solve it until `?γ` is known, but then it will run
typeclass inference, and take the first solution it finds, for any value of `?α`,
which thereby determines what `?α` should be.
This expresses that in a term like `a ∈ s`, `s` might be a `Set α` or
`List α` or some other type with a membership operation, and in each case
the "member" type `α` is determined by looking at the container type.
-/
@[reducible] def outParam (α : Sort u) : Sort u := α
/--
Gadget for marking semi output parameters in type classes.
Semi-output parameters influence the order in which arguments to type class
instances are processed. Lean determines an order where all non-(semi-)output
parameters to the instance argument have to be figured out before attempting to
synthesize an argument (that is, they do not contain assignable metavariables
created during TC synthesis). This rules out instances such as `[Mul β] : Add
α` (because `β` could be anything). Marking a parameter as semi-output is a
promise that instances of the type class will always fill in a value for that
parameter.
For example, the `Coe` class is defined as:
```
class Coe (α : semiOutParam (Sort u)) (β : Sort v)
```
This means that all `Coe` instances should provide a concrete value for `α`
(i.e., not an assignable metavariable). An instance like `Coe Nat Int` or `Coe
α (Option α)` is fine, but `Coe α Nat` is not since it does not provide a value
for `α`.
-/
@[reducible] def semiOutParam (α : Sort u) : Sort u := α
set_option linter.unusedVariables.funArgs false in
/-- Auxiliary declaration used to implement named patterns like `x@h:p`. -/
@[reducible] def namedPattern {α : Sort u} (x a : α) (h : Eq x a) : α := a
/--
Auxiliary axiom used to implement `sorry`.
The `sorry` term/tactic expands to `sorryAx _ (synthetic := false)`. This is a
proof of anything, which is intended for stubbing out incomplete parts of a
proof while still having a syntactically correct proof skeleton. Lean will give
a warning whenever a proof uses `sorry`, so you aren't likely to miss it, but
you can double check if a theorem depends on `sorry` by using
`#print axioms my_thm` and looking for `sorryAx` in the axiom list.
The `synthetic` flag is false when written explicitly by the user, but it is
set to `true` when a tactic fails to prove a goal, or if there is a type error
in the expression. A synthetic `sorry` acts like a regular one, except that it
suppresses follow-up errors in order to prevent one error from causing a cascade
of other errors because the desired term was not constructed.
-/
@[extern "lean_sorry", never_extract]
axiom sorryAx (α : Sort u) (synthetic := false) : α
theorem eq_false_of_ne_true : {b : Bool} → Not (Eq b true) → Eq b false
| true, h => False.elim (h rfl)
| false, _ => rfl
theorem eq_true_of_ne_false : {b : Bool} → Not (Eq b false) → Eq b true
| true, _ => rfl
| false, h => False.elim (h rfl)
theorem ne_false_of_eq_true : {b : Bool} → Eq b true → Not (Eq b false)
| true, _ => fun h => Bool.noConfusion h
| false, h => Bool.noConfusion h
theorem ne_true_of_eq_false : {b : Bool} → Eq b false → Not (Eq b true)
| true, h => Bool.noConfusion h
| false, _ => fun h => Bool.noConfusion h
/--
`Inhabited α` is a typeclass that says that `α` has a designated element,
called `(default : α)`. This is sometimes referred to as a "pointed type".
This class is used by functions that need to return a value of the type
when called "out of domain". For example, `Array.get! arr i : α` returns
a value of type `α` when `arr : Array α`, but if `i` is not in range of
the array, it reports a panic message, but this does not halt the program,
so it must still return a value of type `α` (and in fact this is required
for logical consistency), so in this case it returns `default`.
-/
class Inhabited (α : Sort u) where
/-- `default` is a function that produces a "default" element of any
`Inhabited` type. This element does not have any particular specified
properties, but it is often an all-zeroes value. -/
default : α
export Inhabited (default)
/--
`Nonempty α` is a typeclass that says that `α` is not an empty type,
that is, there exists an element in the type. It differs from `Inhabited α`
in that `Nonempty α` is a `Prop`, which means that it does not actually carry
an element of `α`, only a proof that *there exists* such an element.
Given `Nonempty α`, you can construct an element of `α` *nonconstructively*
using `Classical.choice`.
-/
class inductive Nonempty (α : Sort u) : Prop where
/-- If `val : α`, then `α` is nonempty. -/
| intro (val : α) : Nonempty α
/--
**The axiom of choice**. `Nonempty α` is a proof that `α` has an element,
but the element itself is erased. The axiom `choice` supplies a particular
element of `α` given only this proof.
The textbook axiom of choice normally makes a family of choices all at once,
but that is implied from this formulation, because if `α : ι → Type` is a
family of types and `h : ∀ i, Nonempty (α i)` is a proof that they are all
nonempty, then `fun i => Classical.choice (h i) : ∀ i, α i` is a family of
chosen elements. This is actually a bit stronger than the ZFC choice axiom;
this is sometimes called "[global choice](https://en.wikipedia.org/wiki/Axiom_of_global_choice)".
In lean, we use the axiom of choice to derive the law of excluded middle
(see `Classical.em`), so it will often show up in axiom listings where you
may not expect. You can use `#print axioms my_thm` to find out if a given
theorem depends on this or other axioms.
This axiom can be used to construct "data", but obviously there is no algorithm
to compute it, so lean will require you to mark any definition that would
involve executing `Classical.choice` or other axioms as `noncomputable`, and
will not produce any executable code for such definitions.
-/
axiom Classical.choice {α : Sort u} : Nonempty α → α
/--
The elimination principle for `Nonempty α`. If `Nonempty α`, and we can
prove `p` given any element `x : α`, then `p` holds. Note that it is essential
that `p` is a `Prop` here; the version with `p` being a `Sort u` is equivalent
to `Classical.choice`.
-/
protected def Nonempty.elim {α : Sort u} {p : Prop} (h₁ : Nonempty α) (h₂ : α → p) : p :=
match h₁ with
| intro a => h₂ a
instance {α : Sort u} [Inhabited α] : Nonempty α :=
⟨default⟩
/--
A variation on `Classical.choice` that uses typeclass inference to
infer the proof of `Nonempty α`.
-/
noncomputable def Classical.ofNonempty {α : Sort u} [Nonempty α] : α :=
Classical.choice inferInstance
instance (α : Sort u) {β : Sort v} [Nonempty β] : Nonempty (α → β) :=
Nonempty.intro fun _ => Classical.ofNonempty
instance (α : Sort u) {β : α → Sort v} [(a : α) → Nonempty (β a)] : Nonempty ((a : α) → β a) :=
Nonempty.intro fun _ => Classical.ofNonempty
instance : Inhabited (Sort u) where
default := PUnit
instance (α : Sort u) {β : Sort v} [Inhabited β] : Inhabited (α → β) where
default := fun _ => default
instance (α : Sort u) {β : α → Sort v} [(a : α) → Inhabited (β a)] : Inhabited ((a : α) → β a) where
default := fun _ => default
deriving instance Inhabited for Bool
/-- Universe lifting operation from `Sort u` to `Type u`. -/
structure PLift (α : Sort u) : Type u where
/-- Lift a value into `PLift α` -/ up ::
/-- Extract a value from `PLift α` -/ down : α
/-- Bijection between `α` and `PLift α` -/
theorem PLift.up_down {α : Sort u} (b : PLift α) : Eq (up (down b)) b := rfl
/-- Bijection between `α` and `PLift α` -/
theorem PLift.down_up {α : Sort u} (a : α) : Eq (down (up a)) a := rfl
/--
`NonemptyType.{u}` is the type of nonempty types in universe `u`.
It is mainly used in constant declarations where we wish to introduce a type
and simultaneously assert that it is nonempty, but otherwise make the type
opaque.
-/
def NonemptyType := Subtype fun α : Type u => Nonempty α
/-- The underlying type of a `NonemptyType`. -/
abbrev NonemptyType.type (type : NonemptyType.{u}) : Type u :=
type.val
/-- `NonemptyType` is inhabited, because `PUnit` is a nonempty type. -/
instance : Inhabited NonemptyType.{u} where
default := ⟨PUnit, ⟨⟨⟩⟩⟩
/--
Universe lifting operation from a lower `Type` universe to a higher one.
To express this using level variables, the input is `Type s` and the output is
`Type (max s r)`, so if `s ≤ r` then the latter is (definitionally) `Type r`.
The universe variable `r` is written first so that `ULift.{r} α` can be used
when `s` can be inferred from the type of `α`.
-/
structure ULift.{r, s} (α : Type s) : Type (max s r) where
/-- Lift a value into `ULift α` -/ up ::
/-- Extract a value from `ULift α` -/ down : α
/-- Bijection between `α` and `ULift.{v} α` -/
theorem ULift.up_down {α : Type u} (b : ULift.{v} α) : Eq (up (down b)) b := rfl
/-- Bijection between `α` and `ULift.{v} α` -/
theorem ULift.down_up {α : Type u} (a : α) : Eq (down (up.{v} a)) a := rfl
/--
`Decidable p` is a data-carrying class that supplies a proof that `p` is
either `true` or `false`. It is equivalent to `Bool` (and in fact it has the
same code generation as `Bool`) together with a proof that the `Bool` is
true iff `p` is.
`Decidable` instances are used to infer "computation strategies" for
propositions, so that you can have the convenience of writing propositions
inside `if` statements and executing them (which actually executes the inferred
decidability instance instead of the proposition, which has no code).
If a proposition `p` is `Decidable`, then `(by decide : p)` will prove it by
evaluating the decidability instance to `isTrue h` and returning `h`.
-/
class inductive Decidable (p : Prop) where
/-- Prove that `p` is decidable by supplying a proof of `¬p` -/
| isFalse (h : Not p) : Decidable p
/-- Prove that `p` is decidable by supplying a proof of `p` -/
| isTrue (h : p) : Decidable p
/--
Convert a decidable proposition into a boolean value.
If `p : Prop` is decidable, then `decide p : Bool` is the boolean value
which is `true` if `p` is true and `false` if `p` is false.
-/
@[inline_if_reduce, nospecialize] def Decidable.decide (p : Prop) [h : Decidable p] : Bool :=
h.casesOn (fun _ => false) (fun _ => true)
export Decidable (isTrue isFalse decide)
/-- A decidable predicate. See `Decidable`. -/
abbrev DecidablePred {α : Sort u} (r : α → Prop) :=
(a : α) → Decidable (r a)
/-- A decidable relation. See `Decidable`. -/
abbrev DecidableRel {α : Sort u} (r : α → α → Prop) :=
(a b : α) → Decidable (r a b)
/--
Asserts that `α` has decidable equality, that is, `a = b` is decidable
for all `a b : α`. See `Decidable`.
-/
abbrev DecidableEq (α : Sort u) :=
(a b : α) → Decidable (Eq a b)
/-- Proves that `a = b` is decidable given `DecidableEq α`. -/
def decEq {α : Sort u} [inst : DecidableEq α] (a b : α) : Decidable (Eq a b) :=
inst a b
set_option linter.unusedVariables false in
theorem decide_eq_true : [inst : Decidable p] → p → Eq (decide p) true
| isTrue _, _ => rfl
| isFalse h₁, h₂ => absurd h₂ h₁
theorem decide_eq_false : [Decidable p] → Not p → Eq (decide p) false
| isTrue h₁, h₂ => absurd h₁ h₂
| isFalse _, _ => rfl
theorem of_decide_eq_true [inst : Decidable p] : Eq (decide p) true → p := fun h =>
match (generalizing := false) inst with
| isTrue h₁ => h₁
| isFalse h₁ => absurd h (ne_true_of_eq_false (decide_eq_false h₁))
theorem of_decide_eq_false [inst : Decidable p] : Eq (decide p) false → Not p := fun h =>
match (generalizing := false) inst with
| isTrue h₁ => absurd h (ne_false_of_eq_true (decide_eq_true h₁))
| isFalse h₁ => h₁
theorem of_decide_eq_self_eq_true [inst : DecidableEq α] (a : α) : Eq (decide (Eq a a)) true :=
match (generalizing := false) inst a a with
| isTrue _ => rfl
| isFalse h₁ => absurd rfl h₁
/-- Decidable equality for Bool -/
@[inline] def Bool.decEq (a b : Bool) : Decidable (Eq a b) :=
match a, b with
| false, false => isTrue rfl
| false, true => isFalse (fun h => Bool.noConfusion h)
| true, false => isFalse (fun h => Bool.noConfusion h)
| true, true => isTrue rfl
@[inline] instance : DecidableEq Bool :=
Bool.decEq
/--
`BEq α` is a typeclass for supplying a boolean-valued equality relation on
`α`, notated as `a == b`. Unlike `DecidableEq α` (which uses `a = b`), this
is `Bool` valued instead of `Prop` valued, and it also does not have any
axioms like being reflexive or agreeing with `=`. It is mainly intended for
programming applications. See `LawfulBEq` for a version that requires that
`==` and `=` coincide.
-/
class BEq (α : Type u) where
/-- Boolean equality, notated as `a == b`. -/
beq : α → α → Bool
open BEq (beq)
instance [DecidableEq α] : BEq α where
beq a b := decide (Eq a b)
/--
"Dependent" if-then-else, normally written via the notation `if h : c then t(h) else e(h)`,
is sugar for `dite c (fun h => t(h)) (fun h => e(h))`, and it is the same as
`if c then t else e` except that `t` is allowed to depend on a proof `h : c`,
and `e` can depend on `h : ¬c`. (Both branches use the same name for the hypothesis,
even though it has different types in the two cases.)
We use this to be able to communicate the if-then-else condition to the branches.
For example, `Array.get arr ⟨i, h⟩` expects a proof `h : i < arr.size` in order to
avoid a bounds check, so you can write `if h : i < arr.size then arr.get ⟨i, h⟩ else ...`
to avoid the bounds check inside the if branch. (Of course in this case we have only
lifted the check into an explicit `if`, but we could also use this proof multiple times
or derive `i < arr.size` from some other proposition that we are checking in the `if`.)
-/
@[macro_inline] def dite {α : Sort u} (c : Prop) [h : Decidable c] (t : c → α) (e : Not c → α) : α :=
h.casesOn e t
/-! # if-then-else -/
/--
`if c then t else e` is notation for `ite c t e`, "if-then-else", which decides to
return `t` or `e` depending on whether `c` is true or false. The explicit argument
`c : Prop` does not have any actual computational content, but there is an additional
`[Decidable c]` argument synthesized by typeclass inference which actually
determines how to evaluate `c` to true or false.
Because lean uses a strict (call-by-value) evaluation strategy, the signature of this
function is problematic in that it would require `t` and `e` to be evaluated before
calling the `ite` function, which would cause both sides of the `if` to be evaluated.
Even if the result is discarded, this would be a big performance problem,
and is undesirable for users in any case. To resolve this, `ite` is marked as
`@[macro_inline]`, which means that it is unfolded during code generation, and
the definition of the function uses `fun _ => t` and `fun _ => e` so this recovers
the expected "lazy" behavior of `if`: the `t` and `e` arguments delay evaluation
until `c` is known.
-/
@[macro_inline] def ite {α : Sort u} (c : Prop) [h : Decidable c] (t e : α) : α :=
h.casesOn (fun _ => e) (fun _ => t)
@[macro_inline] instance {p q} [dp : Decidable p] [dq : Decidable q] : Decidable (And p q) :=
match dp with
| isTrue hp =>
match dq with
| isTrue hq => isTrue ⟨hp, hq⟩
| isFalse hq => isFalse (fun h => hq (And.right h))
| isFalse hp =>
isFalse (fun h => hp (And.left h))
@[macro_inline] instance [dp : Decidable p] [dq : Decidable q] : Decidable (Or p q) :=
match dp with
| isTrue hp => isTrue (Or.inl hp)
| isFalse hp =>
match dq with
| isTrue hq => isTrue (Or.inr hq)
| isFalse hq =>
isFalse fun h => match h with
| Or.inl h => hp h
| Or.inr h => hq h
instance [dp : Decidable p] : Decidable (Not p) :=
match dp with
| isTrue hp => isFalse (absurd hp)
| isFalse hp => isTrue hp
/-! # Boolean operators -/
/--
`cond b x y` is the same as `if b then x else y`, but optimized for a
boolean condition. It can also be written as `bif b then x else y`.
This is `@[macro_inline]` because `x` and `y` should not
be eagerly evaluated (see `ite`).
-/
@[macro_inline] def cond {α : Type u} (c : Bool) (x y : α) : α :=
match c with
| true => x
| false => y
/--
`or x y`, or `x || y`, is the boolean "or" operation (not to be confused
with `Or : Prop → Prop → Prop`, which is the propositional connective).
It is `@[macro_inline]` because it has C-like short-circuiting behavior:
if `x` is true then `y` is not evaluated.
-/
@[macro_inline] def or (x y : Bool) : Bool :=
match x with
| true => true
| false => y
/--
`and x y`, or `x && y`, is the boolean "and" operation (not to be confused
with `And : Prop → Prop → Prop`, which is the propositional connective).
It is `@[macro_inline]` because it has C-like short-circuiting behavior:
if `x` is false then `y` is not evaluated.
-/
@[macro_inline] def and (x y : Bool) : Bool :=
match x with
| false => false
| true => y
/--
`not x`, or `!x`, is the boolean "not" operation (not to be confused
with `Not : Prop → Prop`, which is the propositional connective).
-/
@[inline] def not : Bool → Bool
| true => false
| false => true
/--
The type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
-/
inductive Nat where
/-- `Nat.zero`, normally written `0 : Nat`, is the smallest natural number.
This is one of the two constructors of `Nat`. -/
| zero : Nat
/-- The successor function on natural numbers, `succ n = n + 1`.
This is one of the two constructors of `Nat`. -/
| succ (n : Nat) : Nat
instance : Inhabited Nat where
default := Nat.zero
/--
The class `OfNat α n` powers the numeric literal parser. If you write
`37 : α`, lean will attempt to synthesize `OfNat α 37`, and will generate
the term `(OfNat.ofNat 37 : α)`.
There is a bit of infinite regress here since the desugaring apparently
still contains a literal `37` in it. The type of expressions contains a
primitive constructor for "raw natural number literals", which you can directly
access using the macro `nat_lit 37`. Raw number literals are always of type `Nat`.
So it would be more correct to say that lean looks for an instance of
`OfNat α (nat_lit 37)`, and it generates the term `(OfNat.ofNat (nat_lit 37) : α)`.
-/
class OfNat (α : Type u) (_ : Nat) where
/-- The `OfNat.ofNat` function is automatically inserted by the parser when
the user writes a numeric literal like `1 : α`. Implementations of this
typeclass can therefore customize the behavior of `n : α` based on `n` and
`α`. -/
ofNat : α
@[default_instance 100] /- low prio -/
instance (n : Nat) : OfNat Nat n where
ofNat := n
/-- `LE α` is the typeclass which supports the notation `x ≤ y` where `x y : α`.-/
class LE (α : Type u) where
/-- The less-equal relation: `x ≤ y` -/
le : α → α → Prop
/-- `LT α` is the typeclass which supports the notation `x < y` where `x y : α`.-/
class LT (α : Type u) where
/-- The less-than relation: `x < y` -/
lt : α → α → Prop
/-- `a ≥ b` is an abbreviation for `b ≤ a`. -/
@[reducible] def GE.ge {α : Type u} [LE α] (a b : α) : Prop := LE.le b a
/-- `a > b` is an abbreviation for `b < a`. -/
@[reducible] def GT.gt {α : Type u} [LT α] (a b : α) : Prop := LT.lt b a
/-- `Max α` is the typeclass which supports the operation `max x y` where `x y : α`.-/
class Max (α : Type u) where
/-- The maximum operation: `max x y`. -/
max : α → α → α
export Max (max)
/-- Implementation of the `max` operation using `≤`. -/
-- Marked inline so that `min x y + max x y` can be optimized to a single branch.
@[inline]
def maxOfLe [LE α] [DecidableRel (@LE.le α _)] : Max α where
max x y := ite (LE.le x y) y x
/-- `Min α` is the typeclass which supports the operation `min x y` where `x y : α`.-/
class Min (α : Type u) where
/-- The minimum operation: `min x y`. -/
min : α → α → α
export Min (min)
/-- Implementation of the `min` operation using `≤`. -/
-- Marked inline so that `min x y + max x y` can be optimized to a single branch.
@[inline]
def minOfLe [LE α] [DecidableRel (@LE.le α _)] : Min α where
min x y := ite (LE.le x y) x y
/--
Transitive chaining of proofs, used e.g. by `calc`.
It takes two relations `r` and `s` as "input", and produces an "output"
relation `t`, with the property that `r a b` and `s b c` implies `t a c`.
The `calc` tactic uses this so that when it sees a chain with `a ≤ b` and `b < c`
it knows that this should be a proof of `a < c` because there is an instance
`Trans (·≤·) (·<·) (·<·)`.
-/
class Trans (r : α → β → Sort u) (s : β → γ → Sort v) (t : outParam (α → γ → Sort w)) where
/-- Compose two proofs by transitivity, generalized over the relations involved. -/
trans : r a b → s b c → t a c
export Trans (trans)
instance (r : α → γ → Sort u) : Trans Eq r r where
trans heq h' := heq ▸ h'
instance (r : α → β → Sort u) : Trans r Eq r where
trans h' heq := heq ▸ h'
/--
The notation typeclass for heterogeneous addition.
This enables the notation `a + b : γ` where `a : α`, `b : β`.
-/
class HAdd (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. -/
hAdd : α → β → γ
/--
The notation typeclass for heterogeneous subtraction.
This enables the notation `a - b : γ` where `a : α`, `b : β`.
-/
class HSub (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. -/
hSub : α → β → γ
/--
The notation typeclass for heterogeneous multiplication.
This enables the notation `a * b : γ` where `a : α`, `b : β`.
-/
class HMul (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. -/
hMul : α → β → γ
/--
The notation typeclass for heterogeneous division.
This enables the notation `a / b : γ` where `a : α`, `b : β`.
-/
class HDiv (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a / b` computes the result of dividing `a` by `b`.
The meaning of this notation is type-dependent.
* For most types like `Nat`, `Int`, `Rat`, `Real`, `a / 0` is defined to be `0`.
* For `Nat` and `Int`, `a / b` rounds toward 0.
* For `Float`, `a / 0` follows the IEEE 754 semantics for division,
usually resulting in `inf` or `nan`. -/
hDiv : α → β → γ
/--
The notation typeclass for heterogeneous modulo / remainder.
This enables the notation `a % b : γ` where `a : α`, `b : β`.
-/
class HMod (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a % b` computes the remainder upon dividing `a` by `b`.
The meaning of this notation is type-dependent.
* For `Nat` and `Int`, `a % 0` is defined to be `a`. -/
hMod : α → β → γ
/--
The notation typeclass for heterogeneous exponentiation.
This enables the notation `a ^ b : γ` where `a : α`, `b : β`.
-/
class HPow (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. -/
hPow : α → β → γ
/--
The notation typeclass for heterogeneous append.
This enables the notation `a ++ b : γ` where `a : α`, `b : β`.
-/
class HAppend (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a ++ b` is the result of concatenation of `a` and `b`, usually read "append".
The meaning of this notation is type-dependent. -/
hAppend : α → β → γ
/--
The typeclass behind the notation `a <|> b : γ` where `a : α`, `b : β`.
Because `b` is "lazy" in this notation, it is passed as `Unit → β` to the
implementation so it can decide when to evaluate it.
-/
class HOrElse (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a <|> b` executes `a` and returns the result, unless it fails in which
case it executes and returns `b`. Because `b` is not always executed, it
is passed as a thunk so it can be forced only when needed.
The meaning of this notation is type-dependent. -/
hOrElse : α → (Unit → β) → γ
/--
The typeclass behind the notation `a >> b : γ` where `a : α`, `b : β`.
Because `b` is "lazy" in this notation, it is passed as `Unit → β` to the
implementation so it can decide when to evaluate it.
-/
class HAndThen (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a >> b` executes `a`, ignores the result, and then executes `b`.
If `a` fails then `b` is not executed. Because `b` is not always executed, it
is passed as a thunk so it can be forced only when needed.
The meaning of this notation is type-dependent. -/
hAndThen : α → (Unit → β) → γ
/-- The typeclass behind the notation `a &&& b : γ` where `a : α`, `b : β`. -/
class HAnd (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a &&& b` computes the bitwise AND of `a` and `b`.
The meaning of this notation is type-dependent. -/
hAnd : α → β → γ
/-- The typeclass behind the notation `a ^^^ b : γ` where `a : α`, `b : β`. -/
class HXor (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a ^^^ b` computes the bitwise XOR of `a` and `b`.
The meaning of this notation is type-dependent. -/
hXor : α → β → γ
/-- The typeclass behind the notation `a ||| b : γ` where `a : α`, `b : β`. -/
class HOr (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a ||| b` computes the bitwise OR of `a` and `b`.
The meaning of this notation is type-dependent. -/
hOr : α → β → γ
/-- The typeclass behind the notation `a <<< b : γ` where `a : α`, `b : β`. -/
class HShiftLeft (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a <<< b` computes `a` shifted to the left by `b` places.
The meaning of this notation is type-dependent.
* On `Nat`, this is equivalent to `a * 2 ^ b`.
* On `UInt8` and other fixed width unsigned types, this is the same but
truncated to the bit width. -/
hShiftLeft : α → β → γ
/-- The typeclass behind the notation `a >>> b : γ` where `a : α`, `b : β`. -/
class HShiftRight (α : Type u) (β : Type v) (γ : outParam (Type w)) where
/-- `a >>> b` computes `a` shifted to the right by `b` places.
The meaning of this notation is type-dependent.
* On `Nat` and fixed width unsigned types like `UInt8`,
this is equivalent to `a / 2 ^ b`. -/
hShiftRight : α → β → γ
/-- The homogeneous version of `HAdd`: `a + b : α` where `a b : α`. -/
class Add (α : Type u) where
/-- `a + b` computes the sum of `a` and `b`. See `HAdd`. -/
add : α → α → α
/-- The homogeneous version of `HSub`: `a - b : α` where `a b : α`. -/
class Sub (α : Type u) where
/-- `a - b` computes the difference of `a` and `b`. See `HSub`. -/
sub : α → α → α
/-- The homogeneous version of `HMul`: `a * b : α` where `a b : α`. -/
class Mul (α : Type u) where
/-- `a * b` computes the product of `a` and `b`. See `HMul`. -/
mul : α → α → α
/--
The notation typeclass for negation.
This enables the notation `-a : α` where `a : α`.
-/
class Neg (α : Type u) where
/-- `-a` computes the negative or opposite of `a`.
The meaning of this notation is type-dependent. -/
neg : α → α
/-- The homogeneous version of `HDiv`: `a / b : α` where `a b : α`. -/
class Div (α : Type u) where
/-- `a / b` computes the result of dividing `a` by `b`. See `HDiv`. -/
div : α → α → α
/-- The homogeneous version of `HMod`: `a % b : α` where `a b : α`. -/
class Mod (α : Type u) where
/-- `a % b` computes the remainder upon dividing `a` by `b`. See `HMod`. -/
mod : α → α → α
/--
The homogeneous version of `HPow`: `a ^ b : α` where `a : α`, `b : β`.
(The right argument is not the same as the left since we often want this even
in the homogeneous case.)
-/
class Pow (α : Type u) (β : Type v) where
/-- `a ^ b` computes `a` to the power of `b`. See `HPow`. -/
pow : α → β → α
/-- The homogeneous version of `HAppend`: `a ++ b : α` where `a b : α`. -/
class Append (α : Type u) where
/-- `a ++ b` is the result of concatenation of `a` and `b`. See `HAppend`. -/
append : α → α → α
/--
The homogeneous version of `HOrElse`: `a <|> b : α` where `a b : α`.
Because `b` is "lazy" in this notation, it is passed as `Unit → α` to the
implementation so it can decide when to evaluate it.
-/
class OrElse (α : Type u) where
/-- The implementation of `a <|> b : α`. See `HOrElse`. -/
orElse : α → (Unit → α) → α
/--
The homogeneous version of `HAndThen`: `a >> b : α` where `a b : α`.
Because `b` is "lazy" in this notation, it is passed as `Unit → α` to the
implementation so it can decide when to evaluate it.
-/
class AndThen (α : Type u) where
/-- The implementation of `a >> b : α`. See `HAndThen`. -/
andThen : α → (Unit → α) → α
/--
The homogeneous version of `HAnd`: `a &&& b : α` where `a b : α`.
(It is called `AndOp` because `And` is taken for the propositional connective.)
-/
class AndOp (α : Type u) where
/-- The implementation of `a &&& b : α`. See `HAnd`. -/
and : α → α → α
/-- The homogeneous version of `HXor`: `a ^^^ b : α` where `a b : α`. -/
class Xor (α : Type u) where
/-- The implementation of `a ^^^ b : α`. See `HXor`. -/
xor : α → α → α
/--
The homogeneous version of `HOr`: `a ||| b : α` where `a b : α`.
(It is called `OrOp` because `Or` is taken for the propositional connective.)
-/
class OrOp (α : Type u) where
/-- The implementation of `a ||| b : α`. See `HOr`. -/
or : α → α → α
/-- The typeclass behind the notation `~~~a : α` where `a : α`. -/
class Complement (α : Type u) where
/-- The implementation of `~~~a : α`. -/
complement : α → α
/-- The homogeneous version of `HShiftLeft`: `a <<< b : α` where `a b : α`. -/
class ShiftLeft (α : Type u) where
/-- The implementation of `a <<< b : α`. See `HShiftLeft`. -/
shiftLeft : α → α → α
/-- The homogeneous version of `HShiftRight`: `a >>> b : α` where `a b : α`. -/
class ShiftRight (α : Type u) where
/-- The implementation of `a >>> b : α`. See `HShiftRight`. -/
shiftRight : α → α → α
@[default_instance]
instance [Add α] : HAdd α α α where
hAdd a b := Add.add a b
@[default_instance]
instance [Sub α] : HSub α α α where
hSub a b := Sub.sub a b
@[default_instance]
instance [Mul α] : HMul α α α where
hMul a b := Mul.mul a b
@[default_instance]
instance [Div α] : HDiv α α α where
hDiv a b := Div.div a b
@[default_instance]
instance [Mod α] : HMod α α α where
hMod a b := Mod.mod a b
@[default_instance]
instance [Pow α β] : HPow α β α where
hPow a b := Pow.pow a b
@[default_instance]
instance [Append α] : HAppend α α α where
hAppend a b := Append.append a b
@[default_instance]
instance [OrElse α] : HOrElse α α α where
hOrElse a b := OrElse.orElse a b
@[default_instance]
instance [AndThen α] : HAndThen α α α where
hAndThen a b := AndThen.andThen a b
@[default_instance]
instance [AndOp α] : HAnd α α α where
hAnd a b := AndOp.and a b
@[default_instance]
instance [Xor α] : HXor α α α where
hXor a b := Xor.xor a b
@[default_instance]
instance [OrOp α] : HOr α α α where
hOr a b := OrOp.or a b
@[default_instance]
instance [ShiftLeft α] : HShiftLeft α α α where
hShiftLeft a b := ShiftLeft.shiftLeft a b
@[default_instance]
instance [ShiftRight α] : HShiftRight α α α where
hShiftRight a b := ShiftRight.shiftRight a b
open HAdd (hAdd)
open HMul (hMul)
open HPow (hPow)
open HAppend (hAppend)
/--
The typeclass behind the notation `a ∈ s : Prop` where `a : α`, `s : γ`.
Because `α` is an `outParam`, the "container type" `γ` determines the type
of the elements of the container.
-/
class Membership (α : outParam (Type u)) (γ : Type v) where
/-- The membership relation `a ∈ s : Prop` where `a : α`, `s : γ`. -/
mem : α → γ → Prop
set_option bootstrap.genMatcherCode false in
/--
Addition of natural numbers.
This definition is overridden in both the kernel and the compiler to efficiently
evaluate using the "bignum" representation (see `Nat`). The definition provided
here is the logical model (and it is soundness-critical that they coincide).
-/
@[extern "lean_nat_add"]
protected def Nat.add : (@& Nat) → (@& Nat) → Nat
| a, Nat.zero => a
| a, Nat.succ b => Nat.succ (Nat.add a b)
instance : Add Nat where
add := Nat.add
/- We mark the following definitions as pattern to make sure they can be used in recursive equations,
and reduced by the equation Compiler. -/
attribute [match_pattern] Nat.add Add.add HAdd.hAdd Neg.neg
set_option bootstrap.genMatcherCode false in
/--
Multiplication of natural numbers.
This definition is overridden in both the kernel and the compiler to efficiently
evaluate using the "bignum" representation (see `Nat`). The definition provided
here is the logical model (and it is soundness-critical that they coincide).
-/
@[extern "lean_nat_mul"]
protected def Nat.mul : (@& Nat) → (@& Nat) → Nat
| _, 0 => 0
| a, Nat.succ b => Nat.add (Nat.mul a b) a
instance : Mul Nat where
mul := Nat.mul
set_option bootstrap.genMatcherCode false in
/--
The power operation on natural numbers.
This definition is overridden in both the kernel and the compiler to efficiently
evaluate using the "bignum" representation (see `Nat`). The definition provided
here is the logical model.
-/
@[extern "lean_nat_pow"]
protected def Nat.pow (m : @& Nat) : (@& Nat) → Nat
| 0 => 1
| succ n => Nat.mul (Nat.pow m n) m
instance : Pow Nat Nat where
pow := Nat.pow
set_option bootstrap.genMatcherCode false in
/--
(Boolean) equality of natural numbers.
This definition is overridden in both the kernel and the compiler to efficiently
evaluate using the "bignum" representation (see `Nat`). The definition provided
here is the logical model (and it is soundness-critical that they coincide).
-/
@[extern "lean_nat_dec_eq"]
def Nat.beq : (@& Nat) → (@& Nat) → Bool
| zero, zero => true
| zero, succ _ => false
| succ _, zero => false
| succ n, succ m => beq n m
instance : BEq Nat where
beq := Nat.beq
theorem Nat.eq_of_beq_eq_true : {n m : Nat} → Eq (beq n m) true → Eq n m
| zero, zero, _ => rfl
| zero, succ _, h => Bool.noConfusion h
| succ _, zero, h => Bool.noConfusion h
| succ n, succ m, h =>
have : Eq (beq n m) true := h
have : Eq n m := eq_of_beq_eq_true this
this ▸ rfl
theorem Nat.ne_of_beq_eq_false : {n m : Nat} → Eq (beq n m) false → Not (Eq n m)
| zero, zero, h₁, _ => Bool.noConfusion h₁
| zero, succ _, _, h₂ => Nat.noConfusion h₂
| succ _, zero, _, h₂ => Nat.noConfusion h₂
| succ n, succ m, h₁, h₂ =>
have : Eq (beq n m) false := h₁
Nat.noConfusion h₂ (fun h₂ => absurd h₂ (ne_of_beq_eq_false this))
/--
A decision procedure for equality of natural numbers.
This definition is overridden in the compiler to efficiently
evaluate using the "bignum" representation (see `Nat`). The definition provided
here is the logical model.
-/
@[reducible, extern "lean_nat_dec_eq"]
protected def Nat.decEq (n m : @& Nat) : Decidable (Eq n m) :=
match h:beq n m with
| true => isTrue (eq_of_beq_eq_true h)
| false => isFalse (ne_of_beq_eq_false h)
@[inline] instance : DecidableEq Nat := Nat.decEq
set_option bootstrap.genMatcherCode false in
/--
The (Boolean) less-equal relation on natural numbers.
This definition is overridden in both the kernel and the compiler to efficiently
evaluate using the "bignum" representation (see `Nat`). The definition provided
here is the logical model (and it is soundness-critical that they coincide).
-/
@[extern "lean_nat_dec_le"]
def Nat.ble : @& Nat → @& Nat → Bool
| zero, zero => true
| zero, succ _ => true
| succ _, zero => false
| succ n, succ m => ble n m
/--
An inductive definition of the less-equal relation on natural numbers,
characterized as the least relation `≤` such that `n ≤ n` and `n ≤ m → n ≤ m + 1`.
-/
protected inductive Nat.le (n : Nat) : Nat → Prop
/-- Less-equal is reflexive: `n ≤ n` -/
| refl : Nat.le n n
/-- If `n ≤ m`, then `n ≤ m + 1`. -/
| step {m} : Nat.le n m → Nat.le n (succ m)
instance : LE Nat where
le := Nat.le
/-- The strict less than relation on natural numbers is defined as `n < m := n + 1 ≤ m`. -/
protected def Nat.lt (n m : Nat) : Prop :=
Nat.le (succ n) m
instance : LT Nat where
lt := Nat.lt
theorem Nat.not_succ_le_zero : ∀ (n : Nat), LE.le (succ n) 0 → False
| 0, h => nomatch h
| succ _, h => nomatch h
theorem Nat.not_lt_zero (n : Nat) : Not (LT.lt n 0) :=
not_succ_le_zero n
theorem Nat.zero_le : (n : Nat) → LE.le 0 n
| zero => Nat.le.refl
| succ n => Nat.le.step (zero_le n)
theorem Nat.succ_le_succ : LE.le n m → LE.le (succ n) (succ m)
| Nat.le.refl => Nat.le.refl
| Nat.le.step h => Nat.le.step (succ_le_succ h)
theorem Nat.zero_lt_succ (n : Nat) : LT.lt 0 (succ n) :=
succ_le_succ (zero_le n)
theorem Nat.le_step (h : LE.le n m) : LE.le n (succ m) :=
Nat.le.step h
protected theorem Nat.le_trans {n m k : Nat} : LE.le n m → LE.le m k → LE.le n k
| h, Nat.le.refl => h
| h₁, Nat.le.step h₂ => Nat.le.step (Nat.le_trans h₁ h₂)
protected theorem Nat.lt_trans {n m k : Nat} (h₁ : LT.lt n m) : LT.lt m k → LT.lt n k :=
Nat.le_trans (le_step h₁)
theorem Nat.le_succ (n : Nat) : LE.le n (succ n) :=
Nat.le.step Nat.le.refl
theorem Nat.le_succ_of_le {n m : Nat} (h : LE.le n m) : LE.le n (succ m) :=
Nat.le_trans h (le_succ m)
protected theorem Nat.le_refl (n : Nat) : LE.le n n :=
Nat.le.refl
theorem Nat.succ_pos (n : Nat) : LT.lt 0 (succ n) :=
zero_lt_succ n
set_option bootstrap.genMatcherCode false in
/--
The predecessor function on natural numbers.
This definition is overridden in the compiler to use `n - 1` instead.
The definition provided here is the logical model.
-/
@[extern "lean_nat_pred"]
def Nat.pred : (@& Nat) → Nat
| 0 => 0
| succ a => a
theorem Nat.pred_le_pred : {n m : Nat} → LE.le n m → LE.le (pred n) (pred m)
| _, _, Nat.le.refl => Nat.le.refl
| 0, succ _, Nat.le.step h => h
| succ _, succ _, Nat.le.step h => Nat.le_trans (le_succ _) h
theorem Nat.le_of_succ_le_succ {n m : Nat} : LE.le (succ n) (succ m) → LE.le n m :=
pred_le_pred
theorem Nat.le_of_lt_succ {m n : Nat} : LT.lt m (succ n) → LE.le m n :=
le_of_succ_le_succ
protected theorem Nat.eq_or_lt_of_le : {n m: Nat} → LE.le n m → Or (Eq n m) (LT.lt n m)
| zero, zero, _ => Or.inl rfl
| zero, succ _, _ => Or.inr (Nat.succ_le_succ (Nat.zero_le _))
| succ _, zero, h => absurd h (not_succ_le_zero _)
| succ n, succ m, h =>
have : LE.le n m := Nat.le_of_succ_le_succ h
match Nat.eq_or_lt_of_le this with
| Or.inl h => Or.inl (h ▸ rfl)
| Or.inr h => Or.inr (succ_le_succ h)
protected theorem Nat.lt_or_ge (n m : Nat) : Or (LT.lt n m) (GE.ge n m) :=
match m with
| zero => Or.inr (zero_le n)
| succ m =>
match Nat.lt_or_ge n m with
| Or.inl h => Or.inl (le_succ_of_le h)
| Or.inr h =>
match Nat.eq_or_lt_of_le h with
| Or.inl h1 => Or.inl (h1 ▸ Nat.le_refl _)
| Or.inr h1 => Or.inr h1
theorem Nat.not_succ_le_self : (n : Nat) → Not (LE.le (succ n) n)
| 0 => not_succ_le_zero _
| succ n => fun h => absurd (le_of_succ_le_succ h) (not_succ_le_self n)
protected theorem Nat.lt_irrefl (n : Nat) : Not (LT.lt n n) :=
Nat.not_succ_le_self n
protected theorem Nat.lt_of_le_of_lt {n m k : Nat} (h₁ : LE.le n m) (h₂ : LT.lt m k) : LT.lt n k :=
Nat.le_trans (Nat.succ_le_succ h₁) h₂
protected theorem Nat.le_antisymm {n m : Nat} (h₁ : LE.le n m) (h₂ : LE.le m n) : Eq n m :=
match h₁ with
| Nat.le.refl => rfl
| Nat.le.step h => absurd (Nat.lt_of_le_of_lt h h₂) (Nat.lt_irrefl n)
protected theorem Nat.lt_of_le_of_ne {n m : Nat} (h₁ : LE.le n m) (h₂ : Not (Eq n m)) : LT.lt n m :=
match Nat.lt_or_ge n m with
| Or.inl h₃ => h₃
| Or.inr h₃ => absurd (Nat.le_antisymm h₁ h₃) h₂
theorem Nat.le_of_ble_eq_true (h : Eq (Nat.ble n m) true) : LE.le n m :=
match n, m with
| 0, _ => Nat.zero_le _
| succ _, succ _ => Nat.succ_le_succ (le_of_ble_eq_true h)
theorem Nat.ble_self_eq_true : (n : Nat) → Eq (Nat.ble n n) true
| 0 => rfl
| succ n => ble_self_eq_true n
theorem Nat.ble_succ_eq_true : {n m : Nat} → Eq (Nat.ble n m) true → Eq (Nat.ble n (succ m)) true
| 0, _, _ => rfl
| succ n, succ _, h => ble_succ_eq_true (n := n) h
theorem Nat.ble_eq_true_of_le (h : LE.le n m) : Eq (Nat.ble n m) true :=
match h with
| Nat.le.refl => Nat.ble_self_eq_true n
| Nat.le.step h => Nat.ble_succ_eq_true (ble_eq_true_of_le h)
theorem Nat.not_le_of_not_ble_eq_true (h : Not (Eq (Nat.ble n m) true)) : Not (LE.le n m) :=
fun h' => absurd (Nat.ble_eq_true_of_le h') h
@[extern "lean_nat_dec_le"]
instance Nat.decLe (n m : @& Nat) : Decidable (LE.le n m) :=
dite (Eq (Nat.ble n m) true) (fun h => isTrue (Nat.le_of_ble_eq_true h)) (fun h => isFalse (Nat.not_le_of_not_ble_eq_true h))
@[extern "lean_nat_dec_lt"]
instance Nat.decLt (n m : @& Nat) : Decidable (LT.lt n m) :=
decLe (succ n) m
instance : Min Nat := minOfLe
set_option bootstrap.genMatcherCode false in
/--
(Truncated) subtraction of natural numbers. Because natural numbers are not
closed under subtraction, we define `n - m` to be `0` when `n < m`.
This definition is overridden in both the kernel and the compiler to efficiently
evaluate using the "bignum" representation (see `Nat`). The definition provided
here is the logical model (and it is soundness-critical that they coincide).
-/
@[extern "lean_nat_sub"]
protected def Nat.sub : (@& Nat) → (@& Nat) → Nat
| a, 0 => a
| a, succ b => pred (Nat.sub a b)
instance : Sub Nat where
sub := Nat.sub
/--
Gets the word size of the platform. That is, whether the platform is 64 or 32 bits.
This function is opaque because we cannot guarantee at compile time that the target
will have the same size as the host, and also because we would like to avoid
typechecking being architecture-dependent. Nevertheless, lean only works on
64 and 32 bit systems so we can encode this as a fact available for proof purposes.
-/
@[extern "lean_system_platform_nbits"] opaque System.Platform.getNumBits : Unit → Subtype fun (n : Nat) => Or (Eq n 32) (Eq n 64) :=
fun _ => ⟨64, Or.inr rfl⟩ -- inhabitant
/-- Gets the word size of the platform. That is, whether the platform is 64 or 32 bits. -/
def System.Platform.numBits : Nat :=
(getNumBits ()).val
theorem System.Platform.numBits_eq : Or (Eq numBits 32) (Eq numBits 64) :=
(getNumBits ()).property
/--
`Fin n` is a natural number `i` with the constraint that `0 ≤ i < n`.
It is the "canonical type with `n` elements".
-/
structure Fin (n : Nat) where
/-- If `i : Fin n`, then `i.val : ℕ` is the described number. It can also be
written as `i.1` or just `i` when the target type is known. -/
val : Nat
/-- If `i : Fin n`, then `i.2` is a proof that `i.1 < n`. -/
isLt : LT.lt val n
theorem Fin.eq_of_val_eq {n} : ∀ {i j : Fin n}, Eq i.val j.val → Eq i j
| ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
theorem Fin.val_eq_of_eq {n} {i j : Fin n} (h : Eq i j) : Eq i.val j.val :=
h ▸ rfl
theorem Fin.ne_of_val_ne {n} {i j : Fin n} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
fun h' => absurd (val_eq_of_eq h') h
instance (n : Nat) : DecidableEq (Fin n) :=
fun i j =>
match decEq i.val j.val with
| isTrue h => isTrue (Fin.eq_of_val_eq h)
| isFalse h => isFalse (Fin.ne_of_val_ne h)
instance {n} : LT (Fin n) where
lt a b := LT.lt a.val b.val
instance {n} : LE (Fin n) where
le a b := LE.le a.val b.val
instance Fin.decLt {n} (a b : Fin n) : Decidable (LT.lt a b) := Nat.decLt ..
instance Fin.decLe {n} (a b : Fin n) : Decidable (LE.le a b) := Nat.decLe ..
/-- The size of type `UInt8`, that is, `2^8 = 256`. -/
def UInt8.size : Nat := 256
/--
The type of unsigned 8-bit integers. This type has special support in the
compiler to make it actually 8 bits rather than wrapping a `Nat`.
-/
structure UInt8 where
/-- Unpack a `UInt8` as a `Nat` less than `2^8`.
This function is overridden with a native implementation. -/
val : Fin UInt8.size
attribute [extern "lean_uint8_of_nat_mk"] UInt8.mk
attribute [extern "lean_uint8_to_nat"] UInt8.val
/--
Pack a `Nat` less than `2^8` into a `UInt8`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint8_of_nat"]
def UInt8.ofNatCore (n : @& Nat) (h : LT.lt n UInt8.size) : UInt8 where
val := { val := n, isLt := h }
set_option bootstrap.genMatcherCode false in
/--
Decides equality on `UInt8`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint8_dec_eq"]
def UInt8.decEq (a b : UInt8) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt8.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt8 := UInt8.decEq
instance : Inhabited UInt8 where
default := UInt8.ofNatCore 0 (by decide)
/-- The size of type `UInt16`, that is, `2^16 = 65536`. -/
def UInt16.size : Nat := 65536
/--
The type of unsigned 16-bit integers. This type has special support in the
compiler to make it actually 16 bits rather than wrapping a `Nat`.
-/
structure UInt16 where
/-- Unpack a `UInt16` as a `Nat` less than `2^16`.
This function is overridden with a native implementation. -/
val : Fin UInt16.size
attribute [extern "lean_uint16_of_nat_mk"] UInt16.mk
attribute [extern "lean_uint16_to_nat"] UInt16.val
/--
Pack a `Nat` less than `2^16` into a `UInt16`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint16_of_nat"]
def UInt16.ofNatCore (n : @& Nat) (h : LT.lt n UInt16.size) : UInt16 where
val := { val := n, isLt := h }
set_option bootstrap.genMatcherCode false in
/--
Decides equality on `UInt16`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint16_dec_eq"]
def UInt16.decEq (a b : UInt16) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt16.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt16 := UInt16.decEq
instance : Inhabited UInt16 where
default := UInt16.ofNatCore 0 (by decide)
/-- The size of type `UInt32`, that is, `2^32 = 4294967296`. -/
def UInt32.size : Nat := 4294967296
/--
The type of unsigned 32-bit integers. This type has special support in the
compiler to make it actually 32 bits rather than wrapping a `Nat`.
-/
structure UInt32 where
/-- Unpack a `UInt32` as a `Nat` less than `2^32`.
This function is overridden with a native implementation. -/
val : Fin UInt32.size
attribute [extern "lean_uint32_of_nat_mk"] UInt32.mk
attribute [extern "lean_uint32_to_nat"] UInt32.val
/--
Pack a `Nat` less than `2^32` into a `UInt32`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint32_of_nat"]
def UInt32.ofNatCore (n : @& Nat) (h : LT.lt n UInt32.size) : UInt32 where
val := { val := n, isLt := h }
/--
Unpack a `UInt32` as a `Nat`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint32_to_nat"]
def UInt32.toNat (n : UInt32) : Nat := n.val.val
set_option bootstrap.genMatcherCode false in
/--
Decides equality on `UInt32`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint32_dec_eq"]
def UInt32.decEq (a b : UInt32) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt32.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt32 := UInt32.decEq
instance : Inhabited UInt32 where
default := UInt32.ofNatCore 0 (by decide)
instance : LT UInt32 where
lt a b := LT.lt a.val b.val
instance : LE UInt32 where
le a b := LE.le a.val b.val
set_option bootstrap.genMatcherCode false in
/--
Decides less-equal on `UInt32`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint32_dec_lt"]
def UInt32.decLt (a b : UInt32) : Decidable (LT.lt a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ => inferInstanceAs (Decidable (LT.lt n m))
set_option bootstrap.genMatcherCode false in
/--
Decides less-than on `UInt32`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint32_dec_le"]
def UInt32.decLe (a b : UInt32) : Decidable (LE.le a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ => inferInstanceAs (Decidable (LE.le n m))
instance (a b : UInt32) : Decidable (LT.lt a b) := UInt32.decLt a b
instance (a b : UInt32) : Decidable (LE.le a b) := UInt32.decLe a b
instance : Max UInt32 := maxOfLe
instance : Min UInt32 := minOfLe
/-- The size of type `UInt64`, that is, `2^64 = 18446744073709551616`. -/
def UInt64.size : Nat := 18446744073709551616
/--
The type of unsigned 64-bit integers. This type has special support in the
compiler to make it actually 64 bits rather than wrapping a `Nat`.
-/
structure UInt64 where
/-- Unpack a `UInt64` as a `Nat` less than `2^64`.
This function is overridden with a native implementation. -/
val : Fin UInt64.size
attribute [extern "lean_uint64_of_nat_mk"] UInt64.mk
attribute [extern "lean_uint64_to_nat"] UInt64.val
/--
Pack a `Nat` less than `2^64` into a `UInt64`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint64_of_nat"]
def UInt64.ofNatCore (n : @& Nat) (h : LT.lt n UInt64.size) : UInt64 where
val := { val := n, isLt := h }
set_option bootstrap.genMatcherCode false in
/--
Decides equality on `UInt64`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint64_dec_eq"]
def UInt64.decEq (a b : UInt64) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h => isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => UInt64.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq UInt64 := UInt64.decEq
instance : Inhabited UInt64 where
default := UInt64.ofNatCore 0 (by decide)
/--
The size of type `UInt16`, that is, `2^System.Platform.numBits`, which may
be either `2^32` or `2^64` depending on the platform's architecture.
-/
def USize.size : Nat := hPow 2 System.Platform.numBits
theorem usize_size_eq : Or (Eq USize.size 4294967296) (Eq USize.size 18446744073709551616) :=
show Or (Eq (hPow 2 System.Platform.numBits) 4294967296) (Eq (hPow 2 System.Platform.numBits) 18446744073709551616) from
match System.Platform.numBits, System.Platform.numBits_eq with
| _, Or.inl rfl => Or.inl (by decide)
| _, Or.inr rfl => Or.inr (by decide)
/--
A `USize` is an unsigned integer with the size of a word
for the platform's architecture.
For example, if running on a 32-bit machine, USize is equivalent to UInt32.
Or on a 64-bit machine, UInt64.
-/
structure USize where
/-- Unpack a `USize` as a `Nat` less than `USize.size`.
This function is overridden with a native implementation. -/
val : Fin USize.size
attribute [extern "lean_usize_of_nat_mk"] USize.mk
attribute [extern "lean_usize_to_nat"] USize.val
/--
Pack a `Nat` less than `USize.size` into a `USize`.
This function is overridden with a native implementation.
-/
@[extern "lean_usize_of_nat"]
def USize.ofNatCore (n : @& Nat) (h : LT.lt n USize.size) : USize := {
val := { val := n, isLt := h }
}
set_option bootstrap.genMatcherCode false in
/--
Decides equality on `USize`.
This function is overridden with a native implementation.
-/
@[extern "lean_usize_dec_eq"]
def USize.decEq (a b : USize) : Decidable (Eq a b) :=
match a, b with
| ⟨n⟩, ⟨m⟩ =>
dite (Eq n m) (fun h =>isTrue (h ▸ rfl)) (fun h => isFalse (fun h' => USize.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq USize := USize.decEq
instance : Inhabited USize where
default := USize.ofNatCore 0 (match USize.size, usize_size_eq with
| _, Or.inl rfl => by decide
| _, Or.inr rfl => by decide)
/--
Upcast a `Nat` less than `2^32` to a `USize`.
This is lossless because `USize.size` is either `2^32` or `2^64`.
This function is overridden with a native implementation.
-/
@[extern "lean_usize_of_nat"]
def USize.ofNat32 (n : @& Nat) (h : LT.lt n 4294967296) : USize where
val := {
val := n
isLt := match USize.size, usize_size_eq with
| _, Or.inl rfl => h
| _, Or.inr rfl => Nat.lt_trans h (by decide)
}
/--
A `Nat` denotes a valid unicode codepoint if it is less than `0x110000`, and
it is also not a "surrogate" character (the range `0xd800` to `0xdfff` inclusive).
-/
abbrev Nat.isValidChar (n : Nat) : Prop :=
Or (LT.lt n 0xd800) (And (LT.lt 0xdfff n) (LT.lt n 0x110000))
/--
A `UInt32` denotes a valid unicode codepoint if it is less than `0x110000`, and
it is also not a "surrogate" character (the range `0xd800` to `0xdfff` inclusive).
-/
abbrev UInt32.isValidChar (n : UInt32) : Prop :=
n.toNat.isValidChar
/-- The `Char` Type represents an unicode scalar value.
See http://www.unicode.org/glossary/#unicode_scalar_value). -/
structure Char where
/-- The underlying unicode scalar value as a `UInt32`. -/
val : UInt32
/-- The value must be a legal codepoint. -/
valid : val.isValidChar
private theorem isValidChar_UInt32 {n : Nat} (h : n.isValidChar) : LT.lt n UInt32.size :=
match h with
| Or.inl h => Nat.lt_trans h (by decide)
| Or.inr ⟨_, h⟩ => Nat.lt_trans h (by decide)
/--
Pack a `Nat` encoding a valid codepoint into a `Char`.
This function is overridden with a native implementation.
-/
@[extern "lean_uint32_of_nat"]
def Char.ofNatAux (n : @& Nat) (h : n.isValidChar) : Char :=
{ val := ⟨{ val := n, isLt := isValidChar_UInt32 h }⟩, valid := h }
/--
Convert a `Nat` into a `Char`. If the `Nat` does not encode a valid unicode scalar value,
`'\0'` is returned instead.
-/
@[noinline, match_pattern]
def Char.ofNat (n : Nat) : Char :=
dite (n.isValidChar)
(fun h => Char.ofNatAux n h)
(fun _ => { val := ⟨{ val := 0, isLt := by decide }⟩, valid := Or.inl (by decide) })
theorem Char.eq_of_val_eq : ∀ {c d : Char}, Eq c.val d.val → Eq c d
| ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
theorem Char.val_eq_of_eq : ∀ {c d : Char}, Eq c d → Eq c.val d.val
| _, _, rfl => rfl
theorem Char.ne_of_val_ne {c d : Char} (h : Not (Eq c.val d.val)) : Not (Eq c d) :=
fun h' => absurd (val_eq_of_eq h') h
theorem Char.val_ne_of_ne {c d : Char} (h : Not (Eq c d)) : Not (Eq c.val d.val) :=
fun h' => absurd (eq_of_val_eq h') h
instance : DecidableEq Char :=
fun c d =>
match decEq c.val d.val with
| isTrue h => isTrue (Char.eq_of_val_eq h)
| isFalse h => isFalse (Char.ne_of_val_ne h)
/-- Returns the number of bytes required to encode this `Char` in UTF-8. -/
def Char.utf8Size (c : Char) : UInt32 :=
let v := c.val
ite (LE.le v (UInt32.ofNatCore 0x7F (by decide)))
(UInt32.ofNatCore 1 (by decide))
(ite (LE.le v (UInt32.ofNatCore 0x7FF (by decide)))
(UInt32.ofNatCore 2 (by decide))
(ite (LE.le v (UInt32.ofNatCore 0xFFFF (by decide)))
(UInt32.ofNatCore 3 (by decide))
(UInt32.ofNatCore 4 (by decide))))
/--
`Option α` is the type of values which are either `some a` for some `a : α`,
or `none`. In functional programming languages, this type is used to represent
the possibility of failure, or sometimes nullability.
For example, the function `HashMap.find? : HashMap α β → α → Option β` looks up
a specified key `a : α` inside the map. Because we do not know in advance
whether the key is actually in the map, the return type is `Option β`, where
`none` means the value was not in the map, and `some b` means that the value
was found and `b` is the value retrieved.
To extract a value from an `Option α`, we use pattern matching:
```
def map (f : α → β) (x : Option α) : Option β :=
match x with
| some a => some (f a)
| none => none
```
We can also use `if let` to pattern match on `Option` and get the value
in the branch:
```
def map (f : α → β) (x : Option α) : Option β :=
if let some a := x then
some (f a)
else
none
```
-/
inductive Option (α : Type u) where
/-- No value. -/
| none : Option α
/-- Some value of type `α`. -/
| some (val : α) : Option α
attribute [unbox] Option
export Option (none some)
instance {α} : Inhabited (Option α) where
default := none
/--
Get with default. If `opt : Option α` and `dflt : α`, then `opt.getD dflt`
returns `a` if `opt = some a` and `dflt` otherwise.
This function is `@[macro_inline]`, so `dflt` will not be evaluated unless
`opt` turns out to be `none`.
-/
@[macro_inline] def Option.getD : Option α → α → α
| some x, _ => x
| none, e => e
/--
Map a function over an `Option` by applying the function to the contained
value if present.
-/
@[inline] protected def Option.map (f : α → β) : Option α → Option β
| some x => some (f x)
| none => none
/--
`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
-/
inductive List (α : Type u) where
/-- `[]` is the empty list. -/
| nil : List α
/-- If `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. -/
| cons (head : α) (tail : List α) : List α
instance {α} : Inhabited (List α) where
default := List.nil
/-- Implements decidable equality for `List α`, assuming `α` has decidable equality. -/
protected def List.hasDecEq {α : Type u} [DecidableEq α] : (a b : List α) → Decidable (Eq a b)
| nil, nil => isTrue rfl
| cons _ _, nil => isFalse (fun h => List.noConfusion h)
| nil, cons _ _ => isFalse (fun h => List.noConfusion h)
| cons a as, cons b bs =>
match decEq a b with
| isTrue hab =>
match List.hasDecEq as bs with
| isTrue habs => isTrue (hab ▸ habs ▸ rfl)
| isFalse nabs => isFalse (fun h => List.noConfusion h (fun _ habs => absurd habs nabs))
| isFalse nab => isFalse (fun h => List.noConfusion h (fun hab _ => absurd hab nab))
instance {α : Type u} [DecidableEq α] : DecidableEq (List α) := List.hasDecEq
/--
Folds a function over a list from the left:
`foldl f z [a, b, c] = f (f (f z a) b) c`
-/
@[specialize]
def List.foldl {α : Type u} {β : Type v} (f : α → β → α) : (init : α) → List β → α
| a, nil => a
| a, cons b l => foldl f (f a b) l
/--
`l.set n a` sets the value of list `l` at (zero-based) index `n` to `a`:
`[a, b, c, d].set 1 b' = [a, b', c, d]`
-/
def List.set : List α → Nat → α → List α
| cons _ as, 0, b => cons b as
| cons a as, Nat.succ n, b => cons a (set as n b)
| nil, _, _ => nil
/--
The length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
-/
def List.length : List α → Nat
| nil => 0
| cons _ as => HAdd.hAdd (length as) 1
/-- Auxiliary function for `List.lengthTR`. -/
def List.lengthTRAux : List α → Nat → Nat
| nil, n => n
| cons _ as, n => lengthTRAux as (Nat.succ n)
/--
A tail-recursive version of `List.length`, used to implement `List.length`
without running out of stack space.
-/
def List.lengthTR (as : List α) : Nat :=
lengthTRAux as 0
@[simp] theorem List.length_cons {α} (a : α) (as : List α) : Eq (cons a as).length as.length.succ :=
rfl
/-- `l.concat a` appends `a` at the *end* of `l`, that is, `l ++ [a]`. -/
def List.concat {α : Type u} : List α → α → List α
| nil, b => cons b nil
| cons a as, b => cons a (concat as b)
/--
`as.get i` returns the `i`'th element of the list `as`.
This version of the function uses `i : Fin as.length` to ensure that it will
not index out of bounds.
-/
def List.get {α : Type u} : (as : List α) → Fin as.length → α
| cons a _, ⟨0, _⟩ => a
| cons _ as, ⟨Nat.succ i, h⟩ => get as ⟨i, Nat.le_of_succ_le_succ h⟩
/--
`String` is the type of (UTF-8 encoded) strings.
The compiler overrides the data representation of this type to a byte sequence,
and both `String.utf8ByteSize` and `String.length` are cached and O(1).
-/
structure String where
/-- Pack a `List Char` into a `String`. This function is overridden by the
compiler and is O(n) in the length of the list. -/
mk ::
/-- Unpack `String` into a `List Char`. This function is overridden by the
compiler and is O(n) in the length of the list. -/
data : List Char
attribute [extern "lean_string_mk"] String.mk
attribute [extern "lean_string_data"] String.data
/--
Decides equality on `String`.
This function is overridden with a native implementation.
-/
@[extern "lean_string_dec_eq"]
def String.decEq (s₁ s₂ : @& String) : Decidable (Eq s₁ s₂) :=
match s₁, s₂ with
| ⟨s₁⟩, ⟨s₂⟩ =>
dite (Eq s₁ s₂) (fun h => isTrue (congrArg _ h)) (fun h => isFalse (fun h' => String.noConfusion h' (fun h' => absurd h' h)))
instance : DecidableEq String := String.decEq
/--
A byte position in a `String`. Internally, `String`s are UTF-8 encoded.
Codepoint positions (counting the Unicode codepoints rather than bytes)
are represented by plain `Nat`s instead.
Indexing a `String` by a byte position is constant-time, while codepoint
positions need to be translated internally to byte positions in linear-time.
-/
structure String.Pos where
/-- Get the underlying byte index of a `String.Pos` -/
byteIdx : Nat := 0
instance : Inhabited String.Pos where
default := {}
instance : DecidableEq String.Pos :=
fun ⟨a⟩ ⟨b⟩ => match decEq a b with
| isTrue h => isTrue (h ▸ rfl)
| isFalse h => isFalse (fun he => String.Pos.noConfusion he fun he => absurd he h)
/--
A `Substring` is a view into some subslice of a `String`.
The actual string slicing is deferred because this would require copying the
string; here we only store a reference to the original string for
garbage collection purposes.
-/
structure Substring where
/-- The underlying string to slice. -/
str : String
/-- The byte position of the start of the string slice. -/
startPos : String.Pos
/-- The byte position of the end of the string slice. -/
stopPos : String.Pos
instance : Inhabited Substring where
default := ⟨"", {}, {}⟩
/-- The byte length of the substring. -/
@[inline] def Substring.bsize : Substring → Nat
| ⟨_, b, e⟩ => e.byteIdx.sub b.byteIdx
/-- Returns the number of bytes required to encode this `Char` in UTF-8. -/
def String.csize (c : Char) : Nat :=
c.utf8Size.toNat
/--
The UTF-8 byte length of this string.
This is overridden by the compiler to be cached and O(1).
-/
@[extern "lean_string_utf8_byte_size"]
def String.utf8ByteSize : (@& String) → Nat
| ⟨s⟩ => go s
where
go : List Char → Nat
| .nil => 0
| .cons c cs => hAdd (go cs) (csize c)
instance : HAdd String.Pos String.Pos String.Pos where
hAdd p₁ p₂ := { byteIdx := hAdd p₁.byteIdx p₂.byteIdx }
instance : HSub String.Pos String.Pos String.Pos where
hSub p₁ p₂ := { byteIdx := HSub.hSub p₁.byteIdx p₂.byteIdx }
instance : HAdd String.Pos Char String.Pos where
hAdd p c := { byteIdx := hAdd p.byteIdx (String.csize c) }
instance : HAdd String.Pos String String.Pos where
hAdd p s := { byteIdx := hAdd p.byteIdx s.utf8ByteSize }
instance : LE String.Pos where
le p₁ p₂ := LE.le p₁.byteIdx p₂.byteIdx
instance : LT String.Pos where
lt p₁ p₂ := LT.lt p₁.byteIdx p₂.byteIdx
instance (p₁ p₂ : String.Pos) : Decidable (LE.le p₁ p₂) :=
inferInstanceAs (Decidable (LE.le p₁.byteIdx p₂.byteIdx))
instance (p₁ p₂ : String.Pos) : Decidable (LT.lt p₁ p₂) :=
inferInstanceAs (Decidable (LT.lt p₁.byteIdx p₂.byteIdx))
/-- A `String.Pos` pointing at the end of this string. -/
@[inline] def String.endPos (s : String) : String.Pos where
byteIdx := utf8ByteSize s
/-- Convert a `String` into a `Substring` denoting the entire string. -/
@[inline] def String.toSubstring (s : String) : Substring where
str := s
startPos := {}
stopPos := s.endPos
/-- `String.toSubstring` without `[inline]` annotation. -/
def String.toSubstring' (s : String) : Substring :=
s.toSubstring
/--
This function will cast a value of type `α` to type `β`, and is a no-op in the
compiler. This function is **extremely dangerous** because there is no guarantee
that types `α` and `β` have the same data representation, and this can lead to
memory unsafety. It is also logically unsound, since you could just cast
`True` to `False`. For all those reasons this function is marked as `unsafe`.
It is implemented by lifting both `α` and `β` into a common universe, and then
using `cast (lcProof : ULift (PLift α) = ULift (PLift β))` to actually perform
the cast. All these operations are no-ops in the compiler.
Using this function correctly requires some knowledge of the data representation
of the source and target types. Some general classes of casts which are safe in
the current runtime:
* `Array α` to `Array β` where `α` and `β` have compatible representations,
or more generally for other inductive types.
* `Quot α r` and `α`.
* `@Subtype α p` and `α`, or generally any structure containing only one
non-`Prop` field of type `α`.
* Casting `α` to/from `NonScalar` when `α` is a boxed generic type
(i.e. a function that accepts an arbitrary type `α` and is not specialized to
a scalar type like `UInt8`).
-/
unsafe def unsafeCast {α : Sort u} {β : Sort v} (a : α) : β :=
PLift.down (ULift.down.{max u v} (cast lcProof (ULift.up.{max u v} (PLift.up a))))
/-- Auxiliary definition for `panic`. -/
/-
This is a workaround for `panic` occurring in monadic code. See issue #695.
The `panicCore` definition cannot be specialized since it is an extern.
When `panic` occurs in monadic code, the `Inhabited α` parameter depends on a
`[inst : Monad m]` instance. The `inst` parameter will not be eliminated during
specialization if it occurs inside of a binder (to avoid work duplication), and
will prevent the actual monad from being "copied" to the code being specialized.
When we reimplement the specializer, we may consider copying `inst` if it also
occurs outside binders or if it is an instance.
-/
@[never_extract, extern "lean_panic_fn"]
def panicCore {α : Type u} [Inhabited α] (msg : String) : α := default
/--
`(panic "msg" : α)` has a built-in implementation which prints `msg` to
the error buffer. It *does not* terminate execution, and because it is a safe
function, it still has to return an element of `α`, so it takes `[Inhabited α]`
and returns `default`. It is primarily intended for debugging in pure contexts,
and assertion failures.
Because this is a pure function with side effects, it is marked as
`@[never_extract]` so that the compiler will not perform common sub-expression
elimination and other optimizations that assume that the expression is pure.
-/
@[noinline, never_extract]
def panic {α : Type u} [Inhabited α] (msg : String) : α :=
panicCore msg
-- TODO: this be applied directly to `Inhabited`'s definition when we remove the above workaround
attribute [nospecialize] Inhabited
/--
The class `GetElem cont idx elem dom` implements the `xs[i]` notation.
When you write this, given `xs : cont` and `i : idx`, lean looks for an instance
of `GetElem cont idx elem dom`. Here `elem` is the type of `xs[i]`, while
`dom` is whatever proof side conditions are required to make this applicable.
For example, the instance for arrays looks like
`GetElem (Array α) Nat α (fun xs i => i < xs.size)`.
The proof side-condition `dom xs i` is automatically dispatched by the
`get_elem_tactic` tactic, which can be extended by adding more clauses to
`get_elem_tactic_trivial`.
-/
class GetElem (cont : Type u) (idx : Type v) (elem : outParam (Type w)) (dom : outParam (cont → idx → Prop)) where
/--
The syntax `arr[i]` gets the `i`'th element of the collection `arr`.
If there are proof side conditions to the application, they will be automatically
inferred by the `get_elem_tactic` tactic.
The actual behavior of this class is type-dependent,
but here are some important implementations:
* `arr[i] : α` where `arr : Array α` and `i : Nat` or `i : USize`:
does array indexing with no bounds check and a proof side goal `i < arr.size`.
* `l[i] : α` where `l : List α` and `i : Nat`: index into a list,
with proof side goal `i < l.length`.
* `stx[i] : Syntax` where `stx : Syntax` and `i : Nat`: get a syntax argument,
no side goal (returns `.missing` out of range)
There are other variations on this syntax:
* `arr[i]`: proves the proof side goal by `get_elem_tactic`
* `arr[i]!`: panics if the side goal is false
* `arr[i]?`: returns `none` if the side goal is false
* `arr[i]'h`: uses `h` to prove the side goal
-/
getElem (xs : cont) (i : idx) (h : dom xs i) : elem
export GetElem (getElem)
/--
`Array α` is the type of [dynamic arrays](https://en.wikipedia.org/wiki/Dynamic_array)
with elements from `α`. This type has special support in the runtime.
An array has a size and a capacity; the size is `Array.size` but the capacity
is not observable from lean code. Arrays perform best when unshared; as long
as they are used "linearly" all updates will be performed destructively on the
array, so it has comparable performance to mutable arrays in imperative
programming languages.
-/
structure Array (α : Type u) where
/-- Convert a `List α` into an `Array α`. This function is overridden
to `List.toArray` and is O(n) in the length of the list. -/
mk ::
/-- Convert an `Array α` into a `List α`. This function is overridden
to `Array.toList` and is O(n) in the length of the list. -/
data : List α
attribute [extern "lean_array_data"] Array.data
attribute [extern "lean_array_mk"] Array.mk
/-- Construct a new empty array with initial capacity `c`. -/
@[extern "lean_mk_empty_array_with_capacity"]
def Array.mkEmpty {α : Type u} (c : @& Nat) : Array α where
data := List.nil
/-- Construct a new empty array. -/
def Array.empty {α : Type u} : Array α := mkEmpty 0
/-- Get the size of an array. This is a cached value, so it is O(1) to access. -/
@[reducible, extern "lean_array_get_size"]
def Array.size {α : Type u} (a : @& Array α) : Nat :=
a.data.length
/-- Access an element from an array without bounds checks, using a `Fin` index. -/
@[extern "lean_array_fget"]
def Array.get {α : Type u} (a : @& Array α) (i : @& Fin a.size) : α :=
a.data.get i
/-- Access an element from an array, or return `v₀` if the index is out of bounds. -/
@[inline] abbrev Array.getD (a : Array α) (i : Nat) (v₀ : α) : α :=
dite (LT.lt i a.size) (fun h => a.get ⟨i, h⟩) (fun _ => v₀)
/-- Access an element from an array, or panic if the index is out of bounds. -/
@[extern "lean_array_get"]
def Array.get! {α : Type u} [Inhabited α] (a : @& Array α) (i : @& Nat) : α :=
Array.getD a i default
instance : GetElem (Array α) Nat α fun xs i => LT.lt i xs.size where
getElem xs i h := xs.get ⟨i, h⟩
/--
Push an element onto the end of an array. This is amortized O(1) because
`Array α` is internally a dynamic array.
-/
@[extern "lean_array_push"]
def Array.push {α : Type u} (a : Array α) (v : α) : Array α where
data := List.concat a.data v
/-- Create array `#[]` -/
def Array.mkArray0 {α : Type u} : Array α :=
mkEmpty 0
/-- Create array `#[a₁]` -/
def Array.mkArray1 {α : Type u} (a₁ : α) : Array α :=
(mkEmpty 1).push a₁
/-- Create array `#[a₁, a₂]` -/
def Array.mkArray2 {α : Type u} (a₁ a₂ : α) : Array α :=
((mkEmpty 2).push a₁).push a₂
/-- Create array `#[a₁, a₂, a₃]` -/
def Array.mkArray3 {α : Type u} (a₁ a₂ a₃ : α) : Array α :=
(((mkEmpty 3).push a₁).push a₂).push a₃
/-- Create array `#[a₁, a₂, a₃, a₄]` -/
def Array.mkArray4 {α : Type u} (a₁ a₂ a₃ a₄ : α) : Array α :=
((((mkEmpty 4).push a₁).push a₂).push a₃).push a₄
/-- Create array `#[a₁, a₂, a₃, a₄, a₅]` -/
def Array.mkArray5 {α : Type u} (a₁ a₂ a₃ a₄ a₅ : α) : Array α :=
(((((mkEmpty 5).push a₁).push a₂).push a₃).push a₄).push a₅
/-- Create array `#[a₁, a₂, a₃, a₄, a₅, a₆]` -/
def Array.mkArray6 {α : Type u} (a₁ a₂ a₃ a₄ a₅ a₆ : α) : Array α :=
((((((mkEmpty 6).push a₁).push a₂).push a₃).push a₄).push a₅).push a₆
/-- Create array `#[a₁, a₂, a₃, a₄, a₅, a₆, a₇]` -/
def Array.mkArray7 {α : Type u} (a₁ a₂ a₃ a₄ a₅ a₆ a₇ : α) : Array α :=
(((((((mkEmpty 7).push a₁).push a₂).push a₃).push a₄).push a₅).push a₆).push a₇
/-- Create array `#[a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈]` -/
def Array.mkArray8 {α : Type u} (a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈ : α) : Array α :=
((((((((mkEmpty 8).push a₁).push a₂).push a₃).push a₄).push a₅).push a₆).push a₇).push a₈
/--
Set an element in an array without bounds checks, using a `Fin` index.
This will perform the update destructively provided that `a` has a reference
count of 1 when called.
-/
@[extern "lean_array_fset"]
def Array.set (a : Array α) (i : @& Fin a.size) (v : α) : Array α where
data := a.data.set i.val v
/--
Set an element in an array, or do nothing if the index is out of bounds.
This will perform the update destructively provided that `a` has a reference
count of 1 when called.
-/
@[inline] def Array.setD (a : Array α) (i : Nat) (v : α) : Array α :=
dite (LT.lt i a.size) (fun h => a.set ⟨i, h⟩ v) (fun _ => a)
/--
Set an element in an array, or panic if the index is out of bounds.
This will perform the update destructively provided that `a` has a reference
count of 1 when called.
-/
@[extern "lean_array_set"]
def Array.set! (a : Array α) (i : @& Nat) (v : α) : Array α :=
Array.setD a i v
/-- Slower `Array.append` used in quotations. -/
protected def Array.appendCore {α : Type u} (as : Array α) (bs : Array α) : Array α :=
let rec loop (i : Nat) (j : Nat) (as : Array α) : Array α :=
dite (LT.lt j bs.size)
(fun hlt =>
match i with
| 0 => as
| Nat.succ i' => loop i' (hAdd j 1) (as.push (bs.get ⟨j, hlt⟩)))
(fun _ => as)
loop bs.size 0 as
/--
Returns the slice of `as` from indices `start` to `stop` (exclusive).
If `start` is greater or equal to `stop`, the result is empty.
If `stop` is greater than the length of `as`, the length is used instead. -/
-- NOTE: used in the quotation elaborator output
def Array.extract (as : Array α) (start stop : Nat) : Array α :=
let rec loop (i : Nat) (j : Nat) (bs : Array α) : Array α :=
dite (LT.lt j as.size)
(fun hlt =>
match i with
| 0 => bs
| Nat.succ i' => loop i' (hAdd j 1) (bs.push (as.get ⟨j, hlt⟩)))
(fun _ => bs)
let sz' := Nat.sub (min stop as.size) start
loop sz' start (mkEmpty sz')
/-- Auxiliary definition for `List.toArray`. -/
@[inline_if_reduce]
def List.toArrayAux : List α → Array α → Array α
| nil, r => r
| cons a as, r => toArrayAux as (r.push a)
/-- A non-tail-recursive version of `List.length`, used for `List.toArray`. -/
@[inline_if_reduce]
def List.redLength : List α → Nat
| nil => 0
| cons _ as => as.redLength.succ
/--
Convert a `List α` into an `Array α`. This is O(n) in the length of the list.
This function is exported to C, where it is called by `Array.mk`
(the constructor) to implement this functionality.
-/
@[inline, match_pattern, export lean_list_to_array]
def List.toArray (as : List α) : Array α :=
as.toArrayAux (Array.mkEmpty as.redLength)
/-- The typeclass which supplies the `>>=` "bind" function. See `Monad`. -/
class Bind (m : Type u → Type v) where
/-- If `x : m α` and `f : α → m β`, then `x >>= f : m β` represents the
result of executing `x` to get a value of type `α` and then passing it to `f`. -/
bind : {α β : Type u} → m α → (α → m β) → m β
export Bind (bind)
/-- The typeclass which supplies the `pure` function. See `Monad`. -/
class Pure (f : Type u → Type v) where
/-- If `a : α`, then `pure a : f α` represents a monadic action that does
nothing and returns `a`. -/
pure {α : Type u} : α → f α
export Pure (pure)
/--
In functional programming, a "functor" is a function on types `F : Type u → Type v`
equipped with an operator called `map` or `<$>` such that if `f : α → β` then
`map f : F α → F β`, so `f <$> x : F β` if `x : F α`. This corresponds to the
category-theory notion of [functor](https://en.wikipedia.org/wiki/Functor) in
the special case where the category is the category of types and functions
between them, except that this class supplies only the operations and not the
laws (see `LawfulFunctor`).
-/
class Functor (f : Type u → Type v) : Type (max (u+1) v) where
/-- If `f : α → β` and `x : F α` then `f <$> x : F β`. -/
map : {α β : Type u} → (α → β) → f α → f β
/-- The special case `const a <$> x`, which can sometimes be implemented more
efficiently. -/
mapConst : {α β : Type u} → α → f β → f α := Function.comp map (Function.const _)
/-- The typeclass which supplies the `<*>` "seq" function. See `Applicative`. -/
class Seq (f : Type u → Type v) : Type (max (u+1) v) where
/-- If `mf : F (α → β)` and `mx : F α`, then `mf <*> mx : F β`.
In a monad this is the same as `do let f ← mf; x ← mx; pure (f x)`:
it evaluates first the function, then the argument, and applies one to the other.
To avoid surprising evaluation semantics, `mx` is taken "lazily", using a
`Unit → f α` function. -/
seq : {α β : Type u} → f (α → β) → (Unit → f α) → f β
/-- The typeclass which supplies the `<*` "seqLeft" function. See `Applicative`. -/
class SeqLeft (f : Type u → Type v) : Type (max (u+1) v) where
/-- If `x : F α` and `y : F β`, then `x <* y` evaluates `x`, then `y`,
and returns the result of `x`.
To avoid surprising evaluation semantics, `y` is taken "lazily", using a
`Unit → f β` function. -/
seqLeft : {α β : Type u} → f α → (Unit → f β) → f α
/-- The typeclass which supplies the `*>` "seqRight" function. See `Applicative`. -/
class SeqRight (f : Type u → Type v) : Type (max (u+1) v) where
/-- If `x : F α` and `y : F β`, then `x *> y` evaluates `x`, then `y`,
and returns the result of `y`.
To avoid surprising evaluation semantics, `y` is taken "lazily", using a
`Unit → f β` function. -/
seqRight : {α β : Type u} → f α → (Unit → f β) → f β
/--
An [applicative functor](https://en.wikipedia.org/wiki/Applicative_functor) is
an intermediate structure between `Functor` and `Monad`. It mainly consists of
two operations:
* `pure : α → F α`
* `seq : F (α → β) → F α → F β` (written as `<*>`)
The `seq` operator gives a notion of evaluation order to the effects, where
the first argument is executed before the second, but unlike a monad the results
of earlier computations cannot be used to define later actions.
-/
class Applicative (f : Type u → Type v) extends Functor f, Pure f, Seq f, SeqLeft f, SeqRight f where
map := fun x y => Seq.seq (pure x) fun _ => y
seqLeft := fun a b => Seq.seq (Functor.map (Function.const _) a) b
seqRight := fun a b => Seq.seq (Functor.map (Function.const _ id) a) b
/--
A [monad](https://en.wikipedia.org/wiki/Monad_(functional_programming)) is a
structure which abstracts the concept of sequential control flow.
It mainly consists of two operations:
* `pure : α → F α`
* `bind : F α → (α → F β) → F β` (written as `>>=`)
Like many functional programming languages, Lean makes extensive use of monads
for structuring programs. In particular, the `do` notation is a very powerful
syntax over monad operations, and it depends on a `Monad` instance.
See [the `do` notation](https://leanprover.github.io/lean4/doc/do.html)
chapter of the manual for details.
-/
class Monad (m : Type u → Type v) extends Applicative m, Bind m : Type (max (u+1) v) where
map f x := bind x (Function.comp pure f)
seq f x := bind f fun y => Functor.map y (x ())
seqLeft x y := bind x fun a => bind (y ()) (fun _ => pure a)
seqRight x y := bind x fun _ => y ()
instance {α : Type u} {m : Type u → Type v} [Monad m] : Inhabited (α → m α) where
default := pure
instance {α : Type u} {m : Type u → Type v} [Monad m] [Inhabited α] : Inhabited (m α) where
default := pure default
instance [Monad m] : [Nonempty α] → Nonempty (m α)
| ⟨x⟩ => ⟨pure x⟩
/-- A fusion of Haskell's `sequence` and `map`. Used in syntax quotations. -/
def Array.sequenceMap {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : α → m β) : m (Array β) :=
let rec loop (i : Nat) (j : Nat) (bs : Array β) : m (Array β) :=
dite (LT.lt j as.size)
(fun hlt =>
match i with
| 0 => pure bs
| Nat.succ i' => Bind.bind (f (as.get ⟨j, hlt⟩)) fun b => loop i' (hAdd j 1) (bs.push b))
(fun _ => pure bs)
loop as.size 0 (Array.mkEmpty as.size)
/--
A function for lifting a computation from an inner `Monad` to an outer `Monad`.
Like Haskell's [`MonadTrans`], but `n` does not have to be a monad transformer.
Alternatively, an implementation of [`MonadLayer`] without `layerInvmap` (so far).
[`MonadTrans`]: https://hackage.haskell.org/package/transformers-0.5.5.0/docs/Control-Monad-Trans-Class.html
[`MonadLayer`]: https://hackage.haskell.org/package/layers-0.1/docs/Control-Monad-Layer.html#t:MonadLayer
-/
class MonadLift (m : semiOutParam (Type u → Type v)) (n : Type u → Type w) where
/-- Lifts a value from monad `m` into monad `n`. -/
monadLift : {α : Type u} → m α → n α
/--
The reflexive-transitive closure of `MonadLift`. `monadLift` is used to
transitively lift monadic computations such as `StateT.get` or `StateT.put s`.
Corresponds to Haskell's [`MonadLift`].
[`MonadLift`]: https://hackage.haskell.org/package/layers-0.1/docs/Control-Monad-Layer.html#t:MonadLift
-/
class MonadLiftT (m : Type u → Type v) (n : Type u → Type w) where
/-- Lifts a value from monad `m` into monad `n`. -/
monadLift : {α : Type u} → m α → n α
export MonadLiftT (monadLift)
/-- Lifts a value from monad `m` into monad `n`. -/
abbrev liftM := @monadLift
@[always_inline]
instance (m n o) [MonadLift n o] [MonadLiftT m n] : MonadLiftT m o where
monadLift x := MonadLift.monadLift (m := n) (monadLift x)
instance (m) : MonadLiftT m m where
monadLift x := x
/--
A functor in the category of monads. Can be used to lift monad-transforming functions.
Based on [`MFunctor`] from the `pipes` Haskell package, but not restricted to
monad transformers. Alternatively, an implementation of [`MonadTransFunctor`].
[`MFunctor`]: https://hackage.haskell.org/package/pipes-2.4.0/docs/Control-MFunctor.html
[`MonadTransFunctor`]: http://duairc.netsoc.ie/layers-docs/Control-Monad-Layer.html#t:MonadTransFunctor
-/
class MonadFunctor (m : semiOutParam (Type u → Type v)) (n : Type u → Type w) where
/-- Lifts a monad morphism `f : {β : Type u} → m β → m β` to
`monadMap f : {α : Type u} → n α → n α`. -/
monadMap {α : Type u} : ({β : Type u} → m β → m β) → n α → n α
/-- The reflexive-transitive closure of `MonadFunctor`.
`monadMap` is used to transitively lift `Monad` morphisms. -/
class MonadFunctorT (m : Type u → Type v) (n : Type u → Type w) where
/-- Lifts a monad morphism `f : {β : Type u} → m β → m β` to
`monadMap f : {α : Type u} → n α → n α`. -/
monadMap {α : Type u} : ({β : Type u} → m β → m β) → n α → n α
export MonadFunctorT (monadMap)
@[always_inline]
instance (m n o) [MonadFunctor n o] [MonadFunctorT m n] : MonadFunctorT m o where
monadMap f := MonadFunctor.monadMap (m := n) (monadMap (m := m) f)
instance monadFunctorRefl (m) : MonadFunctorT m m where
monadMap f := f
/--
`Except ε α` is a type which represents either an error of type `ε`, or an "ok"
value of type `α`. The error type is listed first because
`Except ε : Type → Type` is a `Monad`: the pure operation is `ok` and the bind
operation returns the first encountered `error`.
-/
inductive Except (ε : Type u) (α : Type v) where
/-- A failure value of type `ε` -/
| error : ε → Except ε α
/-- A success value of type `α` -/
| ok : α → Except ε α
attribute [unbox] Except
instance {ε : Type u} {α : Type v} [Inhabited ε] : Inhabited (Except ε α) where
default := Except.error default
/--
An implementation of Haskell's [`MonadError`] class. A `MonadError ε m` is a
monad `m` with two operations:
* `throw : ε → m α` "throws an error" of type `ε` to the nearest enclosing
catch block
* `tryCatch (body : m α) (handler : ε → m α) : m α` will catch any errors in
`body` and pass the resulting error to `handler`.
Errors in `handler` will not be caught.
The `try ... catch e => ...` syntax inside `do` blocks is sugar for the
`tryCatch` operation.
[`MonadError`]: https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Except.html#t:MonadError
-/
class MonadExceptOf (ε : semiOutParam (Type u)) (m : Type v → Type w) where
/-- `throw : ε → m α` "throws an error" of type `ε` to the nearest enclosing
catch block. -/
throw {α : Type v} : ε → m α
/-- `tryCatch (body : m α) (handler : ε → m α) : m α` will catch any errors in
`body` and pass the resulting error to `handler`.
Errors in `handler` will not be caught. -/
tryCatch {α : Type v} (body : m α) (handler : ε → m α) : m α
/--
This is the same as `throw`, but allows specifying the particular error type
in case the monad supports throwing more than one type of error.
-/
abbrev throwThe (ε : Type u) {m : Type v → Type w} [MonadExceptOf ε m] {α : Type v} (e : ε) : m α :=
MonadExceptOf.throw e
/--
This is the same as `tryCatch`, but allows specifying the particular error type
in case the monad supports throwing more than one type of error.
-/
abbrev tryCatchThe (ε : Type u) {m : Type v → Type w} [MonadExceptOf ε m] {α : Type v} (x : m α) (handle : ε → m α) : m α :=
MonadExceptOf.tryCatch x handle
/-- Similar to `MonadExceptOf`, but `ε` is an `outParam` for convenience. -/
class MonadExcept (ε : outParam (Type u)) (m : Type v → Type w) where
/-- `throw : ε → m α` "throws an error" of type `ε` to the nearest enclosing
catch block. -/
throw {α : Type v} : ε → m α
/-- `tryCatch (body : m α) (handler : ε → m α) : m α` will catch any errors in
`body` and pass the resulting error to `handler`.
Errors in `handler` will not be caught. -/
tryCatch {α : Type v} : m α → (ε → m α) → m α
/-- "Unwraps" an `Except ε α` to get the `α`, or throws the exception otherwise. -/
def MonadExcept.ofExcept [Monad m] [MonadExcept ε m] : Except ε α → m α
| .ok a => pure a
| .error e => throw e
export MonadExcept (throw tryCatch ofExcept)
instance (ε : outParam (Type u)) (m : Type v → Type w) [MonadExceptOf ε m] : MonadExcept ε m where
throw := throwThe ε
tryCatch := tryCatchThe ε
namespace MonadExcept
variable {ε : Type u} {m : Type v → Type w}
/-- A `MonadExcept` can implement `t₁ <|> t₂` as `try t₁ catch _ => t₂`. -/
@[inline] protected def orElse [MonadExcept ε m] {α : Type v} (t₁ : m α) (t₂ : Unit → m α) : m α :=
tryCatch t₁ fun _ => t₂ ()
instance [MonadExcept ε m] {α : Type v} : OrElse (m α) where
orElse := MonadExcept.orElse
end MonadExcept
/--
An implementation of Haskell's [`ReaderT`]. This is a monad transformer which
equips a monad with additional read-only state, of type `ρ`.
[`ReaderT`]: https://hackage.haskell.org/package/transformers-0.5.5.0/docs/Control-Monad-Trans-Reader.html#t:ReaderT
-/
def ReaderT (ρ : Type u) (m : Type u → Type v) (α : Type u) : Type (max u v) :=
ρ → m α
instance (ρ : Type u) (m : Type u → Type v) (α : Type u) [Inhabited (m α)] : Inhabited (ReaderT ρ m α) where
default := fun _ => default
/--
If `x : ReaderT ρ m α` and `r : ρ`, then `x.run r : ρ` runs the monad with the
given reader state.
-/
@[always_inline, inline]
def ReaderT.run {ρ : Type u} {m : Type u → Type v} {α : Type u} (x : ReaderT ρ m α) (r : ρ) : m α :=
x r
namespace ReaderT
section
variable {ρ : Type u} {m : Type u → Type v} {α : Type u}
instance : MonadLift m (ReaderT ρ m) where
monadLift x := fun _ => x
@[always_inline]
instance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ReaderT ρ m) where
throw e := liftM (m := m) (throw e)
tryCatch := fun x c r => tryCatchThe ε (x r) (fun e => (c e) r)
end
section
variable {ρ : Type u} {m : Type u → Type v}
/-- `(← read) : ρ` gets the read-only state of a `ReaderT ρ`. -/
@[always_inline, inline]
protected def read [Monad m] : ReaderT ρ m ρ :=
pure
/-- The `pure` operation of the `ReaderT` monad. -/
@[always_inline, inline]
protected def pure [Monad m] {α} (a : α) : ReaderT ρ m α :=
fun _ => pure a
/-- The `bind` operation of the `ReaderT` monad. -/
@[always_inline, inline]
protected def bind [Monad m] {α β} (x : ReaderT ρ m α) (f : α → ReaderT ρ m β) : ReaderT ρ m β :=
fun r => bind (x r) fun a => f a r
@[always_inline]
instance [Monad m] : Functor (ReaderT ρ m) where
map f x r := Functor.map f (x r)
mapConst a x r := Functor.mapConst a (x r)
@[always_inline]
instance [Monad m] : Applicative (ReaderT ρ m) where
pure := ReaderT.pure
seq f x r := Seq.seq (f r) fun _ => x () r
seqLeft a b r := SeqLeft.seqLeft (a r) fun _ => b () r
seqRight a b r := SeqRight.seqRight (a r) fun _ => b () r
instance [Monad m] : Monad (ReaderT ρ m) where
bind := ReaderT.bind
instance (ρ m) : MonadFunctor m (ReaderT ρ m) where
monadMap f x := fun ctx => f (x ctx)
/--
`adapt (f : ρ' → ρ)` precomposes function `f` on the reader state of a
`ReaderT ρ`, yielding a `ReaderT ρ'`.
-/
@[always_inline, inline]
protected def adapt {ρ' α : Type u} (f : ρ' → ρ) : ReaderT ρ m α → ReaderT ρ' m α :=
fun x r => x (f r)
end
end ReaderT
/--
An implementation of Haskell's [`MonadReader`] (sans functional dependency; see also `MonadReader`
in this module). It does not contain `local` because this
function cannot be lifted using `monadLift`. `local` is instead provided by
the `MonadWithReader` class as `withReader`.
Note: This class can be seen as a simplification of the more "principled" definition
```
class MonadReaderOf (ρ : Type u) (n : Type u → Type u) where
lift {α : Type u} : ({m : Type u → Type u} → [Monad m] → ReaderT ρ m α) → n α
```
[`MonadReader`]: https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Reader-Class.html#t:MonadReader
-/
class MonadReaderOf (ρ : semiOutParam (Type u)) (m : Type u → Type v) where
/-- `(← read) : ρ` reads the state out of monad `m`. -/
read : m ρ
/--
Like `read`, but with `ρ` explicit. This is useful if a monad supports
`MonadReaderOf` for multiple different types `ρ`.
-/
@[always_inline, inline]
def readThe (ρ : Type u) {m : Type u → Type v} [MonadReaderOf ρ m] : m ρ :=
MonadReaderOf.read
/-- Similar to `MonadReaderOf`, but `ρ` is an `outParam` for convenience. -/
class MonadReader (ρ : outParam (Type u)) (m : Type u → Type v) where
/-- `(← read) : ρ` reads the state out of monad `m`. -/
read : m ρ
export MonadReader (read)
instance (ρ : Type u) (m : Type u → Type v) [MonadReaderOf ρ m] : MonadReader ρ m where
read := readThe ρ
instance {ρ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadLift m n] [MonadReaderOf ρ m] : MonadReaderOf ρ n where
read := liftM (m := m) read
instance {ρ : Type u} {m : Type u → Type v} [Monad m] : MonadReaderOf ρ (ReaderT ρ m) where
read := ReaderT.read
/--
`MonadWithReaderOf ρ` adds the operation `withReader : (ρ → ρ) → m α → m α`.
This runs the inner `x : m α` inside a modified context after applying the
function `f : ρ → ρ`. In addition to `ReaderT` itself, this operation lifts
over most monad transformers, so it allows us to apply `withReader` to monads
deeper in the stack.
-/
class MonadWithReaderOf (ρ : semiOutParam (Type u)) (m : Type u → Type v) where
/-- `withReader (f : ρ → ρ) (x : m α) : m α` runs the inner `x : m α` inside
a modified context after applying the function `f : ρ → ρ`.-/
withReader {α : Type u} : (ρ → ρ) → m α → m α
/--
Like `withReader`, but with `ρ` explicit. This is useful if a monad supports
`MonadWithReaderOf` for multiple different types `ρ`.
-/
@[always_inline, inline]
def withTheReader (ρ : Type u) {m : Type u → Type v} [MonadWithReaderOf ρ m] {α : Type u} (f : ρ → ρ) (x : m α) : m α :=
MonadWithReaderOf.withReader f x
/-- Similar to `MonadWithReaderOf`, but `ρ` is an `outParam` for convenience. -/
class MonadWithReader (ρ : outParam (Type u)) (m : Type u → Type v) where
/-- `withReader (f : ρ → ρ) (x : m α) : m α` runs the inner `x : m α` inside
a modified context after applying the function `f : ρ → ρ`.-/
withReader {α : Type u} : (ρ → ρ) → m α → m α
export MonadWithReader (withReader)
instance (ρ : Type u) (m : Type u → Type v) [MonadWithReaderOf ρ m] : MonadWithReader ρ m where
withReader := withTheReader ρ
instance {ρ : Type u} {m : Type u → Type v} {n : Type u → Type v} [MonadFunctor m n] [MonadWithReaderOf ρ m] : MonadWithReaderOf ρ n where
withReader f := monadMap (m := m) (withTheReader ρ f)
instance {ρ : Type u} {m : Type u → Type v} [Monad m] : MonadWithReaderOf ρ (ReaderT ρ m) where
withReader f x := fun ctx => x (f ctx)
/--
An implementation of [`MonadState`]. In contrast to the Haskell implementation,
we use overlapping instances to derive instances automatically from `monadLift`.
[`MonadState`]: https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-State-Class.html
-/
class MonadStateOf (σ : semiOutParam (Type u)) (m : Type u → Type v) where
/-- `(← get) : σ` gets the state out of a monad `m`. -/
get : m σ
/-- `set (s : σ)` replaces the state with value `s`. -/
set : σ → m PUnit
/-- `modifyGet (f : σ → α × σ)` applies `f` to the current state, replaces
the state with the return value, and returns a computed value.
It is equivalent to `do let (a, s) := f (← get); put s; pure a`, but
`modifyGet f` may be preferable because the former does not use the state
linearly (without sufficient inlining). -/
modifyGet {α : Type u} : (σ → Prod α σ) → m α
export MonadStateOf (set)
/--
Like `withReader`, but with `ρ` explicit. This is useful if a monad supports
`MonadWithReaderOf` for multiple different types `ρ`.
-/
abbrev getThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] : m σ :=
MonadStateOf.get
/--
Like `modify`, but with `σ` explicit. This is useful if a monad supports
`MonadStateOf` for multiple different types `σ`.
-/
@[always_inline, inline]
abbrev modifyThe (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → σ) : m PUnit :=
MonadStateOf.modifyGet fun s => (PUnit.unit, f s)
/--
Like `modifyGet`, but with `σ` explicit. This is useful if a monad supports
`MonadStateOf` for multiple different types `σ`.
-/
@[always_inline, inline]
abbrev modifyGetThe {α : Type u} (σ : Type u) {m : Type u → Type v} [MonadStateOf σ m] (f : σ → Prod α σ) : m α :=
MonadStateOf.modifyGet f
/-- Similar to `MonadStateOf`, but `σ` is an `outParam` for convenience. -/
class MonadState (σ : outParam (Type u)) (m : Type u → Type v) where
/-- `(← get) : σ` gets the state out of a monad `m`. -/
get : m σ
/-- `set (s : σ)` replaces the state with value `s`. -/
set : σ → m PUnit
/-- `modifyGet (f : σ → α × σ)` applies `f` to the current state, replaces
the state with the return value, and returns a computed value.
It is equivalent to `do let (a, s) := f (← get); put s; pure a`, but
`modifyGet f` may be preferable because the former does not use the state
linearly (without sufficient inlining). -/
modifyGet {α : Type u} : (σ → Prod α σ) → m α
export MonadState (get modifyGet)
instance (σ : Type u) (m : Type u → Type v) [MonadStateOf σ m] : MonadState σ m where
set := MonadStateOf.set
get := getThe σ
modifyGet f := MonadStateOf.modifyGet f
/--
`modify (f : σ → σ)` applies the function `f` to the state.
It is equivalent to `do put (f (← get))`, but `modify f` may be preferable
because the former does not use the state linearly (without sufficient inlining).
-/
@[always_inline, inline]
def modify {σ : Type u} {m : Type u → Type v} [MonadState σ m] (f : σ → σ) : m PUnit :=
modifyGet fun s => (PUnit.unit, f s)
/--
`getModify f` gets the state, applies function `f`, and returns the old value
of the state. It is equivalent to `get <* modify f` but may be more efficient.
-/
@[always_inline, inline]
def getModify {σ : Type u} {m : Type u → Type v} [MonadState σ m] [Monad m] (f : σ → σ) : m σ :=
modifyGet fun s => (s, f s)
-- NOTE: The Ordering of the following two instances determines that the top-most `StateT` Monad layer
-- will be picked first
@[always_inline]
instance {σ : Type u} {m : Type u → Type v} {n : Type u → Type w} [MonadLift m n] [MonadStateOf σ m] : MonadStateOf σ n where
get := liftM (m := m) MonadStateOf.get
set s := liftM (m := m) (MonadStateOf.set s)
modifyGet f := monadLift (m := m) (MonadState.modifyGet f)
namespace EStateM
/--
`Result ε σ α` is equivalent to `Except ε α × σ`, but using a single
combined inductive yields a more efficient data representation.
-/
inductive Result (ε σ α : Type u) where
/-- A success value of type `α`, and a new state `σ`. -/
| ok : α → σ → Result ε σ α
/-- A failure value of type `ε`, and a new state `σ`. -/
| error : ε → σ → Result ε σ α
variable {ε σ α : Type u}
instance [Inhabited ε] [Inhabited σ] : Inhabited (Result ε σ α) where
default := Result.error default default
end EStateM
open EStateM (Result) in
/--
`EStateM ε σ` is a combined error and state monad, equivalent to
`ExceptT ε (StateM σ)` but more efficient.
-/
def EStateM (ε σ α : Type u) := σ → Result ε σ α
namespace EStateM
variable {ε σ α β : Type u}
instance [Inhabited ε] : Inhabited (EStateM ε σ α) where
default := fun s => Result.error default s
/-- The `pure` operation of the `EStateM` monad. -/
@[always_inline, inline]
protected def pure (a : α) : EStateM ε σ α := fun s =>
Result.ok a s
/-- The `set` operation of the `EStateM` monad. -/
@[always_inline, inline]
protected def set (s : σ) : EStateM ε σ PUnit := fun _ =>
Result.ok ⟨⟩ s
/-- The `get` operation of the `EStateM` monad. -/
@[always_inline, inline]
protected def get : EStateM ε σ σ := fun s =>
Result.ok s s
/-- The `modifyGet` operation of the `EStateM` monad. -/
@[always_inline, inline]
protected def modifyGet (f : σ → Prod α σ) : EStateM ε σ α := fun s =>
match f s with
| (a, s) => Result.ok a s
/-- The `throw` operation of the `EStateM` monad. -/
@[always_inline, inline]
protected def throw (e : ε) : EStateM ε σ α := fun s =>
Result.error e s
/--
Auxiliary instance for saving/restoring the "backtrackable" part of the state.
Here `σ` is the state, and `δ` is some subpart of it, and we have a
getter and setter for it (a "lens" in the Haskell terminology).
-/
class Backtrackable (δ : outParam (Type u)) (σ : Type u) where
/-- `save s : δ` retrieves a copy of the backtracking state out of the state. -/
save : σ → δ
/-- `restore (s : σ) (x : δ) : σ` applies the old backtracking state `x` to
the state `s` to get a backtracked state `s'`. -/
restore : σ → δ → σ
/-- Implementation of `tryCatch` for `EStateM` where the state is `Backtrackable`. -/
@[always_inline, inline]
protected def tryCatch {δ} [Backtrackable δ σ] {α} (x : EStateM ε σ α) (handle : ε → EStateM ε σ α) : EStateM ε σ α := fun s =>
let d := Backtrackable.save s
match x s with
| Result.error e s => handle e (Backtrackable.restore s d)
| ok => ok
/-- Implementation of `orElse` for `EStateM` where the state is `Backtrackable`. -/
@[always_inline, inline]
protected def orElse {δ} [Backtrackable δ σ] (x₁ : EStateM ε σ α) (x₂ : Unit → EStateM ε σ α) : EStateM ε σ α := fun s =>
let d := Backtrackable.save s;
match x₁ s with
| Result.error _ s => x₂ () (Backtrackable.restore s d)
| ok => ok
/-- Map the exception type of a `EStateM ε σ α` by a function `f : ε → ε'`. -/
@[always_inline, inline]
def adaptExcept {ε' : Type u} (f : ε → ε') (x : EStateM ε σ α) : EStateM ε' σ α := fun s =>
match x s with
| Result.error e s => Result.error (f e) s
| Result.ok a s => Result.ok a s
/-- The `bind` operation of the `EStateM` monad. -/
@[always_inline, inline]
protected def bind (x : EStateM ε σ α) (f : α → EStateM ε σ β) : EStateM ε σ β := fun s =>
match x s with
| Result.ok a s => f a s
| Result.error e s => Result.error e s
/-- The `map` operation of the `EStateM` monad. -/
@[always_inline, inline]
protected def map (f : α → β) (x : EStateM ε σ α) : EStateM ε σ β := fun s =>
match x s with
| Result.ok a s => Result.ok (f a) s
| Result.error e s => Result.error e s
/-- The `seqRight` operation of the `EStateM` monad. -/
@[always_inline, inline]
protected def seqRight (x : EStateM ε σ α) (y : Unit → EStateM ε σ β) : EStateM ε σ β := fun s =>
match x s with
| Result.ok _ s => y () s
| Result.error e s => Result.error e s
@[always_inline]
instance : Monad (EStateM ε σ) where
bind := EStateM.bind
pure := EStateM.pure
map := EStateM.map
seqRight := EStateM.seqRight
instance {δ} [Backtrackable δ σ] : OrElse (EStateM ε σ α) where
orElse := EStateM.orElse
instance : MonadStateOf σ (EStateM ε σ) where
set := EStateM.set
get := EStateM.get
modifyGet := EStateM.modifyGet
instance {δ} [Backtrackable δ σ] : MonadExceptOf ε (EStateM ε σ) where
throw := EStateM.throw
tryCatch := EStateM.tryCatch
/-- Execute an `EStateM` on initial state `s` to get a `Result`. -/
@[always_inline, inline]
def run (x : EStateM ε σ α) (s : σ) : Result ε σ α := x s
/--
Execute an `EStateM` on initial state `s` for the returned value `α`.
If the monadic action throws an exception, returns `none` instead.
-/
@[always_inline, inline]
def run' (x : EStateM ε σ α) (s : σ) : Option α :=
match run x s with
| Result.ok v _ => some v
| Result.error .. => none
/-- The `save` implementation for `Backtrackable PUnit σ`. -/
@[inline] def dummySave : σ → PUnit := fun _ => ⟨⟩
/-- The `restore` implementation for `Backtrackable PUnit σ`. -/
@[inline] def dummyRestore : σ → PUnit → σ := fun s _ => s
/--
Dummy default instance. This makes every `σ` trivially "backtrackable"
by doing nothing on backtrack. Because this is the first declared instance
of `Backtrackable _ σ`, it will be picked only if there are no other
`Backtrackable _ σ` instances registered.
-/
instance nonBacktrackable : Backtrackable PUnit σ where
save := dummySave
restore := dummyRestore
end EStateM
/-- A class for types that can be hashed into a `UInt64`. -/
class Hashable (α : Sort u) where
/-- Hashes the value `a : α` into a `UInt64`. -/
hash : α → UInt64
export Hashable (hash)
/-- Converts a `UInt64` to a `USize` by reducing modulo `USize.size`. -/
@[extern "lean_uint64_to_usize"]
opaque UInt64.toUSize (u : UInt64) : USize
/--
Upcast a `USize` to a `UInt64`.
This is lossless because `USize.size` is either `2^32` or `2^64`.
This function is overridden with a native implementation.
-/
@[extern "lean_usize_to_uint64"]
def USize.toUInt64 (u : USize) : UInt64 where
val := {
val := u.val.val
isLt :=
let ⟨n, h⟩ := u
show LT.lt n _ from
match USize.size, usize_size_eq, h with
| _, Or.inl rfl, h => Nat.lt_trans h (by decide)
| _, Or.inr rfl, h => h
}
/-- An opaque hash mixing operation, used to implement hashing for tuples. -/
@[extern "lean_uint64_mix_hash"]
opaque mixHash (u₁ u₂ : UInt64) : UInt64
instance [Hashable α] {p : α → Prop} : Hashable (Subtype p) where
hash a := hash a.val
/-- An opaque string hash function. -/
@[extern "lean_string_hash"]
protected opaque String.hash (s : @& String) : UInt64
instance : Hashable String where
hash := String.hash
namespace Lean
/--
Hierarchical names. We use hierarchical names to name declarations and
for creating unique identifiers for free variables and metavariables.
You can create hierarchical names using the following quotation notation.
```
`Lean.Meta.whnf
```
It is short for `.str (.str (.str .anonymous "Lean") "Meta") "whnf"`
You can use double quotes to request Lean to statically check whether the name
corresponds to a Lean declaration in scope.
```
``Lean.Meta.whnf
```
If the name is not in scope, Lean will report an error.
-/
inductive Name where
/-- The "anonymous" name. -/
| anonymous : Name
/--
A string name. The name `Lean.Meta.run` is represented at
```lean
.str (.str (.str .anonymous "Lean") "Meta") "run"
```
-/
| str (pre : Name) (str : String)
/--
A numerical name. This kind of name is used, for example, to create hierarchical names for
free variables and metavariables. The identifier `_uniq.231` is represented as
```lean
.num (.str .anonymous "_uniq") 231
```
-/
| num (pre : Name) (i : Nat)
with
/-- A hash function for names, which is stored inside the name itself as a
computed field. -/
@[computed_field] hash : Name → UInt64
| .anonymous => .ofNatCore 1723 (by decide)
| .str p s => mixHash p.hash s.hash
| .num p v => mixHash p.hash (dite (LT.lt v UInt64.size) (fun h => UInt64.ofNatCore v h) (fun _ => UInt64.ofNatCore 17 (by decide)))
instance : Inhabited Name where
default := Name.anonymous
instance : Hashable Name where
hash := Name.hash
namespace Name
/--
`.str p s` is now the preferred form.
-/
@[export lean_name_mk_string]
abbrev mkStr (p : Name) (s : String) : Name :=
Name.str p s
/--
`.num p v` is now the preferred form.
-/
@[export lean_name_mk_numeral]
abbrev mkNum (p : Name) (v : Nat) : Name :=
Name.num p v
/--
Short for `.str .anonymous s`.
-/
abbrev mkSimple (s : String) : Name :=
.str .anonymous s
/-- Make name `s₁` -/
@[reducible] def mkStr1 (s₁ : String) : Name :=
.str .anonymous s₁
/-- Make name `s₁.s₂` -/
@[reducible] def mkStr2 (s₁ s₂ : String) : Name :=
.str (.str .anonymous s₁) s₂
/-- Make name `s₁.s₂.s₃` -/
@[reducible] def mkStr3 (s₁ s₂ s₃ : String) : Name :=
.str (.str (.str .anonymous s₁) s₂) s₃
/-- Make name `s₁.s₂.s₃.s₄` -/
@[reducible] def mkStr4 (s₁ s₂ s₃ s₄ : String) : Name :=
.str (.str (.str (.str .anonymous s₁) s₂) s₃) s₄
/-- Make name `s₁.s₂.s₃.s₄.s₅` -/
@[reducible] def mkStr5 (s₁ s₂ s₃ s₄ s₅ : String) : Name :=
.str (.str (.str (.str (.str .anonymous s₁) s₂) s₃) s₄) s₅
/-- Make name `s₁.s₂.s₃.s₄.s₅.s₆` -/
@[reducible] def mkStr6 (s₁ s₂ s₃ s₄ s₅ s₆ : String) : Name :=
.str (.str (.str (.str (.str (.str .anonymous s₁) s₂) s₃) s₄) s₅) s₆
/-- Make name `s₁.s₂.s₃.s₄.s₅.s₆.s₇` -/
@[reducible] def mkStr7 (s₁ s₂ s₃ s₄ s₅ s₆ s₇ : String) : Name :=
.str (.str (.str (.str (.str (.str (.str .anonymous s₁) s₂) s₃) s₄) s₅) s₆) s₇
/-- Make name `s₁.s₂.s₃.s₄.s₅.s₆.s₇.s₈` -/
@[reducible] def mkStr8 (s₁ s₂ s₃ s₄ s₅ s₆ s₇ s₈ : String) : Name :=
.str (.str (.str (.str (.str (.str (.str (.str .anonymous s₁) s₂) s₃) s₄) s₅) s₆) s₇) s₈
/-- (Boolean) equality comparator for names. -/
@[extern "lean_name_eq"]
protected def beq : (@& Name) → (@& Name) → Bool
| anonymous, anonymous => true
| str p₁ s₁, str p₂ s₂ => and (BEq.beq s₁ s₂) (Name.beq p₁ p₂)
| num p₁ n₁, num p₂ n₂ => and (BEq.beq n₁ n₂) (Name.beq p₁ p₂)
| _, _ => false
instance : BEq Name where
beq := Name.beq
/--
This function does not have special support for macro scopes.
See `Name.append`.
-/
def appendCore : Name → Name → Name
| n, .anonymous => n
| n, .str p s => .str (appendCore n p) s
| n, .num p d => .num (appendCore n p) d
end Name
/-! # Syntax -/
/-- Source information of tokens. -/
inductive SourceInfo where
/--
Token from original input with whitespace and position information.
`leading` will be inferred after parsing by `Syntax.updateLeading`. During parsing,
it is not at all clear what the preceding token was, especially with backtracking.
-/
| original (leading : Substring) (pos : String.Pos) (trailing : Substring) (endPos : String.Pos)
/--
Synthesized syntax (e.g. from a quotation) annotated with a span from the original source.
In the delaborator, we "misuse" this constructor to store synthetic positions identifying
subterms.
The `canonical` flag on synthetic syntax is enabled for syntax that is not literally part
of the original input syntax but should be treated "as if" the user really wrote it
for the purpose of hovers and error messages. This is usually used on identifiers,
to connect the binding site to the user's original syntax even if the name of the identifier
changes during expansion, as well as on tokens where we will attach targeted messages.
The syntax `token%$stx` in a syntax quotation will annotate the token `token` with the span
from `stx` and also mark it as canonical.
As a rough guide, a macro expansion should only use a given piece of input syntax in
a single canonical token, although this is sometimes violated when the same identifier
is used to declare two binders, as in the macro expansion for dependent if:
```
`(if $h : $cond then $t else $e) ~>
`(dite $cond (fun $h => $t) (fun $h => $t))
```
In these cases if the user hovers over `h` they will see information about both binding sites.
-/
| synthetic (pos : String.Pos) (endPos : String.Pos) (canonical := false)
/-- Synthesized token without position information. -/
| protected none
instance : Inhabited SourceInfo := ⟨SourceInfo.none⟩
namespace SourceInfo
/--
Gets the position information from a `SourceInfo`, if available.
If `originalOnly` is true, then `.synthetic` syntax will also return `none`.
-/
def getPos? (info : SourceInfo) (canonicalOnly := false) : Option String.Pos :=
match info, canonicalOnly with
| original (pos := pos) .., _
| synthetic (pos := pos) (canonical := true) .., _
| synthetic (pos := pos) .., false => some pos
| _, _ => none
end SourceInfo
/--
A `SyntaxNodeKind` classifies `Syntax.node` values. It is an abbreviation for
`Name`, and you can use name literals to construct `SyntaxNodeKind`s, but
they need not refer to declarations in the environment. Conventionally, a
`SyntaxNodeKind` will correspond to the `Parser` or `ParserDesc` declaration
that parses it.
-/
abbrev SyntaxNodeKind := Name
/-! # Syntax AST -/
/--
Binding information resolved and stored at compile time of a syntax quotation.
Note: We do not statically know whether a syntax expects a namespace or term name,
so a `Syntax.ident` may contain both preresolution kinds.
-/
inductive Syntax.Preresolved where
/-- A potential namespace reference -/
| namespace (ns : Name)
/-- A potential global constant or section variable reference, with additional field accesses -/
| decl (n : Name) (fields : List String)
/--
Syntax objects used by the parser, macro expander, delaborator, etc.
-/
inductive Syntax where
/-- A `missing` syntax corresponds to a portion of the syntax tree that is
missing because of a parse error. The indexing operator on Syntax also
returns `missing` for indexing out of bounds. -/
| missing : Syntax
/-- Node in the syntax tree.
The `info` field is used by the delaborator to store the position of the
subexpression corresponding to this node. The parser sets the `info` field
to `none`.
The parser sets the `info` field to `none`, with position retrieval continuing recursively.
Nodes created by quotations use the result from `SourceInfo.fromRef` so that they are marked
as synthetic even when the leading/trailing token is not.
The delaborator uses the `info` field to store the position of the subexpression
corresponding to this node.
(Remark: the `node` constructor did not have an `info` field in previous
versions. This caused a bug in the interactive widgets, where the popup for
`a + b` was the same as for `a`. The delaborator used to associate
subexpressions with pretty-printed syntax by setting the (string) position
of the first atom/identifier to the (expression) position of the
subexpression. For example, both `a` and `a + b` have the same first
identifier, and so their infos got mixed up.) -/
| node (info : SourceInfo) (kind : SyntaxNodeKind) (args : Array Syntax) : Syntax
/-- An `atom` corresponds to a keyword or piece of literal unquoted syntax.
These correspond to quoted strings inside `syntax` declarations.
For example, in `(x + y)`, `"("`, `"+"` and `")"` are `atom`
and `x` and `y` are `ident`. -/
| atom (info : SourceInfo) (val : String) : Syntax
/-- An `ident` corresponds to an identifier as parsed by the `ident` or
`rawIdent` parsers.
* `rawVal` is the literal substring from the input file
* `val` is the parsed identifier (with hygiene)
* `preresolved` is the list of possible declarations this could refer to
-/
| ident (info : SourceInfo) (rawVal : Substring) (val : Name) (preresolved : List Syntax.Preresolved) : Syntax
/-- Create syntax node with 1 child -/
def Syntax.node1 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ : Syntax) : Syntax :=
Syntax.node info kind (Array.mkArray1 a₁)
/-- Create syntax node with 2 children -/
def Syntax.node2 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ : Syntax) : Syntax :=
Syntax.node info kind (Array.mkArray2 a₁ a₂)
/-- Create syntax node with 3 children -/
def Syntax.node3 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ : Syntax) : Syntax :=
Syntax.node info kind (Array.mkArray3 a₁ a₂ a₃)
/-- Create syntax node with 4 children -/
def Syntax.node4 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ : Syntax) : Syntax :=
Syntax.node info kind (Array.mkArray4 a₁ a₂ a₃ a₄)
/-- Create syntax node with 5 children -/
def Syntax.node5 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ a₅ : Syntax) : Syntax :=
Syntax.node info kind (Array.mkArray5 a₁ a₂ a₃ a₄ a₅)
/-- Create syntax node with 6 children -/
def Syntax.node6 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ a₅ a₆ : Syntax) : Syntax :=
Syntax.node info kind (Array.mkArray6 a₁ a₂ a₃ a₄ a₅ a₆)
/-- Create syntax node with 7 children -/
def Syntax.node7 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ a₅ a₆ a₇ : Syntax) : Syntax :=
Syntax.node info kind (Array.mkArray7 a₁ a₂ a₃ a₄ a₅ a₆ a₇)
/-- Create syntax node with 8 children -/
def Syntax.node8 (info : SourceInfo) (kind : SyntaxNodeKind) (a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈ : Syntax) : Syntax :=
Syntax.node info kind (Array.mkArray8 a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈)
/-- `SyntaxNodeKinds` is a set of `SyntaxNodeKind` (implemented as a list). -/
def SyntaxNodeKinds := List SyntaxNodeKind
/--
A `Syntax` value of one of the given syntax kinds.
Note that while syntax quotations produce/expect `TSyntax` values of the correct kinds,
this is not otherwise enforced and can easily be circumvented by direct use of the constructor.
The namespace `TSyntax.Compat` can be opened to expose a general coercion from `Syntax` to any
`TSyntax ks` for porting older code.
-/
structure TSyntax (ks : SyntaxNodeKinds) where
/-- The underlying `Syntax` value. -/
raw : Syntax
instance : Inhabited Syntax where
default := Syntax.missing
instance : Inhabited (TSyntax ks) where
default := ⟨default⟩
/-! # Builtin kinds -/
/--
The `choice` kind is used when a piece of syntax has multiple parses, and the
determination of which to use is deferred until typing information is available.
-/
abbrev choiceKind : SyntaxNodeKind := `choice
/-- The null kind is used for raw list parsers like `many`. -/
abbrev nullKind : SyntaxNodeKind := `null
/--
The `group` kind is by the `group` parser, to avoid confusing with the null
kind when used inside `optional`.
-/
abbrev groupKind : SyntaxNodeKind := `group
/--
`ident` is not actually used as a node kind, but it is returned by
`getKind` in the `ident` case so that things that handle different node
kinds can also handle `ident`.
-/
abbrev identKind : SyntaxNodeKind := `ident
/-- `str` is the node kind of string literals like `"foo"`. -/
abbrev strLitKind : SyntaxNodeKind := `str
/-- `char` is the node kind of character literals like `'A'`. -/
abbrev charLitKind : SyntaxNodeKind := `char
/-- `num` is the node kind of number literals like `42`. -/
abbrev numLitKind : SyntaxNodeKind := `num
/-- `scientific` is the node kind of floating point literals like `1.23e-3`. -/
abbrev scientificLitKind : SyntaxNodeKind := `scientific
/-- `name` is the node kind of name literals like `` `foo ``. -/
abbrev nameLitKind : SyntaxNodeKind := `name
/-- `fieldIdx` is the node kind of projection indices like the `2` in `x.2`. -/
abbrev fieldIdxKind : SyntaxNodeKind := `fieldIdx
/--
`hygieneInfo` is the node kind of the `hygieneInfo` parser, which is an
"invisible token" which captures the hygiene information at the current point
without parsing anything.
They can be used to generate identifiers (with `Lean.HygieneInfo.mkIdent`)
as if they were introduced by the calling context, not the called macro.
-/
abbrev hygieneInfoKind : SyntaxNodeKind := `hygieneInfo
/--
`interpolatedStrLitKind` is the node kind of interpolated string literal
fragments like `"value = {` and `}"` in `s!"value = {x}"`.
-/
abbrev interpolatedStrLitKind : SyntaxNodeKind := `interpolatedStrLitKind
/--
`interpolatedStrKind` is the node kind of an interpolated string literal
like `"value = {x}"` in `s!"value = {x}"`.
-/
abbrev interpolatedStrKind : SyntaxNodeKind := `interpolatedStrKind
/-- Creates an info-less node of the given kind and children. -/
@[inline] def mkNode (k : SyntaxNodeKind) (args : Array Syntax) : TSyntax (.cons k .nil) :=
⟨Syntax.node SourceInfo.none k args⟩
/-- Creates an info-less `nullKind` node with the given children, if any. -/
-- NOTE: used by the quotation elaborator output
@[inline] def mkNullNode (args : Array Syntax := Array.empty) : Syntax :=
mkNode nullKind args |>.raw
namespace Syntax
/--
Gets the kind of a `Syntax` node. For non-`node` syntax, we use "pseudo kinds":
`identKind` for `ident`, `missing` for `missing`, and the atom's string literal
for atoms.
-/
def getKind (stx : Syntax) : SyntaxNodeKind :=
match stx with
| Syntax.node _ k _ => k
-- We use these "pseudo kinds" for antiquotation kinds.
-- For example, an antiquotation `$id:ident` (using Lean.Parser.Term.ident)
-- is compiled to ``if stx.isOfKind `ident ...``
| Syntax.missing => `missing
| Syntax.atom _ v => Name.mkSimple v
| Syntax.ident .. => identKind
/--
Changes the kind at the root of a `Syntax` node to `k`.
Does nothing for non-`node` nodes.
-/
def setKind (stx : Syntax) (k : SyntaxNodeKind) : Syntax :=
match stx with
| Syntax.node info _ args => Syntax.node info k args
| _ => stx
/-- Is this a syntax with node kind `k`? -/
def isOfKind (stx : Syntax) (k : SyntaxNodeKind) : Bool :=
beq stx.getKind k
/--
Gets the `i`'th argument of the syntax node. This can also be written `stx[i]`.
Returns `missing` if `i` is out of range.
-/
def getArg (stx : Syntax) (i : Nat) : Syntax :=
match stx with
| Syntax.node _ _ args => args.getD i Syntax.missing
| _ => Syntax.missing
instance : GetElem Syntax Nat Syntax fun _ _ => True where
getElem stx i _ := stx.getArg i
/-- Gets the list of arguments of the syntax node, or `#[]` if it's not a `node`. -/
def getArgs (stx : Syntax) : Array Syntax :=
match stx with
| Syntax.node _ _ args => args
| _ => Array.empty
/-- Gets the number of arguments of the syntax node, or `0` if it's not a `node`. -/
def getNumArgs (stx : Syntax) : Nat :=
match stx with
| Syntax.node _ _ args => args.size
| _ => 0
/--
Assuming `stx` was parsed by `optional`, returns the enclosed syntax
if it parsed something and `none` otherwise.
-/
def getOptional? (stx : Syntax) : Option Syntax :=
match stx with
| Syntax.node _ k args => match and (beq k nullKind) (beq args.size 1) with
| true => some (args.get! 0)
| false => none
| _ => none
/-- Is this syntax `.missing`? -/
def isMissing : Syntax → Bool
| Syntax.missing => true
| _ => false
/-- Is this syntax a `node` with kind `k`? -/
def isNodeOf (stx : Syntax) (k : SyntaxNodeKind) (n : Nat) : Bool :=
and (stx.isOfKind k) (beq stx.getNumArgs n)
/-- `stx.isIdent` is `true` iff `stx` is an identifier. -/
def isIdent : Syntax → Bool
| ident .. => true
| _ => false
/-- If this is an `ident`, return the parsed value, else `.anonymous`. -/
def getId : Syntax → Name
| ident _ _ val _ => val
| _ => Name.anonymous
/--
Updates the argument list without changing the node kind.
Does nothing for non-`node` nodes.
-/
def setArgs (stx : Syntax) (args : Array Syntax) : Syntax :=
match stx with
| node info k _ => node info k args
| stx => stx
/--
Updates the `i`'th argument of the syntax.
Does nothing for non-`node` nodes, or if `i` is out of bounds of the node list.
-/
def setArg (stx : Syntax) (i : Nat) (arg : Syntax) : Syntax :=
match stx with
| node info k args => node info k (args.setD i arg)
| stx => stx
/-- Retrieve the left-most node or leaf's info in the Syntax tree. -/
partial def getHeadInfo? : Syntax → Option SourceInfo
| atom info _ => some info
| ident info .. => some info
| node SourceInfo.none _ args =>
let rec loop (i : Nat) : Option SourceInfo :=
match decide (LT.lt i args.size) with
| true => match getHeadInfo? (args.get! i) with
| some info => some info
| none => loop (hAdd i 1)
| false => none
loop 0
| node info _ _ => some info
| _ => none
/-- Retrieve the left-most leaf's info in the Syntax tree, or `none` if there is no token. -/
partial def getHeadInfo (stx : Syntax) : SourceInfo :=
match stx.getHeadInfo? with
| some info => info
| none => SourceInfo.none
/--
Get the starting position of the syntax, if possible.
If `canonicalOnly` is true, non-canonical `synthetic` nodes are treated as not carrying
position information.
-/
def getPos? (stx : Syntax) (canonicalOnly := false) : Option String.Pos :=
stx.getHeadInfo.getPos? canonicalOnly
/--
Get the ending position of the syntax, if possible.
If `canonicalOnly` is true, non-canonical `synthetic` nodes are treated as not carrying
position information.
-/
partial def getTailPos? (stx : Syntax) (canonicalOnly := false) : Option String.Pos :=
match stx, canonicalOnly with
| atom (SourceInfo.original (endPos := pos) ..) .., _
| atom (SourceInfo.synthetic (endPos := pos) (canonical := true) ..) _, _
| atom (SourceInfo.synthetic (endPos := pos) ..) _, false
| ident (SourceInfo.original (endPos := pos) ..) .., _
| ident (SourceInfo.synthetic (endPos := pos) (canonical := true) ..) .., _
| ident (SourceInfo.synthetic (endPos := pos) ..) .., false
| node (SourceInfo.original (endPos := pos) ..) .., _
| node (SourceInfo.synthetic (endPos := pos) (canonical := true) ..) .., _
| node (SourceInfo.synthetic (endPos := pos) ..) .., false => some pos
| node _ _ args, _ =>
let rec loop (i : Nat) : Option String.Pos :=
match decide (LT.lt i args.size) with
| true => match getTailPos? (args.get! ((args.size.sub i).sub 1)) canonicalOnly with
| some info => some info
| none => loop (hAdd i 1)
| false => none
loop 0
| _, _ => none
/--
An array of syntax elements interspersed with separators. Can be coerced
to/from `Array Syntax` to automatically remove/insert the separators.
-/
structure SepArray (sep : String) where
/-- The array of elements and separators, ordered like
`#[el1, sep1, el2, sep2, el3]`. -/
elemsAndSeps : Array Syntax
/-- A typed version of `SepArray`. -/
structure TSepArray (ks : SyntaxNodeKinds) (sep : String) where
/-- The array of elements and separators, ordered like
`#[el1, sep1, el2, sep2, el3]`. -/
elemsAndSeps : Array Syntax
end Syntax
/-- An array of syntaxes of kind `ks`. -/
abbrev TSyntaxArray (ks : SyntaxNodeKinds) := Array (TSyntax ks)
/-- Implementation of `TSyntaxArray.raw`. -/
unsafe def TSyntaxArray.rawImpl : TSyntaxArray ks → Array Syntax := unsafeCast
/-- Converts a `TSyntaxArray` to an `Array Syntax`, without reallocation. -/
@[implemented_by TSyntaxArray.rawImpl]
opaque TSyntaxArray.raw (as : TSyntaxArray ks) : Array Syntax := Array.empty
/-- Implementation of `TSyntaxArray.mk`. -/
unsafe def TSyntaxArray.mkImpl : Array Syntax → TSyntaxArray ks := unsafeCast
/-- Converts an `Array Syntax` to a `TSyntaxArray`, without reallocation. -/
@[implemented_by TSyntaxArray.mkImpl]
opaque TSyntaxArray.mk (as : Array Syntax) : TSyntaxArray ks := Array.empty
/-- Constructs a synthetic `SourceInfo` using a `ref : Syntax` for the span. -/
def SourceInfo.fromRef (ref : Syntax) (canonical := false) : SourceInfo :=
let noncanonical ref :=
match ref.getPos?, ref.getTailPos? with
| some pos, some tailPos => .synthetic pos tailPos
| _, _ => .none
match canonical with
| true =>
match ref.getPos? true, ref.getTailPos? true with
| some pos, some tailPos => .synthetic pos tailPos true
| _, _ => noncanonical ref
| false => noncanonical ref
/-- Constructs a synthetic `atom` with no source info. -/
def mkAtom (val : String) : Syntax :=
Syntax.atom SourceInfo.none val
/-- Constructs a synthetic `atom` with source info coming from `src`. -/
def mkAtomFrom (src : Syntax) (val : String) (canonical := false) : Syntax :=
Syntax.atom (SourceInfo.fromRef src canonical) val
/-! # Parser descriptions -/
/--
A `ParserDescr` is a grammar for parsers. This is used by the `syntax` command
to produce parsers without having to `import Lean`.
-/
inductive ParserDescr where
/-- A (named) nullary parser, like `ppSpace` -/
| const (name : Name)
/-- A (named) unary parser, like `group(p)` -/
| unary (name : Name) (p : ParserDescr)
/-- A (named) binary parser, like `orelse` or `andthen`
(written as `p1 <|> p2` and `p1 p2` respectively in `syntax`) -/
| binary (name : Name) (p₁ p₂ : ParserDescr)
/-- Parses using `p`, then pops the stack to create a new node with kind `kind`.
The precedence `prec` is used to determine whether the parser should apply given
the current precedence level. -/
| node (kind : SyntaxNodeKind) (prec : Nat) (p : ParserDescr)
/-- Like `node` but for trailing parsers (which start with a nonterminal).
Assumes the lhs is already on the stack, and parses using `p`, then pops the
stack including the lhs to create a new node with kind `kind`.
The precedence `prec` and `lhsPrec` are used to determine whether the parser
should apply. -/
| trailingNode (kind : SyntaxNodeKind) (prec lhsPrec : Nat) (p : ParserDescr)
/-- A literal symbol parser: parses `val` as a literal.
This parser does not work on identifiers, so `symbol` arguments are declared
as "keywords" and cannot be used as identifiers anywhere in the file. -/
| symbol (val : String)
/-- Like `symbol`, but without reserving `val` as a keyword.
If `includeIdent` is true then `ident` will be reinterpreted as `atom` if it matches. -/
| nonReservedSymbol (val : String) (includeIdent : Bool)
/-- Parses using the category parser `catName` with right binding power
(i.e. precedence) `rbp`. -/
| cat (catName : Name) (rbp : Nat)
/-- Parses using another parser `declName`, which can be either
a `Parser` or `ParserDescr`. -/
| parser (declName : Name)
/-- Like `node`, but also declares that the body can be matched using an antiquotation
with name `name`. For example, `def $id:declId := 1` uses an antiquotation with
name `declId` in the place where a `declId` is expected. -/
| nodeWithAntiquot (name : String) (kind : SyntaxNodeKind) (p : ParserDescr)
/-- A `sepBy(p, sep)` parses 0 or more occurrences of `p` separated by `sep`.
`psep` is usually the same as `symbol sep`, but it can be overridden.
`sep` is only used in the antiquot syntax: `$x;*` would match if `sep` is `";"`.
`allowTrailingSep` is true if e.g. `a, b,` is also allowed to match. -/
| sepBy (p : ParserDescr) (sep : String) (psep : ParserDescr) (allowTrailingSep : Bool := false)
/-- `sepBy1` is just like `sepBy`, except it takes 1 or more instead of
0 or more occurrences of `p`. -/
| sepBy1 (p : ParserDescr) (sep : String) (psep : ParserDescr) (allowTrailingSep : Bool := false)
instance : Inhabited ParserDescr where
default := ParserDescr.symbol ""
/--
Although `TrailingParserDescr` is an abbreviation for `ParserDescr`, Lean will
look at the declared type in order to determine whether to add the parser to
the leading or trailing parser table. The determination is done automatically
by the `syntax` command.
-/
abbrev TrailingParserDescr := ParserDescr
/-!
Runtime support for making quotation terms auto-hygienic, by mangling identifiers
introduced by them with a "macro scope" supplied by the context. Details to appear in a
paper soon.
-/
/--
A macro scope identifier is just a `Nat` that gets bumped every time we
enter a new macro scope. Within a macro scope, all occurrences of identifier `x`
parse to the same thing, but `x` parsed from different macro scopes will
produce different identifiers.
-/
abbrev MacroScope := Nat
/-- Macro scope used internally. It is not available for our frontend. -/
def reservedMacroScope := 0
/-- First macro scope available for our frontend -/
def firstFrontendMacroScope := hAdd reservedMacroScope 1
/--
A `MonadRef` is a monad that has a `ref : Syntax` in the read-only state.
This is used to keep track of the location where we are working; if an exception
is thrown, the `ref` gives the location where the error will be reported,
assuming no more specific location is provided.
-/
class MonadRef (m : Type → Type) where
/-- Get the current value of the `ref` -/
getRef : m Syntax
/-- Run `x : m α` with a modified value for the `ref` -/
withRef {α} : Syntax → m α → m α
export MonadRef (getRef)
instance (m n : Type → Type) [MonadLift m n] [MonadFunctor m n] [MonadRef m] : MonadRef n where
getRef := liftM (getRef : m _)
withRef ref x := monadMap (m := m) (MonadRef.withRef ref) x
/--
Replaces `oldRef` with `ref`, unless `ref` has no position info.
This biases us to having a valid span to report an error on.
-/
def replaceRef (ref : Syntax) (oldRef : Syntax) : Syntax :=
match ref.getPos? with
| some _ => ref
| _ => oldRef
/--
Run `x : m α` with a modified value for the `ref`. This is not exactly
the same as `MonadRef.withRef`, because it uses `replaceRef` to avoid putting
syntax with bad spans in the state.
-/
@[always_inline, inline]
def withRef [Monad m] [MonadRef m] {α} (ref : Syntax) (x : m α) : m α :=
bind getRef fun oldRef =>
let ref := replaceRef ref oldRef
MonadRef.withRef ref x
/-- A monad that supports syntax quotations. Syntax quotations (in term
position) are monadic values that when executed retrieve the current "macro
scope" from the monad and apply it to every identifier they introduce
(independent of whether this identifier turns out to be a reference to an
existing declaration, or an actually fresh binding during further
elaboration). We also apply the position of the result of `getRef` to each
introduced symbol, which results in better error positions than not applying
any position. -/
class MonadQuotation (m : Type → Type) extends MonadRef m where
/-- Get the fresh scope of the current macro invocation -/
getCurrMacroScope : m MacroScope
/-- Get the module name of the current file. This is used to ensure that
hygienic names don't clash across multiple files. -/
getMainModule : m Name
/--
Execute action in a new macro invocation context. This transformer should be
used at all places that morally qualify as the beginning of a "macro call",
e.g. `elabCommand` and `elabTerm` in the case of the elaborator. However, it
can also be used internally inside a "macro" if identifiers introduced by
e.g. different recursive calls should be independent and not collide. While
returning an intermediate syntax tree that will recursively be expanded by
the elaborator can be used for the same effect, doing direct recursion inside
the macro guarded by this transformer is often easier because one is not
restricted to passing a single syntax tree. Modelling this helper as a
transformer and not just a monadic action ensures that the current macro
scope before the recursive call is restored after it, as expected.
-/
withFreshMacroScope {α : Type} : m α → m α
export MonadQuotation (getCurrMacroScope getMainModule withFreshMacroScope)
/-- Construct a synthetic `SourceInfo` from the `ref` in the monad state. -/
@[inline]
def MonadRef.mkInfoFromRefPos [Monad m] [MonadRef m] : m SourceInfo :=
return SourceInfo.fromRef (← getRef)
instance [MonadFunctor m n] [MonadLift m n] [MonadQuotation m] : MonadQuotation n where
getCurrMacroScope := liftM (m := m) getCurrMacroScope
getMainModule := liftM (m := m) getMainModule
withFreshMacroScope := monadMap (m := m) withFreshMacroScope
/-!
We represent a name with macro scopes as
```
<actual name>._@.(<module_name>.<scopes>)*.<module_name>._hyg.<scopes>
```
Example: suppose the module name is `Init.Data.List.Basic`, and name is `foo.bla`, and macroscopes [2, 5]
```
foo.bla._@.Init.Data.List.Basic._hyg.2.5
```
We may have to combine scopes from different files/modules.
The main modules being processed is always the right-most one.
This situation may happen when we execute a macro generated in
an imported file in the current file.
```
foo.bla._@.Init.Data.List.Basic.2.1.Init.Lean.Expr._hyg.4
```
The delimiter `_hyg` is used just to improve the `hasMacroScopes` performance.
-/
/-- Does this name have hygienic macro scopes? -/
def Name.hasMacroScopes : Name → Bool
| str _ s => beq s "_hyg"
| num p _ => hasMacroScopes p
| _ => false
private def eraseMacroScopesAux : Name → Name
| .str p s => match beq s "_@" with
| true => p
| false => eraseMacroScopesAux p
| .num p _ => eraseMacroScopesAux p
| .anonymous => Name.anonymous
/-- Remove the macro scopes from the name. -/
@[export lean_erase_macro_scopes]
def Name.eraseMacroScopes (n : Name) : Name :=
match n.hasMacroScopes with
| true => eraseMacroScopesAux n
| false => n
private def simpMacroScopesAux : Name → Name
| .num p i => Name.mkNum (simpMacroScopesAux p) i
| n => eraseMacroScopesAux n
/-- Helper function we use to create binder names that do not need to be unique. -/
@[export lean_simp_macro_scopes]
def Name.simpMacroScopes (n : Name) : Name :=
match n.hasMacroScopes with
| true => simpMacroScopesAux n
| false => n
/--
A `MacroScopesView` represents a parsed hygienic name. `extractMacroScopes`
will decode it from a `Name`, and `.review` will re-encode it. The grammar of a
hygienic name is:
```
<name>._@.(<module_name>.<scopes>)*.<mainModule>._hyg.<scopes>
```
-/
structure MacroScopesView where
/-- The original (unhygienic) name. -/
name : Name
/-- All the name components `(<module_name>.<scopes>)*` from the imports
concatenated together. -/
imported : Name
/-- The main module in which this identifier was parsed. -/
mainModule : Name
/-- The list of macro scopes. -/
scopes : List MacroScope
instance : Inhabited MacroScopesView where
default := ⟨default, default, default, default⟩
/-- Encode a hygienic name from the parsed pieces. -/
def MacroScopesView.review (view : MacroScopesView) : Name :=
match view.scopes with
| List.nil => view.name
| List.cons _ _ =>
let base := (Name.mkStr (Name.appendCore (Name.appendCore (Name.mkStr view.name "_@") view.imported) view.mainModule) "_hyg")
view.scopes.foldl Name.mkNum base
private def assembleParts : List Name → Name → Name
| .nil, acc => acc
| .cons (.str _ s) ps, acc => assembleParts ps (Name.mkStr acc s)
| .cons (.num _ n) ps, acc => assembleParts ps (Name.mkNum acc n)
| _, _ => panic "Error: unreachable @ assembleParts"
private def extractImported (scps : List MacroScope) (mainModule : Name) : Name → List Name → MacroScopesView
| n@(Name.str p str), parts =>
match beq str "_@" with
| true => { name := p, mainModule := mainModule, imported := assembleParts parts Name.anonymous, scopes := scps }
| false => extractImported scps mainModule p (List.cons n parts)
| n@(Name.num p _), parts => extractImported scps mainModule p (List.cons n parts)
| _, _ => panic "Error: unreachable @ extractImported"
private def extractMainModule (scps : List MacroScope) : Name → List Name → MacroScopesView
| n@(Name.str p str), parts =>
match beq str "_@" with
| true => { name := p, mainModule := assembleParts parts Name.anonymous, imported := Name.anonymous, scopes := scps }
| false => extractMainModule scps p (List.cons n parts)
| n@(Name.num _ _), acc => extractImported scps (assembleParts acc Name.anonymous) n List.nil
| _, _ => panic "Error: unreachable @ extractMainModule"
private def extractMacroScopesAux : Name → List MacroScope → MacroScopesView
| Name.num p scp, acc => extractMacroScopesAux p (List.cons scp acc)
| Name.str p _ , acc => extractMainModule acc p List.nil -- str must be "_hyg"
| _, _ => panic "Error: unreachable @ extractMacroScopesAux"
/--
Revert all `addMacroScope` calls. `v = extractMacroScopes n → n = v.review`.
This operation is useful for analyzing/transforming the original identifiers, then adding back
the scopes (via `MacroScopesView.review`). -/
def extractMacroScopes (n : Name) : MacroScopesView :=
match n.hasMacroScopes with
| true => extractMacroScopesAux n List.nil
| false => { name := n, scopes := List.nil, imported := Name.anonymous, mainModule := Name.anonymous }
/-- Add a new macro scope onto the name `n`, in the given `mainModule`. -/
def addMacroScope (mainModule : Name) (n : Name) (scp : MacroScope) : Name :=
match n.hasMacroScopes with
| true =>
let view := extractMacroScopes n
match beq view.mainModule mainModule with
| true => Name.mkNum n scp
| false =>
{ view with
imported := view.scopes.foldl Name.mkNum (Name.appendCore view.imported view.mainModule)
mainModule := mainModule
scopes := List.cons scp List.nil
}.review
| false =>
Name.mkNum (Name.mkStr (Name.appendCore (Name.mkStr n "_@") mainModule) "_hyg") scp
/--
Append two names that may have macro scopes. The macro scopes in `b` are always erased.
If `a` has macro scopes, then they are propagated to the result of `append a b`.
-/
def Name.append (a b : Name) : Name :=
match a.hasMacroScopes, b.hasMacroScopes with
| true, true =>
panic "Error: invalid `Name.append`, both arguments have macro scopes, consider using `eraseMacroScopes`"
| true, false =>
let view := extractMacroScopes a
{ view with name := appendCore view.name b }.review
| false, true =>
let view := extractMacroScopes b
{ view with name := appendCore a view.name }.review
| false, false => appendCore a b
instance : Append Name where
append := Name.append
/--
Add a new macro scope onto the name `n`, using the monad state to supply the
main module and current macro scope.
-/
@[inline] def MonadQuotation.addMacroScope {m : Type → Type} [MonadQuotation m] [Monad m] (n : Name) : m Name :=
bind getMainModule fun mainModule =>
bind getCurrMacroScope fun scp =>
pure (Lean.addMacroScope mainModule n scp)
/-- The default maximum recursion depth. This is adjustable using the `maxRecDepth` option. -/
def defaultMaxRecDepth := 512
/-- The message to display on stack overflow. -/
def maxRecDepthErrorMessage : String :=
"maximum recursion depth has been reached (use `set_option maxRecDepth <num>` to increase limit)"
namespace Syntax
/-- Is this syntax a null `node`? -/
def matchesNull (stx : Syntax) (n : Nat) : Bool :=
stx.isNodeOf nullKind n
/--
Function used for determining whether a syntax pattern `` `(id) `` is matched.
There are various conceivable notions of when two syntactic identifiers should be regarded as identical,
but semantic definitions like whether they refer to the same global name cannot be implemented without
context information (i.e. `MonadResolveName`). Thus in patterns we default to the structural solution
of comparing the identifiers' `Name` values, though we at least do so modulo macro scopes so that
identifiers that "look" the same match. This is particularly useful when dealing with identifiers that
do not actually refer to Lean bindings, e.g. in the `stx` pattern `` `(many($p)) ``. -/
def matchesIdent (stx : Syntax) (id : Name) : Bool :=
and stx.isIdent (beq stx.getId.eraseMacroScopes id.eraseMacroScopes)
/-- Is this syntax a node kind `k` wrapping an `atom _ val`? -/
def matchesLit (stx : Syntax) (k : SyntaxNodeKind) (val : String) : Bool :=
match stx with
| Syntax.node _ k' args => and (beq k k') (match args.getD 0 Syntax.missing with
| Syntax.atom _ val' => beq val val'
| _ => false)
| _ => false
end Syntax
namespace Macro
/-- References -/
private opaque MethodsRefPointed : NonemptyType.{0}
private def MethodsRef : Type := MethodsRefPointed.type
instance : Nonempty MethodsRef := MethodsRefPointed.property
/-- The read-only context for the `MacroM` monad. -/
structure Context where
/-- An opaque reference to the `Methods` object. This is done to break a
dependency cycle: the `Methods` involve `MacroM` which has not been defined yet. -/
methods : MethodsRef
/-- The currently parsing module. -/
mainModule : Name
/-- The current macro scope. -/
currMacroScope : MacroScope
/-- The current recursion depth. -/
currRecDepth : Nat := 0
/-- The maximum recursion depth. -/
maxRecDepth : Nat := defaultMaxRecDepth
/-- The syntax which supplies the position of error messages. -/
ref : Syntax
/-- An exception in the `MacroM` monad. -/
inductive Exception where
/-- A general error, given a message and a span (expressed as a `Syntax`). -/
| error : Syntax → String → Exception
/-- An unsupported syntax exception. We keep this separate because it is
used for control flow: if one macro does not support a syntax then we try
the next one. -/
| unsupportedSyntax : Exception
/-- The mutable state for the `MacroM` monad. -/
structure State where
/-- The global macro scope counter, used for producing fresh scope names. -/
macroScope : MacroScope
/-- The list of trace messages that have been produced, each with a trace
class and a message. -/
traceMsgs : List (Prod Name String) := List.nil
deriving Inhabited
end Macro
/--
The `MacroM` monad is the main monad for macro expansion. It has the
information needed to handle hygienic name generation, and is the monad that
`macro` definitions live in.
Notably, this is a (relatively) pure monad: there is no `IO` and no access to
the `Environment`. That means that things like declaration lookup are
impossible here, as well as `IO.Ref` or other side-effecting operations.
For more capabilities, macros can instead be written as `elab` using `adaptExpander`.
-/
abbrev MacroM := ReaderT Macro.Context (EStateM Macro.Exception Macro.State)
/--
A `macro` has type `Macro`, which is a `Syntax → MacroM Syntax`: it
receives an input syntax and is supposed to "expand" it into another piece of
syntax.
-/
abbrev Macro := Syntax → MacroM Syntax
namespace Macro
instance : MonadRef MacroM where
getRef := bind read fun ctx => pure ctx.ref
withRef := fun ref x => withReader (fun ctx => { ctx with ref := ref }) x
/-- Add a new macro scope to the name `n`. -/
def addMacroScope (n : Name) : MacroM Name :=
bind read fun ctx =>
pure (Lean.addMacroScope ctx.mainModule n ctx.currMacroScope)
/-- Throw an `unsupportedSyntax` exception. -/
def throwUnsupported {α} : MacroM α :=
throw Exception.unsupportedSyntax
/--
Throw an error with the given message,
using the `ref` for the location information.
-/
def throwError {α} (msg : String) : MacroM α :=
bind getRef fun ref =>
throw (Exception.error ref msg)
/-- Throw an error with the given message and location information. -/
def throwErrorAt {α} (ref : Syntax) (msg : String) : MacroM α :=
withRef ref (throwError msg)
/--
Increments the macro scope counter so that inside the body of `x` the macro
scope is fresh.
-/
@[inline] protected def withFreshMacroScope {α} (x : MacroM α) : MacroM α :=
bind (modifyGet (fun s => (s.macroScope, { s with macroScope := hAdd s.macroScope 1 }))) fun fresh =>
withReader (fun ctx => { ctx with currMacroScope := fresh }) x
/-- Run `x` with an incremented recursion depth counter. -/
@[inline] def withIncRecDepth {α} (ref : Syntax) (x : MacroM α) : MacroM α :=
bind read fun ctx =>
match beq ctx.currRecDepth ctx.maxRecDepth with
| true => throw (Exception.error ref maxRecDepthErrorMessage)
| false => withReader (fun ctx => { ctx with currRecDepth := hAdd ctx.currRecDepth 1 }) x
instance : MonadQuotation MacroM where
getCurrMacroScope ctx := pure ctx.currMacroScope
getMainModule ctx := pure ctx.mainModule
withFreshMacroScope := Macro.withFreshMacroScope
/-- The opaque methods that are available to `MacroM`. -/
structure Methods where
/-- Expands macros in the given syntax. A return value of `none` means there
was nothing to expand. -/
expandMacro? : Syntax → MacroM (Option Syntax)
/-- Get the current namespace in the file. -/
getCurrNamespace : MacroM Name
/-- Check if a given name refers to a declaration. -/
hasDecl : Name → MacroM Bool
/-- Resolves the given name to an overload list of namespaces. -/
resolveNamespace : Name → MacroM (List Name)
/-- Resolves the given name to an overload list of global definitions.
The `List String` in each alternative is the deduced list of projections
(which are ambiguous with name components). -/
resolveGlobalName : Name → MacroM (List (Prod Name (List String)))
deriving Inhabited
/-- Implementation of `mkMethods`. -/
unsafe def mkMethodsImp (methods : Methods) : MethodsRef :=
unsafeCast methods
/-- Make an opaque reference to a `Methods`. -/
@[implemented_by mkMethodsImp]
opaque mkMethods (methods : Methods) : MethodsRef
instance : Inhabited MethodsRef where
default := mkMethods default
/-- Implementation of `getMethods`. -/
unsafe def getMethodsImp : MacroM Methods :=
bind read fun ctx => pure (unsafeCast (ctx.methods))
/-- Extract the methods list from the `MacroM` state. -/
@[implemented_by getMethodsImp] opaque getMethods : MacroM Methods
/--
`expandMacro? stx` returns `some stxNew` if `stx` is a macro,
and `stxNew` is its expansion.
-/
def expandMacro? (stx : Syntax) : MacroM (Option Syntax) := do
(← getMethods).expandMacro? stx
/-- Returns `true` if the environment contains a declaration with name `declName` -/
def hasDecl (declName : Name) : MacroM Bool := do
(← getMethods).hasDecl declName
/-- Gets the current namespace given the position in the file. -/
def getCurrNamespace : MacroM Name := do
(← getMethods).getCurrNamespace
/-- Resolves the given name to an overload list of namespaces. -/
def resolveNamespace (n : Name) : MacroM (List Name) := do
(← getMethods).resolveNamespace n
/--
Resolves the given name to an overload list of global definitions.
The `List String` in each alternative is the deduced list of projections
(which are ambiguous with name components).
-/
def resolveGlobalName (n : Name) : MacroM (List (Prod Name (List String))) := do
(← getMethods).resolveGlobalName n
/-- Add a new trace message, with the given trace class and message. -/
def trace (clsName : Name) (msg : String) : MacroM Unit := do
modify fun s => { s with traceMsgs := List.cons (Prod.mk clsName msg) s.traceMsgs }
end Macro
export Macro (expandMacro?)
namespace PrettyPrinter
/--
The unexpander monad, essentially `Syntax → Option α`. The `Syntax` is the `ref`,
and it has the possibility of failure without an error message.
-/
abbrev UnexpandM := ReaderT Syntax (EStateM Unit Unit)
/--
Function that tries to reverse macro expansions as a post-processing step of delaboration.
While less general than an arbitrary delaborator, it can be declared without importing `Lean`.
Used by the `[app_unexpander]` attribute.
-/
-- a `kindUnexpander` could reasonably be added later
abbrev Unexpander := Syntax → UnexpandM Syntax
instance : MonadQuotation UnexpandM where
getRef := read
withRef ref x := withReader (fun _ => ref) x
-- unexpanders should not need to introduce new names
getCurrMacroScope := pure 0
getMainModule := pure `_fakeMod
withFreshMacroScope := id
end PrettyPrinter
end Lean
|
7be47ca50842eb0e7a4871c885c0752ced71791b | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/mutualdef1.lean | c24deb843cff494fcccd668fd04354abb7a2d9f0 | [
"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 | 462 | lean | --
mutual
def f (x : Nat) : Nat → Nat
| 0 => 1
| x+1 => g x
theorem g (x : Nat) : Nat → Nat -- cannot mix theorems and definitions
| 0 => 2
| x+1 => f x
end
mutual
def f : Nat → Nat
| 0 => 1
| x+1 => g x
example : Nat := -- cannot mix examples and definitions
g 10
end
mutual
def f (x : Nat) : Nat → Nat
| 0 => 1
| x+1 => g x
unsafe def g (x : Nat) : Nat → Nat -- cannot mix safe and unsafe definitions
| 0 => 2
| x+1 => f x
end
|
f97b9cb58a8b84226cc1bf072696f2e9f5142023 | 1abd1ed12aa68b375cdef28959f39531c6e95b84 | /src/category_theory/isomorphism.lean | 16206c5ca7ea69c8ace3257d2fae81fe814be43d | [
"Apache-2.0"
] | permissive | jumpy4/mathlib | d3829e75173012833e9f15ac16e481e17596de0f | af36f1a35f279f0e5b3c2a77647c6bf2cfd51a13 | refs/heads/master | 1,693,508,842,818 | 1,636,203,271,000 | 1,636,203,271,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 14,213 | lean | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import category_theory.functor
/-!
# Isomorphisms
This file defines isomorphisms between objects of a category.
## Main definitions
- `structure iso` : a bundled isomorphism between two objects of a category;
- `class is_iso` : an unbundled version of `iso`;
note that `is_iso f` is a `Prop`, and only asserts the existence of an inverse.
Of course, this inverse is unique, so it doesn't cost us much to use choice to retrieve it.
- `inv f`, for the inverse of a morphism with `[is_iso f]`
- `as_iso` : convert from `is_iso` to `iso` (noncomputable);
- `of_iso` : convert from `iso` to `is_iso`;
- standard operations on isomorphisms (composition, inverse etc)
## Notations
- `X ≅ Y` : same as `iso X Y`;
- `α ≪≫ β` : composition of two isomorphisms; it is called `iso.trans`
## Tags
category, category theory, isomorphism
-/
universes v u -- morphism levels before object levels. See note [category_theory universes].
namespace category_theory
open category
/--
An isomorphism (a.k.a. an invertible morphism) between two objects of a category.
The inverse morphism is bundled.
See also `category_theory.core` for the category with the same objects and isomorphisms playing
the role of morphisms.
See https://stacks.math.columbia.edu/tag/0017.
-/
structure iso {C : Type u} [category.{v} C] (X Y : C) :=
(hom : X ⟶ Y)
(inv : Y ⟶ X)
(hom_inv_id' : hom ≫ inv = 𝟙 X . obviously)
(inv_hom_id' : inv ≫ hom = 𝟙 Y . obviously)
restate_axiom iso.hom_inv_id'
restate_axiom iso.inv_hom_id'
attribute [simp, reassoc] iso.hom_inv_id iso.inv_hom_id
infixr ` ≅ `:10 := iso -- type as \cong or \iso
variables {C : Type u} [category.{v} C]
variables {X Y Z : C}
namespace iso
@[ext] lemma ext ⦃α β : X ≅ Y⦄ (w : α.hom = β.hom) : α = β :=
suffices α.inv = β.inv, by cases α; cases β; cc,
calc α.inv
= α.inv ≫ (β.hom ≫ β.inv) : by rw [iso.hom_inv_id, category.comp_id]
... = (α.inv ≫ α.hom) ≫ β.inv : by rw [category.assoc, ←w]
... = β.inv : by rw [iso.inv_hom_id, category.id_comp]
/-- Inverse isomorphism. -/
@[symm] def symm (I : X ≅ Y) : Y ≅ X :=
{ hom := I.inv,
inv := I.hom,
hom_inv_id' := I.inv_hom_id',
inv_hom_id' := I.hom_inv_id' }
@[simp] lemma symm_hom (α : X ≅ Y) : α.symm.hom = α.inv := rfl
@[simp] lemma symm_inv (α : X ≅ Y) : α.symm.inv = α.hom := rfl
@[simp] lemma symm_mk {X Y : C} (hom : X ⟶ Y) (inv : Y ⟶ X) (hom_inv_id) (inv_hom_id) :
iso.symm {hom := hom, inv := inv, hom_inv_id' := hom_inv_id, inv_hom_id' := inv_hom_id} =
{hom := inv, inv := hom, hom_inv_id' := inv_hom_id, inv_hom_id' := hom_inv_id} := rfl
@[simp] lemma symm_symm_eq {X Y : C} (α : X ≅ Y) : α.symm.symm = α :=
by cases α; refl
@[simp] lemma symm_eq_iff {X Y : C} {α β : X ≅ Y} : α.symm = β.symm ↔ α = β :=
⟨λ h, symm_symm_eq α ▸ symm_symm_eq β ▸ congr_arg symm h, congr_arg symm⟩
/-- Identity isomorphism. -/
@[refl, simps] def refl (X : C) : X ≅ X :=
{ hom := 𝟙 X,
inv := 𝟙 X }
instance : inhabited (X ≅ X) := ⟨iso.refl X⟩
@[simp] lemma refl_symm (X : C) : (iso.refl X).symm = iso.refl X := rfl
/-- Composition of two isomorphisms -/
@[trans, simps] def trans (α : X ≅ Y) (β : Y ≅ Z) : X ≅ Z :=
{ hom := α.hom ≫ β.hom,
inv := β.inv ≫ α.inv }
infixr ` ≪≫ `:80 := iso.trans -- type as `\ll \gg`.
@[simp] lemma trans_mk {X Y Z : C}
(hom : X ⟶ Y) (inv : Y ⟶ X) (hom_inv_id) (inv_hom_id)
(hom' : Y ⟶ Z) (inv' : Z ⟶ Y) (hom_inv_id') (inv_hom_id') (hom_inv_id'') (inv_hom_id'') :
iso.trans
{hom := hom, inv := inv, hom_inv_id' := hom_inv_id, inv_hom_id' := inv_hom_id}
{hom := hom', inv := inv', hom_inv_id' := hom_inv_id', inv_hom_id' := inv_hom_id'} =
{ hom := hom ≫ hom', inv := inv' ≫ inv, hom_inv_id' := hom_inv_id'',
inv_hom_id' := inv_hom_id''} :=
rfl
@[simp] lemma trans_symm (α : X ≅ Y) (β : Y ≅ Z) : (α ≪≫ β).symm = β.symm ≪≫ α.symm := rfl
@[simp] lemma trans_assoc {Z' : C} (α : X ≅ Y) (β : Y ≅ Z) (γ : Z ≅ Z') :
(α ≪≫ β) ≪≫ γ = α ≪≫ β ≪≫ γ :=
by ext; simp only [trans_hom, category.assoc]
@[simp] lemma refl_trans (α : X ≅ Y) : (iso.refl X) ≪≫ α = α := by ext; apply category.id_comp
@[simp] lemma trans_refl (α : X ≅ Y) : α ≪≫ (iso.refl Y) = α := by ext; apply category.comp_id
@[simp] lemma symm_self_id (α : X ≅ Y) : α.symm ≪≫ α = iso.refl Y := ext α.inv_hom_id
@[simp] lemma self_symm_id (α : X ≅ Y) : α ≪≫ α.symm = iso.refl X := ext α.hom_inv_id
@[simp] lemma symm_self_id_assoc (α : X ≅ Y) (β : Y ≅ Z) : α.symm ≪≫ α ≪≫ β = β :=
by rw [← trans_assoc, symm_self_id, refl_trans]
@[simp] lemma self_symm_id_assoc (α : X ≅ Y) (β : X ≅ Z) : α ≪≫ α.symm ≪≫ β = β :=
by rw [← trans_assoc, self_symm_id, refl_trans]
lemma inv_comp_eq (α : X ≅ Y) {f : X ⟶ Z} {g : Y ⟶ Z} : α.inv ≫ f = g ↔ f = α.hom ≫ g :=
⟨λ H, by simp [H.symm], λ H, by simp [H]⟩
lemma eq_inv_comp (α : X ≅ Y) {f : X ⟶ Z} {g : Y ⟶ Z} : g = α.inv ≫ f ↔ α.hom ≫ g = f :=
(inv_comp_eq α.symm).symm
lemma comp_inv_eq (α : X ≅ Y) {f : Z ⟶ Y} {g : Z ⟶ X} : f ≫ α.inv = g ↔ f = g ≫ α.hom :=
⟨λ H, by simp [H.symm], λ H, by simp [H]⟩
lemma eq_comp_inv (α : X ≅ Y) {f : Z ⟶ Y} {g : Z ⟶ X} : g = f ≫ α.inv ↔ g ≫ α.hom = f :=
(comp_inv_eq α.symm).symm
lemma inv_eq_inv (f g : X ≅ Y) : f.inv = g.inv ↔ f.hom = g.hom :=
have ∀{X Y : C} (f g : X ≅ Y), f.hom = g.hom → f.inv = g.inv, from λ X Y f g h, by rw [ext h],
⟨this f.symm g.symm, this f g⟩
lemma hom_comp_eq_id (α : X ≅ Y) {f : Y ⟶ X} : α.hom ≫ f = 𝟙 X ↔ f = α.inv :=
by rw [←eq_inv_comp, comp_id]
lemma comp_hom_eq_id (α : X ≅ Y) {f : Y ⟶ X} : f ≫ α.hom = 𝟙 Y ↔ f = α.inv :=
by rw [←eq_comp_inv, id_comp]
lemma hom_eq_inv (α : X ≅ Y) (β : Y ≅ X) : α.hom = β.inv ↔ β.hom = α.inv :=
by { erw [inv_eq_inv α.symm β, eq_comm], refl }
end iso
/-- `is_iso` typeclass expressing that a morphism is invertible. -/
class is_iso (f : X ⟶ Y) : Prop :=
(out : ∃ inv : Y ⟶ X, f ≫ inv = 𝟙 X ∧ inv ≫ f = 𝟙 Y)
/--
The inverse of a morphism `f` when we have `[is_iso f]`.
-/
noncomputable def inv (f : X ⟶ Y) [I : is_iso f] := classical.some I.1
namespace is_iso
@[simp, reassoc] lemma hom_inv_id (f : X ⟶ Y) [I : is_iso f] : f ≫ inv f = 𝟙 X :=
(classical.some_spec I.1).left
@[simp, reassoc] lemma inv_hom_id (f : X ⟶ Y) [I : is_iso f] : inv f ≫ f = 𝟙 Y :=
(classical.some_spec I.1).right
end is_iso
open is_iso
/-- Reinterpret a morphism `f` with an `is_iso f` instance as an `iso`. -/
noncomputable
def as_iso (f : X ⟶ Y) [h : is_iso f] : X ≅ Y := ⟨f, inv f, hom_inv_id f, inv_hom_id f⟩
@[simp] lemma as_iso_hom (f : X ⟶ Y) [is_iso f] : (as_iso f).hom = f := rfl
@[simp] lemma as_iso_inv (f : X ⟶ Y) [is_iso f] : (as_iso f).inv = inv f := rfl
namespace is_iso
@[priority 100] -- see Note [lower instance priority]
instance epi_of_iso (f : X ⟶ Y) [is_iso f] : epi f :=
{ left_cancellation := λ Z g h w,
-- This is an interesting test case for better rewrite automation.
by rw [← is_iso.inv_hom_id_assoc f g, w, is_iso.inv_hom_id_assoc f h] }
@[priority 100] -- see Note [lower instance priority]
instance mono_of_iso (f : X ⟶ Y) [is_iso f] : mono f :=
{ right_cancellation := λ Z g h w,
by rw [← category.comp_id g, ← category.comp_id h, ← is_iso.hom_inv_id f, ← category.assoc, w,
← category.assoc] }
@[ext] lemma inv_eq_of_hom_inv_id {f : X ⟶ Y} [is_iso f] {g : Y ⟶ X}
(hom_inv_id : f ≫ g = 𝟙 X) : inv f = g :=
begin
apply (cancel_epi f).mp,
simp [hom_inv_id],
end
lemma inv_eq_of_inv_hom_id {f : X ⟶ Y} [is_iso f] {g : Y ⟶ X}
(inv_hom_id : g ≫ f = 𝟙 Y) : inv f = g :=
begin
apply (cancel_mono f).mp,
simp [inv_hom_id],
end
@[ext] lemma eq_inv_of_hom_inv_id {f : X ⟶ Y} [is_iso f] {g : Y ⟶ X}
(hom_inv_id : f ≫ g = 𝟙 X) : g = inv f :=
(inv_eq_of_hom_inv_id hom_inv_id).symm
lemma eq_inv_of_inv_hom_id {f : X ⟶ Y} [is_iso f] {g : Y ⟶ X}
(inv_hom_id : g ≫ f = 𝟙 Y) : g = inv f :=
(inv_eq_of_inv_hom_id inv_hom_id).symm
instance id (X : C) : is_iso (𝟙 X) :=
⟨⟨𝟙 X, by simp⟩⟩
instance of_iso (f : X ≅ Y) : is_iso f.hom :=
⟨⟨f.inv, by simp⟩⟩
instance of_iso_inv (f : X ≅ Y) : is_iso f.inv :=
is_iso.of_iso f.symm
variables {f g : X ⟶ Y} {h : Y ⟶ Z}
instance inv_is_iso [is_iso f] : is_iso (inv f) :=
is_iso.of_iso_inv (as_iso f)
instance comp_is_iso [is_iso f] [is_iso h] : is_iso (f ≫ h) :=
is_iso.of_iso $ (as_iso f) ≪≫ (as_iso h)
@[simp] lemma inv_id : inv (𝟙 X) = 𝟙 X := by { ext, simp, }
@[simp] lemma inv_comp [is_iso f] [is_iso h] : inv (f ≫ h) = inv h ≫ inv f := by { ext, simp, }
@[simp] lemma inv_inv [is_iso f] : inv (inv f) = f := by { ext, simp, }
@[simp] lemma iso.inv_inv (f : X ≅ Y) : inv (f.inv) = f.hom := by { ext, simp, }
@[simp] lemma iso.inv_hom (f : X ≅ Y) : inv (f.hom) = f.inv := by { ext, simp, }
@[simp]
lemma inv_comp_eq (α : X ⟶ Y) [is_iso α] {f : X ⟶ Z} {g : Y ⟶ Z} : inv α ≫ f = g ↔ f = α ≫ g :=
(as_iso α).inv_comp_eq
@[simp]
lemma eq_inv_comp (α : X ⟶ Y) [is_iso α] {f : X ⟶ Z} {g : Y ⟶ Z} : g = inv α ≫ f ↔ α ≫ g = f :=
(as_iso α).eq_inv_comp
@[simp]
lemma comp_inv_eq (α : X ⟶ Y) [is_iso α] {f : Z ⟶ Y} {g : Z ⟶ X} : f ≫ inv α = g ↔ f = g ≫ α :=
(as_iso α).comp_inv_eq
@[simp]
lemma eq_comp_inv (α : X ⟶ Y) [is_iso α] {f : Z ⟶ Y} {g : Z ⟶ X} : g = f ≫ inv α ↔ g ≫ α = f :=
(as_iso α).eq_comp_inv
end is_iso
open is_iso
lemma eq_of_inv_eq_inv {f g : X ⟶ Y} [is_iso f] [is_iso g] (p : inv f = inv g) : f = g :=
begin
apply (cancel_epi (inv f)).1,
erw [inv_hom_id, p, inv_hom_id],
end
lemma is_iso.inv_eq_inv {f g : X ⟶ Y} [is_iso f] [is_iso g] : inv f = inv g ↔ f = g :=
iso.inv_eq_inv (as_iso f) (as_iso g)
lemma hom_comp_eq_id (g : X ⟶ Y) [is_iso g] {f : Y ⟶ X} : g ≫ f = 𝟙 X ↔ f = inv g :=
(as_iso g).hom_comp_eq_id
lemma comp_hom_eq_id (g : X ⟶ Y) [is_iso g] {f : Y ⟶ X} : f ≫ g = 𝟙 Y ↔ f = inv g :=
(as_iso g).comp_hom_eq_id
namespace iso
@[ext] lemma inv_ext {f : X ≅ Y} {g : Y ⟶ X}
(hom_inv_id : f.hom ≫ g = 𝟙 X) : f.inv = g :=
begin
apply (cancel_epi f.hom).mp,
simp [hom_inv_id],
end
@[ext] lemma inv_ext' {f : X ≅ Y} {g : Y ⟶ X}
(hom_inv_id : f.hom ≫ g = 𝟙 X) : g = f.inv :=
by { symmetry, ext, assumption, }
/-!
All these cancellation lemmas can be solved by `simp [cancel_mono]` (or `simp [cancel_epi]`),
but with the current design `cancel_mono` is not a good `simp` lemma,
because it generates a typeclass search.
When we can see syntactically that a morphism is a `mono` or an `epi`
because it came from an isomorphism, it's fine to do the cancellation via `simp`.
In the longer term, it might be worth exploring making `mono` and `epi` structures,
rather than typeclasses, with coercions back to `X ⟶ Y`.
Presumably we could write `X ↪ Y` and `X ↠ Y`.
-/
@[simp] lemma cancel_iso_hom_left {X Y Z : C} (f : X ≅ Y) (g g' : Y ⟶ Z) :
f.hom ≫ g = f.hom ≫ g' ↔ g = g' :=
by simp only [cancel_epi]
@[simp] lemma cancel_iso_inv_left {X Y Z : C} (f : Y ≅ X) (g g' : Y ⟶ Z) :
f.inv ≫ g = f.inv ≫ g' ↔ g = g' :=
by simp only [cancel_epi]
@[simp] lemma cancel_iso_hom_right {X Y Z : C} (f f' : X ⟶ Y) (g : Y ≅ Z) :
f ≫ g.hom = f' ≫ g.hom ↔ f = f' :=
by simp only [cancel_mono]
@[simp] lemma cancel_iso_inv_right {X Y Z : C} (f f' : X ⟶ Y) (g : Z ≅ Y) :
f ≫ g.inv = f' ≫ g.inv ↔ f = f' :=
by simp only [cancel_mono]
/-
Unfortunately cancelling an isomorphism from the right of a chain of compositions is awkward.
We would need separate lemmas for each chain length (worse: for each pair of chain lengths).
We provide two more lemmas, for case of three morphisms, because this actually comes up in practice,
but then stop.
-/
@[simp] lemma cancel_iso_hom_right_assoc {W X X' Y Z : C}
(f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X') (g' : X' ⟶ Y)
(h : Y ≅ Z) :
f ≫ g ≫ h.hom = f' ≫ g' ≫ h.hom ↔ f ≫ g = f' ≫ g' :=
by simp only [←category.assoc, cancel_mono]
@[simp] lemma cancel_iso_inv_right_assoc {W X X' Y Z : C}
(f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X') (g' : X' ⟶ Y)
(h : Z ≅ Y) :
f ≫ g ≫ h.inv = f' ≫ g' ≫ h.inv ↔ f ≫ g = f' ≫ g' :=
by simp only [←category.assoc, cancel_mono]
end iso
namespace functor
universes u₁ v₁ u₂ v₂
variables {D : Type u₂}
variables [category.{v₂} D]
/-- A functor `F : C ⥤ D` sends isomorphisms `i : X ≅ Y` to isomorphisms `F.obj X ≅ F.obj Y` -/
@[simps]
def map_iso (F : C ⥤ D) {X Y : C} (i : X ≅ Y) : F.obj X ≅ F.obj Y :=
{ hom := F.map i.hom,
inv := F.map i.inv,
hom_inv_id' := by rw [←map_comp, iso.hom_inv_id, ←map_id],
inv_hom_id' := by rw [←map_comp, iso.inv_hom_id, ←map_id] }
@[simp] lemma map_iso_symm (F : C ⥤ D) {X Y : C} (i : X ≅ Y) :
F.map_iso i.symm = (F.map_iso i).symm :=
rfl
@[simp] lemma map_iso_trans (F : C ⥤ D) {X Y Z : C} (i : X ≅ Y) (j : Y ≅ Z) :
F.map_iso (i ≪≫ j) = (F.map_iso i) ≪≫ (F.map_iso j) :=
by ext; apply functor.map_comp
@[simp] lemma map_iso_refl (F : C ⥤ D) (X : C) : F.map_iso (iso.refl X) = iso.refl (F.obj X) :=
iso.ext $ F.map_id X
instance map_is_iso (F : C ⥤ D) (f : X ⟶ Y) [is_iso f] : is_iso (F.map f) :=
is_iso.of_iso $ F.map_iso (as_iso f)
@[simp] lemma map_inv (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [is_iso f] :
F.map (inv f) = inv (F.map f) :=
by { ext, simp [←F.map_comp], }
lemma map_hom_inv (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [is_iso f] :
F.map f ≫ F.map (inv f) = 𝟙 (F.obj X) :=
by simp
lemma map_inv_hom (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [is_iso f] :
F.map (inv f) ≫ F.map f = 𝟙 (F.obj Y) :=
by simp
end functor
end category_theory
|
0295af615e4e5641936d5754e3354290eb780479 | 217bb195841a8be2d1b4edd2084d6b69ccd62f50 | /tests/playground/parser/syntax.lean | 1e014bf661c9bf26796ef2500f6627c48e5c1814 | [
"Apache-2.0"
] | permissive | frank-lesser/lean4 | 717f56c9bacd5bf3a67542d2f5cea721d4743a30 | 79e2abe33f73162f773ea731265e456dbfe822f9 | refs/heads/master | 1,589,741,267,933 | 1,556,424,200,000 | 1,556,424,281,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 10,625 | 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
-/
prelude
import init.lean.name init.lean.format init.data.array
-- namespace Lean
namespace Parser
open Lean
/-- A hygiene marker introduced by a macro expansion. -/
@[derive DecidableEq HasToFormat]
def MacroScope := Nat
abbrev MacroScopes := List MacroScope
structure SourceInfo :=
/- Will be inferred after parsing by `Syntax.updateLeading`. During parsing,
it is not at all clear what the preceding token was, especially with backtracking. -/
(leading : Substring)
(pos : String.Pos)
(trailing : Substring)
def SourceInfo.updateTrailing : SourceInfo → Substring → SourceInfo
| ⟨leading, pos, _⟩ trailing := ⟨leading, pos, trailing⟩
/- Node kind generation -/
def mkUniqIdRef : IO (IO.Ref Nat) :=
IO.mkRef 0
@[init mkUniqIdRef]
constant nextUniqId : IO.Ref Nat := default _
structure SyntaxNodeKind :=
(name : Name) (id : Nat)
instance stxKindInh : Inhabited SyntaxNodeKind :=
⟨{name := default _, id := default _}⟩
instance stxKindBeq : HasBeq SyntaxNodeKind :=
⟨λ k₁ k₂, k₁.id == k₂.id⟩
def mkNameToKindTable : IO (IO.Ref (NameMap Nat)) :=
IO.mkRef {}
@[init mkNameToKindTable]
constant nameToKindTable : IO.Ref (NameMap Nat) := default _
def nextKind (k : Name) : IO SyntaxNodeKind :=
do m ← nameToKindTable.get,
when (m.contains k) (throw $ IO.userError ("kind '" ++ toString k ++ "' already exists")),
id ← nextUniqId.get,
nameToKindTable.set (m.insert k id),
nextUniqId.set (id+1),
pure { name := k, id := id }
/- Basic node kinds -/
def mkNullKind : IO SyntaxNodeKind := nextKind `null
@[init mkNullKind] constant nullKind : SyntaxNodeKind := default _
def mkChoiceKind : IO SyntaxNodeKind := nextKind `choice
@[init mkChoiceKind] constant choiceKind : SyntaxNodeKind := default _
def mkOptionSomeKind : IO SyntaxNodeKind := nextKind `some
@[init mkOptionSomeKind] constant optionSomeKind : SyntaxNodeKind := default _
def mkOptionNoneKind : IO SyntaxNodeKind := nextKind `none
@[init mkOptionNoneKind] constant optionNoneKind : SyntaxNodeKind := default _
def mkManyKind : IO SyntaxNodeKind := nextKind `many
@[init mkManyKind] constant manyKind : SyntaxNodeKind := default _
def mkHoleKind : IO SyntaxNodeKind := nextKind `hole
/- Syntax AST -/
inductive Syntax
| missing
| node (kind : SyntaxNodeKind) (args : Array Syntax) (scopes : MacroScopes)
| atom (info : Option SourceInfo) (val : String)
| ident (info : Option SourceInfo) (rawVal : Substring) (val : Name) (preresolved : List Name) (scopes : MacroScopes)
instance stxInh : Inhabited Syntax :=
⟨Syntax.missing⟩
def SyntaxNodeKind.fix : SyntaxNodeKind → IO SyntaxNodeKind
| {name := n, ..} := do
m ← nameToKindTable.get,
match m.find n with
| some id := pure {name := n, id := id}
| none := throw $ IO.userError ("Error unknown Syntax kind '" ++ toString n ++ "'")
partial def Syntax.fixKinds : Syntax → IO Syntax
| (Syntax.node k args scopes) := do
k ← k.fix,
args ← args.mmap Syntax.fixKinds,
pure (Syntax.node k args scopes)
| other := pure other
inductive IsNode : Syntax → Prop
| mk (kind : SyntaxNodeKind) (args : Array Syntax) (scopes : MacroScopes) : IsNode (Syntax.node kind args scopes)
def SyntaxNode : Type := {s : Syntax // IsNode s }
def notIsNodeMissing (h : IsNode Syntax.missing) : False := match h with end
def notIsNodeAtom {info val} (h : IsNode (Syntax.atom info val)) : False := match h with end
def notIsNodeIdent {info rawVal val preresolved scopes} (h : IsNode (Syntax.ident info rawVal val preresolved scopes)) : False := match h with end
def unreachIsNodeMissing {α : Type} (h : IsNode Syntax.missing) : α := False.elim (notIsNodeMissing h)
def unreachIsNodeAtom {α : Type} {info val} (h : IsNode (Syntax.atom info val)) : α := False.elim (notIsNodeAtom h)
def unreachIsNodeIdent {α : Type} {info rawVal val preresolved scopes} (h : IsNode (Syntax.ident info rawVal val preresolved scopes)) : α :=
False.elim (match h with end)
@[inline] def withArgs {α : Type} (n : SyntaxNode) (fn : Array Syntax → α) : α :=
match n with
| ⟨Syntax.node _ args _, _⟩ := fn args
| ⟨Syntax.missing, h⟩ := unreachIsNodeMissing h
| ⟨Syntax.atom _ _, h⟩ := unreachIsNodeAtom h
| ⟨Syntax.ident _ _ _ _ _, h⟩ := unreachIsNodeIdent h
@[inline] def updateArgs (n : SyntaxNode) (fn : Array Syntax → Array Syntax) : Syntax :=
match n with
| ⟨Syntax.node kind args scopes, _⟩ := Syntax.node kind (fn args) scopes
| ⟨Syntax.missing, h⟩ := unreachIsNodeMissing h
| ⟨Syntax.atom _ _, h⟩ := unreachIsNodeAtom h
| ⟨Syntax.ident _ _ _ _ _, h⟩ := unreachIsNodeIdent h
-- TODO(Sebastian): exhaustively argue why (if?) this is correct
-- The basic idea is List concatenation with elimination of adjacent identical scopes
def MacroScopes.flip : MacroScopes → MacroScopes → MacroScopes
| ys [] := ys
| ys (x::xs) := match MacroScopes.flip ys xs with
| y::ys := if x == y then ys else x::y::ys
| [] := [x]
namespace Syntax
def isIdent : Syntax → Bool
| (Syntax.ident _ _ _ _ _) := true
| _ := false
def isOfKind : Syntax → SyntaxNodeKind → Bool
| (Syntax.node kind _ _) k := k == kind
| other _ := false
def flipScopes (scopes : MacroScopes) : Syntax → Syntax
| (Syntax.ident info rawVal val pre scopes) := Syntax.ident info rawVal val pre (scopes.flip scopes)
| (Syntax.node kind args scopes) := Syntax.node kind args (scopes.flip scopes)
| other := other
@[inline] def toSyntaxNode {α : Type} (s : Syntax) (base : α) (fn : SyntaxNode → α) : α :=
match s with
| Syntax.node kind args [] := fn ⟨Syntax.node kind args [], IsNode.mk _ _ _⟩
| Syntax.node kind args scopes := fn ⟨Syntax.node kind (args.hmap (flipScopes scopes)) [], IsNode.mk _ _ _⟩
| other := base
local attribute [instance] monadInhabited
@[specialize] partial def mreplace {m : Type → Type} [Monad m] (fn : Syntax → m (Option Syntax)) : Syntax → m Syntax
| stx@(node kind args scopes) := do
o ← fn stx,
(match o with
| some stx := pure stx
| none := do args ← args.mmap mreplace, pure (node kind args scopes))
| stx := do o ← fn stx, pure (o.getOrElse stx)
@[inline] def replace {m : Type → Type} [Monad m] (fn : Syntax → m (Option Syntax)) := @mreplace Id _
private def updateInfo : SourceInfo → String.Pos → SourceInfo
| {leading := {str := s, startPos := _, stopPos := _}, pos := pos, trailing := trailing} last :=
{leading := {str := s, startPos := last, stopPos := pos}, pos := pos, trailing := trailing}
/- Remark: the State `String.Pos` is the `SourceInfo.trailing.endPos` of the previous token,
or the beginning of the String. -/
@[inline]
private def updateLeadingAux : Syntax → State String.Pos (Option Syntax)
| (atom (some info) val) := do
last ← get,
set info.trailing.stopPos,
let newInfo := updateInfo info last in
pure $ some (atom (some newInfo) val)
| (ident (some info) rawVal val pre scopes) := do
last ← get,
set info.trailing.stopPos,
let newInfo := updateInfo info last in
pure $ some (ident (some newInfo) rawVal val pre scopes)
| _ := pure none
/-- Set `SourceInfo.leading` according to the trailing stop of the preceding token.
The Result is a round-tripping Syntax tree IF, in the input Syntax tree,
* all leading stops, atom contents, and trailing starts are correct
* trailing stops are between the trailing start and the next leading stop.
Remark: after parsing all `SourceInfo.leading` fields are Empty.
The Syntax argument is the output produced by the Parser for `source`.
This Function "fixes" the `source.leanding` field.
Note that, the `SourceInfo.trailing` fields are correct.
The implementation of this Function relies on this property. -/
def updateLeading : Syntax → Syntax :=
λ stx, Prod.fst <$> (mreplace updateLeadingAux stx).run 0
partial def updateTrailing (trailing : Substring) : Syntax → Syntax
| (Syntax.atom (some info) val) := Syntax.atom (some (info.updateTrailing trailing)) val
| (Syntax.ident (some info) rawVal val pre scopes) := Syntax.ident (some (info.updateTrailing trailing)) rawVal val pre scopes
| n@(Syntax.node k args scopes) :=
if args.size == 0 then n
else
let i := args.size - 1 in
let last := updateTrailing (args.get i) in
let args := args.set i last in
Syntax.node k args scopes
| other := other
/-- Retrieve the left-most leaf's info in the Syntax tree. -/
partial def getHeadInfo : Syntax → Option SourceInfo
| (atom info _) := info
| (ident info _ _ _ _ ) := info
| (node _ args _) := args.find getHeadInfo
| _ := none
def getPos (stx : Syntax) : Option String.Pos :=
SourceInfo.pos <$> stx.getHeadInfo
private def reprintLeaf : Option SourceInfo → String → String
| none val := val
| (some info) val := info.leading.toString ++ val ++ info.trailing.toString
partial def reprint : Syntax → Option String
| (atom info val) := reprintLeaf info val
| (ident info rawVal _ _ _) := reprintLeaf info rawVal.toString
| (node kind args _) :=
if kind == choiceKind then
if args.size == 0 then failure
else do
s ← reprint (args.get 0),
args.mfoldlFrom s (λ stx s, do s' ← reprint stx, guard (s == s'), pure s) 1
else args.mfoldl "" (λ stx r, do s ← reprint stx, pure $ r ++ s)
| missing := ""
open Lean.Format
protected partial def toFormat : Syntax → Format
| (atom info val) := toFmt $ repr val
| (ident _ _ val pre scopes) :=
let scopes := pre.map toFmt ++ scopes.reverse.map toFmt in
let scopes := match scopes with [] := toFmt "" | _ := bracket "{" (joinSep scopes ", ") "}" in
toFmt "`" ++ toFmt val ++ scopes
| (node kind args scopes) :=
let scopes := match scopes with [] := toFmt "" | _ := bracket "{" (joinSep scopes.reverse ", ") "}" in
if kind.name = `Lean.Parser.noKind then
sbracket $ scopes ++ joinSep (args.toList.map toFormat) line
else
let shorterName := kind.name.replacePrefix `Lean.Parser Name.anonymous in
paren $ joinSep ((toFmt shorterName ++ scopes) :: args.toList.map toFormat) line
| missing := "<missing>"
instance : HasToFormat Syntax := ⟨Syntax.toFormat⟩
instance : HasToString Syntax := ⟨toString ∘ toFmt⟩
end Syntax
end Parser
-- end Lean
|
a94d04e6b1f9b7572d5a6ffe11c186a0d934309f | 9be442d9ec2fcf442516ed6e9e1660aa9071b7bd | /stage0/src/Lean/Meta/Tactic/Util.lean | d7cdc693807713960f7eb57e5105980a7e1aba6f | [
"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 | EdAyers/lean4 | 57ac632d6b0789cb91fab2170e8c9e40441221bd | 37ba0df5841bde51dbc2329da81ac23d4f6a4de4 | refs/heads/master | 1,676,463,245,298 | 1,660,619,433,000 | 1,660,619,433,000 | 183,433,437 | 1 | 0 | Apache-2.0 | 1,657,612,672,000 | 1,556,196,574,000 | Lean | UTF-8 | Lean | false | false | 6,328 | 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
-/
import Lean.Util.ForEachExpr
import Lean.Meta.Basic
import Lean.Meta.AppBuilder
import Lean.Meta.PPGoal
namespace Lean.Meta
/-- Get the user name of the given metavariable. -/
def _root_.Lean.MVarId.getTag (mvarId : MVarId) : MetaM Name :=
return (← mvarId.getDecl).userName
@[deprecated MVarId.getTag]
def getMVarTag (mvarId : MVarId) : MetaM Name :=
mvarId.getTag
def _root_.Lean.MVarId.setTag (mvarId : MVarId) (tag : Name) : MetaM Unit := do
modify fun s => { s with mctx := s.mctx.setMVarUserName mvarId tag }
@[deprecated MVarId.setTag]
def setMVarTag (mvarId : MVarId) (tag : Name) : MetaM Unit := do
mvarId.setTag tag
def appendTag (tag : Name) (suffix : Name) : Name :=
tag.modifyBase (· ++ suffix.eraseMacroScopes)
def appendTagSuffix (mvarId : MVarId) (suffix : Name) : MetaM Unit := do
let tag ← mvarId.getTag
mvarId.setTag (appendTag tag suffix)
def mkFreshExprSyntheticOpaqueMVar (type : Expr) (tag : Name := Name.anonymous) : MetaM Expr :=
mkFreshExprMVar type MetavarKind.syntheticOpaque tag
def throwTacticEx (tacticName : Name) (mvarId : MVarId) (msg : MessageData) : MetaM α :=
withPPForTacticGoal do
if msg.isEmpty then
throwError "tactic '{tacticName}' failed\n{mvarId}"
else
throwError "tactic '{tacticName}' failed, {msg}\n{mvarId}"
def throwNestedTacticEx {α} (tacticName : Name) (ex : Exception) : MetaM α := do
throwError "tactic '{tacticName}' failed, nested error:\n{ex.toMessageData}"
/-- Throw a tactic exception with given tactic name if the given metavariable is assigned. -/
def _root_.Lean.MVarId.checkNotAssigned (mvarId : MVarId) (tacticName : Name) : MetaM Unit := do
if (← mvarId.isAssigned) then
throwTacticEx tacticName mvarId "metavariable has already been assigned"
@[deprecated MVarId.checkNotAssigned]
def checkNotAssigned (mvarId : MVarId) (tacticName : Name) : MetaM Unit := do
mvarId.checkNotAssigned tacticName
/-- Get the type the given metavariable. -/
def _root_.Lean.MVarId.getType (mvarId : MVarId) : MetaM Expr :=
return (← mvarId.getDecl).type
@[deprecated MVarId.getType]
def getMVarType (mvarId : MVarId) : MetaM Expr :=
mvarId.getType
/-- Get the type the given metavariable after instantiating metavariables and reducing to
weak head normal form. -/
def _root_.Lean.MVarId.getType' (mvarId : MVarId) : MetaM Expr := do
whnf (← instantiateMVars (← mvarId.getType))
@[deprecated MVarId.getType']
def getMVarType' (mvarId : MVarId) : MetaM Expr := do
mvarId.getType'
builtin_initialize registerTraceClass `Meta.Tactic
/-- Assign `mvarId` to `sorryAx` -/
def _root_.Lean.MVarId.admit (mvarId : MVarId) (synthetic := true) : MetaM Unit :=
mvarId.withContext do
mvarId.checkNotAssigned `admit
let mvarType ← mvarId.getType
let val ← mkSorry mvarType synthetic
mvarId.assign val
@[deprecated MVarId.admit]
def admit (mvarId : MVarId) (synthetic := true) : MetaM Unit :=
mvarId.admit synthetic
/-- Beta reduce the metavariable type head -/
def _root_.Lean.MVarId.headBetaType (mvarId : MVarId) : MetaM Unit := do
mvarId.setType (← mvarId.getType).headBeta
@[deprecated MVarId.headBetaType]
def headBetaMVarType (mvarId : MVarId) : MetaM Unit := do
mvarId.headBetaType
/-- Collect nondependent hypotheses that are propositions. -/
def _root_.Lean.MVarId.getNondepPropHyps (mvarId : MVarId) : MetaM (Array FVarId) :=
let removeDeps (e : Expr) (candidates : FVarIdHashSet) : MetaM FVarIdHashSet := do
let e ← instantiateMVars e
let visit : StateRefT FVarIdHashSet MetaM FVarIdHashSet := do
e.forEach fun
| Expr.fvar fvarId => modify fun s => s.erase fvarId
| _ => pure ()
get
visit |>.run' candidates
mvarId.withContext do
let mut candidates : FVarIdHashSet := {}
for localDecl in (← getLCtx) do
unless localDecl.isAuxDecl do
candidates ← removeDeps localDecl.type candidates
match localDecl.value? with
| none => pure ()
| some value => candidates ← removeDeps value candidates
if (← isProp localDecl.type) && !localDecl.hasValue then
candidates := candidates.insert localDecl.fvarId
candidates ← removeDeps (← mvarId.getType) candidates
if candidates.isEmpty then
return #[]
else
let mut result := #[]
for localDecl in (← getLCtx) do
if candidates.contains localDecl.fvarId then
result := result.push localDecl.fvarId
return result
@[deprecated MVarId.getNondepPropHyps]
def getNondepPropHyps (mvarId : MVarId) : MetaM (Array FVarId) :=
mvarId.getNondepPropHyps
partial def saturate (mvarId : MVarId) (x : MVarId → MetaM (Option (List MVarId))) : MetaM (List MVarId) := do
let (_, r) ← go mvarId |>.run #[]
return r.toList
where
go (mvarId : MVarId) : StateRefT (Array MVarId) MetaM Unit :=
withIncRecDepth do
match (← x mvarId) with
| none => modify fun s => s.push mvarId
| some mvarIds => mvarIds.forM go
def exactlyOne (mvarIds : List MVarId) (msg : MessageData := "unexpected number of goals") : MetaM MVarId :=
match mvarIds with
| [mvarId] => return mvarId
| _ => throwError msg
def ensureAtMostOne (mvarIds : List MVarId) (msg : MessageData := "unexpected number of goals") : MetaM (Option MVarId) :=
match mvarIds with
| [] => return none
| [mvarId] => return some mvarId
| _ => throwError msg
/-- Return all propositions in the local context. -/
def getPropHyps : MetaM (Array FVarId) := do
let mut result := #[]
for localDecl in (← getLCtx) do
unless localDecl.isAuxDecl do
if (← isProp localDecl.type) then
result := result.push localDecl.fvarId
return result
def _root_.Lean.MVarId.inferInstance (mvarId : MVarId) : MetaM Unit := mvarId.withContext do
mvarId.checkNotAssigned `infer_instance
let synthVal ← synthInstance (← mvarId.getType)
unless (← isDefEq (mkMVar mvarId) synthVal) do
throwTacticEx `infer_instance mvarId "`infer_instance` tactic failed to assign instance"
inductive TacticResultCNM where
| closed
| noChange
| modified (mvarId : MVarId)
end Lean.Meta
|
d70228a107e11ce7597ebfa854d7f0abcc0d5083 | b7f22e51856f4989b970961f794f1c435f9b8f78 | /library/init/reserved_notation.lean | 55cad3d402002f1456ff1a623866b27e1fd61bc6 | [
"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 | 7,421 | lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
-/
prelude
import init.datatypes
notation `assume` binders `,` r:(scoped f, f) := r
notation `take` binders `,` r:(scoped f, f) := r
structure has_zero [class] (A : Type) := (zero : A)
structure has_one [class] (A : Type) := (one : A)
structure has_add [class] (A : Type) := (add : A → A → A)
structure has_mul [class] (A : Type) := (mul : A → A → A)
structure has_inv [class] (A : Type) := (inv : A → A)
structure has_neg [class] (A : Type) := (neg : A → A)
structure has_sub [class] (A : Type) := (sub : A → A → A)
structure has_div [class] (A : Type) := (div : A → A → A)
structure has_dvd [class] (A : Type) := (dvd : A → A → Prop)
structure has_mod [class] (A : Type) := (mod : A → A → A)
structure has_le [class] (A : Type) := (le : A → A → Prop)
structure has_lt [class] (A : Type) := (lt : A → A → Prop)
definition zero {A : Type} [s : has_zero A] : A := has_zero.zero A
definition one {A : Type} [s : has_one A] : A := has_one.one A
definition add {A : Type} [s : has_add A] : A → A → A := has_add.add
definition mul {A : Type} [s : has_mul A] : A → A → A := has_mul.mul
definition sub {A : Type} [s : has_sub A] : A → A → A := has_sub.sub
definition div {A : Type} [s : has_div A] : A → A → A := has_div.div
definition dvd {A : Type} [s : has_dvd A] : A → A → Prop := has_dvd.dvd
definition mod {A : Type} [s : has_mod A] : A → A → A := has_mod.mod
definition neg {A : Type} [s : has_neg A] : A → A := has_neg.neg
definition inv {A : Type} [s : has_inv A] : A → A := has_inv.inv
definition le {A : Type} [s : has_le A] : A → A → Prop := has_le.le
definition lt {A : Type} [s : has_lt A] : A → A → Prop := has_lt.lt
definition ge [reducible] {A : Type} [s : has_le A] (a b : A) : Prop := le b a
definition gt [reducible] {A : Type} [s : has_lt A] (a b : A) : Prop := lt b a
definition bit0 {A : Type} [s : has_add A] (a : A) : A := add a a
definition bit1 {A : Type} [s₁ : has_one A] [s₂ : has_add A] (a : A) : A := add (bit0 a) one
definition num_has_zero [instance] : has_zero num :=
has_zero.mk num.zero
definition num_has_one [instance] : has_one num :=
has_one.mk (num.pos pos_num.one)
definition pos_num_has_one [instance] : has_one pos_num :=
has_one.mk (pos_num.one)
namespace pos_num
open bool
definition is_one (a : pos_num) : bool :=
pos_num.rec_on a tt (λn r, ff) (λn r, ff)
definition pred (a : pos_num) : pos_num :=
pos_num.rec_on a one (λn r, bit0 n) (λn r, bool.rec_on (is_one n) (bit1 r) one)
definition size (a : pos_num) : pos_num :=
pos_num.rec_on a one (λn r, succ r) (λn r, succ r)
definition add (a b : pos_num) : pos_num :=
pos_num.rec_on a
succ
(λn f b, pos_num.rec_on b
(succ (bit1 n))
(λm r, succ (bit1 (f m)))
(λm r, bit1 (f m)))
(λn f b, pos_num.rec_on b
(bit1 n)
(λm r, bit1 (f m))
(λm r, bit0 (f m)))
b
end pos_num
definition pos_num_has_add [instance] : has_add pos_num :=
has_add.mk pos_num.add
namespace num
open pos_num
definition add (a b : num) : num :=
num.rec_on a b (λpa, num.rec_on b (pos pa) (λpb, pos (pos_num.add pa pb)))
end num
definition num_has_add [instance] : has_add num :=
has_add.mk num.add
definition std.priority.default : num := 1000
definition std.priority.max : num := 4294967295
namespace nat
protected definition prio := num.add std.priority.default 100
protected definition add (a b : nat) : nat :=
nat.rec a (λ b₁ r, succ r) b
definition of_num (n : num) : nat :=
num.rec zero
(λ n, pos_num.rec (succ zero) (λ n r, nat.add (nat.add r r) (succ zero)) (λ n r, nat.add r r) n) n
end nat
attribute pos_num_has_add pos_num_has_one num_has_zero num_has_one num_has_add
[instance] [priority nat.prio]
definition nat_has_zero [instance] [priority nat.prio] : has_zero nat :=
has_zero.mk nat.zero
definition nat_has_one [instance] [priority nat.prio] : has_one nat :=
has_one.mk (nat.succ (nat.zero))
definition nat_has_add [instance] [priority nat.prio] : has_add nat :=
has_add.mk nat.add
/-
Global declarations of right binding strength
If a module reassigns these, it will be incompatible with other modules that adhere to these
conventions.
When hovering over a symbol, use "C-c C-k" to see how to input it.
-/
definition std.prec.max : num := 1024 -- the strength of application, identifiers, (, [, etc.
definition std.prec.arrow : num := 25
/-
The next definition is "max + 10". It can be used e.g. for postfix operations that should
be stronger than application.
-/
definition std.prec.max_plus :=
num.succ (num.succ (num.succ (num.succ (num.succ (num.succ (num.succ (num.succ (num.succ
(num.succ std.prec.max)))))))))
/- Logical operations and relations -/
reserve prefix `¬`:40
reserve prefix `~`:40
reserve infixr ` ∧ `:35
reserve infixr ` /\ `:35
reserve infixr ` \/ `:30
reserve infixr ` ∨ `:30
reserve infix ` <-> `:20
reserve infix ` ↔ `:20
reserve infix ` = `:50
reserve infix ` ≠ `:50
reserve infix ` ≈ `:50
reserve infix ` ~ `:50
reserve infix ` ≡ `:50
reserve infixr ` ∘ `:60 -- input with \comp
reserve postfix `⁻¹`:std.prec.max_plus -- input with \sy or \-1 or \inv
reserve infixl ` ⬝ `:75
reserve infixr ` ▸ `:75
reserve infixr ` ▹ `:75
/- types and type constructors -/
reserve infixl ` ⊎ `:25
reserve infixl ` × `:30
/- arithmetic operations -/
reserve infixl ` + `:65
reserve infixl ` - `:65
reserve infixl ` * `:70
reserve infixl ` / `:70
reserve infixl ` % `:70
reserve prefix `-`:100
reserve infix ` ^ `:80
reserve infix ` <= `:50
reserve infix ` ≤ `:50
reserve infix ` < `:50
reserve infix ` >= `:50
reserve infix ` ≥ `:50
reserve infix ` > `:50
/- boolean operations -/
reserve infixl ` && `:70
reserve infixl ` || `:65
/- set operations -/
reserve infix ` ∈ `:50
reserve infix ` ∉ `:50
reserve infixl ` ∩ `:70
reserve infixl ` ∪ `:65
reserve infix ` ⊆ `:50
reserve infix ` ⊇ `:50
reserve infix ` ' `:75 -- for the image of a set under a function
reserve infix ` '- `:75 -- for the preimage of a set under a function
/- other symbols -/
reserve infix ` ∣ `:50
reserve infixl ` ++ `:65
reserve infixr ` :: `:67
infix + := add
infix * := mul
infix - := sub
infix / := div
infix ∣ := dvd
infix % := mod
prefix - := neg
postfix ⁻¹ := inv
infix ≤ := le
infix ≥ := ge
infix < := lt
infix > := gt
notation [parsing_only] x ` +[`:65 A:0 `] `:0 y:65 := @add A _ x y
notation [parsing_only] x ` -[`:65 A:0 `] `:0 y:65 := @sub A _ x y
notation [parsing_only] x ` *[`:70 A:0 `] `:0 y:70 := @mul A _ x y
notation [parsing_only] x ` /[`:70 A:0 `] `:0 y:70 := @div A _ x y
notation [parsing_only] x ` ∣[`:70 A:0 `] `:0 y:70 := @dvd A _ x y
notation [parsing_only] x ` %[`:70 A:0 `] `:0 y:70 := @mod A _ x y
notation [parsing_only] x ` ≤[`:50 A:0 `] `:0 y:50 := @le A _ x y
notation [parsing_only] x ` ≥[`:50 A:0 `] `:0 y:50 := @ge A _ x y
notation [parsing_only] x ` <[`:50 A:0 `] `:0 y:50 := @lt A _ x y
notation [parsing_only] x ` >[`:50 A:0 `] `:0 y:50 := @gt A _ x y
|
014bf410eec7a01229000c201e4305eafdb06a6e | 32025d5c2d6e33ad3b6dd8a3c91e1e838066a7f7 | /src/Lean/Compiler/IR/Basic.lean | 6b34ef8551987975a5dd35b329200bdfc2e65525 | [
"Apache-2.0"
] | permissive | walterhu1015/lean4 | b2c71b688975177402758924eaa513475ed6ce72 | 2214d81e84646a905d0b20b032c89caf89c737ad | refs/heads/master | 1,671,342,096,906 | 1,599,695,985,000 | 1,599,695,985,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 25,708 | 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
-/
import Lean.Data.KVMap
import Lean.Data.Name
import Lean.Data.Format
import Lean.Compiler.ExternAttr
/-
Implements (extended) λPure and λRc proposed in the article
"Counting Immutable Beans", Sebastian Ullrich and Leonardo de Moura.
The Lean to IR transformation produces λPure code, and
this part is implemented in C++. The procedures described in the paper
above are implemented in Lean.
-/
namespace Lean
namespace IR
/- Function identifier -/
abbrev FunId := Name
abbrev Index := Nat
/- Variable identifier -/
structure VarId :=
(idx : Index)
/- Join point identifier -/
structure JoinPointId :=
(idx : Index)
abbrev Index.lt (a b : Index) : Bool := a < b
namespace VarId
instance : HasBeq VarId := ⟨fun a b => a.idx == b.idx⟩
instance : HasToString VarId := ⟨fun a => "x_" ++ toString a.idx⟩
instance : HasFormat VarId := ⟨fun a => toString a⟩
instance : Hashable VarId := ⟨fun a => hash a.idx⟩
end VarId
namespace JoinPointId
instance : HasBeq JoinPointId := ⟨fun a b => a.idx == b.idx⟩
instance : HasToString JoinPointId := ⟨fun a => "block_" ++ toString a.idx⟩
instance : HasFormat JoinPointId := ⟨fun a => toString a⟩
instance : Hashable JoinPointId := ⟨fun a => hash a.idx⟩
end JoinPointId
abbrev MData := KVMap
namespace MData
abbrev empty : MData := { : KVMap }
instance : HasEmptyc MData := ⟨empty⟩
end MData
/- Low Level IR types. Most are self explanatory.
- `usize` represents the C++ `size_t` Type. We have it here
because it is 32-bit in 32-bit machines, and 64-bit in 64-bit machines,
and we want the C++ backend for our Compiler to generate platform independent code.
- `irrelevant` for Lean types, propositions and proofs.
- `object` a pointer to a value in the heap.
- `tobject` a pointer to a value in the heap or tagged pointer
(i.e., the least significant bit is 1) storing a scalar value.
- `struct` and `union` are used to return small values (e.g., `Option`, `Prod`, `Except`)
on the stack.
Remark: the RC operations for `tobject` are slightly more expensive because we
first need to test whether the `tobject` is really a pointer or not.
Remark: the Lean runtime assumes that sizeof(void*) == sizeof(sizeT).
Lean cannot be compiled on old platforms where this is not True.
Since values of type `struct` and `union` are only used to return values,
We assume they must be used/consumed "linearly". We use the term "linear" here
to mean "exactly once" in each execution. That is, given `x : S`, where `S` is a struct,
then one of the following must hold in each (execution) branch.
1- `x` occurs only at a single `ret x` instruction. That is, it is being consumed by being returned.
2- `x` occurs only at a single `ctor`. That is, it is being "consumed" by being stored into another `struct/union`.
3- We extract (aka project) every single field of `x` exactly once. That is, we are consuming `x` by consuming each
of one of its components. Minor refinement: we don't need to consume scalar fields or struct/union
fields that do not contain object fields.
-/
inductive IRType
| float | uint8 | uint16 | uint32 | uint64 | usize
| irrelevant | object | tobject
| struct (leanTypeName : Option Name) (types : Array IRType) : IRType
| union (leanTypeName : Name) (types : Array IRType) : IRType
namespace IRType
partial def beq : IRType → IRType → Bool
| float, float => true
| uint8, uint8 => true
| uint16, uint16 => true
| uint32, uint32 => true
| uint64, uint64 => true
| usize, usize => true
| irrelevant, irrelevant => true
| object, object => true
| tobject, tobject => true
| struct n₁ tys₁, struct n₂ tys₂ => n₁ == n₂ && Array.isEqv tys₁ tys₂ beq
| union n₁ tys₁, union n₂ tys₂ => n₁ == n₂ && Array.isEqv tys₁ tys₂ beq
| _, _ => false
instance HasBeq : HasBeq IRType := ⟨beq⟩
def isScalar : IRType → Bool
| float => true
| uint8 => true
| uint16 => true
| uint32 => true
| uint64 => true
| usize => true
| _ => false
def isObj : IRType → Bool
| object => true
| tobject => true
| _ => false
def isIrrelevant : IRType → Bool
| irrelevant => true
| _ => false
def isStruct : IRType → Bool
| struct _ _ => true
| _ => false
def isUnion : IRType → Bool
| union _ _ => true
| _ => false
end IRType
/- Arguments to applications, constructors, etc.
We use `irrelevant` for Lean types, propositions and proofs that have been erased.
Recall that for a Function `f`, we also generate `f._rarg` which does not take
`irrelevant` arguments. However, `f._rarg` is only safe to be used in full applications. -/
inductive Arg
| var (id : VarId)
| irrelevant
namespace Arg
protected def beq : Arg → Arg → Bool
| var x, var y => x == y
| irrelevant, irrelevant => true
| _, _ => false
instance : HasBeq Arg := ⟨Arg.beq⟩
instance : Inhabited Arg := ⟨irrelevant⟩
end Arg
@[export lean_ir_mk_var_arg] def mkVarArg (id : VarId) : Arg := Arg.var id
inductive LitVal
| num (v : Nat)
| str (v : String)
def LitVal.beq : LitVal → LitVal → Bool
| LitVal.num v₁, LitVal.num v₂ => v₁ == v₂
| LitVal.str v₁, LitVal.str v₂ => v₁ == v₂
| _, _ => false
instance LitVal.HasBeq : HasBeq LitVal := ⟨LitVal.beq⟩
/- Constructor information.
- `name` is the Name of the Constructor in Lean.
- `cidx` is the Constructor index (aka tag).
- `size` is the number of arguments of type `object/tobject`.
- `usize` is the number of arguments of type `usize`.
- `ssize` is the number of bytes used to store scalar values.
Recall that a Constructor object contains a header, then a sequence of
pointers to other Lean objects, a sequence of `USize` (i.e., `size_t`)
scalar values, and a sequence of other scalar values. -/
structure CtorInfo :=
(name : Name) (cidx : Nat) (size : Nat) (usize : Nat) (ssize : Nat)
def CtorInfo.beq : CtorInfo → CtorInfo → Bool
| ⟨n₁, cidx₁, size₁, usize₁, ssize₁⟩, ⟨n₂, cidx₂, size₂, usize₂, ssize₂⟩ =>
n₁ == n₂ && cidx₁ == cidx₂ && size₁ == size₂ && usize₁ == usize₂ && ssize₁ == ssize₂
instance CtorInfo.HasBeq : HasBeq CtorInfo := ⟨CtorInfo.beq⟩
def CtorInfo.isRef (info : CtorInfo) : Bool :=
info.size > 0 || info.usize > 0 || info.ssize > 0
def CtorInfo.isScalar (info : CtorInfo) : Bool :=
!info.isRef
inductive Expr
/- We use `ctor` mainly for constructing Lean object/tobject values `lean_ctor_object` in the runtime.
This instruction is also used to creat `struct` and `union` return values.
For `union`, only `i.cidx` is relevant. For `struct`, `i` is irrelevant. -/
| ctor (i : CtorInfo) (ys : Array Arg)
| reset (n : Nat) (x : VarId)
/- `reuse x in ctor_i ys` instruction in the paper. -/
| reuse (x : VarId) (i : CtorInfo) (updtHeader : Bool) (ys : Array Arg)
/- Extract the `tobject` value at Position `sizeof(void*)*i` from `x`.
We also use `proj` for extracting fields from `struct` return values, and casting `union` return values. -/
| proj (i : Nat) (x : VarId)
/- Extract the `Usize` value at Position `sizeof(void*)*i` from `x`. -/
| uproj (i : Nat) (x : VarId)
/- Extract the scalar value at Position `sizeof(void*)*n + offset` from `x`. -/
| sproj (n : Nat) (offset : Nat) (x : VarId)
/- Full application. -/
| fap (c : FunId) (ys : Array Arg)
/- Partial application that creates a `pap` value (aka closure in our nonstandard terminology). -/
| pap (c : FunId) (ys : Array Arg)
/- Application. `x` must be a `pap` value. -/
| ap (x : VarId) (ys : Array Arg)
/- Given `x : ty` where `ty` is a scalar type, this operation returns a value of Type `tobject`.
For small scalar values, the Result is a tagged pointer, and no memory allocation is performed. -/
| box (ty : IRType) (x : VarId)
/- Given `x : [t]object`, obtain the scalar value. -/
| unbox (x : VarId)
| lit (v : LitVal)
/- Return `1 : uint8` Iff `RC(x) > 1` -/
| isShared (x : VarId)
/- Return `1 : uint8` Iff `x : tobject` is a tagged pointer (storing a scalar value). -/
| isTaggedPtr (x : VarId)
@[export lean_ir_mk_ctor_expr] def mkCtorExpr (n : Name) (cidx : Nat) (size : Nat) (usize : Nat) (ssize : Nat) (ys : Array Arg) : Expr := Expr.ctor ⟨n, cidx, size, usize, ssize⟩ ys
@[export lean_ir_mk_proj_expr] def mkProjExpr (i : Nat) (x : VarId) : Expr := Expr.proj i x
@[export lean_ir_mk_uproj_expr] def mkUProjExpr (i : Nat) (x : VarId) : Expr := Expr.uproj i x
@[export lean_ir_mk_sproj_expr] def mkSProjExpr (n : Nat) (offset : Nat) (x : VarId) : Expr := Expr.sproj n offset x
@[export lean_ir_mk_fapp_expr] def mkFAppExpr (c : FunId) (ys : Array Arg) : Expr := Expr.fap c ys
@[export lean_ir_mk_papp_expr] def mkPAppExpr (c : FunId) (ys : Array Arg) : Expr := Expr.pap c ys
@[export lean_ir_mk_app_expr] def mkAppExpr (x : VarId) (ys : Array Arg) : Expr := Expr.ap x ys
@[export lean_ir_mk_num_expr] def mkNumExpr (v : Nat) : Expr := Expr.lit (LitVal.num v)
@[export lean_ir_mk_str_expr] def mkStrExpr (v : String) : Expr := Expr.lit (LitVal.str v)
structure Param :=
(x : VarId) (borrow : Bool) (ty : IRType)
instance paramInh : Inhabited Param := ⟨{ x := { idx := 0 }, borrow := false, ty := IRType.object }⟩
@[export lean_ir_mk_param] def mkParam (x : VarId) (borrow : Bool) (ty : IRType) : Param := ⟨x, borrow, ty⟩
inductive AltCore (FnBody : Type) : Type
| ctor (info : CtorInfo) (b : FnBody) : AltCore
| default (b : FnBody) : AltCore
inductive FnBody
/- `let x : ty := e; b` -/
| vdecl (x : VarId) (ty : IRType) (e : Expr) (b : FnBody)
/- Join point Declaration `block_j (xs) := e; b` -/
| jdecl (j : JoinPointId) (xs : Array Param) (v : FnBody) (b : FnBody)
/- Store `y` at Position `sizeof(void*)*i` in `x`. `x` must be a Constructor object and `RC(x)` must be 1.
This operation is not part of λPure is only used during optimization. -/
| set (x : VarId) (i : Nat) (y : Arg) (b : FnBody)
| setTag (x : VarId) (cidx : Nat) (b : FnBody)
/- Store `y : Usize` at Position `sizeof(void*)*i` in `x`. `x` must be a Constructor object and `RC(x)` must be 1. -/
| uset (x : VarId) (i : Nat) (y : VarId) (b : FnBody)
/- Store `y : ty` at Position `sizeof(void*)*i + offset` in `x`. `x` must be a Constructor object and `RC(x)` must be 1.
`ty` must not be `object`, `tobject`, `irrelevant` nor `Usize`. -/
| sset (x : VarId) (i : Nat) (offset : Nat) (y : VarId) (ty : IRType) (b : FnBody)
/- RC increment for `object`. If c == `true`, then `inc` must check whether `x` is a tagged pointer or not.
If `persistent == true` then `x` is statically known to be a persistent object. -/
| inc (x : VarId) (n : Nat) (c : Bool) (persistent : Bool) (b : FnBody)
/- RC decrement for `object`. If c == `true`, then `inc` must check whether `x` is a tagged pointer or not.
If `persistent == true` then `x` is statically known to be a persistent object. -/
| dec (x : VarId) (n : Nat) (c : Bool) (persistent : Bool) (b : FnBody)
| del (x : VarId) (b : FnBody)
| mdata (d : MData) (b : FnBody)
| case (tid : Name) (x : VarId) (xType : IRType) (cs : Array (AltCore FnBody))
| ret (x : Arg)
/- Jump to join point `j` -/
| jmp (j : JoinPointId) (ys : Array Arg)
| unreachable
instance : Inhabited FnBody := ⟨FnBody.unreachable⟩
abbrev FnBody.nil := FnBody.unreachable
@[export lean_ir_mk_vdecl] def mkVDecl (x : VarId) (ty : IRType) (e : Expr) (b : FnBody) : FnBody := FnBody.vdecl x ty e b
@[export lean_ir_mk_jdecl] def mkJDecl (j : JoinPointId) (xs : Array Param) (v : FnBody) (b : FnBody) : FnBody := FnBody.jdecl j xs v b
@[export lean_ir_mk_uset] def mkUSet (x : VarId) (i : Nat) (y : VarId) (b : FnBody) : FnBody := FnBody.uset x i y b
@[export lean_ir_mk_sset] def mkSSet (x : VarId) (i : Nat) (offset : Nat) (y : VarId) (ty : IRType) (b : FnBody) : FnBody := FnBody.sset x i offset y ty b
@[export lean_ir_mk_case] def mkCase (tid : Name) (x : VarId) (cs : Array (AltCore FnBody)) : FnBody :=
-- Tyhe field `xType` is set by `explicitBoxing` compiler pass.
FnBody.case tid x IRType.object cs
@[export lean_ir_mk_ret] def mkRet (x : Arg) : FnBody := FnBody.ret x
@[export lean_ir_mk_jmp] def mkJmp (j : JoinPointId) (ys : Array Arg) : FnBody := FnBody.jmp j ys
@[export lean_ir_mk_unreachable] def mkUnreachable : Unit → FnBody := fun _ => FnBody.unreachable
abbrev Alt := AltCore FnBody
@[matchPattern] abbrev Alt.ctor := @AltCore.ctor FnBody
@[matchPattern] abbrev Alt.default := @AltCore.default FnBody
instance altInh : Inhabited Alt :=
⟨Alt.default (arbitrary _)⟩
def FnBody.isTerminal : FnBody → Bool
| FnBody.case _ _ _ _ => true
| FnBody.ret _ => true
| FnBody.jmp _ _ => true
| FnBody.unreachable => true
| _ => false
def FnBody.body : FnBody → FnBody
| FnBody.vdecl _ _ _ b => b
| FnBody.jdecl _ _ _ b => b
| FnBody.set _ _ _ b => b
| FnBody.uset _ _ _ b => b
| FnBody.sset _ _ _ _ _ b => b
| FnBody.setTag _ _ b => b
| FnBody.inc _ _ _ _ b => b
| FnBody.dec _ _ _ _ b => b
| FnBody.del _ b => b
| FnBody.mdata _ b => b
| other => other
def FnBody.setBody : FnBody → FnBody → FnBody
| FnBody.vdecl x t v _, b => FnBody.vdecl x t v b
| FnBody.jdecl j xs v _, b => FnBody.jdecl j xs v b
| FnBody.set x i y _, b => FnBody.set x i y b
| FnBody.uset x i y _, b => FnBody.uset x i y b
| FnBody.sset x i o y t _, b => FnBody.sset x i o y t b
| FnBody.setTag x i _, b => FnBody.setTag x i b
| FnBody.inc x n c p _, b => FnBody.inc x n c p b
| FnBody.dec x n c p _, b => FnBody.dec x n c p b
| FnBody.del x _, b => FnBody.del x b
| FnBody.mdata d _, b => FnBody.mdata d b
| other, b => other
@[inline] def FnBody.resetBody (b : FnBody) : FnBody :=
b.setBody FnBody.nil
/- If b is a non terminal, then return a pair `(c, b')` s.t. `b == c <;> b'`,
and c.body == FnBody.nil -/
@[inline] def FnBody.split (b : FnBody) : FnBody × FnBody :=
let b' := b.body;
let c := b.resetBody;
(c, b')
def AltCore.body : Alt → FnBody
| Alt.ctor _ b => b
| Alt.default b => b
def AltCore.setBody : Alt → FnBody → Alt
| Alt.ctor c _, b => Alt.ctor c b
| Alt.default _, b => Alt.default b
@[inline] def AltCore.modifyBody (f : FnBody → FnBody) : AltCore FnBody → Alt
| Alt.ctor c b => Alt.ctor c (f b)
| Alt.default b => Alt.default (f b)
@[inline] def AltCore.mmodifyBody {m : Type → Type} [Monad m] (f : FnBody → m FnBody) : AltCore FnBody → m Alt
| Alt.ctor c b => Alt.ctor c <$> f b
| Alt.default b => Alt.default <$> f b
def Alt.isDefault : Alt → Bool
| Alt.ctor _ _ => false
| Alt.default _ => true
def push (bs : Array FnBody) (b : FnBody) : Array FnBody :=
let b := b.resetBody;
bs.push b
partial def flattenAux : FnBody → Array FnBody → (Array FnBody) × FnBody
| b, r =>
if b.isTerminal then (r, b)
else flattenAux b.body (push r b)
def FnBody.flatten (b : FnBody) : (Array FnBody) × FnBody :=
flattenAux b #[]
partial def reshapeAux : Array FnBody → Nat → FnBody → FnBody
| a, i, b =>
if i == 0 then b
else
let i := i - 1;
let (curr, a) := a.swapAt! i (arbitrary _);
let b := curr.setBody b;
reshapeAux a i b
def reshape (bs : Array FnBody) (term : FnBody) : FnBody :=
reshapeAux bs bs.size term
@[inline] def modifyJPs (bs : Array FnBody) (f : FnBody → FnBody) : Array FnBody :=
bs.map $ fun b => match b with
| FnBody.jdecl j xs v k => FnBody.jdecl j xs (f v) k
| other => other
@[inline] def mmodifyJPs {m : Type → Type} [Monad m] (bs : Array FnBody) (f : FnBody → m FnBody) : m (Array FnBody) :=
bs.mapM $ fun b => match b with
| FnBody.jdecl j xs v k => do v ← f v; pure $ FnBody.jdecl j xs v k
| other => pure other
@[export lean_ir_mk_alt] def mkAlt (n : Name) (cidx : Nat) (size : Nat) (usize : Nat) (ssize : Nat) (b : FnBody) : Alt := Alt.ctor ⟨n, cidx, size, usize, ssize⟩ b
inductive Decl
| fdecl (f : FunId) (xs : Array Param) (ty : IRType) (b : FnBody)
| extern (f : FunId) (xs : Array Param) (ty : IRType) (ext : ExternAttrData)
namespace Decl
instance : Inhabited Decl :=
⟨fdecl (arbitrary _) (arbitrary _) IRType.irrelevant (arbitrary _)⟩
def name : Decl → FunId
| Decl.fdecl f _ _ _ => f
| Decl.extern f _ _ _ => f
def params : Decl → Array Param
| Decl.fdecl _ xs _ _ => xs
| Decl.extern _ xs _ _ => xs
def resultType : Decl → IRType
| Decl.fdecl _ _ t _ => t
| Decl.extern _ _ t _ => t
def isExtern : Decl → Bool
| Decl.extern _ _ _ _ => true
| _ => false
end Decl
@[export lean_ir_mk_decl] def mkDecl (f : FunId) (xs : Array Param) (ty : IRType) (b : FnBody) : Decl := Decl.fdecl f xs ty b
@[export lean_ir_mk_extern_decl] def mkExternDecl (f : FunId) (xs : Array Param) (ty : IRType) (e : ExternAttrData) : Decl :=
Decl.extern f xs ty e
open Std (RBTree RBTree.empty RBMap)
/-- Set of variable and join point names -/
abbrev IndexSet := RBTree Index Index.lt
instance vsetInh : Inhabited IndexSet := ⟨{}⟩
def mkIndexSet (idx : Index) : IndexSet :=
RBTree.empty.insert idx
inductive LocalContextEntry
| param : IRType → LocalContextEntry
| localVar : IRType → Expr → LocalContextEntry
| joinPoint : Array Param → FnBody → LocalContextEntry
abbrev LocalContext := RBMap Index LocalContextEntry Index.lt
def LocalContext.addLocal (ctx : LocalContext) (x : VarId) (t : IRType) (v : Expr) : LocalContext :=
ctx.insert x.idx (LocalContextEntry.localVar t v)
def LocalContext.addJP (ctx : LocalContext) (j : JoinPointId) (xs : Array Param) (b : FnBody) : LocalContext :=
ctx.insert j.idx (LocalContextEntry.joinPoint xs b)
def LocalContext.addParam (ctx : LocalContext) (p : Param) : LocalContext :=
ctx.insert p.x.idx (LocalContextEntry.param p.ty)
def LocalContext.addParams (ctx : LocalContext) (ps : Array Param) : LocalContext :=
ps.foldl LocalContext.addParam ctx
def LocalContext.isJP (ctx : LocalContext) (idx : Index) : Bool :=
match ctx.find? idx with
| some (LocalContextEntry.joinPoint _ _) => true
| other => false
def LocalContext.getJPBody (ctx : LocalContext) (j : JoinPointId) : Option FnBody :=
match ctx.find? j.idx with
| some (LocalContextEntry.joinPoint _ b) => some b
| other => none
def LocalContext.getJPParams (ctx : LocalContext) (j : JoinPointId) : Option (Array Param) :=
match ctx.find? j.idx with
| some (LocalContextEntry.joinPoint ys _) => some ys
| other => none
def LocalContext.isParam (ctx : LocalContext) (idx : Index) : Bool :=
match ctx.find? idx with
| some (LocalContextEntry.param _) => true
| other => false
def LocalContext.isLocalVar (ctx : LocalContext) (idx : Index) : Bool :=
match ctx.find? idx with
| some (LocalContextEntry.localVar _ _) => true
| other => false
def LocalContext.contains (ctx : LocalContext) (idx : Index) : Bool :=
ctx.contains idx
def LocalContext.eraseJoinPointDecl (ctx : LocalContext) (j : JoinPointId) : LocalContext :=
ctx.erase j.idx
def LocalContext.getType (ctx : LocalContext) (x : VarId) : Option IRType :=
match ctx.find? x.idx with
| some (LocalContextEntry.param t) => some t
| some (LocalContextEntry.localVar t _) => some t
| other => none
def LocalContext.getValue (ctx : LocalContext) (x : VarId) : Option Expr :=
match ctx.find? x.idx with
| some (LocalContextEntry.localVar _ v) => some v
| other => none
abbrev IndexRenaming := RBMap Index Index Index.lt
class HasAlphaEqv (α : Type) :=
(aeqv : IndexRenaming → α → α → Bool)
export HasAlphaEqv (aeqv)
def VarId.alphaEqv (ρ : IndexRenaming) (v₁ v₂ : VarId) : Bool :=
match ρ.find? v₁.idx with
| some v => v == v₂.idx
| none => v₁ == v₂
instance VarId.hasAeqv : HasAlphaEqv VarId := ⟨VarId.alphaEqv⟩
def Arg.alphaEqv (ρ : IndexRenaming) : Arg → Arg → Bool
| Arg.var v₁, Arg.var v₂ => aeqv ρ v₁ v₂
| Arg.irrelevant, Arg.irrelevant => true
| _, _ => false
instance Arg.hasAeqv : HasAlphaEqv Arg := ⟨Arg.alphaEqv⟩
def args.alphaEqv (ρ : IndexRenaming) (args₁ args₂ : Array Arg) : Bool :=
Array.isEqv args₁ args₂ (fun a b => aeqv ρ a b)
instance args.hasAeqv : HasAlphaEqv (Array Arg) := ⟨args.alphaEqv⟩
def Expr.alphaEqv (ρ : IndexRenaming) : Expr → Expr → Bool
| Expr.ctor i₁ ys₁, Expr.ctor i₂ ys₂ => i₁ == i₂ && aeqv ρ ys₁ ys₂
| Expr.reset n₁ x₁, Expr.reset n₂ x₂ => n₁ == n₂ && aeqv ρ x₁ x₂
| Expr.reuse x₁ i₁ u₁ ys₁, Expr.reuse x₂ i₂ u₂ ys₂ => aeqv ρ x₁ x₂ && i₁ == i₂ && u₁ == u₂ && aeqv ρ ys₁ ys₂
| Expr.proj i₁ x₁, Expr.proj i₂ x₂ => i₁ == i₂ && aeqv ρ x₁ x₂
| Expr.uproj i₁ x₁, Expr.uproj i₂ x₂ => i₁ == i₂ && aeqv ρ x₁ x₂
| Expr.sproj n₁ o₁ x₁, Expr.sproj n₂ o₂ x₂ => n₁ == n₂ && o₁ == o₂ && aeqv ρ x₁ x₂
| Expr.fap c₁ ys₁, Expr.fap c₂ ys₂ => c₁ == c₂ && aeqv ρ ys₁ ys₂
| Expr.pap c₁ ys₁, Expr.pap c₂ ys₂ => c₁ == c₂ && aeqv ρ ys₁ ys₂
| Expr.ap x₁ ys₁, Expr.ap x₂ ys₂ => aeqv ρ x₁ x₂ && aeqv ρ ys₁ ys₂
| Expr.box ty₁ x₁, Expr.box ty₂ x₂ => ty₁ == ty₂ && aeqv ρ x₁ x₂
| Expr.unbox x₁, Expr.unbox x₂ => aeqv ρ x₁ x₂
| Expr.lit v₁, Expr.lit v₂ => v₁ == v₂
| Expr.isShared x₁, Expr.isShared x₂ => aeqv ρ x₁ x₂
| Expr.isTaggedPtr x₁, Expr.isTaggedPtr x₂ => aeqv ρ x₁ x₂
| _, _ => false
instance Expr.hasAeqv : HasAlphaEqv Expr:= ⟨Expr.alphaEqv⟩
def addVarRename (ρ : IndexRenaming) (x₁ x₂ : Nat) :=
if x₁ == x₂ then ρ else ρ.insert x₁ x₂
def addParamRename (ρ : IndexRenaming) (p₁ p₂ : Param) : Option IndexRenaming :=
if p₁.ty == p₂.ty && p₁.borrow = p₂.borrow then some (addVarRename ρ p₁.x.idx p₂.x.idx)
else none
def addParamsRename (ρ : IndexRenaming) (ps₁ ps₂ : Array Param) : Option IndexRenaming :=
if ps₁.size != ps₂.size then none
else Array.foldl₂ (fun ρ p₁ p₂ => do ρ ← ρ; addParamRename ρ p₁ p₂) (some ρ) ps₁ ps₂
partial def FnBody.alphaEqv : IndexRenaming → FnBody → FnBody → Bool
| ρ, FnBody.vdecl x₁ t₁ v₁ b₁, FnBody.vdecl x₂ t₂ v₂ b₂ => t₁ == t₂ && aeqv ρ v₁ v₂ && FnBody.alphaEqv (addVarRename ρ x₁.idx x₂.idx) b₁ b₂
| ρ, FnBody.jdecl j₁ ys₁ v₁ b₁, FnBody.jdecl j₂ ys₂ v₂ b₂ => match addParamsRename ρ ys₁ ys₂ with
| some ρ' => FnBody.alphaEqv ρ' v₁ v₂ && FnBody.alphaEqv (addVarRename ρ j₁.idx j₂.idx) b₁ b₂
| none => false
| ρ, FnBody.set x₁ i₁ y₁ b₁, FnBody.set x₂ i₂ y₂ b₂ => aeqv ρ x₁ x₂ && i₁ == i₂ && aeqv ρ y₁ y₂ && FnBody.alphaEqv ρ b₁ b₂
| ρ, FnBody.uset x₁ i₁ y₁ b₁, FnBody.uset x₂ i₂ y₂ b₂ => aeqv ρ x₁ x₂ && i₁ == i₂ && aeqv ρ y₁ y₂ && FnBody.alphaEqv ρ b₁ b₂
| ρ, FnBody.sset x₁ i₁ o₁ y₁ t₁ b₁, FnBody.sset x₂ i₂ o₂ y₂ t₂ b₂ =>
aeqv ρ x₁ x₂ && i₁ = i₂ && o₁ = o₂ && aeqv ρ y₁ y₂ && t₁ == t₂ && FnBody.alphaEqv ρ b₁ b₂
| ρ, FnBody.setTag x₁ i₁ b₁, FnBody.setTag x₂ i₂ b₂ => aeqv ρ x₁ x₂ && i₁ == i₂ && FnBody.alphaEqv ρ b₁ b₂
| ρ, FnBody.inc x₁ n₁ c₁ p₁ b₁, FnBody.inc x₂ n₂ c₂ p₂ b₂ => aeqv ρ x₁ x₂ && n₁ == n₂ && c₁ == c₂ && p₁ == p₂ && FnBody.alphaEqv ρ b₁ b₂
| ρ, FnBody.dec x₁ n₁ c₁ p₁ b₁, FnBody.dec x₂ n₂ c₂ p₂ b₂ => aeqv ρ x₁ x₂ && n₁ == n₂ && c₁ == c₂ && p₁ == p₂ && FnBody.alphaEqv ρ b₁ b₂
| ρ, FnBody.del x₁ b₁, FnBody.del x₂ b₂ => aeqv ρ x₁ x₂ && FnBody.alphaEqv ρ b₁ b₂
| ρ, FnBody.mdata m₁ b₁, FnBody.mdata m₂ b₂ => m₁ == m₂ && FnBody.alphaEqv ρ b₁ b₂
| ρ, FnBody.case n₁ x₁ _ alts₁, FnBody.case n₂ x₂ _ alts₂ => n₁ == n₂ && aeqv ρ x₁ x₂ && Array.isEqv alts₁ alts₂ (fun alt₁ alt₂ =>
match alt₁, alt₂ with
| Alt.ctor i₁ b₁, Alt.ctor i₂ b₂ => i₁ == i₂ && FnBody.alphaEqv ρ b₁ b₂
| Alt.default b₁, Alt.default b₂ => FnBody.alphaEqv ρ b₁ b₂
| _, _ => false)
| ρ, FnBody.jmp j₁ ys₁, FnBody.jmp j₂ ys₂ => j₁ == j₂ && aeqv ρ ys₁ ys₂
| ρ, FnBody.ret x₁, FnBody.ret x₂ => aeqv ρ x₁ x₂
| _, FnBody.unreachable, FnBody.unreachable => true
| _, _, _ => false
def FnBody.beq (b₁ b₂ : FnBody) : Bool :=
FnBody.alphaEqv ∅ b₁ b₂
instance FnBody.HasBeq : HasBeq FnBody := ⟨FnBody.beq⟩
abbrev VarIdSet := RBTree VarId (fun x y => x.idx < y.idx)
namespace VarIdSet
instance : Inhabited VarIdSet := ⟨{}⟩
end VarIdSet
def mkIf (x : VarId) (t e : FnBody) : FnBody :=
FnBody.case `Bool x IRType.uint8 #[
Alt.ctor {name := `Bool.false, cidx := 0, size := 0, usize := 0, ssize := 0} e,
Alt.ctor {name := `Bool.true, cidx := 1, size := 0, usize := 0, ssize := 0} t
]
end IR
end Lean
|
d605ac7fc040bcf2316c491c817f3c88565cfa5b | 9be442d9ec2fcf442516ed6e9e1660aa9071b7bd | /stage0/src/Lean/Meta/Match/Basic.lean | 8a7109ce18eb50f74a5e4591c542ca707eddcfd7 | [
"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 | EdAyers/lean4 | 57ac632d6b0789cb91fab2170e8c9e40441221bd | 37ba0df5841bde51dbc2329da81ac23d4f6a4de4 | refs/heads/master | 1,676,463,245,298 | 1,660,619,433,000 | 1,660,619,433,000 | 183,433,437 | 1 | 0 | Apache-2.0 | 1,657,612,672,000 | 1,556,196,574,000 | Lean | UTF-8 | Lean | false | false | 14,436 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Meta.Check
import Lean.Meta.CollectFVars
import Lean.Meta.Match.MatcherInfo
import Lean.Meta.Match.CaseArraySizes
namespace Lean.Meta.Match
def mkNamedPattern (x h p : Expr) : MetaM Expr :=
mkAppM ``namedPattern #[x, p, h]
def isNamedPattern? (e : Expr) : Option Expr :=
let e := e.consumeMData
if e.getAppNumArgs == 4 && e.getAppFn.consumeMData.isConstOf ``namedPattern then
some e
else
none
inductive Pattern : Type where
| inaccessible (e : Expr) : Pattern
| var (fvarId : FVarId) : Pattern
| ctor (ctorName : Name) (us : List Level) (params : List Expr) (fields : List Pattern) : Pattern
| val (e : Expr) : Pattern
| arrayLit (type : Expr) (xs : List Pattern) : Pattern
| as (varId : FVarId) (p : Pattern) (hId : FVarId) : Pattern
deriving Inhabited
namespace Pattern
partial def toMessageData : Pattern → MessageData
| inaccessible e => m!".({e})"
| var varId => mkFVar varId
| ctor ctorName _ _ [] => ctorName
| ctor ctorName _ _ pats => m!"({ctorName}{pats.foldl (fun (msg : MessageData) pat => msg ++ " " ++ toMessageData pat) Format.nil})"
| val e => e
| arrayLit _ pats => m!"#[{MessageData.joinSep (pats.map toMessageData) ", "}]"
| as varId p _ => m!"{mkFVar varId}@{toMessageData p}"
partial def toExpr (p : Pattern) (annotate := false) : MetaM Expr :=
visit p
where
visit (p : Pattern) := do
match p with
| inaccessible e =>
if annotate then
pure (mkInaccessible e)
else
pure e
| var fvarId => pure $ mkFVar fvarId
| val e => pure e
| as fvarId p hId =>
-- TODO
if annotate then
mkNamedPattern (mkFVar fvarId) (mkFVar hId) (← visit p)
else
visit p
| arrayLit type xs =>
let xs ← xs.mapM visit
mkArrayLit type xs
| ctor ctorName us params fields =>
let fields ← fields.mapM visit
pure $ mkAppN (mkConst ctorName us) (params ++ fields).toArray
/-- Apply the free variable substitution `s` to the given pattern -/
partial def applyFVarSubst (s : FVarSubst) : Pattern → Pattern
| inaccessible e => inaccessible $ s.apply e
| ctor n us ps fs => ctor n us (ps.map s.apply) $ fs.map (applyFVarSubst s)
| val e => val $ s.apply e
| arrayLit t xs => arrayLit (s.apply t) $ xs.map (applyFVarSubst s)
| var fvarId => match s.find? fvarId with
| some e => inaccessible e
| none => var fvarId
| as fvarId p hId => match s.find? fvarId with
| none => as fvarId (applyFVarSubst s p) hId
| some _ => applyFVarSubst s p
def replaceFVarId (fvarId : FVarId) (v : Expr) (p : Pattern) : Pattern :=
let s : FVarSubst := {}
p.applyFVarSubst (s.insert fvarId v)
partial def hasExprMVar : Pattern → Bool
| inaccessible e => e.hasExprMVar
| ctor _ _ ps fs => ps.any (·.hasExprMVar) || fs.any hasExprMVar
| val e => e.hasExprMVar
| as _ p _ => hasExprMVar p
| arrayLit t xs => t.hasExprMVar || xs.any hasExprMVar
| _ => false
partial def collectFVars (p : Pattern) : StateRefT CollectFVars.State MetaM Unit := do
match p with
| inaccessible e => e.collectFVars
| ctor _ _ ps fs =>
ps.forM fun p => p.collectFVars
fs.forM collectFVars
| val e => e.collectFVars
| arrayLit t xs => t.collectFVars; xs.forM collectFVars
| as fvarId₁ p fvarId₂ => modify (·.add fvarId₁ |>.add fvarId₂); p.collectFVars
| var fvarId => modify (·.add fvarId)
end Pattern
partial def instantiatePatternMVars : Pattern → MetaM Pattern
| Pattern.inaccessible e => return Pattern.inaccessible (← instantiateMVars e)
| Pattern.val e => return Pattern.val (← instantiateMVars e)
| Pattern.ctor n us ps fields => return Pattern.ctor n us (← ps.mapM instantiateMVars) (← fields.mapM instantiatePatternMVars)
| Pattern.as x p h => return Pattern.as x (← instantiatePatternMVars p) h
| Pattern.arrayLit t xs => return Pattern.arrayLit (← instantiateMVars t) (← xs.mapM instantiatePatternMVars)
| p => return p
structure AltLHS where
ref : Syntax
fvarDecls : List LocalDecl -- Free variables used in the patterns.
patterns : List Pattern -- We use `List Pattern` since we have nary match-expressions.
def AltLHS.collectFVars (altLHS: AltLHS) : StateRefT CollectFVars.State MetaM Unit := do
altLHS.fvarDecls.forM fun fvarDecl => fvarDecl.collectFVars
altLHS.patterns.forM fun p => p.collectFVars
def instantiateAltLHSMVars (altLHS : AltLHS) : MetaM AltLHS :=
return { altLHS with
fvarDecls := (← altLHS.fvarDecls.mapM instantiateLocalDeclMVars),
patterns := (← altLHS.patterns.mapM instantiatePatternMVars)
}
/-- `Match` alternative -/
structure Alt where
/-- `Syntax` object for providing position information -/
ref : Syntax
/--
Orginal alternative index. Alternatives can be split, this index is the original
position of the alternative that generated this one.
-/
idx : Nat
/--
Right-hand-side of the alternative.
-/
rhs : Expr
/--
Alternative pattern variables.
-/
fvarDecls : List LocalDecl
/--
Alternative patterns.
-/
patterns : List Pattern
/--
Pending constraints `lhs ≋ rhs` that need to be solved before the alternative
is considered acceptable. We generate them when processing inaccessible patterns.
Note that `lhs` and `rhs` often have different types.
After we perform additional case analysis, their types become definitionally equal.
-/
cnstrs : List (Expr × Expr)
deriving Inhabited
namespace Alt
partial def toMessageData (alt : Alt) : MetaM MessageData := do
withExistingLocalDecls alt.fvarDecls do
let msg := alt.fvarDecls.map fun d => m!"{d.toExpr}:({d.type})"
let mut msg := m!"{msg} |- {alt.patterns.map Pattern.toMessageData} => {alt.rhs}"
for (lhs, rhs) in alt.cnstrs do
msg := m!"{msg}\n | {lhs} ≋ {rhs}"
addMessageContext msg
def applyFVarSubst (s : FVarSubst) (alt : Alt) : Alt :=
{ alt with
patterns := alt.patterns.map fun p => p.applyFVarSubst s,
fvarDecls := alt.fvarDecls.map fun d => d.applyFVarSubst s,
rhs := alt.rhs.applyFVarSubst s
cnstrs := alt.cnstrs.map fun (lhs, rhs) => (lhs.applyFVarSubst s, rhs.applyFVarSubst s) }
def replaceFVarId (fvarId : FVarId) (v : Expr) (alt : Alt) : Alt :=
{ alt with
patterns := alt.patterns.map fun p => p.replaceFVarId fvarId v,
rhs := alt.rhs.replaceFVarId fvarId v
fvarDecls :=
let decls := alt.fvarDecls.filter fun d => d.fvarId != fvarId
decls.map (·.replaceFVarId fvarId v)
cnstrs := alt.cnstrs.map fun (lhs, rhs) => (lhs.replaceFVarId fvarId v, rhs.replaceFVarId fvarId v) }
/-- Return `true` if `fvarId` is one of the alternative pattern variables -/
def isLocalDecl (fvarId : FVarId) (alt : Alt) : Bool :=
alt.fvarDecls.any fun d => d.fvarId == fvarId
/--
Similar to `checkAndReplaceFVarId`, but ensures type of `v` is definitionally equal to type of `fvarId`.
This extra check is necessary when performing dependent elimination and inaccessible terms have been used.
For example, consider the following code fragment:
```
inductive Vec (α : Type u) : Nat → Type u where
| nil : Vec α 0
| cons {n} (head : α) (tail : Vec α n) : Vec α (n+1)
inductive VecPred {α : Type u} (P : α → Prop) : {n : Nat} → Vec α n → Prop where
| nil : VecPred P Vec.nil
| cons {n : Nat} {head : α} {tail : Vec α n} : P head → VecPred P tail → VecPred P (Vec.cons head tail)
theorem ex {α : Type u} (P : α → Prop) : {n : Nat} → (v : Vec α (n+1)) → VecPred P v → Exists P
| _, Vec.cons head _, VecPred.cons h (w : VecPred P Vec.nil) => ⟨head, h⟩
```
Recall that `_` in a pattern can be elaborated into pattern variable or an inaccessible term.
The elaborator uses an inaccessible term when typing constraints restrict its value.
Thus, in the example above, the `_` at `Vec.cons head _` becomes the inaccessible pattern `.(Vec.nil)`
because the type ascription `(w : VecPred P Vec.nil)` propagates typing constraints that restrict its value to be `Vec.nil`.
After elaboration the alternative becomes:
```
| .(0), @Vec.cons .(α) .(0) head .(Vec.nil), @VecPred.cons .(α) .(P) .(0) .(head) .(Vec.nil) h w => ⟨head, h⟩
```
where
```
(head : α), (h: P head), (w : VecPred P Vec.nil)
```
Then, when we process this alternative in this module, the following check will detect that
`w` has type `VecPred P Vec.nil`, when it is supposed to have type `VecPred P tail`.
Note that if we had written
```
theorem ex {α : Type u} (P : α → Prop) : {n : Nat} → (v : Vec α (n+1)) → VecPred P v → Exists P
| _, Vec.cons head Vec.nil, VecPred.cons h (w : VecPred P Vec.nil) => ⟨head, h⟩
```
we would get the easier to digest error message
```
missing cases:
_, (Vec.cons _ _ (Vec.cons _ _ _)), _
```
-/
def checkAndReplaceFVarId (fvarId : FVarId) (v : Expr) (alt : Alt) : MetaM Alt := do
match alt.fvarDecls.find? fun (fvarDecl : LocalDecl) => fvarDecl.fvarId == fvarId with
| none => throwErrorAt alt.ref "unknown free pattern variable"
| some fvarDecl => do
let vType ← inferType v
unless (← isDefEqGuarded fvarDecl.type vType) do
withExistingLocalDecls alt.fvarDecls do
let (expectedType, givenType) ← addPPExplicitToExposeDiff vType fvarDecl.type
throwErrorAt alt.ref "type mismatch during dependent match-elimination at pattern variable '{mkFVar fvarDecl.fvarId}' with type{indentExpr givenType}\nexpected type{indentExpr expectedType}"
return replaceFVarId fvarId v alt
end Alt
inductive Example where
| var : FVarId → Example
| underscore : Example
| ctor : Name → List Example → Example
| val : Expr → Example
| arrayLit : List Example → Example
namespace Example
partial def replaceFVarId (fvarId : FVarId) (ex : Example) : Example → Example
| var x => if x == fvarId then ex else var x
| ctor n exs => ctor n $ exs.map (replaceFVarId fvarId ex)
| arrayLit exs => arrayLit $ exs.map (replaceFVarId fvarId ex)
| ex => ex
partial def applyFVarSubst (s : FVarSubst) : Example → Example
| var fvarId =>
match s.get fvarId with
| Expr.fvar fvarId' => var fvarId'
| _ => underscore
| ctor n exs => ctor n $ exs.map (applyFVarSubst s)
| arrayLit exs => arrayLit $ exs.map (applyFVarSubst s)
| ex => ex
partial def varsToUnderscore : Example → Example
| var _ => underscore
| ctor n exs => ctor n $ exs.map varsToUnderscore
| arrayLit exs => arrayLit $ exs.map varsToUnderscore
| ex => ex
partial def toMessageData : Example → MessageData
| var fvarId => mkFVar fvarId
| ctor ctorName [] => mkConst ctorName
| ctor ctorName exs => m!"({mkConst ctorName}{exs.foldl (fun msg pat => m!"{msg} {toMessageData pat}") Format.nil})"
| arrayLit exs => "#" ++ MessageData.ofList (exs.map toMessageData)
| val e => e
| underscore => "_"
end Example
def examplesToMessageData (cex : List Example) : MessageData :=
MessageData.joinSep (cex.map (Example.toMessageData ∘ Example.varsToUnderscore)) ", "
structure Problem where
mvarId : MVarId
vars : List Expr
alts : List Alt
examples : List Example
deriving Inhabited
def withGoalOf {α} (p : Problem) (x : MetaM α) : MetaM α :=
p.mvarId.withContext x
def Problem.toMessageData (p : Problem) : MetaM MessageData :=
withGoalOf p do
let alts ← p.alts.mapM Alt.toMessageData
let vars ← p.vars.mapM fun x => do let xType ← inferType x; pure m!"{x}:({xType})"
return m!"remaining variables: {vars}\nalternatives:{indentD (MessageData.joinSep alts Format.line)}\nexamples:{examplesToMessageData p.examples}\n"
abbrev CounterExample := List Example
def counterExampleToMessageData (cex : CounterExample) : MessageData :=
examplesToMessageData cex
def counterExamplesToMessageData (cexs : List CounterExample) : MessageData :=
MessageData.joinSep (cexs.map counterExampleToMessageData) Format.line
structure MatcherResult where
matcher : Expr -- The matcher. It is not just `Expr.const matcherName` because the type of the major premises may contain free variables.
counterExamples : List CounterExample
unusedAltIdxs : List Nat
addMatcher : MetaM Unit
/--
Convert a expression occurring as the argument of a `match` motive application back into a `Pattern`
For example, we can use this method to convert `x::y::xs` at
```
...
(motive : List Nat → Sort u_1) (xs : List Nat) (h_1 : (x y : Nat) → (xs : List Nat) → motive (x :: y :: xs))
...
```
into a pattern object
-/
partial def toPattern (e : Expr) : MetaM Pattern := do
match inaccessible? e with
| some t => return Pattern.inaccessible t
| none =>
match e.arrayLit? with
| some (α, lits) =>
return Pattern.arrayLit α (← lits.mapM toPattern)
| none =>
if let some e := isNamedPattern? e then
let p ← toPattern <| e.getArg! 2
match e.getArg! 1, e.getArg! 3 with
| Expr.fvar x, Expr.fvar h => return Pattern.as x p h
| _, _ => throwError "unexpected occurrence of auxiliary declaration 'namedPattern'"
else if isMatchValue e then
return Pattern.val e
else if e.isFVar then
return Pattern.var e.fvarId!
else
let newE ← whnf e
if newE != e then
toPattern newE
else matchConstCtor e.getAppFn (fun _ => throwError "unexpected pattern{indentExpr e}") fun v us => do
let args := e.getAppArgs
unless args.size == v.numParams + v.numFields do
throwError "unexpected pattern{indentExpr e}"
let params := args.extract 0 v.numParams
let fields := args.extract v.numParams args.size
let fields ← fields.mapM toPattern
return Pattern.ctor v.name us params.toList fields.toList
end Lean.Meta.Match
|
abb314b2c12adb6af08d27c9faadbcefacf5ef07 | 3268ab3a126f0fef71459fbf170dc38efe5d0506 | /algebra/left_module.hlean | 83ee6eaa0b258470b9b7c18749b6de381d6a8516 | [
"Apache-2.0"
] | permissive | soraismus/Spectral | f043fed1a4e02ddfeba531769b2980eb817471f4 | 32512bf47db3a1b932856e7ed7c7830b1fc07ef0 | refs/heads/master | 1,585,628,705,579 | 1,538,609,948,000 | 1,538,609,974,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 22,818 | hlean | /-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Floris van Doorn
Modules prod vector spaces over a ring.
(We use "left_module," which is more precise, because "module" is a keyword.)
-/
import algebra.field ..move_to_lib .exactness algebra.group_power
open is_trunc pointed function sigma eq algebra prod is_equiv equiv group
structure has_scalar [class] (F V : Type) :=
(smul : F → V → V)
infixl ` • `:73 := has_scalar.smul
/- modules over a ring -/
namespace left_module
structure left_module (R M : Type) [ringR : ring R] extends has_scalar R M, ab_group M renaming
mul → add mul_assoc → add_assoc one → zero one_mul → zero_add mul_one → add_zero inv → neg
mul_left_inv → add_left_inv mul_comm → add_comm :=
(smul_left_distrib : Π (r : R) (x y : M), smul r (add x y) = (add (smul r x) (smul r y)))
(smul_right_distrib : Π (r s : R) (x : M), smul (ring.add _ r s) x = (add (smul r x) (smul s x)))
(mul_smul : Π r s x, smul (mul r s) x = smul r (smul s x))
(one_smul : Π x, smul one x = x)
/- we make it a class now (and not as part of the structure) to avoid
left_module.to_ab_group to be an instance -/
attribute left_module [class]
definition add_ab_group_of_left_module [reducible] [trans_instance] (R M : Type) [K : ring R]
[H : left_module R M] : add_ab_group M :=
@left_module.to_ab_group R M K H
definition has_scalar_of_left_module [reducible] [trans_instance] (R M : Type) [K : ring R]
[H : left_module R M] : has_scalar R M :=
@left_module.to_has_scalar R M K H
section left_module
variables {R M : Type}
variable [ringR : ring R]
variable [moduleRM : left_module R M]
include ringR moduleRM
-- Note: the anonymous include does not work in the propositions below.
proposition smul_left_distrib (a : R) (u v : M) : a • (u + v) = a • u + a • v :=
!left_module.smul_left_distrib
proposition smul_right_distrib (a b : R) (u : M) : (a + b) • u = a • u + b • u :=
!left_module.smul_right_distrib
proposition mul_smul (a : R) (b : R) (u : M) : (a * b) • u = a • (b • u) :=
!left_module.mul_smul
proposition one_smul (u : M) : (1 : R) • u = u := !left_module.one_smul
proposition zero_smul (u : M) : (0 : R) • u = 0 :=
have (0 : R) • u + 0 • u = 0 • u + 0, by rewrite [-smul_right_distrib, *add_zero],
!add.left_cancel this
proposition smul_zero (a : R) : a • (0 : M) = 0 :=
have a • (0:M) + a • 0 = a • 0 + 0, by rewrite [-smul_left_distrib, *add_zero],
!add.left_cancel this
proposition neg_smul (a : R) (u : M) : (-a) • u = - (a • u) :=
eq_neg_of_add_eq_zero (by rewrite [-smul_right_distrib, add.left_inv, zero_smul])
proposition neg_one_smul (u : M) : -(1 : R) • u = -u :=
by rewrite [neg_smul, one_smul]
proposition smul_neg (a : R) (u : M) : a • (-u) = -(a • u) :=
by rewrite [-neg_one_smul, -mul_smul, mul_neg_one_eq_neg, neg_smul]
proposition smul_sub_left_distrib (a : R) (u v : M) : a • (u - v) = a • u - a • v :=
by rewrite [sub_eq_add_neg, smul_left_distrib, smul_neg]
proposition sub_smul_right_distrib (a b : R) (v : M) : (a - b) • v = a • v - b • v :=
by rewrite [sub_eq_add_neg, smul_right_distrib, neg_smul]
end left_module
/- vector spaces -/
structure vector_space [class] (F V : Type) [fieldF : field F]
extends left_module F V
/- homomorphisms -/
definition is_smul_hom [class] (R : Type) {M₁ M₂ : Type} [has_scalar R M₁] [has_scalar R M₂]
(f : M₁ → M₂) : Type :=
∀ r : R, ∀ a : M₁, f (r • a) = r • f a
definition is_prop_is_smul_hom [instance] (R : Type) {M₁ M₂ : Type} [is_set M₂]
[has_scalar R M₁] [has_scalar R M₂] (f : M₁ → M₂) : is_prop (is_smul_hom R f) :=
begin unfold is_smul_hom, apply _ end
definition respect_smul (R : Type) {M₁ M₂ : Type} [has_scalar R M₁] [has_scalar R M₂]
(f : M₁ → M₂) [H : is_smul_hom R f] :
∀ r : R, ∀ a : M₁, f (r • a) = r • f a :=
H
definition is_module_hom [class] (R : Type) {M₁ M₂ : Type}
[has_scalar R M₁] [has_scalar R M₂] [add_group M₁] [add_group M₂]
(f : M₁ → M₂) :=
is_add_hom f × is_smul_hom R f
definition is_add_hom_of_is_module_hom [instance] (R : Type) {M₁ M₂ : Type}
[has_scalar R M₁] [has_scalar R M₂] [add_group M₁] [add_group M₂]
(f : M₁ → M₂) [H : is_module_hom R f] : is_add_hom f :=
prod.pr1 H
definition is_smul_hom_of_is_module_hom [instance] {R : Type} {M₁ M₂ : Type}
[has_scalar R M₁] [has_scalar R M₂] [add_group M₁] [add_group M₂]
(f : M₁ → M₂) [H : is_module_hom R f] : is_smul_hom R f :=
prod.pr2 H
-- Why do we have to give the instance explicitly?
definition is_prop_is_module_hom [instance] (R : Type) {M₁ M₂ : Type}
[has_scalar R M₁] [has_scalar R M₂] [add_group M₁] [add_group M₂]
(f : M₁ → M₂) : is_prop (is_module_hom R f) :=
have h₁ : is_prop (is_add_hom f), from is_prop_is_add_hom f,
begin unfold is_module_hom, apply _ end
section module_hom
variables {R : Type} {M₁ M₂ M₃ : Type}
variables [has_scalar R M₁] [has_scalar R M₂] [has_scalar R M₃]
variables [add_group M₁] [add_group M₂] [add_group M₃]
variables (g : M₂ → M₃) (f : M₁ → M₂) [is_module_hom R g] [is_module_hom R f]
proposition is_module_hom_id : is_module_hom R (@id M₁) :=
pair (λ a₁ a₂, rfl) (λ r a, rfl)
proposition is_module_hom_comp : is_module_hom R (g ∘ f) :=
pair
(take a₁ a₂, begin esimp, rewrite [respect_add f, respect_add g] end)
(take r a, by esimp; rewrite [respect_smul R f, respect_smul R g])
proposition respect_smul_add_smul (a b : R) (u v : M₁) : f (a • u + b • v) = a • f u + b • f v :=
by rewrite [respect_add f, +respect_smul R f]
end module_hom
section hom_constant
variables {R : Type} {M₁ M₂ : Type}
variables [ring R] [has_scalar R M₁] [add_group M₁] [left_module R M₂]
proposition is_module_hom_constant : is_module_hom R (const M₁ (0 : M₂)) :=
(λm₁ m₂, !add_zero⁻¹, λr m, (smul_zero r)⁻¹ᵖ)
end hom_constant
structure LeftModule (R : Ring) :=
(carrier : Type) (struct : left_module R carrier)
attribute LeftModule.struct [instance]
section
local attribute LeftModule.carrier [coercion]
definition AddAbGroup_of_LeftModule [coercion] {R : Ring} (M : LeftModule R) : AddAbGroup :=
AddAbGroup.mk M (LeftModule.struct M)
end
definition LeftModule.struct2 [instance] {R : Ring} (M : LeftModule R) : left_module R M :=
LeftModule.struct M
-- definition LeftModule.struct3 [instance] {R : Ring} (M : LeftModule R) :
-- left_module R (AddAbGroup_of_LeftModule M) :=
-- _
definition pointed_LeftModule_carrier [instance] {R : Ring} (M : LeftModule R) :
pointed (LeftModule.carrier M) :=
pointed.mk zero
definition pSet_of_LeftModule {R : Ring} (M : LeftModule R) : Set* :=
pSet.mk' (LeftModule.carrier M)
definition left_module_AddAbGroup_of_LeftModule [instance] {R : Ring} (M : LeftModule R) :
left_module R (AddAbGroup_of_LeftModule M) :=
LeftModule.struct M
definition left_module_of_ab_group {G : Type} [gG : add_ab_group G] {R : Type} [ring R]
(smul : R → G → G)
(h1 : Π (r : R) (x y : G), smul r (x + y) = (smul r x + smul r y))
(h2 : Π (r s : R) (x : G), smul (r + s) x = (smul r x + smul s x))
(h3 : Π r s x, smul (r * s) x = smul r (smul s x))
(h4 : Π x, smul 1 x = x) : left_module R G :=
left_module.mk smul _ add add.assoc 0 zero_add add_zero neg add.left_inv add.comm h1 h2 h3 h4
definition LeftModule_of_AddAbGroup {R : Ring} (G : AddAbGroup) (smul : R → G → G)
(h1 h2 h3 h4) : LeftModule R :=
LeftModule.mk G (left_module_of_ab_group smul h1 h2 h3 h4)
open unit
definition trivial_LeftModule [constructor] (R : Ring) : LeftModule R :=
LeftModule_of_AddAbGroup trivial_ab_group (λr u, star)
(λr u₁ u₂, idp) (λr₁ r₂ u, idp) (λr₁ r₂ u, idp) unit.eta
section
variables {R : Ring} {M M₁ M₂ M₃ : LeftModule R}
definition smul_homomorphism [constructor] (M : LeftModule R) (r : R) : M →a M :=
add_homomorphism.mk (λg, r • g) (smul_left_distrib r)
proposition to_smul_left_distrib (a : R) (u v : M) : a • (u + v) = a • u + a • v :=
!smul_left_distrib
proposition to_smul_right_distrib (a b : R) (u : M) : (a + b) • u = a • u + b • u :=
!smul_right_distrib
proposition to_mul_smul (a : R) (b : R) (u : M) : (a * b) • u = a • (b • u) :=
!mul_smul
proposition to_one_smul (u : M) : (1 : R) • u = u := !one_smul
structure homomorphism (M₁ M₂ : LeftModule R) : Type :=
(fn : LeftModule.carrier M₁ → LeftModule.carrier M₂)
(p : is_module_hom R fn)
infix ` →lm `:55 := homomorphism
definition homomorphism_fn [unfold 4] [coercion] := @homomorphism.fn
definition is_module_hom_of_homomorphism [unfold 4] [instance] [priority 900]
{M₁ M₂ : LeftModule R} (φ : M₁ →lm M₂) : is_module_hom R φ :=
homomorphism.p φ
section
variable (φ : M₁ →lm M₂)
definition to_respect_add (x y : M₁) : φ (x + y) = φ x + φ y :=
respect_add φ x y
definition to_respect_smul (a : R) (x : M₁) : φ (a • x) = a • (φ x) :=
respect_smul R φ a x
definition to_respect_sub (x y : M₁) : φ (x - y) = φ x - φ y :=
respect_sub φ x y
definition is_embedding_of_homomorphism /- φ -/ (H : Π{x}, φ x = 0 → x = 0) : is_embedding φ :=
is_embedding_of_is_add_hom φ @H
variables (M₁ M₂)
definition is_set_homomorphism [instance] : is_set (M₁ →lm M₂) :=
begin
have H : M₁ →lm M₂ ≃ Σ(f : LeftModule.carrier M₁ → LeftModule.carrier M₂),
is_module_hom (Ring.carrier R) f,
begin
fapply equiv.MK,
{ intro φ, induction φ, constructor, exact p},
{ intro v, induction v with f H, constructor, exact H},
{ intro v, induction v, reflexivity},
{ intro φ, induction φ, reflexivity}
end,
have ∀ f : LeftModule.carrier M₁ → LeftModule.carrier M₂,
is_set (is_module_hom (Ring.carrier R) f), from _,
exact is_trunc_equiv_closed_rev _ H _
end
variables {M₁ M₂}
definition pmap_of_homomorphism [constructor] /- φ -/ :
pSet_of_LeftModule M₁ →* pSet_of_LeftModule M₂ :=
have H : φ 0 = 0, from respect_zero φ,
pmap.mk φ begin esimp, exact H end
definition homomorphism_change_fun [constructor]
(φ : M₁ →lm M₂) (f : M₁ → M₂) (p : φ ~ f) : M₁ →lm M₂ :=
homomorphism.mk f
(prod.mk
(λx₁ x₂, (p (x₁ + x₂))⁻¹ ⬝ to_respect_add φ x₁ x₂ ⬝ ap011 _ (p x₁) (p x₂))
(λ a x, (p (a • x))⁻¹ ⬝ to_respect_smul φ a x ⬝ ap01 _ (p x)))
definition homomorphism_eq (φ₁ φ₂ : M₁ →lm M₂) (p : φ₁ ~ φ₂) : φ₁ = φ₂ :=
begin
induction φ₁ with φ₁ q₁, induction φ₂ with φ₂ q₂, esimp at p, induction p,
exact ap (homomorphism.mk φ₁) !is_prop.elim
end
end
section
definition homomorphism.mk' [constructor] (φ : M₁ → M₂)
(p : Π(g₁ g₂ : M₁), φ (g₁ + g₂) = φ g₁ + φ g₂)
(q : Π(r : R) x, φ (r • x) = r • φ x) : M₁ →lm M₂ :=
homomorphism.mk φ (p, q)
definition to_respect_zero (φ : M₁ →lm M₂) : φ 0 = 0 :=
respect_zero φ
definition homomorphism_compose [reducible] [constructor] (f' : M₂ →lm M₃) (f : M₁ →lm M₂) :
M₁ →lm M₃ :=
homomorphism.mk (f' ∘ f) !is_module_hom_comp
variable (M)
definition homomorphism_id [reducible] [constructor] [refl] : M →lm M :=
homomorphism.mk (@id M) !is_module_hom_id
variable {M}
abbreviation lmid [constructor] := homomorphism_id M
infixr ` ∘lm `:75 := homomorphism_compose
definition lm_constant [constructor] (M₁ M₂ : LeftModule R) : M₁ →lm M₂ :=
homomorphism.mk (const M₁ 0) !is_module_hom_constant
structure isomorphism (M₁ M₂ : LeftModule R) :=
(to_hom : M₁ →lm M₂)
(is_equiv_to_hom : is_equiv to_hom)
infix ` ≃lm `:25 := isomorphism
attribute isomorphism.to_hom [coercion]
attribute isomorphism.is_equiv_to_hom [instance]
attribute isomorphism._trans_of_to_hom [unfold 4]
definition equiv_of_isomorphism [constructor] (φ : M₁ ≃lm M₂) : M₁ ≃ M₂ :=
equiv.mk φ !isomorphism.is_equiv_to_hom
section
local attribute pSet_of_LeftModule [coercion]
definition pequiv_of_isomorphism [constructor] (φ : M₁ ≃lm M₂) : M₁ ≃* M₂ :=
pequiv_of_equiv (equiv_of_isomorphism φ) (to_respect_zero φ)
end
definition isomorphism_of_equiv [constructor] (φ : M₁ ≃ M₂)
(p : Π(g₁ g₂ : M₁), φ (g₁ + g₂) = φ g₁ + φ g₂)
(q : Πr x, φ (r • x) = r • φ x) : M₁ ≃lm M₂ :=
isomorphism.mk (homomorphism.mk φ (p, q)) !to_is_equiv
definition isomorphism_of_eq [constructor] {M₁ M₂ : LeftModule R} (p : M₁ = M₂ :> LeftModule R)
: M₁ ≃lm M₂ :=
isomorphism_of_equiv (equiv_of_eq (ap LeftModule.carrier p))
begin intros, induction p, reflexivity end
begin intros, induction p, reflexivity end
-- definition pequiv_of_isomorphism_of_eq {M₁ M₂ : LeftModule R} (p : M₁ = M₂ :> LeftModule R) :
-- pequiv_of_isomorphism (isomorphism_of_eq p) = pequiv_of_eq (ap pType_of_LeftModule p) :=
-- begin
-- induction p,
-- apply pequiv_eq,
-- fapply pmap_eq,
-- { intro g, reflexivity},
-- { apply is_prop.elim}
-- end
definition to_lminv [constructor] (φ : M₁ ≃lm M₂) : M₂ →lm M₁ :=
homomorphism.mk φ⁻¹
abstract begin
split,
intro g₁ g₂, apply inj' φ,
rewrite [respect_add φ, +right_inv φ],
intro r x, apply inj' φ,
rewrite [to_respect_smul φ, +right_inv φ],
end end
variable (M)
definition isomorphism.refl [refl] [constructor] : M ≃lm M :=
isomorphism.mk lmid !is_equiv_id
variable {M}
definition isomorphism.rfl [refl] [constructor] : M ≃lm M := isomorphism.refl M
definition isomorphism.symm [symm] [constructor] (φ : M₁ ≃lm M₂) : M₂ ≃lm M₁ :=
isomorphism.mk (to_lminv φ) !is_equiv_inv
definition isomorphism.trans [trans] [constructor] (φ : M₁ ≃lm M₂) (ψ : M₂ ≃lm M₃) : M₁ ≃lm M₃ :=
isomorphism.mk (ψ ∘lm φ) (is_equiv_compose ψ φ _ _)
definition isomorphism.eq_trans [trans] [constructor]
{M₁ M₂ : LeftModule R} {M₃ : LeftModule R} (φ : M₁ = M₂) (ψ : M₂ ≃lm M₃) : M₁ ≃lm M₃ :=
proof isomorphism.trans (isomorphism_of_eq φ) ψ qed
definition isomorphism.trans_eq [trans] [constructor]
{M₁ : LeftModule R} {M₂ M₃ : LeftModule R} (φ : M₁ ≃lm M₂) (ψ : M₂ = M₃) : M₁ ≃lm M₃ :=
isomorphism.trans φ (isomorphism_of_eq ψ)
postfix `⁻¹ˡᵐ`:(max + 1) := isomorphism.symm
infixl ` ⬝lm `:75 := isomorphism.trans
infixl ` ⬝lmp `:75 := isomorphism.trans_eq
infixl ` ⬝plm `:75 := isomorphism.eq_trans
definition homomorphism_of_eq [constructor] {M₁ M₂ : LeftModule R} (p : M₁ = M₂ :> LeftModule R)
: M₁ →lm M₂ :=
isomorphism_of_eq p
definition group_homomorphism_of_lm_homomorphism [constructor] {M₁ M₂ : LeftModule R}
(φ : M₁ →lm M₂) : M₁ →a M₂ :=
add_homomorphism.mk φ (to_respect_add φ)
definition lm_homomorphism_of_group_homomorphism [constructor] {M₁ M₂ : LeftModule R}
(φ : M₁ →a M₂) (h : Π(r : R) g, φ (r • g) = r • φ g) : M₁ →lm M₂ :=
homomorphism.mk' φ (group.to_respect_add φ) h
definition group_isomorphism_of_lm_isomorphism [constructor] {M₁ M₂ : LeftModule R}
(φ : M₁ ≃lm M₂) : AddGroup_of_AddAbGroup M₁ ≃g AddGroup_of_AddAbGroup M₂ :=
group.isomorphism.mk (group_homomorphism_of_lm_homomorphism φ) (isomorphism.is_equiv_to_hom φ)
definition lm_isomorphism_of_group_isomorphism [constructor] {M₁ M₂ : LeftModule R}
(φ : AddGroup_of_AddAbGroup M₁ ≃g AddGroup_of_AddAbGroup M₂)
(h : Π(r : R) g, φ (r • g) = r • φ g) : M₁ ≃lm M₂ :=
isomorphism.mk (lm_homomorphism_of_group_homomorphism φ h) (group.isomorphism.is_equiv_to_hom φ)
section
local attribute pSet_of_LeftModule [coercion]
definition is_exact_mod (f : M₁ →lm M₂) (f' : M₂ →lm M₃) : Type :=
@is_exact M₁ M₂ M₃ (homomorphism_fn f) (homomorphism_fn f')
definition is_exact_mod.mk {f : M₁ →lm M₂} {f' : M₂ →lm M₃}
(h₁ : Πm, f' (f m) = 0) (h₂ : Πm, f' m = 0 → image f m) : is_exact_mod f f' :=
is_exact.mk h₁ h₂
structure short_exact_mod (A B C : LeftModule R) :=
(f : A →lm B)
(g : B →lm C)
(h : @is_short_exact A B C f g)
local abbreviation g_of_lm := @group_homomorphism_of_lm_homomorphism
definition short_exact_mod_of_is_exact {X A B C Y : LeftModule R}
(k : X →lm A) (f : A →lm B) (g : B →lm C) (l : C →lm Y)
(hX : is_contr X) (hY : is_contr Y)
(kf : is_exact_mod k f) (fg : is_exact_mod f g) (gl : is_exact_mod g l) :
short_exact_mod A B C :=
short_exact_mod.mk f g
(is_short_exact_of_is_exact (g_of_lm k) (g_of_lm f) (g_of_lm g) (g_of_lm l) hX hY kf fg gl)
definition short_exact_mod_isomorphism {A B A' B' C C' : LeftModule R}
(eA : A ≃lm A') (eB : B ≃lm B') (eC : C ≃lm C')
(H : short_exact_mod A' B' C') : short_exact_mod A B C :=
short_exact_mod.mk (eB⁻¹ˡᵐ ∘lm short_exact_mod.f H ∘lm eA) (eC⁻¹ˡᵐ ∘lm short_exact_mod.g H ∘lm eB)
(is_short_exact_equiv _ _
(equiv_of_isomorphism eA) (equiv_of_isomorphism eB) (pequiv_of_isomorphism eC)
(λa, to_right_inv (equiv_of_isomorphism eB) _) (λb, to_right_inv (equiv_of_isomorphism eC) _)
(short_exact_mod.h H))
definition is_contr_middle_of_short_exact_mod {A B C : LeftModule R} (H : short_exact_mod A B C)
(HA : is_contr A) (HC : is_contr C) : is_contr B :=
is_contr_middle_of_is_exact (is_exact_of_is_short_exact (short_exact_mod.h H))
definition is_contr_right_of_short_exact_mod {A B C : LeftModule R} (H : short_exact_mod A B C)
(HB : is_contr B) : is_contr C :=
is_contr_right_of_is_short_exact (short_exact_mod.h H) _ _
definition is_contr_left_of_short_exact_mod {A B C : LeftModule R} (H : short_exact_mod A B C)
(HB : is_contr B) : is_contr A :=
is_contr_left_of_is_short_exact (short_exact_mod.h H) _ pt
definition isomorphism_of_is_contr_left {A B C : LeftModule R} (H : short_exact_mod A B C)
(HA : is_contr A) : B ≃lm C :=
isomorphism.mk (short_exact_mod.g H)
begin
apply @is_equiv_right_of_is_short_exact _ _ _
(group_homomorphism_of_lm_homomorphism (short_exact_mod.f H))
(group_homomorphism_of_lm_homomorphism (short_exact_mod.g H)),
rexact short_exact_mod.h H, exact HA,
end
definition isomorphism_of_is_contr_right {A B C : LeftModule R} (H : short_exact_mod A B C)
(HC : is_contr C) : A ≃lm B :=
isomorphism.mk (short_exact_mod.f H)
begin
apply @is_equiv_left_of_is_short_exact _ _ _
(group_homomorphism_of_lm_homomorphism (short_exact_mod.f H))
(group_homomorphism_of_lm_homomorphism (short_exact_mod.g H)),
rexact short_exact_mod.h H, exact HC,
end
end
end
end
/- we say that an left module D is built from the sequence E if
D is a "twisted sum" of the E, and E has only finitely many nontrivial values -/
open nat
structure is_built_from.{u v w} {R : Ring} (D : LeftModule.{u v} R)
(E : ℕ → LeftModule.{u w} R) : Type.{max u (v+1) w} :=
(part : ℕ → LeftModule.{u v} R)
(ses : Πn, short_exact_mod (E n) (part n) (part (n+1)))
(e0 : part 0 ≃lm D)
(n₀ : ℕ)
(HD' : Π(s : ℕ), n₀ ≤ s → is_contr (part s))
open is_built_from
universe variables u v w
variables {R : Ring.{u}} {D D' : LeftModule.{u v} R} {E E' : ℕ → LeftModule.{u w} R}
definition is_built_from_shift (H : is_built_from D E) :
is_built_from (part H 1) (λn, E (n+1)) :=
is_built_from.mk (λn, part H (n+1)) (λn, ses H (n+1)) isomorphism.rfl (pred (n₀ H))
(λs Hle, HD' H _ (le_succ_of_pred_le Hle))
definition isomorphism_of_is_contr_part (H : is_built_from D E) (n : ℕ) (HE : is_contr (E n)) :
part H n ≃lm part H (n+1) :=
isomorphism_of_is_contr_left (ses H n) HE
definition is_contr_submodules (H : is_built_from D E) (HD : is_contr D) (n : ℕ) :
is_contr (part H n) :=
begin
induction n with n IH,
{ exact is_trunc_equiv_closed_rev -2 (equiv_of_isomorphism (e0 H)) _ },
{ exact is_contr_right_of_short_exact_mod (ses H n) IH }
end
definition is_contr_quotients (H : is_built_from D E) (HD : is_contr D) (n : ℕ) :
is_contr (E n) :=
begin
apply is_contr_left_of_short_exact_mod (ses H n),
exact is_contr_submodules H HD n
end
definition is_contr_total (H : is_built_from D E) (HE : Πn, is_contr (E n)) : is_contr D :=
have is_contr (part H 0), from
nat.rec_down (λn, is_contr (part H n)) _ (HD' H _ !le.refl)
(λn H2, is_contr_middle_of_short_exact_mod (ses H n) (HE n) H2),
is_contr_equiv_closed (equiv_of_isomorphism (e0 H)) _
definition is_built_from_isomorphism (e : D ≃lm D') (f : Πn, E n ≃lm E' n)
(H : is_built_from D E) : is_built_from D' E' :=
⦃is_built_from, H, e0 := e0 H ⬝lm e,
ses := λn, short_exact_mod_isomorphism (f n)⁻¹ˡᵐ isomorphism.rfl isomorphism.rfl (ses H n)⦄
definition is_built_from_isomorphism_left (e : D ≃lm D') (H : is_built_from D E) :
is_built_from D' E :=
⦃is_built_from, H, e0 := e0 H ⬝lm e⦄
definition isomorphism_zero_of_is_built_from (H : is_built_from D E) (p : n₀ H = 1) : E 0 ≃lm D :=
isomorphism_of_is_contr_right (ses H 0) (HD' H 1 (le_of_eq p)) ⬝lm e0 H
section int
open int
definition left_module_int_of_ab_group [constructor] (A : Type) [add_ab_group A] : left_module rℤ A :=
left_module_of_ab_group imul imul_add add_imul mul_imul one_imul
definition LeftModule_int_of_AbGroup [constructor] (A : AddAbGroup) : LeftModule rℤ :=
LeftModule.mk A (left_module_int_of_ab_group A)
definition lm_hom_int.mk [constructor] {A B : AbGroup} (φ : A →g B) :
LeftModule_int_of_AbGroup A →lm LeftModule_int_of_AbGroup B :=
lm_homomorphism_of_group_homomorphism φ (to_respect_imul φ)
definition lm_iso_int.mk [constructor] {A B : AbGroup} (φ : A ≃g B) :
LeftModule_int_of_AbGroup A ≃lm LeftModule_int_of_AbGroup B :=
isomorphism.mk (lm_hom_int.mk φ) (group.isomorphism.is_equiv_to_hom φ)
definition group_isomorphism_of_lm_isomorphism_int [constructor] {A B : AbGroup}
(φ : LeftModule_int_of_AbGroup A ≃lm LeftModule_int_of_AbGroup B) : A ≃g B :=
group_isomorphism_of_lm_isomorphism φ
end int
end left_module
|
7bfc15237dee80244c795577cca001b4fe52523d | 302c785c90d40ad3d6be43d33bc6a558354cc2cf | /src/ring_theory/finiteness.lean | 01258563d0d99e5b73f50be13f5a372bd2ce98fa | [
"Apache-2.0"
] | permissive | ilitzroth/mathlib | ea647e67f1fdfd19a0f7bdc5504e8acec6180011 | 5254ef14e3465f6504306132fe3ba9cec9ffff16 | refs/heads/master | 1,680,086,661,182 | 1,617,715,647,000 | 1,617,715,647,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 20,185 | 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 ring_theory.noetherian
import ring_theory.ideal.operations
import ring_theory.algebra_tower
/-!
# Finiteness conditions in commutative algebra
In this file we define several notions of finiteness that are common in commutative algebra.
## Main declarations
- `module.finite`, `algebra.finite`, `ring_hom.finite`, `alg_hom.finite`
all of these express that some object is finitely generated *as module* over some base ring.
- `algebra.finite_type`, `ring_hom.finite_type`, `alg_hom.finite_type`
all of these express that some object is finitely generated *as algebra* over some base ring.
- `algebra.finite_presentation`, `ring_hom.finite_presentation`, `alg_hom.finite_presentation`
all of these express that some object is finitely presented *as algebra* over some base ring.
-/
open function (surjective)
open_locale big_operators
section module_and_algebra
variables (R A B M N : Type*) [comm_ring R]
variables [comm_ring A] [algebra R A] [comm_ring B] [algebra R B]
variables [add_comm_group M] [module R M]
variables [add_comm_group N] [module R N]
/-- A module over a commutative ring is `finite` if it is finitely generated as a module. -/
class module.finite : Prop := (out : (⊤ : submodule R M).fg)
/-- An algebra over a commutative ring is of `finite_type` if it is finitely generated
over the base ring as algebra. -/
class algebra.finite_type : Prop := (out : (⊤ : subalgebra R A).fg)
/-- An algebra over a commutative ring is `finite_presentation` if it is the quotient of a
polynomial ring in `n` variables by a finitely generated ideal. -/
def algebra.finite_presentation : Prop :=
∃ (n : ℕ) (f : mv_polynomial (fin n) R →ₐ[R] A),
surjective f ∧ f.to_ring_hom.ker.fg
namespace module
variables {R M N}
lemma finite_def : finite R M ↔ (⊤ : submodule R M).fg := ⟨λ h, h.1, λ h, ⟨h⟩⟩
variables (R M N)
@[priority 100] -- see Note [lower instance priority]
instance is_noetherian.finite [is_noetherian R M] : finite R M :=
⟨is_noetherian.noetherian ⊤⟩
namespace finite
variables {R M N}
lemma of_surjective [hM : finite R M] (f : M →ₗ[R] N) (hf : surjective f) :
finite R N :=
⟨begin
rw [← linear_map.range_eq_top.2 hf, ← submodule.map_top],
exact submodule.fg_map hM.1
end⟩
lemma of_injective [is_noetherian R N] (f : M →ₗ[R] N)
(hf : function.injective f) : finite R M :=
⟨fg_of_injective f $ linear_map.ker_eq_bot.2 hf⟩
variables (R)
instance self : finite R R :=
⟨⟨{1}, by simpa only [finset.coe_singleton] using ideal.span_singleton_one⟩⟩
variables {R}
instance prod [hM : finite R M] [hN : finite R N] : finite R (M × N) :=
⟨begin
rw ← submodule.prod_top,
exact submodule.fg_prod hM.1 hN.1
end⟩
lemma equiv [hM : finite R M] (e : M ≃ₗ[R] N) : finite R N :=
of_surjective (e : M →ₗ[R] N) e.surjective
section algebra
lemma trans [algebra A B] [is_scalar_tower R A B] :
∀ [finite R A] [finite A B], finite R B
| ⟨⟨s, hs⟩⟩ ⟨⟨t, ht⟩⟩ := ⟨submodule.fg_def.2
⟨set.image2 (•) (↑s : set A) (↑t : set B),
set.finite.image2 _ s.finite_to_set t.finite_to_set,
by rw [set.image2_smul, submodule.span_smul hs (↑t : set B),
ht, submodule.restrict_scalars_top]⟩⟩
@[priority 100] -- see Note [lower instance priority]
instance finite_type [hRA : finite R A] : algebra.finite_type R A :=
⟨subalgebra.fg_of_submodule_fg hRA.1⟩
end algebra
end finite
end module
namespace algebra
namespace finite_type
lemma self : finite_type R R := ⟨⟨{1}, subsingleton.elim _ _⟩⟩
section
open_locale classical
protected lemma mv_polynomial (ι : Type*) [fintype ι] : finite_type R (mv_polynomial ι R) :=
⟨⟨finset.univ.image mv_polynomial.X, begin
rw eq_top_iff, refine λ p, mv_polynomial.induction_on' p
(λ u x, finsupp.induction u (subalgebra.algebra_map_mem _ x)
(λ i n f hif hn ih, _))
(λ p q ihp ihq, subalgebra.add_mem _ ihp ihq),
rw [add_comm, mv_polynomial.monomial_add_single],
exact subalgebra.mul_mem _ ih
(subalgebra.pow_mem _ (subset_adjoin $ finset.mem_image_of_mem _ $ finset.mem_univ _) _)
end⟩⟩
end
variables {R A B}
lemma of_surjective (hRA : finite_type R A) (f : A →ₐ[R] B) (hf : surjective f) :
finite_type R B :=
⟨begin
convert subalgebra.fg_map _ f hRA.1,
simpa only [map_top f, @eq_comm _ ⊤, eq_top_iff, alg_hom.mem_range] using hf
end⟩
lemma equiv (hRA : finite_type R A) (e : A ≃ₐ[R] B) : finite_type R B :=
hRA.of_surjective e e.surjective
lemma trans [algebra A B] [is_scalar_tower R A B] (hRA : finite_type R A) (hAB : finite_type A B) :
finite_type R B :=
⟨fg_trans' hRA.1 hAB.1⟩
/-- An algebra is finitely generated if and only if it is a quotient
of a polynomial ring whose variables are indexed by a finset. -/
lemma iff_quotient_mv_polynomial : (finite_type R A) ↔ ∃ (s : finset A)
(f : (mv_polynomial {x // x ∈ s} R) →ₐ[R] A), (surjective f) :=
begin
split,
{ rintro ⟨s, hs⟩,
use [s, mv_polynomial.aeval coe],
intro x,
have hrw : (↑s : set A) = (λ (x : A), x ∈ s.val) := rfl,
rw [← set.mem_range, ← alg_hom.coe_range, ← adjoin_eq_range, ← hrw, hs],
exact mem_top },
{ rintro ⟨s, ⟨f, hsur⟩⟩,
exact finite_type.of_surjective (finite_type.mv_polynomial R {x // x ∈ s}) f hsur }
end
/-- An algebra is finitely generated if and only if it is a quotient
of a polynomial ring whose variables are indexed by a fintype. -/
lemma iff_quotient_mv_polynomial' : (finite_type R A) ↔ ∃ (ι : Type u_2) [fintype ι]
(f : (mv_polynomial ι R) →ₐ[R] A), (surjective f) :=
begin
split,
{ rw iff_quotient_mv_polynomial,
rintro ⟨s, ⟨f, hsur⟩⟩,
use [{x // x ∈ s}, by apply_instance, f, hsur] },
{ rintro ⟨ι, ⟨hfintype, ⟨f, hsur⟩⟩⟩,
letI : fintype ι := hfintype,
exact finite_type.of_surjective (finite_type.mv_polynomial R ι) f hsur }
end
/-- An algebra is finitely generated if and only if it is a quotient of a polynomial ring in `n`
variables. -/
lemma iff_quotient_mv_polynomial'' : (finite_type R A) ↔ ∃ (n : ℕ)
(f : (mv_polynomial (fin n) R) →ₐ[R] A), (surjective f) :=
begin
split,
{ rw iff_quotient_mv_polynomial',
rintro ⟨ι, hfintype, ⟨f, hsur⟩⟩,
obtain ⟨n, equiv⟩ := @fintype.exists_equiv_fin ι hfintype,
replace equiv := mv_polynomial.rename_equiv R (nonempty.some equiv),
exact ⟨n, alg_hom.comp f equiv.symm, function.surjective.comp hsur
(alg_equiv.symm equiv).surjective⟩ },
{ rintro ⟨n, ⟨f, hsur⟩⟩,
exact finite_type.of_surjective (finite_type.mv_polynomial R (fin n)) f hsur }
end
/-- A finitely presented algebra is of finite type. -/
lemma of_finite_presentation : finite_presentation R A → finite_type R A :=
begin
rintro ⟨n, f, hf⟩,
apply (finite_type.iff_quotient_mv_polynomial'').2,
exact ⟨n, f, hf.1⟩
end
end finite_type
namespace finite_presentation
variables {R A B}
/-- An algebra over a Noetherian ring is finitely generated if and only if it is finitely
presented. -/
lemma of_finite_type [is_noetherian_ring R] : finite_type R A ↔ finite_presentation R A :=
begin
refine ⟨λ h, _, algebra.finite_type.of_finite_presentation⟩,
obtain ⟨n, f, hf⟩ := algebra.finite_type.iff_quotient_mv_polynomial''.1 h,
refine ⟨n, f, hf, _⟩,
have hnoet : is_noetherian_ring (mv_polynomial (fin n) R) := by apply_instance,
replace hnoet := (is_noetherian_ring_iff.1 hnoet).noetherian,
exact hnoet f.to_ring_hom.ker,
end
/-- If `e : A ≃ₐ[R] B` and `A` is finitely presented, then so is `B`. -/
lemma equiv (hfp : finite_presentation R A) (e : A ≃ₐ[R] B) : finite_presentation R B :=
begin
obtain ⟨n, f, hf⟩ := hfp,
use [n, alg_hom.comp ↑e f],
split,
{ exact function.surjective.comp e.surjective hf.1 },
suffices hker : (alg_hom.comp ↑e f).to_ring_hom.ker = f.to_ring_hom.ker,
{ rw hker, exact hf.2 },
{ have hco : (alg_hom.comp ↑e f).to_ring_hom = ring_hom.comp ↑e.to_ring_equiv f.to_ring_hom,
{ have h : (alg_hom.comp ↑e f).to_ring_hom = e.to_alg_hom.to_ring_hom.comp f.to_ring_hom := rfl,
have h1 : ↑(e.to_ring_equiv) = (e.to_alg_hom).to_ring_hom := rfl,
rw [h, h1] },
rw [ring_hom.ker_eq_comap_bot, hco, ← ideal.comap_comap, ← ring_hom.ker_eq_comap_bot,
ring_hom.ker_coe_equiv (alg_equiv.to_ring_equiv e), ring_hom.ker_eq_comap_bot] }
end
variable (R)
/-- The ring of polynomials in finitely many variables is finitely presented. -/
lemma mv_polynomial (ι : Type u_2) [fintype ι] : finite_presentation R (mv_polynomial ι R) :=
begin
obtain ⟨n, equiv⟩ := @fintype.exists_equiv_fin ι _,
replace equiv := mv_polynomial.rename_equiv R (nonempty.some equiv),
use [n, alg_equiv.to_alg_hom equiv.symm],
split,
{ exact (alg_equiv.symm equiv).surjective },
suffices hinj : function.injective equiv.symm.to_alg_hom.to_ring_hom,
{ rw [(ring_hom.injective_iff_ker_eq_bot _).1 hinj],
exact submodule.fg_bot },
exact (alg_equiv.symm equiv).injective
end
/-- `R` is finitely presented as `R`-algebra. -/
lemma self : finite_presentation R R :=
equiv (mv_polynomial R pempty) (mv_polynomial.pempty_alg_equiv R)
variable {R}
/-- The quotient of a finitely presented algebra by a finitely generated ideal is finitely
presented. -/
lemma quotient {I : ideal A} (h : submodule.fg I) (hfp : finite_presentation R A) :
finite_presentation R I.quotient :=
begin
obtain ⟨n, f, hf⟩ := hfp,
refine ⟨n, (ideal.quotient.mkₐ R I).comp f, _, _⟩,
{ exact (ideal.quotient.mkₐ_surjective R I).comp hf.1 },
{ refine submodule.fg_ker_ring_hom_comp _ _ hf.2 _ hf.1,
rwa ideal.quotient.mkₐ_ker R I }
end
/-- If `f : A →ₐ[R] B` is surjective with finitely generated kernel and `A` is finitely presented,
then so is `B`. -/
lemma of_surjective {f : A →ₐ[R] B} (hf : function.surjective f) (hker : f.to_ring_hom.ker.fg)
(hfp : finite_presentation R A) : finite_presentation R B :=
equiv (quotient hker hfp) (ideal.quotient_ker_alg_equiv_of_surjective hf)
lemma iff : finite_presentation R A ↔
∃ n (I : ideal (_root_.mv_polynomial (fin n) R)) (e : I.quotient ≃ₐ[R] A), I.fg :=
begin
refine ⟨λ h,_, λ h, _⟩,
{ obtain ⟨n, f, hf⟩ := h,
use [n, f.to_ring_hom.ker, ideal.quotient_ker_alg_equiv_of_surjective hf.1, hf.2] },
{ obtain ⟨n, I, e, hfg⟩ := h,
exact equiv (quotient hfg (mv_polynomial R _)) e }
end
/-- An algebra is finitely presented if and only if it is a quotient of a polynomial ring whose
variables are indexed by a fintype by a finitely generated ideal. -/
lemma iff_quotient_mv_polynomial' : finite_presentation R A ↔ ∃ (ι : Type u_2) [fintype ι]
(f : (_root_.mv_polynomial ι R) →ₐ[R] A), (surjective f) ∧ f.to_ring_hom.ker.fg :=
begin
split,
{ rintro ⟨n, f, hfs, hfk⟩,
set ulift_var := mv_polynomial.rename_equiv R equiv.ulift,
refine ⟨ulift (fin n), infer_instance, f.comp ulift_var.to_alg_hom,
hfs.comp ulift_var.surjective,
submodule.fg_ker_ring_hom_comp _ _ _ hfk ulift_var.surjective⟩,
convert submodule.fg_bot,
exact ring_hom.ker_coe_equiv ulift_var.to_ring_equiv, },
{ rintro ⟨ι, hfintype, f, hf⟩,
haveI : fintype ι := hfintype,
obtain ⟨n, equiv⟩ := fintype.exists_equiv_fin ι,
replace equiv := mv_polynomial.rename_equiv R (nonempty.some equiv),
refine ⟨n, f.comp equiv.symm,
hf.1.comp (alg_equiv.symm equiv).surjective,
submodule.fg_ker_ring_hom_comp _ f _ hf.2 equiv.symm.surjective⟩,
convert submodule.fg_bot,
exact ring_hom.ker_coe_equiv (equiv.symm.to_ring_equiv), }
end
/-- If `A` is a finitely presented `R`-algebra, then `mv_polynomial (fin n) A` is finitely presented
as `R`-algebra. -/
lemma mv_polynomial_of_finite_presentation (hfp : finite_presentation R A) (ι : Type*)
[fintype ι] : finite_presentation R (_root_.mv_polynomial ι A) :=
begin
obtain ⟨n, e⟩ := fintype.exists_equiv_fin ι,
replace e := (mv_polynomial.rename_equiv A (nonempty.some e)).restrict_scalars R,
refine equiv _ e.symm,
obtain ⟨m, I, e, hfg⟩ := iff.1 hfp,
refine equiv _ (mv_polynomial.map_alg_equiv (fin n) e),
-- typeclass inference seems to struggle to find this path
letI : is_scalar_tower R
(_root_.mv_polynomial (fin m) R) (_root_.mv_polynomial (fin m) R) :=
is_scalar_tower.right,
letI : is_scalar_tower R
(_root_.mv_polynomial (fin m) R)
(_root_.mv_polynomial (fin n) (_root_.mv_polynomial (fin m) R)) :=
mv_polynomial.is_scalar_tower,
refine equiv _ ((@mv_polynomial.quotient_equiv_quotient_mv_polynomial
_ (fin n) _ I).restrict_scalars R).symm,
refine quotient (submodule.map_fg_of_fg I hfg _) _,
refine equiv _ (mv_polynomial.sum_alg_equiv _ _ _),
exact equiv (mv_polynomial R (fin (n + m))) (mv_polynomial.rename_equiv R fin_sum_fin_equiv).symm
end
/-- If `A` is an `R`-algebra and `S` is an `A`-algebra, both finitely presented, then `S` is
finitely presented as `R`-algebra. -/
lemma trans [algebra A B] [is_scalar_tower R A B] (hfpA : finite_presentation R A)
(hfpB : finite_presentation A B) : finite_presentation R B :=
begin
obtain ⟨n, I, e, hfg⟩ := iff.1 hfpB,
exact equiv (quotient hfg (mv_polynomial_of_finite_presentation hfpA _)) (e.restrict_scalars R)
end
end finite_presentation
end algebra
end module_and_algebra
namespace ring_hom
variables {A B C : Type*} [comm_ring A] [comm_ring B] [comm_ring C]
/-- A ring morphism `A →+* B` is `finite` if `B` is finitely generated as `A`-module. -/
def finite (f : A →+* B) : Prop :=
by letI : algebra A B := f.to_algebra; exact module.finite A B
/-- A ring morphism `A →+* B` is of `finite_type` if `B` is finitely generated as `A`-algebra. -/
def finite_type (f : A →+* B) : Prop := @algebra.finite_type A B _ _ f.to_algebra
/-- A ring morphism `A →+* B` is of `finite_presentation` if `B` is finitely presented as
`A`-algebra. -/
def finite_presentation (f : A →+* B) : Prop := @algebra.finite_presentation A B _ _ f.to_algebra
namespace finite
variables (A)
lemma id : finite (ring_hom.id A) := module.finite.self A
variables {A}
lemma of_surjective (f : A →+* B) (hf : surjective f) : f.finite :=
begin
letI := f.to_algebra,
exact module.finite.of_surjective (algebra.of_id A B).to_linear_map hf
end
lemma comp {g : B →+* C} {f : A →+* B} (hg : g.finite) (hf : f.finite) : (g.comp f).finite :=
@module.finite.trans A B C _ _ f.to_algebra _ (g.comp f).to_algebra g.to_algebra
begin
fconstructor,
intros a b c,
simp only [algebra.smul_def, ring_hom.map_mul, mul_assoc],
refl
end
hf hg
lemma finite_type {f : A →+* B} (hf : f.finite) : finite_type f :=
@module.finite.finite_type _ _ _ _ f.to_algebra hf
end finite
namespace finite_type
variables (A)
lemma id : finite_type (ring_hom.id A) := algebra.finite_type.self A
variables {A}
lemma comp_surjective {f : A →+* B} {g : B →+* C} (hf : f.finite_type) (hg : surjective g) :
(g.comp f).finite_type :=
@algebra.finite_type.of_surjective A B C _ _ f.to_algebra _ (g.comp f).to_algebra hf
{ to_fun := g, commutes' := λ a, rfl, .. g } hg
lemma of_surjective (f : A →+* B) (hf : surjective f) : f.finite_type :=
by { rw ← f.comp_id, exact (id A).comp_surjective hf }
lemma comp {g : B →+* C} {f : A →+* B} (hg : g.finite_type) (hf : f.finite_type) :
(g.comp f).finite_type :=
@algebra.finite_type.trans A B C _ _ f.to_algebra _ (g.comp f).to_algebra g.to_algebra
begin
fconstructor,
intros a b c,
simp only [algebra.smul_def, ring_hom.map_mul, mul_assoc],
refl
end
hf hg
lemma of_finite_presentation {f : A →+* B} (hf : f.finite_presentation) : f.finite_type :=
@algebra.finite_type.of_finite_presentation A B _ _ f.to_algebra hf
end finite_type
namespace finite_presentation
variables (A)
lemma id : finite_presentation (ring_hom.id A) := algebra.finite_presentation.self A
variables {A}
lemma comp_surjective {f : A →+* B} {g : B →+* C} (hf : f.finite_presentation) (hg : surjective g)
(hker : g.ker.fg) : (g.comp f).finite_presentation :=
@algebra.finite_presentation.of_surjective A B C _ _ f.to_algebra _ (g.comp f).to_algebra
{ to_fun := g, commutes' := λ a, rfl, .. g } hg hker hf
lemma of_surjective (f : A →+* B) (hf : surjective f) (hker : f.ker.fg) : f.finite_presentation :=
by { rw ← f.comp_id, exact (id A).comp_surjective hf hker}
lemma of_finite_type [is_noetherian_ring A] {f : A →+* B} : f.finite_type ↔ f.finite_presentation :=
@algebra.finite_presentation.of_finite_type A B _ _ f.to_algebra _
lemma comp {g : B →+* C} {f : A →+* B} (hg : g.finite_presentation) (hf : f.finite_presentation) :
(g.comp f).finite_presentation :=
@algebra.finite_presentation.trans A B C _ _ f.to_algebra _ (g.comp f).to_algebra g.to_algebra
{ smul_assoc := λ a b c, begin
simp only [algebra.smul_def, ring_hom.map_mul, mul_assoc],
refl
end }
hf hg
end finite_presentation
end ring_hom
namespace alg_hom
variables {R A B C : Type*} [comm_ring R]
variables [comm_ring A] [comm_ring B] [comm_ring C]
variables [algebra R A] [algebra R B] [algebra R C]
/-- An algebra morphism `A →ₐ[R] B` is finite if it is finite as ring morphism.
In other words, if `B` is finitely generated as `A`-module. -/
def finite (f : A →ₐ[R] B) : Prop := f.to_ring_hom.finite
/-- An algebra morphism `A →ₐ[R] B` is of `finite_type` if it is of finite type as ring morphism.
In other words, if `B` is finitely generated as `A`-algebra. -/
def finite_type (f : A →ₐ[R] B) : Prop := f.to_ring_hom.finite_type
/-- An algebra morphism `A →ₐ[R] B` is of `finite_presentation` if it is of finite presentation as
ring morphism. In other words, if `B` is finitely presented as `A`-algebra. -/
def finite_presentation (f : A →ₐ[R] B) : Prop := f.to_ring_hom.finite_presentation
namespace finite
variables (R A)
lemma id : finite (alg_hom.id R A) := ring_hom.finite.id A
variables {R A}
lemma comp {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.finite) (hf : f.finite) : (g.comp f).finite :=
ring_hom.finite.comp hg hf
lemma of_surjective (f : A →ₐ[R] B) (hf : surjective f) : f.finite :=
ring_hom.finite.of_surjective f hf
lemma finite_type {f : A →ₐ[R] B} (hf : f.finite) : finite_type f :=
ring_hom.finite.finite_type hf
end finite
namespace finite_type
variables (R A)
lemma id : finite_type (alg_hom.id R A) := ring_hom.finite_type.id A
variables {R A}
lemma comp {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.finite_type) (hf : f.finite_type) :
(g.comp f).finite_type :=
ring_hom.finite_type.comp hg hf
lemma comp_surjective {f : A →ₐ[R] B} {g : B →ₐ[R] C} (hf : f.finite_type) (hg : surjective g) :
(g.comp f).finite_type :=
ring_hom.finite_type.comp_surjective hf hg
lemma of_surjective (f : A →ₐ[R] B) (hf : surjective f) : f.finite_type :=
ring_hom.finite_type.of_surjective f hf
lemma of_finite_presentation {f : A →ₐ[R] B} (hf : f.finite_presentation) : f.finite_type :=
ring_hom.finite_type.of_finite_presentation hf
end finite_type
namespace finite_presentation
variables (R A)
lemma id : finite_presentation (alg_hom.id R A) := ring_hom.finite_presentation.id A
variables {R A}
lemma comp {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.finite_presentation)
(hf : f.finite_presentation) : (g.comp f).finite_presentation :=
ring_hom.finite_presentation.comp hg hf
lemma comp_surjective {f : A →ₐ[R] B} {g : B →ₐ[R] C} (hf : f.finite_presentation)
(hg : surjective g) (hker : g.to_ring_hom.ker.fg) : (g.comp f).finite_presentation :=
ring_hom.finite_presentation.comp_surjective hf hg hker
lemma of_surjective (f : A →ₐ[R] B) (hf : surjective f) (hker : f.to_ring_hom.ker.fg) :
f.finite_presentation :=
ring_hom.finite_presentation.of_surjective f hf hker
lemma of_finite_type [is_noetherian_ring A] {f : A →ₐ[R] B} :
f.finite_type ↔ f.finite_presentation :=
ring_hom.finite_presentation.of_finite_type
end finite_presentation
end alg_hom
|
403ad20ba8da08c88edfbe619266d0748a5def8b | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/measure_theory/function/uniform_integrable.lean | 789d0c95688e220d7657817d3acce57301e84620 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 48,695 | lean | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import measure_theory.function.convergence_in_measure
import measure_theory.function.l1_space
/-!
# Uniform integrability
This file contains the definitions for uniform integrability (both in the measure theory sense
as well as the probability theory sense). This file also contains the Vitali convergence theorem
which estabishes a relation between uniform integrability, convergence in measure and
Lp convergence.
Uniform integrability plays a vital role in the theory of martingales most notably is used to
fomulate the martingale convergence theorem.
## Main definitions
* `measure_theory.unif_integrable`: uniform integrability in the measure theory sense.
In particular, a sequence of functions `f` is uniformly integrable if for all `ε > 0`, there
exists some `δ > 0` such that for all sets `s` of smaller measure than `δ`, the Lp-norm of
`f i` restricted `s` is smaller than `ε` for all `i`.
* `measure_theory.uniform_integrable`: uniform integrability in the probability theory sense.
In particular, a sequence of measurable functions `f` is uniformly integrable in the
probability theory sense if it is uniformly integrable in the measure theory sense and
has uniformly bounded Lp-norm.
# Main results
* `measure_theory.unif_integrable_fintype`: a finite sequence of Lp functions is uniformly
integrable.
* `measure_theory.tendsto_Lp_of_tendsto_ae`: a sequence of Lp functions which is uniformly
integrable converges in Lp if they converge almost everywhere.
* `measure_theory.tendsto_in_measure_iff_tendsto_Lp`: Vitali convergence theorem:
a sequence of Lp functions converges in Lp if and only if it is uniformly integrable
and converges in measure.
## Tags
uniform integrable, uniformly absolutely continuous integral, Vitali convergence theorem
-/
noncomputable theory
open_locale classical measure_theory nnreal ennreal topological_space big_operators
namespace measure_theory
open set filter topological_space
variables {α β ι : Type*} {m : measurable_space α} {μ : measure α} [normed_add_comm_group β]
/-- Uniform integrability in the measure theory sense.
A sequence of functions `f` is said to be uniformly integrable if for all `ε > 0`, there exists
some `δ > 0` such that for all sets `s` with measure less than `δ`, the Lp-norm of `f i`
restricted on `s` is less than `ε`.
Uniform integrablility is also known as uniformly absolutely continuous integrals. -/
def unif_integrable {m : measurable_space α} (f : ι → α → β) (p : ℝ≥0∞) (μ : measure α) : Prop :=
∀ ⦃ε : ℝ⦄ (hε : 0 < ε), ∃ (δ : ℝ) (hδ : 0 < δ), ∀ i s, measurable_set s → μ s ≤ ennreal.of_real δ →
snorm (s.indicator (f i)) p μ ≤ ennreal.of_real ε
/-- In probability theory, a family of measurable functions is uniformly integrable if it is
uniformly integrable in the measure theory sense and is uniformly bounded. -/
def uniform_integrable {m : measurable_space α}
(f : ι → α → β) (p : ℝ≥0∞) (μ : measure α) : Prop :=
(∀ i, ae_strongly_measurable (f i) μ) ∧ unif_integrable f p μ ∧ ∃ C : ℝ≥0, ∀ i, snorm (f i) p μ ≤ C
namespace uniform_integrable
protected lemma ae_strongly_measurable {f : ι → α → β} {p : ℝ≥0∞}
(hf : uniform_integrable f p μ) (i : ι) : ae_strongly_measurable (f i) μ :=
hf.1 i
protected lemma unif_integrable {f : ι → α → β} {p : ℝ≥0∞}
(hf : uniform_integrable f p μ) : unif_integrable f p μ :=
hf.2.1
protected lemma mem_ℒp {f : ι → α → β} {p : ℝ≥0∞}
(hf : uniform_integrable f p μ) (i : ι) :
mem_ℒp (f i) p μ :=
⟨hf.1 i, let ⟨_, _, hC⟩ := hf.2 in lt_of_le_of_lt (hC i) ennreal.coe_lt_top⟩
end uniform_integrable
section unif_integrable
/-! ### `unif_integrable`
This section deals with uniform integrability in the measure theory sense. -/
namespace unif_integrable
variables {f g : ι → α → β} {p : ℝ≥0∞}
protected lemma add
(hf : unif_integrable f p μ) (hg : unif_integrable g p μ) (hp : 1 ≤ p)
(hf_meas : ∀ i, ae_strongly_measurable (f i) μ) (hg_meas : ∀ i, ae_strongly_measurable (g i) μ) :
unif_integrable (f + g) p μ :=
begin
intros ε hε,
have hε2 : 0 < ε / 2 := half_pos hε,
obtain ⟨δ₁, hδ₁_pos, hfδ₁⟩ := hf hε2,
obtain ⟨δ₂, hδ₂_pos, hgδ₂⟩ := hg hε2,
refine ⟨min δ₁ δ₂, lt_min hδ₁_pos hδ₂_pos, λ i s hs hμs, _⟩,
simp_rw [pi.add_apply, indicator_add'],
refine (snorm_add_le ((hf_meas i).indicator hs) ((hg_meas i).indicator hs) hp).trans _,
have hε_halves : ennreal.of_real ε = ennreal.of_real (ε / 2) + ennreal.of_real (ε / 2),
by rw [← ennreal.of_real_add hε2.le hε2.le, add_halves],
rw hε_halves,
exact add_le_add (hfδ₁ i s hs (hμs.trans (ennreal.of_real_le_of_real (min_le_left _ _))))
(hgδ₂ i s hs (hμs.trans (ennreal.of_real_le_of_real (min_le_right _ _)))),
end
protected lemma neg (hf : unif_integrable f p μ) : unif_integrable (-f) p μ :=
by { simp_rw [unif_integrable, pi.neg_apply, indicator_neg', snorm_neg], exact hf, }
protected lemma sub
(hf : unif_integrable f p μ) (hg : unif_integrable g p μ) (hp : 1 ≤ p)
(hf_meas : ∀ i, ae_strongly_measurable (f i) μ) (hg_meas : ∀ i, ae_strongly_measurable (g i) μ) :
unif_integrable (f - g) p μ :=
by { rw sub_eq_add_neg, exact hf.add hg.neg hp hf_meas (λ i, (hg_meas i).neg), }
protected lemma ae_eq (hf : unif_integrable f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :
unif_integrable g p μ :=
begin
intros ε hε,
obtain ⟨δ, hδ_pos, hfδ⟩ := hf hε,
refine ⟨δ, hδ_pos, λ n s hs hμs, (le_of_eq $ snorm_congr_ae _).trans (hfδ n s hs hμs)⟩,
filter_upwards [hfg n] with x hx,
simp_rw [indicator_apply, hx],
end
end unif_integrable
lemma unif_integrable_zero_meas [measurable_space α] {p : ℝ≥0∞} {f : ι → α → β} :
unif_integrable f p (0 : measure α) :=
λ ε hε, ⟨1, one_pos, λ i s hs hμs, by simp⟩
lemma unif_integrable_congr_ae {p : ℝ≥0∞} {f g : ι → α → β} (hfg : ∀ n, f n =ᵐ[μ] g n) :
unif_integrable f p μ ↔ unif_integrable g p μ :=
⟨λ hf, hf.ae_eq hfg, λ hg, hg.ae_eq (λ n, (hfg n).symm)⟩
lemma tendsto_indicator_ge (f : α → β) (x : α):
tendsto (λ M : ℕ, {x | (M : ℝ) ≤ ‖f x‖₊}.indicator f x) at_top (𝓝 0) :=
begin
refine @tendsto_at_top_of_eventually_const _ _ _ _ _ _ _ (nat.ceil (‖f x‖₊ : ℝ) + 1) (λ n hn, _),
rw indicator_of_not_mem,
simp only [not_le, mem_set_of_eq],
refine lt_of_le_of_lt (nat.le_ceil _) _,
refine lt_of_lt_of_le (lt_add_one _) _,
norm_cast,
rwa [ge_iff_le, coe_nnnorm] at hn,
end
variables (μ) {p : ℝ≥0∞}
section
variables {f : α → β}
/-- This lemma is weaker than `measure_theory.mem_ℒp.integral_indicator_norm_ge_nonneg_le`
as the latter provides `0 ≤ M` and does not require the measurability of `f`. -/
lemma mem_ℒp.integral_indicator_norm_ge_le
(hf : mem_ℒp f 1 μ) (hmeas : strongly_measurable f) {ε : ℝ} (hε : 0 < ε) :
∃ M : ℝ, ∫⁻ x, ‖{x | M ≤ ‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ennreal.of_real ε :=
begin
have htendsto : ∀ᵐ x ∂μ, tendsto (λ M : ℕ, {x | (M : ℝ) ≤ ‖f x‖₊}.indicator f x) at_top (𝓝 0) :=
univ_mem' (id $ λ x, tendsto_indicator_ge f x),
have hmeas : ∀ M : ℕ, ae_strongly_measurable ({x | (M : ℝ) ≤ ‖f x‖₊}.indicator f) μ,
{ assume M,
apply hf.1.indicator,
apply strongly_measurable.measurable_set_le strongly_measurable_const
hmeas.nnnorm.measurable.coe_nnreal_real.strongly_measurable },
have hbound : has_finite_integral (λ x, ‖f x‖) μ,
{ rw mem_ℒp_one_iff_integrable at hf,
exact hf.norm.2 },
have := tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto,
{ rw ennreal.tendsto_at_top_zero at this,
obtain ⟨M, hM⟩ := this (ennreal.of_real ε) (ennreal.of_real_pos.2 hε),
simp only [true_and, ge_iff_le, zero_tsub, zero_le,
sub_zero, zero_add, coe_nnnorm, mem_Icc] at hM,
refine ⟨M, _⟩,
convert hM M le_rfl,
ext1 x,
simp only [coe_nnnorm, ennreal.of_real_eq_coe_nnreal (norm_nonneg _)],
refl },
{ refine λ n, univ_mem' (id $ λ x, _),
by_cases hx : (n : ℝ) ≤ ‖f x‖,
{ dsimp,
rwa indicator_of_mem },
{ dsimp,
rw [indicator_of_not_mem, norm_zero],
{ exact norm_nonneg _ },
{ assumption } } }
end
/-- This lemma is superceded by `measure_theory.mem_ℒp.integral_indicator_norm_ge_nonneg_le`
which does not require measurability. -/
lemma mem_ℒp.integral_indicator_norm_ge_nonneg_le_of_meas
(hf : mem_ℒp f 1 μ) (hmeas : strongly_measurable f) {ε : ℝ} (hε : 0 < ε) :
∃ M : ℝ, 0 ≤ M ∧ ∫⁻ x, ‖{x | M ≤ ‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ennreal.of_real ε :=
let ⟨M, hM⟩ := hf.integral_indicator_norm_ge_le μ hmeas hε in ⟨max M 0, le_max_right _ _, by simpa⟩
lemma mem_ℒp.integral_indicator_norm_ge_nonneg_le
(hf : mem_ℒp f 1 μ) {ε : ℝ} (hε : 0 < ε) :
∃ M : ℝ, 0 ≤ M ∧ ∫⁻ x, ‖{x | M ≤ ‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ennreal.of_real ε :=
begin
have hf_mk : mem_ℒp (hf.1.mk f) 1 μ := (mem_ℒp_congr_ae hf.1.ae_eq_mk).mp hf,
obtain ⟨M, hM_pos, hfM⟩ := hf_mk.integral_indicator_norm_ge_nonneg_le_of_meas μ
hf.1.strongly_measurable_mk hε,
refine ⟨M, hM_pos, (le_of_eq _).trans hfM⟩,
refine lintegral_congr_ae _,
filter_upwards [hf.1.ae_eq_mk] with x hx,
simp only [indicator_apply, coe_nnnorm, mem_set_of_eq, ennreal.coe_eq_coe, hx.symm],
end
lemma mem_ℒp.snorm_ess_sup_indicator_norm_ge_eq_zero
(hf : mem_ℒp f ∞ μ) (hmeas : strongly_measurable f) :
∃ M : ℝ, snorm_ess_sup ({x | M ≤ ‖f x‖₊}.indicator f) μ = 0 :=
begin
have hbdd : snorm_ess_sup f μ < ∞ := hf.snorm_lt_top,
refine ⟨(snorm f ∞ μ + 1).to_real, _⟩,
rw snorm_ess_sup_indicator_eq_snorm_ess_sup_restrict,
have : μ.restrict {x : α | (snorm f ⊤ μ + 1).to_real ≤ ‖f x‖₊} = 0,
{ simp only [coe_nnnorm, snorm_exponent_top, measure.restrict_eq_zero],
have : {x : α | (snorm_ess_sup f μ + 1).to_real ≤ ‖f x‖} ⊆
{x : α | snorm_ess_sup f μ < ‖f x‖₊},
{ intros x hx,
rw [mem_set_of_eq, ← ennreal.to_real_lt_to_real hbdd.ne ennreal.coe_lt_top.ne,
ennreal.coe_to_real, coe_nnnorm],
refine lt_of_lt_of_le _ hx,
rw ennreal.to_real_lt_to_real hbdd.ne,
{ exact ennreal.lt_add_right hbdd.ne one_ne_zero },
{ exact (ennreal.add_lt_top.2 ⟨hbdd, ennreal.one_lt_top⟩).ne } },
rw ← nonpos_iff_eq_zero,
refine (measure_mono this).trans _,
have hle := coe_nnnorm_ae_le_snorm_ess_sup f μ,
simp_rw [ae_iff, not_le] at hle,
exact nonpos_iff_eq_zero.2 hle },
rw [this, snorm_ess_sup_measure_zero],
exact measurable_set_le measurable_const hmeas.nnnorm.measurable.subtype_coe,
end
/- This lemma is slightly weaker than `measure_theory.mem_ℒp.snorm_indicator_norm_ge_pos_le` as the
latter provides `0 < M`. -/
lemma mem_ℒp.snorm_indicator_norm_ge_le
(hf : mem_ℒp f p μ) (hmeas : strongly_measurable f) {ε : ℝ} (hε : 0 < ε) :
∃ M : ℝ, snorm ({x | M ≤ ‖f x‖₊}.indicator f) p μ ≤ ennreal.of_real ε :=
begin
by_cases hp_ne_zero : p = 0,
{ refine ⟨1, hp_ne_zero.symm ▸ _⟩,
simp [snorm_exponent_zero] },
by_cases hp_ne_top : p = ∞,
{ subst hp_ne_top,
obtain ⟨M, hM⟩ := hf.snorm_ess_sup_indicator_norm_ge_eq_zero μ hmeas,
refine ⟨M, _⟩,
simp only [snorm_exponent_top, hM, zero_le] },
obtain ⟨M, hM', hM⟩ := @mem_ℒp.integral_indicator_norm_ge_nonneg_le _ _ _ μ _
(λ x, ‖f x‖^p.to_real) (hf.norm_rpow hp_ne_zero hp_ne_top) _
(real.rpow_pos_of_pos hε p.to_real),
refine ⟨M ^(1 / p.to_real), _⟩,
rw [snorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top,
← ennreal.rpow_one (ennreal.of_real ε)],
conv_rhs { rw ← mul_one_div_cancel (ennreal.to_real_pos hp_ne_zero hp_ne_top).ne.symm },
rw [ennreal.rpow_mul,
ennreal.rpow_le_rpow_iff (one_div_pos.2 $ ennreal.to_real_pos hp_ne_zero hp_ne_top),
ennreal.of_real_rpow_of_pos hε],
convert hM,
ext1 x,
rw [ennreal.coe_rpow_of_nonneg _ ennreal.to_real_nonneg,
nnnorm_indicator_eq_indicator_nnnorm, nnnorm_indicator_eq_indicator_nnnorm],
have hiff : M ^ (1 / p.to_real) ≤ ‖f x‖₊ ↔ M ≤ ‖‖f x‖ ^ p.to_real‖₊,
{ rw [coe_nnnorm, coe_nnnorm, real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm,
← real.rpow_le_rpow_iff hM' (real.rpow_nonneg_of_nonneg (norm_nonneg _) _)
(one_div_pos.2 $ ennreal.to_real_pos hp_ne_zero hp_ne_top),
← real.rpow_mul (norm_nonneg _),
mul_one_div_cancel (ennreal.to_real_pos hp_ne_zero hp_ne_top).ne.symm, real.rpow_one] },
by_cases hx : x ∈ {x : α | M ^ (1 / p.to_real) ≤ ‖f x‖₊},
{ rw [set.indicator_of_mem hx,set.indicator_of_mem, real.nnnorm_of_nonneg], refl,
change _ ≤ _,
rwa ← hiff },
{ rw [set.indicator_of_not_mem hx, set.indicator_of_not_mem],
{ simp [(ennreal.to_real_pos hp_ne_zero hp_ne_top).ne.symm] },
{ change ¬ _ ≤ _,
rwa ← hiff } }
end
/-- This lemma implies that a single function is uniformly integrable (in the probability sense). -/
lemma mem_ℒp.snorm_indicator_norm_ge_pos_le
(hf : mem_ℒp f p μ) (hmeas : strongly_measurable f) {ε : ℝ} (hε : 0 < ε) :
∃ M : ℝ, 0 < M ∧ snorm ({x | M ≤ ‖f x‖₊}.indicator f) p μ ≤ ennreal.of_real ε :=
begin
obtain ⟨M, hM⟩ := hf.snorm_indicator_norm_ge_le μ hmeas hε,
refine ⟨max M 1, lt_of_lt_of_le zero_lt_one (le_max_right _ _),
le_trans (snorm_mono (λ x, _)) hM⟩,
rw [norm_indicator_eq_indicator_norm, norm_indicator_eq_indicator_norm],
refine indicator_le_indicator_of_subset (λ x hx, _) (λ x, norm_nonneg _) x,
change max _ _ ≤ _ at hx, -- removing the `change` breaks the proof!
exact (max_le_iff.1 hx).1,
end
end
lemma snorm_indicator_le_of_bound {f : α → β} (hp_top : p ≠ ∞)
{ε : ℝ} (hε : 0 < ε) {M : ℝ} (hf : ∀ x, ‖f x‖ < M) :
∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, measurable_set s → μ s ≤ ennreal.of_real δ →
snorm (s.indicator f) p μ ≤ ennreal.of_real ε :=
begin
by_cases hM : M ≤ 0,
{ refine ⟨1, zero_lt_one, λ s hs hμ, _⟩,
rw (_ : f = 0),
{ simp [hε.le] },
{ ext x,
rw [pi.zero_apply, ← norm_le_zero_iff],
exact (lt_of_lt_of_le (hf x) hM).le } },
rw not_le at hM,
refine ⟨(ε / M) ^ p.to_real, real.rpow_pos_of_pos (div_pos hε hM) _, λ s hs hμ, _⟩,
by_cases hp : p = 0,
{ simp [hp] },
rw snorm_indicator_eq_snorm_restrict hs,
have haebdd : ∀ᵐ x ∂μ.restrict s, ‖f x‖ ≤ M,
{ filter_upwards,
exact (λ x, (hf x).le) },
refine le_trans (snorm_le_of_ae_bound haebdd) _,
rw [measure.restrict_apply measurable_set.univ, univ_inter,
← ennreal.le_div_iff_mul_le (or.inl _) (or.inl ennreal.of_real_ne_top)],
{ rw [← one_div, ennreal.rpow_one_div_le_iff (ennreal.to_real_pos hp hp_top)],
refine le_trans hμ _,
rw [← ennreal.of_real_rpow_of_pos (div_pos hε hM),
ennreal.rpow_le_rpow_iff (ennreal.to_real_pos hp hp_top), ennreal.of_real_div_of_pos hM],
exact le_rfl },
{ simpa only [ennreal.of_real_eq_zero, not_le, ne.def] },
end
section
variables {f : α → β}
/-- Auxiliary lemma for `measure_theory.mem_ℒp.snorm_indicator_le`. -/
lemma mem_ℒp.snorm_indicator_le' (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
(hf : mem_ℒp f p μ) (hmeas : strongly_measurable f) {ε : ℝ} (hε : 0 < ε) :
∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, measurable_set s → μ s ≤ ennreal.of_real δ →
snorm (s.indicator f) p μ ≤ 2 * ennreal.of_real ε :=
begin
obtain ⟨M, hMpos, hM⟩ := hf.snorm_indicator_norm_ge_pos_le μ hmeas hε,
obtain ⟨δ, hδpos, hδ⟩ := @snorm_indicator_le_of_bound _ _ _ μ _ _
({x | ‖f x‖ < M}.indicator f) hp_top _ hε M _,
{ refine ⟨δ, hδpos, λ s hs hμs, _⟩,
rw (_ : f = {x : α | M ≤ ‖f x‖₊}.indicator f + {x : α | ‖f x‖ < M}.indicator f),
{ rw snorm_indicator_eq_snorm_restrict hs,
refine le_trans (snorm_add_le _ _ hp_one) _,
{ exact strongly_measurable.ae_strongly_measurable (hmeas.indicator
(measurable_set_le measurable_const hmeas.nnnorm.measurable.subtype_coe)) },
{ exact strongly_measurable.ae_strongly_measurable (hmeas.indicator
(measurable_set_lt hmeas.nnnorm.measurable.subtype_coe measurable_const)) },
{ rw two_mul,
refine add_le_add (le_trans (snorm_mono_measure _ measure.restrict_le_self) hM) _,
rw ← snorm_indicator_eq_snorm_restrict hs,
exact hδ s hs hμs } },
{ ext x,
by_cases hx : M ≤ ‖f x‖,
{ rw [pi.add_apply, indicator_of_mem, indicator_of_not_mem, add_zero];
simpa },
{ rw [pi.add_apply, indicator_of_not_mem, indicator_of_mem, zero_add];
simpa using hx } } },
{ intros x,
rw [norm_indicator_eq_indicator_norm, indicator_apply],
split_ifs,
exacts [h, hMpos] }
end
/-- This lemma is superceded by `measure_theory.mem_ℒp.snorm_indicator_le` which does not require
measurability on `f`. -/
lemma mem_ℒp.snorm_indicator_le_of_meas (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
(hf : mem_ℒp f p μ) (hmeas : strongly_measurable f) {ε : ℝ} (hε : 0 < ε) :
∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, measurable_set s → μ s ≤ ennreal.of_real δ →
snorm (s.indicator f) p μ ≤ ennreal.of_real ε :=
begin
obtain ⟨δ, hδpos, hδ⟩ := hf.snorm_indicator_le' μ hp_one hp_top hmeas (half_pos hε),
refine ⟨δ, hδpos, λ s hs hμs, le_trans (hδ s hs hμs) _⟩,
rw [ennreal.of_real_div_of_pos zero_lt_two, (by norm_num : ennreal.of_real 2 = 2),
ennreal.mul_div_cancel'];
norm_num,
end
lemma mem_ℒp.snorm_indicator_le (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
(hf : mem_ℒp f p μ) {ε : ℝ} (hε : 0 < ε) :
∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, measurable_set s → μ s ≤ ennreal.of_real δ →
snorm (s.indicator f) p μ ≤ ennreal.of_real ε :=
begin
have hℒp := hf,
obtain ⟨⟨f', hf', heq⟩, hnorm⟩ := hf,
obtain ⟨δ, hδpos, hδ⟩ := (hℒp.ae_eq heq).snorm_indicator_le_of_meas μ hp_one hp_top hf' hε,
refine ⟨δ, hδpos, λ s hs hμs, _⟩,
convert hδ s hs hμs using 1,
rw [snorm_indicator_eq_snorm_restrict hs, snorm_indicator_eq_snorm_restrict hs],
refine snorm_congr_ae heq.restrict,
end
/-- A constant function is uniformly integrable. -/
lemma unif_integrable_const {g : α → β} (hp : 1 ≤ p) (hp_ne_top : p ≠ ∞) (hg : mem_ℒp g p μ) :
unif_integrable (λ n : ι, g) p μ :=
begin
intros ε hε,
obtain ⟨δ, hδ_pos, hgδ⟩ := hg.snorm_indicator_le μ hp hp_ne_top hε,
exact ⟨δ, hδ_pos, λ i, hgδ⟩,
end
/-- A single function is uniformly integrable. -/
lemma unif_integrable_subsingleton [subsingleton ι]
(hp_one : 1 ≤ p) (hp_top : p ≠ ∞) {f : ι → α → β} (hf : ∀ i, mem_ℒp (f i) p μ) :
unif_integrable f p μ :=
begin
intros ε hε,
by_cases hι : nonempty ι,
{ cases hι with i,
obtain ⟨δ, hδpos, hδ⟩ := (hf i).snorm_indicator_le μ hp_one hp_top hε,
refine ⟨δ, hδpos, λ j s hs hμs, _⟩,
convert hδ s hs hμs },
{ exact ⟨1, zero_lt_one, λ i, false.elim $ hι $ nonempty.intro i⟩ }
end
/-- This lemma is less general than `measure_theory.unif_integrable_fintype` which applies to
all sequences indexed by a finite type. -/
lemma unif_integrable_fin (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
{n : ℕ} {f : fin n → α → β} (hf : ∀ i, mem_ℒp (f i) p μ) :
unif_integrable f p μ :=
begin
revert f,
induction n with n h,
{ exact (λ f hf, unif_integrable_subsingleton μ hp_one hp_top hf) },
intros f hfLp ε hε,
set g : fin n → α → β := λ k, f k with hg,
have hgLp : ∀ i, mem_ℒp (g i) p μ := λ i, hfLp i,
obtain ⟨δ₁, hδ₁pos, hδ₁⟩ := h hgLp hε,
obtain ⟨δ₂, hδ₂pos, hδ₂⟩ := (hfLp n).snorm_indicator_le μ hp_one hp_top hε,
refine ⟨min δ₁ δ₂, lt_min hδ₁pos hδ₂pos, λ i s hs hμs, _⟩,
by_cases hi : i.val < n,
{ rw (_ : f i = g ⟨i.val, hi⟩),
{ exact hδ₁ _ s hs (le_trans hμs $ ennreal.of_real_le_of_real $ min_le_left _ _) },
{ rw hg, simp } },
{ rw (_ : i = n),
{ exact hδ₂ _ hs (le_trans hμs $ ennreal.of_real_le_of_real $ min_le_right _ _) },
{ have hi' := fin.is_lt i,
rw nat.lt_succ_iff at hi',
rw not_lt at hi,
simp [← le_antisymm hi' hi] } }
end
/-- A finite sequence of Lp functions is uniformly integrable. -/
lemma unif_integrable_finite [finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
{f : ι → α → β} (hf : ∀ i, mem_ℒp (f i) p μ) :
unif_integrable f p μ :=
begin
obtain ⟨n, hn⟩ := finite.exists_equiv_fin ι,
intros ε hε,
set g : fin n → α → β := f ∘ hn.some.symm with hgeq,
have hg : ∀ i, mem_ℒp (g i) p μ := λ _, hf _,
obtain ⟨δ, hδpos, hδ⟩ := unif_integrable_fin μ hp_one hp_top hg hε,
refine ⟨δ, hδpos, λ i s hs hμs, _⟩,
specialize hδ (hn.some i) s hs hμs,
simp_rw [hgeq, function.comp_app, equiv.symm_apply_apply] at hδ,
assumption,
end
end
lemma snorm_sub_le_of_dist_bdd
{p : ℝ≥0∞} (hp' : p ≠ ∞) {s : set α} (hs : measurable_set[m] s)
{f g : α → β} {c : ℝ} (hc : 0 ≤ c) (hf : ∀ x ∈ s, dist (f x) (g x) ≤ c) :
snorm (s.indicator (f - g)) p μ ≤ ennreal.of_real c * μ s ^ (1 / p.to_real) :=
begin
by_cases hp : p = 0,
{ simp [hp], },
have : ∀ x, ‖s.indicator (f - g) x‖ ≤ ‖s.indicator (λ x, c) x‖,
{ intro x,
by_cases hx : x ∈ s,
{ rw [indicator_of_mem hx, indicator_of_mem hx, pi.sub_apply, ← dist_eq_norm,
real.norm_eq_abs, abs_of_nonneg hc],
exact hf x hx },
{ simp [indicator_of_not_mem hx] } },
refine le_trans (snorm_mono this) _,
rw snorm_indicator_const hs hp hp',
refine ennreal.mul_le_mul (le_of_eq _) le_rfl,
rw [← of_real_norm_eq_coe_nnnorm, real.norm_eq_abs, abs_of_nonneg hc],
end
/-- A sequence of uniformly integrable functions which converges μ-a.e. converges in Lp. -/
lemma tendsto_Lp_of_tendsto_ae_of_meas [is_finite_measure μ]
(hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ℕ → α → β} {g : α → β}
(hf : ∀ n, strongly_measurable (f n)) (hg : strongly_measurable g)
(hg' : mem_ℒp g p μ) (hui : unif_integrable f p μ)
(hfg : ∀ᵐ x ∂μ, tendsto (λ n, f n x) at_top (𝓝 (g x))) :
tendsto (λ n, snorm (f n - g) p μ) at_top (𝓝 0) :=
begin
rw ennreal.tendsto_at_top_zero,
intros ε hε,
by_cases ε < ∞, swap,
{ rw [not_lt, top_le_iff] at h,
exact ⟨0, λ n hn, by simp [h]⟩ },
by_cases hμ : μ = 0,
{ exact ⟨0, λ n hn, by simp [hμ]⟩ },
have hε' : 0 < ε.to_real / 3 :=
div_pos (ennreal.to_real_pos (gt_iff_lt.1 hε).ne.symm h.ne) (by norm_num),
have hdivp : 0 ≤ 1 / p.to_real,
{ refine one_div_nonneg.2 _,
rw [← ennreal.zero_to_real, ennreal.to_real_le_to_real ennreal.zero_ne_top hp'],
exact le_trans ennreal.zero_lt_one.le hp },
have hpow : 0 < (measure_univ_nnreal μ) ^ (1 / p.to_real) :=
real.rpow_pos_of_pos (measure_univ_nnreal_pos hμ) _,
obtain ⟨δ₁, hδ₁, hsnorm₁⟩ := hui hε',
obtain ⟨δ₂, hδ₂, hsnorm₂⟩ := hg'.snorm_indicator_le μ hp hp' hε',
obtain ⟨t, htm, ht₁, ht₂⟩ := tendsto_uniformly_on_of_ae_tendsto' hf
hg hfg (lt_min hδ₁ hδ₂),
rw metric.tendsto_uniformly_on_iff at ht₂,
specialize ht₂ (ε.to_real / (3 * measure_univ_nnreal μ ^ (1 / p.to_real)))
(div_pos (ennreal.to_real_pos (gt_iff_lt.1 hε).ne.symm h.ne) (mul_pos (by norm_num) hpow)),
obtain ⟨N, hN⟩ := eventually_at_top.1 ht₂, clear ht₂,
refine ⟨N, λ n hn, _⟩,
rw [← t.indicator_self_add_compl (f n - g)],
refine le_trans (snorm_add_le ((((hf n).sub hg).indicator htm).ae_strongly_measurable)
(((hf n).sub hg).indicator htm.compl).ae_strongly_measurable hp) _,
rw [sub_eq_add_neg, indicator_add' t, indicator_neg'],
refine le_trans (add_le_add_right (snorm_add_le ((hf n).indicator htm).ae_strongly_measurable
(hg.indicator htm).neg.ae_strongly_measurable hp) _) _,
have hnf : snorm (t.indicator (f n)) p μ ≤ ennreal.of_real (ε.to_real / 3),
{ refine hsnorm₁ n t htm (le_trans ht₁ _),
rw ennreal.of_real_le_of_real_iff hδ₁.le,
exact min_le_left _ _ },
have hng : snorm (t.indicator g) p μ ≤ ennreal.of_real (ε.to_real / 3),
{ refine hsnorm₂ t htm (le_trans ht₁ _),
rw ennreal.of_real_le_of_real_iff hδ₂.le,
exact min_le_right _ _ },
have hlt : snorm (tᶜ.indicator (f n - g)) p μ ≤ ennreal.of_real (ε.to_real / 3),
{ specialize hN n hn,
have := snorm_sub_le_of_dist_bdd μ
hp' htm.compl _ (λ x hx, (dist_comm (g x) (f n x) ▸ (hN x hx).le :
dist (f n x) (g x) ≤ ε.to_real / (3 * measure_univ_nnreal μ ^ (1 / p.to_real)))),
refine le_trans this _,
rw [div_mul_eq_div_mul_one_div, ← ennreal.of_real_to_real (measure_lt_top μ tᶜ).ne,
ennreal.of_real_rpow_of_nonneg ennreal.to_real_nonneg hdivp, ← ennreal.of_real_mul,
mul_assoc],
{ refine ennreal.of_real_le_of_real (mul_le_of_le_one_right hε'.le _),
rw [mul_comm, mul_one_div, div_le_one],
{ refine real.rpow_le_rpow ennreal.to_real_nonneg
(ennreal.to_real_le_of_le_of_real (measure_univ_nnreal_pos hμ).le _) hdivp,
rw [ennreal.of_real_coe_nnreal, coe_measure_univ_nnreal],
exact measure_mono (subset_univ _) },
{ exact real.rpow_pos_of_pos (measure_univ_nnreal_pos hμ) _ } },
{ refine mul_nonneg (hε').le (one_div_nonneg.2 hpow.le) },
{ rw div_mul_eq_div_mul_one_div,
exact mul_nonneg hε'.le (one_div_nonneg.2 hpow.le) } },
have : ennreal.of_real (ε.to_real / 3) = ε / 3,
{ rw [ennreal.of_real_div_of_pos (show (0 : ℝ) < 3, by norm_num), ennreal.of_real_to_real h.ne],
simp },
rw this at hnf hng hlt,
rw [snorm_neg, ← ennreal.add_thirds ε, ← sub_eq_add_neg],
exact add_le_add_three hnf hng hlt
end
/-- A sequence of uniformly integrable functions which converges μ-a.e. converges in Lp. -/
lemma tendsto_Lp_of_tendsto_ae [is_finite_measure μ]
(hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ℕ → α → β} {g : α → β}
(hf : ∀ n, ae_strongly_measurable (f n) μ) (hg : mem_ℒp g p μ) (hui : unif_integrable f p μ)
(hfg : ∀ᵐ x ∂μ, tendsto (λ n, f n x) at_top (𝓝 (g x))) :
tendsto (λ n, snorm (f n - g) p μ) at_top (𝓝 0) :=
begin
suffices : tendsto (λ (n : ℕ), snorm ((hf n).mk (f n) - (hg.1.mk g)) p μ) at_top (𝓝 0),
{ convert this,
exact funext (λ n, snorm_congr_ae ((hf n).ae_eq_mk.sub hg.1.ae_eq_mk)), },
refine tendsto_Lp_of_tendsto_ae_of_meas μ hp hp' (λ n, (hf n).strongly_measurable_mk)
hg.1.strongly_measurable_mk (hg.ae_eq hg.1.ae_eq_mk) (hui.ae_eq (λ n, (hf n).ae_eq_mk)) _,
have h_ae_forall_eq : ∀ᵐ x ∂μ, ∀ n, f n x = (hf n).mk (f n) x,
{ rw ae_all_iff,
exact λ n, (hf n).ae_eq_mk, },
filter_upwards [hfg, h_ae_forall_eq, hg.1.ae_eq_mk] with x hx_tendsto hxf_eq hxg_eq,
rw ← hxg_eq,
convert hx_tendsto,
ext1 n,
exact (hxf_eq n).symm,
end
variables {f : ℕ → α → β} {g : α → β}
lemma unif_integrable_of_tendsto_Lp_zero (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ n, mem_ℒp (f n) p μ) (hf_tendsto : tendsto (λ n, snorm (f n) p μ) at_top (𝓝 0)) :
unif_integrable f p μ :=
begin
intros ε hε,
rw ennreal.tendsto_at_top_zero at hf_tendsto,
obtain ⟨N, hN⟩ := hf_tendsto (ennreal.of_real ε) (by simpa),
set F : fin N → α → β := λ n, f n,
have hF : ∀ n, mem_ℒp (F n) p μ := λ n, hf n,
obtain ⟨δ₁, hδpos₁, hδ₁⟩ := unif_integrable_fin μ hp hp' hF hε,
refine ⟨δ₁, hδpos₁, λ n s hs hμs, _⟩,
by_cases hn : n < N,
{ exact hδ₁ ⟨n, hn⟩ s hs hμs },
{ exact (snorm_indicator_le _).trans (hN n (not_lt.1 hn)) },
end
/-- Convergence in Lp implies uniform integrability. -/
lemma unif_integrable_of_tendsto_Lp
(hp : 1 ≤ p) (hp' : p ≠ ∞) (hf : ∀ n, mem_ℒp (f n) p μ) (hg : mem_ℒp g p μ)
(hfg : tendsto (λ n, snorm (f n - g) p μ) at_top (𝓝 0)) :
unif_integrable f p μ :=
begin
have : f = (λ n, g) + λ n, f n - g, by { ext1 n, simp, },
rw this,
refine unif_integrable.add _ _ hp
(λ _, hg.ae_strongly_measurable) (λ n, (hf n).1.sub hg.ae_strongly_measurable),
{ exact unif_integrable_const μ hp hp' hg, },
{ exact unif_integrable_of_tendsto_Lp_zero μ hp hp' (λ n, (hf n).sub hg) hfg, },
end
/-- Forward direction of Vitali's convergence theorem: if `f` is a sequence of uniformly integrable
functions that converge in measure to some function `g` in a finite measure space, then `f`
converge in Lp to `g`. -/
lemma tendsto_Lp_of_tendsto_in_measure [is_finite_measure μ]
(hp : 1 ≤ p) (hp' : p ≠ ∞) (hf : ∀ n, ae_strongly_measurable (f n) μ)
(hg : mem_ℒp g p μ) (hui : unif_integrable f p μ)
(hfg : tendsto_in_measure μ f at_top g) :
tendsto (λ n, snorm (f n - g) p μ) at_top (𝓝 0) :=
begin
refine tendsto_of_subseq_tendsto (λ ns hns, _),
obtain ⟨ms, hms, hms'⟩ := tendsto_in_measure.exists_seq_tendsto_ae (λ ε hε, (hfg ε hε).comp hns),
exact ⟨ms, tendsto_Lp_of_tendsto_ae μ hp hp' (λ _, hf _) hg
(λ ε hε, let ⟨δ, hδ, hδ'⟩ := hui hε in ⟨δ, hδ, λ i s hs hμs, hδ' _ s hs hμs⟩) hms'⟩,
end
/-- **Vitali's convergence theorem**: A sequence of functions `f` converges to `g` in Lp if and
only if it is uniformly integrable and converges to `g` in measure. -/
lemma tendsto_in_measure_iff_tendsto_Lp [is_finite_measure μ]
(hp : 1 ≤ p) (hp' : p ≠ ∞) (hf : ∀ n, mem_ℒp (f n) p μ) (hg : mem_ℒp g p μ) :
tendsto_in_measure μ f at_top g ∧ unif_integrable f p μ ↔
tendsto (λ n, snorm (f n - g) p μ) at_top (𝓝 0) :=
⟨λ h, tendsto_Lp_of_tendsto_in_measure μ hp hp' (λ n, (hf n).1) hg h.2 h.1,
λ h, ⟨tendsto_in_measure_of_tendsto_snorm
(lt_of_lt_of_le ennreal.zero_lt_one hp).ne.symm
(λ n, (hf n).ae_strongly_measurable)
hg.ae_strongly_measurable h, unif_integrable_of_tendsto_Lp μ hp hp' hf hg h⟩⟩
/-- This lemma is superceded by `unif_integrable_of` which do not require `C` to be positive. -/
lemma unif_integrable_of' (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
(hf : ∀ i, strongly_measurable (f i))
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0, 0 < C ∧
∀ i, snorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ennreal.of_real ε) :
unif_integrable f p μ :=
begin
have hpzero := (lt_of_lt_of_le ennreal.zero_lt_one hp).ne.symm,
by_cases hμ : μ set.univ = 0,
{ rw measure.measure_univ_eq_zero at hμ,
exact hμ.symm ▸ unif_integrable_zero_meas },
intros ε hε,
obtain ⟨C, hCpos, hC⟩ := h (ε / 2) (half_pos hε),
refine ⟨(ε / (2 * C)) ^ ennreal.to_real p, real.rpow_pos_of_pos
(div_pos hε (mul_pos two_pos (nnreal.coe_pos.2 hCpos))) _, λ i s hs hμs, _⟩,
by_cases hμs' : μ s = 0,
{ rw (snorm_eq_zero_iff ((hf i).indicator hs).ae_strongly_measurable hpzero).2
(indicator_meas_zero hμs'),
norm_num },
calc snorm (indicator s (f i)) p μ ≤ snorm (indicator (s ∩ {x | C ≤ ‖f i x‖₊}) (f i)) p μ +
snorm (indicator (s ∩ {x | ‖f i x‖₊ < C}) (f i)) p μ :
begin
refine le_trans (eq.le _) (snorm_add_le (strongly_measurable.ae_strongly_measurable
((hf i).indicator (hs.inter (strongly_measurable_const.measurable_set_le (hf i).nnnorm))))
(strongly_measurable.ae_strongly_measurable ((hf i).indicator (hs.inter
((hf i).nnnorm.measurable_set_lt strongly_measurable_const)))) hp),
congr,
change _ = λ x, (s ∩ {x : α | C ≤ ‖f i x‖₊}).indicator (f i) x +
(s ∩ {x : α | ‖f i x‖₊ < C}).indicator (f i) x,
rw ← set.indicator_union_of_disjoint,
{ congr,
rw [← inter_union_distrib_left, (by { ext, simp [le_or_lt] } :
{x : α | C ≤ ‖f i x‖₊} ∪ {x : α | ‖f i x‖₊ < C} = set.univ), inter_univ] },
{ refine (disjoint.inf_right' _ _).inf_left' _,
rw disjoint_iff_inf_le,
rintro x ⟨hx₁ : _ ≤ _, hx₂ : _ < _⟩,
exact false.elim (hx₂.ne (eq_of_le_of_not_lt hx₁ (not_lt.2 hx₂.le)).symm) }
end
... ≤ snorm (indicator ({x | C ≤ ‖f i x‖₊}) (f i)) p μ + C * μ s ^ (1 / ennreal.to_real p) :
begin
refine add_le_add (snorm_mono $ λ x, norm_indicator_le_of_subset
(inter_subset_right _ _) _ _) _,
rw ← indicator_indicator,
rw snorm_indicator_eq_snorm_restrict,
have : ∀ᵐ x ∂(μ.restrict s), ‖({x : α | ‖f i x‖₊ < C}).indicator (f i) x‖ ≤ C,
{ refine ae_of_all _ _,
simp_rw norm_indicator_eq_indicator_norm,
exact indicator_le' (λ x (hx : _ < _), hx.le) (λ _ _, nnreal.coe_nonneg _) },
refine le_trans (snorm_le_of_ae_bound this) _,
rw [mul_comm, measure.restrict_apply' hs, univ_inter,
ennreal.of_real_coe_nnreal, one_div],
exacts [le_rfl, hs],
end
... ≤ ennreal.of_real (ε / 2) + C * ennreal.of_real (ε / (2 * C)) :
begin
refine add_le_add (hC i) (mul_le_mul_left' _ _),
rwa [ennreal.rpow_one_div_le_iff (ennreal.to_real_pos hpzero hp'),
ennreal.of_real_rpow_of_pos (div_pos hε (mul_pos two_pos (nnreal.coe_pos.2 hCpos)))]
end
... ≤ ennreal.of_real (ε / 2) + ennreal.of_real (ε / 2) :
begin
refine add_le_add_left _ _,
rw [← ennreal.of_real_coe_nnreal, ← ennreal.of_real_mul (nnreal.coe_nonneg _),
← div_div, mul_div_cancel' _ (nnreal.coe_pos.2 hCpos).ne.symm],
exact le_rfl,
end
... ≤ ennreal.of_real ε :
begin
rw [← ennreal.of_real_add (half_pos hε).le (half_pos hε).le, add_halves],
exact le_rfl,
end
end
lemma unif_integrable_of (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
(hf : ∀ i, ae_strongly_measurable (f i) μ)
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
∀ i, snorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ennreal.of_real ε) :
unif_integrable f p μ :=
begin
set g : ι → α → β := λ i, (hf i).some,
refine (unif_integrable_of' μ hp hp' (λ i, (Exists.some_spec $hf i).1) (λ ε hε, _)).ae_eq
(λ i, (Exists.some_spec $ hf i).2.symm),
obtain ⟨C, hC⟩ := h ε hε,
have hCg : ∀ i, snorm ({x | C ≤ ‖g i x‖₊}.indicator (g i)) p μ ≤ ennreal.of_real ε,
{ intro i,
refine le_trans (le_of_eq $ snorm_congr_ae _) (hC i),
filter_upwards [(Exists.some_spec $ hf i).2] with x hx,
by_cases hfx : x ∈ {x | C ≤ ‖f i x‖₊},
{ rw [indicator_of_mem hfx, indicator_of_mem, hx],
rwa [mem_set_of, hx] at hfx },
{ rw [indicator_of_not_mem hfx, indicator_of_not_mem],
rwa [mem_set_of, hx] at hfx } },
refine ⟨max C 1, lt_max_of_lt_right one_pos, λ i, le_trans (snorm_mono (λ x, _)) (hCg i)⟩,
rw [norm_indicator_eq_indicator_norm, norm_indicator_eq_indicator_norm],
exact indicator_le_indicator_of_subset
(λ x hx, le_trans (le_max_left _ _) hx) (λ _, norm_nonneg _) _,
end
end unif_integrable
section uniform_integrable
/-! `uniform_integrable`
In probability theory, uniform integrability normally refers to the condition that a sequence
of function `(fₙ)` satisfies for all `ε > 0`, there exists some `C ≥ 0` such that
`∫ x in {|fₙ| ≥ C}, fₙ x ∂μ ≤ ε` for all `n`.
In this section, we will develope some API for `uniform_integrable` and prove that
`uniform_integrable` is equivalent to this definition of uniform integrability.
-/
variables {p : ℝ≥0∞} {f : ι → α → β}
lemma uniform_integrable_zero_meas [measurable_space α] :
uniform_integrable f p (0 : measure α) :=
⟨λ n, ae_strongly_measurable_zero_measure _,
unif_integrable_zero_meas, 0, λ i, snorm_measure_zero.le⟩
lemma uniform_integrable.ae_eq {g : ι → α → β}
(hf : uniform_integrable f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :
uniform_integrable g p μ :=
begin
obtain ⟨hfm, hunif, C, hC⟩ := hf,
refine ⟨λ i, (hfm i).congr (hfg i), (unif_integrable_congr_ae hfg).1 hunif, C, λ i, _⟩,
rw ← snorm_congr_ae (hfg i),
exact hC i
end
lemma uniform_integrable_congr_ae {g : ι → α → β}
(hfg : ∀ n, f n =ᵐ[μ] g n) :
uniform_integrable f p μ ↔ uniform_integrable g p μ :=
⟨λ h, h.ae_eq hfg, λ h, h.ae_eq (λ i, (hfg i).symm)⟩
/-- A finite sequence of Lp functions is uniformly integrable in the probability sense. -/
lemma uniform_integrable_finite [finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
(hf : ∀ i, mem_ℒp (f i) p μ) :
uniform_integrable f p μ :=
begin
casesI nonempty_fintype ι,
refine ⟨λ n, (hf n).1, unif_integrable_finite μ hp_one hp_top hf, _⟩,
by_cases hι : nonempty ι,
{ choose ae_meas hf using hf,
set C := (finset.univ.image (λ i : ι, snorm (f i) p μ)).max'
⟨snorm (f hι.some) p μ, finset.mem_image.2 ⟨hι.some, finset.mem_univ _, rfl⟩⟩,
refine ⟨C.to_nnreal, λ i, _⟩,
rw ennreal.coe_to_nnreal,
{ exact finset.le_max' _ _ (finset.mem_image.2 ⟨i, finset.mem_univ _, rfl⟩) },
{ refine ne_of_lt ((finset.max'_lt_iff _ _).2 (λ y hy, _)),
rw finset.mem_image at hy,
obtain ⟨i, -, rfl⟩ := hy,
exact hf i } },
{ exact ⟨0, λ i, false.elim $ hι $ nonempty.intro i⟩ }
end
/-- A single function is uniformly integrable in the probability sense. -/
lemma uniform_integrable_subsingleton [subsingleton ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
(hf : ∀ i, mem_ℒp (f i) p μ) :
uniform_integrable f p μ :=
uniform_integrable_finite hp_one hp_top hf
/-- A constant sequence of functions is uniformly integrable in the probability sense. -/
lemma uniform_integrable_const {g : α → β} (hp : 1 ≤ p) (hp_ne_top : p ≠ ∞)
(hg : mem_ℒp g p μ) :
uniform_integrable (λ n : ι, g) p μ :=
⟨λ i, hg.1, unif_integrable_const μ hp hp_ne_top hg,
⟨(snorm g p μ).to_nnreal, λ i, le_of_eq (ennreal.coe_to_nnreal hg.2.ne).symm⟩⟩
/-- This lemma is superceded by `uniform_integrable_of` which only requires
`ae_strongly_measurable`. -/
lemma uniform_integrable_of' [is_finite_measure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ i, strongly_measurable (f i))
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
∀ i, snorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ennreal.of_real ε) :
uniform_integrable f p μ :=
begin
refine ⟨λ i, (hf i).ae_strongly_measurable,
unif_integrable_of μ hp hp' (λ i, (hf i).ae_strongly_measurable) h, _⟩,
obtain ⟨C, hC⟩ := h 1 one_pos,
refine ⟨(C * (μ univ ^ (p.to_real⁻¹)) + 1 : ℝ≥0∞).to_nnreal, λ i, _⟩,
calc snorm (f i) p μ ≤ snorm ({x : α | ‖f i x‖₊ < C}.indicator (f i)) p μ +
snorm ({x : α | C ≤ ‖f i x‖₊}.indicator (f i)) p μ :
begin
refine le_trans (snorm_mono (λ x, _)) (snorm_add_le
(strongly_measurable.ae_strongly_measurable ((hf i).indicator
((hf i).nnnorm.measurable_set_lt strongly_measurable_const)))
(strongly_measurable.ae_strongly_measurable ((hf i).indicator
(strongly_measurable_const.measurable_set_le (hf i).nnnorm))) hp),
{ rw [pi.add_apply, indicator_apply],
split_ifs with hx,
{ rw [indicator_of_not_mem, add_zero],
simpa using hx },
{ rw [indicator_of_mem, zero_add],
simpa using hx } }
end
... ≤ C * μ univ ^ (p.to_real⁻¹) + 1 :
begin
have : ∀ᵐ x ∂μ, ‖{x : α | ‖f i x‖₊ < C}.indicator (f i) x‖₊ ≤ C,
{ refine eventually_of_forall _,
simp_rw nnnorm_indicator_eq_indicator_nnnorm,
exact indicator_le (λ x (hx : _ < _), hx.le) },
refine add_le_add (le_trans (snorm_le_of_ae_bound this) _) (ennreal.of_real_one ▸ (hC i)),
rw [ennreal.of_real_coe_nnreal, mul_comm],
exact le_rfl,
end
... = (C * (μ univ ^ (p.to_real⁻¹)) + 1 : ℝ≥0∞).to_nnreal :
begin
rw ennreal.coe_to_nnreal,
exact ennreal.add_ne_top.2 ⟨ennreal.mul_ne_top ennreal.coe_ne_top
(ennreal.rpow_ne_top_of_nonneg (inv_nonneg.2 ennreal.to_real_nonneg)
(measure_lt_top _ _).ne), ennreal.one_ne_top⟩,
end
end
/-- A sequene of functions `(fₙ)` is uniformly integrable in the probability sense if for all
`ε > 0`, there exists some `C` such that `∫ x in {|fₙ| ≥ C}, fₙ x ∂μ ≤ ε` for all `n`. -/
lemma uniform_integrable_of [is_finite_measure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ i, ae_strongly_measurable (f i) μ)
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
∀ i, snorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ennreal.of_real ε) :
uniform_integrable f p μ :=
begin
set g : ι → α → β := λ i, (hf i).some,
have hgmeas : ∀ i, strongly_measurable (g i) := λ i, (Exists.some_spec $ hf i).1,
have hgeq : ∀ i, g i =ᵐ[μ] f i := λ i, (Exists.some_spec $ hf i).2.symm,
refine (uniform_integrable_of' hp hp' hgmeas $ λ ε hε, _).ae_eq hgeq,
obtain ⟨C, hC⟩ := h ε hε,
refine ⟨C, λ i, le_trans (le_of_eq $ snorm_congr_ae _) (hC i)⟩,
filter_upwards [(Exists.some_spec $ hf i).2] with x hx,
by_cases hfx : x ∈ {x | C ≤ ‖f i x‖₊},
{ rw [indicator_of_mem hfx, indicator_of_mem, hx],
rwa [mem_set_of, hx] at hfx },
{ rw [indicator_of_not_mem hfx, indicator_of_not_mem],
rwa [mem_set_of, hx] at hfx }
end
/-- This lemma is superceded by `uniform_integrable.spec` which does not require measurability. -/
lemma uniform_integrable.spec' (hp : p ≠ 0) (hp' : p ≠ ∞)
(hf : ∀ i, strongly_measurable (f i))
(hfu : uniform_integrable f p μ) {ε : ℝ} (hε : 0 < ε) :
∃ C : ℝ≥0, ∀ i, snorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ennreal.of_real ε :=
begin
obtain ⟨-, hfu, M, hM⟩ := hfu,
obtain ⟨δ, hδpos, hδ⟩ := hfu hε,
obtain ⟨C, hC⟩ : ∃ C : ℝ≥0, ∀ i, μ {x | C ≤ ‖f i x‖₊} ≤ ennreal.of_real δ,
{ by_contra hcon, push_neg at hcon,
choose ℐ hℐ using hcon,
lift δ to ℝ≥0 using hδpos.le,
have : ∀ C : ℝ≥0, C • (δ : ℝ≥0∞) ^ (1 / p.to_real) ≤ snorm (f (ℐ C)) p μ,
{ intros C,
calc C • (δ : ℝ≥0∞) ^ (1 / p.to_real) ≤ C • μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / p.to_real):
begin
rw [ennreal.smul_def, ennreal.smul_def, smul_eq_mul, smul_eq_mul],
simp_rw ennreal.of_real_coe_nnreal at hℐ,
refine ennreal.mul_le_mul le_rfl (ennreal.rpow_le_rpow (hℐ C).le
(one_div_nonneg.2 ennreal.to_real_nonneg)),
end
... ≤ snorm ({x | C ≤ ‖f (ℐ C) x‖₊}.indicator (f (ℐ C))) p μ :
begin
refine snorm_indicator_ge_of_bdd_below hp hp' _
(measurable_set_le measurable_const (hf _).nnnorm.measurable)
(eventually_of_forall $ λ x hx, _),
rwa [nnnorm_indicator_eq_indicator_nnnorm, indicator_of_mem hx],
end
... ≤ snorm (f (ℐ C)) p μ : snorm_indicator_le _ },
specialize this ((2 * (max M 1) * (δ⁻¹ ^ (1 / p.to_real)))),
rw [ennreal.coe_rpow_of_nonneg _ (one_div_nonneg.2 ennreal.to_real_nonneg),
← ennreal.coe_smul, smul_eq_mul, mul_assoc, nnreal.inv_rpow,
inv_mul_cancel (nnreal.rpow_pos (nnreal.coe_pos.1 hδpos)).ne.symm, mul_one,
ennreal.coe_mul, ← nnreal.inv_rpow] at this,
refine (lt_of_le_of_lt (le_trans (hM $ ℐ $ 2 * (max M 1) * (δ⁻¹ ^ (1 / p.to_real)))
(le_max_left M 1)) (lt_of_lt_of_le _ this)).ne rfl,
rw [← ennreal.coe_one, ← with_top.coe_max, ← ennreal.coe_mul, ennreal.coe_lt_coe],
exact lt_two_mul_self (lt_max_of_lt_right one_pos) },
exact ⟨C, λ i, hδ i _ (measurable_set_le measurable_const (hf i).nnnorm.measurable) (hC i)⟩,
end
lemma uniform_integrable.spec (hp : p ≠ 0) (hp' : p ≠ ∞)
(hfu : uniform_integrable f p μ) {ε : ℝ} (hε : 0 < ε) :
∃ C : ℝ≥0, ∀ i, snorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ennreal.of_real ε :=
begin
set g : ι → α → β := λ i, (hfu.1 i).some,
have hgmeas : ∀ i, strongly_measurable (g i) := λ i, (Exists.some_spec $ hfu.1 i).1,
have hgunif : uniform_integrable g p μ := hfu.ae_eq (λ i, (Exists.some_spec $ hfu.1 i).2),
obtain ⟨C, hC⟩ := hgunif.spec' hp hp' hgmeas hε,
refine ⟨C, λ i, le_trans (le_of_eq $ snorm_congr_ae _) (hC i)⟩,
filter_upwards [(Exists.some_spec $ hfu.1 i).2] with x hx,
by_cases hfx : x ∈ {x | C ≤ ‖f i x‖₊},
{ rw [indicator_of_mem hfx, indicator_of_mem, hx],
rwa [mem_set_of, hx] at hfx },
{ rw [indicator_of_not_mem hfx, indicator_of_not_mem],
rwa [mem_set_of, hx] at hfx }
end
/-- The definition of uniform integrable in mathlib is equivalent to the definition commonly
found in literature. -/
lemma uniform_integrable_iff [is_finite_measure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞) :
uniform_integrable f p μ ↔ (∀ i, ae_strongly_measurable (f i) μ) ∧
∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
∀ i, snorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ennreal.of_real ε :=
⟨λ h, ⟨h.1, λ ε, h.spec (lt_of_lt_of_le ennreal.zero_lt_one hp).ne.symm hp'⟩,
λ h, uniform_integrable_of hp hp' h.1 h.2⟩
/-- The averaging of a uniformly integrable sequence is also uniformly integrable. -/
lemma uniform_integrable_average (hp : 1 ≤ p)
{f : ℕ → α → ℝ} (hf : uniform_integrable f p μ) :
uniform_integrable (λ n, (∑ i in finset.range n, f i) / n) p μ :=
begin
obtain ⟨hf₁, hf₂, hf₃⟩ := hf,
refine ⟨λ n, _, λ ε hε, _, _⟩,
{ simp_rw div_eq_mul_inv,
exact (finset.ae_strongly_measurable_sum' _ (λ i _, hf₁ i)).mul
(ae_strongly_measurable_const : ae_strongly_measurable (λ x, (↑n : ℝ)⁻¹) μ) },
{ obtain ⟨δ, hδ₁, hδ₂⟩ := hf₂ hε,
refine ⟨δ, hδ₁, λ n s hs hle, _⟩,
simp_rw [div_eq_mul_inv, finset.sum_mul, set.indicator_finset_sum],
refine le_trans (snorm_sum_le (λ i hi, ((hf₁ i).mul_const (↑n)⁻¹).indicator hs) hp) _,
have : ∀ i, s.indicator (f i * (↑n)⁻¹) = (↑n : ℝ)⁻¹ • s.indicator (f i),
{ intro i,
rw [mul_comm, (_ : (↑n)⁻¹ * f i = λ ω, (↑n : ℝ)⁻¹ • f i ω)],
{ rw set.indicator_const_smul s (↑n)⁻¹ (f i),
refl },
{ refl } },
simp_rw [this, snorm_const_smul, ← finset.mul_sum, nnnorm_inv, real.nnnorm_coe_nat],
by_cases hn : (↑(↑n : ℝ≥0)⁻¹ : ℝ≥0∞) = 0,
{ simp only [hn, zero_mul, zero_le] },
refine le_trans _ (_ : ↑(↑n : ℝ≥0)⁻¹ * (n • ennreal.of_real ε) ≤ ennreal.of_real ε),
{ refine (ennreal.mul_le_mul_left hn ennreal.coe_ne_top).2 _,
conv_rhs { rw ← finset.card_range n },
exact finset.sum_le_card_nsmul _ _ _ (λ i hi, hδ₂ _ _ hs hle) },
{ simp only [ennreal.coe_eq_zero, inv_eq_zero, nat.cast_eq_zero] at hn,
rw [nsmul_eq_mul, ← mul_assoc, ennreal.coe_inv, ennreal.coe_nat,
ennreal.inv_mul_cancel _ (ennreal.nat_ne_top _), one_mul],
{ exact le_rfl },
all_goals { simpa only [ne.def, nat.cast_eq_zero] } } },
{ obtain ⟨C, hC⟩ := hf₃,
simp_rw [div_eq_mul_inv, finset.sum_mul],
refine ⟨C, λ n, (snorm_sum_le (λ i hi, (hf₁ i).mul_const (↑n)⁻¹) hp).trans _⟩,
have : ∀ i, (λ ω, f i ω * (↑n)⁻¹) = (↑n : ℝ)⁻¹ • λ ω, f i ω,
{ intro i,
ext ω,
simp only [mul_comm, pi.smul_apply, algebra.id.smul_eq_mul] },
simp_rw [this, snorm_const_smul, ← finset.mul_sum, nnnorm_inv, real.nnnorm_coe_nat],
by_cases hn : (↑(↑n : ℝ≥0)⁻¹ : ℝ≥0∞) = 0,
{ simp only [hn, zero_mul, zero_le] },
refine le_trans _ (_ : ↑(↑n : ℝ≥0)⁻¹ * (n • C : ℝ≥0∞) ≤ C),
{ refine (ennreal.mul_le_mul_left hn ennreal.coe_ne_top).2 _,
conv_rhs { rw ← finset.card_range n },
exact finset.sum_le_card_nsmul _ _ _ (λ i hi, hC i) },
{ simp only [ennreal.coe_eq_zero, inv_eq_zero, nat.cast_eq_zero] at hn,
rw [nsmul_eq_mul, ← mul_assoc, ennreal.coe_inv, ennreal.coe_nat,
ennreal.inv_mul_cancel _ (ennreal.nat_ne_top _), one_mul],
{ exact le_rfl },
all_goals { simpa only [ne.def, nat.cast_eq_zero] } } }
end
end uniform_integrable
end measure_theory
|
3a60b527b2dc7873f896d334524dff1dddbcedc4 | 8930e38ac0fae2e5e55c28d0577a8e44e2639a6d | /data/equiv/encodable.lean | 120a0f47e4e44f4c868275b66f3209392a8dc16e | [
"Apache-2.0"
] | permissive | SG4316/mathlib | 3d64035d02a97f8556ad9ff249a81a0a51a3321a | a7846022507b531a8ab53b8af8a91953fceafd3a | refs/heads/master | 1,584,869,960,527 | 1,530,718,645,000 | 1,530,724,110,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 9,923 | lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Mario Carneiro
Type class for encodable Types.
Note that every encodable Type is countable.
-/
import data.equiv.nat
open option list nat function
/-- An encodable type is a "constructively countable" type. This is where
we have an explicit injection `encode : α → nat` and a partial inverse
`decode : nat → option α`. This makes the range of `encode` decidable,
although it is not decidable if `α` is finite or not. -/
class encodable (α : Type*) :=
(encode : α → nat) (decode : nat → option α) (encodek : ∀ a, decode (encode a) = some a)
namespace encodable
variables {α : Type*} {β : Type*}
universe u
open encodable
theorem encode_injective [encodable α] : function.injective (@encode α _)
| x y e := option.some.inj $ by rw [← encodek, e, encodek]
/- This is not set as an instance because this is usually not the best way
to infer decidability. -/
def decidable_eq_of_encodable (α) [encodable α] : decidable_eq α
| a b := decidable_of_iff _ encode_injective.eq_iff
def of_left_injection [encodable α]
(f : β → α) (finv : α → option β) (linv : ∀ b, finv (f b) = some b) : encodable β :=
⟨λ b, encode (f b),
λ n, (decode α n).bind finv,
λ b, by simp [encodable.encodek, option.bind, linv]⟩
def of_left_inverse [encodable α]
(f : β → α) (finv : α → β) (linv : ∀ b, finv (f b) = b) : encodable β :=
of_left_injection f (some ∘ finv) (λ b, congr_arg some (linv b))
def of_equiv (α) [encodable α] (e : β ≃ α) : encodable β :=
of_left_inverse e e.symm e.left_inv
@[simp] theorem encode_of_equiv {α β} [encodable α] (e : β ≃ α) (b : β) :
@encode _ (of_equiv _ e) b = encode (e b) := rfl
@[simp] theorem decode_of_equiv {α β} [encodable α] (e : β ≃ α) (n : ℕ) :
@decode _ (of_equiv _ e) n = (decode α n).map e.symm := rfl
instance nat : encodable nat :=
⟨id, some, λ a, rfl⟩
@[simp] theorem encode_nat (n : ℕ) : encode n = n := rfl
@[simp] theorem decode_nat (n : ℕ) : decode ℕ n = some n := rfl
instance empty : encodable empty :=
⟨λ a, a.rec _, λ n, none, λ a, a.rec _⟩
instance unit : encodable punit :=
⟨λ_, zero, λn, nat.cases_on n (some punit.star) (λ _, none), λ⟨⟩, by simp⟩
@[simp] theorem encode_star : encode punit.star = 0 := rfl
@[simp] theorem decode_unit_zero : decode punit 0 = some punit.star := rfl
@[simp] theorem decode_unit_succ (n) : decode punit (succ n) = none := rfl
instance option {α : Type*} [h : encodable α] : encodable (option α) :=
⟨λ o, option.cases_on o nat.zero (λ a, succ (encode a)),
λ n, nat.cases_on n (some none) (λ m, (decode α m).map some),
λ o, by cases o; dsimp; simp [encodek, nat.succ_ne_zero]⟩
@[simp] theorem encode_none [encodable α] : encode (@none α) = 0 := rfl
@[simp] theorem encode_some [encodable α] (a : α) :
encode (some a) = succ (encode a) := rfl
@[simp] theorem decode_option_zero [encodable α] : decode (option α) 0 = some none := rfl
@[simp] theorem decode_option_succ [encodable α] (n) :
decode (option α) (succ n) = (decode α n).map some := rfl
def decode2 (α) [encodable α] (n : ℕ) : option α :=
(decode α n).bind (option.guard (λ a, encode a = n))
theorem mem_decode2 [encodable α] {n : ℕ} {a : α} :
a ∈ decode2 α n ↔ a ∈ decode α n ∧ encode a = n :=
by simp [decode2]; exact
⟨λ ⟨_, h₁, rfl, h₂⟩, ⟨h₁, h₂⟩, λ ⟨h₁, h₂⟩, ⟨_, h₁, rfl, h₂⟩⟩
theorem decode2_inj [encodable α] {n : ℕ} {a₁ a₂ : α}
(h₁ : a₁ ∈ decode2 α n) (h₂ : a₂ ∈ decode2 α n) : a₁ = a₂ :=
encode_injective $ (mem_decode2.1 h₁).2.trans (mem_decode2.1 h₂).2.symm
theorem encodek2 [encodable α] (a : α) : decode2 α (encode a) = some a :=
mem_decode2.2 ⟨encodek _, rfl⟩
section sum
variables [encodable α] [encodable β]
def encode_sum : α ⊕ β → nat
| (sum.inl a) := bit0 $ encode a
| (sum.inr b) := bit1 $ encode b
def decode_sum (n : nat) : option (α ⊕ β) :=
match bodd_div2 n with
| (ff, m) := (decode α m).map sum.inl
| (tt, m) := (decode β m).map sum.inr
end
instance sum : encodable (α ⊕ β) :=
⟨encode_sum, decode_sum, λ s,
by cases s; simp [encode_sum, decode_sum, encodek]; refl⟩
@[simp] theorem encode_inl (a : α) :
@encode (α ⊕ β) _ (sum.inl a) = bit0 (encode a) := rfl
@[simp] theorem encode_inr (b : β) :
@encode (α ⊕ β) _ (sum.inr b) = bit1 (encode b) := rfl
@[simp] theorem decode_sum_val (n : ℕ) :
decode (α ⊕ β) n = decode_sum n := rfl
end sum
instance bool : encodable bool :=
of_equiv (unit ⊕ unit) equiv.bool_equiv_unit_sum_unit
@[simp] theorem encode_tt : encode tt = 1 := rfl
@[simp] theorem encode_ff : encode ff = 0 := rfl
@[simp] theorem decode_zero : decode bool 0 = some ff := rfl
@[simp] theorem decode_one : decode bool 1 = some tt := rfl
theorem decode_ge_two (n) (h : 2 ≤ n) : decode bool n = none :=
begin
suffices : decode_sum n = none,
{ change (decode_sum n).map _ = none, rw this, refl },
have : 1 ≤ div2 n,
{ rw [div2_val, nat.le_div_iff_mul_le],
exacts [h, dec_trivial] },
cases exists_eq_succ_of_ne_zero (ne_of_gt this) with m e,
simp [decode_sum]; cases bodd n; simp [decode_sum]; rw e; refl
end
section sigma
variables {γ : α → Type*} [encodable α] [∀ a, encodable (γ a)]
def encode_sigma : sigma γ → ℕ
| ⟨a, b⟩ := mkpair (encode a) (encode b)
def decode_sigma (n : ℕ) : option (sigma γ) :=
let (n₁, n₂) := unpair n in
(decode α n₁).bind $ λ a, (decode (γ a) n₂).map $ sigma.mk a
instance sigma : encodable (sigma γ) :=
⟨encode_sigma, decode_sigma, λ ⟨a, b⟩,
by simp [encode_sigma, decode_sigma, option.bind, option.map, unpair_mkpair, encodek]⟩
@[simp] theorem decode_sigma_val (n : ℕ) : decode (sigma γ) n =
(decode α n.unpair.1).bind (λ a, (decode (γ a) n.unpair.2).map $ sigma.mk a) :=
show decode_sigma._match_1 _ = _, by cases n.unpair; refl
@[simp] theorem encode_sigma_val (a b) : @encode (sigma γ) _ ⟨a, b⟩ =
mkpair (encode a) (encode b) := rfl
end sigma
section prod
variables [encodable α] [encodable β]
instance prod : encodable (α × β) :=
of_equiv _ (equiv.sigma_equiv_prod α β).symm
@[simp] theorem decode_prod_val (n : ℕ) : decode (α × β) n =
(decode α n.unpair.1).bind (λ a, (decode β n.unpair.2).map $ prod.mk a) :=
show (decode (sigma (λ _, β)) n).map (equiv.sigma_equiv_prod α β) = _,
by simp; cases decode α n.unpair.1; simp [option.bind];
cases decode β n.unpair.2; refl
@[simp] theorem encode_prod_val (a b) : @encode (α × β) _ (a, b) =
mkpair (encode a) (encode b) := rfl
end prod
section subtype
open subtype decidable
variable {P : α → Prop}
variable [encA : encodable α]
variable [decP : decidable_pred P]
include encA
def encode_subtype : {a : α // P a} → nat
| ⟨v, h⟩ := encode v
include decP
def decode_subtype (v : nat) : option {a : α // P a} :=
(decode α v).bind $ λ a,
if h : P a then some ⟨a, h⟩ else none
instance subtype : encodable {a : α // P a} :=
⟨encode_subtype, decode_subtype,
λ ⟨v, h⟩, by simp [encode_subtype, decode_subtype, encodek, h]⟩
end subtype
instance fin (n) : encodable (fin n) :=
of_equiv _ (equiv.fin_equiv_subtype _)
instance int : encodable ℤ :=
of_equiv _ equiv.int_equiv_nat
instance ulift [encodable α] : encodable (ulift α) :=
of_equiv _ equiv.ulift
instance plift [encodable α] : encodable (plift α) :=
of_equiv _ equiv.plift
noncomputable def of_inj [encodable β] (f : α → β) (hf : injective f) : encodable α :=
of_left_injection f (partial_inv f) (λ x, (partial_inv_of_injective hf _ _).2 rfl)
end encodable
/-
Choice function for encodable types and decidable predicates.
We provide the following API
choose {α : Type*} {p : α → Prop} [c : encodable α] [d : decidable_pred p] : (∃ x, p x) → α :=
choose_spec {α : Type*} {p : α → Prop} [c : encodable α] [d : decidable_pred p] (ex : ∃ x, p x) : p (choose ex) :=
-/
namespace encodable
section find_a
variables {α : Type*} (p : α → Prop) [encodable α] [decidable_pred p]
private def good : option α → Prop
| (some a) := p a
| none := false
private def decidable_good : decidable_pred (good p)
| n := by cases n; unfold good; apply_instance
local attribute [instance] decidable_good
open encodable
variable {p}
def choose_x (h : ∃ x, p x) : {a:α // p a} :=
have ∃ n, good p (decode α n), from
let ⟨w, pw⟩ := h in ⟨encode w, by simp [good, encodek, pw]⟩,
match _, nat.find_spec this : ∀ o, good p o → {a // p a} with
| some a, h := ⟨a, h⟩
end
def choose (h : ∃ x, p x) : α := (choose_x h).1
lemma choose_spec (h : ∃ x, p x) : p (choose h) := (choose_x h).2
end find_a
theorem axiom_of_choice {α : Type*} {β : α → Type*} {R : Π x, β x → Prop}
[Π a, encodable (β a)] [∀ x y, decidable (R x y)]
(H : ∀x, ∃y, R x y) : ∃f:Πa, β a, ∀x, R x (f x) :=
⟨λ x, choose (H x), λ x, choose_spec (H x)⟩
theorem skolem {α : Type*} {β : α → Type*} {P : Π x, β x → Prop}
[c : Π a, encodable (β a)] [d : ∀ x y, decidable (P x y)] :
(∀x, ∃y, P x y) ↔ ∃f : Π a, β a, (∀x, P x (f x)) :=
⟨axiom_of_choice, λ ⟨f, H⟩ x, ⟨_, H x⟩⟩
end encodable
namespace quot
open encodable
variables {α : Type*} {s : setoid α} [@decidable_rel α (≈)] [encodable α]
-- Choose equivalence class representative
def rep (q : quotient s) : α :=
choose (exists_rep q)
theorem rep_spec (q : quotient s) : ⟦rep q⟧ = q :=
choose_spec (exists_rep q)
def encodable_quotient : encodable (quotient s) :=
⟨λ q, encode (rep q),
λ n, quotient.mk <$> decode α n,
λ q, quot.induction_on q $ λ l,
by rw encodek; exact congr_arg some (rep_spec _)⟩
end quot
|
db36eec42aeaedb4552a5a285317be10bc4d5b7c | 367134ba5a65885e863bdc4507601606690974c1 | /src/control/bitraversable/instances.lean | 37b46e8e5e6a6c3ea84c63aefd238da2ea9f71cd | [
"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 | 4,396 | lean | /-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author(s): Simon Hudon
-/
import control.bitraversable.lemmas
import control.traversable.lemmas
/-!
# bitraversable instances
## Instances
* prod
* sum
* const
* flip
* bicompl
* bicompr
## References
* Hackage: <https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>
## Tags
traversable bitraversable functor bifunctor applicative
-/
universes u v w
variables {t : Type u → Type u → Type u} [bitraversable t]
section
variables {F : Type u → Type u} [applicative F]
def prod.bitraverse {α α' β β'} (f : α → F α') (f' : β → F β') : α × β → F (α' × β')
| (x,y) := prod.mk <$> f x <*> f' y
instance : bitraversable prod :=
{ bitraverse := @prod.bitraverse }
instance : is_lawful_bitraversable prod :=
by constructor; introsI; cases x;
simp [bitraverse,prod.bitraverse] with functor_norm; refl
open functor
def sum.bitraverse {α α' β β'} (f : α → F α') (f' : β → F β') : α ⊕ β → F (α' ⊕ β')
| (sum.inl x) := sum.inl <$> f x
| (sum.inr x) := sum.inr <$> f' x
instance : bitraversable sum :=
{ bitraverse := @sum.bitraverse }
instance : is_lawful_bitraversable sum :=
by constructor; introsI; cases x;
simp [bitraverse,sum.bitraverse] with functor_norm; refl
def const.bitraverse {α α' β β'} (f : α → F α') (f' : β → F β') : const α β → F (const α' β') := f
instance bitraversable.const : bitraversable const :=
{ bitraverse := @const.bitraverse }
instance is_lawful_bitraversable.const : is_lawful_bitraversable const :=
by constructor; introsI;
simp [bitraverse,const.bitraverse] with functor_norm; refl
def flip.bitraverse {α α' β β'} (f : α → F α') (f' : β → F β') : flip t α β → F (flip t α' β') :=
(bitraverse f' f : t β α → F (t β' α'))
instance bitraversable.flip : bitraversable (flip t) :=
{ bitraverse := @flip.bitraverse t _ }
open is_lawful_bitraversable
instance is_lawful_bitraversable.flip [is_lawful_bitraversable t]
: is_lawful_bitraversable (flip t) :=
by constructor; intros; unfreezingI { casesm is_lawful_bitraversable t }; tactic.apply_assumption
open bitraversable functor
@[priority 10]
instance bitraversable.traversable {α} : traversable (t α) :=
{ traverse := @tsnd t _ _ }
@[priority 10]
instance bitraversable.is_lawful_traversable [is_lawful_bitraversable t] {α} :
is_lawful_traversable (t α) :=
by { constructor; introsI; simp [traverse,comp_tsnd] with functor_norm,
{ refl },
{ simp [tsnd_eq_snd_id], refl },
{ simp [tsnd,binaturality,function.comp] with functor_norm } }
end
open bifunctor traversable is_lawful_traversable is_lawful_bitraversable
open function (bicompl bicompr)
section bicompl
variables (F G : Type u → Type u) [traversable F] [traversable G]
def bicompl.bitraverse {m} [applicative m] {α β α' β'} (f : α → m β) (f' : α' → m β') :
bicompl t F G α α' → m (bicompl t F G β β') :=
(bitraverse (traverse f) (traverse f') : t (F α) (G α') → m _)
instance : bitraversable (bicompl t F G) :=
{ bitraverse := @bicompl.bitraverse t _ F G _ _ }
instance [is_lawful_traversable F] [is_lawful_traversable G] [is_lawful_bitraversable t] :
is_lawful_bitraversable (bicompl t F G) :=
begin
constructor; introsI;
simp [bitraverse, bicompl.bitraverse, bimap, traverse_id, bitraverse_id_id, comp_bitraverse]
with functor_norm,
{ simp [traverse_eq_map_id',bitraverse_eq_bimap_id], },
{ revert x, dunfold bicompl,
simp [binaturality,naturality_pf] }
end
end bicompl
section bicompr
variables (F : Type u → Type u) [traversable F]
def bicompr.bitraverse {m} [applicative m] {α β α' β'} (f : α → m β) (f' : α' → m β') :
bicompr F t α α' → m (bicompr F t β β') :=
(traverse (bitraverse f f') : F (t α α') → m _)
instance : bitraversable (bicompr F t) :=
{ bitraverse := @bicompr.bitraverse t _ F _ }
instance [is_lawful_traversable F] [is_lawful_bitraversable t] :
is_lawful_bitraversable (bicompr F t) :=
begin
constructor; introsI;
simp [bitraverse,bicompr.bitraverse,bitraverse_id_id] with functor_norm,
{ simp [bitraverse_eq_bimap_id',traverse_eq_map_id'], refl },
{ revert x, dunfold bicompr, intro,
simp [naturality,binaturality'] }
end
end bicompr
|
a7c634fccec3c326289b17c733569c8520a5b027 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/topology/metric_space/completion.lean | f9271f26ee2b8318f89aca9f277155284e8cc27d | [
"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 | 8,096 | lean | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import topology.uniform_space.completion
import topology.metric_space.isometry
import topology.instances.real
/-!
# The completion of a metric space
Completion of uniform spaces are already defined in `topology.uniform_space.completion`. We show
here that the uniform space completion of a metric space inherits a metric space structure,
by extending the distance to the completion and checking that it is indeed a distance, and that
it defines the same uniformity as the already defined uniform structure on the completion
-/
open set filter uniform_space metric
open_locale filter topological_space uniformity
noncomputable theory
universes u v
variables {α : Type u} {β : Type v} [pseudo_metric_space α]
namespace uniform_space.completion
/-- The distance on the completion is obtained by extending the distance on the original space,
by uniform continuity. -/
instance : has_dist (completion α) :=
⟨completion.extension₂ dist⟩
/-- The new distance is uniformly continuous. -/
protected lemma uniform_continuous_dist :
uniform_continuous (λp:completion α × completion α, dist p.1 p.2) :=
uniform_continuous_extension₂ dist
/-- The new distance is continuous. -/
protected lemma continuous_dist [topological_space β] {f g : β → completion α} (hf : continuous f)
(hg : continuous g) :
continuous (λ x, dist (f x) (g x)) :=
completion.uniform_continuous_dist.continuous.comp (hf.prod_mk hg : _)
/-- The new distance is an extension of the original distance. -/
@[simp] protected lemma dist_eq (x y : α) : dist (x : completion α) y = dist x y :=
completion.extension₂_coe_coe uniform_continuous_dist _ _
/- Let us check that the new distance satisfies the axioms of a distance, by starting from the
properties on α and extending them to `completion α` by continuity. -/
protected lemma dist_self (x : completion α) : dist x x = 0 :=
begin
apply induction_on x,
{ refine is_closed_eq _ continuous_const,
exact completion.continuous_dist continuous_id continuous_id },
{ assume a,
rw [completion.dist_eq, dist_self] }
end
protected lemma dist_comm (x y : completion α) : dist x y = dist y x :=
begin
apply induction_on₂ x y,
{ exact is_closed_eq (completion.continuous_dist continuous_fst continuous_snd)
(completion.continuous_dist continuous_snd continuous_fst) },
{ assume a b,
rw [completion.dist_eq, completion.dist_eq, dist_comm] }
end
protected lemma dist_triangle (x y z : completion α) : dist x z ≤ dist x y + dist y z :=
begin
apply induction_on₃ x y z,
{ refine is_closed_le _ (continuous.add _ _);
apply_rules [completion.continuous_dist, continuous.fst, continuous.snd, continuous_id] },
{ assume a b c,
rw [completion.dist_eq, completion.dist_eq, completion.dist_eq],
exact dist_triangle a b c }
end
/-- Elements of the uniformity (defined generally for completions) can be characterized in terms
of the distance. -/
protected lemma mem_uniformity_dist (s : set (completion α × completion α)) :
s ∈ 𝓤 (completion α) ↔ (∃ε>0, ∀{a b}, dist a b < ε → (a, b) ∈ s) :=
begin
split,
{ /- Start from an entourage `s`. It contains a closed entourage `t`. Its pullback in `α` is an
entourage, so it contains an `ε`-neighborhood of the diagonal by definition of the entourages
in metric spaces. Then `t` contains an `ε`-neighborhood of the diagonal in `completion α`, as
closed properties pass to the completion. -/
assume hs,
rcases mem_uniformity_is_closed hs with ⟨t, ht, ⟨tclosed, ts⟩⟩,
have A : {x : α × α | (coe (x.1), coe (x.2)) ∈ t} ∈ uniformity α :=
uniform_continuous_def.1 (uniform_continuous_coe α) t ht,
rcases mem_uniformity_dist.1 A with ⟨ε, εpos, hε⟩,
refine ⟨ε, εpos, λx y hxy, _⟩,
have : ε ≤ dist x y ∨ (x, y) ∈ t,
{ apply induction_on₂ x y,
{ have : {x : completion α × completion α | ε ≤ dist (x.fst) (x.snd) ∨ (x.fst, x.snd) ∈ t}
= {p : completion α × completion α | ε ≤ dist p.1 p.2} ∪ t, by ext; simp,
rw this,
apply is_closed.union _ tclosed,
exact is_closed_le continuous_const completion.uniform_continuous_dist.continuous },
{ assume x y,
rw completion.dist_eq,
by_cases h : ε ≤ dist x y,
{ exact or.inl h },
{ have Z := hε (not_le.1 h),
simp only [set.mem_set_of_eq] at Z,
exact or.inr Z }}},
simp only [not_le.mpr hxy, false_or, not_le] at this,
exact ts this },
{ /- Start from a set `s` containing an ε-neighborhood of the diagonal in `completion α`. To show
that it is an entourage, we use the fact that `dist` is uniformly continuous on
`completion α × completion α` (this is a general property of the extension of uniformly
continuous functions). Therefore, the preimage of the ε-neighborhood of the diagonal in ℝ
is an entourage in `completion α × completion α`. Massaging this property, it follows that
the ε-neighborhood of the diagonal is an entourage in `completion α`, and therefore this is
also the case of `s`. -/
rintros ⟨ε, εpos, hε⟩,
let r : set (ℝ × ℝ) := {p | dist p.1 p.2 < ε},
have : r ∈ uniformity ℝ := metric.dist_mem_uniformity εpos,
have T := uniform_continuous_def.1 (@completion.uniform_continuous_dist α _) r this,
simp only [uniformity_prod_eq_prod, mem_prod_iff, exists_prop,
filter.mem_map, set.mem_set_of_eq] at T,
rcases T with ⟨t1, ht1, t2, ht2, ht⟩,
refine mem_of_superset ht1 _,
have A : ∀a b : completion α, (a, b) ∈ t1 → dist a b < ε,
{ assume a b hab,
have : ((a, b), (a, a)) ∈ t1 ×ˢ t2 := ⟨hab, refl_mem_uniformity ht2⟩,
have I := ht this,
simp [completion.dist_self, real.dist_eq, completion.dist_comm] at I,
exact lt_of_le_of_lt (le_abs_self _) I },
show t1 ⊆ s,
{ rintros ⟨a, b⟩ hp,
have : dist a b < ε := A a b hp,
exact hε this }}
end
/-- If two points are at distance 0, then they coincide. -/
protected lemma eq_of_dist_eq_zero (x y : completion α) (h : dist x y = 0) : x = y :=
begin
/- This follows from the separation of `completion α` and from the description of
entourages in terms of the distance. -/
have : separated_space (completion α) := by apply_instance,
refine separated_def.1 this x y (λs hs, _),
rcases (completion.mem_uniformity_dist s).1 hs with ⟨ε, εpos, hε⟩,
rw ← h at εpos,
exact hε εpos
end
/-- Reformulate `completion.mem_uniformity_dist` in terms that are suitable for the definition
of the metric space structure. -/
protected lemma uniformity_dist' :
𝓤 (completion α) = (⨅ε:{ε : ℝ // 0 < ε}, 𝓟 {p | dist p.1 p.2 < ε.val}) :=
begin
ext s, rw mem_infi_of_directed,
{ simp [completion.mem_uniformity_dist, subset_def] },
{ rintro ⟨r, hr⟩ ⟨p, hp⟩, use ⟨min r p, lt_min hr hp⟩,
simp [lt_min_iff, (≥)] {contextual := tt} }
end
protected lemma uniformity_dist :
𝓤 (completion α) = (⨅ ε>0, 𝓟 {p | dist p.1 p.2 < ε}) :=
by simpa [infi_subtype] using @completion.uniformity_dist' α _
/-- Metric space structure on the completion of a pseudo_metric space. -/
instance : metric_space (completion α) :=
{ dist_self := completion.dist_self,
eq_of_dist_eq_zero := completion.eq_of_dist_eq_zero,
dist_comm := completion.dist_comm,
dist_triangle := completion.dist_triangle,
to_uniform_space := by apply_instance,
uniformity_dist := completion.uniformity_dist }
/-- The embedding of a metric space in its completion is an isometry. -/
lemma coe_isometry : isometry (coe : α → completion α) :=
isometry_emetric_iff_metric.2 completion.dist_eq
@[simp] protected lemma edist_eq (x y : α) : edist (x : completion α) y = edist x y :=
coe_isometry x y
end uniform_space.completion
|
a6c109f8653d56b8ca87dc7f2a98f960ad287ff9 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/data/polynomial/induction.lean | 10f7f2a91162fe554df8fa1939154667fe4e270d | [
"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 | 3,067 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import data.polynomial.coeff
import ring_theory.ideal.basic
/-!
# Induction on polynomials
This file contains lemmas dealing with different flavours of induction on polynomials.
See also `data/polynomial/inductions.lean` (with an `s`!).
The main result is `polynomial.induction_on`.
-/
noncomputable theory
open finsupp finset
namespace polynomial
open_locale polynomial
universes u v w x y z
variables {R : Type u} {S : Type v} {T : Type w} {ι : Type x} {k : Type y} {A : Type z}
{a b : R} {m n : ℕ}
section semiring
variables [semiring R] {p q r : R[X]}
@[elab_as_eliminator] protected lemma induction_on {M : R[X] → Prop} (p : R[X])
(h_C : ∀a, M (C a))
(h_add : ∀p q, M p → M q → M (p + q))
(h_monomial : ∀(n : ℕ) (a : R), M (C a * X^n) → M (C a * X^(n+1))) :
M p :=
begin
have A : ∀{n:ℕ} {a}, M (C a * X^n),
{ assume n a,
induction n with n ih,
{ simp only [pow_zero, mul_one, h_C] },
{ exact h_monomial _ _ ih } },
have B : ∀ (s : finset ℕ), M (s.sum (λ (n : ℕ), C (p.coeff n) * X ^ n)),
{ apply finset.induction,
{ convert h_C 0, exact C_0.symm },
{ assume n s ns ih, rw sum_insert ns, exact h_add _ _ A ih, } },
rw [← sum_C_mul_X_eq p, polynomial.sum],
exact B _
end
/--
To prove something about polynomials,
it suffices to show the condition is closed under taking sums,
and it holds for monomials.
-/
@[elab_as_eliminator] protected lemma induction_on' {M : R[X] → Prop} (p : R[X])
(h_add : ∀p q, M p → M q → M (p + q))
(h_monomial : ∀(n : ℕ) (a : R), M (monomial n a)) :
M p :=
polynomial.induction_on p (h_monomial 0) h_add
(λ n a h, by { rw ← monomial_eq_C_mul_X at ⊢, exact h_monomial _ _ })
open submodule polynomial set
variables {f : R[X]} {I : ideal R[X]}
/-- If the coefficients of a polynomial belong to an ideal, then that ideal contains
the ideal spanned by the coefficients of the polynomial. -/
lemma span_le_of_C_coeff_mem (cf : ∀ (i : ℕ), C (f.coeff i) ∈ I) :
ideal.span {g | ∃ i, g = C (f.coeff i)} ≤ I :=
begin
simp only [@eq_comm _ _ (C _)] {single_pass := tt},
exact (ideal.span_le.trans range_subset_iff).mpr cf,
end
lemma mem_span_C_coeff : f ∈ ideal.span {g : R[X] | ∃ i : ℕ, g = C (coeff f i)} :=
begin
let p := ideal.span {g : R[X] | ∃ i : ℕ, g = C (coeff f i)},
nth_rewrite 0 (sum_C_mul_X_eq f).symm,
refine submodule.sum_mem _ (λ n hn, _),
dsimp,
have : C (coeff f n) ∈ p, by { apply subset_span, simp },
have : (monomial n (1 : R)) • C (coeff f n) ∈ p := p.smul_mem _ this,
convert this using 1,
simp only [monomial_mul_C, one_mul, smul_eq_mul],
rw monomial_eq_C_mul_X,
end
lemma exists_C_coeff_not_mem : f ∉ I → ∃ i : ℕ, C (coeff f i) ∉ I :=
not.imp_symm $ λ cf, span_le_of_C_coeff_mem (not_exists_not.mp cf) mem_span_C_coeff
end semiring
end polynomial
|
ac27f3448f0462b6fd4724fabdbc558387e14e04 | 2eab05920d6eeb06665e1a6df77b3157354316ad | /src/linear_algebra/matrix/determinant.lean | 01ca391f7a7ccfa833435ec0fc0a2e0fd3684cb5 | [
"Apache-2.0"
] | permissive | ayush1801/mathlib | 78949b9f789f488148142221606bf15c02b960d2 | ce164e28f262acbb3de6281b3b03660a9f744e3c | refs/heads/master | 1,692,886,907,941 | 1,635,270,866,000 | 1,635,270,866,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 30,361 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Tim Baanen
-/
import data.matrix.pequiv
import data.matrix.block
import data.fintype.card
import group_theory.perm.fin
import group_theory.perm.sign
import algebra.algebra.basic
import tactic.ring
import linear_algebra.alternating
import linear_algebra.pi
/-!
# Determinant of a matrix
This file defines the determinant of a matrix, `matrix.det`, and its essential properties.
## Main definitions
- `matrix.det`: the determinant of a square matrix, as a sum over permutations
- `matrix.det_row_multilinear`: the determinant, as an `alternating_map` in the rows of the matrix
## Main results
- `det_mul`: the determinant of `A ⬝ B` is the product of determinants
- `det_zero_of_row_eq`: the determinant is zero if there is a repeated row
- `det_block_diagonal`: the determinant of a block diagonal matrix is a product
of the blocks' determinants
## Implementation notes
It is possible to configure `simp` to compute determinants. See the file
`test/matrix.lean` for some examples.
-/
universes u v w z
open equiv equiv.perm finset function
namespace matrix
open_locale matrix big_operators
variables {m n : Type*} [decidable_eq n] [fintype n] [decidable_eq m] [fintype m]
variables {R : Type v} [comm_ring R]
local notation `ε` σ:max := ((sign σ : ℤ ) : R)
/-- `det` is an `alternating_map` in the rows of the matrix. -/
def det_row_multilinear : alternating_map R (n → R) R n :=
((multilinear_map.mk_pi_algebra R n R).comp_linear_map (linear_map.proj)).alternatization
/-- The determinant of a matrix given by the Leibniz formula. -/
abbreviation det (M : matrix n n R) : R :=
det_row_multilinear M
lemma det_apply (M : matrix n n R) :
M.det = ∑ σ : perm n, σ.sign • ∏ i, M (σ i) i :=
multilinear_map.alternatization_apply _ M
-- This is what the old definition was. We use it to avoid having to change the old proofs below
lemma det_apply' (M : matrix n n R) :
M.det = ∑ σ : perm n, ε σ * ∏ i, M (σ i) i :=
by simp [det_apply, units.smul_def]
@[simp] lemma det_diagonal {d : n → R} : det (diagonal d) = ∏ i, d i :=
begin
rw det_apply',
refine (finset.sum_eq_single 1 _ _).trans _,
{ intros σ h1 h2,
cases not_forall.1 (mt equiv.ext h2) with x h3,
convert mul_zero _,
apply finset.prod_eq_zero,
{ change x ∈ _, simp },
exact if_neg h3 },
{ simp },
{ simp }
end
@[simp] lemma det_zero (h : nonempty n) : det (0 : matrix n n R) = 0 :=
(det_row_multilinear : alternating_map R (n → R) R n).map_zero
@[simp] lemma det_one : det (1 : matrix n n R) = 1 :=
by rw [← diagonal_one]; simp [-diagonal_one]
@[simp]
lemma det_is_empty [is_empty n] {A : matrix n n R} : det A = 1 :=
by simp [det_apply]
lemma det_eq_one_of_card_eq_zero {A : matrix n n R} (h : fintype.card n = 0) : det A = 1 :=
begin
haveI : is_empty n := fintype.card_eq_zero_iff.mp h,
exact det_is_empty,
end
/-- If `n` has only one element, the determinant of an `n` by `n` matrix is just that element.
Although `unique` implies `decidable_eq` and `fintype`, the instances might
not be syntactically equal. Thus, we need to fill in the args explicitly. -/
@[simp]
lemma det_unique {n : Type*} [unique n] [decidable_eq n] [fintype n] (A : matrix n n R) :
det A = A (default n) (default n) :=
by simp [det_apply, univ_unique]
lemma det_eq_elem_of_subsingleton [subsingleton n] (A : matrix n n R) (k : n) :
det A = A k k :=
begin
convert det_unique _,
exact unique_of_subsingleton k
end
lemma det_eq_elem_of_card_eq_one {A : matrix n n R} (h : fintype.card n = 1) (k : n) :
det A = A k k :=
begin
haveI : subsingleton n := fintype.card_le_one_iff_subsingleton.mp h.le,
exact det_eq_elem_of_subsingleton _ _
end
lemma det_mul_aux {M N : matrix n n R} {p : n → n} (H : ¬bijective p) :
∑ σ : perm n, (ε σ) * ∏ x, (M (σ x) (p x) * N (p x) x) = 0 :=
begin
obtain ⟨i, j, hpij, hij⟩ : ∃ i j, p i = p j ∧ i ≠ j,
{ rw [← fintype.injective_iff_bijective, injective] at H,
push_neg at H,
exact H },
exact sum_involution
(λ σ _, σ * swap i j)
(λ σ _,
have ∏ x, M (σ x) (p x) = ∏ x, M ((σ * swap i j) x) (p x),
from fintype.prod_equiv (swap i j) _ _ (by simp [apply_swap_eq_self hpij]),
by simp [this, sign_swap hij, prod_mul_distrib])
(λ σ _ _, (not_congr mul_swap_eq_iff).mpr hij)
(λ _ _, mem_univ _)
(λ σ _, mul_swap_involutive i j σ)
end
@[simp] lemma det_mul (M N : matrix n n R) : det (M ⬝ N) = det M * det N :=
calc det (M ⬝ N) = ∑ p : n → n, ∑ σ : perm n, ε σ * ∏ i, (M (σ i) (p i) * N (p i) i) :
by simp only [det_apply', mul_apply, prod_univ_sum, mul_sum,
fintype.pi_finset_univ]; rw [finset.sum_comm]
... = ∑ p in (@univ (n → n) _).filter bijective, ∑ σ : perm n,
ε σ * ∏ i, (M (σ i) (p i) * N (p i) i) :
eq.symm $ sum_subset (filter_subset _ _)
(λ f _ hbij, det_mul_aux $ by simpa only [true_and, mem_filter, mem_univ] using hbij)
... = ∑ τ : perm n, ∑ σ : perm n, ε σ * ∏ i, (M (σ i) (τ i) * N (τ i) i) :
sum_bij (λ p h, equiv.of_bijective p (mem_filter.1 h).2) (λ _ _, mem_univ _)
(λ _ _, rfl) (λ _ _ _ _ h, by injection h)
(λ b _, ⟨b, mem_filter.2 ⟨mem_univ _, b.bijective⟩, coe_fn_injective rfl⟩)
... = ∑ σ : perm n, ∑ τ : perm n, (∏ i, N (σ i) i) * ε τ * (∏ j, M (τ j) (σ j)) :
by simp only [mul_comm, mul_left_comm, prod_mul_distrib, mul_assoc]
... = ∑ σ : perm n, ∑ τ : perm n, (((∏ i, N (σ i) i) * (ε σ * ε τ)) * ∏ i, M (τ i) i) :
sum_congr rfl (λ σ _, fintype.sum_equiv (equiv.mul_right σ⁻¹) _ _
(λ τ,
have ∏ j, M (τ j) (σ j) = ∏ j, M ((τ * σ⁻¹) j) j,
by { rw ← σ⁻¹.prod_comp, simp only [equiv.perm.coe_mul, apply_inv_self] },
have h : ε σ * ε (τ * σ⁻¹) = ε τ :=
calc ε σ * ε (τ * σ⁻¹) = ε ((τ * σ⁻¹) * σ) :
by { rw [mul_comm, sign_mul (τ * σ⁻¹)], simp only [int.cast_mul, units.coe_mul] }
... = ε τ : by simp only [inv_mul_cancel_right],
by { simp_rw [equiv.coe_mul_right, h], simp only [this] }))
... = det M * det N : by simp only [det_apply', finset.mul_sum, mul_comm, mul_left_comm]
/-- The determinant of a matrix, as a monoid homomorphism. -/
def det_monoid_hom : matrix n n R →* R :=
{ to_fun := det,
map_one' := det_one,
map_mul' := det_mul }
@[simp] lemma coe_det_monoid_hom : (det_monoid_hom : matrix n n R → R) = det := rfl
/-- On square matrices, `mul_comm` applies under `det`. -/
lemma det_mul_comm (M N : matrix m m R) : det (M ⬝ N) = det (N ⬝ M) :=
by rw [det_mul, det_mul, mul_comm]
/-- On square matrices, `mul_left_comm` applies under `det`. -/
lemma det_mul_left_comm (M N P : matrix m m R) : det (M ⬝ (N ⬝ P)) = det (N ⬝ (M ⬝ P)) :=
by rw [←matrix.mul_assoc, ←matrix.mul_assoc, det_mul, det_mul_comm M N, ←det_mul]
/-- On square matrices, `mul_right_comm` applies under `det`. -/
lemma det_mul_right_comm (M N P : matrix m m R) :
det (M ⬝ N ⬝ P) = det (M ⬝ P ⬝ N) :=
by rw [matrix.mul_assoc, matrix.mul_assoc, det_mul, det_mul_comm N P, ←det_mul]
lemma det_units_conj (M : units (matrix m m R)) (N : matrix m m R) :
det (↑M ⬝ N ⬝ ↑M⁻¹ : matrix m m R) = det N :=
by rw [det_mul_right_comm, ←mul_eq_mul, ←mul_eq_mul, units.mul_inv, one_mul]
lemma det_units_conj' (M : units (matrix m m R)) (N : matrix m m R) :
det (↑M⁻¹ ⬝ N ⬝ ↑M : matrix m m R) = det N := det_units_conj M⁻¹ N
/-- Transposing a matrix preserves the determinant. -/
@[simp] lemma det_transpose (M : matrix n n R) : Mᵀ.det = M.det :=
begin
rw [det_apply', det_apply'],
refine fintype.sum_bijective _ inv_involutive.bijective _ _ _,
intros σ,
rw sign_inv,
congr' 1,
apply fintype.prod_equiv σ,
intros,
simp
end
/-- Permuting the columns changes the sign of the determinant. -/
lemma det_permute (σ : perm n) (M : matrix n n R) : matrix.det (λ i, M (σ i)) = σ.sign * M.det :=
((det_row_multilinear : alternating_map R (n → R) R n).map_perm M σ).trans
(by simp [units.smul_def])
/-- Permuting rows and columns with the same equivalence has no effect. -/
@[simp]
lemma det_minor_equiv_self (e : n ≃ m) (A : matrix m m R) :
det (A.minor e e) = det A :=
begin
rw [det_apply', det_apply'],
apply fintype.sum_equiv (equiv.perm_congr e),
intro σ,
rw equiv.perm.sign_perm_congr e σ,
congr' 1,
apply fintype.prod_equiv e,
intro i,
rw [equiv.perm_congr_apply, equiv.symm_apply_apply, minor_apply],
end
/-- Reindexing both indices along the same equivalence preserves the determinant.
For the `simp` version of this lemma, see `det_minor_equiv_self`; this one is unsuitable because
`matrix.reindex_apply` unfolds `reindex` first.
-/
lemma det_reindex_self (e : m ≃ n) (A : matrix m m R) : det (reindex e e A) = det A :=
det_minor_equiv_self e.symm A
/-- The determinant of a permutation matrix equals its sign. -/
@[simp] lemma det_permutation (σ : perm n) :
matrix.det (σ.to_pequiv.to_matrix : matrix n n R) = σ.sign :=
by rw [←matrix.mul_one (σ.to_pequiv.to_matrix : matrix n n R), pequiv.to_pequiv_mul_matrix,
det_permute, det_one, mul_one]
@[simp] lemma det_smul (A : matrix n n R) (c : R) : det (c • A) = c ^ fintype.card n * det A :=
calc det (c • A) = det (matrix.mul (diagonal (λ _, c)) A) : by rw [smul_eq_diagonal_mul]
... = det (diagonal (λ _, c)) * det A : det_mul _ _
... = c ^ fintype.card n * det A : by simp [card_univ]
/-- Multiplying each row by a fixed `v i` multiplies the determinant by
the product of the `v`s. -/
lemma det_mul_row (v : n → R) (A : matrix n n R) :
det (λ i j, v j * A i j) = (∏ i, v i) * det A :=
calc det (λ i j, v j * A i j) = det (A ⬝ diagonal v) : congr_arg det $ by { ext, simp [mul_comm] }
... = (∏ i, v i) * det A : by rw [det_mul, det_diagonal, mul_comm]
/-- Multiplying each column by a fixed `v j` multiplies the determinant by
the product of the `v`s. -/
lemma det_mul_column (v : n → R) (A : matrix n n R) :
det (λ i j, v i * A i j) = (∏ i, v i) * det A :=
multilinear_map.map_smul_univ _ v A
@[simp] lemma det_pow (M : matrix m m R) (n : ℕ) : det (M ^ n) = (det M) ^ n :=
(det_monoid_hom : matrix m m R →* R).map_pow M n
section hom_map
variables {S : Type w} [comm_ring S]
lemma _root_.ring_hom.map_det (f : R →+* S) (M : matrix n n R) :
f M.det = matrix.det (f.map_matrix M) :=
by simp [matrix.det_apply', f.map_sum, f.map_prod]
lemma _root_.ring_equiv.map_det (f : R ≃+* S) (M : matrix n n R) :
f M.det = matrix.det (f.map_matrix M) :=
f.to_ring_hom.map_det _
lemma _root_.alg_hom.map_det [algebra R S] {T : Type z} [comm_ring T] [algebra R T]
(f : S →ₐ[R] T) (M : matrix n n S) :
f M.det = matrix.det ((f : S →+* T).map_matrix M) :=
f.to_ring_hom.map_det _
lemma _root_.alg_equiv.map_det [algebra R S] {T : Type z} [comm_ring T] [algebra R T]
(f : S ≃ₐ[R] T) (M : matrix n n S) :
f M.det = matrix.det ((f : S →+* T).map_matrix M) :=
f.to_alg_hom.map_det _
end hom_map
@[simp] lemma det_conj_transpose [star_ring R] (M : matrix m m R) : det (Mᴴ) = star (det M) :=
((star_ring_aut : ring_aut R).map_det _).symm.trans $ congr_arg star M.det_transpose
section det_zero
/-!
### `det_zero` section
Prove that a matrix with a repeated column has determinant equal to zero.
-/
lemma det_eq_zero_of_row_eq_zero {A : matrix n n R} (i : n) (h : ∀ j, A i j = 0) : det A = 0 :=
(det_row_multilinear : alternating_map R (n → R) R n).map_coord_zero i (funext h)
lemma det_eq_zero_of_column_eq_zero {A : matrix n n R} (j : n) (h : ∀ i, A i j = 0) : det A = 0 :=
by { rw ← det_transpose, exact det_eq_zero_of_row_eq_zero j h, }
variables {M : matrix n n R} {i j : n}
/-- If a matrix has a repeated row, the determinant will be zero. -/
theorem det_zero_of_row_eq (i_ne_j : i ≠ j) (hij : M i = M j) : M.det = 0 :=
(det_row_multilinear : alternating_map R (n → R) R n).map_eq_zero_of_eq M hij i_ne_j
/-- If a matrix has a repeated column, the determinant will be zero. -/
theorem det_zero_of_column_eq (i_ne_j : i ≠ j) (hij : ∀ k, M k i = M k j) : M.det = 0 :=
by { rw [← det_transpose, det_zero_of_row_eq i_ne_j], exact funext hij }
end det_zero
lemma det_update_row_add (M : matrix n n R) (j : n) (u v : n → R) :
det (update_row M j $ u + v) = det (update_row M j u) + det (update_row M j v) :=
(det_row_multilinear : alternating_map R (n → R) R n).map_add M j u v
lemma det_update_column_add (M : matrix n n R) (j : n) (u v : n → R) :
det (update_column M j $ u + v) = det (update_column M j u) + det (update_column M j v) :=
begin
rw [← det_transpose, ← update_row_transpose, det_update_row_add],
simp [update_row_transpose, det_transpose]
end
lemma det_update_row_smul (M : matrix n n R) (j : n) (s : R) (u : n → R) :
det (update_row M j $ s • u) = s * det (update_row M j u) :=
(det_row_multilinear : alternating_map R (n → R) R n).map_smul M j s u
lemma det_update_column_smul (M : matrix n n R) (j : n) (s : R) (u : n → R) :
det (update_column M j $ s • u) = s * det (update_column M j u) :=
begin
rw [← det_transpose, ← update_row_transpose, det_update_row_smul],
simp [update_row_transpose, det_transpose]
end
lemma det_update_row_smul' (M : matrix n n R) (j : n) (s : R) (u : n → R) :
det (update_row (s • M) j u) = s ^ (fintype.card n - 1) * det (update_row M j u) :=
multilinear_map.map_update_smul _ M j s u
lemma det_update_column_smul' (M : matrix n n R) (j : n) (s : R) (u : n → R) :
det (update_column (s • M) j u) = s ^ (fintype.card n - 1) * det (update_column M j u) :=
begin
rw [← det_transpose, ← update_row_transpose, transpose_smul, det_update_row_smul'],
simp [update_row_transpose, det_transpose]
end
section det_eq
/-! ### `det_eq` section
Lemmas showing the determinant is invariant under a variety of operations.
-/
lemma det_eq_of_eq_mul_det_one {A B : matrix n n R}
(C : matrix n n R) (hC : det C = 1) (hA : A = B ⬝ C) : det A = det B :=
calc det A = det (B ⬝ C) : congr_arg _ hA
... = det B * det C : det_mul _ _
... = det B : by rw [hC, mul_one]
lemma det_eq_of_eq_det_one_mul {A B : matrix n n R}
(C : matrix n n R) (hC : det C = 1) (hA : A = C ⬝ B) : det A = det B :=
calc det A = det (C ⬝ B) : congr_arg _ hA
... = det C * det B : det_mul _ _
... = det B : by rw [hC, one_mul]
lemma det_update_row_add_self (A : matrix n n R) {i j : n} (hij : i ≠ j) :
det (update_row A i (A i + A j)) = det A :=
by simp [det_update_row_add,
det_zero_of_row_eq hij ((update_row_self).trans (update_row_ne hij.symm).symm)]
lemma det_update_column_add_self (A : matrix n n R) {i j : n} (hij : i ≠ j) :
det (update_column A i (λ k, A k i + A k j)) = det A :=
by { rw [← det_transpose, ← update_row_transpose, ← det_transpose A],
exact det_update_row_add_self Aᵀ hij }
lemma det_update_row_add_smul_self (A : matrix n n R) {i j : n} (hij : i ≠ j) (c : R) :
det (update_row A i (A i + c • A j)) = det A :=
by simp [det_update_row_add, det_update_row_smul,
det_zero_of_row_eq hij ((update_row_self).trans (update_row_ne hij.symm).symm)]
lemma det_update_column_add_smul_self (A : matrix n n R) {i j : n} (hij : i ≠ j) (c : R) :
det (update_column A i (λ k, A k i + c • A k j)) = det A :=
by { rw [← det_transpose, ← update_row_transpose, ← det_transpose A],
exact det_update_row_add_smul_self Aᵀ hij c }
lemma det_eq_of_forall_row_eq_smul_add_const_aux
{A B : matrix n n R} {s : finset n} : ∀ (c : n → R) (hs : ∀ i, i ∉ s → c i = 0)
(k : n) (hk : k ∉ s) (A_eq : ∀ i j, A i j = B i j + c i * B k j),
det A = det B :=
begin
revert B,
refine s.induction_on _ _,
{ intros A c hs k hk A_eq,
have : ∀ i, c i = 0,
{ intros i,
specialize hs i,
contrapose! hs,
simp [hs] },
congr,
ext i j,
rw [A_eq, this, zero_mul, add_zero], },
{ intros i s hi ih B c hs k hk A_eq,
have hAi : A i = B i + c i • B k := funext (A_eq i),
rw [@ih (update_row B i (A i)) (function.update c i 0), hAi,
det_update_row_add_smul_self],
{ exact mt (λ h, show k ∈ insert i s, from h ▸ finset.mem_insert_self _ _) hk },
{ intros i' hi',
rw function.update_apply,
split_ifs with hi'i, { refl },
{ exact hs i' (λ h, hi' ((finset.mem_insert.mp h).resolve_left hi'i)) } },
{ exact λ h, hk (finset.mem_insert_of_mem h) },
{ intros i' j',
rw [update_row_apply, function.update_apply],
split_ifs with hi'i,
{ simp [hi'i] },
rw [A_eq, update_row_ne (λ (h : k = i), hk $ h ▸ finset.mem_insert_self k s)] } }
end
/-- If you add multiples of row `B k` to other rows, the determinant doesn't change. -/
lemma det_eq_of_forall_row_eq_smul_add_const
{A B : matrix n n R} (c : n → R) (k : n) (hk : c k = 0)
(A_eq : ∀ i j, A i j = B i j + c i * B k j) :
det A = det B :=
det_eq_of_forall_row_eq_smul_add_const_aux c
(λ i, not_imp_comm.mp $ λ hi, finset.mem_erase.mpr
⟨mt (λ (h : i = k), show c i = 0, from h.symm ▸ hk) hi, finset.mem_univ i⟩)
k (finset.not_mem_erase k finset.univ) A_eq
lemma det_eq_of_forall_row_eq_smul_add_pred_aux {n : ℕ} (k : fin (n + 1)) :
∀ (c : fin n → R) (hc : ∀ (i : fin n), k < i.succ → c i = 0)
{M N : matrix (fin n.succ) (fin n.succ) R}
(h0 : ∀ j, M 0 j = N 0 j)
(hsucc : ∀ (i : fin n) j, M i.succ j = N i.succ j + c i * M i.cast_succ j),
det M = det N :=
begin
refine fin.induction _ (λ k ih, _) k;
intros c hc M N h0 hsucc,
{ congr,
ext i j,
refine fin.cases (h0 j) (λ i, _) i,
rw [hsucc, hc i (fin.succ_pos _), zero_mul, add_zero] },
set M' := update_row M k.succ (N k.succ) with hM',
have hM : M = update_row M' k.succ (M' k.succ + c k • M k.cast_succ),
{ ext i j,
by_cases hi : i = k.succ,
{ simp [hi, hM', hsucc, update_row_self] },
rw [update_row_ne hi, hM', update_row_ne hi] },
have k_ne_succ : k.cast_succ ≠ k.succ := (fin.cast_succ_lt_succ k).ne,
have M_k : M k.cast_succ = M' k.cast_succ := (update_row_ne k_ne_succ).symm,
rw [hM, M_k, det_update_row_add_smul_self M' k_ne_succ.symm, ih (function.update c k 0)],
{ intros i hi,
rw [fin.lt_iff_coe_lt_coe, fin.coe_cast_succ, fin.coe_succ, nat.lt_succ_iff] at hi,
rw function.update_apply,
split_ifs with hik, { refl },
exact hc _ (fin.succ_lt_succ_iff.mpr (lt_of_le_of_ne hi (ne.symm hik))) },
{ rwa [hM', update_row_ne (fin.succ_ne_zero _).symm] },
intros i j,
rw function.update_apply,
split_ifs with hik,
{ rw [zero_mul, add_zero, hM', hik, update_row_self] },
rw [hM', update_row_ne ((fin.succ_injective _).ne hik), hsucc],
by_cases hik2 : k < i,
{ simp [hc i (fin.succ_lt_succ_iff.mpr hik2)] },
rw update_row_ne,
apply ne_of_lt,
rwa [fin.lt_iff_coe_lt_coe, fin.coe_cast_succ, fin.coe_succ, nat.lt_succ_iff, ← not_lt]
end
/-- If you add multiples of previous rows to the next row, the determinant doesn't change. -/
lemma det_eq_of_forall_row_eq_smul_add_pred {n : ℕ}
{A B : matrix (fin (n + 1)) (fin (n + 1)) R} (c : fin n → R)
(A_zero : ∀ j, A 0 j = B 0 j)
(A_succ : ∀ (i : fin n) j, A i.succ j = B i.succ j + c i * A i.cast_succ j) :
det A = det B :=
det_eq_of_forall_row_eq_smul_add_pred_aux (fin.last _) c
(λ i hi, absurd hi (not_lt_of_ge (fin.le_last _)))
A_zero A_succ
/-- If you add multiples of previous columns to the next columns, the determinant doesn't change. -/
lemma det_eq_of_forall_col_eq_smul_add_pred {n : ℕ}
{A B : matrix (fin (n + 1)) (fin (n + 1)) R} (c : fin n → R)
(A_zero : ∀ i, A i 0 = B i 0)
(A_succ : ∀ i (j : fin n), A i j.succ = B i j.succ + c j * A i j.cast_succ) :
det A = det B :=
by { rw [← det_transpose A, ← det_transpose B],
exact det_eq_of_forall_row_eq_smul_add_pred c A_zero (λ i j, A_succ j i) }
end det_eq
@[simp] lemma det_block_diagonal {o : Type*} [fintype o] [decidable_eq o] (M : o → matrix n n R) :
(block_diagonal M).det = ∏ k, (M k).det :=
begin
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply'],
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw finset.prod_sum,
simp_rw [finset.mem_univ, finset.prod_attach_univ, finset.univ_pi_univ],
-- We claim that the only permutations contributing to the sum are those that
-- preserve their second component.
let preserving_snd : finset (equiv.perm (n × o)) :=
finset.univ.filter (λ σ, ∀ x, (σ x).snd = x.snd),
have mem_preserving_snd : ∀ {σ : equiv.perm (n × o)},
σ ∈ preserving_snd ↔ ∀ x, (σ x).snd = x.snd :=
λ σ, finset.mem_filter.trans ⟨λ h, h.2, λ h, ⟨finset.mem_univ _, h⟩⟩,
rw ← finset.sum_subset (finset.subset_univ preserving_snd) _,
-- And that these are in bijection with `o → equiv.perm m`.
rw (finset.sum_bij (λ (σ : ∀ (k : o), k ∈ finset.univ → equiv.perm n) _,
prod_congr_left (λ k, σ k (finset.mem_univ k))) _ _ _ _).symm,
{ intros σ _,
rw mem_preserving_snd,
rintros ⟨k, x⟩,
simp only [prod_congr_left_apply] },
{ intros σ _,
rw [finset.prod_mul_distrib, ←finset.univ_product_univ, finset.prod_product, finset.prod_comm],
simp only [sign_prod_congr_left, units.coe_prod, int.cast_prod, block_diagonal_apply_eq,
prod_congr_left_apply] },
{ intros σ σ' _ _ eq,
ext x hx k,
simp only at eq,
have : ∀ k x, prod_congr_left (λ k, σ k (finset.mem_univ _)) (k, x) =
prod_congr_left (λ k, σ' k (finset.mem_univ _)) (k, x) :=
λ k x, by rw eq,
simp only [prod_congr_left_apply, prod.mk.inj_iff] at this,
exact (this k x).1 },
{ intros σ hσ,
rw mem_preserving_snd at hσ,
have hσ' : ∀ x, (σ⁻¹ x).snd = x.snd,
{ intro x, conv_rhs { rw [← perm.apply_inv_self σ x, hσ] } },
have mk_apply_eq : ∀ k x, ((σ (x, k)).fst, k) = σ (x, k),
{ intros k x,
ext,
{ simp only},
{ simp only [hσ] } },
have mk_inv_apply_eq : ∀ k x, ((σ⁻¹ (x, k)).fst, k) = σ⁻¹ (x, k),
{ intros k x,
conv_lhs { rw ← perm.apply_inv_self σ (x, k) },
ext,
{ simp only [apply_inv_self] },
{ simp only [hσ'] } },
refine ⟨λ k _, ⟨λ x, (σ (x, k)).fst, λ x, (σ⁻¹ (x, k)).fst, _, _⟩, _, _⟩,
{ intro x,
simp only [mk_apply_eq, inv_apply_self] },
{ intro x,
simp only [mk_inv_apply_eq, apply_inv_self] },
{ apply finset.mem_univ },
{ ext ⟨k, x⟩,
{ simp only [coe_fn_mk, prod_congr_left_apply] },
{ simp only [prod_congr_left_apply, hσ] } } },
{ intros σ _ hσ,
rw mem_preserving_snd at hσ,
obtain ⟨⟨k, x⟩, hkx⟩ := not_forall.mp hσ,
rw [finset.prod_eq_zero (finset.mem_univ (k, x)), mul_zero],
rw [← @prod.mk.eta _ _ (σ (k, x)), block_diagonal_apply_ne],
exact hkx }
end
/-- The determinant of a 2x2 block matrix with the lower-left block equal to zero is the product of
the determinants of the diagonal blocks. For the generalization to any number of blocks, see
`matrix.upper_block_triangular_det`. -/
lemma upper_two_block_triangular_det
(A : matrix m m R) (B : matrix m n R) (D : matrix n n R) :
(matrix.from_blocks A B 0 D).det = A.det * D.det :=
begin
classical,
simp_rw det_apply',
convert
(sum_subset (subset_univ ((sum_congr_hom m n).range : set (perm (m ⊕ n))).to_finset) _).symm,
rw sum_mul_sum,
simp_rw univ_product_univ,
rw (sum_bij (λ (σ : perm m × perm n) _, equiv.sum_congr σ.fst σ.snd) _ _ _ _).symm,
{ intros σ₁₂ h,
simp only [],
erw [set.mem_to_finset, monoid_hom.mem_range],
use σ₁₂,
simp only [sum_congr_hom_apply] },
{ simp only [forall_prop_of_true, prod.forall, mem_univ],
intros σ₁ σ₂,
rw fintype.prod_sum_type,
simp_rw [equiv.sum_congr_apply, sum.map_inr, sum.map_inl, from_blocks_apply₁₁,
from_blocks_apply₂₂],
have hr : ∀ (a b c d : R), (a * b) * (c * d) = a * c * (b * d), { intros, ac_refl },
rw hr,
congr,
rw [sign_sum_congr, units.coe_mul, int.cast_mul] },
{ intros σ₁ σ₂ h₁ h₂,
dsimp only [],
intro h,
have h2 : ∀ x, perm.sum_congr σ₁.fst σ₁.snd x = perm.sum_congr σ₂.fst σ₂.snd x,
{ intro x, exact congr_fun (congr_arg to_fun h) x },
simp only [sum.map_inr, sum.map_inl, perm.sum_congr_apply, sum.forall] at h2,
ext,
{ exact h2.left x },
{ exact h2.right x }},
{ intros σ hσ,
erw [set.mem_to_finset, monoid_hom.mem_range] at hσ,
obtain ⟨σ₁₂, hσ₁₂⟩ := hσ,
use σ₁₂,
rw ←hσ₁₂,
simp },
{ intros σ hσ hσn,
have h1 : ¬ (∀ x, ∃ y, sum.inl y = σ (sum.inl x)),
{ by_contradiction,
rw set.mem_to_finset at hσn,
apply absurd (mem_sum_congr_hom_range_of_perm_maps_to_inl _) hσn,
rintros x ⟨a, ha⟩,
rw [←ha], exact h a },
obtain ⟨a, ha⟩ := not_forall.mp h1,
cases hx : σ (sum.inl a) with a2 b,
{ have hn := (not_exists.mp ha) a2,
exact absurd hx.symm hn },
{ rw [finset.prod_eq_zero (finset.mem_univ (sum.inl a)), mul_zero],
rw [hx, from_blocks_apply₂₁], refl }}
end
/-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column 0. -/
lemma det_succ_column_zero {n : ℕ} (A : matrix (fin n.succ) (fin n.succ) R) :
det A = ∑ i : fin n.succ, (-1) ^ (i : ℕ) * A i 0 *
det (A.minor i.succ_above fin.succ) :=
begin
rw [matrix.det_apply, finset.univ_perm_fin_succ, ← finset.univ_product_univ],
simp only [finset.sum_map, equiv.to_embedding_apply, finset.sum_product, matrix.minor],
refine finset.sum_congr rfl (λ i _, fin.cases _ (λ i, _) i),
{ simp only [fin.prod_univ_succ, matrix.det_apply, finset.mul_sum,
equiv.perm.decompose_fin_symm_apply_zero, fin.coe_zero, one_mul,
equiv.perm.decompose_fin.symm_sign, equiv.swap_self, if_true, id.def, eq_self_iff_true,
equiv.perm.decompose_fin_symm_apply_succ, fin.succ_above_zero, equiv.coe_refl, pow_zero,
mul_smul_comm] },
-- `univ_perm_fin_succ` gives a different embedding of `perm (fin n)` into
-- `perm (fin n.succ)` than the determinant of the submatrix we want,
-- permute `A` so that we get the correct one.
have : (-1 : R) ^ (i : ℕ) = i.cycle_range.sign,
{ simp [fin.sign_cycle_range] },
rw [fin.coe_succ, pow_succ, this, mul_assoc, mul_assoc, mul_left_comm ↑(equiv.perm.sign _),
← det_permute, matrix.det_apply, finset.mul_sum, finset.mul_sum],
-- now we just need to move the corresponding parts to the same place
refine finset.sum_congr rfl (λ σ _, _),
rw [equiv.perm.decompose_fin.symm_sign, if_neg (fin.succ_ne_zero i)],
calc ((-1) * σ.sign : ℤ) • ∏ i', A (equiv.perm.decompose_fin.symm (fin.succ i, σ) i') i'
= ((-1) * σ.sign : ℤ) • (A (fin.succ i) 0 *
∏ i', A (((fin.succ i).succ_above) (fin.cycle_range i (σ i'))) i'.succ) :
by simp only [fin.prod_univ_succ, fin.succ_above_cycle_range,
equiv.perm.decompose_fin_symm_apply_zero, equiv.perm.decompose_fin_symm_apply_succ]
... = (-1) * (A (fin.succ i) 0 * (σ.sign : ℤ) •
∏ i', A (((fin.succ i).succ_above) (fin.cycle_range i (σ i'))) i'.succ) :
by simp only [mul_assoc, mul_comm, neg_mul_eq_neg_mul_symm, one_mul, gsmul_eq_mul, neg_inj,
neg_smul, fin.succ_above_cycle_range],
end
/-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row 0. -/
lemma det_succ_row_zero {n : ℕ} (A : matrix (fin n.succ) (fin n.succ) R) :
det A = ∑ j : fin n.succ, (-1) ^ (j : ℕ) * A 0 j *
det (A.minor fin.succ j.succ_above) :=
by { rw [← det_transpose A, det_succ_column_zero],
refine finset.sum_congr rfl (λ i _, _),
rw [← det_transpose],
simp only [transpose_apply, transpose_minor, transpose_transpose] }
/-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along row `i`. -/
lemma det_succ_row {n : ℕ} (A : matrix (fin n.succ) (fin n.succ) R) (i : fin n.succ) :
det A = ∑ j : fin n.succ, (-1) ^ (i + j : ℕ) * A i j *
det (A.minor i.succ_above j.succ_above) :=
begin
simp_rw [pow_add, mul_assoc, ← mul_sum],
have : det A = (-1 : R) ^ (i : ℕ) * (i.cycle_range⁻¹).sign * det A,
{ calc det A = ↑((-1 : units ℤ) ^ (i : ℕ) * (-1 : units ℤ) ^ (i : ℕ) : units ℤ) * det A :
by simp
... = (-1 : R) ^ (i : ℕ) * (i.cycle_range⁻¹).sign * det A :
by simp [-int.units_mul_self] },
rw [this, mul_assoc],
congr,
rw [← det_permute, det_succ_row_zero],
refine finset.sum_congr rfl (λ j _, _),
rw [mul_assoc, matrix.minor, matrix.minor],
congr,
{ rw [equiv.perm.inv_def, fin.cycle_range_symm_zero] },
{ ext i' j',
rw [equiv.perm.inv_def, fin.cycle_range_symm_succ] },
end
/-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column `j`. -/
lemma det_succ_column {n : ℕ} (A : matrix (fin n.succ) (fin n.succ) R) (j : fin n.succ) :
det A = ∑ i : fin n.succ, (-1) ^ (i + j : ℕ) * A i j *
det (A.minor i.succ_above j.succ_above) :=
by { rw [← det_transpose, det_succ_row _ j],
refine finset.sum_congr rfl (λ i _, _),
rw [add_comm, ← det_transpose, transpose_apply, transpose_minor, transpose_transpose] }
/-- Determinant of 0x0 matrix -/
@[simp] lemma det_fin_zero {A : matrix (fin 0) (fin 0) R} : det A = 1 :=
det_is_empty
/-- Determinant of 1x1 matrix -/
lemma det_fin_one (A : matrix (fin 1) (fin 1) R) : det A = A 0 0 := det_unique A
/-- Determinant of 2x2 matrix -/
lemma det_fin_two (A : matrix (fin 2) (fin 2) R) :
det A = A 0 0 * A 1 1 - A 0 1 * A 1 0 :=
begin
simp [matrix.det_succ_row_zero, fin.sum_univ_succ],
ring
end
/-- Determinant of 3x3 matrix -/
lemma det_fin_three (A : matrix (fin 3) (fin 3) R) :
det A = A 0 0 * A 1 1 * A 2 2 - A 0 0 * A 1 2 * A 2 1 - A 0 1 * A 1 0 * A 2 2
+ A 0 1 * A 1 2 * A 2 0 + A 0 2 * A 1 0 * A 2 1 - A 0 2 * A 1 1 * A 2 0 :=
begin
simp [matrix.det_succ_row_zero, fin.sum_univ_succ],
ring
end
end matrix
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.