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33bdaa2972651c531b2071e108d770e8ad328393 | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /tests/lean/t13.lean | cef8024b67b5acd19aa5fcac33203dbc2dac1600 | [
"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 | 327 | lean | prelude constant A : Type.{1}
constant f : A → A → A
constant g : A → A → A
precedence `+` : 65
infixl + := f
infixl + := g
constant a : A
constant b : A
print raw a+b -- + is overloaded
check fun (h : A → A → A)
(infixl + := h), -- Like local declarations, local notation "shadows" global one.
a+b
|
e1d086be1777cbea86b66e37c9766d087fe0def0 | a45212b1526d532e6e83c44ddca6a05795113ddc | /src/ring_theory/adjoin_root.lean | 83445af23e45ad3ccf6e2360371617cfffc2ccf3 | [
"Apache-2.0"
] | permissive | fpvandoorn/mathlib | b21ab4068db079cbb8590b58fda9cc4bc1f35df4 | b3433a51ea8bc07c4159c1073838fc0ee9b8f227 | refs/heads/master | 1,624,791,089,608 | 1,556,715,231,000 | 1,556,715,231,000 | 165,722,980 | 5 | 0 | Apache-2.0 | 1,552,657,455,000 | 1,547,494,646,000 | Lean | UTF-8 | Lean | false | false | 3,691 | lean | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
Adjoining roots of polynomials
-/
import data.polynomial ring_theory.principal_ideal_domain
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w}
open polynomial ideal
def adjoin_root [comm_ring α] [decidable_eq α] (f : polynomial α) : Type u :=
ideal.quotient (span {f} : ideal (polynomial α))
namespace adjoin_root
section comm_ring
variables [comm_ring α] [decidable_eq α] (f : polynomial α)
instance : comm_ring (adjoin_root f) := ideal.quotient.comm_ring _
noncomputable instance : decidable_eq (adjoin_root f) := classical.dec_eq _
variable {f}
def mk : polynomial α → adjoin_root f := ideal.quotient.mk _
def root : adjoin_root f := mk X
def of (x : α) : adjoin_root f := mk (C x)
instance adjoin_root.has_coe_t : has_coe_t α (adjoin_root f) := ⟨of⟩
instance mk.is_ring_hom : is_ring_hom (mk : polynomial α → adjoin_root f) :=
ideal.quotient.is_ring_hom_mk _
@[simp] lemma mk_self : (mk f : adjoin_root f) = 0 :=
quotient.sound' (mem_span_singleton.2 $ by simp)
instance : is_ring_hom (coe : α → adjoin_root f) :=
@is_ring_hom.comp _ _ _ _ C _ _ _ mk mk.is_ring_hom
lemma eval₂_root (f : polynomial α) : f.eval₂ coe (root : adjoin_root f) = 0 :=
quotient.induction_on' (root : adjoin_root f)
(λ (g : polynomial α) (hg : mk g = mk X),
show finsupp.sum f (λ (e : ℕ) (a : α), mk (C a) * mk g ^ e) = 0,
by simp only [hg, (is_semiring_hom.map_pow (mk : polynomial α → adjoin_root f) _ _).symm,
(is_ring_hom.map_mul (mk : polynomial α → adjoin_root f)).symm];
rw [finsupp.sum, finset.sum_hom (mk : polynomial α → adjoin_root f),
show finset.sum _ _ = _, from sum_C_mul_X_eq _, mk_self])
(show (root : adjoin_root f) = mk X, from rfl)
lemma is_root_root (f : polynomial α) : is_root (f.map coe) (root : adjoin_root f) :=
by rw [is_root, eval_map, eval₂_root]
variables [comm_ring β]
def lift (i : α → β) [is_ring_hom i] (x : β) (h : f.eval₂ i x = 0) : (adjoin_root f) → β :=
ideal.quotient.lift _ (eval₂ i x) $ λ g H,
begin
simp [mem_span_singleton] at H,
cases H with y H,
dsimp at H,
rw [H, eval₂_mul],
simp [h]
end
variables {i : α → β} [is_ring_hom i] {a : β} {h : f.eval₂ i a = 0}
@[simp] lemma lift_mk {g : polynomial α} : lift i a h (mk g) = g.eval₂ i a :=
ideal.quotient.lift_mk
@[simp] lemma lift_root : lift i a h root = a := by simp [root, h]
@[simp] lemma lift_of {x : α} : lift i a h x = i x :=
by show lift i a h (ideal.quotient.mk _ (C x)) = i x;
convert ideal.quotient.lift_mk; simp
instance is_ring_hom_lift : is_ring_hom (lift i a h) :=
by unfold lift; apply_instance
end comm_ring
variables [discrete_field α] {f : polynomial α} [irreducible f]
instance is_maximal_span : is_maximal (span {f} : ideal (polynomial α)) :=
principal_ideal_domain.is_maximal_of_irreducible ‹irreducible f›
noncomputable instance field : discrete_field (adjoin_root f) :=
ideal.quotient.field (span {f} : ideal (polynomial α))
instance : is_field_hom (coe : α → adjoin_root f) := by apply_instance
instance lift_is_field_hom [field β] {i : α → β} [is_ring_hom i] {a : β}
{h : f.eval₂ i a = 0} : is_field_hom (lift i a h) := by apply_instance
lemma coe_injective : function.injective (coe : α → adjoin_root f) :=
is_field_hom.injective _
lemma mul_div_root_cancel (f : polynomial α) [irreducible f] :
(X - C (root : adjoin_root f)) * (f.map coe / (X - C root)) = f.map coe :=
mul_div_eq_iff_is_root.2 $ is_root_root _
end adjoin_root
|
f0cdf48916a94106c2efc4fc0da228f0b4998d63 | 38bf3fd2bb651ab70511408fcf70e2029e2ba310 | /src/category_theory/groupoid.lean | 23fd1e92bdde7aecedfb4cddffe78165c28f1f39 | [
"Apache-2.0"
] | permissive | JaredCorduan/mathlib | 130392594844f15dad65a9308c242551bae6cd2e | d5de80376088954d592a59326c14404f538050a1 | refs/heads/master | 1,595,862,206,333 | 1,570,816,457,000 | 1,570,816,457,000 | 209,134,499 | 0 | 0 | Apache-2.0 | 1,568,746,811,000 | 1,568,746,811,000 | null | UTF-8 | Lean | false | false | 1,034 | lean | /-
Copyright (c) 2018 Reid Barton All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import category_theory.category
import category_theory.isomorphism
namespace category_theory
universes v u -- declare the `v`'s first; see `category_theory.category` for an explanation
class groupoid (obj : Type u) extends category.{v} obj : Type (max u (v+1)) :=
(inv : Π {X Y : obj}, (X ⟶ Y) → (Y ⟶ X))
(inv_comp' : ∀ {X Y : obj} (f : X ⟶ Y), comp (inv f) f = id Y . obviously)
(comp_inv' : ∀ {X Y : obj} (f : X ⟶ Y), comp f (inv f) = id X . obviously)
restate_axiom groupoid.inv_comp'
restate_axiom groupoid.comp_inv'
attribute [simp] groupoid.inv_comp groupoid.comp_inv
abbreviation large_groupoid (C : Type (u+1)) : Type (u+1) := groupoid.{u} C
abbreviation small_groupoid (C : Type u) : Type (u+1) := groupoid.{u} C
instance of_groupoid {C : Type u} [groupoid.{v} C] {X Y : C} (f : X ⟶ Y) : is_iso f :=
{ inv := groupoid.inv f }
end category_theory
|
6243aa796b92dfd2dc30f1b17418a4b24ab856dc | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /src/Lean/Data/SSet.lean | 578340f86402462562d3a02c867fb09298b73ca9 | [
"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 | 1,379 | lean | /-
Copyright (c) 2021 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Data.SMap
namespace Lean
/-- Staged set. It is just a simple wrapper on top of Staged maps. -/
def SSet (α : Type u) [BEq α] [Hashable α] := SMap α Unit
namespace SSet
variable {α : Type u} [BEq α] [Hashable α]
instance : Inhabited (SSet α) := inferInstanceAs (Inhabited (SMap ..))
abbrev empty : SSet α := SMap.empty
abbrev insert (s : SSet α) (a : α) : SSet α :=
SMap.insert s a ()
abbrev contains (s : SSet α) (a : α) : Bool :=
SMap.contains s a
abbrev forM [Monad m] (s : SSet α) (f : α → m PUnit) : m PUnit :=
SMap.forM s fun a _ => f a
/-- Move from stage 1 into stage 2. -/
abbrev switch (s : SSet α) : SSet α :=
SMap.switch s
abbrev fold (f : σ → α → σ) (init : σ) (s : SSet α) : σ :=
SMap.fold (fun d a _ => f d a) init s
abbrev size (s : SSet α) : Nat :=
SMap.size s
def toList (m : SSet α) : List α :=
m.fold (init := []) fun es a => a::es
end SSet
end Lean
def List.toSSet [BEq α] [Hashable α] (es : List α) : Lean.SSet α :=
es.foldl (init := {}) fun s a => s.insert a
instance {_ : BEq α} {_ : Hashable α} [Repr α] : Repr (Lean.SSet α) where
reprPrec v prec := Repr.addAppParen (reprArg v.toList ++ ".toSSet") prec
|
725950a956f5d7c420363a2f4c7c88c0787851d2 | 5719a16e23dfc08cdea7a5bf035b81690f307965 | /stage0/src/Init/Data/Array/Basic.lean | 875106bbdc49935e96cf3fd5d2b95dcb2c0e091d | [
"Apache-2.0"
] | permissive | postmasters/lean4 | 488b03969a371e1507e1e8a4df9ebf63c7cbe7ac | f3976fc53a883ac7606fc59357d43f4b51016ca7 | refs/heads/master | 1,655,582,707,480 | 1,588,682,595,000 | 1,588,682,595,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 24,820 | lean | /-
Copyright (c) 2018 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.Data.Nat.Basic
import Init.Data.Fin.Basic
import Init.Data.UInt
import Init.Data.Repr
import Init.Data.ToString
import Init.Control.Id
import Init.Util
universes u v w
/-
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
@[reducible, extern "lean_array_get_size"]
def Array.size {α : Type u} (a : @& Array α) : Nat :=
a.sz
namespace Array
variables {α : Type u}
/- The parameter `c` is the initial capacity -/
@[extern "lean_mk_empty_array_with_capacity"]
def mkEmpty (c : @& Nat) : Array α :=
{ sz := 0,
data := fun ⟨x, h⟩ => absurd h (Nat.notLtZero x) }
@[extern "lean_array_push"]
def push (a : Array α) (v : α) : Array α :=
{ sz := Nat.succ a.sz,
data := fun ⟨j, h₁⟩ =>
if h₂ : j = a.sz then v
else a.data ⟨j, Nat.ltOfLeOfNe (Nat.leOfLtSucc h₁) h₂⟩ }
@[extern "lean_mk_array"]
def mkArray {α : Type u} (n : Nat) (v : α) : Array α :=
{ sz := n,
data := fun _ => v}
theorem szMkArrayEq {α : Type u} (n : Nat) (v : α) : (mkArray n v).sz = n :=
rfl
def empty : Array α :=
mkEmpty 0
instance : HasEmptyc (Array α) :=
⟨Array.empty⟩
instance : Inhabited (Array α) :=
⟨Array.empty⟩
def isEmpty (a : Array α) : Bool :=
a.size = 0
def singleton (v : α) : Array α :=
mkArray 1 v
@[extern "lean_array_fget"]
def get (a : @& Array α) (i : @& Fin a.size) : α :=
a.data i
/- Low-level version of `fget` which is as fast as a C array read.
`Fin` values are represented as tag pointers in the Lean runtime. Thus,
`fget` may be slightly slower than `uget`. -/
@[extern "lean_array_uget"]
def uget (a : @& Array α) (i : USize) (h : i.toNat < a.size) : α :=
a.get ⟨i.toNat, h⟩
/- "Comfortable" version of `fget`. It performs a bound check at runtime. -/
@[extern "lean_array_get"]
def get! [Inhabited α] (a : @& Array α) (i : @& Nat) : α :=
if h : i < a.size then a.get ⟨i, h⟩ else arbitrary α
def back [Inhabited α] (a : Array α) : α :=
a.get! (a.size - 1)
def get? (a : Array α) (i : Nat) : Option α :=
if h : i < a.size then some (a.get ⟨i, h⟩) else none
def getD (a : Array α) (i : Nat) (v₀ : α) : α :=
if h : i < a.size then a.get ⟨i, h⟩ else v₀
def getOp [Inhabited α] (self : Array α) (idx : Nat) : α :=
self.get! idx
-- auxiliary declaration used in the equation compiler when pattern matching array literals.
abbrev getLit {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size = n) (h₂ : i < n) : α :=
a.get ⟨i, h₁.symm ▸ h₂⟩
@[extern "lean_array_fset"]
def set (a : Array α) (i : @& Fin a.size) (v : α) : Array α :=
{ sz := a.sz,
data := fun j => if h : i = j then v else a.data j }
theorem szFSetEq (a : Array α) (i : Fin a.size) (v : α) : (set a i v).size = a.size :=
rfl
theorem szPushEq (a : Array α) (v : α) : (push a v).size = a.size + 1 :=
rfl
/- Low-level version of `fset` which is as fast as a C array fset.
`Fin` values are represented as tag pointers in the Lean runtime. Thus,
`fset` may be slightly slower than `uset`. -/
@[extern "lean_array_uset"]
def uset (a : Array α) (i : USize) (v : α) (h : i.toNat < a.size) : Array α :=
a.set ⟨i.toNat, h⟩ v
/- "Comfortable" version of `fset`. It performs a bound check at runtime. -/
@[extern "lean_array_set"]
def set! (a : Array α) (i : @& Nat) (v : α) : Array α :=
if h : i < a.size then a.set ⟨i, h⟩ v else panic! "index out of bounds"
@[extern "lean_array_fswap"]
def swap (a : Array α) (i j : @& Fin a.size) : Array α :=
let v₁ := a.get i;
let v₂ := a.get j;
let a := a.set i v₂;
a.set j v₁
@[extern "lean_array_swap"]
def swap! (a : Array α) (i j : @& Nat) : Array α :=
if h₁ : i < a.size then
if h₂ : j < a.size then swap a ⟨i, h₁⟩ ⟨j, h₂⟩
else panic! "index out of bounds"
else panic! "index out of bounds"
@[inline] def swapAt {α : Type} (a : Array α) (i : Fin a.size) (v : α) : α × Array α :=
let e := a.get i;
let a := a.set i v;
(e, a)
-- TODO: delete as soon as we can define local instances
@[neverExtract] private def swapAtPanic! [Inhabited α] (i : Nat) : α × Array α :=
panic! ("index " ++ toString i ++ " out of bounds")
@[inline] def swapAt! {α : Type} (a : Array α) (i : Nat) (v : α) : α × Array α :=
if h : i < a.size then swapAt a ⟨i, h⟩ v else @swapAtPanic! _ ⟨v⟩ i
@[extern "lean_array_pop"]
def pop (a : Array α) : Array α :=
{ sz := Nat.pred a.size,
data := fun ⟨j, h⟩ => a.get ⟨j, Nat.ltOfLtOfLe h (Nat.predLe _)⟩ }
-- TODO(Leo): justify termination using wf-rec
partial def shrink : Array α → Nat → Array α
| a, n => if n ≥ a.size then a else shrink a.pop n
section
variables {m : Type v → Type w} [Monad m]
variables {β : Type v} {σ : Type u}
-- TODO(Leo): justify termination using wf-rec
@[specialize] partial def iterateMAux (a : Array α) (f : ∀ (i : Fin a.size), α → β → m β) : Nat → β → m β
| i, b =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
f idx (a.get idx) b >>= iterateMAux (i+1)
else pure b
@[inline] def iterateM (a : Array α) (b : β) (f : ∀ (i : Fin a.size), α → β → m β) : m β :=
iterateMAux a f 0 b
@[inline] def foldlM (f : β → α → m β) (init : β) (a : Array α) : m β :=
iterateM a init (fun _ b a => f a b)
@[inline] def foldlFromM (f : β → α → m β) (init : β) (a : Array α) (beginIdx : Nat := 0) : m β :=
iterateMAux a (fun _ b a => f a b) beginIdx init
-- TODO(Leo): justify termination using wf-rec
@[specialize] partial def iterateM₂Aux (a₁ : Array α) (a₂ : Array σ) (f : ∀ (i : Fin a₁.size), α → σ → β → m β) : Nat → β → m β
| i, b =>
if h₁ : i < a₁.size then
let idx₁ : Fin a₁.size := ⟨i, h₁⟩;
if h₂ : i < a₂.size then
let idx₂ : Fin a₂.size := ⟨i, h₂⟩;
f idx₁ (a₁.get idx₁) (a₂.get idx₂) b >>= iterateM₂Aux (i+1)
else pure b
else pure b
@[inline] def iterateM₂ (a₁ : Array α) (a₂ : Array σ) (b : β) (f : ∀ (i : Fin a₁.size), α → σ → β → m β) : m β :=
iterateM₂Aux a₁ a₂ f 0 b
@[inline] def foldlM₂ (f : β → α → σ → m β) (b : β) (a₁ : Array α) (a₂ : Array σ): m β :=
iterateM₂ a₁ a₂ b (fun _ a₁ a₂ b => f b a₁ a₂)
@[specialize] partial def findSomeMAux (a : Array α) (f : α → m (Option β)) : Nat → m (Option β)
| i =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
do r ← f (a.get idx);
match r with
| some v => pure r
| none => findSomeMAux (i+1)
else pure none
@[inline] def findSomeM? (a : Array α) (f : α → m (Option β)) : m (Option β) :=
findSomeMAux a f 0
@[specialize] partial def findSomeRevMAux (a : Array α) (f : α → m (Option β)) : ∀ (idx : Nat), idx ≤ a.size → m (Option β)
| i, h =>
if hLt : 0 < i then
have i - 1 < i from Nat.subLt hLt (Nat.zeroLtSucc 0);
have i - 1 < a.size from Nat.ltOfLtOfLe this h;
let idx : Fin a.size := ⟨i - 1, this⟩;
do
r ← f (a.get idx);
match r with
| some v => pure r
| none =>
have i - 1 ≤ a.size from Nat.leOfLt this;
findSomeRevMAux (i-1) this
else pure none
@[inline] def findSomeRevM? (a : Array α) (f : α → m (Option β)) : m (Option β) :=
findSomeRevMAux a f a.size (Nat.leRefl _)
end
-- TODO(Leo): justify termination using wf-rec
@[specialize] partial def findMAux {α : Type} {m : Type → Type} [Monad m] (a : Array α) (p : α → m Bool) : Nat → m (Option α)
| i =>
if h : i < a.size then do
let v := a.get ⟨i, h⟩;
condM (p v) (pure (some v)) (findMAux (i+1))
else pure none
@[inline] def findM? {α : Type} {m : Type → Type} [Monad m] (a : Array α) (p : α → m Bool) : m (Option α) :=
findMAux a p 0
@[inline] def find? {α : Type} (a : Array α) (p : α → Bool) : Option α :=
Id.run $ a.findM? p
@[specialize] partial def findRevMAux {α : Type} {m : Type → Type} [Monad m] (a : Array α) (p : α → m Bool) : ∀ (idx : Nat), idx ≤ a.size → m (Option α)
| i, h =>
if hLt : 0 < i then
have i - 1 < i from Nat.subLt hLt (Nat.zeroLtSucc 0);
have i - 1 < a.size from Nat.ltOfLtOfLe this h;
let v := a.get ⟨i - 1, this⟩;
do {
condM (p v) (pure (some v)) $
have i - 1 ≤ a.size from Nat.leOfLt this;
findRevMAux (i-1) this
}
else pure none
@[inline] def findRevM? {α : Type} {m : Type → Type} [Monad m] (a : Array α) (p : α → m Bool) : m (Option α) :=
findRevMAux a p a.size (Nat.leRefl _)
@[inline] def findRev? {α : Type} (a : Array α) (p : α → Bool) : Option α :=
Id.run $ a.findRevM? p
section
variables {β : Type w} {σ : Type u}
@[inline] def iterate (a : Array α) (b : β) (f : ∀ (i : Fin a.size), α → β → β) : β :=
Id.run $ iterateMAux a f 0 b
@[inline] def iterateFrom (a : Array α) (b : β) (i : Nat) (f : ∀ (i : Fin a.size), α → β → β) : β :=
Id.run $ iterateMAux a f i b
@[inline] def foldl (f : β → α → β) (init : β) (a : Array α) : β :=
iterate a init (fun _ a b => f b a)
@[inline] def foldlFrom (f : β → α → β) (init : β) (a : Array α) (beginIdx : Nat := 0) : β :=
Id.run $ foldlFromM f init a beginIdx
@[inline] def iterate₂ (a₁ : Array α) (a₂ : Array σ) (b : β) (f : ∀ (i : Fin a₁.size), α → σ → β → β) : β :=
Id.run $ iterateM₂Aux a₁ a₂ f 0 b
@[inline] def foldl₂ (f : β → α → σ → β) (b : β) (a₁ : Array α) (a₂ : Array σ) : β :=
iterate₂ a₁ a₂ b (fun _ a₁ a₂ b => f b a₁ a₂)
@[inline] def findSome? (a : Array α) (f : α → Option β) : Option β :=
Id.run $ findSomeMAux a f 0
@[inline] def findSome! [Inhabited β] (a : Array α) (f : α → Option β) : β :=
match findSome? a f with
| some b => b
| none => panic! "failed to find element"
@[inline] def findSomeRev? (a : Array α) (f : α → Option β) : Option β :=
Id.run $ findSomeRevMAux a f a.size (Nat.leRefl _)
@[inline] def findSomeRev! [Inhabited β] (a : Array α) (f : α → Option β) : β :=
match findSomeRev? a f with
| some b => b
| none => panic! "failed to find element"
@[specialize] partial def findIdxMAux {m : Type → Type u} [Monad m] (a : Array α) (p : α → m Bool) : Nat → m (Option Nat)
| i =>
if h : i < a.size then
condM (p (a.get ⟨i, h⟩)) (pure (some i)) (findIdxMAux (i+1))
else
pure none
@[inline] def findIdxM? {m : Type → Type u} [Monad m] (a : Array α) (p : α → m Bool) : m (Option Nat) :=
findIdxMAux a p 0
@[specialize] partial def findIdxAux (a : Array α) (p : α → Bool) : Nat → Option Nat
| i =>
if h : i < a.size then
if p (a.get ⟨i, h⟩) then some i else findIdxAux (i+1)
else
none
@[inline] def findIdx? (a : Array α) (p : α → Bool) : Option Nat :=
findIdxAux a p 0
@[inline] def findIdx! (a : Array α) (p : α → Bool) : Nat :=
match findIdxAux a p 0 with
| some i => i
| none => panic! "failed to find element"
def getIdx? [HasBeq α] (a : Array α) (v : α) : Option Nat :=
a.findIdx? $ fun a => a == v
end
section
variables {m : Type → Type w} [Monad m]
@[specialize] partial def anyRangeMAux (a : Array α) (endIdx : Nat) (hlt : endIdx ≤ a.size) (p : α → m Bool) : Nat → m Bool
| i =>
if h : i < endIdx then
let idx : Fin a.size := ⟨i, Nat.ltOfLtOfLe h hlt⟩;
do b ← p (a.get idx);
match b with
| true => pure true
| false => anyRangeMAux (i+1)
else pure false
@[inline] def anyM (a : Array α) (p : α → m Bool) : m Bool :=
anyRangeMAux a a.size (Nat.leRefl _) p 0
@[inline] def allM (a : Array α) (p : α → m Bool) : m Bool := do
b ← anyM a (fun v => do b ← p v; pure (!b)); pure (!b)
@[inline] def anyRangeM (a : Array α) (beginIdx endIdx : Nat) (p : α → m Bool) : m Bool :=
if h : endIdx ≤ a.size then
anyRangeMAux a endIdx h p beginIdx
else
anyRangeMAux a a.size (Nat.leRefl _) p beginIdx
@[inline] def allRangeM (a : Array α) (beginIdx endIdx : Nat) (p : α → m Bool) : m Bool := do
b ← anyRangeM a beginIdx endIdx (fun v => do b ← p v; pure b); pure (!b)
end
@[inline] def any (a : Array α) (p : α → Bool) : Bool :=
Id.run $ anyM a p
@[inline] def anyRange (a : Array α) (beginIdx endIdx : Nat) (p : α → Bool) : Bool :=
Id.run $ anyRangeM a beginIdx endIdx p
@[inline] def anyFrom (a : Array α) (beginIdx : Nat) (p : α → Bool) : Bool :=
Id.run $ anyRangeM a beginIdx a.size p
@[inline] def all (a : Array α) (p : α → Bool) : Bool :=
!any a (fun v => !p v)
@[inline] def allRange (a : Array α) (beginIdx endIdx : Nat) (p : α → Bool) : Bool :=
!anyRange a beginIdx endIdx (fun v => !p v)
section
variables {m : Type v → Type w} [Monad m]
variable {β : Type v}
@[specialize] private def iterateRevMAux (a : Array α) (f : ∀ (i : Fin a.size), α → β → m β) : ∀ (i : Nat), i ≤ a.size → β → m β
| 0, h, b => pure b
| j+1, h, b => do
let i : Fin a.size := ⟨j, h⟩;
b ← f i (a.get i) b;
iterateRevMAux j (Nat.leOfLt h) b
@[inline] def iterateRevM (a : Array α) (b : β) (f : ∀ (i : Fin a.size), α → β → m β) : m β :=
iterateRevMAux a f a.size (Nat.leRefl _) b
@[inline] def foldrM (f : α → β → m β) (init : β) (a : Array α) : m β :=
iterateRevM a init (fun _ => f)
@[specialize] private def foldrRangeMAux (a : Array α) (f : α → β → m β) (beginIdx : Nat) : ∀ (i : Nat), i ≤ a.size → β → m β
| 0, h, b => pure b
| j+1, h, b => do
let i : Fin a.size := ⟨j, h⟩;
b ← f (a.get i) b;
if j == beginIdx then pure b else foldrRangeMAux j (Nat.leOfLt h) b
@[inline] def foldrRangeM (beginIdx endIdx : Nat) (f : α → β → m β) (ini : β) (a : Array α) : m β :=
if h : endIdx ≤ a.size then
foldrRangeMAux a f beginIdx endIdx h ini
else
foldrRangeMAux a f beginIdx a.size (Nat.leRefl _) ini
@[specialize] partial def foldlStepMAux (step : Nat) (a : Array α) (f : β → α → m β) : Nat → β → m β
| i, b =>
if h : i < a.size then do
let curr := a.get ⟨i, h⟩;
b ← f b curr;
foldlStepMAux (i+step) b
else
pure b
@[inline] def foldlStepM (f : β → α → m β) (init : β) (step : Nat) (a : Array α) : m β :=
foldlStepMAux step a f 0 init
end
@[inline] def iterateRev {β} (a : Array α) (b : β) (f : ∀ (i : Fin a.size), α → β → β) : β :=
Id.run $ iterateRevM a b f
@[inline] def foldr {β} (f : α → β → β) (init : β) (a : Array α) : β :=
Id.run $ foldrM f init a
@[inline] def foldrRange {β} (beginIdx endIdx : Nat) (f : α → β → β) (init : β) (a : Array α) : β :=
Id.run $ foldrRangeM beginIdx endIdx f init a
@[inline] def foldlStep {β} (f : β → α → β) (init : β) (step : Nat) (a : Array α) : β :=
Id.run $ foldlStepM f init step a
@[inline] def getEvenElems (a : Array α) : Array α :=
a.foldlStep (fun r a => Array.push r a) empty 2
def toList (a : Array α) : List α :=
a.foldr List.cons []
instance [HasRepr α] : HasRepr (Array α) :=
⟨fun a => "#" ++ repr a.toList⟩
instance [HasToString α] : HasToString (Array α) :=
⟨fun a => "#" ++ toString a.toList⟩
section
variables {m : Type v → Type w} [Monad m]
variable {β : Type v}
@[specialize] unsafe partial def umapMAux (f : Nat → α → m β) : Nat → Array NonScalar → m (Array PNonScalar.{v})
| i, a =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
let v : NonScalar := a.get idx;
let a := a.set idx (arbitrary _);
do newV ← f i (unsafeCast v); umapMAux (i+1) (a.set idx (unsafeCast newV))
else
pure (unsafeCast a)
@[inline] unsafe partial def umapM (f : α → m β) (as : Array α) : m (Array β) :=
@unsafeCast (m (Array PNonScalar.{v})) (m (Array β)) $ umapMAux (fun i a => f a) 0 (unsafeCast as)
@[inline] unsafe partial def umapIdxM (f : Nat → α → m β) (as : Array α) : m (Array β) :=
@unsafeCast (m (Array PNonScalar.{v})) (m (Array β)) $ umapMAux f 0 (unsafeCast as)
@[implementedBy Array.umapM] def mapM (f : α → m β) (as : Array α) : m (Array β) :=
as.foldlM (fun bs a => do b ← f a; pure (bs.push b)) (mkEmpty as.size)
@[implementedBy Array.umapIdxM] def mapIdxM (f : Nat → α → m β) (as : Array α) : m (Array β) :=
as.iterateM (mkEmpty as.size) (fun i a bs => do b ← f i.val a; pure (bs.push b))
end
section
variables {m : Type u → Type v} [Monad m]
@[inline] def modifyM [Inhabited α] (a : Array α) (i : Nat) (f : α → m α) : m (Array α) :=
if h : i < a.size then do
let idx : Fin a.size := ⟨i, h⟩;
let v := a.get idx;
let a := a.set idx (arbitrary α);
v ← f v;
pure $ (a.set idx v)
else
pure a
end
section
variable {β : Type v}
@[inline] def modify [Inhabited α] (a : Array α) (i : Nat) (f : α → α) : Array α :=
Id.run $ a.modifyM i f
@[inline] def modifyOp [Inhabited α] (self : Array α) (idx : Nat) (f : α → α) : Array α :=
self.modify idx f
@[inline] def mapIdx (f : Nat → α → β) (a : Array α) : Array β :=
Id.run $ mapIdxM f a
@[inline] def map (f : α → β) (as : Array α) : Array β :=
Id.run $ mapM f as
end
section
variables {m : Type v → Type w} [Monad m]
variable {β : Type v}
@[specialize]
partial def forMAux (f : α → m β) (a : Array α) : Nat → m PUnit
| i =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
let v : α := a.get idx;
do f v; forMAux (i+1)
else
pure ⟨⟩
@[inline] def forM (f : α → m β) (a : Array α) : m PUnit :=
a.forMAux f 0
@[specialize]
partial def forRevMAux (f : α → m β) (a : Array α) : forall (i : Nat), i ≤ a.size → m PUnit
| i, h =>
if hLt : 0 < i then
have i - 1 < i from Nat.subLt hLt (Nat.zeroLtSucc 0);
have i - 1 < a.size from Nat.ltOfLtOfLe this h;
let v : α := a.get ⟨i-1, this⟩;
have i - 1 ≤ a.size from Nat.leOfLt this;
do f v; forRevMAux (i-1) this
else
pure ⟨⟩
@[inline] def forRevM (f : α → m β) (a : Array α) : m PUnit :=
a.forRevMAux f a.size (Nat.leRefl _)
end
-- TODO(Leo): justify termination using wf-rec
partial def extractAux (a : Array α) : Nat → ∀ (e : Nat), e ≤ a.size → Array α → Array α
| i, e, hle, r =>
if hlt : i < e then
let idx : Fin a.size := ⟨i, Nat.ltOfLtOfLe hlt hle⟩;
extractAux (i+1) e hle (r.push (a.get idx))
else r
def extract (a : Array α) (b e : Nat) : Array α :=
let r : Array α := mkEmpty (e - b);
if h : e ≤ a.size then extractAux a b e h r
else r
protected def append (a : Array α) (b : Array α) : Array α :=
b.foldl (fun a v => a.push v) a
instance : HasAppend (Array α) := ⟨Array.append⟩
-- TODO(Leo): justify termination using wf-rec
@[specialize] partial def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) : Nat → Bool
| i =>
if h : i < a.size then
let aidx : Fin a.size := ⟨i, h⟩;
let bidx : Fin b.size := ⟨i, hsz ▸ h⟩;
match p (a.get aidx) (b.get bidx) with
| true => isEqvAux (i+1)
| false => false
else
true
@[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
if h : a.size = b.size then
isEqvAux a b h p 0
else
false
instance [HasBeq α] : HasBeq (Array α) :=
⟨fun a b => isEqv a b HasBeq.beq⟩
-- TODO(Leo): justify termination using wf-rec, and use `swap`
partial def reverseAux : Array α → Nat → Array α
| a, i =>
let n := a.size;
if i < n / 2 then
reverseAux (a.swap! i (n - i - 1)) (i+1)
else
a
def reverse (a : Array α) : Array α :=
reverseAux a 0
-- TODO(Leo): justify termination using wf-rec
@[specialize] partial def filterAux (p : α → Bool) : Array α → Nat → Nat → Array α
| a, i, j =>
if h₁ : i < a.size then
if p (a.get ⟨i, h₁⟩) then
if h₂ : j < i then
filterAux (a.swap ⟨i, h₁⟩ ⟨j, Nat.ltTrans h₂ h₁⟩) (i+1) (j+1)
else
filterAux a (i+1) (j+1)
else
filterAux a (i+1) j
else
a.shrink j
@[inline] def filter (p : α → Bool) (as : Array α) : Array α :=
filterAux p as 0 0
@[specialize] partial def filterMAux {m : Type → Type} [Monad m] {α : Type} (p : α → m Bool) : Array α → Nat → Nat → m (Array α)
| a, i, j =>
if h₁ : i < a.size then
condM (p (a.get ⟨i, h₁⟩))
(if h₂ : j < i then
filterMAux (a.swap ⟨i, h₁⟩ ⟨j, Nat.ltTrans h₂ h₁⟩) (i+1) (j+1)
else
filterMAux a (i+1) (j+1))
(filterMAux a (i+1) j)
else
pure $ a.shrink j
@[inline] def filterM {m : Type → Type} [Monad m] {α : Type} (p : α → m Bool) (as : Array α) : m (Array α) :=
filterMAux p as 0 0
partial def indexOfAux {α} [HasBeq α] (a : Array α) (v : α) : Nat → Option (Fin a.size)
| i =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
if a.get idx == v then some idx
else indexOfAux (i+1)
else none
def indexOf {α} [HasBeq α] (a : Array α) (v : α) : Option (Fin a.size) :=
indexOfAux a v 0
partial def eraseIdxAux {α} : Nat → Array α → Array α
| i, a =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
let idx1 : Fin a.size := ⟨i - 1, Nat.ltOfLeOfLt (Nat.predLe i) h⟩;
eraseIdxAux (i+1) (a.swap idx idx1)
else
a.pop
def feraseIdx {α} (a : Array α) (i : Fin a.size) : Array α :=
eraseIdxAux (i.val + 1) a
def eraseIdx {α} (a : Array α) (i : Nat) : Array α :=
if i < a.size then eraseIdxAux (i+1) a else a
theorem szFSwapEq (a : Array α) (i j : Fin a.size) : (a.swap i j).size = a.size :=
rfl
theorem szPopEq (a : Array α) : a.pop.size = a.size - 1 :=
rfl
section
/- Instance for justifying `partial` declaration.
We should be able to delete it as soon as we restore support for well-founded recursion. -/
instance eraseIdxSzAuxInstance (a : Array α) : Inhabited { r : Array α // r.size = a.size - 1 } :=
⟨⟨a.pop, szPopEq a⟩⟩
partial def eraseIdxSzAux {α} (a : Array α) : ∀ (i : Nat) (r : Array α), r.size = a.size → { r : Array α // r.size = a.size - 1 }
| i, r, heq =>
if h : i < r.size then
let idx : Fin r.size := ⟨i, h⟩;
let idx1 : Fin r.size := ⟨i - 1, Nat.ltOfLeOfLt (Nat.predLe i) h⟩;
eraseIdxSzAux (i+1) (r.swap idx idx1) ((szFSwapEq r idx idx1).trans heq)
else
⟨r.pop, (szPopEq r).trans (heq ▸ rfl)⟩
end
def eraseIdx' {α} (a : Array α) (i : Fin a.size) : { r : Array α // r.size = a.size - 1 } :=
eraseIdxSzAux a (i.val + 1) a rfl
def contains [HasBeq α] (as : Array α) (a : α) : Bool :=
as.any $ fun b => a == b
def elem [HasBeq α] (a : α) (as : Array α) : Bool :=
as.contains a
partial def insertAtAux {α} (i : Nat) : Array α → Nat → Array α
| as, j =>
if i == j then as
else
let as := as.swap! (j-1) j;
insertAtAux as (j-1)
/--
Insert element `a` at position `i`.
Pre: `i < as.size` -/
def insertAt {α} (as : Array α) (i : Nat) (a : α) : Array α :=
if i > as.size then panic! "invalid index"
else
let as := as.push a;
as.insertAtAux i as.size
theorem ext {α : Type u} (a b : Array α) : a.size = b.size → (∀ (i : Nat) (hi₁ : i < a.size) (hi₂ : i < b.size) , a.get ⟨i, hi₁⟩ = b.get ⟨i, hi₂⟩) → a = b :=
match a, b with
| ⟨sz₁, f₁⟩, ⟨sz₂, f₂⟩ =>
show sz₁ = sz₂ → (∀ (i : Nat) (hi₁ : i < sz₁) (hi₂ : i < sz₂) , f₁ ⟨i, hi₁⟩ = f₂ ⟨i, hi₂⟩) → Array.mk sz₁ f₁ = Array.mk sz₂ f₂ from
fun h₁ h₂ =>
match sz₁, sz₂, f₁, f₂, h₁, h₂ with
| sz, _, f₁, f₂, rfl, h₂ =>
have f₁ = f₂ from funext $ fun ⟨i, hi₁⟩ => h₂ i hi₁ hi₁;
congrArg _ this
theorem extLit {α : Type u} {n : Nat}
(a b : Array α)
(hsz₁ : a.size = n) (hsz₂ : b.size = n)
(h : ∀ (i : Nat) (hi : i < n), a.getLit i hsz₁ hi = b.getLit i hsz₂ hi) : a = b :=
Array.ext a b (hsz₁.trans hsz₂.symm) $ fun i hi₁ hi₂ => h i (hsz₁ ▸ hi₁)
end Array
export Array (mkArray)
@[inlineIfReduce] def List.toArrayAux {α : Type u} : List α → Array α → Array α
| [], r => r
| a::as, r => List.toArrayAux as (r.push a)
@[inlineIfReduce] def List.redLength {α : Type u} : List α → Nat
| [] => 0
| _::as => as.redLength + 1
@[inline] def List.toArray {α : Type u} (as : List α) : Array α :=
as.toArrayAux (Array.mkEmpty as.redLength)
|
2707ebccf7f0b9ae1c5468e56fefb5903dcc6ddb | d1a52c3f208fa42c41df8278c3d280f075eb020c | /tests/lean/run/270.lean | 6d6b122b6d67cad36f978981bd50549d0cfe25c3 | [
"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 | 1,083 | lean | class CommAddSemigroup (α : Type u) extends Add α where
addComm : {a b : α} → a + b = b + a
addAssoc : {a b c : α} → a + b + c = a + (b + c)
open CommAddSemigroup
theorem addComm3 [CommAddSemigroup α] {a b c : α} : a + b + c = a + c + b := by {
have h : b + c = c + b := addComm;
have h' := congrArg (a + .) h;
simp at h';
rw [←addAssoc] at h';
rw [←addAssoc (a := a)] at h';
exact h';
}
theorem addComm4 [CommAddSemigroup α] {a b c : α} : a + b + c = a + c + b := by {
rw [addAssoc, addAssoc];
rw [addComm (a := b)];
}
theorem addComm5 [CommAddSemigroup α] {a b c : α} : a + b + c = a + c + b := by {
have h : b + c = c + b := addComm;
have h' := congrArg (a + .) h;
simp at h';
rw [←addAssoc] at h';
rw [←@addAssoc (a := a)] at h';
exact h';
}
theorem addComm6 [CommAddSemigroup α] {a b c : α} : a + b + c = a + c + b := by {
have h : b + c = c + b := addComm;
have h' := congrArg (a + .) h;
simp at h';
rw [←addAssoc] at h';
rw [←addAssoc] at h';
exact h';
}
|
1834d5e85c8740fe1c118a07c3fd9c1fe8bcaedc | dd0f5513e11c52db157d2fcc8456d9401a6cd9da | /03_Propositions_and_Proofs.org.22.lean | da22fad0928e1b41d20c90199a96657752b04ea1 | [] | no_license | cjmazey/lean-tutorial | ba559a49f82aa6c5848b9bf17b7389bf7f4ba645 | 381f61c9fcac56d01d959ae0fa6e376f2c4e3b34 | refs/heads/master | 1,610,286,098,832 | 1,447,124,923,000 | 1,447,124,923,000 | 43,082,433 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 225 | lean | /- page 40 -/
import standard
variables p q r: Prop
-- BEGIN
example (H : p ∨ q) : q ∨ p :=
or.elim H
(assume Hp : p,
show q ∨ p, from or.intro_right q Hp)
(assume Hq : q,
show q ∨ p, from or.intro_left p Hq)
-- END
|
fc29a24fd0c3b8068cd718cedbc11a619cdebd49 | 8d65764a9e5f0923a67fc435eb1a5a1d02fd80e3 | /src/algebra/ordered_ring.lean | b61bae03643de571aa7339a53df69f81d346a229 | [
"Apache-2.0"
] | permissive | troyjlee/mathlib | e18d4b8026e32062ab9e89bc3b003a5d1cfec3f5 | 45e7eb8447555247246e3fe91c87066506c14875 | refs/heads/master | 1,689,248,035,046 | 1,629,470,528,000 | 1,629,470,528,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 65,617 | 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
-/
import algebra.ordered_group
import algebra.invertible
import data.set.intervals.basic
set_option old_structure_cmd true
universe u
variable {α : Type u}
lemma add_one_le_two_mul [preorder α] [semiring α] [covariant_class α α (+) (≤)]
{a : α} (a1 : 1 ≤ a) :
a + 1 ≤ 2 * a :=
calc a + 1 ≤ a + a : add_le_add_left a1 a
... = 2 * a : (two_mul _).symm
/-- An `ordered_semiring α` is a semiring `α` with a partial order such that
multiplication with a positive number and addition are monotone. -/
@[protect_proj]
class ordered_semiring (α : Type u) extends semiring α, ordered_cancel_add_comm_monoid α :=
(zero_le_one : 0 ≤ (1 : α))
(mul_lt_mul_of_pos_left : ∀ a b c : α, a < b → 0 < c → c * a < c * b)
(mul_lt_mul_of_pos_right : ∀ a b c : α, a < b → 0 < c → a * c < b * c)
section ordered_semiring
variables [ordered_semiring α] {a b c d : α}
@[simp] lemma zero_le_one : 0 ≤ (1:α) :=
ordered_semiring.zero_le_one
lemma zero_le_two : 0 ≤ (2:α) :=
add_nonneg zero_le_one zero_le_one
lemma one_le_two : 1 ≤ (2:α) :=
calc (1:α) = 0 + 1 : (zero_add _).symm
... ≤ 1 + 1 : add_le_add_right zero_le_one _
section nontrivial
variables [nontrivial α]
@[simp] lemma zero_lt_one : 0 < (1 : α) :=
lt_of_le_of_ne zero_le_one zero_ne_one
lemma zero_lt_two : 0 < (2:α) := add_pos zero_lt_one zero_lt_one
@[field_simps] lemma two_ne_zero : (2:α) ≠ 0 :=
ne.symm (ne_of_lt zero_lt_two)
lemma one_lt_two : 1 < (2:α) :=
calc (2:α) = 1+1 : one_add_one_eq_two
... > 1+0 : add_lt_add_left zero_lt_one _
... = 1 : add_zero 1
lemma zero_lt_three : 0 < (3:α) := add_pos zero_lt_two zero_lt_one
lemma zero_lt_four : 0 < (4:α) := add_pos zero_lt_two zero_lt_two
end nontrivial
lemma mul_lt_mul_of_pos_left (h₁ : a < b) (h₂ : 0 < c) : c * a < c * b :=
ordered_semiring.mul_lt_mul_of_pos_left a b c h₁ h₂
lemma mul_lt_mul_of_pos_right (h₁ : a < b) (h₂ : 0 < c) : a * c < b * c :=
ordered_semiring.mul_lt_mul_of_pos_right a b c h₁ h₂
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul_of_nonneg_left [@decidable_rel α (≤)]
(h₁ : a ≤ b) (h₂ : 0 ≤ c) : c * a ≤ c * b :=
begin
by_cases ba : b ≤ a, { simp [ba.antisymm h₁] },
by_cases c0 : c ≤ 0, { simp [c0.antisymm h₂] },
exact (mul_lt_mul_of_pos_left (h₁.lt_of_not_le ba) (h₂.lt_of_not_le c0)).le,
end
lemma mul_le_mul_of_nonneg_left : a ≤ b → 0 ≤ c → c * a ≤ c * b :=
by classical; exact decidable.mul_le_mul_of_nonneg_left
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul_of_nonneg_right [@decidable_rel α (≤)]
(h₁ : a ≤ b) (h₂ : 0 ≤ c) : a * c ≤ b * c :=
begin
by_cases ba : b ≤ a, { simp [ba.antisymm h₁] },
by_cases c0 : c ≤ 0, { simp [c0.antisymm h₂] },
exact (mul_lt_mul_of_pos_right (h₁.lt_of_not_le ba) (h₂.lt_of_not_le c0)).le,
end
lemma mul_le_mul_of_nonneg_right : a ≤ b → 0 ≤ c → a * c ≤ b * c :=
by classical; exact decidable.mul_le_mul_of_nonneg_right
-- TODO: there are four variations, depending on which variables we assume to be nonneg
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul [@decidable_rel α (≤)]
(hac : a ≤ c) (hbd : b ≤ d) (nn_b : 0 ≤ b) (nn_c : 0 ≤ c) : a * b ≤ c * d :=
calc
a * b ≤ c * b : decidable.mul_le_mul_of_nonneg_right hac nn_b
... ≤ c * d : decidable.mul_le_mul_of_nonneg_left hbd nn_c
lemma mul_le_mul : a ≤ c → b ≤ d → 0 ≤ b → 0 ≤ c → a * b ≤ c * d :=
by classical; exact decidable.mul_le_mul
-- See Note [decidable namespace]
protected lemma decidable.mul_nonneg_le_one_le {α : Type*} [ordered_semiring α]
[@decidable_rel α (≤)] {a b c : α}
(h₁ : 0 ≤ c) (h₂ : a ≤ c) (h₃ : 0 ≤ b) (h₄ : b ≤ 1) : a * b ≤ c :=
by simpa only [mul_one] using decidable.mul_le_mul h₂ h₄ h₃ h₁
lemma mul_nonneg_le_one_le {α : Type*} [ordered_semiring α] {a b c : α} :
0 ≤ c → a ≤ c → 0 ≤ b → b ≤ 1 → a * b ≤ c :=
by classical; exact decidable.mul_nonneg_le_one_le
-- See Note [decidable namespace]
protected lemma decidable.mul_nonneg [@decidable_rel α (≤)]
(ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b :=
have h : 0 * b ≤ a * b, from decidable.mul_le_mul_of_nonneg_right ha hb,
by rwa [zero_mul] at h
lemma mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b := by classical; exact decidable.mul_nonneg
-- See Note [decidable namespace]
protected lemma decidable.mul_nonpos_of_nonneg_of_nonpos [@decidable_rel α (≤)]
(ha : 0 ≤ a) (hb : b ≤ 0) : a * b ≤ 0 :=
have h : a * b ≤ a * 0, from decidable.mul_le_mul_of_nonneg_left hb ha,
by rwa mul_zero at h
lemma mul_nonpos_of_nonneg_of_nonpos : 0 ≤ a → b ≤ 0 → a * b ≤ 0 :=
by classical; exact decidable.mul_nonpos_of_nonneg_of_nonpos
-- See Note [decidable namespace]
protected lemma decidable.mul_nonpos_of_nonpos_of_nonneg [@decidable_rel α (≤)]
(ha : a ≤ 0) (hb : 0 ≤ b) : a * b ≤ 0 :=
have h : a * b ≤ 0 * b, from decidable.mul_le_mul_of_nonneg_right ha hb,
by rwa zero_mul at h
lemma mul_nonpos_of_nonpos_of_nonneg : a ≤ 0 → 0 ≤ b → a * b ≤ 0 :=
by classical; exact decidable.mul_nonpos_of_nonpos_of_nonneg
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_mul [@decidable_rel α (≤)]
(hac : a < c) (hbd : b ≤ d) (pos_b : 0 < b) (nn_c : 0 ≤ c) : a * b < c * d :=
calc
a * b < c * b : mul_lt_mul_of_pos_right hac pos_b
... ≤ c * d : decidable.mul_le_mul_of_nonneg_left hbd nn_c
lemma mul_lt_mul : a < c → b ≤ d → 0 < b → 0 ≤ c → a * b < c * d :=
by classical; exact decidable.mul_lt_mul
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_mul' [@decidable_rel α (≤)]
(h1 : a ≤ c) (h2 : b < d) (h3 : 0 ≤ b) (h4 : 0 < c) : a * b < c * d :=
calc
a * b ≤ c * b : decidable.mul_le_mul_of_nonneg_right h1 h3
... < c * d : mul_lt_mul_of_pos_left h2 h4
lemma mul_lt_mul' : a ≤ c → b < d → 0 ≤ b → 0 < c → a * b < c * d :=
by classical; exact decidable.mul_lt_mul'
lemma mul_pos (ha : 0 < a) (hb : 0 < b) : 0 < a * b :=
have h : 0 * b < a * b, from mul_lt_mul_of_pos_right ha hb,
by rwa zero_mul at h
lemma mul_neg_of_pos_of_neg (ha : 0 < a) (hb : b < 0) : a * b < 0 :=
have h : a * b < a * 0, from mul_lt_mul_of_pos_left hb ha,
by rwa mul_zero at h
lemma mul_neg_of_neg_of_pos (ha : a < 0) (hb : 0 < b) : a * b < 0 :=
have h : a * b < 0 * b, from mul_lt_mul_of_pos_right ha hb,
by rwa zero_mul at h
-- See Note [decidable namespace]
protected lemma decidable.mul_self_lt_mul_self [@decidable_rel α (≤)]
(h1 : 0 ≤ a) (h2 : a < b) : a * a < b * b :=
decidable.mul_lt_mul' h2.le h2 h1 $ h1.trans_lt h2
lemma mul_self_lt_mul_self (h1 : 0 ≤ a) (h2 : a < b) : a * a < b * b :=
mul_lt_mul' h2.le h2 h1 $ h1.trans_lt h2
-- See Note [decidable namespace]
protected lemma decidable.strict_mono_incr_on_mul_self [@decidable_rel α (≤)] :
strict_mono_incr_on (λ x : α, x * x) (set.Ici 0) :=
λ x hx y hy hxy, decidable.mul_self_lt_mul_self hx hxy
lemma strict_mono_incr_on_mul_self : strict_mono_incr_on (λ x : α, x * x) (set.Ici 0) :=
λ x hx y hy hxy, mul_self_lt_mul_self hx hxy
-- See Note [decidable namespace]
protected lemma decidable.mul_self_le_mul_self [@decidable_rel α (≤)]
(h1 : 0 ≤ a) (h2 : a ≤ b) : a * a ≤ b * b :=
decidable.mul_le_mul h2 h2 h1 $ h1.trans h2
lemma mul_self_le_mul_self (h1 : 0 ≤ a) (h2 : a ≤ b) : a * a ≤ b * b :=
mul_le_mul h2 h2 h1 $ h1.trans h2
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_mul'' [@decidable_rel α (≤)]
(h1 : a < c) (h2 : b < d) (h3 : 0 ≤ a) (h4 : 0 ≤ b) : a * b < c * d :=
h4.lt_or_eq_dec.elim
(λ b0, decidable.mul_lt_mul h1 h2.le b0 $ h3.trans h1.le)
(λ b0, by rw [← b0, mul_zero]; exact
mul_pos (h3.trans_lt h1) (h4.trans_lt h2))
lemma mul_lt_mul'' : a < c → b < d → 0 ≤ a → 0 ≤ b → a * b < c * d :=
by classical; exact decidable.mul_lt_mul''
-- See Note [decidable namespace]
protected lemma decidable.le_mul_of_one_le_right [@decidable_rel α (≤)]
(hb : 0 ≤ b) (h : 1 ≤ a) : b ≤ b * a :=
suffices b * 1 ≤ b * a, by rwa mul_one at this,
decidable.mul_le_mul_of_nonneg_left h hb
lemma le_mul_of_one_le_right : 0 ≤ b → 1 ≤ a → b ≤ b * a :=
by classical; exact decidable.le_mul_of_one_le_right
-- See Note [decidable namespace]
protected lemma decidable.le_mul_of_one_le_left [@decidable_rel α (≤)]
(hb : 0 ≤ b) (h : 1 ≤ a) : b ≤ a * b :=
suffices 1 * b ≤ a * b, by rwa one_mul at this,
decidable.mul_le_mul_of_nonneg_right h hb
lemma le_mul_of_one_le_left : 0 ≤ b → 1 ≤ a → b ≤ a * b :=
by classical; exact decidable.le_mul_of_one_le_left
-- See Note [decidable namespace]
protected lemma decidable.lt_mul_of_one_lt_right [@decidable_rel α (≤)]
(hb : 0 < b) (h : 1 < a) : b < b * a :=
suffices b * 1 < b * a, by rwa mul_one at this,
decidable.mul_lt_mul' (le_refl _) h zero_le_one hb
lemma lt_mul_of_one_lt_right : 0 < b → 1 < a → b < b * a :=
by classical; exact decidable.lt_mul_of_one_lt_right
-- See Note [decidable namespace]
protected lemma decidable.lt_mul_of_one_lt_left [@decidable_rel α (≤)]
(hb : 0 < b) (h : 1 < a) : b < a * b :=
suffices 1 * b < a * b, by rwa one_mul at this,
decidable.mul_lt_mul h (le_refl _) hb (zero_le_one.trans h.le)
lemma lt_mul_of_one_lt_left : 0 < b → 1 < a → b < a * b :=
by classical; exact decidable.lt_mul_of_one_lt_left
-- See Note [decidable namespace]
protected lemma decidable.add_le_mul_two_add [@decidable_rel α (≤)] {a b : α}
(a2 : 2 ≤ a) (b0 : 0 ≤ b) : a + (2 + b) ≤ a * (2 + b) :=
calc a + (2 + b) ≤ a + (a + a * b) :
add_le_add_left (add_le_add a2 (decidable.le_mul_of_one_le_left b0 (one_le_two.trans a2))) a
... ≤ a * (2 + b) : by rw [mul_add, mul_two, add_assoc]
lemma add_le_mul_two_add {a b : α} : 2 ≤ a → 0 ≤ b → a + (2 + b) ≤ a * (2 + b) :=
by classical; exact decidable.add_le_mul_two_add
-- See Note [decidable namespace]
protected lemma decidable.one_le_mul_of_one_le_of_one_le [@decidable_rel α (≤)]
{a b : α} (a1 : 1 ≤ a) (b1 : 1 ≤ b) : (1 : α) ≤ a * b :=
(mul_one (1 : α)).symm.le.trans (decidable.mul_le_mul a1 b1 zero_le_one (zero_le_one.trans a1))
lemma one_le_mul_of_one_le_of_one_le {a b : α} : 1 ≤ a → 1 ≤ b → (1 : α) ≤ a * b :=
by classical; exact decidable.one_le_mul_of_one_le_of_one_le
/-- Pullback an `ordered_semiring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.ordered_semiring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
ordered_semiring β :=
{ zero_le_one := show f 0 ≤ f 1, by simp only [zero, one, zero_le_one],
mul_lt_mul_of_pos_left := λ a b c ab c0, show f (c * a) < f (c * b),
begin
rw [mul, mul],
refine mul_lt_mul_of_pos_left ab _,
rwa ← zero,
end,
mul_lt_mul_of_pos_right := λ a b c ab c0, show f (a * c) < f (b * c),
begin
rw [mul, mul],
refine mul_lt_mul_of_pos_right ab _,
rwa ← zero,
end,
..hf.ordered_cancel_add_comm_monoid f zero add,
..hf.semiring f zero one add mul }
section
variable [nontrivial α]
lemma bit1_pos (h : 0 ≤ a) : 0 < bit1 a :=
lt_add_of_le_of_pos (add_nonneg h h) zero_lt_one
lemma lt_add_one (a : α) : a < a + 1 :=
lt_add_of_le_of_pos le_rfl zero_lt_one
lemma lt_one_add (a : α) : a < 1 + a :=
by { rw [add_comm], apply lt_add_one }
end
lemma bit1_pos' (h : 0 < a) : 0 < bit1 a :=
begin
nontriviality,
exact bit1_pos h.le,
end
-- See Note [decidable namespace]
protected lemma decidable.one_lt_mul [@decidable_rel α (≤)]
(ha : 1 ≤ a) (hb : 1 < b) : 1 < a * b :=
begin
nontriviality,
exact (one_mul (1 : α)) ▸ decidable.mul_lt_mul' ha hb zero_le_one (zero_lt_one.trans_le ha)
end
lemma one_lt_mul : 1 ≤ a → 1 < b → 1 < a * b :=
by classical; exact decidable.one_lt_mul
-- See Note [decidable namespace]
protected lemma decidable.mul_le_one [@decidable_rel α (≤)]
(ha : a ≤ 1) (hb' : 0 ≤ b) (hb : b ≤ 1) : a * b ≤ 1 :=
begin rw ← one_mul (1 : α), apply decidable.mul_le_mul; {assumption <|> apply zero_le_one} end
lemma mul_le_one : a ≤ 1 → 0 ≤ b → b ≤ 1 → a * b ≤ 1 :=
by classical; exact decidable.mul_le_one
-- See Note [decidable namespace]
protected lemma decidable.one_lt_mul_of_le_of_lt [@decidable_rel α (≤)]
(ha : 1 ≤ a) (hb : 1 < b) : 1 < a * b :=
begin
nontriviality,
calc 1 = 1 * 1 : by rw one_mul
... < a * b : decidable.mul_lt_mul' ha hb zero_le_one (zero_lt_one.trans_le ha)
end
lemma one_lt_mul_of_le_of_lt : 1 ≤ a → 1 < b → 1 < a * b :=
by classical; exact decidable.one_lt_mul_of_le_of_lt
-- See Note [decidable namespace]
protected lemma decidable.one_lt_mul_of_lt_of_le [@decidable_rel α (≤)]
(ha : 1 < a) (hb : 1 ≤ b) : 1 < a * b :=
begin
nontriviality,
calc 1 = 1 * 1 : by rw one_mul
... < a * b : decidable.mul_lt_mul ha hb zero_lt_one $ zero_le_one.trans ha.le
end
lemma one_lt_mul_of_lt_of_le : 1 < a → 1 ≤ b → 1 < a * b :=
by classical; exact decidable.one_lt_mul_of_lt_of_le
-- See Note [decidable namespace]
protected lemma decidable.mul_le_of_le_one_right [@decidable_rel α (≤)]
(ha : 0 ≤ a) (hb1 : b ≤ 1) : a * b ≤ a :=
calc a * b ≤ a * 1 : decidable.mul_le_mul_of_nonneg_left hb1 ha
... = a : mul_one a
lemma mul_le_of_le_one_right : 0 ≤ a → b ≤ 1 → a * b ≤ a :=
by classical; exact decidable.mul_le_of_le_one_right
-- See Note [decidable namespace]
protected lemma decidable.mul_le_of_le_one_left [@decidable_rel α (≤)]
(hb : 0 ≤ b) (ha1 : a ≤ 1) : a * b ≤ b :=
calc a * b ≤ 1 * b : decidable.mul_le_mul ha1 le_rfl hb zero_le_one
... = b : one_mul b
lemma mul_le_of_le_one_left : 0 ≤ b → a ≤ 1 → a * b ≤ b :=
by classical; exact decidable.mul_le_of_le_one_left
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_one_of_nonneg_of_lt_one_left [@decidable_rel α (≤)]
(ha0 : 0 ≤ a) (ha : a < 1) (hb : b ≤ 1) : a * b < 1 :=
calc a * b ≤ a : decidable.mul_le_of_le_one_right ha0 hb
... < 1 : ha
lemma mul_lt_one_of_nonneg_of_lt_one_left : 0 ≤ a → a < 1 → b ≤ 1 → a * b < 1 :=
by classical; exact decidable.mul_lt_one_of_nonneg_of_lt_one_left
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_one_of_nonneg_of_lt_one_right [@decidable_rel α (≤)]
(ha : a ≤ 1) (hb0 : 0 ≤ b) (hb : b < 1) : a * b < 1 :=
calc a * b ≤ b : decidable.mul_le_of_le_one_left hb0 ha
... < 1 : hb
lemma mul_lt_one_of_nonneg_of_lt_one_right : a ≤ 1 → 0 ≤ b → b < 1 → a * b < 1 :=
by classical; exact decidable.mul_lt_one_of_nonneg_of_lt_one_right
end ordered_semiring
section ordered_comm_semiring
/-- An `ordered_comm_semiring α` is a commutative semiring `α` with a partial order such that
multiplication with a positive number and addition are monotone. -/
@[protect_proj]
class ordered_comm_semiring (α : Type u) extends ordered_semiring α, comm_semiring α
/-- Pullback an `ordered_comm_semiring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.ordered_comm_semiring [ordered_comm_semiring α] {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
ordered_comm_semiring β :=
{ ..hf.comm_semiring f zero one add mul,
..hf.ordered_semiring f zero one add mul }
end ordered_comm_semiring
/--
A `linear_ordered_semiring α` is a nontrivial semiring `α` with a linear order
such that multiplication with a positive number and addition are monotone.
-/
-- It's not entirely clear we should assume `nontrivial` at this point;
-- it would be reasonable to explore changing this,
-- but be warned that the instances involving `domain` may cause
-- typeclass search loops.
@[protect_proj]
class linear_ordered_semiring (α : Type u) extends ordered_semiring α, linear_order α, nontrivial α
section linear_ordered_semiring
variables [linear_ordered_semiring α] {a b c d : α}
-- `norm_num` expects the lemma stating `0 < 1` to have a single typeclass argument
-- (see `norm_num.prove_pos_nat`).
-- Rather than working out how to relax that assumption,
-- we provide a synonym for `zero_lt_one` (which needs both `ordered_semiring α` and `nontrivial α`)
-- with only a `linear_ordered_semiring` typeclass argument.
lemma zero_lt_one' : 0 < (1 : α) := zero_lt_one
lemma lt_of_mul_lt_mul_left (h : c * a < c * b) (hc : 0 ≤ c) : a < b :=
by haveI := @linear_order.decidable_le α _; exact lt_of_not_ge
(assume h1 : b ≤ a,
have h2 : c * b ≤ c * a, from decidable.mul_le_mul_of_nonneg_left h1 hc,
h2.not_lt h)
lemma lt_of_mul_lt_mul_right (h : a * c < b * c) (hc : 0 ≤ c) : a < b :=
by haveI := @linear_order.decidable_le α _; exact lt_of_not_ge
(assume h1 : b ≤ a,
have h2 : b * c ≤ a * c, from decidable.mul_le_mul_of_nonneg_right h1 hc,
h2.not_lt h)
lemma le_of_mul_le_mul_left (h : c * a ≤ c * b) (hc : 0 < c) : a ≤ b :=
le_of_not_gt
(assume h1 : b < a,
have h2 : c * b < c * a, from mul_lt_mul_of_pos_left h1 hc,
h2.not_le h)
lemma le_of_mul_le_mul_right (h : a * c ≤ b * c) (hc : 0 < c) : a ≤ b :=
le_of_not_gt
(assume h1 : b < a,
have h2 : b * c < a * c, from mul_lt_mul_of_pos_right h1 hc,
h2.not_le h)
lemma pos_and_pos_or_neg_and_neg_of_mul_pos (hab : 0 < a * b) :
(0 < a ∧ 0 < b) ∨ (a < 0 ∧ b < 0) :=
begin
haveI := @linear_order.decidable_le α _,
rcases lt_trichotomy 0 a with (ha|rfl|ha),
{ refine or.inl ⟨ha, lt_imp_lt_of_le_imp_le (λ hb, _) hab⟩,
exact decidable.mul_nonpos_of_nonneg_of_nonpos ha.le hb },
{ rw [zero_mul] at hab, exact hab.false.elim },
{ refine or.inr ⟨ha, lt_imp_lt_of_le_imp_le (λ hb, _) hab⟩,
exact decidable.mul_nonpos_of_nonpos_of_nonneg ha.le hb }
end
lemma nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nnonneg (hab : 0 ≤ a * b) :
(0 ≤ a ∧ 0 ≤ b) ∨ (a ≤ 0 ∧ b ≤ 0) :=
begin
haveI := @linear_order.decidable_le α _,
refine decidable.or_iff_not_and_not.2 _,
simp only [not_and, not_le], intros ab nab, apply not_lt_of_le hab _,
rcases lt_trichotomy 0 a with (ha|rfl|ha),
exacts [mul_neg_of_pos_of_neg ha (ab ha.le), ((ab le_rfl).asymm (nab le_rfl)).elim,
mul_neg_of_neg_of_pos ha (nab ha.le)]
end
lemma pos_of_mul_pos_left (h : 0 < a * b) (ha : 0 ≤ a) : 0 < b :=
((pos_and_pos_or_neg_and_neg_of_mul_pos h).resolve_right $ λ h, h.1.not_le ha).2
lemma pos_of_mul_pos_right (h : 0 < a * b) (hb : 0 ≤ b) : 0 < a :=
((pos_and_pos_or_neg_and_neg_of_mul_pos h).resolve_right $ λ h, h.2.not_le hb).1
@[simp] lemma inv_of_pos [invertible a] : 0 < ⅟a ↔ 0 < a :=
begin
have : 0 < a * ⅟a, by simp only [mul_inv_of_self, zero_lt_one],
exact ⟨λ h, pos_of_mul_pos_right this h.le, λ h, pos_of_mul_pos_left this h.le⟩
end
@[simp] lemma inv_of_nonpos [invertible a] : ⅟a ≤ 0 ↔ a ≤ 0 :=
by simp only [← not_lt, inv_of_pos]
lemma nonneg_of_mul_nonneg_left (h : 0 ≤ a * b) (h1 : 0 < a) : 0 ≤ b :=
le_of_not_gt (assume h2 : b < 0, (mul_neg_of_pos_of_neg h1 h2).not_le h)
lemma nonneg_of_mul_nonneg_right (h : 0 ≤ a * b) (h1 : 0 < b) : 0 ≤ a :=
le_of_not_gt (assume h2 : a < 0, (mul_neg_of_neg_of_pos h2 h1).not_le h)
@[simp] lemma inv_of_nonneg [invertible a] : 0 ≤ ⅟a ↔ 0 ≤ a :=
begin
have : 0 < a * ⅟a, by simp only [mul_inv_of_self, zero_lt_one],
exact ⟨λ h, (pos_of_mul_pos_right this h).le, λ h, (pos_of_mul_pos_left this h).le⟩
end
@[simp] lemma inv_of_lt_zero [invertible a] : ⅟a < 0 ↔ a < 0 :=
by simp only [← not_le, inv_of_nonneg]
@[simp] lemma inv_of_le_one [invertible a] (h : 1 ≤ a) : ⅟a ≤ 1 :=
by haveI := @linear_order.decidable_le α _; exact
mul_inv_of_self a ▸ decidable.le_mul_of_one_le_left (inv_of_nonneg.2 $ zero_le_one.trans h) h
lemma neg_of_mul_neg_left (h : a * b < 0) (h1 : 0 ≤ a) : b < 0 :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (assume h2 : b ≥ 0, (decidable.mul_nonneg h1 h2).not_lt h)
lemma neg_of_mul_neg_right (h : a * b < 0) (h1 : 0 ≤ b) : a < 0 :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (assume h2 : a ≥ 0, (decidable.mul_nonneg h2 h1).not_lt h)
lemma nonpos_of_mul_nonpos_left (h : a * b ≤ 0) (h1 : 0 < a) : b ≤ 0 :=
le_of_not_gt (assume h2 : b > 0, (mul_pos h1 h2).not_le h)
lemma nonpos_of_mul_nonpos_right (h : a * b ≤ 0) (h1 : 0 < b) : a ≤ 0 :=
le_of_not_gt (assume h2 : a > 0, (mul_pos h2 h1).not_le h)
@[simp] lemma mul_le_mul_left (h : 0 < c) : c * a ≤ c * b ↔ a ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
⟨λ h', le_of_mul_le_mul_left h' h, λ h', decidable.mul_le_mul_of_nonneg_left h' h.le⟩
@[simp] lemma mul_le_mul_right (h : 0 < c) : a * c ≤ b * c ↔ a ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
⟨λ h', le_of_mul_le_mul_right h' h, λ h', decidable.mul_le_mul_of_nonneg_right h' h.le⟩
@[simp] lemma mul_lt_mul_left (h : 0 < c) : c * a < c * b ↔ a < b :=
by haveI := @linear_order.decidable_le α _; exact
⟨lt_imp_lt_of_le_imp_le $ λ h', decidable.mul_le_mul_of_nonneg_left h' h.le,
λ h', mul_lt_mul_of_pos_left h' h⟩
@[simp] lemma mul_lt_mul_right (h : 0 < c) : a * c < b * c ↔ a < b :=
by haveI := @linear_order.decidable_le α _; exact
⟨lt_imp_lt_of_le_imp_le $ λ h', decidable.mul_le_mul_of_nonneg_right h' h.le,
λ h', mul_lt_mul_of_pos_right h' h⟩
@[simp] lemma zero_le_mul_left (h : 0 < c) : 0 ≤ c * b ↔ 0 ≤ b :=
by { convert mul_le_mul_left h, simp }
@[simp] lemma zero_le_mul_right (h : 0 < c) : 0 ≤ b * c ↔ 0 ≤ b :=
by { convert mul_le_mul_right h, simp }
@[simp] lemma zero_lt_mul_left (h : 0 < c) : 0 < c * b ↔ 0 < b :=
by { convert mul_lt_mul_left h, simp }
@[simp] lemma zero_lt_mul_right (h : 0 < c) : 0 < b * c ↔ 0 < b :=
by { convert mul_lt_mul_right h, simp }
lemma add_le_mul_of_left_le_right (a2 : 2 ≤ a) (ab : a ≤ b) : a + b ≤ a * b :=
have 0 < b, from
calc 0 < 2 : zero_lt_two
... ≤ a : a2
... ≤ b : ab,
calc a + b ≤ b + b : add_le_add_right ab b
... = 2 * b : (two_mul b).symm
... ≤ a * b : (mul_le_mul_right this).mpr a2
lemma add_le_mul_of_right_le_left (b2 : 2 ≤ b) (ba : b ≤ a) : a + b ≤ a * b :=
have 0 < a, from
calc 0 < 2 : zero_lt_two
... ≤ b : b2
... ≤ a : ba,
calc a + b ≤ a + a : add_le_add_left ba a
... = a * 2 : (mul_two a).symm
... ≤ a * b : (mul_le_mul_left this).mpr b2
lemma add_le_mul (a2 : 2 ≤ a) (b2 : 2 ≤ b) : a + b ≤ a * b :=
if hab : a ≤ b then add_le_mul_of_left_le_right a2 hab
else add_le_mul_of_right_le_left b2 (le_of_not_le hab)
lemma add_le_mul' (a2 : 2 ≤ a) (b2 : 2 ≤ b) : a + b ≤ b * a :=
(le_of_eq (add_comm _ _)).trans (add_le_mul b2 a2)
section
variables [nontrivial α]
@[simp] lemma bit0_le_bit0 : bit0 a ≤ bit0 b ↔ a ≤ b :=
by rw [bit0, bit0, ← two_mul, ← two_mul, mul_le_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma bit0_lt_bit0 : bit0 a < bit0 b ↔ a < b :=
by rw [bit0, bit0, ← two_mul, ← two_mul, mul_lt_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma bit1_le_bit1 : bit1 a ≤ bit1 b ↔ a ≤ b :=
(add_le_add_iff_right 1).trans bit0_le_bit0
@[simp] lemma bit1_lt_bit1 : bit1 a < bit1 b ↔ a < b :=
(add_lt_add_iff_right 1).trans bit0_lt_bit0
@[simp] lemma one_le_bit1 : (1 : α) ≤ bit1 a ↔ 0 ≤ a :=
by rw [bit1, le_add_iff_nonneg_left, bit0, ← two_mul, zero_le_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma one_lt_bit1 : (1 : α) < bit1 a ↔ 0 < a :=
by rw [bit1, lt_add_iff_pos_left, bit0, ← two_mul, zero_lt_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma zero_le_bit0 : (0 : α) ≤ bit0 a ↔ 0 ≤ a :=
by rw [bit0, ← two_mul, zero_le_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma zero_lt_bit0 : (0 : α) < bit0 a ↔ 0 < a :=
by rw [bit0, ← two_mul, zero_lt_mul_left (zero_lt_two : 0 < (2:α))]
end
lemma le_mul_iff_one_le_left (hb : 0 < b) : b ≤ a * b ↔ 1 ≤ a :=
suffices 1 * b ≤ a * b ↔ 1 ≤ a, by rwa one_mul at this,
mul_le_mul_right hb
lemma lt_mul_iff_one_lt_left (hb : 0 < b) : b < a * b ↔ 1 < a :=
suffices 1 * b < a * b ↔ 1 < a, by rwa one_mul at this,
mul_lt_mul_right hb
lemma le_mul_iff_one_le_right (hb : 0 < b) : b ≤ b * a ↔ 1 ≤ a :=
suffices b * 1 ≤ b * a ↔ 1 ≤ a, by rwa mul_one at this,
mul_le_mul_left hb
lemma lt_mul_iff_one_lt_right (hb : 0 < b) : b < b * a ↔ 1 < a :=
suffices b * 1 < b * a ↔ 1 < a, by rwa mul_one at this,
mul_lt_mul_left hb
theorem mul_nonneg_iff_right_nonneg_of_pos (ha : 0 < a) : 0 ≤ b * a ↔ 0 ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
⟨λ h, nonneg_of_mul_nonneg_right h ha, λ h, decidable.mul_nonneg h ha.le⟩
lemma mul_le_iff_le_one_left (hb : 0 < b) : a * b ≤ b ↔ a ≤ 1 :=
⟨ λ h, le_of_not_lt (mt (lt_mul_iff_one_lt_left hb).2 h.not_lt),
λ h, le_of_not_lt (mt (lt_mul_iff_one_lt_left hb).1 h.not_lt) ⟩
lemma mul_lt_iff_lt_one_left (hb : 0 < b) : a * b < b ↔ a < 1 :=
⟨ λ h, lt_of_not_ge (mt (le_mul_iff_one_le_left hb).2 h.not_le),
λ h, lt_of_not_ge (mt (le_mul_iff_one_le_left hb).1 h.not_le) ⟩
lemma mul_le_iff_le_one_right (hb : 0 < b) : b * a ≤ b ↔ a ≤ 1 :=
⟨ λ h, le_of_not_lt (mt (lt_mul_iff_one_lt_right hb).2 h.not_lt),
λ h, le_of_not_lt (mt (lt_mul_iff_one_lt_right hb).1 h.not_lt) ⟩
lemma mul_lt_iff_lt_one_right (hb : 0 < b) : b * a < b ↔ a < 1 :=
⟨ λ h, lt_of_not_ge (mt (le_mul_iff_one_le_right hb).2 h.not_le),
λ h, lt_of_not_ge (mt (le_mul_iff_one_le_right hb).1 h.not_le) ⟩
lemma nonpos_of_mul_nonneg_left (h : 0 ≤ a * b) (hb : b < 0) : a ≤ 0 :=
le_of_not_gt (λ ha, absurd h (mul_neg_of_pos_of_neg ha hb).not_le)
lemma nonpos_of_mul_nonneg_right (h : 0 ≤ a * b) (ha : a < 0) : b ≤ 0 :=
le_of_not_gt (λ hb, absurd h (mul_neg_of_neg_of_pos ha hb).not_le)
lemma neg_of_mul_pos_left (h : 0 < a * b) (hb : b ≤ 0) : a < 0 :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (λ ha, absurd h (decidable.mul_nonpos_of_nonneg_of_nonpos ha hb).not_lt)
lemma neg_of_mul_pos_right (h : 0 < a * b) (ha : a ≤ 0) : b < 0 :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (λ hb, absurd h (decidable.mul_nonpos_of_nonpos_of_nonneg ha hb).not_lt)
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_semiring.to_no_top_order {α : Type*} [linear_ordered_semiring α] :
no_top_order α :=
⟨assume a, ⟨a + 1, lt_add_of_pos_right _ zero_lt_one⟩⟩
/-- Pullback a `linear_ordered_semiring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.linear_ordered_semiring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [nontrivial β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
linear_ordered_semiring β :=
{ ..linear_order.lift f hf,
..‹nontrivial β›,
..hf.ordered_semiring f zero one add mul }
end linear_ordered_semiring
section mono
variables {β : Type*} [linear_ordered_semiring α] [preorder β] {f g : β → α} {a : α}
lemma monotone_mul_left_of_nonneg (ha : 0 ≤ a) : monotone (λ x, a*x) :=
by haveI := @linear_order.decidable_le α _; exact
assume b c b_le_c, decidable.mul_le_mul_of_nonneg_left b_le_c ha
lemma monotone_mul_right_of_nonneg (ha : 0 ≤ a) : monotone (λ x, x*a) :=
by haveI := @linear_order.decidable_le α _; exact
assume b c b_le_c, decidable.mul_le_mul_of_nonneg_right b_le_c ha
lemma monotone.mul_const (hf : monotone f) (ha : 0 ≤ a) :
monotone (λ x, (f x) * a) :=
(monotone_mul_right_of_nonneg ha).comp hf
lemma monotone.const_mul (hf : monotone f) (ha : 0 ≤ a) :
monotone (λ x, a * (f x)) :=
(monotone_mul_left_of_nonneg ha).comp hf
lemma monotone.mul (hf : monotone f) (hg : monotone g) (hf0 : ∀ x, 0 ≤ f x) (hg0 : ∀ x, 0 ≤ g x) :
monotone (λ x, f x * g x) :=
by haveI := @linear_order.decidable_le α _; exact
λ x y h, decidable.mul_le_mul (hf h) (hg h) (hg0 x) (hf0 y)
lemma strict_mono_mul_left_of_pos (ha : 0 < a) : strict_mono (λ x, a * x) :=
assume b c b_lt_c, (mul_lt_mul_left ha).2 b_lt_c
lemma strict_mono_mul_right_of_pos (ha : 0 < a) : strict_mono (λ x, x * a) :=
assume b c b_lt_c, (mul_lt_mul_right ha).2 b_lt_c
lemma strict_mono.mul_const (hf : strict_mono f) (ha : 0 < a) :
strict_mono (λ x, (f x) * a) :=
(strict_mono_mul_right_of_pos ha).comp hf
lemma strict_mono.const_mul (hf : strict_mono f) (ha : 0 < a) :
strict_mono (λ x, a * (f x)) :=
(strict_mono_mul_left_of_pos ha).comp hf
lemma strict_mono.mul_monotone (hf : strict_mono f) (hg : monotone g) (hf0 : ∀ x, 0 ≤ f x)
(hg0 : ∀ x, 0 < g x) :
strict_mono (λ x, f x * g x) :=
by haveI := @linear_order.decidable_le α _; exact
λ x y h, decidable.mul_lt_mul (hf h) (hg h.le) (hg0 x) (hf0 y)
lemma monotone.mul_strict_mono (hf : monotone f) (hg : strict_mono g) (hf0 : ∀ x, 0 < f x)
(hg0 : ∀ x, 0 ≤ g x) :
strict_mono (λ x, f x * g x) :=
by haveI := @linear_order.decidable_le α _; exact
λ x y h, decidable.mul_lt_mul' (hf h.le) (hg h) (hg0 x) (hf0 y)
lemma strict_mono.mul (hf : strict_mono f) (hg : strict_mono g) (hf0 : ∀ x, 0 ≤ f x)
(hg0 : ∀ x, 0 ≤ g x) :
strict_mono (λ x, f x * g x) :=
by haveI := @linear_order.decidable_le α _; exact
λ x y h, decidable.mul_lt_mul'' (hf h) (hg h) (hf0 x) (hg0 x)
end mono
section linear_ordered_semiring
variables [linear_ordered_semiring α] {a b c : α}
lemma mul_max_of_nonneg (b c : α) (ha : 0 ≤ a) : a * max b c = max (a * b) (a * c) :=
(monotone_mul_left_of_nonneg ha).map_max
lemma mul_min_of_nonneg (b c : α) (ha : 0 ≤ a) : a * min b c = min (a * b) (a * c) :=
(monotone_mul_left_of_nonneg ha).map_min
lemma max_mul_of_nonneg (a b : α) (hc : 0 ≤ c) : max a b * c = max (a * c) (b * c) :=
(monotone_mul_right_of_nonneg hc).map_max
lemma min_mul_of_nonneg (a b : α) (hc : 0 ≤ c) : min a b * c = min (a * c) (b * c) :=
(monotone_mul_right_of_nonneg hc).map_min
end linear_ordered_semiring
/-- An `ordered_ring α` is a ring `α` with a partial order such that
multiplication with a positive number and addition are monotone. -/
@[protect_proj]
class ordered_ring (α : Type u) extends ring α, ordered_add_comm_group α :=
(zero_le_one : 0 ≤ (1 : α))
(mul_pos : ∀ a b : α, 0 < a → 0 < b → 0 < a * b)
section ordered_ring
variables [ordered_ring α] {a b c : α}
-- See Note [decidable namespace]
protected lemma decidable.ordered_ring.mul_nonneg [@decidable_rel α (≤)]
{a b : α} (h₁ : 0 ≤ a) (h₂ : 0 ≤ b) : 0 ≤ a * b :=
begin
by_cases ha : a ≤ 0, { simp [le_antisymm ha h₁] },
by_cases hb : b ≤ 0, { simp [le_antisymm hb h₂] },
exact (le_not_le_of_lt (ordered_ring.mul_pos a b (h₁.lt_of_not_le ha) (h₂.lt_of_not_le hb))).1,
end
lemma ordered_ring.mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b :=
by classical; exact decidable.ordered_ring.mul_nonneg
-- See Note [decidable namespace]
protected lemma decidable.ordered_ring.mul_le_mul_of_nonneg_left
[@decidable_rel α (≤)] (h₁ : a ≤ b) (h₂ : 0 ≤ c) : c * a ≤ c * b :=
begin
rw [← sub_nonneg, ← mul_sub],
exact decidable.ordered_ring.mul_nonneg h₂ (sub_nonneg.2 h₁),
end
lemma ordered_ring.mul_le_mul_of_nonneg_left : a ≤ b → 0 ≤ c → c * a ≤ c * b :=
by classical; exact decidable.ordered_ring.mul_le_mul_of_nonneg_left
-- See Note [decidable namespace]
protected lemma decidable.ordered_ring.mul_le_mul_of_nonneg_right
[@decidable_rel α (≤)] (h₁ : a ≤ b) (h₂ : 0 ≤ c) : a * c ≤ b * c :=
begin
rw [← sub_nonneg, ← sub_mul],
exact decidable.ordered_ring.mul_nonneg (sub_nonneg.2 h₁) h₂,
end
lemma ordered_ring.mul_le_mul_of_nonneg_right : a ≤ b → 0 ≤ c → a * c ≤ b * c :=
by classical; exact decidable.ordered_ring.mul_le_mul_of_nonneg_right
lemma ordered_ring.mul_lt_mul_of_pos_left (h₁ : a < b) (h₂ : 0 < c) : c * a < c * b :=
begin
rw [← sub_pos, ← mul_sub],
exact ordered_ring.mul_pos _ _ h₂ (sub_pos.2 h₁),
end
lemma ordered_ring.mul_lt_mul_of_pos_right (h₁ : a < b) (h₂ : 0 < c) : a * c < b * c :=
begin
rw [← sub_pos, ← sub_mul],
exact ordered_ring.mul_pos _ _ (sub_pos.2 h₁) h₂,
end
@[priority 100] -- see Note [lower instance priority]
instance ordered_ring.to_ordered_semiring : ordered_semiring α :=
{ mul_zero := mul_zero,
zero_mul := zero_mul,
add_left_cancel := @add_left_cancel α _,
le_of_add_le_add_left := @le_of_add_le_add_left α _ _ _,
mul_lt_mul_of_pos_left := @ordered_ring.mul_lt_mul_of_pos_left α _,
mul_lt_mul_of_pos_right := @ordered_ring.mul_lt_mul_of_pos_right α _,
..‹ordered_ring α› }
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul_of_nonpos_left [@decidable_rel α (≤)]
{a b c : α} (h : b ≤ a) (hc : c ≤ 0) : c * a ≤ c * b :=
have -c ≥ 0, from neg_nonneg_of_nonpos hc,
have -c * b ≤ -c * a, from decidable.mul_le_mul_of_nonneg_left h this,
have -(c * b) ≤ -(c * a), by rwa [← neg_mul_eq_neg_mul, ← neg_mul_eq_neg_mul] at this,
le_of_neg_le_neg this
lemma mul_le_mul_of_nonpos_left {a b c : α} : b ≤ a → c ≤ 0 → c * a ≤ c * b :=
by classical; exact decidable.mul_le_mul_of_nonpos_left
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul_of_nonpos_right [@decidable_rel α (≤)]
{a b c : α} (h : b ≤ a) (hc : c ≤ 0) : a * c ≤ b * c :=
have -c ≥ 0, from neg_nonneg_of_nonpos hc,
have b * -c ≤ a * -c, from decidable.mul_le_mul_of_nonneg_right h this,
have -(b * c) ≤ -(a * c), by rwa [← neg_mul_eq_mul_neg, ← neg_mul_eq_mul_neg] at this,
le_of_neg_le_neg this
lemma mul_le_mul_of_nonpos_right {a b c : α} : b ≤ a → c ≤ 0 → a * c ≤ b * c :=
by classical; exact decidable.mul_le_mul_of_nonpos_right
-- See Note [decidable namespace]
protected lemma decidable.mul_nonneg_of_nonpos_of_nonpos [@decidable_rel α (≤)]
{a b : α} (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a * b :=
have 0 * b ≤ a * b, from decidable.mul_le_mul_of_nonpos_right ha hb,
by rwa zero_mul at this
lemma mul_nonneg_of_nonpos_of_nonpos {a b : α} : a ≤ 0 → b ≤ 0 → 0 ≤ a * b :=
by classical; exact decidable.mul_nonneg_of_nonpos_of_nonpos
lemma mul_lt_mul_of_neg_left {a b c : α} (h : b < a) (hc : c < 0) : c * a < c * b :=
have -c > 0, from neg_pos_of_neg hc,
have -c * b < -c * a, from mul_lt_mul_of_pos_left h this,
have -(c * b) < -(c * a), by rwa [← neg_mul_eq_neg_mul, ← neg_mul_eq_neg_mul] at this,
lt_of_neg_lt_neg this
lemma mul_lt_mul_of_neg_right {a b c : α} (h : b < a) (hc : c < 0) : a * c < b * c :=
have -c > 0, from neg_pos_of_neg hc,
have b * -c < a * -c, from mul_lt_mul_of_pos_right h this,
have -(b * c) < -(a * c), by rwa [← neg_mul_eq_mul_neg, ← neg_mul_eq_mul_neg] at this,
lt_of_neg_lt_neg this
lemma mul_pos_of_neg_of_neg {a b : α} (ha : a < 0) (hb : b < 0) : 0 < a * b :=
have 0 * b < a * b, from mul_lt_mul_of_neg_right ha hb,
by rwa zero_mul at this
/-- Pullback an `ordered_ring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.ordered_ring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (- x) = - f x) (sub : ∀ x y, f (x - y) = f x - f y) :
ordered_ring β :=
{ mul_pos := λ a b a0 b0, show f 0 < f (a * b), by { rw [zero, mul], apply mul_pos; rwa ← zero },
..hf.ordered_semiring f zero one add mul,
..hf.ring f zero one add mul neg sub }
end ordered_ring
section ordered_comm_ring
/-- An `ordered_comm_ring α` is a commutative ring `α` with a partial order such that
multiplication with a positive number and addition are monotone. -/
@[protect_proj]
class ordered_comm_ring (α : Type u) extends ordered_ring α, ordered_comm_semiring α, comm_ring α
/-- Pullback an `ordered_comm_ring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.ordered_comm_ring [ordered_comm_ring α] {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (- x) = - f x) (sub : ∀ x y, f (x - y) = f x - f y) :
ordered_comm_ring β :=
{ ..hf.ordered_comm_semiring f zero one add mul,
..hf.ordered_ring f zero one add mul neg sub,
..hf.comm_ring f zero one add mul neg sub }
end ordered_comm_ring
/-- A `linear_ordered_ring α` is a ring `α` with a linear order such that
multiplication with a positive number and addition are monotone. -/
@[protect_proj] class linear_ordered_ring (α : Type u)
extends ordered_ring α, linear_order α, nontrivial α
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_ring.to_linear_ordered_add_comm_group [s : linear_ordered_ring α] :
linear_ordered_add_comm_group α :=
{ .. s }
section linear_ordered_ring
variables [linear_ordered_ring α] {a b c : α}
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_ring.to_linear_ordered_semiring : linear_ordered_semiring α :=
{ mul_zero := mul_zero,
zero_mul := zero_mul,
add_left_cancel := @add_left_cancel α _,
le_of_add_le_add_left := @le_of_add_le_add_left α _ _ _,
mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left α _,
mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right α _,
le_total := linear_ordered_ring.le_total,
..‹linear_ordered_ring α› }
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_ring.to_domain : domain α :=
{ eq_zero_or_eq_zero_of_mul_eq_zero :=
begin
intros a b hab,
refine decidable.or_iff_not_and_not.2 (λ h, _), revert hab,
cases lt_or_gt_of_ne h.1 with ha ha; cases lt_or_gt_of_ne h.2 with hb hb,
exacts [(mul_pos_of_neg_of_neg ha hb).ne.symm, (mul_neg_of_neg_of_pos ha hb).ne,
(mul_neg_of_pos_of_neg ha hb).ne, (mul_pos ha hb).ne.symm]
end,
.. ‹linear_ordered_ring α› }
@[simp] lemma abs_one : abs (1 : α) = 1 := abs_of_pos zero_lt_one
@[simp] lemma abs_two : abs (2 : α) = 2 := abs_of_pos zero_lt_two
lemma abs_mul (a b : α) : abs (a * b) = abs a * abs b :=
begin
haveI := @linear_order.decidable_le α _,
rw [abs_eq (decidable.mul_nonneg (abs_nonneg a) (abs_nonneg b))],
cases le_total a 0 with ha ha; cases le_total b 0 with hb hb;
simp only [abs_of_nonpos, abs_of_nonneg, true_or, or_true, eq_self_iff_true,
neg_mul_eq_neg_mul_symm, mul_neg_eq_neg_mul_symm, neg_neg, *]
end
/-- `abs` as a `monoid_with_zero_hom`. -/
def abs_hom : monoid_with_zero_hom α α := ⟨abs, abs_zero, abs_one, abs_mul⟩
@[simp] lemma abs_mul_abs_self (a : α) : abs a * abs a = a * a :=
abs_by_cases (λ x, x * x = a * a) rfl (neg_mul_neg a a)
@[simp] lemma abs_mul_self (a : α) : abs (a * a) = a * a :=
by rw [abs_mul, abs_mul_abs_self]
lemma mul_pos_iff : 0 < a * b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 :=
⟨pos_and_pos_or_neg_and_neg_of_mul_pos,
λ h, h.elim (and_imp.2 mul_pos) (and_imp.2 mul_pos_of_neg_of_neg)⟩
lemma mul_neg_iff : a * b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b :=
by rw [← neg_pos, neg_mul_eq_mul_neg, mul_pos_iff, neg_pos, neg_lt_zero]
lemma mul_nonneg_iff : 0 ≤ a * b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 :=
by haveI := @linear_order.decidable_le α _; exact
⟨nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nnonneg,
λ h, h.elim (and_imp.2 decidable.mul_nonneg) (and_imp.2 decidable.mul_nonneg_of_nonpos_of_nonpos)⟩
/-- Out of three elements of a `linear_ordered_ring`, two must have the same sign. -/
lemma mul_nonneg_of_three (a b c : α) :
0 ≤ a * b ∨ 0 ≤ b * c ∨ 0 ≤ c * a :=
by iterate 3 { rw mul_nonneg_iff };
have := le_total 0 a; have := le_total 0 b; have := le_total 0 c; itauto
lemma mul_nonpos_iff : a * b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b :=
by rw [← neg_nonneg, neg_mul_eq_mul_neg, mul_nonneg_iff, neg_nonneg, neg_nonpos]
lemma mul_self_nonneg (a : α) : 0 ≤ a * a :=
abs_mul_self a ▸ abs_nonneg _
@[simp] lemma neg_le_self_iff : -a ≤ a ↔ 0 ≤ a :=
by simp [neg_le_iff_add_nonneg, ← two_mul, mul_nonneg_iff, zero_le_one, (@zero_lt_two α _ _).not_le]
@[simp] lemma neg_lt_self_iff : -a < a ↔ 0 < a :=
by simp [neg_lt_iff_pos_add, ← two_mul, mul_pos_iff, zero_lt_one, (@zero_lt_two α _ _).not_lt]
@[simp] lemma le_neg_self_iff : a ≤ -a ↔ a ≤ 0 :=
calc a ≤ -a ↔ -(-a) ≤ -a : by rw neg_neg
... ↔ 0 ≤ -a : neg_le_self_iff
... ↔ a ≤ 0 : neg_nonneg
@[simp] lemma lt_neg_self_iff : a < -a ↔ a < 0 :=
calc a < -a ↔ -(-a) < -a : by rw neg_neg
... ↔ 0 < -a : neg_lt_self_iff
... ↔ a < 0 : neg_pos
@[simp] lemma abs_eq_self : abs a = a ↔ 0 ≤ a := by simp [abs]
@[simp] lemma abs_eq_neg_self : abs a = -a ↔ a ≤ 0 := by simp [abs]
/-- For an element `a` of a linear ordered ring, either `abs a = a` and `0 ≤ a`,
or `abs a = -a` and `a < 0`.
Use cases on this lemma to automate linarith in inequalities -/
lemma abs_cases (a : α) : (abs a = a ∧ 0 ≤ a) ∨ (abs a = -a ∧ a < 0) :=
begin
by_cases 0 ≤ a,
{ left,
exact ⟨abs_eq_self.mpr h, h⟩ },
{ right,
push_neg at h,
exact ⟨abs_eq_neg_self.mpr (le_of_lt h), h⟩ }
end
lemma gt_of_mul_lt_mul_neg_left (h : c * a < c * b) (hc : c ≤ 0) : b < a :=
have nhc : 0 ≤ -c, from neg_nonneg_of_nonpos hc,
have h2 : -(c * b) < -(c * a), from neg_lt_neg h,
have h3 : (-c) * b < (-c) * a, from calc
(-c) * b = - (c * b) : by rewrite neg_mul_eq_neg_mul
... < -(c * a) : h2
... = (-c) * a : by rewrite neg_mul_eq_neg_mul,
lt_of_mul_lt_mul_left h3 nhc
lemma neg_one_lt_zero : -1 < (0:α) := neg_lt_zero.2 zero_lt_one
lemma le_of_mul_le_of_one_le {a b c : α} (h : a * c ≤ b) (hb : 0 ≤ b) (hc : 1 ≤ c) : a ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
have h' : a * c ≤ b * c, from calc
a * c ≤ b : h
... = b * 1 : by rewrite mul_one
... ≤ b * c : decidable.mul_le_mul_of_nonneg_left hc hb,
le_of_mul_le_mul_right h' (zero_lt_one.trans_le hc)
lemma nonneg_le_nonneg_of_sq_le_sq {a b : α} (hb : 0 ≤ b) (h : a * a ≤ b * b) : a ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
le_of_not_gt (λhab, (decidable.mul_self_lt_mul_self hb hab).not_le h)
lemma mul_self_le_mul_self_iff {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a ≤ b ↔ a * a ≤ b * b :=
by haveI := @linear_order.decidable_le α _; exact
⟨decidable.mul_self_le_mul_self h1, nonneg_le_nonneg_of_sq_le_sq h2⟩
lemma mul_self_lt_mul_self_iff {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a < b ↔ a * a < b * b :=
by haveI := @linear_order.decidable_le α _; exact
((@decidable.strict_mono_incr_on_mul_self α _ _).lt_iff_lt h1 h2).symm
lemma mul_self_inj {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a * a = b * b ↔ a = b :=
by haveI := @linear_order.decidable_le α _; exact
(@decidable.strict_mono_incr_on_mul_self α _ _).inj_on.eq_iff h1 h2
@[simp] lemma mul_le_mul_left_of_neg {a b c : α} (h : c < 0) : c * a ≤ c * b ↔ b ≤ a :=
by haveI := @linear_order.decidable_le α _; exact
⟨le_imp_le_of_lt_imp_lt $ λ h', mul_lt_mul_of_neg_left h' h,
λ h', decidable.mul_le_mul_of_nonpos_left h' h.le⟩
@[simp] lemma mul_le_mul_right_of_neg {a b c : α} (h : c < 0) : a * c ≤ b * c ↔ b ≤ a :=
by haveI := @linear_order.decidable_le α _; exact
⟨le_imp_le_of_lt_imp_lt $ λ h', mul_lt_mul_of_neg_right h' h,
λ h', decidable.mul_le_mul_of_nonpos_right h' h.le⟩
@[simp] lemma mul_lt_mul_left_of_neg {a b c : α} (h : c < 0) : c * a < c * b ↔ b < a :=
lt_iff_lt_of_le_iff_le (mul_le_mul_left_of_neg h)
@[simp] lemma mul_lt_mul_right_of_neg {a b c : α} (h : c < 0) : a * c < b * c ↔ b < a :=
lt_iff_lt_of_le_iff_le (mul_le_mul_right_of_neg h)
lemma sub_one_lt (a : α) : a - 1 < a :=
sub_lt_iff_lt_add.2 (lt_add_one a)
lemma mul_self_pos {a : α} (ha : a ≠ 0) : 0 < a * a :=
by rcases lt_trichotomy a 0 with h|h|h;
[exact mul_pos_of_neg_of_neg h h, exact (ha h).elim, exact mul_pos h h]
lemma mul_self_le_mul_self_of_le_of_neg_le {x y : α} (h₁ : x ≤ y) (h₂ : -x ≤ y) : x * x ≤ y * y :=
begin
haveI := @linear_order.decidable_le α _,
rw [← abs_mul_abs_self x],
exact decidable.mul_self_le_mul_self (abs_nonneg x) (abs_le.2 ⟨neg_le.2 h₂, h₁⟩)
end
lemma nonneg_of_mul_nonpos_left {a b : α} (h : a * b ≤ 0) (hb : b < 0) : 0 ≤ a :=
le_of_not_gt (λ ha, absurd h (mul_pos_of_neg_of_neg ha hb).not_le)
lemma nonneg_of_mul_nonpos_right {a b : α} (h : a * b ≤ 0) (ha : a < 0) : 0 ≤ b :=
le_of_not_gt (λ hb, absurd h (mul_pos_of_neg_of_neg ha hb).not_le)
lemma pos_of_mul_neg_left {a b : α} (h : a * b < 0) (hb : b ≤ 0) : 0 < a :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (λ ha, absurd h (decidable.mul_nonneg_of_nonpos_of_nonpos ha hb).not_lt)
lemma pos_of_mul_neg_right {a b : α} (h : a * b < 0) (ha : a ≤ 0) : 0 < b :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (λ hb, absurd h (decidable.mul_nonneg_of_nonpos_of_nonpos ha hb).not_lt)
/-- The sum of two squares is zero iff both elements are zero. -/
lemma mul_self_add_mul_self_eq_zero {x y : α} : x * x + y * y = 0 ↔ x = 0 ∧ y = 0 :=
by rw [add_eq_zero_iff', mul_self_eq_zero, mul_self_eq_zero]; apply mul_self_nonneg
lemma eq_zero_of_mul_self_add_mul_self_eq_zero (h : a * a + b * b = 0) : a = 0 :=
(mul_self_add_mul_self_eq_zero.mp h).left
lemma abs_eq_iff_mul_self_eq : abs a = abs b ↔ a * a = b * b :=
begin
rw [← abs_mul_abs_self, ← abs_mul_abs_self b],
exact (mul_self_inj (abs_nonneg a) (abs_nonneg b)).symm,
end
lemma abs_lt_iff_mul_self_lt : abs a < abs b ↔ a * a < b * b :=
begin
rw [← abs_mul_abs_self, ← abs_mul_abs_self b],
exact mul_self_lt_mul_self_iff (abs_nonneg a) (abs_nonneg b)
end
lemma abs_le_iff_mul_self_le : abs a ≤ abs b ↔ a * a ≤ b * b :=
begin
rw [← abs_mul_abs_self, ← abs_mul_abs_self b],
exact mul_self_le_mul_self_iff (abs_nonneg a) (abs_nonneg b)
end
lemma abs_le_one_iff_mul_self_le_one : abs a ≤ 1 ↔ a * a ≤ 1 :=
by simpa only [abs_one, one_mul] using @abs_le_iff_mul_self_le α _ a 1
/-- Pullback a `linear_ordered_ring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.linear_ordered_ring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [nontrivial β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
linear_ordered_ring β :=
{ ..linear_order.lift f hf,
..‹nontrivial β›,
..hf.ordered_ring f zero one add mul neg sub }
end linear_ordered_ring
/-- A `linear_ordered_comm_ring α` is a commutative ring `α` with a linear order
such that multiplication with a positive number and addition are monotone. -/
@[protect_proj]
class linear_ordered_comm_ring (α : Type u) extends linear_ordered_ring α, comm_monoid α
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_comm_ring.to_ordered_comm_ring [d : linear_ordered_comm_ring α] :
ordered_comm_ring α :=
-- One might hope that `{ ..linear_ordered_ring.to_linear_ordered_semiring, ..d }`
-- achieved the same result here.
-- Unfortunately with that definition we see mismatched instances in `algebra.star.chsh`.
let s : linear_ordered_semiring α := @linear_ordered_ring.to_linear_ordered_semiring α _ in
{ zero_mul := @linear_ordered_semiring.zero_mul α s,
mul_zero := @linear_ordered_semiring.mul_zero α s,
add_left_cancel := @linear_ordered_semiring.add_left_cancel α s,
le_of_add_le_add_left := @linear_ordered_semiring.le_of_add_le_add_left α s,
mul_lt_mul_of_pos_left := @linear_ordered_semiring.mul_lt_mul_of_pos_left α s,
mul_lt_mul_of_pos_right := @linear_ordered_semiring.mul_lt_mul_of_pos_right α s,
..d }
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_comm_ring.to_integral_domain [s : linear_ordered_comm_ring α] :
integral_domain α :=
{ ..linear_ordered_ring.to_domain, ..s }
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_comm_ring.to_linear_ordered_semiring [d : linear_ordered_comm_ring α] :
linear_ordered_semiring α :=
-- One might hope that `{ ..linear_ordered_ring.to_linear_ordered_semiring, ..d }`
-- achieved the same result here.
-- Unfortunately with that definition we see mismatched `preorder ℝ` instances in
-- `topology.metric_space.basic`.
let s : linear_ordered_semiring α := @linear_ordered_ring.to_linear_ordered_semiring α _ in
{ zero_mul := @linear_ordered_semiring.zero_mul α s,
mul_zero := @linear_ordered_semiring.mul_zero α s,
add_left_cancel := @linear_ordered_semiring.add_left_cancel α s,
le_of_add_le_add_left := @linear_ordered_semiring.le_of_add_le_add_left α s,
mul_lt_mul_of_pos_left := @linear_ordered_semiring.mul_lt_mul_of_pos_left α s,
mul_lt_mul_of_pos_right := @linear_ordered_semiring.mul_lt_mul_of_pos_right α s,
..d }
section linear_ordered_comm_ring
variables [linear_ordered_comm_ring α] {a b c d : α}
lemma max_mul_mul_le_max_mul_max (b c : α) (ha : 0 ≤ a) (hd: 0 ≤ d) :
max (a * b) (d * c) ≤ max a c * max d b :=
by haveI := @linear_order.decidable_le α _; exact
have ba : b * a ≤ max d b * max c a, from
decidable.mul_le_mul (le_max_right d b) (le_max_right c a) ha (le_trans hd (le_max_left d b)),
have cd : c * d ≤ max a c * max b d, from
decidable.mul_le_mul (le_max_right a c) (le_max_right b d) hd (le_trans ha (le_max_left a c)),
max_le
(by simpa [mul_comm, max_comm] using ba)
(by simpa [mul_comm, max_comm] using cd)
lemma abs_sub_sq (a b : α) : abs (a - b) * abs (a - b) = a * a + b * b - (1 + 1) * a * b :=
begin
rw abs_mul_abs_self,
simp only [mul_add, add_comm, add_left_comm, mul_comm, sub_eq_add_neg,
mul_one, mul_neg_eq_neg_mul_symm, neg_add_rev, neg_neg],
end
@[simp] lemma abs_dvd (a b : α) : abs a ∣ b ↔ a ∣ b :=
by { cases abs_choice a with h h; simp only [h, neg_dvd] }
lemma abs_dvd_self (a : α) : abs a ∣ a :=
(abs_dvd a a).mpr (dvd_refl a)
@[simp] lemma dvd_abs (a b : α) : a ∣ abs b ↔ a ∣ b :=
by { cases abs_choice b with h h; simp only [h, dvd_neg] }
lemma self_dvd_abs (a : α) : a ∣ abs a :=
(dvd_abs a a).mpr (dvd_refl a)
lemma abs_dvd_abs (a b : α) : abs a ∣ abs b ↔ a ∣ b :=
(abs_dvd _ _).trans (dvd_abs _ _)
lemma even_abs {a : α} : even (abs a) ↔ even a :=
dvd_abs _ _
/-- Pullback a `linear_ordered_comm_ring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.linear_ordered_comm_ring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [nontrivial β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
linear_ordered_comm_ring β :=
{ ..linear_order.lift f hf,
..‹nontrivial β›,
..hf.ordered_comm_ring f zero one add mul neg sub }
end linear_ordered_comm_ring
/-- Extend `nonneg_add_comm_group` to support ordered rings
specified by their nonnegative elements -/
class nonneg_ring (α : Type*) extends ring α, nonneg_add_comm_group α :=
(one_nonneg : nonneg 1)
(mul_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a * b))
(mul_pos : ∀ {a b}, pos a → pos b → pos (a * b))
/-- Extend `nonneg_add_comm_group` to support linearly ordered rings
specified by their nonnegative elements -/
class linear_nonneg_ring (α : Type*) extends domain α, nonneg_add_comm_group α :=
(one_pos : pos 1)
(mul_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a * b))
(nonneg_total : ∀ a, nonneg a ∨ nonneg (-a))
[dec_nonneg : decidable_pred nonneg]
namespace nonneg_ring
open nonneg_add_comm_group
variable [nonneg_ring α]
/-- `to_linear_nonneg_ring` shows that a `nonneg_ring` with a total order is a `domain`,
hence a `linear_nonneg_ring`. -/
def to_linear_nonneg_ring [nontrivial α] [decidable_pred (@nonneg α _)]
(nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a))
: linear_nonneg_ring α :=
{ one_pos := (pos_iff 1).mpr ⟨one_nonneg, λ h, zero_ne_one (nonneg_antisymm one_nonneg h).symm⟩,
nonneg_total := nonneg_total,
eq_zero_or_eq_zero_of_mul_eq_zero :=
suffices ∀ {a} b : α, nonneg a → a * b = 0 → a = 0 ∨ b = 0,
from λ a b, (nonneg_total a).elim (this b)
(λ na, by simpa using this b na),
suffices ∀ {a b : α}, nonneg a → nonneg b → a * b = 0 → a = 0 ∨ b = 0,
from λ a b na, (nonneg_total b).elim (this na)
(λ nb, by simpa using this na nb),
λ a b na nb z, decidable.by_cases
(λ nna : nonneg (-a), or.inl (nonneg_antisymm na nna))
(λ pa, decidable.by_cases
(λ nnb : nonneg (-b), or.inr (nonneg_antisymm nb nnb))
(λ pb, absurd z $ ne_of_gt $ pos_def.1 $ mul_pos
((pos_iff _).2 ⟨na, pa⟩)
((pos_iff _).2 ⟨nb, pb⟩))),
..‹nontrivial α›,
..‹nonneg_ring α› }
end nonneg_ring
namespace linear_nonneg_ring
open nonneg_add_comm_group
variable [linear_nonneg_ring α]
@[priority 100] -- see Note [lower instance priority]
instance to_nonneg_ring : nonneg_ring α :=
{ one_nonneg := ((pos_iff _).mp one_pos).1,
mul_pos := λ a b pa pb,
let ⟨a1, a2⟩ := (pos_iff a).1 pa,
⟨b1, b2⟩ := (pos_iff b).1 pb in
have ab : nonneg (a * b), from mul_nonneg a1 b1,
(pos_iff _).2 ⟨ab, λ hn,
have a * b = 0, from nonneg_antisymm ab hn,
(eq_zero_or_eq_zero_of_mul_eq_zero _ _ this).elim
(ne_of_gt (pos_def.1 pa))
(ne_of_gt (pos_def.1 pb))⟩,
..‹linear_nonneg_ring α› }
/-- Construct `linear_order` from `linear_nonneg_ring`. This is not an instance
because we don't use it in `mathlib`. -/
local attribute [instance]
def to_linear_order [decidable_pred (nonneg : α → Prop)] : linear_order α :=
{ le_total := nonneg_total_iff.1 nonneg_total,
decidable_le := by apply_instance,
decidable_lt := by apply_instance,
..‹linear_nonneg_ring α›, ..(infer_instance : ordered_add_comm_group α) }
/-- Construct `linear_ordered_ring` from `linear_nonneg_ring`.
This is not an instance because we don't use it in `mathlib`. -/
local attribute [instance]
def to_linear_ordered_ring [decidable_pred (nonneg : α → Prop)] : linear_ordered_ring α :=
{ mul_pos := by simp [pos_def.symm]; exact @nonneg_ring.mul_pos _ _,
zero_le_one := le_of_lt $ lt_of_not_ge $ λ (h : nonneg (0 - 1)), begin
rw [zero_sub] at h,
have := mul_nonneg h h, simp at this,
exact zero_ne_one (nonneg_antisymm this h).symm
end,
..‹linear_nonneg_ring α›, ..(infer_instance : ordered_add_comm_group α),
..(infer_instance : linear_order α) }
/-- Convert a `linear_nonneg_ring` with a commutative multiplication and
decidable non-negativity into a `linear_ordered_comm_ring` -/
def to_linear_ordered_comm_ring
[decidable_pred (@nonneg α _)]
[comm : @is_commutative α (*)]
: linear_ordered_comm_ring α :=
{ mul_comm := is_commutative.comm,
..@linear_nonneg_ring.to_linear_ordered_ring _ _ _ }
end linear_nonneg_ring
/-- A canonically ordered commutative semiring is an ordered, commutative semiring
in which `a ≤ b` iff there exists `c` with `b = a + c`. This is satisfied by the
natural numbers, for example, but not the integers or other ordered groups. -/
@[protect_proj]
class canonically_ordered_comm_semiring (α : Type*) extends
canonically_ordered_add_monoid α, comm_semiring α :=
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀ a b : α, a * b = 0 → a = 0 ∨ b = 0)
namespace canonically_ordered_comm_semiring
variables [canonically_ordered_comm_semiring α] {a b : α}
@[priority 100] -- see Note [lower instance priority]
instance to_no_zero_divisors : no_zero_divisors α :=
⟨canonically_ordered_comm_semiring.eq_zero_or_eq_zero_of_mul_eq_zero⟩
@[priority 100] -- see Note [lower instance priority]
instance to_covariant_mul_le : covariant_class α α (*) (≤) :=
begin
refine ⟨λ a b c h, _⟩,
rcases le_iff_exists_add.1 h with ⟨c, rfl⟩,
rw mul_add,
apply self_le_add_right
end
/-- A version of `zero_lt_one : 0 < 1` for a `canonically_ordered_comm_semiring`. -/
lemma zero_lt_one [nontrivial α] : (0:α) < 1 := (zero_le 1).lt_of_ne zero_ne_one
lemma mul_pos : 0 < a * b ↔ (0 < a) ∧ (0 < b) :=
by simp only [pos_iff_ne_zero, ne.def, mul_eq_zero, not_or_distrib]
end canonically_ordered_comm_semiring
/-! ### Structures involving `*` and `0` on `with_top` and `with_bot`
The main results of this section are `with_top.canonically_ordered_comm_semiring` and
`with_bot.comm_monoid_with_zero`.
-/
namespace with_top
instance [nonempty α] : nontrivial (with_top α) :=
option.nontrivial
variable [decidable_eq α]
section has_mul
variables [has_zero α] [has_mul α]
instance : mul_zero_class (with_top α) :=
{ zero := 0,
mul := λm n, if m = 0 ∨ n = 0 then 0 else m.bind (λa, n.bind $ λb, ↑(a * b)),
zero_mul := assume a, if_pos $ or.inl rfl,
mul_zero := assume a, if_pos $ or.inr rfl }
lemma mul_def {a b : with_top α} :
a * b = if a = 0 ∨ b = 0 then 0 else a.bind (λa, b.bind $ λb, ↑(a * b)) := rfl
@[simp] lemma mul_top {a : with_top α} (h : a ≠ 0) : a * ⊤ = ⊤ :=
by cases a; simp [mul_def, h]; refl
@[simp] lemma top_mul {a : with_top α} (h : a ≠ 0) : ⊤ * a = ⊤ :=
by cases a; simp [mul_def, h]; refl
@[simp] lemma top_mul_top : (⊤ * ⊤ : with_top α) = ⊤ :=
top_mul top_ne_zero
end has_mul
section mul_zero_class
variables [mul_zero_class α]
@[norm_cast] lemma coe_mul {a b : α} : (↑(a * b) : with_top α) = a * b :=
decidable.by_cases (assume : a = 0, by simp [this]) $ assume ha,
decidable.by_cases (assume : b = 0, by simp [this]) $ assume hb,
by { simp [*, mul_def], refl }
lemma mul_coe {b : α} (hb : b ≠ 0) : ∀{a : with_top α}, a * b = a.bind (λa:α, ↑(a * b))
| none := show (if (⊤:with_top α) = 0 ∨ (b:with_top α) = 0 then 0 else ⊤ : with_top α) = ⊤,
by simp [hb]
| (some a) := show ↑a * ↑b = ↑(a * b), from coe_mul.symm
@[simp] lemma mul_eq_top_iff {a b : with_top α} : a * b = ⊤ ↔ (a ≠ 0 ∧ b = ⊤) ∨ (a = ⊤ ∧ b ≠ 0) :=
begin
cases a; cases b; simp only [none_eq_top, some_eq_coe],
{ simp [← coe_mul] },
{ suffices : ⊤ * (b : with_top α) = ⊤ ↔ b ≠ 0, by simpa,
by_cases hb : b = 0; simp [hb] },
{ suffices : (a : with_top α) * ⊤ = ⊤ ↔ a ≠ 0, by simpa,
by_cases ha : a = 0; simp [ha] },
{ simp [← coe_mul] }
end
lemma mul_lt_top [partial_order α] {a b : with_top α} (ha : a < ⊤) (hb : b < ⊤) : a * b < ⊤ :=
begin
lift a to α using ne_top_of_lt ha,
lift b to α using ne_top_of_lt hb,
simp only [← coe_mul, coe_lt_top]
end
end mul_zero_class
/-- `nontrivial α` is needed here as otherwise we have `1 * ⊤ = ⊤` but also `= 0 * ⊤ = 0`. -/
instance [mul_zero_one_class α] [nontrivial α] : mul_zero_one_class (with_top α) :=
{ mul := (*),
one := 1,
zero := 0,
one_mul := λ a, match a with
| none := show ((1:α) : with_top α) * ⊤ = ⊤, by simp [-with_top.coe_one]
| (some a) := show ((1:α) : with_top α) * a = a, by simp [coe_mul.symm, -with_top.coe_one]
end,
mul_one := λ a, match a with
| none := show ⊤ * ((1:α) : with_top α) = ⊤, by simp [-with_top.coe_one]
| (some a) := show ↑a * ((1:α) : with_top α) = a, by simp [coe_mul.symm, -with_top.coe_one]
end,
.. with_top.mul_zero_class }
instance [mul_zero_class α] [no_zero_divisors α] : no_zero_divisors (with_top α) :=
⟨λ a b, by cases a; cases b; dsimp [mul_def]; split_ifs;
simp [*, none_eq_top, some_eq_coe, mul_eq_zero] at *⟩
instance [semigroup_with_zero α] [no_zero_divisors α] : semigroup_with_zero (with_top α) :=
{ mul := (*),
zero := 0,
mul_assoc := λ a b c, begin
cases a,
{ by_cases hb : b = 0; by_cases hc : c = 0;
simp [*, none_eq_top] },
cases b,
{ by_cases ha : a = 0; by_cases hc : c = 0;
simp [*, none_eq_top, some_eq_coe] },
cases c,
{ by_cases ha : a = 0; by_cases hb : b = 0;
simp [*, none_eq_top, some_eq_coe] },
simp [some_eq_coe, coe_mul.symm, mul_assoc]
end,
.. with_top.mul_zero_class }
instance [monoid_with_zero α] [no_zero_divisors α] [nontrivial α] : monoid_with_zero (with_top α) :=
{ .. with_top.mul_zero_one_class, .. with_top.semigroup_with_zero }
instance [comm_monoid_with_zero α] [no_zero_divisors α] [nontrivial α] :
comm_monoid_with_zero (with_top α) :=
{ mul := (*),
zero := 0,
mul_comm := λ a b, begin
by_cases ha : a = 0, { simp [ha] },
by_cases hb : b = 0, { simp [hb] },
simp [ha, hb, mul_def, option.bind_comm a b, mul_comm]
end,
.. with_top.monoid_with_zero }
variables [canonically_ordered_comm_semiring α]
private lemma distrib' (a b c : with_top α) : (a + b) * c = a * c + b * c :=
begin
cases c,
{ show (a + b) * ⊤ = a * ⊤ + b * ⊤,
by_cases ha : a = 0; simp [ha] },
{ show (a + b) * c = a * c + b * c,
by_cases hc : c = 0, { simp [hc] },
simp [mul_coe hc], cases a; cases b,
repeat { refl <|> exact congr_arg some (add_mul _ _ _) } }
end
/-- This instance requires `canonically_ordered_comm_semiring` as it is the smallest class
that derives from both `non_assoc_non_unital_semiring` and `canonically_ordered_add_monoid`, both
of which are required for distributivity. -/
instance [nontrivial α] : comm_semiring (with_top α) :=
{ right_distrib := distrib',
left_distrib := assume a b c, by rw [mul_comm, distrib', mul_comm b, mul_comm c]; refl,
.. with_top.add_comm_monoid, .. with_top.comm_monoid_with_zero,}
instance [nontrivial α] : canonically_ordered_comm_semiring (with_top α) :=
{ .. with_top.comm_semiring,
.. with_top.canonically_ordered_add_monoid,
.. with_top.no_zero_divisors, }
end with_top
namespace with_bot
instance [nonempty α] : nontrivial (with_bot α) :=
option.nontrivial
variable [decidable_eq α]
section has_mul
variables [has_zero α] [has_mul α]
instance : mul_zero_class (with_bot α) :=
with_top.mul_zero_class
lemma mul_def {a b : with_bot α} :
a * b = if a = 0 ∨ b = 0 then 0 else a.bind (λa, b.bind $ λb, ↑(a * b)) := rfl
@[simp] lemma mul_bot {a : with_bot α} (h : a ≠ 0) : a * ⊥ = ⊥ :=
with_top.mul_top h
@[simp] lemma bot_mul {a : with_bot α} (h : a ≠ 0) : ⊥ * a = ⊥ :=
with_top.top_mul h
@[simp] lemma bot_mul_bot : (⊥ * ⊥ : with_bot α) = ⊥ :=
with_top.top_mul_top
end has_mul
section mul_zero_class
variables [mul_zero_class α]
@[norm_cast] lemma coe_mul {a b : α} : (↑(a * b) : with_bot α) = a * b :=
decidable.by_cases (assume : a = 0, by simp [this]) $ assume ha,
decidable.by_cases (assume : b = 0, by simp [this]) $ assume hb,
by { simp [*, mul_def], refl }
lemma mul_coe {b : α} (hb : b ≠ 0) {a : with_bot α} : a * b = a.bind (λa:α, ↑(a * b)) :=
with_top.mul_coe hb
@[simp] lemma mul_eq_bot_iff {a b : with_bot α} : a * b = ⊥ ↔ (a ≠ 0 ∧ b = ⊥) ∨ (a = ⊥ ∧ b ≠ 0) :=
with_top.mul_eq_top_iff
lemma bot_lt_mul [partial_order α] {a b : with_bot α} (ha : ⊥ < a) (hb : ⊥ < b) : ⊥ < a * b :=
begin
lift a to α using ne_bot_of_gt ha,
lift b to α using ne_bot_of_gt hb,
simp only [← coe_mul, bot_lt_coe],
end
end mul_zero_class
/-- `nontrivial α` is needed here as otherwise we have `1 * ⊥ = ⊥` but also `= 0 * ⊥ = 0`. -/
instance [mul_zero_one_class α] [nontrivial α] : mul_zero_one_class (with_bot α) :=
with_top.mul_zero_one_class
instance [mul_zero_class α] [no_zero_divisors α] : no_zero_divisors (with_bot α) :=
with_top.no_zero_divisors
instance [semigroup_with_zero α] [no_zero_divisors α] : semigroup_with_zero (with_bot α) :=
with_top.semigroup_with_zero
instance [monoid_with_zero α] [no_zero_divisors α] [nontrivial α] : monoid_with_zero (with_bot α) :=
with_top.monoid_with_zero
instance [comm_monoid_with_zero α] [no_zero_divisors α] [nontrivial α] :
comm_monoid_with_zero (with_bot α) :=
with_top.comm_monoid_with_zero
instance [canonically_ordered_comm_semiring α] [nontrivial α] : comm_semiring (with_bot α) :=
with_top.comm_semiring
end with_bot
|
606fb8687b71cbfeb760b567668448a8acd0bc88 | 624f6f2ae8b3b1adc5f8f67a365c51d5126be45a | /src/Init/Control/Applicative.lean | 2de9cd3cdff8df095e8579427a01c224962bf026 | [
"Apache-2.0"
] | permissive | mhuisi/lean4 | 28d35a4febc2e251c7f05492e13f3b05d6f9b7af | dda44bc47f3e5d024508060dac2bcb59fd12e4c0 | refs/heads/master | 1,621,225,489,283 | 1,585,142,689,000 | 1,585,142,689,000 | 250,590,438 | 0 | 2 | Apache-2.0 | 1,602,443,220,000 | 1,585,327,814,000 | C | UTF-8 | Lean | false | false | 1,366 | 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, Sebastian Ullrich
-/
prelude
import Init.Control.Functor
open Function
universes u v
class HasPure (f : Type u → Type v) :=
(pure {} {α : Type u} : α → f α)
export HasPure (pure)
class HasSeq (f : Type u → Type v) : Type (max (u+1) v) :=
(seq : ∀ {α β : Type u}, f (α → β) → f α → f β)
infixl <*> := HasSeq.seq
class HasSeqLeft (f : Type u → Type v) : Type (max (u+1) v) :=
(seqLeft : ∀ {α β : Type u}, f α → f β → f α)
infixl <* := HasSeqLeft.seqLeft
class HasSeqRight (f : Type u → Type v) : Type (max (u+1) v) :=
(seqRight : ∀ {α β : Type u}, f α → f β → f β)
infixr *> := HasSeqRight.seqRight
class Applicative (f : Type u → Type v) extends Functor f, HasPure f, HasSeq f, HasSeqLeft f, HasSeqRight f :=
(map := fun _ _ x y => pure x <*> y)
(seqLeft := fun α β a b => const β <$> a <*> b)
(seqRight := fun α β a b => const α id <$> a <*> b)
@[macroInline]
def when {m : Type → Type u} [Applicative m] (c : Prop) [h : Decidable c] (t : m Unit) : m Unit :=
if c then t else pure ()
@[macroInline]
def unless {m : Type → Type u} [Applicative m] (c : Prop) [h : Decidable c] (e : m Unit) : m Unit :=
if c then pure () else e
|
0edf1cffa9f13e2970387ef622e3faa72efce9ef | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/tactic/zify_auto.lean | 698bbc46ce8ad5c80eea526b0342df983717859e | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,060 | lean | /-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.tactic.norm_cast
import Mathlib.data.int.cast
import Mathlib.PostPort
namespace Mathlib
/-!
# A tactic to shift `ℕ` goals to `ℤ`
It is often easier to work in `ℤ`, where subtraction is well behaved, than in `ℕ` where it isn't.
`zify` is a tactic that casts goals and hypotheses about natural numbers to ones about integers.
It makes use of `push_cast`, part of the `norm_cast` family, to simplify these goals.
## Implementation notes
`zify` is extensible, using the attribute `@[zify]` to label lemmas used for moving propositions
from `ℕ` to `ℤ`.
`zify` lemmas should have the form `∀ a₁ ... aₙ : ℕ, Pz (a₁ : ℤ) ... (aₙ : ℤ) ↔ Pn a₁ ... aₙ`.
For example, `int.coe_nat_le_coe_nat_iff : ∀ (m n : ℕ), ↑m ≤ ↑n ↔ m ≤ n` is a `zify` lemma.
`zify` is very nearly just `simp only with zify push_cast`. There are a few minor differences:
* `zify` lemmas are used in the opposite order of the standard simp form.
E.g. we will rewrite with `int.coe_nat_le_coe_nat_iff` from right to left.
* `zify` should fail if no `zify` lemma applies (i.e. it was unable to shift any proposition to ℤ).
However, once this succeeds, it does not necessarily need to rewrite with any `push_cast` rules.
-/
namespace zify
/--
The `zify` attribute is used by the `zify` tactic. It applies to lemmas that shift propositions
between `nat` and `int`.
`zify` lemmas should have the form `∀ a₁ ... aₙ : ℕ, Pz (a₁ : ℤ) ... (aₙ : ℤ) ↔ Pn a₁ ... aₙ`.
For example, `int.coe_nat_le_coe_nat_iff : ∀ (m n : ℕ), ↑m ≤ ↑n ↔ m ≤ n` is a `zify` lemma.
-/
/--
Given an expression `e`, `lift_to_z e` looks for subterms of `e` that are propositions "about"
natural numbers and change them to propositions about integers.
Returns an expression `e'` and a proof that `e = e'`.
Includes `ge_iff_le` and `gt_iff_lt` in the simp set. These can't be tagged with `zify` as we
want to use them in the "forward", not "backward", direction.
-/
end zify
theorem int.coe_nat_ne_coe_nat_iff (a : ℕ) (b : ℕ) : ↑a ≠ ↑b ↔ a ≠ b := sorry
/--
`zify extra_lems e` is used to shift propositions in `e` from `ℕ` to `ℤ`.
This is often useful since `ℤ` has well-behaved subtraction.
The list of extra lemmas is used in the `push_cast` step.
Returns an expression `e'` and a proof that `e = e'`.-/
/--
A variant of `tactic.zify` that takes `h`, a proof of a proposition about natural numbers,
and returns a proof of the zified version of that propositon.
-/
/--
The `zify` tactic is used to shift propositions from `ℕ` to `ℤ`.
This is often useful since `ℤ` has well-behaved subtraction.
```lean
example (a b c x y z : ℕ) (h : ¬ x*y*z < 0) : c < a + 3*b :=
begin
zify,
zify at h,
/-
h : ¬↑x * ↑y * ↑z < 0
⊢ ↑c < ↑a + 3 * ↑b
-/
end Mathlib |
2b53d8f100f7bd879fc5b083d9c0cf5ea42fb99c | fe25de614feb5587799621c41487aaee0d083b08 | /src/Init/Data/Array/Basic.lean | d1f5f73a3d97036d5980b8ced6e1601598903e5f | [
"Apache-2.0"
] | permissive | pollend/lean4 | e8469c2f5fb8779b773618c3267883cf21fb9fac | c913886938c4b3b83238a3f99673c6c5a9cec270 | refs/heads/master | 1,687,973,251,481 | 1,628,039,739,000 | 1,628,039,739,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 27,617 | lean | /-
Copyright (c) 2018 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.Data.Nat.Basic
import Init.Data.Fin.Basic
import Init.Data.UInt
import Init.Data.Repr
import Init.Data.ToString.Basic
import Init.Util
universe u v w
namespace Array
variable {α : Type u}
@[extern "lean_mk_array"]
def mkArray {α : Type u} (n : Nat) (v : α) : Array α := {
data := List.replicate n v
}
@[simp] theorem size_mkArray (n : Nat) (v : α) : (mkArray n v).size = n :=
List.length_replicate ..
instance : EmptyCollection (Array α) := ⟨Array.empty⟩
instance : Inhabited (Array α) where
default := Array.empty
def isEmpty (a : Array α) : Bool :=
a.size = 0
def singleton (v : α) : Array α :=
mkArray 1 v
/- Low-level version of `fget` which is as fast as a C array read.
`Fin` values are represented as tag pointers in the Lean runtime. Thus,
`fget` may be slightly slower than `uget`. -/
@[extern "lean_array_uget"]
def uget (a : @& Array α) (i : USize) (h : i.toNat < a.size) : α :=
a.get ⟨i.toNat, h⟩
def back [Inhabited α] (a : Array α) : α :=
a.get! (a.size - 1)
def get? (a : Array α) (i : Nat) : Option α :=
if h : i < a.size then some (a.get ⟨i, h⟩) else none
def back? (a : Array α) : Option α :=
a.get? (a.size - 1)
-- auxiliary declaration used in the equation compiler when pattern matching array literals.
abbrev getLit {α : Type u} {n : Nat} (a : Array α) (i : Nat) (h₁ : a.size = n) (h₂ : i < n) : α :=
a.get ⟨i, h₁.symm ▸ h₂⟩
@[simp] theorem size_set (a : Array α) (i : Fin a.size) (v : α) : (set a i v).size = a.size :=
List.length_set ..
@[simp] theorem size_push (a : Array α) (v : α) : (push a v).size = a.size + 1 :=
List.length_concat ..
/- Low-level version of `fset` which is as fast as a C array fset.
`Fin` values are represented as tag pointers in the Lean runtime. Thus,
`fset` may be slightly slower than `uset`. -/
@[extern "lean_array_uset"]
def uset (a : Array α) (i : USize) (v : α) (h : i.toNat < a.size) : Array α :=
a.set ⟨i.toNat, h⟩ v
@[extern "lean_array_fswap"]
def swap (a : Array α) (i j : @& Fin a.size) : Array α :=
let v₁ := a.get i
let v₂ := a.get j
let a' := a.set i v₂
a'.set (size_set a i v₂ ▸ j) v₁
@[extern "lean_array_swap"]
def swap! (a : Array α) (i j : @& Nat) : Array α :=
if h₁ : i < a.size then
if h₂ : j < a.size then swap a ⟨i, h₁⟩ ⟨j, h₂⟩
else panic! "index out of bounds"
else panic! "index out of bounds"
@[inline] def swapAt (a : Array α) (i : Fin a.size) (v : α) : α × Array α :=
let e := a.get i
let a := a.set i v
(e, a)
@[inline]
def swapAt! (a : Array α) (i : Nat) (v : α) : α × Array α :=
if h : i < a.size then
swapAt a ⟨i, h⟩ v
else
have : Inhabited α := ⟨v⟩
panic! ("index " ++ toString i ++ " out of bounds")
@[extern "lean_array_pop"]
def pop (a : Array α) : Array α := {
data := a.data.dropLast
}
def shrink (a : Array α) (n : Nat) : Array α :=
let rec loop
| 0, a => a
| n+1, a => loop n a.pop
loop (a.size - n) a
@[inline]
def modifyM [Monad m] [Inhabited α] (a : Array α) (i : Nat) (f : α → m α) : m (Array α) := do
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩
let v := a.get idx
let a' := a.set idx arbitrary
let v ← f v
pure <| a'.set (size_set a .. ▸ idx) v
else
pure a
@[inline]
def modify [Inhabited α] (a : Array α) (i : Nat) (f : α → α) : Array α :=
Id.run <| a.modifyM i f
@[inline]
def modifyOp [Inhabited α] (self : Array α) (idx : Nat) (f : α → α) : Array α :=
self.modify idx f
/-
We claim this unsafe implementation is correct because an array cannot have more than `usizeSz` elements in our runtime.
This kind of low level trick can be removed with a little bit of compiler support. For example, if the compiler simplifies `as.size < usizeSz` to true. -/
@[inline] unsafe def forInUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (b : β) (f : α → β → m (ForInStep β)) : m β :=
let sz := USize.ofNat as.size
let rec @[specialize] loop (i : USize) (b : β) : m β := do
if i < sz then
let a := as.uget i lcProof
match (← f a b) with
| ForInStep.done b => pure b
| ForInStep.yield b => loop (i+1) b
else
pure b
loop 0 b
-- Move?
private theorem zeroLtOfLt : {a b : Nat} → a < b → 0 < b
| 0, _, h => h
| a+1, b, h =>
have : a < b := Nat.ltTrans (Nat.ltSuccSelf _) h
zeroLtOfLt this
/- Reference implementation for `forIn` -/
@[implementedBy Array.forInUnsafe]
protected def forIn {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (b : β) (f : α → β → m (ForInStep β)) : m β :=
let rec loop (i : Nat) (h : i ≤ as.size) (b : β) : m β := do
match i, h with
| 0, _ => pure b
| i+1, h =>
have h' : i < as.size := Nat.ltOfLtOfLe (Nat.ltSuccSelf i) h
have : as.size - 1 < as.size := Nat.subLt (zeroLtOfLt h') (by decide)
have : as.size - 1 - i < as.size := Nat.ltOfLeOfLt (Nat.subLe (as.size - 1) i) this
match (← f (as.get ⟨as.size - 1 - i, this⟩) b) with
| ForInStep.done b => pure b
| ForInStep.yield b => loop i (Nat.leOfLt h') b
loop as.size (Nat.leRefl _) b
instance : ForIn m (Array α) α where
forIn := Array.forIn
/- See comment at forInUnsafe -/
@[inline]
unsafe def foldlMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (init : β) (as : Array α) (start := 0) (stop := as.size) : m β :=
let rec @[specialize] fold (i : USize) (stop : USize) (b : β) : m β := do
if i == stop then
pure b
else
fold (i+1) stop (← f b (as.uget i lcProof))
if start < stop then
if stop ≤ as.size then
fold (USize.ofNat start) (USize.ofNat stop) init
else
pure init
else
pure init
/- Reference implementation for `foldlM` -/
@[implementedBy foldlMUnsafe]
def foldlM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) (init : β) (as : Array α) (start := 0) (stop := as.size) : m β :=
let fold (stop : Nat) (h : stop ≤ as.size) :=
let rec loop (i : Nat) (j : Nat) (b : β) : m β := do
if hlt : j < stop then
match i with
| 0 => pure b
| i'+1 =>
loop i' (j+1) (← f b (as.get ⟨j, Nat.ltOfLtOfLe hlt h⟩))
else
pure b
loop (stop - start) start init
if h : stop ≤ as.size then
fold stop h
else
fold as.size (Nat.leRefl _)
/- See comment at forInUnsafe -/
@[inline]
unsafe def foldrMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (init : β) (as : Array α) (start := as.size) (stop := 0) : m β :=
let rec @[specialize] fold (i : USize) (stop : USize) (b : β) : m β := do
if i == stop then
pure b
else
fold (i-1) stop (← f (as.uget (i-1) lcProof) b)
if start ≤ as.size then
if stop < start then
fold (USize.ofNat start) (USize.ofNat stop) init
else
pure init
else if stop < as.size then
fold (USize.ofNat as.size) (USize.ofNat stop) init
else
pure init
/- Reference implementation for `foldrM` -/
@[implementedBy foldrMUnsafe]
def foldrM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (init : β) (as : Array α) (start := as.size) (stop := 0) : m β :=
let rec fold (i : Nat) (h : i ≤ as.size) (b : β) : m β := do
if i == stop then
pure b
else match i, h with
| 0, _ => pure b
| i+1, h =>
have : i < as.size := Nat.ltOfLtOfLe (Nat.ltSuccSelf _) h
fold i (Nat.leOfLt this) (← f (as.get ⟨i, this⟩) b)
if h : start ≤ as.size then
if stop < start then
fold start h init
else
pure init
else if stop < as.size then
fold as.size (Nat.leRefl _) init
else
pure init
/- See comment at forInUnsafe -/
@[inline]
unsafe def mapMUnsafe {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β) :=
let sz := USize.ofNat as.size
let rec @[specialize] map (i : USize) (r : Array NonScalar) : m (Array PNonScalar.{v}) := do
if i < sz then
let v := r.uget i lcProof
let r := r.uset i arbitrary lcProof
let vNew ← f (unsafeCast v)
map (i+1) (r.uset i (unsafeCast vNew) lcProof)
else
pure (unsafeCast r)
unsafeCast <| map 0 (unsafeCast as)
/- Reference implementation for `mapM` -/
@[implementedBy mapMUnsafe]
def mapM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → m β) (as : Array α) : m (Array β) :=
as.foldlM (fun bs a => do let b ← f a; pure (bs.push b)) (mkEmpty as.size)
@[inline]
def mapIdxM {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : Fin as.size → α → m β) : m (Array β) :=
let rec @[specialize] map (i : Nat) (j : Nat) (inv : i + j = as.size) (bs : Array β) : m (Array β) := do
match i, inv with
| 0, _ => pure bs
| i+1, inv =>
have : j < as.size := by rw [← inv, Nat.add_assoc, Nat.add_comm 1 j, Nat.add_left_comm]; apply Nat.leAddRight
let idx : Fin as.size := ⟨j, this⟩
have : i + (j + 1) = as.size := by rw [← inv, Nat.add_comm j 1, Nat.add_assoc]
map i (j+1) this (bs.push (← f idx (as.get idx)))
map as.size 0 rfl (mkEmpty as.size)
@[inline]
def findSomeM? {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : α → m (Option β)) : m (Option β) := do
for a in as do
match (← f a) with
| some b => return b
| _ => pure ⟨⟩
return none
@[inline]
def findM? {α : Type} {m : Type → Type} [Monad m] (as : Array α) (p : α → m Bool) : m (Option α) := do
for a in as do
if (← p a) then
return a
return none
@[inline]
def findIdxM? [Monad m] (as : Array α) (p : α → m Bool) : m (Option Nat) := do
let mut i := 0
for a in as do
if (← p a) then
return some i
i := i + 1
return none
@[inline]
unsafe def anyMUnsafe {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m Bool :=
let rec @[specialize] any (i : USize) (stop : USize) : m Bool := do
if i == stop then
pure false
else
if (← p (as.uget i lcProof)) then
pure true
else
any (i+1) stop
if start < stop then
if stop ≤ as.size then
any (USize.ofNat start) (USize.ofNat stop)
else
pure false
else
pure false
@[implementedBy anyMUnsafe]
def anyM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m Bool :=
let any (stop : Nat) (h : stop ≤ as.size) :=
let rec loop (i : Nat) (j : Nat) : m Bool := do
if hlt : j < stop then
match i with
| 0 => pure false
| i'+1 =>
if (← p (as.get ⟨j, Nat.ltOfLtOfLe hlt h⟩)) then
pure true
else
loop i' (j+1)
else
pure false
loop (stop - start) start
if h : stop ≤ as.size then
any stop h
else
any as.size (Nat.leRefl _)
@[inline]
def allM {α : Type u} {m : Type → Type w} [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m Bool :=
return !(← as.anyM fun v => return !(← p v))
@[inline]
def findSomeRevM? {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (as : Array α) (f : α → m (Option β)) : m (Option β) :=
let rec @[specialize] find : (i : Nat) → i ≤ as.size → m (Option β)
| 0, h => pure none
| i+1, h => do
have : i < as.size := Nat.ltOfLtOfLe (Nat.ltSuccSelf _) h
let r ← f (as.get ⟨i, this⟩)
match r with
| some v => pure r
| none =>
have : i ≤ as.size := Nat.leOfLt this
find i this
find as.size (Nat.leRefl _)
@[inline]
def findRevM? {α : Type} {m : Type → Type w} [Monad m] (as : Array α) (p : α → m Bool) : m (Option α) :=
as.findSomeRevM? fun a => return if (← p a) then some a else none
@[inline]
def forM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Array α) (start := 0) (stop := as.size) : m PUnit :=
as.foldlM (fun _ => f) ⟨⟩ start stop
@[inline]
def forRevM {α : Type u} {m : Type v → Type w} [Monad m] (f : α → m PUnit) (as : Array α) (start := as.size) (stop := 0) : m PUnit :=
as.foldrM (fun a _ => f a) ⟨⟩ start stop
@[inline]
def foldl {α : Type u} {β : Type v} (f : β → α → β) (init : β) (as : Array α) (start := 0) (stop := as.size) : β :=
Id.run <| as.foldlM f init start stop
@[inline]
def foldr {α : Type u} {β : Type v} (f : α → β → β) (init : β) (as : Array α) (start := as.size) (stop := 0) : β :=
Id.run <| as.foldrM f init start stop
@[inline]
def map {α : Type u} {β : Type v} (f : α → β) (as : Array α) : Array β :=
Id.run <| as.mapM f
@[inline]
def mapIdx {α : Type u} {β : Type v} (as : Array α) (f : Fin as.size → α → β) : Array β :=
Id.run <| as.mapIdxM f
@[inline]
def find? {α : Type} (as : Array α) (p : α → Bool) : Option α :=
Id.run <| as.findM? p
@[inline]
def findSome? {α : Type u} {β : Type v} (as : Array α) (f : α → Option β) : Option β :=
Id.run <| as.findSomeM? f
@[inline]
def findSome! {α : Type u} {β : Type v} [Inhabited β] (a : Array α) (f : α → Option β) : β :=
match findSome? a f with
| some b => b
| none => panic! "failed to find element"
@[inline]
def findSomeRev? {α : Type u} {β : Type v} (as : Array α) (f : α → Option β) : Option β :=
Id.run <| as.findSomeRevM? f
@[inline]
def findRev? {α : Type} (as : Array α) (p : α → Bool) : Option α :=
Id.run <| as.findRevM? p
@[inline]
def findIdx? {α : Type u} (as : Array α) (p : α → Bool) : Option Nat :=
let rec loop (i : Nat) (j : Nat) (inv : i + j = as.size) : Option Nat :=
if hlt : j < as.size then
match i, inv with
| 0, inv => by
apply False.elim
rw [Nat.zero_add] at inv
rw [inv] at hlt
exact absurd hlt (Nat.ltIrrefl _)
| i+1, inv =>
if p (as.get ⟨j, hlt⟩) then
some j
else
have : i + (j+1) = as.size := by
rw [← inv, Nat.add_comm j 1, Nat.add_assoc]
loop i (j+1) this
else
none
loop as.size 0 rfl
def getIdx? [BEq α] (a : Array α) (v : α) : Option Nat :=
a.findIdx? fun a => a == v
@[inline]
def any (as : Array α) (p : α → Bool) (start := 0) (stop := as.size) : Bool :=
Id.run <| as.anyM p start stop
@[inline]
def all (as : Array α) (p : α → Bool) (start := 0) (stop := as.size) : Bool :=
Id.run <| as.allM p start stop
def contains [BEq α] (as : Array α) (a : α) : Bool :=
as.any fun b => a == b
def elem [BEq α] (a : α) (as : Array α) : Bool :=
as.contains a
-- TODO(Leo): justify termination using wf-rec, and use `swap`
partial def reverse (as : Array α) : Array α :=
let n := as.size
let mid := n / 2
let rec rev (as : Array α) (i : Nat) :=
if i < mid then
rev (as.swap! i (n - i - 1)) (i+1)
else
as
rev as 0
@[inline] def getEvenElems (as : Array α) : Array α :=
(·.2) <| as.foldl (init := (true, Array.empty)) fun (even, r) a =>
if even then
(false, r.push a)
else
(true, r)
@[export lean_array_to_list]
def toList (as : Array α) : List α :=
as.foldr List.cons []
instance {α : Type u} [Repr α] : Repr (Array α) where
reprPrec a n :=
let _ : Std.ToFormat α := ⟨repr⟩
if a.size == 0 then
"#[]"
else
Std.Format.bracketFill "#[" (Std.Format.joinSep (toList a) ("," ++ Std.Format.line)) "]"
instance [ToString α] : ToString (Array α) where
toString a := "#" ++ toString a.toList
protected def append (as : Array α) (bs : Array α) : Array α :=
bs.foldl (init := as) fun r v => r.push v
instance : Append (Array α) := ⟨Array.append⟩
protected def appendList (as : Array α) (bs : List α) : Array α :=
bs.foldl (init := as) fun r v => r.push v
instance : HAppend (Array α) (List α) (Array α) := ⟨Array.appendList⟩
@[inline]
def concatMapM [Monad m] (f : α → m (Array β)) (as : Array α) : m (Array β) :=
as.foldlM (init := empty) fun bs a => do return bs ++ (← f a)
@[inline]
def concatMap (f : α → Array β) (as : Array α) : Array β :=
as.foldl (init := empty) fun bs a => bs ++ f a
end Array
export Array (mkArray)
syntax "#[" sepBy(term, ", ") "]" : term
macro_rules
| `(#[ $elems,* ]) => `(List.toArray [ $elems,* ])
namespace Array
-- TODO(Leo): cleanup
@[specialize]
partial def isEqvAux (a b : Array α) (hsz : a.size = b.size) (p : α → α → Bool) (i : Nat) : Bool :=
if h : i < a.size then
let aidx : Fin a.size := ⟨i, h⟩;
let bidx : Fin b.size := ⟨i, hsz ▸ h⟩;
match p (a.get aidx) (b.get bidx) with
| true => isEqvAux a b hsz p (i+1)
| false => false
else
true
@[inline] def isEqv (a b : Array α) (p : α → α → Bool) : Bool :=
if h : a.size = b.size then
isEqvAux a b h p 0
else
false
instance [BEq α] : BEq (Array α) :=
⟨fun a b => isEqv a b BEq.beq⟩
@[inline]
def filter (p : α → Bool) (as : Array α) (start := 0) (stop := as.size) : Array α :=
as.foldl (init := #[]) (start := start) (stop := stop) fun r a =>
if p a then r.push a else r
@[inline]
def filterM [Monad m] (p : α → m Bool) (as : Array α) (start := 0) (stop := as.size) : m (Array α) :=
as.foldlM (init := #[]) (start := start) (stop := stop) fun r a => do
if (← p a) then r.push a else r
@[specialize]
def filterMapM [Monad m] (f : α → m (Option β)) (as : Array α) (start := 0) (stop := as.size) : m (Array β) :=
as.foldlM (init := #[]) (start := start) (stop := stop) fun bs a => do
match (← f a) with
| some b => pure (bs.push b)
| none => pure bs
@[inline]
def filterMap (f : α → Option β) (as : Array α) (start := 0) (stop := as.size) : Array β :=
Id.run <| as.filterMapM f (start := start) (stop := stop)
@[specialize]
def getMax? (as : Array α) (lt : α → α → Bool) : Option α :=
if h : 0 < as.size then
let a0 := as.get ⟨0, h⟩
some <| as.foldl (init := a0) (start := 1) fun best a =>
if lt best a then a else best
else
none
@[inline]
def partition (p : α → Bool) (as : Array α) : Array α × Array α := do
let mut bs := #[]
let mut cs := #[]
for a in as do
if p a then
bs := bs.push a
else
cs := cs.push a
return (bs, cs)
theorem ext (a b : Array α)
(h₁ : a.size = b.size)
(h₂ : (i : Nat) → (hi₁ : i < a.size) → (hi₂ : i < b.size) → a.get ⟨i, hi₁⟩ = b.get ⟨i, hi₂⟩)
: a = b := by
let rec extAux (a b : List α)
(h₁ : a.length = b.length)
(h₂ : (i : Nat) → (hi₁ : i < a.length) → (hi₂ : i < b.length) → a.get i hi₁ = b.get i hi₂)
: a = b := by
induction a generalizing b with
| nil =>
cases b with
| nil => rfl
| cons b bs => rw [List.length_cons] at h₁; injection h₁
| cons a as ih =>
cases b with
| nil => rw [List.length_cons] at h₁; injection h₁
| cons b bs =>
have hz₁ : 0 < (a::as).length := by rw [List.length_cons]; apply Nat.zeroLtSucc
have hz₂ : 0 < (b::bs).length := by rw [List.length_cons]; apply Nat.zeroLtSucc
have headEq : a = b := h₂ 0 hz₁ hz₂
have h₁' : as.length = bs.length := by rw [List.length_cons, List.length_cons] at h₁; injection h₁; assumption
have h₂' : (i : Nat) → (hi₁ : i < as.length) → (hi₂ : i < bs.length) → as.get i hi₁ = bs.get i hi₂ := by
intro i hi₁ hi₂
have hi₁' : i+1 < (a::as).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
have hi₂' : i+1 < (b::bs).length := by rw [List.length_cons]; apply Nat.succ_lt_succ; assumption
have : (a::as).get (i+1) hi₁' = (b::bs).get (i+1) hi₂' := h₂ (i+1) hi₁' hi₂'
apply this
have tailEq : as = bs := ih bs h₁' h₂'
rw [headEq, tailEq]
cases a; cases b
apply congrArg
apply extAux
assumption
assumption
theorem extLit {n : Nat}
(a b : Array α)
(hsz₁ : a.size = n) (hsz₂ : b.size = n)
(h : (i : Nat) → (hi : i < n) → a.getLit i hsz₁ hi = b.getLit i hsz₂ hi) : a = b :=
Array.ext a b (hsz₁.trans hsz₂.symm) fun i hi₁ hi₂ => h i (hsz₁ ▸ hi₁)
end Array
-- CLEANUP the following code
namespace Array
partial def indexOfAux [BEq α] (a : Array α) (v : α) : Nat → Option (Fin a.size)
| i =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
if a.get idx == v then some idx
else indexOfAux a v (i+1)
else none
def indexOf? [BEq α] (a : Array α) (v : α) : Option (Fin a.size) :=
indexOfAux a v 0
partial def eraseIdxAux : Nat → Array α → Array α
| i, a =>
if h : i < a.size then
let idx : Fin a.size := ⟨i, h⟩;
let idx1 : Fin a.size := ⟨i - 1, by exact Nat.ltOfLeOfLt (Nat.predLe i) h⟩;
eraseIdxAux (i+1) (a.swap idx idx1)
else
a.pop
def feraseIdx (a : Array α) (i : Fin a.size) : Array α :=
eraseIdxAux (i.val + 1) a
def eraseIdx (a : Array α) (i : Nat) : Array α :=
if i < a.size then eraseIdxAux (i+1) a else a
@[simp] theorem size_swap (a : Array α) (i j : Fin a.size) : (a.swap i j).size = a.size := by
show ((a.set i (a.get j)).set (size_set a i _ ▸ j) (a.get i)).size = a.size
rw [size_set, size_set]
@[simp] theorem size_pop (a : Array α) : a.pop.size = a.size - 1 :=
List.length_dropLast ..
section
/- Instance for justifying `partial` declaration.
We should be able to delete it as soon as we restore support for well-founded recursion. -/
instance eraseIdxSzAuxInstance (a : Array α) : Inhabited { r : Array α // r.size = a.size - 1 } where
default := ⟨a.pop, size_pop a⟩
partial def eraseIdxSzAux (a : Array α) : ∀ (i : Nat) (r : Array α), r.size = a.size → { r : Array α // r.size = a.size - 1 }
| i, r, heq =>
if h : i < r.size then
let idx : Fin r.size := ⟨i, h⟩;
let idx1 : Fin r.size := ⟨i - 1, by exact Nat.ltOfLeOfLt (Nat.predLe i) h⟩;
eraseIdxSzAux a (i+1) (r.swap idx idx1) ((size_swap r idx idx1).trans heq)
else
⟨r.pop, (size_pop r).trans (heq ▸ rfl)⟩
end
def eraseIdx' (a : Array α) (i : Fin a.size) : { r : Array α // r.size = a.size - 1 } :=
eraseIdxSzAux a (i.val + 1) a rfl
def erase [BEq α] (as : Array α) (a : α) : Array α :=
match as.indexOf? a with
| none => as
| some i => as.feraseIdx i
partial def insertAtAux (i : Nat) : Array α → Nat → Array α
| as, j =>
if i == j then as
else
let as := as.swap! (j-1) j;
insertAtAux i as (j-1)
/--
Insert element `a` at position `i`.
Pre: `i < as.size` -/
def insertAt (as : Array α) (i : Nat) (a : α) : Array α :=
if i > as.size then panic! "invalid index"
else
let as := as.push a;
as.insertAtAux i as.size
def toListLitAux (a : Array α) (n : Nat) (hsz : a.size = n) : ∀ (i : Nat), i ≤ a.size → List α → List α
| 0, hi, acc => acc
| (i+1), hi, acc => toListLitAux a n hsz i (Nat.leOfSuccLe hi) (a.getLit i hsz (Nat.ltOfLtOfEq (Nat.ltOfLtOfLe (Nat.ltSuccSelf i) hi) hsz) :: acc)
def toArrayLit (a : Array α) (n : Nat) (hsz : a.size = n) : Array α :=
List.toArray <| toListLitAux a n hsz n (hsz ▸ Nat.leRefl _) []
theorem toArrayLitEq (a : Array α) (n : Nat) (hsz : a.size = n) : a = toArrayLit a n hsz :=
-- TODO: this is painful to prove without proper automation
sorry
/-
First, we need to prove
∀ i j acc, i ≤ a.size → (toListLitAux a n hsz (i+1) hi acc).index j = if j < i then a.getLit j hsz _ else acc.index (j - i)
by induction
Base case is trivial
(j : Nat) (acc : List α) (hi : 0 ≤ a.size)
|- (toListLitAux a n hsz 0 hi acc).index j = if j < 0 then a.getLit j hsz _ else acc.index (j - 0)
... |- acc.index j = acc.index j
Induction
(j : Nat) (acc : List α) (hi : i+1 ≤ a.size)
|- (toListLitAux a n hsz (i+1) hi acc).index j = if j < i + 1 then a.getLit j hsz _ else acc.index (j - (i + 1))
... |- (toListLitAux a n hsz i hi' (a.getLit i hsz _ :: acc)).index j = if j < i + 1 then a.getLit j hsz _ else acc.index (j - (i + 1)) * by def
... |- if j < i then a.getLit j hsz _ else (a.getLit i hsz _ :: acc).index (j-i) * by induction hypothesis
=
if j < i + 1 then a.getLit j hsz _ else acc.index (j - (i + 1))
If j < i, then both are a.getLit j hsz _
If j = i, then lhs reduces else-branch to (a.getLit i hsz _) and rhs is then-brachn (a.getLit i hsz _)
If j >= i + 1, we use
- j - i >= 1 > 0
- (a::as).index k = as.index (k-1) If k > 0
- j - (i + 1) = (j - i) - 1
Then lhs = (a.getLit i hsz _ :: acc).index (j-i) = acc.index (j-i-1) = acc.index (j-(i+1)) = rhs
With this proof, we have
∀ j, j < n → (toListLitAux a n hsz n _ []).index j = a.getLit j hsz _
We also need
- (toListLitAux a n hsz n _ []).length = n
- j < n -> (List.toArray as).getLit j _ _ = as.index j
Then using Array.extLit, we have that a = List.toArray <| toListLitAux a n hsz n _ []
-/
partial def isPrefixOfAux [BEq α] (as bs : Array α) (hle : as.size ≤ bs.size) : Nat → Bool
| i =>
if h : i < as.size then
let a := as.get ⟨i, h⟩;
let b := bs.get ⟨i, Nat.ltOfLtOfLe h hle⟩;
if a == b then
isPrefixOfAux as bs hle (i+1)
else
false
else
true
/- Return true iff `as` is a prefix of `bs` -/
def isPrefixOf [BEq α] (as bs : Array α) : Bool :=
if h : as.size ≤ bs.size then
isPrefixOfAux as bs h 0
else
false
private def allDiffAuxAux [BEq α] (as : Array α) (a : α) : forall (i : Nat), i < as.size → Bool
| 0, h => true
| i+1, h =>
have : i < as.size := Nat.ltTrans (Nat.ltSuccSelf _) h;
a != as.get ⟨i, this⟩ && allDiffAuxAux as a i this
private partial def allDiffAux [BEq α] (as : Array α) : Nat → Bool
| i =>
if h : i < as.size then
allDiffAuxAux as (as.get ⟨i, h⟩) i h && allDiffAux as (i+1)
else
true
def allDiff [BEq α] (as : Array α) : Bool :=
allDiffAux as 0
@[specialize] partial def zipWithAux (f : α → β → γ) (as : Array α) (bs : Array β) : Nat → Array γ → Array γ
| i, cs =>
if h : i < as.size then
let a := as.get ⟨i, h⟩;
if h : i < bs.size then
let b := bs.get ⟨i, h⟩;
zipWithAux f as bs (i+1) <| cs.push <| f a b
else
cs
else
cs
@[inline] def zipWith (as : Array α) (bs : Array β) (f : α → β → γ) : Array γ :=
zipWithAux f as bs 0 #[]
def zip (as : Array α) (bs : Array β) : Array (α × β) :=
zipWith as bs Prod.mk
def unzip (as : Array (α × β)) : Array α × Array β :=
as.foldl (init := (#[], #[])) fun (as, bs) (a, b) => (as.push a, bs.push b)
def split (as : Array α) (p : α → Bool) : Array α × Array α :=
as.foldl (init := (#[], #[])) fun (as, bs) a =>
if p a then (as.push a, bs) else (as, bs.push a)
end Array
|
637e060a4cdd2be7889a4fcd8053745cc1404bbe | 69d4931b605e11ca61881fc4f66db50a0a875e39 | /src/algebra/smul_with_zero.lean | 6e7dc416dd4993bf4eeb793ae0ff16ea6770fbbc | [
"Apache-2.0"
] | permissive | abentkamp/mathlib | d9a75d291ec09f4637b0f30cc3880ffb07549ee5 | 5360e476391508e092b5a1e5210bd0ed22dc0755 | refs/heads/master | 1,682,382,954,948 | 1,622,106,077,000 | 1,622,106,077,000 | 149,285,665 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 5,251 | lean | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import group_theory.group_action.defs
import algebra.group_power.basic
/-!
# Introduce `smul_with_zero`
In analogy with the usual monoid action on a Type `M`, we introduce an action of a
`monoid_with_zero` on a Type with `0`.
In particular, for Types `R` and `M`, both containing `0`, we define `smul_with_zero R M` to
be the typeclass where the products `r • 0` and `0 • m` vanish for all `r : R` and all `m : M`.
Moreover, in the case in which `R` is a `monoid_with_zero`, we introduce the typeclass
`mul_action_with_zero R M`, mimicking group actions and having an absorbing `0` in `R`.
Thus, the action is required to be compatible with
* the unit of the monoid, acting as the identity;
* the zero of the monoid_with_zero, acting as zero;
* associativity of the monoid.
We also add an `instance`:
* any `monoid_with_zero` has a `mul_action_with_zero R R` acting on itself.
-/
variables {R R' M M' : Type*}
section has_zero
variables (R M)
/-- `smul_with_zero` is a class consisting of a Type `R` with `0 ∈ R` and a scalar multiplication
of `R` on a Type `M` with `0`, such that the equality `r • m = 0` holds if at least one among `r`
or `m` equals `0`. -/
class smul_with_zero [has_zero R] [has_zero M] extends has_scalar R M :=
(smul_zero : ∀ r : R, r • (0 : M) = 0)
(zero_smul : ∀ m : M, (0 : R) • m = 0)
instance mul_zero_class.to_smul_with_zero [mul_zero_class R] : smul_with_zero R R :=
{ smul := (*),
smul_zero := mul_zero,
zero_smul := zero_mul }
instance add_monoid.to_smul_with_zero [add_monoid M] : smul_with_zero ℕ M :=
{ smul_zero := nsmul_zero,
zero_smul := zero_nsmul }
variables (R) {M} [has_zero R] [has_zero M] [smul_with_zero R M]
@[simp] lemma zero_smul (m : M) : (0 : R) • m = 0 := smul_with_zero.zero_smul m
variables {R} (M)
/-- Note that this lemma has different typeclass assumptions to `smul_zero`. -/
@[simp] lemma smul_zero' (r : R) : r • (0 : M) = 0 := smul_with_zero.smul_zero r
variables {R M} [has_zero R'] [has_zero M'] [has_scalar R M']
/-- Pullback a `smul_with_zero` structure along an injective zero-preserving homomorphism. -/
protected def function.injective.smul_with_zero
(f : zero_hom M' M) (hf : function.injective f) (smul : ∀ (a : R) b, f (a • b) = a • f b) :
smul_with_zero R M' :=
{ smul := (•),
zero_smul := λ a, hf $ by simp [smul],
smul_zero := λ a, hf $ by simp [smul]}
/-- Pushforward a `smul_with_zero` structure along a surjective zero-preserving homomorphism. -/
protected def function.surjective.smul_with_zero
(f : zero_hom M M') (hf : function.surjective f) (smul : ∀ (a : R) b, f (a • b) = a • f b) :
smul_with_zero R M' :=
{ smul := (•),
zero_smul := λ m, by { rcases hf m with ⟨x, rfl⟩, simp [←smul] },
smul_zero := λ c, by simp only [← f.map_zero, ← smul, smul_zero'] }
variables (M)
/-- Compose a `smul_with_zero` with a `zero_hom`, with action `f r' • m` -/
def smul_with_zero.comp_hom (f : zero_hom R' R) : smul_with_zero R' M :=
{ smul := (•) ∘ f,
smul_zero := λ m, by simp,
zero_smul := λ m, by simp }
end has_zero
section monoid_with_zero
variables [monoid_with_zero R] [monoid_with_zero R'] [has_zero M]
variables (R M)
/-- An action of a monoid with zero `R` on a Type `M`, also with `0`, extends `mul_action` and
is compatible with `0` (both in `R` and in `M`), with `1 ∈ R`, and with associativity of
multiplication on the monoid `M`. -/
class mul_action_with_zero extends mul_action R M :=
-- these fields are copied from `smul_with_zero`, as `extends` behaves poorly
(smul_zero : ∀ r : R, r • (0 : M) = 0)
(zero_smul : ∀ m : M, (0 : R) • m = 0)
@[priority 100] -- see Note [lower instance priority]
instance mul_action_with_zero.to_smul_with_zero [m : mul_action_with_zero R M] :
smul_with_zero R M :=
{..m}
instance monoid_with_zero.to_mul_action_with_zero : mul_action_with_zero R R :=
{ ..mul_zero_class.to_smul_with_zero R,
..monoid.to_mul_action R }
variables {R M} [mul_action_with_zero R M] [has_zero M'] [has_scalar R M']
/-- Pullback a `mul_action_with_zero` structure along an injective zero-preserving homomorphism. -/
protected def function.injective.mul_action_with_zero
(f : zero_hom M' M) (hf : function.injective f) (smul : ∀ (a : R) b, f (a • b) = a • f b) :
mul_action_with_zero R M' :=
{ ..hf.mul_action f smul, ..hf.smul_with_zero f smul }
/-- Pushforward a `mul_action_with_zero` structure along a surjective zero-preserving homomorphism.
-/
protected def function.surjective.mul_action_with_zero
(f : zero_hom M M') (hf : function.surjective f) (smul : ∀ (a : R) b, f (a • b) = a • f b) :
mul_action_with_zero R M' :=
{ ..hf.mul_action f smul, ..hf.smul_with_zero f smul }
variables (M)
/-- Compose a `mul_action_with_zero` with a `monoid_with_zero_hom`, with action `f r' • m` -/
def mul_action_with_zero.comp_hom (f : monoid_with_zero_hom R' R) :
mul_action_with_zero R' M :=
{ smul := (•) ∘ f,
mul_smul := λ r s m, by simp [mul_smul],
one_smul := λ m, by simp,
.. smul_with_zero.comp_hom M f.to_zero_hom}
end monoid_with_zero
|
dd246ed8b460d46eee667217f455c28934a9aead | 87fd6b43d22688237c02b87c30d2a524f53bab24 | /src/game/sets/sqrtOfTwo.lean | 5a65e871c9572b49759280cd6b01c3a521b95ead | [
"Apache-2.0"
] | permissive | grthomson/real-number-game | 66142fedf0987db90f66daed52f9c8b42b70f909 | 8ddc15fdddc241c246653f7bb341df36e4e880a8 | refs/heads/master | 1,668,059,330,605 | 1,592,873,454,000 | 1,592,873,454,000 | 262,025,764 | 0 | 0 | null | 1,588,849,107,000 | 1,588,849,106,000 | null | UTF-8 | Lean | false | false | 3,895 | lean | import data.nat.prime
open nat
theorem sqrt_two_irrational_V1 {a b : ℕ} (co : gcd a b = 1) : a^2 ≠ 2 * b^2 :=
assume h : a^2 = 2 * b^2,
have 2 ∣ a^2,
by simp [h],
have ha : 2 ∣ a,
from prime.dvd_of_dvd_pow prime_two this,
-- usually after exists.elim the rest of the proof should be inside
-- parantheses
exists.elim ha (assume c : ℕ, assume aeq : a = 2 * c,
have 2 * (2 * c^2) = 2 * b^2,
by simp [eq.symm h, aeq];
simp [nat.pow_succ, mul_comm, mul_assoc, mul_left_comm],
have 2 * c^2 = b^2,
from eq_of_mul_eq_mul_left dec_trivial this,
have 2 ∣ b^2,
by simp [eq.symm this],
have hb : 2 ∣ b,
from prime.dvd_of_dvd_pow prime_two this,
have 2 ∣ gcd a b,
from dvd_gcd ha hb,
have habs : 2 ∣ (1:ℕ),
by simp * at *,
-- therefore the ) below!!!!
show false, from absurd habs dec_trivial )
#check sqrt_two_irrational_V1
-- tactics mode
theorem sqrt_two_irrational_V2 {a b : ℕ} (co : gcd a b = 1) : a^2 ≠ 2 * b^2 :=
begin
intro h,
have h1 : 2 ∣ a^2, by simp [h],
have h2 : 2 ∣ a, from prime.dvd_of_dvd_pow prime_two h1,
cases h2 with c aeq,
have h3 : 2 * (2 * c^2) = 2 * b^2,
by simp [eq.symm h, aeq];
simp [nat.pow_succ, mul_comm, mul_assoc, mul_left_comm],
have h4 : 2 * c^2 = b^2,
from eq_of_mul_eq_mul_left dec_trivial h3,
have h5 : 2 ∣ b^2,
by simp [eq.symm h4],
have hb : 2 ∣ b,
from prime.dvd_of_dvd_pow prime_two h5,
have h6 : 2 ∣ gcd a b, from dvd_gcd (exists.intro c aeq) hb,
have habs : 2 ∣ (1:ℕ), by simp * at *,
exact absurd habs dec_trivial, done
end
-- using let (it may introduce a bug due to _let_match hypothesis)
theorem sqrt_two_irrational_V3 {a b : ℕ} (co : gcd a b = 1) : a^2 ≠ 2 * b^2 :=
assume h : a^2 = 2 * b^2,
have 2 ∣ a^2,
by simp [h],
have ha : 2 ∣ a,
from prime.dvd_of_dvd_pow prime_two this,
let ⟨c, aeq⟩ := ha in
have 2 * (2 * c^2) = 2 * b^2,
by simp [eq.symm h, aeq];
simp [nat.pow_succ, mul_comm, mul_assoc, mul_left_comm],
have 2 * c^2 = b^2,
from eq_of_mul_eq_mul_left dec_trivial this,
have 2 ∣ b^2,
by simp [eq.symm this],
have hb : 2 ∣ b,
from prime.dvd_of_dvd_pow prime_two this,
have 2 ∣ gcd a b,
from dvd_gcd ha hb,
have habs : 2 ∣ (1:ℕ),
by clear_aux_decl; simp * at *, -- replaces simp * at *,
show false, from absurd habs dec_trivial
-- variant with match, needs end to finish
theorem sqrt_two_irrational_V4 {a b : ℕ} (co : gcd a b = 1) : a^2 ≠ 2 * b^2 :=
assume h : a^2 = 2 * b^2,
have 2 ∣ a^2,
by simp [h],
have ha : 2 ∣ a,
from prime.dvd_of_dvd_pow prime_two this,
match ha with | ⟨c, aeq⟩ :=
have 2 * (2 * c^2) = 2 * b^2,
by simp [eq.symm h, aeq];
simp [nat.pow_succ, mul_comm, mul_assoc, mul_left_comm],
have 2 * c^2 = b^2,
from eq_of_mul_eq_mul_left dec_trivial this,
have 2 ∣ b^2,
by simp [eq.symm this],
have hb : 2 ∣ b,
from prime.dvd_of_dvd_pow prime_two this,
have 2 ∣ gcd a b,
from dvd_gcd ha hb,
have habs : 2 ∣ (1:ℕ),
by clear_aux_decl; simp * at *, -- replaces simp * at *,
show false, from absurd habs dec_trivial
end
theorem sqrt_two_irrational {a b : ℕ} (co : gcd a b = 1) :
a^2 ≠ 2 * b^2 :=
assume h : a^2 = 2 * b^2,
have 2 ∣ a^2,
by simp [h],
have 2 ∣ a,
from prime.dvd_of_dvd_pow prime_two this,
exists.elim this $
assume (c : nat) (aeq : a = 2 * c),
have 2 * (2 * c^2) = 2 * b^2,
by simp [eq.symm h, aeq];
simp [nat.pow_succ, mul_comm, mul_assoc, mul_left_comm],
have 2 * c^2 = b^2,
from eq_of_mul_eq_mul_left dec_trivial this,
have 2 ∣ b^2,
by simp [eq.symm this],
have 2 ∣ b,
from prime.dvd_of_dvd_pow prime_two this,
have 2 ∣ gcd a b,
from dvd_gcd ‹2 ∣ a› ‹2 ∣ b›,
have 2 ∣ (1 : ℕ),
by simp * at *,
show false, from absurd ‹2 ∣ 1› dec_trivial
#check sqrt_two_irrational
|
b900dc735393a5cf55a8864037bf8ec78b5ad49f | 4d2583807a5ac6caaffd3d7a5f646d61ca85d532 | /src/algebra/field_power.lean | bb35ad0880d844bd5f370942e8b7cd6ab5622bb6 | [
"Apache-2.0"
] | permissive | AntoineChambert-Loir/mathlib | 64aabb896129885f12296a799818061bc90da1ff | 07be904260ab6e36a5769680b6012f03a4727134 | refs/heads/master | 1,693,187,631,771 | 1,636,719,886,000 | 1,636,719,886,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 8,735 | lean | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import algebra.group_with_zero.power
import tactic.linarith
import data.equiv.ring
/-!
# Integer power operation on fields and division rings
This file collects basic facts about the operation of raising an element of a `division_ring` to an
integer power. More specialised results are provided in the case of a linearly ordered field.
-/
universe u
@[simp] lemma ring_hom.map_zpow {K L : Type*} [division_ring K] [division_ring L] (f : K →+* L) :
∀ (a : K) (n : ℤ), f (a ^ n) = f a ^ n :=
f.to_monoid_with_zero_hom.map_zpow
@[simp] lemma ring_equiv.map_zpow {K L : Type*} [division_ring K] [division_ring L] (f : K ≃+* L) :
∀ (a : K) (n : ℤ), f (a ^ n) = f a ^ n :=
f.to_ring_hom.map_zpow
@[simp] lemma zpow_bit0_neg {K : Type*} [division_ring K] (x : K) (n : ℤ) :
(-x) ^ (bit0 n) = x ^ bit0 n :=
by rw [zpow_bit0', zpow_bit0', neg_mul_neg]
lemma even.zpow_neg {K : Type*} [division_ring K] {n : ℤ} (h : even n) (a : K) :
(-a) ^ n = a ^ n :=
begin
obtain ⟨k, rfl⟩ := h,
rw [←bit0_eq_two_mul, zpow_bit0_neg],
end
@[simp] lemma zpow_bit1_neg {K : Type*} [division_ring K] (x : K) (n : ℤ) :
(-x) ^ (bit1 n) = - x ^ bit1 n :=
by rw [zpow_bit1', zpow_bit1', neg_mul_neg, neg_mul_eq_mul_neg]
section ordered_field_power
open int
variables {K : Type u} [linear_ordered_field K] {a : K} {n : ℤ}
lemma zpow_eq_zero_iff (hn : 0 < n) :
a ^ n = 0 ↔ a = 0 :=
begin
refine ⟨zpow_eq_zero, _⟩,
rintros rfl,
exact zero_zpow _ hn.ne'
end
lemma zpow_nonneg {a : K} (ha : 0 ≤ a) : ∀ (z : ℤ), 0 ≤ a ^ z
| (n : ℕ) := by { rw zpow_coe_nat, exact pow_nonneg ha _ }
| -[1+n] := by { rw zpow_neg_succ_of_nat, exact inv_nonneg.2 (pow_nonneg ha _) }
lemma zpow_pos_of_pos {a : K} (ha : 0 < a) : ∀ (z : ℤ), 0 < a ^ z
| (n : ℕ) := by { rw zpow_coe_nat, exact pow_pos ha _ }
| -[1+n] := by { rw zpow_neg_succ_of_nat, exact inv_pos.2 (pow_pos ha _) }
lemma zpow_le_of_le {x : K} (hx : 1 ≤ x) {a b : ℤ} (h : a ≤ b) : x ^ a ≤ x ^ b :=
begin
induction a with a a; induction b with b b,
{ simp only [of_nat_eq_coe, zpow_coe_nat],
apply pow_le_pow hx,
apply le_of_coe_nat_le_coe_nat h },
{ apply absurd h,
apply not_le_of_gt,
exact lt_of_lt_of_le (neg_succ_lt_zero _) (of_nat_nonneg _) },
{ simp only [zpow_neg_succ_of_nat, one_div, of_nat_eq_coe, zpow_coe_nat],
apply le_trans (inv_le_one _); apply one_le_pow_of_one_le hx },
{ simp only [zpow_neg_succ_of_nat],
apply (inv_le_inv _ _).2,
{ apply pow_le_pow hx,
have : -(↑(a+1) : ℤ) ≤ -(↑(b+1) : ℤ), from h,
have h' := le_of_neg_le_neg this,
apply le_of_coe_nat_le_coe_nat h' },
repeat { apply pow_pos (lt_of_lt_of_le zero_lt_one hx) } }
end
lemma pow_le_max_of_min_le {x : K} (hx : 1 ≤ x) {a b c : ℤ} (h : min a b ≤ c) :
x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b)) :=
begin
wlog hle : a ≤ b,
have hnle : -b ≤ -a, from neg_le_neg hle,
have hfle : x ^ (-b) ≤ x ^ (-a), from zpow_le_of_le hx hnle,
have : x ^ (-c) ≤ x ^ (-a),
{ apply zpow_le_of_le hx,
simpa only [min_eq_left hle, neg_le_neg_iff] using h },
simpa only [max_eq_left hfle]
end
lemma zpow_le_one_of_nonpos {p : K} (hp : 1 ≤ p) {z : ℤ} (hz : z ≤ 0) : p ^ z ≤ 1 :=
calc p ^ z ≤ p ^ 0 : zpow_le_of_le hp hz
... = 1 : by simp
lemma one_le_zpow_of_nonneg {p : K} (hp : 1 ≤ p) {z : ℤ} (hz : 0 ≤ z) : 1 ≤ p ^ z :=
calc p ^ z ≥ p ^ 0 : zpow_le_of_le hp hz
... = 1 : by simp
theorem zpow_bit0_nonneg (a : K) (n : ℤ) : 0 ≤ a ^ bit0 n :=
by { rw zpow_bit0₀, exact mul_self_nonneg _ }
theorem zpow_two_nonneg (a : K) : 0 ≤ a ^ 2 :=
pow_bit0_nonneg a 1
theorem zpow_bit0_pos {a : K} (h : a ≠ 0) (n : ℤ) : 0 < a ^ bit0 n :=
(zpow_bit0_nonneg a n).lt_of_ne (zpow_ne_zero _ h).symm
theorem zpow_two_pos_of_ne_zero (a : K) (h : a ≠ 0) : 0 < a ^ 2 :=
pow_bit0_pos h 1
@[simp] theorem zpow_bit1_neg_iff : a ^ bit1 n < 0 ↔ a < 0 :=
⟨λ h, not_le.1 $ λ h', not_le.2 h $ zpow_nonneg h' _,
λ h, by rw [bit1, zpow_add_one₀ h.ne]; exact mul_neg_of_pos_of_neg (zpow_bit0_pos h.ne _) h⟩
@[simp] theorem zpow_bit1_nonneg_iff : 0 ≤ a ^ bit1 n ↔ 0 ≤ a :=
le_iff_le_iff_lt_iff_lt.2 zpow_bit1_neg_iff
@[simp] theorem zpow_bit1_nonpos_iff : a ^ bit1 n ≤ 0 ↔ a ≤ 0 :=
begin
rw [le_iff_lt_or_eq, zpow_bit1_neg_iff],
split,
{ rintro (h | h),
{ exact h.le },
{ exact (zpow_eq_zero h).le } },
{ intro h,
rcases eq_or_lt_of_le h with rfl|h,
{ exact or.inr (zero_zpow _ (bit1_ne_zero n)) },
{ exact or.inl h } }
end
@[simp] theorem zpow_bit1_pos_iff : 0 < a ^ bit1 n ↔ 0 < a :=
lt_iff_lt_of_le_iff_le zpow_bit1_nonpos_iff
lemma even.zpow_nonneg {n : ℤ} (hn : even n) (a : K) :
0 ≤ a ^ n :=
begin
cases le_or_lt 0 a with h h,
{ exact zpow_nonneg h _ },
{ exact (hn.zpow_neg a).subst (zpow_nonneg (neg_nonneg_of_nonpos h.le) _) }
end
theorem even.zpow_pos (hn : even n) (ha : a ≠ 0) : 0 < a ^ n :=
by cases hn with k hk; simpa only [hk, two_mul] using zpow_bit0_pos ha k
theorem odd.zpow_nonneg (hn : odd n) (ha : 0 ≤ a) : 0 ≤ a ^ n :=
by cases hn with k hk; simpa only [hk, two_mul] using zpow_bit1_nonneg_iff.mpr ha
theorem odd.zpow_pos (hn : odd n) (ha : 0 < a) : 0 < a ^ n :=
by cases hn with k hk; simpa only [hk, two_mul] using zpow_bit1_pos_iff.mpr ha
theorem odd.zpow_nonpos (hn : odd n) (ha : a ≤ 0) : a ^ n ≤ 0:=
by cases hn with k hk; simpa only [hk, two_mul] using zpow_bit1_nonpos_iff.mpr ha
theorem odd.zpow_neg (hn : odd n) (ha : a < 0) : a ^ n < 0:=
by cases hn with k hk; simpa only [hk, two_mul] using zpow_bit1_neg_iff.mpr ha
lemma even.zpow_abs {p : ℤ} (hp : even p) (a : K) : |a| ^ p = a ^ p :=
begin
cases abs_choice a with h h;
simp only [h, hp.zpow_neg _],
end
@[simp] lemma zpow_bit0_abs (a : K) (p : ℤ) : |a| ^ bit0 p = a ^ bit0 p :=
(even_bit0 _).zpow_abs _
lemma even.abs_zpow {p : ℤ} (hp : even p) (a : K) : |a ^ p| = a ^ p :=
begin
rw [abs_eq_self],
exact hp.zpow_nonneg _
end
@[simp] lemma abs_zpow_bit0 (a : K) (p : ℤ) :
|a ^ bit0 p| = a ^ bit0 p :=
(even_bit0 _).abs_zpow _
end ordered_field_power
lemma one_lt_zpow {K} [linear_ordered_field K] {p : K} (hp : 1 < p) :
∀ z : ℤ, 0 < z → 1 < p ^ z
| (n : ℕ) h := (zpow_coe_nat p n).symm.subst (one_lt_pow hp $ int.coe_nat_ne_zero.mp h.ne')
| -[1+ n] h := ((int.neg_succ_not_pos _).mp h).elim
section ordered
variables {K : Type*} [linear_ordered_field K]
lemma nat.zpow_pos_of_pos {p : ℕ} (h : 0 < p) (n:ℤ) : 0 < (p:K)^n :=
by { apply zpow_pos_of_pos, exact_mod_cast h }
lemma nat.zpow_ne_zero_of_pos {p : ℕ} (h : 0 < p) (n:ℤ) : (p:K)^n ≠ 0 :=
ne_of_gt (nat.zpow_pos_of_pos h n)
lemma zpow_strict_mono {x : K} (hx : 1 < x) :
strict_mono (λ n:ℤ, x ^ n) :=
λ m n h, show x ^ m < x ^ n,
begin
have xpos : 0 < x := zero_lt_one.trans hx,
have h₀ : x ≠ 0 := xpos.ne',
have hxm : 0 < x^m := zpow_pos_of_pos xpos m,
have h : 1 < x ^ (n - m) := one_lt_zpow hx _ (sub_pos_of_lt h),
replace h := mul_lt_mul_of_pos_right h hxm,
rwa [sub_eq_add_neg, zpow_add₀ h₀, mul_assoc, zpow_neg_mul_zpow_self _ h₀, one_mul, mul_one] at h,
end
@[simp] lemma zpow_lt_iff_lt {x : K} (hx : 1 < x) {m n : ℤ} :
x ^ m < x ^ n ↔ m < n :=
(zpow_strict_mono hx).lt_iff_lt
@[simp] lemma zpow_le_iff_le {x : K} (hx : 1 < x) {m n : ℤ} :
x ^ m ≤ x ^ n ↔ m ≤ n :=
(zpow_strict_mono hx).le_iff_le
@[simp] lemma pos_div_pow_pos {a b : K} (ha : 0 < a) (hb : 0 < b) (k : ℕ) : 0 < a/b^k :=
div_pos ha (pow_pos hb k)
@[simp] lemma div_pow_le {a b : K} (ha : 0 < a) (hb : 1 ≤ b) (k : ℕ) : a/b^k ≤ a :=
(div_le_iff $ pow_pos (lt_of_lt_of_le zero_lt_one hb) k).mpr
(calc a = a * 1 : (mul_one a).symm
... ≤ a*b^k : (mul_le_mul_left ha).mpr $ one_le_pow_of_one_le hb _)
lemma zpow_injective {x : K} (h₀ : 0 < x) (h₁ : x ≠ 1) :
function.injective ((^) x : ℤ → K) :=
begin
intros m n h,
rcases h₁.lt_or_lt with H|H,
{ apply (zpow_strict_mono (one_lt_inv h₀ H)).injective,
show x⁻¹ ^ m = x⁻¹ ^ n,
rw [← zpow_neg_one₀, ← zpow_mul₀, ← zpow_mul₀, mul_comm _ m, mul_comm _ n, zpow_mul₀, zpow_mul₀,
h], },
{ exact (zpow_strict_mono H).injective h, },
end
@[simp] lemma zpow_inj {x : K} (h₀ : 0 < x) (h₁ : x ≠ 1) {m n : ℤ} :
x ^ m = x ^ n ↔ m = n :=
(zpow_injective h₀ h₁).eq_iff
end ordered
section
variables {K : Type*} [field K]
@[simp, norm_cast] theorem rat.cast_zpow [char_zero K] (q : ℚ) (n : ℤ) :
((q ^ n : ℚ) : K) = q ^ n :=
(rat.cast_hom K).map_zpow q n
end
|
5b4e959c149be6357e933dc51e74ce1c74868460 | 8e6cad62ec62c6c348e5faaa3c3f2079012bdd69 | /src/topology/instances/real.lean | 17b963588a19e227a0b70bea3f21d3eb86eea478 | [
"Apache-2.0"
] | permissive | benjamindavidson/mathlib | 8cc81c865aa8e7cf4462245f58d35ae9a56b150d | fad44b9f670670d87c8e25ff9cdf63af87ad731e | refs/heads/master | 1,679,545,578,362 | 1,615,343,014,000 | 1,615,343,014,000 | 312,926,983 | 0 | 0 | Apache-2.0 | 1,615,360,301,000 | 1,605,399,418,000 | Lean | UTF-8 | Lean | false | false | 16,892 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import topology.metric_space.basic
import topology.algebra.uniform_group
import topology.algebra.ring
import topology.algebra.continuous_functions
import ring_theory.subring
import group_theory.archimedean
/-!
# Topological properties of ℝ
-/
noncomputable theory
open classical set filter topological_space metric
open_locale classical
open_locale topological_space
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w}
instance : metric_space ℚ :=
metric_space.induced coe rat.cast_injective real.metric_space
theorem rat.dist_eq (x y : ℚ) : dist x y = abs (x - y) := rfl
@[norm_cast, simp] lemma rat.dist_cast (x y : ℚ) : dist (x : ℝ) y = dist x y := rfl
section low_prio
-- we want to ignore this instance for the next declaration
local attribute [instance, priority 10] int.uniform_space
instance : metric_space ℤ :=
begin
letI M := metric_space.induced coe int.cast_injective real.metric_space,
refine @metric_space.replace_uniformity _ int.uniform_space M
(le_antisymm refl_le_uniformity $ λ r ru,
mem_uniformity_dist.2 ⟨1, zero_lt_one, λ a b h,
mem_principal_sets.1 ru $ dist_le_zero.1 (_ : (abs (a - b) : ℝ) ≤ 0)⟩),
have : (abs (↑a - ↑b) : ℝ) < 1 := h,
have : abs (a - b) < 1, by norm_cast at this; assumption,
have : abs (a - b) ≤ 0 := (@int.lt_add_one_iff _ 0).mp this,
norm_cast, assumption
end
end low_prio
theorem int.dist_eq (x y : ℤ) : dist x y = abs (x - y) := rfl
@[norm_cast, simp] theorem int.dist_cast_real (x y : ℤ) : dist (x : ℝ) y = dist x y := rfl
@[norm_cast, simp] theorem int.dist_cast_rat (x y : ℤ) : dist (x : ℚ) y = dist x y :=
by rw [← int.dist_cast_real, ← rat.dist_cast]; congr' 1; norm_cast
instance : proper_space ℤ :=
⟨ begin
intros x r,
apply set.finite.is_compact,
have : closed_ball x r = coe ⁻¹' (closed_ball (x:ℝ) r) := rfl,
simp [this, closed_ball_Icc, set.Icc_ℤ_finite],
end ⟩
theorem uniform_continuous_of_rat : uniform_continuous (coe : ℚ → ℝ) :=
uniform_continuous_comap
theorem uniform_embedding_of_rat : uniform_embedding (coe : ℚ → ℝ) :=
uniform_embedding_comap rat.cast_injective
theorem dense_embedding_of_rat : dense_embedding (coe : ℚ → ℝ) :=
uniform_embedding_of_rat.dense_embedding $
λ x, mem_closure_iff_nhds.2 $ λ t ht,
let ⟨ε,ε0, hε⟩ := mem_nhds_iff.1 ht in
let ⟨q, h⟩ := exists_rat_near x ε0 in
⟨_, hε (mem_ball'.2 h), q, rfl⟩
theorem embedding_of_rat : embedding (coe : ℚ → ℝ) := dense_embedding_of_rat.to_embedding
theorem continuous_of_rat : continuous (coe : ℚ → ℝ) := uniform_continuous_of_rat.continuous
theorem real.uniform_continuous_add : uniform_continuous (λp : ℝ × ℝ, p.1 + p.2) :=
metric.uniform_continuous_iff.2 $ λ ε ε0,
let ⟨δ, δ0, Hδ⟩ := rat_add_continuous_lemma abs ε0 in
⟨δ, δ0, λ a b h, let ⟨h₁, h₂⟩ := max_lt_iff.1 h in Hδ h₁ h₂⟩
-- TODO(Mario): Find a way to use rat_add_continuous_lemma
theorem rat.uniform_continuous_add : uniform_continuous (λp : ℚ × ℚ, p.1 + p.2) :=
uniform_embedding_of_rat.to_uniform_inducing.uniform_continuous_iff.2 $ by simp [(∘)]; exact
real.uniform_continuous_add.comp ((uniform_continuous_of_rat.comp uniform_continuous_fst).prod_mk
(uniform_continuous_of_rat.comp uniform_continuous_snd))
theorem real.uniform_continuous_neg : uniform_continuous (@has_neg.neg ℝ _) :=
metric.uniform_continuous_iff.2 $ λ ε ε0, ⟨_, ε0, λ a b h,
by rw dist_comm at h; simpa [real.dist_eq] using h⟩
theorem rat.uniform_continuous_neg : uniform_continuous (@has_neg.neg ℚ _) :=
metric.uniform_continuous_iff.2 $ λ ε ε0, ⟨_, ε0, λ a b h,
by rw dist_comm at h; simpa [rat.dist_eq] using h⟩
instance : uniform_add_group ℝ :=
uniform_add_group.mk' real.uniform_continuous_add real.uniform_continuous_neg
instance : uniform_add_group ℚ :=
uniform_add_group.mk' rat.uniform_continuous_add rat.uniform_continuous_neg
-- short-circuit type class inference
instance : topological_add_group ℝ := by apply_instance
instance : topological_add_group ℚ := by apply_instance
instance : order_topology ℚ :=
induced_order_topology _ (λ x y, rat.cast_lt) (@exists_rat_btwn _ _ _)
lemma real.is_topological_basis_Ioo_rat :
@is_topological_basis ℝ _ (⋃(a b : ℚ) (h : a < b), {Ioo a b}) :=
is_topological_basis_of_open_of_nhds
(by simp [is_open_Ioo] {contextual:=tt})
(assume a v hav hv,
let ⟨l, u, ⟨hl, hu⟩, h⟩ := mem_nhds_iff_exists_Ioo_subset.mp (mem_nhds_sets hv hav),
⟨q, hlq, hqa⟩ := exists_rat_btwn hl,
⟨p, hap, hpu⟩ := exists_rat_btwn hu in
⟨Ioo q p,
by { simp only [mem_Union], exact ⟨q, p, rat.cast_lt.1 $ hqa.trans hap, rfl⟩ },
⟨hqa, hap⟩, assume a' ⟨hqa', ha'p⟩, h ⟨hlq.trans hqa', ha'p.trans hpu⟩⟩)
instance : second_countable_topology ℝ :=
⟨⟨(⋃(a b : ℚ) (h : a < b), {Ioo a b}),
by simp [countable_Union, countable_Union_Prop],
real.is_topological_basis_Ioo_rat.2.2⟩⟩
/- TODO(Mario): Prove that these are uniform isomorphisms instead of uniform embeddings
lemma uniform_embedding_add_rat {r : ℚ} : uniform_embedding (λp:ℚ, p + r) :=
_
lemma uniform_embedding_mul_rat {q : ℚ} (hq : q ≠ 0) : uniform_embedding ((*) q) :=
_ -/
lemma real.mem_closure_iff {s : set ℝ} {x : ℝ} : x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, abs (y - x) < ε :=
by simp [mem_closure_iff_nhds_basis nhds_basis_ball, real.dist_eq]
lemma real.uniform_continuous_inv (s : set ℝ) {r : ℝ} (r0 : 0 < r) (H : ∀ x ∈ s, r ≤ abs x) :
uniform_continuous (λp:s, p.1⁻¹) :=
metric.uniform_continuous_iff.2 $ λ ε ε0,
let ⟨δ, δ0, Hδ⟩ := rat_inv_continuous_lemma abs ε0 r0 in
⟨δ, δ0, λ a b h, Hδ (H _ a.2) (H _ b.2) h⟩
lemma real.uniform_continuous_abs : uniform_continuous (abs : ℝ → ℝ) :=
metric.uniform_continuous_iff.2 $ λ ε ε0,
⟨ε, ε0, λ a b, lt_of_le_of_lt (abs_abs_sub_abs_le_abs_sub _ _)⟩
lemma rat.uniform_continuous_abs : uniform_continuous (abs : ℚ → ℚ) :=
metric.uniform_continuous_iff.2 $ λ ε ε0,
⟨ε, ε0, λ a b h, lt_of_le_of_lt
(by simpa [rat.dist_eq] using abs_abs_sub_abs_le_abs_sub _ _) h⟩
lemma real.tendsto_inv {r : ℝ} (r0 : r ≠ 0) : tendsto (λq, q⁻¹) (𝓝 r) (𝓝 r⁻¹) :=
by rw ← abs_pos at r0; exact
tendsto_of_uniform_continuous_subtype
(real.uniform_continuous_inv {x | abs r / 2 < abs x} (half_pos r0) (λ x h, le_of_lt h))
(mem_nhds_sets ((is_open_lt' (abs r / 2)).preimage continuous_abs) (half_lt_self r0))
lemma real.continuous_inv : continuous (λa:{r:ℝ // r ≠ 0}, a.val⁻¹) :=
continuous_iff_continuous_at.mpr $ assume ⟨r, hr⟩,
tendsto.comp (real.tendsto_inv hr) (continuous_iff_continuous_at.mp continuous_subtype_val _)
lemma real.continuous.inv [topological_space α] {f : α → ℝ} (h : ∀a, f a ≠ 0) (hf : continuous f) :
continuous (λa, (f a)⁻¹) :=
show continuous ((has_inv.inv ∘ @subtype.val ℝ (λr, r ≠ 0)) ∘ λa, ⟨f a, h a⟩),
from real.continuous_inv.comp (continuous_subtype_mk _ hf)
lemma real.uniform_continuous_mul_const {x : ℝ} : uniform_continuous ((*) x) :=
metric.uniform_continuous_iff.2 $ λ ε ε0, begin
cases no_top (abs x) with y xy,
have y0 := lt_of_le_of_lt (abs_nonneg _) xy,
refine ⟨_, div_pos ε0 y0, λ a b h, _⟩,
rw [real.dist_eq, ← mul_sub, abs_mul, ← mul_div_cancel' ε (ne_of_gt y0)],
exact mul_lt_mul' (le_of_lt xy) h (abs_nonneg _) y0
end
lemma real.uniform_continuous_mul (s : set (ℝ × ℝ))
{r₁ r₂ : ℝ} (H : ∀ x ∈ s, abs (x : ℝ × ℝ).1 < r₁ ∧ abs x.2 < r₂) :
uniform_continuous (λp:s, p.1.1 * p.1.2) :=
metric.uniform_continuous_iff.2 $ λ ε ε0,
let ⟨δ, δ0, Hδ⟩ := rat_mul_continuous_lemma abs ε0 in
⟨δ, δ0, λ a b h,
let ⟨h₁, h₂⟩ := max_lt_iff.1 h in Hδ (H _ a.2).1 (H _ b.2).2 h₁ h₂⟩
protected lemma real.continuous_mul : continuous (λp : ℝ × ℝ, p.1 * p.2) :=
continuous_iff_continuous_at.2 $ λ ⟨a₁, a₂⟩,
tendsto_of_uniform_continuous_subtype
(real.uniform_continuous_mul
({x | abs x < abs a₁ + 1}.prod {x | abs x < abs a₂ + 1})
(λ x, id))
(mem_nhds_sets
(((is_open_gt' (abs a₁ + 1)).preimage continuous_abs).prod
((is_open_gt' (abs a₂ + 1)).preimage continuous_abs ))
⟨lt_add_one (abs a₁), lt_add_one (abs a₂)⟩)
instance : topological_ring ℝ :=
{ continuous_mul := real.continuous_mul, ..real.topological_add_group }
instance : topological_semiring ℝ := by apply_instance -- short-circuit type class inference
lemma rat.continuous_mul : continuous (λp : ℚ × ℚ, p.1 * p.2) :=
embedding_of_rat.continuous_iff.2 $ by simp [(∘)]; exact
real.continuous_mul.comp ((continuous_of_rat.comp continuous_fst).prod_mk
(continuous_of_rat.comp continuous_snd))
instance : topological_ring ℚ :=
{ continuous_mul := rat.continuous_mul, ..rat.topological_add_group }
theorem real.ball_eq_Ioo (x ε : ℝ) : ball x ε = Ioo (x - ε) (x + ε) :=
set.ext $ λ y, by rw [mem_ball, real.dist_eq,
abs_sub_lt_iff, sub_lt_iff_lt_add', and_comm, sub_lt]; refl
theorem real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) :=
by rw [real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add,
add_sub_cancel', add_self_div_two, ← add_div,
add_assoc, add_sub_cancel'_right, add_self_div_two]
lemma real.totally_bounded_Ioo (a b : ℝ) : totally_bounded (Ioo a b) :=
metric.totally_bounded_iff.2 $ λ ε ε0, begin
rcases exists_nat_gt ((b - a) / ε) with ⟨n, ba⟩,
rw [div_lt_iff' ε0, sub_lt_iff_lt_add'] at ba,
let s := (λ i:ℕ, a + ε * i) '' {i:ℕ | i < n},
refine ⟨s, (set.finite_lt_nat _).image _, _⟩,
rintro x ⟨ax, xb⟩,
let i : ℕ := ⌊(x - a) / ε⌋.to_nat,
have : (i : ℤ) = ⌊(x - a) / ε⌋ :=
int.to_nat_of_nonneg (floor_nonneg.2 $ le_of_lt (div_pos (sub_pos.2 ax) ε0)),
simp, use i, split,
{ rw [← int.coe_nat_lt, this],
refine int.cast_lt.1 (lt_of_le_of_lt (floor_le _) _),
rw [int.cast_coe_nat, div_lt_iff' ε0, sub_lt_iff_lt_add'],
exact lt_trans xb ba },
{ rw [real.dist_eq, ← int.cast_coe_nat, this, abs_of_nonneg,
← sub_sub, sub_lt_iff_lt_add'],
{ have := lt_floor_add_one ((x - a) / ε),
rwa [div_lt_iff' ε0, mul_add, mul_one] at this },
{ have := floor_le ((x - a) / ε),
rwa [sub_nonneg, ← le_sub_iff_add_le', ← le_div_iff' ε0] } }
end
lemma real.totally_bounded_ball (x ε : ℝ) : totally_bounded (ball x ε) :=
by rw real.ball_eq_Ioo; apply real.totally_bounded_Ioo
lemma real.totally_bounded_Ico (a b : ℝ) : totally_bounded (Ico a b) :=
let ⟨c, ac⟩ := no_bot a in totally_bounded_subset
(by exact λ x ⟨h₁, h₂⟩, ⟨lt_of_lt_of_le ac h₁, h₂⟩)
(real.totally_bounded_Ioo c b)
lemma real.totally_bounded_Icc (a b : ℝ) : totally_bounded (Icc a b) :=
let ⟨c, bc⟩ := no_top b in totally_bounded_subset
(by exact λ x ⟨h₁, h₂⟩, ⟨h₁, lt_of_le_of_lt h₂ bc⟩)
(real.totally_bounded_Ico a c)
lemma rat.totally_bounded_Icc (a b : ℚ) : totally_bounded (Icc a b) :=
begin
have := totally_bounded_preimage uniform_embedding_of_rat (real.totally_bounded_Icc a b),
rwa (set.ext (λ q, _) : Icc _ _ = _), simp
end
instance : complete_space ℝ :=
begin
apply complete_of_cauchy_seq_tendsto,
intros u hu,
let c : cau_seq ℝ abs := ⟨u, cauchy_seq_iff'.1 hu⟩,
refine ⟨c.lim, λ s h, _⟩,
rcases metric.mem_nhds_iff.1 h with ⟨ε, ε0, hε⟩,
have := c.equiv_lim ε ε0,
simp only [mem_map, mem_at_top_sets, mem_set_of_eq],
refine this.imp (λ N hN n hn, hε (hN n hn))
end
section
lemma closure_of_rat_image_lt {q : ℚ} : closure ((coe:ℚ → ℝ) '' {x | q < x}) = {r | ↑q ≤ r} :=
subset.antisymm
((is_closed_ge' _).closure_subset_iff.2
(image_subset_iff.2 $ λ p h, le_of_lt $ (@rat.cast_lt ℝ _ _ _).2 h)) $
λ x hx, mem_closure_iff_nhds.2 $ λ t ht,
let ⟨ε, ε0, hε⟩ := metric.mem_nhds_iff.1 ht in
let ⟨p, h₁, h₂⟩ := exists_rat_btwn ((lt_add_iff_pos_right x).2 ε0) in
⟨_, hε (show abs _ < _,
by rwa [abs_of_nonneg (le_of_lt $ sub_pos.2 h₁), sub_lt_iff_lt_add']),
p, rat.cast_lt.1 (@lt_of_le_of_lt ℝ _ _ _ _ hx h₁), rfl⟩
/- TODO(Mario): Put these back only if needed later
lemma closure_of_rat_image_le_eq {q : ℚ} : closure ((coe:ℚ → ℝ) '' {x | q ≤ x}) = {r | ↑q ≤ r} :=
_
lemma closure_of_rat_image_le_le_eq {a b : ℚ} (hab : a ≤ b) :
closure (of_rat '' {q:ℚ | a ≤ q ∧ q ≤ b}) = {r:ℝ | of_rat a ≤ r ∧ r ≤ of_rat b} :=
_-/
lemma compact_Icc {a b : ℝ} : is_compact (Icc a b) :=
compact_of_totally_bounded_is_closed
(real.totally_bounded_Icc a b)
(is_closed_inter (is_closed_ge' a) (is_closed_le' b))
instance {a b : ℝ} : compact_space (Icc a b) :=
compact_iff_compact_space.mp compact_Icc
lemma compact_pi_Icc {ι : Type*} {a b : ι → ℝ} : is_compact (Icc a b) :=
pi_univ_Icc a b ▸ compact_univ_pi $ λ i, compact_Icc
instance compact_space_pi_Icc {ι : Type*} {a b : ι → ℝ} : compact_space (Icc a b) :=
compact_iff_compact_space.mp compact_pi_Icc
instance : proper_space ℝ :=
{ compact_ball := λx r, by rw closed_ball_Icc; apply compact_Icc }
lemma real.bounded_iff_bdd_below_bdd_above {s : set ℝ} : bounded s ↔ bdd_below s ∧ bdd_above s :=
⟨begin
assume bdd,
rcases (bounded_iff_subset_ball 0).1 bdd with ⟨r, hr⟩, -- hr : s ⊆ closed_ball 0 r
rw closed_ball_Icc at hr, -- hr : s ⊆ Icc (0 - r) (0 + r)
exact ⟨⟨-r, λy hy, by simpa using (hr hy).1⟩, ⟨r, λy hy, by simpa using (hr hy).2⟩⟩
end,
begin
rintros ⟨⟨m, hm⟩, ⟨M, hM⟩⟩,
have I : s ⊆ Icc m M := λx hx, ⟨hm hx, hM hx⟩,
have : Icc m M = closed_ball ((m+M)/2) ((M-m)/2) :=
by rw closed_ball_Icc; congr; ring,
rw this at I,
exact bounded.subset I bounded_closed_ball
end⟩
lemma real.image_Icc {f : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b) (h : continuous_on f $ Icc a b) :
f '' Icc a b = Icc (Inf $ f '' Icc a b) (Sup $ f '' Icc a b) :=
eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, is_preconnected_Icc.image f h⟩
(compact_Icc.image_of_continuous_on h)
end
instance reals_semimodule : topological_semimodule ℝ ℝ := ⟨continuous_mul⟩
instance real_maps_algebra {α : Type*} [topological_space α] :
algebra ℝ C(α, ℝ) := continuous_map_algebra
section subgroups
/-- Given a nontrivial subgroup `G ⊆ ℝ`, if `G ∩ ℝ_{>0}` has no minimum then `G` is dense. -/
lemma real.subgroup_dense_of_no_min {G : add_subgroup ℝ} {g₀ : ℝ} (g₀_in : g₀ ∈ G) (g₀_ne : g₀ ≠ 0)
(H' : ¬ ∃ a : ℝ, is_least {g : ℝ | g ∈ G ∧ 0 < g} a) :
dense (G : set ℝ) :=
begin
let G_pos := {g : ℝ | g ∈ G ∧ 0 < g},
push_neg at H',
intros x,
suffices : ∀ ε > (0 : ℝ), ∃ g ∈ G, abs (x - g) < ε,
by simpa only [real.mem_closure_iff, abs_sub],
intros ε ε_pos,
obtain ⟨g₁, g₁_in, g₁_pos⟩ : ∃ g₁ : ℝ, g₁ ∈ G ∧ 0 < g₁,
{ cases lt_or_gt_of_ne g₀_ne with Hg₀ Hg₀,
{ exact ⟨-g₀, G.neg_mem g₀_in, neg_pos.mpr Hg₀⟩ },
{ exact ⟨g₀, g₀_in, Hg₀⟩ } },
obtain ⟨a, ha⟩ : ∃ a, is_glb G_pos a :=
⟨Inf G_pos, is_glb_cInf ⟨g₁, g₁_in, g₁_pos⟩ ⟨0, λ _ hx, le_of_lt hx.2⟩⟩,
have a_notin : a ∉ G_pos,
{ intros H,
exact H' a ⟨H, ha.1⟩ },
obtain ⟨g₂, g₂_in, g₂_pos, g₂_lt⟩ : ∃ g₂ : ℝ, g₂ ∈ G ∧ 0 < g₂ ∧ g₂ < ε,
{ obtain ⟨b, hb, hb', hb''⟩ := ha.exists_between_self_add' a_notin ε_pos,
obtain ⟨c, hc, hc', hc''⟩ := ha.exists_between_self_add' a_notin (sub_pos.2 hb'),
refine ⟨b - c, G.sub_mem hb.1 hc.1, _, _⟩ ;
linarith },
refine ⟨floor (x/g₂) * g₂, _, _⟩,
{ exact add_subgroup.int_mul_mem _ g₂_in },
{ rw abs_of_nonneg (sub_floor_div_mul_nonneg x g₂_pos),
linarith [sub_floor_div_mul_lt x g₂_pos] }
end
/-- Subgroups of `ℝ` are either dense or cyclic. See `real.subgroup_dense_of_no_min` and
`subgroup_cyclic_of_min` for more precise statements. -/
lemma real.subgroup_dense_or_cyclic (G : add_subgroup ℝ) :
dense (G : set ℝ) ∨ ∃ a : ℝ, G = add_subgroup.closure {a} :=
begin
cases add_subgroup.bot_or_exists_ne_zero G with H H,
{ right,
use 0,
rw [H, add_subgroup.closure_singleton_zero] },
{ let G_pos := {g : ℝ | g ∈ G ∧ 0 < g},
by_cases H' : ∃ a, is_least G_pos a,
{ right,
rcases H' with ⟨a, ha⟩,
exact ⟨a, add_subgroup.cyclic_of_min ha⟩ },
{ left,
rcases H with ⟨g₀, g₀_in, g₀_ne⟩,
exact real.subgroup_dense_of_no_min g₀_in g₀_ne H' } }
end
end subgroups
|
db5c99292b7b9c65f68721369617b42d2e6e8be7 | 57aec6ee746bc7e3a3dd5e767e53bd95beb82f6d | /tests/lean/run/elab_cmd.lean | c835a715370cab4dc00910b2ba1cc876b697fbbf | [
"Apache-2.0"
] | permissive | collares/lean4 | 861a9269c4592bce49b71059e232ff0bfe4594cc | 52a4f535d853a2c7c7eea5fee8a4fa04c682c1ee | refs/heads/master | 1,691,419,031,324 | 1,618,678,138,000 | 1,618,678,138,000 | 358,989,750 | 0 | 0 | Apache-2.0 | 1,618,696,333,000 | 1,618,696,333,000 | null | UTF-8 | Lean | false | false | 652 | lean | import Lean
open Lean.Elab.Term
open Lean.Elab.Command
elab "∃'" b:term "," P:term : term => do
let ex ← `(Exists (fun $b => $P));
elabTerm ex none
elab "#check2" b:term : command => do
let cmd ← `(#check $b #check $b);
elabCommand cmd
#check ∃' x, x > 0
#check ∃' (x : UInt32), x > 0
#check2 10
elab "try" t:tactic : tactic => do
let t' ← `(tactic| first $t | skip);
Lean.Elab.Tactic.evalTactic t'
theorem tst (x y z : Nat) : y = z → x = x → x = y → x = z :=
by {
intro h1; intro h2; intro h3;
apply @Eq.trans;
try exact h1; -- `exact h1` fails
traceState;
try exact h3;
traceState;
try exact h1;
}
|
73a91713cf6be41c3deba387719e1fc3c3c6d778 | 69d4931b605e11ca61881fc4f66db50a0a875e39 | /src/ring_theory/finiteness.lean | 98242602f66305c1502562c1ca51d8861a9b0a35 | [
"Apache-2.0"
] | permissive | abentkamp/mathlib | d9a75d291ec09f4637b0f30cc3880ffb07549ee5 | 5360e476391508e092b5a1e5210bd0ed22dc0755 | refs/heads/master | 1,682,382,954,948 | 1,622,106,077,000 | 1,622,106,077,000 | 149,285,665 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 33,018 | 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
import group_theory.finiteness
/-!
# 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 (R : Type*) (M : Type*) [semiring R] [add_comm_monoid M] [module R M] :
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
lemma finite_def {R : Type*} {M : Type*} [semiring R] [add_comm_monoid M] [module R M] :
finite R M ↔ (⊤ : submodule R M).fg := ⟨λ h, h.1, λ h, ⟨h⟩⟩
@[priority 100] -- see Note [lower instance priority]
instance is_noetherian.finite [is_noetherian R M] : finite R M :=
⟨is_noetherian.noetherian ⊤⟩
namespace finite
open submodule set
lemma iff_add_monoid_fg {M : Type*} [add_comm_monoid M] : module.finite ℕ M ↔ add_monoid.fg M :=
⟨λ h, add_monoid.fg_def.2 $ (fg_iff_add_submonoid_fg ⊤).1 (finite_def.1 h),
λ h, finite_def.2 $ (fg_iff_add_submonoid_fg ⊤).2 (add_monoid.fg_def.1 h)⟩
lemma iff_add_group_fg {G : Type*} [add_comm_group G] : module.finite ℤ G ↔ add_group.fg G :=
⟨λ h, add_group.fg_def.2 $ (fg_iff_add_subgroup_fg ⊤).1 (finite_def.1 h),
λ h, finite_def.2 $ (fg_iff_add_subgroup_fg ⊤).2 (add_group.fg_def.1 h)⟩
variables {R M N}
lemma exists_fin [finite R M] : ∃ (n : ℕ) (s : fin n → M), span R (range s) = ⊤ :=
submodule.fg_iff_exists_fin_generating_family.mp out
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⟩⟩,
letI := hfintype,
obtain ⟨equiv⟩ := @fintype.trunc_equiv_fin ι (classical.dec_eq ι) hfintype,
replace equiv := mv_polynomial.rename_equiv R equiv,
exact ⟨fintype.card ι, 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 ⟨equiv⟩ := @fintype.trunc_equiv_fin ι (classical.dec_eq ι) _,
replace equiv := mv_polynomial.rename_equiv R equiv,
refine ⟨_, 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 ⟨equiv⟩ := @fintype.trunc_equiv_fin ι (classical.dec_eq ι) _,
replace equiv := mv_polynomial.rename_equiv R equiv,
refine ⟨fintype.card ι, 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
classical,
let n := fintype.card ι,
obtain ⟨e⟩ := fintype.trunc_equiv_fin ι,
replace e := (mv_polynomial.rename_equiv A 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 _) _,
let := mv_polynomial.sum_alg_equiv R (fin n) (fin m),
refine equiv _ this,
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
section monoid_algebra
variables {R : Type*} {M : Type*}
namespace add_monoid_algebra
open algebra add_submonoid submodule
section span
section semiring
variables [comm_semiring R] [add_monoid M]
/-- An element of `add_monoid_algebra R M` is in the subalgebra generated by its support. -/
lemma mem_adjoin_support (f : add_monoid_algebra R M) : f ∈ adjoin R (of' R M '' f.support) :=
begin
suffices : span R (of' R M '' f.support) ≤ (adjoin R (of' R M '' f.support)).to_submodule,
{ exact this (mem_span_support f) },
rw submodule.span_le,
exact subset_adjoin
end
/-- If a set `S` generates, as algebra, `add_monoid_algebra R M`, then the set of supports of
elements of `S` generates `add_monoid_algebra R M`. -/
lemma support_gen_of_gen {S : set (add_monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :
algebra.adjoin R (⋃ f ∈ S, (of' R M '' (f.support : set M))) = ⊤ :=
begin
refine le_antisymm le_top _,
rw [← hS, adjoin_le_iff],
intros f hf,
have hincl : of' R M '' f.support ⊆
⋃ (g : add_monoid_algebra R M) (H : g ∈ S), of' R M '' g.support,
{ intros s hs,
exact set.mem_bUnion_iff.2 ⟨f, ⟨hf, hs⟩⟩ },
exact adjoin_mono hincl (mem_adjoin_support f)
end
/-- If a set `S` generates, as algebra, `add_monoid_algebra R M`, then the image of the union of
the supports of elements of `S` generates `add_monoid_algebra R M`. -/
lemma support_gen_of_gen' {S : set (add_monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :
algebra.adjoin R (of' R M '' (⋃ f ∈ S, (f.support : set M))) = ⊤ :=
begin
suffices : of' R M '' (⋃ f ∈ S, (f.support : set M)) = ⋃ f ∈ S, (of' R M '' (f.support : set M)),
{ rw this,
exact support_gen_of_gen hS },
simp only [set.image_Union]
end
end semiring
section ring
variables [comm_ring R] [add_comm_monoid M]
/-- If `add_monoid_algebra R M` is of finite type, there there is a `G : finset M` such that its
image generates, as algera, `add_monoid_algebra R M`. -/
lemma exists_finset_adjoin_eq_top [h : finite_type R (add_monoid_algebra R M)] :
∃ G : finset M, algebra.adjoin R (of' R M '' G) = ⊤ :=
begin
unfreezingI { obtain ⟨S, hS⟩ := h },
letI : decidable_eq M := classical.dec_eq M,
use finset.bUnion S (λ f, f.support),
have : (finset.bUnion S (λ f, f.support) : set M) = ⋃ f ∈ S, (f.support : set M),
{ simp only [finset.set_bUnion_coe, finset.coe_bUnion] },
rw [this],
exact support_gen_of_gen' hS
end
/-- The image of an element `m : M` in `add_monoid_algebra R M` belongs the submodule generated by
`S : set M` if and only if `m ∈ S`. -/
lemma of'_mem_span [nontrivial R] {m : M} {S : set M} :
of' R M m ∈ span R (of' R M '' S) ↔ m ∈ S :=
begin
refine ⟨λ h, _, λ h, submodule.subset_span $ set.mem_image_of_mem (of R M) h⟩,
rw [of', ← finsupp.supported_eq_span_single, finsupp.mem_supported,
finsupp.support_single_ne_zero (@one_ne_zero R _ (by apply_instance))] at h,
simpa using h
end
/--If the image of an element `m : M` in `add_monoid_algebra R M` belongs the submodule generated by
the closure of some `S : set M` then `m ∈ closure S`. -/
lemma mem_closure_of_mem_span_closure [nontrivial R] {m : M} {S : set M}
(h : of' R M m ∈ span R (submonoid.closure (of' R M '' S) : set (add_monoid_algebra R M))) :
m ∈ closure S :=
begin
suffices : multiplicative.of_add m ∈ submonoid.closure (multiplicative.to_add ⁻¹' S),
{ simpa [← to_submonoid_closure] },
rw [set.image_congr' (show ∀ x, of' R M x = of R M x, from λ x, of'_eq_of x),
← monoid_hom.map_mclosure] at h,
simpa using of'_mem_span.1 h
end
end ring
end span
variables [add_comm_monoid M]
/-- If a set `S` generates an additive monoid `M`, then the image of `M` generates, as algebra,
`add_monoid_algebra R M`. -/
lemma mv_polynomial_aeval_of_surjective_of_closure [comm_semiring R] {S : set M}
(hS : closure S = ⊤) : function.surjective (mv_polynomial.aeval
(λ (s : S), of' R M ↑s) : mv_polynomial S R → add_monoid_algebra R M) :=
begin
refine λ f, induction_on f (λ m, _) _ _,
{ have : m ∈ closure S := hS.symm ▸ mem_top _,
refine closure_induction this (λ m hm, _) _ _,
{ exact ⟨mv_polynomial.X ⟨m, hm⟩, mv_polynomial.aeval_X _ _⟩ },
{ exact ⟨1, alg_hom.map_one _⟩ },
{ rintro m₁ m₂ ⟨P₁, hP₁⟩ ⟨P₂, hP₂⟩,
exact ⟨P₁ * P₂, by rw [alg_hom.map_mul, hP₁, hP₂, of_apply, of_apply, of_apply,
single_mul_single, one_mul]; refl⟩ } },
{ rintro f g ⟨P, rfl⟩ ⟨Q, rfl⟩,
exact ⟨P + Q, alg_hom.map_add _ _ _⟩ },
{ rintro r f ⟨P, rfl⟩,
exact ⟨r • P, alg_hom.map_smul _ _ _⟩ }
end
/-- If an additive monoid `M` is finitely generated then `add_monoid_algebra R M` is of finite
type. -/
instance finite_type_of_fg [comm_ring R] [h : add_monoid.fg M] :
finite_type R (add_monoid_algebra R M) :=
begin
obtain ⟨S, hS⟩ := h.out,
exact (finite_type.mv_polynomial R (S : set M)).of_surjective (mv_polynomial.aeval
(λ (s : (S : set M)), of' R M ↑s)) (mv_polynomial_aeval_of_surjective_of_closure hS)
end
/-- An additive monoid `M` is finitely generated if and only if `add_monoid_algebra R M` is of
finite type. -/
lemma finite_type_iff_fg [comm_ring R] [nontrivial R] :
finite_type R (add_monoid_algebra R M) ↔ add_monoid.fg M :=
begin
refine ⟨λ h, _, λ h, @add_monoid_algebra.finite_type_of_fg _ _ _ _ h⟩,
obtain ⟨S, hS⟩ := @exists_finset_adjoin_eq_top R M _ _ h,
refine add_monoid.fg_def.2 ⟨S, (eq_top_iff' _).2 (λ m, _)⟩,
have hm : of' R M m ∈ (adjoin R (of' R M '' ↑S)).to_submodule,
{ simp only [hS, top_to_submodule, submodule.mem_top], },
rw [adjoin_eq_span] at hm,
exact mem_closure_of_mem_span_closure hm
end
/-- If `add_monoid_algebra R M` is of finite type then `M` is finitely generated. -/
lemma fg_of_finite_type [comm_ring R] [nontrivial R] [h : finite_type R (add_monoid_algebra R M)] :
add_monoid.fg M :=
finite_type_iff_fg.1 h
/-- An additive group `G` is finitely generated if and only if `add_monoid_algebra R G` is of
finite type. -/
lemma finite_type_iff_group_fg {G : Type*} [add_comm_group G] [comm_ring R] [nontrivial R] :
finite_type R (add_monoid_algebra R G) ↔ add_group.fg G :=
by simpa [add_group.fg_iff_add_monoid.fg] using finite_type_iff_fg
end add_monoid_algebra
namespace monoid_algebra
open algebra submonoid submodule
section span
section semiring
variables [comm_semiring R] [monoid M]
/-- An element of `monoid_algebra R M` is in the subalgebra generated by its support. -/
lemma mem_adjoint_support (f : monoid_algebra R M) : f ∈ adjoin R (of R M '' f.support) :=
begin
suffices : span R (of R M '' f.support) ≤ (adjoin R (of R M '' f.support)).to_submodule,
{ exact this (mem_span_support f) },
rw submodule.span_le,
exact subset_adjoin
end
/-- If a set `S` generates, as algebra, `monoid_algebra R M`, then the set of supports of elements
of `S` generates `monoid_algebra R M`. -/
lemma support_gen_of_gen {S : set (monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :
algebra.adjoin R (⋃ f ∈ S, (of R M '' (f.support : set M))) = ⊤ :=
begin
refine le_antisymm le_top _,
rw [← hS, adjoin_le_iff],
intros f hf,
have hincl : (of R M) '' f.support ⊆
⋃ (g : monoid_algebra R M) (H : g ∈ S), of R M '' g.support,
{ intros s hs,
exact set.mem_bUnion_iff.2 ⟨f, ⟨hf, hs⟩⟩ },
exact adjoin_mono hincl (mem_adjoint_support f)
end
/-- If a set `S` generates, as algebra, `monoid_algebra R M`, then the image of the union of the
supports of elements of `S` generates `monoid_algebra R M`. -/
lemma support_gen_of_gen' {S : set (monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :
algebra.adjoin R (of R M '' (⋃ f ∈ S, (f.support : set M))) = ⊤ :=
begin
suffices : of R M '' (⋃ f ∈ S, (f.support : set M)) = ⋃ f ∈ S, (of R M '' (f.support : set M)),
{ rw this,
exact support_gen_of_gen hS },
simp only [set.image_Union]
end
end semiring
section ring
variables [comm_ring R] [comm_monoid M]
/-- If `monoid_algebra R M` is of finite type, there there is a `G : finset M` such that its image
generates, as algera, `monoid_algebra R M`. -/
lemma exists_finset_adjoin_eq_top [h :finite_type R (monoid_algebra R M)] :
∃ G : finset M, algebra.adjoin R (of R M '' G) = ⊤ :=
begin
unfreezingI { obtain ⟨S, hS⟩ := h },
letI : decidable_eq M := classical.dec_eq M,
use finset.bUnion S (λ f, f.support),
have : (finset.bUnion S (λ f, f.support) : set M) = ⋃ f ∈ S, (f.support : set M),
{ simp only [finset.set_bUnion_coe, finset.coe_bUnion] },
rw [this],
exact support_gen_of_gen' hS
end
/-- The image of an element `m : M` in `monoid_algebra R M` belongs the submodule generated by
`S : set M` if and only if `m ∈ S`. -/
lemma of_mem_span_of_iff [nontrivial R] {m : M} {S : set M} :
of R M m ∈ span R (of R M '' S) ↔ m ∈ S :=
begin
refine ⟨λ h, _, λ h, submodule.subset_span $ set.mem_image_of_mem (of R M) h⟩,
rw [of, monoid_hom.coe_mk, ← finsupp.supported_eq_span_single, finsupp.mem_supported,
finsupp.support_single_ne_zero (@one_ne_zero R _ (by apply_instance))] at h,
simpa using h
end
/--If the image of an element `m : M` in `monoid_algebra R M` belongs the submodule generated by the
closure of some `S : set M` then `m ∈ closure S`. -/
lemma mem_closure_of_mem_span_closure [nontrivial R] {m : M} {S : set M}
(h : of R M m ∈ span R (submonoid.closure (of R M '' S) : set (monoid_algebra R M))) :
m ∈ closure S :=
begin
rw ← monoid_hom.map_mclosure at h,
simpa using of_mem_span_of_iff.1 h
end
end ring
end span
variables [comm_monoid M]
/-- If a set `S` generates a monoid `M`, then the image of `M` generates, as algebra,
`monoid_algebra R M`. -/
lemma mv_polynomial_aeval_of_surjective_of_closure [comm_semiring R] {S : set M}
(hS : closure S = ⊤) : function.surjective (mv_polynomial.aeval
(λ (s : S), of R M ↑s) : mv_polynomial S R → monoid_algebra R M) :=
begin
refine λ f, induction_on f (λ m, _) _ _,
{ have : m ∈ closure S := hS.symm ▸ mem_top _,
refine closure_induction this (λ m hm, _) _ _,
{ exact ⟨mv_polynomial.X ⟨m, hm⟩, mv_polynomial.aeval_X _ _⟩ },
{ exact ⟨1, alg_hom.map_one _⟩ },
{ rintro m₁ m₂ ⟨P₁, hP₁⟩ ⟨P₂, hP₂⟩,
exact ⟨P₁ * P₂, by rw [alg_hom.map_mul, hP₁, hP₂, of_apply, of_apply, of_apply,
single_mul_single, one_mul]⟩ } },
{ rintro f g ⟨P, rfl⟩ ⟨Q, rfl⟩,
exact ⟨P + Q, alg_hom.map_add _ _ _⟩ },
{ rintro r f ⟨P, rfl⟩,
exact ⟨r • P, alg_hom.map_smul _ _ _⟩ }
end
/-- If a monoid `M` is finitely generated then `monoid_algebra R M` is of finite type. -/
instance finite_type_of_fg [comm_ring R] [monoid.fg M] : finite_type R (monoid_algebra R M) :=
add_monoid_algebra.finite_type_of_fg.equiv (to_additive_alg_equiv R M).symm
/-- A monoid `M` is finitely generated if and only if `monoid_algebra R M` is of finite type. -/
lemma finite_type_iff_fg [comm_ring R] [nontrivial R] :
finite_type R (monoid_algebra R M) ↔ monoid.fg M :=
⟨λ h, monoid.fg_iff_add_fg.2 $ add_monoid_algebra.finite_type_iff_fg.1 $ h.equiv $
to_additive_alg_equiv R M, λ h, @monoid_algebra.finite_type_of_fg _ _ _ _ h⟩
/-- If `monoid_algebra R M` is of finite type then `M` is finitely generated. -/
lemma fg_of_finite_type [comm_ring R] [nontrivial R] [h : finite_type R (monoid_algebra R M)] :
monoid.fg M :=
finite_type_iff_fg.1 h
/-- A group `G` is finitely generated if and only if `add_monoid_algebra R G` is of finite type. -/
lemma finite_type_iff_group_fg {G : Type*} [comm_group G] [comm_ring R] [nontrivial R] :
finite_type R (monoid_algebra R G) ↔ group.fg G :=
by simpa [group.fg_iff_monoid.fg] using finite_type_iff_fg
end monoid_algebra
end monoid_algebra
|
06a9bbe9cdf66220a47844414d5545e0976a98e0 | 1a61aba1b67cddccce19532a9596efe44be4285f | /tests/lean/hott/614.hlean | bd35214793bae6eacf7b911596574efc78bc1462 | [
"Apache-2.0"
] | permissive | eigengrau/lean | 07986a0f2548688c13ba36231f6cdbee82abf4c6 | f8a773be1112015e2d232661ce616d23f12874d0 | refs/heads/master | 1,610,939,198,566 | 1,441,352,386,000 | 1,441,352,494,000 | 41,903,576 | 0 | 0 | null | 1,441,352,210,000 | 1,441,352,210,000 | null | UTF-8 | Lean | false | false | 588 | hlean | import hit.circle
open circle eq int pi
attribute circle.rec [recursor]
protected definition my_decode {x : circle} (c : circle.code x) : base = x :=
begin
induction x,
{ revert c, exact power loop },
{ apply arrow_pathover_left, intro b, apply concato_eq, apply pathover_eq_r,
rewrite [power_con,transport_code_loop]},
end
protected definition my_decode' {x : circle} : circle.code x → base = x :=
begin
induction x,
{ exact power loop},
{ apply arrow_pathover_left, intro b, apply concato_eq, apply pathover_eq_r,
rewrite [power_con,transport_code_loop]},
end
|
6ad0abb78d657a6a9f3df88d2a7cee88304d0562 | aa3f8992ef7806974bc1ffd468baa0c79f4d6643 | /tests/lean/run/proj.lean | 724bdc01e851ad4e271462ce7df39b6be1493f4d | [
"Apache-2.0"
] | permissive | codyroux/lean | 7f8dff750722c5382bdd0a9a9275dc4bb2c58dd3 | 0cca265db19f7296531e339192e9b9bae4a31f8b | refs/heads/master | 1,610,909,964,159 | 1,407,084,399,000 | 1,416,857,075,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 341 | lean | import logic
inductive sigma {A : Type} (B : A → Type) :=
mk : Π (a : A), B a → sigma B
#projections sigma :: proj1 proj2
check sigma.proj1
check sigma.proj2
variables {A : Type} {B : A → Type}
variables (a : A) (b : B a)
theorem tst1 : sigma.proj1 (sigma.mk a b) = a :=
rfl
theorem tst2 : sigma.proj2 (sigma.mk a b) = b :=
rfl
|
fe1e47ff5180c64c363ab780f7dc49814ee72df4 | 367134ba5a65885e863bdc4507601606690974c1 | /src/probability_theory/integration.lean | ecdb312b921ffb8bffa3df02bc71d7ed3353ce62 | [
"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 | 5,750 | lean | /-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich
-/
import measure_theory.integration
import probability_theory.independence
/-!
# Integration in Probability Theory
Integration results for independent random variables. Specifically, for two
independent random variables X and Y over the extended non-negative
reals, `E[X * Y] = E[X] * E[Y]`, and similar results.
-/
noncomputable theory
open set measure_theory
open_locale ennreal
variables {α : Type*}
namespace probability_theory
/-- This (roughly) proves that if a random variable `f` is independent of an event `T`,
then if you restrict the random variable to `T`, then
`E[f * indicator T c 0]=E[f] * E[indicator T c 0]`. It is useful for
`lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurable_space`. -/
lemma lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator
{Mf : measurable_space α} [M : measurable_space α] {μ : measure α}
(hMf : Mf ≤ M) (c : ℝ≥0∞) {T : set α} (h_meas_T : M.measurable_set' T)
(h_ind : indep_sets Mf.measurable_set' {T} μ)
{f : α → ℝ≥0∞} (h_meas_f : @measurable α ℝ≥0∞ Mf _ f) :
∫⁻ a, f a * T.indicator (λ _, c) a ∂μ =
∫⁻ a, f a ∂μ * ∫⁻ a, T.indicator (λ _, c) a ∂μ :=
begin
revert f,
have h_mul_indicator : ∀ g, measurable g → measurable (λ a, g a * T.indicator (λ x, c) a) :=
λ g h_mg, h_mg.ennreal_mul (measurable_const.indicator h_meas_T),
apply measurable.ennreal_induction,
{ intros c' s' h_meas_s',
simp_rw [← inter_indicator_mul],
rw [lintegral_indicator _ (measurable_set.inter (hMf _ h_meas_s') (h_meas_T)),
lintegral_indicator _ (hMf _ h_meas_s'),
lintegral_indicator _ h_meas_T],
simp only [measurable_const, lintegral_const, univ_inter, lintegral_const_mul,
measurable_set.univ, measure.restrict_apply],
rw h_ind, { ring }, { apply h_meas_s' }, { simp } },
{ intros f' g h_univ h_meas_f' h_meas_g h_ind_f' h_ind_g,
have h_measM_f' := h_meas_f'.mono hMf le_rfl,
have h_measM_g := h_meas_g.mono hMf le_rfl,
simp_rw [pi.add_apply, right_distrib],
rw [lintegral_add (h_mul_indicator _ h_measM_f') (h_mul_indicator _ h_measM_g),
lintegral_add h_measM_f' h_measM_g, right_distrib, h_ind_f', h_ind_g] },
{ intros f h_meas_f h_mono_f h_ind_f,
have h_measM_f := λ n, (h_meas_f n).mono hMf le_rfl,
simp_rw [ennreal.supr_mul],
rw [lintegral_supr h_measM_f h_mono_f, lintegral_supr, ennreal.supr_mul],
{ simp_rw [← h_ind_f] },
{ exact λ n, h_mul_indicator _ (h_measM_f n) },
{ intros m n h_le a, apply ennreal.mul_le_mul _ le_rfl, apply h_mono_f h_le } },
end
/-- This (roughly) proves that if `f` and `g` are independent random variables,
then `E[f * g] = E[f] * E[g]`. However, instead of directly using the independence
of the random variables, it uses the independence of measurable spaces for the
domains of `f` and `g`. This is similar to the sigma-algebra approach to
independence. See `lintegral_mul_eq_lintegral_mul_lintegral_of_independent_fn` for
a more common variant of the product of independent variables. -/
lemma lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurable_space
{Mf : measurable_space α} {Mg : measurable_space α} [M : measurable_space α]
{μ : measure α} (hMf : Mf ≤ M) (hMg : Mg ≤ M)
(h_ind : indep Mf Mg μ)
(f g : α → ℝ≥0∞) (h_meas_f : @measurable α ℝ≥0∞ Mf _ f)
(h_meas_g : @measurable α ℝ≥0∞ Mg _ g) :
∫⁻ a, f a * g a ∂μ = ∫⁻ a, f a ∂μ * ∫⁻ a, g a ∂μ :=
begin
revert g,
have h_meas_Mg : ∀ ⦃f : α → ℝ≥0∞⦄, @measurable α ℝ≥0∞ Mg _ f → measurable f,
{ intros f' h_meas_f', apply h_meas_f'.mono hMg le_rfl },
have h_measM_f := h_meas_f.mono hMf le_rfl,
apply measurable.ennreal_induction,
{ intros c s h_s,
apply lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator hMf _ (hMg _ h_s) _ h_meas_f,
apply probability_theory.indep_sets_of_indep_sets_of_le_right h_ind,
rw singleton_subset_iff, apply h_s },
{ intros f' g h_univ h_measMg_f' h_measMg_g h_ind_f' h_ind_g',
have h_measM_f' := h_meas_Mg h_measMg_f',
have h_measM_g := h_meas_Mg h_measMg_g,
simp_rw [pi.add_apply, left_distrib],
rw [lintegral_add h_measM_f' h_measM_g,
lintegral_add (measurable.ennreal_mul h_measM_f h_measM_f')
(measurable.ennreal_mul h_measM_f h_measM_g),
left_distrib, h_ind_f', h_ind_g'] },
{ intros f' h_meas_f' h_mono_f' h_ind_f',
have h_measM_f' := λ n, h_meas_Mg (h_meas_f' n),
simp_rw [ennreal.mul_supr],
rw [lintegral_supr, lintegral_supr h_measM_f' h_mono_f', ennreal.mul_supr],
{ simp_rw [← h_ind_f'] },
{ apply λ (n : ℕ), measurable.ennreal_mul h_measM_f (h_measM_f' n) },
{ apply λ (n : ℕ) (m : ℕ) (h_le : n ≤ m) a, ennreal.mul_le_mul le_rfl
(h_mono_f' h_le a) } }
end
/-- This proves that if `f` and `g` are independent random variables,
then `E[f * g] = E[f] * E[g]`. -/
lemma lintegral_mul_eq_lintegral_mul_lintegral_of_indep_fun [M : measurable_space α]
(μ : measure α) (f g : α → ℝ≥0∞) (h_meas_f : measurable f) (h_meas_g : measurable g)
(h_indep_fun : indep_fun (borel ennreal) (borel ennreal) f g μ) :
∫⁻ (a : α), (f * g) a ∂μ = ∫⁻ (a : α), f a ∂μ * ∫⁻ (a : α), g a ∂μ :=
begin
apply lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurable_space
(measurable_iff_comap_le.1 h_meas_f) (measurable_iff_comap_le.1 h_meas_g),
apply h_indep_fun,
repeat { apply measurable.of_comap_le le_rfl },
end
end probability_theory
|
560b85de160befc201ad745fc1c9ff1a170ab325 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/analysis/normed_space/add_torsor_bases.lean | 5e162d8f02c820269a316023e0706e0779060608 | [
"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 | 7,168 | lean | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import analysis.normed_space.finite_dimension
import analysis.calculus.affine_map
import analysis.convex.combination
import linear_algebra.affine_space.finite_dimensional
/-!
# Bases in normed affine spaces.
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file contains results about bases in normed affine spaces.
## Main definitions:
* `continuous_barycentric_coord`
* `is_open_map_barycentric_coord`
* `affine_basis.interior_convex_hull`
* `exists_subset_affine_independent_span_eq_top_of_open`
* `interior_convex_hull_nonempty_iff_affine_span_eq_top`
-/
section barycentric
variables {ι 𝕜 E P : Type*} [nontrivially_normed_field 𝕜] [complete_space 𝕜]
variables [normed_add_comm_group E] [normed_space 𝕜 E]
variables [metric_space P] [normed_add_torsor E P]
include E
lemma is_open_map_barycentric_coord [nontrivial ι] (b : affine_basis ι 𝕜 P) (i : ι) :
is_open_map (b.coord i) :=
affine_map.is_open_map_linear_iff.mp $ (b.coord i).linear.is_open_map_of_finite_dimensional $
(b.coord i).linear_surjective_iff.mpr (b.surjective_coord i)
variables [finite_dimensional 𝕜 E] (b : affine_basis ι 𝕜 P)
@[continuity]
lemma continuous_barycentric_coord (i : ι) : continuous (b.coord i) :=
(b.coord i).continuous_of_finite_dimensional
lemma smooth_barycentric_coord (b : affine_basis ι 𝕜 E) (i : ι) : cont_diff 𝕜 ⊤ (b.coord i) :=
(⟨b.coord i, continuous_barycentric_coord b i⟩ : E →A[𝕜] 𝕜).cont_diff
end barycentric
open set
/-- Given a finite-dimensional normed real vector space, the interior of the convex hull of an
affine basis is the set of points whose barycentric coordinates are strictly positive with respect
to this basis.
TODO Restate this result for affine spaces (instead of vector spaces) once the definition of
convexity is generalised to this setting. -/
lemma affine_basis.interior_convex_hull {ι E : Type*} [finite ι] [normed_add_comm_group E]
[normed_space ℝ E] (b : affine_basis ι ℝ E) :
interior (convex_hull ℝ (range b)) = {x | ∀ i, 0 < b.coord i x} :=
begin
casesI subsingleton_or_nontrivial ι,
{ -- The zero-dimensional case.
have : range b = univ,
from affine_subspace.eq_univ_of_subsingleton_span_eq_top (subsingleton_range _) b.tot,
simp [this] },
{ -- The positive-dimensional case.
haveI : finite_dimensional ℝ E := b.finite_dimensional,
have : convex_hull ℝ (range b) = ⋂ i, (b.coord i)⁻¹' Ici 0,
{ rw [b.convex_hull_eq_nonneg_coord, set_of_forall], refl },
ext,
simp only [this, interior_Inter, ← is_open_map.preimage_interior_eq_interior_preimage
(is_open_map_barycentric_coord b _) (continuous_barycentric_coord b _),
interior_Ici, mem_Inter, mem_set_of_eq, mem_Ioi, mem_preimage], },
end
variables {V P : Type*} [normed_add_comm_group V] [normed_space ℝ V] [metric_space P]
[normed_add_torsor V P]
include V
open affine_map
/-- Given a set `s` of affine-independent points belonging to an open set `u`, we may extend `s` to
an affine basis, all of whose elements belong to `u`. -/
lemma is_open.exists_between_affine_independent_span_eq_top {s u : set P} (hu : is_open u)
(hsu : s ⊆ u) (hne : s.nonempty) (h : affine_independent ℝ (coe : s → P)) :
∃ t : set P, s ⊆ t ∧ t ⊆ u ∧ affine_independent ℝ (coe : t → P) ∧ affine_span ℝ t = ⊤ :=
begin
obtain ⟨q, hq⟩ := hne,
obtain ⟨ε, ε0, hεu⟩ := metric.nhds_basis_closed_ball.mem_iff.1 (hu.mem_nhds $ hsu hq),
obtain ⟨t, ht₁, ht₂, ht₃⟩ := exists_subset_affine_independent_affine_span_eq_top h,
let f : P → P := λ y, line_map q y (ε / dist y q),
have hf : ∀ y, f y ∈ u,
{ refine λ y, hεu _,
simp only [f],
rw [metric.mem_closed_ball, line_map_apply, dist_vadd_left, norm_smul, real.norm_eq_abs,
dist_eq_norm_vsub V y q, abs_div, abs_of_pos ε0, abs_of_nonneg (norm_nonneg _), div_mul_comm],
exact mul_le_of_le_one_left ε0.le (div_self_le_one _) },
have hεyq : ∀ y ∉ s, ε / dist y q ≠ 0,
from λ y hy, div_ne_zero ε0.ne' (dist_ne_zero.2 (ne_of_mem_of_not_mem hq hy).symm),
classical,
let w : t → ℝˣ := λ p, if hp : (p : P) ∈ s then 1 else units.mk0 _ (hεyq ↑p hp),
refine ⟨set.range (λ (p : t), line_map q p (w p : ℝ)), _, _, _, _⟩,
{ intros p hp, use ⟨p, ht₁ hp⟩, simp [w, hp], },
{ rintros y ⟨⟨p, hp⟩, rfl⟩,
by_cases hps : p ∈ s; simp only [w, hps, line_map_apply_one, units.coe_mk0, dif_neg, dif_pos,
not_false_iff, units.coe_one, subtype.coe_mk];
[exact hsu hps, exact hf p], },
{ exact (ht₂.units_line_map ⟨q, ht₁ hq⟩ w).range, },
{ rw [affine_span_eq_affine_span_line_map_units (ht₁ hq) w, ht₃], },
end
lemma is_open.exists_subset_affine_independent_span_eq_top {u : set P} (hu : is_open u)
(hne : u.nonempty) :
∃ s ⊆ u, affine_independent ℝ (coe : s → P) ∧ affine_span ℝ s = ⊤ :=
begin
rcases hne with ⟨x, hx⟩,
rcases hu.exists_between_affine_independent_span_eq_top (singleton_subset_iff.mpr hx)
(singleton_nonempty _) (affine_independent_of_subsingleton _ _) with ⟨s, -, hsu, hs⟩,
exact ⟨s, hsu, hs⟩
end
/-- The affine span of a nonempty open set is `⊤`. -/
lemma is_open.affine_span_eq_top {u : set P} (hu : is_open u) (hne : u.nonempty) :
affine_span ℝ u = ⊤ :=
let ⟨s, hsu, hs, hs'⟩ := hu.exists_subset_affine_independent_span_eq_top hne
in top_unique $ hs' ▸ affine_span_mono _ hsu
lemma affine_span_eq_top_of_nonempty_interior {s : set V}
(hs : (interior $ convex_hull ℝ s).nonempty) :
affine_span ℝ s = ⊤ :=
top_unique $ is_open_interior.affine_span_eq_top hs ▸
(affine_span_mono _ interior_subset).trans_eq (affine_span_convex_hull _)
lemma affine_basis.centroid_mem_interior_convex_hull {ι} [fintype ι] (b : affine_basis ι ℝ V) :
finset.univ.centroid ℝ b ∈ interior (convex_hull ℝ (range b)) :=
begin
haveI := b.nonempty,
simp only [b.interior_convex_hull, mem_set_of_eq, b.coord_apply_centroid (finset.mem_univ _),
inv_pos, nat.cast_pos, finset.card_pos, finset.univ_nonempty, forall_true_iff]
end
lemma interior_convex_hull_nonempty_iff_affine_span_eq_top [finite_dimensional ℝ V] {s : set V} :
(interior (convex_hull ℝ s)).nonempty ↔ affine_span ℝ s = ⊤ :=
begin
refine ⟨affine_span_eq_top_of_nonempty_interior, λ h, _⟩,
obtain ⟨t, hts, b, hb⟩ := affine_basis.exists_affine_subbasis h,
suffices : (interior (convex_hull ℝ (range b))).nonempty,
{ rw [hb, subtype.range_coe_subtype, set_of_mem_eq] at this,
refine this.mono _,
mono* },
lift t to finset V using b.finite_set,
exact ⟨_, b.centroid_mem_interior_convex_hull⟩
end
lemma convex.interior_nonempty_iff_affine_span_eq_top [finite_dimensional ℝ V] {s : set V}
(hs : convex ℝ s) : (interior s).nonempty ↔ affine_span ℝ s = ⊤ :=
by rw [← interior_convex_hull_nonempty_iff_affine_span_eq_top, hs.convex_hull_eq]
|
5b81d0f24ba46b24134bfb0f537a3381500d3531 | 9be442d9ec2fcf442516ed6e9e1660aa9071b7bd | /tests/lean/calcErrors.lean | c83e2edc9eb34744d6d3f0e92328669e96eb4a65 | [
"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 | 1,566 | lean | theorem ex1 (a b c : Nat) (h₁ : a = b) (h₂ : b = c) : a + b = 0 + c + b :=
calc a + b = b + b := by rw [h₁]
_ = 0 + c + b := rfl
theorem ex2 (a b c : Nat) (h₁ : a = b) (h₂ : b = c) : a + b = 0 + c + b :=
calc a + b = b + b := by rw [h₁]
0 + c + b = 0 + c + b := rfl
theorem ex3 (a b c : Nat) (h₁ : a = b) (h₂ : b = c) : a + b = 0 + c + b :=
calc a + b = b + b := by rw [h₁]
_ = 0 + b + b := by rw [Nat.zero_add]
_ = 0 + c + b := by rw [h₂]
theorem ex4 (p : Nat → Prop) (a b : Nat) (h₁ : p a) (h₂ : p b) : p c :=
calc p a := h₁
_ := h₂
theorem ex5 (p : Nat → Nat → Prop) (a b : Nat) (h₁ : p a b) (h₂ : p b c) : p a c :=
calc p a b := h₁
p _ c := h₂
infix:50 " ≅ " => HEq
instance {α β γ} : Trans (· ≅ · : α → β → Prop) (· ≅ · : β → γ → Prop) (· ≅ · : α → γ → Prop) where
trans h₁ h₂ := HEq.trans h₁ h₂
theorem ex6 {a : α} {b : β} {c : γ} (h₁ : HEq a b) (h₂ : b ≅ c) : a ≅ c :=
calc a ≅ b := h₁
_ ≅ c := h₂ -- Error because the last two arguments of HEq are not explicit
abbrev HEqRel {α β} (a : α) (b : β) := HEq a b
infix:50 "===" => HEqRel
instance {α β γ} : Trans (HEqRel : α → β → Prop) (HEqRel : β → γ → Prop) (HEqRel : α → γ → Prop) where
trans h₁ h₂ := HEq.trans h₁ h₂
theorem ex7 {a : α} {b : β} {c : γ} (h₁ : a ≅ b) (h₂ : b ≅ c) : a ≅ c :=
calc a === b := h₁
_ === c := h₂
|
1c07a78c9dda155704f41f29891a690f284f32be | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/algebra/homology/additive.lean | 3eb6121fe66855fcdd594290903ec71a140aa0e7 | [
"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 | 8,810 | lean | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import algebra.homology.homology
import algebra.homology.single
import category_theory.preadditive.additive_functor
/-!
# Homology is an additive functor
When `V` is preadditive, `homological_complex V c` is also preadditive,
and `homology_functor` is additive.
TODO: similarly for `R`-linear.
-/
universes v u
open_locale classical
noncomputable theory
open category_theory category_theory.limits homological_complex
variables {ι : Type*}
variables {V : Type u} [category.{v} V] [preadditive V]
variables {c : complex_shape ι} {C D E : homological_complex V c}
variables (f g : C ⟶ D) (h k : D ⟶ E) (i : ι)
namespace homological_complex
instance : has_zero (C ⟶ D) := ⟨{ f := λ i, 0 }⟩
instance : has_add (C ⟶ D) := ⟨λ f g, { f := λ i, f.f i + g.f i, }⟩
instance : has_neg (C ⟶ D) := ⟨λ f, { f := λ i, -(f.f i), }⟩
instance : has_sub (C ⟶ D) := ⟨λ f g, { f := λ i, f.f i - g.f i, }⟩
@[simp] lemma zero_f_apply (i : ι) : (0 : C ⟶ D).f i = 0 := rfl
@[simp] lemma add_f_apply (f g : C ⟶ D) (i : ι) : (f + g).f i = f.f i + g.f i := rfl
@[simp] lemma neg_f_apply (f : C ⟶ D) (i : ι) : (-f).f i = -(f.f i) := rfl
@[simp] lemma sub_f_apply (f g : C ⟶ D) (i : ι) : (f - g).f i = f.f i - g.f i := rfl
/- TODO(jmc/Scott): the instance below doesn't have the correct defeq for `nsmul` and `gsmul`.
We should generalize `function.injective.add_comm_group` and friends.
For the `R`-linear version, it will be very convenient to have
a good definition of `nsmul` and `gsmul` that matches `smul`. -/
instance : add_comm_group (C ⟶ D) :=
function.injective.add_comm_group hom.f
homological_complex.hom_f_injective (by tidy) (by tidy) (by tidy) (by tidy)
instance : preadditive (homological_complex V c) := {}
/-- The `i`-th component of a chain map, as an additive map from chain maps to morphisms. -/
@[simps]
def hom.f_add_monoid_hom {C₁ C₂ : homological_complex V c} (i : ι) :
(C₁ ⟶ C₂) →+ (C₁.X i ⟶ C₂.X i) :=
add_monoid_hom.mk' (λ f, hom.f f i) (λ _ _, rfl)
end homological_complex
namespace homological_complex
instance eval_additive (i : ι) : (eval V c i).additive := {}
variables [has_zero_object V]
instance cycles_additive [has_equalizers V] : (cycles_functor V c i).additive := {}
variables [has_images V] [has_image_maps V]
instance boundaries_additive : (boundaries_functor V c i).additive := {}
variables [has_equalizers V] [has_cokernels V]
instance homology_additive : (homology_functor V c i).additive :=
{ map_zero' := λ C D, begin
dsimp [homology_functor],
ext,
simp only [limits.cokernel.π_desc, limits.comp_zero, homology.π_map],
convert zero_comp,
ext,
simp,
end,
map_add' := λ C D f g, begin
dsimp [homology_functor],
ext,
simp only [homology.π_map, preadditive.comp_add, ←preadditive.add_comp],
congr,
ext, simp,
end }
end homological_complex
namespace category_theory
variables {W : Type*} [category W] [preadditive W]
/--
An additive functor induces a functor between homological complexes.
This is sometimes called the "prolongation".
-/
@[simps]
def functor.map_homological_complex (F : V ⥤ W) [F.additive] (c : complex_shape ι) :
homological_complex V c ⥤ homological_complex W c :=
{ obj := λ C,
{ X := λ i, F.obj (C.X i),
d := λ i j, F.map (C.d i j),
shape' := λ i j w, by rw [C.shape _ _ w, F.map_zero],
d_comp_d' := λ i j k _ _, by rw [←F.map_comp, C.d_comp_d, F.map_zero], },
map := λ C D f,
{ f := λ i, F.map (f.f i),
comm' := λ i j h, by { dsimp, rw [←F.map_comp, ←F.map_comp, f.comm], }, }, }.
instance functor.map_homogical_complex_additive
(F : V ⥤ W) [F.additive] (c : complex_shape ι) : (F.map_homological_complex c).additive := {}
/--
A natural transformation between functors induces a natural transformation
between those functors applied to homological complexes.
-/
@[simps]
def nat_trans.map_homological_complex {F G : V ⥤ W} [F.additive] [G.additive]
(α : F ⟶ G) (c : complex_shape ι) : F.map_homological_complex c ⟶ G.map_homological_complex c :=
{ app := λ C, { f := λ i, α.app _, }, }
@[simp] lemma nat_trans.map_homological_complex_id (c : complex_shape ι) (F : V ⥤ W) [F.additive] :
nat_trans.map_homological_complex (𝟙 F) c = 𝟙 (F.map_homological_complex c) :=
by tidy
@[simp] lemma nat_trans.map_homological_complex_comp (c : complex_shape ι)
{F G H : V ⥤ W} [F.additive] [G.additive] [H.additive]
(α : F ⟶ G) (β : G ⟶ H):
nat_trans.map_homological_complex (α ≫ β) c =
nat_trans.map_homological_complex α c ≫ nat_trans.map_homological_complex β c :=
by tidy
@[simp, reassoc] lemma nat_trans.map_homological_complex_naturality {c : complex_shape ι}
{F G : V ⥤ W} [F.additive] [G.additive] (α : F ⟶ G) {C D : homological_complex V c} (f : C ⟶ D) :
(F.map_homological_complex c).map f ≫ (nat_trans.map_homological_complex α c).app D =
(nat_trans.map_homological_complex α c).app C ≫ (G.map_homological_complex c).map f :=
by tidy
end category_theory
variables [has_zero_object V] {W : Type*} [category W] [preadditive W] [has_zero_object W]
namespace homological_complex
/--
Turning an object into a complex supported at `j` then applying a functor is
the same as applying the functor then forming the complex.
-/
def single_map_homological_complex (F : V ⥤ W) [F.additive] (c : complex_shape ι) (j : ι):
single V c j ⋙ F.map_homological_complex _ ≅ F ⋙ single W c j :=
nat_iso.of_components (λ X,
{ hom := { f := λ i, if h : i = j then
eq_to_hom (by simp [h])
else
0, },
inv := { f := λ i, if h : i = j then
eq_to_hom (by simp [h])
else
0, },
hom_inv_id' := begin
ext i,
dsimp,
split_ifs with h,
{ simp [h] },
{ rw [zero_comp, if_neg h],
exact (zero_of_source_iso_zero _ F.map_zero_object).symm, },
end,
inv_hom_id' := begin
ext i,
dsimp,
split_ifs with h,
{ simp [h] },
{ rw [zero_comp, if_neg h],
simp, },
end, })
(λ X Y f, begin
ext i,
dsimp,
split_ifs with h; simp [h],
end).
variables (F : V ⥤ W) [functor.additive F] (c)
@[simp] lemma single_map_homological_complex_hom_app_self (j : ι) (X : V) :
((single_map_homological_complex F c j).hom.app X).f j = eq_to_hom (by simp) :=
by simp [single_map_homological_complex]
@[simp] lemma single_map_homological_complex_hom_app_ne
{i j : ι} (h : i ≠ j) (X : V) :
((single_map_homological_complex F c j).hom.app X).f i = 0 :=
by simp [single_map_homological_complex, h]
@[simp] lemma single_map_homological_complex_inv_app_self (j : ι) (X : V) :
((single_map_homological_complex F c j).inv.app X).f j = eq_to_hom (by simp) :=
by simp [single_map_homological_complex]
@[simp] lemma single_map_homological_complex_inv_app_ne
{i j : ι} (h : i ≠ j) (X : V):
((single_map_homological_complex F c j).inv.app X).f i = 0 :=
by simp [single_map_homological_complex, h]
end homological_complex
namespace chain_complex
-- TODO: dualize to cochain complexes
/--
Turning an object into a chain complex supported at zero then applying a functor is
the same as applying the functor then forming the complex.
-/
def single₀_map_homological_complex (F : V ⥤ W) [F.additive] :
single₀ V ⋙ F.map_homological_complex _ ≅ F ⋙ single₀ W :=
nat_iso.of_components (λ X,
{ hom := { f := λ i, match i with
| 0 := 𝟙 _
| (i+1) := F.map_zero_object.hom
end, },
inv := { f := λ i, match i with
| 0 := 𝟙 _
| (i+1) := F.map_zero_object.inv
end, },
hom_inv_id' := begin
ext (_|i),
{ unfold_aux, simp, },
{ unfold_aux,
dsimp,
simp only [comp_f, id_f, zero_comp],
exact (zero_of_source_iso_zero _ F.map_zero_object).symm, }
end,
inv_hom_id' := by { ext (_|i); { unfold_aux, dsimp, simp, }, }, })
(λ X Y f, by { ext (_|i); { unfold_aux, dsimp, simp, }, }).
@[simp] lemma single₀_map_homological_complex_hom_app_zero (F : V ⥤ W) [F.additive] (X : V) :
((single₀_map_homological_complex F).hom.app X).f 0 = 𝟙 _ := rfl
@[simp] lemma single₀_map_homological_complex_hom_app_succ
(F : V ⥤ W) [F.additive] (X : V) (n : ℕ) :
((single₀_map_homological_complex F).hom.app X).f (n+1) = 0 := rfl
@[simp] lemma single₀_map_homological_complex_inv_app_zero (F : V ⥤ W) [F.additive] (X : V) :
((single₀_map_homological_complex F).inv.app X).f 0 = 𝟙 _ := rfl
@[simp] lemma single₀_map_homological_complex_inv_app_succ
(F : V ⥤ W) [F.additive] (X : V) (n : ℕ) :
((single₀_map_homological_complex F).inv.app X).f (n+1) = 0 := rfl
end chain_complex
|
50305d9ca023b7ea223f011e4725ff3d1e2068a2 | 96e44fc78cabfc9d646dc37d0e756189b6b79181 | /library/init/meta/tactic.lean | d4803e27b54bcdd9f99b39fb6a5f011c65923364 | [
"Apache-2.0"
] | permissive | TwoFX/lean | 23c73c10a340f5a381f6abf27a27f53f1fb7e2e3 | 7e3f336714055869690b7309b6bb651fbc67e76e | refs/heads/master | 1,612,504,908,183 | 1,594,641,622,000 | 1,594,641,622,000 | 243,750,847 | 0 | 0 | Apache-2.0 | 1,582,890,661,000 | 1,582,890,661,000 | null | UTF-8 | Lean | false | false | 72,322 | 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
-/
prelude
import init.function init.data.option.basic init.util
import init.control.combinators init.control.monad init.control.alternative init.control.monad_fail
import init.data.nat.div init.meta.exceptional init.meta.format init.meta.environment
import init.meta.pexpr init.data.repr init.data.string.basic init.meta.interaction_monad
open native
meta constant tactic_state : Type
universes u v
namespace tactic_state
meta constant env : tactic_state → environment
/-- Format the given tactic state. If `target_lhs_only` is true and the target
is of the form `lhs ~ rhs`, where `~` is a simplification relation,
then only the `lhs` is displayed.
Remark: the parameter `target_lhs_only` is a temporary hack used to implement
the `conv` monad. It will be removed in the future. -/
meta constant to_format (s : tactic_state) (target_lhs_only : bool := ff) : format
/-- Format expression with respect to the main goal in the tactic state.
If the tactic state does not contain any goals, then format expression
using an empty local context. -/
meta constant format_expr : tactic_state → expr → format
meta constant get_options : tactic_state → options
meta constant set_options : tactic_state → options → tactic_state
end tactic_state
meta instance : has_to_format tactic_state :=
⟨tactic_state.to_format⟩
meta instance : has_to_string tactic_state :=
⟨λ s, (to_fmt s).to_string s.get_options⟩
/-- `tactic` is the monad for building tactics.
You use this to:
- View and modify the local goals and hypotheses in the prover's state.
- Invoke type checking and elaboration of terms.
- View and modify the environment.
- Build new tactics out of existing ones such as `simp` and `rewrite`.
-/
@[reducible] meta def tactic := interaction_monad tactic_state
@[reducible] meta def tactic_result := interaction_monad.result tactic_state
namespace tactic
export interaction_monad (hiding failed fail)
/-- Cause the tactic to fail with no error message. -/
meta def failed {α : Type} : tactic α := interaction_monad.failed
meta def fail {α : Type u} {β : Type v} [has_to_format β] (msg : β) : tactic α :=
interaction_monad.fail msg
end tactic
namespace tactic_result
export interaction_monad.result
end tactic_result
open tactic
open tactic_result
infixl ` >>=[tactic] `:2 := interaction_monad_bind
infixl ` >>[tactic] `:2 := interaction_monad_seq
meta instance : alternative tactic :=
{ failure := @interaction_monad.failed _,
orelse := @interaction_monad_orelse _,
..interaction_monad.monad }
meta def {u₁ u₂} tactic.up {α : Type u₂} (t : tactic α) : tactic (ulift.{u₁} α) :=
λ s, match t s with
| success a s' := success (ulift.up a) s'
| exception t ref s := exception t ref s
end
meta def {u₁ u₂} tactic.down {α : Type u₂} (t : tactic (ulift.{u₁} α)) : tactic α :=
λ s, match t s with
| success (ulift.up a) s' := success a s'
| exception t ref s := exception t ref s
end
namespace interactive
/-- Typeclass for custom interaction monads, which provides
the information required to convert an interactive-mode
construction to a `tactic` which can actually be executed.
Given a `[monad m]`, `execute_with` explains how to turn a `begin ... end`
block, or a `by ...` statement into a `tactic α` which can actually be
executed. The `inhabited` first argument facilitates the passing of an
optional configuration parameter `config`, using the syntax:
```
begin [custom_monad] with config,
...
end
```
-/
meta class executor (m : Type → Type u) [monad m] :=
(config_type : Type)
[inhabited : inhabited config_type]
(execute_with : config_type → m unit → tactic unit)
attribute [inline] executor.execute_with
@[inline]
meta def executor.execute_explicit (m : Type → Type u)
[monad m] [e : executor m] : m unit → tactic unit :=
executor.execute_with e.inhabited.default
@[inline]
meta def executor.execute_with_explicit (m : Type → Type u)
[monad m] [executor m] : executor.config_type m → m unit → tactic unit :=
executor.execute_with
/-- Default `executor` instance for `tactic`s themselves -/
meta instance : executor tactic :=
{ config_type := unit,
inhabited := ⟨()⟩,
execute_with := λ _, id }
end interactive
namespace tactic
open interaction_monad.result
variables {α : Type u}
/-- Does nothing. -/
meta def skip : tactic unit :=
success ()
/--
`try_core t` acts like `t`, but succeeds even if `t` fails. It returns the
result of `t` if `t` succeeded and `none` otherwise.
-/
meta def try_core (t : tactic α) : tactic (option α) := λ s,
match t s with
| (exception _ _ _) := success none s
| (success a s') := success (some a) s'
end
/--
`try t` acts like `t`, but succeeds even if `t` fails.
-/
meta def try (t : tactic α) : tactic unit := λ s,
match t s with
| (exception _ _ _) := success () s
| (success _ s') := success () s'
end
meta def try_lst : list (tactic unit) → tactic unit
| [] := failed
| (tac :: tacs) := λ s,
match tac s with
| success _ s' := try (try_lst tacs) s'
| exception e p s' :=
match try_lst tacs s' with
| exception _ _ _ := exception e p s'
| r := r
end
end
/--
`fail_if_success t` acts like `t`, but succeeds if `t` fails and fails if `t`
succeeds. Changes made by `t` to the `tactic_state` are preserved only if `t`
succeeds.
-/
meta def fail_if_success {α : Type u} (t : tactic α) : tactic unit := λ s,
match (t s) with
| (success a s) := mk_exception "fail_if_success combinator failed, given tactic succeeded" none s
| (exception _ _ _) := success () s
end
/--
`success_if_fail t` acts like `t`, but succeeds if `t` fails and fails if `t`
succeeds. Changes made by `t` to the `tactic_state` are preserved only if `t`
succeeds.
-/
meta def success_if_fail {α : Type u} (t : tactic α) : tactic unit := λ s,
match t s with
| (success a s) :=
mk_exception "success_if_fail combinator failed, given tactic succeeded" none s
| (exception _ _ _) := success () s
end
open nat
/--
`iterate_at_most n t` iterates `t` `n` times or until `t` fails, returning the
result of each successful iteration.
-/
meta def iterate_at_most : nat → tactic α → tactic (list α)
| 0 t := pure []
| (n + 1) t := do
(some a) ← try_core t | pure [],
as ← iterate_at_most n t,
pure $ a :: as
/--
`iterate_at_most' n t` repeats `t` `n` times or until `t` fails.
-/
meta def iterate_at_most' : nat → tactic unit → tactic unit
| 0 t := skip
| (succ n) t := do
(some _) ← try_core t | skip,
iterate_at_most' n t
/--
`iterate_exactly n t` iterates `t` `n` times, returning the result of
each iteration. If any iteration fails, the whole tactic fails.
-/
meta def iterate_exactly : nat → tactic α → tactic (list α)
| 0 t := pure []
| (n + 1) t := do
a ← t,
as ← iterate_exactly n t,
pure $ a ::as
/--
`iterate_exactly' n t` executes `t` `n` times. If any iteration fails, the whole
tactic fails.
-/
meta def iterate_exactly' : nat → tactic unit → tactic unit
| 0 t := skip
| (n + 1) t := t *> iterate_exactly' n t
/--
`iterate t` repeats `t` 100.000 times or until `t` fails, returning the
result of each iteration.
-/
meta def iterate : tactic α → tactic (list α) :=
iterate_at_most 100000
/--
`iterate' t` repeats `t` 100.000 times or until `t` fails.
-/
meta def iterate' : tactic unit → tactic unit :=
iterate_at_most' 100000
meta def returnopt (e : option α) : tactic α :=
λ s, match e with
| (some a) := success a s
| none := mk_exception "failed" none s
end
meta instance opt_to_tac : has_coe (option α) (tactic α) :=
⟨returnopt⟩
/-- Decorate t's exceptions with msg. -/
meta def decorate_ex (msg : format) (t : tactic α) : tactic α :=
λ s, result.cases_on (t s)
success
(λ opt_thunk,
match opt_thunk with
| some e := exception (some (λ u, msg ++ format.nest 2 (format.line ++ e u)))
| none := exception none
end)
/-- Set the tactic_state. -/
@[inline] meta def write (s' : tactic_state) : tactic unit :=
λ s, success () s'
/-- Get the tactic_state. -/
@[inline] meta def read : tactic tactic_state :=
λ s, success s s
meta def get_options : tactic options :=
do s ← read, return s.get_options
meta def set_options (o : options) : tactic unit :=
do s ← read, write (s.set_options o)
meta def save_options {α : Type} (t : tactic α) : tactic α :=
do o ← get_options,
a ← t,
set_options o,
return a
meta def returnex {α : Type} (e : exceptional α) : tactic α :=
λ s, match e with
| exceptional.success a := success a s
| exceptional.exception f :=
match get_options s with
| success opt _ := exception (some (λ u, f opt)) none s
| exception _ _ _ := exception (some (λ u, f options.mk)) none s
end
end
meta instance ex_to_tac {α : Type} : has_coe (exceptional α) (tactic α) :=
⟨returnex⟩
end tactic
meta def tactic_format_expr (e : expr) : tactic format :=
do s ← tactic.read, return (tactic_state.format_expr s e)
meta class has_to_tactic_format (α : Type u) :=
(to_tactic_format : α → tactic format)
meta instance : has_to_tactic_format expr :=
⟨tactic_format_expr⟩
meta def tactic.pp {α : Type u} [has_to_tactic_format α] : α → tactic format :=
has_to_tactic_format.to_tactic_format
open tactic format
meta instance {α : Type u} [has_to_tactic_format α] : has_to_tactic_format (list α) :=
⟨λ l, to_fmt <$> l.mmap pp⟩
meta instance (α : Type u) (β : Type v) [has_to_tactic_format α] [has_to_tactic_format β] :
has_to_tactic_format (α × β) :=
⟨λ ⟨a, b⟩, to_fmt <$> (prod.mk <$> pp a <*> pp b)⟩
meta def option_to_tactic_format {α : Type u} [has_to_tactic_format α] : option α → tactic format
| (some a) := do fa ← pp a, return (to_fmt "(some " ++ fa ++ ")")
| none := return "none"
meta instance {α : Type u} [has_to_tactic_format α] : has_to_tactic_format (option α) :=
⟨option_to_tactic_format⟩
meta instance {α} (a : α) : has_to_tactic_format (reflected a) :=
⟨λ h, pp h.to_expr⟩
@[priority 10] meta instance has_to_format_to_has_to_tactic_format (α : Type) [has_to_format α] : has_to_tactic_format α :=
⟨(λ x, return x) ∘ to_fmt⟩
namespace tactic
open tactic_state
meta def get_env : tactic environment :=
do s ← read,
return $ env s
meta def get_decl (n : name) : tactic declaration :=
do s ← read,
(env s).get n
meta constant get_trace_msg_pos : tactic pos
meta def trace {α : Type u} [has_to_tactic_format α] (a : α) : tactic unit :=
do fmt ← pp a,
return $ _root_.trace_fmt fmt (λ u, ())
meta def trace_call_stack : tactic unit :=
assume state, _root_.trace_call_stack (success () state)
meta def timetac {α : Type u} (desc : string) (t : thunk (tactic α)) : tactic α :=
λ s, timeit desc (t () s)
meta def trace_state : tactic unit :=
do s ← read,
trace $ to_fmt s
/-- A parameter representing how aggressively definitions should be unfolded when trying to decide if two terms match, unify or are definitionally equal.
By default, theorem declarations are never unfolded.
- `all` will unfold everything, including macros and theorems. Except projection macros.
- `semireducible` will unfold everything except theorems and definitions tagged as irreducible.
- `instances` will unfold all class instance definitions and definitions tagged with reducible.
- `reducible` will only unfold definitions tagged with the `reducible` attribute.
- `none` will never unfold anything.
[NOTE] You are not allowed to tag a definition with more than one of `reducible`, `irreducible`, `semireducible` attributes.
[NOTE] there is a config flag `m_unfold_lemmas`that will make it unfold theorems.
-/
inductive transparency
| all | semireducible | instances | reducible | none
export transparency (reducible semireducible)
/-- (eval_expr α e) evaluates 'e' IF 'e' has type 'α'. -/
meta constant eval_expr (α : Type u) [reflected α] : expr → tactic α
/-- Return the partial term/proof constructed so far. Note that the resultant expression
may contain variables that are not declarate in the current main goal. -/
meta constant result : tactic expr
/-- Display the partial term/proof constructed so far. This tactic is *not* equivalent to
`do { r ← result, s ← read, return (format_expr s r) }` because this one will format the result with respect
to the current goal, and trace_result will do it with respect to the initial goal. -/
meta constant format_result : tactic format
/-- Return target type of the main goal. Fail if tactic_state does not have any goal left. -/
meta constant target : tactic expr
meta constant intro_core : name → tactic expr
meta constant intron : nat → tactic unit
/-- Clear the given local constant. The tactic fails if the given expression is not a local constant. -/
meta constant clear : expr → tactic unit
/-- `revert_lst : list expr → tactic nat` is the reverse of `intron`. It takes a local constant `c` and puts it back as bound by a `pi` or `elet` of the main target.
If there are other local constants that depend on `c`, these are also reverted. Because of this, the `nat` that is returned is the actual number of reverted local constants.
Example: with `x : ℕ, h : P(x) ⊢ T(x)`, `revert_lst [x]` returns `2` and produces the state ` ⊢ Π x, P(x) → T(x)`.
-/
meta constant revert_lst : list expr → tactic nat
/-- Return `e` in weak head normal form with respect to the given transparency setting.
If `unfold_ginductive` is `tt`, then nested and/or mutually recursive inductive datatype constructors
and types are unfolded. Recall that nested and mutually recursive inductive datatype declarations
are compiled into primitive datatypes accepted by the Kernel. -/
meta constant whnf (e : expr) (md := semireducible) (unfold_ginductive := tt) : tactic expr
/-- (head) eta expand the given expression. `f : α → β` head-eta-expands to `λ a, f a`. If `f` isn't a function then it just returns `f`. -/
meta constant head_eta_expand : expr → tactic expr
/-- (head) beta reduction. `(λ x, B) c` reduces to `B[x/c]`. -/
meta constant head_beta : expr → tactic expr
/-- (head) zeta reduction. Reduction of let bindings at the head of the expression. `let x : a := b in c` reduces to `c[x/b]`. -/
meta constant head_zeta : expr → tactic expr
/-- Zeta reduction. Reduction of let bindings. `let x : a := b in c` reduces to `c[x/b]`. -/
meta constant zeta : expr → tactic expr
/-- (head) eta reduction. `(λ x, f x)` reduces to `f`. -/
meta constant head_eta : expr → tactic expr
/-- Succeeds if `t` and `s` can be unified using the given transparency setting. -/
meta constant unify (t s : expr) (md := semireducible) (approx := ff) : tactic unit
/-- Similar to `unify`, but it treats metavariables as constants. -/
meta constant is_def_eq (t s : expr) (md := semireducible) (approx := ff) : tactic unit
/-- Infer the type of the given expression.
Remark: transparency does not affect type inference -/
meta constant infer_type : expr → tactic expr
/-- Get the `local_const` expr for the given `name`. -/
meta constant get_local : name → tactic expr
/-- Resolve a name using the current local context, environment, aliases, etc. -/
meta constant resolve_name : name → tactic pexpr
/-- Return the hypothesis in the main goal. Fail if tactic_state does not have any goal left. -/
meta constant local_context : tactic (list expr)
/-- Get a fresh name that is guaranteed to not be in use in the local context.
If `n` is provided and `n` is not in use, then `n` is returned.
Otherwise a number `i` is appended to give `"n_i"`.
-/
meta constant get_unused_name (n : name := `_x) (i : option nat := none) : tactic name
/-- Helper tactic for creating simple applications where some arguments are inferred using
type inference.
Example, given
```
rel.{l_1 l_2} : Pi (α : Type.{l_1}) (β : α -> Type.{l_2}), (Pi x : α, β x) -> (Pi x : α, β x) -> , Prop
nat : Type
real : Type
vec.{l} : Pi (α : Type l) (n : nat), Type.{l1}
f g : Pi (n : nat), vec real n
```
then
```
mk_app_core semireducible "rel" [f, g]
```
returns the application
```
rel.{1 2} nat (fun n : nat, vec real n) f g
```
The unification constraints due to type inference are solved using the transparency `md`.
-/
meta constant mk_app (fn : name) (args : list expr) (md := semireducible) : tactic expr
/-- Similar to `mk_app`, but allows to specify which arguments are explicit/implicit.
Example, given `(a b : nat)` then
```
mk_mapp "ite" [some (a > b), none, none, some a, some b]
```
returns the application
```
@ite.{1} (a > b) (nat.decidable_gt a b) nat a b
```
-/
meta constant mk_mapp (fn : name) (args : list (option expr)) (md := semireducible) : tactic expr
/-- (mk_congr_arg h₁ h₂) is a more efficient version of (mk_app `congr_arg [h₁, h₂]) -/
meta constant mk_congr_arg : expr → expr → tactic expr
/-- (mk_congr_fun h₁ h₂) is a more efficient version of (mk_app `congr_fun [h₁, h₂]) -/
meta constant mk_congr_fun : expr → expr → tactic expr
/-- (mk_congr h₁ h₂) is a more efficient version of (mk_app `congr [h₁, h₂]) -/
meta constant mk_congr : expr → expr → tactic expr
/-- (mk_eq_refl h) is a more efficient version of (mk_app `eq.refl [h]) -/
meta constant mk_eq_refl : expr → tactic expr
/-- (mk_eq_symm h) is a more efficient version of (mk_app `eq.symm [h]) -/
meta constant mk_eq_symm : expr → tactic expr
/-- (mk_eq_trans h₁ h₂) is a more efficient version of (mk_app `eq.trans [h₁, h₂]) -/
meta constant mk_eq_trans : expr → expr → tactic expr
/-- (mk_eq_mp h₁ h₂) is a more efficient version of (mk_app `eq.mp [h₁, h₂]) -/
meta constant mk_eq_mp : expr → expr → tactic expr
/-- (mk_eq_mpr h₁ h₂) is a more efficient version of (mk_app `eq.mpr [h₁, h₂]) -/
meta constant mk_eq_mpr : expr → expr → tactic expr
/- Given a local constant t, if t has type (lhs = rhs) apply substitution.
Otherwise, try to find a local constant that has type of the form (t = t') or (t' = t).
The tactic fails if the given expression is not a local constant. -/
meta constant subst_core : expr → tactic unit
/-- Close the current goal using `e`. Fail is the type of `e` is not definitionally equal to
the target type. -/
meta constant exact (e : expr) (md := semireducible) : tactic unit
/-- Elaborate the given quoted expression with respect to the current main goal.
Note that this means that any implicit arguments for the given `pexpr` will be applied with fresh metavariables.
If `allow_mvars` is tt, then metavariables are tolerated and become new goals if `subgoals` is tt. -/
meta constant to_expr (q : pexpr) (allow_mvars := tt) (subgoals := tt) : tactic expr
/-- Return true if the given expression is a type class. -/
meta constant is_class : expr → tactic bool
/-- Try to create an instance of the given type class. -/
meta constant mk_instance : expr → tactic expr
/-- Change the target of the main goal.
The input expression must be definitionally equal to the current target.
If `check` is `ff`, then the tactic does not check whether `e`
is definitionally equal to the current target. If it is not,
then the error will only be detected by the kernel type checker. -/
meta constant change (e : expr) (check : bool := tt): tactic unit
/-- `assert_core H T`, adds a new goal for T, and change target to `T -> target`. -/
meta constant assert_core : name → expr → tactic unit
/-- `assertv_core H T P`, change target to (T -> target) if P has type T. -/
meta constant assertv_core : name → expr → expr → tactic unit
/-- `define_core H T`, adds a new goal for T, and change target to `let H : T := ?M in target` in the current goal. -/
meta constant define_core : name → expr → tactic unit
/-- `definev_core H T P`, change target to `let H : T := P in target` if P has type T. -/
meta constant definev_core : name → expr → expr → tactic unit
/-- Rotate goals to the left. That is, `rotate_left 1` takes the main goal and puts it to the back of the subgoal list. -/
meta constant rotate_left : nat → tactic unit
/-- Gets a list of metavariables, one for each goal. -/
meta constant get_goals : tactic (list expr)
/-- Replace the current list of goals with the given one. Each expr in the list should be a metavariable. Any assigned metavariables will be ignored.-/
meta constant set_goals : list expr → tactic unit
/-- How to order the new goals made from an `apply` tactic.
Supposing we were applying `e : ∀ (a:α) (p : P(a)), Q`
- `non_dep_first` would produce goals `⊢ P(?m)`, `⊢ α`. It puts the P goal at the front because none of the arguments after `p` in `e` depend on `p`. It doesn't matter what the result `Q` depends on.
- `non_dep_only` would produce goal `⊢ P(?m)`.
- `all` would produce goals `⊢ α`, `⊢ P(?m)`.
-/
inductive new_goals
| non_dep_first | non_dep_only | all
/-- Configuration options for the `apply` tactic.
- `md` sets how aggressively definitions are unfolded.
- `new_goals` is the strategy for ordering new goals.
- `instances` if `tt`, then `apply` tries to synthesize unresolved `[...]` arguments using type class resolution.
- `auto_param` if `tt`, then `apply` tries to synthesize unresolved `(h : p . tac_id)` arguments using tactic `tac_id`.
- `opt_param` if `tt`, then `apply` tries to synthesize unresolved `(a : t := v)` arguments by setting them to `v`.
- `unify` if `tt`, then `apply` is free to assign existing metavariables in the goal when solving unification constraints.
For example, in the goal `|- ?x < succ 0`, the tactic `apply succ_lt_succ` succeeds with the default configuration,
but `apply_with succ_lt_succ {unify := ff}` doesn't since it would require Lean to assign `?x` to `succ ?y` where
`?y` is a fresh metavariable.
-/
structure apply_cfg :=
(md := semireducible)
(approx := tt)
(new_goals := new_goals.non_dep_first)
(instances := tt)
(auto_param := tt)
(opt_param := tt)
(unify := tt)
/-- Apply the expression `e` to the main goal, the unification is performed using the transparency mode in `cfg`.
Supposing `e : Π (a₁:α₁) ... (aₙ:αₙ), P(a₁,...,aₙ)` and the target is `Q`, `apply` will attempt to unify `Q` with `P(?a₁,...?aₙ)`.
All of the metavariables that are not assigned are added as new metavariables.
If `cfg.approx` is `tt`, then fallback to first-order unification, and approximate context during unification.
`cfg.new_goals` specifies which unassigned metavariables become new goals, and their order.
If `cfg.instances` is `tt`, then use type class resolution to instantiate unassigned meta-variables.
The fields `cfg.auto_param` and `cfg.opt_param` are ignored by this tactic (See `tactic.apply`).
It returns a list of all introduced meta variables and the parameter name associated with them, even the assigned ones. -/
meta constant apply_core (e : expr) (cfg : apply_cfg := {}) : tactic (list (name × expr))
/- Create a fresh meta universe variable. -/
meta constant mk_meta_univ : tactic level
/- Create a fresh meta-variable with the given type.
The scope of the new meta-variable is the local context of the main goal. -/
meta constant mk_meta_var : expr → tactic expr
/-- Return the value assigned to the given universe meta-variable.
Fail if argument is not an universe meta-variable or if it is not assigned. -/
meta constant get_univ_assignment : level → tactic level
/-- Return the value assigned to the given meta-variable.
Fail if argument is not a meta-variable or if it is not assigned. -/
meta constant get_assignment : expr → tactic expr
/-- Return true if the given meta-variable is assigned.
Fail if argument is not a meta-variable. -/
meta constant is_assigned : expr → tactic bool
/-- Make a name that is guaranteed to be unique. Eg `_fresh.1001.4667`. These will be different for each run of the tactic. -/
meta constant mk_fresh_name : tactic name
/-- Induction on `h` using recursor `rec`, names for the new hypotheses
are retrieved from `ns`. If `ns` does not have sufficient names, then use the internal binder names
in the recursor.
It returns for each new goal the name of the constructor (if `rec_name` is a builtin recursor),
a list of new hypotheses, and a list of substitutions for hypotheses
depending on `h`. The substitutions map internal names to their replacement terms. If the
replacement is again a hypothesis the user name stays the same. The internal names are only valid
in the original goal, not in the type context of the new goal.
Remark: if `rec_name` is not a builtin recursor, we use parameter names of `rec_name` instead of
constructor names.
If `rec` is none, then the type of `h` is inferred, if it is of the form `C ...`, tactic uses `C.rec` -/
meta constant induction (h : expr) (ns : list name := []) (rec : option name := none) (md := semireducible) : tactic (list (name × list expr × list (name × expr)))
/-- Apply `cases_on` recursor, names for the new hypotheses are retrieved from `ns`.
`h` must be a local constant. It returns for each new goal the name of the constructor, a list of new hypotheses, and a list of
substitutions for hypotheses depending on `h`. The number of new goals may be smaller than the
number of constructors. Some goals may be discarded when the indices to not match.
See `induction` for information on the list of substitutions.
The `cases` tactic is implemented using this one, and it relaxes the restriction of `h`.
Note: There is one "new hypothesis" for every constructor argument. These are
usually local constants, but due to dependent pattern matching, they can also
be arbitrary terms. -/
meta constant cases_core (h : expr) (ns : list name := []) (md := semireducible) : tactic (list (name × list expr × list (name × expr)))
/-- Similar to cases tactic, but does not revert/intro/clear hypotheses. -/
meta constant destruct (e : expr) (md := semireducible) : tactic unit
/-- Generalizes the target with respect to `e`. -/
meta constant generalize (e : expr) (n : name := `_x) (md := semireducible) : tactic unit
/-- instantiate assigned metavariables in the given expression -/
meta constant instantiate_mvars : expr → tactic expr
/-- Add the given declaration to the environment -/
meta constant add_decl : declaration → tactic unit
/--
Changes the environment to the `new_env`.
The new environment does not need to be a descendant of the old one.
Use with care.
-/
meta constant set_env_core : environment → tactic unit
/-- Changes the environment to the `new_env`. `new_env` needs to be a descendant from the current environment. -/
meta constant set_env : environment → tactic unit
/-- `doc_string env d k` returns the doc string for `d` (if available) -/
meta constant doc_string : name → tactic string
/-- Set the docstring for the given declaration. -/
meta constant add_doc_string : name → string → tactic unit
/--
Create an auxiliary definition with name `c` where `type` and `value` may contain local constants and
meta-variables. This function collects all dependencies (universe parameters, universe metavariables,
local constants (aka hypotheses) and metavariables).
It updates the environment in the tactic_state, and returns an expression of the form
(c.{l_1 ... l_n} a_1 ... a_m)
where l_i's and a_j's are the collected dependencies.
-/
meta constant add_aux_decl (c : name) (type : expr) (val : expr) (is_lemma : bool) : tactic expr
/-- Returns a list of all top-level (`/-! ... -/`) docstrings in the active module and imported ones.
The returned object is a list of modules, indexed by `(some filename)` for imported modules
and `none` for the active one, where each module in the list is paired with a list
of `(position_in_file, docstring)` pairs. -/
meta constant olean_doc_strings : tactic (list (option string × (list (pos × string))))
/-- Returns a list of docstrings in the active module. An entry in the list can be either:
- a top-level (`/-! ... -/`) docstring, represented as `(none, docstring)`
- a declaration-specific (`/-- ... -/`) docstring, represented as `(some decl_name, docstring)` -/
meta def module_doc_strings : tactic (list (option name × string)) :=
do
/- Obtain a list of top-level docs in current module. -/
mod_docs ← olean_doc_strings,
let mod_docs: list (list (option name × string)) :=
mod_docs.filter_map (λ d,
if d.1.is_none
then some (d.2.map
(λ pos_doc, ⟨none, pos_doc.2⟩))
else none),
let mod_docs := mod_docs.join,
/- Obtain list of declarations in current module. -/
e ← get_env,
let decls := environment.fold e ([]: list name)
(λ d acc, let n := d.to_name in
if (environment.decl_olean e n).is_none
then n::acc else acc),
/- Map declarations to those which have docstrings. -/
decls ← decls.mfoldl (λa n,
(doc_string n >>=
λ doc, pure $ (some n, doc) :: a)
<|> pure a) [],
pure (mod_docs ++ decls)
/-- Set attribute `attr_name` for constant `c_name` with the given priority.
If the priority is none, then use default -/
meta constant set_basic_attribute (attr_name : name) (c_name : name) (persistent := ff) (prio : option nat := none) : tactic unit
/-- `unset_attribute attr_name c_name` -/
meta constant unset_attribute : name → name → tactic unit
/-- `has_attribute attr_name c_name` succeeds if the declaration `decl_name`
has the attribute `attr_name`. The result is the priority and whether or not
the attribute is persistent. -/
meta constant has_attribute : name → name → tactic (bool × nat)
/-- `copy_attribute attr_name c_name p d_name` copy attribute `attr_name` from
`src` to `tgt` if it is defined for `src`; make it persistent if `p` is `tt`;
if `p` is `none`, the copied attribute is made persistent iff it is persistent on `src` -/
meta def copy_attribute (attr_name : name) (src : name) (tgt : name) (p : option bool := none) : tactic unit :=
try $ do
(p', prio) ← has_attribute attr_name src,
let p := p.get_or_else p',
set_basic_attribute attr_name tgt p (some prio)
/-- Name of the declaration currently being elaborated. -/
meta constant decl_name : tactic name
/-- `save_type_info e ref` save (typeof e) at position associated with ref -/
meta constant save_type_info {elab : bool} : expr → expr elab → tactic unit
meta constant save_info_thunk : pos → (unit → format) → tactic unit
/-- Return list of currently open namespaces -/
meta constant open_namespaces : tactic (list name)
/-- Return tt iff `t` "occurs" in `e`. The occurrence checking is performed using
keyed matching with the given transparency setting.
We say `t` occurs in `e` by keyed matching iff there is a subterm `s`
s.t. `t` and `s` have the same head, and `is_def_eq t s md`
The main idea is to minimize the number of `is_def_eq` checks
performed. -/
meta constant kdepends_on (e t : expr) (md := reducible) : tactic bool
/-- Abstracts all occurrences of the term `t` in `e` using keyed matching.
If `unify` is `ff`, then matching is used instead of unification.
That is, metavariables occurring in `e` are not assigned. -/
meta constant kabstract (e t : expr) (md := reducible) (unify := tt) : tactic expr
/-- Blocks the execution of the current thread for at least `msecs` milliseconds.
This tactic is used mainly for debugging purposes. -/
meta constant sleep (msecs : nat) : tactic unit
/-- Type check `e` with respect to the current goal.
Fails if `e` is not type correct. -/
meta constant type_check (e : expr) (md := semireducible) : tactic unit
open list nat
/-- A `tag` is a list of `names`. These are attached to goals to help tactics track them.-/
def tag : Type := list name
/-- Enable/disable goal tagging. -/
meta constant enable_tags (b : bool) : tactic unit
/-- Return tt iff goal tagging is enabled. -/
meta constant tags_enabled : tactic bool
/-- Tag goal `g` with tag `t`. It does nothing if goal tagging is disabled.
Remark: `set_goal g []` removes the tag -/
meta constant set_tag (g : expr) (t : tag) : tactic unit
/-- Return tag associated with `g`. Return `[]` if there is no tag. -/
meta constant get_tag (g : expr) : tactic tag
/-- By default, Lean only considers local instances in the header of declarations.
This has two main benefits.
1- Results produced by the type class resolution procedure can be easily cached.
2- The set of local instances does not have to be recomputed.
This approach has the following disadvantages:
1- Frozen local instances cannot be reverted.
2- Local instances defined inside of a declaration are not considered during type
class resolution.
This tactic resets the set of local instances. After executing this tactic,
the set of local instances will be recomputed and the cache will be frequently
reset. Note that, the cache is still used when executing a single tactic that
may generate many type class resolution problems (e.g., `simp`). -/
meta constant unfreeze_local_instances : tactic unit
/--
Freeze the current set of local instances.
-/
meta constant freeze_local_instances : tactic unit
/- Return the list of frozen local instances. Return `none` if local instances were not frozen. -/
meta constant frozen_local_instances : tactic (option (list expr))
meta def induction' (h : expr) (ns : list name := []) (rec : option name := none) (md := semireducible) : tactic unit :=
induction h ns rec md >> return ()
/-- Remark: set_goals will erase any solved goal -/
meta def cleanup : tactic unit :=
get_goals >>= set_goals
/-- Auxiliary definition used to implement begin ... end blocks -/
meta def step {α : Type u} (t : tactic α) : tactic unit :=
t >>[tactic] cleanup
meta def istep {α : Type u} (line0 col0 : ℕ) (line col : ℕ) (t : tactic α) : tactic unit :=
λ s, (@scope_trace _ line col (λ _, step t s)).clamp_pos line0 line col
meta def is_prop (e : expr) : tactic bool :=
do t ← infer_type e,
return (t = `(Prop))
/-- Return true iff n is the name of declaration that is a proposition. -/
meta def is_prop_decl (n : name) : tactic bool :=
do env ← get_env,
d ← env.get n,
t ← return $ d.type,
is_prop t
meta def is_proof (e : expr) : tactic bool :=
infer_type e >>= is_prop
meta def whnf_no_delta (e : expr) : tactic expr :=
whnf e transparency.none
/-- Return `e` in weak head normal form with respect to the given transparency setting,
or `e` head is a generalized constructor or inductive datatype. -/
meta def whnf_ginductive (e : expr) (md := semireducible) : tactic expr :=
whnf e md ff
meta def whnf_target : tactic unit :=
target >>= whnf >>= change
/-- Change the target of the main goal.
The input expression must be definitionally equal to the current target.
The tactic does not check whether `e`
is definitionally equal to the current target. The error will only be detected by the kernel type checker. -/
meta def unsafe_change (e : expr) : tactic unit :=
change e ff
/-- Pi or elet introduction.
Given the tactic state `⊢ Π x : α, Y`, ``intro `hello`` will produce the state `hello : α ⊢ Y[x/hello]`.
Returns the new local constant. Similarly for `elet` expressions.
If the target is not a Pi or elet it will try to put it in WHNF.
-/
meta def intro (n : name) : tactic expr :=
do t ← target,
if expr.is_pi t ∨ expr.is_let t then intro_core n
else whnf_target >> intro_core n
/-- Like `intro` except the name is derived from the bound name in the Π. -/
meta def intro1 : tactic expr :=
intro `_
/-- Repeatedly apply `intro1` and return the list of new local constants in order of introduction.-/
meta def intros : tactic (list expr) :=
do t ← target,
match t with
| expr.pi _ _ _ _ := do H ← intro1, Hs ← intros, return (H :: Hs)
| expr.elet _ _ _ _ := do H ← intro1, Hs ← intros, return (H :: Hs)
| _ := return []
end
/-- Same as `intros`, except with the given names for the new hypotheses. Use the name ```_``` to instead use the binder's name.-/
meta def intro_lst : list name → tactic (list expr)
| [] := return []
| (n::ns) := do H ← intro n, Hs ← intro_lst ns, return (H :: Hs)
/-- Introduces new hypotheses with forward dependencies. -/
meta def intros_dep : tactic (list expr) :=
do t ← target,
let proc (b : expr) :=
if b.has_var_idx 0 then
do h ← intro1, hs ← intros_dep, return (h::hs)
else
-- body doesn't depend on new hypothesis
return [],
match t with
| expr.pi _ _ _ b := proc b
| expr.elet _ _ _ b := proc b
| _ := return []
end
meta def introv : list name → tactic (list expr)
| [] := intros_dep
| (n::ns) := do hs ← intros_dep, h ← intro n, hs' ← introv ns, return (hs ++ h :: hs')
/--
`intron' n` introduces `n` hypotheses and returns the resulting local
constants. Fails if there are not at least `n` arguments to introduce. If you do
not need the return value, use `intron`.
-/
meta def intron' : ℕ → tactic (list expr)
| 0 := pure []
| (n + 1) := do
h ← intro1,
hs ← intron' n,
pure $ h :: hs
/-- Returns n fully qualified if it refers to a constant, or else fails. -/
meta def resolve_constant (n : name) : tactic name :=
do (expr.const n _) ← resolve_name n,
pure n
meta def to_expr_strict (q : pexpr) : tactic expr :=
to_expr q
/--
Example: with `x : ℕ, h : P(x) ⊢ T(x)`, `revert x` returns `2` and produces the state ` ⊢ Π x, P(x) → T(x)`.
-/
meta def revert (l : expr) : tactic nat :=
revert_lst [l]
/- Revert "all" hypotheses. Actually, the tactic only reverts
hypotheses occurring after the last frozen local instance.
Recall that frozen local instances cannot be reverted.
We can use `unfreeze_local_instances` to workaround this limitation. -/
meta def revert_all : tactic nat :=
do lctx ← local_context,
lis ← frozen_local_instances,
match lis with
| none := revert_lst lctx
| some [] := revert_lst lctx
/- `hi` is the last local instance. We shoul truncate `lctx` at `hi`. -/
| some (hi::his) := revert_lst $ lctx.foldl (λ r h, if h.local_uniq_name = hi.local_uniq_name then [] else h :: r) []
end
meta def clear_lst : list name → tactic unit
| [] := skip
| (n::ns) := do H ← get_local n, clear H, clear_lst ns
meta def match_not (e : expr) : tactic expr :=
match (expr.is_not e) with
| (some a) := return a
| none := fail "expression is not a negation"
end
meta def match_and (e : expr) : tactic (expr × expr) :=
match (expr.is_and e) with
| (some (α, β)) := return (α, β)
| none := fail "expression is not a conjunction"
end
meta def match_or (e : expr) : tactic (expr × expr) :=
match (expr.is_or e) with
| (some (α, β)) := return (α, β)
| none := fail "expression is not a disjunction"
end
meta def match_iff (e : expr) : tactic (expr × expr) :=
match (expr.is_iff e) with
| (some (lhs, rhs)) := return (lhs, rhs)
| none := fail "expression is not an iff"
end
meta def match_eq (e : expr) : tactic (expr × expr) :=
match (expr.is_eq e) with
| (some (lhs, rhs)) := return (lhs, rhs)
| none := fail "expression is not an equality"
end
meta def match_ne (e : expr) : tactic (expr × expr) :=
match (expr.is_ne e) with
| (some (lhs, rhs)) := return (lhs, rhs)
| none := fail "expression is not a disequality"
end
meta def match_heq (e : expr) : tactic (expr × expr × expr × expr) :=
do match (expr.is_heq e) with
| (some (α, lhs, β, rhs)) := return (α, lhs, β, rhs)
| none := fail "expression is not a heterogeneous equality"
end
meta def match_refl_app (e : expr) : tactic (name × expr × expr) :=
do env ← get_env,
match (environment.is_refl_app env e) with
| (some (R, lhs, rhs)) := return (R, lhs, rhs)
| none := fail "expression is not an application of a reflexive relation"
end
meta def match_app_of (e : expr) (n : name) : tactic (list expr) :=
guard (expr.is_app_of e n) >> return e.get_app_args
meta def get_local_type (n : name) : tactic expr :=
get_local n >>= infer_type
meta def trace_result : tactic unit :=
format_result >>= trace
meta def rexact (e : expr) : tactic unit :=
exact e reducible
meta def any_hyp_aux {α : Type} (f : expr → tactic α) : list expr → tactic α
| [] := failed
| (h :: hs) := f h <|> any_hyp_aux hs
meta def any_hyp {α : Type} (f : expr → tactic α) : tactic α :=
local_context >>= any_hyp_aux f
/-- `find_same_type t es` tries to find in es an expression with type definitionally equal to t -/
meta def find_same_type : expr → list expr → tactic expr
| e [] := failed
| e (H :: Hs) :=
do t ← infer_type H,
(unify e t >> return H) <|> find_same_type e Hs
meta def find_assumption (e : expr) : tactic expr :=
do ctx ← local_context, find_same_type e ctx
meta def assumption : tactic unit :=
do { ctx ← local_context,
t ← target,
H ← find_same_type t ctx,
exact H }
<|> fail "assumption tactic failed"
meta def save_info (p : pos) : tactic unit :=
do s ← read,
tactic.save_info_thunk p (λ _, tactic_state.to_format s)
notation `‹` p `›` := (by assumption : p)
/-- Swap first two goals, do nothing if tactic state does not have at least two goals. -/
meta def swap : tactic unit :=
do gs ← get_goals,
match gs with
| (g₁ :: g₂ :: rs) := set_goals (g₂ :: g₁ :: rs)
| e := skip
end
/-- `assert h t`, adds a new goal for t, and the hypothesis `h : t` in the current goal. -/
meta def assert (h : name) (t : expr) : tactic expr :=
do assert_core h t, swap, e ← intro h, swap, return e
/-- `assertv h t v`, adds the hypothesis `h : t` in the current goal if v has type t. -/
meta def assertv (h : name) (t : expr) (v : expr) : tactic expr :=
assertv_core h t v >> intro h
/-- `define h t`, adds a new goal for t, and the hypothesis `h : t := ?M` in the current goal. -/
meta def define (h : name) (t : expr) : tactic expr :=
do define_core h t, swap, e ← intro h, swap, return e
/-- `definev h t v`, adds the hypothesis (h : t := v) in the current goal if v has type t. -/
meta def definev (h : name) (t : expr) (v : expr) : tactic expr :=
definev_core h t v >> intro h
/-- Add `h : t := pr` to the current goal -/
meta def pose (h : name) (t : option expr := none) (pr : expr) : tactic expr :=
let dv := λt, definev h t pr in
option.cases_on t (infer_type pr >>= dv) dv
/-- Add `h : t` to the current goal, given a proof `pr : t` -/
meta def note (h : name) (t : option expr := none) (pr : expr) : tactic expr :=
let dv := λt, assertv h t pr in
option.cases_on t (infer_type pr >>= dv) dv
/-- Return the number of goals that need to be solved -/
meta def num_goals : tactic nat :=
do gs ← get_goals,
return (length gs)
/-- Rotate the goals to the right by `n`. That is, take the goal at the back and push it to the front `n` times.
[NOTE] We have to provide the instance argument `[has_mod nat]` because
mod for nat was not defined yet -/
meta def rotate_right (n : nat) [has_mod nat] : tactic unit :=
do ng ← num_goals,
if ng = 0 then skip
else rotate_left (ng - n % ng)
/-- Rotate the goals to the left by `n`. That is, put the main goal to the back `n` times. -/
meta def rotate : nat → tactic unit :=
rotate_left
private meta def repeat_aux (t : tactic unit) : list expr → list expr → tactic unit
| [] r := set_goals r.reverse
| (g::gs) r := do
ok ← try_core (set_goals [g] >> t),
match ok with
| none := repeat_aux gs (g::r)
| _ := do
gs' ← get_goals,
repeat_aux (gs' ++ gs) r
end
/-- This tactic is applied to each goal. If the application succeeds,
the tactic is applied recursively to all the generated subgoals until it eventually fails.
The recursion stops in a subgoal when the tactic has failed to make progress.
The tactic `repeat` never fails. -/
meta def repeat (t : tactic unit) : tactic unit :=
do gs ← get_goals, repeat_aux t gs []
/-- `first [t_1, ..., t_n]` applies the first tactic that doesn't fail.
The tactic fails if all t_i's fail. -/
meta def first {α : Type u} : list (tactic α) → tactic α
| [] := fail "first tactic failed, no more alternatives"
| (t::ts) := t <|> first ts
/-- Applies the given tactic to the main goal and fails if it is not solved. -/
meta def solve1 {α} (tac : tactic α) : tactic α :=
do gs ← get_goals,
match gs with
| [] := fail "solve1 tactic failed, there isn't any goal left to focus"
| (g::rs) :=
do set_goals [g],
a ← tac,
gs' ← get_goals,
match gs' with
| [] := set_goals rs >> pure a
| gs := fail "solve1 tactic failed, focused goal has not been solved"
end
end
/-- `solve [t_1, ... t_n]` applies the first tactic that solves the main goal. -/
meta def solve {α} (ts : list (tactic α)) : tactic α :=
first $ map solve1 ts
private meta def focus_aux {α} : list (tactic α) → list expr → list expr → tactic (list α)
| [] [] rs := set_goals rs *> pure []
| (t::ts) [] rs := fail "focus' tactic failed, insufficient number of goals"
| tts (g::gs) rs :=
mcond (is_assigned g) (focus_aux tts gs rs) $
do set_goals [g],
t::ts ← pure tts | fail "focus' tactic failed, insufficient number of tactics",
a ← t,
rs' ← get_goals,
as ← focus_aux ts gs (rs ++ rs'),
pure $ a :: as
/--
`focus [t_1, ..., t_n]` applies t_i to the i-th goal. Fails if the number of
goals is not n. Returns the results of t_i (one per goal).
-/
meta def focus {α} (ts : list (tactic α)) : tactic (list α) :=
do gs ← get_goals, focus_aux ts gs []
private meta def focus'_aux : list (tactic unit) → list expr → list expr → tactic unit
| [] [] rs := set_goals rs
| (t::ts) [] rs := fail "focus tactic failed, insufficient number of goals"
| tts (g::gs) rs :=
mcond (is_assigned g) (focus'_aux tts gs rs) $
do set_goals [g],
t::ts ← pure tts | fail "focus tactic failed, insufficient number of tactics",
t,
rs' ← get_goals,
focus'_aux ts gs (rs ++ rs')
/-- `focus' [t_1, ..., t_n]` applies t_i to the i-th goal. Fails if the number of goals is not n. -/
meta def focus' (ts : list (tactic unit)) : tactic unit :=
do gs ← get_goals, focus'_aux ts gs []
meta def focus1 {α} (tac : tactic α) : tactic α :=
do g::gs ← get_goals,
match gs with
| [] := tac
| _ := do
set_goals [g],
a ← tac,
gs' ← get_goals,
set_goals (gs' ++ gs),
return a
end
private meta def all_goals_core {α} (tac : tactic α)
: list expr → list expr → tactic (list α)
| [] ac := set_goals ac *> pure []
| (g :: gs) ac :=
mcond (is_assigned g) (all_goals_core gs ac) $
do set_goals [g],
a ← tac,
new_gs ← get_goals,
as ← all_goals_core gs (ac ++ new_gs),
pure $ a :: as
/--
Apply the given tactic to all goals. Return one result per goal.
-/
meta def all_goals {α} (tac : tactic α) : tactic (list α) :=
do gs ← get_goals,
all_goals_core tac gs []
private meta def all_goals'_core (tac : tactic unit) : list expr → list expr → tactic unit
| [] ac := set_goals ac
| (g :: gs) ac :=
mcond (is_assigned g) (all_goals'_core gs ac) $
do set_goals [g],
tac,
new_gs ← get_goals,
all_goals'_core gs (ac ++ new_gs)
/-- Apply the given tactic to all goals. -/
meta def all_goals' (tac : tactic unit) : tactic unit :=
do gs ← get_goals,
all_goals'_core tac gs []
private meta def any_goals_core {α} (tac : tactic α) : list expr → list expr → bool → tactic (list (option α))
| [] ac progress := guard progress *> set_goals ac *> pure []
| (g :: gs) ac progress :=
mcond (is_assigned g) (any_goals_core gs ac progress) $
do set_goals [g],
res ← try_core tac,
new_gs ← get_goals,
ress ← any_goals_core gs (ac ++ new_gs) (res.is_some || progress),
pure $ res :: ress
/--
Apply `tac` to any goal where it succeeds. The tactic succeeds if `tac`
succeeds for at least one goal. The returned list contains the result of `tac`
for each goal: `some a` if tac succeeded, or `none` if it did not.
-/
meta def any_goals {α} (tac : tactic α) : tactic (list (option α)) :=
do gs ← get_goals,
any_goals_core tac gs [] ff
private meta def any_goals'_core (tac : tactic unit) : list expr → list expr → bool → tactic unit
| [] ac progress := guard progress >> set_goals ac
| (g :: gs) ac progress :=
mcond (is_assigned g) (any_goals'_core gs ac progress) $
do set_goals [g],
succeeded ← try_core tac,
new_gs ← get_goals,
any_goals'_core gs (ac ++ new_gs) (succeeded.is_some || progress)
/-- Apply the given tactic to any goal where it succeeds. The tactic succeeds only if
tac succeeds for at least one goal. -/
meta def any_goals' (tac : tactic unit) : tactic unit :=
do gs ← get_goals,
any_goals'_core tac gs [] ff
/--
LCF-style AND_THEN tactic. It applies `tac1` to the main goal, then applies
`tac2` to each goal produced by `tac1`.
-/
meta def seq {α β} (tac1 : tactic α) (tac2 : α → tactic β) : tactic (list β) :=
do g::gs ← get_goals,
set_goals [g],
a ← tac1,
bs ← all_goals $ tac2 a,
gs' ← get_goals,
set_goals (gs' ++ gs),
pure bs
/-- LCF-style AND_THEN tactic. It applies tac1, and if succeed applies tac2 to each subgoal produced by tac1 -/
meta def seq' (tac1 : tactic unit) (tac2 : tactic unit) : tactic unit :=
do g::gs ← get_goals,
set_goals [g],
tac1, all_goals' tac2,
gs' ← get_goals,
set_goals (gs' ++ gs)
/--
Applies `tac1` to the main goal, then applies each of the tactics in `tacs2` to
one of the produced subgoals (like `focus'`).
-/
meta def seq_focus {α β} (tac1 : tactic α) (tacs2 : α → list (tactic β)) : tactic (list β) :=
do g::gs ← get_goals,
set_goals [g],
a ← tac1,
bs ← focus $ tacs2 a,
gs' ← get_goals,
set_goals (gs' ++ gs),
pure bs
/--
Applies `tac1` to the main goal, then applies each of the tactics in `tacs2` to
one of the produced subgoals (like `focus`).
-/
meta def seq_focus' (tac1 : tactic unit) (tacs2 : list (tactic unit)) : tactic unit :=
do g::gs ← get_goals,
set_goals [g],
tac1, focus tacs2,
gs' ← get_goals,
set_goals (gs' ++ gs)
meta instance andthen_seq : has_andthen (tactic unit) (tactic unit) (tactic unit) :=
⟨seq'⟩
meta instance andthen_seq_focus : has_andthen (tactic unit) (list (tactic unit)) (tactic unit) :=
⟨seq_focus'⟩
meta constant is_trace_enabled_for : name → bool
/-- Execute tac only if option trace.n is set to true. -/
meta def when_tracing (n : name) (tac : tactic unit) : tactic unit :=
when (is_trace_enabled_for n = tt) tac
/-- Fail if there are no remaining goals. -/
meta def fail_if_no_goals : tactic unit :=
do n ← num_goals,
when (n = 0) (fail "tactic failed, there are no goals to be solved")
/-- Fail if there are unsolved goals. -/
meta def done : tactic unit :=
do n ← num_goals,
when (n ≠ 0) (fail "done tactic failed, there are unsolved goals")
meta def apply_opt_param : tactic unit :=
do `(opt_param %%t %%v) ← target,
exact v
meta def apply_auto_param : tactic unit :=
do `(auto_param %%type %%tac_name_expr) ← target,
change type,
tac_name ← eval_expr name tac_name_expr,
tac ← eval_expr (tactic unit) (expr.const tac_name []),
tac
meta def has_opt_auto_param (ms : list expr) : tactic bool :=
ms.mfoldl
(λ r m, do type ← infer_type m,
return $ r || type.is_napp_of `opt_param 2 || type.is_napp_of `auto_param 2)
ff
meta def try_apply_opt_auto_param (cfg : apply_cfg) (ms : list expr) : tactic unit :=
when (cfg.auto_param || cfg.opt_param) $
mwhen (has_opt_auto_param ms) $ do
gs ← get_goals,
ms.mmap' (λ m, mwhen (bnot <$> is_assigned m) $
set_goals [m] >>
when cfg.opt_param (try apply_opt_param) >>
when cfg.auto_param (try apply_auto_param)),
set_goals gs
meta def has_opt_auto_param_for_apply (ms : list (name × expr)) : tactic bool :=
ms.mfoldl
(λ r m, do type ← infer_type m.2,
return $ r || type.is_napp_of `opt_param 2 || type.is_napp_of `auto_param 2)
ff
meta def try_apply_opt_auto_param_for_apply (cfg : apply_cfg) (ms : list (name × expr)) : tactic unit :=
mwhen (has_opt_auto_param_for_apply ms) $ do
gs ← get_goals,
ms.mmap' (λ m, mwhen (bnot <$> (is_assigned m.2)) $
set_goals [m.2] >>
when cfg.opt_param (try apply_opt_param) >>
when cfg.auto_param (try apply_auto_param)),
set_goals gs
meta def apply (e : expr) (cfg : apply_cfg := {}) : tactic (list (name × expr)) :=
do r ← apply_core e cfg,
try_apply_opt_auto_param_for_apply cfg r,
return r
/-- Same as `apply` but __all__ arguments that weren't inferred are added to goal list. -/
meta def fapply (e : expr) : tactic (list (name × expr)) :=
apply e {new_goals := new_goals.all}
/-- Same as `apply` but only goals that don't depend on other goals are added to goal list. -/
meta def eapply (e : expr) : tactic (list (name × expr)) :=
apply e {new_goals := new_goals.non_dep_only}
/-- Try to solve the main goal using type class resolution. -/
meta def apply_instance : tactic unit :=
do tgt ← target >>= instantiate_mvars,
b ← is_class tgt,
if b then mk_instance tgt >>= exact
else fail "apply_instance tactic fail, target is not a type class"
/-- Create a list of universe meta-variables of the given size. -/
meta def mk_num_meta_univs : nat → tactic (list level)
| 0 := return []
| (succ n) := do
l ← mk_meta_univ,
ls ← mk_num_meta_univs n,
return (l::ls)
/-- Return `expr.const c [l_1, ..., l_n]` where l_i's are fresh universe meta-variables. -/
meta def mk_const (c : name) : tactic expr :=
do env ← get_env,
decl ← env.get c,
let num := decl.univ_params.length,
ls ← mk_num_meta_univs num,
return (expr.const c ls)
/-- Apply the constant `c` -/
meta def applyc (c : name) (cfg : apply_cfg := {}) : tactic unit :=
do c ← mk_const c, apply c cfg, skip
meta def eapplyc (c : name) : tactic unit :=
do c ← mk_const c, eapply c, skip
meta def save_const_type_info (n : name) {elab : bool} (ref : expr elab) : tactic unit :=
try (do c ← mk_const n, save_type_info c ref)
/-- Create a fresh universe `?u`, a metavariable `?T : Type.{?u}`,
and return metavariable `?M : ?T`.
This action can be used to create a meta-variable when
we don't know its type at creation time -/
meta def mk_mvar : tactic expr :=
do u ← mk_meta_univ,
t ← mk_meta_var (expr.sort u),
mk_meta_var t
/-- Makes a sorry macro with a meta-variable as its type. -/
meta def mk_sorry : tactic expr := do
u ← mk_meta_univ,
t ← mk_meta_var (expr.sort u),
return $ expr.mk_sorry t
/-- Closes the main goal using sorry. -/
meta def admit : tactic unit :=
target >>= exact ∘ expr.mk_sorry
meta def mk_local' (pp_name : name) (bi : binder_info) (type : expr) : tactic expr := do
uniq_name ← mk_fresh_name,
return $ expr.local_const uniq_name pp_name bi type
meta def mk_local_def (pp_name : name) (type : expr) : tactic expr :=
mk_local' pp_name binder_info.default type
meta def mk_local_pis : expr → tactic (list expr × expr)
| (expr.pi n bi d b) := do
p ← mk_local' n bi d,
(ps, r) ← mk_local_pis (expr.instantiate_var b p),
return ((p :: ps), r)
| e := return ([], e)
private meta def get_pi_arity_aux : expr → tactic nat
| (expr.pi n bi d b) :=
do m ← mk_fresh_name,
let l := expr.local_const m n bi d,
new_b ← whnf (expr.instantiate_var b l),
r ← get_pi_arity_aux new_b,
return (r + 1)
| e := return 0
/-- Compute the arity of the given (Pi-)type -/
meta def get_pi_arity (type : expr) : tactic nat :=
whnf type >>= get_pi_arity_aux
/-- Compute the arity of the given function -/
meta def get_arity (fn : expr) : tactic nat :=
infer_type fn >>= get_pi_arity
meta def triv : tactic unit := mk_const `trivial >>= exact
notation `dec_trivial` := of_as_true (by tactic.triv)
meta def by_contradiction (H : option name := none) : tactic expr :=
do tgt : expr ← target,
(match_not tgt >> return ())
<|>
(mk_mapp `decidable.by_contradiction [some tgt, none] >>= eapply >> skip)
<|>
fail "tactic by_contradiction failed, target is not a negation nor a decidable proposition (remark: when 'local attribute [instance] classical.prop_decidable' is used, all propositions are decidable)",
match H with
| some n := intro n
| none := intro1
end
private meta def generalizes_aux (md : transparency) : list expr → tactic unit
| [] := skip
| (e::es) := generalize e `x md >> generalizes_aux es
meta def generalizes (es : list expr) (md := semireducible) : tactic unit :=
generalizes_aux md es
private meta def kdependencies_core (e : expr) (md : transparency) : list expr → list expr → tactic (list expr)
| [] r := return r
| (h::hs) r :=
do type ← infer_type h,
d ← kdepends_on type e md,
if d then kdependencies_core hs (h::r)
else kdependencies_core hs r
/-- Return all hypotheses that depends on `e`
The dependency test is performed using `kdepends_on` with the given transparency setting. -/
meta def kdependencies (e : expr) (md := reducible) : tactic (list expr) :=
do ctx ← local_context, kdependencies_core e md ctx []
/-- Revert all hypotheses that depend on `e` -/
meta def revert_kdependencies (e : expr) (md := reducible) : tactic nat :=
kdependencies e md >>= revert_lst
meta def revert_kdeps (e : expr) (md := reducible) :=
revert_kdependencies e md
/-- Postprocess the output of `cases_core`:
- The third component of each tuple in the input list (the list of
substitutions) is dropped since we don't use it anywhere.
- The second component (the list of new hypotheses) is filtered: any expression
that is not a local constant is dropped. We only use the new hypotheses for
the renaming functionality of `case`, so we want to keep only those
"new hypotheses" that are, in fact, local constants. -/
private meta def cases_postprocess (hs : list (name × list expr × list (name × expr)))
: list (name × list expr) :=
hs.map $ λ ⟨n, hs, _⟩, (n, hs.filter (λ h, h.is_local_constant))
/-- Similar to `cases_core`, but `e` doesn't need to be a hypothesis.
Remark, it reverts dependencies using `revert_kdeps`.
Two different transparency modes are used `md` and `dmd`.
The mode `md` is used with `cases_core` and `dmd` with `generalize` and `revert_kdeps`.
It returns the constructor names associated with each new goal and the newly
introduced hypotheses. Note that while `cases_core` may return "new
hypotheses" that are not local constants, this tactic only returns local
constants.
-/
meta def cases (e : expr) (ids : list name := []) (md := semireducible) (dmd := semireducible) : tactic (list (name × list expr)) :=
if e.is_local_constant then
do r ← cases_core e ids md, return $ cases_postprocess r
else do
n ← revert_kdependencies e dmd,
x ← get_unused_name,
(tactic.generalize e x dmd)
<|>
(do t ← infer_type e,
tactic.assertv x t e,
get_local x >>= tactic.revert,
return ()),
h ← tactic.intro1,
focus1 $ do
r ← cases_core h ids md,
hs' ← all_goals (intron' n),
return $ cases_postprocess $ r.map₂ (λ ⟨n, hs, x⟩ hs', (n, hs ++ hs', x)) hs'
/-- The same as `exact` except you can add proof holes. -/
meta def refine (e : pexpr) : tactic unit :=
do tgt : expr ← target,
to_expr ``(%%e : %%tgt) tt >>= exact
meta def by_cases (e : expr) (h : name) : tactic unit :=
do dec_e ← (mk_app `decidable [e] <|> fail "by_cases tactic failed, type is not a proposition"),
inst ← (mk_instance dec_e <|> fail "by_cases tactic failed, type of given expression is not decidable"),
t ← target,
tm ← mk_mapp `dite [some e, some inst, some t],
seq' (apply tm >> skip) (intro h >> skip)
meta def funext_core : list name → bool → tactic unit
| [] tt := return ()
| ids only_ids := try $
do some (lhs, rhs) ← expr.is_eq <$> (target >>= whnf),
applyc `funext,
id ← if ids.empty ∨ ids.head = `_ then do
(expr.lam n _ _ _) ← whnf lhs
| pure `_,
return n
else return ids.head,
intro id,
funext_core ids.tail only_ids
meta def funext : tactic unit :=
funext_core [] ff
meta def funext_lst (ids : list name) : tactic unit :=
funext_core ids tt
private meta def get_undeclared_const (env : environment) (base : name) : ℕ → name | i :=
let n := base <.> ("_aux_" ++ repr i) in
if ¬env.contains n then n
else get_undeclared_const (i+1)
meta def new_aux_decl_name : tactic name := do
env ← get_env, n ← decl_name,
return $ get_undeclared_const env n 1
private meta def mk_aux_decl_name : option name → tactic name
| none := new_aux_decl_name
| (some suffix) := do p ← decl_name, return $ p ++ suffix
meta def abstract (tac : tactic unit) (suffix : option name := none) (zeta_reduce := tt) : tactic unit :=
do fail_if_no_goals,
gs ← get_goals,
type ← if zeta_reduce then target >>= zeta else target,
is_lemma ← is_prop type,
m ← mk_meta_var type,
set_goals [m],
tac,
n ← num_goals,
when (n ≠ 0) (fail "abstract tactic failed, there are unsolved goals"),
set_goals gs,
val ← instantiate_mvars m,
val ← if zeta_reduce then zeta val else return val,
c ← mk_aux_decl_name suffix,
e ← add_aux_decl c type val is_lemma,
exact e
/-- `solve_aux type tac` synthesize an element of 'type' using tactic 'tac' -/
meta def solve_aux {α : Type} (type : expr) (tac : tactic α) : tactic (α × expr) :=
do m ← mk_meta_var type,
gs ← get_goals,
set_goals [m],
a ← tac,
set_goals gs,
return (a, m)
/-- Return tt iff 'd' is a declaration in one of the current open namespaces -/
meta def in_open_namespaces (d : name) : tactic bool :=
do ns ← open_namespaces,
env ← get_env,
return $ ns.any (λ n, n.is_prefix_of d) && env.contains d
/-- Execute tac for 'max' "heartbeats". The heartbeat is approx. the maximum number of
memory allocations (in thousands) performed by 'tac'. This is a deterministic way of interrupting
long running tactics. -/
meta def try_for {α} (max : nat) (tac : tactic α) : tactic α :=
λ s,
match _root_.try_for max (tac s) with
| some r := r
| none := mk_exception "try_for tactic failed, timeout" none s
end
meta def updateex_env (f : environment → exceptional environment) : tactic unit :=
do env ← get_env,
env ← returnex $ f env,
set_env env
/- Add a new inductive datatype to the environment
name, universe parameters, number of parameters, type, constructors (name and type), is_meta -/
meta def add_inductive (n : name) (ls : list name) (p : nat) (ty : expr) (is : list (name × expr))
(is_meta : bool := ff) : tactic unit :=
updateex_env $ λe, e.add_inductive n ls p ty is is_meta
meta def add_meta_definition (n : name) (lvls : list name) (type value : expr) : tactic unit :=
add_decl (declaration.defn n lvls type value reducibility_hints.abbrev ff)
/-- add declaration `d` as a protected declaration -/
meta def add_protected_decl (d : declaration) : tactic unit :=
updateex_env $ λ e, e.add_protected d
/-- check if `n` is the name of a protected declaration -/
meta def is_protected_decl (n : name) : tactic bool :=
do env ← get_env,
return $ env.is_protected n
/-- `add_defn_equations` adds a definition specified by a list of equations.
The arguments:
* `lp`: list of universe parameters
* `params`: list of parameters (binders before the colon);
* `fn`: a local constant giving the name and type of the declaration
(with `params` in the local context);
* `eqns`: a list of equations, each of which is a list of patterns
(constructors applied to new local constants) and the branch
expression;
* `is_meta`: is the definition meta?
`add_defn_equations` can be used as:
do my_add ← mk_local_def `my_add `(ℕ → ℕ),
a ← mk_local_def `a ℕ,
b ← mk_local_def `b ℕ,
add_defn_equations [a] my_add
[ ([``(nat.zero)], a),
([``(nat.succ %%b)], my_add b) ])
ff -- non-meta
to create the following definition:
def my_add (a : ℕ) : ℕ → ℕ
| nat.zero := a
| (nat.succ b) := my_add b
-/
meta def add_defn_equations (lp : list name) (params : list expr) (fn : expr)
(eqns : list (list pexpr × expr)) (is_meta : bool) : tactic unit :=
do opt ← get_options,
updateex_env $ λ e, e.add_defn_eqns opt lp params fn eqns is_meta
/-- Get the revertible part of the local context. These are the hypotheses that
appear after the last frozen local instance in the local context. We call them
revertible because `revert` can revert them, unlike those hypotheses which occur
before a frozen instance. -/
meta def revertible_local_context : tactic (list expr) :=
do ctx ← local_context,
frozen ← frozen_local_instances,
pure $
match frozen with
| none := ctx
| some [] := ctx
| some (h :: _) := ctx.after (eq h)
end
/--
Rename local hypotheses according to the given `name_map`. The `name_map`
contains as keys those hypotheses that should be renamed; the associated values
are the new names.
This tactic can only rename hypotheses which occur after the last frozen local
instance. If you need to rename earlier hypotheses, try
`unfreeze_local_instances`.
If `strict` is true, we fail if `name_map` refers to hypotheses that do not
appear in the local context or that appear before a frozen local instance.
Conversely, if `strict` is false, some entries of `name_map` may be silently
ignored.
If `use_unique_names` is true, the keys of `name_map` should be the unique names
of hypotheses to be renamed. Otherwise, the keys should be display names.
Note that we allow shadowing, so renamed hypotheses may have the same name
as other hypotheses in the context. If `use_unique_names` is false and there are
multiple hypotheses with the same display name in the context, they are all
renamed.
-/
meta def rename_many (renames : name_map name) (strict := tt) (use_unique_names := ff)
: tactic unit :=
do let hyp_name : expr → name :=
if use_unique_names then expr.local_uniq_name else expr.local_pp_name,
ctx ← revertible_local_context,
-- The part of the context after (but including) the first hypthesis that
-- must be renamed.
let ctx_suffix := ctx.drop_while (λ h, (renames.find $ hyp_name h).is_none),
when strict $ do {
let ctx_names := rb_map.set_of_list (ctx_suffix.map hyp_name),
let invalid_renames :=
(renames.to_list.map prod.fst).filter (λ h, ¬ ctx_names.contains h),
when ¬ invalid_renames.empty $ fail $ format.join
[ "Cannot rename these hypotheses:\n"
, format.join $ (invalid_renames.map to_fmt).intersperse ", "
, format.line
, "This is because these hypotheses either do not occur in the\n"
, "context or they occur before a frozen local instance.\n"
, "In the latter case, try `tactic.unfreeze_local_instances`."
]
},
-- The new names for all hypotheses in ctx_suffix.
let new_names :=
ctx_suffix.map $ λ h,
(renames.find $ hyp_name h).get_or_else h.local_pp_name,
revert_lst ctx_suffix,
intro_lst new_names,
pure ()
/--
Rename a local hypothesis. This is a special case of `rename_many`;
see there for caveats.
-/
meta def rename (curr : name) (new : name) : tactic unit :=
rename_many (rb_map.of_list [⟨curr, new⟩])
/--
Rename a local hypothesis. Unlike `rename` and `rename_many`, this tactic does
not preserve the order of hypotheses. Its implementation is simpler (and
therefore probably faster) than that of `rename`.
-/
meta def rename_unstable (curr : name) (new : name) : tactic unit :=
do h ← get_local curr,
n ← revert h,
intro new,
intron (n - 1)
/--
"Replace" hypothesis `h : type` with `h : new_type` where `eq_pr` is a proof
that (type = new_type). The tactic actually creates a new hypothesis
with the same user facing name, and (tries to) clear `h`.
The `clear` step fails if `h` has forward dependencies. In this case, the old `h`
will remain in the local context. The tactic returns the new hypothesis. -/
meta def replace_hyp (h : expr) (new_type : expr) (eq_pr : expr) : tactic expr :=
do h_type ← infer_type h,
new_h ← assert h.local_pp_name new_type,
mk_eq_mp eq_pr h >>= exact,
try $ clear h,
return new_h
meta def main_goal : tactic expr :=
do g::gs ← get_goals, return g
/- Goal tagging support -/
meta def with_enable_tags {α : Type} (t : tactic α) (b := tt) : tactic α :=
do old ← tags_enabled,
enable_tags b,
r ← t,
enable_tags old,
return r
meta def get_main_tag : tactic tag :=
main_goal >>= get_tag
meta def set_main_tag (t : tag) : tactic unit :=
do g ← main_goal, set_tag g t
meta def subst (h : expr) : tactic unit :=
(do guard h.is_local_constant,
some (α, lhs, β, rhs) ← expr.is_heq <$> infer_type h,
is_def_eq α β,
new_h_type ← mk_app `eq [lhs, rhs],
new_h_pr ← mk_app `eq_of_heq [h],
new_h ← assertv h.local_pp_name new_h_type new_h_pr,
try (clear h),
subst_core new_h)
<|> subst_core h
end tactic
notation [parsing_only] `command`:max := tactic unit
open tactic
namespace list
meta def for_each {α} : list α → (α → tactic unit) → tactic unit
| [] fn := skip
| (e::es) fn := do fn e, for_each es fn
meta def any_of {α β} : list α → (α → tactic β) → tactic β
| [] fn := failed
| (e::es) fn := do opt_b ← try_core (fn e),
match opt_b with
| some b := return b
| none := any_of es fn
end
end list
/- Install monad laws tactic and use it to prove some instances. -/
/-- Try to prove with `iff.refl`.-/
meta def order_laws_tac := whnf_target >> intros >> to_expr ``(iff.refl _) >>= exact
meta def monad_from_pure_bind {m : Type u → Type v}
(pure : Π {α : Type u}, α → m α)
(bind : Π {α β : Type u}, m α → (α → m β) → m β) : monad m :=
{pure := @pure, bind := @bind}
meta instance : monad task :=
{map := @task.map, bind := @task.bind, pure := @task.pure}
namespace tactic
meta def mk_id_proof (prop : expr) (pr : expr) : expr :=
expr.app (expr.app (expr.const ``id [level.zero]) prop) pr
meta def mk_id_eq (lhs : expr) (rhs : expr) (pr : expr) : tactic expr :=
do prop ← mk_app `eq [lhs, rhs],
return $ mk_id_proof prop pr
meta def replace_target (new_target : expr) (pr : expr) : tactic unit :=
do t ← target,
assert `htarget new_target, swap,
ht ← get_local `htarget,
locked_pr ← mk_id_eq t new_target pr,
mk_eq_mpr locked_pr ht >>= exact
meta def eval_pexpr (α) [reflected α] (e : pexpr) : tactic α :=
to_expr ``(%%e : %%(reflect α)) ff ff >>= eval_expr α
meta def run_simple {α} : tactic_state → tactic α → option α
| ts t := match t ts with
| (interaction_monad.result.success a ts') := some a
| (interaction_monad.result.exception _ _ _) := none
end
end tactic
|
f8f858a233214742bb95ba81edc63a3a221f40b7 | 302c785c90d40ad3d6be43d33bc6a558354cc2cf | /src/ring_theory/perfection.lean | 1555bfd49b631d7f06ad2da26d6836ad6b064853 | [
"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 | 24,280 | lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import algebra.char_p
import algebra.ring.pi
import analysis.special_functions.pow
import field_theory.perfect_closure
import ring_theory.localization
import ring_theory.subring
import ring_theory.valuation.integers
/-!
# Ring Perfection and Tilt
In this file we define the perfection of a ring of characteristic p, and the tilt of a field
given a valuation to `ℝ≥0`.
## TODO
Define the valuation on the tilt, and define a characteristic predicate for the tilt.
-/
universes u₁ u₂ u₃ u₄
open_locale nnreal
/-- The perfection of a monoid `M`, defined to be the projective limit of `M`
using the `p`-th power maps `M → M` indexed by the natural numbers, implemented as
`{ f : ℕ → M | ∀ n, f (n + 1) ^ p = f n }`. -/
def monoid.perfection (M : Type u₁) [comm_monoid M] (p : ℕ) : submonoid (ℕ → M) :=
{ carrier := { f | ∀ n, f (n + 1) ^ p = f n },
one_mem' := λ n, one_pow _,
mul_mem' := λ f g hf hg n, (mul_pow _ _ _).trans $ congr_arg2 _ (hf n) (hg n) }
/-- The perfection of a ring `R` with characteristic `p`,
defined to be the projective limit of `R` using the Frobenius maps `R → R`
indexed by the natural numbers, implemented as `{ f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }`. -/
def ring.perfection (R : Type u₁) [comm_semiring R]
(p : ℕ) [hp : fact p.prime] [char_p R p] :
subsemiring (ℕ → R) :=
{ zero_mem' := λ n, zero_pow $ hp.1.pos,
add_mem' := λ f g hf hg n, (frobenius_add R p _ _).trans $ congr_arg2 _ (hf n) (hg n),
.. monoid.perfection R p }
namespace perfection
variables (R : Type u₁) [comm_semiring R] (p : ℕ) [hp : fact p.prime] [char_p R p]
include hp
/-- The `n`-th coefficient of an element of the perfection. -/
def coeff (n : ℕ) : ring.perfection R p →+* R :=
{ to_fun := λ f, f.1 n,
map_one' := rfl,
map_mul' := λ f g, rfl,
map_zero' := rfl,
map_add' := λ f g, rfl }
variables {R p}
@[ext] lemma ext {f g : ring.perfection R p} (h : ∀ n, coeff R p n f = coeff R p n g) : f = g :=
subtype.eq $ funext h
variables (R p)
/-- The `p`-th root of an element of the perfection. -/
def pth_root : ring.perfection R p →+* ring.perfection R p :=
{ to_fun := λ f, ⟨λ n, coeff R p (n + 1) f, λ n, f.2 _⟩,
map_one' := rfl,
map_mul' := λ f g, rfl,
map_zero' := rfl,
map_add' := λ f g, rfl }
variables {R p}
@[simp] lemma coeff_mk (f : ℕ → R) (hf) (n : ℕ) : coeff R p n ⟨f, hf⟩ = f n := rfl
lemma coeff_pth_root (f : ring.perfection R p) (n : ℕ) :
coeff R p n (pth_root R p f) = coeff R p (n + 1) f :=
rfl
lemma coeff_pow_p (f : ring.perfection R p) (n : ℕ) :
coeff R p (n + 1) (f ^ p) = coeff R p n f :=
by { rw ring_hom.map_pow, exact f.2 n }
lemma coeff_pow_p' (f : ring.perfection R p) (n : ℕ) :
coeff R p (n + 1) f ^ p = coeff R p n f :=
f.2 n
lemma coeff_frobenius (f : ring.perfection R p) (n : ℕ) :
coeff R p (n + 1) (frobenius _ p f) = coeff R p n f :=
by apply coeff_pow_p f n -- `coeff_pow_p f n` also works but is slow!
lemma coeff_iterate_frobenius (f : ring.perfection R p) (n m : ℕ) :
coeff R p (n + m) (frobenius _ p ^[m] f) = coeff R p n f :=
nat.rec_on m rfl $ λ m ih, by erw [function.iterate_succ_apply', coeff_frobenius, ih]
lemma coeff_iterate_frobenius' (f : ring.perfection R p) (n m : ℕ) (hmn : m ≤ n) :
coeff R p n (frobenius _ p ^[m] f) = coeff R p (n - m) f :=
eq.symm $ (coeff_iterate_frobenius _ _ m).symm.trans $ (nat.sub_add_cancel hmn).symm ▸ rfl
lemma pth_root_frobenius : (pth_root R p).comp (frobenius _ p) = ring_hom.id _ :=
ring_hom.ext $ λ x, ext $ λ n,
by rw [ring_hom.comp_apply, ring_hom.id_apply, coeff_pth_root, coeff_frobenius]
lemma frobenius_pth_root : (frobenius _ p).comp (pth_root R p) = ring_hom.id _ :=
ring_hom.ext $ λ x, ext $ λ n,
by rw [ring_hom.comp_apply, ring_hom.id_apply, ring_hom.map_frobenius, coeff_pth_root,
← ring_hom.map_frobenius, coeff_frobenius]
lemma coeff_add_ne_zero {f : ring.perfection R p} {n : ℕ} (hfn : coeff R p n f ≠ 0) (k : ℕ) :
coeff R p (n + k) f ≠ 0 :=
nat.rec_on k hfn $ λ k ih h, ih $ by erw [← coeff_pow_p, ring_hom.map_pow, h, zero_pow hp.1.pos]
lemma coeff_ne_zero_of_le {f : ring.perfection R p} {m n : ℕ} (hfm : coeff R p m f ≠ 0)
(hmn : m ≤ n) : coeff R p n f ≠ 0 :=
let ⟨k, hk⟩ := nat.exists_eq_add_of_le hmn in hk.symm ▸ coeff_add_ne_zero hfm k
variables (R p)
instance perfect_ring : perfect_ring (ring.perfection R p) p :=
{ pth_root' := pth_root R p,
frobenius_pth_root' := congr_fun $ congr_arg ring_hom.to_fun $ @frobenius_pth_root R _ p _ _,
pth_root_frobenius' := congr_fun $ congr_arg ring_hom.to_fun $ @pth_root_frobenius R _ p _ _ }
instance ring (R : Type u₁) [comm_ring R] [char_p R p] : ring (ring.perfection R p) :=
((ring.perfection R p).to_subring $ λ n, by simp_rw [← frobenius_def, pi.neg_apply,
pi.one_apply, ring_hom.map_neg, ring_hom.map_one]).to_ring
instance comm_ring (R : Type u₁) [comm_ring R] [char_p R p] : comm_ring (ring.perfection R p) :=
((ring.perfection R p).to_subring $ λ n, by simp_rw [← frobenius_def, pi.neg_apply,
pi.one_apply, ring_hom.map_neg, ring_hom.map_one]).to_comm_ring
/-- Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* perfection S p`. -/
@[simps] def lift (R : Type u₁) [comm_semiring R] [char_p R p] [perfect_ring R p]
(S : Type u₂) [comm_semiring S] [char_p S p] :
(R →+* S) ≃ (R →+* ring.perfection S p) :=
{ to_fun := λ f,
{ to_fun := λ r, ⟨λ n, f $ _root_.pth_root R p ^[n] r,
λ n, by rw [← f.map_pow, function.iterate_succ_apply', pth_root_pow_p]⟩,
map_one' := ext $ λ n, (congr_arg f $ ring_hom.iterate_map_one _ _).trans f.map_one,
map_mul' := λ x y, ext $ λ n, (congr_arg f $ ring_hom.iterate_map_mul _ _ _ _).trans $
f.map_mul _ _,
map_zero' := ext $ λ n, (congr_arg f $ ring_hom.iterate_map_zero _ _).trans f.map_zero,
map_add' := λ x y, ext $ λ n, (congr_arg f $ ring_hom.iterate_map_add _ _ _ _).trans $
f.map_add _ _ },
inv_fun := ring_hom.comp $ coeff S p 0,
left_inv := λ f, ring_hom.ext $ λ r, rfl,
right_inv := λ f, ring_hom.ext $ λ r, ext $ λ n,
show coeff S p 0 (f (_root_.pth_root R p ^[n] r)) = coeff S p n (f r),
by rw [← coeff_iterate_frobenius _ 0 n, zero_add, ← ring_hom.map_iterate_frobenius,
right_inverse_pth_root_frobenius.iterate] }
lemma hom_ext {R : Type u₁} [comm_semiring R] [char_p R p] [perfect_ring R p]
{S : Type u₂} [comm_semiring S] [char_p S p] {f g : R →+* ring.perfection S p}
(hfg : ∀ x, coeff S p 0 (f x) = coeff S p 0 (g x)) : f = g :=
(lift p R S).symm.injective $ ring_hom.ext hfg
variables {R} {S : Type u₂} [comm_semiring S] [char_p S p]
/-- A ring homomorphism `R →+* S` induces `perfection R p →+* perfection S p` -/
@[simps] def map (φ : R →+* S) : ring.perfection R p →+* ring.perfection S p :=
{ to_fun := λ f, ⟨λ n, φ (coeff R p n f), λ n, by rw [← φ.map_pow, coeff_pow_p']⟩,
map_one' := subtype.eq $ funext $ λ n, φ.map_one,
map_mul' := λ f g, subtype.eq $ funext $ λ n, φ.map_mul _ _,
map_zero' := subtype.eq $ funext $ λ n, φ.map_zero,
map_add' := λ f g, subtype.eq $ funext $ λ n, φ.map_add _ _ }
lemma coeff_map (φ : R →+* S) (f : ring.perfection R p) (n : ℕ) :
coeff S p n (map p φ f) = φ (coeff R p n f) :=
rfl
end perfection
/-- A perfection map to a ring of characteristic `p` is a map that is isomorphic
to its perfection. -/
@[nolint has_inhabited_instance] structure perfection_map (p : ℕ) [fact p.prime]
{R : Type u₁} [comm_semiring R] [char_p R p]
{P : Type u₂} [comm_semiring P] [char_p P p] [perfect_ring P p] (π : P →+* R) : Prop :=
(injective : ∀ ⦃x y : P⦄, (∀ n, π (pth_root P p ^[n] x) = π (pth_root P p ^[n] y)) → x = y)
(surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) →
∃ x : P, ∀ n, π (pth_root P p ^[n] x) = f n)
namespace perfection_map
variables {p : ℕ} [fact p.prime]
variables {R : Type u₁} [comm_semiring R] [char_p R p]
variables {P : Type u₃} [comm_semiring P] [char_p P p] [perfect_ring P p]
/-- Create a `perfection_map` from an isomorphism to the perfection. -/
@[simps] lemma mk' {f : P →+* R} (g : P ≃+* ring.perfection R p)
(hfg : perfection.lift p P R f = g) :
perfection_map p f :=
{ injective := λ x y hxy, g.injective $ (ring_hom.ext_iff.1 hfg x).symm.trans $
eq.symm $ (ring_hom.ext_iff.1 hfg y).symm.trans $ perfection.ext $ λ n, (hxy n).symm,
surjective := λ y hy, let ⟨x, hx⟩ := g.surjective ⟨y, hy⟩ in
⟨x, λ n, show perfection.coeff R p n (perfection.lift p P R f x) =
perfection.coeff R p n ⟨y, hy⟩,
by rw [hfg, ← coe_fn_coe_base, hx]⟩ }
variables (p R P)
/-- The canonical perfection map from the perfection of a ring. -/
lemma of : perfection_map p (perfection.coeff R p 0) :=
mk' (ring_equiv.refl _) $ (equiv.apply_eq_iff_eq_symm_apply _).2 rfl
/-- For a perfect ring, it itself is the perfection. -/
lemma id [perfect_ring R p] : perfection_map p (ring_hom.id R) :=
{ injective := λ x y hxy, hxy 0,
surjective := λ f hf, ⟨f 0, λ n, show pth_root R p ^[n] (f 0) = f n,
from nat.rec_on n rfl $ λ n ih, injective_pow_p p $
by rw [function.iterate_succ_apply', pth_root_pow_p _, ih, hf]⟩ }
variables {p R P}
/-- A perfection map induces an isomorphism to the prefection. -/
noncomputable def equiv {π : P →+* R} (m : perfection_map p π) : P ≃+* ring.perfection R p :=
ring_equiv.of_bijective (perfection.lift p P R π)
⟨λ x y hxy, m.injective $ λ n, (congr_arg (perfection.coeff R p n) hxy : _),
λ f, let ⟨x, hx⟩ := m.surjective f.1 f.2 in ⟨x, perfection.ext $ hx⟩⟩
lemma equiv_apply {π : P →+* R} (m : perfection_map p π) (x : P) :
m.equiv x = perfection.lift p P R π x :=
rfl
lemma comp_equiv {π : P →+* R} (m : perfection_map p π) (x : P) :
perfection.coeff R p 0 (m.equiv x) = π x :=
rfl
lemma comp_equiv' {π : P →+* R} (m : perfection_map p π) :
(perfection.coeff R p 0).comp ↑m.equiv = π :=
ring_hom.ext $ λ x, rfl
lemma comp_symm_equiv {π : P →+* R} (m : perfection_map p π) (f : ring.perfection R p) :
π (m.equiv.symm f) = perfection.coeff R p 0 f :=
(m.comp_equiv _).symm.trans $ congr_arg _ $ m.equiv.apply_symm_apply f
lemma comp_symm_equiv' {π : P →+* R} (m : perfection_map p π) :
π.comp ↑m.equiv.symm = perfection.coeff R p 0 :=
ring_hom.ext m.comp_symm_equiv
variables (p R P)
/-- Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* P`,
where `P` is any perfection of `S`. -/
@[simps] noncomputable def lift [perfect_ring R p] (S : Type u₂) [comm_semiring S] [char_p S p]
(P : Type u₃) [comm_semiring P] [char_p P p] [perfect_ring P p]
(π : P →+* S) (m : perfection_map p π) :
(R →+* S) ≃ (R →+* P) :=
{ to_fun := λ f, ring_hom.comp ↑m.equiv.symm $ perfection.lift p R S f,
inv_fun := λ f, π.comp f,
left_inv := λ f, by { simp_rw [← ring_hom.comp_assoc, comp_symm_equiv'],
exact (perfection.lift p R S).symm_apply_apply f },
right_inv := λ f, ring_hom.ext $ λ x, m.equiv.injective $ (m.equiv.apply_symm_apply _).trans $
show perfection.lift p R S (π.comp f) x = ring_hom.comp ↑m.equiv f x,
from ring_hom.ext_iff.1 ((perfection.lift p R S).apply_eq_iff_eq_symm_apply.2 rfl) _ }
variables {R p}
lemma hom_ext [perfect_ring R p] {S : Type u₂} [comm_semiring S] [char_p S p]
{P : Type u₃} [comm_semiring P] [char_p P p] [perfect_ring P p]
(π : P →+* S) (m : perfection_map p π) {f g : R →+* P}
(hfg : ∀ x, π (f x) = π (g x)) : f = g :=
(lift p R S P π m).symm.injective $ ring_hom.ext hfg
variables {R P} (p) {S : Type u₂} [comm_semiring S] [char_p S p]
variables {Q : Type u₄} [comm_semiring Q] [char_p Q p] [perfect_ring Q p]
/-- A ring homomorphism `R →+* S` induces `P →+* Q`, a map of the respective perfections. -/
@[nolint unused_arguments]
noncomputable def map {π : P →+* R} (m : perfection_map p π) {σ : Q →+* S} (n : perfection_map p σ)
(φ : R →+* S) : P →+* Q :=
lift p P S Q σ n $ φ.comp π
lemma comp_map {π : P →+* R} (m : perfection_map p π) {σ : Q →+* S} (n : perfection_map p σ)
(φ : R →+* S) : σ.comp (map p m n φ) = φ.comp π :=
(lift p P S Q σ n).symm_apply_apply _
lemma map_map {π : P →+* R} (m : perfection_map p π) {σ : Q →+* S} (n : perfection_map p σ)
(φ : R →+* S) (x : P) : σ (map p m n φ x) = φ (π x) :=
ring_hom.ext_iff.1 (comp_map p m n φ) x
-- Why is this slow?
lemma map_eq_map (φ : R →+* S) :
@map p _ R _ _ _ _ _ _ S _ _ _ _ _ _ _ (of p R) _ (of p S) φ = perfection.map p φ :=
hom_ext _ (of p S) $ λ f, by rw [map_map, perfection.coeff_map]
end perfection_map
section perfectoid
variables (K : Type u₁) [field K] (v : valuation K ℝ≥0)
variables (O : Type u₂) [comm_ring O] [algebra O K] (hv : v.integers O)
variables (p : ℕ)
include hv
/-- `O/(p)` for `O`, ring of integers of `K`. -/
@[nolint unused_arguments has_inhabited_instance] def mod_p :=
(ideal.span {p} : ideal O).quotient
variables [hp : fact p.prime] [hvp : fact (v p ≠ 1)]
namespace mod_p
instance : comm_ring (mod_p K v O hv p) :=
ideal.quotient.comm_ring _
include hp hvp
instance : char_p (mod_p K v O hv p) p :=
char_p.quotient O p $ mt hv.one_of_is_unit $ ((algebra_map O K).map_nat_cast p).symm ▸ hvp.1
instance : nontrivial (mod_p K v O hv p) :=
char_p.nontrivial_of_char_ne_one hp.1.ne_one
section classical
local attribute [instance] classical.dec
omit hp hvp
/-- For a field `K` with valuation `v : K → ℝ≥0` and ring of integers `O`,
a function `O/(p) → ℝ≥0` that sends `0` to `0` and `x + (p)` to `v(x)` as long as `x ∉ (p)`. -/
noncomputable def pre_val (x : mod_p K v O hv p) : ℝ≥0 :=
if x = 0 then 0 else v (algebra_map O K x.out')
variables {K v O hv p}
lemma pre_val_mk {x : O} (hx : (ideal.quotient.mk _ x : mod_p K v O hv p) ≠ 0) :
pre_val K v O hv p (ideal.quotient.mk _ x) = v (algebra_map O K x) :=
begin
obtain ⟨r, hr⟩ := ideal.mem_span_singleton'.1 (ideal.quotient.eq.1 $ quotient.sound' $
@quotient.mk_out' O (ideal.span {p} : ideal O).quotient_rel x),
refine (if_neg hx).trans (v.map_eq_of_sub_lt $ lt_of_not_ge' _),
erw [← ring_hom.map_sub, ← hr, hv.le_iff_dvd],
exact λ hprx, hx (ideal.quotient.eq_zero_iff_mem.2 $ ideal.mem_span_singleton.2 $
dvd_of_mul_left_dvd hprx),
end
lemma pre_val_zero : pre_val K v O hv p 0 = 0 :=
if_pos rfl
lemma pre_val_mul {x y : mod_p K v O hv p} (hxy0 : x * y ≠ 0) :
pre_val K v O hv p (x * y) = pre_val K v O hv p x * pre_val K v O hv p y :=
begin
have hx0 : x ≠ 0 := mt (by { rintro rfl, rw zero_mul }) hxy0,
have hy0 : y ≠ 0 := mt (by { rintro rfl, rw mul_zero }) hxy0,
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surjective y,
rw ← ring_hom.map_mul at hxy0 ⊢,
rw [pre_val_mk hx0, pre_val_mk hy0, pre_val_mk hxy0, ring_hom.map_mul, v.map_mul]
end
lemma pre_val_add (x y : mod_p K v O hv p) :
pre_val K v O hv p (x + y) ≤ max (pre_val K v O hv p x) (pre_val K v O hv p y) :=
begin
by_cases hx0 : x = 0, { rw [hx0, zero_add], exact le_max_right _ _ },
by_cases hy0 : y = 0, { rw [hy0, add_zero], exact le_max_left _ _ },
by_cases hxy0 : x + y = 0, { rw [hxy0, pre_val_zero], exact zero_le _ },
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surjective y,
rw ← ring_hom.map_add at hxy0 ⊢,
rw [pre_val_mk hx0, pre_val_mk hy0, pre_val_mk hxy0, ring_hom.map_add], exact v.map_add _ _
end
lemma v_p_lt_pre_val {x : mod_p K v O hv p} : v p < pre_val K v O hv p x ↔ x ≠ 0 :=
begin
refine ⟨λ h hx, by { rw [hx, pre_val_zero] at h, exact not_lt_zero' h },
λ h, lt_of_not_ge' $ λ hp, h _⟩,
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
rw [pre_val_mk h, ← (algebra_map O K).map_nat_cast p, hv.le_iff_dvd] at hp,
rw [ideal.quotient.eq_zero_iff_mem, ideal.mem_span_singleton], exact hp
end
lemma pre_val_eq_zero {x : mod_p K v O hv p} : pre_val K v O hv p x = 0 ↔ x = 0 :=
⟨λ hvx, classical.by_contradiction $ λ hx0 : x ≠ 0,
by { rw [← v_p_lt_pre_val, hvx] at hx0, exact not_lt_zero' hx0 },
λ hx, hx.symm ▸ pre_val_zero⟩
variables (hv hvp)
lemma v_p_lt_val {x : O} :
v p < v (algebra_map O K x) ↔ (ideal.quotient.mk _ x : mod_p K v O hv p) ≠ 0 :=
by rw [lt_iff_not_ge', not_iff_not, ← (algebra_map O K).map_nat_cast p, hv.le_iff_dvd,
ideal.quotient.eq_zero_iff_mem, ideal.mem_span_singleton]
open nnreal
variables {hv} [hvp]
include hp
lemma mul_ne_zero_of_pow_p_ne_zero {x y : mod_p K v O hv p} (hx : x ^ p ≠ 0) (hy : y ^ p ≠ 0) :
x * y ≠ 0 :=
begin
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surjective y,
have h1p : (0 : ℝ) < 1 / p := one_div_pos.2 (nat.cast_pos.2 hp.1.pos),
rw ← ring_hom.map_mul, rw ← ring_hom.map_pow at hx hy,
rw ← v_p_lt_val hv at hx hy ⊢,
rw [ring_hom.map_pow, v.map_pow, ← rpow_lt_rpow_iff h1p, ← rpow_nat_cast, ← rpow_mul,
mul_one_div_cancel (nat.cast_ne_zero.2 hp.1.ne_zero : (p : ℝ) ≠ 0), rpow_one] at hx hy,
rw [ring_hom.map_mul, v.map_mul], refine lt_of_le_of_lt _ (mul_lt_mul'''' hx hy),
by_cases hvp : v p = 0, { rw hvp, exact zero_le _ }, replace hvp := zero_lt_iff.2 hvp,
conv_lhs { rw ← rpow_one (v p) }, rw ← rpow_add (ne_of_gt hvp),
refine rpow_le_rpow_of_exponent_ge hvp ((algebra_map O K).map_nat_cast p ▸ hv.2 _) _,
rw [← add_div, div_le_one (nat.cast_pos.2 hp.1.pos : 0 < (p : ℝ))], exact_mod_cast hp.1.two_le
end
end classical
end mod_p
include hp hvp
/-- Perfection of `O/(p)` where `O` is the ring of integers of `K`. -/
@[nolint has_inhabited_instance] def pre_tilt :=
ring.perfection (mod_p K v O hv p) p
namespace pre_tilt
instance : comm_ring (pre_tilt K v O hv p) :=
perfection.comm_ring p _
section classical
local attribute [instance] classical.dec
open perfection
/-- The valuation `Perfection(O/(p)) → ℝ≥0` as a function.
Given `f ∈ Perfection(O/(p))`, if `f = 0` then output `0`;
otherwise output `pre_val(f(n))^(p^n)` for any `n` such that `f(n) ≠ 0`. -/
noncomputable def val_aux (f : pre_tilt K v O hv p) : ℝ≥0 :=
if h : ∃ n, coeff _ _ n f ≠ 0
then mod_p.pre_val K v O hv p (coeff _ _ (nat.find h) f) ^ (p ^ nat.find h)
else 0
variables {K v O hv p}
lemma coeff_nat_find_add_ne_zero {f : pre_tilt K v O hv p} {h : ∃ n, coeff _ _ n f ≠ 0} (k : ℕ) :
coeff _ _ (nat.find h + k) f ≠ 0 :=
coeff_add_ne_zero (nat.find_spec h) k
lemma val_aux_eq {f : pre_tilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0) :
val_aux K v O hv p f = mod_p.pre_val K v O hv p (coeff _ _ n f) ^ (p ^ n) :=
begin
have h : ∃ n, coeff _ _ n f ≠ 0 := ⟨n, hfn⟩,
rw [val_aux, dif_pos h],
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le (nat.find_min' h hfn),
induction k with k ih, { refl },
obtain ⟨x, hx⟩ := ideal.quotient.mk_surjective (coeff _ _ (nat.find h + k + 1) f),
have h1 : (ideal.quotient.mk _ x : mod_p K v O hv p) ≠ 0 := hx.symm ▸ hfn,
have h2 : (ideal.quotient.mk _ (x ^ p) : mod_p K v O hv p) ≠ 0,
by { erw [ring_hom.map_pow, hx, ← ring_hom.map_pow, coeff_pow_p],
exact coeff_nat_find_add_ne_zero k },
erw [ih (coeff_nat_find_add_ne_zero k), ← hx, ← coeff_pow_p, ring_hom.map_pow, ← hx,
← ring_hom.map_pow, mod_p.pre_val_mk h1, mod_p.pre_val_mk h2,
ring_hom.map_pow, v.map_pow, ← pow_mul, pow_succ], refl
end
lemma val_aux_zero : val_aux K v O hv p 0 = 0 :=
dif_neg $ λ ⟨n, hn⟩, hn rfl
lemma val_aux_one : val_aux K v O hv p 1 = 1 :=
(val_aux_eq $ by exact one_ne_zero).trans $
by { rw [pow_zero, pow_one, ring_hom.map_one, ← (ideal.quotient.mk _).map_one, mod_p.pre_val_mk,
ring_hom.map_one, v.map_one], exact @one_ne_zero (mod_p K v O hv p) _ _ }
lemma val_aux_mul (f g : pre_tilt K v O hv p) :
val_aux K v O hv p (f * g) = val_aux K v O hv p f * val_aux K v O hv p g :=
begin
by_cases hf : f = 0, { rw [hf, zero_mul, val_aux_zero, zero_mul] },
by_cases hg : g = 0, { rw [hg, mul_zero, val_aux_zero, mul_zero] },
replace hf : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 (λ h, hf $ perfection.ext h),
replace hg : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 (λ h, hg $ perfection.ext h),
obtain ⟨m, hm⟩ := hf, obtain ⟨n, hn⟩ := hg,
replace hm := coeff_ne_zero_of_le hm (le_max_left m n),
replace hn := coeff_ne_zero_of_le hn (le_max_right m n),
have hfg : coeff _ _ (max m n + 1) (f * g) ≠ 0,
{ rw ring_hom.map_mul, refine mod_p.mul_ne_zero_of_pow_p_ne_zero _ _;
rw [← ring_hom.map_pow, coeff_pow_p]; assumption },
rw [val_aux_eq (coeff_add_ne_zero hm 1), val_aux_eq (coeff_add_ne_zero hn 1), val_aux_eq hfg],
rw ring_hom.map_mul at hfg ⊢, rw [mod_p.pre_val_mul hfg, mul_pow]
end
lemma val_aux_add (f g : pre_tilt K v O hv p) :
val_aux K v O hv p (f + g) ≤ max (val_aux K v O hv p f) (val_aux K v O hv p g) :=
begin
by_cases hf : f = 0, { rw [hf, zero_add, val_aux_zero, max_eq_right], exact zero_le _ },
by_cases hg : g = 0, { rw [hg, add_zero, val_aux_zero, max_eq_left], exact zero_le _ },
by_cases hfg : f + g = 0, { rw [hfg, val_aux_zero], exact zero_le _ },
replace hf : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 (λ h, hf $ perfection.ext h),
replace hg : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 (λ h, hg $ perfection.ext h),
replace hfg : ∃ n, coeff _ _ n (f + g) ≠ 0 := not_forall.1 (λ h, hfg $ perfection.ext h),
obtain ⟨m, hm⟩ := hf, obtain ⟨n, hn⟩ := hg, obtain ⟨k, hk⟩ := hfg,
replace hm := coeff_ne_zero_of_le hm (le_trans (le_max_left m n) (le_max_left _ k)),
replace hn := coeff_ne_zero_of_le hn (le_trans (le_max_right m n) (le_max_left _ k)),
replace hk := coeff_ne_zero_of_le hk (le_max_right (max m n) k),
rw [val_aux_eq hm, val_aux_eq hn, val_aux_eq hk, ring_hom.map_add],
cases le_max_iff.1
(mod_p.pre_val_add (coeff _ _ (max (max m n) k) f) (coeff _ _ (max (max m n) k) g)) with h h,
{ exact le_max_left_of_le (canonically_ordered_semiring.pow_le_pow_of_le_left h _) },
{ exact le_max_right_of_le (canonically_ordered_semiring.pow_le_pow_of_le_left h _) }
end
variables (K v O hv p)
/-- The valuation `Perfection(O/(p)) → ℝ≥0`.
Given `f ∈ Perfection(O/(p))`, if `f = 0` then output `0`;
otherwise output `pre_val(f(n))^(p^n)` for any `n` such that `f(n) ≠ 0`. -/
noncomputable def val : valuation (pre_tilt K v O hv p) ℝ≥0 :=
{ to_fun := val_aux K v O hv p,
map_one' := val_aux_one,
map_mul' := val_aux_mul,
map_zero' := val_aux_zero,
map_add' := val_aux_add }
variables {K v O hv p}
lemma map_eq_zero {f : pre_tilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 :=
begin
by_cases hf0 : f = 0, { rw hf0, exact iff_of_true (valuation.map_zero _) rfl },
obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 (λ h, hf0 $ perfection.ext h),
show val_aux K v O hv p f = 0 ↔ f = 0, refine iff_of_false (λ hvf, hn _) hf0,
rw val_aux_eq hn at hvf, replace hvf := pow_eq_zero hvf, rwa mod_p.pre_val_eq_zero at hvf
end
end classical
instance : integral_domain (pre_tilt K v O hv p) :=
{ exists_pair_ne := (char_p.nontrivial_of_char_ne_one hp.1.ne_one).1,
eq_zero_or_eq_zero_of_mul_eq_zero := λ f g hfg,
by { simp_rw ← map_eq_zero at hfg ⊢, contrapose! hfg, rw valuation.map_mul,
exact mul_ne_zero hfg.1 hfg.2 },
.. (infer_instance : comm_ring (pre_tilt K v O hv p)) }
end pre_tilt
/-- The tilt of a field, as defined in Perfectoid Spaces by Peter Scholze, as in
[scholze2011perfectoid]. Given a field `K` with valuation `K → ℝ≥0` and ring of integers `O`,
this is implemented as the fraction field of the perfection of `O/(p)`. -/
@[nolint has_inhabited_instance] def tilt :=
fraction_ring (pre_tilt K v O hv p)
namespace tilt
noncomputable instance : field (tilt K v O hv p) :=
fraction_ring.field
end tilt
end perfectoid
|
b9612fc2b04dfd2c45fdad6f73ea6935528c5ee4 | b70031c8e2c5337b91d7e70f1e0c5f528f7b0e77 | /src/analysis/special_functions/exp_log.lean | 2ea61436da3b327da5fb57994ad0a8ee9fb0845e | [
"Apache-2.0"
] | permissive | molodiuc/mathlib | cae2ba3ef1601c1f42ca0b625c79b061b63fef5b | 98ebe5a6739fbe254f9ee9d401882d4388f91035 | refs/heads/master | 1,674,237,127,059 | 1,606,353,533,000 | 1,606,353,533,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 28,774 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import data.complex.exponential
import analysis.complex.basic
import analysis.calculus.mean_value
import measure_theory.borel_space
/-!
# Complex and real exponential, real logarithm
## Main statements
This file establishes the basic analytical properties of the complex and real exponential functions
(continuity, differentiability, computation of the derivative).
It also contains the definition of the real logarithm function (as the inverse of the
exponential on `(0, +∞)`, extended to `ℝ` by setting `log (-x) = log x`) and its basic
properties (continuity, differentiability, formula for the derivative).
The complex logarithm is *not* defined in this file as it relies on trigonometric functions. See
instead `trigonometric.lean`.
## Tags
exp, log
-/
noncomputable theory
open finset filter metric asymptotics
open_locale classical topological_space
namespace complex
/-- The complex exponential is everywhere differentiable, with the derivative `exp x`. -/
lemma has_deriv_at_exp (x : ℂ) : has_deriv_at exp (exp x) x :=
begin
rw has_deriv_at_iff_is_o_nhds_zero,
have : (1 : ℕ) < 2 := by norm_num,
refine (is_O.of_bound (∥exp x∥) _).trans_is_o (is_o_pow_id this),
have : metric.ball (0 : ℂ) 1 ∈ nhds (0 : ℂ) := metric.ball_mem_nhds 0 zero_lt_one,
apply filter.mem_sets_of_superset this (λz hz, _),
simp only [metric.mem_ball, dist_zero_right] at hz,
simp only [exp_zero, mul_one, one_mul, add_comm, normed_field.norm_pow,
zero_add, set.mem_set_of_eq],
calc ∥exp (x + z) - exp x - z * exp x∥
= ∥exp x * (exp z - 1 - z)∥ : by { congr, rw [exp_add], ring }
... = ∥exp x∥ * ∥exp z - 1 - z∥ : normed_field.norm_mul _ _
... ≤ ∥exp x∥ * ∥z∥^2 :
mul_le_mul_of_nonneg_left (abs_exp_sub_one_sub_id_le (le_of_lt hz)) (norm_nonneg _)
end
lemma differentiable_exp : differentiable ℂ exp :=
λx, (has_deriv_at_exp x).differentiable_at
lemma differentiable_at_exp {x : ℂ} : differentiable_at ℂ exp x :=
differentiable_exp x
@[simp] lemma deriv_exp : deriv exp = exp :=
funext $ λ x, (has_deriv_at_exp x).deriv
@[simp] lemma iter_deriv_exp : ∀ n : ℕ, (deriv^[n] exp) = exp
| 0 := rfl
| (n+1) := by rw [function.iterate_succ_apply, deriv_exp, iter_deriv_exp n]
lemma continuous_exp : continuous exp :=
differentiable_exp.continuous
lemma times_cont_diff_exp : ∀ {n}, times_cont_diff ℂ n exp :=
begin
refine times_cont_diff_all_iff_nat.2 (λ n, _),
induction n with n ihn,
{ exact times_cont_diff_zero.2 continuous_exp },
{ rw times_cont_diff_succ_iff_deriv,
use differentiable_exp,
rwa deriv_exp }
end
lemma measurable_exp : measurable exp := continuous_exp.measurable
end complex
section
variables {f : ℂ → ℂ} {f' x : ℂ} {s : set ℂ}
lemma has_deriv_at.cexp (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, complex.exp (f x)) (complex.exp (f x) * f') x :=
(complex.has_deriv_at_exp (f x)).comp x hf
lemma has_deriv_within_at.cexp (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, complex.exp (f x)) (complex.exp (f x) * f') s x :=
(complex.has_deriv_at_exp (f x)).comp_has_deriv_within_at x hf
lemma deriv_within_cexp (hf : differentiable_within_at ℂ f s x)
(hxs : unique_diff_within_at ℂ s x) :
deriv_within (λx, complex.exp (f x)) s x = complex.exp (f x) * (deriv_within f s x) :=
hf.has_deriv_within_at.cexp.deriv_within hxs
@[simp] lemma deriv_cexp (hc : differentiable_at ℂ f x) :
deriv (λx, complex.exp (f x)) x = complex.exp (f x) * (deriv f x) :=
hc.has_deriv_at.cexp.deriv
end
section
variables {E : Type*} [normed_group E] [normed_space ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ}
{x : E} {s : set E}
lemma measurable.cexp {α : Type*} [measurable_space α] {f : α → ℂ} (hf : measurable f) :
measurable (λ x, complex.exp (f x)) :=
complex.measurable_exp.comp hf
lemma has_fderiv_within_at.cexp (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, complex.exp (f x)) (complex.exp (f x) • f') s x :=
(complex.has_deriv_at_exp (f x)).comp_has_fderiv_within_at x hf
lemma has_fderiv_at.cexp (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, complex.exp (f x)) (complex.exp (f x) • f') x :=
has_fderiv_within_at_univ.1 $ hf.has_fderiv_within_at.cexp
lemma differentiable_within_at.cexp (hf : differentiable_within_at ℂ f s x) :
differentiable_within_at ℂ (λ x, complex.exp (f x)) s x :=
hf.has_fderiv_within_at.cexp.differentiable_within_at
@[simp] lemma differentiable_at.cexp (hc : differentiable_at ℂ f x) :
differentiable_at ℂ (λx, complex.exp (f x)) x :=
hc.has_fderiv_at.cexp.differentiable_at
lemma differentiable_on.cexp (hc : differentiable_on ℂ f s) :
differentiable_on ℂ (λx, complex.exp (f x)) s :=
λx h, (hc x h).cexp
@[simp] lemma differentiable.cexp (hc : differentiable ℂ f) :
differentiable ℂ (λx, complex.exp (f x)) :=
λx, (hc x).cexp
lemma times_cont_diff.cexp {n} (h : times_cont_diff ℂ n f) :
times_cont_diff ℂ n (λ x, complex.exp (f x)) :=
complex.times_cont_diff_exp.comp h
lemma times_cont_diff_at.cexp {n} (hf : times_cont_diff_at ℂ n f x) :
times_cont_diff_at ℂ n (λ x, complex.exp (f x)) x :=
complex.times_cont_diff_exp.times_cont_diff_at.comp x hf
lemma times_cont_diff_on.cexp {n} (hf : times_cont_diff_on ℂ n f s) :
times_cont_diff_on ℂ n (λ x, complex.exp (f x)) s :=
complex.times_cont_diff_exp.comp_times_cont_diff_on hf
lemma times_cont_diff_within_at.cexp {n} (hf : times_cont_diff_within_at ℂ n f s x) :
times_cont_diff_within_at ℂ n (λ x, complex.exp (f x)) s x :=
complex.times_cont_diff_exp.times_cont_diff_at.comp_times_cont_diff_within_at x hf
end
namespace real
variables {x y z : ℝ}
lemma has_deriv_at_exp (x : ℝ) : has_deriv_at exp (exp x) x :=
(complex.has_deriv_at_exp x).real_of_complex
lemma times_cont_diff_exp {n} : times_cont_diff ℝ n exp :=
complex.times_cont_diff_exp.real_of_complex
lemma differentiable_exp : differentiable ℝ exp :=
λx, (has_deriv_at_exp x).differentiable_at
lemma differentiable_at_exp : differentiable_at ℝ exp x :=
differentiable_exp x
@[simp] lemma deriv_exp : deriv exp = exp :=
funext $ λ x, (has_deriv_at_exp x).deriv
@[simp] lemma iter_deriv_exp : ∀ n : ℕ, (deriv^[n] exp) = exp
| 0 := rfl
| (n+1) := by rw [function.iterate_succ_apply, deriv_exp, iter_deriv_exp n]
lemma continuous_exp : continuous exp :=
differentiable_exp.continuous
lemma measurable_exp : measurable exp := continuous_exp.measurable
end real
section
/-! Register lemmas for the derivatives of the composition of `real.exp` with a differentiable
function, for standalone use and use with `simp`. -/
variables {f : ℝ → ℝ} {f' x : ℝ} {s : set ℝ}
lemma has_deriv_at.exp (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, real.exp (f x)) (real.exp (f x) * f') x :=
(real.has_deriv_at_exp (f x)).comp x hf
lemma has_deriv_within_at.exp (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, real.exp (f x)) (real.exp (f x) * f') s x :=
(real.has_deriv_at_exp (f x)).comp_has_deriv_within_at x hf
lemma deriv_within_exp (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, real.exp (f x)) s x = real.exp (f x) * (deriv_within f s x) :=
hf.has_deriv_within_at.exp.deriv_within hxs
@[simp] lemma deriv_exp (hc : differentiable_at ℝ f x) :
deriv (λx, real.exp (f x)) x = real.exp (f x) * (deriv f x) :=
hc.has_deriv_at.exp.deriv
end
section
/-! Register lemmas for the derivatives of the composition of `real.exp` with a differentiable
function, for standalone use and use with `simp`. -/
variables {E : Type*} [normed_group E] [normed_space ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ}
{x : E} {s : set E}
lemma measurable.exp {α : Type*} [measurable_space α] {f : α → ℝ} (hf : measurable f) :
measurable (λ x, real.exp (f x)) :=
real.measurable_exp.comp hf
lemma times_cont_diff.exp {n} (hf : times_cont_diff ℝ n f) :
times_cont_diff ℝ n (λ x, real.exp (f x)) :=
real.times_cont_diff_exp.comp hf
lemma times_cont_diff_at.exp {n} (hf : times_cont_diff_at ℝ n f x) :
times_cont_diff_at ℝ n (λ x, real.exp (f x)) x :=
real.times_cont_diff_exp.times_cont_diff_at.comp x hf
lemma times_cont_diff_on.exp {n} (hf : times_cont_diff_on ℝ n f s) :
times_cont_diff_on ℝ n (λ x, real.exp (f x)) s :=
real.times_cont_diff_exp.comp_times_cont_diff_on hf
lemma times_cont_diff_within_at.exp {n} (hf : times_cont_diff_within_at ℝ n f s x) :
times_cont_diff_within_at ℝ n (λ x, real.exp (f x)) s x :=
real.times_cont_diff_exp.times_cont_diff_at.comp_times_cont_diff_within_at x hf
lemma has_fderiv_within_at.exp (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, real.exp (f x)) (real.exp (f x) • f') s x :=
begin
convert (has_deriv_at_iff_has_fderiv_at.1 $
real.has_deriv_at_exp (f x)).comp_has_fderiv_within_at x hf,
ext y, simp [mul_comm]
end
lemma has_fderiv_at.exp (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, real.exp (f x)) (real.exp (f x) • f') x :=
has_fderiv_within_at_univ.1 $ hf.has_fderiv_within_at.exp
lemma differentiable_within_at.exp (hf : differentiable_within_at ℝ f s x) :
differentiable_within_at ℝ (λ x, real.exp (f x)) s x :=
hf.has_fderiv_within_at.exp.differentiable_within_at
@[simp] lemma differentiable_at.exp (hc : differentiable_at ℝ f x) :
differentiable_at ℝ (λx, real.exp (f x)) x :=
hc.has_fderiv_at.exp.differentiable_at
lemma differentiable_on.exp (hc : differentiable_on ℝ f s) :
differentiable_on ℝ (λx, real.exp (f x)) s :=
λx h, (hc x h).exp
@[simp] lemma differentiable.exp (hc : differentiable ℝ f) :
differentiable ℝ (λx, real.exp (f x)) :=
λx, (hc x).exp
lemma fderiv_within_exp (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λx, real.exp (f x)) s x = real.exp (f x) • (fderiv_within ℝ f s x) :=
hf.has_fderiv_within_at.exp.fderiv_within hxs
@[simp] lemma fderiv_exp (hc : differentiable_at ℝ f x) :
fderiv ℝ (λx, real.exp (f x)) x = real.exp (f x) • (fderiv ℝ f x) :=
hc.has_fderiv_at.exp.fderiv
end
namespace real
variables {x y z : ℝ}
lemma exists_exp_eq_of_pos {x : ℝ} (hx : 0 < x) : ∃ y, exp y = x :=
have ∀ {z:ℝ}, 1 ≤ z → z ∈ set.range exp,
from λ z hz, intermediate_value_univ 0 (z - 1) continuous_exp
⟨by simpa, by simpa using add_one_le_exp_of_nonneg (sub_nonneg.2 hz)⟩,
match le_total x 1 with
| (or.inl hx1) := let ⟨y, hy⟩ := this (one_le_inv hx hx1) in
⟨-y, by rw [exp_neg, hy, inv_inv']⟩
| (or.inr hx1) := this hx1
end
/-- The real logarithm function, equal to the inverse of the exponential for `x > 0`,
to `log |x|` for `x < 0`, and to `0` for `0`. We use this unconventional extension to
`(-∞, 0]` as it gives the formula `log (x * y) = log x + log y` for all nonzero `x` and `y`, and
the derivative of `log` is `1/x` away from `0`. -/
@[pp_nodot] noncomputable def log (x : ℝ) : ℝ :=
if hx : x ≠ 0 then classical.some (exists_exp_eq_of_pos (abs_pos.mpr hx)) else 0
lemma exp_log_eq_abs (hx : x ≠ 0) : exp (log x) = abs x :=
by { rw [log, dif_pos hx], exact classical.some_spec (exists_exp_eq_of_pos ((abs_pos.mpr hx))) }
lemma exp_log (hx : 0 < x) : exp (log x) = x :=
by { rw exp_log_eq_abs (ne_of_gt hx), exact abs_of_pos hx }
lemma range_exp : set.range exp = {x | 0 < x} :=
set.ext $ λ x, ⟨by { rintro ⟨x, rfl⟩, exact exp_pos x }, λ hx, ⟨log x, exp_log hx⟩⟩
lemma exp_log_of_neg (hx : x < 0) : exp (log x) = -x :=
by { rw exp_log_eq_abs (ne_of_lt hx), exact abs_of_neg hx }
@[simp] lemma log_exp (x : ℝ) : log (exp x) = x :=
exp_injective $ exp_log (exp_pos x)
lemma log_surjective : function.surjective log :=
λ x, ⟨exp x, log_exp x⟩
@[simp] lemma range_log : set.range log = set.univ :=
log_surjective.range_eq
@[simp] lemma log_zero : log 0 = 0 :=
by simp [log]
@[simp] lemma log_one : log 1 = 0 :=
exp_injective $ by rw [exp_log zero_lt_one, exp_zero]
@[simp] lemma log_abs (x : ℝ) : log (abs x) = log x :=
begin
by_cases h : x = 0,
{ simp [h] },
{ apply exp_injective,
rw [exp_log_eq_abs h, exp_log_eq_abs, abs_abs],
simp [h] }
end
@[simp] lemma log_neg_eq_log (x : ℝ) : log (-x) = log x :=
by rw [← log_abs x, ← log_abs (-x), abs_neg]
lemma log_mul (hx : x ≠ 0) (hy : y ≠ 0) : log (x * y) = log x + log y :=
exp_injective $
by rw [exp_log_eq_abs (mul_ne_zero hx hy), exp_add, exp_log_eq_abs hx, exp_log_eq_abs hy, abs_mul]
@[simp] lemma log_inv (x : ℝ) : log (x⁻¹) = -log x :=
begin
by_cases hx : x = 0, { simp [hx] },
apply eq_neg_of_add_eq_zero,
rw [← log_mul (inv_ne_zero hx) hx, inv_mul_cancel hx, log_one]
end
lemma log_le_log (h : 0 < x) (h₁ : 0 < y) : real.log x ≤ real.log y ↔ x ≤ y :=
⟨λ h₂, by rwa [←real.exp_le_exp, real.exp_log h, real.exp_log h₁] at h₂, λ h₂,
(real.exp_le_exp).1 $ by rwa [real.exp_log h₁, real.exp_log h]⟩
lemma log_lt_log (hx : 0 < x) : x < y → log x < log y :=
by { intro h, rwa [← exp_lt_exp, exp_log hx, exp_log (lt_trans hx h)] }
lemma log_lt_log_iff (hx : 0 < x) (hy : 0 < y) : log x < log y ↔ x < y :=
by { rw [← exp_lt_exp, exp_log hx, exp_log hy] }
lemma log_pos_iff (hx : 0 < x) : 0 < log x ↔ 1 < x :=
by { rw ← log_one, exact log_lt_log_iff (by norm_num) hx }
lemma log_pos (hx : 1 < x) : 0 < log x :=
(log_pos_iff (lt_trans zero_lt_one hx)).2 hx
lemma log_neg_iff (h : 0 < x) : log x < 0 ↔ x < 1 :=
by { rw ← log_one, exact log_lt_log_iff h (by norm_num) }
lemma log_neg (h0 : 0 < x) (h1 : x < 1) : log x < 0 := (log_neg_iff h0).2 h1
lemma log_nonneg_iff (hx : 0 < x) : 0 ≤ log x ↔ 1 ≤ x :=
by rw [← not_lt, log_neg_iff hx, not_lt]
lemma log_nonneg (hx : 1 ≤ x) : 0 ≤ log x :=
(log_nonneg_iff (zero_lt_one.trans_le hx)).2 hx
lemma log_nonpos_iff (hx : 0 < x) : log x ≤ 0 ↔ x ≤ 1 :=
by rw [← not_lt, log_pos_iff hx, not_lt]
lemma log_nonpos_iff' (hx : 0 ≤ x) : log x ≤ 0 ↔ x ≤ 1 :=
begin
rcases hx.eq_or_lt with (rfl|hx),
{ simp [le_refl, zero_le_one] },
exact log_nonpos_iff hx
end
lemma log_nonpos (hx : 0 ≤ x) (h'x : x ≤ 1) : log x ≤ 0 :=
(log_nonpos_iff' hx).2 h'x
section prove_log_is_continuous
lemma tendsto_log_one_zero : tendsto log (𝓝 1) (𝓝 0) :=
begin
rw tendsto_nhds_nhds, assume ε ε0,
let δ := min (exp ε - 1) (1 - exp (-ε)),
have : 0 < δ,
refine lt_min (sub_pos_of_lt (by rwa one_lt_exp_iff)) (sub_pos_of_lt _),
by { rw exp_lt_one_iff, linarith },
use [δ, this], assume x h,
cases le_total 1 x with hx hx,
{ have h : x < exp ε,
rw [dist_eq, abs_of_nonneg (sub_nonneg_of_le hx)] at h,
linarith [(min_le_left _ _ : δ ≤ exp ε - 1)],
calc abs (log x - 0) = abs (log x) : by simp
... = log x : abs_of_nonneg $ log_nonneg hx
... < ε : by { rwa [← exp_lt_exp, exp_log], linarith }},
{ have h : exp (-ε) < x,
rw [dist_eq, abs_of_nonpos (sub_nonpos_of_le hx)] at h,
linarith [(min_le_right _ _ : δ ≤ 1 - exp (-ε))],
have : 0 < x := lt_trans (exp_pos _) h,
calc abs (log x - 0) = abs (log x) : by simp
... = -log x : abs_of_nonpos $ log_nonpos (le_of_lt this) hx
... < ε : by { rw [neg_lt, ← exp_lt_exp, exp_log], assumption' } }
end
lemma continuous_log' : continuous (λx : {x:ℝ // 0 < x}, log x) :=
continuous_iff_continuous_at.2 $ λ x,
begin
rw continuous_at,
let f₁ := λ h:{h:ℝ // 0 < h}, log (x.1 * h.1),
let f₂ := λ y:{y:ℝ // 0 < y}, subtype.mk (x.1 ⁻¹ * y.1) (mul_pos (inv_pos.2 x.2) y.2),
have H1 : tendsto f₁ (𝓝 ⟨1, zero_lt_one⟩) (𝓝 (log (x.1*1))),
have : f₁ = λ h:{h:ℝ // 0 < h}, log x.1 + log h.1,
ext h, rw ← log_mul (ne_of_gt x.2) (ne_of_gt h.2),
simp only [this, log_mul (ne_of_gt x.2) one_ne_zero, log_one],
exact tendsto_const_nhds.add (tendsto.comp tendsto_log_one_zero continuous_at_subtype_coe),
have H2 : tendsto f₂ (𝓝 x) (𝓝 ⟨x.1⁻¹ * x.1, mul_pos (inv_pos.2 x.2) x.2⟩),
rw tendsto_subtype_rng, exact tendsto_const_nhds.mul continuous_at_subtype_coe,
suffices h : tendsto (f₁ ∘ f₂) (𝓝 x) (𝓝 (log x.1)),
begin
convert h, ext y,
have : x.val * (x.val⁻¹ * y.val) = y.val,
rw [← mul_assoc, mul_inv_cancel (ne_of_gt x.2), one_mul],
show log (y.val) = log (x.val * (x.val⁻¹ * y.val)), rw this
end,
exact tendsto.comp (by rwa mul_one at H1)
(by { simp only [inv_mul_cancel (ne_of_gt x.2)] at H2, assumption })
end
lemma continuous_at_log (hx : 0 < x) : continuous_at log x :=
continuous_within_at.continuous_at (continuous_on_iff_continuous_restrict.2 continuous_log' _ hx)
(mem_nhds_sets (is_open_lt' _) hx)
/--
Three forms of the continuity of `real.log` are provided.
For the other two forms, see `real.continuous_log'` and `real.continuous_at_log`
-/
lemma continuous_log {α : Type*} [topological_space α] {f : α → ℝ} (h : ∀a, 0 < f a)
(hf : continuous f) : continuous (λa, log (f a)) :=
show continuous ((log ∘ @subtype.val ℝ (λr, 0 < r)) ∘ λa, ⟨f a, h a⟩),
from continuous_log'.comp (continuous_subtype_mk _ hf)
end prove_log_is_continuous
lemma has_deriv_at_log_of_pos (hx : 0 < x) : has_deriv_at log x⁻¹ x :=
have has_deriv_at log (exp $ log x)⁻¹ x,
from (has_deriv_at_exp $ log x).of_local_left_inverse (continuous_at_log hx)
(ne_of_gt $ exp_pos _) $ eventually.mono (mem_nhds_sets is_open_Ioi hx) @exp_log,
by rwa [exp_log hx] at this
lemma has_deriv_at_log (hx : x ≠ 0) : has_deriv_at log x⁻¹ x :=
begin
by_cases h : 0 < x, { exact has_deriv_at_log_of_pos h },
push_neg at h,
convert ((has_deriv_at_log_of_pos (neg_pos.mpr (lt_of_le_of_ne h hx)))
.comp x (has_deriv_at_id x).neg),
{ ext y, exact (log_neg_eq_log y).symm },
{ field_simp [hx] }
end
lemma measurable_log : measurable log :=
measurable_of_measurable_on_compl_singleton 0 $ continuous.measurable $
continuous_iff_continuous_at.2 $ λ x, (real.has_deriv_at_log x.2).continuous_at.comp
continuous_at_subtype_coe
end real
section log_differentiable
open real
variables {f : ℝ → ℝ} {x f' : ℝ} {s : set ℝ}
lemma measurable.log (hf : measurable f) : measurable (λ x, log (f x)) :=
measurable_log.comp hf
lemma has_deriv_within_at.log (hf : has_deriv_within_at f f' s x) (hx : f x ≠ 0) :
has_deriv_within_at (λ y, log (f y)) (f' / (f x)) s x :=
begin
convert (has_deriv_at_log hx).comp_has_deriv_within_at x hf,
field_simp
end
lemma has_deriv_at.log (hf : has_deriv_at f f' x) (hx : f x ≠ 0) :
has_deriv_at (λ y, log (f y)) (f' / f x) x :=
begin
rw ← has_deriv_within_at_univ at *,
exact hf.log hx
end
lemma differentiable_within_at.log (hf : differentiable_within_at ℝ f s x) (hx : f x ≠ 0) :
differentiable_within_at ℝ (λx, log (f x)) s x :=
(hf.has_deriv_within_at.log hx).differentiable_within_at
@[simp] lemma differentiable_at.log (hf : differentiable_at ℝ f x) (hx : f x ≠ 0) :
differentiable_at ℝ (λx, log (f x)) x :=
(hf.has_deriv_at.log hx).differentiable_at
lemma differentiable_on.log (hf : differentiable_on ℝ f s) (hx : ∀ x ∈ s, f x ≠ 0) :
differentiable_on ℝ (λx, log (f x)) s :=
λx h, (hf x h).log (hx x h)
@[simp] lemma differentiable.log (hf : differentiable ℝ f) (hx : ∀ x, f x ≠ 0) :
differentiable ℝ (λx, log (f x)) :=
λx, (hf x).log (hx x)
lemma deriv_within_log' (hf : differentiable_within_at ℝ f s x) (hx : f x ≠ 0)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, log (f x)) s x = (deriv_within f s x) / (f x) :=
(hf.has_deriv_within_at.log hx).deriv_within hxs
@[simp] lemma deriv_log' (hf : differentiable_at ℝ f x) (hx : f x ≠ 0) :
deriv (λx, log (f x)) x = (deriv f x) / (f x) :=
(hf.has_deriv_at.log hx).deriv
end log_differentiable
namespace real
/-- The real exponential function tends to `+∞` at `+∞`. -/
lemma tendsto_exp_at_top : tendsto exp at_top at_top :=
begin
have A : tendsto (λx:ℝ, x + 1) at_top at_top :=
tendsto_at_top_add_const_right at_top 1 tendsto_id,
have B : ∀ᶠ x in at_top, x + 1 ≤ exp x :=
eventually_at_top.2 ⟨0, λx hx, add_one_le_exp_of_nonneg hx⟩,
exact tendsto_at_top_mono' at_top B A
end
/-- The real exponential function tends to `0` at `-∞` or, equivalently, `exp(-x)` tends to `0`
at `+∞` -/
lemma tendsto_exp_neg_at_top_nhds_0 : tendsto (λx, exp (-x)) at_top (𝓝 0) :=
(tendsto_inv_at_top_zero.comp (tendsto_exp_at_top)).congr (λx, (exp_neg x).symm)
/-- The real exponential function tends to `1` at `0`. -/
lemma tendsto_exp_nhds_0_nhds_1 : tendsto exp (𝓝 0) (𝓝 1) :=
by { convert continuous_exp.tendsto 0, simp }
/-- The function `exp(x)/x^n` tends to `+∞` at `+∞`, for any natural number `n` -/
lemma tendsto_exp_div_pow_at_top (n : ℕ) : tendsto (λx, exp x / x^n) at_top at_top :=
begin
have n_pos : (0 : ℝ) < n + 1 := nat.cast_add_one_pos n,
have n_ne_zero : (n : ℝ) + 1 ≠ 0 := ne_of_gt n_pos,
have A : ∀x:ℝ, 0 < x → exp (x / (n+1)) / (n+1)^n ≤ exp x / x^n,
{ assume x hx,
let y := x / (n+1),
have y_pos : 0 < y := div_pos hx n_pos,
have : exp (x / (n+1)) ≤ (n+1)^n * (exp x / x^n), from calc
exp y = exp y * 1 : by simp
... ≤ exp y * (exp y / y)^n : begin
apply mul_le_mul_of_nonneg_left (one_le_pow_of_one_le _ n) (le_of_lt (exp_pos _)),
rw one_le_div y_pos,
apply le_trans _ (add_one_le_exp_of_nonneg (le_of_lt y_pos)),
exact le_add_of_le_of_nonneg (le_refl _) (zero_le_one)
end
... = exp y * exp (n * y) / y^n :
by rw [div_pow, exp_nat_mul, mul_div_assoc]
... = exp ((n + 1) * y) / y^n :
by rw [← exp_add, add_mul, one_mul, add_comm]
... = exp x / (x / (n+1))^n :
by { dsimp [y], rw mul_div_cancel' _ n_ne_zero }
... = (n+1)^n * (exp x / x^n) :
by rw [← mul_div_assoc, div_pow, div_div_eq_mul_div, mul_comm],
rwa div_le_iff' (pow_pos n_pos n) },
have B : ∀ᶠ x in at_top, exp (x / (n+1)) / (n+1)^n ≤ exp x / x^n :=
mem_at_top_sets.2 ⟨1, λx hx, A _ (lt_of_lt_of_le zero_lt_one hx)⟩,
have C : tendsto (λx, exp (x / (n+1)) / (n+1)^n) at_top at_top :=
tendsto_at_top_div (pow_pos n_pos n)
(tendsto_exp_at_top.comp (tendsto_at_top_div (nat.cast_add_one_pos n) tendsto_id)),
exact tendsto_at_top_mono' at_top B C
end
/-- The function `x^n * exp(-x)` tends to `0` at `+∞`, for any natural number `n`. -/
lemma tendsto_pow_mul_exp_neg_at_top_nhds_0 (n : ℕ) : tendsto (λx, x^n * exp (-x)) at_top (𝓝 0) :=
(tendsto_inv_at_top_zero.comp (tendsto_exp_div_pow_at_top n)).congr $ λx,
by rw [function.comp_app, inv_eq_one_div, div_div_eq_mul_div, one_mul, div_eq_mul_inv, exp_neg]
/-- The function `(b * exp x + c) / (x ^ n)` tends to `+∞` at `+∞`, for any positive natural number
`n` and any real numbers `b` and `c` such that `b` is positive. -/
lemma tendsto_mul_exp_add_div_pow_at_top (b c : ℝ) (n : ℕ) (hb : 0 < b) (hn : 1 ≤ n) :
tendsto (λ x, (b * (exp x) + c) / (x^n)) at_top at_top :=
begin
refine tendsto.congr' (eventually_eq_of_mem (Ioi_mem_at_top 0) _)
(tendsto_at_top_add_tendsto_right (tendsto_at_top_mul_left hb (tendsto_exp_div_pow_at_top n))
((tendsto_pow_neg_at_top hn).mul (@tendsto_const_nhds _ _ _ c _))),
intros x hx,
simp only [fpow_neg x n],
ring,
end
/-- The function `(x ^ n) / (b * exp x + c)` tends to `0` at `+∞`, for any positive natural number
`n` and any real numbers `b` and `c` such that `b` is nonzero. -/
lemma tendsto_div_pow_mul_exp_add_at_top (b c : ℝ) (n : ℕ) (hb : 0 ≠ b) (hn : 1 ≤ n) :
tendsto (λ x, x^n / (b * (exp x) + c)) at_top (𝓝 0) :=
begin
have H : ∀ d e, 0 < d → tendsto (λ (x:ℝ), x^n / (d * (exp x) + e)) at_top (𝓝 0),
{ intros b' c' h,
convert tendsto.inv_tendsto_at_top (tendsto_mul_exp_add_div_pow_at_top b' c' n h hn),
ext x,
simpa only [pi.inv_apply] using inv_div.symm },
cases lt_or_gt_of_ne hb,
{ exact H b c h },
{ convert (H (-b) (-c) (neg_pos.mpr h)).neg,
{ ext x,
field_simp,
rw [← neg_add (b * exp x) c, neg_div_neg_eq] },
{ exact neg_zero.symm } },
end
/-- The real logarithm function tends to `+∞` at `+∞`. -/
lemma tendsto_log_at_top : tendsto log at_top at_top :=
begin
rw tendsto_at_top_at_top,
intro b,
use exp b,
intros a hab,
rw [← exp_le_exp, exp_log_eq_abs (ne_of_gt $ lt_of_lt_of_le (exp_pos b) hab)],
exact le_trans hab (le_abs_self a)
end
open_locale big_operators
/-- A crude lemma estimating the difference between `log (1-x)` and its Taylor series at `0`,
where the main point of the bound is that it tends to `0`. The goal is to deduce the series
expansion of the logarithm, in `has_sum_pow_div_log_of_abs_lt_1`.
-/
lemma abs_log_sub_add_sum_range_le {x : ℝ} (h : abs x < 1) (n : ℕ) :
abs ((∑ i in range n, x^(i+1)/(i+1)) + log (1-x)) ≤ (abs x)^(n+1) / (1 - abs x) :=
begin
/- For the proof, we show that the derivative of the function to be estimated is small,
and then apply the mean value inequality. -/
let F : ℝ → ℝ := λ x, ∑ i in range n, x^(i+1)/(i+1) + log (1-x),
-- First step: compute the derivative of `F`
have A : ∀ y ∈ set.Ioo (-1 : ℝ) 1, deriv F y = - (y^n) / (1 - y),
{ assume y hy,
have : (∑ i in range n, (↑i + 1) * y ^ i / (↑i + 1)) = (∑ i in range n, y ^ i),
{ congr' with i,
have : (i : ℝ) + 1 ≠ 0 := ne_of_gt (nat.cast_add_one_pos i),
field_simp [this, mul_comm] },
field_simp [F, this, ← geom_series_def, geom_sum (ne_of_lt hy.2),
sub_ne_zero_of_ne (ne_of_gt hy.2), sub_ne_zero_of_ne (ne_of_lt hy.2)],
ring },
-- second step: show that the derivative of `F` is small
have B : ∀ y ∈ set.Icc (-abs x) (abs x), abs (deriv F y) ≤ (abs x)^n / (1 - abs x),
{ assume y hy,
have : y ∈ set.Ioo (-(1 : ℝ)) 1 := ⟨lt_of_lt_of_le (neg_lt_neg h) hy.1, lt_of_le_of_lt hy.2 h⟩,
calc abs (deriv F y) = abs (-(y^n) / (1 - y)) : by rw [A y this]
... ≤ (abs x)^n / (1 - abs x) :
begin
have : abs y ≤ abs x := abs_le.2 hy,
have : 0 < 1 - abs x, by linarith,
have : 1 - abs x ≤ abs (1 - y) := le_trans (by linarith [hy.2]) (le_abs_self _),
simp only [← pow_abs, abs_div, abs_neg],
apply_rules [div_le_div, pow_nonneg, abs_nonneg, pow_le_pow_of_le_left]
end },
-- third step: apply the mean value inequality
have C : ∥F x - F 0∥ ≤ ((abs x)^n / (1 - abs x)) * ∥x - 0∥,
{ have : ∀ y ∈ set.Icc (- abs x) (abs x), differentiable_at ℝ F y,
{ assume y hy,
have : 1 - y ≠ 0 := sub_ne_zero_of_ne (ne_of_gt (lt_of_le_of_lt hy.2 h)),
simp [F, this] },
apply convex.norm_image_sub_le_of_norm_deriv_le this B (convex_Icc _ _) _ _,
{ simpa using abs_nonneg x },
{ simp [le_abs_self x, neg_le.mp (neg_le_abs_self x)] } },
-- fourth step: conclude by massaging the inequality of the third step
simpa [F, norm_eq_abs, div_mul_eq_mul_div, pow_succ'] using C
end
/-- Power series expansion of the logarithm around `1`. -/
theorem has_sum_pow_div_log_of_abs_lt_1 {x : ℝ} (h : abs x < 1) :
has_sum (λ (n : ℕ), x ^ (n + 1) / (n + 1)) (-log (1 - x)) :=
begin
rw summable.has_sum_iff_tendsto_nat,
show tendsto (λ (n : ℕ), ∑ (i : ℕ) in range n, x ^ (i + 1) / (i + 1)) at_top (𝓝 (-log (1 - x))),
{ rw [tendsto_iff_norm_tendsto_zero],
simp only [norm_eq_abs, sub_neg_eq_add],
refine squeeze_zero (λ n, abs_nonneg _) (abs_log_sub_add_sum_range_le h) _,
suffices : tendsto (λ (t : ℕ), abs x ^ (t + 1) / (1 - abs x)) at_top
(𝓝 (abs x * 0 / (1 - abs x))), by simpa,
simp only [pow_succ],
refine (tendsto_const_nhds.mul _).div_const,
exact tendsto_pow_at_top_nhds_0_of_lt_1 (abs_nonneg _) h },
show summable (λ (n : ℕ), x ^ (n + 1) / (n + 1)),
{ refine summable_of_norm_bounded _ (summable_geometric_of_lt_1 (abs_nonneg _) h) (λ i, _),
calc ∥x ^ (i + 1) / (i + 1)∥
= abs x ^ (i+1) / (i+1) :
begin
have : (0 : ℝ) ≤ i + 1 := le_of_lt (nat.cast_add_one_pos i),
rw [norm_eq_abs, abs_div, ← pow_abs, abs_of_nonneg this],
end
... ≤ abs x ^ (i+1) / (0 + 1) :
begin
apply_rules [div_le_div_of_le_left, pow_nonneg, abs_nonneg, add_le_add_right,
i.cast_nonneg],
norm_num,
end
... ≤ abs x ^ i :
by simpa [pow_succ'] using mul_le_of_le_one_right (pow_nonneg (abs_nonneg x) i) (le_of_lt h) }
end
end real
|
e918f7dbbc7dc4ab6a5068d0550d34ea3ebedfef | 94e33a31faa76775069b071adea97e86e218a8ee | /src/analysis/special_functions/bernstein.lean | 9965481e753404db10801a8701ae1b3f7f35d5db | [
"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 | 12,949 | lean | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import algebra.order.field
import ring_theory.polynomial.bernstein
import topology.continuous_function.polynomial
/-!
# Bernstein approximations and Weierstrass' theorem
We prove that the Bernstein approximations
```
∑ k : fin (n+1), f (k/n : ℝ) * n.choose k * x^k * (1-x)^(n-k)
```
for a continuous function `f : C([0,1], ℝ)` converge uniformly to `f` as `n` tends to infinity.
Our proof follows [Richard Beals' *Analysis, an introduction*][beals-analysis], §7D.
The original proof, due to [Bernstein](bernstein1912) in 1912, is probabilistic,
and relies on Bernoulli's theorem,
which gives bounds for how quickly the observed frequencies in a
Bernoulli trial approach the underlying probability.
The proof here does not directly rely on Bernoulli's theorem,
but can also be given a probabilistic account.
* Consider a weighted coin which with probability `x` produces heads,
and with probability `1-x` produces tails.
* The value of `bernstein n k x` is the probability that
such a coin gives exactly `k` heads in a sequence of `n` tosses.
* If such an appearance of `k` heads results in a payoff of `f(k / n)`,
the `n`-th Bernstein approximation for `f` evaluated at `x` is the expected payoff.
* The main estimate in the proof bounds the probability that
the observed frequency of heads differs from `x` by more than some `δ`,
obtaining a bound of `(4 * n * δ^2)⁻¹`, irrespective of `x`.
* This ensures that for `n` large, the Bernstein approximation is (uniformly) close to the
payoff function `f`.
(You don't need to think in these terms to follow the proof below: it's a giant `calc` block!)
This result proves Weierstrass' theorem that polynomials are dense in `C([0,1], ℝ)`,
although we defer an abstract statement of this until later.
-/
noncomputable theory
open_locale classical
open_locale big_operators
open_locale bounded_continuous_function
open_locale unit_interval
/--
The Bernstein polynomials, as continuous functions on `[0,1]`.
-/
def bernstein (n ν : ℕ) : C(I, ℝ) :=
(bernstein_polynomial ℝ n ν).to_continuous_map_on I
@[simp] lemma bernstein_apply (n ν : ℕ) (x : I) :
bernstein n ν x = n.choose ν * x^ν * (1-x)^(n-ν) :=
begin
dsimp [bernstein, polynomial.to_continuous_map_on, polynomial.to_continuous_map,
bernstein_polynomial],
simp,
end
lemma bernstein_nonneg {n ν : ℕ} {x : I} :
0 ≤ bernstein n ν x :=
begin
simp only [bernstein_apply],
exact mul_nonneg
(mul_nonneg (nat.cast_nonneg _) (pow_nonneg (by unit_interval) _))
(pow_nonneg (by unit_interval) _),
end
/-!
We now give a slight reformulation of `bernstein_polynomial.variance`.
-/
namespace bernstein
/--
Send `k : fin (n+1)` to the equally spaced points `k/n` in the unit interval.
-/
def z {n : ℕ} (k : fin (n+1)) : I :=
⟨(k : ℝ) / n,
begin
cases n,
{ norm_num },
{ have h₁ : 0 < (n.succ : ℝ) := by exact_mod_cast (nat.succ_pos _),
have h₂ : ↑k ≤ n.succ := by exact_mod_cast (fin.le_last k),
rw [set.mem_Icc, le_div_iff h₁, div_le_iff h₁],
norm_cast,
simp [h₂], },
end⟩
local postfix `/ₙ`:90 := z
lemma probability (n : ℕ) (x : I) :
∑ k : fin (n+1), bernstein n k x = 1 :=
begin
have := bernstein_polynomial.sum ℝ n,
apply_fun (λ p, polynomial.aeval (x : ℝ) p) at this,
simp [alg_hom.map_sum, finset.sum_range] at this,
exact this,
end
lemma variance {n : ℕ} (h : 0 < (n : ℝ)) (x : I) :
∑ k : fin (n+1), (x - k/ₙ : ℝ)^2 * bernstein n k x = x * (1-x) / n :=
begin
have h' : (n : ℝ) ≠ 0 := ne_of_gt h,
apply_fun (λ x : ℝ, x * n) using group_with_zero.mul_right_injective h',
apply_fun (λ x : ℝ, x * n) using group_with_zero.mul_right_injective h',
dsimp,
conv_lhs { simp only [finset.sum_mul, z], },
conv_rhs { rw div_mul_cancel _ h', },
have := bernstein_polynomial.variance ℝ n,
apply_fun (λ p, polynomial.aeval (x : ℝ) p) at this,
simp [alg_hom.map_sum, finset.sum_range, ←polynomial.nat_cast_mul] at this,
convert this using 1,
{ congr' 1, funext k,
rw [mul_comm _ (n : ℝ), mul_comm _ (n : ℝ), ←mul_assoc, ←mul_assoc],
congr' 1,
field_simp [h],
ring, },
{ ring, },
end
end bernstein
open bernstein
local postfix `/ₙ`:2000 := z
/--
The `n`-th approximation of a continuous function on `[0,1]` by Bernstein polynomials,
given by `∑ k, f (k/n) * bernstein n k x`.
-/
def bernstein_approximation (n : ℕ) (f : C(I, ℝ)) : C(I, ℝ) :=
∑ k : fin (n+1), f k/ₙ • bernstein n k
/-!
We now set up some of the basic machinery of the proof that the Bernstein approximations
converge uniformly.
A key player is the set `S f ε h n x`,
for some function `f : C(I, ℝ)`, `h : 0 < ε`, `n : ℕ` and `x : I`.
This is the set of points `k` in `fin (n+1)` such that
`k/n` is within `δ` of `x`, where `δ` is the modulus of uniform continuity for `f`,
chosen so `|f x - f y| < ε/2` when `|x - y| < δ`.
We show that if `k ∉ S`, then `1 ≤ δ^-2 * (x - k/n)^2`.
-/
namespace bernstein_approximation
@[simp] lemma apply (n : ℕ) (f : C(I, ℝ)) (x : I) :
bernstein_approximation n f x = ∑ k : fin (n+1), f k/ₙ * bernstein n k x :=
by simp [bernstein_approximation]
/--
The modulus of (uniform) continuity for `f`, chosen so `|f x - f y| < ε/2` when `|x - y| < δ`.
-/
def δ (f : C(I, ℝ)) (ε : ℝ) (h : 0 < ε) : ℝ := f.modulus (ε/2) (half_pos h)
lemma δ_pos {f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} : 0 < δ f ε h := f.modulus_pos
/--
The set of points `k` so `k/n` is within `δ` of `x`.
-/
def S (f : C(I, ℝ)) (ε : ℝ) (h : 0 < ε) (n : ℕ) (x : I) : finset (fin (n+1)) :=
{ k : fin (n+1) | dist k/ₙ x < δ f ε h }.to_finset
/--
If `k ∈ S`, then `f(k/n)` is close to `f x`.
-/
lemma lt_of_mem_S
{f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} {n : ℕ} {x : I} {k : fin (n+1)} (m : k ∈ S f ε h n x) :
|f k/ₙ - f x| < ε/2 :=
begin
apply f.dist_lt_of_dist_lt_modulus (ε/2) (half_pos h),
simpa [S] using m,
end
/--
If `k ∉ S`, then as `δ ≤ |x - k/n|`, we have the inequality `1 ≤ δ^-2 * (x - k/n)^2`.
This particular formulation will be helpful later.
-/
lemma le_of_mem_S_compl
{f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} {n : ℕ} {x : I} {k : fin (n+1)} (m : k ∈ (S f ε h n x)ᶜ) :
(1 : ℝ) ≤ (δ f ε h)^(-2 : ℤ) * (x - k/ₙ) ^ 2 :=
begin
simp only [finset.mem_compl, not_lt, set.mem_to_finset, set.mem_set_of_eq, S] at m,
erw [zpow_neg, ← div_eq_inv_mul, one_le_div (pow_pos δ_pos 2), sq_le_sq, abs_of_pos δ_pos],
rwa [dist_comm] at m
end
end bernstein_approximation
open bernstein_approximation
open bounded_continuous_function
open filter
open_locale topological_space
/--
The Bernstein approximations
```
∑ k : fin (n+1), f (k/n : ℝ) * n.choose k * x^k * (1-x)^(n-k)
```
for a continuous function `f : C([0,1], ℝ)` converge uniformly to `f` as `n` tends to infinity.
This is the proof given in [Richard Beals' *Analysis, an introduction*][beals-analysis], §7D,
and reproduced on wikipedia.
-/
theorem bernstein_approximation_uniform (f : C(I, ℝ)) :
tendsto (λ n : ℕ, bernstein_approximation n f) at_top (𝓝 f) :=
begin
simp only [metric.nhds_basis_ball.tendsto_right_iff, metric.mem_ball, dist_eq_norm],
intros ε h,
let δ := δ f ε h,
have nhds_zero := tendsto_const_div_at_top_nhds_0_nat (2 * ∥f∥ * δ ^ (-2 : ℤ)),
filter_upwards [nhds_zero.eventually (gt_mem_nhds (half_pos h)), eventually_gt_at_top 0]
with n nh npos',
have npos : 0 < (n:ℝ) := by exact_mod_cast npos',
-- Two easy inequalities we'll need later:
have w₁ : 0 ≤ 2 * ∥f∥ := mul_nonneg (by norm_num) (norm_nonneg f),
have w₂ : 0 ≤ 2 * ∥f∥ * δ^(-2 : ℤ) := mul_nonneg w₁ pow_minus_two_nonneg,
-- As `[0,1]` is compact, it suffices to check the inequality pointwise.
rw (continuous_map.norm_lt_iff _ h),
intro x,
-- The idea is to split up the sum over `k` into two sets,
-- `S`, where `x - k/n < δ`, and its complement.
let S := S f ε h n x,
calc
|(bernstein_approximation n f - f) x|
= |bernstein_approximation n f x - f x|
: rfl
... = |bernstein_approximation n f x - f x * 1|
: by rw mul_one
... = |bernstein_approximation n f x - f x * (∑ k : fin (n+1), bernstein n k x)|
: by rw bernstein.probability
... = |∑ k : fin (n+1), (f k/ₙ - f x) * bernstein n k x|
: by simp [bernstein_approximation, finset.mul_sum, sub_mul]
... ≤ ∑ k : fin (n+1), |(f k/ₙ - f x) * bernstein n k x|
: finset.abs_sum_le_sum_abs _ _
... = ∑ k : fin (n+1), |f k/ₙ - f x| * bernstein n k x
: by simp_rw [abs_mul, abs_eq_self.mpr bernstein_nonneg]
... = ∑ k in S, |f k/ₙ - f x| * bernstein n k x +
∑ k in Sᶜ, |f k/ₙ - f x| * bernstein n k x
: (S.sum_add_sum_compl _).symm
-- We'll now deal with the terms in `S` and the terms in `Sᶜ` in separate calc blocks.
... < ε/2 + ε/2 : add_lt_add_of_le_of_lt _ _
... = ε : add_halves ε,
{ -- We now work on the terms in `S`: uniform continuity and `bernstein.probability`
-- quickly give us a bound.
calc ∑ k in S, |f k/ₙ - f x| * bernstein n k x
≤ ∑ k in S, ε/2 * bernstein n k x
: finset.sum_le_sum
(λ k m, (mul_le_mul_of_nonneg_right (le_of_lt (lt_of_mem_S m))
bernstein_nonneg))
... = ε/2 * ∑ k in S, bernstein n k x
: by rw finset.mul_sum
-- In this step we increase the sum over `S` back to a sum over all of `fin (n+1)`,
-- so that we can use `bernstein.probability`.
... ≤ ε/2 * ∑ k : fin (n+1), bernstein n k x
: mul_le_mul_of_nonneg_left
(finset.sum_le_univ_sum_of_nonneg (λ k, bernstein_nonneg))
(le_of_lt (half_pos h))
... = ε/2 : by rw [bernstein.probability, mul_one] },
{ -- We now turn to working on `Sᶜ`: we control the difference term just using `∥f∥`,
-- and then insert a `δ^(-2) * (x - k/n)^2` factor
-- (which is at least one because we are not in `S`).
calc ∑ k in Sᶜ, |f k/ₙ - f x| * bernstein n k x
≤ ∑ k in Sᶜ, (2 * ∥f∥) * bernstein n k x
: finset.sum_le_sum
(λ k m, mul_le_mul_of_nonneg_right (f.dist_le_two_norm _ _)
bernstein_nonneg)
... = (2 * ∥f∥) * ∑ k in Sᶜ, bernstein n k x
: by rw finset.mul_sum
... ≤ (2 * ∥f∥) * ∑ k in Sᶜ, δ^(-2 : ℤ) * (x - k/ₙ)^2 * bernstein n k x
: mul_le_mul_of_nonneg_left
(finset.sum_le_sum (λ k m, begin
conv_lhs { rw ←one_mul (bernstein _ _ _), },
exact mul_le_mul_of_nonneg_right
(le_of_mem_S_compl m) bernstein_nonneg,
end)) w₁
-- Again enlarging the sum from `Sᶜ` to all of `fin (n+1)`
... ≤ (2 * ∥f∥) * ∑ k : fin (n+1), δ^(-2 : ℤ) * (x - k/ₙ)^2 * bernstein n k x
: mul_le_mul_of_nonneg_left
(finset.sum_le_univ_sum_of_nonneg
(λ k, mul_nonneg
(mul_nonneg pow_minus_two_nonneg (sq_nonneg _))
bernstein_nonneg)) w₁
... = (2 * ∥f∥) * δ^(-2 : ℤ) * ∑ k : fin (n+1), (x - k/ₙ)^2 * bernstein n k x
: by conv_rhs
{ rw [mul_assoc, finset.mul_sum], simp only [←mul_assoc], }
-- `bernstein.variance` and `x ∈ [0,1]` gives the uniform bound
... = (2 * ∥f∥) * δ^(-2 : ℤ) * x * (1-x) / n
: by { rw variance npos, ring, }
... ≤ (2 * ∥f∥) * δ^(-2 : ℤ) / n
: (div_le_div_right npos).mpr
begin
apply mul_nonneg_le_one_le w₂,
apply mul_nonneg_le_one_le w₂ le_rfl,
all_goals { unit_interval, },
end
... < ε/2 : nh, }
end
|
ae154ddbbac95a2da8379b4b238c27f00cedb70a | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/data/bitvec/basic.lean | 2b7cc5a1f1cfb2332edd78762727bd010a8c6906 | [
"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 | 3,679 | lean | /-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import data.bitvec.core
import data.fin
import tactic.norm_num
import tactic.monotonicity
namespace bitvec
instance (n : ℕ) : preorder (bitvec n) :=
preorder.lift bitvec.to_nat
/-- convert `fin` to `bitvec` -/
def of_fin {n : ℕ} (i : fin $ 2^n) : bitvec n :=
bitvec.of_nat _ i.val
lemma of_fin_val {n : ℕ} (i : fin $ 2^n) : (of_fin i).to_nat = i.val :=
by rw [of_fin,to_nat_of_nat,nat.mod_eq_of_lt]; apply i.is_lt
/-- convert `bitvec` to `fin` -/
def to_fin {n : ℕ} (i : bitvec n) : fin $ 2^n :=
@fin.of_nat' _ ⟨pow_pos (by norm_num) _⟩ i.to_nat
lemma add_lsb_eq_twice_add_one {x b} :
add_lsb x b = 2 * x + cond b 1 0 :=
by simp [add_lsb,two_mul]
lemma to_nat_eq_foldr_reverse {n : ℕ} (v : bitvec n) :
v.to_nat = v.to_list.reverse.foldr (flip add_lsb) 0 :=
by rw [list.foldr_reverse, flip]; refl
lemma to_nat_lt {n : ℕ} (v : bitvec n) : v.to_nat < 2^n :=
begin
suffices : v.to_nat + 1 ≤ 2 ^ n, { simpa },
rw to_nat_eq_foldr_reverse,
cases v with xs h,
dsimp [bitvec.to_nat,bits_to_nat],
rw ← list.length_reverse at h,
generalize_hyp : xs.reverse = ys at ⊢ h, clear xs,
induction ys generalizing n,
{ simp [← h] },
{ simp only [←h, pow_add, flip, list.length, list.foldr, pow_one],
rw [add_lsb_eq_twice_add_one],
transitivity 2 * list.foldr (λ (x : bool) (y : ℕ), add_lsb y x) 0 ys_tl + 2 * 1,
{ ac_mono, rw two_mul, mono, cases ys_hd; simp },
{ rw ← left_distrib, ac_mono, norm_num,
apply ys_ih, refl } },
end
lemma add_lsb_div_two {x b} : add_lsb x b / 2 = x :=
by cases b; simp only [nat.add_mul_div_left, add_lsb, ←two_mul, add_comm, nat.succ_pos',
nat.mul_div_right, gt_iff_lt, zero_add, cond]; norm_num
lemma to_bool_add_lsb_mod_two {x b} : to_bool (add_lsb x b % 2 = 1) = b :=
by cases b; simp only [to_bool_iff, nat.add_mul_mod_self_left, add_lsb, ←two_mul, add_comm,
bool.to_bool_false, nat.mul_mod_right, zero_add, cond, zero_ne_one]; norm_num
lemma of_nat_to_nat {n : ℕ} (v : bitvec n) : bitvec.of_nat _ v.to_nat = v :=
begin
cases v with xs h,
ext1,
change vector.to_list _ = xs,
dsimp [bitvec.to_nat,bits_to_nat],
rw ← list.length_reverse at h,
rw [← list.reverse_reverse xs,list.foldl_reverse],
generalize_hyp : xs.reverse = ys at ⊢ h, clear xs,
induction ys generalizing n,
{ cases h, simp [bitvec.of_nat] },
{ simp only [←nat.succ_eq_add_one, list.length] at h, subst n,
simp only [bitvec.of_nat, vector.to_list_cons, vector.to_list_nil, list.reverse_cons,
vector.to_list_append, list.foldr],
erw [add_lsb_div_two, to_bool_add_lsb_mod_two],
congr, apply ys_ih, refl }
end
lemma to_fin_val {n : ℕ} (v : bitvec n) : (to_fin v : ℕ) = v.to_nat :=
by rw [to_fin, fin.coe_of_nat_eq_mod', nat.mod_eq_of_lt]; apply to_nat_lt
lemma to_fin_le_to_fin_of_le {n} {v₀ v₁ : bitvec n} (h : v₀ ≤ v₁) : v₀.to_fin ≤ v₁.to_fin :=
show (v₀.to_fin : ℕ) ≤ v₁.to_fin,
by rw [to_fin_val,to_fin_val]; exact h
lemma of_fin_le_of_fin_of_le {n : ℕ} {i j : fin (2^n)} (h : i ≤ j) : of_fin i ≤ of_fin j :=
show (bitvec.of_nat n i).to_nat ≤ (bitvec.of_nat n j).to_nat,
by { simp only [to_nat_of_nat, nat.mod_eq_of_lt, fin.is_lt], exact h }
lemma to_fin_of_fin {n} (i : fin $ 2^n) : (of_fin i).to_fin = i :=
fin.eq_of_veq (by simp [to_fin_val, of_fin, to_nat_of_nat, nat.mod_eq_of_lt, i.is_lt])
lemma of_fin_to_fin {n} (v : bitvec n) : of_fin (to_fin v) = v :=
by dsimp [of_fin]; rw [to_fin_val, of_nat_to_nat]
end bitvec
|
5699b395840f7bcaad00475c5aeeabd86a802790 | a45212b1526d532e6e83c44ddca6a05795113ddc | /src/group_theory/sylow.lean | 249864b8cddcfa8bd5a784227d1c26bf92311442 | [
"Apache-2.0"
] | permissive | fpvandoorn/mathlib | b21ab4068db079cbb8590b58fda9cc4bc1f35df4 | b3433a51ea8bc07c4159c1073838fc0ee9b8f227 | refs/heads/master | 1,624,791,089,608 | 1,556,715,231,000 | 1,556,715,231,000 | 165,722,980 | 5 | 0 | Apache-2.0 | 1,552,657,455,000 | 1,547,494,646,000 | Lean | UTF-8 | Lean | false | false | 11,385 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import group_theory.group_action group_theory.quotient_group
import group_theory.order_of_element data.zmod.basic
open equiv fintype finset mul_action function
open equiv.perm is_subgroup list quotient_group
universes u v w
variables {G : Type u} {α : Type v} {β : Type w} [group G]
local attribute [instance, priority 0] subtype.fintype set_fintype classical.prop_decidable
namespace mul_action
variables [mul_action G α]
lemma mem_fixed_points_iff_card_orbit_eq_one {a : α}
[fintype (orbit G a)] : a ∈ fixed_points G α ↔ card (orbit G a) = 1 :=
begin
rw [fintype.card_eq_one_iff, mem_fixed_points],
split,
{ exact λ h, ⟨⟨a, mem_orbit_self _⟩, λ ⟨b, ⟨x, hx⟩⟩, subtype.eq $ by simp [h x, hx.symm]⟩ },
{ assume h x,
rcases h with ⟨⟨z, hz⟩, hz₁⟩,
exact calc x • a = z : subtype.mk.inj (hz₁ ⟨x • a, mem_orbit _ _⟩)
... = a : (subtype.mk.inj (hz₁ ⟨a, mem_orbit_self _⟩)).symm }
end
lemma card_modeq_card_fixed_points [fintype α] [fintype G] [fintype (fixed_points G α)]
{p n : ℕ} (hp : nat.prime p) (h : card G = p ^ n) : card α ≡ card (fixed_points G α) [MOD p] :=
calc card α = card (Σ y : quotient (orbit_rel G α), {x // quotient.mk' x = y}) :
card_congr (equiv_fib (@quotient.mk' _ (orbit_rel G α)))
... = univ.sum (λ a : quotient (orbit_rel G α), card {x // quotient.mk' x = a}) : card_sigma _
... ≡ (@univ (fixed_points G α) _).sum (λ _, 1) [MOD p] :
begin
rw [← zmodp.eq_iff_modeq_nat hp, sum_nat_cast, sum_nat_cast],
refine eq.symm (sum_bij_ne_zero (λ a _ _, quotient.mk' a.1)
(λ _ _ _, mem_univ _)
(λ a₁ a₂ _ _ _ _ h,
subtype.eq ((mem_fixed_points' α).1 a₂.2 a₁.1 (quotient.exact' h)))
(λ b, _)
(λ a ha _, by rw [← mem_fixed_points_iff_card_orbit_eq_one.1 a.2];
simp only [quotient.eq']; congr)),
{ refine quotient.induction_on' b (λ b _ hb, _),
have : card (orbit G b) ∣ p ^ n,
{ rw [← h, fintype.card_congr (orbit_equiv_quotient_stabilizer G b)];
exact card_quotient_dvd_card _ },
rcases (nat.dvd_prime_pow hp).1 this with ⟨k, _, hk⟩,
have hb' :¬ p ^ 1 ∣ p ^ k,
{ rw [nat.pow_one, ← hk, ← nat.modeq.modeq_zero_iff, ← zmodp.eq_iff_modeq_nat hp,
nat.cast_zero, ← ne.def],
exact eq.mpr (by simp only [quotient.eq']; congr) hb },
have : k = 0 := nat.le_zero_iff.1 (nat.le_of_lt_succ (lt_of_not_ge (mt (nat.pow_dvd_pow p) hb'))),
refine ⟨⟨b, mem_fixed_points_iff_card_orbit_eq_one.2 $ by rw [hk, this, nat.pow_zero]⟩, mem_univ _,
by simp [zero_ne_one], rfl⟩ }
end
... = _ : by simp; refl
end mul_action
lemma quotient_group.card_preimage_mk [fintype G] (s : set G) [is_subgroup s]
(t : set (quotient s)) : fintype.card (quotient_group.mk ⁻¹' t) =
fintype.card s * fintype.card t :=
by rw [← fintype.card_prod, fintype.card_congr
(preimage_mk_equiv_subgroup_times_set _ _)]
namespace sylow
def mk_vector_prod_eq_one (n : ℕ) (v : vector G n) : vector G (n+1) :=
v.to_list.prod⁻¹ :: v
lemma mk_vector_prod_eq_one_inj (n : ℕ) : injective (@mk_vector_prod_eq_one G _ n) :=
λ ⟨v, _⟩ ⟨w, _⟩ h, subtype.eq (show v = w, by injection h with h; injection h)
def vectors_prod_eq_one (G : Type*) [group G] (n : ℕ) : set (vector G n) :=
{v | v.to_list.prod = 1}
lemma mem_vectors_prod_eq_one {n : ℕ} (v : vector G n) :
v ∈ vectors_prod_eq_one G n ↔ v.to_list.prod = 1 := iff.rfl
lemma mem_vectors_prod_eq_one_iff {n : ℕ} (v : vector G (n + 1)) :
v ∈ vectors_prod_eq_one G (n + 1) ↔ v ∈ set.range (@mk_vector_prod_eq_one G _ n) :=
⟨λ (h : v.to_list.prod = 1), ⟨v.tail,
begin
unfold mk_vector_prod_eq_one,
conv {to_rhs, rw ← vector.cons_head_tail v},
suffices : (v.tail.to_list.prod)⁻¹ = v.head,
{ rw this },
rw [← mul_right_inj v.tail.to_list.prod, inv_mul_self, ← list.prod_cons,
← vector.to_list_cons, vector.cons_head_tail, h]
end⟩,
λ ⟨w, hw⟩, by rw [mem_vectors_prod_eq_one, ← hw, mk_vector_prod_eq_one,
vector.to_list_cons, list.prod_cons, inv_mul_self]⟩
def rotate_vectors_prod_eq_one (G : Type*) [group G] (n : ℕ+) (m : multiplicative (zmod n))
(v : vectors_prod_eq_one G n) : vectors_prod_eq_one G n :=
⟨⟨v.1.to_list.rotate m.1, by simp⟩, prod_rotate_eq_one_of_prod_eq_one v.2 _⟩
instance rotate_vectors_prod_eq_one.mul_action (n : ℕ+) :
mul_action (multiplicative (zmod n)) (vectors_prod_eq_one G n) :=
{ smul := (rotate_vectors_prod_eq_one G n),
one_smul := λ v, subtype.eq $ vector.eq _ _ $ rotate_zero v.1.to_list,
mul_smul := λ a b ⟨⟨v, hv₁⟩, hv₂⟩, subtype.eq $ vector.eq _ _ $
show v.rotate ((a + b : zmod n).val) = list.rotate (list.rotate v (b.val)) (a.val),
by rw [zmod.add_val, rotate_rotate, ← rotate_mod _ (b.1 + a.1), add_comm, hv₁] }
lemma one_mem_vectors_prod_eq_one (n : ℕ) : vector.repeat (1 : G) n ∈ vectors_prod_eq_one G n :=
by simp [vector.repeat, vectors_prod_eq_one]
lemma one_mem_fixed_points_rotate (n : ℕ+) :
(⟨vector.repeat (1 : G) n, one_mem_vectors_prod_eq_one n⟩ : vectors_prod_eq_one G n) ∈
fixed_points (multiplicative (zmod n)) (vectors_prod_eq_one G n) :=
λ m, subtype.eq $ vector.eq _ _ $
by haveI : nonempty G := ⟨1⟩; exact
rotate_eq_self_iff_eq_repeat.2 ⟨(1 : G),
show list.repeat (1 : G) n = list.repeat 1 (list.repeat (1 : G) n).length, by simp⟩ _
/-- Cauchy's theorem -/
lemma exists_prime_order_of_dvd_card [fintype G] {p : ℕ} (hp : nat.prime p)
(hdvd : p ∣ card G) : ∃ x : G, order_of x = p :=
let n : ℕ+ := ⟨p - 1, nat.sub_pos_of_lt hp.gt_one⟩ in
let p' : ℕ+ := ⟨p, hp.pos⟩ in
have hn : p' = n + 1 := subtype.eq (nat.succ_sub hp.pos),
have hcard : card (vectors_prod_eq_one G (n + 1)) = card G ^ (n : ℕ),
by rw [set.ext mem_vectors_prod_eq_one_iff,
set.card_range_of_injective (mk_vector_prod_eq_one_inj _), card_vector],
have hzmod : fintype.card (multiplicative (zmod p')) =
(p' : ℕ) ^ 1 := (nat.pow_one p').symm ▸ card_fin _,
have hmodeq : _ = _ := @mul_action.card_modeq_card_fixed_points
(multiplicative (zmod p')) (vectors_prod_eq_one G p') _ _ _ _ _ _ 1 hp hzmod,
have hdvdcard : p ∣ fintype.card (vectors_prod_eq_one G (n + 1)) :=
calc p ∣ card G ^ 1 : by rwa nat.pow_one
... ∣ card G ^ (n : ℕ) : nat.pow_dvd_pow _ n.2
... = card (vectors_prod_eq_one G (n + 1)) : hcard.symm,
have hdvdcard₂ : p ∣ card (fixed_points (multiplicative (zmod p')) (vectors_prod_eq_one G p')) :=
nat.dvd_of_mod_eq_zero (hmodeq ▸ hn.symm ▸ nat.mod_eq_zero_of_dvd hdvdcard),
have hcard_pos : 0 < card (fixed_points (multiplicative (zmod p')) (vectors_prod_eq_one G p')) :=
fintype.card_pos_iff.2 ⟨⟨⟨vector.repeat 1 p', one_mem_vectors_prod_eq_one _⟩,
one_mem_fixed_points_rotate _⟩⟩,
have hlt : 1 < card (fixed_points (multiplicative (zmod p')) (vectors_prod_eq_one G p')) :=
calc (1 : ℕ) < p' : hp.gt_one
... ≤ _ : nat.le_of_dvd hcard_pos hdvdcard₂,
let ⟨⟨⟨⟨x, hx₁⟩, hx₂⟩, hx₃⟩, hx₄⟩ := fintype.exists_ne_of_card_gt_one hlt
⟨_, one_mem_fixed_points_rotate p'⟩ in
have hx : x ≠ list.repeat (1 : G) p', from λ h, by simpa [h, vector.repeat] using hx₄,
have nG : nonempty G, from ⟨1⟩,
have ∃ a, x = list.repeat a x.length := by exactI rotate_eq_self_iff_eq_repeat.1 (λ n,
have list.rotate x (n : zmod p').val = x :=
subtype.mk.inj (subtype.mk.inj (hx₃ (n : zmod p'))),
by rwa [zmod.val_cast_nat, ← hx₁, rotate_mod] at this),
let ⟨a, ha⟩ := this in
⟨a, have hx1 : x.prod = 1 := hx₂,
have ha1: a ≠ 1, from λ h, hx (ha.symm ▸ h ▸ hx₁ ▸ rfl),
have a ^ p = 1, by rwa [ha, list.prod_repeat, hx₁] at hx1,
(hp.2 _ (order_of_dvd_of_pow_eq_one this)).resolve_left
(λ h, ha1 (order_of_eq_one_iff.1 h))⟩
open is_subgroup is_submonoid is_group_hom mul_action
lemma mem_fixed_points_mul_left_cosets_iff_mem_normalizer {H : set G} [is_subgroup H] [fintype H]
{x : G} : (x : quotient H) ∈ fixed_points H (quotient H) ↔ x ∈ normalizer H :=
⟨λ hx, have ha : ∀ {y : quotient H}, y ∈ orbit H (x : quotient H) → y = x,
from λ _, ((mem_fixed_points' _).1 hx _),
(inv_mem_iff _).1 (mem_normalizer_fintype (λ n hn,
have (n⁻¹ * x)⁻¹ * x ∈ H := quotient_group.eq.1 (ha (mem_orbit _ ⟨n⁻¹, inv_mem hn⟩)),
by simpa only [mul_inv_rev, inv_inv] using this)),
λ (hx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H),
(mem_fixed_points' _).2 $ λ y, quotient.induction_on' y $ λ y hy, quotient_group.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy in
have hb₂ : (b * x)⁻¹ * y ∈ H := quotient_group.eq.1 hb₂,
(inv_mem_iff H).1 $ (hx _).2 $ (mul_mem_cancel_right H (inv_mem hb₁)).1
$ by rw hx at hb₂;
simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
lemma fixed_points_mul_left_cosets_equiv_quotient (H : set G) [is_subgroup H] [fintype H] :
fixed_points H (quotient H) ≃ quotient (subtype.val ⁻¹' H : set (normalizer H)) :=
@subtype_quotient_equiv_quotient_subtype G (normalizer H) (id _) (id _) (fixed_points _ _)
(λ a, mem_fixed_points_mul_left_cosets_iff_mem_normalizer.symm) (by intros; refl)
local attribute [instance] set_fintype
lemma exists_subgroup_card_pow_prime [fintype G] {p : ℕ} : ∀ {n : ℕ} (hp : nat.prime p)
(hdvd : p ^ n ∣ card G), ∃ H : set G, is_subgroup H ∧ fintype.card H = p ^ n
| 0 := λ _ _, ⟨trivial G, by apply_instance, by simp⟩
| (n+1) := λ hp hdvd,
let ⟨H, ⟨hH1, hH2⟩⟩ := exists_subgroup_card_pow_prime hp
(dvd.trans (nat.pow_dvd_pow _ (nat.le_succ _)) hdvd) in
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd in
by exactI
have hcard : card (quotient H) = s * p :=
(nat.mul_right_inj (show card H > 0, from fintype.card_pos_iff.2
⟨⟨1, is_submonoid.one_mem H⟩⟩)).1
(by rwa [← card_eq_card_quotient_mul_card_subgroup, hH2, hs,
nat.pow_succ, mul_assoc, mul_comm p]),
have hm : s * p % p = card (quotient (subtype.val ⁻¹' H : set (normalizer H))) % p :=
card_congr (fixed_points_mul_left_cosets_equiv_quotient H) ▸ hcard ▸
card_modeq_card_fixed_points hp hH2,
have hm' : p ∣ card (quotient (subtype.val ⁻¹' H : set (normalizer H))) :=
nat.dvd_of_mod_eq_zero
(by rwa [nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm),
let ⟨x, hx⟩ := @exists_prime_order_of_dvd_card _ (quotient_group.group _) _ _ hp hm' in
have hxcard : ∀ {f : fintype (gpowers x)}, card (gpowers x) = p,
from λ f, by rw [← hx, order_eq_card_gpowers]; congr,
have is_subgroup (mk ⁻¹' gpowers x),
from is_group_hom.preimage _ _,
have fintype (mk ⁻¹' gpowers x), by apply_instance,
have hequiv : H ≃ (subtype.val ⁻¹' H : set (normalizer H)):=
⟨λ a, ⟨⟨a.1, subset_normalizer _ a.2⟩, a.2⟩, λ a, ⟨a.1.1, a.2⟩,
λ ⟨_, _⟩, rfl, λ ⟨⟨_, _⟩, _⟩, rfl⟩,
⟨subtype.val '' (mk ⁻¹' gpowers x), by apply_instance,
by rw [set.card_image_of_injective (mk ⁻¹' gpowers x) subtype.val_injective,
nat.pow_succ, ← hH2, fintype.card_congr hequiv, ← hx, order_eq_card_gpowers,
← fintype.card_prod];
exact @fintype.card_congr _ _ (id _) (id _) (preimage_mk_equiv_subgroup_times_set _ _)⟩
end sylow
|
970f5779fa0f4c5c3581c33331a449f1e6d6d989 | dd0f5513e11c52db157d2fcc8456d9401a6cd9da | /06_Inductive_Types.org.44.lean | d565171c28ab01036fd95afe39953afce20464ca | [] | no_license | cjmazey/lean-tutorial | ba559a49f82aa6c5848b9bf17b7389bf7f4ba645 | 381f61c9fcac56d01d959ae0fa6e376f2c4e3b34 | refs/heads/master | 1,610,286,098,832 | 1,447,124,923,000 | 1,447,124,923,000 | 43,082,433 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 243 | lean | /- page 92 -/
import standard
namespace hide
inductive eq {A : Type} (a : A) : A → Prop :=
refl : eq a a
-- BEGIN
theorem subst {A : Type} {a b : A} {P : A → Prop}
(H₁ : eq a b) (H₂ : P a) : P b :=
eq.rec H₂ H₁
-- END
end hide
|
9f73db18641603941b430194a1a5ed70cd3f77d8 | 9be442d9ec2fcf442516ed6e9e1660aa9071b7bd | /tests/lean/986.lean | 3f88415f8622a2009bf2a10a8c3721ac21abd932 | [
"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 | 133 | lean | attribute [simp] Array.insertionSort.swapLoop
#check @Array.insertionSort.swapLoop._eq_1
#check @Array.insertionSort.swapLoop._eq_2
|
b30f4fe5d428f41b249944dbdf56baf4d25780ba | 491068d2ad28831e7dade8d6dff871c3e49d9431 | /tests/lean/unzip_error.lean | 25fb95ce239b81d2b90ffd7a726aebb2f4216ad9 | [
"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 | 364 | lean | import data.examples.vector
open nat vector prod
variables {A B : Type}
definition unzip : Π {n : nat}, vector (A × B) n → vector A n × vector B n
| unzip nil := (nil, nil)
| unzip ((a, b) :: v) :=
match unzip v with
(va, vb) := (a :: va, b :: vb)
end
example : unzip ((1, 20) :: (2, 30) :: nil) = (1 :: 2 :: nil, 20 :: 30 :: nil) :=
rfl
|
f5015aaff2daff6f1085d103fc3ab1253841b108 | d1a52c3f208fa42c41df8278c3d280f075eb020c | /stage0/src/Lean/PrettyPrinter/Delaborator/Builtins.lean | 7d5fcda18b5406021d11495a0224426b96aaf974 | [
"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 | 31,987 | lean | /-
Copyright (c) 2020 Sebastian Ullrich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sebastian Ullrich
-/
import Lean.PrettyPrinter.Delaborator.Basic
import Lean.PrettyPrinter.Delaborator.SubExpr
import Lean.PrettyPrinter.Delaborator.TopDownAnalyze
import Lean.Parser
namespace Lean.PrettyPrinter.Delaborator
open Lean.Meta
open Lean.Parser.Term
open SubExpr
def maybeAddBlockImplicit (ident : Syntax) : DelabM Syntax := do
if ← getPPOption getPPAnalysisBlockImplicit then `(@$ident:ident) else ident
def unfoldMDatas : Expr → Expr
| Expr.mdata _ e _ => unfoldMDatas e
| e => e
@[builtinDelab fvar]
def delabFVar : Delab := do
let Expr.fvar id _ ← getExpr | unreachable!
try
let l ← getLocalDecl id
maybeAddBlockImplicit (mkIdent l.userName)
catch _ =>
-- loose free variable, use internal name
maybeAddBlockImplicit $ mkIdent id.name
-- loose bound variable, use pseudo syntax
@[builtinDelab bvar]
def delabBVar : Delab := do
let Expr.bvar idx _ ← getExpr | unreachable!
pure $ mkIdent $ Name.mkSimple $ "#" ++ toString idx
@[builtinDelab mvar]
def delabMVar : Delab := do
let Expr.mvar n _ ← getExpr | unreachable!
let mvarDecl ← getMVarDecl n
let n :=
match mvarDecl.userName with
| Name.anonymous => n.name.replacePrefix `_uniq `m
| n => n
`(?$(mkIdent n))
@[builtinDelab sort]
def delabSort : Delab := do
let Expr.sort l _ ← getExpr | unreachable!
match l with
| Level.zero _ => `(Prop)
| Level.succ (Level.zero _) _ => `(Type)
| _ => match l.dec with
| some l' => `(Type $(Level.quote l' max_prec))
| none => `(Sort $(Level.quote l max_prec))
def unresolveNameGlobal (n₀ : Name) : DelabM Name := do
if n₀.hasMacroScopes then return n₀
if (← getPPOption getPPFullNames) then
match (← resolveGlobalName n₀) with
| [(potentialMatch, _)] => if potentialMatch == n₀ then return n₀ else return rootNamespace ++ n₀
| _ => return n₀ -- if can't resolve, return the original
let mut initialNames := (getRevAliases (← getEnv) n₀).toArray
initialNames := initialNames.push (rootNamespace ++ n₀)
for initialName in initialNames do
match (← unresolveNameCore initialName) with
| none => continue
| some n => return n
return n₀ -- if can't resolve, return the original
where
unresolveNameCore (n : Name) : DelabM (Option Name) := do
let mut revComponents := n.components'
let mut candidate := Name.anonymous
for i in [:revComponents.length] do
match revComponents with
| [] => return none
| cmpt::rest => candidate := cmpt ++ candidate; revComponents := rest
match (← resolveGlobalName candidate) with
| [(potentialMatch, _)] => if potentialMatch == n₀ then return some candidate else continue
| _ => continue
return none
-- NOTE: not a registered delaborator, as `const` is never called (see [delab] description)
def delabConst : Delab := do
let Expr.const c₀ ls _ ← getExpr | unreachable!
let ctx ← read
let c₀ := if (← getPPOption getPPPrivateNames) then c₀ else (privateToUserName? c₀).getD c₀
let mut c ← unresolveNameGlobal c₀
let stx ←
if ls.isEmpty || !(← getPPOption getPPUniverses) then
if (← getLCtx).usesUserName c then
-- `c` is also a local declaration
if c == c₀ && !(← read).inPattern then
-- `c` is the fully qualified named. So, we append the `_root_` prefix
c := `_root_ ++ c
else
c := c₀
return mkIdent c
else
`($(mkIdent c).{$[$(ls.toArray.map quote)],*})
maybeAddBlockImplicit stx
structure ParamKind where
name : Name
bInfo : BinderInfo
defVal : Option Expr := none
isAutoParam : Bool := false
def ParamKind.isRegularExplicit (param : ParamKind) : Bool :=
param.bInfo.isExplicit && !param.isAutoParam && param.defVal.isNone
/-- Return array with n-th element set to kind of n-th parameter of `e`. -/
partial def getParamKinds : DelabM (Array ParamKind) := do
let e ← getExpr
try
withTransparency TransparencyMode.all do
forallTelescopeArgs e.getAppFn e.getAppArgs fun params _ => do
params.mapM fun param => do
let l ← getLocalDecl param.fvarId!
pure { name := l.userName, bInfo := l.binderInfo, defVal := l.type.getOptParamDefault?, isAutoParam := l.type.isAutoParam }
catch _ => pure #[] -- recall that expr may be nonsensical
where
forallTelescopeArgs f args k := do
forallBoundedTelescope (← inferType f) args.size fun xs b =>
if xs.isEmpty || xs.size == args.size then
-- we still want to consider optParams
forallTelescopeReducing b fun ys b => k (xs ++ ys) b
else
forallTelescopeArgs (mkAppN f $ args.shrink xs.size) (args.extract xs.size args.size) fun ys b =>
k (xs ++ ys) b
@[builtinDelab app]
def delabAppExplicit : Delab := whenPPOption getPPExplicit do
let paramKinds ← getParamKinds
let (fnStx, _, argStxs) ← withAppFnArgs
(do
let fn ← getExpr
let stx ← if fn.isConst then delabConst else delab
let needsExplicit := paramKinds.any (fun param => !param.isRegularExplicit) && stx.getKind != `Lean.Parser.Term.explicit
let stx ← if needsExplicit then `(@$stx) else pure stx
pure (stx, paramKinds.toList, #[]))
(fun ⟨fnStx, paramKinds, argStxs⟩ => do
let isInstImplicit := match paramKinds with
| [] => false
| param :: _ => param.bInfo == BinderInfo.instImplicit
let argStx ← if ← getPPOption getPPAnalysisHole then `(_)
else if isInstImplicit == true then
let stx ← if ← getPPOption getPPInstances then delab else `(_)
if ← getPPOption getPPInstanceTypes then
let typeStx ← withType delab
`(($stx : $typeStx))
else stx
else delab
pure (fnStx, paramKinds.tailD [], argStxs.push argStx))
Syntax.mkApp fnStx argStxs
def shouldShowMotive (motive : Expr) (opts : Options) : MetaM Bool := do
getPPMotivesAll opts
<||> (← getPPMotivesPi opts <&&> returnsPi motive)
<||> (← getPPMotivesNonConst opts <&&> isNonConstFun motive)
def withMDataOptions [Inhabited α] (x : DelabM α) : DelabM α := do
match ← getExpr with
| Expr.mdata m .. =>
let mut posOpts := (← read).optionsPerPos
let pos ← getPos
for (k, v) in m do
if (`pp).isPrefixOf k then
let opts := posOpts.find? pos |>.getD {}
posOpts := posOpts.insert pos (opts.insert k v)
withReader ({ · with optionsPerPos := posOpts }) $ withMDataExpr x
| _ => x
partial def withMDatasOptions [Inhabited α] (x : DelabM α) : DelabM α := do
if (← getExpr).isMData then withMDataOptions (withMDatasOptions x) else x
def isRegularApp : DelabM Bool := do
let e ← getExpr
if not (unfoldMDatas e.getAppFn).isConst then return false
if ← withNaryFn (withMDatasOptions (getPPOption getPPUniverses <||> getPPOption getPPAnalysisBlockImplicit)) then return false
for i in [:e.getAppNumArgs] do
if ← withNaryArg i (getPPOption getPPAnalysisNamedArg) then return false
return true
def unexpandRegularApp (stx : Syntax) : Delab := do
let Expr.const c .. ← pure (unfoldMDatas (← getExpr).getAppFn) | unreachable!
let fs ← appUnexpanderAttribute.getValues (← getEnv) c
fs.firstM fun f =>
match f stx |>.run () with
| EStateM.Result.ok stx _ => pure stx
| _ => failure
-- abbrev coe {α : Sort u} {β : Sort v} (a : α) [CoeT α a β] : β
-- abbrev coeFun {α : Sort u} {γ : α → Sort v} (a : α) [CoeFun α γ] : γ a
def unexpandCoe (stx : Syntax) : Delab := whenPPOption getPPCoercions do
if not (← isCoe (← getExpr)) then failure
let e ← getExpr
match stx with
| `($fn $arg) => arg
| `($fn $args*) => `($(args.get! 0) $(args.eraseIdx 0)*)
| _ => failure
def unexpandStructureInstance (stx : Syntax) : Delab := whenPPOption getPPStructureInstances do
let env ← getEnv
let e ← getExpr
let some s ← pure $ e.isConstructorApp? env | failure
guard $ isStructure env s.induct;
/- If implicit arguments should be shown, and the structure has parameters, we should not
pretty print using { ... }, because we will not be able to see the parameters. -/
let fieldNames := getStructureFields env s.induct
let mut fields := #[]
guard $ fieldNames.size == stx[1].getNumArgs
let args := e.getAppArgs
let fieldVals := args.extract s.numParams args.size
for idx in [:fieldNames.size] do
let fieldName := fieldNames[idx]
let fieldId := mkIdent fieldName
let fieldPos ← nextExtraPos
let fieldId := annotatePos fieldPos fieldId
addFieldInfo fieldPos (s.induct ++ fieldName) fieldName fieldId fieldVals[idx]
let field ← `(structInstField|$fieldId:ident := $(stx[1][idx]):term)
fields := fields.push field
let tyStx ← withType do
if (← getPPOption getPPStructureInstanceType) then delab >>= pure ∘ some else pure none
if fields.isEmpty then
`({ $[: $tyStx]? })
else
let lastField := fields.back
fields := fields.pop
`({ $[$fields, ]* $lastField $[: $tyStx]? })
@[builtinDelab app]
def delabAppImplicit : Delab := do
-- TODO: always call the unexpanders, make them guard on the right # args?
let paramKinds ← getParamKinds
if ← getPPOption getPPExplicit then
if paramKinds.any (fun param => !param.isRegularExplicit) then failure
let (fnStx, _, argStxs) ← withAppFnArgs
(do
let fn ← getExpr
let stx ← if fn.isConst then delabConst else delab
pure (stx, paramKinds.toList, #[]))
(fun (fnStx, paramKinds, argStxs) => do
let arg ← getExpr
let opts ← getOptions
let mkNamedArg (name : Name) (argStx : Syntax) : DelabM Syntax := do
`(Parser.Term.namedArgument| ($(← mkIdent name):ident := $argStx:term))
let argStx? : Option Syntax ←
if ← getPPOption getPPAnalysisSkip then pure none
else if ← getPPOption getPPAnalysisHole then `(_)
else
match paramKinds with
| [] => delab
| param :: rest =>
if param.defVal.isSome && rest.isEmpty then
let v := param.defVal.get!
if !v.hasLooseBVars && v == arg then none else delab
else if !param.isRegularExplicit && param.defVal.isNone then
if ← getPPOption getPPAnalysisNamedArg <||> (param.name == `motive <&&> shouldShowMotive arg opts) then mkNamedArg param.name (← delab) else none
else delab
let argStxs := match argStx? with
| none => argStxs
| some stx => argStxs.push stx
pure (fnStx, paramKinds.tailD [], argStxs))
let stx := Syntax.mkApp fnStx argStxs
if ← isRegularApp then
(guard (← getPPOption getPPNotation) *> unexpandRegularApp stx)
<|> (guard (← getPPOption getPPStructureInstances) *> unexpandStructureInstance stx)
<|> (guard (← getPPOption getPPNotation) *> unexpandCoe stx)
<|> pure stx
else pure stx
/-- State for `delabAppMatch` and helpers. -/
structure AppMatchState where
info : MatcherInfo
matcherTy : Expr
params : Array Expr := #[]
motive : Option (Syntax × Expr) := none
motiveNamed : Bool := false
discrs : Array Syntax := #[]
varNames : Array (Array Name) := #[]
rhss : Array Syntax := #[]
-- additional arguments applied to the result of the `match` expression
moreArgs : Array Syntax := #[]
/--
Extract arguments of motive applications from the matcher type.
For the example below: `#[#[`([])], #[`(a::as)]]` -/
private partial def delabPatterns (st : AppMatchState) : DelabM (Array (Array Syntax)) :=
withReader (fun ctx => { ctx with inPattern := true, optionsPerPos := {} }) do
let ty ← instantiateForall st.matcherTy st.params
forallTelescope ty fun params _ => do
-- skip motive and discriminators
let alts := Array.ofSubarray params[1 + st.discrs.size:]
alts.mapIdxM fun idx alt => do
let ty ← inferType alt
-- TODO: this is a hack; we are accessing the expression out-of-sync with the position
-- Currently, we reset `optionsPerPos` at the beginning of `delabPatterns` to avoid
-- incorrectly considering annotations.
withTheReader SubExpr ({ · with expr := ty }) $
usingNames st.varNames[idx] do
withAppFnArgs (pure #[]) (fun pats => do pure $ pats.push (← delab))
where
usingNames {α} (varNames : Array Name) (x : DelabM α) : DelabM α :=
usingNamesAux 0 varNames x
usingNamesAux {α} (i : Nat) (varNames : Array Name) (x : DelabM α) : DelabM α :=
if i < varNames.size then
withBindingBody varNames[i] <| usingNamesAux (i+1) varNames x
else
x
/-- Skip `numParams` binders, and execute `x varNames` where `varNames` contains the new binder names. -/
private partial def skippingBinders {α} (numParams : Nat) (x : Array Name → DelabM α) : DelabM α :=
loop numParams #[]
where
loop : Nat → Array Name → DelabM α
| 0, varNames => x varNames
| n+1, varNames => do
let rec visitLambda : DelabM α := do
let varName ← (← getExpr).bindingName!.eraseMacroScopes
-- Pattern variables cannot shadow each other
if varNames.contains varName then
let varName := (← getLCtx).getUnusedName varName
withBindingBody varName do
loop n (varNames.push varName)
else
withBindingBodyUnusedName fun id => do
loop n (varNames.push id.getId)
let e ← getExpr
if e.isLambda then
visitLambda
else
-- eta expand `e`
let e ← forallTelescopeReducing (← inferType e) fun xs _ => do
if xs.size == 1 && (← inferType xs[0]).isConstOf ``Unit then
-- `e` might be a thunk create by the dependent pattern matching compiler, and `xs[0]` may not even be a pattern variable.
-- If it is a pattern variable, it doesn't look too bad to use `()` instead of the pattern variable.
-- If it becomes a problem in the future, we should modify the dependent pattern matching compiler, and make sure
-- it adds an annotation to distinguish these two cases.
mkLambdaFVars xs (mkApp e (mkConst ``Unit.unit))
else
mkLambdaFVars xs (mkAppN e xs)
withTheReader SubExpr (fun ctx => { ctx with expr := e }) visitLambda
/--
Delaborate applications of "matchers" such as
```
List.map.match_1 : {α : Type _} →
(motive : List α → Sort _) →
(x : List α) → (Unit → motive List.nil) → ((a : α) → (as : List α) → motive (a :: as)) → motive x
```
-/
@[builtinDelab app]
def delabAppMatch : Delab := whenPPOption getPPNotation <| whenPPOption getPPMatch do
-- incrementally fill `AppMatchState` from arguments
let st ← withAppFnArgs
(do
let (Expr.const c us _) ← getExpr | failure
let (some info) ← getMatcherInfo? c | failure
{ matcherTy := (← getConstInfo c).instantiateTypeLevelParams us, info := info : AppMatchState })
(fun st => do
if st.params.size < st.info.numParams then
pure { st with params := st.params.push (← getExpr) }
else if st.motive.isNone then
-- store motive argument separately
let lamMotive ← getExpr
let piMotive ← lambdaTelescope lamMotive fun xs body => mkForallFVars xs body
-- TODO: pp.analyze has not analyzed `piMotive`, only `lamMotive`
-- Thus the binder types won't have any annotations
let piStx ← withTheReader SubExpr (fun cfg => { cfg with expr := piMotive }) delab
let named ← getPPOption getPPAnalysisNamedArg
pure { st with motive := (piStx, lamMotive), motiveNamed := named }
else if st.discrs.size < st.info.numDiscrs then
pure { st with discrs := st.discrs.push (← delab) }
else if st.rhss.size < st.info.altNumParams.size then
/- We save the variables names here to be able to implement safe_shadowing.
The pattern delaboration must use the names saved here. -/
let (varNames, rhs) ← skippingBinders st.info.altNumParams[st.rhss.size] fun varNames => do
let rhs ← delab
return (varNames, rhs)
pure { st with rhss := st.rhss.push rhs, varNames := st.varNames.push varNames }
else
pure { st with moreArgs := st.moreArgs.push (← delab) })
if st.discrs.size < st.info.numDiscrs || st.rhss.size < st.info.altNumParams.size then
-- underapplied
failure
match st.discrs, st.rhss with
| #[discr], #[] =>
let stx ← `(nomatch $discr)
Syntax.mkApp stx st.moreArgs
| _, #[] => failure
| _, _ =>
let pats ← delabPatterns st
let stx ← do
let (piStx, lamMotive) := st.motive.get!
let opts ← getOptions
-- TODO: disable the match if other implicits are needed?
if ← st.motiveNamed <||> shouldShowMotive lamMotive opts then
`(match $[$st.discrs:term],* : $piStx with $[| $pats,* => $st.rhss]*)
else
`(match $[$st.discrs:term],* with $[| $pats,* => $st.rhss]*)
Syntax.mkApp stx st.moreArgs
/--
Delaborate applications of the form `(fun x => b) v` as `let_fun x := v; b`
-/
def delabLetFun : Delab := do
let stxV ← withAppArg delab
withAppFn do
let Expr.lam n t b _ ← getExpr | unreachable!
let n ← getUnusedName n b
let stxB ← withBindingBody n delab
if ← getPPOption getPPLetVarTypes <||> getPPOption getPPAnalysisLetVarType then
let stxT ← withBindingDomain delab
`(let_fun $(mkIdent n) : $stxT := $stxV; $stxB)
else
`(let_fun $(mkIdent n) := $stxV; $stxB)
@[builtinDelab mdata]
def delabMData : Delab := do
if let some _ := inaccessible? (← getExpr) then
let s ← withMDataExpr delab
if (← read).inPattern then
`(.($s)) -- We only include the inaccessible annotation when we are delaborating patterns
else
return s
else if isLetFun (← getExpr) && getPPNotation (← getOptions) then
withMDataExpr <| delabLetFun
else if let some _ := isLHSGoal? (← getExpr) then
withMDataExpr <| withAppFn <| withAppArg <| delab
else
withMDataOptions delab
/--
Check for a `Syntax.ident` of the given name anywhere in the tree.
This is usually a bad idea since it does not check for shadowing bindings,
but in the delaborator we assume that bindings are never shadowed.
-/
partial def hasIdent (id : Name) : Syntax → Bool
| Syntax.ident _ _ id' _ => id == id'
| Syntax.node _ _ args => args.any (hasIdent id)
| _ => false
/--
Return `true` iff current binder should be merged with the nested
binder, if any, into a single binder group:
* both binders must have same binder info and domain
* they cannot be inst-implicit (`[a b : A]` is not valid syntax)
* `pp.binderTypes` must be the same value for both terms
* prefer `fun a b` over `fun (a b)`
-/
private def shouldGroupWithNext : DelabM Bool := do
let e ← getExpr
let ppEType ← getPPOption (getPPBinderTypes e)
let go (e' : Expr) := do
let ppE'Type ← withBindingBody `_ $ getPPOption (getPPBinderTypes e)
pure $ e.binderInfo == e'.binderInfo &&
e.bindingDomain! == e'.bindingDomain! &&
e'.binderInfo != BinderInfo.instImplicit &&
ppEType == ppE'Type &&
(e'.binderInfo != BinderInfo.default || ppE'Type)
match e with
| Expr.lam _ _ e'@(Expr.lam _ _ _ _) _ => go e'
| Expr.forallE _ _ e'@(Expr.forallE _ _ _ _) _ => go e'
| _ => pure false
where
getPPBinderTypes (e : Expr) :=
if e.isForall then getPPPiBinderTypes else getPPFunBinderTypes
private partial def delabBinders (delabGroup : Array Syntax → Syntax → Delab) : optParam (Array Syntax) #[] → Delab
-- Accumulate names (`Syntax.ident`s with position information) of the current, unfinished
-- binder group `(d e ...)` as determined by `shouldGroupWithNext`. We cannot do grouping
-- inside-out, on the Syntax level, because it depends on comparing the Expr binder types.
| curNames => do
if ← shouldGroupWithNext then
-- group with nested binder => recurse immediately
withBindingBodyUnusedName fun stxN => delabBinders delabGroup (curNames.push stxN)
else
-- don't group => delab body and prepend current binder group
let (stx, stxN) ← withBindingBodyUnusedName fun stxN => do (← delab, stxN)
delabGroup (curNames.push stxN) stx
@[builtinDelab lam]
def delabLam : Delab :=
delabBinders fun curNames stxBody => do
let e ← getExpr
let stxT ← withBindingDomain delab
let ppTypes ← getPPOption getPPFunBinderTypes
let expl ← getPPOption getPPExplicit
let usedDownstream ← curNames.any (fun n => hasIdent n.getId stxBody)
-- leave lambda implicit if possible
-- TODO: for now we just always block implicit lambdas when delaborating. We can revisit.
-- Note: the current issue is that it requires state, i.e. if *any* previous binder was implicit,
-- it doesn't seem like we can leave a subsequent binder implicit.
let blockImplicitLambda := true
/-
let blockImplicitLambda := expl ||
e.binderInfo == BinderInfo.default ||
-- Note: the following restriction fixes many issues with roundtripping,
-- but this condition may still not be perfectly in sync with the elaborator.
e.binderInfo == BinderInfo.instImplicit ||
Elab.Term.blockImplicitLambda stxBody ||
usedDownstream
-/
if !blockImplicitLambda then
pure stxBody
else
let group ← match e.binderInfo, ppTypes with
| BinderInfo.default, true =>
-- "default" binder group is the only one that expects binder names
-- as a term, i.e. a single `Syntax.ident` or an application thereof
let stxCurNames ←
if curNames.size > 1 then
`($(curNames.get! 0) $(curNames.eraseIdx 0)*)
else
pure $ curNames.get! 0;
`(funBinder| ($stxCurNames : $stxT))
| BinderInfo.default, false => pure curNames.back -- here `curNames.size == 1`
| BinderInfo.implicit, true => `(funBinder| {$curNames* : $stxT})
| BinderInfo.implicit, false => `(funBinder| {$curNames*})
| BinderInfo.strictImplicit, true => `(funBinder| ⦃$curNames* : $stxT⦄)
| BinderInfo.strictImplicit, false => `(funBinder| ⦃$curNames*⦄)
| BinderInfo.instImplicit, _ =>
if usedDownstream then `(funBinder| [$curNames.back : $stxT]) -- here `curNames.size == 1`
else `(funBinder| [$stxT])
| _ , _ => unreachable!;
match stxBody with
| `(fun $binderGroups* => $stxBody) => `(fun $group $binderGroups* => $stxBody)
| _ => `(fun $group => $stxBody)
@[builtinDelab forallE]
def delabForall : Delab :=
delabBinders fun curNames stxBody => do
let e ← getExpr
let prop ← try isProp e catch _ => false
let stxT ← withBindingDomain delab
let group ← match e.binderInfo with
| BinderInfo.implicit => `(bracketedBinderF|{$curNames* : $stxT})
| BinderInfo.strictImplicit => `(bracketedBinderF|⦃$curNames* : $stxT⦄)
-- here `curNames.size == 1`
| BinderInfo.instImplicit => `(bracketedBinderF|[$curNames.back : $stxT])
| _ =>
-- heuristic: use non-dependent arrows only if possible for whole group to avoid
-- noisy mix like `(α : Type) → Type → (γ : Type) → ...`.
let dependent := curNames.any $ fun n => hasIdent n.getId stxBody
-- NOTE: non-dependent arrows are available only for the default binder info
if dependent then
if prop && !(← getPPOption getPPPiBinderTypes) then
return ← `(∀ $curNames:ident*, $stxBody)
else
`(bracketedBinderF|($curNames* : $stxT))
else
return ← curNames.foldrM (fun _ stxBody => `($stxT → $stxBody)) stxBody
if prop then
match stxBody with
| `(∀ $groups*, $stxBody) => `(∀ $group $groups*, $stxBody)
| _ => `(∀ $group, $stxBody)
else
`($group:bracketedBinder → $stxBody)
@[builtinDelab letE]
def delabLetE : Delab := do
let Expr.letE n t v b _ ← getExpr | unreachable!
let n ← getUnusedName n b
let stxV ← descend v 1 delab
let stxB ← withLetDecl n t v fun fvar =>
let b := b.instantiate1 fvar
descend b 2 delab
if ← getPPOption getPPLetVarTypes <||> getPPOption getPPAnalysisLetVarType then
let stxT ← descend t 0 delab
`(let $(mkIdent n) : $stxT := $stxV; $stxB)
else `(let $(mkIdent n) := $stxV; $stxB)
@[builtinDelab lit]
def delabLit : Delab := do
let Expr.lit l _ ← getExpr | unreachable!
match l with
| Literal.natVal n => pure $ quote n
| Literal.strVal s => pure $ quote s
-- `@OfNat.ofNat _ n _` ~> `n`
@[builtinDelab app.OfNat.ofNat]
def delabOfNat : Delab := whenPPOption getPPCoercions do
let (Expr.app (Expr.app _ (Expr.lit (Literal.natVal n) _) _) _ _) ← getExpr | failure
return quote n
-- `@OfDecimal.ofDecimal _ _ m s e` ~> `m*10^(sign * e)` where `sign == 1` if `s = false` and `sign = -1` if `s = true`
@[builtinDelab app.OfScientific.ofScientific]
def delabOfScientific : Delab := whenPPOption getPPCoercions do
let expr ← getExpr
guard <| expr.getAppNumArgs == 5
let Expr.lit (Literal.natVal m) _ ← pure (expr.getArg! 2) | failure
let Expr.lit (Literal.natVal e) _ ← pure (expr.getArg! 4) | failure
let s ← match expr.getArg! 3 with
| Expr.const `Bool.true _ _ => pure true
| Expr.const `Bool.false _ _ => pure false
| _ => failure
let str := toString m
if s && e == str.length then
return Syntax.mkScientificLit ("0." ++ str)
else if s && e < str.length then
let mStr := str.extract 0 (str.length - e)
let eStr := str.extract (str.length - e) str.length
return Syntax.mkScientificLit (mStr ++ "." ++ eStr)
else
return Syntax.mkScientificLit (str ++ "e" ++ (if s then "-" else "") ++ toString e)
/--
Delaborate a projection primitive. These do not usually occur in
user code, but are pretty-printed when e.g. `#print`ing a projection
function.
-/
@[builtinDelab proj]
def delabProj : Delab := do
let Expr.proj _ idx _ _ ← getExpr | unreachable!
let e ← withProj delab
-- not perfectly authentic: elaborates to the `idx`-th named projection
-- function (e.g. `e.1` is `Prod.fst e`), which unfolds to the actual
-- `proj`.
let idx := Syntax.mkLit fieldIdxKind (toString (idx + 1));
`($(e).$idx:fieldIdx)
/-- Delaborate a call to a projection function such as `Prod.fst`. -/
@[builtinDelab app]
def delabProjectionApp : Delab := whenPPOption getPPStructureProjections $ do
let e@(Expr.app fn _ _) ← getExpr | failure
let Expr.const c@(Name.str _ f _) _ _ ← pure fn.getAppFn | failure
let env ← getEnv
let some info ← pure $ env.getProjectionFnInfo? c | failure
-- can't use with classes since the instance parameter is implicit
guard $ !info.fromClass
-- projection function should be fully applied (#struct params + 1 instance parameter)
-- TODO: support over-application
guard $ e.getAppNumArgs == info.numParams + 1
-- If pp.explicit is true, and the structure has parameters, we should not
-- use field notation because we will not be able to see the parameters.
let expl ← getPPOption getPPExplicit
guard $ !expl || info.numParams == 0
let appStx ← withAppArg delab
`($(appStx).$(mkIdent f):ident)
@[builtinDelab app.dite]
def delabDIte : Delab := whenPPOption getPPNotation do
-- Note: we keep this as a delaborator for now because it actually accesses the expression.
guard $ (← getExpr).getAppNumArgs == 5
let c ← withAppFn $ withAppFn $ withAppFn $ withAppArg delab
let (t, h) ← withAppFn $ withAppArg $ delabBranch none
let (e, _) ← withAppArg $ delabBranch h
`(if $(mkIdent h):ident : $c then $t else $e)
where
delabBranch (h? : Option Name) : DelabM (Syntax × Name) := do
let e ← getExpr
guard e.isLambda
let h ← match h? with
| some h => return (← withBindingBody h delab, h)
| none => withBindingBodyUnusedName fun h => do
return (← delab, h.getId)
@[builtinDelab app.namedPattern]
def delabNamedPattern : Delab := do
-- Note: we keep this as a delaborator because it accesses the DelabM context
guard (← read).inPattern
guard $ (← getExpr).getAppNumArgs == 3
let x ← withAppFn $ withAppArg delab
let p ← withAppArg delab
guard x.isIdent
`($x:ident@$p:term)
-- Sigma and PSigma delaborators
def delabSigmaCore (sigma : Bool) : Delab := whenPPOption getPPNotation do
guard $ (← getExpr).getAppNumArgs == 2
guard $ (← getExpr).appArg!.isLambda
withAppArg do
let α ← withBindingDomain delab
let bodyExpr := (← getExpr).bindingBody!
withBindingBodyUnusedName fun n => do
let b ← delab
if bodyExpr.hasLooseBVars then
if sigma then `(($n:ident : $α) × $b) else `(($n:ident : $α) ×' $b)
else
if sigma then `((_ : $α) × $b) else `((_ : $α) ×' $b)
@[builtinDelab app.Sigma]
def delabSigma : Delab := delabSigmaCore (sigma := true)
@[builtinDelab app.PSigma]
def delabPSigma : Delab := delabSigmaCore (sigma := false)
partial def delabDoElems : DelabM (List Syntax) := do
let e ← getExpr
if e.isAppOfArity `Bind.bind 6 then
-- Bind.bind.{u, v} : {m : Type u → Type v} → [self : Bind m] → {α β : Type u} → m α → (α → m β) → m β
let α := e.getAppArgs[2]
let ma ← withAppFn $ withAppArg delab
withAppArg do
match (← getExpr) with
| Expr.lam _ _ body _ =>
withBindingBodyUnusedName fun n => do
if body.hasLooseBVars then
prependAndRec `(doElem|let $n:term ← $ma:term)
else if α.isConstOf `Unit || α.isConstOf `PUnit then
prependAndRec `(doElem|$ma:term)
else
prependAndRec `(doElem|let _ ← $ma:term)
| _ => failure
else if e.isLet then
let Expr.letE n t v b _ ← getExpr | unreachable!
let n ← getUnusedName n b
let stxT ← descend t 0 delab
let stxV ← descend v 1 delab
withLetDecl n t v fun fvar =>
let b := b.instantiate1 fvar
descend b 2 $
prependAndRec `(doElem|let $(mkIdent n) : $stxT := $stxV)
else
let stx ← delab
[←`(doElem|$stx:term)]
where
prependAndRec x : DelabM _ := List.cons <$> x <*> delabDoElems
@[builtinDelab app.Bind.bind]
def delabDo : Delab := whenPPOption getPPNotation do
guard <| (← getExpr).isAppOfArity `Bind.bind 6
let elems ← delabDoElems
let items ← elems.toArray.mapM (`(doSeqItem|$(·):doElem))
`(do $items:doSeqItem*)
def reifyName : Expr → DelabM Name
| Expr.const ``Lean.Name.anonymous .. => Name.anonymous
| Expr.app (Expr.app (Expr.const ``Lean.Name.mkStr ..) n _) (Expr.lit (Literal.strVal s) _) _ => do
(← reifyName n).mkStr s
| Expr.app (Expr.app (Expr.const ``Lean.Name.mkNum ..) n _) (Expr.lit (Literal.natVal i) _) _ => do
(← reifyName n).mkNum i
| _ => failure
@[builtinDelab app.Lean.Name.mkStr]
def delabNameMkStr : Delab := whenPPOption getPPNotation do
let n ← reifyName (← getExpr)
-- not guaranteed to be a syntactically valid name, but usually more helpful than the explicit version
mkNode ``Lean.Parser.Term.quotedName #[Syntax.mkNameLit s!"`{n}"]
@[builtinDelab app.Lean.Name.mkNum]
def delabNameMkNum : Delab := delabNameMkStr
end Lean.PrettyPrinter.Delaborator
|
e1b6c3975fcfb116ab432987c53113bd9985250d | c777c32c8e484e195053731103c5e52af26a25d1 | /counterexamples/linear_order_with_pos_mul_pos_eq_zero.lean | 6e08c5f9f847f451787435f5987042be48c19607 | [
"Apache-2.0"
] | permissive | kbuzzard/mathlib | 2ff9e85dfe2a46f4b291927f983afec17e946eb8 | 58537299e922f9c77df76cb613910914a479c1f7 | refs/heads/master | 1,685,313,702,744 | 1,683,974,212,000 | 1,683,974,212,000 | 128,185,277 | 1 | 0 | null | 1,522,920,600,000 | 1,522,920,600,000 | null | UTF-8 | Lean | false | false | 2,190 | lean | /-
Copyright (c) 2021 Johan Commelin.
All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Damiano Testa, Kevin Buzzard
-/
import algebra.order.monoid.defs
import algebra.order.monoid.with_zero.defs
/-!
An example of a `linear_ordered_comm_monoid_with_zero` in which the product of two positive
elements vanishes.
This is the monoid with 3 elements `0, ε, 1` where `ε ^ 2 = 0` and everything else is forced.
The order is `0 < ε < 1`. Since `ε ^ 2 = 0`, the product of strictly positive elements can vanish.
Relevant Zulip chat:
https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/mul_pos
-/
/-- The three element monoid. -/
@[derive [decidable_eq]]
inductive foo
| zero
| eps
| one
namespace foo
instance inhabited : inhabited foo := ⟨zero⟩
instance : has_zero foo := ⟨zero⟩
instance : has_one foo := ⟨one⟩
local notation `ε` := eps
/-- The order on `foo` is the one induced by the natural order on the image of `aux1`. -/
def aux1 : foo → ℕ
| 0 := 0
| ε := 1
| 1 := 2
/-- A tactic to prove facts by cases. -/
meta def boom : tactic unit :=
`[repeat {rintro ⟨⟩}; dec_trivial]
lemma aux1_inj : function.injective aux1 :=
by boom
instance : linear_order foo :=
linear_order.lift' aux1 aux1_inj
/-- Multiplication on `foo`: the only external input is that `ε ^ 2 = 0`. -/
def mul : foo → foo → foo
| 1 x := x
| x 1 := x
| _ _ := 0
instance : comm_monoid foo :=
{ mul := mul,
one := 1,
one_mul := by boom,
mul_one := by boom,
mul_comm := by boom,
mul_assoc := by boom }
instance : linear_ordered_comm_monoid_with_zero foo :=
{ zero := 0,
zero_mul := by boom,
mul_zero := by boom,
mul_le_mul_left := by { rintro ⟨⟩ ⟨⟩ h ⟨⟩; revert h; dec_trivial },
zero_le_one := dec_trivial,
.. foo.linear_order,
.. foo.comm_monoid }
lemma not_mul_pos : ¬ ∀ {M : Type} [linear_ordered_comm_monoid_with_zero M], by exactI ∀
(a b : M) (ha : 0 < a) (hb : 0 < b),
0 < a * b :=
begin
intros h,
specialize h ε ε (by boom) (by boom),
exact (lt_irrefl 0 (h.trans_le (by boom))).elim,
end
example : 0 < ε ∧ ε * ε = 0 := by boom
end foo
|
cbb1699ea004d10086a3951e78e605eec5da6a77 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /counterexamples/linear_order_with_pos_mul_pos_eq_zero.lean | 8484b0d9cf9172ffb3611f5dc0daca8a8b2fe9ac | [
"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 | 2,136 | lean | /-
Copyright (c) 2021 Johan Commelin.
All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Damiano Testa, Kevin Buzzard
-/
import algebra.order.monoid
/-!
An example of a `linear_ordered_comm_monoid_with_zero` in which the product of two positive
elements vanishes.
This is the monoid with 3 elements `0, ε, 1` where `ε ^ 2 = 0` and everything else is forced.
The order is `0 < ε < 1`. Since `ε ^ 2 = 0`, the product of strictly positive elements can vanish.
Relevant Zulip chat:
https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/mul_pos
-/
/-- The three element monoid. -/
@[derive [decidable_eq]]
inductive foo
| zero
| eps
| one
namespace foo
instance inhabited : inhabited foo := ⟨zero⟩
instance : has_zero foo := ⟨zero⟩
instance : has_one foo := ⟨one⟩
notation `ε` := eps
/-- The order on `foo` is the one induced by the natural order on the image of `aux1`. -/
def aux1 : foo → ℕ
| 0 := 0
| ε := 1
| 1 := 2
/-- A tactic to prove facts by cases. -/
meta def boom : tactic unit :=
`[repeat {rintro ⟨⟩}; dec_trivial]
lemma aux1_inj : function.injective aux1 :=
by boom
instance : linear_order foo :=
linear_order.lift' aux1 aux1_inj
/-- Multiplication on `foo`: the only external input is that `ε ^ 2 = 0`. -/
def mul : foo → foo → foo
| 1 x := x
| x 1 := x
| _ _ := 0
instance : comm_monoid foo :=
{ mul := mul,
one := 1,
one_mul := by boom,
mul_one := by boom,
mul_comm := by boom,
mul_assoc := by boom }
instance : linear_ordered_comm_monoid_with_zero foo :=
{ zero := 0,
zero_mul := by boom,
mul_zero := by boom,
mul_le_mul_left := by { rintro ⟨⟩ ⟨⟩ h ⟨⟩; revert h; dec_trivial },
zero_le_one := dec_trivial,
.. foo.linear_order,
.. foo.comm_monoid }
lemma not_mul_pos : ¬ ∀ {M : Type} [linear_ordered_comm_monoid_with_zero M], by exactI ∀
(a b : M) (ha : 0 < a) (hb : 0 < b),
0 < a * b :=
begin
intros h,
specialize h ε ε (by boom) (by boom),
exact (lt_irrefl 0 (h.trans_le (by boom))).elim,
end
example : 0 < ε ∧ ε * ε = 0 := by boom
end foo
|
9d40f45f80273b5e726a5f77f50cff948b9417d3 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/ring_theory/eisenstein_criterion.lean | 9dcce2f4139fe98a522dccba21746f271deacbd8 | [
"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 | 5,397 | lean | /-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import ring_theory.ideal.operations
import data.polynomial.ring_division
import tactic.apply_fun
import ring_theory.prime
/-!
# Eisenstein's criterion
A proof of a slight generalisation of Eisenstein's criterion for the irreducibility of
a polynomial over an integral domain.
-/
open polynomial ideal.quotient
variables {R : Type*} [integral_domain R]
namespace polynomial
namespace eisenstein_criterion_aux
/- Section for auxiliary lemmas used in the proof of `irreducible_of_eisenstein_criterion`-/
lemma map_eq_C_mul_X_pow_of_forall_coeff_mem {f : polynomial R} {P : ideal R}
(hfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P) (hf0 : f ≠ 0) :
map (mk P) f = C ((mk P) f.leading_coeff) * X ^ f.nat_degree :=
polynomial.ext (λ n, begin
rcases lt_trichotomy ↑n (degree f) with h | h | h,
{ erw [coeff_map, eq_zero_iff_mem.2 (hfP n h), coeff_C_mul, coeff_X_pow, if_neg, mul_zero],
rintro rfl, exact not_lt_of_ge degree_le_nat_degree h },
{ have : nat_degree f = n, from nat_degree_eq_of_degree_eq_some h.symm,
rw [coeff_C_mul, coeff_X_pow, if_pos this.symm, mul_one, leading_coeff, this, coeff_map] },
{ rw [coeff_eq_zero_of_degree_lt, coeff_eq_zero_of_degree_lt],
{ refine lt_of_le_of_lt (degree_C_mul_X_pow_le _ _) _,
rwa ← degree_eq_nat_degree hf0 },
{ exact lt_of_le_of_lt (degree_map_le _) h } }
end)
lemma le_nat_degree_of_map_eq_mul_X_pow {n : ℕ}
{P : ideal R} (hP : P.is_prime) {q : polynomial R} {c : polynomial P.quotient}
(hq : map (mk P) q = c * X ^ n) (hc0 : c.degree = 0) : n ≤ q.nat_degree :=
with_bot.coe_le_coe.1
(calc ↑n = degree (q.map (mk P)) :
by rw [hq, degree_mul, hc0, zero_add, degree_pow, degree_X, nsmul_one, nat.cast_with_bot]
... ≤ degree q : degree_map_le _
... ≤ nat_degree q : degree_le_nat_degree)
lemma eval_zero_mem_ideal_of_eq_mul_X_pow {n : ℕ} {P : ideal R}
{q : polynomial R} {c : polynomial P.quotient}
(hq : map (mk P) q = c * X ^ n) (hn0 : 0 < n) : eval 0 q ∈ P :=
by rw [← coeff_zero_eq_eval_zero, ← eq_zero_iff_mem, ← coeff_map,
coeff_zero_eq_eval_zero, hq, eval_mul, eval_pow, eval_X, zero_pow hn0, mul_zero]
lemma is_unit_of_nat_degree_eq_zero_of_forall_dvd_is_unit {p q : polynomial R}
(hu : ∀ (x : R), C x ∣ p * q → is_unit x) (hpm : p.nat_degree = 0) :
is_unit p :=
begin
rw [eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpm), is_unit_C],
refine hu _ _,
rw [← eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpm)],
exact dvd_mul_right _ _
end
end eisenstein_criterion_aux
open eisenstein_criterion_aux
/-- If `f` is a non constant polynomial with coefficients in `R`, and `P` is a prime ideal in `R`,
then if every coefficient in `R` except the leading coefficient is in `P`, and
the trailing coefficient is not in `P^2` and no non units in `R` divide `f`, then `f` is
irreducible. -/
theorem irreducible_of_eisenstein_criterion {f : polynomial R} {P : ideal R} (hP : P.is_prime)
(hfl : f.leading_coeff ∉ P)
(hfP : ∀ n : ℕ, ↑n < degree f → f.coeff n ∈ P)
(hfd0 : 0 < degree f) (h0 : f.coeff 0 ∉ P^2)
(hu : ∀ x : R, C x ∣ f → is_unit x) : irreducible f :=
have hf0 : f ≠ 0, from λ _, by simp only [*, not_true, submodule.zero_mem, coeff_zero] at *,
have hf : f.map (mk P) = C (mk P (leading_coeff f)) * X ^ nat_degree f,
from map_eq_C_mul_X_pow_of_forall_coeff_mem hfP hf0,
have hfd0 : 0 < f.nat_degree, from with_bot.coe_lt_coe.1
(lt_of_lt_of_le hfd0 degree_le_nat_degree),
⟨mt degree_eq_zero_of_is_unit (λ h, by simp only [*, lt_irrefl] at *),
begin
rintros p q rfl,
rw [map_mul] at hf,
rcases mul_eq_mul_prime_pow (show prime (X : polynomial (ideal.quotient P)),
from prime_of_degree_eq_one_of_monic degree_X monic_X) hf with
⟨m, n, b, c, hmnd, hbc, hp, hq⟩,
have hmn : 0 < m → 0 < n → false,
{ assume hm0 hn0,
refine h0 _,
rw [coeff_zero_eq_eval_zero, eval_mul, pow_two],
exact ideal.mul_mem_mul
(eval_zero_mem_ideal_of_eq_mul_X_pow hp hm0)
(eval_zero_mem_ideal_of_eq_mul_X_pow hq hn0) },
have hpql0 : (mk P) (p * q).leading_coeff ≠ 0,
{ rwa [ne.def, eq_zero_iff_mem] },
have hp0 : p ≠ 0, from λ h, by simp only [*, zero_mul, eq_self_iff_true, not_true, ne.def] at *,
have hq0 : q ≠ 0, from λ h, by simp only [*, eq_self_iff_true, not_true, ne.def, mul_zero] at *,
have hbc0 : degree b = 0 ∧ degree c = 0,
{ apply_fun degree at hbc,
rwa [degree_C hpql0, degree_mul, eq_comm, nat.with_bot.add_eq_zero_iff] at hbc },
have hmp : m ≤ nat_degree p,
from le_nat_degree_of_map_eq_mul_X_pow hP hp hbc0.1,
have hnq : n ≤ nat_degree q,
from le_nat_degree_of_map_eq_mul_X_pow hP hq hbc0.2,
have hpmqn : p.nat_degree = m ∧ q.nat_degree = n,
{ rw [nat_degree_mul hp0 hq0] at hmnd,
clear_except hmnd hmp hnq,
omega },
obtain rfl | rfl : m = 0 ∨ n = 0,
{ rwa [nat.pos_iff_ne_zero, nat.pos_iff_ne_zero, imp_false, not_not,
← or_iff_not_imp_left] at hmn },
{ exact or.inl (is_unit_of_nat_degree_eq_zero_of_forall_dvd_is_unit hu hpmqn.1) },
{ exact or.inr (is_unit_of_nat_degree_eq_zero_of_forall_dvd_is_unit
(by simpa only [mul_comm] using hu) hpmqn.2) }
end⟩
end polynomial
|
8960748776ab7c0f649ddce21bef6ef2bbb49bbc | 4fa118f6209450d4e8d058790e2967337811b2b5 | /src/sheaves/f_map.lean | 488dfc5fa39d1a5608056993029a97bcd1aa3c34 | [
"Apache-2.0"
] | permissive | leanprover-community/lean-perfectoid-spaces | 16ab697a220ed3669bf76311daa8c466382207f7 | 95a6520ce578b30a80b4c36e36ab2d559a842690 | refs/heads/master | 1,639,557,829,139 | 1,638,797,866,000 | 1,638,797,866,000 | 135,769,296 | 96 | 10 | Apache-2.0 | 1,638,797,866,000 | 1,527,892,754,000 | Lean | UTF-8 | Lean | false | false | 10,099 | lean | import for_mathlib.open_embeddings
import sheaves.sheaf_of_topological_rings
import sheaves.stalk_of_rings
universe variable u
open topological_space
variables {X : Type u} {Y : Type u} {Z : Type u}
variables [topological_space X] [topological_space Y] [topological_space Z]
namespace presheaf_of_rings
variables {F : presheaf_of_rings X} {G : presheaf_of_rings Y} {H : presheaf_of_rings Z}
structure f_map
(F : presheaf_of_rings X) (G : presheaf_of_rings Y) :=
(f : X → Y)
(hf : continuous f)
(f_flat : ∀ V : opens Y, G V → F (hf.comap V))
(f_flat_is_ring_hom : ∀ V : opens Y, is_ring_hom (f_flat V))
(presheaf_f_flat : ∀ V W : opens Y, ∀ (hWV : W ⊆ V),
∀ s : G V, F.res _ _ (hf.comap_mono hWV) (f_flat V s) = f_flat W (G.res V W hWV s))
attribute [instance] f_map.f_flat_is_ring_hom
def f_map_id (F : presheaf_of_rings X) : presheaf_of_rings.f_map F F :=
{ f := λ x, x,
hf := continuous_id,
f_flat := λ U, F.res _ _ (λ _ hx, hx),
f_flat_is_ring_hom := λ U, presheaf_of_rings.res_is_ring_hom _ _ _ _,
presheaf_f_flat := λ U V hVU s, begin
rw ←F.to_presheaf.Hcomp',
rw ←F.to_presheaf.Hcomp',
end }
def f_map.comp (a : presheaf_of_rings.f_map F G) (b : presheaf_of_rings.f_map G H) :
presheaf_of_rings.f_map F H :=
{ f := λ x, b.f (a.f x),
hf := b.hf.comp a.hf,
f_flat := λ V s, (a.f_flat (b.hf.comap V)) ((b.f_flat V) s),
f_flat_is_ring_hom := λ V, show (is_ring_hom ((a.f_flat (b.hf.comap V)) ∘ (b.f_flat V))), from is_ring_hom.comp _ _,
presheaf_f_flat := λ V W hWV s,
begin
rw ←b.presheaf_f_flat V W hWV s,
rw ←a.presheaf_f_flat (b.hf.comap V) (b.hf.comap W) (b.hf.comap_mono hWV),
refl,
end }
@[simp] lemma f_map.id_comp (a : presheaf_of_rings.f_map F G) :
(presheaf_of_rings.f_map_id F).comp a = a :=
begin
cases a, delta presheaf_of_rings.f_map_id presheaf_of_rings.f_map.comp,
congr, funext V s, dsimp,
show _ = id _, apply congr_fun, exact F.to_presheaf.Hid _,
end
@[simp] lemma f_map.comp_id (a : presheaf_of_rings.f_map F G) :
a.comp (presheaf_of_rings.f_map_id G) = a :=
begin
cases a with f hf f_flat f_flat_is_ring_hom presheaf_f_flat,
delta presheaf_of_rings.f_map_id presheaf_of_rings.f_map.comp,
congr, funext V s, dsimp,
rw ← presheaf_f_flat,
show _ = id _, apply congr_fun, exact F.to_presheaf.Hid _,
end
end presheaf_of_rings
namespace presheaf_of_topological_rings
variables {F : presheaf_of_topological_rings X} {G : presheaf_of_topological_rings Y} {H : presheaf_of_topological_rings Z}
structure f_map (F : presheaf_of_topological_rings X) (G : presheaf_of_topological_rings Y) :=
(f : X → Y)
(hf : continuous f)
(f_flat : ∀ V : opens Y, G V → F (hf.comap V))
[f_flat_is_ring_hom : ∀ V : opens Y, is_ring_hom (f_flat V)]
(cont_f_flat : ∀ V : opens Y, continuous (f_flat V))
(presheaf_f_flat : ∀ V W : opens Y, ∀ (hWV : W ⊆ V),
∀ s : G V, F.res _ _ (hf.comap_mono hWV) (f_flat V s) = f_flat W (G.res V W hWV s))
instance f_map_flat.is_ring_hom (f : presheaf_of_topological_rings.f_map F G) (V : opens Y) :
is_ring_hom (f.f_flat V) := f.f_flat_is_ring_hom V
attribute [instance] presheaf_of_topological_rings.f_map.f_flat_is_ring_hom
def f_map.to_presheaf_of_rings_f_map
{X : Type u} [topological_space X] {Y : Type u} [topological_space Y]
{F : presheaf_of_topological_rings X} {G : presheaf_of_topological_rings Y}
(f : presheaf_of_topological_rings.f_map F G) :
presheaf_of_rings.f_map F.to_presheaf_of_rings G.to_presheaf_of_rings :=
{ ..f}
@[ext]
lemma presheaf_of_topological_rings.f_map.ext
{X : Type u} [topological_space X] {Y : Type u} [topological_space Y]
{F : presheaf_of_topological_rings X} {G : presheaf_of_topological_rings Y}
(a b : F.f_map G) (h : a.to_presheaf_of_rings_f_map = b.to_presheaf_of_rings_f_map) :
a = b :=
begin
cases a, cases b,
dsimp [f_map.to_presheaf_of_rings_f_map] at h,
injections,
simp [*]
end
@[simp] lemma f_map.to_presheaf_of_rings_f_map_f (f : presheaf_of_topological_rings.f_map F G) :
f.to_presheaf_of_rings_f_map.f = f.f := rfl
def f_map_id (F : presheaf_of_topological_rings X) :
presheaf_of_topological_rings.f_map F F :=
{ cont_f_flat := λ U, begin
show continuous (((F.to_presheaf_of_rings).to_presheaf).res U (continuous.comap continuous_id U) _),
convert continuous_id,
{ simp [continuous.comap_id U] },
{ simp [continuous.comap_id U] },
convert heq_of_eq (F.Hid U),
rw continuous.comap_id U,
exact continuous.comap_id U,
end,
..presheaf_of_rings.f_map_id F.to_presheaf_of_rings }
@[simp] lemma f_map.to_presheaf_of_rings_f_map_id (F : presheaf_of_topological_rings X) :
(presheaf_of_topological_rings.f_map_id F).to_presheaf_of_rings_f_map =
presheaf_of_rings.f_map_id F.to_presheaf_of_rings := rfl
@[simp] lemma f_map_id_apply (F : presheaf_of_topological_rings X) (x : X) :
(presheaf_of_topological_rings.f_map_id F).f x = x := rfl
def f_map.comp (a : presheaf_of_topological_rings.f_map F G) (b : presheaf_of_topological_rings.f_map G H) :
presheaf_of_topological_rings.f_map F H :=
{ cont_f_flat := λ V, (a.cont_f_flat _).comp (b.cont_f_flat _),
.. a.to_presheaf_of_rings_f_map.comp b.to_presheaf_of_rings_f_map }
@[simp] lemma f_map.comp_f (a : presheaf_of_topological_rings.f_map F G) (b : presheaf_of_topological_rings.f_map G H) :
(a.comp b).f = b.f ∘ a.f := rfl
@[simp] lemma f_map.comp_to_presheaf_of_rings_f_map
(a : presheaf_of_topological_rings.f_map F G) (b : presheaf_of_topological_rings.f_map G H) :
(a.comp b).to_presheaf_of_rings_f_map =
a.to_presheaf_of_rings_f_map.comp b.to_presheaf_of_rings_f_map := rfl
@[simp] lemma f_map.id_comp (a : presheaf_of_topological_rings.f_map F G) :
(presheaf_of_topological_rings.f_map_id F).comp a = a :=
by ext; simp
@[simp] lemma f_map.comp_id (a : presheaf_of_topological_rings.f_map F G) :
a.comp (presheaf_of_topological_rings.f_map_id G) = a :=
by ext; simp
end presheaf_of_topological_rings
open_locale classical
/-- The map on stalks induced from an f-map -/
noncomputable def stalk_map {F : presheaf_of_rings X} {G : presheaf_of_rings Y}
(f : F.f_map G) (x : X) :
stalk_of_rings G (f.f x) → stalk_of_rings F x :=
to_stalk.rec G (f.f x) (stalk_of_rings F x)
(λ V hfx s, ⟦⟨f.hf.comap V, hfx, f.f_flat V s⟩⟧)
(λ V W H r hfx, quotient.sound begin
use [f.hf.comap V, hfx, set.subset.refl _, f.hf.comap_mono H],
erw F.to_presheaf.Hid,
symmetry,
apply f.presheaf_f_flat
end )
namespace stalk_map
variables {F : presheaf_of_rings X} {G : presheaf_of_rings Y} {H : presheaf_of_rings Z}
variables (f : F.f_map G) (g : G.f_map H)
instance (F : presheaf_of_rings X) (x : X) :
comm_ring (quotient (stalk.setoid (F.to_presheaf) x)) :=
stalk_of_rings_is_comm_ring F x
instance f_flat_is_ring_hom (x : X) (V : opens Y) (hfx : f.f x ∈ V) :
is_ring_hom (λ (s : G.F V), (⟦⟨f.hf.comap V, hfx, f.f_flat V s⟩⟧ : stalk_of_rings F x)) :=
begin
show is_ring_hom ((to_stalk F x (f.hf.comap V) hfx) ∘ (f.f_flat V)),
refine is_ring_hom.comp _ _,
end
instance (x : X) : is_ring_hom (stalk_map f x) := to_stalk.rec_is_ring_hom _ _ _ _ _
@[simp] lemma stalk_map_id (F : presheaf_of_rings X) (x : X) (s : stalk_of_rings F x) :
stalk_map (presheaf_of_rings.f_map_id F) x s = s :=
begin
induction s,
apply quotient.sound,
use s.U,
use s.HxU,
use (le_refl s.U),
use (le_refl s.U),
symmetry,
convert (F.to_presheaf.Hcomp' _ _ _ _ _ s.s),
refl,
end
@[simp] lemma stalk_map_id' (F : presheaf_of_rings X) (x : X) :
stalk_map (presheaf_of_rings.f_map_id F) x = id := by ext; apply stalk_map_id
lemma stalk_map_comp (x : X) (s : stalk_of_rings H (g.f (f.f x))) :
stalk_map (f.comp g) x s = stalk_map f x (stalk_map g (f.f x) s) :=
begin
induction s,
apply quotient.sound,
use f.hf.comap (g.hf.comap s.U),
use s.HxU,
existsi _, swap, intros t ht, exact ht,
existsi _, swap, intros t ht, exact ht,
refl,
refl,
end
@[simp] lemma stalk_map_comp' (x : X) :
stalk_map (f.comp g) x = (stalk_map f x) ∘ (stalk_map g (f.f x)) :=
by ext; apply stalk_map_comp
end stalk_map
namespace presheaf_of_rings
def restrict (U : opens X) (G : presheaf_of_rings X) :
presheaf_of_rings U :=
{ F := λ V, G.F (topological_space.opens.map U V),
res := λ V W HWV, G.res _ _ (topological_space.opens.map_mono HWV),
Hid := λ V, G.Hid (topological_space.opens.map U V),
Hcomp := λ V₁ V₂ V₃ H12 H23, G.Hcomp (topological_space.opens.map U V₁)
(topological_space.opens.map U V₂) (topological_space.opens.map U V₃)
(topological_space.opens.map_mono H12) (topological_space.opens.map_mono H23),
Fring := λ V, G.Fring (topological_space.opens.map U V),
res_is_ring_hom := λ V W HWV, G.res_is_ring_hom (topological_space.opens.map U V)
(topological_space.opens.map U W) (topological_space.opens.map_mono HWV) }
variables {U : opens X} (G : presheaf_of_rings X) (u : U)
noncomputable def restrict_stalk_map :
stalk_of_rings (G.restrict U) u → stalk_of_rings G u :=
to_stalk.rec (G.restrict U) u (stalk_of_rings G u)
(λ V hu, to_stalk G u (topological_space.opens.map U V) ( opens.map_mem_of_mem hu))
(λ W V HWV s huW, quotient.sound (begin
use [(topological_space.opens.map U W), opens.map_mem_of_mem huW],
use [(set.subset.refl (topological_space.opens.map U W)), topological_space.opens.map_mono HWV],
rw G.Hid (topological_space.opens.map U W),
refl,
end))
instance : is_ring_hom (G.restrict_stalk_map u) :=
by delta restrict_stalk_map; apply_instance
end presheaf_of_rings
def presheaf_of_topological_rings.restrict (U : opens X) (G : presheaf_of_topological_rings X) :
presheaf_of_topological_rings U :=
{ Ftop := λ V, G.Ftop (topological_space.opens.map U V),
Ftop_ring := λ V, G.Ftop_ring (topological_space.opens.map U V),
res_continuous := λ V W HWV, G.res_continuous (topological_space.opens.map U V)
(topological_space.opens.map U W) (topological_space.opens.map_mono HWV),
..presheaf_of_rings.restrict U G.to_presheaf_of_rings }
|
594c219cb039bdeb202e45741287c48a5a11d625 | e39f04f6ff425fe3b3f5e26a8998b817d1dba80f | /category_theory/limits/functor_category.lean | d077e795c172673fc3bfd7aaff0260e509e880dc | [
"Apache-2.0"
] | permissive | kristychoi/mathlib | c504b5e8f84e272ea1d8966693c42de7523bf0ec | 257fd84fe98927ff4a5ffe044f68c4e9d235cc75 | refs/heads/master | 1,586,520,722,896 | 1,544,030,145,000 | 1,544,031,933,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 4,788 | lean | -- Copyright (c) 2018 Scott Morrison. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Scott Morrison
import category_theory.products
import category_theory.limits.preserves
open category_theory category_theory.category
namespace category_theory.limits
universes u v
variables {C : Type u} [𝒞 : category.{u v} C]
include 𝒞
variables {J K : Type v} [small_category J] [small_category K]
@[simp] lemma cone.functor_w {F : J ⥤ (K ⥤ C)} (c : cone F) {j j' : J} (f : j ⟶ j') (k : K) :
(c.π.app j).app k ≫ (F.map f).app k = (c.π.app j').app k :=
by convert ←nat_trans.congr_app (c.π.naturality f).symm k; apply id_comp
@[simp] lemma cocone.functor_w {F : J ⥤ (K ⥤ C)} (c : cocone F) {j j' : J} (f : j ⟶ j') (k : K) :
(F.map f).app k ≫ (c.ι.app j').app k = (c.ι.app j).app k :=
by convert ←nat_trans.congr_app (c.ι.naturality f) k; apply comp_id
@[simp] def functor_category_limit_cone [has_limits_of_shape J C] (F : J ⥤ K ⥤ C) :
cone F :=
{ X := F.flip ⋙ lim,
π :=
{ app := λ j,
{ app := λ k, limit.π (F.flip.obj k) j },
naturality' := λ j j' f,
by ext k; convert (limit.w (F.flip.obj k) _).symm using 1; apply id_comp } }
@[simp] def functor_category_colimit_cocone [has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) :
cocone F :=
{ X := F.flip ⋙ colim,
ι :=
{ app := λ j,
{ app := λ k, colimit.ι (F.flip.obj k) j },
naturality' := λ j j' f,
by ext k; convert (colimit.w (F.flip.obj k) _) using 1; apply comp_id } }
@[simp] def evaluate_functor_category_limit_cone
[has_limits_of_shape J C] (F : J ⥤ K ⥤ C) (k : K) :
((evaluation K C).obj k).map_cone (functor_category_limit_cone F) ≅
limit.cone (F.flip.obj k) :=
cones.ext (iso.refl _) (by tidy)
@[simp] def evaluate_functor_category_colimit_cocone
[has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) (k : K) :
((evaluation K C).obj k).map_cocone (functor_category_colimit_cocone F) ≅
colimit.cocone (F.flip.obj k) :=
cocones.ext (iso.refl _) (by tidy)
def functor_category_is_limit_cone [has_limits_of_shape J C] (F : J ⥤ K ⥤ C) :
is_limit (functor_category_limit_cone F) :=
{ lift := λ s,
{ app := λ k, limit.lift (F.flip.obj k) (((evaluation K C).obj k).map_cone s),
naturality' := λ k k' f,
by ext; dsimp; simpa using (s.π.app j).naturality f },
uniq' := λ s m w,
begin
ext1 k,
exact is_limit.uniq _
(((evaluation K C).obj k).map_cone s) (m.app k) (λ j, nat_trans.congr_app (w j) k)
end }
def functor_category_is_colimit_cocone [has_colimits_of_shape.{u v} J C] (F : J ⥤ K ⥤ C) :
is_colimit (functor_category_colimit_cocone F) :=
{ desc := λ s,
{ app := λ k, colimit.desc (F.flip.obj k) (((evaluation K C).obj k).map_cocone s),
naturality' := λ k k' f,
begin
ext,
rw [←assoc, ←assoc],
dsimp [functor.flip],
simpa using (s.ι.app j).naturality f
end },
uniq' := λ s m w,
begin
ext1 k,
exact is_colimit.uniq _
(((evaluation K C).obj k).map_cocone s) (m.app k) (λ j, nat_trans.congr_app (w j) k)
end }
instance functor_category_has_limits_of_shape
[has_limits_of_shape J C] : has_limits_of_shape J (K ⥤ C) :=
λ F,
{ cone := functor_category_limit_cone F,
is_limit := functor_category_is_limit_cone F }
instance functor_category_has_colimits_of_shape
[has_colimits_of_shape J C] : has_colimits_of_shape J (K ⥤ C) :=
λ F,
{ cocone := functor_category_colimit_cocone F,
is_colimit := functor_category_is_colimit_cocone F }
instance functor_category_has_limits [has_limits C] : has_limits (K ⥤ C) :=
λ J 𝒥, by resetI; apply_instance
instance functor_category_has_colimits [has_colimits C] : has_colimits (K ⥤ C) :=
λ J 𝒥, by resetI; apply_instance
instance evaluation_preserves_limits_of_shape [has_limits_of_shape J C] (k : K) :
preserves_limits_of_shape J ((evaluation K C).obj k) :=
λ F, preserves_limit_of_preserves_limit_cone (limit.is_limit _) $
is_limit.of_iso_limit (limit.is_limit _)
(evaluate_functor_category_limit_cone F k).symm
instance evaluation_preserves_colimits_of_shape [has_colimits_of_shape J C] (k : K) :
preserves_colimits_of_shape J ((evaluation K C).obj k) :=
λ F, preserves_colimit_of_preserves_colimit_cocone (colimit.is_colimit _) $
is_colimit.of_iso_colimit (colimit.is_colimit _)
(evaluate_functor_category_colimit_cocone F k).symm
instance evaluation_preserves_limits [has_limits C] (k : K) :
preserves_limits ((evaluation K C).obj k) :=
λ J 𝒥, by resetI; apply_instance
instance evaluation_preserves_colimits [has_colimits C] (k : K) :
preserves_colimits ((evaluation K C).obj k) :=
λ J 𝒥, by resetI; apply_instance
end category_theory.limits
|
64110e384ba30dc38ff9d651a1e3d922dc4cab67 | 2d34dfb0a1cc250584282618dc10ea03d3fa858e | /src/for_mathlib/add_monoid_hom.lean | 44bcce7198410d803aceef30b26d0f3fcbfeec1e | [] | no_license | zeta1999/lean-liquid | 61e294ec5adae959d8ee1b65d015775484ff58c2 | 96bb0fa3afc3b451bcd1fb7d974348de2f290541 | refs/heads/master | 1,676,579,150,248 | 1,610,771,445,000 | 1,610,771,445,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 815 | lean | import algebra.group.hom
import algebra.big_operators
open_locale big_operators
namespace add_monoid_hom
variables {M₁ M₂ : Type*}
section
variables [add_monoid M₁] [add_comm_monoid M₂]
@[simp] lemma sum_apply {ι : Type*} (s : finset ι) (f : ι → (M₁ →+ M₂)) (m : M₁) :
(∑ i in s, f i) m = ∑ i in s, (f i m) :=
begin
classical,
apply finset.induction_on s,
{ simp only [finset.sum_empty, zero_apply] },
{ intros i s his IH, simp only [his, IH, finset.sum_insert, not_false_iff, add_apply] }
end
end
section
variables [add_monoid M₁] [add_comm_group M₂]
@[simp] lemma gsmul_apply (n : ℤ) (f : (M₁ →+ M₂)) (m : M₁) :
(n •ℤ f) m = n •ℤ (f m) :=
show eval m (n •ℤ f) = n •ℤ eval m f, from add_monoid_hom.map_gsmul _ _ _
end
end add_monoid_hom
|
0705441c73e04dbc341c7758a02438917853dabd | 5883d9218e6f144e20eee6ca1dab8529fa1a97c0 | /src/aeq/id.lean | 1ebcdbdadd3ac37af5cd244f29c2092c25a90d0f | [] | no_license | spl/alpha-conversion-is-easy | 0d035bc570e52a6345d4890e4d0c9e3f9b8126c1 | ed937fe85d8495daffd9412a5524c77b9fcda094 | refs/heads/master | 1,607,649,280,020 | 1,517,380,240,000 | 1,517,380,240,000 | 52,174,747 | 4 | 0 | null | 1,456,052,226,000 | 1,456,001,163,000 | Lean | UTF-8 | Lean | false | false | 1,853 | lean | /-
This file contains declarations related to `aeq` identity or reflexivity.
-/
import .equiv
namespace acie -----------------------------------------------------------------
namespace aeq ------------------------------------------------------------------
variables {V : Type} [decidable_eq V] -- Type of variable names
variables {vs : Type → Type} [vset vs V] -- Type of variable name sets
variables {X : vs V} -- Variable name sets
-- Identity `aeq` with an implicit `vset`.
@[inline, reducible]
protected
def id : exp X → exp X → Prop :=
aeq (vrel.id X)
-- Notation for `aeq.id`.
infix ` ≡α `:50 := aeq.id
namespace id -------------------------------------------------------------------
-- Paper: Corollary 3
-- Reflexivity of `aeq.id`
@[refl]
protected
theorem refl : ∀ (e : exp X), aeq (vrel.id X) e e :=
aeq.refl
-- Symmetry of `aeq.id`
@[symm]
protected
theorem symm ⦃e₁ e₂ : exp X⦄
: aeq (vrel.id X) e₁ e₂ → aeq (vrel.id X) e₂ e₁ :=
λ a, map.simple (λ x₁ x₂, vrel.inv.of_id) (aeq.symm a)
-- Transitivity of `aeq.id`
@[trans]
protected
theorem trans ⦃e₁ e₂ e₃ : exp X⦄
: aeq (vrel.id X) e₁ e₂ → aeq (vrel.id X) e₂ e₃ → aeq (vrel.id X) e₁ e₃ :=
λ a₁ a₂, map.simple (λ x₁ x₂, vrel.id.of_comp) (aeq.trans a₁ a₂)
-- Equivalence
protected
theorem equiv (X : vs V) : equivalence (aeq (vrel.id X)) :=
mk_equivalence (aeq (vrel.id X))
(@id.refl _ _ _ _ X)
(@id.symm _ _ _ _ X)
(@id.trans _ _ _ _ X)
-- Setoid
instance setoid (X : vs V) : setoid (exp X) :=
setoid.mk (aeq (vrel.id X)) (id.equiv X)
end /- namespace -/ id ---------------------------------------------------------
end /- namespace -/ aeq --------------------------------------------------------
end /- namespace -/ acie -------------------------------------------------------
|
776c32829432509c8de622912bd3a436d6178183 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/struct1.lean | f6cc3037f4e9b21fd7485df6dc19822abaef0959 | [
"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 | 983 | lean | --
structure A (α : Type) :=
(x : α)
structure B (α : Type) :=
(x : α)
structure S : Nat := -- error expected Type
(x : Nat)
structure S extends Nat → Nat := -- error expected structure
(x : Nat)
set_option structureDiamondWarning true in
structure S' extends A Nat, A Bool := -- error field toA already declared
(x : Nat)
structure S extends A Nat, B Bool := -- error field `x` from `B` has already been declared
(x : Nat)
structure S1 :=
(_x : Nat)
structure S2 :=
(x _y : Nat)
structure S :=
(x : Nat)
(x : Nat) -- error
structure S extends A Nat :=
(x : Nat) -- error
structure S extends A Nat :=
(x := true) -- error type mismatch
structure S extends A Nat :=
(x : Bool := true) -- error omit type
structure S :=
(x : Nat := true) -- error type mismatch
private structure S :=
private mk :: (x : Nat)
private structure S :=
protected mk :: (x : Nat)
private structure S :=
protected (x : Nat)
private structure S :=
mk2 :: (x : Nat)
#check S
#check S.mk2
|
8bc69d36aa0c03cc0f2c217cbe71138cb9540584 | 08bd4ba4ca87dba1f09d2c96a26f5d65da81f4b4 | /src/Lean/Elab/InfoTree/Main.lean | bad0d0a48ffc22c6cc5a5a9802ef3d2f38225802 | [
"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",
"Apache-2.0",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | gebner/lean4 | d51c4922640a52a6f7426536ea669ef18a1d9af5 | 8cd9ce06843c9d42d6d6dc43d3e81e3b49dfc20f | refs/heads/master | 1,685,732,780,391 | 1,672,962,627,000 | 1,673,459,398,000 | 373,307,283 | 0 | 0 | Apache-2.0 | 1,691,316,730,000 | 1,622,669,271,000 | Lean | UTF-8 | Lean | false | false | 15,844 | lean | /-
Copyright (c) 2020 Wojciech Nawrocki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wojciech Nawrocki, Leonardo de Moura, Sebastian Ullrich
-/
import Lean.Meta.PPGoal
namespace Lean.Elab.ContextInfo
variable [Monad m] [MonadEnv m] [MonadMCtx m] [MonadOptions m] [MonadResolveName m] [MonadNameGenerator m]
def saveNoFileMap : m ContextInfo := return {
env := (← getEnv)
fileMap := default
mctx := (← getMCtx)
options := (← getOptions)
currNamespace := (← getCurrNamespace)
openDecls := (← getOpenDecls)
ngen := (← getNGen)
}
def save [MonadFileMap m] : m ContextInfo := do
let ctx ← saveNoFileMap
return { ctx with fileMap := (← getFileMap) }
end ContextInfo
def CompletionInfo.stx : CompletionInfo → Syntax
| dot i .. => i.stx
| id stx .. => stx
| dotId stx .. => stx
| fieldId stx .. => stx
| namespaceId stx => stx
| option stx => stx
| endSection stx .. => stx
| tactic stx .. => stx
def CustomInfo.format : CustomInfo → Format
| i => f!"CustomInfo({i.value.typeName})"
instance : ToFormat CustomInfo := ⟨CustomInfo.format⟩
partial def InfoTree.findInfo? (p : Info → Bool) (t : InfoTree) : Option Info :=
match t with
| context _ t => findInfo? p t
| node i ts =>
if p i then
some i
else
ts.findSome? (findInfo? p)
| _ => none
/-- Instantiate the holes on the given `tree` with the assignment table.
(analoguous to instantiating the metavariables in an expression) -/
partial def InfoTree.substitute (tree : InfoTree) (assignment : PersistentHashMap MVarId InfoTree) : InfoTree :=
match tree with
| node i c => node i <| c.map (substitute · assignment)
| context i t => context i (substitute t assignment)
| hole id => match assignment.find? id with
| none => hole id
| some tree => substitute tree assignment
def ContextInfo.runMetaM (info : ContextInfo) (lctx : LocalContext) (x : MetaM α) : IO α := do
let x := x.run { lctx := lctx } { mctx := info.mctx }
/-
We must execute `x` using the `ngen` stored in `info`. Otherwise, we may create `MVarId`s and `FVarId`s that
have been used in `lctx` and `info.mctx`.
-/
let ((a, _), _) ←
x.toIO { options := info.options, currNamespace := info.currNamespace, openDecls := info.openDecls, fileName := "<InfoTree>", fileMap := default }
{ env := info.env, ngen := info.ngen }
return a
def ContextInfo.toPPContext (info : ContextInfo) (lctx : LocalContext) : PPContext :=
{ env := info.env, mctx := info.mctx, lctx := lctx,
opts := info.options, currNamespace := info.currNamespace, openDecls := info.openDecls }
def ContextInfo.ppSyntax (info : ContextInfo) (lctx : LocalContext) (stx : Syntax) : IO Format := do
ppTerm (info.toPPContext lctx) ⟨stx⟩ -- HACK: might not be a term
private def formatStxRange (ctx : ContextInfo) (stx : Syntax) : Format :=
let pos := stx.getPos?.getD 0
let endPos := stx.getTailPos?.getD pos
f!"{fmtPos pos stx.getHeadInfo}-{fmtPos endPos stx.getTailInfo}"
where fmtPos pos info :=
let pos := format <| ctx.fileMap.toPosition pos
match info with
| .original .. => pos
| .synthetic (canonical := true) .. => f!"{pos}†!"
| _ => f!"{pos}†"
private def formatElabInfo (ctx : ContextInfo) (info : ElabInfo) : Format :=
if info.elaborator.isAnonymous then
formatStxRange ctx info.stx
else
f!"{formatStxRange ctx info.stx} @ {info.elaborator}"
def TermInfo.runMetaM (info : TermInfo) (ctx : ContextInfo) (x : MetaM α) : IO α :=
ctx.runMetaM info.lctx x
def TermInfo.format (ctx : ContextInfo) (info : TermInfo) : IO Format := do
info.runMetaM ctx do
let ty : Format ← try
Meta.ppExpr (← Meta.inferType info.expr)
catch _ =>
pure "<failed-to-infer-type>"
return f!"{← Meta.ppExpr info.expr} {if info.isBinder then "(isBinder := true) " else ""}: {ty} @ {formatElabInfo ctx info.toElabInfo}"
def CompletionInfo.format (ctx : ContextInfo) (info : CompletionInfo) : IO Format :=
match info with
| .dot i (expectedType? := expectedType?) .. => return f!"[.] {← i.format ctx} : {expectedType?}"
| .id stx _ _ lctx expectedType? => ctx.runMetaM lctx do return f!"[.] {stx} : {expectedType?} @ {formatStxRange ctx info.stx}"
| _ => return f!"[.] {info.stx} @ {formatStxRange ctx info.stx}"
def CommandInfo.format (ctx : ContextInfo) (info : CommandInfo) : IO Format := do
return f!"command @ {formatElabInfo ctx info.toElabInfo}"
def OptionInfo.format (ctx : ContextInfo) (info : OptionInfo) : IO Format := do
return f!"option {info.optionName} @ {formatStxRange ctx info.stx}"
def FieldInfo.format (ctx : ContextInfo) (info : FieldInfo) : IO Format := do
ctx.runMetaM info.lctx do
return f!"{info.fieldName} : {← Meta.ppExpr (← Meta.inferType info.val)} := {← Meta.ppExpr info.val} @ {formatStxRange ctx info.stx}"
def ContextInfo.ppGoals (ctx : ContextInfo) (goals : List MVarId) : IO Format :=
if goals.isEmpty then
return "no goals"
else
ctx.runMetaM {} (return Std.Format.prefixJoin "\n" (← goals.mapM (Meta.ppGoal ·)))
def TacticInfo.format (ctx : ContextInfo) (info : TacticInfo) : IO Format := do
let ctxB := { ctx with mctx := info.mctxBefore }
let ctxA := { ctx with mctx := info.mctxAfter }
let goalsBefore ← ctxB.ppGoals info.goalsBefore
let goalsAfter ← ctxA.ppGoals info.goalsAfter
return f!"Tactic @ {formatElabInfo ctx info.toElabInfo}\n{info.stx}\nbefore {goalsBefore}\nafter {goalsAfter}"
def MacroExpansionInfo.format (ctx : ContextInfo) (info : MacroExpansionInfo) : IO Format := do
let stx ← ctx.ppSyntax info.lctx info.stx
let output ← ctx.ppSyntax info.lctx info.output
return f!"Macro expansion\n{stx}\n===>\n{output}"
def UserWidgetInfo.format (info : UserWidgetInfo) : Format :=
f!"UserWidget {info.widgetId}\n{Std.ToFormat.format info.props}"
def FVarAliasInfo.format (info : FVarAliasInfo) : Format :=
f!"FVarAlias {info.userName.eraseMacroScopes}"
def FieldRedeclInfo.format (ctx : ContextInfo) (info : FieldRedeclInfo) : Format :=
f!"FieldRedecl @ {formatStxRange ctx info.stx}"
def Info.format (ctx : ContextInfo) : Info → IO Format
| ofTacticInfo i => i.format ctx
| ofTermInfo i => i.format ctx
| ofCommandInfo i => i.format ctx
| ofMacroExpansionInfo i => i.format ctx
| ofOptionInfo i => i.format ctx
| ofFieldInfo i => i.format ctx
| ofCompletionInfo i => i.format ctx
| ofUserWidgetInfo i => pure <| i.format
| ofCustomInfo i => pure <| Std.ToFormat.format i
| ofFVarAliasInfo i => pure <| i.format
| ofFieldRedeclInfo i => pure <| i.format ctx
def Info.toElabInfo? : Info → Option ElabInfo
| ofTacticInfo i => some i.toElabInfo
| ofTermInfo i => some i.toElabInfo
| ofCommandInfo i => some i.toElabInfo
| ofMacroExpansionInfo _ => none
| ofOptionInfo _ => none
| ofFieldInfo _ => none
| ofCompletionInfo _ => none
| ofUserWidgetInfo _ => none
| ofCustomInfo _ => none
| ofFVarAliasInfo _ => none
| ofFieldRedeclInfo _ => none
/--
Helper function for propagating the tactic metavariable context to its children nodes.
We need this function because we preserve `TacticInfo` nodes during backtracking *and* their
children. Moreover, we backtrack the metavariable context to undo metavariable assignments.
`TacticInfo` nodes save the metavariable context before/after the tactic application, and
can be pretty printed without any extra information. This is not the case for `TermInfo` nodes.
Without this function, the formatting method would often fail when processing `TermInfo` nodes
that are children of `TacticInfo` nodes that have been preserved during backtracking.
Saving the metavariable context at `TermInfo` nodes is also not a good option because
at `TermInfo` creation time, the metavariable context often miss information, e.g.,
a TC problem has not been resolved, a postponed subterm has not been elaborated, etc.
See `Term.SavedState.restore`.
-/
def Info.updateContext? : Option ContextInfo → Info → Option ContextInfo
| some ctx, ofTacticInfo i => some { ctx with mctx := i.mctxAfter }
| ctx?, _ => ctx?
partial def InfoTree.format (tree : InfoTree) (ctx? : Option ContextInfo := none) : IO Format := do
match tree with
| hole id => return .nestD f!"• ?{toString id.name}"
| context i t => format t i
| node i cs => match ctx? with
| none => return "• <context-not-available>"
| some ctx =>
let fmt ← i.format ctx
if cs.size == 0 then
return .nestD f!"• {fmt}"
else
let ctx? := i.updateContext? ctx?
return .nestD f!"• {fmt}{Std.Format.prefixJoin .line (← cs.toList.mapM fun c => format c ctx?)}"
section
variable [Monad m] [MonadInfoTree m]
@[inline] private def modifyInfoTrees (f : PersistentArray InfoTree → PersistentArray InfoTree) : m Unit :=
modifyInfoState fun s => { s with trees := f s.trees }
/-- Returns the current array of InfoTrees and resets it to an empty array. -/
def getResetInfoTrees : m (PersistentArray InfoTree) := do
let trees := (← getInfoState).trees
modifyInfoTrees fun _ => {}
return trees
def pushInfoTree (t : InfoTree) : m Unit := do
if (← getInfoState).enabled then
modifyInfoTrees fun ts => ts.push t
def pushInfoLeaf (t : Info) : m Unit := do
if (← getInfoState).enabled then
pushInfoTree <| InfoTree.node (children := {}) t
def addCompletionInfo (info : CompletionInfo) : m Unit := do
pushInfoLeaf <| Info.ofCompletionInfo info
def addConstInfo [MonadEnv m] [MonadError m]
(stx : Syntax) (n : Name) (expectedType? : Option Expr := none) : m Unit := do
pushInfoLeaf <| .ofTermInfo {
elaborator := .anonymous
lctx := .empty
expr := (← mkConstWithLevelParams n)
stx
expectedType?
}
/-- This does the same job as `resolveGlobalConstNoOverload`; resolving an identifier
syntax to a unique fully resolved name or throwing if there are ambiguities.
But also adds this resolved name to the infotree. This means that when you hover
over a name in the sourcefile you will see the fully resolved name in the hover info.-/
def resolveGlobalConstNoOverloadWithInfo [MonadResolveName m] [MonadEnv m] [MonadError m]
(id : Syntax) (expectedType? : Option Expr := none) : m Name := do
let n ← resolveGlobalConstNoOverload id
if (← getInfoState).enabled then
-- we do not store a specific elaborator since identifiers are special-cased by the server anyway
addConstInfo id n expectedType?
return n
/-- Similar to `resolveGlobalConstNoOverloadWithInfo`, except if there are multiple name resolutions then it returns them as a list. -/
def resolveGlobalConstWithInfos [MonadResolveName m] [MonadEnv m] [MonadError m]
(id : Syntax) (expectedType? : Option Expr := none) : m (List Name) := do
let ns ← resolveGlobalConst id
if (← getInfoState).enabled then
for n in ns do
addConstInfo id n expectedType?
return ns
/-- Similar to `resolveGlobalName`, but it also adds the resolved name to the info tree. -/
def resolveGlobalNameWithInfos [MonadResolveName m] [MonadEnv m] [MonadError m]
(ref : Syntax) (id : Name) : m (List (Name × List String)) := do
let ns ← resolveGlobalName id
if (← getInfoState).enabled then
for (n, _) in ns do
addConstInfo ref n
return ns
/-- Use this to descend a node on the infotree that is being built.
It saves the current list of trees `t₀` and resets it and then runs `x >>= mkInfo`, producing either an `i : Info` or a hole id.
Running `x >>= mkInfo` will modify the trees state and produce a new list of trees `t₁`.
In the `i : Info` case, `t₁` become the children of a node `node i t₁` that is appended to `t₀`.
-/
def withInfoContext' [MonadFinally m] (x : m α) (mkInfo : α → m (Sum Info MVarId)) : m α := do
if (← getInfoState).enabled then
let treesSaved ← getResetInfoTrees
Prod.fst <$> MonadFinally.tryFinally' x fun a? => do
match a? with
| none => modifyInfoTrees fun _ => treesSaved
| some a =>
let info ← mkInfo a
modifyInfoTrees fun trees =>
match info with
| Sum.inl info => treesSaved.push <| InfoTree.node info trees
| Sum.inr mvarId => treesSaved.push <| InfoTree.hole mvarId
else
x
/-- Saves the current list of trees `t₀`, runs `x` to produce a new tree list `t₁` and
runs `mkInfoTree t₁` to get `n : InfoTree` and then restores the trees to be `t₀ ++ [n]`.-/
def withInfoTreeContext [MonadFinally m] (x : m α) (mkInfoTree : PersistentArray InfoTree → m InfoTree) : m α := do
if (← getInfoState).enabled then
let treesSaved ← getResetInfoTrees
Prod.fst <$> MonadFinally.tryFinally' x fun _ => do
let st ← getInfoState
let tree ← mkInfoTree st.trees
modifyInfoTrees fun _ => treesSaved.push tree
else
x
/-- Run `x` as a new child infotree node with header given by `mkInfo`. -/
@[inline] def withInfoContext [MonadFinally m] (x : m α) (mkInfo : m Info) : m α := do
withInfoTreeContext x (fun trees => do return InfoTree.node (← mkInfo) trees)
/-- Resets the trees state `t₀`, runs `x` to produce a new trees
state `t₁` and sets the state to be `t₀ ++ (InfoTree.context Γ <$> t₁)`
where `Γ` is the context derived from the monad state. -/
def withSaveInfoContext [MonadNameGenerator m] [MonadFinally m] [MonadEnv m] [MonadOptions m] [MonadMCtx m] [MonadResolveName m] [MonadFileMap m] (x : m α) : m α := do
if (← getInfoState).enabled then
let treesSaved ← getResetInfoTrees
Prod.fst <$> MonadFinally.tryFinally' x fun _ => do
let st ← getInfoState
let trees ← st.trees.mapM fun tree => do
let tree := tree.substitute st.assignment
pure <| InfoTree.context (← ContextInfo.save) tree
modifyInfoTrees fun _ => treesSaved ++ trees
else
x
def getInfoHoleIdAssignment? (mvarId : MVarId) : m (Option InfoTree) :=
return (← getInfoState).assignment[mvarId]
def assignInfoHoleId (mvarId : MVarId) (infoTree : InfoTree) : m Unit := do
assert! (← getInfoHoleIdAssignment? mvarId).isNone
modifyInfoState fun s => { s with assignment := s.assignment.insert mvarId infoTree }
end
def withMacroExpansionInfo [MonadFinally m] [Monad m] [MonadInfoTree m] [MonadLCtx m] (stx output : Syntax) (x : m α) : m α :=
let mkInfo : m Info := do
return Info.ofMacroExpansionInfo {
lctx := (← getLCtx)
stx, output
}
withInfoContext x mkInfo
@[inline] def withInfoHole [MonadFinally m] [Monad m] [MonadInfoTree m] (mvarId : MVarId) (x : m α) : m α := do
if (← getInfoState).enabled then
let treesSaved ← getResetInfoTrees
Prod.fst <$> MonadFinally.tryFinally' x fun _ => modifyInfoState fun s =>
if h : s.trees.size > 0 then
have : s.trees.size - 1 < s.trees.size := Nat.sub_lt h (by decide)
{ s with trees := treesSaved, assignment := s.assignment.insert mvarId s.trees[s.trees.size - 1] }
else
{ s with trees := treesSaved }
else
x
def enableInfoTree [MonadInfoTree m] (flag := true) : m Unit :=
modifyInfoState fun s => { s with enabled := flag }
def withEnableInfoTree [Monad m] [MonadInfoTree m] [MonadFinally m] (flag : Bool) (x : m α) : m α := do
let saved := (← getInfoState).enabled
try
enableInfoTree flag
x
finally
enableInfoTree saved
def getInfoTrees [MonadInfoTree m] [Monad m] : m (PersistentArray InfoTree) :=
return (← getInfoState).trees
end Lean.Elab
|
b6b5ce4e410ab6f0f7330b2c752b81127ff1a530 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/category_theory/limits/yoneda.lean | 65dd54a9e3b4489fb63e383b005e7413ea5afa87 | [
"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 | 5,498 | lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import category_theory.limits.functor_category
import tactic.assert_exists
/-!
# Limit properties relating to the (co)yoneda embedding.
We calculate the colimit of `Y ↦ (X ⟶ Y)`, which is just `punit`.
(This is used in characterising cofinal functors.)
We also show the (co)yoneda embeddings preserve limits and jointly reflect them.
-/
open opposite
open category_theory
open category_theory.limits
universes w v u
namespace category_theory
namespace coyoneda
variables {C : Type v} [small_category C]
/--
The colimit cocone over `coyoneda.obj X`, with cocone point `punit`.
-/
@[simps]
def colimit_cocone (X : Cᵒᵖ) : cocone (coyoneda.obj X) :=
{ X := punit,
ι := { app := by tidy, } }
/--
The proposed colimit cocone over `coyoneda.obj X` is a colimit cocone.
-/
@[simps]
def colimit_cocone_is_colimit (X : Cᵒᵖ) : is_colimit (colimit_cocone X) :=
{ desc := λ s x, s.ι.app (unop X) (𝟙 _),
fac' := λ s Y, by { ext f, convert congr_fun (s.w f).symm (𝟙 (unop X)), simp, },
uniq' := λ s m w, by { ext ⟨⟩, rw ← w, simp, } }
instance (X : Cᵒᵖ) : has_colimit (coyoneda.obj X) :=
has_colimit.mk { cocone := _, is_colimit := colimit_cocone_is_colimit X }
/--
The colimit of `coyoneda.obj X` is isomorphic to `punit`.
-/
noncomputable
def colimit_coyoneda_iso (X : Cᵒᵖ) : colimit (coyoneda.obj X) ≅ punit :=
colimit.iso_colimit_cocone { cocone := _, is_colimit := colimit_cocone_is_colimit X }
end coyoneda
variables {C : Type u} [category.{v} C]
open limits
/-- The yoneda embedding `yoneda.obj X : Cᵒᵖ ⥤ Type v` for `X : C` preserves limits. -/
instance yoneda_preserves_limits (X : C) : preserves_limits (yoneda.obj X) :=
{ preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ K,
{ preserves := λ c t,
{ lift := λ s x, quiver.hom.unop (t.lift ⟨op X, λ j, (s.π.app j x).op, λ j₁ j₂ α, _⟩),
fac' := λ s j, funext $ λ x, quiver.hom.op_inj (t.fac _ _),
uniq' := λ s m w, funext $ λ x,
begin
refine quiver.hom.op_inj (t.uniq ⟨op X, _, _⟩ _ (λ j, _)),
{ dsimp, simp [← s.w α] }, -- See library note [dsimp, simp]
{ exact quiver.hom.unop_inj (congr_fun (w j) x) },
end } } } }
/-- The coyoneda embedding `coyoneda.obj X : C ⥤ Type v` for `X : Cᵒᵖ` preserves limits. -/
instance coyoneda_preserves_limits (X : Cᵒᵖ) : preserves_limits (coyoneda.obj X) :=
{ preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ K,
{ preserves := λ c t,
{ lift := λ s x, t.lift ⟨unop X, λ j, s.π.app j x, λ j₁ j₂ α, by { dsimp, simp [← s.w α]}⟩,
-- See library note [dsimp, simp]
fac' := λ s j, funext $ λ x, t.fac _ _,
uniq' := λ s m w, funext $ λ x,
begin
refine (t.uniq ⟨unop X, _⟩ _ (λ j, _)),
exact congr_fun (w j) x,
end } } } }
/-- The yoneda embeddings jointly reflect limits. -/
def yoneda_jointly_reflects_limits (J : Type w) [small_category J] (K : J ⥤ Cᵒᵖ) (c : cone K)
(t : Π (X : C), is_limit ((yoneda.obj X).map_cone c)) : is_limit c :=
let s' : Π (s : cone K), cone (K ⋙ yoneda.obj s.X.unop) :=
λ s, ⟨punit, λ j _, (s.π.app j).unop, λ j₁ j₂ α, funext $ λ _, quiver.hom.op_inj (s.w α).symm⟩
in
{ lift := λ s, ((t s.X.unop).lift (s' s) punit.star).op,
fac' := λ s j, quiver.hom.unop_inj (congr_fun ((t s.X.unop).fac (s' s) j) punit.star),
uniq' := λ s m w,
begin
apply quiver.hom.unop_inj,
suffices : (λ (x : punit), m.unop) = (t s.X.unop).lift (s' s),
{ apply congr_fun this punit.star },
apply (t _).uniq (s' s) _ (λ j, _),
ext,
exact quiver.hom.op_inj (w j),
end }
/-- The coyoneda embeddings jointly reflect limits. -/
def coyoneda_jointly_reflects_limits (J : Type w) [small_category J] (K : J ⥤ C) (c : cone K)
(t : Π (X : Cᵒᵖ), is_limit ((coyoneda.obj X).map_cone c)) : is_limit c :=
let s' : Π (s : cone K), cone (K ⋙ coyoneda.obj (op s.X)) :=
λ s, ⟨punit, λ j _, s.π.app j, λ j₁ j₂ α, funext $ λ _, (s.w α).symm⟩
in
{ lift := λ s, (t (op s.X)).lift (s' s) punit.star,
fac' := λ s j, congr_fun ((t _).fac (s' s) j) punit.star,
uniq' := λ s m w,
begin
suffices : (λ (x : punit), m) = (t _).lift (s' s),
{ apply congr_fun this punit.star },
apply (t _).uniq (s' s) _ (λ j, _),
ext,
exact (w j),
end }
variables {D : Type u} [small_category D]
instance yoneda_functor_preserves_limits : preserves_limits (@yoneda D _) :=
begin
apply preserves_limits_of_evaluation,
intro K,
change preserves_limits (coyoneda.obj K),
apply_instance
end
instance coyoneda_functor_preserves_limits : preserves_limits (@coyoneda D _) :=
begin
apply preserves_limits_of_evaluation,
intro K,
change preserves_limits (yoneda.obj K),
apply_instance
end
instance yoneda_functor_reflects_limits : reflects_limits (@yoneda D _) :=
limits.fully_faithful_reflects_limits _
instance coyoneda_functor_reflects_limits : reflects_limits (@coyoneda D _) :=
limits.fully_faithful_reflects_limits _
end category_theory
-- We don't need to have developed any algebra or set theory to reach (at least) this point
-- in the category theory hierarchy.
assert_not_exists set.range
assert_not_exists add_comm_monoid
|
e06b994e6443a23c0b0285da4ec2228534dbb0a9 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/ring_theory/witt_vector/witt_polynomial.lean | 5d2ba835a6a6d59559841072a8bf7806b977fcce | [
"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 | 10,592 | lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import algebra.invertible
import data.mv_polynomial.variables
import data.mv_polynomial.comm_ring
import data.mv_polynomial.expand
import data.zmod.basic
open mv_polynomial
open finset (hiding map)
open finsupp (single)
open_locale big_operators
local attribute [-simp] coe_eval₂_hom
variables (p : ℕ)
variables (R : Type*) [comm_ring R]
/-!
# Witt polynomials
To endow `witt_vector p R` with a ring structure,
we need to study the so-called Witt polynomials.
Fix a base value `p : ℕ`.
The `p`-adic Witt polynomials are an infinite family of polynomials
indexed by a natural number `n`, taking values in an arbitrary ring `R`.
The variables of these polynomials are represented by natural numbers.
The variable set of the `n`th Witt polynomial contains at most `n+1` elements `{0, ..., n}`,
with exactly these variables when `R` has characteristic `0`.
These polynomials are used to define the addition and multiplication operators
on the type of Witt vectors. (While this type itself is not complicated,
the ring operations are what make it interesting.)
When the base `p` is invertible in `R`, the `p`-adic Witt polynomials
form a basis for `mv_polynomial ℕ R`, equivalent to the standard basis.
## Main declarations
* `witt_polynomial p R n`: the `n`-th Witt polynomial, viewed as polynomial over the ring `R`
* `X_in_terms_of_W p R n`: if `p` is invertible, the polynomial `X n` is contained in the subalgebra
generated by the Witt polynomials. `X_in_terms_of_W p R n` is the explicit polynomial,
which upon being bound to the Witt polynomials yields `X n`.
* `bind₁_witt_polynomial_X_in_terms_of_W`: the proof of the claim that
`bind₁ (X_in_terms_of_W p R) (W_ R n) = X n`
* `bind₁_X_in_terms_of_W_witt_polynomial`: the converse of the above statement
## Notation
In this file we use the following notation
* `p` is a natural number, typically assumed to be prime.
* `R` and `S` are commutative rings
* `W n` (and `W_ R n` when the ring needs to be explicit) denotes the `n`th Witt polynomial
-/
/-- `witt_polynomial p R n` is the `n`-th Witt polynomial
with respect to a prime `p` with coefficients in a commutative ring `R`.
It is defined as:
`∑_{i ≤ n} p^i X_i^{p^{n-i}} ∈ R[X_0, X_1, X_2, …]`. -/
noncomputable def witt_polynomial (n : ℕ) : mv_polynomial ℕ R :=
∑ i in range (n+1), monomial (single i (p ^ (n - i))) (p ^ i)
lemma witt_polynomial_eq_sum_C_mul_X_pow (n : ℕ) :
witt_polynomial p R n = ∑ i in range (n+1), C (p ^ i : R) * X i ^ (p ^ (n - i)) :=
begin
apply sum_congr rfl,
rintro i -,
rw [monomial_eq, finsupp.prod_single_index],
rw pow_zero,
end
/-! We set up notation locally to this file, to keep statements short and comprehensible.
This allows us to simply write `W n` or `W_ ℤ n`. -/
-- Notation with ring of coefficients explicit
localized "notation `W_` := witt_polynomial p" in witt
-- Notation with ring of coefficients implicit
localized "notation `W` := witt_polynomial p _" in witt
open_locale witt
open mv_polynomial
/- The first observation is that the Witt polynomial doesn't really depend on the coefficient ring.
If we map the coefficients through a ring homomorphism, we obtain the corresponding Witt polynomial
over the target ring. -/
section
variables {R} {S : Type*} [comm_ring S]
@[simp] lemma map_witt_polynomial (f : R →+* S) (n : ℕ) :
map f (W n) = W n :=
begin
rw [witt_polynomial, ring_hom.map_sum, witt_polynomial, sum_congr rfl],
intros i hi,
rw [map_monomial, ring_hom.map_pow, ring_hom.map_nat_cast],
end
variables (R)
@[simp] lemma constant_coeff_witt_polynomial [hp : fact p.prime] (n : ℕ) :
constant_coeff (witt_polynomial p R n) = 0 :=
begin
simp only [witt_polynomial, ring_hom.map_sum, constant_coeff_monomial],
rw [sum_eq_zero],
rintro i hi,
rw [if_neg],
rw [finsupp.single_eq_zero],
exact ne_of_gt (pow_pos hp.pos _)
end
@[simp] lemma witt_polynomial_zero : witt_polynomial p R 0 = X 0 :=
by simp only [witt_polynomial, X, sum_singleton, range_one, pow_zero]
@[simp] lemma witt_polynomial_one : witt_polynomial p R 1 = C ↑p * X 1 + (X 0) ^ p :=
by simp only [witt_polynomial_eq_sum_C_mul_X_pow, sum_range_succ, range_one,
sum_singleton, one_mul, pow_one, C_1, pow_zero]
lemma aeval_witt_polynomial {A : Type*} [comm_ring A] [algebra R A] (f : ℕ → A) (n : ℕ) :
aeval f (W_ R n) = ∑ i in range (n+1), p^i * (f i) ^ (p ^ (n-i)) :=
by simp [witt_polynomial, alg_hom.map_sum, aeval_monomial, finsupp.prod_single_index]
/--
Over the ring `zmod (p^(n+1))`, we produce the `n+1`st Witt polynomial
by expanding the `n`th witt polynomial by `p`.
-/
@[simp] lemma witt_polynomial_zmod_self (n : ℕ) :
W_ (zmod (p ^ (n + 1))) (n + 1) = expand p (W_ (zmod (p^(n + 1))) n) :=
begin
simp only [witt_polynomial_eq_sum_C_mul_X_pow],
rw [sum_range_succ, ← nat.cast_pow, char_p.cast_eq_zero (zmod (p^(n+1))) (p^(n+1)), C_0, zero_mul, zero_add,
alg_hom.map_sum, sum_congr rfl],
intros k hk,
rw [alg_hom.map_mul, alg_hom.map_pow, expand_X, alg_hom_C, ← pow_mul, ← pow_succ],
congr,
rw mem_range at hk,
omega
end
section p_prime
-- in fact, `0 < p` would be sufficient
variables [hp : fact p.prime]
include hp
lemma witt_polynomial_vars [char_zero R] (n : ℕ) :
(witt_polynomial p R n).vars = range (n + 1) :=
begin
have : ∀ i, (monomial (finsupp.single i (p ^ (n - i))) (p ^ i : R)).vars = {i},
{ intro i,
rw vars_monomial_single,
{ rw ← nat.pos_iff_ne_zero,
apply pow_pos hp.pos },
{ rw [← nat.cast_pow, nat.cast_ne_zero],
apply ne_of_gt,
apply pow_pos hp.pos i } },
rw [witt_polynomial, vars_sum_of_disjoint],
{ simp only [this, int.nat_cast_eq_coe_nat, bind_singleton_eq_self], },
{ simp only [this, int.nat_cast_eq_coe_nat],
intros a b h,
apply singleton_disjoint.mpr,
rwa mem_singleton, },
end
lemma witt_polynomial_vars_subset (n : ℕ) :
(witt_polynomial p R n).vars ⊆ range (n + 1) :=
begin
rw [← map_witt_polynomial p (int.cast_ring_hom R), ← witt_polynomial_vars p ℤ],
apply vars_map,
end
end p_prime
end
/-!
## Witt polynomials as a basis of the polynomial algebra
If `p` is invertible in `R`, then the Witt polynomials form a basis
of the polynomial algebra `mv_polynomial ℕ R`.
The polynomials `X_in_terms_of_W` give the coordinate transformation in the backwards direction.
-/
/-- The `X_in_terms_of_W p R n` is the polynomial on the basis of Witt polynomials
that corresponds to the ordinary `X n`. -/
noncomputable def X_in_terms_of_W [invertible (p : R)] :
ℕ → mv_polynomial ℕ R
| n := (X n - (∑ i : fin n,
have _ := i.2, (C (p^(i : ℕ) : R) * (X_in_terms_of_W i)^(p^(n-i))))) * C (⅟p ^ n : R)
lemma X_in_terms_of_W_eq [invertible (p : R)] {n : ℕ} :
X_in_terms_of_W p R n =
(X n - (∑ i in range n, C (p^i : R) * X_in_terms_of_W p R i ^ p ^ (n - i))) * C (⅟p ^ n : R) :=
by { rw [X_in_terms_of_W, ← fin.sum_univ_eq_sum_range] }
@[simp] lemma constant_coeff_X_in_terms_of_W [hp : fact p.prime] [invertible (p : R)] (n : ℕ) :
constant_coeff (X_in_terms_of_W p R n) = 0 :=
begin
apply nat.strong_induction_on n; clear n,
intros n IH,
rw [X_in_terms_of_W_eq, mul_comm, ring_hom.map_mul, ring_hom.map_sub, ring_hom.map_sum,
constant_coeff_C, sum_eq_zero],
{ simp only [constant_coeff_X, sub_zero, mul_zero] },
{ intros m H,
rw mem_range at H,
simp only [ring_hom.map_mul, ring_hom.map_pow, constant_coeff_C, IH m H],
rw [zero_pow, mul_zero],
apply pow_pos hp.pos, }
end
@[simp] lemma X_in_terms_of_W_zero [invertible (p : R)] :
X_in_terms_of_W p R 0 = X 0 :=
by rw [X_in_terms_of_W_eq, range_zero, sum_empty, pow_zero, C_1, mul_one, sub_zero]
section p_prime
variables [hp : fact p.prime]
include hp
lemma X_in_terms_of_W_vars_aux (n : ℕ) :
n ∈ (X_in_terms_of_W p ℚ n).vars ∧
(X_in_terms_of_W p ℚ n).vars ⊆ range (n + 1) :=
begin
apply nat.strong_induction_on n, clear n,
intros n ih,
rw [X_in_terms_of_W_eq, mul_comm, vars_C_mul, vars_sub_of_disjoint, vars_X, range_succ, insert_eq],
swap 3, { apply nonzero_of_invertible },
work_on_goal 0 {
simp only [true_and, true_or, eq_self_iff_true,
mem_union, mem_singleton],
intro i,
rw [mem_union, mem_union],
apply or.imp id },
work_on_goal 1 { rw [vars_X, singleton_disjoint] },
all_goals {
intro H,
replace H := vars_sum_subset _ _ H,
rw mem_bind at H,
rcases H with ⟨j, hj, H⟩,
rw vars_C_mul at H,
swap,
{ apply pow_ne_zero, exact_mod_cast hp.ne_zero },
rw mem_range at hj,
replace H := (ih j hj).2 (vars_pow _ _ H),
rw mem_range at H },
{ rw mem_range,
exact lt_of_lt_of_le H hj },
{ exact lt_irrefl n (lt_of_lt_of_le H hj) },
end
lemma X_in_terms_of_W_vars_subset (n : ℕ) :
(X_in_terms_of_W p ℚ n).vars ⊆ range (n + 1) :=
(X_in_terms_of_W_vars_aux p n).2
end p_prime
lemma X_in_terms_of_W_aux [invertible (p : R)] (n : ℕ) :
X_in_terms_of_W p R n * C (p^n : R) =
X n - ∑ i in range n, C (p^i : R) * (X_in_terms_of_W p R i)^p^(n-i) :=
by rw [X_in_terms_of_W_eq, mul_assoc, ← C_mul, ← mul_pow, inv_of_mul_self, one_pow, C_1, mul_one]
@[simp] lemma bind₁_X_in_terms_of_W_witt_polynomial [invertible (p : R)] (k : ℕ) :
bind₁ (X_in_terms_of_W p R) (W_ R k) = X k :=
begin
rw [witt_polynomial_eq_sum_C_mul_X_pow, alg_hom.map_sum],
simp only [alg_hom.map_pow, C_pow, alg_hom.map_mul, alg_hom_C],
rw [sum_range_succ, nat.sub_self, pow_zero, pow_one, bind₁_X_right,
mul_comm, ← C_pow, X_in_terms_of_W_aux],
simp only [C_pow, bind₁_X_right, sub_add_cancel],
end
@[simp] lemma bind₁_witt_polynomial_X_in_terms_of_W [invertible (p : R)] (n : ℕ) :
bind₁ (W_ R) (X_in_terms_of_W p R n) = X n :=
begin
apply nat.strong_induction_on n,
clear n, intros n H,
rw [X_in_terms_of_W_eq, alg_hom.map_mul, alg_hom.map_sub, bind₁_X_right, alg_hom_C, alg_hom.map_sum],
have : W_ R n - ∑ i in range n, C (p ^ i : R) * (X i) ^ p ^ (n - i) = C (p ^ n : R) * X n,
by simp only [witt_polynomial_eq_sum_C_mul_X_pow, nat.sub_self, sum_range_succ,
pow_one, add_sub_cancel, pow_zero],
rw [sum_congr rfl, this],
{ -- this is really slow for some reason
rw [mul_right_comm, ← C_mul, ← mul_pow, mul_inv_of_self, one_pow, C_1, one_mul] },
{ intros i h,
rw mem_range at h,
simp only [alg_hom.map_mul, alg_hom.map_pow, alg_hom_C, H i h] },
end
|
af5382d5d1a811d12d81d13ab724da544716c90c | 6df8d5ae3acf20ad0d7f0247d2cee1957ef96df1 | /ExamPractice/quiz1.lean | 91bc5d613617fefe6ef9fd562ee2b9114869deb2 | [] | no_license | derekjohnsonva/CS2102 | 8ed45daa6658e6121bac0f6691eac6147d08246d | b3f507d4be824a2511838a1054d04fc9aef3304c | refs/heads/master | 1,648,529,162,527 | 1,578,851,859,000 | 1,578,851,859,000 | 233,433,207 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 416 | lean | -- Derek Johnson
def s1 := "Hello, "
def s2 := "Nifty!"
def s3 := s1 ++ s2
theorem t1 : (s1 ++ s2) = s3 := eq.refl s3
theorem t2 : 4^2 = 16 := eq.refl 16
theorem t3 : (s1 ++ s2) = s3 ∧ (5^2 = 25) := and.intro
(eq.refl s3)
(eq.refl 25)
theorem t4 :
∀ (P Q R : Prop), (P ∧ Q) ∧ (Q ∧ R) → (P ∧ R) :=
λ (P Q R : Prop),
λ h,
and.intro (h.left.left)(h.right.right) |
b44ef53d9b436b4d95e3b8bd69e1ac09fb2fb662 | be30445afb85fcb2041b437059d21f9a84dff894 | /move_to_main.hlean | aa2a6d005a7e456e5948e66475cc8466b9fda79a | [
"Apache-2.0"
] | permissive | EgbertRijke/GraphModel | 784efde97299c4f3cb1760369e8a260e02caafd5 | 2a473880764093dd151554a913292ed465e80e62 | refs/heads/master | 1,609,477,731,119 | 1,476,299,056,000 | 1,476,299,056,000 | 58,213,801 | 6 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,026 | hlean | import types
open eq
inductive pathover2 {A B : Type} (C : A → B → Type) :
Π {a a' : A} (p : a=a') {b b' : B} (q : b=b') (x : C a b) (y : C a' b'), Type :=
| idpo2 : Π {a : A} {b : B} (x : C a b), pathover2 C (idpath a) (idpath b) x x
definition tr_eq_of_homotopy {A : Type} {B : A → Type} (x y : A) (C : B x → B y → Type) :
Π (f g : Π (a : A), B a) (H : f ~ g) (c : C (f x) (f y)), (eq_of_homotopy H) ▸ c = (H x) ▸ ( (H y) ▸ c) :=
begin
intros f g H,
induction H,
rewrite eq_of_homotopy_idp,
intro c,
reflexivity
end
local attribute homotopy2.rec_on [recursor]
definition tr_eq_of_homotopy2 {A : Type} {B : A → Type} {C : Π {a : A}, B a → Type} {a a' : A}
(x : B a) (y : B a') (D : C x → C y → Type) :
Π (f g : Π {{a : A}} (b : B a), C b) (H : f ~2 g) (d : D (f x) (f y)),
(eq_of_homotopy2 H) ▸ d = (H a x) ▸ ((H a' y) ▸ d) :=
begin
intros f g H,
induction H,
induction q,
intro d,
krewrite eq_of_homotopy2_id,
end
|
0727642dc573820eb3fb5392873ccc8a91f02880 | 947b78d97130d56365ae2ec264df196ce769371a | /tests/lean/macroscopes.lean | 37ea91c864792a62c771c9492aec8079da25e9df | [
"Apache-2.0"
] | permissive | shyamalschandra/lean4 | 27044812be8698f0c79147615b1d5090b9f4b037 | 6e7a883b21eaf62831e8111b251dc9b18f40e604 | refs/heads/master | 1,671,417,126,371 | 1,601,859,995,000 | 1,601,860,020,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,136 | lean | new_frontend
open Lean
def check (b : Bool) : IO Unit :=
«unless» b $ throw $ IO.userError "check failed"
def test1 : IO Unit := do
let x := `x;
let x := addMacroScope `main x 1;
IO.println $ x;
let v := extractMacroScopes x;
let x := { v with name := `y }.review;
IO.println $ x;
let v := extractMacroScopes x;
let x := { v with name := `x }.review;
IO.println $ x;
let x := addMacroScope `main x 2;
IO.println $ x;
let v := extractMacroScopes x;
let x := { v with name := `y }.review;
IO.println $ x;
let v := extractMacroScopes x;
let x := { v with name := `x }.review;
IO.println $ x;
let x := addMacroScope `main x 3;
IO.println $ x;
let x := addMacroScope `foo x 4;
IO.println $ x;
let x := addMacroScope `foo x 5;
let v := extractMacroScopes x;
check (v.mainModule == `foo);
IO.println $ x;
let x := addMacroScope `bla.bla x 6;
IO.println $ x;
let v := extractMacroScopes x;
check (v.mainModule == `bla.bla);
let x := addMacroScope `bla.bla x 7;
IO.println $ x;
let v := extractMacroScopes x;
let x := { v with name := `y }.review;
IO.println $ x;
let x := { v with name := `z.w }.review;
IO.println $ x;
pure ()
#eval test1
|
8c9159d58439e7f5d4e57a5a2a82923b1683ead7 | aa5a655c05e5359a70646b7154e7cac59f0b4132 | /tests/lean/mvar3.lean | 77033e4d434e3f7b7b3f66261ecafb76e5a4a9d2 | [
"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 | 2,853 | lean | import Lean.MetavarContext
open Lean
def mkLambdaTest (mctx : MetavarContext) (ngen : NameGenerator) (lctx : LocalContext) (xs : Array Expr) (e : Expr)
: Except MetavarContext.MkBinding.Exception (MetavarContext × NameGenerator × Expr) :=
match MetavarContext.mkLambda xs e lctx { mctx := mctx, ngen := ngen } with
| EStateM.Result.ok e s => Except.ok (s.mctx, s.ngen, e)
| EStateM.Result.error e s => Except.error e
def check (b : Bool) : IO Unit :=
«unless» b (throw $ IO.userError "error")
def f := mkConst `f
def g := mkConst `g
def a := mkConst `a
def b := mkConst `b
def c := mkConst `c
def b0 := mkBVar 0
def b1 := mkBVar 1
def b2 := mkBVar 2
def u := mkLevelParam `u
def typeE := mkSort levelOne
def natE := mkConst `Nat
def boolE := mkConst `Bool
def vecE := mkConst `Vec [levelZero]
def α := mkFVar `α
def x := mkFVar `x
def y := mkFVar `y
def z := mkFVar `z
def w := mkFVar `w
def m1 := mkMVar `m1
def m2 := mkMVar `m2
def m3 := mkMVar `m3
def bi := BinderInfo.default
def arrow (d b : Expr) := mkForall `_ bi d b
def lctx1 : LocalContext := {}
def lctx2 := lctx1.mkLocalDecl `α `α typeE
def lctx3 := lctx2.mkLocalDecl `x `x m1
def lctx4 := lctx3.mkLocalDecl `y `y (arrow natE m2)
def mctx1 : MetavarContext := {}
def mctx2 := mctx1.addExprMVarDecl `m1 `m1 lctx2 #[] typeE
def mctx3 := mctx2.addExprMVarDecl `m2 `m2 lctx3 #[] natE
def mctx4 := mctx3.addExprMVarDecl `m3 `m3 lctx3 #[] natE
def mctx4' := mctx3.addExprMVarDecl `m3 `m3 lctx3 #[] natE MetavarKind.syntheticOpaque
def R1 :=
match mkLambdaTest mctx4 {namePrefix := `n} lctx4 #[α, x, y] $ mkAppN f #[m3, x] with
| Except.ok s => s
| Except.error e => panic! (toString e)
def e1 := R1.2.2
def mctx5 := R1.1
def sortNames (xs : List Name) : List Name :=
(xs.toArray.qsort Name.lt).toList
def sortNamePairs {α} [Inhabited α] (xs : List (Name × α)) : List (Name × α) :=
(xs.toArray.qsort (fun a b => Name.lt a.1 b.1)).toList
#eval toString $ sortNames $ mctx5.decls.toList.map Prod.fst
#eval toString $ sortNamePairs $ mctx5.eAssignment.toList
#eval e1
#eval check (!e1.hasFVar)
def R2 :=
match mkLambdaTest mctx4' {namePrefix := `n} lctx4 #[α, x, y] $ mkAppN f #[m3, y] with
| Except.ok s => s
| Except.error e => panic! (toString e)
def e2 := R2.2.2
def mctx6 := R2.1
#eval toString $ sortNames $ mctx6.decls.toList.map Prod.fst
#eval toString $ sortNamePairs $ mctx6.eAssignment.toList
-- ?n.2 was delayed assigned because ?m.3 is synthetic
#eval toString $ sortNames $ mctx6.dAssignment.toList.map Prod.fst
#eval e2
#print "assigning ?m1 and ?n.1"
def R3 :=
let mctx := mctx6.assignExpr `m3 x;
let mctx := mctx.assignExpr (Name.mkNum `n 1) (mkLambda `_ bi typeE natE);
-- ?n.2 is instantiated because we have the delayed assignment `?n.2 α x := ?m1`
(mctx.instantiateMVars e2)
def e3 := R3.1
def mctx7 := R3.2
#eval e3
|
d619eaab696dad63010ca0eec5347cea7d9026b0 | 29cc89d6158dd3b90acbdbcab4d2c7eb9a7dbf0f | /Exercises week 5/22_homework_sheet.lean | 3d2c22b3762be204a671061e5245afcfe7af2c1d | [] | no_license | KjellZijlemaker/Logical_Verification_VU | ced0ba95316a30e3c94ba8eebd58ea004fa6f53b | 4578b93bf1615466996157bb333c84122b201d99 | refs/heads/master | 1,585,966,086,108 | 1,549,187,704,000 | 1,549,187,704,000 | 155,690,284 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 5,177 | lean | /- Homework 2.2: Functional Programming — Trees -/
/- Question 1: Huffman's algorithm -/
/- Recall the following definition from the lecture. -/
inductive tree (α : Type)
| leaf : ℕ → α → tree
| inner : ℕ → tree → tree → tree
export tree (leaf inner)
/- To make our lives easy, we will fix a type α of symbols equipped with decidable equality
throughout this question. -/
section
parameter (α : Type)
parameter [decidable_eq α]
/- We start by defining the alphabet of a tree: the set of its labels. We use lists to represent
sets to get executability for free. The cost is that we must ignore the order or number of
occurrences of elements in a list when comparing alphabets. -/
def alphabet : tree α → list α
| (leaf _ a) := [a]
| (inner _ l r) := alphabet l ++ alphabet r
/- 1.1. Define the following function, so that we can obtain the alphabet of an entire forest
(i.e. list of trees).
Hint: Through this exercise, try to keep your definitions as simple as possible, because this will
affect proving ease further down. -/
def alphabet_list : list (tree α) → list α
| [] := []
| (x :: xs) := alphabet_list xs ++ alphabet x
/- A tree is consistent if it contains no duplicate labels. This can be defined as an inductive
predicate. -/
inductive consistent : tree α → Prop
| consistent_leaf (w : ℕ) (a : α) : consistent (leaf w a)
| consistent_inner (w : ℕ) (l : tree α) (r : tree α) :
consistent l → consistent r → (∀a, a ∈ alphabet l → a ∈ alphabet r → false) →
consistent (inner w l r)
/- 1.2. A forest is consistent if it contains no duplicate labels (even in different trees). Define
an inductive predicate `consistent_list` that captures this property. -/
inductive consistent_trees : list (tree α) → Prop
| consistent_nil (w: ℕ) (a: α) : consistent_trees[]
| consistent_cons (w: ℕ) (t: tree α) (ts: list (tree α) : consistent t → consistent_trees ts → (∀a, a ∈ alphabet t → a ∈ alphabet_list ts → false) →
consistent_trees ts
/- 1.3. The height of a tree is the length of the longest path from its root node to a leaf. A tree
consisting of a single node has height 0 by convention. Define a recursive function `height` that
computes this.
Hint: You may use the built-in `max` function. -/
def height {α: Type} : tree α → ℕ
| (leaf _ a) := 1
| (inner _ l r) := 1 + max (height l) (height r)
/- 1.4. Now define the same function on forests, where the height of a forest is defined as the
height of its tallest tree. -/
def height_forest {α: Type}: list(tree α) → ℕ
| [] := 0
| (x :: xs) := 1 + max (height x) (height_forest xs)
/- 1.5. Define a function that returns the depth of a symbol in a tree. For example, if the symbol
is stored in a leaf node immediately below the root, it would be at depth 1. Symbols that do not
occur in the tree are conventionally put at depth 0. For symbols that occur several times as labels,
you can arbitrarily choose which occurrence to consider (e.g. the leftmost one). -/
def depth : tree α → α → ℕ
| (leaf _ a) x := 1
| (inner _ l r) x := depth l x + depth r x
/- 1.6. State and prove that the depth of a symbol in a tree is less than or equal to the height of
the tree.
Hint: You might find the lemmas `le_max_left`, `add_le_add_left`, and similarly named ones useful.
Use Visual Studio Code's Command Palette to explore Lean's library (e.g. `#le_max`). -/
lemma depthlessequal :∀t : tree α, consistent t → ∀a ∈ alphabet t, depth t a <= height t
| (leaf _ a) ct := begin repeat {apply exists.intro}, simp[alphabet], simp[depth, height], intro s, simp[le_of_eq] end
| (inner _ l r) ct :=
begin
repeat{apply exists.intro},
simp[alphabet],
simp[depth, height],
intro s,
intro a,
simp[le_max],
end
/- 1.7. A tree's height is not only an upper bound on the depth of symbols stored in the tree. It is
also a tight bound. Complete the following proof. -/
lemma exists_at_height : ∀t : tree α, consistent t → ∃a ∈ alphabet t, depth t a = height t
| (leaf _ a) ct :=
begin
repeat {apply exists.intro},
simp[alphabet],
simp[depth, height]
end
| t@(inner _ l r) ct := -- `t@` introduces the alias `t` for `inner _ l r`
have cl : consistent l :=
begin
have bla: ∀ (t : tree α), consistent t → (∃ (a : α) (H : a ∈ alphabet t), depth t a = height t) ,
intro a,
intro b,
apply exists_at_height,
assumption,
end
have cr : consistent r :=
sorry,
let ⟨b, b_in_alpha_l, depth_l_b_eq_height_l⟩ := exists_at_height l cl in
let ⟨c, c_in_alpha_r, depth_r_c_eq_height_r⟩ := exists_at_height r cr in
let b_or_c := if height r ≤ height l then b else c in
have borc_in_alpha_t : b_or_c ∈ alphabet t :=
begin
repeat {apply exists.intro},
simp[alphabet],
apply or.inr,
simp [b_or_c],
by_cases (ite (height r ≤ height l) b c ∈ alphabet r),
end
have depth_t_borc_eq_height_t : depth t b_or_c = height t :=
sorry, -- **optional**
⟨b_or_c, borc_in_alpha_t, depth_t_borc_eq_height_t⟩
end
/- Question 2: Enter your question here. -/
|
13f88bc647f83a337126b9419ae6d85c28f316a4 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/algebra/homology/exact_auto.lean | f92b32fd408632ec89c527c66a05276edcc10fcf | [] | 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 | 5,713 | lean | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.algebra.homology.image_to_kernel_map
import Mathlib.PostPort
universes v u l
namespace Mathlib
/-!
# Exact sequences
In a category with zero morphisms, images, and equalizers we say that `f : A ⟶ B` and `g : B ⟶ C`
are exact if `f ≫ g = 0` and the natural map `image f ⟶ kernel g` is an epimorphism.
This definition is equivalent to the homology at `B` vanishing (at least for preadditive
categories). At this level of generality, this is not necessarily equivalent to other reasonable
definitions of exactness, for example that the inclusion map `image.ι f` is a kernel of `g` or that
the map `image f ⟶ kernel g` is an isomorphism. By adding more assumptions on our category, we get
these equivalences and more. Currently, there is one particular set of assumptions mathlib knows
about: abelian categories. Consequently, many interesting results about exact sequences are found in
`category_theory/abelian/exact.lean`.
# Main results
* Suppose that cokernels exist and that `f` and `g` are exact. If `s` is any kernel fork over `g`
and `t` is any cokernel cofork over `f`, then `fork.ι s ≫ cofork.π t = 0`.
* Precomposing the first morphism with an epimorphism retains exactness. Postcomposing the second
morphism with a monomorphism retains exactness.
* If `f` and `g` are exact and `i` is an isomorphism, then `f ≫ i.hom` and `i.inv ≫ g` are also
exact.
# Future work
* Short exact sequences, split exact sequences, the splitting lemma (maybe only for abelian
categories?)
* Two adjacent maps in a chain complex are exact iff the homology vanishes
-/
namespace category_theory
/-- Two morphisms `f : A ⟶ B`, `g : B ⟶ C` are called exact if `f ≫ g = 0` and the natural map
`image f ⟶ kernel g` is an epimorphism. -/
class exact {V : Type u} [category V] [limits.has_zero_morphisms V] [limits.has_equalizers V]
[limits.has_images V] {A : V} {B : V} {C : V} (f : A ⟶ B) (g : B ⟶ C)
where
w : f ≫ g = 0
epi : epi (image_to_kernel_map f g w)
@[simp] theorem exact.w_assoc {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] {A : V} {B : V} {C : V} {f : A ⟶ B} {g : B ⟶ C}
[c : exact f g] {X' : V} (f' : C ⟶ X') : f ≫ g ≫ f' = 0 ≫ f' :=
sorry
theorem exact_comp_hom_inv_comp {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] {A : V} {B : V} {C : V} {D : V} {f : A ⟶ B}
{g : B ⟶ C} [exact f g] (i : B ≅ D) : exact (f ≫ iso.hom i) (iso.inv i ≫ g) :=
sorry
theorem exact_comp_hom_inv_comp_iff {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] {A : V} {B : V} {C : V} {D : V} {f : A ⟶ B}
{g : B ⟶ C} (i : B ≅ D) : exact (f ≫ iso.hom i) (iso.inv i ≫ g) ↔ exact f g :=
sorry
theorem exact_epi_comp {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] {A : V} {B : V} {C : V} {D : V} {f : A ⟶ B}
{g : B ⟶ C} {h : C ⟶ D} [exact g h] [epi f] : exact (f ≫ g) h :=
sorry
theorem exact_comp_mono {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] {A : V} {B : V} {C : V} {D : V} {f : A ⟶ B}
{g : B ⟶ C} {h : C ⟶ D} [exact f g] [mono h] : exact f (g ≫ h) :=
sorry
theorem exact_kernel {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] {A : V} {B : V} {f : A ⟶ B} :
exact (limits.kernel.ι f) f :=
sorry
theorem kernel_ι_eq_zero_of_exact_zero_left {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] (A : V) {B : V} {C : V} {g : B ⟶ C}
[exact 0 g] : limits.kernel.ι g = 0 :=
sorry
theorem exact_zero_left_of_mono {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] (A : V) {B : V} {C : V} {g : B ⟶ C}
[limits.has_zero_object V] [mono g] : exact 0 g :=
sorry
@[simp] theorem kernel_comp_cokernel_assoc {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] [limits.has_cokernels V] {A : V} {B : V} {C : V}
(f : A ⟶ B) (g : B ⟶ C) [exact f g] {X' : V} (f' : limits.cokernel f ⟶ X') :
limits.kernel.ι g ≫ limits.cokernel.π f ≫ f' = 0 ≫ f' :=
sorry
theorem comp_eq_zero_of_exact {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] [limits.has_cokernels V] {A : V} {B : V} {C : V}
(f : A ⟶ B) (g : B ⟶ C) [exact f g] {X : V} {Y : V} {ι : X ⟶ B} (hι : ι ≫ g = 0) {π : B ⟶ Y}
(hπ : f ≫ π = 0) : ι ≫ π = 0 :=
sorry
@[simp] theorem fork_ι_comp_cofork_π {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] [limits.has_cokernels V] {A : V} {B : V} {C : V}
(f : A ⟶ B) (g : B ⟶ C) [exact f g] (s : limits.kernel_fork g) (t : limits.cokernel_cofork f) :
limits.fork.ι s ≫ limits.cofork.π t = 0 :=
comp_eq_zero_of_exact f g (limits.kernel_fork.condition s) (limits.cokernel_cofork.condition t)
theorem exact_of_zero {V : Type u} [category V] [limits.has_zero_morphisms V]
[limits.has_equalizers V] [limits.has_images V] [limits.has_zero_object V] {A : V} {C : V}
(f : A ⟶ 0) (g : 0 ⟶ C) : exact f g :=
sorry
end Mathlib |
11842e33b48acf54d70f35af8dda59e61f1b04bf | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/data/qpf/multivariate/constructions/prj_auto.lean | 8bb64c4813618f90404fb24cda7d6c00814a743b | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,816 | lean | /-
Copyright (c) 2020 Simon Hudon All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Simon Hudon
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.control.functor.multivariate
import Mathlib.data.qpf.multivariate.basic
import Mathlib.PostPort
universes u u_1 u_2
namespace Mathlib
/-!
Projection functors are QPFs. The `n`-ary projection functors on `i` is an `n`-ary
functor `F` such that `F (α₀..αᵢ₋₁, αᵢ, αᵢ₊₁..αₙ₋₁) = αᵢ`
-/
namespace mvqpf
/-- The projection `i` functor -/
def prj {n : ℕ} (i : fin2 n) (v : typevec n) := v i
protected instance prj.inhabited {n : ℕ} (i : fin2 n) {v : typevec n} [Inhabited (v i)] :
Inhabited (prj i v) :=
{ default := Inhabited.default }
/-- `map` on functor `prj i` -/
def prj.map {n : ℕ} (i : fin2 n) {α : typevec n} {β : typevec n} (f : typevec.arrow α β) :
prj i α → prj i β :=
f i
protected instance prj.mvfunctor {n : ℕ} (i : fin2 n) : mvfunctor (prj i) :=
mvfunctor.mk (prj.map i)
/-- Polynomial representation of the projection functor -/
def prj.P {n : ℕ} (i : fin2 n) : mvpfunctor n :=
mvpfunctor.mk PUnit fun (_x : PUnit) (j : fin2 n) => ulift (plift (i = j))
/-- Abstraction function of the `qpf` instance -/
def prj.abs {n : ℕ} (i : fin2 n) {α : typevec n} : mvpfunctor.obj (prj.P i) α → prj i α := sorry
/-- Representation function of the `qpf` instance -/
def prj.repr {n : ℕ} (i : fin2 n) {α : typevec n} : prj i α → mvpfunctor.obj (prj.P i) α :=
fun (x : α i) =>
sigma.mk PUnit.unit fun (j : fin2 n) (_x : mvpfunctor.B (prj.P i) PUnit.unit j) => sorry
protected instance prj.mvqpf {n : ℕ} (i : fin2 n) : mvqpf (prj i) :=
mk (prj.P i) (prj.abs i) (prj.repr i) sorry sorry
end Mathlib |
c18e0fdd790908046d2306cfcc95399e9da780ba | 7490bf5d40d31857a58062614642bb5a41c36154 | /dm_box.lean | a634a587ec8d3ac90c090c9582f81afedcd54236 | [] | 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 | 513 | lean | /-
HOMEWORK 3
Reese Allen (rga2uz)
CS2102 - Sullivan
-/
-- defining new polymorphic abstract data type
inductive dm_box (α : Type) : Type
| mk : (α : Type) → dm_box
-- function defintions using lambda expressions, c-style (') and by cases ('')
def unbox {α : Type} : (dm_box α) → α :=
λ (b : dm_box α),
match b with
| dm_box.mk x := x
end
def unbox' {α : Type } (b : dm_box α) :=
match b with
| dm_box.mk x := x
end
def unbox'' {α : Type} : (dm_box α) → α
| (dm_box.mk x) := x
|
845f4afdd8c2f03b49c75dd076f709d5a42a8a9d | 6b02ce66658141f3e0aa3dfa88cd30bbbb24d17b | /src/Lean/Elab/SetOption.lean | 60c0880b2dfe3ad414d7fdae98756673c654fe5f | [
"Apache-2.0"
] | permissive | pbrinkmeier/lean4 | d31991fd64095e64490cb7157bcc6803f9c48af4 | 32fd82efc2eaf1232299e930ec16624b370eac39 | refs/heads/master | 1,681,364,001,662 | 1,618,425,427,000 | 1,618,425,427,000 | 358,314,562 | 0 | 0 | Apache-2.0 | 1,618,504,558,000 | 1,618,501,999,000 | null | UTF-8 | Lean | false | false | 1,338 | lean | /-
Copyright (c) 2021 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Elab.Log
import Lean.Elab.InfoTree
namespace Lean.Elab
variable [Monad m] [MonadOptions m] [MonadExceptOf Exception m] [MonadRef m]
variable [AddErrorMessageContext m] [MonadLiftT (EIO Exception) m] [MonadInfoTree m]
def elabSetOption (id : Syntax) (val : Syntax) : m Options := do
let optionName := id.getId.eraseMacroScopes
match val.isStrLit? with
| some str => setOption optionName (DataValue.ofString str)
| none =>
match val.isNatLit? with
| some num => setOption optionName (DataValue.ofNat num)
| none =>
match val with
| Syntax.atom _ "true" => setOption optionName (DataValue.ofBool true)
| Syntax.atom _ "false" => setOption optionName (DataValue.ofBool false)
| _ =>
addCompletionInfo <| CompletionInfo.option (← getRef)
throwError "unexpected set_option value {val}"
where
setOption (optionName : Name) (val : DataValue) : m Options := do
let decl ← IO.toEIO (fun (ex : IO.Error) => Exception.error (← getRef) ex.toString) (getOptionDecl optionName)
unless decl.defValue.sameCtor val do throwError "type mismatch at set_option"
return (← getOptions).insert optionName val
end Lean.Elab |
ad84b28b025c0edc780cfb20df420ff18c3789d4 | 80cc5bf14c8ea85ff340d1d747a127dcadeb966f | /src/ring_theory/discrete_valuation_ring.lean | c8433e754c71085c5377d2bc6f623a25e0c12bb2 | [
"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 | 13,139 | lean | /-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import ring_theory.principal_ideal_domain
import order.conditionally_complete_lattice
import ring_theory.multiplicity
import ring_theory.valuation.basic
/-!
# Discrete valuation rings
This file defines discrete valuation rings (DVRs) and develops a basic interface
for them.
## Important definitions
There are various definitions of a DVR in the literature; we define a DVR to be a local PID
which is not a field (the first definition in Wikipedia) and prove that this is equivalent
to being a PID with a unique non-zero prime ideal (the definition in Serre's
book "Local Fields").
Let R be an integral domain, assumed to be a principal ideal ring and a local ring.
* `discrete_valuation_ring R` : a predicate expressing that R is a DVR
### Definitions
## Implementation notes
It's a theorem that an element of a DVR is a uniformizer if and only if it's irreducible.
We do not hence define `uniformizer` at all, because we can use `irreducible` instead.
## Tags
discrete valuation ring
-/
open_locale classical
universe u
open ideal local_ring
/-- An integral domain is a discrete valuation ring if it's a local PID which is not a field -/
class discrete_valuation_ring (R : Type u) [integral_domain R]
extends is_principal_ideal_ring R, local_ring R : Prop :=
(not_a_field' : maximal_ideal R ≠ ⊥)
namespace discrete_valuation_ring
variables (R : Type u) [integral_domain R] [discrete_valuation_ring R]
lemma not_a_field : maximal_ideal R ≠ ⊥ := not_a_field'
variable {R}
open principal_ideal_ring
/-- An element of a DVR is irreducible iff it is a uniformizer, that is, generates the
maximal ideal of R -/
theorem irreducible_iff_uniformizer (ϖ : R) :
irreducible ϖ ↔ maximal_ideal R = ideal.span {ϖ} :=
⟨λ hϖ, (eq_maximal_ideal (is_maximal_of_irreducible hϖ)).symm,
begin
intro h,
have h2 : ¬(is_unit ϖ) := show ϖ ∈ maximal_ideal R,
from h.symm ▸ submodule.mem_span_singleton_self ϖ,
refine ⟨h2, _⟩,
intros a b hab,
by_contra h,
push_neg at h,
obtain ⟨ha : a ∈ maximal_ideal R, hb : b ∈ maximal_ideal R⟩ := h,
rw h at ha hb,
rw mem_span_singleton' at ha hb,
rcases ha with ⟨a, rfl⟩,
rcases hb with ⟨b, rfl⟩,
rw (show a * ϖ * (b * ϖ) = ϖ * (ϖ * (a * b)), by ring) at hab,
have h3 := eq_zero_of_mul_eq_self_right _ hab.symm,
{ apply not_a_field R,
simp [h, h3] },
{ intro hh, apply h2,
refine is_unit_of_dvd_one ϖ _,
use a * b, exact hh.symm }
end⟩
variable (R)
/-- Uniformisers exist in a DVR -/
theorem exists_irreducible : ∃ ϖ : R, irreducible ϖ :=
by {simp_rw [irreducible_iff_uniformizer],
exact (is_principal_ideal_ring.principal $ maximal_ideal R).principal}
/-- an integral domain is a DVR iff it's a PID with a unique non-zero prime ideal -/
theorem iff_pid_with_one_nonzero_prime (R : Type u) [integral_domain R] :
discrete_valuation_ring R ↔ is_principal_ideal_ring R ∧ ∃! P : ideal R, P ≠ ⊥ ∧ is_prime P :=
begin
split,
{ intro RDVR,
rcases id RDVR with ⟨RPID, Rlocal, Rnotafield⟩,
split, assumption,
resetI,
use local_ring.maximal_ideal R,
split, split,
{ assumption },
{ apply_instance } ,
{ rintro Q ⟨hQ1, hQ2⟩,
obtain ⟨q, rfl⟩ := (is_principal_ideal_ring.principal Q).1,
have hq : q ≠ 0,
{ rintro rfl,
apply hQ1,
simp },
erw span_singleton_prime hq at hQ2,
replace hQ2 := irreducible_of_prime hQ2,
rw irreducible_iff_uniformizer at hQ2,
exact hQ2.symm } },
{ rintro ⟨RPID, Punique⟩,
haveI : local_ring R := local_of_unique_nonzero_prime R Punique,
refine {not_a_field' := _},
rcases Punique with ⟨P, ⟨hP1, hP2⟩, hP3⟩,
have hPM : P ≤ maximal_ideal R := le_maximal_ideal (hP2.1),
intro h, rw [h, le_bot_iff] at hPM, exact hP1 hPM }
end
lemma associated_of_irreducible {a b : R} (ha : irreducible a) (hb : irreducible b) :
associated a b :=
begin
rw irreducible_iff_uniformizer at ha hb,
rw [←span_singleton_eq_span_singleton, ←ha, hb],
end
end discrete_valuation_ring
namespace discrete_valuation_ring
variable (R : Type*)
/-- Alternative characterisation of discrete valuation rings. -/
def has_unit_mul_pow_irreducible_factorization [integral_domain R] : Prop :=
∃ p : R, irreducible p ∧ ∀ {x : R}, x ≠ 0 → ∃ (n : ℕ), associated (p ^ n) x
namespace has_unit_mul_pow_irreducible_factorization
variables {R} [integral_domain R] (hR : has_unit_mul_pow_irreducible_factorization R)
include hR
lemma unique_irreducible ⦃p q : R⦄ (hp : irreducible p) (hq : irreducible q) :
associated p q :=
begin
rcases hR with ⟨ϖ, hϖ, hR⟩,
suffices : ∀ {p : R} (hp : irreducible p), associated p ϖ,
{ apply associated.trans (this hp) (this hq).symm, },
clear hp hq p q,
intros p hp,
obtain ⟨n, hn⟩ := hR hp.ne_zero,
have : irreducible (ϖ ^ n) := irreducible_of_associated hn.symm hp,
rcases lt_trichotomy n 1 with (H|rfl|H),
{ obtain rfl : n = 0, { clear hn this, revert H n, exact dec_trivial },
simpa only [not_irreducible_one, pow_zero] using this, },
{ simpa only [pow_one] using hn.symm, },
{ obtain ⟨n, rfl⟩ : ∃ k, n = 1 + k + 1 := nat.exists_eq_add_of_lt H,
rw pow_succ at this,
rcases this.2 _ _ rfl with H0|H0,
{ exact (hϖ.not_unit H0).elim, },
{ rw [add_comm, pow_succ] at H0,
exact (hϖ.not_unit (is_unit_of_mul_is_unit_left H0)).elim } }
end
/-- Implementation detail: an integral domain in which there is a unit `p`
such that every nonzero element is associated to a power of `p` is a unique factorization domain.
See `discrete_valuation_ring.of_has_unit_mul_pow_irreducible_factorization`. -/
theorem ufd : unique_factorization_monoid R :=
let p := classical.some hR in
let spec := classical.some_spec hR in
unique_factorization_monoid_of_exists_prime_of_factor $ λ x hx,
begin
use multiset.repeat p (classical.some (spec.2 hx)),
split,
{ intros q hq,
have hpq := multiset.eq_of_mem_repeat hq,
rw hpq,
refine ⟨spec.1.ne_zero, spec.1.not_unit, _⟩,
intros a b h,
by_cases ha : a = 0,
{ rw ha, simp only [true_or, dvd_zero], },
by_cases hb : b = 0,
{ rw hb, simp only [or_true, dvd_zero], },
obtain ⟨m, u, rfl⟩ := spec.2 ha,
rw [mul_assoc, mul_left_comm, is_unit.dvd_mul_left _ _ _ (is_unit_unit _)] at h,
rw is_unit.dvd_mul_right (is_unit_unit _),
by_cases hm : m = 0,
{ simp only [hm, one_mul, pow_zero] at h ⊢, right, exact h },
left,
obtain ⟨m, rfl⟩ := nat.exists_eq_succ_of_ne_zero hm,
apply dvd_mul_of_dvd_left (dvd_refl _) _ },
{ rw [multiset.prod_repeat], exact (classical.some_spec (spec.2 hx)), }
end
omit hR
lemma of_ufd_of_unique_irreducible [unique_factorization_monoid R]
(h₁ : ∃ p : R, irreducible p)
(h₂ : ∀ ⦃p q : R⦄, irreducible p → irreducible q → associated p q) :
has_unit_mul_pow_irreducible_factorization R :=
begin
obtain ⟨p, hp⟩ := h₁,
refine ⟨p, hp, _⟩,
intros x hx,
cases wf_dvd_monoid.exists_factors x hx with fx hfx,
refine ⟨fx.card, _⟩,
have H := hfx.2,
rw ← associates.mk_eq_mk_iff_associated at H ⊢,
rw [← H, ← associates.prod_mk, associates.mk_pow, ← multiset.prod_repeat],
congr' 1,
symmetry,
rw multiset.eq_repeat,
simp only [true_and, and_imp, multiset.card_map, eq_self_iff_true,
multiset.mem_map, exists_imp_distrib],
rintros _ q hq rfl,
rw associates.mk_eq_mk_iff_associated,
apply h₂ (hfx.1 _ hq) hp,
end
end has_unit_mul_pow_irreducible_factorization
lemma aux_pid_of_ufd_of_unique_irreducible
(R : Type u) [integral_domain R] [unique_factorization_monoid R]
(h₁ : ∃ p : R, irreducible p)
(h₂ : ∀ ⦃p q : R⦄, irreducible p → irreducible q → associated p q) :
is_principal_ideal_ring R :=
begin
constructor,
intro I,
by_cases I0 : I = ⊥, { rw I0, use 0, simp only [set.singleton_zero, submodule.span_zero], },
obtain ⟨x, hxI, hx0⟩ : ∃ x ∈ I, x ≠ (0:R) := I.ne_bot_iff.mp I0,
obtain ⟨p, hp, H⟩ :=
has_unit_mul_pow_irreducible_factorization.of_ufd_of_unique_irreducible h₁ h₂,
have ex : ∃ n : ℕ, p ^ n ∈ I,
{ obtain ⟨n, u, rfl⟩ := H hx0,
refine ⟨n, _⟩,
simpa only [units.mul_inv_cancel_right] using @ideal.mul_mem_right _ _ I _ ↑u⁻¹ hxI, },
constructor,
use p ^ (nat.find ex),
show I = ideal.span _,
apply le_antisymm,
{ intros r hr,
by_cases hr0 : r = 0,
{ simp only [hr0, submodule.zero_mem], },
obtain ⟨n, u, rfl⟩ := H hr0,
simp only [mem_span_singleton, is_unit_unit, is_unit.dvd_mul_right],
apply pow_dvd_pow,
apply nat.find_min',
simpa only [units.mul_inv_cancel_right] using @ideal.mul_mem_right _ _ I _ ↑u⁻¹ hr, },
{ erw submodule.span_singleton_le_iff_mem,
exact nat.find_spec ex, },
end
/--
A unique factorization domain with at least one irreducible element
in which all irreducible elements are associated
is a discrete valuation ring.
-/
lemma of_ufd_of_unique_irreducible {R : Type u} [integral_domain R] [unique_factorization_monoid R]
(h₁ : ∃ p : R, irreducible p)
(h₂ : ∀ ⦃p q : R⦄, irreducible p → irreducible q → associated p q) :
discrete_valuation_ring R :=
begin
rw iff_pid_with_one_nonzero_prime,
haveI PID : is_principal_ideal_ring R := aux_pid_of_ufd_of_unique_irreducible R h₁ h₂,
obtain ⟨p, hp⟩ := h₁,
refine ⟨PID, ⟨ideal.span {p}, ⟨_, _⟩, _⟩⟩,
{ rw submodule.ne_bot_iff,
refine ⟨p, ideal.mem_span_singleton.mpr (dvd_refl p), hp.ne_zero⟩, },
{ rwa [ideal.span_singleton_prime hp.ne_zero,
← unique_factorization_monoid.irreducible_iff_prime], },
{ intro I,
rw ← submodule.is_principal.span_singleton_generator I,
rintro ⟨I0, hI⟩,
apply span_singleton_eq_span_singleton.mpr,
apply h₂ _ hp,
erw [ne.def, span_singleton_eq_bot] at I0,
rwa [unique_factorization_monoid.irreducible_iff_prime, ← ideal.span_singleton_prime I0], },
end
/--
An integral domain in which there is a unit `p`
such that every nonzero element is associated to a power of `p`
is a discrete valuation ring.
-/
lemma of_has_unit_mul_pow_irreducible_factorization {R : Type u} [integral_domain R]
(hR : has_unit_mul_pow_irreducible_factorization R) :
discrete_valuation_ring R :=
begin
letI : unique_factorization_monoid R := hR.ufd,
apply of_ufd_of_unique_irreducible _ hR.unique_irreducible,
unfreezingI { obtain ⟨p, hp, H⟩ := hR, exact ⟨p, hp⟩, },
end
section
variables [integral_domain R] [discrete_valuation_ring R]
variable {R}
lemma associated_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : irreducible ϖ) :
∃ (n : ℕ), associated x (ϖ ^ n) :=
begin
have : wf_dvd_monoid R := is_noetherian_ring.wf_dvd_monoid,
cases wf_dvd_monoid.exists_factors x hx with fx hfx,
unfreezingI { use fx.card },
have H := hfx.2,
rw ← associates.mk_eq_mk_iff_associated at H ⊢,
rw [← H, ← associates.prod_mk, associates.mk_pow, ← multiset.prod_repeat],
congr' 1,
rw multiset.eq_repeat,
simp only [true_and, and_imp, multiset.card_map, eq_self_iff_true,
multiset.mem_map, exists_imp_distrib],
rintros _ _ _ rfl,
rw associates.mk_eq_mk_iff_associated,
refine associated_of_irreducible _ _ hirr,
apply hfx.1,
assumption
end
open submodule.is_principal
lemma ideal_eq_span_pow_irreducible {s : ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : irreducible ϖ) :
∃ n : ℕ, s = ideal.span {ϖ ^ n} :=
begin
have gen_ne_zero : generator s ≠ 0,
{ rw [ne.def, ← eq_bot_iff_generator_eq_zero], assumption },
rcases associated_pow_irreducible gen_ne_zero hirr with ⟨n, u, hnu⟩,
use n,
have : span _ = _ := span_singleton_generator s,
rw [← this, ← hnu, span_singleton_eq_span_singleton],
use u
end
lemma unit_mul_pow_congr_pow {p q : R} (hp : irreducible p) (hq : irreducible q)
(u v : units R) (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) :
m = n :=
begin
have key : associated (multiset.repeat p m).prod (multiset.repeat q n).prod,
{ rw [multiset.prod_repeat, multiset.prod_repeat, associated],
refine ⟨u * v⁻¹, _⟩,
simp only [units.coe_mul],
rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, units.mul_inv, one_mul], },
have := multiset.card_eq_card_of_rel (unique_factorization_monoid.factors_unique _ _ key),
{ simpa only [multiset.card_repeat] },
all_goals
{ intros x hx, replace hx := multiset.eq_of_mem_repeat hx,
unfreezingI { subst hx, assumption } },
end
lemma unit_mul_pow_congr_unit {ϖ : R} (hirr : irreducible ϖ) (u v : units R) (m n : ℕ)
(h : ↑u * ϖ ^ m = v * ϖ ^ n) :
u = v :=
begin
obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h,
rw ← sub_eq_zero at h,
rw [← sub_mul, mul_eq_zero] at h,
cases h,
{ rw sub_eq_zero at h, exact_mod_cast h },
{ apply (hirr.ne_zero (pow_eq_zero h)).elim, }
end
end
end discrete_valuation_ring
|
92a9c556ebfd8532cb9cfe1918132ebaf73df455 | bb31430994044506fa42fd667e2d556327e18dfe | /src/data/finset/card.lean | 8a6f1f2e0e381e91ff227e813bcac0d1967c5b78 | [
"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 | 24,538 | lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import data.finset.image
import tactic.by_contra
/-!
# Cardinality of a finite set
This defines the cardinality of a `finset` and provides induction principles for finsets.
## Main declarations
* `finset.card`: `s.card : ℕ` returns the cardinality of `s : finset α`.
### Induction principles
* `finset.strong_induction`: Strong induction
* `finset.strong_induction_on`
* `finset.strong_downward_induction`
* `finset.strong_downward_induction_on`
* `finset.case_strong_induction_on`
## TODO
Should we add a noncomputable version?
-/
open function multiset nat
variables {α β : Type*}
namespace finset
variables {s t : finset α} {a b : α}
/-- `s.card` is the number of elements of `s`, aka its cardinality. -/
def card (s : finset α) : ℕ := s.1.card
lemma card_def (s : finset α) : s.card = s.1.card := rfl
@[simp] lemma card_mk {m nodup} : (⟨m, nodup⟩ : finset α).card = m.card := rfl
@[simp] lemma card_empty : card (∅ : finset α) = 0 := rfl
lemma card_le_of_subset : s ⊆ t → s.card ≤ t.card := multiset.card_le_of_le ∘ val_le_iff.mpr
@[mono] lemma card_mono : monotone (@card α) := by apply card_le_of_subset
@[simp] lemma card_eq_zero : s.card = 0 ↔ s = ∅ := card_eq_zero.trans val_eq_zero
lemma card_pos : 0 < s.card ↔ s.nonempty :=
pos_iff_ne_zero.trans $ (not_congr card_eq_zero).trans nonempty_iff_ne_empty.symm
alias card_pos ↔ _ nonempty.card_pos
lemma card_ne_zero_of_mem (h : a ∈ s) : s.card ≠ 0 := (not_congr card_eq_zero).2 $ ne_empty_of_mem h
@[simp] lemma card_singleton (a : α) : card ({a} : finset α) = 1 := card_singleton _
lemma card_singleton_inter [decidable_eq α] : ({a} ∩ s).card ≤ 1 :=
begin
cases (finset.decidable_mem a s),
{ simp [finset.singleton_inter_of_not_mem h] },
{ simp [finset.singleton_inter_of_mem h] }
end
@[simp] lemma card_cons (h : a ∉ s) : (s.cons a h).card = s.card + 1 := card_cons _ _
section insert_erase
variables [decidable_eq α]
@[simp] lemma card_insert_of_not_mem (h : a ∉ s) : (insert a s).card = s.card + 1 :=
by rw [←cons_eq_insert _ _ h, card_cons]
lemma card_insert_of_mem (h : a ∈ s) : card (insert a s) = s.card := by rw insert_eq_of_mem h
lemma card_insert_le (a : α) (s : finset α) : card (insert a s) ≤ s.card + 1 :=
by by_cases a ∈ s; [{rw insert_eq_of_mem h, exact nat.le_succ _ }, rw card_insert_of_not_mem h]
/-- If `a ∈ s` is known, see also `finset.card_insert_of_mem` and `finset.card_insert_of_not_mem`.
-/
lemma card_insert_eq_ite : card (insert a s) = if a ∈ s then s.card else s.card + 1 :=
begin
by_cases h : a ∈ s,
{ rw [card_insert_of_mem h, if_pos h] },
{ rw [card_insert_of_not_mem h, if_neg h] }
end
@[simp] lemma card_doubleton (h : a ≠ b) : ({a, b} : finset α).card = 2 :=
by rw [card_insert_of_not_mem (not_mem_singleton.2 h), card_singleton]
@[simp] lemma card_erase_of_mem : a ∈ s → (s.erase a).card = s.card - 1 := card_erase_of_mem
@[simp] lemma card_erase_add_one : a ∈ s → (s.erase a).card + 1 = s.card := card_erase_add_one
lemma card_erase_lt_of_mem : a ∈ s → (s.erase a).card < s.card := card_erase_lt_of_mem
lemma card_erase_le : (s.erase a).card ≤ s.card := card_erase_le
lemma pred_card_le_card_erase : s.card - 1 ≤ (s.erase a).card :=
begin
by_cases h : a ∈ s,
{ exact (card_erase_of_mem h).ge },
{ rw erase_eq_of_not_mem h,
exact nat.sub_le _ _ }
end
/-- If `a ∈ s` is known, see also `finset.card_erase_of_mem` and `finset.erase_eq_of_not_mem`. -/
lemma card_erase_eq_ite : (s.erase a).card = if a ∈ s then s.card - 1 else s.card :=
card_erase_eq_ite
end insert_erase
@[simp] lemma card_range (n : ℕ) : (range n).card = n := card_range n
@[simp] lemma card_attach : s.attach.card = s.card := multiset.card_attach
end finset
section to_list_multiset
variables [decidable_eq α] (m : multiset α) (l : list α)
lemma multiset.card_to_finset : m.to_finset.card = m.dedup.card := rfl
lemma multiset.to_finset_card_le : m.to_finset.card ≤ m.card := card_le_of_le $ dedup_le _
lemma multiset.to_finset_card_of_nodup {m : multiset α} (h : m.nodup) : m.to_finset.card = m.card :=
congr_arg card $ multiset.dedup_eq_self.mpr h
lemma list.card_to_finset : l.to_finset.card = l.dedup.length := rfl
lemma list.to_finset_card_le : l.to_finset.card ≤ l.length := multiset.to_finset_card_le ⟦l⟧
lemma list.to_finset_card_of_nodup {l : list α} (h : l.nodup) : l.to_finset.card = l.length :=
multiset.to_finset_card_of_nodup h
end to_list_multiset
namespace finset
variables {s t : finset α} {f : α → β} {n : ℕ}
@[simp] lemma length_to_list (s : finset α) : s.to_list.length = s.card :=
by { rw [to_list, ←multiset.coe_card, multiset.coe_to_list], refl }
lemma card_image_le [decidable_eq β] : (s.image f).card ≤ s.card :=
by simpa only [card_map] using (s.1.map f).to_finset_card_le
lemma card_image_of_inj_on [decidable_eq β] (H : set.inj_on f s) : (s.image f).card = s.card :=
by simp only [card, image_val_of_inj_on H, card_map]
lemma inj_on_of_card_image_eq [decidable_eq β] (H : (s.image f).card = s.card) : set.inj_on f s :=
begin
change (s.1.map f).dedup.card = s.1.card at H,
have : (s.1.map f).dedup = s.1.map f,
{ refine multiset.eq_of_le_of_card_le (multiset.dedup_le _) _,
rw H,
simp only [multiset.card_map] },
rw multiset.dedup_eq_self at this,
exact inj_on_of_nodup_map this,
end
lemma card_image_iff [decidable_eq β] : (s.image f).card = s.card ↔ set.inj_on f s :=
⟨inj_on_of_card_image_eq, card_image_of_inj_on⟩
lemma card_image_of_injective [decidable_eq β] (s : finset α) (H : injective f) :
(s.image f).card = s.card :=
card_image_of_inj_on $ λ x _ y _ h, H h
lemma fiber_card_ne_zero_iff_mem_image (s : finset α) (f : α → β) [decidable_eq β] (y : β) :
(s.filter (λ x, f x = y)).card ≠ 0 ↔ y ∈ s.image f :=
by { rw [←pos_iff_ne_zero, card_pos, fiber_nonempty_iff_mem_image] }
@[simp] lemma card_map (f : α ↪ β) : (s.map f).card = s.card := multiset.card_map _ _
@[simp] lemma card_subtype (p : α → Prop) [decidable_pred p] (s : finset α) :
(s.subtype p).card = (s.filter p).card :=
by simp [finset.subtype]
lemma card_filter_le (s : finset α) (p : α → Prop) [decidable_pred p] :
(s.filter p).card ≤ s.card :=
card_le_of_subset $ filter_subset _ _
lemma eq_of_subset_of_card_le {s t : finset α} (h : s ⊆ t) (h₂ : t.card ≤ s.card) : s = t :=
eq_of_veq $ multiset.eq_of_le_of_card_le (val_le_iff.mpr h) h₂
lemma eq_of_superset_of_card_ge (hst : s ⊆ t) (hts : t.card ≤ s.card) : t = s :=
(eq_of_subset_of_card_le hst hts).symm
lemma subset_iff_eq_of_card_le (h : t.card ≤ s.card) : s ⊆ t ↔ s = t :=
⟨λ hst, eq_of_subset_of_card_le hst h, eq.subset'⟩
lemma map_eq_of_subset {f : α ↪ α} (hs : s.map f ⊆ s) : s.map f = s :=
eq_of_subset_of_card_le hs (card_map _).ge
lemma filter_card_eq {p : α → Prop} [decidable_pred p] (h : (s.filter p).card = s.card) (x : α)
(hx : x ∈ s) :
p x :=
begin
rw [←eq_of_subset_of_card_le (s.filter_subset p) h.ge, mem_filter] at hx,
exact hx.2,
end
lemma card_lt_card (h : s ⊂ t) : s.card < t.card := card_lt_of_lt $ val_lt_iff.2 h
lemma card_eq_of_bijective (f : ∀ i, i < n → α) (hf : ∀ a ∈ s, ∃ i, ∃ h : i < n, f i h = a)
(hf' : ∀ i (h : i < n), f i h ∈ s) (f_inj : ∀ i j (hi : i < n)
(hj : j < n), f i hi = f j hj → i = j) :
s.card = n :=
begin
classical,
have : ∀ (a : α), a ∈ s ↔ ∃ i (hi : i ∈ range n), f i (mem_range.1 hi) = a,
from λ a, ⟨λ ha, let ⟨i, hi, eq⟩ := hf a ha in ⟨i, mem_range.2 hi, eq⟩,
λ ⟨i, hi, eq⟩, eq ▸ hf' i (mem_range.1 hi)⟩,
have : s = ((range n).attach.image $ λi, f i.1 (mem_range.1 i.2)),
by simpa only [ext_iff, mem_image, exists_prop, subtype.exists, mem_attach, true_and],
calc s.card = card ((range n).attach.image $ λ i, f i.1 (mem_range.1 i.2)) :
by rw this
... = card ((range n).attach) :
card_image_of_injective _ $ λ ⟨i, hi⟩ ⟨j, hj⟩ eq,
subtype.eq $ f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq
... = card (range n) : card_attach
... = n : card_range n
end
lemma card_congr {t : finset β} (f : Π a ∈ s, β) (h₁ : ∀ a ha, f a ha ∈ t)
(h₂ : ∀ a b ha hb, f a ha = f b hb → a = b) (h₃ : ∀ b ∈ t, ∃ a ha, f a ha = b) :
s.card = t.card :=
by classical;
calc s.card = s.attach.card : card_attach.symm
... = (s.attach.image (λ (a : {a // a ∈ s}), f a.1 a.2)).card
: eq.symm (card_image_of_injective _ $ λ a b h, subtype.eq $ h₂ _ _ _ _ h)
... = t.card : congr_arg card (finset.ext $ λ b,
⟨λ h, let ⟨a, ha₁, ha₂⟩ := mem_image.1 h in ha₂ ▸ h₁ _ _,
λ h, let ⟨a, ha₁, ha₂⟩ := h₃ b h in mem_image.2 ⟨⟨a, ha₁⟩, by simp [ha₂]⟩⟩)
lemma card_le_card_of_inj_on {t : finset β} (f : α → β) (hf : ∀ a ∈ s, f a ∈ t)
(f_inj : ∀ a₁ ∈ s, ∀ a₂ ∈ s, f a₁ = f a₂ → a₁ = a₂) :
s.card ≤ t.card :=
by classical;
calc s.card = (s.image f).card : (card_image_of_inj_on f_inj).symm
... ≤ t.card : card_le_of_subset $ image_subset_iff.2 hf
/-- If there are more pigeons than pigeonholes, then there are two pigeons in the same pigeonhole.
-/
lemma exists_ne_map_eq_of_card_lt_of_maps_to {t : finset β} (hc : t.card < s.card)
{f : α → β} (hf : ∀ a ∈ s, f a ∈ t) :
∃ (x ∈ s) (y ∈ s), x ≠ y ∧ f x = f y :=
begin
classical,
by_contra' hz,
refine hc.not_le (card_le_card_of_inj_on f hf _),
intros x hx y hy, contrapose, exact hz x hx y hy,
end
lemma le_card_of_inj_on_range (f : ℕ → α) (hf : ∀ i < n, f i ∈ s)
(f_inj : ∀ (i < n) (j < n), f i = f j → i = j) :
n ≤ s.card :=
calc n = card (range n) : (card_range n).symm
... ≤ s.card : card_le_card_of_inj_on f (by simpa only [mem_range]) (by simpa only [mem_range])
lemma surj_on_of_inj_on_of_card_le {t : finset β} (f : Π a ∈ s, β) (hf : ∀ a ha, f a ha ∈ t)
(hinj : ∀ a₁ a₂ ha₁ ha₂, f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂) (hst : t.card ≤ s.card) :
∀ b ∈ t, ∃ a ha, b = f a ha :=
begin
classical,
intros b hb,
have h : (s.attach.image $ λ (a : {a // a ∈ s}), f a a.prop).card = s.card,
{ exact @card_attach _ s ▸ card_image_of_injective _
(λ ⟨a₁, ha₁⟩ ⟨a₂, ha₂⟩ h, subtype.eq $ hinj _ _ _ _ h) },
have h' : image (λ a : {a // a ∈ s}, f a a.prop) s.attach = t,
{ exact eq_of_subset_of_card_le (λ b h, let ⟨a, ha₁, ha₂⟩ := mem_image.1 h in
ha₂ ▸ hf _ _) (by simp [hst, h]) },
rw ←h' at hb,
obtain ⟨a, ha₁, ha₂⟩ := mem_image.1 hb,
exact ⟨a, a.2, ha₂.symm⟩,
end
lemma inj_on_of_surj_on_of_card_le {t : finset β} (f : Π a ∈ s, β) (hf : ∀ a ha, f a ha ∈ t)
(hsurj : ∀ b ∈ t, ∃ a ha, b = f a ha) (hst : s.card ≤ t.card) ⦃a₁ a₂⦄ (ha₁ : a₁ ∈ s)
(ha₂ : a₂ ∈ s) (ha₁a₂: f a₁ ha₁ = f a₂ ha₂) :
a₁ = a₂ :=
by haveI : inhabited {x // x ∈ s} := ⟨⟨a₁, ha₁⟩⟩; exact
let f' : {x // x ∈ s} → {x // x ∈ t} := λ x, ⟨f x.1 x.2, hf x.1 x.2⟩ in
let g : {x // x ∈ t} → {x // x ∈ s} :=
@surj_inv _ _ f'
(λ x, let ⟨y, hy₁, hy₂⟩ := hsurj x.1 x.2 in ⟨⟨y, hy₁⟩, subtype.eq hy₂.symm⟩) in
have hg : injective g, from injective_surj_inv _,
have hsg : surjective g, from λ x,
let ⟨y, hy⟩ := surj_on_of_inj_on_of_card_le (λ (x : {x // x ∈ t}) (hx : x ∈ t.attach), g x)
(λ x _, show (g x) ∈ s.attach, from mem_attach _ _)
(λ x y _ _ hxy, hg hxy) (by simpa) x (mem_attach _ _) in
⟨y, hy.snd.symm⟩,
have hif : injective f',
from (left_inverse_of_surjective_of_right_inverse hsg
(right_inverse_surj_inv _)).injective,
subtype.ext_iff_val.1 (@hif ⟨a₁, ha₁⟩ ⟨a₂, ha₂⟩ (subtype.eq ha₁a₂))
@[simp] lemma card_disj_union (s t : finset α) (h) : (s.disj_union t h).card = s.card + t.card :=
multiset.card_add _ _
/-! ### Lattice structure -/
section lattice
variables [decidable_eq α]
lemma card_union_add_card_inter (s t : finset α) : (s ∪ t).card + (s ∩ t).card = s.card + t.card :=
finset.induction_on t (by simp) $ λ a r har, by by_cases a ∈ s; simp *; cc
lemma card_inter_add_card_union (s t : finset α) : (s ∩ t).card + (s ∪ t).card = s.card + t.card :=
by rw [add_comm, card_union_add_card_inter]
lemma card_union_le (s t : finset α) : (s ∪ t).card ≤ s.card + t.card :=
card_union_add_card_inter s t ▸ nat.le_add_right _ _
lemma card_union_eq (h : disjoint s t) : (s ∪ t).card = s.card + t.card :=
by rw [←disj_union_eq_union s t h, card_disj_union _ _ _]
@[simp] lemma card_disjoint_union (h : disjoint s t) : card (s ∪ t) = s.card + t.card :=
card_union_eq h
lemma card_sdiff (h : s ⊆ t) : card (t \ s) = t.card - s.card :=
suffices card (t \ s) = card ((t \ s) ∪ s) - s.card, by rwa sdiff_union_of_subset h at this,
by rw [card_disjoint_union sdiff_disjoint, add_tsub_cancel_right]
lemma card_sdiff_add_card_eq_card {s t : finset α} (h : s ⊆ t) : card (t \ s) + card s = card t :=
((nat.sub_eq_iff_eq_add (card_le_of_subset h)).mp (card_sdiff h).symm).symm
lemma le_card_sdiff (s t : finset α) : t.card - s.card ≤ card (t \ s) :=
calc card t - card s
≤ card t - card (s ∩ t) : tsub_le_tsub_left (card_le_of_subset (inter_subset_left s t)) _
... = card (t \ (s ∩ t)) : (card_sdiff (inter_subset_right s t)).symm
... ≤ card (t \ s) : by rw sdiff_inter_self_right t s
lemma card_le_card_sdiff_add_card : s.card ≤ (s \ t).card + t.card :=
tsub_le_iff_right.1 $ le_card_sdiff _ _
lemma card_sdiff_add_card : (s \ t).card + t.card = (s ∪ t).card :=
by rw [←card_disjoint_union sdiff_disjoint, sdiff_union_self_eq_union]
end lattice
lemma filter_card_add_filter_neg_card_eq_card (p : α → Prop) [decidable_pred p] :
(s.filter p).card + (s.filter (not ∘ p)).card = s.card :=
by { classical, simp [←card_union_eq, filter_union_filter_neg_eq, disjoint_filter] }
/-- Given a set `A` and a set `B` inside it, we can shrink `A` to any appropriate size, and keep `B`
inside it. -/
lemma exists_intermediate_set {A B : finset α} (i : ℕ) (h₁ : i + card B ≤ card A) (h₂ : B ⊆ A) :
∃ (C : finset α), B ⊆ C ∧ C ⊆ A ∧ card C = i + card B :=
begin
classical,
rcases nat.le.dest h₁ with ⟨k, _⟩,
clear h₁,
induction k with k ih generalizing A,
{ exact ⟨A, h₂, subset.refl _, h.symm⟩ },
have : (A \ B).nonempty,
{ rw [←card_pos, card_sdiff h₂, ←h, nat.add_right_comm,
add_tsub_cancel_right, nat.add_succ],
apply nat.succ_pos },
rcases this with ⟨a, ha⟩,
have z : i + card B + k = card (erase A a),
{ rw [card_erase_of_mem (mem_sdiff.1 ha).1, ←h],
refl },
rcases ih _ z with ⟨B', hB', B'subA', cards⟩,
{ exact ⟨B', hB', trans B'subA' (erase_subset _ _), cards⟩ },
{ rintro t th,
apply mem_erase_of_ne_of_mem _ (h₂ th),
rintro rfl,
exact not_mem_sdiff_of_mem_right th ha }
end
/-- We can shrink `A` to any smaller size. -/
lemma exists_smaller_set (A : finset α) (i : ℕ) (h₁ : i ≤ card A) :
∃ (B : finset α), B ⊆ A ∧ card B = i :=
let ⟨B, _, x₁, x₂⟩ := exists_intermediate_set i (by simpa) (empty_subset A) in ⟨B, x₁, x₂⟩
lemma exists_subset_or_subset_of_two_mul_lt_card [decidable_eq α] {X Y : finset α} {n : ℕ}
(hXY : 2 * n < (X ∪ Y).card) :
∃ C : finset α, n < C.card ∧ (C ⊆ X ∨ C ⊆ Y) :=
begin
have h₁ : (X ∩ (Y \ X)).card = 0 := finset.card_eq_zero.mpr (finset.inter_sdiff_self X Y),
have h₂ : (X ∪ Y).card = X.card + (Y \ X).card,
{ rw [←card_union_add_card_inter X (Y \ X), finset.union_sdiff_self_eq_union, h₁, add_zero] },
rw [h₂, two_mul] at hXY,
rcases lt_or_lt_of_add_lt_add hXY with h|h,
{ exact ⟨X, h, or.inl (finset.subset.refl X)⟩ },
{ exact ⟨Y \ X, h, or.inr (finset.sdiff_subset Y X)⟩ }
end
/-! ### Explicit description of a finset from its card -/
lemma card_eq_one : s.card = 1 ↔ ∃ a, s = {a} :=
by cases s; simp only [multiset.card_eq_one, finset.card, ←val_inj, singleton_val]
lemma exists_eq_insert_iff [decidable_eq α] {s t : finset α} :
(∃ a ∉ s, insert a s = t) ↔ s ⊆ t ∧ s.card + 1 = t.card :=
begin
split,
{ rintro ⟨a, ha, rfl⟩,
exact ⟨subset_insert _ _, (card_insert_of_not_mem ha).symm⟩ },
{ rintro ⟨hst, h⟩,
obtain ⟨a, ha⟩ : ∃ a, t \ s = {a},
{ exact card_eq_one.1 (by rw [card_sdiff hst, ←h, add_tsub_cancel_left]) },
refine ⟨a, λ hs, (_ : a ∉ {a}) $ mem_singleton_self _,
by rw [insert_eq, ←ha, sdiff_union_of_subset hst]⟩,
rw ←ha,
exact not_mem_sdiff_of_mem_right hs }
end
lemma card_le_one : s.card ≤ 1 ↔ ∀ (a ∈ s) (b ∈ s), a = b :=
begin
obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty,
{ simp },
refine (nat.succ_le_of_lt (card_pos.2 ⟨x, hx⟩)).le_iff_eq.trans (card_eq_one.trans ⟨_, _⟩),
{ rintro ⟨y, rfl⟩,
simp },
{ exact λ h, ⟨x, eq_singleton_iff_unique_mem.2 ⟨hx, λ y hy, h _ hy _ hx⟩⟩ }
end
lemma card_le_one_iff : s.card ≤ 1 ↔ ∀ {a b}, a ∈ s → b ∈ s → a = b := by { rw card_le_one, tauto }
lemma card_le_one_iff_subset_singleton [nonempty α] : s.card ≤ 1 ↔ ∃ (x : α), s ⊆ {x} :=
begin
refine ⟨λ H, _, _⟩,
{ obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty,
{ exact ⟨classical.arbitrary α, empty_subset _⟩ },
{ exact ⟨x, λ y hy, by rw [card_le_one.1 H y hy x hx, mem_singleton]⟩ } },
{ rintro ⟨x, hx⟩,
rw ←card_singleton x,
exact card_le_of_subset hx }
end
/-- A `finset` of a subsingleton type has cardinality at most one. -/
lemma card_le_one_of_subsingleton [subsingleton α] (s : finset α) : s.card ≤ 1 :=
finset.card_le_one_iff.2 $ λ _ _ _ _, subsingleton.elim _ _
lemma one_lt_card : 1 < s.card ↔ ∃ (a ∈ s) (b ∈ s), a ≠ b :=
by { rw ←not_iff_not, push_neg, exact card_le_one }
lemma one_lt_card_iff : 1 < s.card ↔ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b :=
by { rw one_lt_card, simp only [exists_prop, exists_and_distrib_left] }
lemma two_lt_card_iff : 2 < s.card ↔ ∃ a b c, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c :=
begin
classical,
refine ⟨λ h, _, _⟩,
{ obtain ⟨c, hc⟩ := card_pos.mp (zero_lt_two.trans h),
have : 1 < (s.erase c).card := by rwa [←add_lt_add_iff_right 1, card_erase_add_one hc],
obtain ⟨a, b, ha, hb, hab⟩ := one_lt_card_iff.mp this,
exact ⟨a, b, c, mem_of_mem_erase ha, mem_of_mem_erase hb, hc,
hab, ne_of_mem_erase ha, ne_of_mem_erase hb⟩ },
{ rintros ⟨a, b, c, ha, hb, hc, hab, hac, hbc⟩,
rw [←card_erase_add_one hc, ←card_erase_add_one (mem_erase_of_ne_of_mem hbc hb),
←card_erase_add_one (mem_erase_of_ne_of_mem hab (mem_erase_of_ne_of_mem hac ha))],
apply nat.le_add_left },
end
lemma two_lt_card : 2 < s.card ↔ ∃ (a ∈ s) (b ∈ s) (c ∈ s), a ≠ b ∧ a ≠ c ∧ b ≠ c :=
by simp_rw [two_lt_card_iff, exists_prop, exists_and_distrib_left]
lemma exists_ne_of_one_lt_card (hs : 1 < s.card) (a : α) : ∃ b, b ∈ s ∧ b ≠ a :=
begin
obtain ⟨x, hx, y, hy, hxy⟩ := finset.one_lt_card.mp hs,
by_cases ha : y = a,
{ exact ⟨x, hx, ne_of_ne_of_eq hxy ha⟩ },
{ exact ⟨y, hy, ha⟩ }
end
lemma card_eq_succ [decidable_eq α] : s.card = n + 1 ↔ ∃ a t, a ∉ t ∧ insert a t = s ∧ t.card = n :=
⟨λ h,
let ⟨a, has⟩ := card_pos.mp (h.symm ▸ nat.zero_lt_succ _ : 0 < s.card) in
⟨a, s.erase a, s.not_mem_erase a, insert_erase has,
by simp only [h, card_erase_of_mem has, add_tsub_cancel_right]⟩,
λ ⟨a, t, hat, s_eq, n_eq⟩, s_eq ▸ n_eq ▸ card_insert_of_not_mem hat⟩
lemma card_eq_two [decidable_eq α] : s.card = 2 ↔ ∃ x y, x ≠ y ∧ s = {x, y} :=
begin
split,
{ rw card_eq_succ,
simp_rw [card_eq_one],
rintro ⟨a, _, hab, rfl, b, rfl⟩,
exact ⟨a, b, not_mem_singleton.1 hab, rfl⟩ },
{ rintro ⟨x, y, h, rfl⟩,
exact card_doubleton h }
end
lemma card_eq_three [decidable_eq α] :
s.card = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z} :=
begin
split,
{ rw card_eq_succ,
simp_rw [card_eq_two],
rintro ⟨a, _, abc, rfl, b, c, bc, rfl⟩,
rw [mem_insert, mem_singleton, not_or_distrib] at abc,
exact ⟨a, b, c, abc.1, abc.2, bc, rfl⟩ },
{ rintro ⟨x, y, z, xy, xz, yz, rfl⟩,
simp only [xy, xz, yz, mem_insert, card_insert_of_not_mem, not_false_iff, mem_singleton,
or_self, card_singleton] }
end
/-! ### Inductions -/
/-- Suppose that, given objects defined on all strict subsets of any finset `s`, one knows how to
define an object on `s`. Then one can inductively define an object on all finsets, starting from
the empty set and iterating. This can be used either to define data, or to prove properties. -/
def strong_induction {p : finset α → Sort*} (H : ∀ s, (∀ t ⊂ s, p t) → p s) :
∀ (s : finset α), p s
| s := H s (λ t h, have t.card < s.card, from card_lt_card h, strong_induction t)
using_well_founded {rel_tac := λ _ _, `[exact ⟨_, measure_wf card⟩]}
lemma strong_induction_eq {p : finset α → Sort*} (H : ∀ s, (∀ t ⊂ s, p t) → p s) (s : finset α) :
strong_induction H s = H s (λ t h, strong_induction H t) :=
by rw strong_induction
/-- Analogue of `strong_induction` with order of arguments swapped. -/
@[elab_as_eliminator] def strong_induction_on {p : finset α → Sort*} (s : finset α) :
(∀ s, (∀ t ⊂ s, p t) → p s) → p s :=
λ H, strong_induction H s
lemma strong_induction_on_eq {p : finset α → Sort*} (s : finset α) (H : ∀ s, (∀ t ⊂ s, p t) → p s) :
s.strong_induction_on H = H s (λ t h, t.strong_induction_on H) :=
by { dunfold strong_induction_on, rw strong_induction }
@[elab_as_eliminator] lemma case_strong_induction_on [decidable_eq α] {p : finset α → Prop}
(s : finset α) (h₀ : p ∅) (h₁ : ∀ a s, a ∉ s → (∀ t ⊆ s, p t) → p (insert a s)) :
p s :=
finset.strong_induction_on s $ λ s,
finset.induction_on s (λ _, h₀) $ λ a s n _ ih, h₁ a s n $
λ t ss, ih _ (lt_of_le_of_lt ss (ssubset_insert n) : t < _)
/-- Suppose that, given that `p t` can be defined on all supersets of `s` of cardinality less than
`n`, one knows how to define `p s`. Then one can inductively define `p s` for all finsets `s` of
cardinality less than `n`, starting from finsets of card `n` and iterating. This
can be used either to define data, or to prove properties. -/
def strong_downward_induction {p : finset α → Sort*} {n : ℕ} (H : ∀ t₁, (∀ {t₂ : finset α},
t₂.card ≤ n → t₁ ⊂ t₂ → p t₂) → t₁.card ≤ n → p t₁) :
∀ (s : finset α), s.card ≤ n → p s
| s := H s (λ t ht h, have n - t.card < n - s.card,
from (tsub_lt_tsub_iff_left_of_le ht).2 (finset.card_lt_card h),
strong_downward_induction t ht)
using_well_founded {rel_tac := λ _ _, `[exact ⟨_, measure_wf (λ (t : finset α), n - t.card)⟩]}
lemma strong_downward_induction_eq {p : finset α → Sort*}
(H : ∀ t₁, (∀ {t₂ : finset α}, t₂.card ≤ n → t₁ ⊂ t₂ → p t₂) → t₁.card ≤ n → p t₁)
(s : finset α) :
strong_downward_induction H s = H s (λ t ht hst, strong_downward_induction H t ht) :=
by rw strong_downward_induction
/-- Analogue of `strong_downward_induction` with order of arguments swapped. -/
@[elab_as_eliminator] def strong_downward_induction_on {p : finset α → Sort*} (s : finset α)
(H : ∀ t₁, (∀ {t₂ : finset α}, t₂.card ≤ n → t₁ ⊂ t₂ → p t₂) → t₁.card ≤ n → p t₁) :
s.card ≤ n → p s :=
strong_downward_induction H s
lemma strong_downward_induction_on_eq {p : finset α → Sort*} (s : finset α) (H : ∀ t₁,
(∀ {t₂ : finset α}, t₂.card ≤ n → t₁ ⊂ t₂ → p t₂) → t₁.card ≤ n → p t₁) :
s.strong_downward_induction_on H = H s (λ t ht h, t.strong_downward_induction_on H ht) :=
by { dunfold strong_downward_induction_on, rw strong_downward_induction }
lemma lt_wf {α} : well_founded (@has_lt.lt (finset α) _) :=
have H : subrelation (@has_lt.lt (finset α) _)
(inv_image ( < ) card),
from λ x y hxy, card_lt_card hxy,
subrelation.wf H $ inv_image.wf _ $ nat.lt_wf
end finset
|
55ba3bdee59e06e80f74773457e47120e90539d4 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/data/set/intervals/iso_Ioo.lean | 1ce9692e865e7818d13a7ddf5ed2ccb57e68c221 | [
"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 | 1,693 | 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 order.monotone.odd
import tactic.field_simp
/-!
# Order isomorphism between a linear ordered field and `(-1, 1)`
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
In this file we provide an order isomorphism `order_iso_Ioo_neg_one_one` between the open interval
`(-1, 1)` in a linear ordered field and the whole field.
-/
open set
/-- In a linear ordered field, the whole field is order isomorphic to the open interval `(-1, 1)`.
We consider the actual implementation to be a "black box", so it is irreducible.
-/
@[irreducible] def order_iso_Ioo_neg_one_one (k : Type*) [linear_ordered_field k] :
k ≃o Ioo (-1 : k) 1 :=
begin
refine strict_mono.order_iso_of_right_inverse _ _ (λ x, x / (1 - |x|)) _,
{ refine cod_restrict (λ x, x / (1 + |x|)) _ (λ x, abs_lt.1 _),
have H : 0 < 1 + |x|, from (abs_nonneg x).trans_lt (lt_one_add _),
calc |x / (1 + |x|)| = |x| / (1 + |x|) : by rw [abs_div, abs_of_pos H]
... < 1 : (div_lt_one H).2 (lt_one_add _) },
{ refine (strict_mono_of_odd_strict_mono_on_nonneg _ _).cod_restrict _,
{ intro x, simp only [abs_neg, neg_div] },
{ rintros x (hx : 0 ≤ x) y (hy : 0 ≤ y) hxy,
simp [abs_of_nonneg, mul_add, mul_comm x y, div_lt_div_iff,
hx.trans_lt (lt_one_add _), hy.trans_lt (lt_one_add _), *] } },
{ refine λ x, subtype.ext _,
have : 0 < 1 - |(x : k)|, from sub_pos.2 (abs_lt.2 x.2),
field_simp [abs_div, this.ne', abs_of_pos this] }
end
|
46a8949b35cde5affb6a7892e0f958505f62266a | cf39355caa609c0f33405126beee2739aa3cb77e | /library/init/meta/name.lean | 46d5d6a052866ba36934b3138398b205fc65f117 | [
"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 | 4,104 | 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
-/
prelude
import init.data.ordering.basic init.coe init.data.to_string
/-- Reflect a C++ name object. The VM replaces it with the C++ implementation. -/
inductive name
| anonymous : name
| mk_string : string → name → name
| mk_numeral : unsigned → name → name
/-- Gadget for automatic parameter support. This is similar to the opt_param gadget, but it uses
the tactic declaration names tac_name to synthesize the argument.
Like opt_param, this gadget only affects elaboration.
For example, the tactic will *not* be invoked during type class resolution. -/
@[reducible] def {u} auto_param (α : Sort u) (tac_name : name) : Sort u :=
α
@[simp] lemma {u} auto_param_eq (α : Sort u) (n : name) : auto_param α n = α :=
rfl
instance : inhabited name :=
⟨name.anonymous⟩
def mk_str_name (n : name) (s : string) : name :=
name.mk_string s n
def mk_num_name (n : name) (v : nat) : name :=
name.mk_numeral (unsigned.of_nat' v) n
def mk_simple_name (s : string) : name :=
mk_str_name name.anonymous s
instance string_to_name : has_coe string name :=
⟨mk_simple_name⟩
infix ` <.> `:65 := mk_str_name
open name
def name.get_prefix : name → name
| anonymous := anonymous
| (mk_string s p) := p
| (mk_numeral s p) := p
def name.update_prefix : name → name → name
| anonymous new_p := anonymous
| (mk_string s p) new_p := mk_string s new_p
| (mk_numeral s p) new_p := mk_numeral s new_p
-- Without this option, we get errors when defining the following definitions.
set_option eqn_compiler.ite false
def name.to_string_with_sep (sep : string) : name → string
| anonymous := "[anonymous]"
| (mk_string s anonymous) := s
| (mk_numeral v anonymous) := repr v
| (mk_string s n) := name.to_string_with_sep n ++ sep ++ s
| (mk_numeral v n) := name.to_string_with_sep n ++ sep ++ repr v
private def name.components' : name -> list name
| anonymous := []
| (mk_string s n) := mk_string s anonymous :: name.components' n
| (mk_numeral v n) := mk_numeral v anonymous :: name.components' n
def name.components (n : name) : list name :=
(name.components' n).reverse
protected def name.to_string : name → string :=
name.to_string_with_sep "."
protected def name.repr (n : name) : string :=
"`" ++ n.to_string
instance : has_to_string name :=
⟨name.to_string⟩
instance : has_repr name :=
⟨name.repr⟩
/- TODO(Leo): provide a definition in Lean. -/
meta constant name.has_decidable_eq : decidable_eq name
/-! Both cmp and lex_cmp are total orders, but lex_cmp implements a lexicographical order. -/
meta constant name.cmp : name → name → ordering
meta constant name.lex_cmp : name → name → ordering
meta constant name.append : name → name → name
meta constant name.is_internal : name → bool
protected meta def name.lt (a b : name) : Prop :=
name.cmp a b = ordering.lt
meta instance : decidable_rel name.lt :=
λ a b, ordering.decidable_eq _ _
meta instance : has_lt name :=
⟨name.lt⟩
attribute [instance] name.has_decidable_eq
meta instance : has_append name :=
⟨name.append⟩
/-- `name.append_after n i` return a name of the form n_i -/
meta constant name.append_after : name → nat → name
meta def name.is_prefix_of : name → name → bool
| p name.anonymous := ff
| p n :=
if p = n then tt else name.is_prefix_of p n.get_prefix
meta def name.is_suffix_of : name → name → bool
| anonymous _ := tt
| (mk_string s n) (mk_string s' n') := (s = s') && name.is_suffix_of n n'
| (mk_numeral v n) (mk_numeral v' n') := (v = v') && name.is_suffix_of n n'
| _ _ := ff
meta def name.replace_prefix : name → name → name → name
| anonymous p p' := anonymous
| (mk_string s c) p p' := if c = p then mk_string s p' else mk_string s (name.replace_prefix c p p')
| (mk_numeral v c) p p' := if c = p then mk_numeral v p' else mk_numeral v (name.replace_prefix c p p')
|
cf313373a9872fe3334a3620ab9b4e75c5ead83f | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/protected.lean | 66aa2130683e70b691f721902fa72108c4fe8ec0 | [
"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 | 1,022 | lean |
namespace Foo
protected def x := 10
end Foo
open Foo
#check x -- error unknown identifier `x`
#check Foo.x
namespace Bla.Foo
protected def y := 20
def z := 30
end Bla.Foo
open Bla
#check Foo.y
open Bla.Foo
#check y -- error unknown identifier `y`
#check z
protected def x := 0 -- Error
protected partial def Foo.f (x : Nat) :=
if x > 0 then f (x-1) * 2 else 1 -- Error
protected partial def Bla.f (x : Nat) :=
if x > 0 then Bla.f (x-1) * 2 else 1
#eval Bla.f 3
namespace Foo
protected partial def g (x : Nat) :=
if x > 0 then g (x-1) * 2 else 1 -- error
end Foo
namespace Bla
protected partial def g (x : Nat) :=
if x > 0 then Bla.g (x-1) * 2 else 1
#eval g 3 -- error
#eval Bla.g 3
end Bla
def S (σ : Type) (m : Type → Type) (α : Type) :=
σ → m (α × σ)
namespace S
variable {σ : Type} {m : Type → Type} [Monad m] {α : Type}
protected def pure (a : α) : S σ m α :=
fun s => pure (a, s) -- This `pure` is the top-level one. The `pure` being defined here is called `S.pure`.
end S
|
480cb0cfd39bcc5db323099e687c2eb48db2529f | 4727251e0cd73359b15b664c3170e5d754078599 | /src/model_theory/direct_limit.lean | b228a91cfa896ee54d3d341268f0f4d198178fb7 | [
"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 | 14,744 | lean | /-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import data.fintype.order
import algebra.direct_limit
import model_theory.quotients
import model_theory.finitely_generated
/-!
# Direct Limits of First-Order Structures
This file constructs the direct limit of a directed system of first-order embeddings.
## Main Definitions
* `first_order.language.direct_limit G f` is the direct limit of the directed system `f` of
first-order embeddings between the structures indexed by `G`.
-/
universes v w u₁ u₂
open_locale first_order
namespace first_order
namespace language
open Structure set
variables {L : language} {ι : Type v} [preorder ι]
variables {G : ι → Type w} [Π i, L.Structure (G i)]
variables (f : Π i j, i ≤ j → G i ↪[L] G j)
namespace directed_system
/-- A copy of `directed_system.map_self` specialized to `L`-embeddings, as otherwise the
`λ i j h, f i j h` can confuse the simplifier. -/
lemma map_self [directed_system G (λ i j h, f i j h)] (i x h) :
f i i h x = x :=
directed_system.map_self (λ i j h, f i j h) i x h
/-- A copy of `directed_system.map_map` specialized to `L`-embeddings, as otherwise the
`λ i j h, f i j h` can confuse the simplifier. -/
lemma map_map [directed_system G (λ i j h, f i j h)] {i j k} (hij hjk x) :
f j k hjk (f i j hij x) = f i k (le_trans hij hjk) x :=
directed_system.map_map (λ i j h, f i j h) hij hjk x
variables {G' : ℕ → Type w} [Π i, L.Structure (G' i)] (f' : Π (n : ℕ), G' n ↪[L] G' (n + 1))
/-- Given a chain of embeddings of structures indexed by `ℕ`, defines a `directed_system` by
composing them. -/
def nat_le_rec (m n : ℕ) (h : m ≤ n) : G' m ↪[L] G' n :=
nat.le_rec_on h (λ k g, (f' k).comp g) (embedding.refl L _)
@[simp] lemma coe_nat_le_rec (m n : ℕ) (h : m ≤ n) :
(nat_le_rec f' m n h : G' m → G' n) = nat.le_rec_on h (λ n, f' n) :=
begin
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le h,
ext x,
induction k with k ih,
{ rw [nat_le_rec, nat.le_rec_on_self, embedding.refl_apply, nat.le_rec_on_self] },
{ rw [nat.le_rec_on_succ le_self_add, nat_le_rec, nat.le_rec_on_succ le_self_add, ← nat_le_rec,
embedding.comp_apply, ih] }
end
instance nat_le_rec.directed_system : directed_system G' (λ i j h, nat_le_rec f' i j h) :=
⟨λ i x h, congr (congr rfl (nat.le_rec_on_self _)) rfl,
λ i j k ij jk, by simp [nat.le_rec_on_trans ij jk]⟩
end directed_system
namespace direct_limit
/-- Raises a family of elements in the `Σ`-type to the same level along the embeddings. -/
def unify {α : Type*} (x : α → (Σ i, G i)) (i : ι)
(h : i ∈ upper_bounds (range (sigma.fst ∘ x))) (a : α) : G i :=
f (x a).1 i (h (mem_range_self a)) (x a).2
variable [directed_system G (λ i j h, f i j h)]
@[simp] lemma unify_sigma_mk_self {α : Type*} {i : ι}
{x : α → G i} :
unify f (sigma.mk i ∘ x) i (λ j ⟨a, hj⟩, trans (le_of_eq hj.symm) (refl _)) = x :=
begin
ext a,
simp only [unify, directed_system.map_self],
end
lemma comp_unify {α : Type*} {x : α → (Σ i, G i)} {i j : ι}
(ij : i ≤ j) (h : i ∈ upper_bounds (range (sigma.fst ∘ x))) :
(f i j ij) ∘ (unify f x i h) = unify f x j (λ k hk, trans (mem_upper_bounds.1 h k hk) ij) :=
begin
ext a,
simp [unify, directed_system.map_map],
end
end direct_limit
variables (G)
namespace direct_limit
/-- The directed limit glues together the structures along the embeddings. -/
def setoid [directed_system G (λ i j h, f i j h)] [is_directed ι (≤)] :
setoid (Σ i, G i) :=
{ r := λ ⟨i, x⟩ ⟨j, y⟩, ∃ (k : ι) (ik : i ≤ k) (jk : j ≤ k), f i k ik x = f j k jk y,
iseqv := ⟨λ ⟨i, x⟩, ⟨i, refl i, refl i, rfl⟩,
λ ⟨i, x⟩ ⟨j, y⟩ ⟨k, ik, jk, h⟩, ⟨k, jk, ik, h.symm⟩,
λ ⟨i, x⟩ ⟨j, y⟩ ⟨k, z⟩ ⟨ij, hiij, hjij, hij⟩ ⟨jk, hjjk, hkjk, hjk⟩,
begin
obtain ⟨ijk, hijijk, hjkijk⟩ := (directed_of (≤) ij jk),
refine ⟨ijk, le_trans hiij hijijk, le_trans hkjk hjkijk, _⟩,
rw [← directed_system.map_map, hij, directed_system.map_map],
symmetry,
rw [← directed_system.map_map, ← hjk, directed_system.map_map],
end ⟩ }
/-- The structure on the `Σ`-type which becomes the structure on the direct limit after quotienting.
-/
noncomputable def sigma_structure [is_directed ι (≤)] [nonempty ι] : L.Structure (Σ i, G i) :=
{ fun_map := λ n F x, ⟨_, fun_map F (unify f x
(classical.some (fintype.bdd_above_range (λ a, (x a).1)))
(classical.some_spec (fintype.bdd_above_range (λ a, (x a).1))))⟩,
rel_map := λ n R x, rel_map R (unify f x
(classical.some (fintype.bdd_above_range (λ a, (x a).1)))
(classical.some_spec (fintype.bdd_above_range (λ a, (x a).1)))) }
end direct_limit
/-- The direct limit of a directed system is the structures glued together along the embeddings. -/
def direct_limit [directed_system G (λ i j h, f i j h)] [is_directed ι (≤)] :=
quotient (direct_limit.setoid G f)
local attribute [instance] direct_limit.setoid
instance [directed_system G (λ i j h, f i j h)] [is_directed ι (≤)] [inhabited ι]
[inhabited (G default)] : inhabited (direct_limit G f) :=
⟨⟦⟨default, default⟩⟧⟩
namespace direct_limit
variables [is_directed ι (≤)] [directed_system G (λ i j h, f i j h)]
lemma equiv_iff {x y : Σ i, G i} {i : ι} (hx : x.1 ≤ i) (hy : y.1 ≤ i) :
x ≈ y ↔ (f x.1 i hx) x.2 = (f y.1 i hy) y.2 :=
begin
cases x,
cases y,
refine ⟨λ xy, _, λ xy, ⟨i, hx, hy, xy⟩⟩,
obtain ⟨j, _, _, h⟩ := xy,
obtain ⟨k, ik, jk⟩ := directed_of (≤) i j,
have h := congr_arg (f j k jk) h,
apply (f i k ik).injective,
rw [directed_system.map_map, directed_system.map_map] at *,
exact h
end
lemma fun_map_unify_equiv {n : ℕ} (F : L.functions n) (x : (fin n) → (Σ i, G i)) (i j : ι)
(hi : i ∈ upper_bounds (range (sigma.fst ∘ x))) (hj : j ∈ upper_bounds (range (sigma.fst ∘ x))) :
(⟨i, fun_map F (unify f x i hi)⟩ : Σ i, G i) ≈ ⟨j, fun_map F (unify f x j hj)⟩ :=
begin
obtain ⟨k, ik, jk⟩ := directed_of (≤) i j,
refine ⟨k, ik, jk, _⟩,
rw [(f i k ik).map_fun, (f j k jk).map_fun, comp_unify, comp_unify],
end
lemma rel_map_unify_equiv {n : ℕ} (R : L.relations n) (x : (fin n) → (Σ i, G i)) (i j : ι)
(hi : i ∈ upper_bounds (range (sigma.fst ∘ x))) (hj : j ∈ upper_bounds (range (sigma.fst ∘ x))) :
rel_map R (unify f x i hi) = rel_map R (unify f x j hj) :=
begin
obtain ⟨k, ik, jk⟩ := directed_of (≤) i j,
rw [← (f i k ik).map_rel, comp_unify, ← (f j k jk).map_rel, comp_unify],
end
variable [nonempty ι]
lemma exists_unify_eq {α : Type*} [fintype α] {x y : α → (Σ i, G i)} (xy : x ≈ y) :
∃ (i : ι) (hx : i ∈ upper_bounds (range (sigma.fst ∘ x)))
(hy : i ∈ upper_bounds (range (sigma.fst ∘ y))),
unify f x i hx = unify f y i hy :=
begin
obtain ⟨i, hi⟩ := fintype.bdd_above_range (sum.elim (λ a, (x a).1) (λ a, (y a).1)),
rw [sum.elim_range, upper_bounds_union] at hi,
simp_rw [← function.comp_apply sigma.fst _] at hi,
exact ⟨i, hi.1, hi.2, funext (λ a, (equiv_iff G f _ _).1 (xy a))⟩,
end
lemma fun_map_equiv_unify {n : ℕ} (F : L.functions n) (x : (fin n) → (Σ i, G i)) (i : ι)
(hi : i ∈ upper_bounds (range (sigma.fst ∘ x))) :
@fun_map _ _ (sigma_structure G f) _ F x ≈ ⟨_, fun_map F (unify f x i hi)⟩ :=
fun_map_unify_equiv G f F x (classical.some (fintype.bdd_above_range (λ a, (x a).1))) i _ hi
lemma rel_map_equiv_unify {n : ℕ} (R : L.relations n) (x : (fin n) → (Σ i, G i)) (i : ι)
(hi : i ∈ upper_bounds (range (sigma.fst ∘ x))) :
@rel_map _ _ (sigma_structure G f) _ R x = rel_map R (unify f x i hi) :=
rel_map_unify_equiv G f R x (classical.some (fintype.bdd_above_range (λ a, (x a).1))) i _ hi
/-- The direct limit `setoid` respects the structure `sigma_structure`, so quotienting by it
gives rise to a valid structure. -/
noncomputable instance prestructure : L.prestructure (direct_limit.setoid G f) :=
{ to_structure := sigma_structure G f,
fun_equiv := λ n F x y xy, begin
obtain ⟨i, hx, hy, h⟩ := exists_unify_eq G f xy,
refine setoid.trans (fun_map_equiv_unify G f F x i hx)
(setoid.trans _ (setoid.symm (fun_map_equiv_unify G f F y i hy))),
rw h,
end,
rel_equiv := λ n R x y xy, begin
obtain ⟨i, hx, hy, h⟩ := exists_unify_eq G f xy,
refine trans (rel_map_equiv_unify G f R x i hx)
(trans _ (symm (rel_map_equiv_unify G f R y i hy))),
rw h,
end }
/-- The `L.Structure` on a direct limit of `L.Structure`s. -/
noncomputable instance Structure : L.Structure (direct_limit G f) := language.quotient_structure
@[simp] lemma fun_map_quotient_mk_sigma_mk {n : ℕ} {F : L.functions n} {i : ι} {x : fin n → G i} :
fun_map F (λ a, (⟦⟨i, x a⟩⟧ : direct_limit G f)) = ⟦⟨i, fun_map F x⟩⟧ :=
begin
simp only [function.comp_app, fun_map_quotient_mk, quotient.eq],
obtain ⟨k, ik, jk⟩ := directed_of (≤) i (classical.some (fintype.bdd_above_range
(λ (a : fin n), i))),
refine ⟨k, jk, ik, _⟩,
simp only [embedding.map_fun, comp_unify],
refl
end
@[simp] lemma rel_map_quotient_mk_sigma_mk {n : ℕ} {R : L.relations n} {i : ι} {x : fin n → G i} :
rel_map R (λ a, (⟦⟨i, x a⟩⟧ : direct_limit G f)) = rel_map R x :=
begin
rw [rel_map_quotient_mk],
obtain ⟨k, ik, jk⟩ := directed_of (≤) i (classical.some (fintype.bdd_above_range
(λ (a : fin n), i))),
rw [rel_map_equiv_unify G f R (λ a, ⟨i, x a⟩) i, unify_sigma_mk_self],
end
lemma exists_quotient_mk_sigma_mk_eq {α : Type*} [fintype α] (x : α → direct_limit G f) :
∃ (i : ι) (y : α → G i), x = quotient.mk ∘ (sigma.mk i) ∘ y :=
begin
obtain ⟨i, hi⟩ := fintype.bdd_above_range (λ a, (x a).out.1),
refine ⟨i, unify f (quotient.out ∘ x) i hi, _⟩,
ext a,
rw [quotient.eq_mk_iff_out, function.comp_app, unify, equiv_iff G f _],
{ simp only [directed_system.map_self] },
{ refl }
end
variables (L ι)
/-- The canonical map from a component to the direct limit. -/
def of (i : ι) : G i ↪[L] direct_limit G f :=
{ to_fun := quotient.mk ∘ sigma.mk i,
inj' := λ x y h, begin
simp only [quotient.eq] at h,
obtain ⟨j, h1, h2, h3⟩ := h,
exact (f i j h1).injective h3,
end }
variables {L ι G f}
@[simp] lemma of_apply {i : ι} {x : G i} : of L ι G f i x = ⟦⟨i, x⟩⟧ := rfl
@[simp] lemma of_f {i j : ι} {hij : i ≤ j} {x : G i} :
(of L ι G f j (f i j hij x)) = of L ι G f i x :=
begin
simp only [of_apply, quotient.eq],
refine setoid.symm ⟨j, hij, refl j, _⟩,
simp only [directed_system.map_self],
end
/-- Every element of the direct limit corresponds to some element in
some component of the directed system. -/
theorem exists_of (z : direct_limit G f) :
∃ i x, of L ι G f i x = z :=
⟨z.out.1, z.out.2, by simp⟩
@[elab_as_eliminator]
protected theorem induction_on {C : direct_limit G f → Prop}
(z : direct_limit G f)
(ih : ∀ i x, C (of L ι G f i x)) : C z :=
let ⟨i, x, h⟩ := exists_of z in h ▸ ih i x
variables {P : Type u₁} [L.Structure P] (g : Π i, G i ↪[L] P)
variables (Hg : ∀ i j hij x, g j (f i j hij x) = g i x)
include Hg
variables (L ι G f)
/-- The universal property of the direct limit: maps from the components to another module
that respect the directed system structure (i.e. make some diagram commute) give rise
to a unique map out of the direct limit. -/
def lift : direct_limit G f ↪[L] P :=
{ to_fun := quotient.lift (λ (x : Σ i, G i), (g x.1) x.2) (λ x y xy, begin
simp only,
obtain ⟨i, hx, hy⟩ := directed_of (≤) x.1 y.1,
rw [← Hg x.1 i hx, ← Hg y.1 i hy],
exact congr_arg _ ((equiv_iff _ _ _ _).1 xy),
end),
inj' := λ x y xy, begin
rw [← quotient.out_eq x, ← quotient.out_eq y, quotient.lift_mk, quotient.lift_mk] at xy,
obtain ⟨i, hx, hy⟩ := directed_of (≤) x.out.1 y.out.1,
rw [← Hg x.out.1 i hx, ← Hg y.out.1 i hy] at xy,
rw [← quotient.out_eq x, ← quotient.out_eq y, quotient.eq, equiv_iff G f hx hy],
exact (g i).injective xy,
end,
map_fun' := λ n F x, begin
obtain ⟨i, y, rfl⟩ := exists_quotient_mk_sigma_mk_eq G f x,
rw [fun_map_quotient_mk_sigma_mk, ← function.comp.assoc, quotient.lift_comp_mk],
simp only [quotient.lift_mk, embedding.map_fun],
end,
map_rel' := λ n R x, begin
obtain ⟨i, y, rfl⟩ := exists_quotient_mk_sigma_mk_eq G f x,
rw [rel_map_quotient_mk_sigma_mk G f, ← (g i).map_rel R y,
← function.comp.assoc, quotient.lift_comp_mk],
end }
variables {L ι G f}
omit Hg
@[simp] lemma lift_quotient_mk_sigma_mk {i} (x : G i) :
lift L ι G f g Hg (⟦⟨i, x⟩⟧) = (g i) x :=
begin
change (lift L ι G f g Hg).to_fun (⟦⟨i, x⟩⟧) = _,
simp only [lift, quotient.lift_mk],
end
lemma lift_of {i} (x : G i) : lift L ι G f g Hg (of L ι G f i x) = g i x :=
by simp
theorem lift_unique (F : direct_limit G f ↪[L] P) (x) :
F x = lift L ι G f (λ i, F.comp $ of L ι G f i)
(λ i j hij x, by rw [F.comp_apply, F.comp_apply, of_f]) x :=
direct_limit.induction_on x $ λ i x, by rw lift_of; refl
/-- The direct limit of countably many countably generated structures is countably generated. -/
theorem cg {ι : Type*} [encodable ι] [preorder ι] [is_directed ι (≤)] [nonempty ι]
{G : ι → Type w} [Π i, L.Structure (G i)]
(f : Π i j, i ≤ j → G i ↪[L] G j) (h : ∀ i, Structure.cg L (G i))
[directed_system G (λ i j h, f i j h)] :
Structure.cg L (direct_limit G f) :=
begin
refine ⟨⟨⋃ i, direct_limit.of L ι G f i '' (classical.some (h i).out), _, _⟩⟩,
{ exact set.countable_Union (λ i, set.countable.image (classical.some_spec (h i).out).1 _) },
{ rw [eq_top_iff, substructure.closure_Union],
simp_rw [← embedding.coe_to_hom, substructure.closure_image],
rw le_supr_iff,
intros S hS x hx,
let out := @quotient.out _ (direct_limit.setoid G f),
refine hS (out x).1 ⟨(out x).2, _, _⟩,
{ rw [(classical.some_spec (h (out x).1).out).2],
simp only [substructure.coe_top] },
{ simp only [embedding.coe_to_hom, direct_limit.of_apply, sigma.eta, quotient.out_eq] } }
end
instance cg' {ι : Type*} [encodable ι] [preorder ι] [is_directed ι (≤)] [nonempty ι]
{G : ι → Type w} [Π i, L.Structure (G i)]
(f : Π i j, i ≤ j → G i ↪[L] G j) [h : ∀ i, Structure.cg L (G i)]
[directed_system G (λ i j h, f i j h)] :
Structure.cg L (direct_limit G f) :=
cg f h
end direct_limit
end language
end first_order
|
a61f73c1437c8d5b32e593983cbaab6add4bf354 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/category_theory/products/bifunctor.lean | 805e65ffc4b02e7faab316bd8e6cbb17bcaa7166 | [
"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 | 1,819 | lean | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison
-/
import category_theory.products.basic
open category_theory
namespace category_theory.bifunctor
universes v₁ v₂ v₃ u₁ u₂ u₃
variables {C : Type u₁} {D : Type u₂} {E : Type u₃}
variables [category.{v₁} C] [category.{v₂} D] [category.{v₃} E]
@[simp] lemma map_id (F : (C × D) ⥤ E) (X : C) (Y : D) :
F.map ((𝟙 X, 𝟙 Y) : (X, Y) ⟶ (X, Y)) = 𝟙 (F.obj (X, Y)) :=
F.map_id (X, Y)
@[simp] lemma map_id_comp (F : (C × D) ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((𝟙 W, f ≫ g) : (W, X) ⟶ (W, Z)) =
F.map ((𝟙 W, f) : (W, X) ⟶ (W, Y)) ≫ F.map ((𝟙 W, g) : (W, Y) ⟶ (W, Z)) :=
by rw [←functor.map_comp,prod_comp,category.comp_id]
@[simp] lemma map_comp_id (F : (C × D) ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((f ≫ g, 𝟙 W) : (X, W) ⟶ (Z, W)) =
F.map ((f, 𝟙 W) : (X, W) ⟶ (Y, W)) ≫ F.map ((g, 𝟙 W) : (Y, W) ⟶ (Z, W)) :=
by rw [←functor.map_comp,prod_comp,category.comp_id]
@[simp] lemma diagonal (F : (C × D) ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((𝟙 X, g) : (X, Y) ⟶ (X, Y')) ≫ F.map ((f, 𝟙 Y') : (X, Y') ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) :=
by rw [←functor.map_comp, prod_comp, category.id_comp, category.comp_id]
@[simp] lemma diagonal' (F : (C × D) ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((f, 𝟙 Y) : (X, Y) ⟶ (X', Y)) ≫ F.map ((𝟙 X', g) : (X', Y) ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) :=
by rw [←functor.map_comp, prod_comp, category.id_comp, category.comp_id]
end category_theory.bifunctor
|
f097ed3e78336ef622262611686f478f48b0ec82 | 94e33a31faa76775069b071adea97e86e218a8ee | /src/order/filter/small_sets.lean | 96269fff4611391672d1b47368052c063cac9b6a | [
"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 | 5,257 | lean | /-
Copyright (c) 2022 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Floris van Doorn, Yury Kudryashov
-/
import order.filter.lift
import order.filter.at_top_bot
/-!
# The filter of small sets
This file defines the filter of small sets w.r.t. a filter `f`, which is the largest filter
containing all powersets of members of `f`.
`g` converges to `f.small_sets` if for all `s ∈ f`, eventually we have `g x ⊆ s`.
An example usage is that if `f : ι → E → ℝ` is a family of nonnegative functions with integral 1,
then saying that `λ i, support (f i)` tendsto `(𝓝 0).small_sets` is a way of saying that
`f` tends to the Dirac delta distribution.
-/
open_locale filter
open filter set
variables {α β : Type*} {ι : Sort*}
namespace filter
variables {l l' la : filter α} {lb : filter β}
/-- The filter `l.small_sets` is the largest filter containing all powersets of members of `l`. -/
def small_sets (l : filter α) : filter (set α) := l.lift' powerset
lemma small_sets_eq_generate {f : filter α} : f.small_sets = generate (powerset '' f.sets) :=
by { simp_rw [generate_eq_binfi, small_sets, infi_image], refl }
lemma has_basis.small_sets {p : ι → Prop} {s : ι → set α}
(h : has_basis l p s) : has_basis l.small_sets p (λ i, 𝒫 (s i)) :=
h.lift' monotone_powerset
lemma has_basis_small_sets (l : filter α) :
has_basis l.small_sets (λ t : set α, t ∈ l) powerset :=
l.basis_sets.small_sets
/-- `g` converges to `f.small_sets` if for all `s ∈ f`, eventually we have `g x ⊆ s`. -/
lemma tendsto_small_sets_iff {f : α → set β} :
tendsto f la lb.small_sets ↔ ∀ t ∈ lb, ∀ᶠ x in la, f x ⊆ t :=
(has_basis_small_sets lb).tendsto_right_iff
lemma eventually_small_sets {p : set α → Prop} :
(∀ᶠ s in l.small_sets, p s) ↔ ∃ s ∈ l, ∀ t ⊆ s, p t :=
eventually_lift'_iff monotone_powerset
lemma eventually_small_sets' {p : set α → Prop} (hp : ∀ ⦃s t⦄, s ⊆ t → p t → p s) :
(∀ᶠ s in l.small_sets, p s) ↔ ∃ s ∈ l, p s :=
eventually_small_sets.trans $ exists₂_congr $ λ s hsf,
⟨λ H, H s subset.rfl, λ hs t ht, hp ht hs⟩
lemma frequently_small_sets {p : set α → Prop} :
(∃ᶠ s in l.small_sets, p s) ↔ ∀ t ∈ l, ∃ s ⊆ t, p s :=
l.has_basis_small_sets.frequently_iff
lemma frequently_small_sets_mem (l : filter α) : ∃ᶠ s in l.small_sets, s ∈ l :=
frequently_small_sets.2 $ λ t ht, ⟨t, subset.rfl, ht⟩
lemma has_antitone_basis.tendsto_small_sets {ι} [preorder ι] {s : ι → set α}
(hl : l.has_antitone_basis s) : tendsto s at_top l.small_sets :=
tendsto_small_sets_iff.2 $ λ t ht, hl.eventually_subset ht
@[mono] lemma monotone_small_sets : monotone (@small_sets α) :=
monotone_lift' monotone_id monotone_const
@[simp] lemma small_sets_bot : (⊥ : filter α).small_sets = pure ∅ :=
by rw [small_sets, lift'_bot monotone_powerset, powerset_empty, principal_singleton]
@[simp] lemma small_sets_top : (⊤ : filter α).small_sets = ⊤ :=
by rw [small_sets, lift'_top, powerset_univ, principal_univ]
@[simp] lemma small_sets_principal (s : set α) : (𝓟 s).small_sets = 𝓟(𝒫 s) :=
lift'_principal monotone_powerset
lemma small_sets_comap (l : filter β) (f : α → β) :
(comap f l).small_sets = l.lift' (powerset ∘ preimage f) :=
comap_lift'_eq2 monotone_powerset
lemma comap_small_sets (l : filter β) (f : α → set β) :
comap f l.small_sets = l.lift' (preimage f ∘ powerset) :=
comap_lift'_eq
lemma small_sets_infi {f : ι → filter α} :
(infi f).small_sets = (⨅ i, (f i).small_sets) :=
lift'_infi_of_map_univ powerset_inter powerset_univ
lemma small_sets_inf (l₁ l₂ : filter α) :
(l₁ ⊓ l₂).small_sets = l₁.small_sets ⊓ l₂.small_sets :=
lift'_inf _ _ powerset_inter
instance small_sets_ne_bot (l : filter α) : ne_bot l.small_sets :=
(lift'_ne_bot_iff monotone_powerset).2 $ λ _ _, powerset_nonempty
lemma tendsto.small_sets_mono {s t : α → set β}
(ht : tendsto t la lb.small_sets) (hst : ∀ᶠ x in la, s x ⊆ t x) :
tendsto s la lb.small_sets :=
begin
rw [tendsto_small_sets_iff] at ht ⊢,
exact λ u hu, (ht u hu).mp (hst.mono $ λ a hst ht, subset.trans hst ht)
end
@[simp] lemma eventually_small_sets_eventually {p : α → Prop} :
(∀ᶠ s in l.small_sets, ∀ᶠ x in l', x ∈ s → p x) ↔ ∀ᶠ x in l ⊓ l', p x :=
calc _ ↔ ∃ s ∈ l, ∀ᶠ x in l', x ∈ s → p x :
eventually_small_sets' $ λ s t hst ht, ht.mono $ λ x hx hs, hx (hst hs)
... ↔ ∃ (s ∈ l) (t ∈ l'), ∀ x, x ∈ t → x ∈ s → p x :
by simp only [eventually_iff_exists_mem]
... ↔ ∀ᶠ x in l ⊓ l', p x : by simp only [eventually_inf, and_comm, mem_inter_iff, ← and_imp]
@[simp] lemma eventually_small_sets_forall {p : α → Prop} :
(∀ᶠ s in l.small_sets, ∀ x ∈ s, p x) ↔ ∀ᶠ x in l, p x :=
by simpa only [inf_top_eq, eventually_top] using @eventually_small_sets_eventually α l ⊤ p
alias eventually_small_sets_forall ↔ eventually.of_small_sets eventually.small_sets
@[simp] lemma eventually_small_sets_subset {s : set α} :
(∀ᶠ t in l.small_sets, t ⊆ s) ↔ s ∈ l :=
eventually_small_sets_forall
end filter
|
3d1df3ef69d78c2db35db786aa3a4e0fd5f9edc8 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/analysis/complex/polynomial.lean | 7a7bd7cbe7eaa1a04a253e397000bc2003495490 | [
"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 | 5,622 | lean | /-
Copyright (c) 2019 Chris Hughes All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import analysis.special_functions.pow
import field_theory.algebraic_closure
import topology.algebra.polynomial
/-!
# The fundamental theorem of algebra
This file proves that every nonconstant complex polynomial has a root.
As a consequence, the complex numbers are algebraically closed.
-/
open complex polynomial metric filter is_absolute_value set
open_locale classical
namespace complex
/- The following proof uses the method given at
<https://ncatlab.org/nlab/show/fundamental+theorem+of+algebra#classical_fta_via_advanced_calculus>
-/
/-- The fundamental theorem of algebra. Every non constant complex polynomial
has a root. -/
lemma exists_root {f : polynomial ℂ} (hf : 0 < degree f) : ∃ z : ℂ, is_root f z :=
let ⟨z₀, hz₀⟩ := f.exists_forall_norm_le in
exists.intro z₀ $ classical.by_contradiction $ λ hf0,
have hfX : f - C (f.eval z₀) ≠ 0,
from mt sub_eq_zero.1 (λ h, not_le_of_gt hf (h.symm ▸ degree_C_le)),
let n := root_multiplicity z₀ (f - C (f.eval z₀)) in
let g := (f - C (f.eval z₀)) /ₘ ((X - C z₀) ^ n) in
have hg0 : g.eval z₀ ≠ 0, from eval_div_by_monic_pow_root_multiplicity_ne_zero _ hfX,
have hg : g * (X - C z₀) ^ n = f - C (f.eval z₀),
from div_by_monic_mul_pow_root_multiplicity_eq _ _,
have hn0 : 0 < n, from nat.pos_of_ne_zero $ λ hn0, by simpa [g, hn0] using hg0,
let ⟨δ', hδ'₁, hδ'₂⟩ := continuous_iff.1 (polynomial.continuous g) z₀
((g.eval z₀).abs) (complex.abs_pos.2 hg0) in
let δ := min (min (δ' / 2) 1) (((f.eval z₀).abs / (g.eval z₀).abs) / 2) in
have hf0' : 0 < (f.eval z₀).abs, from complex.abs_pos.2 hf0,
have hg0' : 0 < abs (eval z₀ g), from complex.abs_pos.2 hg0,
have hfg0 : 0 < (f.eval z₀).abs / abs (eval z₀ g), from div_pos hf0' hg0',
have hδ0 : 0 < δ, from lt_min (lt_min (half_pos hδ'₁) (by norm_num)) (half_pos hfg0),
have hδ : ∀ z : ℂ, abs (z - z₀) = δ → abs (g.eval z - g.eval z₀) < (g.eval z₀).abs,
from λ z hz, hδ'₂ z (by rw [complex.dist_eq, hz];
exact ((min_le_left _ _).trans (min_le_left _ _)).trans_lt (half_lt_self hδ'₁)),
have hδ1 : δ ≤ 1, from le_trans (min_le_left _ _) (min_le_right _ _),
let F : polynomial ℂ := C (f.eval z₀) + C (g.eval z₀) * (X - C z₀) ^ n in
let z' := (-f.eval z₀ * (g.eval z₀).abs * δ ^ n /
((f.eval z₀).abs * g.eval z₀)) ^ (n⁻¹ : ℂ) + z₀ in
have hF₁ : F.eval z' = f.eval z₀ - f.eval z₀ * (g.eval z₀).abs * δ ^ n / (f.eval z₀).abs,
by simp only [F, cpow_nat_inv_pow _ hn0, div_eq_mul_inv, eval_pow, mul_assoc,
mul_comm (g.eval z₀), mul_left_comm (g.eval z₀), mul_left_comm (g.eval z₀)⁻¹, mul_inv',
inv_mul_cancel hg0, eval_C, eval_add, eval_neg, sub_eq_add_neg, eval_mul, eval_X,
add_neg_cancel_right, neg_mul_eq_neg_mul_symm, mul_one, div_eq_mul_inv];
simp only [mul_comm, mul_left_comm, mul_assoc],
have hδs : (g.eval z₀).abs * δ ^ n / (f.eval z₀).abs < 1,
from (div_lt_one hf0').2 $ (lt_div_iff' hg0').1 $
calc δ ^ n ≤ δ ^ 1 : pow_le_pow_of_le_one (le_of_lt hδ0) hδ1 hn0
... = δ : pow_one _
... ≤ ((f.eval z₀).abs / (g.eval z₀).abs) / 2 : min_le_right _ _
... < _ : half_lt_self (div_pos hf0' hg0'),
have hF₂ : (F.eval z').abs = (f.eval z₀).abs - (g.eval z₀).abs * δ ^ n,
from calc (F.eval z').abs = (f.eval z₀ - f.eval z₀ * (g.eval z₀).abs
* δ ^ n / (f.eval z₀).abs).abs : congr_arg abs hF₁
... = abs (f.eval z₀) * complex.abs (1 - (g.eval z₀).abs * δ ^ n /
(f.eval z₀).abs : ℝ) : by rw [← complex.abs_mul];
exact congr_arg complex.abs
(by simp [mul_add, add_mul, mul_assoc, div_eq_mul_inv, sub_eq_add_neg])
... = _ : by rw [complex.abs_of_nonneg (sub_nonneg.2 (le_of_lt hδs)),
mul_sub, mul_div_cancel' _ (ne.symm (ne_of_lt hf0')), mul_one],
have hef0 : abs (eval z₀ g) * (eval z₀ f).abs ≠ 0,
from mul_ne_zero (mt complex.abs_eq_zero.1 hg0) (mt complex.abs_eq_zero.1 hf0),
have hz'z₀ : abs (z' - z₀) = δ,
by simp [z', mul_assoc, mul_left_comm _ (_ ^ n), mul_comm _ (_ ^ n),
mul_comm (eval z₀ f).abs, _root_.mul_div_cancel _ hef0, of_real_mul,
neg_mul_eq_neg_mul_symm, neg_div, is_absolute_value.abv_pow complex.abs,
complex.abs_of_nonneg (le_of_lt hδ0), real.pow_nat_rpow_nat_inv (le_of_lt hδ0) hn0],
have hF₃ : (f.eval z' - F.eval z').abs < (g.eval z₀).abs * δ ^ n,
from calc (f.eval z' - F.eval z').abs
= (g.eval z' - g.eval z₀).abs * (z' - z₀).abs ^ n :
by rw [← eq_sub_iff_add_eq.1 hg, ← is_absolute_value.abv_pow complex.abs,
← complex.abs_mul, sub_mul];
simp [F, eval_pow, eval_add, eval_mul, eval_sub, eval_C, eval_X, eval_neg, add_sub_cancel,
sub_eq_add_neg, add_assoc]
... = (g.eval z' - g.eval z₀).abs * δ ^ n : by rw hz'z₀
... < _ : (mul_lt_mul_right (pow_pos hδ0 _)).2 (hδ _ hz'z₀),
lt_irrefl (f.eval z₀).abs $
calc (f.eval z₀).abs ≤ (f.eval z').abs : hz₀ _
... = (F.eval z' + (f.eval z' - F.eval z')).abs : by simp
... ≤ (F.eval z').abs + (f.eval z' - F.eval z').abs : complex.abs_add _ _
... < (f.eval z₀).abs - (g.eval z₀).abs * δ ^ n + (g.eval z₀).abs * δ ^ n :
add_lt_add_of_le_of_lt (by rw hF₂) hF₃
... = (f.eval z₀).abs : sub_add_cancel _ _
instance is_alg_closed : is_alg_closed ℂ :=
is_alg_closed.of_exists_root _ $ λ p _ hp, complex.exists_root $ degree_pos_of_irreducible hp
end complex
|
218c930ab8b3348fb0c34de1926a293b0f88e55b | 4727251e0cd73359b15b664c3170e5d754078599 | /archive/examples/prop_encodable.lean | 64da2837b4f0bad5499490f2f3bb90a74516be79 | [
"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 | 2,982 | lean | /-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import data.W.basic
/-!
# W types
The file `data/W.lean` shows that if `α` is an an encodable fintype and for every `a : α`,
`β a` is encodable, then `W β` is encodable.
As an example of how this can be used, we show that the type of propositional formulas with
variables labeled from an encodable type is encodable.
The strategy is to define a type of labels corresponding to the constructors.
From the definition (using `sum`, `unit`, and an encodable type), Lean can infer
that it is encodable. We then define a map from propositional formulas to the
corresponding `Wfin` type, and show that map has a left inverse.
We mark the auxiliary constructions `private`, since their only purpose is to
show encodability.
-/
/-- Propositional formulas with labels from `α`. -/
inductive prop_form (α : Type*)
| var : α → prop_form
| not : prop_form → prop_form
| and : prop_form → prop_form → prop_form
| or : prop_form → prop_form → prop_form
/-!
The next three functions make it easier to construct functions from a small
`fin`.
-/
section
variable {α : Type*}
/-- the trivial function out of `fin 0`. -/
def mk_fn0 : fin 0 → α
| ⟨_, h⟩ := absurd h dec_trivial
/-- defines a function out of `fin 1` -/
def mk_fn1 (t : α) : fin 1 → α
| ⟨0, _⟩ := t
| ⟨n+1, h⟩ := absurd h dec_trivial
/-- defines a function out of `fin 2` -/
def mk_fn2 (s t : α) : fin 2 → α
| ⟨0, _⟩ := s
| ⟨1, _⟩ := t
| ⟨n+2, h⟩ := absurd h dec_trivial
attribute [simp] mk_fn0 mk_fn1 mk_fn2
end
namespace prop_form
private def constructors (α : Type*) := α ⊕ unit ⊕ unit ⊕ unit
local notation `cvar` a := sum.inl a
local notation `cnot` := sum.inr (sum.inl unit.star)
local notation `cand` := sum.inr (sum.inr (sum.inr unit.star))
local notation `cor` := sum.inr (sum.inr (sum.inl unit.star))
@[simp]
private def arity (α : Type*) : constructors α → nat
| (cvar a) := 0
| cnot := 1
| cand := 2
| cor := 2
variable {α : Type*}
private def f : prop_form α → W_type (λ i, fin (arity α i))
| (var a) := ⟨cvar a, mk_fn0⟩
| (not p) := ⟨cnot, mk_fn1 (f p)⟩
| (and p q) := ⟨cand, mk_fn2 (f p) (f q)⟩
| (or p q) := ⟨cor, mk_fn2 (f p) (f q)⟩
private def finv : W_type (λ i, fin (arity α i)) → prop_form α
| ⟨cvar a, fn⟩ := var a
| ⟨cnot, fn⟩ := not (finv (fn ⟨0, dec_trivial⟩))
| ⟨cand, fn⟩ := and (finv (fn ⟨0, dec_trivial⟩)) (finv (fn ⟨1, dec_trivial⟩))
| ⟨cor, fn⟩ := or (finv (fn ⟨0, dec_trivial⟩)) (finv (fn ⟨1, dec_trivial⟩))
instance [encodable α] : encodable (prop_form α) :=
begin
haveI : encodable (constructors α),
{ unfold constructors, apply_instance },
exact encodable.of_left_inverse f finv
(by { intro p, induction p; simp [f, finv, *] })
end
end prop_form
|
4023e20deba7d981dd1c1d61bc49ad778b710bd9 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/intNegSucc.lean | d9e9089290b572d762f5fb10c9af7cd1903e0878 | [
"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 | 570 | lean | def Int64Min : Int := -1*(Int.ofNat ((UInt64.size / 2)))
#eval Int.negSucc 9223372036854775807
#reduce Int.negSucc 9223372036854775807
#print "---"
#eval - Int.negSucc 9223372036854775807
#reduce - Int.negSucc 9223372036854775807
#print "---"
#eval Int.negSucc 9223372036854775807 == 0
#reduce Int.negSucc 9223372036854775807 == 0
#print "---"
#eval Int.negSucc 9223372036854775807 == Int64Min
#reduce Int.negSucc 9223372036854775807 == Int64Min
#print "---"
#eval (Int.negSucc 9223372036854775807) + 1 == 1
#reduce (Int.negSucc 9223372036854775807) + 1 == 1
|
886d1a7e190098ea0f07330934ccbddbd9b4d211 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/algebra/big_operators/pi.lean | 7ad8db101e32df8e28f6e07aaf03a6361ffbb8ad | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 3,658 | lean | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import algebra.ring.pi
import algebra.big_operators.basic
import data.fintype.basic
import algebra.group.prod
/-!
# Big operators for Pi Types
This file contains theorems relevant to big operators in binary and arbitrary product
of monoids and groups
-/
open_locale big_operators
namespace pi
@[to_additive]
lemma list_prod_apply {α : Type*} {β : α → Type*} [Πa, monoid (β a)] (a : α) (l : list (Πa, β a)) :
l.prod a = (l.map (λf:Πa, β a, f a)).prod :=
(monoid_hom.apply β a).map_list_prod _
@[to_additive]
lemma multiset_prod_apply {α : Type*} {β : α → Type*} [∀a, comm_monoid (β a)] (a : α)
(s : multiset (Πa, β a)) : s.prod a = (s.map (λf:Πa, β a, f a)).prod :=
(monoid_hom.apply β a).map_multiset_prod _
end pi
@[simp, to_additive]
lemma finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀a, comm_monoid (β a)] (a : α)
(s : finset γ) (g : γ → Πa, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=
(monoid_hom.apply β a).map_prod _ _
@[simp, to_additive]
lemma fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [fintype γ]
[∀a, comm_monoid (β a)] (a : α) (g : γ → Πa, β a) : (∏ c, g c) a = ∏ c, g c a :=
finset.prod_apply a finset.univ g
@[simp, to_additive]
lemma monoid_hom.finset_prod_apply {ι M N : Type*} [monoid M] [comm_monoid N] (f : ι → M →* N)
(s : finset ι) (a : M) :
(∏ i in s, f i) a = ∏ i in s, f i a :=
(monoid_hom.eval a).map_prod _ _
@[to_additive prod_mk_sum]
lemma prod_mk_prod {α β γ : Type*} [comm_monoid α] [comm_monoid β] (s : finset γ)
(f : γ → α) (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=
by haveI := classical.dec_eq γ; exact
finset.induction_on s rfl (by simp [prod.ext_iff] {contextual := tt})
section single
variables {I : Type*} [decidable_eq I] {Z : I → Type*}
variables [Π i, add_comm_monoid (Z i)]
-- As we only defined `single` into `add_monoid`, we only prove the `finset.sum` version here.
lemma finset.univ_sum_single [fintype I] (f : Π i, Z i) :
∑ i, pi.single i (f i) = f :=
begin
ext a,
rw [finset.sum_apply, finset.sum_eq_single a],
{ simp, },
{ intros b _ h, simp [h.symm], },
{ intro h, exfalso, simpa using h, },
end
@[ext]
lemma add_monoid_hom.functions_ext [fintype I] (G : Type*)
[add_comm_monoid G] (g h : (Π i, Z i) →+ G)
(w : ∀ (i : I) (x : Z i), g (pi.single i x) = h (pi.single i x)) : g = h :=
begin
ext k,
rw [←finset.univ_sum_single k, add_monoid_hom.map_sum, add_monoid_hom.map_sum],
apply finset.sum_congr rfl,
intros,
apply w,
end
end single
section ring_hom
open pi
variables {I : Type*} [decidable_eq I] {f : I → Type*}
variables [Π i, semiring (f i)]
-- we need `apply`+`convert` because Lean fails to unify different `add_monoid` instances
-- on `Π i, f i`
@[ext]
lemma ring_hom.functions_ext [fintype I] (G : Type*) [semiring G] (g h : (Π i, f i) →+* G)
(w : ∀ (i : I) (x : f i), g (single i x) = h (single i x)) : g = h :=
begin
apply ring_hom.coe_add_monoid_hom_injective,
convert add_monoid_hom.functions_ext _ _ _ _; assumption
end
end ring_hom
namespace prod
variables {α β γ : Type*} [comm_monoid α] [comm_monoid β] {s : finset γ} {f : γ → α × β}
@[to_additive]
lemma fst_prod : (∏ c in s, f c).1 = ∏ c in s, (f c).1 :=
(monoid_hom.fst α β).map_prod f s
@[to_additive]
lemma snd_prod : (∏ c in s, f c).2 = ∏ c in s, (f c).2 :=
(monoid_hom.snd α β).map_prod f s
end prod
|
d847ea1902ad7dfd4b716564805810bdce7328a1 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/algebra/module/ordered_auto.lean | 6b623ac69e89f7036513e423949e764af51d7262 | [] | 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 | 8,901 | 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 Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.algebra.module.pi
import Mathlib.algebra.ordered_pi
import Mathlib.algebra.module.prod
import Mathlib.algebra.ordered_field
import Mathlib.PostPort
universes u_1 u_2 l u_3
namespace Mathlib
/-!
# Ordered semimodules
In this file we define
* `ordered_semimodule R M` : an ordered additive commutative monoid `M` is an `ordered_semimodule`
over an `ordered_semiring` `R` if the scalar product respects the order relation on the
monoid and on the ring. There is a correspondence between this structure and convex cones,
which is proven in `analysis/convex/cone.lean`.
## Implementation notes
* We choose to define `ordered_semimodule` as a `Prop`-valued mixin, so that it can be
used for both modules and algebras
(the axioms for an "ordered algebra" are exactly that the algebra is ordered as a module).
* To get ordered modules and ordered vector spaces, it suffices to the replace the
`order_add_comm_monoid` and the `ordered_semiring` as desired.
## References
* https://en.wikipedia.org/wiki/Ordered_vector_space
## Tags
ordered semimodule, ordered module, ordered vector space
-/
/--
An ordered semimodule is an ordered additive commutative monoid
with a partial order in which the scalar multiplication is compatible with the order.
-/
class ordered_semimodule (R : Type u_1) (M : Type u_2) [ordered_semiring R]
[ordered_add_comm_monoid M] [semimodule R M]
where
smul_lt_smul_of_pos : ∀ {a b : M} {c : R}, a < b → 0 < c → c • a < c • b
lt_of_smul_lt_smul_of_pos : ∀ {a b : M} {c : R}, c • a < c • b → 0 < c → a < b
theorem smul_lt_smul_of_pos {R : Type u_1} {M : Type u_2} [ordered_semiring R]
[ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R} :
a < b → 0 < c → c • a < c • b :=
ordered_semimodule.smul_lt_smul_of_pos
theorem smul_le_smul_of_nonneg {R : Type u_1} {M : Type u_2} [ordered_semiring R]
[ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R}
(h₁ : a ≤ b) (h₂ : 0 ≤ c) : c • a ≤ c • b :=
sorry
theorem eq_of_smul_eq_smul_of_pos_of_le {R : Type u_1} {M : Type u_2} [ordered_semiring R]
[ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R}
(h₁ : c • a = c • b) (hc : 0 < c) (hle : a ≤ b) : a = b :=
or.resolve_left (has_le.le.lt_or_eq hle)
fun (hlt : a < b) => has_lt.lt.ne (smul_lt_smul_of_pos hlt hc) h₁
theorem lt_of_smul_lt_smul_of_nonneg {R : Type u_1} {M : Type u_2} [ordered_semiring R]
[ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R}
(h : c • a < c • b) (hc : 0 ≤ c) : a < b :=
sorry
theorem smul_lt_smul_iff_of_pos {R : Type u_1} {M : Type u_2} [ordered_semiring R]
[ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R}
(hc : 0 < c) : c • a < c • b ↔ a < b :=
{ mp := fun (h : c • a < c • b) => lt_of_smul_lt_smul_of_nonneg h (has_lt.lt.le hc),
mpr := fun (h : a < b) => smul_lt_smul_of_pos h hc }
theorem smul_pos_iff_of_pos {R : Type u_1} {M : Type u_2} [ordered_semiring R]
[ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {c : R}
(hc : 0 < c) : 0 < c • a ↔ 0 < a :=
iff.trans
(eq.mpr (id (Eq._oldrec (Eq.refl (0 < c • a ↔ c • 0 < c • a)) (smul_zero c)))
(iff.refl (0 < c • a)))
(smul_lt_smul_iff_of_pos hc)
/-- If `R` is a linear ordered semifield, then it suffices to verify only the first axiom of
`ordered_semimodule`. Moreover, it suffices to verify that `a < b` and `0 < c` imply
`c • a ≤ c • b`. We have no semifields in `mathlib`, so we use the assumption `∀ c ≠ 0, is_unit c`
instead. -/
theorem ordered_semimodule.mk'' {R : Type u_1} {M : Type u_2} [linear_ordered_semiring R]
[ordered_add_comm_monoid M] [semimodule R M] (hR : ∀ {c : R}, c ≠ 0 → is_unit c)
(hlt : ∀ {a b : M} {c : R}, a < b → 0 < c → c • a ≤ c • b) : ordered_semimodule R M :=
sorry
/-- If `R` is a linear ordered field, then it suffices to verify only the first axiom of
`ordered_semimodule`. -/
theorem ordered_semimodule.mk' {k : Type u_1} {M : Type u_2} [linear_ordered_field k]
[ordered_add_comm_monoid M] [semimodule k M]
(hlt : ∀ {a b : M} {c : k}, a < b → 0 < c → c • a ≤ c • b) : ordered_semimodule k M :=
ordered_semimodule.mk'' (fun (c : k) (hc : c ≠ 0) => is_unit.mk0 c hc) hlt
protected instance linear_ordered_semiring.to_ordered_semimodule {R : Type u_1}
[linear_ordered_semiring R] : ordered_semimodule R R :=
ordered_semimodule.mk ordered_semiring.mul_lt_mul_of_pos_left
fun (_x _x_1 _x_2 : R) (h : _x_2 • _x < _x_2 • _x_1) (hc : 0 < _x_2) =>
lt_of_mul_lt_mul_left h (has_lt.lt.le hc)
theorem smul_le_smul_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k]
[ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}
(hc : 0 < c) : c • a ≤ c • b ↔ a ≤ b :=
sorry
theorem smul_le_smul_iff_of_neg {k : Type u_1} {M : Type u_2} [linear_ordered_field k]
[ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}
(hc : c < 0) : c • a ≤ c • b ↔ b ≤ a :=
sorry
theorem smul_lt_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k]
[ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}
(hc : 0 < c) : c • a < b ↔ a < c⁻¹ • b :=
sorry
theorem smul_le_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k]
[ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}
(hc : 0 < c) : c • a ≤ b ↔ a ≤ c⁻¹ • b :=
sorry
theorem le_smul_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k]
[ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}
(hc : 0 < c) : a ≤ c • b ↔ c⁻¹ • a ≤ b :=
sorry
protected instance prod.ordered_semimodule {k : Type u_1} {M : Type u_2} {N : Type u_3}
[linear_ordered_field k] [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M]
[ordered_add_comm_group N] [semimodule k N] [ordered_semimodule k N] :
ordered_semimodule k (M × N) :=
ordered_semimodule.mk'
fun (v u : M × N) (c : k) (h : v < u) (hc : 0 < c) =>
{ left := smul_le_smul_of_nonneg (and.left (and.left h)) (has_lt.lt.le hc),
right := smul_le_smul_of_nonneg (and.right (and.left h)) (has_lt.lt.le hc) }
protected instance pi.ordered_semimodule {k : Type u_1} [linear_ordered_field k] {ι : Type u_2}
{M : ι → Type u_3} [(i : ι) → ordered_add_comm_group (M i)] [(i : ι) → semimodule k (M i)]
[∀ (i : ι), ordered_semimodule k (M i)] : ordered_semimodule k ((i : ι) → M i) :=
ordered_semimodule.mk'
fun (v u : (i : ι) → M i) (c : k) (h : v < u) (hc : 0 < c) (i : ι) =>
id (smul_le_smul_of_nonneg (has_lt.lt.le h i) (has_lt.lt.le hc))
-- Sometimes Lean fails to apply the dependent version to non-dependent functions,
-- so we define another instance
protected instance pi.ordered_semimodule' {k : Type u_1} [linear_ordered_field k] {ι : Type u_2}
{M : Type u_3} [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] :
ordered_semimodule k (ι → M) :=
pi.ordered_semimodule
protected instance order_dual.has_scalar {R : Type u_1} {M : Type u_2} [semiring R]
[ordered_add_comm_monoid M] [semimodule R M] : has_scalar R (order_dual M) :=
has_scalar.mk has_scalar.smul
protected instance order_dual.mul_action {R : Type u_1} {M : Type u_2} [semiring R]
[ordered_add_comm_monoid M] [semimodule R M] : mul_action R (order_dual M) :=
mul_action.mk sorry sorry
protected instance order_dual.distrib_mul_action {R : Type u_1} {M : Type u_2} [semiring R]
[ordered_add_comm_monoid M] [semimodule R M] : distrib_mul_action R (order_dual M) :=
distrib_mul_action.mk sorry sorry
protected instance order_dual.semimodule {R : Type u_1} {M : Type u_2} [semiring R]
[ordered_add_comm_monoid M] [semimodule R M] : semimodule R (order_dual M) :=
semimodule.mk sorry sorry
protected instance order_dual.ordered_semimodule {R : Type u_1} {M : Type u_2} [ordered_semiring R]
[ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] :
ordered_semimodule R (order_dual M) :=
ordered_semimodule.mk (fun (a b : order_dual M) => ordered_semimodule.smul_lt_smul_of_pos)
fun (a b : order_dual M) => ordered_semimodule.lt_of_smul_lt_smul_of_pos
end Mathlib |
35b773d1b6a15596cb53acfd489c0d07b63422c3 | 9cba98daa30c0804090f963f9024147a50292fa0 | /old/src/classical_geometry_test.lean | aee05ab64abc460025f1daaa13ce3259141b8135 | [] | no_license | kevinsullivan/phys | dcb192f7b3033797541b980f0b4a7e75d84cea1a | ebc2df3779d3605ff7a9b47eeda25c2a551e011f | refs/heads/master | 1,637,490,575,500 | 1,629,899,064,000 | 1,629,899,064,000 | 168,012,884 | 0 | 3 | null | 1,629,644,436,000 | 1,548,699,832,000 | Lean | UTF-8 | Lean | false | false | 107 | lean | import .classical_geometry
noncomputable def aGeom1DSpace : classicalGeometry := classicalGeometry.mk 0 1
|
364e80023a08e810b0e604e3a84b35ffe3338697 | 6094e25ea0b7699e642463b48e51b2ead6ddc23f | /tests/lean/unifier_bug.lean | af6f1136d45464c3e191ae5842e211a32ea3ba06 | [
"Apache-2.0"
] | permissive | gbaz/lean | a7835c4e3006fbbb079e8f8ffe18aacc45adebfb | a501c308be3acaa50a2c0610ce2e0d71becf8032 | refs/heads/master | 1,611,198,791,433 | 1,451,339,111,000 | 1,451,339,111,000 | 48,713,797 | 0 | 0 | null | 1,451,338,939,000 | 1,451,338,939,000 | null | UTF-8 | Lean | false | false | 200 | lean | import logic
theorem test {A B : Type} {a : A} {b : B} (H : a == b) :
eq.rec_on (heq.type_eq H) a = b
:=
-- Remark the error message should not occur in the token theorem
heq.rec_on H rfl
|
3edfe055151f41ed906a9d0bb75cc30c242605ab | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/topology/continuous_function/algebra.lean | 1fdde95a636b9637b3011350f3ac31de229bd547 | [
"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 | 33,360 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Nicolò Cavalleri
-/
import topology.algebra.module.basic
import topology.continuous_function.ordered
import topology.algebra.uniform_group
import topology.uniform_space.compact_convergence
import topology.algebra.star
import algebra.algebra.subalgebra.basic
import tactic.field_simp
/-!
# Algebraic structures over continuous functions
In this file we define instances of algebraic structures over the type `continuous_map α β`
(denoted `C(α, β)`) of **bundled** continuous maps from `α` to `β`. For example, `C(α, β)`
is a group when `β` is a group, a ring when `β` is a ring, etc.
For each type of algebraic structure, we also define an appropriate subobject of `α → β`
with carrier `{ f : α → β | continuous f }`. For example, when `β` is a group, a subgroup
`continuous_subgroup α β` of `α → β` is constructed with carrier `{ f : α → β | continuous f }`.
Note that, rather than using the derived algebraic structures on these subobjects
(for example, when `β` is a group, the derived group structure on `continuous_subgroup α β`),
one should use `C(α, β)` with the appropriate instance of the structure.
-/
local attribute [elab_simple] continuous.comp
namespace continuous_functions
variables {α : Type*} {β : Type*} [topological_space α] [topological_space β]
variables {f g : {f : α → β | continuous f }}
instance : has_coe_to_fun {f : α → β | continuous f} (λ _, α → β) := ⟨subtype.val⟩
end continuous_functions
namespace continuous_map
variables {α : Type*} {β : Type*} {γ : Type*}
variables [topological_space α] [topological_space β] [topological_space γ]
@[to_additive]
instance has_mul [has_mul β] [has_continuous_mul β] : has_mul C(α, β) :=
⟨λ f g, ⟨f * g, continuous_mul.comp (f.continuous.prod_mk g.continuous : _)⟩⟩
@[simp, norm_cast, to_additive]
lemma coe_mul [has_mul β] [has_continuous_mul β] (f g : C(α, β)) : ⇑(f * g) = f * g := rfl
@[simp, to_additive] lemma mul_comp [has_mul γ] [has_continuous_mul γ]
(f₁ f₂ : C(β, γ)) (g : C(α, β)) :
(f₁ * f₂).comp g = f₁.comp g * f₂.comp g :=
rfl
@[to_additive]
instance [has_one β] : has_one C(α, β) := ⟨const α 1⟩
@[simp, norm_cast, to_additive]
lemma coe_one [has_one β] : ⇑(1 : C(α, β)) = 1 := rfl
@[simp, to_additive] lemma one_comp [has_one γ] (g : C(α, β)) : (1 : C(β, γ)).comp g = 1 := rfl
instance [has_nat_cast β] : has_nat_cast C(α, β) :=
⟨λ n, continuous_map.const _ n⟩
@[simp, norm_cast]
lemma coe_nat_cast [has_nat_cast β] (n : ℕ) : ((n : C(α, β)) : α → β) = n := rfl
instance [has_int_cast β] : has_int_cast C(α, β) :=
⟨λ n, continuous_map.const _ n⟩
@[simp, norm_cast]
lemma coe_int_cast [has_int_cast β] (n : ℤ) : ((n : C(α, β)) : α → β) = n := rfl
instance has_nsmul [add_monoid β] [has_continuous_add β] : has_smul ℕ C(α, β) :=
⟨λ n f, ⟨n • f, f.continuous.nsmul n⟩⟩
@[to_additive]
instance has_pow [monoid β] [has_continuous_mul β] : has_pow C(α, β) ℕ :=
⟨λ f n, ⟨f ^ n, f.continuous.pow n⟩⟩
@[norm_cast, to_additive]
lemma coe_pow [monoid β] [has_continuous_mul β] (f : C(α, β)) (n : ℕ) :
⇑(f ^ n) = f ^ n := rfl
-- don't make `coe_nsmul` simp as the linter complains it's redundant WRT `coe_smul`
attribute [simp] coe_pow
@[to_additive] lemma pow_comp [monoid γ] [has_continuous_mul γ]
(f : C(β, γ)) (n : ℕ) (g : C(α, β)) :
(f^n).comp g = (f.comp g)^n :=
rfl
-- don't make `nsmul_comp` simp as the linter complains it's redundant WRT `smul_comp`
attribute [simp] pow_comp
@[to_additive]
instance [group β] [topological_group β] : has_inv C(α, β) :=
{ inv := λ f, ⟨f⁻¹, f.continuous.inv⟩ }
@[simp, norm_cast, to_additive]
lemma coe_inv [group β] [topological_group β] (f : C(α, β)) :
⇑(f⁻¹) = f⁻¹ :=
rfl
@[simp, to_additive] lemma inv_comp [group γ] [topological_group γ] (f : C(β, γ)) (g : C(α, β)) :
(f⁻¹).comp g = (f.comp g)⁻¹ :=
rfl
@[to_additive]
instance [has_div β] [has_continuous_div β] : has_div C(α, β) :=
{ div := λ f g, ⟨f / g, f.continuous.div' g.continuous⟩ }
@[simp, norm_cast, to_additive]
lemma coe_div [has_div β] [has_continuous_div β] (f g : C(α, β)) : ⇑(f / g) = f / g :=
rfl
@[simp, to_additive] lemma div_comp [has_div γ] [has_continuous_div γ]
(f g : C(β, γ)) (h : C(α, β)) :
(f / g).comp h = (f.comp h) / (g.comp h) :=
rfl
instance has_zsmul [add_group β] [topological_add_group β] : has_smul ℤ C(α, β) :=
{ smul := λ z f, ⟨z • f, f.continuous.zsmul z⟩ }
@[to_additive]
instance has_zpow [group β] [topological_group β] :
has_pow C(α, β) ℤ :=
{ pow := λ f z, ⟨f ^ z, f.continuous.zpow z⟩ }
@[norm_cast, to_additive]
lemma coe_zpow [group β] [topological_group β] (f : C(α, β)) (z : ℤ) :
⇑(f ^ z) = f ^ z :=
rfl
-- don't make `coe_zsmul` simp as the linter complains it's redundant WRT `coe_smul`
attribute [simp] coe_zpow
@[to_additive]
lemma zpow_comp [group γ] [topological_group γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β)) :
(f^z).comp g = (f.comp g)^z :=
rfl
-- don't make `zsmul_comp` simp as the linter complains it's redundant WRT `smul_comp`
attribute [simp] zpow_comp
end continuous_map
section group_structure
/-!
### Group stucture
In this section we show that continuous functions valued in a topological group inherit
the structure of a group.
-/
section subtype
/-- The `submonoid` of continuous maps `α → β`. -/
@[to_additive "The `add_submonoid` of continuous maps `α → β`. "]
def continuous_submonoid (α : Type*) (β : Type*) [topological_space α] [topological_space β]
[monoid β] [has_continuous_mul β] : submonoid (α → β) :=
{ carrier := { f : α → β | continuous f },
one_mem' := @continuous_const _ _ _ _ 1,
mul_mem' := λ f g fc gc, fc.mul gc }
/-- The subgroup of continuous maps `α → β`. -/
@[to_additive "The `add_subgroup` of continuous maps `α → β`. "]
def continuous_subgroup (α : Type*) (β : Type*) [topological_space α] [topological_space β]
[group β] [topological_group β] : subgroup (α → β) :=
{ inv_mem' := λ f fc, continuous.inv fc,
..continuous_submonoid α β, }.
end subtype
namespace continuous_map
@[to_additive]
instance {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [semigroup β] [has_continuous_mul β] : semigroup C(α, β) :=
coe_injective.semigroup _ coe_mul
@[to_additive]
instance {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [comm_semigroup β] [has_continuous_mul β] : comm_semigroup C(α, β) :=
coe_injective.comm_semigroup _ coe_mul
@[to_additive]
instance {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [mul_one_class β] [has_continuous_mul β] : mul_one_class C(α, β) :=
coe_injective.mul_one_class _ coe_one coe_mul
instance {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [mul_zero_class β] [has_continuous_mul β] : mul_zero_class C(α, β) :=
coe_injective.mul_zero_class _ coe_zero coe_mul
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[semigroup_with_zero β] [has_continuous_mul β] : semigroup_with_zero C(α, β) :=
coe_injective.semigroup_with_zero _ coe_zero coe_mul
@[to_additive]
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[monoid β] [has_continuous_mul β] : monoid C(α, β) :=
coe_injective.monoid _ coe_one coe_mul coe_pow
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[monoid_with_zero β] [has_continuous_mul β] : monoid_with_zero C(α, β) :=
coe_injective.monoid_with_zero _ coe_zero coe_one coe_mul coe_pow
@[to_additive]
instance {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [comm_monoid β] [has_continuous_mul β] : comm_monoid C(α, β) :=
coe_injective.comm_monoid _ coe_one coe_mul coe_pow
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[comm_monoid_with_zero β] [has_continuous_mul β] : comm_monoid_with_zero C(α, β) :=
coe_injective.comm_monoid_with_zero _ coe_zero coe_one coe_mul coe_pow
@[to_additive]
instance {α : Type*} {β : Type*} [topological_space α]
[locally_compact_space α] [topological_space β]
[has_mul β] [has_continuous_mul β] : has_continuous_mul C(α, β) :=
⟨begin
refine continuous_of_continuous_uncurry _ _,
have h1 : continuous (λ x : (C(α, β) × C(α, β)) × α, x.fst.fst x.snd) :=
continuous_eval'.comp (continuous_fst.prod_map continuous_id),
have h2 : continuous (λ x : (C(α, β) × C(α, β)) × α, x.fst.snd x.snd) :=
continuous_eval'.comp (continuous_snd.prod_map continuous_id),
exact h1.mul h2,
end⟩
/-- Coercion to a function as an `monoid_hom`. Similar to `monoid_hom.coe_fn`. -/
@[to_additive "Coercion to a function as an `add_monoid_hom`. Similar to `add_monoid_hom.coe_fn`.",
simps]
def coe_fn_monoid_hom {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[monoid β] [has_continuous_mul β] : C(α, β) →* (α → β) :=
{ to_fun := coe_fn, map_one' := coe_one, map_mul' := coe_mul }
/-- Composition on the left by a (continuous) homomorphism of topological monoids, as a
`monoid_hom`. Similar to `monoid_hom.comp_left`. -/
@[to_additive "Composition on the left by a (continuous) homomorphism of topological `add_monoid`s,
as an `add_monoid_hom`. Similar to `add_monoid_hom.comp_left`.", simps]
protected def _root_.monoid_hom.comp_left_continuous (α : Type*) {β : Type*} {γ : Type*}
[topological_space α] [topological_space β] [monoid β] [has_continuous_mul β]
[topological_space γ] [monoid γ] [has_continuous_mul γ] (g : β →* γ) (hg : continuous g) :
C(α, β) →* C(α, γ) :=
{ to_fun := λ f, (⟨g, hg⟩ : C(β, γ)).comp f,
map_one' := ext $ λ x, g.map_one,
map_mul' := λ f₁ f₂, ext $ λ x, g.map_mul _ _ }
/-- Composition on the right as a `monoid_hom`. Similar to `monoid_hom.comp_hom'`. -/
@[to_additive "Composition on the right as an `add_monoid_hom`. Similar to
`add_monoid_hom.comp_hom'`.", simps]
def comp_monoid_hom' {α : Type*} {β : Type*} {γ : Type*}
[topological_space α] [topological_space β] [topological_space γ]
[mul_one_class γ] [has_continuous_mul γ] (g : C(α, β)) : C(β, γ) →* C(α, γ) :=
{ to_fun := λ f, f.comp g, map_one' := one_comp g, map_mul' := λ f₁ f₂, mul_comp f₁ f₂ g }
open_locale big_operators
@[simp, to_additive] lemma coe_prod {α : Type*} {β : Type*} [comm_monoid β]
[topological_space α] [topological_space β] [has_continuous_mul β]
{ι : Type*} (s : finset ι) (f : ι → C(α, β)) :
⇑(∏ i in s, f i) = (∏ i in s, (f i : α → β)) :=
(coe_fn_monoid_hom : C(α, β) →* _).map_prod f s
@[to_additive]
lemma prod_apply {α : Type*} {β : Type*} [comm_monoid β]
[topological_space α] [topological_space β] [has_continuous_mul β]
{ι : Type*} (s : finset ι) (f : ι → C(α, β)) (a : α) :
(∏ i in s, f i) a = (∏ i in s, f i a) :=
by simp
@[to_additive]
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[group β] [topological_group β] : group C(α, β) :=
coe_injective.group _ coe_one coe_mul coe_inv coe_div coe_pow coe_zpow
@[to_additive]
instance {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [comm_group β] [topological_group β] : comm_group C(α, β) :=
coe_injective.comm_group _ coe_one coe_mul coe_inv coe_div coe_pow coe_zpow
@[to_additive] instance {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [comm_group β] [topological_group β] : topological_group C(α, β) :=
{ continuous_mul := by
{ letI : uniform_space β := topological_group.to_uniform_space β,
have : uniform_group β := topological_comm_group_is_uniform,
rw continuous_iff_continuous_at,
rintros ⟨f, g⟩,
rw [continuous_at, tendsto_iff_forall_compact_tendsto_uniformly_on, nhds_prod_eq],
exactI λ K hK, uniform_continuous_mul.comp_tendsto_uniformly_on
((tendsto_iff_forall_compact_tendsto_uniformly_on.mp filter.tendsto_id K hK).prod
(tendsto_iff_forall_compact_tendsto_uniformly_on.mp filter.tendsto_id K hK)), },
continuous_inv := by
{ letI : uniform_space β := topological_group.to_uniform_space β,
have : uniform_group β := topological_comm_group_is_uniform,
rw continuous_iff_continuous_at,
intro f,
rw [continuous_at, tendsto_iff_forall_compact_tendsto_uniformly_on],
exactI λ K hK, uniform_continuous_inv.comp_tendsto_uniformly_on
(tendsto_iff_forall_compact_tendsto_uniformly_on.mp filter.tendsto_id K hK), } }
end continuous_map
end group_structure
section ring_structure
/-!
### Ring stucture
In this section we show that continuous functions valued in a topological semiring `R` inherit
the structure of a ring.
-/
section subtype
/-- The subsemiring of continuous maps `α → β`. -/
def continuous_subsemiring (α : Type*) (R : Type*) [topological_space α] [topological_space R]
[semiring R] [topological_semiring R] : subsemiring (α → R) :=
{ ..continuous_add_submonoid α R,
..continuous_submonoid α R }
/-- The subring of continuous maps `α → β`. -/
def continuous_subring (α : Type*) (R : Type*) [topological_space α] [topological_space R]
[ring R] [topological_ring R] : subring (α → R) :=
{ ..continuous_subsemiring α R,
..continuous_add_subgroup α R }
end subtype
namespace continuous_map
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[non_unital_non_assoc_semiring β] [topological_semiring β] :
non_unital_non_assoc_semiring C(α, β) :=
coe_injective.non_unital_non_assoc_semiring _ coe_zero coe_add coe_mul coe_nsmul
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[non_unital_semiring β] [topological_semiring β] :
non_unital_semiring C(α, β) :=
coe_injective.non_unital_semiring _ coe_zero coe_add coe_mul coe_nsmul
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[add_monoid_with_one β] [has_continuous_add β] :
add_monoid_with_one C(α, β) :=
coe_injective.add_monoid_with_one _ coe_zero coe_one coe_add coe_nsmul coe_nat_cast
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[non_assoc_semiring β] [topological_semiring β] :
non_assoc_semiring C(α, β) :=
coe_injective.non_assoc_semiring _ coe_zero coe_one coe_add coe_mul coe_nsmul coe_nat_cast
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[semiring β] [topological_semiring β] : semiring C(α, β) :=
coe_injective.semiring _ coe_zero coe_one coe_add coe_mul coe_nsmul coe_pow coe_nat_cast
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[non_unital_non_assoc_ring β] [topological_ring β] : non_unital_non_assoc_ring C(α, β) :=
coe_injective.non_unital_non_assoc_ring _ coe_zero coe_add coe_mul coe_neg coe_sub
coe_nsmul coe_zsmul
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[non_unital_ring β] [topological_ring β] : non_unital_ring C(α, β) :=
coe_injective.non_unital_ring _ coe_zero coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[non_assoc_ring β] [topological_ring β] : non_assoc_ring C(α, β) :=
coe_injective.non_assoc_ring _ coe_zero coe_one coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul
coe_nat_cast coe_int_cast
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[ring β] [topological_ring β] : ring C(α, β) :=
coe_injective.ring _ coe_zero coe_one coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul coe_pow
coe_nat_cast coe_int_cast
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[non_unital_comm_semiring β] [topological_semiring β] : non_unital_comm_semiring C(α, β) :=
coe_injective.non_unital_comm_semiring _ coe_zero coe_add coe_mul coe_nsmul
instance {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [comm_semiring β] [topological_semiring β] : comm_semiring C(α, β) :=
coe_injective.comm_semiring _ coe_zero coe_one coe_add coe_mul coe_nsmul coe_pow coe_nat_cast
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[non_unital_comm_ring β] [topological_ring β] : non_unital_comm_ring C(α, β) :=
coe_injective.non_unital_comm_ring _ coe_zero coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul
instance {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [comm_ring β] [topological_ring β] : comm_ring C(α, β) :=
coe_injective.comm_ring _ coe_zero coe_one coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul
coe_pow coe_nat_cast coe_int_cast
/-- Composition on the left by a (continuous) homomorphism of topological semirings, as a
`ring_hom`. Similar to `ring_hom.comp_left`. -/
@[simps] protected def _root_.ring_hom.comp_left_continuous (α : Type*) {β : Type*} {γ : Type*}
[topological_space α] [topological_space β] [semiring β] [topological_semiring β]
[topological_space γ] [semiring γ] [topological_semiring γ] (g : β →+* γ) (hg : continuous g) :
C(α, β) →+* C(α, γ) :=
{ .. g.to_monoid_hom.comp_left_continuous α hg,
.. g.to_add_monoid_hom.comp_left_continuous α hg }
/-- Coercion to a function as a `ring_hom`. -/
@[simps]
def coe_fn_ring_hom {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[ring β] [topological_ring β] : C(α, β) →+* (α → β) :=
{ to_fun := coe_fn,
..(coe_fn_monoid_hom : C(α, β) →* _),
..(coe_fn_add_monoid_hom : C(α, β) →+ _) }
end continuous_map
end ring_structure
local attribute [ext] subtype.eq
section module_structure
/-!
### Semiodule stucture
In this section we show that continuous functions valued in a topological module `M` over a
topological semiring `R` inherit the structure of a module.
-/
section subtype
variables (α : Type*) [topological_space α]
variables (R : Type*) [semiring R]
variables (M : Type*) [topological_space M] [add_comm_group M]
variables [module R M] [has_continuous_const_smul R M] [topological_add_group M]
/-- The `R`-submodule of continuous maps `α → M`. -/
def continuous_submodule : submodule R (α → M) :=
{ carrier := { f : α → M | continuous f },
smul_mem' := λ c f hf, hf.const_smul c,
..continuous_add_subgroup α M }
end subtype
namespace continuous_map
variables {α β : Type*} [topological_space α] [topological_space β]
{R R₁ : Type*}
{M : Type*} [topological_space M]
{M₂ : Type*} [topological_space M₂]
@[to_additive continuous_map.has_vadd]
instance [has_smul R M] [has_continuous_const_smul R M] : has_smul R C(α, M) :=
⟨λ r f, ⟨r • f, f.continuous.const_smul r⟩⟩
@[to_additive]
instance [locally_compact_space α] [has_smul R M] [has_continuous_const_smul R M] :
has_continuous_const_smul R C(α, M) :=
⟨λ γ, continuous_of_continuous_uncurry _ (continuous_eval'.const_smul γ)⟩
@[to_additive]
instance [locally_compact_space α] [topological_space R] [has_smul R M]
[has_continuous_smul R M] : has_continuous_smul R C(α, M) :=
⟨begin
refine continuous_of_continuous_uncurry _ _,
have h : continuous (λ x : (R × C(α, M)) × α, x.fst.snd x.snd) :=
continuous_eval'.comp (continuous_snd.prod_map continuous_id),
exact (continuous_fst.comp continuous_fst).smul h,
end⟩
@[simp, to_additive, norm_cast]
lemma coe_smul [has_smul R M] [has_continuous_const_smul R M]
(c : R) (f : C(α, M)) : ⇑(c • f) = c • f := rfl
@[to_additive]
lemma smul_apply [has_smul R M] [has_continuous_const_smul R M]
(c : R) (f : C(α, M)) (a : α) : (c • f) a = c • (f a) :=
rfl
@[simp, to_additive] lemma smul_comp [has_smul R M] [has_continuous_const_smul R M]
(r : R) (f : C(β, M)) (g : C(α, β)) :
(r • f).comp g = r • (f.comp g) :=
rfl
@[to_additive]
instance [has_smul R M] [has_continuous_const_smul R M]
[has_smul R₁ M] [has_continuous_const_smul R₁ M]
[smul_comm_class R R₁ M] : smul_comm_class R R₁ C(α, M) :=
{ smul_comm := λ _ _ _, ext $ λ _, smul_comm _ _ _ }
instance [has_smul R M] [has_continuous_const_smul R M]
[has_smul R₁ M] [has_continuous_const_smul R₁ M]
[has_smul R R₁] [is_scalar_tower R R₁ M] : is_scalar_tower R R₁ C(α, M) :=
{ smul_assoc := λ _ _ _, ext $ λ _, smul_assoc _ _ _ }
instance [has_smul R M] [has_smul Rᵐᵒᵖ M] [has_continuous_const_smul R M]
[is_central_scalar R M] : is_central_scalar R C(α, M) :=
{ op_smul_eq_smul := λ _ _, ext $ λ _, op_smul_eq_smul _ _ }
instance [monoid R] [mul_action R M] [has_continuous_const_smul R M] : mul_action R C(α, M) :=
function.injective.mul_action _ coe_injective coe_smul
instance [monoid R] [add_monoid M] [distrib_mul_action R M]
[has_continuous_add M] [has_continuous_const_smul R M] :
distrib_mul_action R C(α, M) :=
function.injective.distrib_mul_action coe_fn_add_monoid_hom coe_injective coe_smul
variables [semiring R] [add_comm_monoid M] [add_comm_monoid M₂]
variables [has_continuous_add M] [module R M] [has_continuous_const_smul R M]
variables [has_continuous_add M₂] [module R M₂] [has_continuous_const_smul R M₂]
instance module : module R C(α, M) :=
function.injective.module R coe_fn_add_monoid_hom coe_injective coe_smul
variables (R)
/-- Composition on the left by a continuous linear map, as a `linear_map`.
Similar to `linear_map.comp_left`. -/
@[simps] protected def _root_.continuous_linear_map.comp_left_continuous (α : Type*)
[topological_space α] (g : M →L[R] M₂) :
C(α, M) →ₗ[R] C(α, M₂) :=
{ map_smul' := λ c f, ext $ λ x, g.map_smul' c _,
.. g.to_linear_map.to_add_monoid_hom.comp_left_continuous α g.continuous }
/-- Coercion to a function as a `linear_map`. -/
@[simps]
def coe_fn_linear_map : C(α, M) →ₗ[R] (α → M) :=
{ to_fun := coe_fn,
map_smul' := coe_smul,
..(coe_fn_add_monoid_hom : C(α, M) →+ _) }
end continuous_map
end module_structure
section algebra_structure
/-!
### Algebra structure
In this section we show that continuous functions valued in a topological algebra `A` over a ring
`R` inherit the structure of an algebra. Note that the hypothesis that `A` is a topological algebra
is obtained by requiring that `A` be both a `has_continuous_smul` and a `topological_semiring`.-/
section subtype
variables {α : Type*} [topological_space α]
{R : Type*} [comm_semiring R]
{A : Type*} [topological_space A] [semiring A]
[algebra R A] [topological_semiring A]
/-- The `R`-subalgebra of continuous maps `α → A`. -/
def continuous_subalgebra : subalgebra R (α → A) :=
{ carrier := { f : α → A | continuous f },
algebra_map_mem' := λ r, (continuous_const : continuous $ λ (x : α), algebra_map R A r),
..continuous_subsemiring α A }
end subtype
section continuous_map
variables {α : Type*} [topological_space α]
{R : Type*} [comm_semiring R]
{A : Type*} [topological_space A] [semiring A]
[algebra R A] [topological_semiring A]
{A₂ : Type*} [topological_space A₂] [semiring A₂]
[algebra R A₂] [topological_semiring A₂]
/-- Continuous constant functions as a `ring_hom`. -/
def continuous_map.C : R →+* C(α, A) :=
{ to_fun := λ c : R, ⟨λ x: α, ((algebra_map R A) c), continuous_const⟩,
map_one' := by ext x; exact (algebra_map R A).map_one,
map_mul' := λ c₁ c₂, by ext x; exact (algebra_map R A).map_mul _ _,
map_zero' := by ext x; exact (algebra_map R A).map_zero,
map_add' := λ c₁ c₂, by ext x; exact (algebra_map R A).map_add _ _ }
@[simp] lemma continuous_map.C_apply (r : R) (a : α) : continuous_map.C r a = algebra_map R A r :=
rfl
variables [has_continuous_const_smul R A] [has_continuous_const_smul R A₂]
instance continuous_map.algebra : algebra R C(α, A) :=
{ to_ring_hom := continuous_map.C,
commutes' := λ c f, by ext x; exact algebra.commutes' _ _,
smul_def' := λ c f, by ext x; exact algebra.smul_def' _ _, }
variables (R)
/-- Composition on the left by a (continuous) homomorphism of topological `R`-algebras, as an
`alg_hom`. Similar to `alg_hom.comp_left`. -/
@[simps] protected def alg_hom.comp_left_continuous {α : Type*} [topological_space α]
(g : A →ₐ[R] A₂) (hg : continuous g) :
C(α, A) →ₐ[R] C(α, A₂) :=
{ commutes' := λ c, continuous_map.ext $ λ _, g.commutes' _,
.. g.to_ring_hom.comp_left_continuous α hg }
/-- Coercion to a function as an `alg_hom`. -/
@[simps]
def continuous_map.coe_fn_alg_hom : C(α, A) →ₐ[R] (α → A) :=
{ to_fun := coe_fn,
commutes' := λ r, rfl,
-- `..(continuous_map.coe_fn_ring_hom : C(α, A) →+* _)` times out for some reason
map_zero' := continuous_map.coe_zero,
map_one' := continuous_map.coe_one,
map_add' := continuous_map.coe_add,
map_mul' := continuous_map.coe_mul }
variables {R}
/--
A version of `separates_points` for subalgebras of the continuous functions,
used for stating the Stone-Weierstrass theorem.
-/
abbreviation subalgebra.separates_points (s : subalgebra R C(α, A)) : Prop :=
set.separates_points ((λ f : C(α, A), (f : α → A)) '' (s : set C(α, A)))
lemma subalgebra.separates_points_monotone :
monotone (λ s : subalgebra R C(α, A), s.separates_points) :=
λ s s' r h x y n,
begin
obtain ⟨f, m, w⟩ := h n,
rcases m with ⟨f, ⟨m, rfl⟩⟩,
exact ⟨_, ⟨f, ⟨r m, rfl⟩⟩, w⟩,
end
@[simp] lemma algebra_map_apply (k : R) (a : α) :
algebra_map R C(α, A) k a = k • 1 :=
by { rw algebra.algebra_map_eq_smul_one, refl, }
variables {𝕜 : Type*} [topological_space 𝕜]
/--
A set of continuous maps "separates points strongly"
if for each pair of distinct points there is a function with specified values on them.
We give a slightly unusual formulation, where the specified values are given by some
function `v`, and we ask `f x = v x ∧ f y = v y`. This avoids needing a hypothesis `x ≠ y`.
In fact, this definition would work perfectly well for a set of non-continuous functions,
but as the only current use case is in the Stone-Weierstrass theorem,
writing it this way avoids having to deal with casts inside the set.
(This may need to change if we do Stone-Weierstrass on non-compact spaces,
where the functions would be continuous functions vanishing at infinity.)
-/
def set.separates_points_strongly (s : set C(α, 𝕜)) : Prop :=
∀ (v : α → 𝕜) (x y : α), ∃ f : s, (f x : 𝕜) = v x ∧ f y = v y
variables [field 𝕜] [topological_ring 𝕜]
/--
Working in continuous functions into a topological field,
a subalgebra of functions that separates points also separates points strongly.
By the hypothesis, we can find a function `f` so `f x ≠ f y`.
By an affine transformation in the field we can arrange so that `f x = a` and `f x = b`.
-/
lemma subalgebra.separates_points.strongly {s : subalgebra 𝕜 C(α, 𝕜)} (h : s.separates_points) :
(s : set C(α, 𝕜)).separates_points_strongly :=
λ v x y,
begin
by_cases n : x = y,
{ subst n,
use ((v x) • 1 : C(α, 𝕜)),
{ apply s.smul_mem,
apply s.one_mem, },
{ simp [coe_fn_coe_base'] }, },
obtain ⟨f, ⟨f, ⟨m, rfl⟩⟩, w⟩ := h n,
replace w : f x - f y ≠ 0 := sub_ne_zero_of_ne w,
let a := v x,
let b := v y,
let f' := ((b - a) * (f x - f y)⁻¹) • (continuous_map.C (f x) - f) + continuous_map.C a,
refine ⟨⟨f', _⟩, _, _⟩,
{ simp only [f', set_like.mem_coe, subalgebra.mem_to_submodule],
-- TODO should there be a tactic for this?
-- We could add an attribute `@[subobject_mem]`, and a tactic
-- ``def subobject_mem := `[solve_by_elim with subobject_mem { max_depth := 10 }]``
solve_by_elim
[subalgebra.add_mem, subalgebra.smul_mem, subalgebra.sub_mem, subalgebra.algebra_map_mem]
{ max_depth := 6 }, },
{ simp [f', coe_fn_coe_base'], },
{ simp [f', coe_fn_coe_base', inv_mul_cancel_right₀ w], },
end
end continuous_map
instance continuous_map.subsingleton_subalgebra (α : Type*) [topological_space α]
(R : Type*) [comm_semiring R] [topological_space R] [topological_semiring R]
[subsingleton α] : subsingleton (subalgebra R C(α, R)) :=
begin
fsplit,
intros s₁ s₂,
by_cases n : nonempty α,
{ obtain ⟨x⟩ := n,
ext f,
have h : f = algebra_map R C(α, R) (f x),
{ ext x', simp only [mul_one, algebra.id.smul_eq_mul, algebra_map_apply], congr, },
rw h,
simp only [subalgebra.algebra_map_mem], },
{ ext f,
have h : f = 0,
{ ext x', exact false.elim (n ⟨x'⟩), },
subst h,
simp only [subalgebra.zero_mem], },
end
end algebra_structure
section module_over_continuous_functions
/-!
### Structure as module over scalar functions
If `M` is a module over `R`, then we show that the space of continuous functions from `α` to `M`
is naturally a module over the ring of continuous functions from `α` to `R`. -/
namespace continuous_map
instance has_smul' {α : Type*} [topological_space α]
{R : Type*} [semiring R] [topological_space R]
{M : Type*} [topological_space M] [add_comm_monoid M]
[module R M] [has_continuous_smul R M] :
has_smul C(α, R) C(α, M) :=
⟨λ f g, ⟨λ x, (f x) • (g x), (continuous.smul f.2 g.2)⟩⟩
instance module' {α : Type*} [topological_space α]
(R : Type*) [ring R] [topological_space R] [topological_ring R]
(M : Type*) [topological_space M] [add_comm_monoid M] [has_continuous_add M]
[module R M] [has_continuous_smul R M] :
module C(α, R) C(α, M) :=
{ smul := (•),
smul_add := λ c f g, by ext x; exact smul_add (c x) (f x) (g x),
add_smul := λ c₁ c₂ f, by ext x; exact add_smul (c₁ x) (c₂ x) (f x),
mul_smul := λ c₁ c₂ f, by ext x; exact mul_smul (c₁ x) (c₂ x) (f x),
one_smul := λ f, by ext x; exact one_smul R (f x),
zero_smul := λ f, by ext x; exact zero_smul _ _,
smul_zero := λ r, by ext x; exact smul_zero _, }
end continuous_map
end module_over_continuous_functions
/-!
We now provide formulas for `f ⊓ g` and `f ⊔ g`, where `f g : C(α, β)`,
in terms of `continuous_map.abs`.
-/
section
variables {R : Type*} [linear_ordered_field R]
-- TODO:
-- This lemma (and the next) could go all the way back in `algebra.order.field`,
-- except that it is tedious to prove without tactics.
-- Rather than stranding it at some intermediate location,
-- it's here, immediately prior to the point of use.
lemma min_eq_half_add_sub_abs_sub {x y : R} : min x y = 2⁻¹ * (x + y - |x - y|) :=
by cases le_total x y with h h; field_simp [h, abs_of_nonneg, abs_of_nonpos, mul_two]; abel
lemma max_eq_half_add_add_abs_sub {x y : R} : max x y = 2⁻¹ * (x + y + |x - y|) :=
by cases le_total x y with h h; field_simp [h, abs_of_nonneg, abs_of_nonpos, mul_two]; abel
end
namespace continuous_map
section lattice
variables {α : Type*} [topological_space α]
variables {β : Type*} [linear_ordered_field β] [topological_space β]
[order_topology β] [topological_ring β]
lemma inf_eq (f g : C(α, β)) : f ⊓ g = (2⁻¹ : β) • (f + g - |f - g|) :=
ext (λ x, by simpa using min_eq_half_add_sub_abs_sub)
-- Not sure why this is grosser than `inf_eq`:
lemma sup_eq (f g : C(α, β)) : f ⊔ g = (2⁻¹ : β) • (f + g + |f - g|) :=
ext (λ x, by simpa [mul_add] using @max_eq_half_add_add_abs_sub _ _ (f x) (g x))
end lattice
/-!
### Star structure
If `β` has a continuous star operation, we put a star structure on `C(α, β)` by using the
star operation pointwise.
If `β` is a ⋆-ring, then `C(α, β)` inherits a ⋆-ring structure.
If `β` is a ⋆-ring and a ⋆-module over `R`, then the space of continuous functions from `α` to `β`
is a ⋆-module over `R`.
-/
section star_structure
variables {R α β : Type*}
variables [topological_space α] [topological_space β]
section has_star
variables [has_star β] [has_continuous_star β]
instance : has_star C(α, β) :=
{ star := λ f, star_continuous_map.comp f }
@[simp] lemma coe_star (f : C(α, β)) : ⇑(star f) = star f := rfl
@[simp] lemma star_apply (f : C(α, β)) (x : α) : star f x = star (f x) := rfl
end has_star
instance [has_involutive_star β] [has_continuous_star β] : has_involutive_star C(α, β) :=
{ star_involutive := λ f, ext $ λ x, star_star _ }
instance [add_monoid β] [has_continuous_add β] [star_add_monoid β] [has_continuous_star β] :
star_add_monoid C(α, β) :=
{ star_add := λ f g, ext $ λ x, star_add _ _ }
instance [semigroup β] [has_continuous_mul β] [star_semigroup β] [has_continuous_star β] :
star_semigroup C(α, β) :=
{ star_mul := λ f g, ext $ λ x, star_mul _ _ }
instance [non_unital_semiring β] [topological_semiring β] [star_ring β] [has_continuous_star β] :
star_ring C(α, β) :=
{ ..continuous_map.star_add_monoid }
instance [has_star R] [has_star β] [has_smul R β] [star_module R β]
[has_continuous_star β] [has_continuous_const_smul R β] :
star_module R C(α, β) :=
{ star_smul := λ k f, ext $ λ x, star_smul _ _ }
end star_structure
end continuous_map
|
92ec93dde275324a165910c86ba9b9fcdab4631c | 49bd2218ae088932d847f9030c8dbff1c5607bb7 | /src/analysis/normed_space/basic.lean | 50f9c4f81c6ec4e69be23253d7a838ca7f15e2fc | [
"Apache-2.0"
] | permissive | FredericLeRoux/mathlib | e8f696421dd3e4edc8c7edb3369421c8463d7bac | 3645bf8fb426757e0a20af110d1fdded281d286e | refs/heads/master | 1,607,062,870,732 | 1,578,513,186,000 | 1,578,513,186,000 | 231,653,181 | 0 | 0 | Apache-2.0 | 1,578,080,327,000 | 1,578,080,326,000 | null | UTF-8 | Lean | false | false | 34,264 | lean | /-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Normed spaces.
Authors: Patrick Massot, Johannes Hölzl
-/
import algebra.pi_instances
import linear_algebra.basic
import topology.instances.nnreal topology.instances.complex
import topology.algebra.module
variables {α : Type*} {β : Type*} {γ : Type*} {ι : Type*}
noncomputable theory
open filter metric
open_locale topological_space
localized "notation f `→_{`:50 a `}`:0 b := filter.tendsto f (_root_.nhds a) (_root_.nhds b)" in filter
/-- Auxiliary class, endowing a type `α` with a function `norm : α → ℝ`. This class is designed to
be extended in more interesting classes specifying the properties of the norm. -/
class has_norm (α : Type*) := (norm : α → ℝ)
export has_norm (norm)
notation `∥`:1024 e:1 `∥`:1 := norm e
section prio
set_option default_priority 100 -- see Note [default priority]
/-- A normed group is an additive group endowed with a norm for which `dist x y = ∥x - y∥` defines
a metric space structure. -/
class normed_group (α : Type*) extends has_norm α, add_comm_group α, metric_space α :=
(dist_eq : ∀ x y, dist x y = norm (x - y))
end prio
/-- Construct a normed group from a translation invariant distance -/
def normed_group.of_add_dist [has_norm α] [add_comm_group α] [metric_space α]
(H1 : ∀ x:α, ∥x∥ = dist x 0)
(H2 : ∀ x y z : α, dist x y ≤ dist (x + z) (y + z)) : normed_group α :=
{ dist_eq := λ x y, begin
rw H1, apply le_antisymm,
{ rw [sub_eq_add_neg, ← add_right_neg y], apply H2 },
{ have := H2 (x-y) 0 y, rwa [sub_add_cancel, zero_add] at this }
end }
/-- Construct a normed group from a translation invariant distance -/
def normed_group.of_add_dist' [has_norm α] [add_comm_group α] [metric_space α]
(H1 : ∀ x:α, ∥x∥ = dist x 0)
(H2 : ∀ x y z : α, dist (x + z) (y + z) ≤ dist x y) : normed_group α :=
{ dist_eq := λ x y, begin
rw H1, apply le_antisymm,
{ have := H2 (x-y) 0 y, rwa [sub_add_cancel, zero_add] at this },
{ rw [sub_eq_add_neg, ← add_right_neg y], apply H2 }
end }
/-- A normed group can be built from a norm that satisfies algebraic properties. This is
formalised in this structure. -/
structure normed_group.core (α : Type*) [add_comm_group α] [has_norm α] :=
(norm_eq_zero_iff : ∀ x : α, ∥x∥ = 0 ↔ x = 0)
(triangle : ∀ x y : α, ∥x + y∥ ≤ ∥x∥ + ∥y∥)
(norm_neg : ∀ x : α, ∥-x∥ = ∥x∥)
/-- Constructing a normed group from core properties of a norm, i.e., registering the distance and
the metric space structure from the norm properties. -/
noncomputable def normed_group.of_core (α : Type*) [add_comm_group α] [has_norm α]
(C : normed_group.core α) : normed_group α :=
{ dist := λ x y, ∥x - y∥,
dist_eq := assume x y, by refl,
dist_self := assume x, (C.norm_eq_zero_iff (x - x)).mpr (show x - x = 0, by simp),
eq_of_dist_eq_zero := assume x y h, show (x = y), from sub_eq_zero.mp $ (C.norm_eq_zero_iff (x - y)).mp h,
dist_triangle := assume x y z,
calc ∥x - z∥ = ∥x - y + (y - z)∥ : by simp
... ≤ ∥x - y∥ + ∥y - z∥ : C.triangle _ _,
dist_comm := assume x y,
calc ∥x - y∥ = ∥ -(y - x)∥ : by simp
... = ∥y - x∥ : by { rw [C.norm_neg] } }
section normed_group
variables [normed_group α] [normed_group β]
lemma dist_eq_norm (g h : α) : dist g h = ∥g - h∥ :=
normed_group.dist_eq _ _
@[simp] lemma dist_zero_right (g : α) : dist g 0 = ∥g∥ :=
by rw [dist_eq_norm, sub_zero]
lemma norm_sub_rev (g h : α) : ∥g - h∥ = ∥h - g∥ :=
by simpa only [dist_eq_norm] using dist_comm g h
@[simp] lemma norm_neg (g : α) : ∥-g∥ = ∥g∥ :=
by simpa using norm_sub_rev 0 g
@[simp] lemma dist_add_left (g h₁ h₂ : α) : dist (g + h₁) (g + h₂) = dist h₁ h₂ :=
by simp [dist_eq_norm]
@[simp] lemma dist_add_right (g₁ g₂ h : α) : dist (g₁ + h) (g₂ + h) = dist g₁ g₂ :=
by simp [dist_eq_norm]
@[simp] lemma dist_neg_neg (g h : α) : dist (-g) (-h) = dist g h :=
by simp only [dist_eq_norm, neg_sub_neg, norm_sub_rev]
@[simp] lemma dist_sub_left (g h₁ h₂ : α) : dist (g - h₁) (g - h₂) = dist h₁ h₂ :=
by simp only [sub_eq_add_neg, dist_add_left, dist_neg_neg]
@[simp] lemma dist_sub_right (g₁ g₂ h : α) : dist (g₁ - h) (g₂ - h) = dist g₁ g₂ :=
dist_add_right _ _ _
/-- Triangle inequality for the norm. -/
lemma norm_add_le (g h : α) : ∥g + h∥ ≤ ∥g∥ + ∥h∥ :=
by simpa [dist_eq_norm] using dist_triangle g 0 (-h)
lemma norm_add_le_of_le {g₁ g₂ : α} {n₁ n₂ : ℝ} (H₁ : ∥g₁∥ ≤ n₁) (H₂ : ∥g₂∥ ≤ n₂) :
∥g₁ + g₂∥ ≤ n₁ + n₂ :=
le_trans (norm_add_le g₁ g₂) (add_le_add H₁ H₂)
lemma dist_add_add_le (g₁ g₂ h₁ h₂ : α) :
dist (g₁ + g₂) (h₁ + h₂) ≤ dist g₁ h₁ + dist g₂ h₂ :=
by simpa only [dist_add_left, dist_add_right] using dist_triangle (g₁ + g₂) (h₁ + g₂) (h₁ + h₂)
lemma dist_add_add_le_of_le {g₁ g₂ h₁ h₂ : α} {d₁ d₂ : ℝ}
(H₁ : dist g₁ h₁ ≤ d₁) (H₂ : dist g₂ h₂ ≤ d₂) :
dist (g₁ + g₂) (h₁ + h₂) ≤ d₁ + d₂ :=
le_trans (dist_add_add_le g₁ g₂ h₁ h₂) (add_le_add H₁ H₂)
lemma dist_sub_sub_le (g₁ g₂ h₁ h₂ : α) :
dist (g₁ - g₂) (h₁ - h₂) ≤ dist g₁ h₁ + dist g₂ h₂ :=
dist_neg_neg g₂ h₂ ▸ dist_add_add_le _ _ _ _
lemma dist_sub_sub_le_of_le {g₁ g₂ h₁ h₂ : α} {d₁ d₂ : ℝ}
(H₁ : dist g₁ h₁ ≤ d₁) (H₂ : dist g₂ h₂ ≤ d₂) :
dist (g₁ - g₂) (h₁ - h₂) ≤ d₁ + d₂ :=
le_trans (dist_sub_sub_le g₁ g₂ h₁ h₂) (add_le_add H₁ H₂)
@[simp] lemma norm_nonneg (g : α) : 0 ≤ ∥g∥ :=
by { rw[←dist_zero_right], exact dist_nonneg }
lemma norm_eq_zero (g : α) : ∥g∥ = 0 ↔ g = 0 :=
by { rw[←dist_zero_right], exact dist_eq_zero }
@[simp] lemma norm_zero : ∥(0:α)∥ = 0 := (norm_eq_zero _).2 rfl
lemma norm_sum_le {β} : ∀(s : finset β) (f : β → α), ∥s.sum f∥ ≤ s.sum (λa, ∥ f a ∥) :=
finset.le_sum_of_subadditive norm norm_zero norm_add_le
lemma norm_sum_le_of_le {β} (s : finset β) {f : β → α} {n : β → ℝ} (h : ∀ b ∈ s, ∥f b∥ ≤ n b) :
∥s.sum f∥ ≤ s.sum n :=
by { haveI := classical.dec_eq β, exact le_trans (norm_sum_le s f) (finset.sum_le_sum h) }
lemma norm_pos_iff (g : α) : 0 < ∥ g ∥ ↔ g ≠ 0 :=
dist_zero_right g ▸ dist_pos
lemma norm_le_zero_iff (g : α) : ∥g∥ ≤ 0 ↔ g = 0 :=
by { rw[←dist_zero_right], exact dist_le_zero }
lemma norm_sub_le (g h : α) : ∥g - h∥ ≤ ∥g∥ + ∥h∥ :=
by simpa [dist_eq_norm] using dist_triangle g 0 h
lemma norm_sub_le_of_le {g₁ g₂ : α} {n₁ n₂ : ℝ} (H₁ : ∥g₁∥ ≤ n₁) (H₂ : ∥g₂∥ ≤ n₂) :
∥g₁ - g₂∥ ≤ n₁ + n₂ :=
le_trans (norm_sub_le g₁ g₂) (add_le_add H₁ H₂)
lemma dist_le_norm_add_norm (g h : α) : dist g h ≤ ∥g∥ + ∥h∥ :=
by { rw dist_eq_norm, apply norm_sub_le }
lemma abs_norm_sub_norm_le (g h : α) : abs(∥g∥ - ∥h∥) ≤ ∥g - h∥ :=
by simpa [dist_eq_norm] using abs_dist_sub_le g h 0
lemma norm_sub_norm_le (g h : α) : ∥g∥ - ∥h∥ ≤ ∥g - h∥ :=
le_trans (le_abs_self _) (abs_norm_sub_norm_le g h)
lemma dist_norm_norm_le (g h : α) : dist ∥g∥ ∥h∥ ≤ ∥g - h∥ :=
abs_norm_sub_norm_le g h
lemma ball_0_eq (ε : ℝ) : ball (0:α) ε = {x | ∥x∥ < ε} :=
set.ext $ assume a, by simp
theorem normed_group.tendsto_nhds_zero {f : γ → α} {l : filter γ} :
tendsto f l (𝓝 0) ↔ ∀ ε > 0, { x | ∥ f x ∥ < ε } ∈ l :=
metric.tendsto_nhds.trans $ forall_congr $ λ ε, forall_congr $ λ εgt0,
begin
simp only [dist_zero_right],
exact exists_sets_subset_iff
end
section nnnorm
/-- Version of the norm taking values in nonnegative reals. -/
def nnnorm (a : α) : nnreal := ⟨norm a, norm_nonneg a⟩
@[simp] lemma coe_nnnorm (a : α) : (nnnorm a : ℝ) = norm a := rfl
lemma nndist_eq_nnnorm (a b : α) : nndist a b = nnnorm (a - b) := nnreal.eq $ dist_eq_norm _ _
lemma nnnorm_eq_zero (a : α) : nnnorm a = 0 ↔ a = 0 :=
by simp only [nnreal.eq_iff.symm, nnreal.coe_zero, coe_nnnorm, norm_eq_zero]
@[simp] lemma nnnorm_zero : nnnorm (0 : α) = 0 :=
nnreal.eq norm_zero
lemma nnnorm_add_le (g h : α) : nnnorm (g + h) ≤ nnnorm g + nnnorm h :=
nnreal.coe_le.2 $ norm_add_le g h
@[simp] lemma nnnorm_neg (g : α) : nnnorm (-g) = nnnorm g :=
nnreal.eq $ norm_neg g
lemma nndist_nnnorm_nnnorm_le (g h : α) : nndist (nnnorm g) (nnnorm h) ≤ nnnorm (g - h) :=
nnreal.coe_le.2 $ dist_norm_norm_le g h
lemma of_real_norm_eq_coe_nnnorm (x : β) : ennreal.of_real ∥x∥ = (nnnorm x : ennreal) :=
ennreal.of_real_eq_coe_nnreal _
lemma edist_eq_coe_nnnorm (x : β) : edist x 0 = (nnnorm x : ennreal) :=
by { rw [edist_dist, dist_eq_norm, _root_.sub_zero, of_real_norm_eq_coe_nnnorm] }
end nnnorm
/-- A submodule of a normed group is also a normed group, with the restriction of the norm.
As all instances can be inferred from the submodule `s`, they are put as implicit instead of
typeclasses. -/
instance submodule.normed_group {𝕜 : Type*} {_ : ring 𝕜}
{E : Type*} [normed_group E] {_ : module 𝕜 E} (s : submodule 𝕜 E) : normed_group s :=
{ norm := λx, norm (x : E),
dist_eq := λx y, dist_eq_norm (x : E) (y : E) }
/-- normed group instance on the product of two normed groups, using the sup norm. -/
instance prod.normed_group : normed_group (α × β) :=
{ norm := λx, max ∥x.1∥ ∥x.2∥,
dist_eq := assume (x y : α × β),
show max (dist x.1 y.1) (dist x.2 y.2) = (max ∥(x - y).1∥ ∥(x - y).2∥), by simp [dist_eq_norm] }
lemma norm_fst_le (x : α × β) : ∥x.1∥ ≤ ∥x∥ :=
by simp [norm, le_max_left]
lemma norm_snd_le (x : α × β) : ∥x.2∥ ≤ ∥x∥ :=
by simp [norm, le_max_right]
/-- normed group instance on the product of finitely many normed groups, using the sup norm. -/
instance pi.normed_group {π : ι → Type*} [fintype ι] [∀i, normed_group (π i)] :
normed_group (Πb, π b) :=
{ norm := λf, ((finset.sup finset.univ (λ b, nnnorm (f b)) : nnreal) : ℝ),
dist_eq := assume x y,
congr_arg (coe : nnreal → ℝ) $ congr_arg (finset.sup finset.univ) $ funext $ assume a,
show nndist (x a) (y a) = nnnorm (x a - y a), from nndist_eq_nnnorm _ _ }
/-- The norm of an element in a product space is `≤ r` if and only if the norm of each
component is. -/
lemma pi_norm_le_iff {π : ι → Type*} [fintype ι] [∀i, normed_group (π i)] {r : ℝ} (hr : 0 ≤ r)
{x : Πb, π b} : ∥x∥ ≤ r ↔ ∀i, ∥x i∥ ≤ r :=
by { simp only [(dist_zero_right _).symm, dist_pi_le_iff hr], refl }
lemma tendsto_iff_norm_tendsto_zero {f : ι → β} {a : filter ι} {b : β} :
tendsto f a (𝓝 b) ↔ tendsto (λ e, ∥ f e - b ∥) a (𝓝 0) :=
by rw tendsto_iff_dist_tendsto_zero ; simp only [(dist_eq_norm _ _).symm]
lemma tendsto_zero_iff_norm_tendsto_zero {f : γ → β} {a : filter γ} :
tendsto f a (𝓝 0) ↔ tendsto (λ e, ∥ f e ∥) a (𝓝 0) :=
have tendsto f a (𝓝 0) ↔ tendsto (λ e, ∥ f e - 0 ∥) a (𝓝 0) :=
tendsto_iff_norm_tendsto_zero,
by simpa
lemma lim_norm (x : α) : (λg:α, ∥g - x∥) →_{x} 0 :=
tendsto_iff_norm_tendsto_zero.1 (continuous_iff_continuous_at.1 continuous_id x)
lemma lim_norm_zero : (λg:α, ∥g∥) →_{0} 0 :=
by simpa using lim_norm (0:α)
lemma continuous_norm : continuous (λg:α, ∥g∥) :=
begin
rw continuous_iff_continuous_at,
intro x,
rw [continuous_at, tendsto_iff_dist_tendsto_zero],
exact squeeze_zero (λ t, abs_nonneg _) (λ t, abs_norm_sub_norm_le _ _) (lim_norm x)
end
lemma continuous_nnnorm : continuous (nnnorm : α → nnreal) :=
continuous_subtype_mk _ continuous_norm
/-- If `∥y∥→∞`, then we can assume `y≠x` for any fixed `x`. -/
lemma ne_mem_of_tendsto_norm_at_top {l : filter γ} {f : γ → α}
(h : tendsto (λ y, ∥f y∥) l at_top) (x : α) :
{y | f y ≠ x} ∈ l :=
begin
have : {y | 1 + ∥x∥ ≤ ∥f y∥} ∈ l := h (mem_at_top (1 + ∥x∥)),
apply mem_sets_of_superset this,
assume y hy hxy,
subst x,
simp at hy,
exact not_le_of_lt zero_lt_one hy
end
/-- A normed group is a uniform additive group, i.e., addition and subtraction are uniformly
continuous. -/
@[priority 100] -- see Note [lower instance priority]
instance normed_uniform_group : uniform_add_group α :=
begin
refine ⟨metric.uniform_continuous_iff.2 $ assume ε hε, ⟨ε / 2, half_pos hε, assume a b h, _⟩⟩,
rw [prod.dist_eq, max_lt_iff, dist_eq_norm, dist_eq_norm] at h,
calc dist (a.1 - a.2) (b.1 - b.2) = ∥(a.1 - b.1) - (a.2 - b.2)∥ : by simp [dist_eq_norm]
... ≤ ∥a.1 - b.1∥ + ∥a.2 - b.2∥ : norm_sub_le _ _
... < ε / 2 + ε / 2 : add_lt_add h.1 h.2
... = ε : add_halves _
end
@[priority 100] -- see Note [lower instance priority]
instance normed_top_monoid : topological_add_monoid α := by apply_instance -- short-circuit type class inference
@[priority 100] -- see Note [lower instance priority]
instance normed_top_group : topological_add_group α := by apply_instance -- short-circuit type class inference
end normed_group
section normed_ring
section prio
set_option default_priority 100 -- see Note [default priority]
/-- A normed ring is a ring endowed with a norm which satisfies the inequality `∥x y∥ ≤ ∥x∥ ∥y∥`. -/
class normed_ring (α : Type*) extends has_norm α, ring α, metric_space α :=
(dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b)
end prio
@[priority 100] -- see Note [lower instance priority]
instance normed_ring.to_normed_group [β : normed_ring α] : normed_group α := { ..β }
lemma norm_mul_le {α : Type*} [normed_ring α] (a b : α) : (∥a*b∥) ≤ (∥a∥) * (∥b∥) :=
normed_ring.norm_mul _ _
lemma norm_pow_le {α : Type*} [normed_ring α] (a : α) : ∀ {n : ℕ}, 0 < n → ∥a^n∥ ≤ ∥a∥^n
| 1 h := by simp
| (n+2) h :=
le_trans (norm_mul_le a (a^(n+1)))
(mul_le_mul (le_refl _)
(norm_pow_le (nat.succ_pos _)) (norm_nonneg _) (norm_nonneg _))
/-- Normed ring structure on the product of two normed rings, using the sup norm. -/
instance prod.normed_ring [normed_ring α] [normed_ring β] : normed_ring (α × β) :=
{ norm_mul := assume x y,
calc
∥x * y∥ = ∥(x.1*y.1, x.2*y.2)∥ : rfl
... = (max ∥x.1*y.1∥ ∥x.2*y.2∥) : rfl
... ≤ (max (∥x.1∥*∥y.1∥) (∥x.2∥*∥y.2∥)) :
max_le_max (norm_mul_le (x.1) (y.1)) (norm_mul_le (x.2) (y.2))
... = (max (∥x.1∥*∥y.1∥) (∥y.2∥*∥x.2∥)) : by simp[mul_comm]
... ≤ (max (∥x.1∥) (∥x.2∥)) * (max (∥y.2∥) (∥y.1∥)) : by { apply max_mul_mul_le_max_mul_max; simp [norm_nonneg] }
... = (max (∥x.1∥) (∥x.2∥)) * (max (∥y.1∥) (∥y.2∥)) : by simp[max_comm]
... = (∥x∥*∥y∥) : rfl,
..prod.normed_group }
end normed_ring
@[priority 100] -- see Note [lower instance priority]
instance normed_ring_top_monoid [normed_ring α] : topological_monoid α :=
⟨ continuous_iff_continuous_at.2 $ λ x, tendsto_iff_norm_tendsto_zero.2 $
have ∀ e : α × α, e.fst * e.snd - x.fst * x.snd =
e.fst * e.snd - e.fst * x.snd + (e.fst * x.snd - x.fst * x.snd), by intro; rw sub_add_sub_cancel,
begin
apply squeeze_zero,
{ intro, apply norm_nonneg },
{ simp only [this], intro, apply norm_add_le },
{ rw ←zero_add (0 : ℝ), apply tendsto.add,
{ apply squeeze_zero,
{ intro, apply norm_nonneg },
{ intro t, show ∥t.fst * t.snd - t.fst * x.snd∥ ≤ ∥t.fst∥ * ∥t.snd - x.snd∥,
rw ←mul_sub, apply norm_mul_le },
{ rw ←mul_zero (∥x.fst∥), apply tendsto.mul,
{ apply continuous_iff_continuous_at.1,
apply continuous_norm.comp continuous_fst },
{ apply tendsto_iff_norm_tendsto_zero.1,
apply continuous_iff_continuous_at.1,
apply continuous_snd }}},
{ apply squeeze_zero,
{ intro, apply norm_nonneg },
{ intro t, show ∥t.fst * x.snd - x.fst * x.snd∥ ≤ ∥t.fst - x.fst∥ * ∥x.snd∥,
rw ←sub_mul, apply norm_mul_le },
{ rw ←zero_mul (∥x.snd∥), apply tendsto.mul,
{ apply tendsto_iff_norm_tendsto_zero.1,
apply continuous_iff_continuous_at.1,
apply continuous_fst },
{ apply tendsto_const_nhds }}}}
end ⟩
/-- A normed ring is a topological ring. -/
@[priority 100] -- see Note [lower instance priority]
instance normed_top_ring [normed_ring α] : topological_ring α :=
⟨ continuous_iff_continuous_at.2 $ λ x, tendsto_iff_norm_tendsto_zero.2 $
have ∀ e : α, -e - -x = -(e - x), by intro; simp,
by simp only [this, norm_neg]; apply lim_norm ⟩
section prio
set_option default_priority 100 -- see Note [default priority]
/-- A normed field is a field with a norm satisfying ∥x y∥ = ∥x∥ ∥y∥. -/
class normed_field (α : Type*) extends has_norm α, discrete_field α, metric_space α :=
(dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul' : ∀ a b, norm (a * b) = norm a * norm b)
/-- A nondiscrete normed field is a normed field in which there is an element of norm different from
`0` and `1`. This makes it possible to bring any element arbitrarily close to `0` by multiplication
by the powers of any element, and thus to relate algebra and topology. -/
class nondiscrete_normed_field (α : Type*) extends normed_field α :=
(non_trivial : ∃x:α, 1<∥x∥)
end prio
@[priority 100] -- see Note [lower instance priority]
instance normed_field.to_normed_ring [i : normed_field α] : normed_ring α :=
{ norm_mul := by finish [i.norm_mul'], ..i }
namespace normed_field
@[simp] lemma norm_one {α : Type*} [normed_field α] : ∥(1 : α)∥ = 1 :=
have ∥(1 : α)∥ * ∥(1 : α)∥ = ∥(1 : α)∥ * 1, by calc
∥(1 : α)∥ * ∥(1 : α)∥ = ∥(1 : α) * (1 : α)∥ : by rw normed_field.norm_mul'
... = ∥(1 : α)∥ * 1 : by simp,
eq_of_mul_eq_mul_left (ne_of_gt ((norm_pos_iff _).2 (by simp))) this
@[simp] lemma norm_mul [normed_field α] (a b : α) : ∥a * b∥ = ∥a∥ * ∥b∥ :=
normed_field.norm_mul' a b
instance normed_field.is_monoid_hom_norm [normed_field α] : is_monoid_hom (norm : α → ℝ) :=
{ map_one := norm_one, map_mul := norm_mul }
@[simp] lemma norm_pow [normed_field α] (a : α) : ∀ (n : ℕ), ∥a^n∥ = ∥a∥^n :=
is_monoid_hom.map_pow norm a
@[simp] lemma norm_prod {β : Type*} [normed_field α] (s : finset β) (f : β → α) :
∥s.prod f∥ = s.prod (λb, ∥f b∥) :=
eq.symm (s.prod_hom norm)
@[simp] lemma norm_div {α : Type*} [normed_field α] (a b : α) : ∥a/b∥ = ∥a∥/∥b∥ :=
if hb : b = 0 then by simp [hb] else
begin
apply eq_div_of_mul_eq,
{ apply ne_of_gt, apply (norm_pos_iff _).mpr hb },
{ rw [←normed_field.norm_mul, div_mul_cancel _ hb] }
end
@[simp] lemma norm_inv {α : Type*} [normed_field α] (a : α) : ∥a⁻¹∥ = ∥a∥⁻¹ :=
by simp only [inv_eq_one_div, norm_div, norm_one]
@[simp] lemma norm_fpow {α : Type*} [normed_field α] (a : α) : ∀n : ℤ,
∥a^n∥ = ∥a∥^n
| (n : ℕ) := norm_pow a n
| -[1+ n] := by simp [fpow_neg_succ_of_nat]
lemma exists_one_lt_norm (α : Type*) [i : nondiscrete_normed_field α] : ∃x : α, 1 < ∥x∥ :=
i.non_trivial
lemma exists_norm_lt_one (α : Type*) [nondiscrete_normed_field α] : ∃x : α, 0 < ∥x∥ ∧ ∥x∥ < 1 :=
begin
rcases exists_one_lt_norm α with ⟨y, hy⟩,
refine ⟨y⁻¹, _, _⟩,
{ simp only [inv_eq_zero, ne.def, norm_pos_iff],
assume h,
rw ← norm_eq_zero at h,
rw h at hy,
exact lt_irrefl _ (lt_trans zero_lt_one hy) },
{ simp [inv_lt_one hy] }
end
lemma exists_lt_norm (α : Type*) [nondiscrete_normed_field α]
(r : ℝ) : ∃ x : α, r < ∥x∥ :=
let ⟨w, hw⟩ := exists_one_lt_norm α in
let ⟨n, hn⟩ := pow_unbounded_of_one_lt r hw in
⟨w^n, by rwa norm_pow⟩
lemma exists_norm_lt (α : Type*) [nondiscrete_normed_field α]
{r : ℝ} (hr : 0 < r) : ∃ x : α, 0 < ∥x∥ ∧ ∥x∥ < r :=
let ⟨w, hw⟩ := exists_one_lt_norm α in
let ⟨n, hle, hlt⟩ := exists_int_pow_near' hr hw in
⟨w^n, by { rw norm_fpow; exact fpow_pos_of_pos (lt_trans zero_lt_one hw) _},
by rwa norm_fpow⟩
lemma tendsto_inv [normed_field α] {r : α} (r0 : r ≠ 0) : tendsto (λq, q⁻¹) (𝓝 r) (𝓝 r⁻¹) :=
begin
refine metric.tendsto_nhds.2 (λε εpos, _),
let δ := min (ε/2/2 * ∥r∥^2) (∥r∥/2),
have norm_r_pos : 0 < ∥r∥ := (norm_pos_iff r).mpr r0,
have A : 0 < ε / 2 / 2 * ∥r∥ ^ 2 := mul_pos' (half_pos (half_pos εpos)) (pow_pos norm_r_pos 2),
have δpos : 0 < δ, by simp [half_pos norm_r_pos, A],
refine ⟨ball r δ, ball_mem_nhds r δpos, λx hx, _⟩,
have rx : ∥r∥/2 ≤ ∥x∥ := calc
∥r∥/2 = ∥r∥ - ∥r∥/2 : by ring
... ≤ ∥r∥ - ∥r - x∥ :
begin
apply sub_le_sub (le_refl _),
rw ← dist_eq_norm,
exact le_trans (le_of_lt (mem_ball'.1 hx)) (min_le_right _ _)
end
... ≤ ∥r - (r - x)∥ : norm_sub_norm_le r (r - x)
... = ∥x∥ : by simp,
have norm_x_pos : 0 < ∥x∥ := lt_of_lt_of_le (half_pos norm_r_pos) rx,
have : x⁻¹ - r⁻¹ = (r - x) * x⁻¹ * r⁻¹,
by rw [sub_mul, sub_mul, mul_inv_cancel ((norm_pos_iff x).mp norm_x_pos), one_mul, mul_comm,
← mul_assoc, inv_mul_cancel r0, one_mul],
calc dist x⁻¹ r⁻¹ = ∥x⁻¹ - r⁻¹∥ : dist_eq_norm _ _
... ≤ ∥r-x∥ * ∥x∥⁻¹ * ∥r∥⁻¹ : by rw [this, norm_mul, norm_mul, norm_inv, norm_inv]
... ≤ (ε/2/2 * ∥r∥^2) * (2 * ∥r∥⁻¹) * (∥r∥⁻¹) : begin
apply_rules [mul_le_mul, inv_nonneg.2, le_of_lt A, norm_nonneg, inv_nonneg.2, mul_nonneg,
(inv_le_inv norm_x_pos norm_r_pos).2, le_refl],
show ∥r - x∥ ≤ ε / 2 / 2 * ∥r∥ ^ 2,
by { rw ← dist_eq_norm, exact le_trans (le_of_lt (mem_ball'.1 hx)) (min_le_left _ _) },
show ∥x∥⁻¹ ≤ 2 * ∥r∥⁻¹,
{ convert (inv_le_inv norm_x_pos (half_pos norm_r_pos)).2 rx,
rw [inv_div (ne.symm (ne_of_lt norm_r_pos)), div_eq_inv_mul, mul_comm],
norm_num },
show (0 : ℝ) ≤ 2, by norm_num
end
... = ε/2 * (∥r∥ * ∥r∥⁻¹)^2 : by { generalize : ∥r∥⁻¹ = u, ring }
... = ε/2 : by { rw [mul_inv_cancel (ne.symm (ne_of_lt norm_r_pos))], simp }
... < ε : half_lt_self εpos
end
lemma continuous_on_inv [normed_field α] : continuous_on (λ(x:α), x⁻¹) {x | x ≠ 0} :=
begin
assume x hx,
apply continuous_at.continuous_within_at,
exact (tendsto_inv hx)
end
instance : normed_field ℝ :=
{ norm := λ x, abs x,
dist_eq := assume x y, rfl,
norm_mul' := abs_mul }
instance : nondiscrete_normed_field ℝ :=
{ non_trivial := ⟨2, by { unfold norm, rw abs_of_nonneg; norm_num }⟩ }
end normed_field
/-- If a function converges to a nonzero value, its inverse converges to the inverse of this value.
We use the name `tendsto.inv'` as `tendsto.inv` is already used in multiplicative topological
groups. -/
lemma filter.tendsto.inv' [normed_field α] {l : filter β} {f : β → α} {y : α}
(hy : y ≠ 0) (h : tendsto f l (𝓝 y)) :
tendsto (λx, (f x)⁻¹) l (𝓝 y⁻¹) :=
(normed_field.tendsto_inv hy).comp h
lemma filter.tendsto.div [normed_field α] {l : filter β} {f g : β → α} {x y : α}
(hf : tendsto f l (𝓝 x)) (hg : tendsto g l (𝓝 y)) (hy : y ≠ 0) :
tendsto (λa, f a / g a) l (𝓝 (x / y)) :=
hf.mul (hg.inv' hy)
lemma real.norm_eq_abs (r : ℝ) : norm r = abs r := rfl
@[simp] lemma norm_norm [normed_group α] (x : α) : ∥∥x∥∥ = ∥x∥ :=
by rw [real.norm_eq_abs, abs_of_nonneg (norm_nonneg _)]
@[simp] lemma nnnorm_norm [normed_group α] (a : α) : nnnorm ∥a∥ = nnnorm a :=
by simp only [nnnorm, norm_norm]
instance : normed_ring ℤ :=
{ norm := λ n, ∥(n : ℝ)∥,
norm_mul := λ m n, le_of_eq $ by simp only [norm, int.cast_mul, abs_mul],
dist_eq := λ m n, by simp only [int.dist_eq, norm, int.cast_sub] }
@[elim_cast] lemma int.norm_cast_real (m : ℤ) : ∥(m : ℝ)∥ = ∥m∥ := rfl
instance : normed_field ℚ :=
{ norm := λ r, ∥(r : ℝ)∥,
norm_mul' := λ r₁ r₂, by simp only [norm, rat.cast_mul, abs_mul],
dist_eq := λ r₁ r₂, by simp only [rat.dist_eq, norm, rat.cast_sub] }
instance : nondiscrete_normed_field ℚ :=
{ non_trivial := ⟨2, by { unfold norm, rw abs_of_nonneg; norm_num }⟩ }
@[elim_cast, simp] lemma rat.norm_cast_real (r : ℚ) : ∥(r : ℝ)∥ = ∥r∥ := rfl
@[elim_cast, simp] lemma int.norm_cast_rat (m : ℤ) : ∥(m : ℚ)∥ = ∥m∥ :=
by rw [← rat.norm_cast_real, ← int.norm_cast_real]; congr' 1; norm_cast
section normed_space
section prio
set_option default_priority 100 -- see Note [default priority]
-- see Note[vector space definition] for why we extend `module`.
/-- A normed space over a normed field is a vector space endowed with a norm which satisfies the
equality `∥c • x∥ = ∥c∥ ∥x∥`. -/
class normed_space (α : Type*) (β : Type*) [normed_field α] [normed_group β]
extends module α β :=
(norm_smul : ∀ (a:α) (b:β), norm (a • b) = has_norm.norm a * norm b)
end prio
variables [normed_field α] [normed_group β]
instance normed_field.to_normed_space : normed_space α α :=
{ norm_smul := normed_field.norm_mul }
set_option class.instance_max_depth 43
lemma norm_smul [normed_space α β] (s : α) (x : β) : ∥s • x∥ = ∥s∥ * ∥x∥ :=
normed_space.norm_smul s x
lemma dist_smul [normed_space α β] (s : α) (x y : β) : dist (s • x) (s • y) = ∥s∥ * dist x y :=
by simp only [dist_eq_norm, (norm_smul _ _).symm, smul_sub]
lemma nnnorm_smul [normed_space α β] (s : α) (x : β) : nnnorm (s • x) = nnnorm s * nnnorm x :=
nnreal.eq $ norm_smul s x
variables {E : Type*} {F : Type*}
[normed_group E] [normed_space α E] [normed_group F] [normed_space α F]
lemma tendsto_smul {f : γ → α} { g : γ → F} {e : filter γ} {s : α} {b : F} :
(tendsto f e (𝓝 s)) → (tendsto g e (𝓝 b)) → tendsto (λ x, (f x) • (g x)) e (𝓝 (s • b)) :=
begin
intros limf limg,
rw tendsto_iff_norm_tendsto_zero,
have ineq := λ x : γ, calc
∥f x • g x - s • b∥ = ∥(f x • g x - s • g x) + (s • g x - s • b)∥ : by simp[add_assoc]
... ≤ ∥f x • g x - s • g x∥ + ∥s • g x - s • b∥ : norm_add_le (f x • g x - s • g x) (s • g x - s • b)
... ≤ ∥f x - s∥*∥g x∥ + ∥s∥*∥g x - b∥ : by { rw [←smul_sub, ←sub_smul, norm_smul, norm_smul] },
apply squeeze_zero,
{ intro t, exact norm_nonneg _ },
{ exact ineq },
{ clear ineq,
have limf': tendsto (λ x, ∥f x - s∥) e (𝓝 0) := tendsto_iff_norm_tendsto_zero.1 limf,
have limg' : tendsto (λ x, ∥g x∥) e (𝓝 ∥b∥) := filter.tendsto.comp (continuous_iff_continuous_at.1 continuous_norm _) limg,
have lim1 := limf'.mul limg',
simp only [zero_mul, sub_eq_add_neg] at lim1,
have limg3 := tendsto_iff_norm_tendsto_zero.1 limg,
have lim2 := (tendsto_const_nhds : tendsto _ _ (𝓝 ∥ s ∥)).mul limg3,
simp only [sub_eq_add_neg, mul_zero] at lim2,
rw [show (0:ℝ) = 0 + 0, by simp],
exact lim1.add lim2 }
end
lemma tendsto_smul_const {g : γ → F} {e : filter γ} (s : α) {b : F} :
(tendsto g e (𝓝 b)) → tendsto (λ x, s • (g x)) e (𝓝 (s • b)) :=
tendsto_smul tendsto_const_nhds
@[priority 100] -- see Note [lower instance priority]
instance normed_space.topological_vector_space : topological_vector_space α E :=
{ continuous_smul := continuous_iff_continuous_at.2 $ λp, tendsto_smul
(continuous_iff_continuous_at.1 continuous_fst _) (continuous_iff_continuous_at.1 continuous_snd _) }
open normed_field
/-- If there is a scalar `c` with `∥c∥>1`, then any element can be moved by scalar multiplication to
any shell of width `∥c∥`. Also recap information on the norm of the rescaling element that shows
up in applications. -/
lemma rescale_to_shell {c : α} (hc : 1 < ∥c∥) {ε : ℝ} (εpos : 0 < ε) {x : E} (hx : x ≠ 0) :
∃d:α, d ≠ 0 ∧ ∥d • x∥ ≤ ε ∧ (ε/∥c∥ ≤ ∥d • x∥) ∧ (∥d∥⁻¹ ≤ ε⁻¹ * ∥c∥ * ∥x∥) :=
begin
have xεpos : 0 < ∥x∥/ε := div_pos_of_pos_of_pos ((norm_pos_iff _).2 hx) εpos,
rcases exists_int_pow_near xεpos hc with ⟨n, hn⟩,
have cpos : 0 < ∥c∥ := lt_trans (zero_lt_one : (0 :ℝ) < 1) hc,
have cnpos : 0 < ∥c^(n+1)∥ := by { rw norm_fpow, exact lt_trans xεpos hn.2 },
refine ⟨(c^(n+1))⁻¹, _, _, _, _⟩,
show (c ^ (n + 1))⁻¹ ≠ 0,
by rwa [ne.def, inv_eq_zero, ← ne.def, ← norm_pos_iff],
show ∥(c ^ (n + 1))⁻¹ • x∥ ≤ ε,
{ rw [norm_smul, norm_inv, ← div_eq_inv_mul, div_le_iff cnpos, mul_comm, norm_fpow],
exact (div_le_iff εpos).1 (le_of_lt (hn.2)) },
show ε / ∥c∥ ≤ ∥(c ^ (n + 1))⁻¹ • x∥,
{ rw [div_le_iff cpos, norm_smul, norm_inv, norm_fpow, fpow_add (ne_of_gt cpos),
fpow_one, mul_inv', mul_comm, ← mul_assoc, ← mul_assoc, mul_inv_cancel (ne_of_gt cpos),
one_mul, ← div_eq_inv_mul, le_div_iff (fpow_pos_of_pos cpos _), mul_comm],
exact (le_div_iff εpos).1 hn.1 },
show ∥(c ^ (n + 1))⁻¹∥⁻¹ ≤ ε⁻¹ * ∥c∥ * ∥x∥,
{ have : ε⁻¹ * ∥c∥ * ∥x∥ = ε⁻¹ * ∥x∥ * ∥c∥, by ring,
rw [norm_inv, inv_inv', norm_fpow, fpow_add (ne_of_gt cpos), fpow_one, this, ← div_eq_inv_mul],
exact mul_le_mul_of_nonneg_right hn.1 (norm_nonneg _) }
end
/-- The product of two normed spaces is a normed space, with the sup norm. -/
instance : normed_space α (E × F) :=
{ norm_smul :=
begin
intros s x,
cases x with x₁ x₂,
change max (∥s • x₁∥) (∥s • x₂∥) = ∥s∥ * max (∥x₁∥) (∥x₂∥),
rw [norm_smul, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg _)]
end,
add_smul := λ r x y, prod.ext (add_smul _ _ _) (add_smul _ _ _),
smul_add := λ r x y, prod.ext (smul_add _ _ _) (smul_add _ _ _),
..prod.normed_group,
..prod.module }
/-- The product of finitely many normed spaces is a normed space, with the sup norm. -/
instance pi.normed_space {E : ι → Type*} [fintype ι] [∀i, normed_group (E i)]
[∀i, normed_space α (E i)] : normed_space α (Πi, E i) :=
{ norm_smul := λ a f,
show (↑(finset.sup finset.univ (λ (b : ι), nnnorm (a • f b))) : ℝ) =
nnnorm a * ↑(finset.sup finset.univ (λ (b : ι), nnnorm (f b))),
by simp only [(nnreal.coe_mul _ _).symm, nnreal.mul_finset_sup, nnnorm_smul] }
/-- A subspace of a normed space is also a normed space, with the restriction of the norm. -/
instance submodule.normed_space {𝕜 : Type*} [normed_field 𝕜]
{E : Type*} [normed_group E] [normed_space 𝕜 E] (s : submodule 𝕜 E) : normed_space 𝕜 s :=
{ norm_smul := λc x, norm_smul c (x : E) }
end normed_space
section normed_algebra
/-- A normed algebra `𝕜'` over `𝕜` is an algebra endowed with a norm for which the embedding of
`𝕜` in `𝕜'` is an isometry. -/
class normed_algebra (𝕜 : Type*) (𝕜' : Type*) [normed_field 𝕜] [normed_ring 𝕜']
extends algebra 𝕜 𝕜' :=
(norm_algebra_map_eq : ∀x:𝕜, ∥algebra_map 𝕜' x∥ = ∥x∥)
@[simp] lemma norm_algebra_map_eq {𝕜 : Type*} (𝕜' : Type*) [normed_field 𝕜] [normed_ring 𝕜']
[h : normed_algebra 𝕜 𝕜'] (x : 𝕜) : ∥algebra_map 𝕜' x∥ = ∥x∥ :=
normed_algebra.norm_algebra_map_eq _ _
end normed_algebra
section restrict_scalars
set_option class.instance_max_depth 40
variables (𝕜 : Type*) (𝕜' : Type*) [normed_field 𝕜] [normed_field 𝕜'] [normed_algebra 𝕜 𝕜']
{E : Type*} [normed_group E] [normed_space 𝕜' E]
/-- `𝕜`-normed space structure induced by a `𝕜'`-normed space structure when `𝕜'` is a
normed algebra over `𝕜`. Not registered as an instance as `𝕜'` can not be inferred. -/
def normed_space.restrict_scalars : normed_space 𝕜 E :=
{ norm_smul := λc x, begin
change ∥(algebra_map 𝕜' c) • x∥ = ∥c∥ * ∥x∥,
simp [norm_smul]
end,
..module.restrict_scalars 𝕜 𝕜' E }
end restrict_scalars
section summable
open_locale classical
open finset filter
variables [normed_group α] [complete_space α]
lemma summable_iff_vanishing_norm {f : ι → α} :
summable f ↔ ∀ε > 0, ∃s:finset ι, ∀t, disjoint t s → ∥ t.sum f ∥ < ε :=
begin
simp only [summable_iff_vanishing, metric.mem_nhds_iff, exists_imp_distrib],
split,
{ assume h ε hε, refine h {x | ∥x∥ < ε} ε hε _, rw [ball_0_eq ε] },
{ assume h s ε hε hs,
rcases h ε hε with ⟨t, ht⟩,
refine ⟨t, assume u hu, hs _⟩,
rw [ball_0_eq],
exact ht u hu }
end
lemma summable_of_norm_bounded {f : ι → α} (g : ι → ℝ) (hf : summable g) (h : ∀i, ∥f i∥ ≤ g i) :
summable f :=
summable_iff_vanishing_norm.2 $ assume ε hε,
let ⟨s, hs⟩ := summable_iff_vanishing_norm.1 hf ε hε in
⟨s, assume t ht,
have ∥t.sum g∥ < ε := hs t ht,
have nn : 0 ≤ t.sum g := finset.sum_nonneg (assume a _, le_trans (norm_nonneg _) (h a)),
lt_of_le_of_lt (norm_sum_le_of_le t (λ i _, h i)) $
by rwa [real.norm_eq_abs, abs_of_nonneg nn] at this⟩
lemma summable_of_summable_norm {f : ι → α} (hf : summable (λa, ∥f a∥)) : summable f :=
summable_of_norm_bounded _ hf (assume i, le_refl _)
lemma norm_tsum_le_tsum_norm {f : ι → α} (hf : summable (λi, ∥f i∥)) : ∥(∑i, f i)∥ ≤ (∑ i, ∥f i∥) :=
have h₁ : tendsto (λs:finset ι, ∥s.sum f∥) at_top (𝓝 ∥(∑ i, f i)∥) :=
(continuous_norm.tendsto _).comp (has_sum_tsum $ summable_of_summable_norm hf),
have h₂ : tendsto (λs:finset ι, s.sum (λi, ∥f i∥)) at_top (𝓝 (∑ i, ∥f i∥)) :=
has_sum_tsum hf,
le_of_tendsto_of_tendsto at_top_ne_bot h₁ h₂ $ univ_mem_sets' $ assume s, norm_sum_le _ _
end summable
|
741a51d972ff7ddff7a8cd1fc93cf897d105addf | 26ac254ecb57ffcb886ff709cf018390161a9225 | /src/algebra/ordered_group.lean | 48dd292eef73cd08a49d238a0722bcd2d2a3056a | [
"Apache-2.0"
] | permissive | eric-wieser/mathlib | 42842584f584359bbe1fc8b88b3ff937c8acd72d | d0df6b81cd0920ad569158c06a3fd5abb9e63301 | refs/heads/master | 1,669,546,404,255 | 1,595,254,668,000 | 1,595,254,668,000 | 281,173,504 | 0 | 0 | Apache-2.0 | 1,595,263,582,000 | 1,595,263,581,000 | null | UTF-8 | Lean | false | false | 65,167 | 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.group.with_one
import algebra.group.type_tags
import order.bounded_lattice
set_option old_structure_cmd true
set_option default_priority 100 -- see Note [default priority]
/-!
# Ordered monoids and groups
This file develops the basics of ordered monoids and 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.
-/
universe u
variable {α : Type u}
/-- An ordered commutative monoid is a commutative monoid
with a partial order such that
* `a ≤ b → c * a ≤ c * b` (multiplication is monotone)
* `a * b < a * c → b < c`.
-/
@[protect_proj, ancestor comm_monoid partial_order]
class ordered_comm_monoid (α : Type*) extends comm_monoid α, partial_order α :=
(mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b)
(lt_of_mul_lt_mul_left : ∀ a b c : α, a * b < a * c → b < c)
/-- An ordered (additive) commutative monoid is a commutative monoid
with a partial order such that
* `a ≤ b → c + a ≤ c + b` (addition is monotone)
* `a + b < a + c → b < c`.
-/
@[protect_proj, ancestor add_comm_monoid partial_order]
class ordered_add_comm_monoid (α : Type*) extends add_comm_monoid α, partial_order α :=
(add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b)
(lt_of_add_lt_add_left : ∀ a b c : α, a + b < a + c → b < c)
attribute [to_additive ordered_add_comm_monoid] ordered_comm_monoid
section ordered_comm_monoid
variables [ordered_comm_monoid α] {a b c d : α}
@[to_additive add_le_add_left]
lemma mul_le_mul_left' (h : a ≤ b) (c) : c * a ≤ c * b :=
ordered_comm_monoid.mul_le_mul_left a b h c
@[to_additive add_le_add_right]
lemma mul_le_mul_right' (h : a ≤ b) (c) : a * c ≤ b * c :=
mul_comm c a ▸ mul_comm c b ▸ mul_le_mul_left' h c
@[to_additive lt_of_add_lt_add_left]
lemma lt_of_mul_lt_mul_left' : a * b < a * c → b < c :=
ordered_comm_monoid.lt_of_mul_lt_mul_left a b c
@[to_additive add_le_add]
lemma mul_le_mul' (h₁ : a ≤ b) (h₂ : c ≤ d) : a * c ≤ b * d :=
le_trans (mul_le_mul_right' h₁ _) (mul_le_mul_left' h₂ _)
@[to_additive]
lemma mul_le_mul_three {a b c d e f : α} (h₁ : a ≤ d) (h₂ : b ≤ e) (h₃ : c ≤ f) :
a * b * c ≤ d * e * f :=
mul_le_mul' (mul_le_mul' h₁ h₂) h₃
@[to_additive]
lemma le_mul_of_one_le_right (h : 1 ≤ b) : a ≤ a * b :=
have a * 1 ≤ a * b, from mul_le_mul_left' h _,
by rwa mul_one at this
@[to_additive]
lemma le_mul_of_one_le_left (h : 1 ≤ b) : a ≤ b * a :=
have 1 * a ≤ b * a, from mul_le_mul_right' h a,
by rwa one_mul at this
@[to_additive lt_of_add_lt_add_right]
lemma lt_of_mul_lt_mul_right' (h : a * b < c * b) : a < c :=
lt_of_mul_lt_mul_left'
(show b * a < b * c, begin rw [mul_comm b a, mul_comm b c], assumption end)
-- here we start using properties of one.
@[to_additive]
lemma le_mul_of_one_le_of_le (ha : 1 ≤ a) (hbc : b ≤ c) : b ≤ a * c :=
one_mul b ▸ mul_le_mul' ha hbc
@[to_additive]
lemma le_mul_of_le_of_one_le (hbc : b ≤ c) (ha : 1 ≤ a) : b ≤ c * a :=
mul_one b ▸ mul_le_mul' hbc ha
@[to_additive add_nonneg]
lemma one_le_mul (ha : 1 ≤ a) (hb : 1 ≤ b) : 1 ≤ a * b :=
le_mul_of_one_le_of_le ha hb
@[to_additive add_pos_of_pos_of_nonneg]
lemma one_lt_mul_of_lt_of_le' (ha : 1 < a) (hb : 1 ≤ b) : 1 < a * b :=
lt_of_lt_of_le ha $ le_mul_of_one_le_right hb
@[to_additive add_pos_of_nonneg_of_pos]
lemma one_lt_mul_of_le_of_lt' (ha : 1 ≤ a) (hb : 1 < b) : 1 < a * b :=
lt_of_lt_of_le hb $ le_mul_of_one_le_left ha
@[to_additive add_pos]
lemma one_lt_mul' (ha : 1 < a) (hb : 1 < b) : 1 < a * b :=
one_lt_mul_of_lt_of_le' ha $ le_of_lt hb
@[to_additive add_nonpos]
lemma mul_le_one' (ha : a ≤ 1) (hb : b ≤ 1) : a * b ≤ 1 :=
one_mul (1:α) ▸ (mul_le_mul' ha hb)
@[to_additive]
lemma mul_le_of_le_one_of_le' (ha : a ≤ 1) (hbc : b ≤ c) : a * b ≤ c :=
one_mul c ▸ mul_le_mul' ha hbc
@[to_additive]
lemma mul_le_of_le_of_le_one' (hbc : b ≤ c) (ha : a ≤ 1) : b * a ≤ c :=
mul_one c ▸ mul_le_mul' hbc ha
@[to_additive]
lemma mul_lt_one_of_lt_one_of_le_one' (ha : a < 1) (hb : b ≤ 1) : a * b < 1 :=
lt_of_le_of_lt (mul_le_of_le_of_le_one' (le_refl _) hb) ha
@[to_additive]
lemma mul_lt_one_of_le_one_of_lt_one' (ha : a ≤ 1) (hb : b < 1) : a * b < 1 :=
lt_of_le_of_lt (mul_le_of_le_one_of_le' ha (le_refl _)) hb
@[to_additive]
lemma mul_lt_one' (ha : a < 1) (hb : b < 1) : a * b < 1 :=
mul_lt_one_of_le_one_of_lt_one' (le_of_lt ha) hb
@[to_additive]
lemma lt_mul_of_one_le_of_lt' (ha : 1 ≤ a) (hbc : b < c) : b < a * c :=
lt_of_lt_of_le hbc $ le_mul_of_one_le_left ha
@[to_additive]
lemma lt_mul_of_lt_of_one_le' (hbc : b < c) (ha : 1 ≤ a) : b < c * a :=
lt_of_lt_of_le hbc $ le_mul_of_one_le_right ha
@[to_additive]
lemma lt_mul_of_one_lt_of_lt' (ha : 1 < a) (hbc : b < c) : b < a * c :=
lt_mul_of_one_le_of_lt' (le_of_lt ha) hbc
@[to_additive]
lemma lt_mul_of_lt_of_one_lt' (hbc : b < c) (ha : 1 < a) : b < c * a :=
lt_mul_of_lt_of_one_le' hbc (le_of_lt ha)
@[to_additive]
lemma mul_lt_of_le_one_of_lt' (ha : a ≤ 1) (hbc : b < c) : a * b < c :=
lt_of_le_of_lt (mul_le_of_le_one_of_le' ha (le_refl _)) hbc
@[to_additive]
lemma mul_lt_of_lt_of_le_one' (hbc : b < c) (ha : a ≤ 1) : b * a < c :=
lt_of_le_of_lt (mul_le_of_le_of_le_one' (le_refl _) ha) hbc
@[to_additive]
lemma mul_lt_of_lt_one_of_lt' (ha : a < 1) (hbc : b < c) : a * b < c :=
mul_lt_of_le_one_of_lt' (le_of_lt ha) hbc
@[to_additive]
lemma mul_lt_of_lt_of_lt_one' (hbc : b < c) (ha : a < 1) : b * a < c :=
mul_lt_of_lt_of_le_one' hbc (le_of_lt ha)
@[to_additive]
lemma mul_eq_one_iff' (ha : 1 ≤ a) (hb : 1 ≤ b) : a * b = 1 ↔ a = 1 ∧ b = 1 :=
iff.intro
(assume hab : a * b = 1,
have a ≤ 1, from hab ▸ le_mul_of_le_of_one_le (le_refl _) hb,
have a = 1, from le_antisymm this ha,
have b ≤ 1, from hab ▸ le_mul_of_one_le_of_le ha (le_refl _),
have b = 1, from le_antisymm this hb,
and.intro ‹a = 1› ‹b = 1›)
(assume ⟨ha', hb'⟩, by rw [ha', hb', mul_one])
section mono
variables {β : Type*} [preorder β] {f g : β → α}
@[to_additive monotone.add]
lemma monotone.mul' (hf : monotone f) (hg : monotone g) : monotone (λ x, f x * g x) :=
λ x y h, mul_le_mul' (hf h) (hg h)
@[to_additive monotone.add_const]
lemma monotone.mul_const' (hf : monotone f) (a : α) : monotone (λ x, f x * a) :=
hf.mul' monotone_const
@[to_additive monotone.const_add]
lemma monotone.const_mul' (hf : monotone f) (a : α) : monotone (λ x, a * f x) :=
monotone_const.mul' hf
end mono
end ordered_comm_monoid
lemma bit0_pos [ordered_add_comm_monoid α] {a : α} (h : 0 < a) : 0 < bit0 a :=
add_pos h h
namespace units
@[to_additive]
instance [monoid α] [preorder α] : preorder (units α) :=
preorder.lift (coe : units α → α)
@[simp, to_additive, norm_cast]
theorem coe_le_coe [monoid α] [preorder α] {a b : units α} :
(a : α) ≤ b ↔ a ≤ b := iff.rfl
@[simp, to_additive, norm_cast]
theorem coe_lt_coe [monoid α] [preorder α] {a b : units α} :
(a : α) < b ↔ a < b := iff.rfl
@[to_additive]
instance [monoid α] [partial_order α] : partial_order (units α) :=
partial_order.lift coe units.ext
@[to_additive]
instance [monoid α] [linear_order α] : linear_order (units α) :=
linear_order.lift coe units.ext
@[to_additive]
instance [monoid α] [decidable_linear_order α] : decidable_linear_order (units α) :=
decidable_linear_order.lift coe units.ext
@[simp, to_additive, norm_cast]
theorem max_coe [monoid α] [decidable_linear_order α] {a b : units α} :
(↑(max a b) : α) = max a b :=
by by_cases a ≤ b; simp [max, h]
@[simp, to_additive, norm_cast]
theorem min_coe [monoid α] [decidable_linear_order α] {a b : units α} :
(↑(min a b) : α) = min a b :=
by by_cases a ≤ b; simp [min, h]
end units
namespace with_zero
instance [preorder α] : preorder (with_zero α) := with_bot.preorder
instance [partial_order α] : partial_order (with_zero α) := with_bot.partial_order
instance [partial_order α] : order_bot (with_zero α) := with_bot.order_bot
instance [lattice α] : lattice (with_zero α) := with_bot.lattice
instance [linear_order α] : linear_order (with_zero α) := with_bot.linear_order
instance [decidable_linear_order α] :
decidable_linear_order (with_zero α) := with_bot.decidable_linear_order
/--
If `0` is the least element in `α`, then `with_zero α` is an `ordered_add_comm_monoid`.
-/
def ordered_add_comm_monoid [ordered_add_comm_monoid α]
(zero_le : ∀ a : α, 0 ≤ a) : ordered_add_comm_monoid (with_zero α) :=
begin
suffices, refine {
add_le_add_left := this,
..with_zero.partial_order,
..with_zero.add_comm_monoid, .. },
{ intros a b c h,
have h' := lt_iff_le_not_le.1 h,
rw lt_iff_le_not_le at ⊢,
refine ⟨λ b h₂, _, λ h₂, h'.2 $ this _ _ h₂ _⟩,
cases h₂, cases c with c,
{ cases h'.2 (this _ _ bot_le a) },
{ refine ⟨_, rfl, _⟩,
cases a with a,
{ exact with_bot.some_le_some.1 h'.1 },
{ exact le_of_lt (lt_of_add_lt_add_left $
with_bot.some_lt_some.1 h), } } },
{ intros a b h c ca h₂,
cases b with b,
{ rw le_antisymm h bot_le at h₂,
exact ⟨_, h₂, le_refl _⟩ },
cases a with a,
{ change c + 0 = some ca at h₂,
simp at h₂, simp [h₂],
exact ⟨_, rfl, by simpa using add_le_add_left (zero_le b) _⟩ },
{ simp at h,
cases c with c; change some _ = _ at h₂;
simp [-add_comm] at h₂; subst ca; refine ⟨_, rfl, _⟩,
{ exact h },
{ exact add_le_add_left h _ } } }
end
end with_zero
namespace with_top
section has_one
variables [has_one α]
@[to_additive] instance : has_one (with_top α) := ⟨(1 : α)⟩
@[simp, to_additive] lemma coe_one : ((1 : α) : with_top α) = 1 := rfl
@[simp, to_additive] lemma coe_eq_one {a : α} : (a : with_top α) = 1 ↔ a = 1 :=
coe_eq_coe
@[simp, to_additive] theorem one_eq_coe {a : α} : 1 = (a : with_top α) ↔ a = 1 :=
by rw [eq_comm, coe_eq_one]
attribute [norm_cast] coe_one coe_eq_one coe_zero coe_eq_zero one_eq_coe zero_eq_coe
@[simp, to_additive] theorem top_ne_one : ⊤ ≠ (1 : with_top α) .
@[simp, to_additive] theorem one_ne_top : (1 : with_top α) ≠ ⊤ .
end has_one
instance [has_add α] : has_add (with_top α) :=
⟨λ o₁ o₂, o₁.bind (λ a, o₂.map (λ b, a + b))⟩
instance [add_semigroup α] : add_semigroup (with_top α) :=
{ add := (+),
..@additive.add_semigroup _ $ @with_zero.semigroup (multiplicative α) _ }
@[norm_cast] lemma coe_add [has_add α] {a b : α} : ((a + b : α) : with_top α) = a + b := rfl
@[norm_cast] lemma coe_bit0 [has_add α] {a : α} : ((bit0 a : α) : with_top α) = bit0 a := rfl
@[norm_cast]
lemma coe_bit1 [has_add α] [has_one α] {a : α} : ((bit1 a : α) : with_top α) = bit1 a := rfl
instance [add_comm_semigroup α] : add_comm_semigroup (with_top α) :=
{ ..@additive.add_comm_semigroup _ $
@with_zero.comm_semigroup (multiplicative α) _ }
instance [add_monoid α] : add_monoid (with_top α) :=
{ zero := some 0,
add := (+),
..@additive.add_monoid _ $ @monoid_with_zero.to_monoid _ $
@with_zero.monoid_with_zero (multiplicative α) _ }
instance [add_comm_monoid α] : add_comm_monoid (with_top α) :=
{ zero := 0,
add := (+),
..@additive.add_comm_monoid _ $ @comm_monoid_with_zero.to_comm_monoid _ $
@with_zero.comm_monoid_with_zero (multiplicative α) _ }
instance [ordered_add_comm_monoid α] : ordered_add_comm_monoid (with_top α) :=
begin
suffices, refine {
add_le_add_left := this,
..with_top.partial_order,
..with_top.add_comm_monoid, ..},
{ intros a b c h,
have h' := h,
rw lt_iff_le_not_le at h' ⊢,
refine ⟨λ c h₂, _, λ h₂, h'.2 $ this _ _ h₂ _⟩,
cases h₂, cases a with a,
{ exact (not_le_of_lt h).elim le_top },
cases b with b,
{ exact (not_le_of_lt h).elim le_top },
{ exact ⟨_, rfl, le_of_lt (lt_of_add_lt_add_left $
with_top.some_lt_some.1 h)⟩ } },
{ intros a b h c ca h₂,
cases c with c, {cases h₂},
cases b with b; cases h₂,
cases a with a, {cases le_antisymm h le_top },
simp at h,
exact ⟨_, rfl, add_le_add_left h _⟩, }
end
@[simp] lemma zero_lt_top [ordered_add_comm_monoid α] : (0 : with_top α) < ⊤ :=
coe_lt_top 0
@[simp, norm_cast] lemma zero_lt_coe [ordered_add_comm_monoid α] (a : α) : (0 : with_top α) < a ↔ 0 < a :=
coe_lt_coe
@[simp] lemma add_top [ordered_add_comm_monoid α] : ∀{a : with_top α}, a + ⊤ = ⊤
| none := rfl
| (some a) := rfl
@[simp] lemma top_add [ordered_add_comm_monoid α] {a : with_top α} : ⊤ + a = ⊤ := rfl
lemma add_eq_top [ordered_add_comm_monoid α] (a b : with_top α) : a + b = ⊤ ↔ a = ⊤ ∨ b = ⊤ :=
by cases a; cases b; simp [none_eq_top, some_eq_coe, coe_add.symm]
lemma add_lt_top [ordered_add_comm_monoid α] (a b : with_top α) : a + b < ⊤ ↔ a < ⊤ ∧ b < ⊤ :=
by simp [lt_top_iff_ne_top, add_eq_top, not_or_distrib]
end with_top
namespace with_bot
instance [has_zero α] : has_zero (with_bot α) := with_top.has_zero
instance [has_one α] : has_one (with_bot α) := with_top.has_one
instance [add_semigroup α] : add_semigroup (with_bot α) := with_top.add_semigroup
instance [add_comm_semigroup α] : add_comm_semigroup (with_bot α) := with_top.add_comm_semigroup
instance [add_monoid α] : add_monoid (with_bot α) := with_top.add_monoid
instance [add_comm_monoid α] : add_comm_monoid (with_bot α) := with_top.add_comm_monoid
instance [ordered_add_comm_monoid α] : ordered_add_comm_monoid (with_bot α) :=
begin
suffices, refine {
add_le_add_left := this,
..with_bot.partial_order,
..with_bot.add_comm_monoid, ..},
{ intros a b c h,
have h' := h,
rw lt_iff_le_not_le at h' ⊢,
refine ⟨λ b h₂, _, λ h₂, h'.2 $ this _ _ h₂ _⟩,
cases h₂, cases a with a,
{ exact (not_le_of_lt h).elim bot_le },
cases c with c,
{ exact (not_le_of_lt h).elim bot_le },
{ exact ⟨_, rfl, le_of_lt (lt_of_add_lt_add_left $
with_bot.some_lt_some.1 h)⟩ } },
{ intros a b h c ca h₂,
cases c with c, {cases h₂},
cases a with a; cases h₂,
cases b with b, {cases le_antisymm h bot_le},
simp at h,
exact ⟨_, rfl, add_le_add_left h _⟩, }
end
-- `by norm_cast` proves this lemma, so I did not tag it with `norm_cast`
lemma coe_zero [has_zero α] : ((0 : α) : with_bot α) = 0 := rfl
-- `by norm_cast` proves this lemma, so I did not tag it with `norm_cast`
lemma coe_one [has_one α] : ((1 : α) : with_bot α) = 1 := rfl
-- `by norm_cast` proves this lemma, so I did not tag it with `norm_cast`
lemma coe_eq_zero {α : Type*}
[add_monoid α] {a : α} : (a : with_bot α) = 0 ↔ a = 0 :=
by norm_cast
-- `by norm_cast` proves this lemma, so I did not tag it with `norm_cast`
lemma coe_add [add_semigroup α] (a b : α) : ((a + b : α) : with_bot α) = a + b := by norm_cast
-- `by norm_cast` proves this lemma, so I did not tag it with `norm_cast`
lemma coe_bit0 [add_semigroup α] {a : α} : ((bit0 a : α) : with_bot α) = bit0 a :=
by norm_cast
-- `by norm_cast` proves this lemma, so I did not tag it with `norm_cast`
lemma coe_bit1 [add_semigroup α] [has_one α] {a : α} : ((bit1 a : α) : with_bot α) = bit1 a :=
by norm_cast
@[simp] lemma bot_add [ordered_add_comm_monoid α] (a : with_bot α) : ⊥ + a = ⊥ := rfl
@[simp] lemma add_bot [ordered_add_comm_monoid α] (a : with_bot α) : a + ⊥ = ⊥ := by cases a; refl
end with_bot
/-- A canonically ordered monoid is an ordered commutative monoid
in which the ordering coincides with the divisibility relation,
which is to say, `a ≤ b` iff there exists `c` with `b = a + c`.
This is satisfied by the natural numbers, for example, but not
the integers or other ordered groups. -/
@[protect_proj]
class canonically_ordered_add_monoid (α : Type*) extends ordered_add_comm_monoid α, order_bot α :=
(le_iff_exists_add : ∀a b:α, a ≤ b ↔ ∃c, b = a + c)
section canonically_ordered_add_monoid
variables [canonically_ordered_add_monoid α] {a b c d : α}
lemma le_iff_exists_add : a ≤ b ↔ ∃c, b = a + c :=
canonically_ordered_add_monoid.le_iff_exists_add a b
@[simp] lemma zero_le (a : α) : 0 ≤ a := le_iff_exists_add.mpr ⟨a, by simp⟩
@[simp] lemma bot_eq_zero : (⊥ : α) = 0 :=
le_antisymm bot_le (zero_le ⊥)
@[simp] lemma add_eq_zero_iff : a + b = 0 ↔ a = 0 ∧ b = 0 :=
add_eq_zero_iff' (zero_le _) (zero_le _)
@[simp] lemma le_zero_iff_eq : a ≤ 0 ↔ a = 0 :=
iff.intro
(assume h, le_antisymm h (zero_le a))
(assume h, h ▸ le_refl a)
lemma zero_lt_iff_ne_zero : 0 < a ↔ a ≠ 0 :=
iff.intro ne_of_gt $ assume hne, lt_of_le_of_ne (zero_le _) hne.symm
@[nolint ge_or_gt]
lemma exists_pos_add_of_lt (h : a < b) : ∃ c > 0, a + c = b :=
begin
obtain ⟨c, hc⟩ := le_iff_exists_add.1 (le_of_lt h),
refine ⟨c, zero_lt_iff_ne_zero.2 _, hc.symm⟩,
rintro rfl,
simpa [hc, lt_irrefl] using h
end
lemma le_add_left (h : a ≤ c) : a ≤ b + c :=
calc a = 0 + a : by simp
... ≤ b + c : add_le_add (zero_le _) h
lemma le_add_right (h : a ≤ b) : a ≤ b + c :=
calc a = a + 0 : by simp
... ≤ b + c : add_le_add h (zero_le _)
instance with_zero.canonically_ordered_add_monoid :
canonically_ordered_add_monoid (with_zero α) :=
{ le_iff_exists_add := λ a b, begin
cases a with a,
{ exact iff_of_true bot_le ⟨b, (zero_add b).symm⟩ },
cases b with b,
{ exact iff_of_false
(mt (le_antisymm bot_le) (by simp))
(λ ⟨c, h⟩, by cases c; cases h) },
{ simp [le_iff_exists_add, -add_comm],
split; intro h; rcases h with ⟨c, h⟩,
{ exact ⟨some c, congr_arg some h⟩ },
{ cases c; cases h,
{ exact ⟨_, (add_zero _).symm⟩ },
{ exact ⟨_, rfl⟩ } } }
end,
bot := 0,
bot_le := assume a a' h, option.no_confusion h,
.. with_zero.ordered_add_comm_monoid zero_le }
instance with_top.canonically_ordered_add_monoid : canonically_ordered_add_monoid (with_top α) :=
{ le_iff_exists_add := assume a b,
match a, b with
| a, none := show a ≤ ⊤ ↔ ∃c, ⊤ = a + c, by simp; refine ⟨⊤, _⟩; cases a; refl
| (some a), (some b) := show (a:with_top α) ≤ ↑b ↔ ∃c:with_top α, ↑b = ↑a + c,
begin
simp [canonically_ordered_add_monoid.le_iff_exists_add, -add_comm],
split,
{ rintro ⟨c, rfl⟩, refine ⟨c, _⟩, norm_cast },
{ exact assume h, match b, h with _, ⟨some c, rfl⟩ := ⟨_, rfl⟩ end }
end
| none, some b := show (⊤ : with_top α) ≤ b ↔ ∃c:with_top α, ↑b = ⊤ + c, by simp
end,
.. with_top.order_bot,
.. with_top.ordered_add_comm_monoid }
end canonically_ordered_add_monoid
/-- An ordered cancellative additive commutative monoid
is an additive commutative monoid with a partial order,
in which addition is cancellative and strictly monotone. -/
@[protect_proj, ancestor add_comm_monoid add_left_cancel_semigroup add_right_cancel_semigroup partial_order]
class ordered_cancel_add_comm_monoid (α : Type u)
extends add_comm_monoid α, add_left_cancel_semigroup α,
add_right_cancel_semigroup α, partial_order α :=
(add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b)
(le_of_add_le_add_left : ∀ a b c : α, a + b ≤ a + c → b ≤ c)
/-- An ordered cancellative commutative monoid
is a commutative monoid with a partial order,
in which multiplication is cancellative and strictly monotone. -/
@[protect_proj, ancestor comm_monoid left_cancel_semigroup right_cancel_semigroup partial_order]
class ordered_cancel_comm_monoid (α : Type u)
extends comm_monoid α, left_cancel_semigroup α,
right_cancel_semigroup α, partial_order α :=
(mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b)
(le_of_mul_le_mul_left : ∀ a b c : α, a * b ≤ a * c → b ≤ c)
attribute [to_additive ordered_cancel_add_comm_monoid] ordered_cancel_comm_monoid
section ordered_cancel_comm_monoid
variables [ordered_cancel_comm_monoid α] {a b c d : α}
@[to_additive]
instance ordered_cancel_comm_monoid.to_left_cancel_monoid :
left_cancel_monoid α := { ..‹ordered_cancel_comm_monoid α› }
@[to_additive le_of_add_le_add_left]
lemma le_of_mul_le_mul_left' : ∀ {a b c : α}, a * b ≤ a * c → b ≤ c :=
ordered_cancel_comm_monoid.le_of_mul_le_mul_left
@[to_additive]
instance ordered_cancel_comm_monoid.to_ordered_comm_monoid : ordered_comm_monoid α :=
{ lt_of_mul_lt_mul_left := λ a b c h, lt_of_le_not_le (le_of_mul_le_mul_left' (le_of_lt h)) $
mt (λ h, ordered_cancel_comm_monoid.mul_le_mul_left _ _ h _) (not_le_of_gt h),
..‹ordered_cancel_comm_monoid α› }
@[to_additive add_lt_add_left]
lemma mul_lt_mul_left' (h : a < b) (c : α) : c * a < c * b :=
lt_of_le_not_le (mul_le_mul_left' (le_of_lt h) _) $
mt le_of_mul_le_mul_left' (not_le_of_gt h)
@[to_additive add_lt_add_right]
lemma mul_lt_mul_right' (h : a < b) (c : α) : a * c < b * c :=
begin
rw [mul_comm a c, mul_comm b c],
exact (mul_lt_mul_left' h c)
end
@[to_additive add_lt_add]
lemma mul_lt_mul''' (h₁ : a < b) (h₂ : c < d) : a * c < b * d :=
lt_trans (mul_lt_mul_right' h₁ c) (mul_lt_mul_left' h₂ b)
@[to_additive]
lemma mul_lt_mul_of_le_of_lt (h₁ : a ≤ b) (h₂ : c < d) : a * c < b * d :=
lt_of_le_of_lt (mul_le_mul_right' h₁ _) (mul_lt_mul_left' h₂ b)
@[to_additive]
lemma mul_lt_mul_of_lt_of_le (h₁ : a < b) (h₂ : c ≤ d) : a * c < b * d :=
lt_of_lt_of_le (mul_lt_mul_right' h₁ c) (mul_le_mul_left' h₂ _)
@[to_additive]
lemma lt_mul_of_one_lt_right (a : α) {b : α} (h : 1 < b) : a < a * b :=
have a * 1 < a * b, from mul_lt_mul_left' h a,
by rwa [mul_one] at this
@[to_additive]
lemma lt_mul_of_one_lt_left (a : α) {b : α} (h : 1 < b) : a < b * a :=
have 1 * a < b * a, from mul_lt_mul_right' h a,
by rwa [one_mul] at this
@[to_additive le_of_add_le_add_right]
lemma le_of_mul_le_mul_right' (h : a * b ≤ c * b) : a ≤ c :=
le_of_mul_le_mul_left'
(show b * a ≤ b * c, begin rw [mul_comm b a, mul_comm b c], assumption end)
@[to_additive]
lemma mul_lt_one (ha : a < 1) (hb : b < 1) : a * b < 1 :=
one_mul (1:α) ▸ (mul_lt_mul''' ha hb)
@[to_additive]
lemma mul_lt_one_of_lt_one_of_le_one (ha : a < 1) (hb : b ≤ 1) : a * b < 1 :=
one_mul (1:α) ▸ (mul_lt_mul_of_lt_of_le ha hb)
@[to_additive]
lemma mul_lt_one_of_le_one_of_lt_one (ha : a ≤ 1) (hb : b < 1) : a * b < 1 :=
one_mul (1:α) ▸ (mul_lt_mul_of_le_of_lt ha hb)
@[to_additive]
lemma lt_mul_of_one_lt_of_le (ha : 1 < a) (hbc : b ≤ c) : b < a * c :=
one_mul b ▸ mul_lt_mul_of_lt_of_le ha hbc
@[to_additive]
lemma lt_mul_of_le_of_one_lt (hbc : b ≤ c) (ha : 1 < a) : b < c * a :=
mul_one b ▸ mul_lt_mul_of_le_of_lt hbc ha
@[to_additive]
lemma mul_le_of_le_one_of_le (ha : a ≤ 1) (hbc : b ≤ c) : a * b ≤ c :=
one_mul c ▸ mul_le_mul' ha hbc
@[to_additive]
lemma mul_le_of_le_of_le_one (hbc : b ≤ c) (ha : a ≤ 1) : b * a ≤ c :=
mul_one c ▸ mul_le_mul' hbc ha
@[to_additive]
lemma mul_lt_of_lt_one_of_le (ha : a < 1) (hbc : b ≤ c) : a * b < c :=
one_mul c ▸ mul_lt_mul_of_lt_of_le ha hbc
@[to_additive]
lemma mul_lt_of_le_of_lt_one (hbc : b ≤ c) (ha : a < 1) : b * a < c :=
mul_one c ▸ mul_lt_mul_of_le_of_lt hbc ha
@[to_additive]
lemma lt_mul_of_one_le_of_lt (ha : 1 ≤ a) (hbc : b < c) : b < a * c :=
one_mul b ▸ mul_lt_mul_of_le_of_lt ha hbc
@[to_additive]
lemma lt_mul_of_lt_of_one_le (hbc : b < c) (ha : 1 ≤ a) : b < c * a :=
mul_one b ▸ mul_lt_mul_of_lt_of_le hbc ha
@[to_additive]
lemma lt_mul_of_one_lt_of_lt (ha : 1 < a) (hbc : b < c) : b < a * c :=
one_mul b ▸ mul_lt_mul''' ha hbc
@[to_additive]
lemma lt_mul_of_lt_of_one_lt (hbc : b < c) (ha : 1 < a) : b < c * a :=
mul_one b ▸ mul_lt_mul''' hbc ha
@[to_additive]
lemma mul_lt_of_le_one_of_lt (ha : a ≤ 1) (hbc : b < c) : a * b < c :=
one_mul c ▸ mul_lt_mul_of_le_of_lt ha hbc
@[to_additive]
lemma mul_lt_of_lt_of_le_one (hbc : b < c) (ha : a ≤ 1) : b * a < c :=
mul_one c ▸ mul_lt_mul_of_lt_of_le hbc ha
@[to_additive]
lemma mul_lt_of_lt_one_of_lt (ha : a < 1) (hbc : b < c) : a * b < c :=
one_mul c ▸ mul_lt_mul''' ha hbc
@[to_additive]
lemma mul_lt_of_lt_of_lt_one (hbc : b < c) (ha : a < 1) : b * a < c :=
mul_one c ▸ mul_lt_mul''' hbc ha
@[simp, to_additive]
lemma mul_le_mul_iff_left (a : α) {b c : α} : a * b ≤ a * c ↔ b ≤ c :=
⟨le_of_mul_le_mul_left', λ h, mul_le_mul_left' h _⟩
@[simp, to_additive]
lemma mul_le_mul_iff_right (c : α) : a * c ≤ b * c ↔ a ≤ b :=
mul_comm c a ▸ mul_comm c b ▸ mul_le_mul_iff_left c
@[simp, to_additive]
lemma mul_lt_mul_iff_left (a : α) {b c : α} : a * b < a * c ↔ b < c :=
⟨lt_of_mul_lt_mul_left', λ h, mul_lt_mul_left' h _⟩
@[simp, to_additive]
lemma mul_lt_mul_iff_right (c : α) : a * c < b * c ↔ a < b :=
mul_comm c a ▸ mul_comm c b ▸ mul_lt_mul_iff_left c
@[simp, to_additive le_add_iff_nonneg_right]
lemma le_mul_iff_one_le_right' (a : α) {b : α} : a ≤ a * b ↔ 1 ≤ b :=
have a * 1 ≤ a * b ↔ 1 ≤ b, from mul_le_mul_iff_left a,
by rwa mul_one at this
@[simp, to_additive le_add_iff_nonneg_left]
lemma le_mul_iff_one_le_left' (a : α) {b : α} : a ≤ b * a ↔ 1 ≤ b :=
by rw [mul_comm, le_mul_iff_one_le_right']
@[simp, to_additive lt_add_iff_pos_right]
lemma lt_mul_iff_one_lt_right' (a : α) {b : α} : a < a * b ↔ 1 < b :=
have a * 1 < a * b ↔ 1 < b, from mul_lt_mul_iff_left a,
by rwa mul_one at this
@[simp, to_additive lt_add_iff_pos_left]
lemma lt_mul_iff_one_lt_left' (a : α) {b : α} : a < b * a ↔ 1 < b :=
by rw [mul_comm, lt_mul_iff_one_lt_right']
@[simp, to_additive add_le_iff_nonpos_left]
lemma mul_le_iff_le_one_left' : a * b ≤ b ↔ a ≤ 1 :=
by { convert mul_le_mul_iff_right b, rw [one_mul] }
@[simp, to_additive add_le_iff_nonpos_right]
lemma mul_le_iff_le_one_right' : a * b ≤ a ↔ b ≤ 1 :=
by { convert mul_le_mul_iff_left a, rw [mul_one] }
@[simp, to_additive add_lt_iff_neg_right]
lemma mul_lt_iff_lt_one_right' : a * b < b ↔ a < 1 :=
by { convert mul_lt_mul_iff_right b, rw [one_mul] }
@[simp, to_additive add_lt_iff_neg_left]
lemma mul_lt_iff_lt_one_left' : a * b < a ↔ b < 1 :=
by { convert mul_lt_mul_iff_left a, rw [mul_one] }
@[to_additive]
lemma mul_eq_one_iff_eq_one_of_one_le
(ha : 1 ≤ a) (hb : 1 ≤ b) : a * b = 1 ↔ a = 1 ∧ b = 1 :=
⟨λ hab : a * b = 1,
by split; apply le_antisymm; try {assumption};
rw ← hab; simp [ha, hb],
λ ⟨ha', hb'⟩, by rw [ha', hb', mul_one]⟩
section mono
variables {β : Type*} [preorder β] {f g : β → α}
@[to_additive monotone.add_strict_mono]
lemma monotone.mul_strict_mono' (hf : monotone f) (hg : strict_mono g) :
strict_mono (λ x, f x * g x) :=
λ x y h, mul_lt_mul_of_le_of_lt (hf $ le_of_lt h) (hg h)
@[to_additive strict_mono.add_monotone]
lemma strict_mono.mul_monotone' (hf : strict_mono f) (hg : monotone g) :
strict_mono (λ x, f x * g x) :=
λ x y h, mul_lt_mul_of_lt_of_le (hf h) (hg $ le_of_lt h)
@[to_additive strict_mono.add_const]
lemma strict_mono.mul_const' (hf : strict_mono f) (c : α) :
strict_mono (λ x, f x * c) :=
hf.mul_monotone' monotone_const
@[to_additive strict_mono.const_add]
lemma strict_mono.const_mul' (hf : strict_mono f) (c : α) :
strict_mono (λ x, c * f x) :=
monotone_const.mul_strict_mono' hf
end mono
end ordered_cancel_comm_monoid
section ordered_cancel_add_comm_monoid
variable [ordered_cancel_add_comm_monoid α]
lemma with_top.add_lt_add_iff_left :
∀{a b c : with_top α}, a < ⊤ → (a + c < a + b ↔ c < b)
| none := assume b c h, (lt_irrefl ⊤ h).elim
| (some a) :=
begin
assume b c h,
cases b; cases c;
simp [with_top.none_eq_top, with_top.some_eq_coe, with_top.coe_lt_top, with_top.coe_lt_coe],
{ norm_cast, exact with_top.coe_lt_top _ },
{ norm_cast, exact add_lt_add_iff_left _ }
end
lemma with_top.add_lt_add_iff_right
{a b c : with_top α} : a < ⊤ → (c + a < b + a ↔ c < b) :=
by simpa [add_comm] using @with_top.add_lt_add_iff_left _ _ a b c
end ordered_cancel_add_comm_monoid
/-- 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_add_comm_group] ordered_comm_group
/--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 _,
.. units.partial_order,
.. (infer_instance : comm_group (units α)) }
section ordered_comm_group
variables [ordered_comm_group α] {a b c d : α}
@[to_additive ordered_add_comm_group.add_lt_add_left]
lemma ordered_comm_group.mul_lt_mul_left' (a b : α) (h : a < b) (c : α) : c * a < c * b :=
begin
rw lt_iff_le_not_le at h ⊢,
split,
{ apply ordered_comm_group.mul_le_mul_left _ _ h.1 },
{ intro w,
replace w : c⁻¹ * (c * b) ≤ c⁻¹ * (c * a) := ordered_comm_group.mul_le_mul_left _ _ w _,
simp only [mul_one, mul_comm, mul_left_inv, mul_left_comm] at w,
exact h.2 w },
end
@[to_additive ordered_add_comm_group.le_of_add_le_add_left]
lemma ordered_comm_group.le_of_mul_le_mul_left (h : a * b ≤ a * c) : b ≤ c :=
have a⁻¹ * (a * b) ≤ a⁻¹ * (a * c), from ordered_comm_group.mul_le_mul_left _ _ h _,
begin simp [inv_mul_cancel_left] at this, assumption end
@[to_additive]
lemma ordered_comm_group.lt_of_mul_lt_mul_left (h : a * b < a * c) : b < c :=
have a⁻¹ * (a * b) < a⁻¹ * (a * c), from ordered_comm_group.mul_lt_mul_left' _ _ h _,
begin simp [inv_mul_cancel_left] at this, assumption end
@[to_additive]
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 := @mul_left_cancel α _,
mul_right_cancel := @mul_right_cancel α _,
le_of_mul_le_mul_left := @ordered_comm_group.le_of_mul_le_mul_left α _,
..s }
@[to_additive neg_le_neg]
lemma inv_le_inv' (h : a ≤ b) : b⁻¹ ≤ a⁻¹ :=
have 1 ≤ a⁻¹ * b, from mul_left_inv a ▸ mul_le_mul_left' h _,
have 1 * b⁻¹ ≤ a⁻¹ * b * b⁻¹, from mul_le_mul_right' this _,
by rwa [mul_inv_cancel_right, one_mul] at this
@[to_additive]
lemma le_of_inv_le_inv (h : b⁻¹ ≤ a⁻¹) : a ≤ b :=
suffices (a⁻¹)⁻¹ ≤ (b⁻¹)⁻¹, from
begin simp [inv_inv] at this, assumption end,
inv_le_inv' h
@[to_additive]
lemma one_le_of_inv_le_one (h : a⁻¹ ≤ 1) : 1 ≤ a :=
have a⁻¹ ≤ 1⁻¹, by rwa one_inv,
le_of_inv_le_inv this
@[to_additive]
lemma inv_le_one_of_one_le (h : 1 ≤ a) : a⁻¹ ≤ 1 :=
have a⁻¹ ≤ 1⁻¹, from inv_le_inv' h,
by rwa one_inv at this
@[to_additive nonpos_of_neg_nonneg]
lemma le_one_of_one_le_inv (h : 1 ≤ a⁻¹) : a ≤ 1 :=
have 1⁻¹ ≤ a⁻¹, by rwa one_inv,
le_of_inv_le_inv this
@[to_additive neg_nonneg_of_nonpos]
lemma one_le_inv_of_le_one (h : a ≤ 1) : 1 ≤ a⁻¹ :=
have 1⁻¹ ≤ a⁻¹, from inv_le_inv' h,
by rwa one_inv at this
@[to_additive neg_lt_neg]
lemma inv_lt_inv' (h : a < b) : b⁻¹ < a⁻¹ :=
have 1 < a⁻¹ * b, from mul_left_inv a ▸ mul_lt_mul_left' h (a⁻¹),
have 1 * b⁻¹ < a⁻¹ * b * b⁻¹, from mul_lt_mul_right' this (b⁻¹),
by rwa [mul_inv_cancel_right, one_mul] at this
@[to_additive]
lemma lt_of_inv_lt_inv (h : b⁻¹ < a⁻¹) : a < b :=
inv_inv a ▸ inv_inv b ▸ inv_lt_inv' h
@[to_additive]
lemma one_lt_of_inv_inv (h : a⁻¹ < 1) : 1 < a :=
have a⁻¹ < 1⁻¹, by rwa one_inv,
lt_of_inv_lt_inv this
@[to_additive]
lemma inv_inv_of_one_lt (h : 1 < a) : a⁻¹ < 1 :=
have a⁻¹ < 1⁻¹, from inv_lt_inv' h,
by rwa one_inv at this
@[to_additive neg_of_neg_pos]
lemma inv_of_one_lt_inv (h : 1 < a⁻¹) : a < 1 :=
have 1⁻¹ < a⁻¹, by rwa one_inv,
lt_of_inv_lt_inv this
@[to_additive neg_pos_of_neg]
lemma one_lt_inv_of_inv (h : a < 1) : 1 < a⁻¹ :=
have 1⁻¹ < a⁻¹, from inv_lt_inv' h,
by rwa one_inv at this
@[to_additive]
lemma le_inv_of_le_inv (h : a ≤ b⁻¹) : b ≤ a⁻¹ :=
begin
have h := inv_le_inv' h,
rwa inv_inv at h
end
@[to_additive]
lemma inv_le_of_inv_le (h : a⁻¹ ≤ b) : b⁻¹ ≤ a :=
begin
have h := inv_le_inv' h,
rwa inv_inv at h
end
@[to_additive]
lemma lt_inv_of_lt_inv (h : a < b⁻¹) : b < a⁻¹ :=
begin
have h := inv_lt_inv' h,
rwa inv_inv at h
end
@[to_additive]
lemma inv_lt_of_inv_lt (h : a⁻¹ < b) : b⁻¹ < a :=
begin
have h := inv_lt_inv' h,
rwa inv_inv at h
end
@[to_additive]
lemma mul_le_of_le_inv_mul (h : b ≤ a⁻¹ * c) : a * b ≤ c :=
begin
have h := mul_le_mul_left' h a,
rwa mul_inv_cancel_left at h
end
@[to_additive]
lemma le_inv_mul_of_mul_le (h : a * b ≤ c) : b ≤ a⁻¹ * c :=
begin
have h := mul_le_mul_left' h a⁻¹,
rwa inv_mul_cancel_left at h
end
@[to_additive]
lemma le_mul_of_inv_mul_le (h : b⁻¹ * a ≤ c) : a ≤ b * c :=
begin
have h := mul_le_mul_left' h b,
rwa mul_inv_cancel_left at h
end
@[to_additive]
lemma inv_mul_le_of_le_mul (h : a ≤ b * c) : b⁻¹ * a ≤ c :=
begin
have h := mul_le_mul_left' h b⁻¹,
rwa inv_mul_cancel_left at h
end
@[to_additive]
lemma le_mul_of_inv_mul_le_left (h : b⁻¹ * a ≤ c) : a ≤ b * c :=
le_mul_of_inv_mul_le h
@[to_additive]
lemma inv_mul_le_left_of_le_mul (h : a ≤ b * c) : b⁻¹ * a ≤ c :=
inv_mul_le_of_le_mul h
@[to_additive]
lemma le_mul_of_inv_mul_le_right (h : c⁻¹ * a ≤ b) : a ≤ b * c :=
by { rw mul_comm, exact le_mul_of_inv_mul_le h }
@[to_additive]
lemma inv_mul_le_right_of_le_mul (h : a ≤ b * c) : c⁻¹ * a ≤ b :=
by { rw mul_comm at h, apply inv_mul_le_left_of_le_mul h }
@[to_additive]
lemma mul_lt_of_lt_inv_mul (h : b < a⁻¹ * c) : a * b < c :=
begin
have h := mul_lt_mul_left' h a,
rwa mul_inv_cancel_left at h
end
@[to_additive]
lemma lt_inv_mul_of_mul_lt (h : a * b < c) : b < a⁻¹ * c :=
begin
have h := mul_lt_mul_left' h (a⁻¹),
rwa inv_mul_cancel_left at h
end
@[to_additive]
lemma lt_mul_of_inv_mul_lt (h : b⁻¹ * a < c) : a < b * c :=
begin
have h := mul_lt_mul_left' h b,
rwa mul_inv_cancel_left at h
end
@[to_additive]
lemma inv_mul_lt_of_lt_mul (h : a < b * c) : b⁻¹ * a < c :=
begin
have h := mul_lt_mul_left' h (b⁻¹),
rwa inv_mul_cancel_left at h
end
@[to_additive]
lemma lt_mul_of_inv_mul_lt_left (h : b⁻¹ * a < c) : a < b * c :=
lt_mul_of_inv_mul_lt h
@[to_additive]
lemma inv_mul_lt_left_of_lt_mul (h : a < b * c) : b⁻¹ * a < c :=
inv_mul_lt_of_lt_mul h
@[to_additive]
lemma lt_mul_of_inv_mul_lt_right (h : c⁻¹ * a < b) : a < b * c :=
by { rw mul_comm, exact lt_mul_of_inv_mul_lt h }
@[to_additive]
lemma inv_mul_lt_right_of_lt_mul (h : a < b * c) : c⁻¹ * a < b :=
by { rw mul_comm at h, exact inv_mul_lt_of_lt_mul h }
@[simp, to_additive]
lemma inv_lt_one_iff_one_lt : a⁻¹ < 1 ↔ 1 < a :=
⟨ one_lt_of_inv_inv, inv_inv_of_one_lt ⟩
@[simp, to_additive]
lemma inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a :=
have a * b * a⁻¹ ≤ a * b * b⁻¹ ↔ a⁻¹ ≤ b⁻¹, from mul_le_mul_iff_left _,
by { rw [mul_inv_cancel_right, mul_comm a, mul_inv_cancel_right] at this, rw [this] }
@[to_additive neg_le]
lemma inv_le' : a⁻¹ ≤ b ↔ b⁻¹ ≤ a :=
have a⁻¹ ≤ (b⁻¹)⁻¹ ↔ b⁻¹ ≤ a, from inv_le_inv_iff,
by rwa inv_inv at this
@[to_additive le_neg]
lemma le_inv' : a ≤ b⁻¹ ↔ b ≤ a⁻¹ :=
have (a⁻¹)⁻¹ ≤ b⁻¹ ↔ b ≤ a⁻¹, from inv_le_inv_iff,
by rwa inv_inv at this
@[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 : a ≤ b⁻¹ ↔ a * b ≤ 1 :=
(mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_self
@[simp, to_additive neg_nonpos]
lemma inv_le_one' : a⁻¹ ≤ 1 ↔ 1 ≤ a :=
have a⁻¹ ≤ 1⁻¹ ↔ 1 ≤ a, from inv_le_inv_iff,
by rwa one_inv at this
@[simp, to_additive neg_nonneg]
lemma one_le_inv' : 1 ≤ a⁻¹ ↔ a ≤ 1 :=
have 1⁻¹ ≤ a⁻¹ ↔ a ≤ 1, from inv_le_inv_iff,
by rwa one_inv at this
@[to_additive]
lemma inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a :=
le_trans (inv_le_one'.2 h) h
@[to_additive]
lemma self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ :=
le_trans h (one_le_inv'.2 h)
@[simp, to_additive]
lemma inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a :=
have a * b * a⁻¹ < a * b * b⁻¹ ↔ a⁻¹ < b⁻¹, from mul_lt_mul_iff_left _,
by { rw [mul_inv_cancel_right, mul_comm a, mul_inv_cancel_right] at this, rw [this] }
@[to_additive neg_lt_zero]
lemma inv_lt_one' : a⁻¹ < 1 ↔ 1 < a :=
have a⁻¹ < 1⁻¹ ↔ 1 < a, from inv_lt_inv_iff,
by rwa one_inv at this
@[to_additive neg_pos]
lemma one_lt_inv' : 1 < a⁻¹ ↔ a < 1 :=
have 1⁻¹ < a⁻¹ ↔ a < 1, from inv_lt_inv_iff,
by rwa one_inv at this
@[to_additive neg_lt]
lemma inv_lt' : a⁻¹ < b ↔ b⁻¹ < a :=
have a⁻¹ < (b⁻¹)⁻¹ ↔ b⁻¹ < a, from inv_lt_inv_iff,
by rwa inv_inv at this
@[to_additive lt_neg]
lemma lt_inv' : a < b⁻¹ ↔ b < a⁻¹ :=
have (a⁻¹)⁻¹ < b⁻¹ ↔ b < a⁻¹, from inv_lt_inv_iff,
by rwa inv_inv at this
@[to_additive]
lemma le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c :=
have a⁻¹ * (a * b) ≤ a⁻¹ * c ↔ a * b ≤ c, from mul_le_mul_iff_left _,
by rwa inv_mul_cancel_left at this
@[simp, to_additive]
lemma inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c :=
have b⁻¹ * a ≤ b⁻¹ * (b * c) ↔ a ≤ b * c, from mul_le_mul_iff_left _,
by rwa inv_mul_cancel_left at this
@[to_additive]
lemma mul_inv_le_iff_le_mul : a * c⁻¹ ≤ b ↔ a ≤ b * c :=
by rw [mul_comm a, mul_comm b, inv_mul_le_iff_le_mul]
@[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]
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 lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c :=
have a⁻¹ * (a * b) < a⁻¹ * c ↔ a * b < c, from mul_lt_mul_iff_left _,
by rwa inv_mul_cancel_left at this
@[simp, to_additive]
lemma inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c :=
have b⁻¹ * a < b⁻¹ * (b * c) ↔ a < b * c, from mul_lt_mul_iff_left _,
by rwa inv_mul_cancel_left at this
@[to_additive]
lemma inv_mul_lt_iff_lt_mul_right : c⁻¹ * a < b ↔ a < b * c :=
by rw [inv_mul_lt_iff_lt_mul, mul_comm]
@[to_additive sub_le_sub_iff]
lemma div_le_div_iff' (a b c d : α) : a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b :=
begin
split ; intro h,
have := mul_le_mul_right' (mul_le_mul_right' h b) d,
rwa [inv_mul_cancel_right, mul_assoc _ _ b, mul_comm _ b, ← mul_assoc, inv_mul_cancel_right] at this,
have := mul_le_mul_right' (mul_le_mul_right' h d⁻¹) b⁻¹,
rwa [mul_inv_cancel_right, _root_.mul_assoc, _root_.mul_comm d⁻¹ b⁻¹, ← mul_assoc, mul_inv_cancel_right] at this,
end
end ordered_comm_group
section ordered_add_comm_group
variables [ordered_add_comm_group α] {a b c d : α}
lemma sub_nonneg_of_le (h : b ≤ a) : 0 ≤ a - b :=
begin
have h := add_le_add_right h (-b),
rwa add_right_neg at h
end
lemma le_of_sub_nonneg (h : 0 ≤ a - b) : b ≤ a :=
begin
have h := add_le_add_right h b,
rwa [sub_add_cancel, zero_add] at h
end
lemma sub_nonpos_of_le (h : a ≤ b) : a - b ≤ 0 :=
begin
have h := add_le_add_right h (-b),
rwa add_right_neg at h
end
lemma le_of_sub_nonpos (h : a - b ≤ 0) : a ≤ b :=
begin
have h := add_le_add_right h b,
rwa [sub_add_cancel, zero_add] at h
end
lemma sub_pos_of_lt (h : b < a) : 0 < a - b :=
begin
have h := add_lt_add_right h (-b),
rwa add_right_neg at h
end
lemma lt_of_sub_pos (h : 0 < a - b) : b < a :=
begin
have h := add_lt_add_right h b,
rwa [sub_add_cancel, zero_add] at h
end
lemma sub_neg_of_lt (h : a < b) : a - b < 0 :=
begin
have h := add_lt_add_right h (-b),
rwa add_right_neg at h
end
lemma lt_of_sub_neg (h : a - b < 0) : a < b :=
begin
have h := add_lt_add_right h b,
rwa [sub_add_cancel, zero_add] at h
end
lemma add_le_of_le_sub_left (h : b ≤ c - a) : a + b ≤ c :=
begin
have h := add_le_add_left h a,
rwa [← add_sub_assoc, add_comm a c, add_sub_cancel] at h
end
lemma le_sub_left_of_add_le (h : a + b ≤ c) : b ≤ c - a :=
begin
have h := add_le_add_right h (-a),
rwa [add_comm a b, add_neg_cancel_right] at h
end
lemma add_le_of_le_sub_right (h : a ≤ c - b) : a + b ≤ c :=
begin
have h := add_le_add_right h b,
rwa sub_add_cancel at h
end
lemma le_sub_right_of_add_le (h : a + b ≤ c) : a ≤ c - b :=
begin
have h := add_le_add_right h (-b),
rwa add_neg_cancel_right at h
end
lemma le_add_of_sub_left_le (h : a - b ≤ c) : a ≤ b + c :=
begin
have h := add_le_add_right h b,
rwa [sub_add_cancel, add_comm] at h
end
lemma sub_left_le_of_le_add (h : a ≤ b + c) : a - b ≤ c :=
begin
have h := add_le_add_right h (-b),
rwa [add_comm b c, add_neg_cancel_right] at h
end
lemma le_add_of_sub_right_le (h : a - c ≤ b) : a ≤ b + c :=
begin
have h := add_le_add_right h c,
rwa sub_add_cancel at h
end
lemma sub_right_le_of_le_add (h : a ≤ b + c) : a - c ≤ b :=
begin
have h := add_le_add_right h (-c),
rwa add_neg_cancel_right at h
end
lemma le_add_of_neg_le_sub_left (h : -a ≤ b - c) : c ≤ a + b :=
le_add_of_neg_add_le_left (add_le_of_le_sub_right h)
lemma neg_le_sub_left_of_le_add (h : c ≤ a + b) : -a ≤ b - c :=
begin
have h := le_neg_add_of_add_le (sub_left_le_of_le_add h),
rwa add_comm at h
end
lemma le_add_of_neg_le_sub_right (h : -b ≤ a - c) : c ≤ a + b :=
le_add_of_sub_right_le (add_le_of_le_sub_left h)
lemma neg_le_sub_right_of_le_add (h : c ≤ a + b) : -b ≤ a - c :=
le_sub_left_of_add_le (sub_right_le_of_le_add h)
lemma sub_le_of_sub_le (h : a - b ≤ c) : a - c ≤ b :=
sub_left_le_of_le_add (le_add_of_sub_right_le h)
lemma sub_le_sub_left (h : a ≤ b) (c : α) : c - b ≤ c - a :=
add_le_add_left (neg_le_neg h) c
lemma sub_le_sub_right (h : a ≤ b) (c : α) : a - c ≤ b - c :=
add_le_add_right h (-c)
lemma sub_le_sub (hab : a ≤ b) (hcd : c ≤ d) : a - d ≤ b - c :=
add_le_add hab (neg_le_neg hcd)
lemma add_lt_of_lt_sub_left (h : b < c - a) : a + b < c :=
begin
have h := add_lt_add_left h a,
rwa [← add_sub_assoc, add_comm a c, add_sub_cancel] at h
end
lemma lt_sub_left_of_add_lt (h : a + b < c) : b < c - a :=
begin
have h := add_lt_add_right h (-a),
rwa [add_comm a b, add_neg_cancel_right] at h
end
lemma add_lt_of_lt_sub_right (h : a < c - b) : a + b < c :=
begin
have h := add_lt_add_right h b,
rwa sub_add_cancel at h
end
lemma lt_sub_right_of_add_lt (h : a + b < c) : a < c - b :=
begin
have h := add_lt_add_right h (-b),
rwa add_neg_cancel_right at h
end
lemma lt_add_of_sub_left_lt (h : a - b < c) : a < b + c :=
begin
have h := add_lt_add_right h b,
rwa [sub_add_cancel, add_comm] at h
end
lemma sub_left_lt_of_lt_add (h : a < b + c) : a - b < c :=
begin
have h := add_lt_add_right h (-b),
rwa [add_comm b c, add_neg_cancel_right] at h
end
lemma lt_add_of_sub_right_lt (h : a - c < b) : a < b + c :=
begin
have h := add_lt_add_right h c,
rwa sub_add_cancel at h
end
lemma sub_right_lt_of_lt_add (h : a < b + c) : a - c < b :=
begin
have h := add_lt_add_right h (-c),
rwa add_neg_cancel_right at h
end
lemma lt_add_of_neg_lt_sub_left (h : -a < b - c) : c < a + b :=
lt_add_of_neg_add_lt_left (add_lt_of_lt_sub_right h)
lemma neg_lt_sub_left_of_lt_add (h : c < a + b) : -a < b - c :=
begin
have h := lt_neg_add_of_add_lt (sub_left_lt_of_lt_add h),
rwa add_comm at h
end
lemma lt_add_of_neg_lt_sub_right (h : -b < a - c) : c < a + b :=
lt_add_of_sub_right_lt (add_lt_of_lt_sub_left h)
lemma neg_lt_sub_right_of_lt_add (h : c < a + b) : -b < a - c :=
lt_sub_left_of_add_lt (sub_right_lt_of_lt_add h)
lemma sub_lt_of_sub_lt (h : a - b < c) : a - c < b :=
sub_left_lt_of_lt_add (lt_add_of_sub_right_lt h)
lemma sub_lt_sub_left (h : a < b) (c : α) : c - b < c - a :=
add_lt_add_left (neg_lt_neg h) c
lemma sub_lt_sub_right (h : a < b) (c : α) : a - c < b - c :=
add_lt_add_right h (-c)
lemma sub_lt_sub (hab : a < b) (hcd : c < d) : a - d < b - c :=
add_lt_add hab (neg_lt_neg hcd)
lemma sub_lt_sub_of_le_of_lt (hab : a ≤ b) (hcd : c < d) : a - d < b - c :=
add_lt_add_of_le_of_lt hab (neg_lt_neg hcd)
lemma sub_lt_sub_of_lt_of_le (hab : a < b) (hcd : c ≤ d) : a - d < b - c :=
add_lt_add_of_lt_of_le hab (neg_le_neg hcd)
lemma sub_le_self (a : α) {b : α} (h : 0 ≤ b) : a - b ≤ a :=
calc
a - b = a + -b : rfl
... ≤ a + 0 : add_le_add_left (neg_nonpos_of_nonneg h) _
... = a : by rw add_zero
lemma sub_lt_self (a : α) {b : α} (h : 0 < b) : a - b < a :=
calc
a - b = a + -b : rfl
... < a + 0 : add_lt_add_left (neg_neg_of_pos h) _
... = a : by rw add_zero
@[simp]
lemma sub_le_sub_iff_left (a : α) {b c : α} : a - b ≤ a - c ↔ c ≤ b :=
(add_le_add_iff_left _).trans neg_le_neg_iff
@[simp]
lemma sub_le_sub_iff_right (c : α) : a - c ≤ b - c ↔ a ≤ b :=
add_le_add_iff_right _
@[simp]
lemma sub_lt_sub_iff_left (a : α) {b c : α} : a - b < a - c ↔ c < b :=
(add_lt_add_iff_left _).trans neg_lt_neg_iff
@[simp]
lemma sub_lt_sub_iff_right (c : α) : a - c < b - c ↔ a < b :=
add_lt_add_iff_right _
@[simp] lemma sub_nonneg : 0 ≤ a - b ↔ b ≤ a :=
have a - a ≤ a - b ↔ b ≤ a, from sub_le_sub_iff_left a,
by rwa sub_self at this
@[simp] lemma sub_nonpos : a - b ≤ 0 ↔ a ≤ b :=
have a - b ≤ b - b ↔ a ≤ b, from sub_le_sub_iff_right b,
by rwa sub_self at this
@[simp] lemma sub_pos : 0 < a - b ↔ b < a :=
have a - a < a - b ↔ b < a, from sub_lt_sub_iff_left a,
by rwa sub_self at this
@[simp] lemma sub_lt_zero : a - b < 0 ↔ a < b :=
have a - b < b - b ↔ a < b, from sub_lt_sub_iff_right b,
by rwa sub_self at this
lemma le_sub_iff_add_le' : b ≤ c - a ↔ a + b ≤ c :=
by rw [sub_eq_add_neg, add_comm, le_neg_add_iff_add_le]
lemma le_sub_iff_add_le : a ≤ c - b ↔ a + b ≤ c :=
by rw [le_sub_iff_add_le', add_comm]
lemma sub_le_iff_le_add' : a - b ≤ c ↔ a ≤ b + c :=
by rw [sub_eq_add_neg, add_comm, neg_add_le_iff_le_add]
lemma sub_le_iff_le_add : a - c ≤ b ↔ a ≤ b + c :=
by rw [sub_le_iff_le_add', add_comm]
@[simp] lemma neg_le_sub_iff_le_add : -b ≤ a - c ↔ c ≤ a + b :=
le_sub_iff_add_le.trans neg_add_le_iff_le_add'
lemma neg_le_sub_iff_le_add' : -a ≤ b - c ↔ c ≤ a + b :=
by rw [neg_le_sub_iff_le_add, add_comm]
lemma sub_le : a - b ≤ c ↔ a - c ≤ b :=
sub_le_iff_le_add'.trans sub_le_iff_le_add.symm
theorem le_sub : a ≤ b - c ↔ c ≤ b - a :=
le_sub_iff_add_le'.trans le_sub_iff_add_le.symm
lemma lt_sub_iff_add_lt' : b < c - a ↔ a + b < c :=
by rw [sub_eq_add_neg, add_comm, lt_neg_add_iff_add_lt]
lemma lt_sub_iff_add_lt : a < c - b ↔ a + b < c :=
by rw [lt_sub_iff_add_lt', add_comm]
lemma sub_lt_iff_lt_add' : a - b < c ↔ a < b + c :=
by rw [sub_eq_add_neg, add_comm, neg_add_lt_iff_lt_add]
lemma sub_lt_iff_lt_add : a - c < b ↔ a < b + c :=
by rw [sub_lt_iff_lt_add', add_comm]
@[simp] lemma neg_lt_sub_iff_lt_add : -b < a - c ↔ c < a + b :=
lt_sub_iff_add_lt.trans neg_add_lt_iff_lt_add_right
lemma neg_lt_sub_iff_lt_add' : -a < b - c ↔ c < a + b :=
by rw [neg_lt_sub_iff_lt_add, add_comm]
lemma sub_lt : a - b < c ↔ a - c < b :=
sub_lt_iff_lt_add'.trans sub_lt_iff_lt_add.symm
theorem lt_sub : a < b - c ↔ c < b - a :=
lt_sub_iff_add_lt'.trans lt_sub_iff_add_lt.symm
lemma sub_le_self_iff (a : α) {b : α} : a - b ≤ a ↔ 0 ≤ b :=
sub_le_iff_le_add'.trans (le_add_iff_nonneg_left _)
lemma sub_lt_self_iff (a : α) {b : α} : a - b < a ↔ 0 < b :=
sub_lt_iff_lt_add'.trans (lt_add_iff_pos_left _)
end ordered_add_comm_group
/-
TODO:
The `add_lt_add_left` field of `ordered_add_comm_group` is redundant,
and it is no longer in core so we can remove it now.
This alternative constructor is a workaround until someone fixes this.
-/
/-- Alternative constructor for ordered commutative groups,
that avoids the field `mul_lt_mul_left`. -/
@[to_additive "Alternative constructor for ordered commutative groups,
that avoids the field `mul_lt_mul_left`."]
def ordered_comm_group.mk' {α : Type u} [comm_group α] [partial_order α]
(mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b) :
ordered_comm_group α :=
{ mul_le_mul_left := mul_le_mul_left,
..(by apply_instance : comm_group α),
..(by apply_instance : partial_order α) }
/-- A decidable linearly ordered cancellative additive commutative monoid
is an additive commutative monoid with a decidable linear order
in which addition is cancellative and strictly monotone. -/
@[protect_proj] class decidable_linear_ordered_cancel_add_comm_monoid (α : Type u)
extends ordered_cancel_add_comm_monoid α, decidable_linear_order α
section decidable_linear_ordered_cancel_add_comm_monoid
variables [decidable_linear_ordered_cancel_add_comm_monoid α]
lemma min_add_add_left (a b c : α) : min (a + b) (a + c) = a + min b c :=
eq.symm (eq_min
(show a + min b c ≤ a + b, from add_le_add_left (min_le_left _ _) _)
(show a + min b c ≤ a + c, from add_le_add_left (min_le_right _ _) _)
(assume d,
assume : d ≤ a + b,
assume : d ≤ a + c,
decidable.by_cases
(assume : b ≤ c, by rwa [min_eq_left this])
(assume : ¬ b ≤ c, by rwa [min_eq_right (le_of_lt (lt_of_not_ge this))])))
lemma min_add_add_right (a b c : α) : min (a + c) (b + c) = min a b + c :=
begin rw [add_comm a c, add_comm b c, add_comm _ c], apply min_add_add_left end
lemma max_add_add_left (a b c : α) : max (a + b) (a + c) = a + max b c :=
eq.symm (eq_max
(add_le_add_left (le_max_left _ _) _)
(add_le_add_left (le_max_right _ _) _)
(assume d,
assume : a + b ≤ d,
assume : a + c ≤ d,
decidable.by_cases
(assume : b ≤ c, by rwa [max_eq_right this])
(assume : ¬ b ≤ c, by rwa [max_eq_left (le_of_lt (lt_of_not_ge this))])))
lemma max_add_add_right (a b c : α) : max (a + c) (b + c) = max a b + c :=
begin rw [add_comm a c, add_comm b c, add_comm _ c], apply max_add_add_left end
end decidable_linear_ordered_cancel_add_comm_monoid
/-- A decidable linearly ordered additive commutative group is an
additive commutative group with a decidable linear order in which
addition is strictly monotone. -/
@[protect_proj] class decidable_linear_ordered_add_comm_group (α : Type u)
extends add_comm_group α, decidable_linear_order α :=
(add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b)
instance decidable_linear_ordered_comm_group.to_ordered_add_comm_group (α : Type u)
[s : decidable_linear_ordered_add_comm_group α] : ordered_add_comm_group α :=
{ add := s.add, ..s }
section decidable_linear_ordered_add_comm_group
variables [decidable_linear_ordered_add_comm_group α]
@[priority 100] -- see Note [lower instance priority]
instance decidable_linear_ordered_add_comm_group.to_decidable_linear_ordered_cancel_add_comm_monoid :
decidable_linear_ordered_cancel_add_comm_monoid α :=
{ le_of_add_le_add_left := λ x y z, le_of_add_le_add_left,
add_left_cancel := λ x y z, add_left_cancel,
add_right_cancel := λ x y z, add_right_cancel,
..‹decidable_linear_ordered_add_comm_group α› }
lemma decidable_linear_ordered_add_comm_group.add_lt_add_left
(a b : α) (h : a < b) (c : α) : c + a < c + b :=
ordered_add_comm_group.add_lt_add_left a b h c
lemma max_neg_neg (a b : α) : max (-a) (-b) = - min a b :=
eq.symm (eq_max
(show -a ≤ -(min a b), from neg_le_neg $ min_le_left a b)
(show -b ≤ -(min a b), from neg_le_neg $ min_le_right a b)
(assume d,
assume H₁ : -a ≤ d,
assume H₂ : -b ≤ d,
have H : -d ≤ min a b,
from le_min (neg_le_of_neg_le H₁) (neg_le_of_neg_le H₂),
show -(min a b) ≤ d, from neg_le_of_neg_le H))
lemma min_eq_neg_max_neg_neg (a b : α) : min a b = - max (-a) (-b) :=
by rw [max_neg_neg, neg_neg]
lemma min_neg_neg (a b : α) : min (-a) (-b) = - max a b :=
by rw [min_eq_neg_max_neg_neg, neg_neg, neg_neg]
lemma max_eq_neg_min_neg_neg (a b : α) : max a b = - min (-a) (-b) :=
by rw [min_neg_neg, neg_neg]
/-- `abs a` is the absolute value of `a`. -/
def abs (a : α) : α := max a (-a)
lemma abs_of_nonneg {a : α} (h : 0 ≤ a) : abs a = a :=
have h' : -a ≤ a, from le_trans (neg_nonpos_of_nonneg h) h,
max_eq_left h'
lemma abs_of_pos {a : α} (h : 0 < a) : abs a = a :=
abs_of_nonneg (le_of_lt h)
lemma abs_of_nonpos {a : α} (h : a ≤ 0) : abs a = -a :=
have h' : a ≤ -a, from le_trans h (neg_nonneg_of_nonpos h),
max_eq_right h'
lemma abs_of_neg {a : α} (h : a < 0) : abs a = -a :=
abs_of_nonpos (le_of_lt h)
lemma abs_zero : abs 0 = (0:α) :=
abs_of_nonneg (le_refl _)
lemma abs_neg (a : α) : abs (-a) = abs a :=
begin unfold abs, rw [max_comm, neg_neg] end
lemma abs_pos_of_pos {a : α} (h : 0 < a) : 0 < abs a :=
by rwa (abs_of_pos h)
lemma abs_pos_of_neg {a : α} (h : a < 0) : 0 < abs a :=
abs_neg a ▸ abs_pos_of_pos (neg_pos_of_neg h)
lemma abs_sub (a b : α) : abs (a - b) = abs (b - a) :=
by rw [← neg_sub, abs_neg]
lemma ne_zero_of_abs_ne_zero {a : α} (h : abs a ≠ 0) : a ≠ 0 :=
assume ha, h (eq.symm ha ▸ abs_zero)
/- these assume a linear order -/
lemma eq_zero_of_neg_eq {a : α} (h : -a = a) : a = 0 :=
match lt_trichotomy a 0 with
| or.inl h₁ :=
have 0 < a, from h ▸ neg_pos_of_neg h₁,
absurd h₁ (lt_asymm this)
| or.inr (or.inl h₁) := h₁
| or.inr (or.inr h₁) :=
have a < 0, from h ▸ neg_neg_of_pos h₁,
absurd h₁ (lt_asymm this)
end
lemma abs_nonneg (a : α) : 0 ≤ abs a :=
or.elim (le_total 0 a)
(assume h : 0 ≤ a, by rwa (abs_of_nonneg h))
(assume h : a ≤ 0, calc
0 ≤ -a : neg_nonneg_of_nonpos h
... = abs a : eq.symm (abs_of_nonpos h))
lemma abs_abs (a : α) : abs (abs a) = abs a :=
abs_of_nonneg $ abs_nonneg a
lemma le_abs_self (a : α) : a ≤ abs a :=
or.elim (le_total 0 a)
(assume h : 0 ≤ a,
begin rw [abs_of_nonneg h] end)
(assume h : a ≤ 0, le_trans h $ abs_nonneg a)
lemma neg_le_abs_self (a : α) : -a ≤ abs a :=
abs_neg a ▸ le_abs_self (-a)
lemma eq_zero_of_abs_eq_zero {a : α} (h : abs a = 0) : a = 0 :=
have h₁ : a ≤ 0, from h ▸ le_abs_self a,
have h₂ : -a ≤ 0, from h ▸ abs_neg a ▸ le_abs_self (-a),
le_antisymm h₁ (nonneg_of_neg_nonpos h₂)
lemma eq_of_abs_sub_eq_zero {a b : α} (h : abs (a - b) = 0) : a = b :=
have a - b = 0, from eq_zero_of_abs_eq_zero h,
show a = b, from eq_of_sub_eq_zero this
lemma abs_pos_of_ne_zero {a : α} (h : a ≠ 0) : 0 < abs a :=
or.elim (lt_or_gt_of_ne h) abs_pos_of_neg abs_pos_of_pos
lemma abs_by_cases (P : α → Prop) {a : α} (h1 : P a) (h2 : P (-a)) : P (abs a) :=
or.elim (le_total 0 a)
(assume h : 0 ≤ a, eq.symm (abs_of_nonneg h) ▸ h1)
(assume h : a ≤ 0, eq.symm (abs_of_nonpos h) ▸ h2)
lemma abs_le_of_le_of_neg_le {a b : α} (h1 : a ≤ b) (h2 : -a ≤ b) : abs a ≤ b :=
abs_by_cases (λ x : α, x ≤ b) h1 h2
lemma abs_lt_of_lt_of_neg_lt {a b : α} (h1 : a < b) (h2 : -a < b) : abs a < b :=
abs_by_cases (λ x : α, x < b) h1 h2
private lemma aux1 {a b : α} (h1 : 0 ≤ a + b) (h2 : 0 ≤ a) : abs (a + b) ≤ abs a + abs b :=
decidable.by_cases
(assume h3 : 0 ≤ b, calc
abs (a + b) ≤ abs (a + b) : by apply le_refl
... = a + b : by rw (abs_of_nonneg h1)
... = abs a + b : by rw (abs_of_nonneg h2)
... = abs a + abs b : by rw (abs_of_nonneg h3))
(assume h3 : ¬ 0 ≤ b,
have h4 : b ≤ 0, from le_of_lt (lt_of_not_ge h3),
calc
abs (a + b) = a + b : by rw (abs_of_nonneg h1)
... = abs a + b : by rw (abs_of_nonneg h2)
... ≤ abs a + 0 : add_le_add_left h4 _
... ≤ abs a + -b : add_le_add_left (neg_nonneg_of_nonpos h4) _
... = abs a + abs b : by rw (abs_of_nonpos h4))
private lemma aux2 {a b : α} (h1 : 0 ≤ a + b) : abs (a + b) ≤ abs a + abs b :=
or.elim (le_total b 0)
(assume h2 : b ≤ 0,
have h3 : ¬ a < 0, from
assume h4 : a < 0,
have h5 : a + b < 0,
begin
have aux := add_lt_add_of_lt_of_le h4 h2,
rwa [add_zero] at aux
end,
not_lt_of_ge h1 h5,
aux1 h1 (le_of_not_gt h3))
(assume h2 : 0 ≤ b,
begin
have h3 : abs (b + a) ≤ abs b + abs a,
begin
rw add_comm at h1,
exact aux1 h1 h2
end,
rw [add_comm, add_comm (abs a)],
exact h3
end)
lemma abs_add_le_abs_add_abs (a b : α) : abs (a + b) ≤ abs a + abs b :=
or.elim (le_total 0 (a + b))
(assume h2 : 0 ≤ a + b, aux2 h2)
(assume h2 : a + b ≤ 0,
have h3 : -a + -b = -(a + b), by rw neg_add,
have h4 : 0 ≤ -(a + b), from neg_nonneg_of_nonpos h2,
have h5 : 0 ≤ -a + -b, begin rw [← h3] at h4, exact h4 end,
calc
abs (a + b) = abs (-a + -b) : by rw [← abs_neg, neg_add]
... ≤ abs (-a) + abs (-b) : aux2 h5
... = abs a + abs b : by rw [abs_neg, abs_neg])
lemma abs_sub_abs_le_abs_sub (a b : α) : abs a - abs b ≤ abs (a - b) :=
have h1 : abs a - abs b + abs b ≤ abs (a - b) + abs b, from
calc
abs a - abs b + abs b = abs a : by rw sub_add_cancel
... = abs (a - b + b) : by rw sub_add_cancel
... ≤ abs (a - b) + abs b : by apply abs_add_le_abs_add_abs,
le_of_add_le_add_right h1
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_eq_add_neg, sub_eq_add_neg, sub_eq_add_neg,
add_assoc, neg_add_cancel_left]
... ≤ abs (a - b) + abs (b - c) : by apply abs_add_le_abs_add_abs
lemma abs_add_three (a b c : α) : abs (a + b + c) ≤ abs a + abs b + abs c :=
begin
apply le_trans,
apply abs_add_le_abs_add_abs,
apply le_trans,
apply add_le_add_right,
apply abs_add_le_abs_add_abs,
apply le_refl
end
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 :=
begin
cases (decidable.em (b ≤ a)) with hba hba,
rw (abs_of_nonneg (sub_nonneg_of_le hba)),
apply sub_le_sub,
apply hau,
apply hbl,
rw [abs_of_neg (sub_neg_of_lt (lt_of_not_ge hba)), neg_sub],
apply sub_le_sub,
apply hbu,
apply hal
end
lemma decidable_linear_ordered_add_comm_group.eq_of_abs_sub_nonpos
{a b : α} (h : abs (a - b) ≤ 0) : a = b :=
eq_of_abs_sub_eq_zero (le_antisymm h (abs_nonneg (a - b)))
end decidable_linear_ordered_add_comm_group
set_option old_structure_cmd true
section prio
set_option default_priority 100 -- see Note [default priority]
/-- 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)
end prio
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 `decidable_linear_ordered_add_comm_group`
if `nonneg` is total and decidable.
-/
def to_decidable_linear_ordered_add_comm_group
[decidable_pred (@nonneg α _)]
(nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a))
: decidable_linear_ordered_add_comm_group α :=
{ le := (≤),
lt := (<),
lt_iff_le_not_le := @lt_iff_le_not_le _ _,
le_refl := @le_refl _ _,
le_trans := @le_trans _ _,
le_antisymm := @le_antisymm _ _,
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 order_dual
instance [ordered_add_comm_monoid α] : ordered_add_comm_monoid (order_dual α) :=
{ add_le_add_left := λ a b h c, @add_le_add_left α _ b a h _,
lt_of_add_lt_add_left := λ a b c h, @lt_of_add_lt_add_left α _ a c b h,
..order_dual.partial_order α,
..show add_comm_monoid α, by apply_instance }
instance [ordered_cancel_add_comm_monoid α] : ordered_cancel_add_comm_monoid (order_dual α) :=
{ le_of_add_le_add_left := λ a b c : α, le_of_add_le_add_left,
add_left_cancel := @add_left_cancel α _,
add_right_cancel := @add_right_cancel α _,
..order_dual.ordered_add_comm_monoid }
instance [ordered_add_comm_group α] : ordered_add_comm_group (order_dual α) :=
{ add_left_neg := λ a : α, add_left_neg a,
..order_dual.ordered_add_comm_monoid,
..show add_comm_group α, by apply_instance }
end order_dual
section type_tags
instance : Π [preorder α], preorder (multiplicative α) := id
instance : Π [preorder α], preorder (additive α) := id
instance : Π [partial_order α], partial_order (multiplicative α) := id
instance : Π [partial_order α], partial_order (additive α) := id
instance : Π [linear_order α], linear_order (multiplicative α) := id
instance : Π [linear_order α], linear_order (additive α) := id
instance : Π [decidable_linear_order α], decidable_linear_order (multiplicative α) := id
instance : Π [decidable_linear_order α], decidable_linear_order (additive α) := id
instance [ordered_add_comm_monoid α] : ordered_comm_monoid (multiplicative α) :=
{ mul_le_mul_left := @ordered_add_comm_monoid.add_le_add_left α _,
lt_of_mul_lt_mul_left := @ordered_add_comm_monoid.lt_of_add_lt_add_left α _,
..multiplicative.partial_order,
..multiplicative.comm_monoid }
instance [ordered_comm_monoid α] : ordered_add_comm_monoid (additive α) :=
{ add_le_add_left := @ordered_comm_monoid.mul_le_mul_left α _,
lt_of_add_lt_add_left := @ordered_comm_monoid.lt_of_mul_lt_mul_left α _,
..additive.partial_order,
..additive.add_comm_monoid }
instance [ordered_cancel_add_comm_monoid α] : ordered_cancel_comm_monoid (multiplicative α) :=
{ le_of_mul_le_mul_left := @ordered_cancel_add_comm_monoid.le_of_add_le_add_left α _,
..multiplicative.right_cancel_semigroup,
..multiplicative.left_cancel_semigroup,
..multiplicative.ordered_comm_monoid }
instance [ordered_cancel_comm_monoid α] : ordered_cancel_add_comm_monoid (additive α) :=
{ le_of_add_le_add_left := @ordered_cancel_comm_monoid.le_of_mul_le_mul_left α _,
..additive.add_right_cancel_semigroup,
..additive.add_left_cancel_semigroup,
..additive.ordered_add_comm_monoid }
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 }
end type_tags
|
e19322cc4f7f3b691a9d9c239ae84166e9eab8f3 | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /src/data/quot.lean | dd78d5304721a1e54564d507fe0f7dc0a1983719 | [
"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 | 16,449 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import logic.relator
/-!
# Quotients -- extends the core library
-/
variables {α : Sort*} {β : Sort*}
namespace setoid
lemma ext {α : Sort*} :
∀{s t : setoid α}, (∀a b, @setoid.r α s a b ↔ @setoid.r α t a b) → s = t
| ⟨r, _⟩ ⟨p, _⟩ eq :=
have r = p, from funext $ assume a, funext $ assume b, propext $ eq a b,
by subst this
end setoid
namespace quot
variables {ra : α → α → Prop} {rb : β → β → Prop} {φ : quot ra → quot rb → Sort*}
local notation `⟦`:max a `⟧` := quot.mk _ a
instance [inhabited α] : inhabited (quot ra) := ⟨⟦default _⟧⟩
/-- Recursion on two `quotient` arguments `a` and `b`, result type depends on `⟦a⟧` and `⟦b⟧`. -/
protected def hrec_on₂ (qa : quot ra) (qb : quot rb) (f : Π a b, φ ⟦a⟧ ⟦b⟧)
(ca : ∀ {b a₁ a₂}, ra a₁ a₂ → f a₁ b == f a₂ b)
(cb : ∀ {a b₁ b₂}, rb b₁ b₂ → f a b₁ == f a b₂) : φ qa qb :=
quot.hrec_on qa (λ a, quot.hrec_on qb (f a) (λ b₁ b₂ pb, cb pb)) $ λ a₁ a₂ pa,
quot.induction_on qb $ λ b,
calc @quot.hrec_on _ _ (φ _) ⟦b⟧ (f a₁) (@cb _)
== f a₁ b : by simp
... == f a₂ b : ca pa
... == @quot.hrec_on _ _ (φ _) ⟦b⟧ (f a₂) (@cb _) : by simp
/-- Map a function `f : α → β` such that `ra x y` implies `rb (f x) (f y)`
to a map `quot ra → quot rb`. -/
protected def map (f : α → β) (h : (ra ⇒ rb) f f) : quot ra → quot rb :=
quot.lift (λ x, ⟦f x⟧) $ assume x y (h₁ : ra x y), quot.sound $ h h₁
/-- If `ra` is a subrelation of `ra'`, then we have a natural map `quot ra → quot ra'`. -/
protected def map_right {ra' : α → α → Prop} (h : ∀a₁ a₂, ra a₁ a₂ → ra' a₁ a₂) :
quot ra → quot ra' :=
quot.map id h
/-- weaken the relation of a quotient -/
def factor {α : Type*} (r s : α → α → Prop) (h : ∀ x y, r x y → s x y) :
quot r → quot s :=
quot.lift (quot.mk s) (λ x y rxy, quot.sound (h x y rxy))
lemma factor_mk_eq {α : Type*} (r s : α → α → Prop) (h : ∀ x y, r x y → s x y) :
factor r s h ∘ quot.mk _ = quot.mk _ := rfl
end quot
namespace quotient
variables [sa : setoid α] [sb : setoid β]
variables {φ : quotient sa → quotient sb → Sort*}
instance [inhabited α] : inhabited (quotient sa) := ⟨⟦default _⟧⟩
/-- Induction on two `quotient` arguments `a` and `b`, result type depends on `⟦a⟧` and `⟦b⟧`. -/
protected def hrec_on₂ (qa : quotient sa) (qb : quotient sb) (f : Π a b, φ ⟦a⟧ ⟦b⟧)
(c : ∀ a₁ b₁ a₂ b₂, a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ == f a₂ b₂) : φ qa qb :=
quot.hrec_on₂ qa qb f
(λ _ _ _ p, c _ _ _ _ p (setoid.refl _))
(λ _ _ _ p, c _ _ _ _ (setoid.refl _) p)
/-- Map a function `f : α → β` that sends equivalent elements to equivalent elements
to a function `quotient sa → quotient sb`. Useful to define unary operations on quotients. -/
protected def map (f : α → β) (h : ((≈) ⇒ (≈)) f f) : quotient sa → quotient sb :=
quot.map f h
@[simp] lemma map_mk (f : α → β) (h : ((≈) ⇒ (≈)) f f) (x : α) :
quotient.map f h (⟦x⟧ : quotient sa) = (⟦f x⟧ : quotient sb) :=
rfl
variables {γ : Sort*} [sc : setoid γ]
/-- Map a function `f : α → β → γ` that sends equivalent elements to equivalent elements
to a function `f : quotient sa → quotient sb → quotient sc`.
Useful to define binary operations on quotients. -/
protected def map₂ (f : α → β → γ) (h : ((≈) ⇒ (≈) ⇒ (≈)) f f) :
quotient sa → quotient sb → quotient sc :=
quotient.lift₂ (λ x y, ⟦f x y⟧) (λ x₁ y₁ x₂ y₂ h₁ h₂, quot.sound $ h h₁ h₂)
end quotient
@[simp] theorem quotient.eq [r : setoid α] {x y : α} : ⟦x⟧ = ⟦y⟧ ↔ x ≈ y :=
⟨quotient.exact, quotient.sound⟩
theorem forall_quotient_iff {α : Type*} [r : setoid α] {p : quotient r → Prop} :
(∀a:quotient r, p a) ↔ (∀a:α, p ⟦a⟧) :=
⟨assume h x, h _, assume h a, a.induction_on h⟩
@[simp] lemma quotient.lift_beta [s : setoid α] (f : α → β) (h : ∀ (a b : α), a ≈ b → f a = f b)
(x : α) :
quotient.lift f h (quotient.mk x) = f x := rfl
@[simp] lemma quotient.lift_on_beta [s : setoid α] (f : α → β) (h : ∀ (a b : α), a ≈ b → f a = f b)
(x : α) :
quotient.lift_on (quotient.mk x) f h = f x := rfl
@[simp] theorem quotient.lift_on_beta₂ {α : Type} {β : Type} [setoid α] (f : α → α → β)
(h : ∀ (a₁ a₂ b₁ b₂ : α), a₁ ≈ b₁ → a₂ ≈ b₂ → f a₁ a₂ = f b₁ b₂) (x y : α) :
quotient.lift_on₂ (quotient.mk x) (quotient.mk y) f h = f x y := rfl
/-- Choose an element of the equivalence class using the axiom of choice.
Sound but noncomputable. -/
noncomputable def quot.out {r : α → α → Prop} (q : quot r) : α :=
classical.some (quot.exists_rep q)
/-- Unwrap the VM representation of a quotient to obtain an element of the equivalence class.
Computable but unsound. -/
meta def quot.unquot {r : α → α → Prop} : quot r → α := unchecked_cast
@[simp] theorem quot.out_eq {r : α → α → Prop} (q : quot r) : quot.mk r q.out = q :=
classical.some_spec (quot.exists_rep q)
/-- Choose an element of the equivalence class using the axiom of choice.
Sound but noncomputable. -/
noncomputable def quotient.out [s : setoid α] : quotient s → α := quot.out
@[simp] theorem quotient.out_eq [s : setoid α] (q : quotient s) : ⟦q.out⟧ = q := q.out_eq
theorem quotient.mk_out [s : setoid α] (a : α) : ⟦a⟧.out ≈ a :=
quotient.exact (quotient.out_eq _)
instance pi_setoid {ι : Sort*} {α : ι → Sort*} [∀ i, setoid (α i)] : setoid (Π i, α i) :=
{ r := λ a b, ∀ i, a i ≈ b i,
iseqv := ⟨
λ a i, setoid.refl _,
λ a b h i, setoid.symm (h _),
λ a b c h₁ h₂ i, setoid.trans (h₁ _) (h₂ _)⟩ }
/-- Given a function `f : Π i, quotient (S i)`, returns the class of functions `Π i, α i` sending
each `i` to an element of the class `f i`. -/
noncomputable def quotient.choice {ι : Type*} {α : ι → Type*} [S : Π i, setoid (α i)]
(f : Π i, quotient (S i)) : @quotient (Π i, α i) (by apply_instance) :=
⟦λ i, (f i).out⟧
theorem quotient.choice_eq {ι : Type*} {α : ι → Type*} [Π i, setoid (α i)]
(f : Π i, α i) : quotient.choice (λ i, ⟦f i⟧) = ⟦f⟧ :=
quotient.sound $ λ i, quotient.mk_out _
lemma nonempty_quotient_iff (s : setoid α) : nonempty (quotient s) ↔ nonempty α :=
⟨assume ⟨a⟩, quotient.induction_on a nonempty.intro, assume ⟨a⟩, ⟨⟦a⟧⟩⟩
/-- `trunc α` is the quotient of `α` by the always-true relation. This
is related to the propositional truncation in HoTT, and is similar
in effect to `nonempty α`, but unlike `nonempty α`, `trunc α` is data,
so the VM representation is the same as `α`, and so this can be used to
maintain computability. -/
def {u} trunc (α : Sort u) : Sort u := @quot α (λ _ _, true)
theorem true_equivalence : @equivalence α (λ _ _, true) :=
⟨λ _, trivial, λ _ _ _, trivial, λ _ _ _ _ _, trivial⟩
namespace trunc
/-- Constructor for `trunc α` -/
def mk (a : α) : trunc α := quot.mk _ a
instance [inhabited α] : inhabited (trunc α) := ⟨mk (default _)⟩
/-- Any constant function lifts to a function out of the truncation -/
def lift (f : α → β) (c : ∀ a b : α, f a = f b) : trunc α → β :=
quot.lift f (λ a b _, c a b)
theorem ind {β : trunc α → Prop} : (∀ a : α, β (mk a)) → ∀ q : trunc α, β q := quot.ind
protected theorem lift_beta (f : α → β) (c) (a : α) : lift f c (mk a) = f a := rfl
/-- Lift a constant function on `q : trunc α`. -/
@[reducible, elab_as_eliminator]
protected def lift_on (q : trunc α) (f : α → β)
(c : ∀ a b : α, f a = f b) : β := lift f c q
@[elab_as_eliminator]
protected theorem induction_on {β : trunc α → Prop} (q : trunc α)
(h : ∀ a, β (mk a)) : β q := ind h q
theorem exists_rep (q : trunc α) : ∃ a : α, mk a = q := quot.exists_rep q
attribute [elab_as_eliminator]
protected theorem induction_on₂ {C : trunc α → trunc β → Prop} (q₁ : trunc α) (q₂ : trunc β)
(h : ∀ a b, C (mk a) (mk b)) : C q₁ q₂ :=
trunc.induction_on q₁ $ λ a₁, trunc.induction_on q₂ (h a₁)
protected theorem eq (a b : trunc α) : a = b :=
trunc.induction_on₂ a b (λ x y, quot.sound trivial)
instance : subsingleton (trunc α) := ⟨trunc.eq⟩
def bind (q : trunc α) (f : α → trunc β) : trunc β :=
trunc.lift_on q f (λ a b, trunc.eq _ _)
/-- A function `f : α → β` defines a function `map f : trunc α → trunc β`. -/
def map (f : α → β) (q : trunc α) : trunc β := bind q (trunc.mk ∘ f)
instance : monad trunc :=
{ pure := @trunc.mk,
bind := @trunc.bind }
instance : is_lawful_monad trunc :=
{ id_map := λ α q, trunc.eq _ _,
pure_bind := λ α β q f, rfl,
bind_assoc := λ α β γ x f g, trunc.eq _ _ }
variable {C : trunc α → Sort*}
/-- Recursion/induction principle for `trunc`. -/
@[reducible, elab_as_eliminator]
protected def rec
(f : Π a, C (mk a)) (h : ∀ (a b : α), (eq.rec (f a) (trunc.eq (mk a) (mk b)) : C (mk b)) = f b)
(q : trunc α) : C q :=
quot.rec f (λ a b _, h a b) q
/-- A version of `trunc.rec` taking `q : trunc α` as the first argument. -/
@[reducible, elab_as_eliminator]
protected def rec_on (q : trunc α) (f : Π a, C (mk a))
(h : ∀ (a b : α), (eq.rec (f a) (trunc.eq (mk a) (mk b)) : C (mk b)) = f b) : C q :=
trunc.rec f h q
/-- A version of `trunc.rec_on` assuming the codomain is a `subsingleton`. -/
@[reducible, elab_as_eliminator]
protected def rec_on_subsingleton
[∀ a, subsingleton (C (mk a))] (q : trunc α) (f : Π a, C (mk a)) : C q :=
trunc.rec f (λ a b, subsingleton.elim _ (f b)) q
/-- Noncomputably extract a representative of `trunc α` (using the axiom of choice). -/
noncomputable def out : trunc α → α := quot.out
@[simp] theorem out_eq (q : trunc α) : mk q.out = q := trunc.eq _ _
end trunc
theorem nonempty_of_trunc (q : trunc α) : nonempty α :=
let ⟨a, _⟩ := q.exists_rep in ⟨a⟩
namespace quotient
variables {γ : Sort*} {φ : Sort*}
{s₁ : setoid α} {s₂ : setoid β} {s₃ : setoid γ}
/- Versions of quotient definitions and lemmas ending in `'` use unification instead
of typeclass inference for inferring the `setoid` argument. This is useful when there are
several different quotient relations on a type, for example quotient groups, rings and modules -/
/-- A version of `quotient.mk` taking `{s : setoid α}` as an implicit argument instead of an
instance argument. -/
protected def mk' (a : α) : quotient s₁ := quot.mk s₁.1 a
/-- A version of `quotient.lift_on` taking `{s : setoid α}` as an implicit argument instead of an
instance argument. -/
@[elab_as_eliminator, reducible]
protected def lift_on' (q : quotient s₁) (f : α → φ)
(h : ∀ a b, @setoid.r α s₁ a b → f a = f b) : φ := quotient.lift_on q f h
/-- A version of `quotient.lift_on₂` taking `{s₁ : setoid α} {s₂ : setoid β}` as implicit arguments
instead of instance arguments. -/
@[elab_as_eliminator, reducible]
protected def lift_on₂' (q₁ : quotient s₁) (q₂ : quotient s₂) (f : α → β → γ)
(h : ∀ a₁ a₂ b₁ b₂, @setoid.r α s₁ a₁ b₁ → @setoid.r β s₂ a₂ b₂ → f a₁ a₂ = f b₁ b₂) : γ :=
quotient.lift_on₂ q₁ q₂ f h
/-- A version of `quotient.ind` taking `{s : setoid α}` as an implicit argument instead of an
instance argument. -/
@[elab_as_eliminator]
protected lemma ind' {p : quotient s₁ → Prop}
(h : ∀ a, p (quotient.mk' a)) (q : quotient s₁) : p q :=
quotient.ind h q
/-- A version of `quotient.ind₂` taking `{s₁ : setoid α} {s₂ : setoid β}` as implicit arguments
instead of instance arguments. -/
@[elab_as_eliminator]
protected lemma ind₂' {p : quotient s₁ → quotient s₂ → Prop}
(h : ∀ a₁ a₂, p (quotient.mk' a₁) (quotient.mk' a₂))
(q₁ : quotient s₁) (q₂ : quotient s₂) : p q₁ q₂ :=
quotient.ind₂ h q₁ q₂
/-- A version of `quotient.induction_on` taking `{s : setoid α}` as an implicit argument instead
of an instance argument. -/
@[elab_as_eliminator]
protected lemma induction_on' {p : quotient s₁ → Prop} (q : quotient s₁)
(h : ∀ a, p (quotient.mk' a)) : p q := quotient.induction_on q h
/-- A version of `quotient.induction_on₂` taking `{s₁ : setoid α} {s₂ : setoid β}` as implicit
arguments instead of instance arguments. -/
@[elab_as_eliminator]
protected lemma induction_on₂' {p : quotient s₁ → quotient s₂ → Prop} (q₁ : quotient s₁)
(q₂ : quotient s₂) (h : ∀ a₁ a₂, p (quotient.mk' a₁) (quotient.mk' a₂)) : p q₁ q₂ :=
quotient.induction_on₂ q₁ q₂ h
/-- A version of `quotient.induction_on₃` taking `{s₁ : setoid α} {s₂ : setoid β} {s₃ : setoid γ}`
as implicit arguments instead of instance arguments. -/
@[elab_as_eliminator]
protected lemma induction_on₃' {p : quotient s₁ → quotient s₂ → quotient s₃ → Prop}
(q₁ : quotient s₁) (q₂ : quotient s₂) (q₃ : quotient s₃)
(h : ∀ a₁ a₂ a₃, p (quotient.mk' a₁) (quotient.mk' a₂) (quotient.mk' a₃)) : p q₁ q₂ q₃ :=
quotient.induction_on₃ q₁ q₂ q₃ h
/-- Recursion on a `quotient` argument `a`, result type depends on `⟦a⟧`. -/
protected def hrec_on' {φ : quotient s₁ → Sort*} (qa : quotient s₁) (f : Π a, φ (quotient.mk' a))
(c : ∀ a₁ a₂, a₁ ≈ a₂ → f a₁ == f a₂) : φ qa :=
quot.hrec_on qa f c
@[simp] lemma hrec_on'_mk' {φ : quotient s₁ → Sort*} (f : Π a, φ (quotient.mk' a))
(c : ∀ a₁ a₂, a₁ ≈ a₂ → f a₁ == f a₂) (x : α) :
(quotient.mk' x).hrec_on' f c = f x :=
rfl
/-- Recursion on two `quotient` arguments `a` and `b`, result type depends on `⟦a⟧` and `⟦b⟧`. -/
protected def hrec_on₂' {φ : quotient s₁ → quotient s₂ → Sort*} (qa : quotient s₁)
(qb : quotient s₂) (f : ∀ a b, φ (quotient.mk' a) (quotient.mk' b))
(c : ∀ a₁ b₁ a₂ b₂, a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ == f a₂ b₂) : φ qa qb :=
quotient.hrec_on₂ qa qb f c
@[simp] lemma hrec_on₂'_mk' {φ : quotient s₁ → quotient s₂ → Sort*}
(f : ∀ a b, φ (quotient.mk' a) (quotient.mk' b))
(c : ∀ a₁ b₁ a₂ b₂, a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ == f a₂ b₂) (x : α) (qb : quotient s₂) :
(quotient.mk' x).hrec_on₂' qb f c = qb.hrec_on' (f x) (λ b₁ b₂, c _ _ _ _ (setoid.refl _)) :=
rfl
/-- Map a function `f : α → β` that sends equivalent elements to equivalent elements
to a function `quotient sa → quotient sb`. Useful to define unary operations on quotients. -/
protected def map' (f : α → β) (h : ((≈) ⇒ (≈)) f f) :
quotient s₁ → quotient s₂ :=
quot.map f h
@[simp] lemma map'_mk' (f : α → β) (h) (x : α) :
(quotient.mk' x : quotient s₁).map' f h = (quotient.mk' (f x) : quotient s₂) :=
rfl
/-- A version of `quotient.map₂` using curly braces and unification. -/
protected def map₂' (f : α → β → γ) (h : ((≈) ⇒ (≈) ⇒ (≈)) f f) :
quotient s₁ → quotient s₂ → quotient s₃ :=
quotient.map₂ f h
@[simp] lemma map₂'_mk' (f : α → β → γ) (h) (x : α) :
(quotient.mk' x : quotient s₁).map₂' f h =
(quotient.map' (f x) (h (setoid.refl x)) : quotient s₂ → quotient s₃) :=
rfl
lemma exact' {a b : α} :
(quotient.mk' a : quotient s₁) = quotient.mk' b → @setoid.r _ s₁ a b :=
quotient.exact
lemma sound' {a b : α} : @setoid.r _ s₁ a b → @quotient.mk' α s₁ a = quotient.mk' b :=
quotient.sound
@[simp]
protected lemma eq' {a b : α} : @quotient.mk' α s₁ a = quotient.mk' b ↔ @setoid.r _ s₁ a b :=
quotient.eq
/-- A version of `quotient.out` taking `{s₁ : setoid α}` as an implicit argument instead of an
instance argument. -/
noncomputable def out' (a : quotient s₁) : α := quotient.out a
@[simp] theorem out_eq' (q : quotient s₁) : quotient.mk' q.out' = q := q.out_eq
theorem mk_out' (a : α) : @setoid.r α s₁ (quotient.mk' a : quotient s₁).out' a :=
quotient.exact (quotient.out_eq _)
end quotient
|
9903f405866ec36e15d902bf4811dcc18e9c04eb | b70031c8e2c5337b91d7e70f1e0c5f528f7b0e77 | /src/analysis/normed_space/basic.lean | 657bfe6632fc6d32bdb41e93b732c7a03ab22291 | [
"Apache-2.0"
] | permissive | molodiuc/mathlib | cae2ba3ef1601c1f42ca0b625c79b061b63fef5b | 98ebe5a6739fbe254f9ee9d401882d4388f91035 | refs/heads/master | 1,674,237,127,059 | 1,606,353,533,000 | 1,606,353,533,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 54,189 | lean | /-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import topology.instances.nnreal
import topology.instances.complex
import topology.algebra.module
import topology.metric_space.antilipschitz
/-!
# Normed spaces
-/
variables {α : Type*} {β : Type*} {γ : Type*} {ι : Type*}
noncomputable theory
open filter metric
open_locale topological_space big_operators nnreal
/-- Auxiliary class, endowing a type `α` with a function `norm : α → ℝ`. This class is designed to
be extended in more interesting classes specifying the properties of the norm. -/
class has_norm (α : Type*) := (norm : α → ℝ)
export has_norm (norm)
notation `∥`:1024 e:1 `∥`:1 := norm e
/-- A normed group is an additive group endowed with a norm for which `dist x y = ∥x - y∥` defines
a metric space structure. -/
class normed_group (α : Type*) extends has_norm α, add_comm_group α, metric_space α :=
(dist_eq : ∀ x y, dist x y = norm (x - y))
/-- Construct a normed group from a translation invariant distance -/
def normed_group.of_add_dist [has_norm α] [add_comm_group α] [metric_space α]
(H1 : ∀ x:α, ∥x∥ = dist x 0)
(H2 : ∀ x y z : α, dist x y ≤ dist (x + z) (y + z)) : normed_group α :=
{ dist_eq := λ x y, begin
rw H1, apply le_antisymm,
{ rw [sub_eq_add_neg, ← add_right_neg y], apply H2 },
{ have := H2 (x-y) 0 y, rwa [sub_add_cancel, zero_add] at this }
end }
/-- Construct a normed group from a translation invariant distance -/
def normed_group.of_add_dist' [has_norm α] [add_comm_group α] [metric_space α]
(H1 : ∀ x:α, ∥x∥ = dist x 0)
(H2 : ∀ x y z : α, dist (x + z) (y + z) ≤ dist x y) : normed_group α :=
{ dist_eq := λ x y, begin
rw H1, apply le_antisymm,
{ have := H2 (x-y) 0 y, rwa [sub_add_cancel, zero_add] at this },
{ rw [sub_eq_add_neg, ← add_right_neg y], apply H2 }
end }
/-- A normed group can be built from a norm that satisfies algebraic properties. This is
formalised in this structure. -/
structure normed_group.core (α : Type*) [add_comm_group α] [has_norm α] : Prop :=
(norm_eq_zero_iff : ∀ x : α, ∥x∥ = 0 ↔ x = 0)
(triangle : ∀ x y : α, ∥x + y∥ ≤ ∥x∥ + ∥y∥)
(norm_neg : ∀ x : α, ∥-x∥ = ∥x∥)
/-- Constructing a normed group from core properties of a norm, i.e., registering the distance and
the metric space structure from the norm properties. -/
noncomputable def normed_group.of_core (α : Type*) [add_comm_group α] [has_norm α]
(C : normed_group.core α) : normed_group α :=
{ dist := λ x y, ∥x - y∥,
dist_eq := assume x y, by refl,
dist_self := assume x, (C.norm_eq_zero_iff (x - x)).mpr (show x - x = 0, by simp),
eq_of_dist_eq_zero := assume x y h, show (x = y), from sub_eq_zero.mp $ (C.norm_eq_zero_iff (x - y)).mp h,
dist_triangle := assume x y z,
calc ∥x - z∥ = ∥x - y + (y - z)∥ : by rw sub_add_sub_cancel
... ≤ ∥x - y∥ + ∥y - z∥ : C.triangle _ _,
dist_comm := assume x y,
calc ∥x - y∥ = ∥ -(y - x)∥ : by simp
... = ∥y - x∥ : by { rw [C.norm_neg] } }
section normed_group
variables [normed_group α] [normed_group β]
lemma dist_eq_norm (g h : α) : dist g h = ∥g - h∥ :=
normed_group.dist_eq _ _
@[simp] lemma dist_zero_right (g : α) : dist g 0 = ∥g∥ :=
by rw [dist_eq_norm, sub_zero]
lemma tendsto_norm_cocompact_at_top [proper_space α] :
tendsto norm (cocompact α) at_top :=
by simpa only [dist_zero_right] using tendsto_dist_right_cocompact_at_top (0:α)
lemma norm_sub_rev (g h : α) : ∥g - h∥ = ∥h - g∥ :=
by simpa only [dist_eq_norm] using dist_comm g h
@[simp] lemma norm_neg (g : α) : ∥-g∥ = ∥g∥ :=
by simpa using norm_sub_rev 0 g
@[simp] lemma dist_add_left (g h₁ h₂ : α) : dist (g + h₁) (g + h₂) = dist h₁ h₂ :=
by simp [dist_eq_norm]
@[simp] lemma dist_add_right (g₁ g₂ h : α) : dist (g₁ + h) (g₂ + h) = dist g₁ g₂ :=
by simp [dist_eq_norm]
@[simp] lemma dist_neg_neg (g h : α) : dist (-g) (-h) = dist g h :=
by simp only [dist_eq_norm, neg_sub_neg, norm_sub_rev]
@[simp] lemma dist_sub_left (g h₁ h₂ : α) : dist (g - h₁) (g - h₂) = dist h₁ h₂ :=
by simp only [sub_eq_add_neg, dist_add_left, dist_neg_neg]
@[simp] lemma dist_sub_right (g₁ g₂ h : α) : dist (g₁ - h) (g₂ - h) = dist g₁ g₂ :=
dist_add_right _ _ _
/-- Triangle inequality for the norm. -/
lemma norm_add_le (g h : α) : ∥g + h∥ ≤ ∥g∥ + ∥h∥ :=
by simpa [dist_eq_norm] using dist_triangle g 0 (-h)
lemma norm_add_le_of_le {g₁ g₂ : α} {n₁ n₂ : ℝ} (H₁ : ∥g₁∥ ≤ n₁) (H₂ : ∥g₂∥ ≤ n₂) :
∥g₁ + g₂∥ ≤ n₁ + n₂ :=
le_trans (norm_add_le g₁ g₂) (add_le_add H₁ H₂)
lemma dist_add_add_le (g₁ g₂ h₁ h₂ : α) :
dist (g₁ + g₂) (h₁ + h₂) ≤ dist g₁ h₁ + dist g₂ h₂ :=
by simpa only [dist_add_left, dist_add_right] using dist_triangle (g₁ + g₂) (h₁ + g₂) (h₁ + h₂)
lemma dist_add_add_le_of_le {g₁ g₂ h₁ h₂ : α} {d₁ d₂ : ℝ}
(H₁ : dist g₁ h₁ ≤ d₁) (H₂ : dist g₂ h₂ ≤ d₂) :
dist (g₁ + g₂) (h₁ + h₂) ≤ d₁ + d₂ :=
le_trans (dist_add_add_le g₁ g₂ h₁ h₂) (add_le_add H₁ H₂)
lemma dist_sub_sub_le (g₁ g₂ h₁ h₂ : α) :
dist (g₁ - g₂) (h₁ - h₂) ≤ dist g₁ h₁ + dist g₂ h₂ :=
dist_neg_neg g₂ h₂ ▸ dist_add_add_le _ _ _ _
lemma dist_sub_sub_le_of_le {g₁ g₂ h₁ h₂ : α} {d₁ d₂ : ℝ}
(H₁ : dist g₁ h₁ ≤ d₁) (H₂ : dist g₂ h₂ ≤ d₂) :
dist (g₁ - g₂) (h₁ - h₂) ≤ d₁ + d₂ :=
le_trans (dist_sub_sub_le g₁ g₂ h₁ h₂) (add_le_add H₁ H₂)
lemma abs_dist_sub_le_dist_add_add (g₁ g₂ h₁ h₂ : α) :
abs (dist g₁ h₁ - dist g₂ h₂) ≤ dist (g₁ + g₂) (h₁ + h₂) :=
by simpa only [dist_add_left, dist_add_right, dist_comm h₂]
using abs_dist_sub_le (g₁ + g₂) (h₁ + h₂) (h₁ + g₂)
@[simp] lemma norm_nonneg (g : α) : 0 ≤ ∥g∥ :=
by { rw[←dist_zero_right], exact dist_nonneg }
@[simp] lemma norm_eq_zero {g : α} : ∥g∥ = 0 ↔ g = 0 :=
dist_zero_right g ▸ dist_eq_zero
@[simp] lemma norm_zero : ∥(0:α)∥ = 0 := norm_eq_zero.2 rfl
@[nontriviality] lemma norm_of_subsingleton [subsingleton α] (x : α) : ∥x∥ = 0 :=
by rw [subsingleton.elim x 0, norm_zero]
lemma norm_sum_le {β} : ∀(s : finset β) (f : β → α), ∥∑ a in s, f a∥ ≤ ∑ a in s, ∥ f a ∥ :=
finset.le_sum_of_subadditive norm norm_zero norm_add_le
lemma norm_sum_le_of_le {β} (s : finset β) {f : β → α} {n : β → ℝ} (h : ∀ b ∈ s, ∥f b∥ ≤ n b) :
∥∑ b in s, f b∥ ≤ ∑ b in s, n b :=
le_trans (norm_sum_le s f) (finset.sum_le_sum h)
@[simp] lemma norm_pos_iff {g : α} : 0 < ∥ g ∥ ↔ g ≠ 0 :=
dist_zero_right g ▸ dist_pos
@[simp] lemma norm_le_zero_iff {g : α} : ∥g∥ ≤ 0 ↔ g = 0 :=
by { rw[←dist_zero_right], exact dist_le_zero }
lemma norm_sub_le (g h : α) : ∥g - h∥ ≤ ∥g∥ + ∥h∥ :=
by simpa [dist_eq_norm] using dist_triangle g 0 h
lemma norm_sub_le_of_le {g₁ g₂ : α} {n₁ n₂ : ℝ} (H₁ : ∥g₁∥ ≤ n₁) (H₂ : ∥g₂∥ ≤ n₂) :
∥g₁ - g₂∥ ≤ n₁ + n₂ :=
le_trans (norm_sub_le g₁ g₂) (add_le_add H₁ H₂)
lemma dist_le_norm_add_norm (g h : α) : dist g h ≤ ∥g∥ + ∥h∥ :=
by { rw dist_eq_norm, apply norm_sub_le }
lemma abs_norm_sub_norm_le (g h : α) : abs(∥g∥ - ∥h∥) ≤ ∥g - h∥ :=
by simpa [dist_eq_norm] using abs_dist_sub_le g h 0
lemma norm_sub_norm_le (g h : α) : ∥g∥ - ∥h∥ ≤ ∥g - h∥ :=
le_trans (le_abs_self _) (abs_norm_sub_norm_le g h)
lemma dist_norm_norm_le (g h : α) : dist ∥g∥ ∥h∥ ≤ ∥g - h∥ :=
abs_norm_sub_norm_le g h
lemma eq_of_norm_sub_eq_zero {u v : α} (h : ∥u - v∥ = 0) : u = v :=
begin
apply eq_of_dist_eq_zero,
rwa dist_eq_norm
end
lemma norm_le_insert (u v : α) : ∥v∥ ≤ ∥u∥ + ∥u - v∥ :=
calc ∥v∥ = ∥u - (u - v)∥ : by abel
... ≤ ∥u∥ + ∥u - v∥ : norm_sub_le u _
lemma ball_0_eq (ε : ℝ) : ball (0:α) ε = {x | ∥x∥ < ε} :=
set.ext $ assume a, by simp
lemma mem_ball_iff_norm {g h : α} {r : ℝ} :
h ∈ ball g r ↔ ∥h - g∥ < r :=
by rw [mem_ball, dist_eq_norm]
lemma mem_ball_iff_norm' {g h : α} {r : ℝ} :
h ∈ ball g r ↔ ∥g - h∥ < r :=
by rw [mem_ball', dist_eq_norm]
lemma mem_closed_ball_iff_norm {g h : α} {r : ℝ} :
h ∈ closed_ball g r ↔ ∥h - g∥ ≤ r :=
by rw [mem_closed_ball, dist_eq_norm]
lemma mem_closed_ball_iff_norm' {g h : α} {r : ℝ} :
h ∈ closed_ball g r ↔ ∥g - h∥ ≤ r :=
by rw [mem_closed_ball', dist_eq_norm]
lemma norm_le_of_mem_closed_ball {g h : α} {r : ℝ} (H : h ∈ closed_ball g r) :
∥h∥ ≤ ∥g∥ + r :=
calc
∥h∥ = ∥g + (h - g)∥ : by rw [add_sub_cancel'_right]
... ≤ ∥g∥ + ∥h - g∥ : norm_add_le _ _
... ≤ ∥g∥ + r : by { apply add_le_add_left, rw ← dist_eq_norm, exact H }
lemma norm_lt_of_mem_ball {g h : α} {r : ℝ} (H : h ∈ ball g r) :
∥h∥ < ∥g∥ + r :=
calc
∥h∥ = ∥g + (h - g)∥ : by rw [add_sub_cancel'_right]
... ≤ ∥g∥ + ∥h - g∥ : norm_add_le _ _
... < ∥g∥ + r : by { apply add_lt_add_left, rw ← dist_eq_norm, exact H }
theorem normed_group.tendsto_nhds_zero {f : γ → α} {l : filter γ} :
tendsto f l (𝓝 0) ↔ ∀ ε > 0, ∀ᶠ x in l, ∥ f x ∥ < ε :=
metric.tendsto_nhds.trans $ by simp only [dist_zero_right]
lemma normed_group.tendsto_nhds_nhds {f : α → β} {x : α} {y : β} :
tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ x', ∥x' - x∥ < δ → ∥f x' - y∥ < ε :=
by simp_rw [metric.tendsto_nhds_nhds, dist_eq_norm]
/-- A homomorphism `f` of normed groups is Lipschitz, if there exists a constant `C` such that for
all `x`, one has `∥f x∥ ≤ C * ∥x∥`.
The analogous condition for a linear map of normed spaces is in `normed_space.operator_norm`. -/
lemma add_monoid_hom.lipschitz_of_bound (f :α →+ β) (C : ℝ) (h : ∀x, ∥f x∥ ≤ C * ∥x∥) :
lipschitz_with (nnreal.of_real C) f :=
lipschitz_with.of_dist_le' $ λ x y, by simpa only [dist_eq_norm, f.map_sub] using h (x - y)
lemma lipschitz_on_with_iff_norm_sub_le {f : α → β} {C : ℝ≥0} {s : set α} :
lipschitz_on_with C f s ↔ ∀ (x ∈ s) (y ∈ s), ∥f x - f y∥ ≤ C * ∥x - y∥ :=
by simp only [lipschitz_on_with_iff_dist_le_mul, dist_eq_norm]
lemma lipschitz_on_with.norm_sub_le {f : α → β} {C : ℝ≥0} {s : set α} (h : lipschitz_on_with C f s)
{x y : α} (x_in : x ∈ s) (y_in : y ∈ s) : ∥f x - f y∥ ≤ C * ∥x - y∥ :=
lipschitz_on_with_iff_norm_sub_le.mp h x x_in y y_in
/-- A homomorphism `f` of normed groups is continuous, if there exists a constant `C` such that for
all `x`, one has `∥f x∥ ≤ C * ∥x∥`.
The analogous condition for a linear map of normed spaces is in `normed_space.operator_norm`. -/
lemma add_monoid_hom.continuous_of_bound (f :α →+ β) (C : ℝ) (h : ∀x, ∥f x∥ ≤ C * ∥x∥) :
continuous f :=
(f.lipschitz_of_bound C h).continuous
section nnnorm
/-- Version of the norm taking values in nonnegative reals. -/
def nnnorm (a : α) : ℝ≥0 := ⟨norm a, norm_nonneg a⟩
@[simp] lemma coe_nnnorm (a : α) : (nnnorm a : ℝ) = norm a := rfl
lemma nndist_eq_nnnorm (a b : α) : nndist a b = nnnorm (a - b) := nnreal.eq $ dist_eq_norm _ _
@[simp] lemma nnnorm_eq_zero {a : α} : nnnorm a = 0 ↔ a = 0 :=
by simp only [nnreal.eq_iff.symm, nnreal.coe_zero, coe_nnnorm, norm_eq_zero]
@[simp] lemma nnnorm_zero : nnnorm (0 : α) = 0 :=
nnreal.eq norm_zero
lemma nnnorm_add_le (g h : α) : nnnorm (g + h) ≤ nnnorm g + nnnorm h :=
nnreal.coe_le_coe.2 $ norm_add_le g h
@[simp] lemma nnnorm_neg (g : α) : nnnorm (-g) = nnnorm g :=
nnreal.eq $ norm_neg g
lemma nndist_nnnorm_nnnorm_le (g h : α) : nndist (nnnorm g) (nnnorm h) ≤ nnnorm (g - h) :=
nnreal.coe_le_coe.2 $ dist_norm_norm_le g h
lemma of_real_norm_eq_coe_nnnorm (x : β) : ennreal.of_real ∥x∥ = (nnnorm x : ennreal) :=
ennreal.of_real_eq_coe_nnreal _
lemma edist_eq_coe_nnnorm_sub (x y : β) : edist x y = (nnnorm (x - y) : ennreal) :=
by rw [edist_dist, dist_eq_norm, of_real_norm_eq_coe_nnnorm]
lemma edist_eq_coe_nnnorm (x : β) : edist x 0 = (nnnorm x : ennreal) :=
by rw [edist_eq_coe_nnnorm_sub, _root_.sub_zero]
lemma nndist_add_add_le (g₁ g₂ h₁ h₂ : α) :
nndist (g₁ + g₂) (h₁ + h₂) ≤ nndist g₁ h₁ + nndist g₂ h₂ :=
nnreal.coe_le_coe.2 $ dist_add_add_le g₁ g₂ h₁ h₂
lemma edist_add_add_le (g₁ g₂ h₁ h₂ : α) :
edist (g₁ + g₂) (h₁ + h₂) ≤ edist g₁ h₁ + edist g₂ h₂ :=
by { simp only [edist_nndist], norm_cast, apply nndist_add_add_le }
lemma nnnorm_sum_le {β} : ∀(s : finset β) (f : β → α), nnnorm (∑ a in s, f a) ≤ ∑ a in s, nnnorm (f a) :=
finset.le_sum_of_subadditive nnnorm nnnorm_zero nnnorm_add_le
end nnnorm
lemma lipschitz_with.neg {α : Type*} [emetric_space α] {K : nnreal} {f : α → β}
(hf : lipschitz_with K f) : lipschitz_with K (λ x, -f x) :=
λ x y, by simpa only [edist_dist, dist_neg_neg] using hf x y
lemma lipschitz_with.add {α : Type*} [emetric_space α] {Kf : nnreal} {f : α → β}
(hf : lipschitz_with Kf f) {Kg : nnreal} {g : α → β} (hg : lipschitz_with Kg g) :
lipschitz_with (Kf + Kg) (λ x, f x + g x) :=
λ x y,
calc edist (f x + g x) (f y + g y) ≤ edist (f x) (f y) + edist (g x) (g y) :
edist_add_add_le _ _ _ _
... ≤ Kf * edist x y + Kg * edist x y :
add_le_add (hf x y) (hg x y)
... = (Kf + Kg) * edist x y :
(add_mul _ _ _).symm
lemma lipschitz_with.sub {α : Type*} [emetric_space α] {Kf : nnreal} {f : α → β}
(hf : lipschitz_with Kf f) {Kg : nnreal} {g : α → β} (hg : lipschitz_with Kg g) :
lipschitz_with (Kf + Kg) (λ x, f x - g x) :=
hf.add hg.neg
lemma antilipschitz_with.add_lipschitz_with {α : Type*} [metric_space α] {Kf : nnreal} {f : α → β}
(hf : antilipschitz_with Kf f) {Kg : nnreal} {g : α → β} (hg : lipschitz_with Kg g)
(hK : Kg < Kf⁻¹) :
antilipschitz_with (Kf⁻¹ - Kg)⁻¹ (λ x, f x + g x) :=
begin
refine antilipschitz_with.of_le_mul_dist (λ x y, _),
rw [nnreal.coe_inv, ← div_eq_inv_mul],
rw le_div_iff (nnreal.coe_pos.2 $ nnreal.sub_pos.2 hK),
rw [mul_comm, nnreal.coe_sub (le_of_lt hK), sub_mul],
calc ↑Kf⁻¹ * dist x y - Kg * dist x y ≤ dist (f x) (f y) - dist (g x) (g y) :
sub_le_sub (hf.mul_le_dist x y) (hg.dist_le_mul x y)
... ≤ _ : le_trans (le_abs_self _) (abs_dist_sub_le_dist_add_add _ _ _ _)
end
/-- A submodule of a normed group is also a normed group, with the restriction of the norm.
As all instances can be inferred from the submodule `s`, they are put as implicit instead of
typeclasses. -/
instance submodule.normed_group {𝕜 : Type*} {_ : ring 𝕜}
{E : Type*} [normed_group E] {_ : module 𝕜 E} (s : submodule 𝕜 E) : normed_group s :=
{ norm := λx, norm (x : E),
dist_eq := λx y, dist_eq_norm (x : E) (y : E) }
/-- normed group instance on the product of two normed groups, using the sup norm. -/
instance prod.normed_group : normed_group (α × β) :=
{ norm := λx, max ∥x.1∥ ∥x.2∥,
dist_eq := assume (x y : α × β),
show max (dist x.1 y.1) (dist x.2 y.2) = (max ∥(x - y).1∥ ∥(x - y).2∥), by simp [dist_eq_norm] }
lemma prod.norm_def (x : α × β) : ∥x∥ = (max ∥x.1∥ ∥x.2∥) := rfl
lemma prod.nnnorm_def (x : α × β) : nnnorm x = max (nnnorm x.1) (nnnorm x.2) :=
by { have := x.norm_def, simp only [← coe_nnnorm] at this, exact_mod_cast this }
lemma norm_fst_le (x : α × β) : ∥x.1∥ ≤ ∥x∥ :=
le_max_left _ _
lemma norm_snd_le (x : α × β) : ∥x.2∥ ≤ ∥x∥ :=
le_max_right _ _
lemma norm_prod_le_iff {x : α × β} {r : ℝ} :
∥x∥ ≤ r ↔ ∥x.1∥ ≤ r ∧ ∥x.2∥ ≤ r :=
max_le_iff
/-- normed group instance on the product of finitely many normed groups, using the sup norm. -/
instance pi.normed_group {π : ι → Type*} [fintype ι] [∀i, normed_group (π i)] :
normed_group (Πi, π i) :=
{ norm := λf, ((finset.sup finset.univ (λ b, nnnorm (f b)) : nnreal) : ℝ),
dist_eq := assume x y,
congr_arg (coe : ℝ≥0 → ℝ) $ congr_arg (finset.sup finset.univ) $ funext $ assume a,
show nndist (x a) (y a) = nnnorm (x a - y a), from nndist_eq_nnnorm _ _ }
/-- The norm of an element in a product space is `≤ r` if and only if the norm of each
component is. -/
lemma pi_norm_le_iff {π : ι → Type*} [fintype ι] [∀i, normed_group (π i)] {r : ℝ} (hr : 0 ≤ r)
{x : Πi, π i} : ∥x∥ ≤ r ↔ ∀i, ∥x i∥ ≤ r :=
by { simp only [(dist_zero_right _).symm, dist_pi_le_iff hr], refl }
lemma norm_le_pi_norm {π : ι → Type*} [fintype ι] [∀i, normed_group (π i)] (x : Πi, π i) (i : ι) :
∥x i∥ ≤ ∥x∥ :=
(pi_norm_le_iff (norm_nonneg x)).1 (le_refl _) i
lemma tendsto_iff_norm_tendsto_zero {f : ι → β} {a : filter ι} {b : β} :
tendsto f a (𝓝 b) ↔ tendsto (λ e, ∥ f e - b ∥) a (𝓝 0) :=
by rw tendsto_iff_dist_tendsto_zero ; simp only [(dist_eq_norm _ _).symm]
lemma tendsto_zero_iff_norm_tendsto_zero {f : γ → β} {a : filter γ} :
tendsto f a (𝓝 0) ↔ tendsto (λ e, ∥ f e ∥) a (𝓝 0) :=
have tendsto f a (𝓝 0) ↔ tendsto (λ e, ∥ f e - 0 ∥) a (𝓝 0) :=
tendsto_iff_norm_tendsto_zero,
by simpa
/-- Special case of the sandwich theorem: if the norm of `f` is eventually bounded by a real
function `g` which tends to `0`, then `f` tends to `0`.
In this pair of lemmas (`squeeze_zero_norm'` and `squeeze_zero_norm`), following a convention of
similar lemmas in `topology.metric_space.basic` and `topology.algebra.ordered`, the `'` version is
phrased using "eventually" and the non-`'` version is phrased absolutely. -/
lemma squeeze_zero_norm' {f : γ → α} {g : γ → ℝ} {t₀ : filter γ}
(h : ∀ᶠ n in t₀, ∥f n∥ ≤ g n)
(h' : tendsto g t₀ (𝓝 0)) : tendsto f t₀ (𝓝 0) :=
tendsto_zero_iff_norm_tendsto_zero.mpr
(squeeze_zero' (eventually_of_forall (λ n, norm_nonneg _)) h h')
/-- Special case of the sandwich theorem: if the norm of `f` is bounded by a real function `g` which
tends to `0`, then `f` tends to `0`. -/
lemma squeeze_zero_norm {f : γ → α} {g : γ → ℝ} {t₀ : filter γ}
(h : ∀ (n:γ), ∥f n∥ ≤ g n)
(h' : tendsto g t₀ (𝓝 0)) :
tendsto f t₀ (𝓝 0) :=
squeeze_zero_norm' (eventually_of_forall h) h'
lemma lim_norm (x : α) : tendsto (λg : α, ∥g - x∥) (𝓝 x) (𝓝 0) :=
tendsto_iff_norm_tendsto_zero.1 (continuous_iff_continuous_at.1 continuous_id x)
lemma lim_norm_zero : tendsto (λg : α, ∥g∥) (𝓝 0) (𝓝 0) :=
by simpa using lim_norm (0:α)
lemma continuous_norm : continuous (λg:α, ∥g∥) :=
begin
rw continuous_iff_continuous_at,
intro x,
rw [continuous_at, tendsto_iff_dist_tendsto_zero],
exact squeeze_zero (λ t, abs_nonneg _) (λ t, abs_norm_sub_norm_le _ _) (lim_norm x)
end
lemma filter.tendsto.norm {β : Type*} {l : filter β} {f : β → α} {a : α} (h : tendsto f l (𝓝 a)) :
tendsto (λ x, ∥f x∥) l (𝓝 ∥a∥) :=
tendsto.comp continuous_norm.continuous_at h
lemma continuous.norm [topological_space γ] {f : γ → α} (hf : continuous f) :
continuous (λ x, ∥f x∥) :=
continuous_norm.comp hf
lemma continuous_nnnorm : continuous (nnnorm : α → nnreal) :=
continuous_subtype_mk _ continuous_norm
lemma filter.tendsto.nnnorm {β : Type*} {l : filter β} {f : β → α} {a : α} (h : tendsto f l (𝓝 a)) :
tendsto (λ x, nnnorm (f x)) l (𝓝 (nnnorm a)) :=
tendsto.comp continuous_nnnorm.continuous_at h
/-- If `∥y∥→∞`, then we can assume `y≠x` for any fixed `x`. -/
lemma eventually_ne_of_tendsto_norm_at_top {l : filter γ} {f : γ → α}
(h : tendsto (λ y, ∥f y∥) l at_top) (x : α) :
∀ᶠ y in l, f y ≠ x :=
begin
have : ∀ᶠ y in l, 1 + ∥x∥ ≤ ∥f y∥ := h (mem_at_top (1 + ∥x∥)),
refine this.mono (λ y hy hxy, _),
subst x,
exact not_le_of_lt zero_lt_one (add_le_iff_nonpos_left.1 hy)
end
/-- A normed group is a uniform additive group, i.e., addition and subtraction are uniformly
continuous. -/
@[priority 100] -- see Note [lower instance priority]
instance normed_uniform_group : uniform_add_group α :=
⟨(lipschitz_with.prod_fst.sub lipschitz_with.prod_snd).uniform_continuous⟩
@[priority 100] -- see Note [lower instance priority]
instance normed_top_monoid : has_continuous_add α := by apply_instance -- short-circuit type class inference
@[priority 100] -- see Note [lower instance priority]
instance normed_top_group : topological_add_group α := by apply_instance -- short-circuit type class inference
end normed_group
section normed_ring
/-- A normed ring is a ring endowed with a norm which satisfies the inequality `∥x y∥ ≤ ∥x∥ ∥y∥`. -/
class normed_ring (α : Type*) extends has_norm α, ring α, metric_space α :=
(dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b)
/-- A normed commutative ring is a commutative ring endowed with a norm which satisfies
the inequality `∥x y∥ ≤ ∥x∥ ∥y∥`. -/
class normed_comm_ring (α : Type*) extends normed_ring α :=
(mul_comm : ∀ x y : α, x * y = y * x)
/-- A mixin class with the axiom `∥1∥ = 1`. Many `normed_ring`s and all `normed_field`s satisfy this
axiom. -/
class norm_one_class (α : Type*) [has_norm α] [has_one α] : Prop :=
(norm_one : ∥(1:α)∥ = 1)
export norm_one_class (norm_one)
attribute [simp] norm_one
@[simp] lemma nnnorm_one [normed_group α] [has_one α] [norm_one_class α] : nnnorm (1:α) = 1 :=
nnreal.eq norm_one
@[priority 100] -- see Note [lower instance priority]
instance normed_comm_ring.to_comm_ring [β : normed_comm_ring α] : comm_ring α := { ..β }
@[priority 100] -- see Note [lower instance priority]
instance normed_ring.to_normed_group [β : normed_ring α] : normed_group α := { ..β }
instance prod.norm_one_class [normed_group α] [has_one α] [norm_one_class α]
[normed_group β] [has_one β] [norm_one_class β] :
norm_one_class (α × β) :=
⟨by simp [prod.norm_def]⟩
variables [normed_ring α]
lemma norm_mul_le (a b : α) : (∥a*b∥) ≤ (∥a∥) * (∥b∥) :=
normed_ring.norm_mul _ _
lemma list.norm_prod_le' : ∀ {l : list α}, l ≠ [] → ∥l.prod∥ ≤ (l.map norm).prod
| [] h := (h rfl).elim
| [a] _ := by simp
| (a :: b :: l) _ :=
begin
rw [list.map_cons, list.prod_cons, @list.prod_cons _ _ _ ∥a∥],
refine le_trans (norm_mul_le _ _) (mul_le_mul_of_nonneg_left _ (norm_nonneg _)),
exact list.norm_prod_le' (list.cons_ne_nil b l)
end
lemma list.norm_prod_le [norm_one_class α] : ∀ l : list α, ∥l.prod∥ ≤ (l.map norm).prod
| [] := by simp
| (a::l) := list.norm_prod_le' (list.cons_ne_nil a l)
lemma finset.norm_prod_le' {α : Type*} [normed_comm_ring α] (s : finset ι) (hs : s.nonempty)
(f : ι → α) :
∥∏ i in s, f i∥ ≤ ∏ i in s, ∥f i∥ :=
begin
rcases s with ⟨⟨l⟩, hl⟩,
have : l.map f ≠ [], by simpa using hs,
simpa using list.norm_prod_le' this
end
lemma finset.norm_prod_le {α : Type*} [normed_comm_ring α] [norm_one_class α] (s : finset ι)
(f : ι → α) :
∥∏ i in s, f i∥ ≤ ∏ i in s, ∥f i∥ :=
begin
rcases s with ⟨⟨l⟩, hl⟩,
simpa using (l.map f).norm_prod_le
end
/-- If `α` is a normed ring, then `∥a^n∥≤ ∥a∥^n` for `n > 0`. See also `norm_pow_le`. -/
lemma norm_pow_le' (a : α) : ∀ {n : ℕ}, 0 < n → ∥a^n∥ ≤ ∥a∥^n
| 1 h := by simp
| (n+2) h :=
le_trans (norm_mul_le a (a^(n+1)))
(mul_le_mul (le_refl _)
(norm_pow_le' (nat.succ_pos _)) (norm_nonneg _) (norm_nonneg _))
/-- If `α` is a normed ring with `∥1∥=1`, then `∥a^n∥≤ ∥a∥^n`. See also `norm_pow_le'`. -/
lemma norm_pow_le [norm_one_class α] (a : α) : ∀ (n : ℕ), ∥a^n∥ ≤ ∥a∥^n
| 0 := by simp
| (n+1) := norm_pow_le' a n.zero_lt_succ
lemma eventually_norm_pow_le (a : α) : ∀ᶠ (n:ℕ) in at_top, ∥a ^ n∥ ≤ ∥a∥ ^ n :=
eventually_at_top.mpr ⟨1, λ b h, norm_pow_le' a (nat.succ_le_iff.mp h)⟩
lemma units.norm_pos [nontrivial α] (x : units α) : 0 < ∥(x:α)∥ :=
norm_pos_iff.mpr (units.ne_zero x)
/-- In a normed ring, the left-multiplication `add_monoid_hom` is bounded. -/
lemma mul_left_bound (x : α) :
∀ (y:α), ∥add_monoid_hom.mul_left x y∥ ≤ ∥x∥ * ∥y∥ :=
norm_mul_le x
/-- In a normed ring, the right-multiplication `add_monoid_hom` is bounded. -/
lemma mul_right_bound (x : α) :
∀ (y:α), ∥add_monoid_hom.mul_right x y∥ ≤ ∥x∥ * ∥y∥ :=
λ y, by {rw mul_comm, convert norm_mul_le y x}
/-- Normed ring structure on the product of two normed rings, using the sup norm. -/
instance prod.normed_ring [normed_ring β] : normed_ring (α × β) :=
{ norm_mul := assume x y,
calc
∥x * y∥ = ∥(x.1*y.1, x.2*y.2)∥ : rfl
... = (max ∥x.1*y.1∥ ∥x.2*y.2∥) : rfl
... ≤ (max (∥x.1∥*∥y.1∥) (∥x.2∥*∥y.2∥)) :
max_le_max (norm_mul_le (x.1) (y.1)) (norm_mul_le (x.2) (y.2))
... = (max (∥x.1∥*∥y.1∥) (∥y.2∥*∥x.2∥)) : by simp[mul_comm]
... ≤ (max (∥x.1∥) (∥x.2∥)) * (max (∥y.2∥) (∥y.1∥)) : by { apply max_mul_mul_le_max_mul_max; simp [norm_nonneg] }
... = (max (∥x.1∥) (∥x.2∥)) * (max (∥y.1∥) (∥y.2∥)) : by simp[max_comm]
... = (∥x∥*∥y∥) : rfl,
..prod.normed_group }
end normed_ring
@[priority 100] -- see Note [lower instance priority]
instance normed_ring_top_monoid [normed_ring α] : has_continuous_mul α :=
⟨ continuous_iff_continuous_at.2 $ λ x, tendsto_iff_norm_tendsto_zero.2 $
have ∀ e : α × α, e.fst * e.snd - x.fst * x.snd =
e.fst * e.snd - e.fst * x.snd + (e.fst * x.snd - x.fst * x.snd), by intro; rw sub_add_sub_cancel,
begin
apply squeeze_zero,
{ intro, apply norm_nonneg },
{ simp only [this], intro, apply norm_add_le },
{ rw ←zero_add (0 : ℝ), apply tendsto.add,
{ apply squeeze_zero,
{ intro, apply norm_nonneg },
{ intro t, show ∥t.fst * t.snd - t.fst * x.snd∥ ≤ ∥t.fst∥ * ∥t.snd - x.snd∥,
rw ←mul_sub, apply norm_mul_le },
{ rw ←mul_zero (∥x.fst∥), apply tendsto.mul,
{ apply continuous_iff_continuous_at.1,
apply continuous_norm.comp continuous_fst },
{ apply tendsto_iff_norm_tendsto_zero.1,
apply continuous_iff_continuous_at.1,
apply continuous_snd }}},
{ apply squeeze_zero,
{ intro, apply norm_nonneg },
{ intro t, show ∥t.fst * x.snd - x.fst * x.snd∥ ≤ ∥t.fst - x.fst∥ * ∥x.snd∥,
rw ←sub_mul, apply norm_mul_le },
{ rw ←zero_mul (∥x.snd∥), apply tendsto.mul,
{ apply tendsto_iff_norm_tendsto_zero.1,
apply continuous_iff_continuous_at.1,
apply continuous_fst },
{ apply tendsto_const_nhds }}}}
end ⟩
/-- A normed ring is a topological ring. -/
@[priority 100] -- see Note [lower instance priority]
instance normed_top_ring [normed_ring α] : topological_ring α :=
⟨ continuous_iff_continuous_at.2 $ λ x, tendsto_iff_norm_tendsto_zero.2 $
have ∀ e : α, -e - -x = -(e - x), by intro; simp,
by simp only [this, norm_neg]; apply lim_norm ⟩
/-- A normed field is a field with a norm satisfying ∥x y∥ = ∥x∥ ∥y∥. -/
class normed_field (α : Type*) extends has_norm α, field α, metric_space α :=
(dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul' : ∀ a b, norm (a * b) = norm a * norm b)
/-- A nondiscrete normed field is a normed field in which there is an element of norm different from
`0` and `1`. This makes it possible to bring any element arbitrarily close to `0` by multiplication
by the powers of any element, and thus to relate algebra and topology. -/
class nondiscrete_normed_field (α : Type*) extends normed_field α :=
(non_trivial : ∃x:α, 1<∥x∥)
namespace normed_field
section normed_field
variables [normed_field α]
@[simp] lemma norm_mul (a b : α) : ∥a * b∥ = ∥a∥ * ∥b∥ :=
normed_field.norm_mul' a b
@[priority 100] -- see Note [lower instance priority]
instance to_normed_comm_ring : normed_comm_ring α :=
{ norm_mul := λ a b, (norm_mul a b).le, ..‹normed_field α› }
@[priority 900]
instance to_norm_one_class : norm_one_class α :=
⟨mul_left_cancel' (mt norm_eq_zero.1 (@one_ne_zero α _ _)) $
by rw [← norm_mul, mul_one, mul_one]⟩
/-- `norm` as a `monoid_hom`. -/
@[simps] def norm_hom : monoid_with_zero_hom α ℝ := ⟨norm, norm_zero, norm_one, norm_mul⟩
@[simp] lemma norm_pow (a : α) : ∀ (n : ℕ), ∥a ^ n∥ = ∥a∥ ^ n :=
norm_hom.to_monoid_hom.map_pow a
@[simp] lemma norm_prod (s : finset β) (f : β → α) :
∥∏ b in s, f b∥ = ∏ b in s, ∥f b∥ :=
(norm_hom.to_monoid_hom : α →* ℝ).map_prod f s
@[simp] lemma norm_div (a b : α) : ∥a / b∥ = ∥a∥ / ∥b∥ :=
(norm_hom : monoid_with_zero_hom α ℝ).map_div a b
@[simp] lemma norm_inv (a : α) : ∥a⁻¹∥ = ∥a∥⁻¹ :=
(norm_hom : monoid_with_zero_hom α ℝ).map_inv' a
@[simp] lemma nnnorm_inv (a : α) : nnnorm (a⁻¹) = (nnnorm a)⁻¹ :=
nnreal.eq $ by simp
@[simp] lemma norm_fpow : ∀ (a : α) (n : ℤ), ∥a^n∥ = ∥a∥^n :=
(norm_hom : monoid_with_zero_hom α ℝ).map_fpow
@[priority 100] -- see Note [lower instance priority]
instance : has_continuous_inv' α :=
begin
refine ⟨λ r r0, (nhds_basis_closed_ball.tendsto_iff nhds_basis_closed_ball).2 (λε εpos, _)⟩,
let δ := min (ε/2 * ∥r∥^2) (∥r∥/2),
have norm_r_pos : 0 < ∥r∥ := norm_pos_iff.mpr r0,
have A : 0 < ε / 2 * ∥r∥ ^ 2 := mul_pos (half_pos εpos) (pow_pos norm_r_pos 2),
have δpos : 0 < δ, by simp [half_pos norm_r_pos, A],
refine ⟨δ, δpos, λ x hx, _⟩,
have rx : ∥r∥/2 ≤ ∥x∥ := calc
∥r∥/2 = ∥r∥ - ∥r∥/2 : by ring
... ≤ ∥r∥ - ∥r - x∥ :
begin
apply sub_le_sub (le_refl _),
rw [← dist_eq_norm, dist_comm],
exact le_trans hx (min_le_right _ _)
end
... ≤ ∥r - (r - x)∥ : norm_sub_norm_le r (r - x)
... = ∥x∥ : by simp [sub_sub_cancel],
have norm_x_pos : 0 < ∥x∥ := lt_of_lt_of_le (half_pos norm_r_pos) rx,
have : x⁻¹ - r⁻¹ = (r - x) * x⁻¹ * r⁻¹,
by rw [sub_mul, sub_mul, mul_inv_cancel (norm_pos_iff.mp norm_x_pos), one_mul, mul_comm,
← mul_assoc, inv_mul_cancel r0, one_mul],
calc dist x⁻¹ r⁻¹ = ∥x⁻¹ - r⁻¹∥ : dist_eq_norm _ _
... ≤ ∥r-x∥ * ∥x∥⁻¹ * ∥r∥⁻¹ : by rw [this, norm_mul, norm_mul, norm_inv, norm_inv]
... ≤ (ε/2 * ∥r∥^2) * (2 * ∥r∥⁻¹) * (∥r∥⁻¹) : begin
apply_rules [mul_le_mul, inv_nonneg.2, le_of_lt A, norm_nonneg, mul_nonneg,
(inv_le_inv norm_x_pos norm_r_pos).2, le_refl],
show ∥r - x∥ ≤ ε / 2 * ∥r∥ ^ 2,
by { rw [← dist_eq_norm, dist_comm], exact le_trans hx (min_le_left _ _) },
show ∥x∥⁻¹ ≤ 2 * ∥r∥⁻¹,
{ convert (inv_le_inv norm_x_pos (half_pos norm_r_pos)).2 rx,
rw [inv_div, div_eq_inv_mul, mul_comm] },
show (0 : ℝ) ≤ 2, by norm_num
end
... = ε * (∥r∥ * ∥r∥⁻¹)^2 : by { generalize : ∥r∥⁻¹ = u, ring }
... = ε : by { rw [mul_inv_cancel (ne.symm (ne_of_lt norm_r_pos))], simp }
end
end normed_field
variables (α) [nondiscrete_normed_field α]
lemma exists_one_lt_norm : ∃x : α, 1 < ∥x∥ := ‹nondiscrete_normed_field α›.non_trivial
lemma exists_norm_lt_one : ∃x : α, 0 < ∥x∥ ∧ ∥x∥ < 1 :=
begin
rcases exists_one_lt_norm α with ⟨y, hy⟩,
refine ⟨y⁻¹, _, _⟩,
{ simp only [inv_eq_zero, ne.def, norm_pos_iff],
rintro rfl,
rw norm_zero at hy,
exact lt_asymm zero_lt_one hy },
{ simp [inv_lt_one hy] }
end
lemma exists_lt_norm (r : ℝ) : ∃ x : α, r < ∥x∥ :=
let ⟨w, hw⟩ := exists_one_lt_norm α in
let ⟨n, hn⟩ := pow_unbounded_of_one_lt r hw in
⟨w^n, by rwa norm_pow⟩
lemma exists_norm_lt {r : ℝ} (hr : 0 < r) : ∃ x : α, 0 < ∥x∥ ∧ ∥x∥ < r :=
let ⟨w, hw⟩ := exists_one_lt_norm α in
let ⟨n, hle, hlt⟩ := exists_int_pow_near' hr hw in
⟨w^n, by { rw norm_fpow; exact fpow_pos_of_pos (lt_trans zero_lt_one hw) _},
by rwa norm_fpow⟩
variable {α}
@[instance]
lemma punctured_nhds_ne_bot (x : α) : ne_bot (𝓝[{x}ᶜ] x) :=
begin
rw [← mem_closure_iff_nhds_within_ne_bot, metric.mem_closure_iff],
rintros ε ε0,
rcases normed_field.exists_norm_lt α ε0 with ⟨b, hb0, hbε⟩,
refine ⟨x + b, mt (set.mem_singleton_iff.trans add_right_eq_self).1 $ norm_pos_iff.1 hb0, _⟩,
rwa [dist_comm, dist_eq_norm, add_sub_cancel'],
end
@[instance]
lemma nhds_within_is_unit_ne_bot : ne_bot (𝓝[{x : α | is_unit x}] 0) :=
by simpa only [is_unit_iff_ne_zero] using punctured_nhds_ne_bot (0:α)
end normed_field
instance : normed_field ℝ :=
{ norm := λ x, abs x,
dist_eq := assume x y, rfl,
norm_mul' := abs_mul }
instance : nondiscrete_normed_field ℝ :=
{ non_trivial := ⟨2, by { unfold norm, rw abs_of_nonneg; norm_num }⟩ }
namespace real
lemma norm_eq_abs (r : ℝ) : ∥r∥ = abs r := rfl
lemma norm_of_nonneg {x : ℝ} (hx : 0 ≤ x) : ∥x∥ = x :=
abs_of_nonneg hx
@[simp] lemma norm_coe_nat (n : ℕ) : ∥(n : ℝ)∥ = n := abs_of_nonneg n.cast_nonneg
@[simp] lemma nnnorm_coe_nat (n : ℕ) : nnnorm (n : ℝ) = n := nnreal.eq $ by simp
@[simp] lemma norm_two : ∥(2:ℝ)∥ = 2 := abs_of_pos (@zero_lt_two ℝ _ _)
@[simp] lemma nnnorm_two : nnnorm (2:ℝ) = 2 := nnreal.eq $ by simp
open_locale nnreal
@[simp] lemma nnreal.norm_eq (x : ℝ≥0) : ∥(x : ℝ)∥ = x :=
by rw [real.norm_eq_abs, x.abs_eq]
lemma nnnorm_coe_eq_self {x : ℝ≥0} : nnnorm (x : ℝ) = x :=
by { ext, exact norm_of_nonneg (zero_le x) }
lemma nnnorm_of_nonneg {x : ℝ} (hx : 0 ≤ x) : nnnorm x = ⟨x, hx⟩ :=
@nnnorm_coe_eq_self ⟨x, hx⟩
lemma ennnorm_eq_of_real {x : ℝ} (hx : 0 ≤ x) : (nnnorm x : ennreal) = ennreal.of_real x :=
by { rw [← of_real_norm_eq_coe_nnnorm, norm_of_nonneg hx] }
end real
@[simp] lemma norm_norm [normed_group α] (x : α) : ∥∥x∥∥ = ∥x∥ :=
by rw [real.norm_of_nonneg (norm_nonneg _)]
@[simp] lemma nnnorm_norm [normed_group α] (a : α) : nnnorm ∥a∥ = nnnorm a :=
by simp only [nnnorm, norm_norm]
instance : normed_comm_ring ℤ :=
{ norm := λ n, ∥(n : ℝ)∥,
norm_mul := λ m n, le_of_eq $ by simp only [norm, int.cast_mul, abs_mul],
dist_eq := λ m n, by simp only [int.dist_eq, norm, int.cast_sub],
mul_comm := mul_comm }
@[norm_cast] lemma int.norm_cast_real (m : ℤ) : ∥(m : ℝ)∥ = ∥m∥ := rfl
instance : norm_one_class ℤ :=
⟨by simp [← int.norm_cast_real]⟩
instance : normed_field ℚ :=
{ norm := λ r, ∥(r : ℝ)∥,
norm_mul' := λ r₁ r₂, by simp only [norm, rat.cast_mul, abs_mul],
dist_eq := λ r₁ r₂, by simp only [rat.dist_eq, norm, rat.cast_sub] }
instance : nondiscrete_normed_field ℚ :=
{ non_trivial := ⟨2, by { unfold norm, rw abs_of_nonneg; norm_num }⟩ }
@[norm_cast, simp] lemma rat.norm_cast_real (r : ℚ) : ∥(r : ℝ)∥ = ∥r∥ := rfl
@[norm_cast, simp] lemma int.norm_cast_rat (m : ℤ) : ∥(m : ℚ)∥ = ∥m∥ :=
by rw [← rat.norm_cast_real, ← int.norm_cast_real]; congr' 1; norm_cast
section normed_space
section prio
set_option extends_priority 920
-- Here, we set a rather high priority for the instance `[normed_space α β] : semimodule α β`
-- to take precedence over `semiring.to_semimodule` as this leads to instance paths with better
-- unification properties.
-- see Note[vector space definition] for why we extend `semimodule`.
/-- A normed space over a normed field is a vector space endowed with a norm which satisfies the
equality `∥c • x∥ = ∥c∥ ∥x∥`. We require only `∥c • x∥ ≤ ∥c∥ ∥x∥` in the definition, then prove
`∥c • x∥ = ∥c∥ ∥x∥` in `norm_smul`. -/
class normed_space (α : Type*) (β : Type*) [normed_field α] [normed_group β]
extends semimodule α β :=
(norm_smul_le : ∀ (a:α) (b:β), ∥a • b∥ ≤ ∥a∥ * ∥b∥)
end prio
variables [normed_field α] [normed_group β]
instance normed_field.to_normed_space : normed_space α α :=
{ norm_smul_le := λ a b, le_of_eq (normed_field.norm_mul a b) }
lemma norm_smul [normed_space α β] (s : α) (x : β) : ∥s • x∥ = ∥s∥ * ∥x∥ :=
begin
classical,
by_cases h : s = 0,
{ simp [h] },
{ refine le_antisymm (normed_space.norm_smul_le s x) _,
calc ∥s∥ * ∥x∥ = ∥s∥ * ∥s⁻¹ • s • x∥ : by rw [inv_smul_smul' h]
... ≤ ∥s∥ * (∥s⁻¹∥ * ∥s • x∥) : _
... = ∥s • x∥ : _,
exact mul_le_mul_of_nonneg_left (normed_space.norm_smul_le _ _) (norm_nonneg _),
rw [normed_field.norm_inv, ← mul_assoc, mul_inv_cancel, one_mul],
rwa [ne.def, norm_eq_zero] }
end
@[simp] lemma abs_norm_eq_norm (z : β) : abs ∥z∥ = ∥z∥ :=
(abs_eq (norm_nonneg z)).mpr (or.inl rfl)
lemma dist_smul [normed_space α β] (s : α) (x y : β) : dist (s • x) (s • y) = ∥s∥ * dist x y :=
by simp only [dist_eq_norm, (norm_smul _ _).symm, smul_sub]
lemma nnnorm_smul [normed_space α β] (s : α) (x : β) : nnnorm (s • x) = nnnorm s * nnnorm x :=
nnreal.eq $ norm_smul s x
lemma nndist_smul [normed_space α β] (s : α) (x y : β) :
nndist (s • x) (s • y) = nnnorm s * nndist x y :=
nnreal.eq $ dist_smul s x y
lemma norm_smul_of_nonneg [normed_space ℝ β] {t : ℝ} (ht : 0 ≤ t) (x : β) : ∥t • x∥ = t * ∥x∥ :=
by rw [norm_smul, real.norm_eq_abs, abs_of_nonneg ht]
variables {E : Type*} {F : Type*}
[normed_group E] [normed_space α E] [normed_group F] [normed_space α F]
@[priority 100] -- see Note [lower instance priority]
instance normed_space.topological_vector_space : topological_vector_space α E :=
begin
refine { continuous_smul := continuous_iff_continuous_at.2 $
λ p, tendsto_iff_norm_tendsto_zero.2 _ },
refine squeeze_zero (λ _, norm_nonneg _) _ _,
{ exact λ q, ∥q.1 - p.1∥ * ∥q.2∥ + ∥p.1∥ * ∥q.2 - p.2∥ },
{ intro q,
rw [← sub_add_sub_cancel, ← norm_smul, ← norm_smul, smul_sub, sub_smul],
exact norm_add_le _ _ },
{ conv { congr, skip, skip, congr, rw [← zero_add (0:ℝ)], congr,
rw [← zero_mul ∥p.2∥], skip, rw [← mul_zero ∥p.1∥] },
exact ((tendsto_iff_norm_tendsto_zero.1 (continuous_fst.tendsto p)).mul
(continuous_snd.tendsto p).norm).add
(tendsto_const_nhds.mul (tendsto_iff_norm_tendsto_zero.1 (continuous_snd.tendsto p))) }
end
theorem closure_ball [normed_space ℝ E] (x : E) {r : ℝ} (hr : 0 < r) :
closure (ball x r) = closed_ball x r :=
begin
refine set.subset.antisymm closure_ball_subset_closed_ball (λ y hy, _),
have : continuous_within_at (λ c : ℝ, c • (y - x) + x) (set.Ico 0 1) 1 :=
((continuous_id.smul continuous_const).add continuous_const).continuous_within_at,
convert this.mem_closure _ _,
{ rw [one_smul, sub_add_cancel] },
{ simp [closure_Ico (@zero_lt_one ℝ _ _), zero_le_one] },
{ rintros c ⟨hc0, hc1⟩,
rw [set.mem_preimage, mem_ball, dist_eq_norm, add_sub_cancel, norm_smul, real.norm_eq_abs,
abs_of_nonneg hc0, mul_comm, ← mul_one r],
rw [mem_closed_ball, dist_eq_norm] at hy,
apply mul_lt_mul'; assumption }
end
theorem frontier_ball [normed_space ℝ E] (x : E) {r : ℝ} (hr : 0 < r) :
frontier (ball x r) = sphere x r :=
begin
rw [frontier, closure_ball x hr, is_open_ball.interior_eq],
ext x, exact (@eq_iff_le_not_lt ℝ _ _ _).symm
end
theorem interior_closed_ball [normed_space ℝ E] (x : E) {r : ℝ} (hr : 0 < r) :
interior (closed_ball x r) = ball x r :=
begin
refine set.subset.antisymm _ ball_subset_interior_closed_ball,
intros y hy,
rcases le_iff_lt_or_eq.1 (mem_closed_ball.1 $ interior_subset hy) with hr|rfl, { exact hr },
set f : ℝ → E := λ c : ℝ, c • (y - x) + x,
suffices : f ⁻¹' closed_ball x (dist y x) ⊆ set.Icc (-1) 1,
{ have hfc : continuous f := (continuous_id.smul continuous_const).add continuous_const,
have hf1 : (1:ℝ) ∈ f ⁻¹' (interior (closed_ball x $ dist y x)), by simpa [f],
have h1 : (1:ℝ) ∈ interior (set.Icc (-1:ℝ) 1) :=
interior_mono this (preimage_interior_subset_interior_preimage hfc hf1),
contrapose h1,
simp },
intros c hc,
rw [set.mem_Icc, ← abs_le, ← real.norm_eq_abs, ← mul_le_mul_right hr],
simpa [f, dist_eq_norm, norm_smul] using hc
end
theorem interior_closed_ball' [normed_space ℝ E] [nontrivial E] (x : E) (r : ℝ) :
interior (closed_ball x r) = ball x r :=
begin
rcases lt_trichotomy r 0 with hr|rfl|hr,
{ simp [closed_ball_eq_empty_iff_neg.2 hr, ball_eq_empty_iff_nonpos.2 (le_of_lt hr)] },
{ suffices : x ∉ interior {x},
{ rw [ball_zero, closed_ball_zero, ← set.subset_empty_iff],
intros y hy,
obtain rfl : y = x := set.mem_singleton_iff.1 (interior_subset hy),
exact this hy },
rw [← set.mem_compl_iff, ← closure_compl],
rcases exists_ne (0 : E) with ⟨z, hz⟩,
suffices : (λ c : ℝ, x + c • z) 0 ∈ closure ({x}ᶜ : set E),
by simpa only [zero_smul, add_zero] using this,
have : (0:ℝ) ∈ closure (set.Ioi (0:ℝ)), by simp [closure_Ioi],
refine (continuous_const.add (continuous_id.smul
continuous_const)).continuous_within_at.mem_closure this _,
intros c hc,
simp [smul_eq_zero, hz, ne_of_gt hc] },
{ exact interior_closed_ball x hr }
end
theorem frontier_closed_ball [normed_space ℝ E] (x : E) {r : ℝ} (hr : 0 < r) :
frontier (closed_ball x r) = sphere x r :=
by rw [frontier, closure_closed_ball, interior_closed_ball x hr,
closed_ball_diff_ball]
theorem frontier_closed_ball' [normed_space ℝ E] [nontrivial E] (x : E) (r : ℝ) :
frontier (closed_ball x r) = sphere x r :=
by rw [frontier, closure_closed_ball, interior_closed_ball' x r, closed_ball_diff_ball]
open normed_field
/-- If there is a scalar `c` with `∥c∥>1`, then any element can be moved by scalar multiplication to
any shell of width `∥c∥`. Also recap information on the norm of the rescaling element that shows
up in applications. -/
lemma rescale_to_shell {c : α} (hc : 1 < ∥c∥) {ε : ℝ} (εpos : 0 < ε) {x : E} (hx : x ≠ 0) :
∃d:α, d ≠ 0 ∧ ∥d • x∥ < ε ∧ (ε/∥c∥ ≤ ∥d • x∥) ∧ (∥d∥⁻¹ ≤ ε⁻¹ * ∥c∥ * ∥x∥) :=
begin
have xεpos : 0 < ∥x∥/ε := div_pos (norm_pos_iff.2 hx) εpos,
rcases exists_int_pow_near xεpos hc with ⟨n, hn⟩,
have cpos : 0 < ∥c∥ := lt_trans (zero_lt_one : (0 :ℝ) < 1) hc,
have cnpos : 0 < ∥c^(n+1)∥ := by { rw norm_fpow, exact lt_trans xεpos hn.2 },
refine ⟨(c^(n+1))⁻¹, _, _, _, _⟩,
show (c ^ (n + 1))⁻¹ ≠ 0,
by rwa [ne.def, inv_eq_zero, ← ne.def, ← norm_pos_iff],
show ∥(c ^ (n + 1))⁻¹ • x∥ < ε,
{ rw [norm_smul, norm_inv, ← div_eq_inv_mul, div_lt_iff cnpos, mul_comm, norm_fpow],
exact (div_lt_iff εpos).1 (hn.2) },
show ε / ∥c∥ ≤ ∥(c ^ (n + 1))⁻¹ • x∥,
{ rw [div_le_iff cpos, norm_smul, norm_inv, norm_fpow, fpow_add (ne_of_gt cpos),
fpow_one, mul_inv_rev', mul_comm, ← mul_assoc, ← mul_assoc, mul_inv_cancel (ne_of_gt cpos),
one_mul, ← div_eq_inv_mul, le_div_iff (fpow_pos_of_pos cpos _), mul_comm],
exact (le_div_iff εpos).1 hn.1 },
show ∥(c ^ (n + 1))⁻¹∥⁻¹ ≤ ε⁻¹ * ∥c∥ * ∥x∥,
{ have : ε⁻¹ * ∥c∥ * ∥x∥ = ε⁻¹ * ∥x∥ * ∥c∥, by ring,
rw [norm_inv, inv_inv', norm_fpow, fpow_add (ne_of_gt cpos), fpow_one, this, ← div_eq_inv_mul],
exact mul_le_mul_of_nonneg_right hn.1 (norm_nonneg _) }
end
/-- The product of two normed spaces is a normed space, with the sup norm. -/
instance : normed_space α (E × F) :=
{ norm_smul_le := λ s x, le_of_eq $ by simp [prod.norm_def, norm_smul, mul_max_of_nonneg],
-- TODO: without the next two lines Lean unfolds `≤` to `real.le`
add_smul := λ r x y, prod.ext (add_smul _ _ _) (add_smul _ _ _),
smul_add := λ r x y, prod.ext (smul_add _ _ _) (smul_add _ _ _),
..prod.normed_group,
..prod.semimodule }
/-- The product of finitely many normed spaces is a normed space, with the sup norm. -/
instance pi.normed_space {E : ι → Type*} [fintype ι] [∀i, normed_group (E i)]
[∀i, normed_space α (E i)] : normed_space α (Πi, E i) :=
{ norm_smul_le := λ a f, le_of_eq $
show (↑(finset.sup finset.univ (λ (b : ι), nnnorm (a • f b))) : ℝ) =
nnnorm a * ↑(finset.sup finset.univ (λ (b : ι), nnnorm (f b))),
by simp only [(nnreal.coe_mul _ _).symm, nnreal.mul_finset_sup, nnnorm_smul] }
/-- A subspace of a normed space is also a normed space, with the restriction of the norm. -/
instance submodule.normed_space {𝕜 : Type*} [normed_field 𝕜]
{E : Type*} [normed_group E] [normed_space 𝕜 E] (s : submodule 𝕜 E) : normed_space 𝕜 s :=
{ norm_smul_le := λc x, le_of_eq $ norm_smul c (x : E) }
end normed_space
section normed_algebra
/-- A normed algebra `𝕜'` over `𝕜` is an algebra endowed with a norm for which the embedding of
`𝕜` in `𝕜'` is an isometry. -/
class normed_algebra (𝕜 : Type*) (𝕜' : Type*) [normed_field 𝕜] [normed_ring 𝕜']
extends algebra 𝕜 𝕜' :=
(norm_algebra_map_eq : ∀x:𝕜, ∥algebra_map 𝕜 𝕜' x∥ = ∥x∥)
@[simp] lemma norm_algebra_map_eq {𝕜 : Type*} (𝕜' : Type*) [normed_field 𝕜] [normed_ring 𝕜']
[h : normed_algebra 𝕜 𝕜'] (x : 𝕜) : ∥algebra_map 𝕜 𝕜' x∥ = ∥x∥ :=
normed_algebra.norm_algebra_map_eq _
variables (𝕜 : Type*) [normed_field 𝕜]
variables (𝕜' : Type*) [normed_ring 𝕜']
@[priority 100]
instance normed_algebra.to_normed_space [h : normed_algebra 𝕜 𝕜'] : normed_space 𝕜 𝕜' :=
{ norm_smul_le := λ s x, calc
∥s • x∥ = ∥((algebra_map 𝕜 𝕜') s) * x∥ : by { rw h.smul_def', refl }
... ≤ ∥algebra_map 𝕜 𝕜' s∥ * ∥x∥ : normed_ring.norm_mul _ _
... = ∥s∥ * ∥x∥ : by rw norm_algebra_map_eq,
..h }
instance normed_algebra.id : normed_algebra 𝕜 𝕜 :=
{ norm_algebra_map_eq := by simp,
.. algebra.id 𝕜}
variables {𝕜'} [normed_algebra 𝕜 𝕜']
include 𝕜
@[simp] lemma normed_algebra.norm_one : ∥(1:𝕜')∥ = 1 :=
by simpa using (norm_algebra_map_eq 𝕜' (1:𝕜))
lemma normed_algebra.norm_one_class : norm_one_class 𝕜' :=
⟨normed_algebra.norm_one 𝕜⟩
lemma normed_algebra.zero_ne_one : (0:𝕜') ≠ 1 :=
begin
refine (norm_pos_iff.mp _).symm,
rw @normed_algebra.norm_one 𝕜, norm_num,
end
lemma normed_algebra.nontrivial : nontrivial 𝕜' :=
⟨⟨0, 1, normed_algebra.zero_ne_one 𝕜⟩⟩
end normed_algebra
section restrict_scalars
variables (𝕜 : Type*) (𝕜' : Type*) [normed_field 𝕜] [normed_field 𝕜'] [normed_algebra 𝕜 𝕜']
(E : Type*) [normed_group E] [normed_space 𝕜' E]
/-- Warning: This declaration should be used judiciously.
Please consider using `is_scalar_tower` instead.
`𝕜`-normed space structure induced by a `𝕜'`-normed space structure when `𝕜'` is a
normed algebra over `𝕜`. Not registered as an instance as `𝕜'` can not be inferred.
The type synonym `semimodule.restrict_scalars 𝕜 𝕜' E` will be endowed with this instance by default.
-/
def normed_space.restrict_scalars : normed_space 𝕜 E :=
{ norm_smul_le := λc x, le_of_eq $ begin
change ∥(algebra_map 𝕜 𝕜' c) • x∥ = ∥c∥ * ∥x∥,
simp [norm_smul]
end,
..restrict_scalars.semimodule 𝕜 𝕜' E }
instance {𝕜 : Type*} {𝕜' : Type*} {E : Type*} [I : normed_group E] :
normed_group (restrict_scalars 𝕜 𝕜' E) := I
instance semimodule.restrict_scalars.normed_space_orig {𝕜 : Type*} {𝕜' : Type*} {E : Type*}
[normed_field 𝕜'] [normed_group E] [I : normed_space 𝕜' E] :
normed_space 𝕜' (restrict_scalars 𝕜 𝕜' E) := I
instance : normed_space 𝕜 (restrict_scalars 𝕜 𝕜' E) :=
(normed_space.restrict_scalars 𝕜 𝕜' E : normed_space 𝕜 E)
end restrict_scalars
section summable
open_locale classical
open finset filter
variables [normed_group α] [normed_group β]
lemma cauchy_seq_finset_iff_vanishing_norm {f : ι → α} :
cauchy_seq (λ s : finset ι, ∑ i in s, f i) ↔
∀ε > (0 : ℝ), ∃s:finset ι, ∀t, disjoint t s → ∥ ∑ i in t, f i ∥ < ε :=
begin
rw [cauchy_seq_finset_iff_vanishing, nhds_basis_ball.forall_iff],
{ simp only [ball_0_eq, set.mem_set_of_eq] },
{ rintros s t hst ⟨s', hs'⟩,
exact ⟨s', λ t' ht', hst $ hs' _ ht'⟩ }
end
lemma summable_iff_vanishing_norm [complete_space α] {f : ι → α} :
summable f ↔ ∀ε > (0 : ℝ), ∃s:finset ι, ∀t, disjoint t s → ∥ ∑ i in t, f i ∥ < ε :=
by rw [summable_iff_cauchy_seq_finset, cauchy_seq_finset_iff_vanishing_norm]
lemma cauchy_seq_finset_of_norm_bounded {f : ι → α} (g : ι → ℝ) (hg : summable g)
(h : ∀i, ∥f i∥ ≤ g i) : cauchy_seq (λ s : finset ι, ∑ i in s, f i) :=
cauchy_seq_finset_iff_vanishing_norm.2 $ assume ε hε,
let ⟨s, hs⟩ := summable_iff_vanishing_norm.1 hg ε hε in
⟨s, assume t ht,
have ∥∑ i in t, g i∥ < ε := hs t ht,
have nn : 0 ≤ ∑ i in t, g i := finset.sum_nonneg (assume a _, le_trans (norm_nonneg _) (h a)),
lt_of_le_of_lt (norm_sum_le_of_le t (λ i _, h i)) $
by rwa [real.norm_eq_abs, abs_of_nonneg nn] at this⟩
lemma cauchy_seq_finset_of_summable_norm {f : ι → α} (hf : summable (λa, ∥f a∥)) :
cauchy_seq (λ s : finset ι, ∑ a in s, f a) :=
cauchy_seq_finset_of_norm_bounded _ hf (assume i, le_refl _)
/-- If a function `f` is summable in norm, and along some sequence of finsets exhausting the space
its sum is converging to a limit `a`, then this holds along all finsets, i.e., `f` is summable
with sum `a`. -/
lemma has_sum_of_subseq_of_summable {f : ι → α} (hf : summable (λa, ∥f a∥))
{s : γ → finset ι} {p : filter γ} [ne_bot p]
(hs : tendsto s p at_top) {a : α} (ha : tendsto (λ b, ∑ i in s b, f i) p (𝓝 a)) :
has_sum f a :=
tendsto_nhds_of_cauchy_seq_of_subseq (cauchy_seq_finset_of_summable_norm hf) hs ha
/-- If `∑' i, ∥f i∥` is summable, then `∥(∑' i, f i)∥ ≤ (∑' i, ∥f i∥)`. Note that we do not assume
that `∑' i, f i` is summable, and it might not be the case if `α` is not a complete space. -/
lemma norm_tsum_le_tsum_norm {f : ι → α} (hf : summable (λi, ∥f i∥)) :
∥(∑'i, f i)∥ ≤ (∑' i, ∥f i∥) :=
begin
by_cases h : summable f,
{ have h₁ : tendsto (λs:finset ι, ∥∑ i in s, f i∥) at_top (𝓝 ∥(∑' i, f i)∥) :=
(continuous_norm.tendsto _).comp h.has_sum,
have h₂ : tendsto (λs:finset ι, ∑ i in s, ∥f i∥) at_top (𝓝 (∑' i, ∥f i∥)) :=
hf.has_sum,
exact le_of_tendsto_of_tendsto' h₁ h₂ (assume s, norm_sum_le _ _) },
{ rw tsum_eq_zero_of_not_summable h,
simp [tsum_nonneg] }
end
lemma has_sum_iff_tendsto_nat_of_summable_norm {f : ℕ → α} {a : α} (hf : summable (λi, ∥f i∥)) :
has_sum f a ↔ tendsto (λn:ℕ, ∑ i in range n, f i) at_top (𝓝 a) :=
⟨λ h, h.tendsto_sum_nat,
λ h, has_sum_of_subseq_of_summable hf tendsto_finset_range h⟩
/-- The direct comparison test for series: if the norm of `f` is bounded by a real function `g`
which is summable, then `f` is summable. -/
lemma summable_of_norm_bounded
[complete_space α] {f : ι → α} (g : ι → ℝ) (hg : summable g) (h : ∀i, ∥f i∥ ≤ g i) :
summable f :=
by { rw summable_iff_cauchy_seq_finset, exact cauchy_seq_finset_of_norm_bounded g hg h }
/-- Quantitative result associated to the direct comparison test for series: If `∑' i, g i` is
summable, and for all `i`, `∥f i∥ ≤ g i`, then `∥(∑' i, f i)∥ ≤ (∑' i, g i)`. Note that we do not
assume that `∑' i, f i` is summable, and it might not be the case if `α` is not a complete space. -/
lemma tsum_of_norm_bounded {f : ι → α} {g : ι → ℝ} {a : ℝ} (hg : has_sum g a) (h : ∀i, ∥f i∥ ≤ g i) :
∥(∑' (i:ι), f i)∥ ≤ a :=
begin
have h' : summable (λ (i : ι), ∥f i∥),
{ let f' : ι → ℝ := λ i, ∥f i∥,
have h'' : ∀ i, ∥f' i∥ ≤ g i,
{ intros i,
convert h i,
simp },
simpa [f'] using summable_of_norm_bounded g hg.summable h'' },
have h1 : ∥(∑' (i:ι), f i)∥ ≤ ∑' (i:ι), ∥f i∥ := by simpa using norm_tsum_le_tsum_norm h',
have h2 := tsum_le_tsum h h' hg.summable,
have h3 : a = ∑' (i:ι), g i := (has_sum.tsum_eq hg).symm,
linarith
end
variable [complete_space α]
/-- Variant of the direct comparison test for series: if the norm of `f` is eventually bounded by a
real function `g` which is summable, then `f` is summable. -/
lemma summable_of_norm_bounded_eventually {f : ι → α} (g : ι → ℝ) (hg : summable g)
(h : ∀ᶠ i in cofinite, ∥f i∥ ≤ g i) : summable f :=
begin
replace h := mem_cofinite.1 h,
refine h.summable_compl_iff.mp _,
refine summable_of_norm_bounded _ (h.summable_compl_iff.mpr hg) _,
rintros ⟨a, h'⟩,
simpa using h'
end
lemma summable_of_nnnorm_bounded {f : ι → α} (g : ι → nnreal) (hg : summable g)
(h : ∀i, nnnorm (f i) ≤ g i) : summable f :=
summable_of_norm_bounded (λ i, (g i : ℝ)) (nnreal.summable_coe.2 hg) (λ i, by exact_mod_cast h i)
lemma summable_of_summable_norm {f : ι → α} (hf : summable (λa, ∥f a∥)) : summable f :=
summable_of_norm_bounded _ hf (assume i, le_refl _)
lemma summable_of_summable_nnnorm {f : ι → α} (hf : summable (λa, nnnorm (f a))) : summable f :=
summable_of_nnnorm_bounded _ hf (assume i, le_refl _)
end summable
|
f111f7dae6ed71d9641f16b8d79e5c4dde836ac6 | 7b02c598aa57070b4cf4fbfe2416d0479220187f | /homotopy/3x3.hlean | 646421b034b2fab9f5d3e894744d7f03e35195f9 | [
"Apache-2.0"
] | permissive | jdchristensen/Spectral | 50d4f0ddaea1484d215ef74be951da6549de221d | 6ded2b94d7ae07c4098d96a68f80a9cd3d433eb8 | refs/heads/master | 1,611,555,010,649 | 1,496,724,191,000 | 1,496,724,191,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,018 | hlean | -- WIP
import ..move_to_lib
open function eq
namespace pushout
section
-- structure span2 : Type :=
-- {A₀₀ A₀₂ A₀₄ A₂₀ A₂₂ A₂₄ A₄₀ A₄₂ A₄₄ : Type}
-- {f₀₁ : A₀₂ → A₀₀} {f₂₁ : A₂₂ → A₂₀} {f₄₁ : A₄₂ → A₄₀}
-- {f₀₃ : A₀₂ → A₀₄} {f₂₃ : A₂₂ → A₂₄} {f₄₃ : A₄₂ → A₄₄}
-- {f₁₀ : A₂₀ → A₀₀} {f₁₂ : A₂₂ → A₀₂} {f₁₄ : A₂₄ → A₀₄}
-- {f₃₀ : A₂₀ → A₄₀} {f₃₂ : A₂₂ → A₄₂} {f₃₄ : A₂₄ → A₄₄}
-- (s₁₁ : f₀₁ ∘ f₁₂ ~ f₁₀ ∘ f₂₁) (s₃₁ : f₄₁ ∘ f₃₂ ~ f₃₀ ∘ f₂₁)
-- (s₁₃ : f₀₃ ∘ f₁₂ ~ f₁₄ ∘ f₂₃) (s₃₃ : f₄₃ ∘ f₃₂ ~ f₃₄ ∘ f₂₃)
parameters {A₀₀ A₀₂ A₀₄ A₂₀ A₂₂ A₂₄ A₄₀ A₄₂ A₄₄ : Type}
{f₀₁ : A₀₂ → A₀₀} {f₂₁ : A₂₂ → A₂₀} {f₄₁ : A₄₂ → A₄₀}
{f₀₃ : A₀₂ → A₀₄} {f₂₃ : A₂₂ → A₂₄} {f₄₃ : A₄₂ → A₄₄}
{f₁₀ : A₂₀ → A₀₀} {f₁₂ : A₂₂ → A₀₂} {f₁₄ : A₂₄ → A₀₄}
{f₃₀ : A₂₀ → A₄₀} {f₃₂ : A₂₂ → A₄₂} {f₃₄ : A₂₄ → A₄₄}
(s₁₁ : f₀₁ ∘ f₁₂ ~ f₁₀ ∘ f₂₁) (s₃₁ : f₄₁ ∘ f₃₂ ~ f₃₀ ∘ f₂₁)
(s₁₃ : f₀₃ ∘ f₁₂ ~ f₁₄ ∘ f₂₃) (s₃₃ : f₄₃ ∘ f₃₂ ~ f₃₄ ∘ f₂₃)
definition pushout2 : Type :=
pushout (pushout.functor f₂₁ f₀₁ f₄₁ s₁₁ s₃₁) (pushout.functor f₂₃ f₀₃ f₄₃ s₁₃ s₃₃)
definition pushout2' : Type :=
pushout (pushout.functor f₁₂ f₁₀ f₁₄ s₁₁⁻¹ʰᵗʸ s₁₃⁻¹ʰᵗʸ) (pushout.functor f₃₂ f₃₀ f₃₄ s₃₁⁻¹ʰᵗʸ s₃₃⁻¹ʰᵗʸ)
end
end pushout
|
1e5281e1543a3a5635ceb820b263cdbdafaeabff | 4efff1f47634ff19e2f786deadd394270a59ecd2 | /src/category_theory/limits/shapes/zero.lean | 931afaf6827a87d315f2475ab4a63fe58e58cc99 | [
"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 | 15,031 | 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 category_theory.limits.shapes.terminal
import category_theory.limits.shapes.binary_products
import category_theory.limits.shapes.products
import category_theory.limits.shapes.images
/-!
# Zero morphisms and zero objects
A category "has zero morphisms" if there is a designated "zero morphism" in each morphism space,
and compositions of zero morphisms with anything give the zero morphism. (Notice this is extra
structure, not merely a property.)
A category "has a zero object" if it has an object which is both initial and terminal. Having a
zero object provides zero morphisms, as the unique morphisms factoring through the zero object.
## References
* https://en.wikipedia.org/wiki/Zero_morphism
* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]
-/
universes v u
open category_theory
open category_theory.category
namespace category_theory.limits
variables (C : Type u) [category.{v} C]
/-- A category "has zero morphisms" if there is a designated "zero morphism" in each morphism space,
and compositions of zero morphisms with anything give the zero morphism. -/
class has_zero_morphisms :=
[has_zero : Π X Y : C, has_zero (X ⟶ Y)]
(comp_zero' : ∀ {X Y : C} (f : X ⟶ Y) (Z : C), f ≫ (0 : Y ⟶ Z) = (0 : X ⟶ Z) . obviously)
(zero_comp' : ∀ (X : C) {Y Z : C} (f : Y ⟶ Z), (0 : X ⟶ Y) ≫ f = (0 : X ⟶ Z) . obviously)
attribute [instance] has_zero_morphisms.has_zero
restate_axiom has_zero_morphisms.comp_zero'
attribute [simp] has_zero_morphisms.comp_zero
restate_axiom has_zero_morphisms.zero_comp'
attribute [simp, reassoc] has_zero_morphisms.zero_comp
instance has_zero_morphisms_pempty : has_zero_morphisms (discrete pempty) :=
{ has_zero := by tidy }
instance has_zero_morphisms_punit : has_zero_morphisms (discrete punit) :=
{ has_zero := by tidy }
namespace has_zero_morphisms
variables {C}
/-- This lemma will be immediately superseded by `ext`, below. -/
private lemma ext_aux (I J : has_zero_morphisms C)
(w : ∀ X Y : C, (@has_zero_morphisms.has_zero _ _ I X Y).zero = (@has_zero_morphisms.has_zero _ _ J X Y).zero) : I = J :=
begin
casesI I, casesI J,
congr,
{ ext X Y,
exact w X Y },
{ apply proof_irrel_heq, },
{ apply proof_irrel_heq, }
end
/--
If you're tempted to use this lemma "in the wild", you should probably
carefully consider whether you've made a mistake in allowing two
instances of `has_zero_morphisms` to exist at all.
See, particularly, the note on `zero_morphisms_of_zero_object` below.
-/
lemma ext (I J : has_zero_morphisms C) : I = J :=
begin
apply ext_aux,
intros X Y,
rw ←@has_zero_morphisms.comp_zero _ _ I X X (@has_zero_morphisms.has_zero _ _ J X X).zero,
rw @has_zero_morphisms.zero_comp _ _ J,
end
instance : subsingleton (has_zero_morphisms C) :=
⟨ext⟩
end has_zero_morphisms
open has_zero_morphisms
section
variables {C} [has_zero_morphisms C]
lemma zero_of_comp_mono {X Y Z : C} {f : X ⟶ Y} (g : Y ⟶ Z) [mono g] (h : f ≫ g = 0) : f = 0 :=
by { rw [←zero_comp X g, cancel_mono] at h, exact h }
lemma zero_of_epi_comp {X Y Z : C} (f : X ⟶ Y) {g : Y ⟶ Z} [epi f] (h : f ≫ g = 0) : g = 0 :=
by { rw [←comp_zero f Z, cancel_epi] at h, exact h }
lemma eq_zero_of_image_eq_zero {X Y : C} {f : X ⟶ Y} [has_image f] (w : image.ι f = 0) : f = 0 :=
by rw [←image.fac f, w, has_zero_morphisms.comp_zero]
lemma nonzero_image_of_nonzero {X Y : C} {f : X ⟶ Y} [has_image f] (w : f ≠ 0) : image.ι f ≠ 0 :=
λ h, w (eq_zero_of_image_eq_zero h)
end
section
universes v' u'
variables (D : Type u') [category.{v'} D]
variables [has_zero_morphisms C] [has_zero_morphisms D]
lemma equivalence_preserves_zero_morphisms (F : C ≌ D) (X Y : C) :
F.functor.map (0 : X ⟶ Y) = (0 : F.functor.obj X ⟶ F.functor.obj Y) :=
begin
have t : F.functor.map (0 : X ⟶ Y) = F.functor.map (0 : X ⟶ Y) ≫ (0 : F.functor.obj Y ⟶ F.functor.obj Y),
{ apply faithful.map_injective (F.inverse),
rw [functor.map_comp, equivalence.inv_fun_map],
dsimp,
rw [zero_comp, comp_zero, zero_comp], },
exact t.trans (by simp)
end
@[simp] lemma is_equivalence_preserves_zero_morphisms (F : C ⥤ D) [is_equivalence F] (X Y : C) :
F.map (0 : X ⟶ Y) = 0 :=
by rw [←functor.as_equivalence_functor F, equivalence_preserves_zero_morphisms]
end
/-- A category "has a zero object" if it has an object which is both initial and terminal. -/
class has_zero_object :=
(zero : C)
(unique_to : Π X : C, unique (zero ⟶ X))
(unique_from : Π X : C, unique (X ⟶ zero))
instance has_zero_object_punit : has_zero_object (discrete punit) :=
{ zero := punit.star,
unique_to := by tidy,
unique_from := by tidy, }
variables {C}
namespace has_zero_object
variables [has_zero_object C]
/--
Construct a `has_zero C` for a category with a zero object.
This can not be a global instance as it will trigger for every `has_zero C` typeclass search.
-/
protected def has_zero : has_zero C :=
{ zero := has_zero_object.zero }
local attribute [instance] has_zero_object.has_zero
local attribute [instance] has_zero_object.unique_to has_zero_object.unique_from
@[ext]
lemma to_zero_ext {X : C} (f g : X ⟶ 0) : f = g :=
by rw [(has_zero_object.unique_from X).uniq f, (has_zero_object.unique_from X).uniq g]
@[ext]
lemma from_zero_ext {X : C} (f g : 0 ⟶ X) : f = g :=
by rw [(has_zero_object.unique_to X).uniq f, (has_zero_object.unique_to X).uniq g]
instance (X : C) : subsingleton (X ≅ 0) := by tidy
instance {X : C} (f : 0 ⟶ X) : mono f :=
{ right_cancellation := λ Z g h w, by ext, }
instance {X : C} (f : X ⟶ 0) : epi f :=
{ left_cancellation := λ Z g h w, by ext, }
/-- A category with a zero object has zero morphisms.
It is rarely a good idea to use this. Many categories that have a zero object have zero
morphisms for some other reason, for example from additivity. Library code that uses
`zero_morphisms_of_zero_object` will then be incompatible with these categories because
the `has_zero_morphisms` instances will not be definitionally equal. For this reason library
code should generally ask for an instance of `has_zero_morphisms` separately, even if it already
asks for an instance of `has_zero_objects`. -/
def zero_morphisms_of_zero_object : has_zero_morphisms C :=
{ has_zero := λ X Y,
{ zero := inhabited.default (X ⟶ 0) ≫ inhabited.default (0 ⟶ Y) },
zero_comp' := λ X Y Z f, by { dunfold has_zero.zero, rw category.assoc, congr, },
comp_zero' := λ X Y Z f, by { dunfold has_zero.zero, rw ←category.assoc, congr, }}
/-- A zero object is in particular initial. -/
def has_initial : has_initial C :=
has_initial_of_unique 0
/-- A zero object is in particular terminal. -/
def has_terminal : has_terminal C :=
has_terminal_of_unique 0
end has_zero_object
section
variables [has_zero_object C] [has_zero_morphisms C]
local attribute [instance] has_zero_object.has_zero
@[simp]
lemma id_zero : 𝟙 (0 : C) = (0 : 0 ⟶ 0) :=
by ext
/-- An arrow ending in the zero object is zero -/
-- This can't be a `simp` lemma because the left hand side would be a metavariable.
lemma zero_of_to_zero {X : C} (f : X ⟶ 0) : f = 0 :=
by ext
lemma zero_of_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : f = 0 :=
begin
have h : f = f ≫ i.hom ≫ 𝟙 0 ≫ i.inv := by simp only [iso.hom_inv_id, id_comp, comp_id],
simpa using h,
end
/-- An arrow starting at the zero object is zero -/
lemma zero_of_from_zero {X : C} (f : 0 ⟶ X) : f = 0 :=
by ext
lemma zero_of_source_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) : f = 0 :=
begin
have h : f = i.hom ≫ 𝟙 0 ≫ i.inv ≫ f := by simp only [iso.hom_inv_id_assoc, id_comp, comp_id],
simpa using h,
end
lemma mono_of_source_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) : mono f :=
⟨λ Z g h w, by rw [zero_of_target_iso_zero g i, zero_of_target_iso_zero h i]⟩
lemma epi_of_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : epi f :=
⟨λ Z g h w, by rw [zero_of_source_iso_zero g i, zero_of_source_iso_zero h i]⟩
/--
An object `X` has `𝟙 X = 0` if and only if it is isomorphic to the zero object.
Because `X ≅ 0` contains data (even if a subsingleton), we express this `↔` as an `≃`.
-/
def id_zero_equiv_iso_zero (X : C) : (𝟙 X = 0) ≃ (X ≅ 0) :=
{ to_fun := λ h, { hom := 0, inv := 0, },
inv_fun := λ i, zero_of_target_iso_zero (𝟙 X) i,
left_inv := by tidy,
right_inv := by tidy, }
@[simp]
lemma id_zero_equiv_iso_zero_apply_hom (X : C) (h : 𝟙 X = 0) :
((id_zero_equiv_iso_zero X) h).hom = 0 := rfl
@[simp]
lemma id_zero_equiv_iso_zero_apply_inv (X : C) (h : 𝟙 X = 0) :
((id_zero_equiv_iso_zero X) h).inv = 0 := rfl
end
section is_iso
variables [has_zero_morphisms C]
/--
A zero morphism `0 : X ⟶ Y` is an isomorphism if and only if
the identities on both `X` and `Y` are zero.
-/
@[simps]
def is_iso_zero_equiv (X Y : C) : is_iso (0 : X ⟶ Y) ≃ (𝟙 X = 0 ∧ 𝟙 Y = 0) :=
{ to_fun := begin introsI i, rw ←is_iso.hom_inv_id (0 : X ⟶ Y), rw ←is_iso.inv_hom_id (0 : X ⟶ Y), simp, end,
inv_fun := λ h, { inv := (0 : Y ⟶ X), },
left_inv := by tidy,
right_inv := by tidy, }
/--
A zero morphism `0 : X ⟶ X` is an isomorphism if and only if
the identity on `X` is zero.
-/
def is_iso_zero_self_equiv (X : C) : is_iso (0 : X ⟶ X) ≃ (𝟙 X = 0) :=
by simpa using is_iso_zero_equiv X X
variables [has_zero_object C]
local attribute [instance] has_zero_object.has_zero
/--
A zero morphism `0 : X ⟶ Y` is an isomorphism if and only if
`X` and `Y` are isomorphic to the zero object.
-/
def is_iso_zero_equiv_iso_zero (X Y : C) : is_iso (0 : X ⟶ Y) ≃ (X ≅ 0) × (Y ≅ 0) :=
begin
-- This is lame, because `prod` can't cope with `Prop`, so we can't use `equiv.prod_congr`.
refine (is_iso_zero_equiv X Y).trans _,
symmetry,
fsplit,
{ rintros ⟨eX, eY⟩, fsplit,
exact (id_zero_equiv_iso_zero X).symm eX,
exact (id_zero_equiv_iso_zero Y).symm eY, },
{ rintros ⟨hX, hY⟩, fsplit,
exact (id_zero_equiv_iso_zero X) hX,
exact (id_zero_equiv_iso_zero Y) hY, },
{ tidy, },
{ tidy, },
end
/--
A zero morphism `0 : X ⟶ X` is an isomorphism if and only if
`X` is isomorphic to the zero object.
-/
def is_iso_zero_self_equiv_iso_zero (X : C) : is_iso (0 : X ⟶ X) ≃ (X ≅ 0) :=
(is_iso_zero_equiv_iso_zero X X).trans subsingleton_prod_self_equiv
end is_iso
/-- If there are zero morphisms, any initial object is a zero object. -/
@[priority 50]
instance has_zero_object_of_has_initial_object
[has_zero_morphisms C] [has_initial C] : has_zero_object C :=
{ zero := ⊥_ C,
unique_to := λ X, ⟨⟨0⟩, by tidy⟩,
unique_from := λ X, ⟨⟨0⟩, λ f,
calc
f = f ≫ 𝟙 _ : (category.comp_id _).symm
... = f ≫ 0 : by congr
... = 0 : has_zero_morphisms.comp_zero _ _
⟩ }
/-- If there are zero morphisms, any terminal object is a zero object. -/
@[priority 50]
instance has_zero_object_of_has_terminal_object
[has_zero_morphisms C] [has_terminal C] : has_zero_object C :=
{ zero := ⊤_ C,
unique_from := λ X, ⟨⟨0⟩, by tidy⟩,
unique_to := λ X, ⟨⟨0⟩, λ f,
calc
f = 𝟙 _ ≫ f : (category.id_comp _).symm
... = 0 ≫ f : by congr
... = 0 : has_zero_morphisms.zero_comp _ _
⟩ }
section image
variable [has_zero_morphisms C]
lemma image_ι_comp_eq_zero {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} [has_image f]
[epi (factor_thru_image f)] (h : f ≫ g = 0) : image.ι f ≫ g = 0 :=
zero_of_epi_comp (factor_thru_image f) $ by simp [h]
variables [has_zero_object C]
local attribute [instance] has_zero_object.has_zero
/--
The zero morphism has a `mono_factorisation` through the zero object.
-/
@[simps]
def mono_factorisation_zero (X Y : C) : mono_factorisation (0 : X ⟶ Y) :=
{ I := 0, m := 0, e := 0, }
/--
Any zero morphism has an image.
We don't set this as an instance, as it is only intended for use inside the following proofs.
-/
def has_image.zero (X Y : C) : has_image (0 : X ⟶ Y) :=
{ F := mono_factorisation_zero X Y,
is_image :=
{ lift := λ F', 0 }}
/-- The image of a zero morphism is the zero object. -/
def image_zero {X Y : C} [has_image (0 : X ⟶ Y)] : image (0 : X ⟶ Y) ≅ 0 :=
is_image.iso_ext (image.is_image (0 : X ⟶ Y)) (has_image.zero X Y).is_image
/-- The image of a morphism which is equal to zero is the zero object. -/
def image_zero' {X Y : C} {f : X ⟶ Y} (h : f = 0) [has_image f] : image f ≅ 0 :=
begin
haveI := has_image.zero X Y,
exact image.eq_to_iso h ≪≫ image_zero,
end
@[simp]
lemma image.ι_zero {X Y : C} [has_image (0 : X ⟶ Y)] : image.ι (0 : X ⟶ Y) = 0 :=
begin
rw ←image.lift_fac (mono_factorisation_zero X Y),
simp,
end
/--
If we know `f = 0`,
it requires a little work to conclude `image.ι f = 0`,
because `f = g` only implies `image f ≅ image g`.
-/
@[simp]
lemma image.ι_zero' [has_equalizers C] {X Y : C} {f : X ⟶ Y} (h : f = 0) [has_image f] :
image.ι f = 0 :=
begin
haveI := has_image.zero X Y,
rw image.eq_fac h,
simp,
end
end image
/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/
instance split_mono_sigma_ι
{β : Type v} [decidable_eq β]
[has_zero_morphisms C]
(f : β → C) [has_colimit (discrete.functor f)] (b : β) : split_mono (sigma.ι f b) :=
{ retraction := sigma.desc (λ b', if h : b' = b then eq_to_hom (congr_arg f h) else 0), }
/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/
instance split_epi_pi_π
{β : Type v} [decidable_eq β]
[has_zero_morphisms C]
(f : β → C) [has_limit (discrete.functor f)] (b : β) : split_epi (pi.π f b) :=
{ section_ := pi.lift (λ b', if h : b = b' then eq_to_hom (congr_arg f h) else 0), }
/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/
instance split_mono_coprod_inl
[has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] :
split_mono (coprod.inl : X ⟶ X ⨿ Y) :=
{ retraction := coprod.desc (𝟙 X) 0, }
/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/
instance split_mono_coprod_inr
[has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] :
split_mono (coprod.inr : Y ⟶ X ⨿ Y) :=
{ retraction := coprod.desc 0 (𝟙 Y), }
/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/
instance split_epi_prod_fst
[has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] :
split_epi (prod.fst : X ⨯ Y ⟶ X) :=
{ section_ := prod.lift (𝟙 X) 0, }
/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/
instance split_epi_prod_snd
[has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] :
split_epi (prod.snd : X ⨯ Y ⟶ Y) :=
{ section_ := prod.lift 0 (𝟙 Y), }
end category_theory.limits
|
88b630cc199edb09c101994185701844225b0e7c | 367134ba5a65885e863bdc4507601606690974c1 | /src/data/setoid/basic.lean | a52a98afd17ac6be01039ec9d397016fbb119670 | [
"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 | 16,337 | lean | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen
-/
import order.galois_connection
/-!
# Equivalence relations
This file defines the complete lattice of equivalence relations on a type, results about the
inductively defined equivalence closure of a binary relation, and the analogues of some isomorphism
theorems for quotients of arbitrary types.
## Implementation notes
The function `rel` and lemmas ending in ' make it easier to talk about different
equivalence relations on the same type.
The complete lattice instance for equivalence relations could have been defined by lifting
the Galois insertion of equivalence relations on α into binary relations on α, and then using
`complete_lattice.copy` to define a complete lattice instance with more appropriate
definitional equalities (a similar example is `filter.complete_lattice` in
`order/filter/basic.lean`). This does not save space, however, and is less clear.
Partitions are not defined as a separate structure here; users are encouraged to
reason about them using the existing `setoid` and its infrastructure.
## Tags
setoid, equivalence, iseqv, relation, equivalence relation
-/
variables {α : Type*} {β : Type*}
/-- A version of `setoid.r` that takes the equivalence relation as an explicit argument. -/
def setoid.rel (r : setoid α) : α → α → Prop := @setoid.r _ r
/-- A version of `quotient.eq'` compatible with `setoid.rel`, to make rewriting possible. -/
lemma quotient.eq_rel {r : setoid α} {x y} : ⟦x⟧ = ⟦y⟧ ↔ r.rel x y := quotient.eq'
namespace setoid
@[ext] lemma ext' {r s : setoid α} (H : ∀ a b, r.rel a b ↔ s.rel a b) :
r = s := ext H
lemma ext_iff {r s : setoid α} : r = s ↔ ∀ a b, r.rel a b ↔ s.rel a b :=
⟨λ h a b, h ▸ iff.rfl, ext'⟩
/-- Two equivalence relations are equal iff their underlying binary operations are equal. -/
theorem eq_iff_rel_eq {r₁ r₂ : setoid α} : r₁ = r₂ ↔ r₁.rel = r₂.rel :=
⟨λ h, h ▸ rfl, λ h, setoid.ext' $ λ x y, h ▸ iff.rfl⟩
/-- Defining `≤` for equivalence relations. -/
instance : has_le (setoid α) := ⟨λ r s, ∀ ⦃x y⦄, r.rel x y → s.rel x y⟩
theorem le_def {r s : setoid α} : r ≤ s ↔ ∀ {x y}, r.rel x y → s.rel x y := iff.rfl
@[refl] lemma refl' (r : setoid α) (x) : r.rel x x := r.2.1 x
@[symm] lemma symm' (r : setoid α) : ∀ {x y}, r.rel x y → r.rel y x := λ _ _ h, r.2.2.1 h
@[trans] lemma trans' (r : setoid α) : ∀ {x y z}, r.rel x y → r.rel y z → r.rel x z :=
λ _ _ _ hx, r.2.2.2 hx
/-- The kernel of a function is an equivalence relation. -/
def ker (f : α → β) : setoid α :=
⟨λ x y, f x = f y, ⟨λ _, rfl, λ _ _ h, h.symm, λ _ _ _ h, h.trans⟩⟩
/-- The kernel of the quotient map induced by an equivalence relation r equals r. -/
@[simp] lemma ker_mk_eq (r : setoid α) : ker (@quotient.mk _ r) = r :=
ext' $ λ x y, quotient.eq
lemma ker_def {f : α → β} {x y : α} : (ker f).rel x y ↔ f x = f y := iff.rfl
/-- Given types `α`, `β`, the product of two equivalence relations `r` on `α` and `s` on `β`:
`(x₁, x₂), (y₁, y₂) ∈ α × β` are related by `r.prod s` iff `x₁` is related to `y₁`
by `r` and `x₂` is related to `y₂` by `s`. -/
protected def prod (r : setoid α) (s : setoid β) : setoid (α × β) :=
{ r := λ x y, r.rel x.1 y.1 ∧ s.rel x.2 y.2,
iseqv := ⟨λ x, ⟨r.refl' x.1, s.refl' x.2⟩, λ _ _ h, ⟨r.symm' h.1, s.symm' h.2⟩,
λ _ _ _ h1 h2, ⟨r.trans' h1.1 h2.1, s.trans' h1.2 h2.2⟩⟩ }
/-- The infimum of two equivalence relations. -/
instance : has_inf (setoid α) :=
⟨λ r s, ⟨λ x y, r.rel x y ∧ s.rel x y, ⟨λ x, ⟨r.refl' x, s.refl' x⟩,
λ _ _ h, ⟨r.symm' h.1, s.symm' h.2⟩,
λ _ _ _ h1 h2, ⟨r.trans' h1.1 h2.1, s.trans' h1.2 h2.2⟩⟩⟩⟩
/-- The infimum of 2 equivalence relations r and s is the same relation as the infimum
of the underlying binary operations. -/
lemma inf_def {r s : setoid α} : (r ⊓ s).rel = r.rel ⊓ s.rel := rfl
theorem inf_iff_and {r s : setoid α} {x y} :
(r ⊓ s).rel x y ↔ r.rel x y ∧ s.rel x y := iff.rfl
/-- The infimum of a set of equivalence relations. -/
instance : has_Inf (setoid α) :=
⟨λ S, ⟨λ x y, ∀ r ∈ S, rel r x y,
⟨λ x r hr, r.refl' x, λ _ _ h r hr, r.symm' $ h r hr,
λ _ _ _ h1 h2 r hr, r.trans' (h1 r hr) $ h2 r hr⟩⟩⟩
/-- The underlying binary operation of the infimum of a set of equivalence relations
is the infimum of the set's image under the map to the underlying binary operation. -/
theorem Inf_def {s : set (setoid α)} : (Inf s).rel = Inf (rel '' s) :=
by { ext, simp only [Inf_image, infi_apply, infi_Prop_eq], refl }
instance : partial_order (setoid α) :=
{ le := (≤),
lt := λ r s, r ≤ s ∧ ¬s ≤ r,
le_refl := λ _ _ _, id,
le_trans := λ _ _ _ hr hs _ _ h, hs $ hr h,
lt_iff_le_not_le := λ _ _, iff.rfl,
le_antisymm := λ r s h1 h2, setoid.ext' $ λ x y, ⟨λ h, h1 h, λ h, h2 h⟩ }
/-- The complete lattice of equivalence relations on a type, with bottom element `=`
and top element the trivial equivalence relation. -/
instance complete_lattice : complete_lattice (setoid α) :=
{ inf := has_inf.inf,
inf_le_left := λ _ _ _ _ h, h.1,
inf_le_right := λ _ _ _ _ h, h.2,
le_inf := λ _ _ _ h1 h2 _ _ h, ⟨h1 h, h2 h⟩,
top := ⟨λ _ _, true, ⟨λ _, trivial, λ _ _ h, h, λ _ _ _ h1 h2, h1⟩⟩,
le_top := λ _ _ _ _, trivial,
bot := ⟨(=), ⟨λ _, rfl, λ _ _ h, h.symm, λ _ _ _ h1 h2, h1.trans h2⟩⟩,
bot_le := λ r x y h, h ▸ r.2.1 x,
.. complete_lattice_of_Inf (setoid α) $ assume s,
⟨λ r hr x y h, h _ hr, λ r hr x y h r' hr', hr hr' h⟩ }
/-- The inductively defined equivalence closure of a binary relation r is the infimum
of the set of all equivalence relations containing r. -/
theorem eqv_gen_eq (r : α → α → Prop) :
eqv_gen.setoid r = Inf {s : setoid α | ∀ ⦃x y⦄, r x y → s.rel x y} :=
le_antisymm
(λ _ _ H, eqv_gen.rec (λ _ _ h _ hs, hs h) (refl' _)
(λ _ _ _, symm' _) (λ _ _ _ _ _, trans' _) H)
(Inf_le $ λ _ _ h, eqv_gen.rel _ _ h)
/-- The supremum of two equivalence relations r and s is the equivalence closure of the binary
relation `x is related to y by r or s`. -/
lemma sup_eq_eqv_gen (r s : setoid α) :
r ⊔ s = eqv_gen.setoid (λ x y, r.rel x y ∨ s.rel x y) :=
begin
rw eqv_gen_eq,
apply congr_arg Inf,
simp only [le_def, or_imp_distrib, ← forall_and_distrib]
end
/-- The supremum of 2 equivalence relations r and s is the equivalence closure of the
supremum of the underlying binary operations. -/
lemma sup_def {r s : setoid α} : r ⊔ s = eqv_gen.setoid (r.rel ⊔ s.rel) :=
by rw sup_eq_eqv_gen; refl
/-- The supremum of a set S of equivalence relations is the equivalence closure of the binary
relation `there exists r ∈ S relating x and y`. -/
lemma Sup_eq_eqv_gen (S : set (setoid α)) :
Sup S = eqv_gen.setoid (λ x y, ∃ r : setoid α, r ∈ S ∧ r.rel x y) :=
begin
rw eqv_gen_eq,
apply congr_arg Inf,
simp only [upper_bounds, le_def, and_imp, exists_imp_distrib],
ext,
exact ⟨λ H x y r hr, H hr, λ H r hr x y, H r hr⟩
end
/-- The supremum of a set of equivalence relations is the equivalence closure of the
supremum of the set's image under the map to the underlying binary operation. -/
lemma Sup_def {s : set (setoid α)} : Sup s = eqv_gen.setoid (Sup (rel '' s)) :=
begin
rw [Sup_eq_eqv_gen, Sup_image],
congr' with x y,
simp only [supr_apply, supr_Prop_eq, exists_prop]
end
/-- The equivalence closure of an equivalence relation r is r. -/
@[simp] lemma eqv_gen_of_setoid (r : setoid α) : eqv_gen.setoid r.r = r :=
le_antisymm (by rw eqv_gen_eq; exact Inf_le (λ _ _, id)) eqv_gen.rel
/-- Equivalence closure is idempotent. -/
@[simp] lemma eqv_gen_idem (r : α → α → Prop) :
eqv_gen.setoid (eqv_gen.setoid r).rel = eqv_gen.setoid r :=
eqv_gen_of_setoid _
/-- The equivalence closure of a binary relation r is contained in any equivalence
relation containing r. -/
theorem eqv_gen_le {r : α → α → Prop} {s : setoid α} (h : ∀ x y, r x y → s.rel x y) :
eqv_gen.setoid r ≤ s :=
by rw eqv_gen_eq; exact Inf_le h
/-- Equivalence closure of binary relations is monotonic. -/
theorem eqv_gen_mono {r s : α → α → Prop} (h : ∀ x y, r x y → s x y) :
eqv_gen.setoid r ≤ eqv_gen.setoid s :=
eqv_gen_le $ λ _ _ hr, eqv_gen.rel _ _ $ h _ _ hr
/-- There is a Galois insertion of equivalence relations on α into binary relations
on α, with equivalence closure the lower adjoint. -/
def gi : @galois_insertion (α → α → Prop) (setoid α) _ _ eqv_gen.setoid rel :=
{ choice := λ r h, eqv_gen.setoid r,
gc := λ r s, ⟨λ H _ _ h, H $ eqv_gen.rel _ _ h, λ H, eqv_gen_of_setoid s ▸ eqv_gen_mono H⟩,
le_l_u := λ x, (eqv_gen_of_setoid x).symm ▸ le_refl x,
choice_eq := λ _ _, rfl }
open function
/-- A function from α to β is injective iff its kernel is the bottom element of the complete lattice
of equivalence relations on α. -/
theorem injective_iff_ker_bot (f : α → β) :
injective f ↔ ker f = ⊥ :=
(@eq_bot_iff (setoid α) _ (ker f)).symm
/-- The elements related to x ∈ α by the kernel of f are those in the preimage of f(x) under f. -/
lemma ker_iff_mem_preimage {f : α → β} {x y} : (ker f).rel x y ↔ x ∈ f ⁻¹' {f y} :=
iff.rfl
/-- Equivalence between functions `α → β` such that `r x y → f x = f y` and functions
`quotient r → β`. -/
def lift_equiv (r : setoid α) : {f : α → β // r ≤ ker f} ≃ (quotient r → β) :=
{ to_fun := λ f, quotient.lift (f : α → β) f.2,
inv_fun := λ f, ⟨f ∘ quotient.mk, λ x y h, by simp [ker_def, quotient.sound h]⟩,
left_inv := λ ⟨f, hf⟩, subtype.eq $ funext $ λ x, rfl,
right_inv := λ f, funext $ λ x, quotient.induction_on' x $ λ x, rfl }
/-- The uniqueness part of the universal property for quotients of an arbitrary type. -/
theorem lift_unique {r : setoid α} {f : α → β} (H : r ≤ ker f) (g : quotient r → β)
(Hg : f = g ∘ quotient.mk) : quotient.lift f H = g :=
begin
ext ⟨x⟩,
erw [quotient.lift_mk f H, Hg],
refl
end
/-- Given a map f from α to β, the natural map from the quotient of α by the kernel of f is
injective. -/
lemma ker_lift_injective (f : α → β) : injective (@quotient.lift _ _ (ker f) f (λ _ _ h, h)) :=
λ x y, quotient.induction_on₂' x y $ λ a b h, quotient.sound' h
/-- Given a map f from α to β, the kernel of f is the unique equivalence relation on α whose
induced map from the quotient of α to β is injective. -/
lemma ker_eq_lift_of_injective {r : setoid α} (f : α → β) (H : ∀ x y, r.rel x y → f x = f y)
(h : injective (quotient.lift f H)) : ker f = r :=
le_antisymm
(λ x y hk, quotient.exact $ h $ show quotient.lift f H ⟦x⟧ = quotient.lift f H ⟦y⟧, from hk)
H
variables (r : setoid α) (f : α → β)
/-- The first isomorphism theorem for sets: the quotient of α by the kernel of a function f
bijects with f's image. -/
noncomputable def quotient_ker_equiv_range :
quotient (ker f) ≃ set.range f :=
equiv.of_bijective (@quotient.lift _ (set.range f) (ker f)
(λ x, ⟨f x, set.mem_range_self x⟩) $ λ _ _ h, subtype.ext_val h)
⟨λ x y h, ker_lift_injective f $ by rcases x; rcases y; injections,
λ ⟨w, z, hz⟩, ⟨@quotient.mk _ (ker f) z, by rw quotient.lift_mk; exact subtype.ext_iff_val.2 hz⟩⟩
/-- The quotient of α by the kernel of a surjective function f bijects with f's codomain. -/
noncomputable def quotient_ker_equiv_of_surjective (hf : surjective f) :
quotient (ker f) ≃ β :=
(quotient_ker_equiv_range f).trans $ equiv.subtype_univ_equiv hf
variables {r f}
/-- Given a function `f : α → β` and equivalence relation `r` on `α`, the equivalence
closure of the relation on `f`'s image defined by '`x ≈ y` iff the elements of `f⁻¹(x)` are
related to the elements of `f⁻¹(y)` by `r`.' -/
def map (r : setoid α) (f : α → β) : setoid β :=
eqv_gen.setoid $ λ x y, ∃ a b, f a = x ∧ f b = y ∧ r.rel a b
/-- Given a surjective function f whose kernel is contained in an equivalence relation r, the
equivalence relation on f's codomain defined by x ≈ y ↔ the elements of f⁻¹(x) are related to
the elements of f⁻¹(y) by r. -/
def map_of_surjective (r) (f : α → β) (h : ker f ≤ r) (hf : surjective f) :
setoid β :=
⟨λ x y, ∃ a b, f a = x ∧ f b = y ∧ r.rel a b,
⟨λ x, let ⟨y, hy⟩ := hf x in ⟨y, y, hy, hy, r.refl' y⟩,
λ _ _ ⟨x, y, hx, hy, h⟩, ⟨y, x, hy, hx, r.symm' h⟩,
λ _ _ _ ⟨x, y, hx, hy, h₁⟩ ⟨y', z, hy', hz, h₂⟩,
⟨x, z, hx, hz, r.trans' h₁ $ r.trans' (h $ by rwa ←hy' at hy) h₂⟩⟩⟩
/-- A special case of the equivalence closure of an equivalence relation r equalling r. -/
lemma map_of_surjective_eq_map (h : ker f ≤ r) (hf : surjective f) :
map r f = map_of_surjective r f h hf :=
by rw ←eqv_gen_of_setoid (map_of_surjective r f h hf); refl
/-- Given a function `f : α → β`, an equivalence relation `r` on `β` induces an equivalence
relation on `α` defined by '`x ≈ y` iff `f(x)` is related to `f(y)` by `r`'. -/
def comap (f : α → β) (r : setoid β) : setoid α :=
⟨λ x y, r.rel (f x) (f y), ⟨λ _, r.refl' _, λ _ _ h, r.symm' h, λ _ _ _ h1, r.trans' h1⟩⟩
/-- Given a map `f : N → M` and an equivalence relation `r` on `β`, the equivalence relation
induced on `α` by `f` equals the kernel of `r`'s quotient map composed with `f`. -/
lemma comap_eq {f : α → β} {r : setoid β} : comap f r = ker (@quotient.mk _ r ∘ f) :=
ext $ λ x y, show _ ↔ ⟦_⟧ = ⟦_⟧, by rw quotient.eq; refl
/-- The second isomorphism theorem for sets. -/
noncomputable def comap_quotient_equiv (f : α → β) (r : setoid β) :
quotient (comap f r) ≃ set.range (@quotient.mk _ r ∘ f) :=
(quotient.congr_right $ ext_iff.1 comap_eq).trans $ quotient_ker_equiv_range $ quotient.mk ∘ f
variables (r f)
/-- The third isomorphism theorem for sets. -/
def quotient_quotient_equiv_quotient (s : setoid α) (h : r ≤ s) :
quotient (ker (quot.map_right h)) ≃ quotient s :=
{ to_fun := λ x, quotient.lift_on' x (λ w, quotient.lift_on' w (@quotient.mk _ s) $
λ x y H, quotient.sound $ h H) $ λ x y, quotient.induction_on₂' x y $ λ w z H,
show @quot.mk _ _ _ = @quot.mk _ _ _, from H,
inv_fun := λ x, quotient.lift_on' x
(λ w, @quotient.mk _ (ker $ quot.map_right h) $ @quotient.mk _ r w) $
λ x y H, quotient.sound' $ show @quot.mk _ _ _ = @quot.mk _ _ _, from quotient.sound H,
left_inv := λ x, quotient.induction_on' x $ λ y, quotient.induction_on' y $
λ w, by show ⟦_⟧ = _; refl,
right_inv := λ x, quotient.induction_on' x $ λ y, by show ⟦_⟧ = _; refl }
variables {r f}
open quotient
/-- Given an equivalence relation `r` on `α`, the order-preserving bijection between the set of
equivalence relations containing `r` and the equivalence relations on the quotient of `α` by `r`. -/
def correspondence (r : setoid α) : {s // r ≤ s} ≃o setoid (quotient r) :=
{ to_fun := λ s, map_of_surjective s.1 quotient.mk ((ker_mk_eq r).symm ▸ s.2) exists_rep,
inv_fun := λ s, ⟨comap quotient.mk s, λ x y h, show s.rel ⟦x⟧ ⟦y⟧, by rw eq_rel.2 h⟩,
left_inv := λ s, subtype.ext_iff_val.2 $ ext' $ λ _ _,
⟨λ h, let ⟨a, b, hx, hy, H⟩ := h in
s.1.trans' (s.1.symm' $ s.2 $ eq_rel.1 hx) $ s.1.trans' H $ s.2 $ eq_rel.1 hy,
λ h, ⟨_, _, rfl, rfl, h⟩⟩,
right_inv := λ s, let Hm : ker quotient.mk ≤ comap quotient.mk s :=
λ x y h, show s.rel ⟦x⟧ ⟦y⟧, by rw (@eq_rel _ r x y).2 ((ker_mk_eq r) ▸ h) in
ext' $ λ x y, ⟨λ h, let ⟨a, b, hx, hy, H⟩ := h in hx ▸ hy ▸ H,
quotient.induction_on₂ x y $ λ w z h, ⟨w, z, rfl, rfl, h⟩⟩,
map_rel_iff' := λ s t, ⟨λ h x y hs, let ⟨a, b, hx, hy, ht⟩ := h ⟨x, y, rfl, rfl, hs⟩ in
t.1.trans' (t.1.symm' $ t.2 $ eq_rel.1 hx) $ t.1.trans' ht $ t.2 $ eq_rel.1 hy,
λ h x y hs, let ⟨a, b, hx, hy, Hs⟩ := hs in ⟨a, b, hx, hy, h Hs⟩⟩ }
end setoid
|
05999bf236e687e8e71a7bfc556174e5c0b8d8b1 | d757fb8bb0d538df3510904d3c09a0da191d06b9 | /sf_weekdays.lean | b8b31b1be398396dd7d63cb558fa03740ef2e21d | [] | no_license | happy-bracket/lean_playground | 188f64c2a431dec5617c75e33c7a75e475b9c23d | 137b76966001091c445f6f472d27bbe5a8cf5d49 | refs/heads/master | 1,600,889,668,807 | 1,575,375,570,000 | 1,575,375,570,000 | 225,569,694 | 2 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 400 | lean | inductive day : Type
| monday
| tuesday
| wednesday
| thursday
| friday
| saturday
| sunday
def next_weekday : day -> day
| day.monday := day.tuesday
| day.tuesday := day.wednesday
| day.wednesday := day.thursday
| day.thursday := day.friday
| day.friday := day.saturday
| day.saturday := day.sunday
| day.sunday := day.monday
example : next_weekday (next_weekday day.saturday) = day.monday := rfl
|
dd4b6099c8fd5280bd48c51d42dcdb66f213d147 | c777c32c8e484e195053731103c5e52af26a25d1 | /src/topology/algebra/order/compact.lean | c7e46e18e122e81b85903b4be221930c8701fd2e | [
"Apache-2.0"
] | permissive | kbuzzard/mathlib | 2ff9e85dfe2a46f4b291927f983afec17e946eb8 | 58537299e922f9c77df76cb613910914a479c1f7 | refs/heads/master | 1,685,313,702,744 | 1,683,974,212,000 | 1,683,974,212,000 | 128,185,277 | 1 | 0 | null | 1,522,920,600,000 | 1,522,920,600,000 | null | UTF-8 | Lean | false | false | 19,243 | lean | /-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
import topology.algebra.order.intermediate_value
import topology.local_extr
/-!
# Compactness of a closed interval
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
In this file we prove that a closed interval in a conditionally complete linear ordered type with
order topology (or a product of such types) is compact.
We prove the extreme value theorem (`is_compact.exists_forall_le`, `is_compact.exists_forall_ge`):
a continuous function on a compact set takes its minimum and maximum values. We provide many
variations of this theorem.
We also prove that the image of a closed interval under a continuous map is a closed interval, see
`continuous_on.image_Icc`.
## Tags
compact, extreme value theorem
-/
open filter order_dual topological_space function set
open_locale filter topology
/-!
### Compactness of a closed interval
In this section we define a typeclass `compact_Icc_space α` saying that all closed intervals in `α`
are compact. Then we provide an instance for a `conditionally_complete_linear_order` and prove that
the product (both `α × β` and an indexed product) of spaces with this property inherits the
property.
We also prove some simple lemmas about spaces with this property.
-/
/-- This typeclass says that all closed intervals in `α` are compact. This is true for all
conditionally complete linear orders with order topology and products (finite or infinite)
of such spaces. -/
class compact_Icc_space (α : Type*) [topological_space α] [preorder α] : Prop :=
(is_compact_Icc : ∀ {a b : α}, is_compact (Icc a b))
export compact_Icc_space (is_compact_Icc)
/-- A closed interval in a conditionally complete linear order is compact. -/
@[priority 100]
instance conditionally_complete_linear_order.to_compact_Icc_space
(α : Type*) [conditionally_complete_linear_order α] [topological_space α] [order_topology α] :
compact_Icc_space α :=
begin
refine ⟨λ a b, _⟩,
cases le_or_lt a b with hab hab, swap, { simp [hab] },
refine is_compact_iff_ultrafilter_le_nhds.2 (λ f hf, _),
contrapose! hf,
rw [le_principal_iff],
have hpt : ∀ x ∈ Icc a b, {x} ∉ f,
from λ x hx hxf, hf x hx ((le_pure_iff.2 hxf).trans (pure_le_nhds x)),
set s := {x ∈ Icc a b | Icc a x ∉ f},
have hsb : b ∈ upper_bounds s, from λ x hx, hx.1.2,
have sbd : bdd_above s, from ⟨b, hsb⟩,
have ha : a ∈ s, by simp [hpt, hab],
rcases hab.eq_or_lt with rfl|hlt, { exact ha.2 },
set c := Sup s,
have hsc : is_lub s c, from is_lub_cSup ⟨a, ha⟩ sbd,
have hc : c ∈ Icc a b, from ⟨hsc.1 ha, hsc.2 hsb⟩,
specialize hf c hc,
have hcs : c ∈ s,
{ cases hc.1.eq_or_lt with heq hlt, { rwa ← heq },
refine ⟨hc, λ hcf, hf (λ U hU, _)⟩,
rcases (mem_nhds_within_Iic_iff_exists_Ioc_subset' hlt).1 (mem_nhds_within_of_mem_nhds hU)
with ⟨x, hxc, hxU⟩,
rcases ((hsc.frequently_mem ⟨a, ha⟩).and_eventually
(Ioc_mem_nhds_within_Iic ⟨hxc, le_rfl⟩)).exists
with ⟨y, ⟨hyab, hyf⟩, hy⟩,
refine mem_of_superset(f.diff_mem_iff.2 ⟨hcf, hyf⟩) (subset.trans _ hxU),
rw diff_subset_iff,
exact subset.trans Icc_subset_Icc_union_Ioc
(union_subset_union subset.rfl $ Ioc_subset_Ioc_left hy.1.le) },
cases hc.2.eq_or_lt with heq hlt, { rw ← heq, exact hcs.2 },
contrapose! hf,
intros U hU,
rcases (mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1 (mem_nhds_within_of_mem_nhds hU)
with ⟨y, hxy, hyU⟩,
refine mem_of_superset _ hyU, clear_dependent U,
have hy : y ∈ Icc a b, from ⟨hc.1.trans hxy.1.le, hxy.2⟩,
by_cases hay : Icc a y ∈ f,
{ refine mem_of_superset (f.diff_mem_iff.2 ⟨f.diff_mem_iff.2 ⟨hay, hcs.2⟩, hpt y hy⟩) _,
rw [diff_subset_iff, union_comm, Ico_union_right hxy.1.le, diff_subset_iff],
exact Icc_subset_Icc_union_Icc },
{ exact ((hsc.1 ⟨hy, hay⟩).not_lt hxy.1).elim },
end
instance {ι : Type*} {α : ι → Type*} [Π i, preorder (α i)] [Π i, topological_space (α i)]
[Π i, compact_Icc_space (α i)] : compact_Icc_space (Π i, α i) :=
⟨λ a b, pi_univ_Icc a b ▸ is_compact_univ_pi $ λ i, is_compact_Icc⟩
instance pi.compact_Icc_space' {α β : Type*} [preorder β] [topological_space β]
[compact_Icc_space β] : compact_Icc_space (α → β) :=
pi.compact_Icc_space
instance {α β : Type*} [preorder α] [topological_space α] [compact_Icc_space α]
[preorder β] [topological_space β] [compact_Icc_space β] :
compact_Icc_space (α × β) :=
⟨λ a b, (Icc_prod_eq a b).symm ▸ is_compact_Icc.prod is_compact_Icc⟩
/-- An unordered closed interval is compact. -/
lemma is_compact_uIcc {α : Type*} [linear_order α] [topological_space α] [compact_Icc_space α]
{a b : α} : is_compact (uIcc a b) :=
is_compact_Icc
/-- A complete linear order is a compact space.
We do not register an instance for a `[compact_Icc_space α]` because this would only add instances
for products (indexed or not) of complete linear orders, and we have instances with higher priority
that cover these cases. -/
@[priority 100] -- See note [lower instance priority]
instance compact_space_of_complete_linear_order {α : Type*} [complete_linear_order α]
[topological_space α] [order_topology α] :
compact_space α :=
⟨by simp only [← Icc_bot_top, is_compact_Icc]⟩
section
variables {α : Type*} [preorder α] [topological_space α] [compact_Icc_space α]
instance compact_space_Icc (a b : α) : compact_space (Icc a b) :=
is_compact_iff_compact_space.mp is_compact_Icc
end
/-!
### Min and max elements of a compact set
-/
variables {α β γ : Type*} [conditionally_complete_linear_order α] [topological_space α]
[order_topology α] [topological_space β] [topological_space γ]
lemma is_compact.Inf_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
Inf s ∈ s :=
hs.is_closed.cInf_mem ne_s hs.bdd_below
lemma is_compact.Sup_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) : Sup s ∈ s :=
@is_compact.Inf_mem αᵒᵈ _ _ _ _ hs ne_s
lemma is_compact.is_glb_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_glb s (Inf s) :=
is_glb_cInf ne_s hs.bdd_below
lemma is_compact.is_lub_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_lub s (Sup s) :=
@is_compact.is_glb_Inf αᵒᵈ _ _ _ _ hs ne_s
lemma is_compact.is_least_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_least s (Inf s) :=
⟨hs.Inf_mem ne_s, (hs.is_glb_Inf ne_s).1⟩
lemma is_compact.is_greatest_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_greatest s (Sup s) :=
@is_compact.is_least_Inf αᵒᵈ _ _ _ _ hs ne_s
lemma is_compact.exists_is_least {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x, is_least s x :=
⟨_, hs.is_least_Inf ne_s⟩
lemma is_compact.exists_is_greatest {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x, is_greatest s x :=
⟨_, hs.is_greatest_Sup ne_s⟩
lemma is_compact.exists_is_glb {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x ∈ s, is_glb s x :=
⟨_, hs.Inf_mem ne_s, hs.is_glb_Inf ne_s⟩
lemma is_compact.exists_is_lub {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x ∈ s, is_lub s x :=
⟨_, hs.Sup_mem ne_s, hs.is_lub_Sup ne_s⟩
lemma is_compact.exists_Inf_image_eq_and_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
{f : β → α} (hf : continuous_on f s) :
∃ x ∈ s, Inf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
let ⟨x, hxs, hx⟩ := (hs.image_of_continuous_on hf).Inf_mem (ne_s.image f)
in ⟨x, hxs, hx.symm, λ y hy,
hx.trans_le $ cInf_le (hs.image_of_continuous_on hf).bdd_below $ mem_image_of_mem f hy⟩
lemma is_compact.exists_Sup_image_eq_and_ge {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
{f : β → α} (hf : continuous_on f s) :
∃ x ∈ s, Sup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
@is_compact.exists_Inf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
lemma is_compact.exists_Inf_image_eq {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
{f : β → α} (hf : continuous_on f s) :
∃ x ∈ s, Inf (f '' s) = f x :=
let ⟨x, hxs, hx, _⟩ := hs.exists_Inf_image_eq_and_le ne_s hf in ⟨x, hxs, hx⟩
lemma is_compact.exists_Sup_image_eq :
∀ {s : set β}, is_compact s → s.nonempty → ∀ {f : β → α}, continuous_on f s →
∃ x ∈ s, Sup (f '' s) = f x :=
@is_compact.exists_Inf_image_eq αᵒᵈ _ _ _ _ _
lemma eq_Icc_of_connected_compact {s : set α} (h₁ : is_connected s) (h₂ : is_compact s) :
s = Icc (Inf s) (Sup s) :=
eq_Icc_cInf_cSup_of_connected_bdd_closed h₁ h₂.bdd_below h₂.bdd_above h₂.is_closed
/-!
### Extreme value theorem
-/
/-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
lemma is_compact.exists_forall_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
{f : β → α} (hf : continuous_on f s) :
∃x∈s, ∀y∈s, f x ≤ f y :=
begin
rcases (hs.image_of_continuous_on hf).exists_is_least (ne_s.image f)
with ⟨_, ⟨x, hxs, rfl⟩, hx⟩,
exact ⟨x, hxs, ball_image_iff.1 hx⟩
end
/-- The **extreme value theorem**: a continuous function realizes its maximum on a compact set. -/
lemma is_compact.exists_forall_ge :
∀ {s : set β}, is_compact s → s.nonempty → ∀ {f : β → α}, continuous_on f s →
∃x∈s, ∀y∈s, f y ≤ f x :=
@is_compact.exists_forall_le αᵒᵈ _ _ _ _ _
/-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
larger than a value in its image away from compact sets, then it has a minimum on this set. -/
lemma continuous_on.exists_forall_le' {s : set β} {f : β → α} (hf : continuous_on f s)
(hsc : is_closed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) :
∃ x ∈ s, ∀ y ∈ s, f x ≤ f y :=
begin
rcases (has_basis_cocompact.inf_principal _).eventually_iff.1 hc with ⟨K, hK, hKf⟩,
have hsub : insert x₀ (K ∩ s) ⊆ s, from insert_subset.2 ⟨h₀, inter_subset_right _ _⟩,
obtain ⟨x, hx, hxf⟩ : ∃ x ∈ insert x₀ (K ∩ s), ∀ y ∈ insert x₀ (K ∩ s), f x ≤ f y :=
((hK.inter_right hsc).insert x₀).exists_forall_le (insert_nonempty _ _) (hf.mono hsub),
refine ⟨x, hsub hx, λ y hy, _⟩,
by_cases hyK : y ∈ K,
exacts [hxf _ (or.inr ⟨hyK, hy⟩), (hxf _ (or.inl rfl)).trans (hKf ⟨hyK, hy⟩)]
end
/-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
smaller than a value in its image away from compact sets, then it has a maximum on this set. -/
lemma continuous_on.exists_forall_ge' {s : set β} {f : β → α} (hf : continuous_on f s)
(hsc : is_closed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) :
∃ x ∈ s, ∀ y ∈ s, f y ≤ f x :=
@continuous_on.exists_forall_le' αᵒᵈ _ _ _ _ _ _ _ hf hsc _ h₀ hc
/-- The **extreme value theorem**: if a continuous function `f` is larger than a value in its range
away from compact sets, then it has a global minimum. -/
lemma _root_.continuous.exists_forall_le' {f : β → α} (hf : continuous f) (x₀ : β)
(h : ∀ᶠ x in cocompact β, f x₀ ≤ f x) : ∃ (x : β), ∀ (y : β), f x ≤ f y :=
let ⟨x, _, hx⟩ := hf.continuous_on.exists_forall_le' is_closed_univ (mem_univ x₀)
(by rwa [principal_univ, inf_top_eq])
in ⟨x, λ y, hx y (mem_univ y)⟩
/-- The **extreme value theorem**: if a continuous function `f` is smaller than a value in its range
away from compact sets, then it has a global maximum. -/
lemma _root_.continuous.exists_forall_ge' {f : β → α} (hf : continuous f) (x₀ : β)
(h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ (x : β), ∀ (y : β), f y ≤ f x :=
@continuous.exists_forall_le' αᵒᵈ _ _ _ _ _ _ hf x₀ h
/-- The **extreme value theorem**: if a continuous function `f` tends to infinity away from compact
sets, then it has a global minimum. -/
lemma _root_.continuous.exists_forall_le [nonempty β] {f : β → α}
(hf : continuous f) (hlim : tendsto f (cocompact β) at_top) :
∃ x, ∀ y, f x ≤ f y :=
by { inhabit β, exact hf.exists_forall_le' default (hlim.eventually $ eventually_ge_at_top _) }
/-- The **extreme value theorem**: if a continuous function `f` tends to negative infinity away from
compact sets, then it has a global maximum. -/
lemma continuous.exists_forall_ge [nonempty β] {f : β → α}
(hf : continuous f) (hlim : tendsto f (cocompact β) at_bot) :
∃ x, ∀ y, f y ≤ f x :=
@continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim
lemma is_compact.Sup_lt_iff_of_continuous {f : β → α}
{K : set β} (hK : is_compact K) (h0K : K.nonempty) (hf : continuous_on f K) (y : α) :
Sup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
begin
refine ⟨λ h x hx, (le_cSup (hK.bdd_above_image hf) $ mem_image_of_mem f hx).trans_lt h, λ h, _⟩,
obtain ⟨x, hx, h2x⟩ := hK.exists_forall_ge h0K hf,
refine (cSup_le (h0K.image f) _).trans_lt (h x hx),
rintro _ ⟨x', hx', rfl⟩, exact h2x x' hx'
end
lemma is_compact.lt_Inf_iff_of_continuous {α β : Type*}
[conditionally_complete_linear_order α] [topological_space α]
[order_topology α] [topological_space β] {f : β → α}
{K : set β} (hK : is_compact K) (h0K : K.nonempty) (hf : continuous_on f K) (y : α) :
y < Inf (f '' K) ↔ ∀ x ∈ K, y < f x :=
@is_compact.Sup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
/-- A continuous function with compact support has a global minimum. -/
@[to_additive "A continuous function with compact support has a global minimum."]
lemma continuous.exists_forall_le_of_has_compact_mul_support [nonempty β] [has_one α]
{f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
∃ (x : β), ∀ (y : β), f x ≤ f y :=
begin
obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_is_least (range_nonempty _),
rw [mem_lower_bounds, forall_range_iff] at hx,
exact ⟨x, hx⟩,
end
/-- A continuous function with compact support has a global maximum. -/
@[to_additive "A continuous function with compact support has a global maximum."]
lemma continuous.exists_forall_ge_of_has_compact_mul_support [nonempty β] [has_one α]
{f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
∃ (x : β), ∀ (y : β), f y ≤ f x :=
@continuous.exists_forall_le_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ _ hf h
lemma is_compact.continuous_Sup {f : γ → β → α}
{K : set β} (hK : is_compact K) (hf : continuous ↿f) :
continuous (λ x, Sup (f x '' K)) :=
begin
rcases eq_empty_or_nonempty K with rfl|h0K,
{ simp_rw [image_empty], exact continuous_const },
rw [continuous_iff_continuous_at],
intro x,
obtain ⟨y, hyK, h2y, hy⟩ :=
hK.exists_Sup_image_eq_and_ge h0K
(show continuous (λ y, f x y), from hf.comp $ continuous.prod.mk x).continuous_on,
rw [continuous_at, h2y, tendsto_order],
have := tendsto_order.mp ((show continuous (λ x, f x y), from
hf.comp $ continuous_id.prod_mk continuous_const).tendsto x),
refine ⟨λ z hz, _, λ z hz, _⟩,
{ refine (this.1 z hz).mono (λ x' hx', hx'.trans_le $ le_cSup _ $ mem_image_of_mem (f x') hyK),
exact hK.bdd_above_image (hf.comp $ continuous.prod.mk x').continuous_on },
{ have h : ({x} : set γ) ×ˢ K ⊆ ↿f ⁻¹' (Iio z),
{ rintro ⟨x', y'⟩ ⟨hx', hy'⟩, cases hx', exact (hy y' hy').trans_lt hz },
obtain ⟨u, v, hu, hv, hxu, hKv, huv⟩ :=
generalized_tube_lemma is_compact_singleton hK (is_open_Iio.preimage hf) h,
refine eventually_of_mem (hu.mem_nhds (singleton_subset_iff.mp hxu)) (λ x' hx', _),
rw [hK.Sup_lt_iff_of_continuous h0K
(show continuous (f x'), from (hf.comp $ continuous.prod.mk x')).continuous_on],
exact λ y' hy', huv (mk_mem_prod hx' (hKv hy')) }
end
lemma is_compact.continuous_Inf {f : γ → β → α}
{K : set β} (hK : is_compact K) (hf : continuous ↿f) :
continuous (λ x, Inf (f x '' K)) :=
@is_compact.continuous_Sup αᵒᵈ β γ _ _ _ _ _ _ _ hK hf
namespace continuous_on
/-!
### Image of a closed interval
-/
variables [densely_ordered α] [conditionally_complete_linear_order β] [order_topology β]
{f : α → β} {a b c : α}
open_locale interval
lemma image_Icc (hab : a ≤ b) (h : continuous_on f $ Icc a b) :
f '' Icc a b = Icc (Inf $ f '' Icc a b) (Sup $ f '' Icc a b) :=
eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, is_preconnected_Icc.image f h⟩
(is_compact_Icc.image_of_continuous_on h)
lemma image_uIcc_eq_Icc (h : continuous_on f $ [a, b]) :
f '' [a, b] = Icc (Inf (f '' [a, b])) (Sup (f '' [a, b])) :=
begin
cases le_total a b with h2 h2,
{ simp_rw [uIcc_of_le h2] at h ⊢, exact h.image_Icc h2 },
{ simp_rw [uIcc_of_ge h2] at h ⊢, exact h.image_Icc h2 },
end
lemma image_uIcc (h : continuous_on f $ [a, b]) :
f '' [a, b] = [Inf (f '' [a, b]), Sup (f '' [a, b])] :=
begin
refine h.image_uIcc_eq_Icc.trans (uIcc_of_le _).symm,
refine cInf_le_cSup _ _ (nonempty_uIcc.image _); rw h.image_uIcc_eq_Icc,
exacts [bdd_below_Icc, bdd_above_Icc]
end
lemma Inf_image_Icc_le (h : continuous_on f $ Icc a b) (hc : c ∈ Icc a b) :
Inf (f '' (Icc a b)) ≤ f c :=
begin
rw h.image_Icc (nonempty_Icc.mp (set.nonempty_of_mem hc)),
exact cInf_le bdd_below_Icc (mem_Icc.mpr ⟨cInf_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
le_cSup (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩),
end
lemma le_Sup_image_Icc (h : continuous_on f $ Icc a b) (hc : c ∈ Icc a b) :
f c ≤ Sup (f '' (Icc a b)) :=
begin
rw h.image_Icc (nonempty_Icc.mp (set.nonempty_of_mem hc)),
exact le_cSup bdd_above_Icc (mem_Icc.mpr ⟨cInf_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
le_cSup (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩),
end
end continuous_on
lemma is_compact.exists_local_min_on_mem_subset {f : β → α} {s t : set β} {z : β}
(ht : is_compact t) (hf : continuous_on f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z') :
∃ x ∈ s, is_local_min_on f t x :=
begin
obtain ⟨x, hx, hfx⟩ : ∃ x ∈ t, ∀ y ∈ t, f x ≤ f y := ht.exists_forall_le ⟨z, hz⟩ hf,
have key : ∀ ⦃y⦄, y ∈ t → (∀ z' ∈ t \ s, f y < f z') → y ∈ s := λ y hy hfy,
by { by_contra; simpa using ((hfy y ((mem_diff y).mpr ⟨hy,h⟩))) },
have h1 : ∀ z' ∈ t \ s, f x < f z' := λ z' hz', (hfx z hz).trans_lt (hfz z' hz'),
have h2 : x ∈ s := key hx h1,
refine ⟨x, h2, eventually_nhds_within_of_forall hfx⟩
end
lemma is_compact.exists_local_min_mem_open {f : β → α} {s t : set β} {z : β} (ht : is_compact t)
(hst : s ⊆ t) (hf : continuous_on f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z')
(hs : is_open s) :
∃ x ∈ s, is_local_min f x :=
begin
obtain ⟨x, hx, hfx⟩ := ht.exists_local_min_on_mem_subset hf hz hfz,
exact ⟨x, hx, hfx.is_local_min (filter.mem_of_superset (hs.mem_nhds hx) hst)⟩
end
|
ddcbdeac7e1b5d60a0f17dd53eb487805f5c3cfa | 222111140d2023e142b07e9cd5119ac2829b36f5 | /src/row_equivalence_fields.lean | fc4e8bd7cf310d163ed001c4b1a58383bc16ea05 | [] | no_license | jjcrawford/lean-gaussian-elimination | 740dfdf2de89dc99c088c53e5221d9dfad3c0b92 | c473d33c07fa6f141d17d9dc42ad07956c33dd03 | refs/heads/master | 1,585,793,867,436 | 1,540,443,865,000 | 1,540,443,865,000 | 153,901,939 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,430 | lean | import ring_theory.matrix
import row_equivalence
import .finset_sum
universes u
variables {m n : ℕ}
variable {α : Type u}
variable [division_ring α]
variable [decidable_eq α]
def elementary.inv : (elementary α m) → (elementary α m) :=
begin
intros e,
cases e with i₁ s hs i₁ i₂ i₁ s i₂ h_ne,
from elementary.scale i₁ s⁻¹ (inv_ne_zero hs),
from @elementary.swap α _ _ _ i₂ i₁,
from @elementary.linear_add α _ _ m i₁ (-s) i₂ h_ne,
end
instance elementary.has_inv : has_inv (elementary α m) := ⟨elementary.inv⟩
@[simp] lemma elementary.inv_inv : Π {e : elementary α m}, (e⁻¹)⁻¹ = e :=
begin
intros e,
cases e;{simp[has_inv.inv, elementary.inv],
try{apply division_ring.inv_inv,
assumption}}
end
theorem elementary.inv_apply_implements : Π {M N : matrix (fin m) (fin n) α} {e : elementary α m}, (elementary.apply e) M = N → M = (elementary.apply e⁻¹) N :=
begin
intros M N e h,
rw ←h,
cases e with i₁ s hs i₁ i₂ i₁ s i₂ i_ne,
-- scale case
{
simp[has_inv.inv, elementary.inv],
simp[elementary.apply],
funext i j,
split_ifs with h₁,
rw ←mul_assoc,
erw inv_mul_cancel hs,
simp,
},
-- swap case
{
simp[has_inv.inv, elementary.inv],
simp[elementary.apply],
funext i j,
split_ifs with h₁ h₂ h₃; {try{simp[h₁]}, try{simp[h₂], try{simp[h₃]}}},
},
-- linear_add case
{
simp[has_inv.inv, elementary.inv],
simp[elementary.apply],
funext i j,
split_ifs with h₁ h₂;try{try{simp[h₁]}, try{simp[h₂]}},
exfalso,
subst h₁,
from i_ne (eq.symm h₂)
}
end
theorem elementary.inv_apply_implements_iff_apply_implements : Π {M N : matrix (fin m) (fin n) α} {e : elementary α m}, (elementary.apply e) M = N ↔ M = (elementary.apply e⁻¹) N :=
begin
intros M N e,
split,
apply elementary.inv_apply_implements,
let e₁ := e⁻¹,
have h₁ : e = e₁⁻¹,
simp[e₁],
rw h₁,
simp,
have H₁, from λ h₂, eq.symm ((@elementary.inv_apply_implements m n α _ _ N M (e⁻¹)) (eq.symm h₂)),
rw elementary.inv_inv at H₁,
from H₁
end
theorem elementary.inv_matrix_implements_iff_matrix_implements : Π {M N : matrix (fin m) (fin n) α} {e : elementary α m}, matrix.mul (elementary.to_matrix e) M = N ↔ M = matrix.mul (elementary.to_matrix e⁻¹) N :=
begin
intros M N e,
simp[elementary.mul_eq_apply],
apply elementary.inv_apply_implements_iff_apply_implements,
end
theorem inv_matrix_implements : Π {M N : matrix (fin m) (fin n) α} {e : elementary α m}, matrix.mul (elementary.to_matrix e) M = N → M = matrix.mul (elementary.to_matrix e⁻¹) N := λ M N e, iff.elim_left elementary.inv_matrix_implements_iff_matrix_implements
def row_equivalent_step.symm : Π {M N : matrix (fin m) (fin n) α}, row_equivalent_step M N → row_equivalent_step N M := begin
intros M N r,
cases r with elem implements,
constructor,
from eq.symm (inv_matrix_implements implements),
end
def row_equivalent.symm : Π {M N : matrix (fin m) (fin n) α}, row_equivalent M N → row_equivalent N M
| M N (row_equivalent.nil) := row_equivalent.nil
| M N (row_equivalent.cons r₁ r₂) := r₁.symm.precons r₂.symm
|
c5f740012c5195ee1f6956a25b5299d454b88aa3 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/algebra/lie/skew_adjoint.lean | 16e91f11ac0ca83d2f38125a8fb9758865ec137e | [
"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 | 7,181 | lean | /-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import algebra.lie.matrix
import linear_algebra.bilinear_form
/-!
# Lie algebras of skew-adjoint endomorphisms of a bilinear form
When a module carries a bilinear form, the Lie algebra of endomorphisms of the module contains a
distinguished Lie subalgebra: the skew-adjoint endomorphisms. Such subalgebras are important
because they provide a simple, explicit construction of the so-called classical Lie algebras.
This file defines the Lie subalgebra of skew-adjoint endomorphims cut out by a bilinear form on
a module and proves some basic related results. It also provides the corresponding definitions and
results for the Lie algebra of square matrices.
## Main definitions
* `skew_adjoint_lie_subalgebra`
* `skew_adjoint_lie_subalgebra_equiv`
* `skew_adjoint_matrices_lie_subalgebra`
* `skew_adjoint_matrices_lie_subalgebra_equiv`
## Tags
lie algebra, skew-adjoint, bilinear form
-/
universes u v w w₁
section skew_adjoint_endomorphisms
open bilin_form
variables {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M]
variables (B : bilin_form R M)
lemma bilin_form.is_skew_adjoint_bracket (f g : module.End R M)
(hf : f ∈ B.skew_adjoint_submodule) (hg : g ∈ B.skew_adjoint_submodule) :
⁅f, g⁆ ∈ B.skew_adjoint_submodule :=
begin
rw mem_skew_adjoint_submodule at *,
have hfg : is_adjoint_pair B B (f * g) (g * f), { rw ←neg_mul_neg g f, exact hf.mul hg, },
have hgf : is_adjoint_pair B B (g * f) (f * g), { rw ←neg_mul_neg f g, exact hg.mul hf, },
change bilin_form.is_adjoint_pair B B (f * g - g * f) (-(f * g - g * f)), rw neg_sub,
exact hfg.sub hgf,
end
/-- Given an `R`-module `M`, equipped with a bilinear form, the skew-adjoint endomorphisms form a
Lie subalgebra of the Lie algebra of endomorphisms. -/
def skew_adjoint_lie_subalgebra : lie_subalgebra R (module.End R M) :=
{ lie_mem' := B.is_skew_adjoint_bracket, ..B.skew_adjoint_submodule }
variables {N : Type w} [add_comm_group N] [module R N] (e : N ≃ₗ[R] M)
/-- An equivalence of modules with bilinear forms gives equivalence of Lie algebras of skew-adjoint
endomorphisms. -/
def skew_adjoint_lie_subalgebra_equiv :
skew_adjoint_lie_subalgebra (B.comp (↑e : N →ₗ[R] M) ↑e) ≃ₗ⁅R⁆ skew_adjoint_lie_subalgebra B :=
begin
apply lie_equiv.of_subalgebras _ _ e.lie_conj,
ext f,
simp only [lie_subalgebra.mem_coe, submodule.mem_map_equiv, lie_subalgebra.mem_map_submodule,
coe_coe],
exact (bilin_form.is_pair_self_adjoint_equiv (-B) B e f).symm,
end
@[simp] lemma skew_adjoint_lie_subalgebra_equiv_apply
(f : skew_adjoint_lie_subalgebra (B.comp ↑e ↑e)) :
↑(skew_adjoint_lie_subalgebra_equiv B e f) = e.lie_conj f :=
by simp [skew_adjoint_lie_subalgebra_equiv]
@[simp] lemma skew_adjoint_lie_subalgebra_equiv_symm_apply (f : skew_adjoint_lie_subalgebra B) :
↑((skew_adjoint_lie_subalgebra_equiv B e).symm f) = e.symm.lie_conj f :=
by simp [skew_adjoint_lie_subalgebra_equiv]
end skew_adjoint_endomorphisms
section skew_adjoint_matrices
open_locale matrix
variables {R : Type u} {n : Type w} [comm_ring R] [decidable_eq n] [fintype n]
variables (J : matrix n n R)
lemma matrix.lie_transpose (A B : matrix n n R) : ⁅A, B⁆ᵀ = ⁅Bᵀ, Aᵀ⁆ :=
show (A * B - B * A)ᵀ = (Bᵀ * Aᵀ - Aᵀ * Bᵀ), by simp
lemma matrix.is_skew_adjoint_bracket (A B : matrix n n R)
(hA : A ∈ skew_adjoint_matrices_submodule J) (hB : B ∈ skew_adjoint_matrices_submodule J) :
⁅A, B⁆ ∈ skew_adjoint_matrices_submodule J :=
begin
simp only [mem_skew_adjoint_matrices_submodule] at *,
change ⁅A, B⁆ᵀ ⬝ J = J ⬝ -⁅A, B⁆, change Aᵀ ⬝ J = J ⬝ -A at hA, change Bᵀ ⬝ J = J ⬝ -B at hB,
simp only [←matrix.mul_eq_mul] at *,
rw [matrix.lie_transpose, lie_ring.of_associative_ring_bracket,
lie_ring.of_associative_ring_bracket, sub_mul, mul_assoc, mul_assoc, hA, hB, ←mul_assoc,
←mul_assoc, hA, hB],
noncomm_ring,
end
/-- The Lie subalgebra of skew-adjoint square matrices corresponding to a square matrix `J`. -/
def skew_adjoint_matrices_lie_subalgebra : lie_subalgebra R (matrix n n R) :=
{ lie_mem' := J.is_skew_adjoint_bracket, ..(skew_adjoint_matrices_submodule J) }
@[simp] lemma mem_skew_adjoint_matrices_lie_subalgebra (A : matrix n n R) :
A ∈ skew_adjoint_matrices_lie_subalgebra J ↔ A ∈ skew_adjoint_matrices_submodule J :=
iff.rfl
/-- An invertible matrix `P` gives a Lie algebra equivalence between those endomorphisms that are
skew-adjoint with respect to a square matrix `J` and those with respect to `PᵀJP`. -/
noncomputable def skew_adjoint_matrices_lie_subalgebra_equiv (P : matrix n n R) (h : is_unit P) :
skew_adjoint_matrices_lie_subalgebra J ≃ₗ⁅R⁆ skew_adjoint_matrices_lie_subalgebra (Pᵀ ⬝ J ⬝ P) :=
lie_equiv.of_subalgebras _ _ (P.lie_conj h).symm
begin
ext A,
suffices : P.lie_conj h A ∈ skew_adjoint_matrices_submodule J ↔
A ∈ skew_adjoint_matrices_submodule (Pᵀ ⬝ J ⬝ P),
{ simp only [lie_subalgebra.mem_coe, submodule.mem_map_equiv, lie_subalgebra.mem_map_submodule,
coe_coe], exact this, },
simp [matrix.is_skew_adjoint, J.is_adjoint_pair_equiv _ _ P h],
end
lemma skew_adjoint_matrices_lie_subalgebra_equiv_apply
(P : matrix n n R) (h : is_unit P) (A : skew_adjoint_matrices_lie_subalgebra J) :
↑(skew_adjoint_matrices_lie_subalgebra_equiv J P h A) = P⁻¹ ⬝ ↑A ⬝ P :=
by simp [skew_adjoint_matrices_lie_subalgebra_equiv]
/-- An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an
equivalence of Lie algebras of skew-adjoint matrices. -/
def skew_adjoint_matrices_lie_subalgebra_equiv_transpose {m : Type w} [decidable_eq m] [fintype m]
(e : matrix n n R ≃ₐ[R] matrix m m R) (h : ∀ A, (e A)ᵀ = e (Aᵀ)) :
skew_adjoint_matrices_lie_subalgebra J ≃ₗ⁅R⁆ skew_adjoint_matrices_lie_subalgebra (e J) :=
lie_equiv.of_subalgebras _ _ e.to_lie_equiv
begin
ext A,
suffices : J.is_skew_adjoint (e.symm A) ↔ (e J).is_skew_adjoint A, by simpa [this],
simp [matrix.is_skew_adjoint, matrix.is_adjoint_pair, ← matrix.mul_eq_mul,
← h, ← function.injective.eq_iff e.injective],
end
@[simp] lemma skew_adjoint_matrices_lie_subalgebra_equiv_transpose_apply
{m : Type w} [decidable_eq m] [fintype m]
(e : matrix n n R ≃ₐ[R] matrix m m R) (h : ∀ A, (e A)ᵀ = e (Aᵀ))
(A : skew_adjoint_matrices_lie_subalgebra J) :
(skew_adjoint_matrices_lie_subalgebra_equiv_transpose J e h A : matrix m m R) = e A :=
rfl
lemma mem_skew_adjoint_matrices_lie_subalgebra_unit_smul (u : units R) (J A : matrix n n R) :
A ∈ skew_adjoint_matrices_lie_subalgebra (u • J) ↔
A ∈ skew_adjoint_matrices_lie_subalgebra J :=
begin
change A ∈ skew_adjoint_matrices_submodule (u • J) ↔ A ∈ skew_adjoint_matrices_submodule J,
simp only [mem_skew_adjoint_matrices_submodule, matrix.is_skew_adjoint, matrix.is_adjoint_pair],
split; intros h,
{ simpa using congr_arg (λ B, u⁻¹ • B) h, },
{ simp [h], },
end
end skew_adjoint_matrices
|
2b6d03b03a8ddf8adac02e639f735dadb38cc2bd | 4727251e0cd73359b15b664c3170e5d754078599 | /src/algebra/direct_sum/basic.lean | 7ffa3730b81b880188294d790849002fc3409fa4 | [
"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 | 12,061 | lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import data.dfinsupp.basic
import group_theory.submonoid.operations
import group_theory.subgroup.basic
/-!
# Direct sum
This file defines the direct sum of abelian groups, indexed by a discrete type.
## Notation
`⨁ i, β i` is the n-ary direct sum `direct_sum`.
This notation is in the `direct_sum` locale, accessible after `open_locale direct_sum`.
## References
* https://en.wikipedia.org/wiki/Direct_sum
-/
open_locale big_operators
universes u v w u₁
variables (ι : Type v) [dec_ι : decidable_eq ι] (β : ι → Type w)
/-- `direct_sum β` is the direct sum of a family of additive commutative monoids `β i`.
Note: `open_locale direct_sum` will enable the notation `⨁ i, β i` for `direct_sum β`. -/
@[derive [add_comm_monoid, inhabited]]
def direct_sum [Π i, add_comm_monoid (β i)] : Type* := Π₀ i, β i
instance [Π i, add_comm_monoid (β i)] : has_coe_to_fun (direct_sum ι β) (λ _, Π i : ι, β i) :=
dfinsupp.has_coe_to_fun
localized "notation `⨁` binders `, ` r:(scoped f, direct_sum _ f) := r" in direct_sum
namespace direct_sum
variables {ι}
section add_comm_group
variables [Π i, add_comm_group (β i)]
instance : add_comm_group (direct_sum ι β) := dfinsupp.add_comm_group
variables {β}
@[simp] lemma sub_apply (g₁ g₂ : ⨁ i, β i) (i : ι) : (g₁ - g₂) i = g₁ i - g₂ i :=
dfinsupp.sub_apply _ _ _
end add_comm_group
variables [Π i, add_comm_monoid (β i)]
@[simp] lemma zero_apply (i : ι) : (0 : ⨁ i, β i) i = 0 := rfl
variables {β}
@[simp] lemma add_apply (g₁ g₂ : ⨁ i, β i) (i : ι) : (g₁ + g₂) i = g₁ i + g₂ i :=
dfinsupp.add_apply _ _ _
variables (β)
include dec_ι
/-- `mk β s x` is the element of `⨁ i, β i` that is zero outside `s`
and has coefficient `x i` for `i` in `s`. -/
def mk (s : finset ι) : (Π i : (↑s : set ι), β i.1) →+ ⨁ i, β i :=
{ to_fun := dfinsupp.mk s,
map_add' := λ _ _, dfinsupp.mk_add,
map_zero' := dfinsupp.mk_zero, }
/-- `of i` is the natural inclusion map from `β i` to `⨁ i, β i`. -/
def of (i : ι) : β i →+ ⨁ i, β i :=
dfinsupp.single_add_hom β i
@[simp] lemma of_eq_same (i : ι) (x : β i) : (of _ i x) i = x :=
dfinsupp.single_eq_same
lemma of_eq_of_ne (i j : ι) (x : β i) (h : i ≠ j) : (of _ i x) j = 0 :=
dfinsupp.single_eq_of_ne h
@[simp] lemma support_zero [Π (i : ι) (x : β i), decidable (x ≠ 0)] :
(0 : ⨁ i, β i).support = ∅ := dfinsupp.support_zero
@[simp] lemma support_of [Π (i : ι) (x : β i), decidable (x ≠ 0)]
(i : ι) (x : β i) (h : x ≠ 0) :
(of _ i x).support = {i} := dfinsupp.support_single_ne_zero h
lemma support_of_subset [Π (i : ι) (x : β i), decidable (x ≠ 0)] {i : ι} {b : β i} :
(of _ i b).support ⊆ {i} := dfinsupp.support_single_subset
lemma sum_support_of [Π (i : ι) (x : β i), decidable (x ≠ 0)] (x : ⨁ i, β i) :
∑ i in x.support, of β i (x i) = x := dfinsupp.sum_single
variables {β}
theorem mk_injective (s : finset ι) : function.injective (mk β s) :=
dfinsupp.mk_injective s
theorem of_injective (i : ι) : function.injective (of β i) :=
dfinsupp.single_injective
@[elab_as_eliminator]
protected theorem induction_on {C : (⨁ i, β i) → Prop}
(x : ⨁ i, β i) (H_zero : C 0)
(H_basic : ∀ (i : ι) (x : β i), C (of β i x))
(H_plus : ∀ x y, C x → C y → C (x + y)) : C x :=
begin
apply dfinsupp.induction x H_zero,
intros i b f h1 h2 ih,
solve_by_elim
end
/-- If two additive homomorphisms from `⨁ i, β i` are equal on each `of β i y`,
then they are equal. -/
lemma add_hom_ext {γ : Type*} [add_monoid γ] ⦃f g : (⨁ i, β i) →+ γ⦄
(H : ∀ (i : ι) (y : β i), f (of _ i y) = g (of _ i y)) : f = g :=
dfinsupp.add_hom_ext H
/-- If two additive homomorphisms from `⨁ i, β i` are equal on each `of β i y`,
then they are equal.
See note [partially-applied ext lemmas]. -/
@[ext] lemma add_hom_ext' {γ : Type*} [add_monoid γ] ⦃f g : (⨁ i, β i) →+ γ⦄
(H : ∀ (i : ι), f.comp (of _ i) = g.comp (of _ i)) : f = g :=
add_hom_ext $ λ i, add_monoid_hom.congr_fun $ H i
variables {γ : Type u₁} [add_comm_monoid γ]
section to_add_monoid
variables (φ : Π i, β i →+ γ) (ψ : (⨁ i, β i) →+ γ)
/-- `to_add_monoid φ` is the natural homomorphism from `⨁ i, β i` to `γ`
induced by a family `φ` of homomorphisms `β i → γ`. -/
def to_add_monoid : (⨁ i, β i) →+ γ :=
(dfinsupp.lift_add_hom φ)
@[simp] lemma to_add_monoid_of (i) (x : β i) : to_add_monoid φ (of β i x) = φ i x :=
dfinsupp.lift_add_hom_apply_single φ i x
theorem to_add_monoid.unique (f : ⨁ i, β i) :
ψ f = to_add_monoid (λ i, ψ.comp (of β i)) f :=
by {congr, ext, simp [to_add_monoid, of]}
end to_add_monoid
section from_add_monoid
/-- `from_add_monoid φ` is the natural homomorphism from `γ` to `⨁ i, β i`
induced by a family `φ` of homomorphisms `γ → β i`.
Note that this is not an isomorphism. Not every homomorphism `γ →+ ⨁ i, β i` arises in this way. -/
def from_add_monoid : (⨁ i, γ →+ β i) →+ (γ →+ ⨁ i, β i) :=
to_add_monoid $ λ i, add_monoid_hom.comp_hom (of β i)
@[simp] lemma from_add_monoid_of (i : ι) (f : γ →+ β i) :
from_add_monoid (of _ i f) = (of _ i).comp f :=
by { rw [from_add_monoid, to_add_monoid_of], refl }
lemma from_add_monoid_of_apply (i : ι) (f : γ →+ β i) (x : γ) :
from_add_monoid (of _ i f) x = of _ i (f x) :=
by rw [from_add_monoid_of, add_monoid_hom.coe_comp]
end from_add_monoid
variables (β)
/-- `set_to_set β S T h` is the natural homomorphism `⨁ (i : S), β i → ⨁ (i : T), β i`,
where `h : S ⊆ T`. -/
-- TODO: generalize this to remove the assumption `S ⊆ T`.
def set_to_set (S T : set ι) (H : S ⊆ T) :
(⨁ (i : S), β i) →+ (⨁ (i : T), β i) :=
to_add_monoid $ λ i, of (λ (i : subtype T), β i) ⟨↑i, H i.prop⟩
variables {β}
omit dec_ι
/-- A direct sum over an empty type is trivial. -/
instance [is_empty ι] : unique (⨁ i, β i) := dfinsupp.unique
/-- The natural equivalence between `⨁ _ : ι, M` and `M` when `unique ι`. -/
protected def id (M : Type v) (ι : Type* := punit) [add_comm_monoid M] [unique ι] :
(⨁ (_ : ι), M) ≃+ M :=
{ to_fun := direct_sum.to_add_monoid (λ _, add_monoid_hom.id M),
inv_fun := of (λ _, M) default,
left_inv := λ x, direct_sum.induction_on x
(by rw [add_monoid_hom.map_zero, add_monoid_hom.map_zero])
(λ p x, by rw [unique.default_eq p, to_add_monoid_of]; refl)
(λ x y ihx ihy, by rw [add_monoid_hom.map_add, add_monoid_hom.map_add, ihx, ihy]),
right_inv := λ x, to_add_monoid_of _ _ _,
..direct_sum.to_add_monoid (λ _, add_monoid_hom.id M) }
section congr_left
variables {κ : Type*}
/--Reindexing terms of a direct sum.-/
def equiv_congr_left (h : ι ≃ κ) : (⨁ i, β i) ≃+ ⨁ k, β (h.symm k) :=
{ map_add' := dfinsupp.comap_domain'_add _ _,
..dfinsupp.equiv_congr_left h }
@[simp] lemma equiv_congr_left_apply (h : ι ≃ κ) (f : ⨁ i, β i) (k : κ) :
equiv_congr_left h f k = f (h.symm k) := dfinsupp.comap_domain'_apply _ _ _ _
end congr_left
section option
variables {α : option ι → Type w} [Π i, add_comm_monoid (α i)]
include dec_ι
/--Isomorphism obtained by separating the term of index `none` of a direct sum over `option ι`.-/
@[simps] noncomputable def add_equiv_prod_direct_sum : (⨁ i, α i) ≃+ α none × ⨁ i, α (some i) :=
{ map_add' := dfinsupp.equiv_prod_dfinsupp_add, ..dfinsupp.equiv_prod_dfinsupp }
end option
section sigma
variables {α : ι → Type u} {δ : Π i, α i → Type w} [Π i j, add_comm_monoid (δ i j)]
/--The natural map between `⨁ (i : Σ i, α i), δ i.1 i.2` and `⨁ i (j : α i), δ i j`.-/
noncomputable def sigma_curry : (⨁ (i : Σ i, _), δ i.1 i.2) →+ ⨁ i j, δ i j :=
{ to_fun := @dfinsupp.sigma_curry _ _ δ _,
map_zero' := dfinsupp.sigma_curry_zero,
map_add' := λ f g, dfinsupp.sigma_curry_add f g }
@[simp] lemma sigma_curry_apply (f : ⨁ (i : Σ i, _), δ i.1 i.2) (i : ι) (j : α i) :
sigma_curry f i j = f ⟨i, j⟩ := dfinsupp.sigma_curry_apply f i j
/--The natural map between `⨁ i (j : α i), δ i j` and `Π₀ (i : Σ i, α i), δ i.1 i.2`, inverse of
`curry`.-/
noncomputable def sigma_uncurry : (⨁ i j, δ i j) →+ ⨁ (i : Σ i, _), δ i.1 i.2 :=
{ to_fun := dfinsupp.sigma_uncurry,
map_zero' := dfinsupp.sigma_uncurry_zero,
map_add' := dfinsupp.sigma_uncurry_add }
@[simp] lemma sigma_uncurry_apply (f : ⨁ i j, δ i j) (i : ι) (j : α i) :
sigma_uncurry f ⟨i, j⟩ = f i j := dfinsupp.sigma_uncurry_apply f i j
/--The natural map between `⨁ (i : Σ i, α i), δ i.1 i.2` and `⨁ i (j : α i), δ i j`.-/
noncomputable def sigma_curry_equiv : (⨁ (i : Σ i, _), δ i.1 i.2) ≃+ ⨁ i j, δ i j :=
{ ..sigma_curry, ..dfinsupp.sigma_curry_equiv }
end sigma
/-- The canonical embedding from `⨁ i, A i` to `M` where `A` is a collection of `add_submonoid M`
indexed by `ι`-/
def add_submonoid_coe {M : Type*} [decidable_eq ι] [add_comm_monoid M]
(A : ι → add_submonoid M) : (⨁ i, A i) →+ M :=
to_add_monoid (λ i, (A i).subtype)
@[simp] lemma add_submonoid_coe_of {M : Type*} [decidable_eq ι] [add_comm_monoid M]
(A : ι → add_submonoid M) (i : ι) (x : A i) :
add_submonoid_coe A (of (λ i, A i) i x) = x :=
to_add_monoid_of _ _ _
lemma coe_of_add_submonoid_apply {M : Type*} [decidable_eq ι] [add_comm_monoid M]
{A : ι → add_submonoid M} (i j : ι) (x : A i) :
(of _ i x j : M) = if i = j then x else 0 :=
begin
obtain rfl | h := decidable.eq_or_ne i j,
{ rw [direct_sum.of_eq_same, if_pos rfl], },
{ rw [direct_sum.of_eq_of_ne _ _ _ _ h, if_neg h, add_submonoid.coe_zero], },
end
/-- The `direct_sum` formed by a collection of `add_submonoid`s of `M` is said to be internal if the
canonical map `(⨁ i, A i) →+ M` is bijective.
See `direct_sum.add_subgroup_is_internal` for the same statement about `add_subgroup`s. -/
def add_submonoid_is_internal {M : Type*} [decidable_eq ι] [add_comm_monoid M]
(A : ι → add_submonoid M) : Prop :=
function.bijective (add_submonoid_coe A)
lemma add_submonoid_is_internal.supr_eq_top {M : Type*} [decidable_eq ι] [add_comm_monoid M]
(A : ι → add_submonoid M)
(h : add_submonoid_is_internal A) : supr A = ⊤ :=
begin
rw [add_submonoid.supr_eq_mrange_dfinsupp_sum_add_hom, add_monoid_hom.mrange_top_iff_surjective],
exact function.bijective.surjective h,
end
/-- The canonical embedding from `⨁ i, A i` to `M` where `A` is a collection of `add_subgroup M`
indexed by `ι`-/
def add_subgroup_coe {M : Type*} [decidable_eq ι] [add_comm_group M]
(A : ι → add_subgroup M) : (⨁ i, A i) →+ M :=
to_add_monoid (λ i, (A i).subtype)
@[simp] lemma add_subgroup_coe_of {M : Type*} [decidable_eq ι] [add_comm_group M]
(A : ι → add_subgroup M) (i : ι) (x : A i) :
add_subgroup_coe A (of (λ i, A i) i x) = x :=
to_add_monoid_of _ _ _
lemma coe_of_add_subgroup_apply {M : Type*} [decidable_eq ι] [add_comm_group M]
{A : ι → add_subgroup M} (i j : ι) (x : A i) :
(of _ i x j : M) = if i = j then x else 0 :=
begin
obtain rfl | h := decidable.eq_or_ne i j,
{ rw [direct_sum.of_eq_same, if_pos rfl], },
{ rw [direct_sum.of_eq_of_ne _ _ _ _ h, if_neg h, add_subgroup.coe_zero], },
end
/-- The `direct_sum` formed by a collection of `add_subgroup`s of `M` is said to be internal if the
canonical map `(⨁ i, A i) →+ M` is bijective.
See `direct_sum.submodule_is_internal` for the same statement about `submodules`s. -/
def add_subgroup_is_internal {M : Type*} [decidable_eq ι] [add_comm_group M]
(A : ι → add_subgroup M) : Prop :=
function.bijective (add_subgroup_coe A)
lemma add_subgroup_is_internal.to_add_submonoid
{M : Type*} [decidable_eq ι] [add_comm_group M] (A : ι → add_subgroup M) :
add_subgroup_is_internal A ↔
add_submonoid_is_internal (λ i, (A i).to_add_submonoid) :=
iff.rfl
end direct_sum
|
a1e03cb2067a8d8664beee652ab9bdf2670dfd7c | e0f9ba56b7fedc16ef8697f6caeef5898b435143 | /src/data/fintype/basic.lean | a5ae272ef743e52e53ce0e56f62ffa36334e0c75 | [
"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 | 40,502 | lean | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Mario Carneiro
Finite types.
-/
import data.finset
import data.array.lemmas
universes u v
variables {α : Type*} {β : Type*} {γ : Type*}
/-- `fintype α` means that `α` is finite, i.e. there are only
finitely many distinct elements of type `α`. The evidence of this
is a finset `elems` (a list up to permutation without duplicates),
together with a proof that everything of type `α` is in the list. -/
class fintype (α : Type*) :=
(elems [] : finset α)
(complete : ∀ x : α, x ∈ elems)
namespace finset
variable [fintype α]
/-- `univ` is the universal finite set of type `finset α` implied from
the assumption `fintype α`. -/
def univ : finset α := fintype.elems α
@[simp] theorem mem_univ (x : α) : x ∈ (univ : finset α) :=
fintype.complete x
@[simp] theorem mem_univ_val : ∀ x, x ∈ (univ : finset α).1 := mem_univ
@[simp] lemma coe_univ : ↑(univ : finset α) = (set.univ : set α) :=
by ext; simp
theorem subset_univ (s : finset α) : s ⊆ univ := λ a _, mem_univ a
theorem eq_univ_iff_forall {s : finset α} : s = univ ↔ ∀ x, x ∈ s :=
by simp [ext]
@[simp] lemma univ_inter [decidable_eq α] (s : finset α) :
univ ∩ s = s := ext' $ λ a, by simp
@[simp] lemma inter_univ [decidable_eq α] (s : finset α) :
s ∩ univ = s :=
by rw [inter_comm, univ_inter]
@[simp] lemma piecewise_univ [∀i : α, decidable (i ∈ (univ : finset α))]
{δ : α → Sort*} (f g : Πi, δ i) : univ.piecewise f g = f :=
by { ext i, simp [piecewise] }
end finset
open finset function
namespace fintype
instance decidable_pi_fintype {α} {β : α → Type*} [∀a, decidable_eq (β a)] [fintype α] :
decidable_eq (Πa, β a) :=
assume f g, decidable_of_iff (∀ a ∈ fintype.elems α, f a = g a)
(by simp [function.funext_iff, fintype.complete])
instance decidable_forall_fintype {p : α → Prop} [decidable_pred p] [fintype α] :
decidable (∀ a, p a) :=
decidable_of_iff (∀ a ∈ @univ α _, p a) (by simp)
instance decidable_exists_fintype {p : α → Prop} [decidable_pred p] [fintype α] :
decidable (∃ a, p a) :=
decidable_of_iff (∃ a ∈ @univ α _, p a) (by simp)
instance decidable_eq_equiv_fintype [decidable_eq β] [fintype α] :
decidable_eq (α ≃ β) :=
λ a b, decidable_of_iff (a.1 = b.1) ⟨λ h, equiv.ext _ _ (congr_fun h), congr_arg _⟩
instance decidable_injective_fintype [decidable_eq α] [decidable_eq β] [fintype α] :
decidable_pred (injective : (α → β) → Prop) := λ x, by unfold injective; apply_instance
instance decidable_surjective_fintype [decidable_eq β] [fintype α] [fintype β] :
decidable_pred (surjective : (α → β) → Prop) := λ x, by unfold surjective; apply_instance
instance decidable_bijective_fintype [decidable_eq α] [decidable_eq β] [fintype α] [fintype β] :
decidable_pred (bijective : (α → β) → Prop) := λ x, by unfold bijective; apply_instance
instance decidable_left_inverse_fintype [decidable_eq α] [fintype α] (f : α → β) (g : β → α) :
decidable (function.right_inverse f g) :=
show decidable (∀ x, g (f x) = x), by apply_instance
instance decidable_right_inverse_fintype [decidable_eq β] [fintype β] (f : α → β) (g : β → α) :
decidable (function.left_inverse f g) :=
show decidable (∀ x, f (g x) = x), by apply_instance
/-- Construct a proof of `fintype α` from a universal multiset -/
def of_multiset [decidable_eq α] (s : multiset α)
(H : ∀ x : α, x ∈ s) : fintype α :=
⟨s.to_finset, by simpa using H⟩
/-- Construct a proof of `fintype α` from a universal list -/
def of_list [decidable_eq α] (l : list α)
(H : ∀ x : α, x ∈ l) : fintype α :=
⟨l.to_finset, by simpa using H⟩
theorem exists_univ_list (α) [fintype α] :
∃ l : list α, l.nodup ∧ ∀ x : α, x ∈ l :=
let ⟨l, e⟩ := quotient.exists_rep (@univ α _).1 in
by have := and.intro univ.2 mem_univ_val;
exact ⟨_, by rwa ← e at this⟩
/-- `card α` is the number of elements in `α`, defined when `α` is a fintype. -/
def card (α) [fintype α] : ℕ := (@univ α _).card
/-- If `l` lists all the elements of `α` without duplicates, then `α ≃ fin (l.length)`. -/
def equiv_fin_of_forall_mem_list {α} [decidable_eq α]
{l : list α} (h : ∀ x:α, x ∈ l) (nd : l.nodup) : α ≃ fin (l.length) :=
⟨λ a, ⟨_, list.index_of_lt_length.2 (h a)⟩,
λ i, l.nth_le i.1 i.2,
λ a, by simp,
λ ⟨i, h⟩, fin.eq_of_veq $ list.nodup_iff_nth_le_inj.1 nd _ _
(list.index_of_lt_length.2 (list.nth_le_mem _ _ _)) h $ by simp⟩
/-- There is (computably) a bijection between `α` and `fin n` where
`n = card α`. Since it is not unique, and depends on which permutation
of the universe list is used, the bijection is wrapped in `trunc` to
preserve computability. -/
def equiv_fin (α) [fintype α] [decidable_eq α] : trunc (α ≃ fin (card α)) :=
by unfold card finset.card; exact
quot.rec_on_subsingleton (@univ α _).1
(λ l (h : ∀ x:α, x ∈ l) (nd : l.nodup), trunc.mk (equiv_fin_of_forall_mem_list h nd))
mem_univ_val univ.2
theorem exists_equiv_fin (α) [fintype α] : ∃ n, nonempty (α ≃ fin n) :=
by haveI := classical.dec_eq α; exact ⟨card α, nonempty_of_trunc (equiv_fin α)⟩
/-- Given a linearly ordered fintype `α` of cardinal `k`, the equiv `mono_equiv_of_fin α h`
is the increasing bijection between `fin k` and `α`. Here, `h` is a proof that
the cardinality of `s` is `k`. We use this instead of a map `fin s.card → α` to avoid
casting issues in further uses of this function. -/
noncomputable def mono_equiv_of_fin (α) [fintype α] [decidable_linear_order α] {k : ℕ}
(h : fintype.card α = k) : fin k ≃ α :=
have A : bijective (mono_of_fin univ h) := begin
apply set.bijective_iff_bij_on_univ.2,
rw ← @coe_univ α _,
exact mono_of_fin_bij_on (univ : finset α) h
end,
equiv.of_bijective A
instance (α : Type*) : subsingleton (fintype α) :=
⟨λ ⟨s₁, h₁⟩ ⟨s₂, h₂⟩, by congr; simp [finset.ext, h₁, h₂]⟩
protected def subtype {p : α → Prop} (s : finset α)
(H : ∀ x : α, x ∈ s ↔ p x) : fintype {x // p x} :=
⟨⟨multiset.pmap subtype.mk s.1 (λ x, (H x).1),
multiset.nodup_pmap (λ a _ b _, congr_arg subtype.val) s.2⟩,
λ ⟨x, px⟩, multiset.mem_pmap.2 ⟨x, (H x).2 px, rfl⟩⟩
theorem subtype_card {p : α → Prop} (s : finset α)
(H : ∀ x : α, x ∈ s ↔ p x) :
@card {x // p x} (fintype.subtype s H) = s.card :=
multiset.card_pmap _ _ _
theorem card_of_subtype {p : α → Prop} (s : finset α)
(H : ∀ x : α, x ∈ s ↔ p x) [fintype {x // p x}] :
card {x // p x} = s.card :=
by rw ← subtype_card s H; congr
/-- Construct a fintype from a finset with the same elements. -/
def of_finset {p : set α} (s : finset α) (H : ∀ x, x ∈ s ↔ x ∈ p) : fintype p :=
fintype.subtype s H
@[simp] theorem card_of_finset {p : set α} (s : finset α) (H : ∀ x, x ∈ s ↔ x ∈ p) :
@fintype.card p (of_finset s H) = s.card :=
fintype.subtype_card s H
theorem card_of_finset' {p : set α} (s : finset α)
(H : ∀ x, x ∈ s ↔ x ∈ p) [fintype p] : fintype.card p = s.card :=
by rw ← card_of_finset s H; congr
/-- If `f : α → β` is a bijection and `α` is a fintype, then `β` is also a fintype. -/
def of_bijective [fintype α] (f : α → β) (H : function.bijective f) : fintype β :=
⟨univ.map ⟨f, H.1⟩,
λ b, let ⟨a, e⟩ := H.2 b in e ▸ mem_map_of_mem _ (mem_univ _)⟩
/-- If `f : α → β` is a surjection and `α` is a fintype, then `β` is also a fintype. -/
def of_surjective [fintype α] [decidable_eq β] (f : α → β) (H : function.surjective f) : fintype β :=
⟨univ.image f, λ b, let ⟨a, e⟩ := H b in e ▸ mem_image_of_mem _ (mem_univ _)⟩
noncomputable def of_injective [fintype β] (f : α → β) (H : function.injective f) : fintype α :=
by letI := classical.dec; exact
if hα : nonempty α then by letI := classical.inhabited_of_nonempty hα;
exact of_surjective (inv_fun f) (inv_fun_surjective H)
else ⟨∅, λ x, (hα ⟨x⟩).elim⟩
/-- If `f : α ≃ β` and `α` is a fintype, then `β` is also a fintype. -/
def of_equiv (α : Type*) [fintype α] (f : α ≃ β) : fintype β := of_bijective _ f.bijective
theorem of_equiv_card [fintype α] (f : α ≃ β) :
@card β (of_equiv α f) = card α :=
multiset.card_map _ _
theorem card_congr {α β} [fintype α] [fintype β] (f : α ≃ β) : card α = card β :=
by rw ← of_equiv_card f; congr
theorem card_eq {α β} [F : fintype α] [G : fintype β] : card α = card β ↔ nonempty (α ≃ β) :=
⟨λ h, ⟨by classical;
calc α ≃ fin (card α) : trunc.out (equiv_fin α)
... ≃ fin (card β) : by rw h
... ≃ β : (trunc.out (equiv_fin β)).symm⟩,
λ ⟨f⟩, card_congr f⟩
def of_subsingleton (a : α) [subsingleton α] : fintype α :=
⟨finset.singleton a, λ b, finset.mem_singleton.2 (subsingleton.elim _ _)⟩
@[simp] theorem univ_of_subsingleton (a : α) [subsingleton α] :
@univ _ (of_subsingleton a) = finset.singleton a := rfl
@[simp] theorem card_of_subsingleton (a : α) [subsingleton α] :
@fintype.card _ (of_subsingleton a) = 1 := rfl
end fintype
namespace set
/-- Construct a finset enumerating a set `s`, given a `fintype` instance. -/
def to_finset (s : set α) [fintype s] : finset α :=
⟨(@finset.univ s _).1.map subtype.val,
multiset.nodup_map (λ a b, subtype.eq) finset.univ.2⟩
@[simp] theorem mem_to_finset {s : set α} [fintype s] {a : α} : a ∈ s.to_finset ↔ a ∈ s :=
by simp [to_finset]
@[simp] theorem mem_to_finset_val {s : set α} [fintype s] {a : α} : a ∈ s.to_finset.1 ↔ a ∈ s :=
mem_to_finset
@[simp] theorem coe_to_finset (s : set α) [fintype s] : (↑s.to_finset : set α) = s :=
set.ext $ λ _, mem_to_finset
end set
lemma finset.card_univ [fintype α] : (finset.univ : finset α).card = fintype.card α :=
rfl
lemma finset.card_univ_diff [fintype α] [decidable_eq α] (s : finset α) :
(finset.univ \ s).card = fintype.card α - s.card :=
finset.card_sdiff (subset_univ s)
instance (n : ℕ) : fintype (fin n) :=
⟨⟨list.fin_range n, list.nodup_fin_range n⟩, list.mem_fin_range⟩
@[simp] theorem fintype.card_fin (n : ℕ) : fintype.card (fin n) = n :=
list.length_fin_range n
@[simp] lemma finset.card_fin (n : ℕ) : finset.card (finset.univ : finset (fin n)) = n :=
by rw [finset.card_univ, fintype.card_fin]
lemma fin.univ_succ (n : ℕ) :
(univ : finset (fin $ n+1)) = insert 0 (univ.image fin.succ) :=
begin
ext m,
simp only [mem_univ, mem_insert, true_iff, mem_image, exists_prop],
exact fin.cases (or.inl rfl) (λ i, or.inr ⟨i, trivial, rfl⟩) m
end
lemma fin.univ_cast_succ (n : ℕ) :
(univ : finset (fin $ n+1)) = insert (fin.last n) (univ.image fin.cast_succ) :=
begin
ext m,
simp only [mem_univ, mem_insert, true_iff, mem_image, exists_prop, true_and],
by_cases h : m.val < n,
{ right,
use fin.cast_lt m h,
rw fin.cast_succ_cast_lt },
{ left,
exact fin.eq_last_of_not_lt h }
end
/-- Any increasing map between `fin k` and a finset of cardinality `k` has to coincide with
the increasing bijection `mono_of_fin s h`. -/
lemma finset.mono_of_fin_unique' [decidable_linear_order α] {s : finset α} {k : ℕ} (h : s.card = k)
{f : fin k → α} (fmap : set.maps_to f set.univ ↑s) (hmono : strict_mono f) :
f = s.mono_of_fin h :=
begin
have finj : set.inj_on f set.univ := hmono.injective.inj_on _,
apply mono_of_fin_unique h (set.bij_on.mk fmap finj (λ y hy, _)) hmono,
simp only [set.image_univ, set.mem_range],
rcases surj_on_of_inj_on_of_card_le (λ i (hi : i ∈ finset.univ), f i)
(λ i hi, fmap (set.mem_univ i)) (λ i j hi hj hij, finj (set.mem_univ i) (set.mem_univ j) hij)
(by simp [h]) y hy with ⟨x, _, hx⟩,
exact ⟨x, hx.symm⟩
end
@[instance, priority 10] def unique.fintype {α : Type*} [unique α] : fintype α :=
⟨finset.singleton (default α), λ x, by rw [unique.eq_default x]; simp⟩
@[simp] lemma univ_unique {α : Type*} [unique α] [f : fintype α] : @finset.univ α _ = {default α} :=
by rw [subsingleton.elim f (@unique.fintype α _)]; refl
instance : fintype empty := ⟨∅, empty.rec _⟩
@[simp] theorem fintype.univ_empty : @univ empty _ = ∅ := rfl
@[simp] theorem fintype.card_empty : fintype.card empty = 0 := rfl
instance : fintype pempty := ⟨∅, pempty.rec _⟩
@[simp] theorem fintype.univ_pempty : @univ pempty _ = ∅ := rfl
@[simp] theorem fintype.card_pempty : fintype.card pempty = 0 := rfl
instance : fintype unit := fintype.of_subsingleton ()
theorem fintype.univ_unit : @univ unit _ = {()} := rfl
theorem fintype.card_unit : fintype.card unit = 1 := rfl
instance : fintype punit := fintype.of_subsingleton punit.star
@[simp] theorem fintype.univ_punit : @univ punit _ = {punit.star} := rfl
@[simp] theorem fintype.card_punit : fintype.card punit = 1 := rfl
instance : fintype bool := ⟨⟨tt::ff::0, by simp⟩, λ x, by cases x; simp⟩
@[simp] theorem fintype.univ_bool : @univ bool _ = {ff, tt} := rfl
instance units_int.fintype : fintype (units ℤ) :=
⟨{1, -1}, λ x, by cases int.units_eq_one_or x; simp *⟩
instance additive.fintype : Π [fintype α], fintype (additive α) := id
instance multiplicative.fintype : Π [fintype α], fintype (multiplicative α) := id
@[simp] theorem fintype.card_units_int : fintype.card (units ℤ) = 2 := rfl
noncomputable instance [monoid α] [fintype α] : fintype (units α) :=
by classical; exact fintype.of_injective units.val units.ext
@[simp] theorem fintype.card_bool : fintype.card bool = 2 := rfl
def finset.insert_none (s : finset α) : finset (option α) :=
⟨none :: s.1.map some, multiset.nodup_cons.2
⟨by simp, multiset.nodup_map (λ a b, option.some.inj) s.2⟩⟩
@[simp] theorem finset.mem_insert_none {s : finset α} : ∀ {o : option α},
o ∈ s.insert_none ↔ ∀ a ∈ o, a ∈ s
| none := iff_of_true (multiset.mem_cons_self _ _) (λ a h, by cases h)
| (some a) := multiset.mem_cons.trans $ by simp; refl
theorem finset.some_mem_insert_none {s : finset α} {a : α} :
some a ∈ s.insert_none ↔ a ∈ s := by simp
instance {α : Type*} [fintype α] : fintype (option α) :=
⟨univ.insert_none, λ a, by simp⟩
@[simp] theorem fintype.card_option {α : Type*} [fintype α] :
fintype.card (option α) = fintype.card α + 1 :=
(multiset.card_cons _ _).trans (by rw multiset.card_map; refl)
instance {α : Type*} (β : α → Type*)
[fintype α] [∀ a, fintype (β a)] : fintype (sigma β) :=
⟨univ.sigma (λ _, univ), λ ⟨a, b⟩, by simp⟩
@[simp] lemma finset.univ_sigma_univ {α : Type*} {β : α → Type*} [fintype α] [∀ a, fintype (β a)] :
(univ : finset α).sigma (λ a, (univ : finset (β a))) = univ := rfl
instance (α β : Type*) [fintype α] [fintype β] : fintype (α × β) :=
⟨univ.product univ, λ ⟨a, b⟩, by simp⟩
@[simp] theorem fintype.card_prod (α β : Type*) [fintype α] [fintype β] :
fintype.card (α × β) = fintype.card α * fintype.card β :=
card_product _ _
def fintype.fintype_prod_left {α β} [decidable_eq α] [fintype (α × β)] [nonempty β] : fintype α :=
⟨(fintype.elems (α × β)).image prod.fst,
assume a, let ⟨b⟩ := ‹nonempty β› in by simp; exact ⟨b, fintype.complete _⟩⟩
def fintype.fintype_prod_right {α β} [decidable_eq β] [fintype (α × β)] [nonempty α] : fintype β :=
⟨(fintype.elems (α × β)).image prod.snd,
assume b, let ⟨a⟩ := ‹nonempty α› in by simp; exact ⟨a, fintype.complete _⟩⟩
instance (α : Type*) [fintype α] : fintype (ulift α) :=
fintype.of_equiv _ equiv.ulift.symm
@[simp] theorem fintype.card_ulift (α : Type*) [fintype α] :
fintype.card (ulift α) = fintype.card α :=
fintype.of_equiv_card _
instance (α : Type u) (β : Type v) [fintype α] [fintype β] : fintype (α ⊕ β) :=
@fintype.of_equiv _ _ (@sigma.fintype _
(λ b, cond b (ulift α) (ulift.{(max u v) v} β)) _
(λ b, by cases b; apply ulift.fintype))
((equiv.sum_equiv_sigma_bool _ _).symm.trans
(equiv.sum_congr equiv.ulift equiv.ulift))
lemma fintype.card_le_of_injective [fintype α] [fintype β] (f : α → β)
(hf : function.injective f) : fintype.card α ≤ fintype.card β :=
by haveI := classical.prop_decidable; exact
finset.card_le_card_of_inj_on f (λ _ _, finset.mem_univ _) (λ _ _ _ _ h, hf h)
lemma fintype.card_eq_one_iff [fintype α] : fintype.card α = 1 ↔ (∃ x : α, ∀ y, y = x) :=
by rw [← fintype.card_unit, fintype.card_eq]; exact
⟨λ ⟨a⟩, ⟨a.symm (), λ y, a.injective (subsingleton.elim _ _)⟩,
λ ⟨x, hx⟩, ⟨⟨λ _, (), λ _, x, λ _, (hx _).trans (hx _).symm,
λ _, subsingleton.elim _ _⟩⟩⟩
lemma fintype.card_eq_zero_iff [fintype α] : fintype.card α = 0 ↔ (α → false) :=
⟨λ h a, have e : α ≃ empty := classical.choice (fintype.card_eq.1 (by simp [h])), (e a).elim,
λ h, have e : α ≃ empty := ⟨λ a, (h a).elim, λ a, a.elim, λ a, (h a).elim, λ a, a.elim⟩,
by simp [fintype.card_congr e]⟩
lemma fintype.card_pos_iff [fintype α] : 0 < fintype.card α ↔ nonempty α :=
⟨λ h, classical.by_contradiction (λ h₁,
have fintype.card α = 0 := fintype.card_eq_zero_iff.2 (λ a, h₁ ⟨a⟩),
lt_irrefl 0 $ by rwa this at h),
λ ⟨a⟩, nat.pos_of_ne_zero (mt fintype.card_eq_zero_iff.1 (λ h, h a))⟩
lemma fintype.card_le_one_iff [fintype α] : fintype.card α ≤ 1 ↔ (∀ a b : α, a = b) :=
let n := fintype.card α in
have hn : n = fintype.card α := rfl,
match n, hn with
| 0 := λ ha, ⟨λ h, λ a, (fintype.card_eq_zero_iff.1 ha.symm a).elim, λ _, ha ▸ nat.le_succ _⟩
| 1 := λ ha, ⟨λ h, λ a b, let ⟨x, hx⟩ := fintype.card_eq_one_iff.1 ha.symm in
by rw [hx a, hx b],
λ _, ha ▸ le_refl _⟩
| (n+2) := λ ha, ⟨λ h, by rw ← ha at h; exact absurd h dec_trivial,
(λ h, fintype.card_unit ▸ fintype.card_le_of_injective (λ _, ())
(λ _ _ _, h _ _))⟩
end
lemma fintype.exists_ne_of_one_lt_card [fintype α] (h : 1 < fintype.card α) (a : α) :
∃ b : α, b ≠ a :=
let ⟨b, hb⟩ := classical.not_forall.1 (mt fintype.card_le_one_iff.2 (not_le_of_gt h)) in
let ⟨c, hc⟩ := classical.not_forall.1 hb in
by haveI := classical.dec_eq α; exact
if hba : b = a then ⟨c, by cc⟩ else ⟨b, hba⟩
lemma fintype.exists_pair_of_one_lt_card [fintype α] (h : 1 < fintype.card α) :
∃ (a b : α), b ≠ a :=
begin
rcases fintype.card_pos_iff.1 (nat.lt_of_succ_lt h) with a,
exact ⟨a, fintype.exists_ne_of_one_lt_card h a⟩,
end
lemma fintype.injective_iff_surjective [fintype α] {f : α → α} : injective f ↔ surjective f :=
by haveI := classical.prop_decidable; exact
have ∀ {f : α → α}, injective f → surjective f,
from λ f hinj x,
have h₁ : image f univ = univ := eq_of_subset_of_card_le (subset_univ _)
((card_image_of_injective univ hinj).symm ▸ le_refl _),
have h₂ : x ∈ image f univ := h₁.symm ▸ mem_univ _,
exists_of_bex (mem_image.1 h₂),
⟨this,
λ hsurj, injective_of_has_left_inverse
⟨surj_inv hsurj, left_inverse_of_surjective_of_right_inverse
(this (injective_surj_inv _)) (right_inverse_surj_inv _)⟩⟩
lemma fintype.injective_iff_bijective [fintype α] {f : α → α} : injective f ↔ bijective f :=
by simp [bijective, fintype.injective_iff_surjective]
lemma fintype.surjective_iff_bijective [fintype α] {f : α → α} : surjective f ↔ bijective f :=
by simp [bijective, fintype.injective_iff_surjective]
lemma fintype.injective_iff_surjective_of_equiv [fintype α] {f : α → β} (e : α ≃ β) :
injective f ↔ surjective f :=
have injective (e.symm ∘ f) ↔ surjective (e.symm ∘ f), from fintype.injective_iff_surjective,
⟨λ hinj, by simpa [function.comp] using
surjective_comp e.surjective (this.1 (injective_comp e.symm.injective hinj)),
λ hsurj, by simpa [function.comp] using
injective_comp e.injective (this.2 (surjective_comp e.symm.surjective hsurj))⟩
lemma fintype.coe_image_univ [fintype α] [decidable_eq β] {f : α → β} :
↑(finset.image f finset.univ) = set.range f :=
by { ext x, simp }
instance list.subtype.fintype [decidable_eq α] (l : list α) : fintype {x // x ∈ l} :=
fintype.of_list l.attach l.mem_attach
instance multiset.subtype.fintype [decidable_eq α] (s : multiset α) : fintype {x // x ∈ s} :=
fintype.of_multiset s.attach s.mem_attach
instance finset.subtype.fintype (s : finset α) : fintype {x // x ∈ s} :=
⟨s.attach, s.mem_attach⟩
instance finset_coe.fintype (s : finset α) : fintype (↑s : set α) :=
finset.subtype.fintype s
@[simp] lemma fintype.card_coe (s : finset α) :
fintype.card (↑s : set α) = s.card := card_attach
lemma finset.card_le_one_iff {s : finset α} :
s.card ≤ 1 ↔ ∀ {x y}, x ∈ s → y ∈ s → x = y :=
begin
let t : set α := ↑s,
letI : fintype t := finset_coe.fintype s,
have : fintype.card t = s.card := fintype.card_coe s,
rw [← this, fintype.card_le_one_iff],
split,
{ assume H x y hx hy,
exact subtype.mk.inj (H ⟨x, hx⟩ ⟨y, hy⟩) },
{ assume H x y,
exact subtype.eq (H x.2 y.2) }
end
lemma finset.one_lt_card_iff {s : finset α} :
1 < s.card ↔ ∃ x y, (x ∈ s) ∧ (y ∈ s) ∧ x ≠ y :=
begin
classical,
rw ← not_iff_not,
push_neg,
simpa [classical.or_iff_not_imp_left] using finset.card_le_one_iff
end
instance plift.fintype (p : Prop) [decidable p] : fintype (plift p) :=
⟨if h : p then finset.singleton ⟨h⟩ else ∅, λ ⟨h⟩, by simp [h]⟩
instance Prop.fintype : fintype Prop :=
⟨⟨true::false::0, by simp [true_ne_false]⟩,
classical.cases (by simp) (by simp)⟩
def set_fintype {α} [fintype α] (s : set α) [decidable_pred s] : fintype s :=
fintype.subtype (univ.filter (∈ s)) (by simp)
namespace fintype
variables [fintype α] [decidable_eq α] {δ : α → Type*}
/-- Given for all `a : α` a finset `t a` of `δ a`, then one can define the
finset `fintype.pi_finset t` of all functions taking values in `t a` for all `a`. This is the
analogue of `finset.pi` where the base finset is `univ` (but formally they are not the same, as
there is an additional condition `i ∈ finset.univ` in the `finset.pi` definition). -/
def pi_finset (t : Πa, finset (δ a)) : finset (Πa, δ a) :=
(finset.univ.pi t).map ⟨λ f a, f a (mem_univ a), λ _ _, by simp [function.funext_iff]⟩
@[simp] lemma mem_pi_finset {t : Πa, finset (δ a)} {f : Πa, δ a} :
f ∈ pi_finset t ↔ (∀a, f a ∈ t a) :=
begin
split,
{ simp only [pi_finset, mem_map, and_imp, forall_prop_of_true, exists_prop, mem_univ,
exists_imp_distrib, mem_pi],
assume g hg hgf a,
rw ← hgf,
exact hg a },
{ simp only [pi_finset, mem_map, forall_prop_of_true, exists_prop, mem_univ, mem_pi],
assume hf,
exact ⟨λ a ha, f a, hf, rfl⟩ }
end
lemma pi_finset_subset (t₁ t₂ : Πa, finset (δ a)) (h : ∀ a, t₁ a ⊆ t₂ a) :
pi_finset t₁ ⊆ pi_finset t₂ :=
λ g hg, mem_pi_finset.2 $ λ a, h a $ mem_pi_finset.1 hg a
lemma pi_finset_disjoint_of_disjoint [∀ a, decidable_eq (δ a)]
(t₁ t₂ : Πa, finset (δ a)) {a : α} (h : disjoint (t₁ a) (t₂ a)) :
disjoint (pi_finset t₁) (pi_finset t₂) :=
disjoint_iff_ne.2 $ λ f₁ hf₁ f₂ hf₂ eq₁₂,
disjoint_iff_ne.1 h (f₁ a) (mem_pi_finset.1 hf₁ a) (f₂ a) (mem_pi_finset.1 hf₂ a) (congr_fun eq₁₂ a)
end fintype
/-! ### pi -/
/-- A dependent product of fintypes, indexed by a fintype, is a fintype. -/
instance pi.fintype {α : Type*} {β : α → Type*}
[decidable_eq α] [fintype α] [∀a, fintype (β a)] : fintype (Πa, β a) :=
⟨fintype.pi_finset (λ _, univ), by simp⟩
@[simp] lemma fintype.pi_finset_univ {α : Type*} {β : α → Type*}
[decidable_eq α] [fintype α] [∀a, fintype (β a)] :
fintype.pi_finset (λ a : α, (finset.univ : finset (β a))) = (finset.univ : finset (Π a, β a)) :=
rfl
instance d_array.fintype {n : ℕ} {α : fin n → Type*}
[∀n, fintype (α n)] : fintype (d_array n α) :=
fintype.of_equiv _ (equiv.d_array_equiv_fin _).symm
instance array.fintype {n : ℕ} {α : Type*} [fintype α] : fintype (array n α) :=
d_array.fintype
instance vector.fintype {α : Type*} [fintype α] {n : ℕ} : fintype (vector α n) :=
fintype.of_equiv _ (equiv.vector_equiv_fin _ _).symm
instance quotient.fintype [fintype α] (s : setoid α)
[decidable_rel ((≈) : α → α → Prop)] : fintype (quotient s) :=
fintype.of_surjective quotient.mk (λ x, quotient.induction_on x (λ x, ⟨x, rfl⟩))
instance finset.fintype [fintype α] : fintype (finset α) :=
⟨univ.powerset, λ x, finset.mem_powerset.2 (finset.subset_univ _)⟩
@[simp] lemma fintype.card_finset [fintype α] :
fintype.card (finset α) = 2 ^ (fintype.card α) :=
finset.card_powerset finset.univ
instance subtype.fintype (p : α → Prop) [decidable_pred p] [fintype α] : fintype {x // p x} :=
set_fintype _
@[simp] lemma set.to_finset_univ [fintype α] :
(set.univ : set α).to_finset = finset.univ :=
by { ext, simp only [set.mem_univ, mem_univ, set.mem_to_finset] }
theorem fintype.card_subtype_le [fintype α] (p : α → Prop) [decidable_pred p] :
fintype.card {x // p x} ≤ fintype.card α :=
by rw fintype.subtype_card; exact card_le_of_subset (subset_univ _)
theorem fintype.card_subtype_lt [fintype α] {p : α → Prop} [decidable_pred p]
{x : α} (hx : ¬ p x) : fintype.card {x // p x} < fintype.card α :=
by rw [fintype.subtype_card]; exact finset.card_lt_card
⟨subset_univ _, classical.not_forall.2 ⟨x, by simp [*, set.mem_def]⟩⟩
instance psigma.fintype {α : Type*} {β : α → Type*} [fintype α] [∀ a, fintype (β a)] :
fintype (Σ' a, β a) :=
fintype.of_equiv _ (equiv.psigma_equiv_sigma _).symm
instance psigma.fintype_prop_left {α : Prop} {β : α → Type*} [decidable α] [∀ a, fintype (β a)] :
fintype (Σ' a, β a) :=
if h : α then fintype.of_equiv (β h) ⟨λ x, ⟨h, x⟩, psigma.snd, λ _, rfl, λ ⟨_, _⟩, rfl⟩
else ⟨∅, λ x, h x.1⟩
instance psigma.fintype_prop_right {α : Type*} {β : α → Prop} [∀ a, decidable (β a)] [fintype α] :
fintype (Σ' a, β a) :=
fintype.of_equiv {a // β a} ⟨λ ⟨x, y⟩, ⟨x, y⟩, λ ⟨x, y⟩, ⟨x, y⟩, λ ⟨x, y⟩, rfl, λ ⟨x, y⟩, rfl⟩
instance psigma.fintype_prop_prop {α : Prop} {β : α → Prop} [decidable α] [∀ a, decidable (β a)] :
fintype (Σ' a, β a) :=
if h : ∃ a, β a then ⟨{⟨h.fst, h.snd⟩}, λ ⟨_, _⟩, by simp⟩ else ⟨∅, λ ⟨x, y⟩, h ⟨x, y⟩⟩
instance set.fintype [fintype α] : fintype (set α) :=
⟨(@finset.univ α _).powerset.map ⟨coe, coe_injective⟩, λ s, begin
classical, refine mem_map.2 ⟨finset.univ.filter s, mem_powerset.2 (subset_univ _), _⟩,
apply (coe_filter _).trans, rw [coe_univ, set.sep_univ], refl
end⟩
instance pfun_fintype (p : Prop) [decidable p] (α : p → Type*)
[Π hp, fintype (α hp)] : fintype (Π hp : p, α hp) :=
if hp : p then fintype.of_equiv (α hp) ⟨λ a _, a, λ f, f hp, λ _, rfl, λ _, rfl⟩
else ⟨singleton (λ h, (hp h).elim), by simp [hp, function.funext_iff]⟩
def quotient.fin_choice_aux {ι : Type*} [decidable_eq ι]
{α : ι → Type*} [S : ∀ i, setoid (α i)] :
∀ (l : list ι), (∀ i ∈ l, quotient (S i)) → @quotient (Π i ∈ l, α i) (by apply_instance)
| [] f := ⟦λ i, false.elim⟧
| (i::l) f := begin
refine quotient.lift_on₂ (f i (list.mem_cons_self _ _))
(quotient.fin_choice_aux l (λ j h, f j (list.mem_cons_of_mem _ h)))
_ _,
exact λ a l, ⟦λ j h,
if e : j = i then by rw e; exact a else
l _ (h.resolve_left e)⟧,
refine λ a₁ l₁ a₂ l₂ h₁ h₂, quotient.sound (λ j h, _),
by_cases e : j = i; simp [e],
{ subst j, exact h₁ },
{ exact h₂ _ _ }
end
theorem quotient.fin_choice_aux_eq {ι : Type*} [decidable_eq ι]
{α : ι → Type*} [S : ∀ i, setoid (α i)] :
∀ (l : list ι) (f : ∀ i ∈ l, α i), quotient.fin_choice_aux l (λ i h, ⟦f i h⟧) = ⟦f⟧
| [] f := quotient.sound (λ i h, h.elim)
| (i::l) f := begin
simp [quotient.fin_choice_aux, quotient.fin_choice_aux_eq l],
refine quotient.sound (λ j h, _),
by_cases e : j = i; simp [e],
subst j, refl
end
def quotient.fin_choice {ι : Type*} [fintype ι] [decidable_eq ι]
{α : ι → Type*} [S : ∀ i, setoid (α i)]
(f : ∀ i, quotient (S i)) : @quotient (Π i, α i) (by apply_instance) :=
quotient.lift_on (@quotient.rec_on _ _ (λ l : multiset ι,
@quotient (Π i ∈ l, α i) (by apply_instance))
finset.univ.1
(λ l, quotient.fin_choice_aux l (λ i _, f i))
(λ a b h, begin
have := λ a, quotient.fin_choice_aux_eq a (λ i h, quotient.out (f i)),
simp [quotient.out_eq] at this,
simp [this],
let g := λ a:multiset ι, ⟦λ (i : ι) (h : i ∈ a), quotient.out (f i)⟧,
refine eq_of_heq ((eq_rec_heq _ _).trans (_ : g a == g b)),
congr' 1, exact quotient.sound h,
end))
(λ f, ⟦λ i, f i (finset.mem_univ _)⟧)
(λ a b h, quotient.sound $ λ i, h _ _)
theorem quotient.fin_choice_eq {ι : Type*} [fintype ι] [decidable_eq ι]
{α : ι → Type*} [∀ i, setoid (α i)]
(f : ∀ i, α i) : quotient.fin_choice (λ i, ⟦f i⟧) = ⟦f⟧ :=
begin
let q, swap, change quotient.lift_on q _ _ = _,
have : q = ⟦λ i h, f i⟧,
{ dsimp [q],
exact quotient.induction_on
(@finset.univ ι _).1 (λ l, quotient.fin_choice_aux_eq _ _) },
simp [this], exact setoid.refl _
end
section equiv
open list equiv equiv.perm
variables [decidable_eq α] [decidable_eq β]
def perms_of_list : list α → list (perm α)
| [] := [1]
| (a :: l) := perms_of_list l ++ l.bind (λ b, (perms_of_list l).map (λ f, swap a b * f))
lemma length_perms_of_list : ∀ l : list α, length (perms_of_list l) = l.length.fact
| [] := rfl
| (a :: l) :=
begin
rw [length_cons, nat.fact_succ],
simp [perms_of_list, length_bind, length_perms_of_list, function.comp, nat.succ_mul],
cc
end
lemma mem_perms_of_list_of_mem : ∀ {l : list α} {f : perm α} (h : ∀ x, f x ≠ x → x ∈ l), f ∈ perms_of_list l
| [] f h := list.mem_singleton.2 $ equiv.ext _ _$ λ x, by simp [imp_false, *] at *
| (a::l) f h :=
if hfa : f a = a
then
mem_append_left _ $ mem_perms_of_list_of_mem
(λ x hx, mem_of_ne_of_mem (λ h, by rw h at hx; exact hx hfa) (h x hx))
else
have hfa' : f (f a) ≠ f a, from mt (λ h, f.injective h) hfa,
have ∀ (x : α), (swap a (f a) * f) x ≠ x → x ∈ l,
from λ x hx, have hxa : x ≠ a, from λ h, by simpa [h, mul_apply] using hx,
have hfxa : f x ≠ f a, from mt (λ h, f.injective h) hxa,
list.mem_of_ne_of_mem hxa
(h x (λ h, by simp [h, mul_apply, swap_apply_def] at hx; split_ifs at hx; cc)),
suffices f ∈ perms_of_list l ∨ ∃ (b : α), b ∈ l ∧ ∃ g : perm α, g ∈ perms_of_list l ∧ swap a b * g = f,
by simpa [perms_of_list],
(@or_iff_not_imp_left _ _ (classical.prop_decidable _)).2
(λ hfl, ⟨f a,
if hffa : f (f a) = a then mem_of_ne_of_mem hfa (h _ (mt (λ h, f.injective h) hfa))
else this _ $ by simp [mul_apply, swap_apply_def]; split_ifs; cc,
⟨swap a (f a) * f, mem_perms_of_list_of_mem this,
by rw [← mul_assoc, mul_def (swap a (f a)) (swap a (f a)), swap_swap, ← equiv.perm.one_def, one_mul]⟩⟩)
lemma mem_of_mem_perms_of_list : ∀ {l : list α} {f : perm α}, f ∈ perms_of_list l → ∀ {x}, f x ≠ x → x ∈ l
| [] f h := have f = 1 := by simpa [perms_of_list] using h, by rw this; simp
| (a::l) f h :=
(mem_append.1 h).elim
(λ h x hx, mem_cons_of_mem _ (mem_of_mem_perms_of_list h hx))
(λ h x hx,
let ⟨y, hy, hy'⟩ := list.mem_bind.1 h in
let ⟨g, hg₁, hg₂⟩ := list.mem_map.1 hy' in
if hxa : x = a then by simp [hxa]
else if hxy : x = y then mem_cons_of_mem _ $ by rwa hxy
else mem_cons_of_mem _ $
mem_of_mem_perms_of_list hg₁ $
by rw [eq_inv_mul_iff_mul_eq.2 hg₂, mul_apply, swap_inv, swap_apply_def];
split_ifs; cc)
lemma mem_perms_of_list_iff {l : list α} {f : perm α} : f ∈ perms_of_list l ↔ ∀ {x}, f x ≠ x → x ∈ l :=
⟨mem_of_mem_perms_of_list, mem_perms_of_list_of_mem⟩
lemma nodup_perms_of_list : ∀ {l : list α} (hl : l.nodup), (perms_of_list l).nodup
| [] hl := by simp [perms_of_list]
| (a::l) hl :=
have hl' : l.nodup, from nodup_of_nodup_cons hl,
have hln' : (perms_of_list l).nodup, from nodup_perms_of_list hl',
have hmeml : ∀ {f : perm α}, f ∈ perms_of_list l → f a = a,
from λ f hf, not_not.1 (mt (mem_of_mem_perms_of_list hf) (nodup_cons.1 hl).1),
by rw [perms_of_list, list.nodup_append, list.nodup_bind, pairwise_iff_nth_le]; exact
⟨hln', ⟨λ _ _, nodup_map (λ _ _, (mul_right_inj _).1) hln',
λ i j hj hij x hx₁ hx₂,
let ⟨f, hf⟩ := list.mem_map.1 hx₁ in
let ⟨g, hg⟩ := list.mem_map.1 hx₂ in
have hix : x a = nth_le l i (lt_trans hij hj),
by rw [← hf.2, mul_apply, hmeml hf.1, swap_apply_left],
have hiy : x a = nth_le l j hj,
by rw [← hg.2, mul_apply, hmeml hg.1, swap_apply_left],
absurd (hf.2.trans (hg.2.symm)) $
λ h, ne_of_lt hij $ nodup_iff_nth_le_inj.1 hl' i j (lt_trans hij hj) hj $
by rw [← hix, hiy]⟩,
λ f hf₁ hf₂,
let ⟨x, hx, hx'⟩ := list.mem_bind.1 hf₂ in
let ⟨g, hg⟩ := list.mem_map.1 hx' in
have hgxa : g⁻¹ x = a, from f.injective $
by rw [hmeml hf₁, ← hg.2]; simp,
have hxa : x ≠ a, from λ h, (list.nodup_cons.1 hl).1 (h ▸ hx),
(list.nodup_cons.1 hl).1 $
hgxa ▸ mem_of_mem_perms_of_list hg.1 (by rwa [apply_inv_self, hgxa])⟩
def perms_of_finset (s : finset α) : finset (perm α) :=
quotient.hrec_on s.1 (λ l hl, ⟨perms_of_list l, nodup_perms_of_list hl⟩)
(λ a b hab, hfunext (congr_arg _ (quotient.sound hab))
(λ ha hb _, heq_of_eq $ finset.ext.2 $
by simp [mem_perms_of_list_iff, hab.mem_iff]))
s.2
lemma mem_perms_of_finset_iff : ∀ {s : finset α} {f : perm α},
f ∈ perms_of_finset s ↔ ∀ {x}, f x ≠ x → x ∈ s :=
by rintros ⟨⟨l⟩, hs⟩ f; exact mem_perms_of_list_iff
lemma card_perms_of_finset : ∀ (s : finset α),
(perms_of_finset s).card = s.card.fact :=
by rintros ⟨⟨l⟩, hs⟩; exact length_perms_of_list l
def fintype_perm [fintype α] : fintype (perm α) :=
⟨perms_of_finset (@finset.univ α _), by simp [mem_perms_of_finset_iff]⟩
instance [fintype α] [fintype β] : fintype (α ≃ β) :=
if h : fintype.card β = fintype.card α
then trunc.rec_on_subsingleton (fintype.equiv_fin α)
(λ eα, trunc.rec_on_subsingleton (fintype.equiv_fin β)
(λ eβ, @fintype.of_equiv _ (perm α) fintype_perm
(equiv_congr (equiv.refl α) (eα.trans (eq.rec_on h eβ.symm)) : (α ≃ α) ≃ (α ≃ β))))
else ⟨∅, λ x, false.elim (h (fintype.card_eq.2 ⟨x.symm⟩))⟩
lemma fintype.card_perm [fintype α] : fintype.card (perm α) = (fintype.card α).fact :=
subsingleton.elim (@fintype_perm α _ _) (@equiv.fintype α α _ _ _ _) ▸
card_perms_of_finset _
lemma fintype.card_equiv [fintype α] [fintype β] (e : α ≃ β) :
fintype.card (α ≃ β) = (fintype.card α).fact :=
fintype.card_congr (equiv_congr (equiv.refl α) e) ▸ fintype.card_perm
lemma univ_eq_singleton_of_card_one {α} [fintype α] (x : α) (h : fintype.card α = 1) :
(univ : finset α) = finset.singleton x :=
begin
apply symm,
apply eq_of_subset_of_card_le (subset_univ (finset.singleton x)),
apply le_of_eq,
simp [h, finset.card_univ]
end
end equiv
namespace fintype
section choose
open fintype
open equiv
variables [fintype α] [decidable_eq α] (p : α → Prop) [decidable_pred p]
def choose_x (hp : ∃! a : α, p a) : {a // p a} :=
⟨finset.choose p univ (by simp; exact hp), finset.choose_property _ _ _⟩
def choose (hp : ∃! a, p a) : α := choose_x p hp
lemma choose_spec (hp : ∃! a, p a) : p (choose p hp) :=
(choose_x p hp).property
end choose
section bijection_inverse
open function
variables [fintype α] [decidable_eq α]
variables [fintype β] [decidable_eq β]
variables {f : α → β}
/-- `
`bij_inv f` is the unique inverse to a bijection `f`. This acts
as a computable alternative to `function.inv_fun`. -/
def bij_inv (f_bij : bijective f) (b : β) : α :=
fintype.choose (λ a, f a = b)
begin
rcases f_bij.right b with ⟨a', fa_eq_b⟩,
rw ← fa_eq_b,
exact ⟨a', ⟨rfl, (λ a h, f_bij.left h)⟩⟩
end
lemma left_inverse_bij_inv (f_bij : bijective f) : left_inverse (bij_inv f_bij) f :=
λ a, f_bij.left (choose_spec (λ a', f a' = f a) _)
lemma right_inverse_bij_inv (f_bij : bijective f) : right_inverse (bij_inv f_bij) f :=
λ b, choose_spec (λ a', f a' = b) _
lemma bijective_bij_inv (f_bij : bijective f) : bijective (bij_inv f_bij) :=
⟨injective_of_left_inverse (right_inverse_bij_inv _),
surjective_of_has_right_inverse ⟨f, left_inverse_bij_inv _⟩⟩
end bijection_inverse
lemma well_founded_of_trans_of_irrefl [fintype α] (r : α → α → Prop)
[is_trans α r] [is_irrefl α r] : well_founded r :=
by classical; exact
have ∀ x y, r x y → (univ.filter (λ z, r z x)).card < (univ.filter (λ z, r z y)).card,
from λ x y hxy, finset.card_lt_card $
by simp only [finset.lt_iff_ssubset.symm, lt_iff_le_not_le,
finset.le_iff_subset, finset.subset_iff, mem_filter, true_and, mem_univ, hxy];
exact ⟨λ z hzx, trans hzx hxy, not_forall_of_exists_not ⟨x, not_imp.2 ⟨hxy, irrefl x⟩⟩⟩,
subrelation.wf this (measure_wf _)
lemma preorder.well_founded [fintype α] [preorder α] : well_founded ((<) : α → α → Prop) :=
well_founded_of_trans_of_irrefl _
@[instance, priority 10] lemma linear_order.is_well_order [fintype α] [linear_order α] :
is_well_order α (<) :=
{ wf := preorder.well_founded }
end fintype
class infinite (α : Type*) : Prop :=
(not_fintype : fintype α → false)
@[simp] lemma not_nonempty_fintype {α : Type*} : ¬nonempty (fintype α) ↔ infinite α :=
⟨λf, ⟨λ x, f ⟨x⟩⟩, λ⟨f⟩ ⟨x⟩, f x⟩
namespace infinite
lemma exists_not_mem_finset [infinite α] (s : finset α) : ∃ x, x ∉ s :=
classical.not_forall.1 $ λ h, not_fintype ⟨s, h⟩
@[priority 100] -- see Note [lower instance priority]
instance nonempty (α : Type*) [infinite α] : nonempty α :=
nonempty_of_exists (exists_not_mem_finset (∅ : finset α))
lemma of_injective [infinite β] (f : β → α) (hf : injective f) : infinite α :=
⟨λ I, by exactI not_fintype (fintype.of_injective f hf)⟩
lemma of_surjective [infinite β] (f : α → β) (hf : surjective f) : infinite α :=
⟨λ I, by classical; exactI not_fintype (fintype.of_surjective f hf)⟩
private noncomputable def nat_embedding_aux (α : Type*) [infinite α] : ℕ → α
| n := by letI := classical.dec_eq α; exact classical.some (exists_not_mem_finset
((multiset.range n).pmap (λ m (hm : m < n), nat_embedding_aux m)
(λ _, multiset.mem_range.1)).to_finset)
private lemma nat_embedding_aux_injective (α : Type*) [infinite α] :
function.injective (nat_embedding_aux α) :=
begin
assume m n h,
letI := classical.dec_eq α,
wlog hmlen : m ≤ n using m n,
by_contradiction hmn,
have hmn : m < n, from lt_of_le_of_ne hmlen hmn,
refine (classical.some_spec (exists_not_mem_finset
((multiset.range n).pmap (λ m (hm : m < n), nat_embedding_aux α m)
(λ _, multiset.mem_range.1)).to_finset)) _,
refine multiset.mem_to_finset.2 (multiset.mem_pmap.2
⟨m, multiset.mem_range.2 hmn, _⟩),
rw [h, nat_embedding_aux]
end
noncomputable def nat_embedding (α : Type*) [infinite α] : ℕ ↪ α :=
⟨_, nat_embedding_aux_injective α⟩
end infinite
lemma not_injective_infinite_fintype [infinite α] [fintype β] (f : α → β) :
¬ injective f :=
assume (hf : injective f),
have H : fintype α := fintype.of_injective f hf,
infinite.not_fintype H
lemma not_surjective_fintype_infinite [fintype α] [infinite β] (f : α → β) :
¬ surjective f :=
assume (hf : surjective f),
have H : infinite α := infinite.of_surjective f hf,
@infinite.not_fintype _ H infer_instance
instance nat.infinite : infinite ℕ :=
⟨λ ⟨s, hs⟩, finset.not_mem_range_self $ s.subset_range_sup_succ (hs _)⟩
instance int.infinite : infinite ℤ :=
infinite.of_injective int.of_nat (λ _ _, int.of_nat_inj)
|
817df3dd2ab6215e27d71e056833230bd7988642 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/category_theory/limits/limits_auto.lean | 2c830b3b1e93ab34ffbb7488c3bddddd05224de6 | [] | 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 | 92,745 | lean | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.category_theory.adjunction.basic
import Mathlib.category_theory.limits.cones
import Mathlib.category_theory.reflects_isomorphisms
import Mathlib.PostPort
universes v u l u' u''
namespace Mathlib
/-!
# Limits and colimits
We set up the general theory of limits and colimits in a category.
In this introduction we only describe the setup for limits;
it is repeated, with slightly different names, for colimits.
The three main structures involved are
* `is_limit c`, for `c : cone F`, `F : J ⥤ C`, expressing that `c` is a limit cone,
* `limit_cone F`, which consists of a choice of cone for `F` and the fact it is a limit cone, and
* `has_limit F`, asserting the mere existence of some limit cone for `F`.
`has_limit` is a propositional typeclass
(it's important that it is a proposition merely asserting the existence of a limit,
as otherwise we would have non-defeq problems from incompatible instances).
Typically there are two different ways one can use the limits library:
1. working with particular cones, and terms of type `is_limit`
2. working solely with `has_limit`.
While `has_limit` only asserts the existence of a limit cone,
we happily use the axiom of choice in mathlib,
so there are convenience functions all depending on `has_limit F`:
* `limit F : C`, producing some limit object (of course all such are isomorphic)
* `limit.π F j : limit F ⟶ F.obj j`, the morphisms out of the limit,
* `limit.lift F c : c.X ⟶ limit F`, the universal morphism from any other `c : cone F`, etc.
Key to using the `has_limit` interface is that there is an `@[ext]` lemma stating that
to check `f = g`, for `f g : Z ⟶ limit F`, it suffices to check `f ≫ limit.π F j = g ≫ limit.π F j`
for every `j`.
This, combined with `@[simp]` lemmas, makes it possible to prove many easy facts about limits using
automation (e.g. `tidy`).
There are abbreviations `has_limits_of_shape J C` and `has_limits C`
asserting the existence of classes of limits.
Later more are introduced, for finite limits, special shapes of limits, etc.
Ideally, many results about limits should be stated first in terms of `is_limit`,
and then a result in terms of `has_limit` derived from this.
At this point, however, this is far from uniformly achieved in mathlib ---
often statements are only written in terms of `has_limit`.
## Implementation
At present we simply say everything twice, in order to handle both limits and colimits.
It would be highly desirable to have some automation support,
e.g. a `@[dualize]` attribute that behaves similarly to `@[to_additive]`.
## References
* [Stacks: Limits and colimits](https://stacks.math.columbia.edu/tag/002D)
-/
namespace category_theory.limits
/--
A cone `t` on `F` is a limit cone if each cone on `F` admits a unique
cone morphism to `t`.
See https://stacks.math.columbia.edu/tag/002E.
-/
structure is_limit {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
(t : cone F)
where
lift : (s : cone F) → cone.X s ⟶ cone.X t
fac' :
autoParam
(∀ (s : cone F) (j : J), lift s ≫ nat_trans.app (cone.π t) j = nat_trans.app (cone.π s) j)
(Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring "Mathlib.obviously")
(Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous "Mathlib") "obviously") [])
uniq' :
autoParam
(∀ (s : cone F) (m : cone.X s ⟶ cone.X t),
(∀ (j : J), m ≫ nat_trans.app (cone.π t) j = nat_trans.app (cone.π s) j) → m = lift s)
(Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring "Mathlib.obviously")
(Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous "Mathlib") "obviously") [])
@[simp] theorem is_limit.fac {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cone F} (c : is_limit t) (s : cone F) (j : J) :
is_limit.lift c s ≫ nat_trans.app (cone.π t) j = nat_trans.app (cone.π s) j :=
sorry
@[simp] theorem is_limit.fac_assoc {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {t : cone F} (c : is_limit t) (s : cone F) (j : J) {X' : C}
(f' : functor.obj F j ⟶ X') :
is_limit.lift c s ≫ nat_trans.app (cone.π t) j ≫ f' = nat_trans.app (cone.π s) j ≫ f' :=
sorry
theorem is_limit.uniq {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cone F} (c : is_limit t) (s : cone F) (m : cone.X s ⟶ cone.X t)
(w : ∀ (j : J), m ≫ nat_trans.app (cone.π t) j = nat_trans.app (cone.π s) j) :
m = is_limit.lift c s :=
sorry
namespace is_limit
protected instance subsingleton {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {t : cone F} : subsingleton (is_limit t) :=
sorry
/-- Given a natural transformation `α : F ⟶ G`, we give a morphism from the cone point
of any cone over `F` to the cone point of a limit cone over `G`. -/
def map {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {G : J ⥤ C}
(s : cone F) {t : cone G} (P : is_limit t) (α : F ⟶ G) : cone.X s ⟶ cone.X t :=
lift P (functor.obj (cones.postcompose α) s)
@[simp] theorem map_π_assoc {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} (c : cone F) {d : cone G} (hd : is_limit d) (α : F ⟶ G) (j : J) {X' : C}
(f' : functor.obj G j ⟶ X') :
map c hd α ≫ nat_trans.app (cone.π d) j ≫ f' =
nat_trans.app (cone.π c) j ≫ nat_trans.app α j ≫ f' :=
sorry
theorem lift_self {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {c : cone F}
(t : is_limit c) : lift t c = 𝟙 :=
Eq.symm (uniq t c 𝟙 fun (j : J) => category.id_comp (nat_trans.app (cone.π c) j))
/- Repackaging the definition in terms of cone morphisms. -/
/-- The universal morphism from any other cone to a limit cone. -/
@[simp] theorem lift_cone_morphism_hom {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {t : cone F} (h : is_limit t) (s : cone F) :
cone_morphism.hom (lift_cone_morphism h s) = lift h s :=
Eq.refl (cone_morphism.hom (lift_cone_morphism h s))
theorem uniq_cone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{s : cone F} {t : cone F} (h : is_limit t) {f : s ⟶ t} {f' : s ⟶ t} : f = f' :=
(fun (this : ∀ {g : s ⟶ t}, g = lift_cone_morphism h s) => Eq.trans this (Eq.symm this))
fun (g : s ⟶ t) =>
cone_morphism.ext g (lift_cone_morphism h s)
(uniq h s (cone_morphism.hom g) (cone_morphism.w g))
/--
Alternative constructor for `is_limit`,
providing a morphism of cones rather than a morphism between the cone points
and separately the factorisation condition.
-/
def mk_cone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cone F} (lift : (s : cone F) → s ⟶ t) (uniq' : ∀ (s : cone F) (m : s ⟶ t), m = lift s) :
is_limit t :=
mk fun (s : cone F) => cone_morphism.hom (lift s)
/-- Limit cones on `F` are unique up to isomorphism. -/
def unique_up_to_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{s : cone F} {t : cone F} (P : is_limit s) (Q : is_limit t) : s ≅ t :=
iso.mk (lift_cone_morphism Q s) (lift_cone_morphism P t)
/-- Any cone morphism between limit cones is an isomorphism. -/
def hom_is_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {s : cone F}
{t : cone F} (P : is_limit s) (Q : is_limit t) (f : s ⟶ t) : is_iso f :=
is_iso.mk (lift_cone_morphism P t)
/-- Limits of `F` are unique up to isomorphism. -/
def cone_point_unique_up_to_iso {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {s : cone F} {t : cone F} (P : is_limit s) (Q : is_limit t) : cone.X s ≅ cone.X t :=
functor.map_iso (cones.forget F) (unique_up_to_iso P Q)
@[simp] theorem cone_point_unique_up_to_iso_hom_comp {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {s : cone F} {t : cone F} (P : is_limit s) (Q : is_limit t) (j : J) :
iso.hom (cone_point_unique_up_to_iso P Q) ≫ nat_trans.app (cone.π t) j =
nat_trans.app (cone.π s) j :=
cone_morphism.w (iso.hom (unique_up_to_iso P Q)) j
@[simp] theorem cone_point_unique_up_to_iso_inv_comp_assoc {J : Type v} [small_category J]
{C : Type u} [category C] {F : J ⥤ C} {s : cone F} {t : cone F} (P : is_limit s)
(Q : is_limit t) (j : J) {X' : C} (f' : functor.obj F j ⟶ X') :
iso.inv (cone_point_unique_up_to_iso P Q) ≫ nat_trans.app (cone.π s) j ≫ f' =
nat_trans.app (cone.π t) j ≫ f' :=
sorry
@[simp] theorem lift_comp_cone_point_unique_up_to_iso_hom {J : Type v} [small_category J]
{C : Type u} [category C] {F : J ⥤ C} {r : cone F} {s : cone F} {t : cone F} (P : is_limit s)
(Q : is_limit t) : lift P r ≫ iso.hom (cone_point_unique_up_to_iso P Q) = lift Q r :=
sorry
@[simp] theorem lift_comp_cone_point_unique_up_to_iso_inv {J : Type v} [small_category J]
{C : Type u} [category C] {F : J ⥤ C} {r : cone F} {s : cone F} {t : cone F} (P : is_limit s)
(Q : is_limit t) : lift Q r ≫ iso.inv (cone_point_unique_up_to_iso P Q) = lift P r :=
sorry
/-- Transport evidence that a cone is a limit cone across an isomorphism of cones. -/
def of_iso_limit {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {r : cone F}
{t : cone F} (P : is_limit r) (i : r ≅ t) : is_limit t :=
mk_cone_morphism (fun (s : cone F) => lift_cone_morphism P s ≫ iso.hom i) sorry
@[simp] theorem of_iso_limit_lift {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {r : cone F} {t : cone F} (P : is_limit r) (i : r ≅ t) (s : cone F) :
lift (of_iso_limit P i) s = lift P s ≫ cone_morphism.hom (iso.hom i) :=
rfl
/-- Isomorphism of cones preserves whether or not they are limiting cones. -/
def equiv_iso_limit {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{r : cone F} {t : cone F} (i : r ≅ t) : is_limit r ≃ is_limit t :=
equiv.mk (fun (h : is_limit r) => of_iso_limit h i)
(fun (h : is_limit t) => of_iso_limit h (iso.symm i)) sorry sorry
@[simp] theorem equiv_iso_limit_apply {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {r : cone F} {t : cone F} (i : r ≅ t) (P : is_limit r) :
coe_fn (equiv_iso_limit i) P = of_iso_limit P i :=
rfl
@[simp] theorem equiv_iso_limit_symm_apply {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {r : cone F} {t : cone F} (i : r ≅ t) (P : is_limit t) :
coe_fn (equiv.symm (equiv_iso_limit i)) P = of_iso_limit P (iso.symm i) :=
rfl
/--
If the canonical morphism from a cone point to a limiting cone point is an iso, then the
first cone was limiting also.
-/
def of_point_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {r : cone F}
{t : cone F} (P : is_limit r) [i : is_iso (lift P t)] : is_limit t :=
of_iso_limit P (iso.symm (as_iso (lift_cone_morphism P t)))
theorem hom_lift {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cone F}
(h : is_limit t) {W : C} (m : W ⟶ cone.X t) :
m = lift h (cone.mk W (nat_trans.mk fun (b : J) => m ≫ nat_trans.app (cone.π t) b)) :=
uniq h (cone.mk W (nat_trans.mk fun (b : J) => m ≫ nat_trans.app (cone.π t) b)) m
fun (b : J) => rfl
/-- Two morphisms into a limit are equal if their compositions with
each cone morphism are equal. -/
theorem hom_ext {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cone F}
(h : is_limit t) {W : C} {f : W ⟶ cone.X t} {f' : W ⟶ cone.X t}
(w : ∀ (j : J), f ≫ nat_trans.app (cone.π t) j = f' ≫ nat_trans.app (cone.π t) j) : f = f' :=
sorry
/--
Given a right adjoint functor between categories of cones,
the image of a limit cone is a limit cone.
-/
def of_right_adjoint {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} {D : Type u'} [category D] {G : K ⥤ D} (h : cone G ⥤ cone F)
[is_right_adjoint h] {c : cone G} (t : is_limit c) : is_limit (functor.obj h c) :=
mk_cone_morphism
(fun (s : cone F) =>
coe_fn (adjunction.hom_equiv (adjunction.of_right_adjoint h) s c)
(lift_cone_morphism t (functor.obj (left_adjoint h) s)))
sorry
/--
Given two functors which have equivalent categories of cones, we can transport a limiting cone across
the equivalence.
-/
def of_cone_equiv {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} {D : Type u'} [category D] {G : K ⥤ D} (h : cone G ≌ cone F)
{c : cone G} : is_limit (functor.obj (equivalence.functor h) c) ≃ is_limit c :=
equiv.mk
(fun (P : is_limit (functor.obj (equivalence.functor h) c)) =>
of_iso_limit (of_right_adjoint (equivalence.inverse h) P)
(iso.app (iso.symm (equivalence.unit_iso h)) c))
(of_right_adjoint (equivalence.functor h)) sorry sorry
@[simp] theorem of_cone_equiv_apply_desc {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} {D : Type u'} [category D] {G : K ⥤ D}
(h : cone G ≌ cone F) {c : cone G} (P : is_limit (functor.obj (equivalence.functor h) c))
(s : cone G) :
lift (coe_fn (of_cone_equiv h) P) s =
(cone_morphism.hom (nat_trans.app (iso.hom (equivalence.unit_iso h)) s) ≫
cone_morphism.hom
(functor.map (functor.inv (equivalence.functor h))
(lift_cone_morphism P (functor.obj (equivalence.functor h) s)))) ≫
cone_morphism.hom (nat_trans.app (iso.inv (equivalence.unit_iso h)) c) :=
rfl
@[simp] theorem of_cone_equiv_symm_apply_desc {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} {D : Type u'} [category D] {G : K ⥤ D}
(h : cone G ≌ cone F) {c : cone G} (P : is_limit c) (s : cone F) :
lift (coe_fn (equiv.symm (of_cone_equiv h)) P) s =
cone_morphism.hom (nat_trans.app (iso.inv (equivalence.counit_iso h)) s) ≫
cone_morphism.hom
(functor.map (equivalence.functor h)
(lift_cone_morphism P (functor.obj (equivalence.inverse h) s))) :=
rfl
/--
A cone postcomposed with a natural isomorphism is a limit cone if and only if the original cone is.
-/
def postcompose_hom_equiv {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} (α : F ≅ G) (c : cone F) :
is_limit (functor.obj (cones.postcompose (iso.hom α)) c) ≃ is_limit c :=
of_cone_equiv (cones.postcompose_equivalence α)
/--
A cone postcomposed with the inverse of a natural isomorphism is a limit cone if and only if
the original cone is.
-/
def postcompose_inv_equiv {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} (α : F ≅ G) (c : cone G) :
is_limit (functor.obj (cones.postcompose (iso.inv α)) c) ≃ is_limit c :=
postcompose_hom_equiv (iso.symm α) c
/--
The cone points of two limit cones for naturally isomorphic functors
are themselves isomorphic.
-/
@[simp] theorem cone_points_iso_of_nat_iso_inv {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} {s : cone F} {t : cone G} (P : is_limit s) (Q : is_limit t)
(w : F ≅ G) : iso.inv (cone_points_iso_of_nat_iso P Q w) = map t P (iso.inv w) :=
Eq.refl (iso.inv (cone_points_iso_of_nat_iso P Q w))
theorem cone_points_iso_of_nat_iso_hom_comp_assoc {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} {s : cone F} {t : cone G} (P : is_limit s) (Q : is_limit t)
(w : F ≅ G) (j : J) {X' : C} (f' : functor.obj G j ⟶ X') :
iso.hom (cone_points_iso_of_nat_iso P Q w) ≫ nat_trans.app (cone.π t) j ≫ f' =
nat_trans.app (cone.π s) j ≫ nat_trans.app (iso.hom w) j ≫ f' :=
sorry
theorem cone_points_iso_of_nat_iso_inv_comp_assoc {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} {s : cone F} {t : cone G} (P : is_limit s) (Q : is_limit t)
(w : F ≅ G) (j : J) {X' : C} (f' : functor.obj F j ⟶ X') :
iso.inv (cone_points_iso_of_nat_iso P Q w) ≫ nat_trans.app (cone.π s) j ≫ f' =
nat_trans.app (cone.π t) j ≫ nat_trans.app (iso.inv w) j ≫ f' :=
sorry
theorem lift_comp_cone_points_iso_of_nat_iso_hom_assoc {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} {r : cone F} {s : cone F} {t : cone G} (P : is_limit s)
(Q : is_limit t) (w : F ≅ G) {X' : C} (f' : cone.X t ⟶ X') :
lift P r ≫ iso.hom (cone_points_iso_of_nat_iso P Q w) ≫ f' = map r Q (iso.hom w) ≫ f' :=
sorry
/--
If `s : cone F` is a limit cone, so is `s` whiskered by an equivalence `e`.
-/
def whisker_equivalence {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} {s : cone F} (P : is_limit s) (e : K ≌ J) :
is_limit (cone.whisker (equivalence.functor e) s) :=
of_right_adjoint (equivalence.functor (cones.whiskering_equivalence e)) P
/--
We can prove two cone points `(s : cone F).X` and `(t.cone F).X` are isomorphic if
* both cones are limit cones
* their indexing categories are equivalent via some `e : J ≌ K`,
* the triangle of functors commutes up to a natural isomorphism: `e.functor ⋙ G ≅ F`.
This is the most general form of uniqueness of cone points,
allowing relabelling of both the indexing category (up to equivalence)
and the functor (up to natural isomorphism).
-/
@[simp] theorem cone_points_iso_of_equivalence_hom {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} {s : cone F} {G : K ⥤ C} {t : cone G}
(P : is_limit s) (Q : is_limit t) (e : J ≌ K) (w : equivalence.functor e ⋙ G ≅ F) :
iso.hom (cone_points_iso_of_equivalence P Q e w) =
lift Q
(functor.obj
(equivalence.functor
(cones.equivalence_of_reindexing (equivalence.symm e)
(iso.symm (iso_whisker_left (equivalence.inverse e) w) ≪≫
equivalence.inv_fun_id_assoc e G)))
s) :=
Eq.refl (iso.hom (cone_points_iso_of_equivalence P Q e w))
/-- The universal property of a limit cone: a map `W ⟶ X` is the same as
a cone on `F` with vertex `W`. -/
def hom_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cone F}
(h : is_limit t) (W : C) : (W ⟶ cone.X t) ≅ functor.obj (functor.const J) W ⟶ F :=
iso.mk (fun (f : W ⟶ cone.X t) => cone.π (cone.extend t f))
fun (π : functor.obj (functor.const J) W ⟶ F) => lift h (cone.mk W π)
@[simp] theorem hom_iso_hom {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cone F} (h : is_limit t) {W : C} (f : W ⟶ cone.X t) :
iso.hom (hom_iso h W) f = cone.π (cone.extend t f) :=
rfl
/-- The limit of `F` represents the functor taking `W` to
the set of cones on `F` with vertex `W`. -/
def nat_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cone F}
(h : is_limit t) : functor.obj yoneda (cone.X t) ≅ functor.cones F :=
nat_iso.of_components (fun (W : Cᵒᵖ) => hom_iso h (opposite.unop W)) sorry
/--
Another, more explicit, formulation of the universal property of a limit cone.
See also `hom_iso`.
-/
def hom_iso' {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cone F}
(h : is_limit t) (W : C) :
(W ⟶ cone.X t) ≅
Subtype
fun (p : (j : J) → W ⟶ functor.obj F j) =>
∀ {j j' : J} (f : j ⟶ j'), p j ≫ functor.map F f = p j' :=
hom_iso h W ≪≫
iso.mk
(fun (π : functor.obj (functor.const J) W ⟶ F) =>
{ val := fun (j : J) => nat_trans.app π j, property := sorry })
fun
(p :
Subtype
fun (p : (j : J) → W ⟶ functor.obj F j) =>
∀ {j j' : J} (f : j ⟶ j'), p j ≫ functor.map F f = p j') =>
nat_trans.mk fun (j : J) => subtype.val p j
/-- If G : C → D is a faithful functor which sends t to a limit cone,
then it suffices to check that the induced maps for the image of t
can be lifted to maps of C. -/
def of_faithful {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cone F}
{D : Type u'} [category D] (G : C ⥤ D) [faithful G] (ht : is_limit (functor.map_cone G t))
(lift : (s : cone F) → cone.X s ⟶ cone.X t)
(h : ∀ (s : cone F), functor.map G (lift s) = lift ht (functor.map_cone G s)) : is_limit t :=
mk lift
/--
If `F` and `G` are naturally isomorphic, then `F.map_cone c` being a limit implies
`G.map_cone c` is also a limit.
-/
def map_cone_equiv {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'}
[category D] {K : J ⥤ C} {F : C ⥤ D} {G : C ⥤ D} (h : F ≅ G) {c : cone K}
(t : is_limit (functor.map_cone F c)) : is_limit (functor.map_cone G c) :=
coe_fn (postcompose_inv_equiv (iso_whisker_left K h) (functor.map_cone G c))
(of_iso_limit t (iso.symm (functor.postcompose_whisker_left_map_cone (iso.symm h) c)))
/--
A cone is a limit cone exactly if
there is a unique cone morphism from any other cone.
-/
def iso_unique_cone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cone F} : is_limit t ≅ (s : cone F) → unique (s ⟶ t) :=
iso.mk
(fun (h : is_limit t) (s : cone F) => unique.mk { default := lift_cone_morphism h s } sorry)
fun (h : (s : cone F) → unique (s ⟶ t)) =>
mk fun (s : cone F) => cone_morphism.hom Inhabited.default
namespace of_nat_iso
/-- If `F.cones` is represented by `X`, each morphism `f : Y ⟶ X` gives a cone with cone point `Y`. -/
def cone_of_hom {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj yoneda X ≅ functor.cones F) {Y : C} (f : Y ⟶ X) : cone F :=
cone.mk Y (nat_trans.app (iso.hom h) (opposite.op Y) f)
/-- If `F.cones` is represented by `X`, each cone `s` gives a morphism `s.X ⟶ X`. -/
def hom_of_cone {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj yoneda X ≅ functor.cones F) (s : cone F) : cone.X s ⟶ X :=
nat_trans.app (iso.inv h) (opposite.op (cone.X s)) (cone.π s)
@[simp] theorem cone_of_hom_of_cone {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {X : C} (h : functor.obj yoneda X ≅ functor.cones F) (s : cone F) :
cone_of_hom h (hom_of_cone h s) = s :=
sorry
@[simp] theorem hom_of_cone_of_hom {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {X : C} (h : functor.obj yoneda X ≅ functor.cones F) {Y : C} (f : Y ⟶ X) :
hom_of_cone h (cone_of_hom h f) = f :=
congr_fun (congr_fun (congr_arg nat_trans.app (iso.hom_inv_id h)) (opposite.op Y)) f
/-- If `F.cones` is represented by `X`, the cone corresponding to the identity morphism on `X`
will be a limit cone. -/
def limit_cone {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj yoneda X ≅ functor.cones F) : cone F :=
cone_of_hom h 𝟙
/-- If `F.cones` is represented by `X`, the cone corresponding to a morphism `f : Y ⟶ X` is
the limit cone extended by `f`. -/
theorem cone_of_hom_fac {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{X : C} (h : functor.obj yoneda X ≅ functor.cones F) {Y : C} (f : Y ⟶ X) :
cone_of_hom h f = cone.extend (limit_cone h) f :=
sorry
/-- If `F.cones` is represented by `X`, any cone is the extension of the limit cone by the
corresponding morphism. -/
theorem cone_fac {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj yoneda X ≅ functor.cones F) (s : cone F) :
cone.extend (limit_cone h) (hom_of_cone h s) = s :=
sorry
end of_nat_iso
/--
If `F.cones` is representable, then the cone corresponding to the identity morphism on
the representing object is a limit cone.
-/
def of_nat_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj yoneda X ≅ functor.cones F) : is_limit (of_nat_iso.limit_cone h) :=
mk fun (s : cone F) => sorry
end is_limit
/--
A cocone `t` on `F` is a colimit cocone if each cocone on `F` admits a unique
cocone morphism from `t`.
See https://stacks.math.columbia.edu/tag/002F.
-/
structure is_colimit {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
(t : cocone F)
where
desc : (s : cocone F) → cocone.X t ⟶ cocone.X s
fac' :
autoParam
(∀ (s : cocone F) (j : J),
nat_trans.app (cocone.ι t) j ≫ desc s = nat_trans.app (cocone.ι s) j)
(Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring "Mathlib.obviously")
(Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous "Mathlib") "obviously") [])
uniq' :
autoParam
(∀ (s : cocone F) (m : cocone.X t ⟶ cocone.X s),
(∀ (j : J), nat_trans.app (cocone.ι t) j ≫ m = nat_trans.app (cocone.ι s) j) → m = desc s)
(Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring "Mathlib.obviously")
(Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous "Mathlib") "obviously") [])
@[simp] theorem is_colimit.fac {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone F} (c : is_colimit t) (s : cocone F) (j : J) :
nat_trans.app (cocone.ι t) j ≫ is_colimit.desc c s = nat_trans.app (cocone.ι s) j :=
sorry
@[simp] theorem is_colimit.fac_assoc {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {t : cocone F} (c : is_colimit t) (s : cocone F) (j : J) {X' : C}
(f' : cocone.X s ⟶ X') :
nat_trans.app (cocone.ι t) j ≫ is_colimit.desc c s ≫ f' = nat_trans.app (cocone.ι s) j ≫ f' :=
sorry
theorem is_colimit.uniq {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone F} (c : is_colimit t) (s : cocone F) (m : cocone.X t ⟶ cocone.X s)
(w : ∀ (j : J), nat_trans.app (cocone.ι t) j ≫ m = nat_trans.app (cocone.ι s) j) :
m = is_colimit.desc c s :=
sorry
namespace is_colimit
protected instance subsingleton {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {t : cocone F} : subsingleton (is_colimit t) :=
sorry
/-- Given a natural transformation `α : F ⟶ G`, we give a morphism from the cocone point
of a colimit cocone over `F` to the cocone point of any cocone over `G`. -/
def map {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {G : J ⥤ C}
{s : cocone F} (P : is_colimit s) (t : cocone G) (α : F ⟶ G) : cocone.X s ⟶ cocone.X t :=
desc P (functor.obj (cocones.precompose α) t)
@[simp] theorem ι_map {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} {c : cocone F} (hc : is_colimit c) (d : cocone G) (α : F ⟶ G) (j : J) :
nat_trans.app (cocone.ι c) j ≫ map hc d α = nat_trans.app α j ≫ nat_trans.app (cocone.ι d) j :=
fac hc (functor.obj (cocones.precompose α) d) j
@[simp] theorem desc_self {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone F} (h : is_colimit t) : desc h t = 𝟙 :=
Eq.symm (uniq h t 𝟙 fun (j : J) => category.comp_id (nat_trans.app (cocone.ι t) j))
/- Repackaging the definition in terms of cocone morphisms. -/
/-- The universal morphism from a colimit cocone to any other cocone. -/
def desc_cocone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone F} (h : is_colimit t) (s : cocone F) : t ⟶ s :=
cocone_morphism.mk (desc h s)
theorem uniq_cocone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{s : cocone F} {t : cocone F} (h : is_colimit t) {f : t ⟶ s} {f' : t ⟶ s} : f = f' :=
(fun (this : ∀ {g : t ⟶ s}, g = desc_cocone_morphism h s) => Eq.trans this (Eq.symm this))
fun (g : t ⟶ s) =>
cocone_morphism.ext g (desc_cocone_morphism h s)
(uniq h s (cocone_morphism.hom g) (cocone_morphism.w g))
/--
Alternative constructor for `is_colimit`,
providing a morphism of cocones rather than a morphism between the cocone points
and separately the factorisation condition.
-/
def mk_cocone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone F} (desc : (s : cocone F) → t ⟶ s)
(uniq' : ∀ (s : cocone F) (m : t ⟶ s), m = desc s) : is_colimit t :=
mk fun (s : cocone F) => cocone_morphism.hom (desc s)
/-- Colimit cocones on `F` are unique up to isomorphism. -/
@[simp] theorem unique_up_to_iso_inv {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {s : cocone F} {t : cocone F} (P : is_colimit s) (Q : is_colimit t) :
iso.inv (unique_up_to_iso P Q) = desc_cocone_morphism Q s :=
Eq.refl (iso.inv (unique_up_to_iso P Q))
/-- Any cocone morphism between colimit cocones is an isomorphism. -/
def hom_is_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {s : cocone F}
{t : cocone F} (P : is_colimit s) (Q : is_colimit t) (f : s ⟶ t) : is_iso f :=
is_iso.mk (desc_cocone_morphism Q s)
/-- Colimits of `F` are unique up to isomorphism. -/
def cocone_point_unique_up_to_iso {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {s : cocone F} {t : cocone F} (P : is_colimit s) (Q : is_colimit t) :
cocone.X s ≅ cocone.X t :=
functor.map_iso (cocones.forget F) (unique_up_to_iso P Q)
@[simp] theorem comp_cocone_point_unique_up_to_iso_hom {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {s : cocone F} {t : cocone F} (P : is_colimit s) (Q : is_colimit t)
(j : J) :
nat_trans.app (cocone.ι s) j ≫ iso.hom (cocone_point_unique_up_to_iso P Q) =
nat_trans.app (cocone.ι t) j :=
cocone_morphism.w (iso.hom (unique_up_to_iso P Q)) j
@[simp] theorem comp_cocone_point_unique_up_to_iso_inv {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {s : cocone F} {t : cocone F} (P : is_colimit s) (Q : is_colimit t)
(j : J) :
nat_trans.app (cocone.ι t) j ≫ iso.inv (cocone_point_unique_up_to_iso P Q) =
nat_trans.app (cocone.ι s) j :=
cocone_morphism.w (iso.inv (unique_up_to_iso P Q)) j
@[simp] theorem cocone_point_unique_up_to_iso_hom_desc_assoc {J : Type v} [small_category J]
{C : Type u} [category C] {F : J ⥤ C} {r : cocone F} {s : cocone F} {t : cocone F}
(P : is_colimit s) (Q : is_colimit t) {X' : C} (f' : cocone.X r ⟶ X') :
iso.hom (cocone_point_unique_up_to_iso P Q) ≫ desc Q r ≫ f' = desc P r ≫ f' :=
sorry
@[simp] theorem cocone_point_unique_up_to_iso_inv_desc {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {r : cocone F} {s : cocone F} {t : cocone F} (P : is_colimit s)
(Q : is_colimit t) : iso.inv (cocone_point_unique_up_to_iso P Q) ≫ desc P r = desc Q r :=
sorry
/-- Transport evidence that a cocone is a colimit cocone across an isomorphism of cocones. -/
def of_iso_colimit {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{r : cocone F} {t : cocone F} (P : is_colimit r) (i : r ≅ t) : is_colimit t :=
mk_cocone_morphism (fun (s : cocone F) => iso.inv i ≫ desc_cocone_morphism P s) sorry
@[simp] theorem of_iso_colimit_desc {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {r : cocone F} {t : cocone F} (P : is_colimit r) (i : r ≅ t) (s : cocone F) :
desc (of_iso_colimit P i) s = cocone_morphism.hom (iso.inv i) ≫ desc P s :=
rfl
/-- Isomorphism of cocones preserves whether or not they are colimiting cocones. -/
def equiv_iso_colimit {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{r : cocone F} {t : cocone F} (i : r ≅ t) : is_colimit r ≃ is_colimit t :=
equiv.mk (fun (h : is_colimit r) => of_iso_colimit h i)
(fun (h : is_colimit t) => of_iso_colimit h (iso.symm i)) sorry sorry
@[simp] theorem equiv_iso_colimit_apply {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {r : cocone F} {t : cocone F} (i : r ≅ t) (P : is_colimit r) :
coe_fn (equiv_iso_colimit i) P = of_iso_colimit P i :=
rfl
@[simp] theorem equiv_iso_colimit_symm_apply {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {r : cocone F} {t : cocone F} (i : r ≅ t) (P : is_colimit t) :
coe_fn (equiv.symm (equiv_iso_colimit i)) P = of_iso_colimit P (iso.symm i) :=
rfl
/--
If the canonical morphism to a cocone point from a colimiting cocone point is an iso, then the
first cocone was colimiting also.
-/
def of_point_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{r : cocone F} {t : cocone F} (P : is_colimit r) [i : is_iso (desc P t)] : is_colimit t :=
of_iso_colimit P (as_iso (desc_cocone_morphism P t))
theorem hom_desc {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone F} (h : is_colimit t) {W : C} (m : cocone.X t ⟶ W) :
m = desc h (cocone.mk W (nat_trans.mk fun (b : J) => nat_trans.app (cocone.ι t) b ≫ m)) :=
uniq h (cocone.mk W (nat_trans.mk fun (b : J) => nat_trans.app (cocone.ι t) b ≫ m)) m
fun (b : J) => rfl
/-- Two morphisms out of a colimit are equal if their compositions with
each cocone morphism are equal. -/
theorem hom_ext {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cocone F}
(h : is_colimit t) {W : C} {f : cocone.X t ⟶ W} {f' : cocone.X t ⟶ W}
(w : ∀ (j : J), nat_trans.app (cocone.ι t) j ≫ f = nat_trans.app (cocone.ι t) j ≫ f') :
f = f' :=
sorry
/--
Given a left adjoint functor between categories of cocones,
the image of a colimit cocone is a colimit cocone.
-/
def of_left_adjoint {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} {D : Type u'} [category D] {G : K ⥤ D} (h : cocone G ⥤ cocone F)
[is_left_adjoint h] {c : cocone G} (t : is_colimit c) : is_colimit (functor.obj h c) :=
mk_cocone_morphism
(fun (s : cocone F) =>
coe_fn (equiv.symm (adjunction.hom_equiv (adjunction.of_left_adjoint h) c s))
(desc_cocone_morphism t (functor.obj (right_adjoint h) s)))
sorry
/--
Given two functors which have equivalent categories of cocones,
we can transport a colimiting cocone across the equivalence.
-/
def of_cocone_equiv {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} {D : Type u'} [category D] {G : K ⥤ D} (h : cocone G ≌ cocone F)
{c : cocone G} : is_colimit (functor.obj (equivalence.functor h) c) ≃ is_colimit c :=
equiv.mk
(fun (P : is_colimit (functor.obj (equivalence.functor h) c)) =>
of_iso_colimit (of_left_adjoint (equivalence.inverse h) P)
(iso.app (iso.symm (equivalence.unit_iso h)) c))
(of_left_adjoint (equivalence.functor h)) sorry sorry
@[simp] theorem of_cocone_equiv_apply_desc {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} {D : Type u'} [category D] {G : K ⥤ D}
(h : cocone G ≌ cocone F) {c : cocone G}
(P : is_colimit (functor.obj (equivalence.functor h) c)) (s : cocone G) :
desc (coe_fn (of_cocone_equiv h) P) s =
cocone_morphism.hom (nat_trans.app (equivalence.unit h) c) ≫
cocone_morphism.hom
(functor.map (equivalence.inverse h)
(desc_cocone_morphism P (functor.obj (equivalence.functor h) s))) ≫
cocone_morphism.hom (nat_trans.app (equivalence.unit_inv h) s) :=
rfl
@[simp] theorem of_cocone_equiv_symm_apply_desc {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} {D : Type u'} [category D] {G : K ⥤ D}
(h : cocone G ≌ cocone F) {c : cocone G} (P : is_colimit c) (s : cocone F) :
desc (coe_fn (equiv.symm (of_cocone_equiv h)) P) s =
cocone_morphism.hom
(functor.map (equivalence.functor h)
(desc_cocone_morphism P (functor.obj (equivalence.inverse h) s))) ≫
cocone_morphism.hom (nat_trans.app (equivalence.counit h) s) :=
rfl
/--
A cocone precomposed with a natural isomorphism is a colimit cocone
if and only if the original cocone is.
-/
def precompose_hom_equiv {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} (α : F ≅ G) (c : cocone G) :
is_colimit (functor.obj (cocones.precompose (iso.hom α)) c) ≃ is_colimit c :=
of_cocone_equiv (cocones.precompose_equivalence α)
/--
A cocone precomposed with the inverse of a natural isomorphism is a colimit cocone
if and only if the original cocone is.
-/
def precompose_inv_equiv {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} (α : F ≅ G) (c : cocone F) :
is_colimit (functor.obj (cocones.precompose (iso.inv α)) c) ≃ is_colimit c :=
precompose_hom_equiv (iso.symm α) c
/--
The cocone points of two colimit cocones for naturally isomorphic functors
are themselves isomorphic.
-/
@[simp] theorem cocone_points_iso_of_nat_iso_inv {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} {s : cocone F} {t : cocone G} (P : is_colimit s)
(Q : is_colimit t) (w : F ≅ G) :
iso.inv (cocone_points_iso_of_nat_iso P Q w) = map Q s (iso.inv w) :=
Eq.refl (iso.inv (cocone_points_iso_of_nat_iso P Q w))
theorem comp_cocone_points_iso_of_nat_iso_hom {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} {s : cocone F} {t : cocone G} (P : is_colimit s)
(Q : is_colimit t) (w : F ≅ G) (j : J) :
nat_trans.app (cocone.ι s) j ≫ iso.hom (cocone_points_iso_of_nat_iso P Q w) =
nat_trans.app (iso.hom w) j ≫ nat_trans.app (cocone.ι t) j :=
sorry
theorem comp_cocone_points_iso_of_nat_iso_inv_assoc {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} {s : cocone F} {t : cocone G} (P : is_colimit s)
(Q : is_colimit t) (w : F ≅ G) (j : J) {X' : C} (f' : cocone.X s ⟶ X') :
nat_trans.app (cocone.ι t) j ≫ iso.inv (cocone_points_iso_of_nat_iso P Q w) ≫ f' =
nat_trans.app (iso.inv w) j ≫ nat_trans.app (cocone.ι s) j ≫ f' :=
sorry
theorem cocone_points_iso_of_nat_iso_hom_desc {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} {s : cocone F} {r : cocone G} {t : cocone G}
(P : is_colimit s) (Q : is_colimit t) (w : F ≅ G) :
iso.hom (cocone_points_iso_of_nat_iso P Q w) ≫ desc Q r = map P r (iso.hom w) :=
sorry
/--
If `s : cone F` is a limit cone, so is `s` whiskered by an equivalence `e`.
-/
def whisker_equivalence {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} {s : cocone F} (P : is_colimit s) (e : K ≌ J) :
is_colimit (cocone.whisker (equivalence.functor e) s) :=
of_left_adjoint (equivalence.functor (cocones.whiskering_equivalence e)) P
/--
We can prove two cocone points `(s : cocone F).X` and `(t.cocone F).X` are isomorphic if
* both cocones are colimit ccoones
* their indexing categories are equivalent via some `e : J ≌ K`,
* the triangle of functors commutes up to a natural isomorphism: `e.functor ⋙ G ≅ F`.
This is the most general form of uniqueness of cocone points,
allowing relabelling of both the indexing category (up to equivalence)
and the functor (up to natural isomorphism).
-/
@[simp] theorem cocone_points_iso_of_equivalence_hom {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} {s : cocone F} {G : K ⥤ C}
{t : cocone G} (P : is_colimit s) (Q : is_colimit t) (e : J ≌ K)
(w : equivalence.functor e ⋙ G ≅ F) :
iso.hom (cocone_points_iso_of_equivalence P Q e w) =
desc P (functor.obj (equivalence.functor (cocones.equivalence_of_reindexing e w)) t) :=
Eq.refl (iso.hom (cocone_points_iso_of_equivalence P Q e w))
/-- The universal property of a colimit cocone: a map `X ⟶ W` is the same as
a cocone on `F` with vertex `W`. -/
def hom_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cocone F}
(h : is_colimit t) (W : C) : (cocone.X t ⟶ W) ≅ F ⟶ functor.obj (functor.const J) W :=
iso.mk (fun (f : cocone.X t ⟶ W) => cocone.ι (cocone.extend t f))
fun (ι : F ⟶ functor.obj (functor.const J) W) => desc h (cocone.mk W ι)
@[simp] theorem hom_iso_hom {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone F} (h : is_colimit t) {W : C} (f : cocone.X t ⟶ W) :
iso.hom (hom_iso h W) f = cocone.ι (cocone.extend t f) :=
rfl
/-- The colimit of `F` represents the functor taking `W` to
the set of cocones on `F` with vertex `W`. -/
def nat_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cocone F}
(h : is_colimit t) : functor.obj coyoneda (opposite.op (cocone.X t)) ≅ functor.cocones F :=
nat_iso.of_components (hom_iso h) sorry
/--
Another, more explicit, formulation of the universal property of a colimit cocone.
See also `hom_iso`.
-/
def hom_iso' {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cocone F}
(h : is_colimit t) (W : C) :
(cocone.X t ⟶ W) ≅
Subtype
fun (p : (j : J) → functor.obj F j ⟶ W) =>
∀ {j j' : J} (f : j ⟶ j'), functor.map F f ≫ p j' = p j :=
hom_iso h W ≪≫
iso.mk
(fun (ι : F ⟶ functor.obj (functor.const J) W) =>
{ val := fun (j : J) => nat_trans.app ι j, property := sorry })
fun
(p :
Subtype
fun (p : (j : J) → functor.obj F j ⟶ W) =>
∀ {j j' : J} (f : j ⟶ j'), functor.map F f ≫ p j' = p j) =>
nat_trans.mk fun (j : J) => subtype.val p j
/-- If G : C → D is a faithful functor which sends t to a colimit cocone,
then it suffices to check that the induced maps for the image of t
can be lifted to maps of C. -/
def of_faithful {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cocone F}
{D : Type u'} [category D] (G : C ⥤ D) [faithful G] (ht : is_colimit (functor.map_cocone G t))
(desc : (s : cocone F) → cocone.X t ⟶ cocone.X s)
(h : ∀ (s : cocone F), functor.map G (desc s) = desc ht (functor.map_cocone G s)) :
is_colimit t :=
mk desc
/--
If `F` and `G` are naturally isomorphic, then `F.map_cone c` being a colimit implies
`G.map_cone c` is also a colimit.
-/
def map_cocone_equiv {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'}
[category D] {K : J ⥤ C} {F : C ⥤ D} {G : C ⥤ D} (h : F ≅ G) {c : cocone K}
(t : is_colimit (functor.map_cocone F c)) : is_colimit (functor.map_cocone G c) :=
of_iso_colimit
(coe_fn (equiv.symm (precompose_inv_equiv (iso_whisker_left K h) (functor.map_cocone F c))) t)
(functor.precompose_whisker_left_map_cocone h c)
/--
A cocone is a colimit cocone exactly if
there is a unique cocone morphism from any other cocone.
-/
def iso_unique_cocone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone F} : is_colimit t ≅ (s : cocone F) → unique (t ⟶ s) :=
iso.mk
(fun (h : is_colimit t) (s : cocone F) =>
unique.mk { default := desc_cocone_morphism h s } sorry)
fun (h : (s : cocone F) → unique (t ⟶ s)) =>
mk fun (s : cocone F) => cocone_morphism.hom Inhabited.default
namespace of_nat_iso
/-- If `F.cocones` is corepresented by `X`, each morphism `f : X ⟶ Y` gives a cocone with cone point `Y`. -/
def cocone_of_hom {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj coyoneda (opposite.op X) ≅ functor.cocones F) {Y : C} (f : X ⟶ Y) : cocone F :=
cocone.mk Y (nat_trans.app (iso.hom h) Y f)
/-- If `F.cocones` is corepresented by `X`, each cocone `s` gives a morphism `X ⟶ s.X`. -/
def hom_of_cocone {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj coyoneda (opposite.op X) ≅ functor.cocones F) (s : cocone F) :
X ⟶ cocone.X s :=
nat_trans.app (iso.inv h) (cocone.X s) (cocone.ι s)
@[simp] theorem cocone_of_hom_of_cocone {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {X : C} (h : functor.obj coyoneda (opposite.op X) ≅ functor.cocones F)
(s : cocone F) : cocone_of_hom h (hom_of_cocone h s) = s :=
sorry
@[simp] theorem hom_of_cocone_of_hom {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {X : C} (h : functor.obj coyoneda (opposite.op X) ≅ functor.cocones F) {Y : C}
(f : X ⟶ Y) : hom_of_cocone h (cocone_of_hom h f) = f :=
congr_fun (congr_fun (congr_arg nat_trans.app (iso.hom_inv_id h)) Y) f
/-- If `F.cocones` is corepresented by `X`, the cocone corresponding to the identity morphism on `X`
will be a colimit cocone. -/
def colimit_cocone {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj coyoneda (opposite.op X) ≅ functor.cocones F) : cocone F :=
cocone_of_hom h 𝟙
/-- If `F.cocones` is corepresented by `X`, the cocone corresponding to a morphism `f : Y ⟶ X` is
the colimit cocone extended by `f`. -/
theorem cocone_of_hom_fac {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{X : C} (h : functor.obj coyoneda (opposite.op X) ≅ functor.cocones F) {Y : C} (f : X ⟶ Y) :
cocone_of_hom h f = cocone.extend (colimit_cocone h) f :=
sorry
/-- If `F.cocones` is corepresented by `X`, any cocone is the extension of the colimit cocone by the
corresponding morphism. -/
theorem cocone_fac {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj coyoneda (opposite.op X) ≅ functor.cocones F) (s : cocone F) :
cocone.extend (colimit_cocone h) (hom_of_cocone h s) = s :=
sorry
end of_nat_iso
/--
If `F.cocones` is corepresentable, then the cocone corresponding to the identity morphism on
the representing object is a colimit cocone.
-/
def of_nat_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {X : C}
(h : functor.obj coyoneda (opposite.op X) ≅ functor.cocones F) :
is_colimit (of_nat_iso.colimit_cocone h) :=
mk fun (s : cocone F) => sorry
end is_colimit
/-- `limit_cone F` contains a cone over `F` together with the information that it is a limit. -/
structure limit_cone {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) where
cone : cone F
is_limit : is_limit cone
/-- `has_limit F` represents the mere existence of a limit for `F`. -/
class has_limit {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) where
mk' :: (exists_limit : Nonempty (limit_cone F))
theorem has_limit.mk {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
(d : limit_cone F) : has_limit F :=
has_limit.mk' (Nonempty.intro d)
/-- Use the axiom of choice to extract explicit `limit_cone F` from `has_limit F`. -/
def get_limit_cone {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_limit F] : limit_cone F :=
Classical.choice has_limit.exists_limit
/-- `C` has limits of shape `J` if there exists a limit for every functor `F : J ⥤ C`. -/
class has_limits_of_shape (J : Type v) [small_category J] (C : Type u) [category C] where
has_limit : ∀ (F : J ⥤ C), has_limit F
/-- `C` has all (small) limits if it has limits of every shape. -/
class has_limits (C : Type u) [category C] where
has_limits_of_shape : ∀ (J : Type v) [𝒥 : small_category J], has_limits_of_shape J C
protected instance has_limit_of_has_limits_of_shape {C : Type u} [category C] {J : Type v}
[small_category J] [H : has_limits_of_shape J C] (F : J ⥤ C) : has_limit F :=
has_limits_of_shape.has_limit F
protected instance has_limits_of_shape_of_has_limits {C : Type u} [category C] {J : Type v}
[small_category J] [H : has_limits C] : has_limits_of_shape J C :=
has_limits.has_limits_of_shape J
/- Interface to the `has_limit` class. -/
/-- An arbitrary choice of limit cone for a functor. -/
def limit.cone {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) [has_limit F] :
cone F :=
limit_cone.cone (get_limit_cone F)
/-- An arbitrary choice of limit object of a functor. -/
def limit {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) [has_limit F] : C :=
cone.X sorry
/-- The projection from the limit object to a value of the functor. -/
def limit.π {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) [has_limit F]
(j : J) : limit F ⟶ functor.obj F j :=
nat_trans.app (cone.π (limit.cone F)) j
@[simp] theorem limit.cone_X {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_limit F] : cone.X (limit.cone F) = limit F :=
rfl
@[simp] theorem limit.cone_π {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_limit F] : nat_trans.app (cone.π (limit.cone F)) = limit.π F :=
rfl
@[simp] theorem limit.w {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_limit F] {j : J} {j' : J} (f : j ⟶ j') : limit.π F j ≫ functor.map F f = limit.π F j' :=
cone.w (limit.cone F) f
/-- Evidence that the arbitrary choice of cone provied by `limit.cone F` is a limit cone. -/
def limit.is_limit {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_limit F] : is_limit (limit.cone F) :=
limit_cone.is_limit (get_limit_cone F)
/-- The morphism from the cone point of any other cone to the limit object. -/
def limit.lift {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) [has_limit F]
(c : cone F) : cone.X c ⟶ limit F :=
is_limit.lift (limit.is_limit F) c
@[simp] theorem limit.is_limit_lift {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_limit F] (c : cone F) : is_limit.lift (limit.is_limit F) c = limit.lift F c :=
rfl
@[simp] theorem limit.lift_π {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_limit F] (c : cone F) (j : J) :
limit.lift F c ≫ limit.π F j = nat_trans.app (cone.π c) j :=
is_limit.fac (limit.is_limit F) c j
/--
Functoriality of limits.
Usually this morphism should be accessed through `lim.map`,
but may be needed separately when you have specified limits for the source and target functors,
but not necessarily for all functors of shape `J`.
-/
def lim_map {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {G : J ⥤ C}
[has_limit F] [has_limit G] (α : F ⟶ G) : limit F ⟶ limit G :=
is_limit.map (limit.cone F) (limit.is_limit G) α
@[simp] theorem lim_map_π {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} [has_limit F] [has_limit G] (α : F ⟶ G) (j : J) :
lim_map α ≫ limit.π G j = limit.π F j ≫ nat_trans.app α j :=
limit.lift_π (functor.obj (cones.postcompose α) (limit.cone F)) j
/-- The cone morphism from any cone to the arbitrary choice of limit cone. -/
def limit.cone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_limit F] (c : cone F) : c ⟶ limit.cone F :=
is_limit.lift_cone_morphism (limit.is_limit F) c
@[simp] theorem limit.cone_morphism_hom {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_limit F] (c : cone F) :
cone_morphism.hom (limit.cone_morphism c) = limit.lift F c :=
rfl
theorem limit.cone_morphism_π {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_limit F] (c : cone F) (j : J) :
cone_morphism.hom (limit.cone_morphism c) ≫ limit.π F j = nat_trans.app (cone.π c) j :=
sorry
@[simp] theorem limit.cone_point_unique_up_to_iso_hom_comp {J : Type v} [small_category J]
{C : Type u} [category C] {F : J ⥤ C} [has_limit F] {c : cone F} (hc : is_limit c) (j : J) :
iso.hom (is_limit.cone_point_unique_up_to_iso hc (limit.is_limit F)) ≫ limit.π F j =
nat_trans.app (cone.π c) j :=
is_limit.cone_point_unique_up_to_iso_hom_comp hc (limit.is_limit F) j
@[simp] theorem limit.cone_point_unique_up_to_iso_inv_comp {J : Type v} [small_category J]
{C : Type u} [category C] {F : J ⥤ C} [has_limit F] {c : cone F} (hc : is_limit c) (j : J) :
iso.inv (is_limit.cone_point_unique_up_to_iso (limit.is_limit F) hc) ≫ limit.π F j =
nat_trans.app (cone.π c) j :=
is_limit.cone_point_unique_up_to_iso_inv_comp (limit.is_limit F) hc j
/--
Given any other limit cone for `F`, the chosen `limit F` is isomorphic to the cone point.
-/
def limit.iso_limit_cone {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_limit F] (t : limit_cone F) : limit F ≅ cone.X (limit_cone.cone t) :=
is_limit.cone_point_unique_up_to_iso (limit.is_limit F) (limit_cone.is_limit t)
@[simp] theorem limit.iso_limit_cone_hom_π_assoc {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} [has_limit F] (t : limit_cone F) (j : J) {X' : C}
(f' : functor.obj F j ⟶ X') :
iso.hom (limit.iso_limit_cone t) ≫ nat_trans.app (cone.π (limit_cone.cone t)) j ≫ f' =
limit.π F j ≫ f' :=
sorry
@[simp] theorem limit.iso_limit_cone_inv_π {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_limit F] (t : limit_cone F) (j : J) :
iso.inv (limit.iso_limit_cone t) ≫ limit.π F j = nat_trans.app (cone.π (limit_cone.cone t)) j :=
sorry
theorem limit.hom_ext {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_limit F] {X : C} {f : X ⟶ limit F} {f' : X ⟶ limit F}
(w : ∀ (j : J), f ≫ limit.π F j = f' ≫ limit.π F j) : f = f' :=
is_limit.hom_ext (limit.is_limit F) w
@[simp] theorem limit.lift_map {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} [has_limit F] [has_limit G] (c : cone F) (α : F ⟶ G) :
limit.lift F c ≫ lim_map α = limit.lift G (functor.obj (cones.postcompose α) c) :=
sorry
@[simp] theorem limit.lift_cone {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_limit F] : limit.lift F (limit.cone F) = 𝟙 :=
is_limit.lift_self (limit.is_limit F)
/--
The isomorphism (in `Type`) between
morphisms from a specified object `W` to the limit object,
and cones with cone point `W`.
-/
def limit.hom_iso {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_limit F] (W : C) : (W ⟶ limit F) ≅ functor.obj (functor.cones F) (opposite.op W) :=
is_limit.hom_iso (limit.is_limit F) W
@[simp] theorem limit.hom_iso_hom {J : Type v} [small_category J] {C : Type u} [category C]
(F : J ⥤ C) [has_limit F] {W : C} (f : W ⟶ limit F) :
iso.hom (limit.hom_iso F W) f = functor.map (functor.const J) f ≫ cone.π (limit.cone F) :=
is_limit.hom_iso_hom (limit.is_limit F) f
/--
The isomorphism (in `Type`) between
morphisms from a specified object `W` to the limit object,
and an explicit componentwise description of cones with cone point `W`.
-/
def limit.hom_iso' {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_limit F] (W : C) :
(W ⟶ limit F) ≅
Subtype
fun (p : (j : J) → W ⟶ functor.obj F j) =>
∀ {j j' : J} (f : j ⟶ j'), p j ≫ functor.map F f = p j' :=
is_limit.hom_iso' (limit.is_limit F) W
theorem limit.lift_extend {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_limit F] (c : cone F) {X : C} (f : X ⟶ cone.X c) :
limit.lift F (cone.extend c f) = f ≫ limit.lift F c :=
sorry
/--
If a functor `F` has a limit, so does any naturally isomorphic functor.
-/
theorem has_limit_of_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} [has_limit F] (α : F ≅ G) : has_limit G :=
has_limit.mk
(limit_cone.mk (functor.obj (cones.postcompose (iso.hom α)) (limit.cone F))
(is_limit.mk
fun (s : cone G) => limit.lift F (functor.obj (cones.postcompose (iso.inv α)) s)))
/-- If a functor `G` has the same collection of cones as a functor `F`
which has a limit, then `G` also has a limit. -/
-- See the construction of limits from products and equalizers
-- for an example usage.
theorem has_limit.of_cones_iso {C : Type u} [category C] {J : Type v} {K : Type v}
[small_category J] [small_category K] (F : J ⥤ C) (G : K ⥤ C)
(h : functor.cones F ≅ functor.cones G) [has_limit F] : has_limit G :=
has_limit.mk
(limit_cone.mk (is_limit.of_nat_iso.limit_cone (is_limit.nat_iso (limit.is_limit F) ≪≫ h))
(is_limit.of_nat_iso (is_limit.nat_iso (limit.is_limit F) ≪≫ h)))
/--
The limits of `F : J ⥤ C` and `G : J ⥤ C` are isomorphic,
if the functors are naturally isomorphic.
-/
def has_limit.iso_of_nat_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} [has_limit F] [has_limit G] (w : F ≅ G) : limit F ≅ limit G :=
is_limit.cone_points_iso_of_nat_iso (limit.is_limit F) (limit.is_limit G) w
@[simp] theorem has_limit.iso_of_nat_iso_hom_π {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} [has_limit F] [has_limit G] (w : F ≅ G) (j : J) :
iso.hom (has_limit.iso_of_nat_iso w) ≫ limit.π G j =
limit.π F j ≫ nat_trans.app (iso.hom w) j :=
is_limit.cone_points_iso_of_nat_iso_hom_comp (limit.is_limit F) (limit.is_limit G) w j
@[simp] theorem has_limit.lift_iso_of_nat_iso_hom_assoc {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} [has_limit F] [has_limit G] (t : cone F) (w : F ≅ G)
{X' : C} (f' : limit G ⟶ X') :
limit.lift F t ≫ iso.hom (has_limit.iso_of_nat_iso w) ≫ f' =
limit.lift G (functor.obj (cones.postcompose (iso.hom w)) t) ≫ f' :=
sorry
/--
The limits of `F : J ⥤ C` and `G : K ⥤ C` are isomorphic,
if there is an equivalence `e : J ≌ K` making the triangle commute up to natural isomorphism.
-/
def has_limit.iso_of_equivalence {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] {F : J ⥤ C} [has_limit F] {G : K ⥤ C} [has_limit G] (e : J ≌ K)
(w : equivalence.functor e ⋙ G ≅ F) : limit F ≅ limit G :=
is_limit.cone_points_iso_of_equivalence (limit.is_limit F) (limit.is_limit G) e w
@[simp] theorem has_limit.iso_of_equivalence_hom_π {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} [has_limit F] {G : K ⥤ C} [has_limit G]
(e : J ≌ K) (w : equivalence.functor e ⋙ G ≅ F) (k : K) :
iso.hom (has_limit.iso_of_equivalence e w) ≫ limit.π G k =
limit.π F (functor.obj (equivalence.inverse e) k) ≫
nat_trans.app (iso.inv w) (functor.obj (equivalence.inverse e) k) ≫
functor.map G (nat_trans.app (equivalence.counit e) k) :=
sorry
@[simp] theorem has_limit.iso_of_equivalence_inv_π {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} [has_limit F] {G : K ⥤ C} [has_limit G]
(e : J ≌ K) (w : equivalence.functor e ⋙ G ≅ F) (j : J) :
iso.inv (has_limit.iso_of_equivalence e w) ≫ limit.π F j =
limit.π G (functor.obj (equivalence.functor e) j) ≫ nat_trans.app (iso.hom w) j :=
sorry
/--
The canonical morphism from the limit of `F` to the limit of `E ⋙ F`.
-/
def limit.pre {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] (F : J ⥤ C) [has_limit F] (E : K ⥤ J) [has_limit (E ⋙ F)] :
limit F ⟶ limit (E ⋙ F) :=
limit.lift (E ⋙ F) (cone.whisker E (limit.cone F))
@[simp] theorem limit.pre_π {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] (F : J ⥤ C) [has_limit F] (E : K ⥤ J) [has_limit (E ⋙ F)] (k : K) :
limit.pre F E ≫ limit.π (E ⋙ F) k = limit.π F (functor.obj E k) :=
sorry
@[simp] theorem limit.lift_pre {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] (F : J ⥤ C) [has_limit F] (E : K ⥤ J) [has_limit (E ⋙ F)]
(c : cone F) : limit.lift F c ≫ limit.pre F E = limit.lift (E ⋙ F) (cone.whisker E c) :=
sorry
@[simp] theorem limit.pre_pre {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] (F : J ⥤ C) [has_limit F] (E : K ⥤ J) [has_limit (E ⋙ F)] {L : Type v}
[small_category L] (D : L ⥤ K) [has_limit (D ⋙ E ⋙ F)] :
limit.pre F E ≫ limit.pre (E ⋙ F) D = limit.pre F (D ⋙ E) :=
sorry
/---
If we have particular limit cones available for `E ⋙ F` and for `F`,
we obtain a formula for `limit.pre F E`.
-/
theorem limit.pre_eq {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} [has_limit F] {E : K ⥤ J} [has_limit (E ⋙ F)] (s : limit_cone (E ⋙ F))
(t : limit_cone F) :
limit.pre F E =
iso.hom (limit.iso_limit_cone t) ≫
is_limit.lift (limit_cone.is_limit s) (cone.whisker E (limit_cone.cone t)) ≫
iso.inv (limit.iso_limit_cone s) :=
sorry
/--
The canonical morphism from `G` applied to the limit of `F` to the limit of `F ⋙ G`.
-/
def limit.post {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) {D : Type u'}
[category D] [has_limit F] (G : C ⥤ D) [has_limit (F ⋙ G)] :
functor.obj G (limit F) ⟶ limit (F ⋙ G) :=
limit.lift (F ⋙ G) (functor.map_cone G (limit.cone F))
@[simp] theorem limit.post_π_assoc {J : Type v} [small_category J] {C : Type u} [category C]
(F : J ⥤ C) {D : Type u'} [category D] [has_limit F] (G : C ⥤ D) [has_limit (F ⋙ G)] (j : J)
{X' : D} (f' : functor.obj (F ⋙ G) j ⟶ X') :
limit.post F G ≫ limit.π (F ⋙ G) j ≫ f' = functor.map G (limit.π F j) ≫ f' :=
sorry
@[simp] theorem limit.lift_post {J : Type v} [small_category J] {C : Type u} [category C]
(F : J ⥤ C) {D : Type u'} [category D] [has_limit F] (G : C ⥤ D) [has_limit (F ⋙ G)]
(c : cone F) :
functor.map G (limit.lift F c) ≫ limit.post F G = limit.lift (F ⋙ G) (functor.map_cone G c) :=
sorry
@[simp] theorem limit.post_post {J : Type v} [small_category J] {C : Type u} [category C]
(F : J ⥤ C) {D : Type u'} [category D] [has_limit F] (G : C ⥤ D) [has_limit (F ⋙ G)]
{E : Type u''} [category E] (H : D ⥤ E) [has_limit ((F ⋙ G) ⋙ H)] :
functor.map H (limit.post F G) ≫ limit.post (F ⋙ G) H = limit.post F (G ⋙ H) :=
sorry
/- H G (limit F) ⟶ H (limit (F ⋙ G)) ⟶ limit ((F ⋙ G) ⋙ H) equals -/
/- H G (limit F) ⟶ limit (F ⋙ (G ⋙ H)) -/
theorem limit.pre_post {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {D : Type u'} [category D] (E : K ⥤ J) (F : J ⥤ C) (G : C ⥤ D) [has_limit F]
[has_limit (E ⋙ F)] [has_limit (F ⋙ G)] [has_limit ((E ⋙ F) ⋙ G)] :
functor.map G (limit.pre F E) ≫ limit.post (E ⋙ F) G = limit.post F G ≫ limit.pre (F ⋙ G) E :=
sorry
/- G (limit F) ⟶ G (limit (E ⋙ F)) ⟶ limit ((E ⋙ F) ⋙ G) vs -/
/- G (limit F) ⟶ limit F ⋙ G ⟶ limit (E ⋙ (F ⋙ G)) or -/
protected instance has_limit_equivalence_comp {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} (e : K ≌ J) [has_limit F] :
has_limit (equivalence.functor e ⋙ F) :=
has_limit.mk
(limit_cone.mk (cone.whisker (equivalence.functor e) (limit.cone F))
(is_limit.whisker_equivalence (limit.is_limit F) e))
/--
If a `E ⋙ F` has a limit, and `E` is an equivalence, we can construct a limit of `F`.
-/
theorem has_limit_of_equivalence_comp {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} (e : K ≌ J)
[has_limit (equivalence.functor e ⋙ F)] : has_limit F :=
has_limit_of_iso (equivalence.inv_fun_id_assoc e F)
-- `has_limit_comp_equivalence` and `has_limit_of_comp_equivalence`
-- are proved in `category_theory/adjunction/limits.lean`.
/-- `limit F` is functorial in `F`, when `C` has all limits of shape `J`. -/
def lim {J : Type v} [small_category J] {C : Type u} [category C] [has_limits_of_shape J C] :
(J ⥤ C) ⥤ C :=
functor.mk (fun (F : J ⥤ C) => limit F) fun (F G : J ⥤ C) (α : F ⟶ G) => lim_map α
-- We generate this manually since `simps` gives it a weird name.
@[simp] theorem lim_map_eq_lim_map {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_limits_of_shape J C] {G : J ⥤ C} (α : F ⟶ G) : functor.map lim α = lim_map α :=
rfl
theorem limit.map_pre {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} [has_limits_of_shape J C] {G : J ⥤ C} (α : F ⟶ G)
[has_limits_of_shape K C] (E : K ⥤ J) :
functor.map lim α ≫ limit.pre G E = limit.pre F E ≫ functor.map lim (whisker_left E α) :=
sorry
theorem limit.map_pre' {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] [has_limits_of_shape J C] [has_limits_of_shape K C] (F : J ⥤ C) {E₁ : K ⥤ J}
{E₂ : K ⥤ J} (α : E₁ ⟶ E₂) :
limit.pre F E₂ = limit.pre F E₁ ≫ functor.map lim (whisker_right α F) :=
sorry
theorem limit.id_pre {J : Type v} [small_category J] {C : Type u} [category C]
[has_limits_of_shape J C] (F : J ⥤ C) :
limit.pre F 𝟭 = functor.map lim (iso.inv (functor.left_unitor F)) :=
sorry
/- H (limit F) ⟶ H (limit G) ⟶ limit (G ⋙ H) vs
theorem limit.map_post {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_limits_of_shape J C] {G : J ⥤ C} (α : F ⟶ G) {D : Type u'} [category D]
[has_limits_of_shape J D] (H : C ⥤ D) :
functor.map H (lim_map α) ≫ limit.post G H = limit.post F H ≫ lim_map (whisker_right α H) :=
sorry
H (limit F) ⟶ limit (F ⋙ H) ⟶ limit (G ⋙ H) -/
/--
The isomorphism between
morphisms from `W` to the cone point of the limit cone for `F`
and cones over `F` with cone point `W`
is natural in `F`.
-/
def lim_yoneda {J : Type v} [small_category J] {C : Type u} [category C] [has_limits_of_shape J C] :
lim ⋙ yoneda ≅ cones J C :=
nat_iso.of_components
(fun (F : J ⥤ C) =>
nat_iso.of_components (fun (W : Cᵒᵖ) => limit.hom_iso F (opposite.unop W)) sorry)
sorry
/--
We can transport limits of shape `J` along an equivalence `J ≌ J'`.
-/
theorem has_limits_of_shape_of_equivalence {J : Type v} [small_category J] {C : Type u} [category C]
{J' : Type v} [small_category J'] (e : J ≌ J') [has_limits_of_shape J C] :
has_limits_of_shape J' C :=
has_limits_of_shape.mk fun (F : J' ⥤ C) => has_limit_of_equivalence_comp e
/-- `colimit_cocone F` contains a cocone over `F` together with the information that it is a
colimit. -/
structure colimit_cocone {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) where
cocone : cocone F
is_colimit : is_colimit cocone
/-- `has_colimit F` represents the mere existence of a colimit for `F`. -/
class has_colimit {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) where
mk' :: (exists_colimit : Nonempty (colimit_cocone F))
theorem has_colimit.mk {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
(d : colimit_cocone F) : has_colimit F :=
has_colimit.mk' (Nonempty.intro d)
/-- Use the axiom of choice to extract explicit `colimit_cocone F` from `has_colimit F`. -/
def get_colimit_cocone {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_colimit F] : colimit_cocone F :=
Classical.choice has_colimit.exists_colimit
/-- `C` has colimits of shape `J` if there exists a colimit for every functor `F : J ⥤ C`. -/
class has_colimits_of_shape (J : Type v) [small_category J] (C : Type u) [category C] where
has_colimit : ∀ (F : J ⥤ C), has_colimit F
/-- `C` has all (small) colimits if it has colimits of every shape. -/
class has_colimits (C : Type u) [category C] where
has_colimits_of_shape : ∀ (J : Type v) [𝒥 : small_category J], has_colimits_of_shape J C
protected instance has_colimit_of_has_colimits_of_shape {C : Type u} [category C] {J : Type v}
[small_category J] [H : has_colimits_of_shape J C] (F : J ⥤ C) : has_colimit F :=
has_colimits_of_shape.has_colimit F
protected instance has_colimits_of_shape_of_has_colimits {C : Type u} [category C] {J : Type v}
[small_category J] [H : has_colimits C] : has_colimits_of_shape J C :=
has_colimits.has_colimits_of_shape J
/- Interface to the `has_colimit` class. -/
/-- An arbitrary choice of colimit cocone of a functor. -/
def colimit.cocone {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_colimit F] : cocone F :=
colimit_cocone.cocone (get_colimit_cocone F)
/-- An arbitrary choice of colimit object of a functor. -/
def colimit {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) [has_colimit F] :
C :=
cocone.X sorry
/-- The coprojection from a value of the functor to the colimit object. -/
def colimit.ι {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) [has_colimit F]
(j : J) : functor.obj F j ⟶ colimit F :=
nat_trans.app (cocone.ι (colimit.cocone F)) j
@[simp] theorem colimit.cocone_ι {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_colimit F] (j : J) :
nat_trans.app (cocone.ι (colimit.cocone F)) j = colimit.ι F j :=
rfl
@[simp] theorem colimit.cocone_X {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_colimit F] : cocone.X (colimit.cocone F) = colimit F :=
rfl
@[simp] theorem colimit.w {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_colimit F] {j : J} {j' : J} (f : j ⟶ j') :
functor.map F f ≫ colimit.ι F j' = colimit.ι F j :=
cocone.w (colimit.cocone F) f
/-- Evidence that the arbitrary choice of cocone is a colimit cocone. -/
def colimit.is_colimit {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_colimit F] : is_colimit (colimit.cocone F) :=
colimit_cocone.is_colimit (get_colimit_cocone F)
/-- The morphism from the colimit object to the cone point of any other cocone. -/
def colimit.desc {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_colimit F] (c : cocone F) : colimit F ⟶ cocone.X c :=
is_colimit.desc (colimit.is_colimit F) c
@[simp] theorem colimit.is_colimit_desc {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_colimit F] (c : cocone F) :
is_colimit.desc (colimit.is_colimit F) c = colimit.desc F c :=
rfl
/--
We have lots of lemmas describing how to simplify `colimit.ι F j ≫ _`,
and combined with `colimit.ext` we rely on these lemmas for many calculations.
However, since `category.assoc` is a `@[simp]` lemma, often expressions are
right associated, and it's hard to apply these lemmas about `colimit.ι`.
We thus use `reassoc` to define additional `@[simp]` lemmas, with an arbitrary extra morphism.
(see `tactic/reassoc_axiom.lean`)
-/
@[simp] theorem colimit.ι_desc {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_colimit F] (c : cocone F) (j : J) :
colimit.ι F j ≫ colimit.desc F c = nat_trans.app (cocone.ι c) j :=
is_colimit.fac (colimit.is_colimit F) c j
/--
Functoriality of colimits.
Usually this morphism should be accessed through `colim.map`,
but may be needed separately when you have specified colimits for the source and target functors,
but not necessarily for all functors of shape `J`.
-/
def colim_map {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {G : J ⥤ C}
[has_colimit F] [has_colimit G] (α : F ⟶ G) : colimit F ⟶ colimit G :=
is_colimit.map (colimit.is_colimit F) (colimit.cocone G) α
@[simp] theorem ι_colim_map_assoc {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {G : J ⥤ C} [has_colimit F] [has_colimit G] (α : F ⟶ G) (j : J) {X' : C}
(f' : colimit G ⟶ X') :
colimit.ι F j ≫ colim_map α ≫ f' = nat_trans.app α j ≫ colimit.ι G j ≫ f' :=
sorry
/-- The cocone morphism from the arbitrary choice of colimit cocone to any cocone. -/
def colimit.cocone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_colimit F] (c : cocone F) : colimit.cocone F ⟶ c :=
is_colimit.desc_cocone_morphism (colimit.is_colimit F) c
@[simp] theorem colimit.cocone_morphism_hom {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} [has_colimit F] (c : cocone F) :
cocone_morphism.hom (colimit.cocone_morphism c) = colimit.desc F c :=
rfl
theorem colimit.ι_cocone_morphism {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_colimit F] (c : cocone F) (j : J) :
colimit.ι F j ≫ cocone_morphism.hom (colimit.cocone_morphism c) =
nat_trans.app (cocone.ι c) j :=
sorry
@[simp] theorem colimit.comp_cocone_point_unique_up_to_iso_hom {J : Type v} [small_category J]
{C : Type u} [category C] {F : J ⥤ C} [has_colimit F] {c : cocone F} (hc : is_colimit c)
(j : J) :
colimit.ι F j ≫ iso.hom (is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit F) hc) =
nat_trans.app (cocone.ι c) j :=
is_colimit.comp_cocone_point_unique_up_to_iso_hom (colimit.is_colimit F) hc j
@[simp] theorem colimit.comp_cocone_point_unique_up_to_iso_inv_assoc {J : Type v} [small_category J]
{C : Type u} [category C] {F : J ⥤ C} [has_colimit F] {c : cocone F} (hc : is_colimit c) (j : J)
{X' : C} (f' : cocone.X c ⟶ X') :
colimit.ι F j ≫
iso.inv (is_colimit.cocone_point_unique_up_to_iso hc (colimit.is_colimit F)) ≫ f' =
nat_trans.app (cocone.ι c) j ≫ f' :=
sorry
/--
Given any other colimit cocone for `F`, the chosen `colimit F` is isomorphic to the cocone point.
-/
def colimit.iso_colimit_cocone {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_colimit F] (t : colimit_cocone F) : colimit F ≅ cocone.X (colimit_cocone.cocone t) :=
is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit F) (colimit_cocone.is_colimit t)
@[simp] theorem colimit.iso_colimit_cocone_ι_hom {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} [has_colimit F] (t : colimit_cocone F) (j : J) :
colimit.ι F j ≫ iso.hom (colimit.iso_colimit_cocone t) =
nat_trans.app (cocone.ι (colimit_cocone.cocone t)) j :=
sorry
@[simp] theorem colimit.iso_colimit_cocone_ι_inv_assoc {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} [has_colimit F] (t : colimit_cocone F) (j : J) {X' : C}
(f' : colimit F ⟶ X') :
nat_trans.app (cocone.ι (colimit_cocone.cocone t)) j ≫
iso.inv (colimit.iso_colimit_cocone t) ≫ f' =
colimit.ι F j ≫ f' :=
sorry
theorem colimit.hom_ext {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_colimit F] {X : C} {f : colimit F ⟶ X} {f' : colimit F ⟶ X}
(w : ∀ (j : J), colimit.ι F j ≫ f = colimit.ι F j ≫ f') : f = f' :=
is_colimit.hom_ext (colimit.is_colimit F) w
@[simp] theorem colimit.desc_cocone {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_colimit F] : colimit.desc F (colimit.cocone F) = 𝟙 :=
is_colimit.desc_self (colimit.is_colimit F)
/--
The isomorphism (in `Type`) between
morphisms from the colimit object to a specified object `W`,
and cocones with cone point `W`.
-/
def colimit.hom_iso {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_colimit F] (W : C) : (colimit F ⟶ W) ≅ functor.obj (functor.cocones F) W :=
is_colimit.hom_iso (colimit.is_colimit F) W
@[simp] theorem colimit.hom_iso_hom {J : Type v} [small_category J] {C : Type u} [category C]
(F : J ⥤ C) [has_colimit F] {W : C} (f : colimit F ⟶ W) :
iso.hom (colimit.hom_iso F W) f =
cocone.ι (colimit.cocone F) ≫ functor.map (functor.const J) f :=
is_colimit.hom_iso_hom (colimit.is_colimit F) f
/--
The isomorphism (in `Type`) between
morphisms from the colimit object to a specified object `W`,
and an explicit componentwise description of cocones with cone point `W`.
-/
def colimit.hom_iso' {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_colimit F] (W : C) :
(colimit F ⟶ W) ≅
Subtype
fun (p : (j : J) → functor.obj F j ⟶ W) =>
∀ {j j' : J} (f : j ⟶ j'), functor.map F f ≫ p j' = p j :=
is_colimit.hom_iso' (colimit.is_colimit F) W
theorem colimit.desc_extend {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C)
[has_colimit F] (c : cocone F) {X : C} (f : cocone.X c ⟶ X) :
colimit.desc F (cocone.extend c f) = colimit.desc F c ≫ f :=
sorry
/--
If `F` has a colimit, so does any naturally isomorphic functor.
-/
-- This has the isomorphism pointing in the opposite direction than in `has_limit_of_iso`.
-- This is intentional; it seems to help with elaboration.
theorem has_colimit_of_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} [has_colimit F] (α : G ≅ F) : has_colimit G :=
has_colimit.mk
(colimit_cocone.mk (functor.obj (cocones.precompose (iso.hom α)) (colimit.cocone F))
(is_colimit.mk
fun (s : cocone G) => colimit.desc F (functor.obj (cocones.precompose (iso.inv α)) s)))
/-- If a functor `G` has the same collection of cocones as a functor `F`
which has a colimit, then `G` also has a colimit. -/
theorem has_colimit.of_cocones_iso {C : Type u} [category C] {J : Type v} {K : Type v}
[small_category J] [small_category K] (F : J ⥤ C) (G : K ⥤ C)
(h : functor.cocones F ≅ functor.cocones G) [has_colimit F] : has_colimit G :=
has_colimit.mk
(colimit_cocone.mk
(is_colimit.of_nat_iso.colimit_cocone (is_colimit.nat_iso (colimit.is_colimit F) ≪≫ h))
(is_colimit.of_nat_iso (is_colimit.nat_iso (colimit.is_colimit F) ≪≫ h)))
/--
The colimits of `F : J ⥤ C` and `G : J ⥤ C` are isomorphic,
if the functors are naturally isomorphic.
-/
def has_colimit.iso_of_nat_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{G : J ⥤ C} [has_colimit F] [has_colimit G] (w : F ≅ G) : colimit F ≅ colimit G :=
is_colimit.cocone_points_iso_of_nat_iso (colimit.is_colimit F) (colimit.is_colimit G) w
@[simp] theorem has_colimit.iso_of_nat_iso_ι_hom {J : Type v} [small_category J] {C : Type u}
[category C] {F : J ⥤ C} {G : J ⥤ C} [has_colimit F] [has_colimit G] (w : F ≅ G) (j : J) :
colimit.ι F j ≫ iso.hom (has_colimit.iso_of_nat_iso w) =
nat_trans.app (iso.hom w) j ≫ colimit.ι G j :=
is_colimit.comp_cocone_points_iso_of_nat_iso_hom (colimit.is_colimit F) (colimit.is_colimit G) w j
@[simp] theorem has_colimit.iso_of_nat_iso_hom_desc_assoc {J : Type v} [small_category J]
{C : Type u} [category C] {F : J ⥤ C} {G : J ⥤ C} [has_colimit F] [has_colimit G] (t : cocone G)
(w : F ≅ G) {X' : C} (f' : cocone.X t ⟶ X') :
iso.hom (has_colimit.iso_of_nat_iso w) ≫ colimit.desc G t ≫ f' =
colimit.desc F (functor.obj (cocones.precompose (iso.hom w)) t) ≫ f' :=
sorry
/--
The colimits of `F : J ⥤ C` and `G : K ⥤ C` are isomorphic,
if there is an equivalence `e : J ≌ K` making the triangle commute up to natural isomorphism.
-/
def has_colimit.iso_of_equivalence {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] {F : J ⥤ C} [has_colimit F] {G : K ⥤ C} [has_colimit G] (e : J ≌ K)
(w : equivalence.functor e ⋙ G ≅ F) : colimit F ≅ colimit G :=
is_colimit.cocone_points_iso_of_equivalence (colimit.is_colimit F) (colimit.is_colimit G) e w
@[simp] theorem has_colimit.iso_of_equivalence_hom_π {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} [has_colimit F] {G : K ⥤ C}
[has_colimit G] (e : J ≌ K) (w : equivalence.functor e ⋙ G ≅ F) (j : J) :
colimit.ι F j ≫ iso.hom (has_colimit.iso_of_equivalence e w) =
functor.map F (nat_trans.app (equivalence.unit e) j) ≫
nat_trans.app (iso.inv w)
(functor.obj (equivalence.functor e ⋙ equivalence.inverse e) j) ≫
colimit.ι G
(functor.obj (equivalence.functor e)
(functor.obj (equivalence.functor e ⋙ equivalence.inverse e) j)) :=
sorry
@[simp] theorem has_colimit.iso_of_equivalence_inv_π {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} [has_colimit F] {G : K ⥤ C}
[has_colimit G] (e : J ≌ K) (w : equivalence.functor e ⋙ G ≅ F) (k : K) :
colimit.ι G k ≫ iso.inv (has_colimit.iso_of_equivalence e w) =
functor.map G (nat_trans.app (equivalence.counit_inv e) k) ≫
nat_trans.app (iso.hom w) (functor.obj (equivalence.inverse e) k) ≫
colimit.ι F (functor.obj (equivalence.inverse e) k) :=
sorry
/--
The canonical morphism from the colimit of `E ⋙ F` to the colimit of `F`.
-/
def colimit.pre {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] (F : J ⥤ C) [has_colimit F] (E : K ⥤ J) [has_colimit (E ⋙ F)] :
colimit (E ⋙ F) ⟶ colimit F :=
colimit.desc (E ⋙ F) (cocone.whisker E (colimit.cocone F))
@[simp] theorem colimit.ι_pre_assoc {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] (F : J ⥤ C) [has_colimit F] (E : K ⥤ J) [has_colimit (E ⋙ F)] (k : K)
{X' : C} (f' : colimit F ⟶ X') :
colimit.ι (E ⋙ F) k ≫ colimit.pre F E ≫ f' = colimit.ι F (functor.obj E k) ≫ f' :=
sorry
@[simp] theorem colimit.pre_desc {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] (F : J ⥤ C) [has_colimit F] (E : K ⥤ J) [has_colimit (E ⋙ F)]
(c : cocone F) :
colimit.pre F E ≫ colimit.desc F c = colimit.desc (E ⋙ F) (cocone.whisker E c) :=
sorry
@[simp] theorem colimit.pre_pre {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] (F : J ⥤ C) [has_colimit F] (E : K ⥤ J) [has_colimit (E ⋙ F)]
{L : Type v} [small_category L] (D : L ⥤ K) [has_colimit (D ⋙ E ⋙ F)] :
colimit.pre (E ⋙ F) D ≫ colimit.pre F E = colimit.pre F (D ⋙ E) :=
sorry
/---
If we have particular colimit cocones available for `E ⋙ F` and for `F`,
we obtain a formula for `colimit.pre F E`.
-/
theorem colimit.pre_eq {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} [has_colimit F] {E : K ⥤ J} [has_colimit (E ⋙ F)]
(s : colimit_cocone (E ⋙ F)) (t : colimit_cocone F) :
colimit.pre F E =
iso.hom (colimit.iso_colimit_cocone s) ≫
is_colimit.desc (colimit_cocone.is_colimit s)
(cocone.whisker E (colimit_cocone.cocone t)) ≫
iso.inv (colimit.iso_colimit_cocone t) :=
sorry
/--
The canonical morphism from `G` applied to the colimit of `F ⋙ G`
to `G` applied to the colimit of `F`.
-/
def colimit.post {J : Type v} [small_category J] {C : Type u} [category C] (F : J ⥤ C) {D : Type u'}
[category D] [has_colimit F] (G : C ⥤ D) [has_colimit (F ⋙ G)] :
colimit (F ⋙ G) ⟶ functor.obj G (colimit F) :=
colimit.desc (F ⋙ G) (functor.map_cocone G (colimit.cocone F))
@[simp] theorem colimit.ι_post_assoc {J : Type v} [small_category J] {C : Type u} [category C]
(F : J ⥤ C) {D : Type u'} [category D] [has_colimit F] (G : C ⥤ D) [has_colimit (F ⋙ G)] (j : J)
{X' : D} (f' : functor.obj G (colimit F) ⟶ X') :
colimit.ι (F ⋙ G) j ≫ colimit.post F G ≫ f' = functor.map G (colimit.ι F j) ≫ f' :=
sorry
@[simp] theorem colimit.post_desc {J : Type v} [small_category J] {C : Type u} [category C]
(F : J ⥤ C) {D : Type u'} [category D] [has_colimit F] (G : C ⥤ D) [has_colimit (F ⋙ G)]
(c : cocone F) :
colimit.post F G ≫ functor.map G (colimit.desc F c) =
colimit.desc (F ⋙ G) (functor.map_cocone G c) :=
sorry
@[simp] theorem colimit.post_post {J : Type v} [small_category J] {C : Type u} [category C]
(F : J ⥤ C) {D : Type u'} [category D] [has_colimit F] (G : C ⥤ D) [has_colimit (F ⋙ G)]
{E : Type u''} [category E] (H : D ⥤ E) [has_colimit ((F ⋙ G) ⋙ H)] :
colimit.post (F ⋙ G) H ≫ functor.map H (colimit.post F G) = colimit.post F (G ⋙ H) :=
sorry
/- H G (colimit F) ⟶ H (colimit (F ⋙ G)) ⟶ colimit ((F ⋙ G) ⋙ H) equals -/
/- H G (colimit F) ⟶ colimit (F ⋙ (G ⋙ H)) -/
theorem colimit.pre_post {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] {D : Type u'} [category D] (E : K ⥤ J) (F : J ⥤ C) (G : C ⥤ D)
[has_colimit F] [has_colimit (E ⋙ F)] [has_colimit (F ⋙ G)] [has_colimit ((E ⋙ F) ⋙ G)] :
colimit.post (E ⋙ F) G ≫ functor.map G (colimit.pre F E) =
colimit.pre (F ⋙ G) E ≫ colimit.post F G :=
sorry
/- G (colimit F) ⟶ G (colimit (E ⋙ F)) ⟶ colimit ((E ⋙ F) ⋙ G) vs -/
/- G (colimit F) ⟶ colimit F ⋙ G ⟶ colimit (E ⋙ (F ⋙ G)) or -/
protected instance has_colimit_equivalence_comp {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} (e : K ≌ J) [has_colimit F] :
has_colimit (equivalence.functor e ⋙ F) :=
has_colimit.mk
(colimit_cocone.mk (cocone.whisker (equivalence.functor e) (colimit.cocone F))
(is_colimit.whisker_equivalence (colimit.is_colimit F) e))
/--
If a `E ⋙ F` has a colimit, and `E` is an equivalence, we can construct a colimit of `F`.
-/
theorem has_colimit_of_equivalence_comp {J : Type v} {K : Type v} [small_category J]
[small_category K] {C : Type u} [category C] {F : J ⥤ C} (e : K ≌ J)
[has_colimit (equivalence.functor e ⋙ F)] : has_colimit F :=
has_colimit_of_iso (iso.symm (equivalence.inv_fun_id_assoc e F))
/-- `colimit F` is functorial in `F`, when `C` has all colimits of shape `J`. -/
def colim {J : Type v} [small_category J] {C : Type u} [category C] [has_colimits_of_shape J C] :
(J ⥤ C) ⥤ C :=
functor.mk (fun (F : J ⥤ C) => colimit F) fun (F G : J ⥤ C) (α : F ⟶ G) => colim_map α
@[simp] theorem colimit.ι_map_assoc {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_colimits_of_shape J C] {G : J ⥤ C} (α : F ⟶ G) (j : J) {X' : C}
(f' : functor.obj colim G ⟶ X') :
colimit.ι F j ≫ functor.map colim α ≫ f' = nat_trans.app α j ≫ colimit.ι G j ≫ f' :=
sorry
@[simp] theorem colimit.map_desc {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} [has_colimits_of_shape J C] {G : J ⥤ C} (α : F ⟶ G) (c : cocone G) :
functor.map colim α ≫ colimit.desc G c =
colimit.desc F (functor.obj (cocones.precompose α) c) :=
sorry
theorem colimit.pre_map {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u}
[category C] {F : J ⥤ C} [has_colimits_of_shape J C] {G : J ⥤ C} (α : F ⟶ G)
[has_colimits_of_shape K C] (E : K ⥤ J) :
colimit.pre F E ≫ functor.map colim α =
functor.map colim (whisker_left E α) ≫ colimit.pre G E :=
sorry
theorem colimit.pre_map' {J : Type v} {K : Type v} [small_category J] [small_category K]
{C : Type u} [category C] [has_colimits_of_shape J C] [has_colimits_of_shape K C] (F : J ⥤ C)
{E₁ : K ⥤ J} {E₂ : K ⥤ J} (α : E₁ ⟶ E₂) :
colimit.pre F E₁ = functor.map colim (whisker_right α F) ≫ colimit.pre F E₂ :=
sorry
theorem colimit.pre_id {J : Type v} [small_category J] {C : Type u} [category C]
[has_colimits_of_shape J C] (F : J ⥤ C) :
colimit.pre F 𝟭 = functor.map colim (iso.hom (functor.left_unitor F)) :=
sorry
/- H (colimit F) ⟶ H (colimit G) ⟶ colimit (G ⋙ H) vs
theorem colimit.map_post {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
[has_colimits_of_shape J C] {G : J ⥤ C} (α : F ⟶ G) {D : Type u'} [category D]
[has_colimits_of_shape J D] (H : C ⥤ D) :
colimit.post F H ≫ functor.map H (functor.map colim α) =
functor.map colim (whisker_right α H) ≫ colimit.post G H :=
sorry
H (colimit F) ⟶ colimit (F ⋙ H) ⟶ colimit (G ⋙ H) -/
/--
The isomorphism between
morphisms from the cone point of the colimit cocone for `F` to `W`
and cocones over `F` with cone point `W`
is natural in `F`.
-/
def colim_coyoneda {J : Type v} [small_category J] {C : Type u} [category C]
[has_colimits_of_shape J C] : functor.op colim ⋙ coyoneda ≅ cocones J C :=
nat_iso.of_components
(fun (F : J ⥤ Cᵒᵖ) => nat_iso.of_components (colimit.hom_iso (opposite.unop F)) sorry) sorry
/--
We can transport colimits of shape `J` along an equivalence `J ≌ J'`.
-/
theorem has_colimits_of_shape_of_equivalence {J : Type v} [small_category J] {C : Type u}
[category C] {J' : Type v} [small_category J'] (e : J ≌ J') [has_colimits_of_shape J C] :
has_colimits_of_shape J' C :=
has_colimits_of_shape.mk fun (F : J' ⥤ C) => has_colimit_of_equivalence_comp e
/--
If `t : cone F` is a limit cone, then `t.op : cocone F.op` is a colimit cocone.
-/
def is_limit.op {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C} {t : cone F}
(P : is_limit t) : is_colimit (cone.op t) :=
is_colimit.mk fun (s : cocone (functor.op F)) => has_hom.hom.op (is_limit.lift P (cocone.unop s))
/--
If `t : cocone F` is a colimit cocone, then `t.op : cone F.op` is a limit cone.
-/
def is_colimit.op {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone F} (P : is_colimit t) : is_limit (cocone.op t) :=
is_limit.mk fun (s : cone (functor.op F)) => has_hom.hom.op (is_colimit.desc P (cone.unop s))
/--
If `t : cone F.op` is a limit cone, then `t.unop : cocone F` is a colimit cocone.
-/
def is_limit.unop {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cone (functor.op F)} (P : is_limit t) : is_colimit (cone.unop t) :=
is_colimit.mk fun (s : cocone F) => has_hom.hom.unop (is_limit.lift P (cocone.op s))
/--
If `t : cocone F.op` is a colimit cocone, then `t.unop : cone F.` is a limit cone.
-/
def is_colimit.unop {J : Type v} [small_category J] {C : Type u} [category C] {F : J ⥤ C}
{t : cocone (functor.op F)} (P : is_colimit t) : is_limit (cocone.unop t) :=
is_limit.mk fun (s : cone F) => has_hom.hom.unop (is_colimit.desc P (cone.op s))
/--
`t : cone F` is a limit cone if and only is `t.op : cocone F.op` is a colimit cocone.
-/
def is_limit_equiv_is_colimit_op {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {t : cone F} : is_limit t ≃ is_colimit (cone.op t) :=
equiv_of_subsingleton_of_subsingleton is_limit.op
fun (P : is_colimit (cone.op t)) =>
is_limit.of_iso_limit (is_colimit.unop P)
(cones.ext (iso.refl (cone.X (cocone.unop (cone.op t)))) sorry)
/--
`t : cocone F` is a colimit cocone if and only is `t.op : cone F.op` is a limit cone.
-/
def is_colimit_equiv_is_limit_op {J : Type v} [small_category J] {C : Type u} [category C]
{F : J ⥤ C} {t : cocone F} : is_colimit t ≃ is_limit (cocone.op t) :=
equiv_of_subsingleton_of_subsingleton is_colimit.op
fun (P : is_limit (cocone.op t)) =>
is_colimit.of_iso_colimit (is_limit.unop P)
(cocones.ext (iso.refl (cocone.X (cone.unop (cocone.op t)))) sorry)
end Mathlib |
792363af4478f89612b35c196adfa938fbd8d309 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/data/bitvec/basic.lean | 354ec842d1a930fdbac31837ccc39c94d50bcccb | [
"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 | 3,652 | lean | /-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import data.bitvec.core
import data.fin.basic
import tactic.norm_num
import tactic.monotonicity
namespace bitvec
instance (n : ℕ) : preorder (bitvec n) :=
preorder.lift bitvec.to_nat
/-- convert `fin` to `bitvec` -/
def of_fin {n : ℕ} (i : fin $ 2^n) : bitvec n :=
bitvec.of_nat _ i.val
lemma of_fin_val {n : ℕ} (i : fin $ 2^n) : (of_fin i).to_nat = i.val :=
by rw [of_fin,to_nat_of_nat,nat.mod_eq_of_lt]; apply i.is_lt
/-- convert `bitvec` to `fin` -/
def to_fin {n : ℕ} (i : bitvec n) : fin $ 2^n :=
fin.of_nat' i.to_nat
lemma add_lsb_eq_twice_add_one {x b} :
add_lsb x b = 2 * x + cond b 1 0 :=
by simp [add_lsb,two_mul]
lemma to_nat_eq_foldr_reverse {n : ℕ} (v : bitvec n) :
v.to_nat = v.to_list.reverse.foldr (flip add_lsb) 0 :=
by rw [list.foldr_reverse, flip]; refl
lemma to_nat_lt {n : ℕ} (v : bitvec n) : v.to_nat < 2^n :=
begin
suffices : v.to_nat + 1 ≤ 2 ^ n, { simpa },
rw to_nat_eq_foldr_reverse,
cases v with xs h,
dsimp [bitvec.to_nat,bits_to_nat],
rw ← list.length_reverse at h,
generalize_hyp : xs.reverse = ys at ⊢ h, clear xs,
induction ys generalizing n,
{ simp [← h] },
{ simp only [←h, pow_add, flip, list.length, list.foldr, pow_one],
rw [add_lsb_eq_twice_add_one],
transitivity 2 * list.foldr (λ (x : bool) (y : ℕ), add_lsb y x) 0 ys_tl + 2 * 1,
{ ac_mono, rw two_mul, mono, cases ys_hd; simp },
{ rw ← left_distrib, ac_mono, norm_num,
apply ys_ih, refl } },
end
lemma add_lsb_div_two {x b} : add_lsb x b / 2 = x :=
by cases b; simp only [nat.add_mul_div_left, add_lsb, ←two_mul, add_comm, nat.succ_pos',
nat.mul_div_right, gt_iff_lt, zero_add, cond]; norm_num
lemma to_bool_add_lsb_mod_two {x b} : to_bool (add_lsb x b % 2 = 1) = b :=
by cases b; simp only [to_bool_iff, nat.add_mul_mod_self_left, add_lsb, ←two_mul, add_comm,
bool.to_bool_false, nat.mul_mod_right, zero_add, cond, zero_ne_one]; norm_num
lemma of_nat_to_nat {n : ℕ} (v : bitvec n) : bitvec.of_nat _ v.to_nat = v :=
begin
cases v with xs h,
ext1,
change vector.to_list _ = xs,
dsimp [bitvec.to_nat,bits_to_nat],
rw ← list.length_reverse at h,
rw [← list.reverse_reverse xs,list.foldl_reverse],
generalize_hyp : xs.reverse = ys at ⊢ h, clear xs,
induction ys generalizing n,
{ cases h, simp [bitvec.of_nat] },
{ simp only [←nat.succ_eq_add_one, list.length] at h, subst n,
simp only [bitvec.of_nat, vector.to_list_cons, vector.to_list_nil, list.reverse_cons,
vector.to_list_append, list.foldr],
erw [add_lsb_div_two, to_bool_add_lsb_mod_two],
congr, apply ys_ih, refl }
end
lemma to_fin_val {n : ℕ} (v : bitvec n) : (to_fin v : ℕ) = v.to_nat :=
by rw [to_fin, fin.coe_of_nat_eq_mod', nat.mod_eq_of_lt]; apply to_nat_lt
lemma to_fin_le_to_fin_of_le {n} {v₀ v₁ : bitvec n} (h : v₀ ≤ v₁) : v₀.to_fin ≤ v₁.to_fin :=
show (v₀.to_fin : ℕ) ≤ v₁.to_fin,
by rw [to_fin_val,to_fin_val]; exact h
lemma of_fin_le_of_fin_of_le {n : ℕ} {i j : fin (2^n)} (h : i ≤ j) : of_fin i ≤ of_fin j :=
show (bitvec.of_nat n i).to_nat ≤ (bitvec.of_nat n j).to_nat,
by { simp only [to_nat_of_nat, nat.mod_eq_of_lt, fin.is_lt], exact h }
lemma to_fin_of_fin {n} (i : fin $ 2^n) : (of_fin i).to_fin = i :=
fin.eq_of_veq (by simp [to_fin_val, of_fin, to_nat_of_nat, nat.mod_eq_of_lt, i.is_lt])
lemma of_fin_to_fin {n} (v : bitvec n) : of_fin (to_fin v) = v :=
by dsimp [of_fin]; rw [to_fin_val, of_nat_to_nat]
end bitvec
|
2fafb771744de8dfab48ad789df0fddd2d1c0bac | 86f6f4f8d827a196a32bfc646234b73328aeb306 | /examples/logic/unnamed_36.lean | 0debe9d82e9f5771e407e9217e7f575ab0928442 | [] | no_license | jamescheuk91/mathematics_in_lean | 09f1f87d2b0dce53464ff0cbe592c568ff59cf5e | 4452499264e2975bca2f42565c0925506ba5dda3 | refs/heads/master | 1,679,716,410,967 | 1,613,957,947,000 | 1,613,957,947,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 137 | lean | import data.real.basic
-- BEGIN
#check ∀ x y ε : ℝ,
0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε
-- END |
b3d3c77c95accba7fe4f5c7624295dcd39db8069 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/order/filter/cofinite.lean | 5d48989dd924b2c83eac4b26f63c373a985a4c70 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,069 | 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 Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.order.filter.at_top_bot
import Mathlib.PostPort
universes u_1
namespace Mathlib
/-!
# 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.
-/
namespace filter
/-- The cofinite filter is the filter of subsets whose complements are finite. -/
def cofinite {α : Type u_1} : filter α :=
mk (set_of fun (s : set α) => set.finite (sᶜ)) sorry sorry sorry
@[simp] theorem mem_cofinite {α : Type u_1} {s : set α} : s ∈ cofinite ↔ set.finite (sᶜ) :=
iff.rfl
@[simp] theorem eventually_cofinite {α : Type u_1} {p : α → Prop} : filter.eventually (fun (x : α) => p x) cofinite ↔ set.finite (set_of fun (x : α) => ¬p x) :=
iff.rfl
protected instance cofinite_ne_bot {α : Type u_1} [infinite α] : ne_bot cofinite :=
mt (iff.mpr empty_in_sets_eq_bot)
(eq.mpr
(id
((fun (a a_1 : Prop) (e_1 : a = a_1) => congr_arg Not e_1) (∅ ∈ cofinite) (set.finite set.univ)
(Eq.trans (propext mem_cofinite)
((fun (s s_1 : set α) (e_1 : s = s_1) => congr_arg set.finite e_1) (∅ᶜ) set.univ set.compl_empty))))
set.infinite_univ)
theorem frequently_cofinite_iff_infinite {α : Type u_1} {p : α → Prop} : filter.frequently (fun (x : α) => p x) cofinite ↔ set.infinite (set_of fun (x : α) => p x) := sorry
end filter
theorem set.finite.compl_mem_cofinite {α : Type u_1} {s : set α} (hs : set.finite s) : sᶜ ∈ filter.cofinite :=
iff.mpr filter.mem_cofinite (Eq.symm (compl_compl s) ▸ hs)
theorem set.finite.eventually_cofinite_nmem {α : Type u_1} {s : set α} (hs : set.finite s) : filter.eventually (fun (x : α) => ¬x ∈ s) filter.cofinite :=
set.finite.compl_mem_cofinite hs
theorem finset.eventually_cofinite_nmem {α : Type u_1} (s : finset α) : filter.eventually (fun (x : α) => ¬x ∈ s) filter.cofinite :=
set.finite.eventually_cofinite_nmem (finset.finite_to_set s)
theorem set.infinite_iff_frequently_cofinite {α : Type u_1} {s : set α} : set.infinite s ↔ filter.frequently (fun (x : α) => x ∈ s) filter.cofinite :=
iff.symm filter.frequently_cofinite_iff_infinite
theorem filter.eventually_cofinite_ne {α : Type u_1} (x : α) : filter.eventually (fun (a : α) => a ≠ x) filter.cofinite :=
set.finite.eventually_cofinite_nmem (set.finite_singleton x)
/-- For natural numbers the filters `cofinite` and `at_top` coincide. -/
theorem nat.cofinite_eq_at_top : filter.cofinite = filter.at_top := sorry
theorem nat.frequently_at_top_iff_infinite {p : ℕ → Prop} : filter.frequently (fun (n : ℕ) => p n) filter.at_top ↔ set.infinite (set_of fun (n : ℕ) => p n) := sorry
|
435e1d3182a42d6ee670d2184df8dedc3d657edf | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /tests/lean/run/opaque_hint_bug.lean | 3c60c133e89e4c594ac53a80d4a3cae8ff7d2d26 | [
"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 | 236 | lean | inductive List (T : Type) : Type
| nil {} : List
| cons : T → List → List
section
variable {T : Type}
definition concat (s t : List T) : List T
:= List.rec t (fun x l u, List.cons x u) s
attribute concat [reducible]
end
|
2fd7b28441059586f955f44c2b64d11cafde9aca | b794ca1df49bc5a3bd3fd5552eed3bc4f63b8b93 | /src/mywork/practice_1.lean | 42045c8f59f0e44062770a378143e9f5e07ad6db | [] | no_license | akraisinger/cs2120f21 | 8235ac98375e04ffcec504cff5cab7833ee69e54 | 4ef83d7151bb6a284028092aa4f1d509c0eb8237 | refs/heads/main | 1,691,714,771,612 | 1,632,889,465,000 | 1,632,889,465,000 | 399,946,508 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 6,204 | lean | /-
EQUALITY (due before class WED! (9/15/2021))
-/
/- #1
Suppose that x, y, z, and w are arbitrary objects of some type,
T; and suppose further that we know (have proofs of the facts)
that x = y, y = z, and w = z. Give a very, very short English
proof of the conjecture that z = w. You can use not only the
axioms of equality, but either of the theorems about properties
of equality that we have proven. Hint: There's something about
this question that makes it much easier to answer than it might
at first appear.
-/
/- The conjecture z = w is true via the symmetric property of equality. -/
/- #2
Give a formal statement of the conjecture (proposition) from
#1 by filling in the "hole" in the following definition. The
def is a keyword. The name you're binding to your proposition
is prop_1. The type of the value is Prop (which is the type of
all propositions in Lean).
-/
def prop_1 : Prop :=
∀ (T : Type)
(x y z w : T),
w = z →
z = w
/- #3 (extra credit)
Give a formal proof of the proposition from #2 by filling in
the hole in this next definition. Hint: Use Lean's versions of
the axioms and basic theorems concerning equality. They are,
again, called eq.refl, eq.subst, eq.symm, eq.trans.
-/
theorem prop_1_proof : prop_1 :=
begin
assume T w z,
assume h1 h2,
apply eq.symm,
end
/-
FOR ALL: ∀.
-/
/- #4
Give a very brief explanation in English of the introduction
rule for ∀. For example, suppose you need to prove (∀ x, P x);
what do you do? (I'm being a little informal in leaving out the
type of X.)
-/
/- When dealing with ∀ statements, you must make an assumption
that if you are able to prove it true for an abitrary object
of that type, then it is true for all objects of that nature. -/
/- #5
Suppose you have a proof, let's call it pf, of the proposition,
(∀ x, P x), and you need a proof of P t, for some particular t.
Write an expression then uses the elimination rule for ∀ to get
such a proof. Complete the answer by replacing the underscores
in the following expression: (∀ t, P t).
-/
/-
IMPLIES: →
In the "code" that follows, we define two predicates, each
taking one natural number as an argument. We call them ev and
odd. When applied to any value, n, ev yields the proposition
that n is even (n % 2 = 0), while odd yields the proposition
that n is odd (n % 2 = 1).
-/
def ev (n : ℕ) := n % 2 = 0
def odd (n : ℕ) := n % 2 = 1
/- #6
Write a formal version of the proposition that, for *any*
natural number n, *if* n is even, *then* n + 1 is odd. Give
your answer by filling the hole in the following definition.
Hint: put parenthesis around "n + 1" in your answer.
-/
def successor_of_even_is_odd : Prop :=
∀ (n : ℕ),
ev n →
odd (n + 1)
/- #7
Suppose that "its_raining" and "the_streets_are_wet" are
propositions. (We formalize these assumptions as axioms in
what follows. Then give a formal definition of the (larger)
proposition, "if it's raining out then the streets are wet")
by filling in the hole
-/
axioms (raining streets_wet : Prop)
axiom if_raining_then_streets_wet :
raining →
streets_wet
/- #9
Now suppose that in addition, its_raining is true, and
we have a proof of it, pf_its_raining. Again, we again give
you this assumption formally as an axiom below. Finish
the formal proof that the streets must be wet. Hint: here
you are asked to use the elimination rule for →.
-/
axiom pf_raining : raining
example : streets_wet :=
begin
apply if_raining_then_streets_wet,
apply pf_raining,
end
/-
AND: ∧
-/
/- #10
In our last class, we proved that "∧ is *commutative*."
That is, for any given *propositions*, P and Q, (P ∧ Q) →
(Q ∧ P). The way we proved it was to *assume* that we're
given such a P, Q, and proof, pq, of (P ∧ Q) -- applying
the introduction rules for ∀ and →). In this context, we
*use* the proof, pq, to derive separate proofs, let's call
them p, a proof of P, and q, a proof of Q. With these in
hand, we then apply the introduction rule for ∧ to put
them back together into a proof of (Q ∧ P). We give you
a formal version of this proof as a reminder, next.
-/
theorem and_commutative : ∀ (P Q : Prop), P ∧ Q → Q ∧ P :=
begin
assume P Q pq,
apply and.intro _ _,
exact (and.elim_right pq),
exact (and.elim_left pq),
end
/-
Your task now is to prove the theorem, "∧ is *associative*."
What this means is that for arbitrary propositions, P, Q, and
R, if (P ∧ (Q ∧ R)) is true, then ((P ∧ Q) ∧ R) is true, *and
vice versa*. You just need to prove it in the first direction.
Hint, if you have a proof, p_qr, of (P ∧ (Q ∧ R)), then the
application of and.elim_left will give you a proof of P, and
and.elim_right will give you a proof of (Q ∧ R).
To help you along, we give you the first part of the proof,
including an example of a new Lean tactic called have, which
allows you to give a name to a new value in the middle of a
proof script.
-/
theorem and_associative :
∀ (P Q R : Prop),
(P ∧ (Q ∧ R)) → ((P ∧ Q) ∧ R) :=
begin
intros P Q R h,
have p : P := and.elim_left h,
have qr := and.elim_right h,
apply and.intro,
apply and.intro,
exact p,
exact and.elim_left qr,
exact and.elim_right qr,
end
/- #11
Give an English language proof of the preceding
theorem. Do it by finishing off the following
partial "proof explanation."
Proof. We assume that P, Q, and R are arbitrary
but specific propositions, and that we have a
proof, let's call it p_qr, of (P ∧ (Q ∧ R)) [by
application of ∧ and → introduction.] What now
remains to be proved is ((P ∧ Q) ∧ R). We can
construct a proof of this proposition by applying
_introduction rule of and_ to a proof of (P ∧ Q)
and a proof of R. What remains, then, is to obtain
these proofs. But this is easily done by the application of
_the elimination rule of and_ to _P ∧ Q ∧ R_. QED.
-/
/-
Note that Lean includes versions of these
theorems (and many, many, many others) in
its extensive library of formalized maths,
as the following check commands reveal.
Note the difference in naming relative to
the definitions we give in this file.
-/
#check @and.comm
#check @and.assoc |
0876925fb3bce8aba28a07d905b8280389405e6c | 82e44445c70db0f03e30d7be725775f122d72f3e | /src/data/mv_polynomial/comm_ring.lean | 1133761292e2b80a1304ae20ff18e68610fcf0b8 | [
"Apache-2.0"
] | permissive | stjordanis/mathlib | 51e286d19140e3788ef2c470bc7b953e4991f0c9 | 2568d41bca08f5d6bf39d915434c8447e21f42ee | refs/heads/master | 1,631,748,053,501 | 1,627,938,886,000 | 1,627,938,886,000 | 228,728,358 | 0 | 0 | Apache-2.0 | 1,576,630,588,000 | 1,576,630,587,000 | null | UTF-8 | Lean | false | false | 5,008 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import data.mv_polynomial.variables
/-!
# Multivariate polynomials over a ring
Many results about polynomials hold when the coefficient ring is a commutative semiring.
Some stronger results can be derived when we assume this semiring is a ring.
This file does not define any new operations, but proves some of these stronger results.
## Notation
As in other polynomial files, we typically use the notation:
+ `σ : Type*` (indexing the variables)
+ `R : Type*` `[comm_ring R]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `mv_polynomial σ R` which mathematicians might call `X^s`
+ `a : R`
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : mv_polynomial σ R`
-/
noncomputable theory
open_locale classical big_operators
open set function finsupp add_monoid_algebra
open_locale big_operators
universes u v
variables {R : Type u} {S : Type v}
namespace mv_polynomial
variables {σ : Type*} {a a' a₁ a₂ : R} {e : ℕ} {n m : σ} {s : σ →₀ ℕ}
section comm_ring
variable [comm_ring R]
variables {p q : mv_polynomial σ R}
instance : comm_ring (mv_polynomial σ R) := add_monoid_algebra.comm_ring
variables (σ a a')
@[simp] lemma C_sub : (C (a - a') : mv_polynomial σ R) = C a - C a' := ring_hom.map_sub _ _ _
@[simp] lemma C_neg : (C (-a) : mv_polynomial σ R) = -C a := ring_hom.map_neg _ _
@[simp] lemma coeff_neg (m : σ →₀ ℕ) (p : mv_polynomial σ R) :
coeff m (-p) = -coeff m p := finsupp.neg_apply _ _
@[simp] lemma coeff_sub (m : σ →₀ ℕ) (p q : mv_polynomial σ R) :
coeff m (p - q) = coeff m p - coeff m q := finsupp.sub_apply _ _ _
@[simp] lemma monomial_neg (m : σ →₀ ℕ) (a : R) : -(monomial m a) = monomial m (-a) :=
single_neg.symm
@[simp] lemma monomial_sub (m : σ →₀ ℕ) (a b : R) :
monomial m a - monomial m b = monomial m (a - b) :=
single_sub.symm
@[simp] lemma support_neg : (- p).support = p.support :=
finsupp.support_neg
variables {σ} (p)
section degrees
lemma degrees_neg (p : mv_polynomial σ R) : (- p).degrees = p.degrees :=
by rw [degrees, support_neg]; refl
lemma degrees_sub (p q : mv_polynomial σ R) :
(p - q).degrees ≤ p.degrees ⊔ q.degrees :=
by simpa only [sub_eq_add_neg] using le_trans (degrees_add p (-q)) (by rw degrees_neg)
end degrees
section vars
variables (p q)
@[simp] lemma vars_neg : (-p).vars = p.vars :=
by simp [vars, degrees_neg]
lemma vars_sub_subset : (p - q).vars ⊆ p.vars ∪ q.vars :=
by convert vars_add_subset p (-q) using 2; simp [sub_eq_add_neg]
variables {p q}
@[simp]
lemma vars_sub_of_disjoint (hpq : disjoint p.vars q.vars) : (p - q).vars = p.vars ∪ q.vars :=
begin
rw ←vars_neg q at hpq,
convert vars_add_of_disjoint hpq using 2;
simp [sub_eq_add_neg]
end
end vars
section eval₂
variables [comm_ring S]
variables (f : R →+* S) (g : σ → S)
@[simp] lemma eval₂_sub : (p - q).eval₂ f g = p.eval₂ f g - q.eval₂ f g :=
(eval₂_hom f g).map_sub _ _
@[simp] lemma eval₂_neg : (-p).eval₂ f g = -(p.eval₂ f g) := (eval₂_hom f g).map_neg _
lemma hom_C (f : mv_polynomial σ ℤ →+* S) (n : ℤ) : f (C n) = (n : S) :=
(f.comp C).eq_int_cast n
/-- A ring homomorphism f : Z[X_1, X_2, ...] → R
is determined by the evaluations f(X_1), f(X_2), ... -/
@[simp] lemma eval₂_hom_X {R : Type u} (c : ℤ →+* S)
(f : mv_polynomial R ℤ →+* S) (x : mv_polynomial R ℤ) :
eval₂ c (f ∘ X) x = f x :=
mv_polynomial.induction_on x
(λ n, by { rw [hom_C f, eval₂_C], exact c.eq_int_cast n })
(λ p q hp hq, by { rw [eval₂_add, hp, hq], exact (f.map_add _ _).symm })
(λ p n hp, by { rw [eval₂_mul, eval₂_X, hp], exact (f.map_mul _ _).symm })
/-- Ring homomorphisms out of integer polynomials on a type `σ` are the same as
functions out of the type `σ`, -/
def hom_equiv : (mv_polynomial σ ℤ →+* S) ≃ (σ → S) :=
{ to_fun := λ f, ⇑f ∘ X,
inv_fun := λ f, eval₂_hom (int.cast_ring_hom S) f,
left_inv := λ f, ring_hom.ext $ eval₂_hom_X _ _,
right_inv := λ f, funext $ λ x, by simp only [coe_eval₂_hom, function.comp_app, eval₂_X] }
end eval₂
section total_degree
@[simp] lemma total_degree_neg (a : mv_polynomial σ R) :
(-a).total_degree = a.total_degree :=
by simp only [total_degree, support_neg]
lemma total_degree_sub (a b : mv_polynomial σ R) :
(a - b).total_degree ≤ max a.total_degree b.total_degree :=
calc (a - b).total_degree = (a + -b).total_degree : by rw sub_eq_add_neg
... ≤ max a.total_degree (-b).total_degree : total_degree_add a (-b)
... = max a.total_degree b.total_degree : by rw total_degree_neg
end total_degree
end comm_ring
end mv_polynomial
|
e38562cca0f1bff2728e5f8d34a081a76e4c59a3 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/algebraic_geometry/limits.lean | 21a2b67b861308832ea10473751697b1fc746a7a | [
"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 | 4,637 | lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import algebraic_geometry.pullbacks
import algebraic_geometry.AffineScheme
import category_theory.limits.constructions.finite_products_of_binary_products
import category_theory.limits.constructions.equalizers
/-!
# (Co)Limits of Schemes
We construct various limits and colimits in the category of schemes.
* The existence of fibred products was shown in `algebraic_geometry/pullbacks.lean`.
* `Spec ℤ` is the terminal object.
* The preceding two results imply that `Scheme` has all finite limits.
* The empty scheme is the (strict) initial object.
## Todo
* Coproducts exists (and the forgetful functors preserve them).
-/
universe u
open category_theory category_theory.limits opposite topological_space
namespace algebraic_geometry
/-- `Spec ℤ` is the terminal object in the category of schemes. -/
noncomputable
def Spec_Z_is_terminal : is_terminal (Scheme.Spec.obj (op $ CommRing.of ℤ)) :=
@@is_terminal.is_terminal_obj _ _ Scheme.Spec _ infer_instance
(terminal_op_of_initial CommRing.Z_is_initial)
instance : has_terminal Scheme := has_terminal_of_has_terminal_of_preserves_limit Scheme.Spec
instance : is_affine (⊤_ Scheme.{u}) :=
is_affine_of_iso (preserves_terminal.iso Scheme.Spec).inv
instance : has_finite_limits Scheme :=
has_finite_limits_of_has_terminal_and_pullbacks
section initial
/-- The map from the empty scheme. -/
@[simps]
def Scheme.empty_to (X : Scheme.{u}) : ∅ ⟶ X :=
⟨{ base := ⟨λ x, pempty.elim x, by continuity⟩,
c := { app := λ U, CommRing.punit_is_terminal.from _ } }, λ x, pempty.elim x⟩
@[ext]
lemma Scheme.empty_ext {X : Scheme.{u}} (f g : ∅ ⟶ X) : f = g :=
by { ext a, exact pempty.elim a }
lemma Scheme.eq_empty_to {X : Scheme.{u}} (f : ∅ ⟶ X) : f = Scheme.empty_to X :=
Scheme.empty_ext f (Scheme.empty_to X)
instance (X : Scheme.{u}) : unique (∅ ⟶ X) :=
⟨⟨Scheme.empty_to _⟩, λ _, Scheme.empty_ext _ _⟩
/-- The empty scheme is the initial object in the category of schemes. -/
def empty_is_initial : is_initial (∅ : Scheme.{u}) :=
is_initial.of_unique _
@[simp]
lemma empty_is_initial_to : empty_is_initial.to = Scheme.empty_to := rfl
instance : is_empty Scheme.empty.carrier :=
show is_empty pempty, by apply_instance
instance Spec_punit_is_empty : is_empty (Scheme.Spec.obj (op $ CommRing.of punit)).carrier :=
⟨prime_spectrum.punit⟩
@[priority 100]
instance is_open_immersion_of_is_empty {X Y : Scheme} (f : X ⟶ Y) [is_empty X.carrier] :
is_open_immersion f :=
begin
apply_with is_open_immersion.of_stalk_iso { instances := ff },
{ apply open_embedding_of_continuous_injective_open,
{ continuity },
{ rintro (i : X.carrier), exact is_empty_elim i },
{ intros U hU, convert is_open_empty, ext, apply (iff_false _).mpr,
exact λ x, is_empty_elim (show X.carrier, from x.some) } },
{ rintro (i : X.carrier), exact is_empty_elim i }
end
@[priority 100]
instance is_iso_of_is_empty {X Y : Scheme} (f : X ⟶ Y) [is_empty Y.carrier] : is_iso f :=
begin
haveI : is_empty X.carrier := ⟨λ x, is_empty_elim (show Y.carrier, from f.1.base x)⟩,
haveI : epi f.1.base,
{ rw Top.epi_iff_surjective, rintro (x : Y.carrier), exact is_empty_elim x },
apply is_open_immersion.to_iso
end
/-- A scheme is initial if its underlying space is empty . -/
noncomputable
def is_initial_of_is_empty {X : Scheme} [is_empty X.carrier] : is_initial X :=
empty_is_initial.of_iso (as_iso $ empty_is_initial.to _)
/-- `Spec 0` is the initial object in the category of schemes. -/
noncomputable
def Spec_punit_is_initial : is_initial (Scheme.Spec.obj (op $ CommRing.of punit)) :=
empty_is_initial.of_iso (as_iso $ empty_is_initial.to _)
@[priority 100]
instance is_affine_of_is_empty {X : Scheme} [is_empty X.carrier] : is_affine X :=
is_affine_of_iso (inv (empty_is_initial.to X) ≫
empty_is_initial.to (Scheme.Spec.obj (op $ CommRing.of punit)))
instance : has_initial Scheme :=
has_initial_of_unique Scheme.empty
instance initial_is_empty : is_empty (⊥_ Scheme).carrier :=
⟨λ x, ((initial.to Scheme.empty : _).1.base x).elim⟩
lemma bot_is_affine_open (X : Scheme) : is_affine_open (⊥ : opens X.carrier) :=
begin
convert range_is_affine_open_of_open_immersion (initial.to X),
ext,
exact (false_iff _).mpr (λ x, is_empty_elim (show (⊥_ Scheme).carrier, from x.some)),
end
instance : has_strict_initial_objects Scheme :=
has_strict_initial_objects_of_initial_is_strict (λ A f, by apply_instance)
end initial
end algebraic_geometry
|
56dad3a3ead3ced901ed20044ba580cecac6d5db | 6fca17f8d5025f89be1b2d9d15c9e0c4b4900cbf | /src/game/world7/level7.lean | fda5f1338c260c49a6600ce5ed5ddc02b1ed5759 | [
"Apache-2.0"
] | permissive | arolihas/natural_number_game | 4f0c93feefec93b8824b2b96adff8b702b8b43ce | 8e4f7b4b42888a3b77429f90cce16292bd288138 | refs/heads/master | 1,621,872,426,808 | 1,586,270,467,000 | 1,586,270,467,000 | 253,648,466 | 0 | 0 | null | 1,586,219,694,000 | 1,586,219,694,000 | null | UTF-8 | Lean | false | false | 768 | lean | import game.world7.level6 -- hide
/-
# Advanced proposition world.
## Level 7: `or_symm`
Proving that $(P\lor Q)\implies(Q\lor P)$ involves an element of danger.
`intro h,` is the obvious start. But now,
even though the goal is an `∨` statement, both `left` and `right` put
you in a situation with an impossible goal. Fortunately, after `intro h,`
you can do `cases h with p q`. Then something new happens: because
there are two ways to prove `P ∨ Q` (namely, proving `P` or proving `Q`),
the `cases` tactic turns one goal into two, one for each case. You should
be able to make it home from there.
-/
/- Lemma
If $P$ and $Q$ are true/false statements, then
$$P\lor Q\implies Q\lor P.$$
-/
lemma or_symm (P Q : Prop) : P ∨ Q → Q ∨ P :=
begin
end
|
f40d45bda9c68383d342013915ba173faa14c8ea | a45212b1526d532e6e83c44ddca6a05795113ddc | /src/data/set/disjointed.lean | c03ae4e53216f89caeecf810dacc5643ff3428cc | [
"Apache-2.0"
] | permissive | fpvandoorn/mathlib | b21ab4068db079cbb8590b58fda9cc4bc1f35df4 | b3433a51ea8bc07c4159c1073838fc0ee9b8f227 | refs/heads/master | 1,624,791,089,608 | 1,556,715,231,000 | 1,556,715,231,000 | 165,722,980 | 5 | 0 | Apache-2.0 | 1,552,657,455,000 | 1,547,494,646,000 | Lean | UTF-8 | Lean | false | false | 3,426 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
Disjointed sets
-/
import data.set.lattice data.nat.basic tactic.wlog
open set classical lattice
local attribute [instance] prop_decidable
universes u v w x
variables {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
{s t u : set α}
/-- A relation `p` holds pairwise if `p i j` for all `i ≠ j`. -/
def pairwise {α : Type*} (p : α → α → Prop) := ∀i j, i ≠ j → p i j
theorem pairwise_on_bool {r} (hr : symmetric r) {a b : α} :
pairwise (r on (λ c, cond c a b)) ↔ r a b :=
by simp [pairwise, bool.forall_bool, function.on_fun];
exact and_iff_right_of_imp (@hr _ _)
theorem pairwise_disjoint_on_bool [semilattice_inf_bot α] {a b : α} :
pairwise (disjoint on (λ c, cond c a b)) ↔ disjoint a b :=
pairwise_on_bool $ λ _ _, disjoint.symm
namespace set
/-- If `f : ℕ → set α` is a sequence of sets, then `disjointed f` is
the sequence formed with each set subtracted from the later ones
in the sequence, to form a disjoint sequence. -/
def disjointed (f : ℕ → set α) (n : ℕ) : set α := f n ∩ (⋂i<n, - f i)
lemma disjoint_disjointed {f : ℕ → set α} : pairwise (disjoint on disjointed f) :=
λ i j h, begin
wlog h' : i ≤ j; [skip, {revert a, exact (this h.symm).symm}],
rintro a ⟨⟨h₁, _⟩, h₂, h₃⟩, simp at h₃,
exact h₃ _ (lt_of_le_of_ne h' h) h₁
end
lemma Union_lt_succ {f : ℕ → set α} {n} : (⋃i < nat.succ n, f i) = f n ∪ (⋃i < n, f i) :=
ext $ λ a, by simp [nat.lt_succ_iff_lt_or_eq, or_and_distrib_right, exists_or_distrib, or_comm]
lemma Inter_lt_succ {f : ℕ → set α} {n} : (⋂i < nat.succ n, f i) = f n ∩ (⋂i < n, f i) :=
ext $ λ a, by simp [nat.lt_succ_iff_lt_or_eq, or_imp_distrib, forall_and_distrib, and_comm]
lemma Union_disjointed {f : ℕ → set α} : (⋃n, disjointed f n) = (⋃n, f n) :=
subset.antisymm (Union_subset_Union $ assume i, inter_subset_left _ _) $
assume x h,
have ∃n, x ∈ f n, from mem_Union.mp h,
have hn : ∀ (i : ℕ), i < nat.find this → x ∉ f i,
from assume i, nat.find_min this,
mem_Union.mpr ⟨nat.find this, nat.find_spec this, by simp; assumption⟩
lemma disjointed_induct {f : ℕ → set α} {n : ℕ} {p : set α → Prop}
(h₁ : p (f n)) (h₂ : ∀t i, p t → p (t \ f i)) :
p (disjointed f n) :=
begin
rw disjointed,
generalize_hyp : f n = t at h₁ ⊢,
induction n,
case nat.zero { simp [nat.not_lt_zero, h₁] },
case nat.succ : n ih {
rw [Inter_lt_succ, inter_comm (-f n), ← inter_assoc],
exact h₂ _ n ih }
end
lemma disjointed_of_mono {f : ℕ → set α} {n : ℕ} (hf : monotone f) :
disjointed f (n + 1) = f (n + 1) \ f n :=
have (⋂i (h : i < n + 1), -f i) = - f n,
from le_antisymm
(infi_le_of_le n $ infi_le_of_le (nat.lt_succ_self _) $ subset.refl _)
(le_infi $ assume i, le_infi $ assume hi, neg_le_neg $ hf $ nat.le_of_succ_le_succ hi),
by simp [disjointed, this, diff_eq]
lemma Union_disjointed_of_mono {f : ℕ → set α} (hf : monotone f) :
∀ n : ℕ, (⋃i<n.succ, disjointed f i) = f n
| 0 := by rw [Union_lt_succ]; simp [disjointed, nat.not_lt_zero]
| (n+1) := by rw [Union_lt_succ,
disjointed_of_mono hf, Union_disjointed_of_mono n,
diff_union_self, union_eq_self_of_subset_right (hf (nat.le_succ _))]
end set
|
5bbfcd0b0edd02260d062190300aeed1e2d1c13d | 33340b3a23ca62ef3c8a7f6a2d4e14c07c6d3354 | /lia/remove_neg.lean | c89530dbb3cc55bad38d5512d5e2e575d62551b3 | [] | no_license | lclem/cooper | 79554e72ced343c64fed24b2d892d24bf9447dfe | 812afc6b158821f2e7dac9c91d3b6123c7a19faf | refs/heads/master | 1,607,554,257,488 | 1,578,694,133,000 | 1,578,694,133,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,579 | lean | import .wf
open list
def remove_neg : atom → atom
| (atom.le i ks) := atom.le (1 - i) (map_neg ks)
| (atom.dvd d i ks) := atom.ndvd d i ks
| (atom.ndvd d i ks) := atom.dvd d i ks
meta def neg_prsv_normal_aux :=
`[unfold remove_neg at hb,
rewrite (eq_of_mem_singleton hb), trivial]
lemma wf_remove_neg : ∀ {a : atom}, a.wf → (remove_neg a).wf
| (atom.le i ks) h := begin simp [remove_neg] end
| (atom.dvd d i ks) h := begin simp [remove_neg], apply h end
| (atom.ndvd d i ks) h := begin simp [remove_neg], apply h end
lemma eval_remove_neg :
∀ (a : atom) (xs : list znum),
(remove_neg a).eval xs ↔ (¬ (a.eval xs))
| (atom.le i ks) xs :=
begin
simp only [remove_neg, atom.eval],
apply calc
(1 - i ≤ znum.dot_prod (map_neg ks) xs)
↔ (-(znum.dot_prod (map_neg ks) xs) ≤ -(1 - i)) :
begin apply iff.intro, apply (@neg_le_neg znum _), apply (@le_of_neg_le_neg znum _) end
... ↔ (znum.dot_prod ks xs ≤ -(1 - i)) :
begin rw [znum.map_neg_dot_prod], simp, end
... ↔ (znum.dot_prod ks xs ≤ i - 1) :
by rewrite neg_sub
... ↔ (znum.dot_prod ks xs < i) :
begin
apply iff.intro,
apply znum.lt_of_le_sub_one,
apply znum.le_sub_one_of_lt
end
... ↔ ¬i ≤ znum.dot_prod ks xs :
begin
apply iff.intro,
apply not_le_of_gt,
apply lt_of_not_ge
end
end
| (atom.dvd d i ks) xs := by refl
| (atom.ndvd d i ks) xs := by apply not_not_iff.symm
|
977542362009c204fd17de3a069387e009b9bb2b | 618003631150032a5676f229d13a079ac875ff77 | /src/algebra/ring.lean | 4a7a27b85ba786c5d9a0f920d86d8cca359938a0 | [
"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 | 37,105 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston
-/
import algebra.group.hom
import algebra.group.units
import tactic.norm_cast
import tactic.split_ifs
/-!
# Properties and homomorphisms of semirings and rings
This file proves simple properties of semirings, rings and domains and their unit groups. It also
defines homomorphisms of semirings and rings, both unbundled (e.g. `is_semiring_hom f`)
and bundled (e.g. `ring_hom a β`, a.k.a. `α →+* β`). The unbundled ones are deprecated
and the plan is to slowly remove them from mathlib.
## Main definitions
is_semiring_hom (deprecated), is_ring_hom (deprecated), ring_hom, nonzero, domain, integral_domain
## Notations
→+* for bundled ring homs (also use for semiring homs)
## Implementation notes
There's a coercion from bundled homs to fun, and the canonical
notation is to use the bundled hom as a function via this coercion.
There is no `semiring_hom` -- the idea is that `ring_hom` is used.
The constructor for a `ring_hom` between semirings needs a proof of `map_zero`, `map_one` and
`map_add` as well as `map_mul`; a separate constructor `ring_hom.mk'` will construct ring homs
between rings from monoid homs given only a proof that addition is preserved.
Throughout the section on `ring_hom` implicit `{}` brackets are often used instead
of type class `[]` brackets. This is done when the instances can be inferred because they are
implicit arguments to the type `ring_hom`. When they can be inferred from the type it is faster
to use this method than to use type class inference.
## Tags
is_ring_hom, is_semiring_hom, ring_hom, semiring_hom, semiring, comm_semiring, ring, comm_ring,
domain, integral_domain, nonzero, units
-/
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w}
set_option default_priority 100 -- see Note [default priority]
set_option old_structure_cmd true
mk_simp_attribute field_simps "The simpset `field_simps` is used by the tactic `field_simp` to
reduce an expression in a field to an expression of the form `n / d` where `n` and `d` are
division-free."
@[protect_proj, ancestor has_mul has_add]
class distrib (α : Type u) extends has_mul α, has_add α :=
(left_distrib : ∀ a b c : α, a * (b + c) = (a * b) + (a * c))
(right_distrib : ∀ a b c : α, (a + b) * c = (a * c) + (b * c))
lemma left_distrib [distrib α] (a b c : α) : a * (b + c) = a * b + a * c :=
distrib.left_distrib a b c
def mul_add := @left_distrib
lemma right_distrib [distrib α] (a b c : α) : (a + b) * c = a * c + b * c :=
distrib.right_distrib a b c
def add_mul := @right_distrib
@[protect_proj, ancestor has_mul has_zero]
class mul_zero_class (α : Type u) extends has_mul α, has_zero α :=
(zero_mul : ∀ a : α, 0 * a = 0)
(mul_zero : ∀ a : α, a * 0 = 0)
@[ematch, simp] lemma zero_mul [mul_zero_class α] (a : α) : 0 * a = 0 :=
mul_zero_class.zero_mul a
@[ematch, simp] lemma mul_zero [mul_zero_class α] (a : α) : a * 0 = 0 :=
mul_zero_class.mul_zero a
/-- Predicate typeclass for expressing that a (semi)ring or similar algebraic structure is nonzero. -/
@[protect_proj] class nonzero (α : Type u) [has_zero α] [has_one α] : Prop :=
(zero_ne_one : 0 ≠ (1:α))
@[simp]
lemma zero_ne_one [has_zero α] [has_one α] [nonzero α] : 0 ≠ (1:α) :=
nonzero.zero_ne_one
@[simp]
lemma one_ne_zero [has_zero α] [has_one α] [nonzero α] : (1:α) ≠ 0 :=
zero_ne_one.symm
/- semiring -/
@[protect_proj, ancestor add_comm_monoid monoid distrib mul_zero_class]
class semiring (α : Type u) extends add_comm_monoid α, monoid α, distrib α, mul_zero_class α
section semiring
variables [semiring α]
lemma one_add_one_eq_two : 1 + 1 = (2 : α) :=
by unfold bit0
theorem two_mul (n : α) : 2 * n = n + n :=
eq.trans (right_distrib 1 1 n) (by simp)
lemma ne_zero_of_mul_ne_zero_right {a b : α} (h : a * b ≠ 0) : a ≠ 0 :=
assume : a = 0,
have a * b = 0, by rw [this, zero_mul],
h this
lemma ne_zero_of_mul_ne_zero_left {a b : α} (h : a * b ≠ 0) : b ≠ 0 :=
assume : b = 0,
have a * b = 0, by rw [this, mul_zero],
h this
lemma distrib_three_right (a b c d : α) : (a + b + c) * d = a * d + b * d + c * d :=
by simp [right_distrib]
theorem mul_two (n : α) : n * 2 = n + n :=
(left_distrib n 1 1).trans (by simp)
theorem bit0_eq_two_mul (n : α) : bit0 n = 2 * n :=
(two_mul _).symm
@[to_additive] lemma mul_ite {α} [has_mul α] (P : Prop) [decidable P] (a b c : α) :
a * (if P then b else c) = if P then a * b else a * c :=
by split_ifs; refl
@[to_additive] lemma ite_mul {α} [has_mul α] (P : Prop) [decidable P] (a b c : α) :
(if P then a else b) * c = if P then a * c else b * c :=
by split_ifs; refl
-- We make `mul_ite` and `ite_mul` simp lemmas,
-- but not `add_ite` or `ite_add`.
-- The problem we're trying to avoid is dealing with
-- summations of the form `s.sum (λ x, f x + ite P 1 0)`,
-- in which `add_ite` followed by `sum_ite` would needlessly slice up
-- the `f x` terms according to whether `P` holds at `x`.
-- There doesn't appear to be a corresponding difficulty so far with
-- `mul_ite` and `ite_mul`.
attribute [simp] mul_ite ite_mul
@[simp] lemma mul_boole {α} [semiring α] (P : Prop) [decidable P] (a : α) :
a * (if P then 1 else 0) = if P then a else 0 :=
by simp
@[simp] lemma boole_mul {α} [semiring α] (P : Prop) [decidable P] (a : α) :
(if P then 1 else 0) * a = if P then a else 0 :=
by simp
variable (α)
/-- Either zero and one are nonequal in a semiring, or the semiring is the zero ring. -/
lemma zero_ne_one_or_forall_eq_0 : (0 : α) ≠ 1 ∨ (∀a:α, a = 0) :=
by haveI := classical.dec;
refine not_or_of_imp (λ h a, _); simpa using congr_arg ((*) a) h.symm
/-- If zero equals one in a semiring, the semiring is the zero ring. -/
lemma eq_zero_of_zero_eq_one (h : (0 : α) = 1) : (∀a:α, a = 0) :=
(zero_ne_one_or_forall_eq_0 α).neg_resolve_left h
/-- If zero equals one in a semiring, all elements of that semiring are equal. -/
theorem subsingleton_of_zero_eq_one (h : (0 : α) = 1) : subsingleton α :=
⟨λa b, by rw [eq_zero_of_zero_eq_one α h a, eq_zero_of_zero_eq_one α h b]⟩
end semiring
namespace add_monoid_hom
/-- Left multiplication by an element of a (semi)ring is an `add_monoid_hom` -/
def mul_left {R : Type*} [semiring R] (r : R) : R →+ R :=
{ to_fun := (*) r,
map_zero' := mul_zero r,
map_add' := mul_add r }
@[simp] lemma coe_mul_left {R : Type*} [semiring R] (r : R) : ⇑(mul_left r) = (*) r := rfl
/-- Right multiplication by an element of a (semi)ring is an `add_monoid_hom` -/
def mul_right {R : Type*} [semiring R] (r : R) : R →+ R :=
{ to_fun := λ a, a * r,
map_zero' := zero_mul r,
map_add' := λ _ _, add_mul _ _ r }
@[simp] lemma mul_right_apply {R : Type*} [semiring R] (a r : R) :
(mul_right r : R → R) a = a * r := rfl
end add_monoid_hom
/-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too. -/
structure ring_hom (α : Type*) (β : Type*) [semiring α] [semiring β]
extends monoid_hom α β, add_monoid_hom α β
infixr ` →+* `:25 := ring_hom
@[priority 1000]
instance {α : Type*} {β : Type*} {rα : semiring α} {rβ : semiring β} : has_coe_to_fun (α →+* β) :=
⟨_, ring_hom.to_fun⟩
@[priority 1000]
instance {α : Type*} {β : Type*} {rα : semiring α} {rβ : semiring β} : has_coe (α →+* β) (α →* β) :=
⟨ring_hom.to_monoid_hom⟩
@[priority 1000]
instance {α : Type*} {β : Type*} {rα : semiring α} {rβ : semiring β} : has_coe (α →+* β) (α →+ β) :=
⟨ring_hom.to_add_monoid_hom⟩
@[simp, norm_cast]
lemma coe_monoid_hom {α : Type*} {β : Type*} {rα : semiring α} {rβ : semiring β} (f : α →+* β) :
⇑(f : α →* β) = f := rfl
@[simp, norm_cast]
lemma coe_add_monoid_hom {α : Type*} {β : Type*} {rα : semiring α} {rβ : semiring β} (f : α →+* β) :
⇑(f : α →+ β) = f := rfl
namespace ring_hom
variables [rα : semiring α] [rβ : semiring β]
section
include rα rβ
@[simp] lemma coe_mk (f : α → β) (h₁ h₂ h₃ h₄) : ⇑(⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) = f := rfl
variables (f : α →+* β) {x y : α} {rα rβ}
theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g :=
by cases f; cases g; cases h; refl
@[ext] theorem ext ⦃f g : α →+* β⦄ (h : ∀ x, f x = g x) : f = g :=
coe_inj (funext h)
theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x :=
⟨λ h x, h ▸ rfl, λ h, ext h⟩
theorem coe_add_monoid_hom_inj : function.injective (coe : (α →+* β) → (α →+ β)) :=
λ f g h, coe_inj $ show ((f : α →+ β) : α → β) = (g : α →+ β), from congr_arg coe_fn h
theorem coe_monoid_hom_inj : function.injective (coe : (α →+* β) → (α →* β)) :=
λ f g h, coe_inj $ show ((f : α →* β) : α → β) = (g : α →* β), from congr_arg coe_fn h
/-- Ring homomorphisms map zero to zero. -/
@[simp] lemma map_zero (f : α →+* β) : f 0 = 0 := f.map_zero'
/-- Ring homomorphisms map one to one. -/
@[simp] lemma map_one (f : α →+* β) : f 1 = 1 := f.map_one'
/-- Ring homomorphisms preserve addition. -/
@[simp] lemma map_add (f : α →+* β) (a b : α) : f (a + b) = f a + f b := f.map_add' a b
/-- Ring homomorphisms preserve multiplication. -/
@[simp] lemma map_mul (f : α →+* β) (a b : α) : f (a * b) = f a * f b := f.map_mul' a b
end
/-- The identity ring homomorphism from a semiring to itself. -/
def id (α : Type*) [semiring α] : α →+* α :=
by refine {to_fun := id, ..}; intros; refl
include rα
@[simp] lemma id_apply (x : α) : ring_hom.id α x = x := rfl
variable {rγ : semiring γ}
include rβ rγ
/-- Composition of ring homomorphisms is a ring homomorphism. -/
def comp (hnp : β →+* γ) (hmn : α →+* β) : α →+* γ :=
{ to_fun := hnp ∘ hmn,
map_zero' := by simp,
map_one' := by simp,
map_add' := λ x y, by simp,
map_mul' := λ x y, by simp}
/-- Composition of semiring homomorphisms is associative. -/
lemma comp_assoc {δ} {rδ: semiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) :
(h.comp g).comp f = h.comp (g.comp f) := rfl
@[simp] lemma coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl
lemma comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x =
(hnp (hmn x)) := rfl
omit rγ
@[simp] lemma comp_id (f : α →+* β) : f.comp (id α) = f := ext $ λ x, rfl
@[simp] lemma id_comp (f : α →+* β) : (id β).comp f = f := ext $ λ x, rfl
omit rβ
instance : monoid (α →+* α) :=
{ one := id α,
mul := comp,
mul_one := comp_id,
one_mul := id_comp,
mul_assoc := λ f g h, comp_assoc _ _ _ }
lemma one_def : (1 : α →+* α) = id α := rfl
@[simp] lemma coe_one : ⇑(1 : α →+* α) = _root_.id := rfl
lemma mul_def (f g : α →+* α) : f * g = f.comp g := rfl
@[simp] lemma coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl
include rβ rγ
lemma cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : function.surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨λ h, ring_hom.ext $ (forall_iff_forall_surj hf).1 (ext_iff.1 h), λ h, h ▸ rfl⟩
lemma cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : function.injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨λ h, ring_hom.ext $ λ x, hg $ by rw [← comp_apply, h, comp_apply], λ h, h ▸ rfl⟩
omit rα rβ rγ
end ring_hom
@[protect_proj, ancestor semiring comm_monoid]
class comm_semiring (α : Type u) extends semiring α, comm_monoid α
section comm_semiring
variables [comm_semiring α] [comm_semiring β] {a b c : α}
instance comm_semiring_has_dvd : has_dvd α :=
has_dvd.mk (λ a b, ∃ c, b = a * c)
-- TODO: this used to not have c explicit, but that seems to be important
-- for use with tactics, similar to exist.intro
theorem dvd.intro (c : α) (h : a * c = b) : a ∣ b :=
exists.intro c h^.symm
def dvd_of_mul_right_eq := @dvd.intro
theorem dvd.intro_left (c : α) (h : c * a = b) : a ∣ b :=
dvd.intro _ (begin rewrite mul_comm at h, apply h end)
def dvd_of_mul_left_eq := @dvd.intro_left
theorem exists_eq_mul_right_of_dvd (h : a ∣ b) : ∃ c, b = a * c := h
theorem dvd.elim {P : Prop} {a b : α} (H₁ : a ∣ b) (H₂ : ∀ c, b = a * c → P) : P :=
exists.elim H₁ H₂
theorem exists_eq_mul_left_of_dvd (h : a ∣ b) : ∃ c, b = c * a :=
dvd.elim h (assume c, assume H1 : b = a * c, exists.intro c (eq.trans H1 (mul_comm a c)))
theorem dvd.elim_left {P : Prop} (h₁ : a ∣ b) (h₂ : ∀ c, b = c * a → P) : P :=
exists.elim (exists_eq_mul_left_of_dvd h₁) (assume c, assume h₃ : b = c * a, h₂ c h₃)
@[refl, simp] theorem dvd_refl (a : α) : a ∣ a :=
dvd.intro 1 (by simp)
local attribute [simp] mul_assoc mul_comm mul_left_comm
@[trans] theorem dvd_trans (h₁ : a ∣ b) (h₂ : b ∣ c) : a ∣ c :=
match h₁, h₂ with
| ⟨d, (h₃ : b = a * d)⟩, ⟨e, (h₄ : c = b * e)⟩ :=
⟨d * e, show c = a * (d * e), by simp [h₃, h₄]⟩
end
def dvd.trans := @dvd_trans
theorem eq_zero_of_zero_dvd (h : 0 ∣ a) : a = 0 :=
dvd.elim h (assume c, assume H' : a = 0 * c, eq.trans H' (zero_mul c))
/-- Given an element a of a commutative semiring, there exists another element whose product
with zero equals a iff a equals zero. -/
@[simp] lemma zero_dvd_iff : 0 ∣ a ↔ a = 0 :=
⟨eq_zero_of_zero_dvd, λ h, by rw h⟩
@[simp] theorem dvd_zero (a : α) : a ∣ 0 := dvd.intro 0 (by simp)
@[simp] theorem one_dvd (a : α) : 1 ∣ a := dvd.intro a (by simp)
@[simp] theorem dvd_mul_right (a b : α) : a ∣ a * b := dvd.intro b rfl
@[simp] theorem dvd_mul_left (a b : α) : a ∣ b * a := dvd.intro b (by simp)
theorem dvd_mul_of_dvd_left (h : a ∣ b) (c : α) : a ∣ b * c :=
dvd.elim h (λ d h', begin rw [h', mul_assoc], apply dvd_mul_right end)
theorem dvd_mul_of_dvd_right (h : a ∣ b) (c : α) : a ∣ c * b :=
begin rw mul_comm, exact dvd_mul_of_dvd_left h _ end
theorem mul_dvd_mul : ∀ {a b c d : α}, a ∣ b → c ∣ d → a * c ∣ b * d
| a ._ c ._ ⟨e, rfl⟩ ⟨f, rfl⟩ := ⟨e * f, by simp⟩
theorem mul_dvd_mul_left (a : α) {b c : α} (h : b ∣ c) : a * b ∣ a * c :=
mul_dvd_mul (dvd_refl a) h
theorem mul_dvd_mul_right (h : a ∣ b) (c : α) : a * c ∣ b * c :=
mul_dvd_mul h (dvd_refl c)
theorem dvd_add (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b + c :=
dvd.elim h₁ (λ d hd, dvd.elim h₂ (λ e he, dvd.intro (d + e) (by simp [left_distrib, hd, he])))
theorem dvd_of_mul_right_dvd (h : a * b ∣ c) : a ∣ c :=
dvd.elim h (begin intros d h₁, rw [h₁, mul_assoc], apply dvd_mul_right end)
theorem dvd_of_mul_left_dvd (h : a * b ∣ c) : b ∣ c :=
dvd.elim h (λ d ceq, dvd.intro (a * d) (by simp [ceq]))
lemma ring_hom.map_dvd (f : α →+* β) {a b : α} : a ∣ b → f a ∣ f b :=
λ ⟨z, hz⟩, ⟨f z, by rw [hz, f.map_mul]⟩
end comm_semiring
/- ring -/
@[protect_proj, ancestor add_comm_group monoid distrib]
class ring (α : Type u) extends add_comm_group α, monoid α, distrib α
section ring
variables [ring α] {a b c d e : α}
lemma ring.mul_zero (a : α) : a * 0 = 0 :=
have a * 0 + 0 = a * 0 + a * 0, from calc
a * 0 + 0 = a * (0 + 0) : by simp
... = a * 0 + a * 0 : by rw left_distrib,
show a * 0 = 0, from (add_left_cancel this).symm
lemma ring.zero_mul (a : α) : 0 * a = 0 :=
have 0 * a + 0 = 0 * a + 0 * a, from calc
0 * a + 0 = (0 + 0) * a : by simp
... = 0 * a + 0 * a : by rewrite right_distrib,
show 0 * a = 0, from (add_left_cancel this).symm
instance ring.to_semiring : semiring α :=
{ mul_zero := ring.mul_zero, zero_mul := ring.zero_mul, ..‹ring α› }
/- The instance from `ring` to `semiring` happens often in linear algebra, for which all the basic
definitions are given in terms of semirings, but many applications use rings or fields. We increase
a little bit its priority above 100 to try it quickly, but remaining below the default 1000 so that
more specific instances are tried first. -/
attribute [instance, priority 200] ring.to_semiring
lemma neg_mul_eq_neg_mul (a b : α) : -(a * b) = -a * b :=
neg_eq_of_add_eq_zero
begin rw [← right_distrib, add_right_neg, zero_mul] end
lemma neg_mul_eq_mul_neg (a b : α) : -(a * b) = a * -b :=
neg_eq_of_add_eq_zero
begin rw [← left_distrib, add_right_neg, mul_zero] end
@[simp] lemma neg_mul_eq_neg_mul_symm (a b : α) : - a * b = - (a * b) :=
eq.symm (neg_mul_eq_neg_mul a b)
@[simp] lemma mul_neg_eq_neg_mul_symm (a b : α) : a * - b = - (a * b) :=
eq.symm (neg_mul_eq_mul_neg a b)
lemma neg_mul_neg (a b : α) : -a * -b = a * b :=
by simp
lemma neg_mul_comm (a b : α) : -a * b = a * -b :=
by simp
theorem neg_eq_neg_one_mul (a : α) : -a = -1 * a :=
by simp
lemma mul_sub_left_distrib (a b c : α) : a * (b - c) = a * b - a * c :=
calc
a * (b - c) = a * b + a * -c : left_distrib a b (-c)
... = a * b - a * c : by simp [sub_eq_add_neg]
def mul_sub := @mul_sub_left_distrib
lemma mul_sub_right_distrib (a b c : α) : (a - b) * c = a * c - b * c :=
calc
(a - b) * c = a * c + -b * c : right_distrib a (-b) c
... = a * c - b * c : by simp [sub_eq_add_neg]
def sub_mul := @mul_sub_right_distrib
/-- An element of a ring multiplied by the additive inverse of one is the element's additive
inverse. -/
lemma mul_neg_one (a : α) : a * -1 = -a := by simp
/-- The additive inverse of one multiplied by an element of a ring is the element's additive
inverse. -/
lemma neg_one_mul (a : α) : -1 * a = -a := by simp
/-- An iff statement following from right distributivity in rings and the definition
of subtraction. -/
theorem mul_add_eq_mul_add_iff_sub_mul_add_eq : a * e + c = b * e + d ↔ (a - b) * e + c = d :=
calc
a * e + c = b * e + d ↔ a * e + c = d + b * e : by simp [add_comm]
... ↔ a * e + c - b * e = d : iff.intro (λ h, begin rw h, simp end) (λ h,
begin rw ← h, simp end)
... ↔ (a - b) * e + c = d : begin simp [sub_mul, sub_add_eq_add_sub] end
/-- A simplification of one side of an equation exploiting right distributivity in rings
and the definition of subtraction. -/
theorem sub_mul_add_eq_of_mul_add_eq_mul_add : a * e + c = b * e + d → (a - b) * e + c = d :=
assume h,
calc
(a - b) * e + c = (a * e + c) - b * e : begin simp [sub_mul, sub_add_eq_add_sub] end
... = d : begin rw h, simp [@add_sub_cancel α] end
/-- If the product of two elements of a ring is nonzero, both elements are nonzero. -/
theorem ne_zero_and_ne_zero_of_mul_ne_zero (h : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 :=
begin
split,
{ intro ha, apply h, simp [ha] },
{ intro hb, apply h, simp [hb] }
end
end ring
namespace units
variables [ring α] {a b : α}
/-- Each element of the group of units of a ring has an additive inverse. -/
instance : has_neg (units α) := ⟨λu, ⟨-↑u, -↑u⁻¹, by simp, by simp⟩ ⟩
/-- Representing an element of a ring's unit group as an element of the ring commutes with
mapping this element to its additive inverse. -/
@[simp] protected theorem coe_neg (u : units α) : (↑-u : α) = -u := rfl
/-- Mapping an element of a ring's unit group to its inverse commutes with mapping this element
to its additive inverse. -/
@[simp] protected theorem neg_inv (u : units α) : (-u)⁻¹ = -u⁻¹ := rfl
/-- An element of a ring's unit group equals the additive inverse of its additive inverse. -/
@[simp] protected theorem neg_neg (u : units α) : - -u = u :=
units.ext $ neg_neg _
/-- Multiplication of elements of a ring's unit group commutes with mapping the first
argument to its additive inverse. -/
@[simp] protected theorem neg_mul (u₁ u₂ : units α) : -u₁ * u₂ = -(u₁ * u₂) :=
units.ext $ neg_mul_eq_neg_mul_symm _ _
/-- Multiplication of elements of a ring's unit group commutes with mapping the second argument
to its additive inverse. -/
@[simp] protected theorem mul_neg (u₁ u₂ : units α) : u₁ * -u₂ = -(u₁ * u₂) :=
units.ext $ (neg_mul_eq_mul_neg _ _).symm
/-- Multiplication of the additive inverses of two elements of a ring's unit group equals
multiplication of the two original elements. -/
@[simp] protected theorem neg_mul_neg (u₁ u₂ : units α) : -u₁ * -u₂ = u₁ * u₂ := by simp
/-- The additive inverse of an element of a ring's unit group equals the additive inverse of
one times the original element. -/
protected theorem neg_eq_neg_one_mul (u : units α) : -u = -1 * u := by simp
end units
namespace ring_hom
/-- Ring homomorphisms preserve additive inverse. -/
@[simp] theorem map_neg {α β} [ring α] [ring β] (f : α →+* β) (x : α) : f (-x) = -(f x) :=
(f : α →+ β).map_neg x
/-- Ring homomorphisms preserve subtraction. -/
@[simp] theorem map_sub {α β} [ring α] [ring β] (f : α →+* β) (x y : α) :
f (x - y) = (f x) - (f y) := (f : α →+ β).map_sub x y
/-- A ring homomorphism is injective iff its kernel is trivial. -/
theorem injective_iff {α β} [ring α] [ring β] (f : α →+* β) :
function.injective f ↔ (∀ a, f a = 0 → a = 0) :=
(f : α →+ β).injective_iff
/-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/
def mk' {γ} [semiring α] [ring γ] (f : α →* γ) (map_add : ∀ a b : α, f (a + b) = f a + f b) :
α →+* γ :=
{ to_fun := f,
.. add_monoid_hom.mk' f map_add, .. f }
end ring_hom
@[protect_proj, ancestor ring comm_semigroup]
class comm_ring (α : Type u) extends ring α, comm_semigroup α
instance comm_ring.to_comm_semiring [s : comm_ring α] : comm_semiring α :=
{ mul_zero := mul_zero, zero_mul := zero_mul, ..s }
section comm_ring
variables [comm_ring α] {a b c : α}
local attribute [simp] add_assoc add_comm add_left_comm mul_comm
lemma mul_self_sub_mul_self_eq (a b : α) : a * a - b * b = (a + b) * (a - b) :=
begin simp [right_distrib, left_distrib, sub_eq_add_neg] end
lemma mul_self_sub_one_eq (a : α) : a * a - 1 = (a + 1) * (a - 1) :=
begin simp [right_distrib, left_distrib, sub_eq_add_neg], rw [add_left_comm, add_comm (-a), add_left_comm a], simp end
lemma add_mul_self_eq (a b : α) : (a + b) * (a + b) = a*a + 2*a*b + b*b :=
calc (a + b)*(a + b) = a*a + (1+1)*a*b + b*b : by simp [right_distrib, left_distrib]
... = a*a + 2*a*b + b*b : by rw one_add_one_eq_two
theorem dvd_neg_of_dvd (h : a ∣ b) : (a ∣ -b) :=
dvd.elim h
(assume c, assume : b = a * c,
dvd.intro (-c) (by simp [this]))
theorem dvd_of_dvd_neg (h : a ∣ -b) : (a ∣ b) :=
let t := dvd_neg_of_dvd h in by rwa neg_neg at t
theorem dvd_neg_iff_dvd (a b : α) : (a ∣ -b) ↔ (a ∣ b) :=
⟨dvd_of_dvd_neg, dvd_neg_of_dvd⟩
theorem neg_dvd_of_dvd (h : a ∣ b) : -a ∣ b :=
dvd.elim h
(assume c, assume : b = a * c,
dvd.intro (-c) (by simp [this]))
theorem dvd_of_neg_dvd (h : -a ∣ b) : a ∣ b :=
let t := neg_dvd_of_dvd h in by rwa neg_neg at t
theorem neg_dvd_iff_dvd (a b : α) : (-a ∣ b) ↔ (a ∣ b) :=
⟨dvd_of_neg_dvd, neg_dvd_of_dvd⟩
theorem dvd_sub (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b - c :=
dvd_add h₁ (dvd_neg_of_dvd h₂)
theorem dvd_add_iff_left (h : a ∣ c) : a ∣ b ↔ a ∣ b + c :=
⟨λh₂, dvd_add h₂ h, λH, by have t := dvd_sub H h; rwa add_sub_cancel at t⟩
theorem dvd_add_iff_right (h : a ∣ b) : a ∣ c ↔ a ∣ b + c :=
by rw add_comm; exact dvd_add_iff_left h
/-- Representation of a difference of two squares in a commutative ring as a product. -/
theorem mul_self_sub_mul_self (a b : α) : a * a - b * b = (a + b) * (a - b) :=
by rw [add_mul, mul_sub, mul_sub, mul_comm a b, sub_add_sub_cancel]
/-- An element a of a commutative ring divides the additive inverse of an element b iff a
divides b. -/
@[simp] lemma dvd_neg (a b : α) : (a ∣ -b) ↔ (a ∣ b) :=
⟨dvd_of_dvd_neg, dvd_neg_of_dvd⟩
/-- The additive inverse of an element a of a commutative ring divides another element b iff a
divides b. -/
@[simp] lemma neg_dvd (a b : α) : (-a ∣ b) ↔ (a ∣ b) :=
⟨dvd_of_neg_dvd, neg_dvd_of_dvd⟩
/-- If an element a divides another element c in a commutative ring, a divides the sum of another
element b with c iff a divides b. -/
theorem dvd_add_left (h : a ∣ c) : a ∣ b + c ↔ a ∣ b :=
(dvd_add_iff_left h).symm
/-- If an element a divides another element b in a commutative ring, a divides the sum of b and
another element c iff a divides c. -/
theorem dvd_add_right (h : a ∣ b) : a ∣ b + c ↔ a ∣ c :=
(dvd_add_iff_right h).symm
/-- An element a divides the sum a + b if and only if a divides b.-/
@[simp] lemma dvd_add_self_left {a b : α} : a ∣ a + b ↔ a ∣ b :=
dvd_add_right (dvd_refl a)
/-- An element a divides the sum b + a if and only if a divides b.-/
@[simp] lemma dvd_add_self_right {a b : α} : a ∣ b + a ↔ a ∣ b :=
dvd_add_left (dvd_refl a)
/-- Vieta's formula for a quadratic equation, relating the coefficients of the polynomial with
its roots. This particular version states that if we have a root `x` of a monic quadratic
polynomial, then there is another root `y` such that `x + y` is negative the `a_1` coefficient
and `x * y` is the `a_0` coefficient. -/
lemma Vieta_formula_quadratic {b c x : α} (h : x * x - b * x + c = 0) :
∃ y : α, y * y - b * y + c = 0 ∧ x + y = b ∧ x * y = c :=
begin
have : c = -(x * x - b * x) := (neg_eq_of_add_eq_zero h).symm,
have : c = x * (b - x), by subst this; simp [mul_sub, mul_comm],
refine ⟨b - x, _, by simp, by rw this⟩,
rw [this, sub_add, ← sub_mul, sub_self]
end
end comm_ring
lemma succ_ne_self [ring α] [nonzero α] (a : α) : a + 1 ≠ a :=
λ h, one_ne_zero ((add_right_inj a).mp (by simp [h]))
lemma pred_ne_self [ring α] [nonzero α] (a : α) : a - 1 ≠ a :=
λ h, one_ne_zero (neg_inj ((add_right_inj a).mp (by { convert h, simp })))
/-- An element of the unit group of a nonzero semiring represented as an element
of the semiring is nonzero. -/
lemma units.coe_ne_zero [semiring α] [nonzero α] (u : units α) : (u : α) ≠ 0 :=
λ h : u.1 = 0, by simpa [h, zero_ne_one] using u.3
/-- Proves that a semiring that contains at least two distinct elements is nonzero. -/
theorem nonzero.of_ne [semiring α] {x y : α} (h : x ≠ y) : nonzero α :=
{ zero_ne_one := λ h01, h $ by rw [← one_mul x, ← one_mul y, ← h01, zero_mul, zero_mul] }
@[protect_proj] class no_zero_divisors (α : Type u) [has_mul α] [has_zero α] : Prop :=
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀ a b : α, a * b = 0 → a = 0 ∨ b = 0)
lemma eq_zero_or_eq_zero_of_mul_eq_zero [has_mul α] [has_zero α] [no_zero_divisors α]
{a b : α} (h : a * b = 0) : a = 0 ∨ b = 0 :=
no_zero_divisors.eq_zero_or_eq_zero_of_mul_eq_zero a b h
lemma eq_zero_of_mul_self_eq_zero [has_mul α] [has_zero α] [no_zero_divisors α]
{a : α} (h : a * a = 0) : a = 0 :=
or.elim (eq_zero_or_eq_zero_of_mul_eq_zero h) (assume h', h') (assume h', h')
/-- A domain is a ring with no zero divisors, i.e. satisfying
the condition `a * b = 0 ↔ a = 0 ∨ b = 0`. Alternatively, a domain
is an integral domain without assuming commutativity of multiplication. -/
@[protect_proj] class domain (α : Type u) extends ring α :=
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀ a b : α, a * b = 0 → a = 0 ∨ b = 0)
(zero_ne_one : (0 : α) ≠ 1)
section domain
variable [domain α]
instance domain.to_no_zero_divisors : no_zero_divisors α :=
⟨domain.eq_zero_or_eq_zero_of_mul_eq_zero⟩
instance domain.to_nonzero : nonzero α :=
⟨domain.zero_ne_one⟩
/-- Simplification theorems for the definition of a domain. -/
@[simp] theorem mul_eq_zero {a b : α} : a * b = 0 ↔ a = 0 ∨ b = 0 :=
⟨eq_zero_or_eq_zero_of_mul_eq_zero, λo,
or.elim o (λh, by rw h; apply zero_mul) (λh, by rw h; apply mul_zero)⟩
@[simp] theorem zero_eq_mul {a b : α} : 0 = a * b ↔ a = 0 ∨ b = 0 :=
by rw [eq_comm, mul_eq_zero]
lemma mul_self_eq_zero {α} [domain α] {x : α} : x * x = 0 ↔ x = 0 := by simp
lemma zero_eq_mul_self {α} [domain α] {x : α} : 0 = x * x ↔ x = 0 := by simp
/-- The product of two nonzero elements of a domain is nonzero. -/
theorem mul_ne_zero' {a b : α} (h₁ : a ≠ 0) (h₂ : b ≠ 0) : a * b ≠ 0 :=
λ h, or.elim (eq_zero_or_eq_zero_of_mul_eq_zero h) h₁ h₂
/-- Right multiplication by a nonzero element in a domain is injective. -/
theorem domain.mul_left_inj {a b c : α} (ha : a ≠ 0) : b * a = c * a ↔ b = c :=
by rw [← sub_eq_zero, ← mul_sub_right_distrib, mul_eq_zero];
simp [ha]; exact sub_eq_zero
/-- Left multiplication by a nonzero element in a domain is injective. -/
theorem domain.mul_right_inj {a b c : α} (ha : a ≠ 0) : a * b = a * c ↔ b = c :=
by rw [← sub_eq_zero, ← mul_sub_left_distrib, mul_eq_zero];
simp [ha]; exact sub_eq_zero
/-- An element of a domain fixed by right multiplication by an element other than one must
be zero. -/
theorem eq_zero_of_mul_eq_self_right' {a b : α} (h₁ : b ≠ 1) (h₂ : a * b = a) : a = 0 :=
by apply (mul_eq_zero.1 _).resolve_right (sub_ne_zero.2 h₁);
rw [mul_sub_left_distrib, mul_one, sub_eq_zero, h₂]
/-- An element of a domain fixed by left multiplication by an element other than one must
be zero. -/
theorem eq_zero_of_mul_eq_self_left' {a b : α} (h₁ : b ≠ 1) (h₂ : b * a = a) : a = 0 :=
by apply (mul_eq_zero.1 _).resolve_left (sub_ne_zero.2 h₁);
rw [mul_sub_right_distrib, one_mul, sub_eq_zero, h₂]
/-- For elements `a`, `b` of a domain, if `a*b` is nonzero, so is `b*a`. -/
theorem mul_ne_zero_comm' {a b : α} (h : a * b ≠ 0) : b * a ≠ 0 :=
mul_ne_zero' (ne_zero_of_mul_ne_zero_left h) (ne_zero_of_mul_ne_zero_right h)
end domain
/- integral domains -/
@[protect_proj, ancestor comm_ring domain]
class integral_domain (α : Type u) extends comm_ring α, domain α
section integral_domain
variables [integral_domain α] {a b c d e : α}
lemma mul_eq_zero_iff_eq_zero_or_eq_zero : a * b = 0 ↔ a = 0 ∨ b = 0 :=
⟨eq_zero_or_eq_zero_of_mul_eq_zero, λo,
or.elim o (λh, by rw h; apply zero_mul) (λh, by rw h; apply mul_zero)⟩
lemma mul_ne_zero (h₁ : a ≠ 0) (h₂ : b ≠ 0) : a * b ≠ 0 :=
λ h, or.elim (eq_zero_or_eq_zero_of_mul_eq_zero h) (assume h₃, h₁ h₃) (assume h₄, h₂ h₄)
lemma eq_of_mul_eq_mul_right (ha : a ≠ 0) (h : b * a = c * a) : b = c :=
have b * a - c * a = 0, from sub_eq_zero_of_eq h,
have (b - c) * a = 0, by rw [mul_sub_right_distrib, this],
have b - c = 0, from (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right ha,
eq_of_sub_eq_zero this
lemma eq_of_mul_eq_mul_left (ha : a ≠ 0) (h : a * b = a * c) : b = c :=
have a * b - a * c = 0, from sub_eq_zero_of_eq h,
have a * (b - c) = 0, by rw [mul_sub_left_distrib, this],
have b - c = 0, from (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_left ha,
eq_of_sub_eq_zero this
lemma eq_zero_of_mul_eq_self_right (h₁ : b ≠ 1) (h₂ : a * b = a) : a = 0 :=
have hb : b - 1 ≠ 0, from
assume : b - 1 = 0,
have b = 0 + 1, from eq_add_of_sub_eq this,
have b = 1, by rwa zero_add at this,
h₁ this,
have a * b - a = 0, by simp [h₂],
have a * (b - 1) = 0, by rwa [mul_sub_left_distrib, mul_one],
show a = 0, from (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hb
lemma eq_zero_of_mul_eq_self_left (h₁ : b ≠ 1) (h₂ : b * a = a) : a = 0 :=
eq_zero_of_mul_eq_self_right h₁ (by rwa mul_comm at h₂)
lemma mul_self_eq_mul_self_iff (a b : α) : a * a = b * b ↔ a = b ∨ a = -b :=
iff.intro
(assume : a * a = b * b,
have (a - b) * (a + b) = 0,
by rewrite [mul_comm, ← mul_self_sub_mul_self_eq, this, sub_self],
have a - b = 0 ∨ a + b = 0, from eq_zero_or_eq_zero_of_mul_eq_zero this,
or.elim this
(assume : a - b = 0, or.inl (eq_of_sub_eq_zero this))
(assume : a + b = 0, or.inr (eq_neg_of_add_eq_zero this)))
(assume : a = b ∨ a = -b, or.elim this
(assume : a = b, by rewrite this)
(assume : a = -b, by rewrite [this, neg_mul_neg]))
lemma mul_self_eq_one_iff (a : α) : a * a = 1 ↔ a = 1 ∨ a = -1 :=
have a * a = 1 * 1 ↔ a = 1 ∨ a = -1, from mul_self_eq_mul_self_iff a 1,
by rwa mul_one at this
/-- Right multiplcation by a nonzero element of an integral domain is injective. -/
theorem eq_of_mul_eq_mul_right_of_ne_zero (ha : a ≠ 0) (h : b * a = c * a) : b = c :=
have b * a - c * a = 0, by simp [h],
have (b - c) * a = 0, by rw [mul_sub_right_distrib, this],
have b - c = 0, from (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right ha,
eq_of_sub_eq_zero this
/-- Left multiplication by a nonzero element of an integral domain is injective. -/
theorem eq_of_mul_eq_mul_left_of_ne_zero (ha : a ≠ 0) (h : a * b = a * c) : b = c :=
have a * b - a * c = 0, by simp [h],
have a * (b - c) = 0, by rw [mul_sub_left_distrib, this],
have b - c = 0, from (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_left ha,
eq_of_sub_eq_zero this
/-- Given two elements b, c of an integral domain and a nonzero element a, a*b divides a*c iff
b divides c. -/
theorem mul_dvd_mul_iff_left (ha : a ≠ 0) : a * b ∣ a * c ↔ b ∣ c :=
exists_congr $ λ d, by rw [mul_assoc, domain.mul_right_inj ha]
/-- Given two elements a, b of an integral domain and a nonzero element c, a*c divides b*c iff
a divides b. -/
theorem mul_dvd_mul_iff_right (hc : c ≠ 0) : a * c ∣ b * c ↔ a ∣ b :=
exists_congr $ λ d, by rw [mul_right_comm, domain.mul_left_inj hc]
/-- In the unit group of an integral domain, a unit is its own inverse iff the unit is one or
one's additive inverse. -/
lemma units.inv_eq_self_iff (u : units α) : u⁻¹ = u ↔ u = 1 ∨ u = -1 :=
by conv {to_lhs, rw [inv_eq_iff_mul_eq_one, ← mul_one (1 : units α), units.ext_iff, units.coe_mul,
units.coe_mul, mul_self_eq_mul_self_iff, ← units.ext_iff, ← units.coe_neg, ← units.ext_iff] }
end integral_domain
/- units in various rings -/
namespace units
section semiring
variables [semiring α]
@[simp] theorem mul_left_eq_zero_iff_eq_zero
{r : α} (u : units α) : r * u = 0 ↔ r = 0 :=
⟨λ h, (mul_left_inj u).1 $ (zero_mul (u : α)).symm ▸ h,
λ h, h.symm ▸ zero_mul (u : α)⟩
@[simp] theorem mul_right_eq_zero_iff_eq_zero
{r : α} (u : units α) : (u : α) * r = 0 ↔ r = 0 :=
⟨λ h, (mul_right_inj u).1 $ (mul_zero (u : α)).symm ▸ h,
λ h, h.symm ▸ mul_zero (u : α)⟩
end semiring
section comm_semiring
variables [comm_semiring α] (a b : α) (u : units α)
/-- Elements of the unit group of a commutative semiring represented as elements of the semiring
divide any element of the semiring. -/
@[simp] lemma coe_dvd : ↑u ∣ a := ⟨↑u⁻¹ * a, by simp⟩
/-- In a commutative semiring, an element a divides an element b iff a divides all
associates of b. -/
@[simp] lemma dvd_coe_mul : a ∣ b * u ↔ a ∣ b :=
iff.intro
(assume ⟨c, eq⟩, ⟨c * ↑u⁻¹, by rw [← mul_assoc, ← eq, units.mul_inv_cancel_right]⟩)
(assume ⟨c, eq⟩, eq.symm ▸ dvd_mul_of_dvd_left (dvd_mul_right _ _) _)
/-- An element of a commutative semiring divides a unit iff the element divides one. -/
@[simp] lemma dvd_coe : a ∣ ↑u ↔ a ∣ 1 :=
suffices a ∣ 1 * ↑u ↔ a ∣ 1, by simpa,
dvd_coe_mul _ _ _
/-- In a commutative semiring, an element a divides an element b iff all associates of a divide b.-/
@[simp] lemma coe_mul_dvd : a * u ∣ b ↔ a ∣ b :=
iff.intro
(assume ⟨c, eq⟩, ⟨c * ↑u, eq.symm ▸ by ac_refl⟩)
(assume h, suffices a * ↑u ∣ b * 1, by simpa, mul_dvd_mul h (coe_dvd _ _))
end comm_semiring
end units
namespace is_unit
section semiring
variables [semiring α]
theorem mul_left_eq_zero_iff_eq_zero {r u : α}
(hu : is_unit u) : r * u = 0 ↔ r = 0 :=
by cases hu with u hu; exact hu ▸ units.mul_left_eq_zero_iff_eq_zero u
theorem mul_right_eq_zero_iff_eq_zero {r u : α}
(hu : is_unit u) : u * r = 0 ↔ r = 0 :=
by cases hu with u hu; exact hu ▸ units.mul_right_eq_zero_iff_eq_zero u
end semiring
end is_unit
/-- A predicate to express that a ring is an integral domain.
This is mainly useful because such a predicate does not contain data,
and can therefore be easily transported along ring isomorphisms. -/
structure is_integral_domain (R : Type u) [ring R] : Prop :=
(mul_comm : ∀ (x y : R), x * y = y * x)
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀ x y : R, x * y = 0 → x = 0 ∨ y = 0)
(zero_ne_one : (0 : R) ≠ 1)
/-- Every integral domain satisfies the predicate for integral domains. -/
lemma integral_domain.to_is_integral_domain (R : Type u) [integral_domain R] :
is_integral_domain R :=
{ .. (‹_› : integral_domain R) }
/-- If a ring satisfies the predicate for integral domains,
then it can be endowed with an `integral_domain` instance
whose data is definitionally equal to the existing data. -/
def is_integral_domain.to_integral_domain (R : Type u) [ring R] (h : is_integral_domain R) :
integral_domain R :=
{ .. (‹_› : ring R), .. (‹_› : is_integral_domain R) }
|
e653ea57fa8ffc89bcdd99a07b453ed666034834 | 6dbfa3fb323bb2e2c3917c61d46053dd0de739cd | /src/lemmas2.lean | 0a2b97786a6e09857a798c20398d76242638681a | [] | no_license | amichaelsen/Lean-Chinese-Remainder-Theorem | b054fc32e0f5383eb83ec7155932c50d92de1038 | 9efb4b97da30e5aedb6af3c95d5132ac300f2902 | refs/heads/master | 1,680,787,277,127 | 1,619,155,524,000 | 1,619,155,524,000 | 275,883,966 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,747 | lean | import data.nat.basic
import data.nat.modeq
import data.nat.gcd
import data.zmod.basic
import tactic
import algebra.euclidean_domain
import data.int.basic
import data.equiv.ring
noncomputable theory
open nat nat.modeq zmod euclidean_domain
namespace lemmas2
/- LEMMAS FOR CHINESE REMAINDER THEOREM WITH 2 CONGRUENCE RELATIONS -/
/--
Two natural numbers are equal if and only if they are mutual divisors
-/
lemma eq_iff_dvd_dvd {n m : ℕ } : n = m ↔ m ∣ n ∧ n ∣ m :=
begin
split,
intro H,
rw H,
split;
refl,
intro H,
rcases H with ⟨⟨c, hc⟩, ⟨d, hd⟩⟩,
rw hd,
rw hc at hd,
induction m with x hx,
rw zero_mul at hc,
rw hc,
ring,
rw mul_assoc at hd,
have hd' : x.succ * (c * d) = x.succ,
linarith,
rw mul_right_eq_self_iff at hd',
have h : d = 1,
rw nat.mul_eq_one_iff at hd',
exact hd'.2,
rw h,
ring,
exact succ_pos',
end
/--
Given coprime natural numbers M1 M2, find the inverse of M2 M1
assuming that both are nonzero so avoid (1,0) case which is silly.
-/
lemma nat_inv (M1 M2 : ℕ ) (M1pos : 0 < M1) (M2pos : 0 < M2) (H : coprime M1 M2) :
∃ b1 : ℕ, modeq M1 (b1 * M2) 1 :=
begin
-- first cast to Z/M1 Z and get the group inverse
have hb1 := mul_inv_eq_gcd (M2 : zmod M1),
have H' := coprime.symm H,
unfold coprime at *,
rw val_nat_cast M2 at hb1,
have H'' : (M2 % M1).gcd M1 = M2.gcd M1,
begin
have qr := div_add_mod (M2 : ℤ) (M1 : ℤ ),
have qr' : (M1 * (M2 / M1) + M2 % M1) = M2,
begin
rw [← int.coe_nat_div M2 M1,
← int.coe_nat_mod M2 M1,
← int.coe_nat_mul _ _ ,
← int.coe_nat_add _ _ ,
int.coe_nat_inj'] at qr,
exact qr,
end,
-- want to show M2.gcd M1 ∣ (M2 % M1).gcd M1,
have div1 : M2.gcd M1 ∣ (M2 % M1).gcd M1,
begin
have f1 := gcd_dvd_left M2 M1,
have f2 := gcd_dvd_right M2 M1,
have f3 : M2.gcd M1 ∣ M1 * (M2 / M1),
{cases f2 with c hc,
use c * (M2 / M1),
rw ← mul_assoc,
rw ← hc,},
have f4 : M2.gcd M1∣ M1 * (M2 / M1) + M2 % M1,
{rw qr',
exact f1,},
rw nat.dvd_add_right f3 at f4,
exact dvd_gcd f4 f2,
end,
-- want to show (M2 % M1).gcd M1 ∣ M2.gcd M1,
have div2 : (M2 % M1).gcd M1 ∣ M2.gcd M1,
begin
have f1 := gcd_dvd_right (M2 % M1) M1,
have f2 := gcd_dvd_left (M2 % M1) M1,
have f3 := dvd_mul_of_dvd_left f1 (M2 / M1),
have f4 : (M2 % M1).gcd M1 ∣ M2,
begin
have k := (nat.dvd_add_right f3).2 f2,
rw qr' at k,
exact k,
end,
exact dvd_gcd f4 f1,
end,
have div : M2.gcd M1 ∣ (M2 % M1).gcd M1 ∧ (M2 % M1).gcd M1 ∣ M2.gcd M1,
exact ⟨div1, div2⟩,
rw ← eq_iff_dvd_dvd at div,
exact div,
end,
-- use coprimeness and equality of gcd's to get as an actual inverse
rw [H'',H'] at hb1,
use (M2 : zmod M1)⁻¹.val,
--translate this to zmod M1
rw ← nat_coe_eq_nat_coe_iff _ _ _,
simp at *,
rw mul_comm,
have fact : (((M2 : zmod M1)⁻¹.val) : zmod M1) = (M2 : zmod M1)⁻¹,
begin
rw @nat_cast_zmod_val _ _,
use M1pos,
end,
rw fact,
exact hb1,
end
/- LEMMAS FOR CHINESE REMAINDER THEOREM WITH K CONGRUENCE RELATIONS -/
end lemmas2
|
ac1631a2baa48dfc06006b6ae70852bd544b812a | a4673261e60b025e2c8c825dfa4ab9108246c32e | /stage0/src/Lean/Meta/DiscrTree.lean | 4c9e6b7aced4d5e3799a72b769e69ea8555a4397 | [
"Apache-2.0"
] | permissive | jcommelin/lean4 | c02dec0cc32c4bccab009285475f265f17d73228 | 2909313475588cc20ac0436e55548a4502050d0a | refs/heads/master | 1,674,129,550,893 | 1,606,415,348,000 | 1,606,415,348,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 14,746 | 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.Meta.Basic
import Lean.Meta.FunInfo
import Lean.Meta.InferType
namespace Lean.Meta.DiscrTree
/-
(Imperfect) discrimination trees.
We use a hybrid representation.
- A `PersistentHashMap` for the root node which usually contains many children.
- A sorted array of key/node pairs for inner nodes.
The edges are labeled by keys:
- Constant names (and arity). Universe levels are ignored.
- Free variables (and arity). Thus, an entry in the discrimination tree
may reference hypotheses from the local context.
- Literals
- Star/Wildcard. We use them to represent metavariables and terms
we want to ignore. We ignore implicit arguments and proofs.
- Other. We use to represent other kinds of terms (e.g., nested lambda, forall, sort, etc).
We reduce terms using `TransparencyMode.reducible`. Thus, all reducible
definitions in an expression `e` are unfolded before we insert it into the
discrimination tree.
Recall that projections from classes are **NOT** reducible.
For example, the expressions `Add.add α (ringAdd ?α ?s) ?x ?x`
and `Add.add Nat Nat.hasAdd a b` generates paths with the following keys
respctively
```
⟨Add.add, 4⟩, *, *, *, *
⟨Add.add, 4⟩, *, *, ⟨a,0⟩, ⟨b,0⟩
```
That is, we don't reduce `Add.add Nat inst a b` into `Nat.add a b`.
We say the `Add.add` applications are the de-facto canonical forms in
the metaprogramming framework.
Moreover, it is the metaprogrammer's responsibility to re-pack applications such as
`Nat.add a b` into `Add.add Nat inst a b`.
Remark: we store the arity in the keys
1- To be able to implement the "skip" operation when retrieving "candidate"
unifiers.
2- Distinguish partial applications `f a`, `f a b`, and `f a b c`.
-/
def Key.ctorIdx : Key → Nat
| Key.star => 0
| Key.other => 1
| Key.lit _ => 2
| Key.fvar _ _ => 3
| Key.const _ _ => 4
def Key.lt : Key → Key → Bool
| Key.lit v₁, Key.lit v₂ => v₁ < v₂
| Key.fvar n₁ a₁, Key.fvar n₂ a₂ => Name.quickLt n₁ n₂ || (n₁ == n₂ && a₁ < a₂)
| Key.const n₁ a₁, Key.const n₂ a₂ => Name.quickLt n₁ n₂ || (n₁ == n₂ && a₁ < a₂)
| k₁, k₂ => k₁.ctorIdx < k₂.ctorIdx
instance : HasLess Key := ⟨fun a b => Key.lt a b⟩
instance (a b : Key) : Decidable (a < b) := inferInstanceAs (Decidable (Key.lt a b))
def Key.format : Key → Format
| Key.star => "*"
| Key.other => "◾"
| Key.lit (Literal.natVal v) => fmt v
| Key.lit (Literal.strVal v) => repr v
| Key.const k _ => fmt k
| Key.fvar k _ => fmt k
instance : ToFormat Key := ⟨Key.format⟩
def Key.arity : Key → Nat
| Key.const _ a => a
| Key.fvar _ a => a
| _ => 0
instance {α} : Inhabited (Trie α) := ⟨Trie.node #[] #[]⟩
def empty {α} : DiscrTree α := { root := {} }
/- The discrimination tree ignores implicit arguments and proofs.
We use the following auxiliary id as a "mark". -/
private def tmpMVarId : MVarId := `_discr_tree_tmp
private def tmpStar := mkMVar tmpMVarId
instance {α} : Inhabited (DiscrTree α) := ⟨{}⟩
/--
Return true iff the argument should be treated as a "wildcard" by the discrimination tree.
- We ignore proofs because of proof irrelevance. It doesn't make sense to try to
index their structure.
- We ignore instance implicit arguments (e.g., `[Add α]`) because they are "morally" canonical.
Moreover, we may have many definitionally equal terms floating around.
Example: `Ring.hasAdd Int Int.isRing` and `Int.hasAdd`.
- We considered ignoring implicit arguments (e.g., `{α : Type}`) since users don't "see" them,
and may not even understand why some simplification rule is not firing.
However, in type class resolution, we have instance such as `Decidable (@Eq Nat x y)`,
where `Nat` is an implicit argument. Thus, we would add the path
```
Decidable -> Eq -> * -> * -> * -> [Nat.decEq]
```
to the discrimination tree IF we ignored the implict `Nat` argument.
This would be BAD since **ALL** decidable equality instances would be in the same path.
So, we index implicit arguments if they are types.
This setting seems sensible for simplification lemmas such as:
```
forall (x y : Unit), (@Eq Unit x y) = true
```
If we ignore the implicit argument `Unit`, the `DiscrTree` will say it is a candidate
simplification lemma for any equality in our goal.
Remark: if users have problems with the solution above, we may provide a `noIndexing` annotation,
and `ignoreArg` would return true for any term of the form `noIndexing t`.
-/
private def ignoreArg (a : Expr) (i : Nat) (infos : Array ParamInfo) : MetaM Bool :=
if h : i < infos.size then
let info := infos.get ⟨i, h⟩
if info.instImplicit then
pure true
else if info.implicit then
not <$> isType a
else
isProof a
else
isProof a
private partial def pushArgsAux (infos : Array ParamInfo) : Nat → Expr → Array Expr → MetaM (Array Expr)
| i, Expr.app f a _, todo => do
if (← ignoreArg a i infos) then
pushArgsAux infos (i-1) f (todo.push tmpStar)
else
pushArgsAux infos (i-1) f (todo.push a)
| _, _, todo => pure todo
private partial def whnfEta (e : Expr) : MetaM Expr := do
let e ← whnf e
match e.etaExpandedStrict? with
| some e => whnfEta e
| none => pure e
/-
TODO: add a parameter (wildcardConsts : NameSet) to `DiscrTree.insert`.
Then, `DiscrTree` users may control which symbols should be treated as wildcards.
Different `DiscrTree` users may populate this set using, for example, attributes. -/
private def shouldAddAsStar (constName : Name) : Bool :=
constName == `Nat.zero || constName == `Nat.succ || constName == `Nat.add || constName == `Add.add
private def pushArgs (todo : Array Expr) (e : Expr) : MetaM (Key × Array Expr) := do
let e ← whnfEta e
let fn := e.getAppFn
let push (k : Key) (nargs : Nat) : MetaM (Key × Array Expr) := do
let info ← getFunInfoNArgs fn nargs
let todo ← pushArgsAux info.paramInfo (nargs-1) e todo
pure (k, todo)
match fn with
| Expr.lit v _ => pure (Key.lit v, todo)
| Expr.const c _ _ =>
if shouldAddAsStar c then
pure (Key.star, todo)
else
let nargs := e.getAppNumArgs
push (Key.const c nargs) nargs
| Expr.fvar fvarId _ =>
let nargs := e.getAppNumArgs
push (Key.fvar fvarId nargs) nargs
| Expr.mvar mvarId _ =>
if mvarId == tmpMVarId then
-- We use `tmp to mark implicit arguments and proofs
pure (Key.star, todo)
else if (← isReadOnlyOrSyntheticOpaqueExprMVar mvarId) then
pure (Key.other, todo)
else
pure (Key.star, todo)
| _ => pure (Key.other, todo)
partial def mkPathAux (todo : Array Expr) (keys : Array Key) : MetaM (Array Key) := do
if todo.isEmpty then
pure keys
else
let e := todo.back
let todo := todo.pop
let (k, todo) ← pushArgs todo e
mkPathAux todo (keys.push k)
private def initCapacity := 8
def mkPath (e : Expr) : MetaM (Array Key) :=
withReducible do
let todo : Array Expr := Array.mkEmpty initCapacity
let keys : Array Key := Array.mkEmpty initCapacity
mkPathAux (todo.push e) keys
private partial def createNodes {α} (keys : Array Key) (v : α) (i : Nat) : Trie α :=
if h : i < keys.size then
let k := keys.get ⟨i, h⟩
let c := createNodes keys v (i+1)
Trie.node #[] #[(k, c)]
else
Trie.node #[v] #[]
private def insertVal {α} [BEq α] (vs : Array α) (v : α) : Array α :=
if vs.contains v then vs else vs.push v
private partial def insertAux {α} [BEq α] (keys : Array Key) (v : α) : Nat → Trie α → Trie α
| i, Trie.node vs cs =>
if h : i < keys.size then
let k := keys.get ⟨i, h⟩
let c := Id.run $ cs.binInsertM
(fun a b => a.1 < b.1)
(fun ⟨_, s⟩ => let c := insertAux keys v (i+1) s; (k, c)) -- merge with existing
(fun _ => let c := createNodes keys v (i+1); (k, c))
(k, arbitrary)
Trie.node vs c
else
Trie.node (insertVal vs v) cs
def insertCore {α} [BEq α] (d : DiscrTree α) (keys : Array Key) (v : α) : DiscrTree α :=
if keys.isEmpty then panic! "invalid key sequence"
else
let k := keys[0]
match d.root.find? k with
| none =>
let c := createNodes keys v 1
{ root := d.root.insert k c }
| some c =>
let c := insertAux keys v 1 c
{ root := d.root.insert k c }
def insert {α} [BEq α] (d : DiscrTree α) (e : Expr) (v : α) : MetaM (DiscrTree α) := do
let keys ← mkPath e
pure $ d.insertCore keys v
partial def Trie.format {α} [ToFormat α] : Trie α → Format
| Trie.node vs cs => Format.group $ Format.paren $
"node" ++ (if vs.isEmpty then Format.nil else " " ++ fmt vs)
++ Format.join (cs.toList.map $ fun ⟨k, c⟩ => Format.line ++ Format.paren (fmt k ++ " => " ++ format c))
instance {α} [ToFormat α] : ToFormat (Trie α) := ⟨Trie.format⟩
partial def format {α} [ToFormat α] (d : DiscrTree α) : Format :=
let (_, r) := d.root.foldl
(fun (p : Bool × Format) k c =>
(false, p.2 ++ (if p.1 == true then Format.nil else Format.line) ++ Format.paren (fmt k ++ " => " ++ fmt c))) -- TODO: fix p.1 == true
(true, Format.nil)
Format.group r
instance {α} [ToFormat α] : ToFormat (DiscrTree α) := ⟨format⟩
private def getKeyArgs (e : Expr) (isMatch? : Bool) : MetaM (Key × Array Expr) := do
let e ← whnfEta e
match e.getAppFn with
| Expr.lit v _ => pure (Key.lit v, #[])
| Expr.const c _ _ =>
let nargs := e.getAppNumArgs
pure (Key.const c nargs, e.getAppRevArgs)
| Expr.fvar fvarId _ =>
let nargs := e.getAppNumArgs
pure (Key.fvar fvarId nargs, e.getAppRevArgs)
| Expr.mvar mvarId _ =>
if isMatch? then
pure (Key.other, #[])
else do
let ctx ← read
if ctx.config.isDefEqStuckEx then
/-
When the configuration flag `isDefEqStuckEx` is set to true,
we want `isDefEq` to throw an exception whenever it tries to assign
a read-only metavariable.
This feature is useful for type class resolution where
we may want to notify the caller that the TC problem may be solveable
later after it assigns `?m`.
The method `DiscrTree.getUnify e` returns candidates `c` that may "unify" with `e`.
That is, `isDefEq c e` may return true. Now, consider `DiscrTree.getUnify d (Add ?m)`
where `?m` is a read-only metavariable, and the discrimination tree contains the keys
`HadAdd Nat` and `Add Int`. If `isDefEqStuckEx` is set to true, we must treat `?m` as
a regular metavariable here, otherwise we return the empty set of candidates.
This is incorrect because it is equivalent to saying that there is no solution even if
the caller assigns `?m` and try again. -/
pure (Key.star, #[])
else if (← isReadOnlyOrSyntheticOpaqueExprMVar mvarId) then
pure (Key.other, #[])
else
pure (Key.star, #[])
| _ => pure (Key.other, #[])
private abbrev getMatchKeyArgs (e : Expr) : MetaM (Key × Array Expr) :=
getKeyArgs e true
private abbrev getUnifyKeyArgs (e : Expr) : MetaM (Key × Array Expr) :=
getKeyArgs e false
private partial def getMatchAux {α} : Array Expr → Trie α → Array α → MetaM (Array α)
| todo, Trie.node vs cs, result =>
if todo.isEmpty then pure $ result ++ vs
else if cs.isEmpty then pure result
else do
let e := todo.back
let todo := todo.pop
let first := cs[0] /- Recall that `Key.star` is the minimal key -/
let (k, args) ← getMatchKeyArgs e
/- We must always visit `Key.star` edges since they are wildcards.
Thus, `todo` is not used linearly when there is `Key.star` edge
and there is an edge for `k` and `k != Key.star`. -/
let visitStarChild (result : Array α) : MetaM (Array α) :=
if first.1 == Key.star then getMatchAux todo first.2 result else pure result
match k with
| Key.star => visitStarChild result
| _ =>
match cs.binSearch (k, arbitrary) (fun a b => a.1 < b.1) with
| none => visitStarChild result
| some c =>
let result ← visitStarChild result
getMatchAux (todo ++ args) c.2 result
private def getStarResult {α} (d : DiscrTree α) : Array α :=
let result : Array α := Array.mkEmpty initCapacity
match d.root.find? Key.star with
| none => result
| some (Trie.node vs _) => result ++ vs
def getMatch {α} (d : DiscrTree α) (e : Expr) : MetaM (Array α) :=
withReducible do
let result := getStarResult d
let (k, args) ← getMatchKeyArgs e
match k with
| Key.star => pure result
| _ =>
match d.root.find? k with
| none => pure result
| some c => getMatchAux args c result
private partial def getUnifyAux {α} : Nat → Array Expr → Trie α → (Array α) → MetaM (Array α)
| skip+1, todo, Trie.node vs cs, result =>
if cs.isEmpty then pure result
else cs.foldlM (fun result ⟨k, c⟩ => getUnifyAux (skip + k.arity) todo c result) result
| 0, todo, Trie.node vs cs, result => do
if todo.isEmpty then pure (result ++ vs)
else if cs.isEmpty then pure result
else
let e := todo.back
let todo := todo.pop
let (k, args) ← getUnifyKeyArgs e
match k with
| Key.star => cs.foldlM (fun result ⟨k, c⟩ => getUnifyAux k.arity todo c result) result
| _ =>
let first := cs[0]
let visitStarChild (result : Array α) : MetaM (Array α) :=
if first.1 == Key.star then getUnifyAux 0 todo first.2 result else pure result
match cs.binSearch (k, arbitrary) (fun a b => a.1 < b.1) with
| none => visitStarChild result
| some c =>
let result ← visitStarChild result
getUnifyAux 0 (todo ++ args) c.2 result
def getUnify {α} (d : DiscrTree α) (e : Expr) : MetaM (Array α) :=
withReducible do
let (k, args) ← getUnifyKeyArgs e
match k with
| Key.star => d.root.foldlM (fun result k c => getUnifyAux k.arity #[] c result) #[]
| _ =>
let result := getStarResult d
match d.root.find? k with
| none => pure result
| some c => getUnifyAux 0 args c result
end Lean.Meta.DiscrTree
|
f112a3ec95bde0d47df97b5a708c2ae7aed028f5 | a0e23cfdd129a671bf3154ee1a8a3a72bf4c7940 | /stage0/src/Lean/Server/FileWorker/Utils.lean | 8b8b33f2def7c541c8a5a6e29ee57fa60ca74306 | [
"Apache-2.0"
] | permissive | WojciechKarpiel/lean4 | 7f89706b8e3c1f942b83a2c91a3a00b05da0e65b | f6e1314fa08293dea66a329e05b6c196a0189163 | refs/heads/master | 1,686,633,402,214 | 1,625,821,189,000 | 1,625,821,258,000 | 384,640,886 | 0 | 0 | Apache-2.0 | 1,625,903,617,000 | 1,625,903,026,000 | null | UTF-8 | Lean | false | false | 1,557 | lean | /-
Copyright (c) 2020 Wojciech Nawrocki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wojciech Nawrocki, Marc Huisinga
-/
import Lean.Server.Utils
import Lean.Server.Snapshots
import Lean.Server.AsyncList
namespace Lean.Server.FileWorker
open Snapshots
open IO
private def logSnapContent (s : Snapshot) (text : FileMap) : IO Unit :=
IO.eprintln s!"[{s.beginPos}, {s.endPos}]: `{text.source.extract s.beginPos (s.endPos-1)}`"
inductive ElabTaskError where
| aborted
| eof
| ioError (e : IO.Error)
instance : Coe IO.Error ElabTaskError := ⟨ElabTaskError.ioError⟩
structure CancelToken where
ref : IO.Ref Bool
deriving Inhabited
namespace CancelToken
def new : IO CancelToken :=
CancelToken.mk <$> IO.mkRef false
def check [MonadExceptOf ElabTaskError m] [MonadLiftT (ST RealWorld) m] [Monad m] (tk : CancelToken) : m Unit := do
let c ← tk.ref.get
if c then
throw ElabTaskError.aborted
def set (tk : CancelToken) : IO Unit :=
tk.ref.set true
end CancelToken
/-- A document editable in the sense that we track the environment
and parser state after each command so that edits can be applied
without recompiling code appearing earlier in the file. -/
structure EditableDocument where
meta : DocumentMeta
/- The first snapshot is that after the header. -/
headerSnap : Snapshot
/- Subsequent snapshots occur after each command. -/
cmdSnaps : AsyncList ElabTaskError Snapshot
cancelTk : CancelToken
deriving Inhabited
end Lean.Server.FileWorker |
11afc48a1a01b61c551fcd7f332f85f8a0cfd7a9 | a7eef317ddec01b9fc6cfbb876fe7ac00f205ac7 | /src/linear_algebra/dimension.lean | 7596f469e8fa96c39476204db73bb8dc008231f2 | [
"Apache-2.0"
] | permissive | kmill/mathlib | ea5a007b67ae4e9e18dd50d31d8aa60f650425ee | 1a419a9fea7b959317eddd556e1bb9639f4dcc05 | refs/heads/master | 1,668,578,197,719 | 1,593,629,163,000 | 1,593,629,163,000 | 276,482,939 | 0 | 0 | null | 1,593,637,960,000 | 1,593,637,959,000 | null | UTF-8 | Lean | false | false | 18,076 | lean | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Mario Carneiro, Johannes Hölzl, Sander Dahmen
-/
import linear_algebra.basis
import set_theory.ordinal
/-!
# Dimension of modules and vector spaces
## Main definitions
* The dimension of a vector space is defined as `vector_space.dim : cardinal`.
## Main statements
* `mk_eq_mk_of_basis`: the dimension theorem, any two bases of the same vector space have the same
cardinality.
* `dim_quotient_add_dim`: if V' is a submodule of V, then dim (V/V') + dim V' = dim V.
* `dim_range_add_dim_ker`: the rank-nullity theorem.
-/
noncomputable theory
universes u u' u'' v' w w'
variables {K : Type u} {V V₂ V₃ V₄ : Type u'}
variables {ι : Type w} {ι' : Type w'} {η : Type u''} {φ : η → Type*}
-- TODO: relax these universe constraints
open_locale classical big_operators
section vector_space
variables [field K] [add_comm_group V] [vector_space K V]
include K
open submodule function set
variables (K V)
/-- the dimension of a vector space, defined as a term of type `cardinal` -/
def vector_space.dim : cardinal :=
cardinal.min
(nonempty_subtype.2 (@exists_is_basis K V _ _ _))
(λ b, cardinal.mk b.1)
variables {K V}
open vector_space
section
theorem is_basis.le_span (zero_ne_one : (0 : K) ≠ 1) {v : ι → V} {J : set V} (hv : is_basis K v)
(hJ : span K J = ⊤) : cardinal.mk (range v) ≤ cardinal.mk J :=
begin
cases le_or_lt cardinal.omega (cardinal.mk J) with oJ oJ,
{ have := cardinal.mk_range_eq_of_injective (linear_independent.injective zero_ne_one hv.1),
let S : J → set ι := λ j, ↑(is_basis.repr hv j).support,
let S' : J → set V := λ j, v '' S j,
have hs : range v ⊆ ⋃ j, S' j,
{ intros b hb,
rcases mem_range.1 hb with ⟨i, hi⟩,
have : span K J ≤ comap hv.repr (finsupp.supported K K (⋃ j, S j)) :=
span_le.2 (λ j hj x hx, ⟨_, ⟨⟨j, hj⟩, rfl⟩, hx⟩),
rw hJ at this,
replace : hv.repr (v i) ∈ (finsupp.supported K K (⋃ j, S j)) := this trivial,
rw [hv.repr_eq_single, finsupp.mem_supported,
finsupp.support_single_ne_zero zero_ne_one.symm] at this,
subst b,
rcases mem_Union.1 (this (finset.mem_singleton_self _)) with ⟨j, hj⟩,
exact mem_Union.2 ⟨j, (mem_image _ _ _).2 ⟨i, hj, rfl⟩⟩ },
refine le_of_not_lt (λ IJ, _),
suffices : cardinal.mk (⋃ j, S' j) < cardinal.mk (range v),
{ exact not_le_of_lt this ⟨set.embedding_of_subset _ _ hs⟩ },
refine lt_of_le_of_lt (le_trans cardinal.mk_Union_le_sum_mk
(cardinal.sum_le_sum _ (λ _, cardinal.omega) _)) _,
{ exact λ j, le_of_lt (cardinal.lt_omega_iff_finite.2 $ (finset.finite_to_set _).image _) },
{ rwa [cardinal.sum_const, cardinal.mul_eq_max oJ (le_refl _), max_eq_left oJ] } },
{ rcases exists_finite_card_le_of_finite_of_linear_independent_of_span
(cardinal.lt_omega_iff_finite.1 oJ) hv.1.to_subtype_range _ with ⟨fI, hi⟩,
{ rwa [← cardinal.nat_cast_le, cardinal.finset_card, set.finite.coe_to_finset,
cardinal.finset_card, set.finite.coe_to_finset] at hi, },
{ rw hJ, apply set.subset_univ } },
end
end
/-- dimension theorem -/
theorem mk_eq_mk_of_basis {v : ι → V} {v' : ι' → V}
(hv : is_basis K v) (hv' : is_basis K v') :
cardinal.lift.{w w'} (cardinal.mk ι) = cardinal.lift.{w' w} (cardinal.mk ι') :=
begin
rw ←cardinal.lift_inj.{(max w w') u'},
rw [cardinal.lift_lift, cardinal.lift_lift],
apply le_antisymm,
{ convert cardinal.lift_le.{u' (max w w')}.2 (hv.le_span zero_ne_one hv'.2),
{ rw cardinal.lift_max.{w u' w'},
apply (cardinal.mk_range_eq_of_injective (hv.injective zero_ne_one)).symm, },
{ rw cardinal.lift_max.{w' u' w},
apply (cardinal.mk_range_eq_of_injective (hv'.injective zero_ne_one)).symm, }, },
{ convert cardinal.lift_le.{u' (max w w')}.2 (hv'.le_span zero_ne_one hv.2),
{ rw cardinal.lift_max.{w' u' w},
apply (cardinal.mk_range_eq_of_injective (hv'.injective zero_ne_one)).symm, },
{ rw cardinal.lift_max.{w u' w'},
apply (cardinal.mk_range_eq_of_injective (hv.injective zero_ne_one)).symm, }, }
end
theorem is_basis.mk_range_eq_dim {v : ι → V} (h : is_basis K v) :
cardinal.mk (range v) = dim K V :=
begin
have := show ∃ v', dim K V = _, from cardinal.min_eq _ _,
rcases this with ⟨v', e⟩,
rw e,
apply cardinal.lift_inj.1,
rw cardinal.mk_range_eq_of_injective (h.injective zero_ne_one),
convert @mk_eq_mk_of_basis _ _ _ _ _ _ _ _ _ h v'.property
end
theorem is_basis.mk_eq_dim {v : ι → V} (h : is_basis K v) :
cardinal.lift.{w u'} (cardinal.mk ι) = cardinal.lift.{u' w} (dim K V) :=
by rw [←h.mk_range_eq_dim, cardinal.mk_range_eq_of_injective (h.injective zero_ne_one)]
variables [add_comm_group V₂] [vector_space K V₂]
/-- Two linearly equivalent vector spaces have the same dimension. -/
theorem linear_equiv.dim_eq (f : V ≃ₗ[K] V₂) :
dim K V = dim K V₂ :=
by letI := classical.dec_eq V;
letI := classical.dec_eq V₂; exact
let ⟨b, hb⟩ := exists_is_basis K V in
cardinal.lift_inj.1 $ hb.mk_eq_dim.symm.trans (f.is_basis hb).mk_eq_dim
@[simp] lemma dim_bot : dim K (⊥ : submodule K V) = 0 :=
by letI := classical.dec_eq V;
rw [← cardinal.lift_inj, ← (@is_basis_empty_bot pempty K V _ _ _ not_nonempty_pempty).mk_eq_dim,
cardinal.mk_pempty]
@[simp] lemma dim_top : dim K (⊤ : submodule K V) = dim K V :=
linear_equiv.dim_eq (linear_equiv.of_top _ rfl)
lemma dim_of_field (K : Type*) [field K] : dim K K = 1 :=
by rw [←cardinal.lift_inj, ← (@is_basis_singleton_one punit K _ _).mk_eq_dim, cardinal.mk_punit]
lemma dim_span {v : ι → V} (hv : linear_independent K v) :
dim K ↥(span K (range v)) = cardinal.mk (range v) :=
by rw [←cardinal.lift_inj, ← (is_basis_span hv).mk_eq_dim,
cardinal.mk_range_eq_of_injective (@linear_independent.injective ι K V v _ _ _ zero_ne_one hv)]
lemma dim_span_set {s : set V} (hs : linear_independent K (λ x, x : s → V)) :
dim K ↥(span K s) = cardinal.mk s :=
by { rw [← @set_of_mem_eq _ s, ← subtype.range_coe_subtype], exact dim_span hs }
lemma cardinal_le_dim_of_linear_independent
{ι : Type u'} {v : ι → V} (hv : linear_independent K v) :
(cardinal.mk ι) ≤ (dim.{u u'} K V) :=
begin
obtain ⟨ι', v', is⟩ := exists_sum_is_basis hv,
simpa using le_trans
(cardinal.lift_mk_le.{u' u' u'}.2 ⟨@function.embedding.inl ι ι'⟩)
(le_of_eq is.mk_eq_dim),
end
lemma cardinal_le_dim_of_linear_independent'
{s : set V} (hs : linear_independent K (λ x, x : s → V)) :
cardinal.mk s ≤ dim K V :=
begin
-- extend s to a basis
obtain ⟨b, ss, h⟩ := exists_subset_is_basis hs,
rw [←h.mk_range_eq_dim, subtype.range_coe],
apply cardinal.mk_le_of_injective (inclusion_injective ss),
end
lemma dim_span_le (s : set V) : dim K (span K s) ≤ cardinal.mk s :=
begin
classical,
rcases
exists_linear_independent (linear_independent_empty K V) (set.empty_subset s)
with ⟨b, hb, _, hsb, hlib⟩,
have hsab : span K s = span K b,
from span_eq_of_le _ hsb (span_le.2 (λ x hx, subset_span (hb hx))),
convert cardinal.mk_le_mk_of_subset hb,
rw [hsab, dim_span_set hlib]
end
lemma dim_span_of_finset (s : finset V) :
dim K (span K (↑s : set V)) < cardinal.omega :=
calc dim K (span K (↑s : set V)) ≤ cardinal.mk (↑s : set V) : dim_span_le ↑s
... = s.card : by rw ←cardinal.finset_card
... < cardinal.omega : cardinal.nat_lt_omega _
theorem dim_prod : dim K (V × V₂) = dim K V + dim K V₂ :=
begin
letI := classical.dec_eq V,
letI := classical.dec_eq V₂,
rcases exists_is_basis K V with ⟨b, hb⟩,
rcases exists_is_basis K V₂ with ⟨c, hc⟩,
rw [← cardinal.lift_inj,
← @is_basis.mk_eq_dim K (V × V₂) _ _ _ _ _ (is_basis_inl_union_inr hb hc),
cardinal.lift_add, cardinal.lift_mk,
← hb.mk_eq_dim, ← hc.mk_eq_dim,
cardinal.lift_mk, cardinal.lift_mk,
cardinal.add_def (ulift b) (ulift c)],
exact cardinal.lift_inj.1 (cardinal.lift_mk_eq.2
⟨equiv.ulift.trans (equiv.sum_congr (@equiv.ulift b) (@equiv.ulift c)).symm ⟩),
end
theorem dim_quotient_add_dim (p : submodule K V) :
dim K p.quotient + dim K p = dim K V :=
by classical; exact let ⟨f⟩ := quotient_prod_linear_equiv p in dim_prod.symm.trans f.dim_eq
theorem dim_quotient_le (p : submodule K V) :
dim K p.quotient ≤ dim K V :=
by { rw ← dim_quotient_add_dim p, exact cardinal.le_add_right _ _ }
/-- rank-nullity theorem -/
theorem dim_range_add_dim_ker (f : V →ₗ[K] V₂) : dim K f.range + dim K f.ker = dim K V :=
begin
haveI := λ (p : submodule K V), classical.dec_eq p.quotient,
rw [← f.quot_ker_equiv_range.dim_eq, dim_quotient_add_dim]
end
lemma dim_range_le (f : V →ₗ[K] V₂) : dim K f.range ≤ dim K V :=
by rw ← dim_range_add_dim_ker f; exact le_add_right (le_refl _)
lemma dim_map_le (f : V →ₗ V₂) (p : submodule K V) : dim K (p.map f) ≤ dim K p :=
begin
have h := dim_range_le (f.comp (submodule.subtype p)),
rwa [linear_map.range_comp, range_subtype] at h,
end
lemma dim_range_of_surjective (f : V →ₗ[K] V₂) (h : surjective f) : dim K f.range = dim K V₂ :=
begin
refine linear_equiv.dim_eq (linear_equiv.of_bijective (submodule.subtype _) _ _),
exact linear_map.ker_eq_bot.2 subtype.val_injective,
rwa [range_subtype, linear_map.range_eq_top]
end
lemma dim_eq_of_surjective (f : V →ₗ[K] V₂) (h : surjective f) : dim K V = dim K V₂ + dim K f.ker :=
by rw [← dim_range_add_dim_ker f, ← dim_range_of_surjective f h]
lemma dim_le_of_surjective (f : V →ₗ[K] V₂) (h : surjective f) : dim K V₂ ≤ dim K V :=
by rw [dim_eq_of_surjective f h]; refine le_add_right (le_refl _)
lemma dim_eq_of_injective (f : V →ₗ[K] V₂) (h : injective f) : dim K V = dim K f.range :=
by rw [← dim_range_add_dim_ker f, linear_map.ker_eq_bot.2 h]; simp [dim_bot]
lemma dim_submodule_le (s : submodule K V) : dim K s ≤ dim K V :=
by { rw ← dim_quotient_add_dim s, exact cardinal.le_add_left _ _ }
lemma dim_le_of_injective (f : V →ₗ[K] V₂) (h : injective f) :
dim K V ≤ dim K V₂ :=
by { rw [dim_eq_of_injective f h], exact dim_submodule_le _ }
lemma dim_le_of_submodule (s t : submodule K V) (h : s ≤ t) : dim K s ≤ dim K t :=
dim_le_of_injective (of_le h) $ assume ⟨x, hx⟩ ⟨y, hy⟩ eq,
subtype.eq $ show x = y, from subtype.ext_iff_val.1 eq
section
variables [add_comm_group V₃] [vector_space K V₃]
variables [add_comm_group V₄] [vector_space K V₄]
open linear_map
/-- This is mostly an auxiliary lemma for `dim_sup_add_dim_inf_eq`. -/
lemma dim_add_dim_split
(db : V₃ →ₗ[K] V) (eb : V₄ →ₗ[K] V) (cd : V₂ →ₗ[K] V₃) (ce : V₂ →ₗ[K] V₄)
(hde : ⊤ ≤ db.range ⊔ eb.range)
(hgd : ker cd = ⊥)
(eq : db.comp cd = eb.comp ce)
(eq₂ : ∀d e, db d = eb e → (∃c, cd c = d ∧ ce c = e)) :
dim K V + dim K V₂ = dim K V₃ + dim K V₄ :=
have hf : surjective (coprod db eb),
begin
refine (range_eq_top.1 $ top_unique $ _),
rwa [← map_top, ← prod_top, map_coprod_prod]
end,
begin
conv {to_rhs, rw [← dim_prod, dim_eq_of_surjective _ hf] },
congr' 1,
apply linear_equiv.dim_eq,
refine linear_equiv.of_bijective _ _ _,
{ refine cod_restrict _ (prod cd (- ce)) _,
{ assume c,
simp only [add_eq_zero_iff_eq_neg, prod_apply, mem_ker,
coprod_apply, neg_neg, map_neg, neg_apply],
exact linear_map.ext_iff.1 eq c } },
{ rw [ker_cod_restrict, ker_prod, hgd, bot_inf_eq] },
{ rw [eq_top_iff, range_cod_restrict, ← map_le_iff_le_comap, map_top, range_subtype],
rintros ⟨d, e⟩,
have h := eq₂ d (-e),
simp only [add_eq_zero_iff_eq_neg, prod_apply, mem_ker, mem_coe, prod.mk.inj_iff,
coprod_apply, map_neg, neg_apply, linear_map.mem_range] at ⊢ h,
assume hde,
rcases h hde with ⟨c, h₁, h₂⟩,
refine ⟨c, h₁, _⟩,
rw [h₂, _root_.neg_neg] }
end
lemma dim_sup_add_dim_inf_eq (s t : submodule K V) :
dim K (s ⊔ t : submodule K V) + dim K (s ⊓ t : submodule K V) = dim K s + dim K t :=
dim_add_dim_split (of_le le_sup_left) (of_le le_sup_right) (of_le inf_le_left) (of_le inf_le_right)
begin
rw [← map_le_map_iff' (ker_subtype $ s ⊔ t), map_sup, map_top,
← linear_map.range_comp, ← linear_map.range_comp, subtype_comp_of_le, subtype_comp_of_le,
range_subtype, range_subtype, range_subtype],
exact le_refl _
end
(ker_of_le _ _ _)
begin ext ⟨x, hx⟩, refl end
begin
rintros ⟨b₁, hb₁⟩ ⟨b₂, hb₂⟩ eq,
have : b₁ = b₂ := congr_arg subtype.val eq,
subst this,
exact ⟨⟨b₁, hb₁, hb₂⟩, rfl, rfl⟩
end
lemma dim_add_le_dim_add_dim (s t : submodule K V) :
dim K (s ⊔ t : submodule K V) ≤ dim K s + dim K t :=
by rw [← dim_sup_add_dim_inf_eq]; exact le_add_right (le_refl _)
end
section fintype
variable [fintype η]
variables [∀i, add_comm_group (φ i)] [∀i, vector_space K (φ i)]
open linear_map
lemma dim_pi : vector_space.dim K (Πi, φ i) = cardinal.sum (λi, vector_space.dim K (φ i)) :=
begin
choose b hb using assume i, exists_is_basis K (φ i),
have : is_basis K (λ (ji : Σ j, b j), std_basis K (λ j, φ j) ji.fst ji.snd.val),
by apply pi.is_basis_std_basis _ hb,
rw [←cardinal.lift_inj, ← this.mk_eq_dim],
simp [λ i, (hb i).mk_range_eq_dim.symm, cardinal.sum_mk]
end
lemma dim_fun {V η : Type u} [fintype η] [add_comm_group V] [vector_space K V] :
vector_space.dim K (η → V) = fintype.card η * vector_space.dim K V :=
by rw [dim_pi, cardinal.sum_const, cardinal.fintype_card]
lemma dim_fun_eq_lift_mul :
vector_space.dim K (η → V) = (fintype.card η : cardinal.{max u'' u'}) *
cardinal.lift.{u' u''} (vector_space.dim K V) :=
by rw [dim_pi, cardinal.sum_const_eq_lift_mul, cardinal.fintype_card, cardinal.lift_nat_cast]
lemma dim_fun' : vector_space.dim K (η → K) = fintype.card η :=
by rw [dim_fun_eq_lift_mul, dim_of_field K, cardinal.lift_one, mul_one, cardinal.nat_cast_inj]
lemma dim_fin_fun (n : ℕ) : dim K (fin n → K) = n :=
by simp [dim_fun']
end fintype
lemma exists_mem_ne_zero_of_ne_bot {s : submodule K V} (h : s ≠ ⊥) : ∃ b : V, b ∈ s ∧ b ≠ 0 :=
begin
classical,
by_contradiction hex,
have : ∀x∈s, (x:V) = 0, { simpa only [not_exists, not_and, not_not, ne.def] using hex },
exact (h $ bot_unique $ assume s hs, (submodule.mem_bot K).2 $ this s hs)
end
lemma exists_mem_ne_zero_of_dim_pos {s : submodule K V} (h : vector_space.dim K s > 0) :
∃ b : V, b ∈ s ∧ b ≠ 0 :=
exists_mem_ne_zero_of_ne_bot $ assume eq, by rw [(>), eq, dim_bot] at h; exact lt_irrefl _ h
lemma exists_is_basis_fintype (h : dim K V < cardinal.omega) :
∃ s : (set V), (is_basis K (subtype.val : s → V)) ∧ nonempty (fintype s) :=
begin
cases exists_is_basis K V with s hs,
rw [←cardinal.lift_lt, ← is_basis.mk_eq_dim hs, cardinal.lift_lt,
cardinal.lt_omega_iff_fintype] at h,
exact ⟨s, hs, h⟩
end
section rank
/-- `rank f` is the rank of a `linear_map f`, defined as the dimension of `f.range`. -/
def rank (f : V →ₗ[K] V₂) : cardinal := dim K f.range
lemma rank_le_domain (f : V →ₗ[K] V₂) : rank f ≤ dim K V :=
by rw [← dim_range_add_dim_ker f]; exact le_add_right (le_refl _)
lemma rank_le_range (f : V →ₗ[K] V₂) : rank f ≤ dim K V₂ :=
dim_submodule_le _
lemma rank_add_le (f g : V →ₗ[K] V₂) : rank (f + g) ≤ rank f + rank g :=
calc rank (f + g) ≤ dim K (f.range ⊔ g.range : submodule K V₂) :
begin
refine dim_le_of_submodule _ _ _,
exact (linear_map.range_le_iff_comap.2 $ eq_top_iff'.2 $
assume x, show f x + g x ∈ (f.range ⊔ g.range : submodule K V₂), from
mem_sup.2 ⟨_, mem_image_of_mem _ (mem_univ _), _, mem_image_of_mem _ (mem_univ _), rfl⟩)
end
... ≤ rank f + rank g : dim_add_le_dim_add_dim _ _
@[simp] lemma rank_zero : rank (0 : V →ₗ[K] V₂) = 0 :=
by rw [rank, linear_map.range_zero, dim_bot]
lemma rank_finset_sum_le {η} (s : finset η) (f : η → V →ₗ[K] V₂) :
rank (∑ d in s, f d) ≤ ∑ d in s, rank (f d) :=
@finset.sum_hom_rel _ _ _ _ _ (λa b, rank a ≤ b) f (λ d, rank (f d)) s (le_of_eq rank_zero)
(λ i g c h, le_trans (rank_add_le _ _) (add_le_add_left' h))
variables [add_comm_group V₃] [vector_space K V₃]
lemma rank_comp_le1 (g : V →ₗ[K] V₂) (f : V₂ →ₗ[K] V₃) : rank (f.comp g) ≤ rank f :=
begin
refine dim_le_of_submodule _ _ _,
rw [linear_map.range_comp],
exact image_subset _ (subset_univ _)
end
lemma rank_comp_le2 (g : V →ₗ[K] V₂) (f : V₂ →ₗ V₃) : rank (f.comp g) ≤ rank g :=
by rw [rank, rank, linear_map.range_comp]; exact dim_map_le _ _
end rank
lemma dim_zero_iff_forall_zero : vector_space.dim K V = 0 ↔ ∀ x : V, x = 0 :=
begin
split,
{ intros h x,
cases exists_is_basis K V with w hw,
have card_mk_range := hw.mk_range_eq_dim,
rw [h, cardinal.mk_emptyc_iff, subtype.range_coe] at card_mk_range,
simpa [card_mk_range] using hw.mem_span x },
{ intro h,
have : (⊤ : submodule K V) = ⊥,
{ ext x, simp [h x] },
rw [←dim_top, this, dim_bot] }
end
lemma dim_pos_iff_exists_ne_zero : 0 < vector_space.dim K V ↔ ∃ x : V, x ≠ 0 :=
begin
rw ←not_iff_not,
simpa using dim_zero_iff_forall_zero
end
end vector_space
section unconstrained_universes
variables {E : Type v'}
variables [field K] [add_comm_group V] [vector_space K V]
[add_comm_group E] [vector_space K E]
open vector_space
/-- Version of linear_equiv.dim_eq without universe constraints. -/
theorem linear_equiv.dim_eq_lift (f : V ≃ₗ[K] E) :
cardinal.lift.{u' v'} (dim K V) = cardinal.lift.{v' u'} (dim K E) :=
begin
cases exists_is_basis K V with b hb,
rw [← cardinal.lift_inj.1 hb.mk_eq_dim, ← (f.is_basis hb).mk_eq_dim, cardinal.lift_mk],
end
end unconstrained_universes
|
e658978c956aec04033d20e70f95cc185b8a4f38 | 7cdf3413c097e5d36492d12cdd07030eb991d394 | /src/game/world5/level4.lean | 5bde5af8d86916ca431a87e98830bee44a3f7416 | [] | no_license | alreadydone/natural_number_game | 3135b9385a9f43e74cfbf79513fc37e69b99e0b3 | 1a39e693df4f4e871eb449890d3c7715a25c2ec9 | refs/heads/master | 1,599,387,390,105 | 1,573,200,587,000 | 1,573,200,691,000 | 220,397,084 | 0 | 0 | null | 1,573,192,734,000 | 1,573,192,733,000 | null | UTF-8 | Lean | false | false | 1,809 | lean | /- Tactic : apply
If you have a function `h : P → Q` and your goal is `⊢ Q`
then `apply h` changes the goal to `⊢ P`. The logic is
simple: if you are trying to create a term of type `Q`,
but `h` is a function which turns terms of type `P` into
terms of type `Q`, then it will suffice to construct a
term of type `P`. A mathematician might say: "we need
to construct an element of $Q$, but we have a function $h:P\to Q$
so it suffices to construct an element of $P$". Or alternatively
"we need to prove $Q$, but we have a proof $h$ that $P\implies Q$
so it suffices to prove $P$".
-/
/-
# Function world.
## Level 4 : `apply`.
Let's do the same level again:
We are given $p \in P$ and our goal is to find an element of $U$, or
in other words to find a path through the maze that links $P$ to $U$.
In level 3 we solved this by using `have`s to move forward, from $P$
to $Q$ to $T$ to $U$. Using the `apply` tactic we can instead construct
the path backwards, moving from $U$ to $T$ to $Q$ to $P$.
Our goal is to construct an element of the set $U$. But $l:T\to U$ is
a function, so it would suffice to construct an element of $T$. Tell
Lean this by starting the proof below with
`apply l,`
and notice that our assumptions don't change but *the goal changes*
from `⊢ U` to `⊢ T`.
Keep `apply`ing functions until your goal is `P`, and try not
to get lost! Now solve this goal
with `exact p`. Note: you will need to learn the difference between
`exact p` (which works) and `exact P` (which doesn't, because $P$ is
not an element of $P$).
-/
/- Lemma : no-side-bar
We can solve a maze.
-/
lemma maze (P Q R S T U: Type)
(p : P)
(h : P → Q)
(i : Q → R)
(j : Q → T)
(k : S → T)
(l : T → U)
: U :=
begin
apply l,
apply j,
apply h,
exact p,
end |
4133c5fedc2644a287cbdcc81960f8162542a4ba | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/run/783.lean | 3d19b694944b64299d8176988ee0014c0cc89c64 | [
"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 | 125 | lean | structure MyStruct where
{α : Type u}
{β : Type v}
a : α
b : β
#check { a := 10, b := true : MyStruct }
|
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