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/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import data.option.defs
import logic.nonempty
import tactic.cache
/-!
# Miscellaneous function constructions and lemmas
-/
universes u v w
namespace function
section
variables {α β γ : Sort*} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`function.eval x : (Π x, β x) → β x`. -/
@[reducible] def eval {β : α → Sort*} (x : α) (f : Π x, β x) : β x := f x
@[simp] lemma eval_apply {β : α → Sort*} (x : α) (f : Π x, β x) : eval x f = f x := rfl
lemma comp_apply {α : Sort u} {β : Sort v} {φ : Sort w} (f : β → φ) (g : α → β) (a : α) :
(f ∘ g) a = f (g a) := rfl
lemma const_def {y : β} : (λ x : α, y) = const α y := rfl
@[simp] lemma const_apply {y : β} {x : α} : const α y x = y := rfl
@[simp] lemma const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c := rfl
@[simp] lemma comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) := rfl
lemma const_injective [nonempty α] : injective (const α : β → α → β) :=
λ y₁ y₂ h, let ⟨x⟩ := ‹nonempty α› in congr_fun h x
@[simp] lemma const_inj [nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨λ h, const_injective h, λ h, h ▸ rfl⟩
lemma id_def : @id α = λ x, x := rfl
lemma hfunext {α α': Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : Πa, β a} {f' : Πa, β' a}
(hα : α = α') (h : ∀a a', a == a' → f a == f' a') : f == f' :=
begin
subst hα,
have : ∀a, f a == f' a,
{ intro a, exact h a a (heq.refl a) },
have : β = β',
{ funext a, exact type_eq_of_heq (this a) },
subst this,
apply heq_of_eq,
funext a,
exact eq_of_heq (this a)
end
lemma funext_iff {β : α → Sort*} {f₁ f₂ : Π (x : α), β x} : f₁ = f₂ ↔ (∀ a, f₁ a = f₂ a) :=
iff.intro (assume h a, h ▸ rfl) funext
lemma ne_iff {β : α → Sort*} {f₁ f₂ : Π a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
protected lemma bijective.injective {f : α → β} (hf : bijective f) : injective f := hf.1
protected lemma bijective.surjective {f : α → β} (hf : bijective f) : surjective f := hf.2
theorem injective.eq_iff (I : injective f) {a b : α} :
f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
theorem injective.eq_iff' (I : injective f) {a b : α} {c : β} (h : f b = c) :
f a = c ↔ a = b :=
h ▸ I.eq_iff
lemma injective.ne (hf : injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt (assume h, hf h)
lemma injective.ne_iff (hf : injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt $ congr_arg f, hf.ne⟩
lemma injective.ne_iff' (hf : injective f) {x y : α} {z : β} (h : f y = z) :
f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def injective.decidable_eq [decidable_eq β] (I : injective f) : decidable_eq α :=
λ a b, decidable_of_iff _ I.eq_iff
lemma injective.of_comp {g : γ → α} (I : injective (f ∘ g)) : injective g :=
λ x y h, I $ show f (g x) = f (g y), from congr_arg f h
lemma injective.of_comp_iff {f : α → β} (hf : injective f) (g : γ → α) :
injective (f ∘ g) ↔ injective g :=
⟨injective.of_comp, hf.comp⟩
lemma injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : bijective g) :
injective (f ∘ g) ↔ injective f :=
⟨ λ h x y, let ⟨x', hx⟩ := hg.surjective x, ⟨y', hy⟩ := hg.surjective y in
hx ▸ hy ▸ λ hf, h hf ▸ rfl,
λ h, h.comp hg.injective⟩
/-- Composition by an injective function on the left is itself injective. -/
lemma injective.comp_left {g : β → γ} (hg : function.injective g) :
function.injective ((∘) g : (α → β) → (α → γ)) :=
λ f₁ f₂ hgf, funext $ λ i, hg $ (congr_fun hgf i : _)
lemma injective_of_subsingleton [subsingleton α] (f : α → β) :
injective f :=
λ a b ab, subsingleton.elim _ _
lemma injective.dite (p : α → Prop) [decidable_pred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : injective f) (hf' : injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
function.injective (λ x, if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
λ x₁ x₂ h, begin
dsimp only at h,
by_cases h₁ : p x₁; by_cases h₂ : p x₂,
{ rw [dif_pos h₁, dif_pos h₂] at h, injection (hf h), },
{ rw [dif_pos h₁, dif_neg h₂] at h, exact (im_disj h).elim, },
{ rw [dif_neg h₁, dif_pos h₂] at h, exact (im_disj h.symm).elim, },
{ rw [dif_neg h₁, dif_neg h₂] at h, injection (hf' h), },
end
lemma surjective.of_comp {g : γ → α} (S : surjective (f ∘ g)) : surjective f :=
λ y, let ⟨x, h⟩ := S y in ⟨g x, h⟩
lemma surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : surjective g) :
surjective (f ∘ g) ↔ surjective f :=
⟨surjective.of_comp, λ h, h.comp hg⟩
lemma surjective.of_comp_iff' (hf : bijective f) (g : γ → α) :
surjective (f ∘ g) ↔ surjective g :=
⟨λ h x, let ⟨x', hx'⟩ := h (f x) in ⟨x', hf.injective hx'⟩, hf.surjective.comp⟩
instance decidable_eq_pfun (p : Prop) [decidable p] (α : p → Type*)
[Π hp, decidable_eq (α hp)] : decidable_eq (Π hp, α hp)
| f g := decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem surjective.forall (hf : surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨λ h x, h (f x), λ h y, let ⟨x, hx⟩ := hf y in hx ▸ h x⟩
protected theorem surjective.forall₂ (hf : surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr $ λ x, hf.forall
protected theorem surjective.forall₃ (hf : surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr $ λ x, hf.forall₂
protected theorem surjective.exists (hf : surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨λ ⟨y, hy⟩, let ⟨x, hx⟩ := hf y in ⟨x, hx.symm ▸ hy⟩, λ ⟨x, hx⟩, ⟨f x, hx⟩⟩
protected theorem surjective.exists₂ (hf : surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans $ exists_congr $ λ x, hf.exists
protected theorem surjective.exists₃ (hf : surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans $ exists_congr $ λ x, hf.exists₂
lemma surjective.injective_comp_right (hf : surjective f) :
injective (λ g : β → γ, g ∘ f) :=
λ g₁ g₂ h, funext $ hf.forall.2 $ congr_fun h
protected lemma surjective.right_cancellable (hf : surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
lemma surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
surjective f :=
begin
specialize h (λ _, true) (λ y, ∃ x, f x = y) (funext $ λ x, _),
{ simp only [(∘), exists_apply_eq_apply] },
{ intro y,
have : true = ∃ x, f x = y, from congr_fun h y,
rw ← this, exact trivial }
end
lemma bijective_iff_exists_unique (f : α → β) : bijective f ↔
∀ b : β, ∃! (a : α), f a = b :=
⟨ λ hf b, let ⟨a, ha⟩ := hf.surjective b in ⟨a, ha, λ a' ha', hf.injective (ha'.trans ha.symm)⟩,
λ he, ⟨
λ a a' h, unique_of_exists_unique (he (f a')) h rfl,
λ b, exists_of_exists_unique (he b) ⟩⟩
/-- Shorthand for using projection notation with `function.bijective_iff_exists_unique`. -/
protected lemma bijective.exists_unique {f : α → β} (hf : bijective f) (b : β) :
∃! (a : α), f a = b :=
(bijective_iff_exists_unique f).mp hf b
lemma bijective.exists_unique_iff {f : α → β} (hf : bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨λ ⟨y, hpy, hy⟩, let ⟨x, hx⟩ := hf.surjective y in ⟨x, by rwa hx,
λ z (hz : p (f z)), hf.injective $ hx.symm ▸ hy _ hz⟩,
λ ⟨x, hpx, hx⟩, ⟨f x, hpx, λ y hy,
let ⟨z, hz⟩ := hf.surjective y in hz ▸ congr_arg f $ hx _ $ by rwa hz⟩⟩
lemma bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : bijective g) :
bijective (f ∘ g) ↔ bijective f :=
and_congr (injective.of_comp_iff' _ hg) (surjective.of_comp_iff _ hg.surjective)
lemma bijective.of_comp_iff' {f : α → β} (hf : bijective f) (g : γ → α) :
function.bijective (f ∘ g) ↔ function.bijective g :=
and_congr (injective.of_comp_iff hf.injective _) (surjective.of_comp_iff' hf _)
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `set α`. -/
theorem cantor_surjective {α} (f : α → set α) : ¬ function.surjective f | h :=
let ⟨D, e⟩ := h {a | ¬ a ∈ f a} in
(iff_not_self (D ∈ f D)).1 $ iff_of_eq (congr_arg ((∈) D) e)
/-- **Cantor's diagonal argument** implies that there are no injective functions from `set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : set α → α) : ¬ function.injective f | i :=
cantor_surjective (λ a, {b | ∀ U, a = f U → b ∈ U}) $
right_inverse.surjective (λ U, funext $ λ a, propext ⟨λ h, h U rfl, λ h' U' e, i e ▸ h'⟩)
/-- There is no surjection from `α : Type u` into `Type u`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type (max u v)) :
¬ surjective f :=
begin
intro hf,
let T : Type (max u v) := sigma f,
cases hf (set T) with U hU,
let g : set T → T := λ s, ⟨U, cast hU.symm s⟩,
have hg : injective g,
{ intros s t h,
suffices : cast hU (g s).2 = cast hU (g t).2,
{ simp only [cast_cast, cast_eq] at this, assumption },
{ congr, assumption } },
exact cantor_injective g hg
end
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def is_partial_inv {α β} (f : α → β) (g : β → option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
theorem is_partial_inv_left {α β} {f : α → β} {g} (H : is_partial_inv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
theorem injective_of_partial_inv {α β} {f : α → β} {g} (H : is_partial_inv f g) : injective f :=
λ a b h, option.some.inj $ ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
theorem injective_of_partial_inv_right {α β} {f : α → β} {g} (H : is_partial_inv f g)
(x y b) (h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
theorem left_inverse.comp_eq_id {f : α → β} {g : β → α} (h : left_inverse f g) : f ∘ g = id :=
funext h
theorem left_inverse_iff_comp {f : α → β} {g : β → α} : left_inverse f g ↔ f ∘ g = id :=
⟨left_inverse.comp_eq_id, congr_fun⟩
theorem right_inverse.comp_eq_id {f : α → β} {g : β → α} (h : right_inverse f g) : g ∘ f = id :=
funext h
theorem right_inverse_iff_comp {f : α → β} {g : β → α} : right_inverse f g ↔ g ∘ f = id :=
⟨right_inverse.comp_eq_id, congr_fun⟩
theorem left_inverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β}
(hf : left_inverse f g) (hh : left_inverse h i) : left_inverse (h ∘ f) (g ∘ i) :=
assume a, show h (f (g (i a))) = a, by rw [hf (i a), hh a]
theorem right_inverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β}
(hf : right_inverse f g) (hh : right_inverse h i) : right_inverse (h ∘ f) (g ∘ i) :=
left_inverse.comp hh hf
theorem left_inverse.right_inverse {f : α → β} {g : β → α} (h : left_inverse g f) :
right_inverse f g := h
theorem right_inverse.left_inverse {f : α → β} {g : β → α} (h : right_inverse g f) :
left_inverse f g := h
theorem left_inverse.surjective {f : α → β} {g : β → α} (h : left_inverse f g) :
surjective f :=
h.right_inverse.surjective
theorem right_inverse.injective {f : α → β} {g : β → α} (h : right_inverse f g) :
injective f :=
h.left_inverse.injective
theorem left_inverse.right_inverse_of_injective {f : α → β} {g : β → α} (h : left_inverse f g)
(hf : injective f) :
right_inverse f g :=
λ x, hf $ h (f x)
theorem left_inverse.right_inverse_of_surjective {f : α → β} {g : β → α} (h : left_inverse f g)
(hg : surjective g) :
right_inverse f g :=
λ x, let ⟨y, hy⟩ := hg x in hy ▸ congr_arg g (h y)
lemma right_inverse.left_inverse_of_surjective {f : α → β} {g : β → α} :
right_inverse f g → surjective f → left_inverse f g :=
left_inverse.right_inverse_of_surjective
lemma right_inverse.left_inverse_of_injective {f : α → β} {g : β → α} :
right_inverse f g → injective g → left_inverse f g :=
left_inverse.right_inverse_of_injective
theorem left_inverse.eq_right_inverse {f : α → β} {g₁ g₂ : β → α} (h₁ : left_inverse g₁ f)
(h₂ : right_inverse g₂ f) :
g₁ = g₂ :=
calc g₁ = g₁ ∘ f ∘ g₂ : by rw [h₂.comp_eq_id, comp.right_id]
... = g₂ : by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
local attribute [instance, priority 10] classical.prop_decidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partial_inv {α β} (f : α → β) (b : β) : option α :=
if h : ∃ a, f a = b then some (classical.some h) else none
theorem partial_inv_of_injective {α β} {f : α → β} (I : injective f) :
is_partial_inv f (partial_inv f) | a b :=
⟨λ h, if h' : ∃ a, f a = b then begin
rw [partial_inv, dif_pos h'] at h,
injection h with h, subst h,
apply classical.some_spec h'
end else by rw [partial_inv, dif_neg h'] at h; contradiction,
λ e, e ▸ have h : ∃ a', f a' = f a, from ⟨_, rfl⟩,
(dif_pos h).trans (congr_arg _ (I $ classical.some_spec h))⟩
theorem partial_inv_left {α β} {f : α → β} (I : injective f) : ∀ x, partial_inv f (f x) = some x :=
is_partial_inv_left (partial_inv_of_injective I)
end
section inv_fun
variables {α β : Sort*} [nonempty α] {f : α → β} {a : α} {b : β}
local attribute [instance, priority 10] classical.prop_decidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
noncomputable def inv_fun (f : α → β) : β → α :=
λ y, if h : ∃ x, f x = y then h.some else classical.arbitrary α
theorem inv_fun_eq (h : ∃ a, f a = b) : f (inv_fun f b) = b :=
by simp only [inv_fun, dif_pos h, h.some_spec]
lemma inv_fun_neg (h : ¬ ∃ a, f a = b) : inv_fun f b = classical.choice ‹_› :=
dif_neg h
theorem inv_fun_eq_of_injective_of_right_inverse {g : β → α}
(hf : injective f) (hg : right_inverse g f) : inv_fun f = g :=
funext $ assume b,
hf begin rw [hg b], exact inv_fun_eq ⟨g b, hg b⟩ end
lemma right_inverse_inv_fun (hf : surjective f) : right_inverse (inv_fun f) f :=
assume b, inv_fun_eq $ hf b
lemma left_inverse_inv_fun (hf : injective f) : left_inverse (inv_fun f) f :=
λ b, hf $ inv_fun_eq ⟨b, rfl⟩
lemma inv_fun_surjective (hf : injective f) : surjective (inv_fun f) :=
(left_inverse_inv_fun hf).surjective
lemma inv_fun_comp (hf : injective f) : inv_fun f ∘ f = id := funext $ left_inverse_inv_fun hf
lemma injective.has_left_inverse (hf : injective f) : has_left_inverse f :=
⟨inv_fun f, left_inverse_inv_fun hf⟩
lemma injective_iff_has_left_inverse : injective f ↔ has_left_inverse f :=
⟨injective.has_left_inverse, has_left_inverse.injective⟩
end inv_fun
section surj_inv
variables {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `inv_fun`, this does not require
`α` to be inhabited.) -/
noncomputable def surj_inv {f : α → β} (h : surjective f) (b : β) : α := classical.some (h b)
lemma surj_inv_eq (h : surjective f) (b) : f (surj_inv h b) = b := classical.some_spec (h b)
lemma right_inverse_surj_inv (hf : surjective f) : right_inverse (surj_inv hf) f :=
surj_inv_eq hf
lemma left_inverse_surj_inv (hf : bijective f) : left_inverse (surj_inv hf.2) f :=
right_inverse_of_injective_of_left_inverse hf.1 (right_inverse_surj_inv hf.2)
lemma surjective.has_right_inverse (hf : surjective f) : has_right_inverse f :=
⟨_, right_inverse_surj_inv hf⟩
lemma surjective_iff_has_right_inverse : surjective f ↔ has_right_inverse f :=
⟨surjective.has_right_inverse, has_right_inverse.surjective⟩
lemma bijective_iff_has_inverse : bijective f ↔ ∃ g, left_inverse g f ∧ right_inverse g f :=
⟨λ hf, ⟨_, left_inverse_surj_inv hf, right_inverse_surj_inv hf.2⟩,
λ ⟨g, gl, gr⟩, ⟨gl.injective, gr.surjective⟩⟩
lemma injective_surj_inv (h : surjective f) : injective (surj_inv h) :=
(right_inverse_surj_inv h).injective
lemma surjective_to_subsingleton [na : nonempty α] [subsingleton β] (f : α → β) :
surjective f :=
λ y, let ⟨a⟩ := na in ⟨a, subsingleton.elim _ _⟩
/-- Composition by an surjective function on the left is itself surjective. -/
lemma surjective.comp_left {g : β → γ} (hg : surjective g) :
surjective ((∘) g : (α → β) → (α → γ)) :=
λ f, ⟨surj_inv hg ∘ f, funext $ λ x, right_inverse_surj_inv _ _⟩
/-- Composition by an bijective function on the left is itself bijective. -/
lemma bijective.comp_left {g : β → γ} (hg : bijective g) :
bijective ((∘) g : (α → β) → (α → γ)) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
end surj_inv
section update
variables {α : Sort u} {β : α → Sort v} {α' : Sort w} [decidable_eq α] [decidable_eq α']
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : Πa, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then eq.rec v h.symm else f a
/-- On non-dependent functions, `function.update` can be expressed as an `ite` -/
lemma update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a :=
begin
dunfold update,
congr,
funext,
rw eq_rec_constant,
end
@[simp] lemma update_same (a : α) (v : β a) (f : Πa, β a) : update f a v a = v :=
dif_pos rfl
lemma surjective_eval {α : Sort u} {β : α → Sort v} [h : Π a, nonempty (β a)] (a : α) :
surjective (eval a : (Π a, β a) → β a) :=
λ b, ⟨@update _ _ (classical.dec_eq α) (λ a, (h a).some) a b,
@update_same _ _ (classical.dec_eq α) _ _ _⟩
lemma update_injective (f : Πa, β a) (a' : α) : injective (update f a') :=
λ v v' h, have _ := congr_fun h a', by rwa [update_same, update_same] at this
@[simp] lemma update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : Πa, β a) :
update f a' v a = f a :=
dif_neg h
lemma forall_update_iff (f : Π a, β a) {a : α} {b : β a} (p : Π a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x ≠ a, p x (f x) :=
by { rw [← and_forall_ne a, update_same], simp { contextual := tt } }
lemma exists_update_iff (f : Π a, β a) {a : α} {b : β a} (p : Π a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ x ≠ a, p x (f x) :=
by { rw [← not_forall_not, forall_update_iff f (λ a b, ¬p a b)], simp [not_and_distrib] }
lemma update_eq_iff {a : α} {b : β a} {f g : Π a, β a} :
update f a b = g ↔ b = g a ∧ ∀ x ≠ a, f x = g x :=
funext_iff.trans $ forall_update_iff _ (λ x y, y = g x)
lemma eq_update_iff {a : α} {b : β a} {f g : Π a, β a} :
g = update f a b ↔ g a = b ∧ ∀ x ≠ a, g x = f x :=
funext_iff.trans $ forall_update_iff _ (λ x y, g x = y)
@[simp] lemma update_eq_self (a : α) (f : Πa, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, λ _ _, rfl⟩
lemma update_comp_eq_of_forall_ne' {α'} (g : Π a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) :
j, (update g i a) (f j)) = (λ j, g (f j)) :=
funext $ λ x, update_noteq (h _) _ _
/-- Non-dependent version of `function.update_comp_eq_of_forall_ne'` -/
lemma update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) :
(update g i a) ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
lemma update_comp_eq_of_injective' (g : Π a, β a) {f : α' → α} (hf : function.injective f)
(i : α') (a : β (f i)) :
j, update g (f i) a (f j)) = update (λ i, g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, λ j hj, update_noteq (hf.ne hj) _ _⟩
/-- Non-dependent version of `function.update_comp_eq_of_injective'` -/
lemma update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : function.injective f) (i : α) (a : β) :
(function.update g (f i) a) ∘ f = function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
lemma apply_update {ι : Sort*} [decidable_eq ι] {α β : ι → Sort*}
(f : Π i, α i → β i) (g : Π i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (λ k, f k (g k)) i (f i v) j :=
begin
by_cases h : j = i,
{ subst j, simp },
{ simp [h] }
end
lemma apply_update₂ {ι : Sort*} [decidable_eq ι] {α β γ : ι → Sort*}
(f : Π i, α i → β i → γ i) (g : Π i, α i) (h : Π i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (λ k, f k (g k) (h k)) i (f i v w) j :=
begin
by_cases h : j = i,
{ subst j, simp },
{ simp [h] }
end
lemma comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ (update g i v) = update (f ∘ g) i (f v) :=
funext $ apply_update _ _ _ _
theorem update_comm {α} [decidable_eq α] {β : α → Sort*}
{a b : α} (h : a ≠ b) (v : β a) (w : β b) (f : Πa, β a) :
update (update f a v) b w = update (update f b w) a v :=
begin
funext c, simp only [update],
by_cases h₁ : c = b; by_cases h₂ : c = a; try {simp [h₁, h₂]},
cases h (h₂.symm.trans h₁),
end
@[simp] theorem update_idem {α} [decidable_eq α] {β : α → Sort*}
{a : α} (v w : β a) (f : Πa, β a) : update (update f a v) a w = update f a w :=
by {funext b, by_cases b = a; simp [update, h]}
end update
section extend
noncomputable theory
local attribute [instance, priority 10] classical.prop_decidable
variables {α β γ : Sort*} {f : α → β}
/-- `extend f g e'` extends a function `g : α → γ`
along a function `f : α → β` to a function → γ`,
by using the values of `g` on the range of `f`
and the values of an auxiliary function `e' : β → γ` elsewhere.
Mostly useful when `f` is injective. -/
def extend (f : α → β) (g : α → γ) (e' : β → γ) : β → γ :=
λ b, if h : ∃ a, f a = b then g (classical.some h) else e' b
lemma extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (classical.some h) else e' b :=
by { unfold extend, congr }
@[simp] lemma extend_apply (hf : injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
begin
simp only [extend_def, dif_pos, exists_apply_eq_apply],
exact congr_arg g (hf $ classical.some_spec (exists_apply_eq_apply f a))
end
@[simp] lemma extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b :=
by simp [function.extend_def, hb]
lemma apply_extend {δ} (hf : injective f) (F : γ → δ) (g : α → γ) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
begin
by_cases hb : ∃ a, f a = b,
{ cases hb with a ha, subst b,
rw [extend_apply hf, extend_apply hf] },
{ rw [extend_apply' _ _ _ hb, extend_apply' _ _ _ hb] }
end
lemma extend_injective (hf : injective f) (e' : β → γ) :
injective (λ g, extend f g e') :=
begin
intros g₁ g₂ hg,
refine funext (λ x, _),
have H := congr_fun hg (f x),
simp only [hf, extend_apply] at H,
exact H
end
@[simp] lemma extend_comp (hf : injective f) (g : α → γ) (e' : β → γ) :
extend f g e' ∘ f = g :=
funext $ λ a, extend_apply hf g e' a
lemma injective.surjective_comp_right' (hf : injective f) (g₀ : β → γ) :
surjective (λ g : β → γ, g ∘ f) :=
λ g, ⟨extend f g g₀, extend_comp hf _ _⟩
lemma injective.surjective_comp_right [nonempty γ] (hf : injective f) :
surjective (λ g : β → γ, g ∘ f) :=
hf.surjective_comp_right' (λ _, classical.choice ‹_›)
lemma bijective.comp_right (hf : bijective f) :
bijective (λ g : β → γ, g ∘ f) :=
⟨hf.surjective.injective_comp_right,
λ g, ⟨g ∘ surj_inv hf.surjective,
by simp only [comp.assoc g _ f, (left_inverse_surj_inv hf).comp_eq_id, comp.right_id]⟩⟩
end extend
lemma uncurry_def {α β γ} (f : α → β → γ) : uncurry f = (λp, f p.1 p.2) :=
rfl
@[simp] lemma uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) :
uncurry f (x, y) = f x y :=
rfl
@[simp] lemma curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) :
curry f x y = f (x, y) :=
rfl
section bicomp
variables {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
/-- Compose an unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
-- Suggested local notation:
local notation f `∘₂` g := bicompr f g
lemma uncurry_bicompr (f : α → β → γ) (g : γ → δ) :
uncurry (g ∘₂ f) = (g ∘ uncurry f) := rfl
lemma uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = (uncurry f) ∘ (prod.map g h) :=
rfl
end bicomp
section uncurry
variables {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class has_uncurry (α : Type*) (β : out_param Type*) (γ : out_param Type*) := (uncurry : α → (β → γ))
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
add_decl_doc has_uncurry.uncurry
notation `↿`:max x:max := has_uncurry.uncurry x
instance has_uncurry_base : has_uncurry (α → β) α β := ⟨id⟩
instance has_uncurry_induction [has_uncurry β γ δ] : has_uncurry (α → β) (α × γ) δ :=
⟨λ f p, ↿(f p.1) p.2
end uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def involutive {α} (f : α → α) : Prop := ∀ x, f (f x) = x
lemma involutive_iff_iter_2_eq_id {α} {f : α → α} : involutive f ↔ (f^[2] = id) :=
funext_iff.symm
namespace involutive
variables {α : Sort u} {f : α → α} (h : involutive f)
include h
@[simp]
lemma comp_self : f ∘ f = id := funext h
protected lemma left_inverse : left_inverse f f := h
protected lemma right_inverse : right_inverse f f := h
protected lemma injective : injective f := h.left_inverse.injective
protected lemma surjective : surjective f := λ x, ⟨f x, h x⟩
protected lemma bijective : bijective f := ⟨h.injective, h.surjective⟩
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected lemma ite_not (P : Prop) [decidable P] (x : α) :
f (ite P x (f x)) = ite (¬ P) x (f x) :=
by rw [apply_ite f, h, ite_not]
/-- An involution commutes across an equality. Compare to `function.injective.eq_iff`. -/
protected lemma eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
end involutive
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function × β → γ` being injective.
-/
def injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
namespace injective2
variables {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected lemma left (hf : injective2 f) (b : β) : function.injective (λ a, f a b) :=
λ a₁ a₂ h, (hf h).left
/-- A binary injective function is injective when only the right argument varies. -/
protected lemma right (hf : injective2 f) (a : α) : function.injective (f a) :=
λ a₁ a₂ h, (hf h).right
protected lemma uncurry {α β γ : Type*} {f : α → β → γ} (hf : injective2 f) :
function.injective (uncurry f) :=
λ ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ h, and.elim (hf h) (congr_arg2 _)
/-- As a map from the left argument to a unary function, `f` is injective. -/
lemma left' (hf : injective2 f) [nonempty β] : function.injective f :=
λ a₁ a₂ h, let ⟨b⟩ := ‹nonempty β› in hf.left b $ (congr_fun h b : _)
/-- As a map from the right argument to a unary function, `f` is injective. -/
lemma right' (hf : injective2 f) [nonempty α] : function.injective (λ b a, f a b) :=
λ b₁ b₂ h, let ⟨a⟩ := ‹nonempty α› in hf.right a $ (congr_fun h a : _)
lemma eq_iff (hf : injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨λ h, hf h, and.rec $ congr_arg2 f⟩
end injective2
section sometimes
local attribute [instance, priority 10] classical.prop_decidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [nonempty β] (f : α → β) : β :=
if h : nonempty α then f (classical.choice h) else classical.choice ‹_›
theorem sometimes_eq {p : Prop} {α} [nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
theorem sometimes_spec {p : Prop} {α} [nonempty α]
(P : α → Prop) (f : p → α) (a : p) (h : P (f a)) : P (sometimes f) :=
by rwa sometimes_eq
end sometimes
end function
/-- `s.piecewise f g` is the function equal to `f` on the set `s`, and to `g` on its complement. -/
def set.piecewise {α : Type u} {β : α → Sort v} (s : set α) (f g : Πi, β i)
[∀j, decidable (j ∈ s)] :
Πi, β i :=
λi, if i ∈ s then f i else g i
/-! ### Bijectivity of `eq.rec`, `eq.mp`, `eq.mpr`, and `cast` -/
lemma eq_rec_on_bijective {α : Sort*} {C : α → Sort*} :
∀ {a a' : α} (h : a = a'), function.bijective (@eq.rec_on _ _ C _ h)
| _ _ rfl := ⟨λ x y, id, λ x, ⟨x, rfl⟩⟩
lemma eq_mp_bijective {α β : Sort*} (h : α = β) : function.bijective (eq.mp h) :=
eq_rec_on_bijective h
lemma eq_mpr_bijective {α β : Sort*} (h : α = β) : function.bijective (eq.mpr h) :=
eq_rec_on_bijective h.symm
lemma cast_bijective {α β : Sort*} (h : α = β) : function.bijective (cast h) :=
eq_rec_on_bijective h
/-! Note these lemmas apply to `Type*` not `Sort*`, as the latter interferes with `simp`, and
is trivial anyway.-/
@[simp]
lemma eq_rec_inj {α : Sort*} {a a' : α} (h : a = a') {C : α → Type*} (x y : C a) :
(eq.rec x h : C a') = eq.rec y h ↔ x = y :=
(eq_rec_on_bijective h).injective.eq_iff
@[simp]
lemma cast_inj {α β : Type*} (h : α = β) {x y : α} : cast h x = cast h y ↔ x = y :=
(cast_bijective h).injective.eq_iff
lemma function.left_inverse.eq_rec_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : function.left_inverse g f) (C : Π a : α, γ (f a)) (a : α) :
(congr_arg f (h a)).rec (C (g (f a))) = C a :=
eq_of_heq $ (eq_rec_heq _ _).trans $ by rw h
lemma function.left_inverse.eq_rec_on_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : function.left_inverse g f) (C : Π a : α, γ (f a)) (a : α) :
(congr_arg f (h a)).rec_on (C (g (f a))) = C a :=
h.eq_rec_eq _ _
lemma function.left_inverse.cast_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : function.left_inverse g f) (C : Π a : α, γ (f a)) (a : α) :
cast (congr_arg (λ a, γ (f a)) (h a)) (C (g (f a))) = C a :=
eq_of_heq $ (eq_rec_heq _ _).trans $ by rw h
/-- A set of functions "separates points"
if for each pair of distinct points there is a function taking different values on them. -/
def set.separates_points {α β : Type*} (A : set (α → β)) : Prop :=
∀ ⦃x y : α⦄, x ≠ y → ∃ f ∈ A, (f x : β) ≠ f y
lemma is_symm_op.flip_eq {α β} (op) [is_symm_op α β op] : flip op = op :=
funext $ λ a, funext $ λ b, (is_symm_op.symm_op a b).symm
lemma inv_image.equivalence {α : Sort u} {β : Sort v} (r : β → β → Prop) (f : α → β)
(h : equivalence r) : equivalence (inv_image r f) :=
⟨λ _, h.1 _, λ _ _ x, h.2.1 x, inv_image.trans r f h.2.2