/- Copyright (c) 2014 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov -/ import data.set.prod import logic.function.conjugate /-! # Functions over sets ## Main definitions ### Predicate * `set.eq_on f₁ f₂ s` : functions `f₁` and `f₂` are equal at every point of `s`; * `set.maps_to f s t` : `f` sends every point of `s` to a point of `t`; * `set.inj_on f s` : restriction of `f` to `s` is injective; * `set.surj_on f s t` : every point in `s` has a preimage in `s`; * `set.bij_on f s t` : `f` is a bijection between `s` and `t`; * `set.left_inv_on f' f s` : for every `x ∈ s` we have `f' (f x) = x`; * `set.right_inv_on f' f t` : for every `y ∈ t` we have `f (f' y) = y`; * `set.inv_on f' f s t` : `f'` is a two-side inverse of `f` on `s` and `t`, i.e. we have `set.left_inv_on f' f s` and `set.right_inv_on f' f t`. ### Functions * `set.restrict f s` : restrict the domain of `f` to the set `s`; * `set.cod_restrict f s h` : given `h : ∀ x, f x ∈ s`, restrict the codomain of `f` to the set `s`; * `set.maps_to.restrict f s t h`: given `h : maps_to f s t`, restrict the domain of `f` to `s` and the codomain to `t`. -/ universes u v w x y variables {α : Type u} {β : Type v} {π : α → Type v} {γ : Type w} {ι : Sort x} open function namespace set /-! ### Restrict -/ /-- Restrict domain of a function `f` to a set `s`. Same as `subtype.restrict` but this version takes an argument `↥s` instead of `subtype s`. -/ def restrict (s : set α) (f : Π a : α, π a) : Π a : s, π a := λ x, f x lemma restrict_eq (f : α → β) (s : set α) : s.restrict f = f ∘ coe := rfl @[simp] lemma restrict_apply (f : α → β) (s : set α) (x : s) : s.restrict f x = f x := rfl lemma restrict_eq_iff {f : Π a, π a} {s : set α} {g : Π a : s, π a} : restrict s f = g ↔ ∀ a (ha : a ∈ s), f a = g ⟨a, ha⟩ := funext_iff.trans subtype.forall lemma eq_restrict_iff {s : set α} {f : Π a : s, π a} {g : Π a, π a} : f = restrict s g ↔ ∀ a (ha : a ∈ s), f ⟨a, ha⟩ = g a := funext_iff.trans subtype.forall @[simp] lemma range_restrict (f : α → β) (s : set α) : set.range (s.restrict f) = f '' s := (range_comp _ _).trans $ congr_arg (('') f) subtype.range_coe lemma image_restrict (f : α → β) (s t : set α) : s.restrict f '' (coe ⁻¹' t) = f '' (t ∩ s) := by rw [restrict, image_comp, image_preimage_eq_inter_range, subtype.range_coe] @[simp] lemma restrict_dite {s : set α} [∀ x, decidable (x ∈ s)] (f : Π a ∈ s, β) (g : Π a ∉ s, β) : s.restrict (λ a, if h : a ∈ s then f a h else g a h) = λ a, f a a.2 := funext $ λ a, dif_pos a.2 @[simp] lemma restrict_dite_compl {s : set α} [∀ x, decidable (x ∈ s)] (f : Π a ∈ s, β) (g : Π a ∉ s, β) : sᶜ.restrict (λ a, if h : a ∈ s then f a h else g a h) = λ a, g a a.2 := funext $ λ a, dif_neg a.2 @[simp] lemma restrict_ite (f g : α → β) (s : set α) [∀ x, decidable (x ∈ s)] : s.restrict (λ a, if a ∈ s then f a else g a) = s.restrict f := restrict_dite _ _ @[simp] lemma restrict_ite_compl (f g : α → β) (s : set α) [∀ x, decidable (x ∈ s)] : sᶜ.restrict (λ a, if a ∈ s then f a else g a) = sᶜ.restrict g := restrict_dite_compl _ _ @[simp] lemma restrict_piecewise (f g : α → β) (s : set α) [∀ x, decidable (x ∈ s)] : s.restrict (piecewise s f g) = s.restrict f := restrict_ite _ _ _ @[simp] lemma restrict_piecewise_compl (f g : α → β) (s : set α) [∀ x, decidable (x ∈ s)] : sᶜ.restrict (piecewise s f g) = sᶜ.restrict g := restrict_ite_compl _ _ _ lemma restrict_extend_range (f : α → β) (g : α → γ) (g' : β → γ) : (range f).restrict (extend f g g') = λ x, g x.coe_prop.some := by convert restrict_dite _ _ @[simp] lemma restrict_extend_compl_range (f : α → β) (g : α → γ) (g' : β → γ) : (range f)ᶜ.restrict (extend f g g') = g' ∘ coe := by convert restrict_dite_compl _ _ lemma range_extend_subset (f : α → β) (g : α → γ) (g' : β → γ) : range (extend f g g') ⊆ range g ∪ g' '' (range f)ᶜ := begin classical, rintro _ ⟨y, rfl⟩, rw extend_def, split_ifs, exacts [or.inl (mem_range_self _), or.inr (mem_image_of_mem _ h)] end lemma range_extend {f : α → β} (hf : injective f) (g : α → γ) (g' : β → γ) : range (extend f g g') = range g ∪ g' '' (range f)ᶜ := begin refine (range_extend_subset _ _ _).antisymm _, rintro z (⟨x, rfl⟩|⟨y, hy, rfl⟩), exacts [⟨f x, extend_apply hf _ _ _⟩, ⟨y, extend_apply' _ _ _ hy⟩] end /-- Restrict codomain of a function `f` to a set `s`. Same as `subtype.coind` but this version has codomain `↥s` instead of `subtype s`. -/ def cod_restrict (f : ι → α) (s : set α) (h : ∀ x, f x ∈ s) : ι → s := λ x, ⟨f x, h x⟩ @[simp] lemma coe_cod_restrict_apply (f : ι → α) (s : set α) (h : ∀ x, f x ∈ s) (x : ι) : (cod_restrict f s h x : α) = f x := rfl @[simp] lemma restrict_comp_cod_restrict {f : ι → α} {g : α → β} {b : set α} (h : ∀ x, f x ∈ b) : (b.restrict g) ∘ (b.cod_restrict f h) = g ∘ f := rfl @[simp] lemma injective_cod_restrict {f : ι → α} {s : set α} (h : ∀ x, f x ∈ s) : injective (cod_restrict f s h) ↔ injective f := by simp only [injective, subtype.ext_iff, coe_cod_restrict_apply] alias injective_cod_restrict ↔ _ _root_.function.injective.cod_restrict variables {s s₁ s₂ : set α} {t t₁ t₂ : set β} {p : set γ} {f f₁ f₂ f₃ : α → β} {g g₁ g₂ : β → γ} {f' f₁' f₂' : β → α} {g' : γ → β} /-! ### Equality on a set -/ /-- Two functions `f₁ f₂ : α → β` are equal on `s` if `f₁ x = f₂ x` for all `x ∈ a`. -/ def eq_on (f₁ f₂ : α → β) (s : set α) : Prop := ∀ ⦃x⦄, x ∈ s → f₁ x = f₂ x @[simp] lemma eq_on_empty (f₁ f₂ : α → β) : eq_on f₁ f₂ ∅ := λ x, false.elim @[simp] lemma restrict_eq_restrict_iff : restrict s f₁ = restrict s f₂ ↔ eq_on f₁ f₂ s := restrict_eq_iff @[symm] lemma eq_on.symm (h : eq_on f₁ f₂ s) : eq_on f₂ f₁ s := λ x hx, (h hx).symm lemma eq_on_comm : eq_on f₁ f₂ s ↔ eq_on f₂ f₁ s := ⟨eq_on.symm, eq_on.symm⟩ @[refl] lemma eq_on_refl (f : α → β) (s : set α) : eq_on f f s := λ _ _, rfl @[trans] lemma eq_on.trans (h₁ : eq_on f₁ f₂ s) (h₂ : eq_on f₂ f₃ s) : eq_on f₁ f₃ s := λ x hx, (h₁ hx).trans (h₂ hx) theorem eq_on.image_eq (heq : eq_on f₁ f₂ s) : f₁ '' s = f₂ '' s := image_congr heq theorem eq_on.inter_preimage_eq (heq : eq_on f₁ f₂ s) (t : set β) : s ∩ f₁ ⁻¹' t = s ∩ f₂ ⁻¹' t := ext $ λ x, and.congr_right_iff.2 $ λ hx, by rw [mem_preimage, mem_preimage, heq hx] lemma eq_on.mono (hs : s₁ ⊆ s₂) (hf : eq_on f₁ f₂ s₂) : eq_on f₁ f₂ s₁ := λ x hx, hf (hs hx) lemma eq_on.comp_left (h : s.eq_on f₁ f₂) : s.eq_on (g ∘ f₁) (g ∘ f₂) := λ a ha, congr_arg _ $ h ha lemma comp_eq_of_eq_on_range {ι : Sort*} {f : ι → α} {g₁ g₂ : α → β} (h : eq_on g₁ g₂ (range f)) : g₁ ∘ f = g₂ ∘ f := funext $ λ x, h $ mem_range_self _ /-! ### Congruence lemmas -/ section order variables [preorder α] [preorder β] lemma _root_.monotone_on.congr (h₁ : monotone_on f₁ s) (h : s.eq_on f₁ f₂) : monotone_on f₂ s := begin intros a ha b hb hab, rw [←h ha, ←h hb], exact h₁ ha hb hab, end lemma _root_.antitone_on.congr (h₁ : antitone_on f₁ s) (h : s.eq_on f₁ f₂) : antitone_on f₂ s := h₁.dual_right.congr h lemma _root_.strict_mono_on.congr (h₁ : strict_mono_on f₁ s) (h : s.eq_on f₁ f₂) : strict_mono_on f₂ s := begin intros a ha b hb hab, rw [←h ha, ←h hb], exact h₁ ha hb hab, end lemma _root_.strict_anti_on.congr (h₁ : strict_anti_on f₁ s) (h : s.eq_on f₁ f₂) : strict_anti_on f₂ s := h₁.dual_right.congr h lemma eq_on.congr_monotone_on (h : s.eq_on f₁ f₂) : monotone_on f₁ s ↔ monotone_on f₂ s := ⟨λ h₁, h₁.congr h, λ h₂, h₂.congr h.symm⟩ lemma eq_on.congr_antitone_on (h : s.eq_on f₁ f₂) : antitone_on f₁ s ↔ antitone_on f₂ s := ⟨λ h₁, h₁.congr h, λ h₂, h₂.congr h.symm⟩ lemma eq_on.congr_strict_mono_on (h : s.eq_on f₁ f₂) : strict_mono_on f₁ s ↔ strict_mono_on f₂ s := ⟨λ h₁, h₁.congr h, λ h₂, h₂.congr h.symm⟩ lemma eq_on.congr_strict_anti_on (h : s.eq_on f₁ f₂) : strict_anti_on f₁ s ↔ strict_anti_on f₂ s := ⟨λ h₁, h₁.congr h, λ h₂, h₂.congr h.symm⟩ end order /-! ### Mono lemmas-/ section mono variables [preorder α] [preorder β] lemma _root_.monotone_on.mono (h : monotone_on f s) (h' : s₂ ⊆ s) : monotone_on f s₂ := λ x hx y hy, h (h' hx) (h' hy) lemma _root_.antitone_on.mono (h : antitone_on f s) (h' : s₂ ⊆ s) : antitone_on f s₂ := λ x hx y hy, h (h' hx) (h' hy) lemma _root_.strict_mono_on.mono (h : strict_mono_on f s) (h' : s₂ ⊆ s) : strict_mono_on f s₂ := λ x hx y hy, h (h' hx) (h' hy) lemma _root_.strict_anti_on.mono (h : strict_anti_on f s) (h' : s₂ ⊆ s) : strict_anti_on f s₂ := λ x hx y hy, h (h' hx) (h' hy) protected lemma _root_.monotone_on.monotone (h : monotone_on f s) : monotone (f ∘ coe : s → β) := λ x y hle, h x.coe_prop y.coe_prop hle protected lemma _root_.antitone_on.monotone (h : antitone_on f s) : antitone (f ∘ coe : s → β) := λ x y hle, h x.coe_prop y.coe_prop hle protected lemma _root_.strict_mono_on.strict_mono (h : strict_mono_on f s) : strict_mono (f ∘ coe : s → β) := λ x y hlt, h x.coe_prop y.coe_prop hlt protected lemma _root_.strict_anti_on.strict_anti (h : strict_anti_on f s) : strict_anti (f ∘ coe : s → β) := λ x y hlt, h x.coe_prop y.coe_prop hlt end mono /-! ### maps to -/ /-- `maps_to f a b` means that the image of `a` is contained in `b`. -/ def maps_to (f : α → β) (s : set α) (t : set β) : Prop := ∀ ⦃x⦄, x ∈ s → f x ∈ t /-- Given a map `f` sending `s : set α` into `t : set β`, restrict domain of `f` to `s` and the codomain to `t`. Same as `subtype.map`. -/ def maps_to.restrict (f : α → β) (s : set α) (t : set β) (h : maps_to f s t) : s → t := subtype.map f h @[simp] lemma maps_to.coe_restrict_apply (h : maps_to f s t) (x : s) : (h.restrict f s t x : β) = f x := rfl lemma maps_to.coe_restrict (h : set.maps_to f s t) : coe ∘ h.restrict f s t = s.restrict f := rfl lemma maps_to.range_restrict (f : α → β) (s : set α) (t : set β) (h : maps_to f s t) : range (h.restrict f s t) = coe ⁻¹' (f '' s) := set.range_subtype_map f h lemma maps_to_iff_exists_map_subtype : maps_to f s t ↔ ∃ g : s → t, ∀ x : s, f x = g x := ⟨λ h, ⟨h.restrict f s t, λ _, rfl⟩, λ ⟨g, hg⟩ x hx, by { erw [hg ⟨x, hx⟩], apply subtype.coe_prop }⟩ theorem maps_to' : maps_to f s t ↔ f '' s ⊆ t := image_subset_iff.symm @[simp] theorem maps_to_singleton {x : α} : maps_to f {x} t ↔ f x ∈ t := singleton_subset_iff theorem maps_to_empty (f : α → β) (t : set β) : maps_to f ∅ t := empty_subset _ theorem maps_to.image_subset (h : maps_to f s t) : f '' s ⊆ t := maps_to'.1 h theorem maps_to.congr (h₁ : maps_to f₁ s t) (h : eq_on f₁ f₂ s) : maps_to f₂ s t := λ x hx, h hx ▸ h₁ hx lemma eq_on.comp_right (hg : t.eq_on g₁ g₂) (hf : s.maps_to f t) : s.eq_on (g₁ ∘ f) (g₂ ∘ f) := λ a ha, hg $ hf ha theorem eq_on.maps_to_iff (H : eq_on f₁ f₂ s) : maps_to f₁ s t ↔ maps_to f₂ s t := ⟨λ h, h.congr H, λ h, h.congr H.symm⟩ theorem maps_to.comp (h₁ : maps_to g t p) (h₂ : maps_to f s t) : maps_to (g ∘ f) s p := λ x h, h₁ (h₂ h) theorem maps_to_id (s : set α) : maps_to id s s := λ x, id theorem maps_to.iterate {f : α → α} {s : set α} (h : maps_to f s s) : ∀ n, maps_to (f^[n]) s s | 0 := λ _, id | (n+1) := (maps_to.iterate n).comp h theorem maps_to.iterate_restrict {f : α → α} {s : set α} (h : maps_to f s s) (n : ℕ) : (h.restrict f s s^[n]) = (h.iterate n).restrict _ _ _ := begin funext x, rw [subtype.ext_iff, maps_to.coe_restrict_apply], induction n with n ihn generalizing x, { refl }, { simp [nat.iterate, ihn] } end theorem maps_to.mono (hf : maps_to f s₁ t₁) (hs : s₂ ⊆ s₁) (ht : t₁ ⊆ t₂) : maps_to f s₂ t₂ := λ x hx, ht (hf $ hs hx) theorem maps_to.mono_left (hf : maps_to f s₁ t) (hs : s₂ ⊆ s₁) : maps_to f s₂ t := λ x hx, hf (hs hx) theorem maps_to.mono_right (hf : maps_to f s t₁) (ht : t₁ ⊆ t₂) : maps_to f s t₂ := λ x hx, ht (hf hx) theorem maps_to.union_union (h₁ : maps_to f s₁ t₁) (h₂ : maps_to f s₂ t₂) : maps_to f (s₁ ∪ s₂) (t₁ ∪ t₂) := λ x hx, hx.elim (λ hx, or.inl $ h₁ hx) (λ hx, or.inr $ h₂ hx) theorem maps_to.union (h₁ : maps_to f s₁ t) (h₂ : maps_to f s₂ t) : maps_to f (s₁ ∪ s₂) t := union_self t ▸ h₁.union_union h₂ @[simp] theorem maps_to_union : maps_to f (s₁ ∪ s₂) t ↔ maps_to f s₁ t ∧ maps_to f s₂ t := ⟨λ h, ⟨h.mono (subset_union_left s₁ s₂) (subset.refl t), h.mono (subset_union_right s₁ s₂) (subset.refl t)⟩, λ h, h.1.union h.2⟩ theorem maps_to.inter (h₁ : maps_to f s t₁) (h₂ : maps_to f s t₂) : maps_to f s (t₁ ∩ t₂) := λ x hx, ⟨h₁ hx, h₂ hx⟩ theorem maps_to.inter_inter (h₁ : maps_to f s₁ t₁) (h₂ : maps_to f s₂ t₂) : maps_to f (s₁ ∩ s₂) (t₁ ∩ t₂) := λ x hx, ⟨h₁ hx.1, h₂ hx.2⟩ @[simp] theorem maps_to_inter : maps_to f s (t₁ ∩ t₂) ↔ maps_to f s t₁ ∧ maps_to f s t₂ := ⟨λ h, ⟨h.mono (subset.refl s) (inter_subset_left t₁ t₂), h.mono (subset.refl s) (inter_subset_right t₁ t₂)⟩, λ h, h.1.inter h.2⟩ theorem maps_to_univ (f : α → β) (s : set α) : maps_to f s univ := λ x h, trivial theorem maps_to_image (f : α → β) (s : set α) : maps_to f s (f '' s) := by rw maps_to' theorem maps_to_preimage (f : α → β) (t : set β) : maps_to f (f ⁻¹' t) t := subset.refl _ theorem maps_to_range (f : α → β) (s : set α) : maps_to f s (range f) := (maps_to_image f s).mono (subset.refl s) (image_subset_range _ _) @[simp] lemma maps_image_to (f : α → β) (g : γ → α) (s : set γ) (t : set β) : maps_to f (g '' s) t ↔ maps_to (f ∘ g) s t := ⟨λ h c hc, h ⟨c, hc, rfl⟩, λ h d ⟨c, hc⟩, hc.2 ▸ h hc.1⟩ @[simp] lemma maps_univ_to (f : α → β) (s : set β) : maps_to f univ s ↔ ∀ a, f a ∈ s := ⟨λ h a, h (mem_univ _), λ h x _, h x⟩ @[simp] lemma maps_range_to (f : α → β) (g : γ → α) (s : set β) : maps_to f (range g) s ↔ maps_to (f ∘ g) univ s := by rw [←image_univ, maps_image_to] theorem surjective_maps_to_image_restrict (f : α → β) (s : set α) : surjective ((maps_to_image f s).restrict f s (f '' s)) := λ ⟨y, x, hs, hxy⟩, ⟨⟨x, hs⟩, subtype.ext hxy⟩ theorem maps_to.mem_iff (h : maps_to f s t) (hc : maps_to f sᶜ tᶜ) {x} : f x ∈ t ↔ x ∈ s := ⟨λ ht, by_contra $ λ hs, hc hs ht, λ hx, h hx⟩ /-! ### Restriction onto preimage -/ section variables (t f) /-- The restriction of a function onto the preimage of a set. -/ @[simps] def restrict_preimage : f ⁻¹' t → t := (set.maps_to_preimage f t).restrict _ _ _ lemma range_restrict_preimage : range (t.restrict_preimage f) = coe ⁻¹' (range f) := begin delta set.restrict_preimage, rw [maps_to.range_restrict, set.image_preimage_eq_inter_range, set.preimage_inter, subtype.coe_preimage_self, set.univ_inter], end end /-! ### Injectivity on a set -/ /-- `f` is injective on `a` if the restriction of `f` to `a` is injective. -/ def inj_on (f : α → β) (s : set α) : Prop := ∀ ⦃x₁ : α⦄, x₁ ∈ s → ∀ ⦃x₂ : α⦄, x₂ ∈ s → f x₁ = f x₂ → x₁ = x₂ theorem subsingleton.inj_on (hs : s.subsingleton) (f : α → β) : inj_on f s := λ x hx y hy h, hs hx hy @[simp] theorem inj_on_empty (f : α → β) : inj_on f ∅ := subsingleton_empty.inj_on f @[simp] theorem inj_on_singleton (f : α → β) (a : α) : inj_on f {a} := subsingleton_singleton.inj_on f theorem inj_on.eq_iff {x y} (h : inj_on f s) (hx : x ∈ s) (hy : y ∈ s) : f x = f y ↔ x = y := ⟨h hx hy, λ h, h ▸ rfl⟩ theorem inj_on.congr (h₁ : inj_on f₁ s) (h : eq_on f₁ f₂ s) : inj_on f₂ s := λ x hx y hy, h hx ▸ h hy ▸ h₁ hx hy theorem eq_on.inj_on_iff (H : eq_on f₁ f₂ s) : inj_on f₁ s ↔ inj_on f₂ s := ⟨λ h, h.congr H, λ h, h.congr H.symm⟩ theorem inj_on.mono (h : s₁ ⊆ s₂) (ht : inj_on f s₂) : inj_on f s₁ := λ x hx y hy H, ht (h hx) (h hy) H theorem inj_on_union (h : disjoint s₁ s₂) : inj_on f (s₁ ∪ s₂) ↔ inj_on f s₁ ∧ inj_on f s₂ ∧ ∀ (x ∈ s₁) (y ∈ s₂), f x ≠ f y := begin refine ⟨λ H, ⟨H.mono $ subset_union_left _ _, H.mono $ subset_union_right _ _, _⟩, _⟩, { intros x hx y hy hxy, obtain rfl : x = y, from H (or.inl hx) (or.inr hy) hxy, exact h ⟨hx, hy⟩ }, { rintro ⟨h₁, h₂, h₁₂⟩, rintro x (hx|hx) y (hy|hy) hxy, exacts [h₁ hx hy hxy, (h₁₂ _ hx _ hy hxy).elim, (h₁₂ _ hy _ hx hxy.symm).elim, h₂ hx hy hxy] } end theorem inj_on_insert {f : α → β} {s : set α} {a : α} (has : a ∉ s) : set.inj_on f (insert a s) ↔ set.inj_on f s ∧ f a ∉ f '' s := have disjoint s {a}, from λ x ⟨hxs, (hxa : x = a)⟩, has (hxa ▸ hxs), by { rw [← union_singleton, inj_on_union this], simp } lemma injective_iff_inj_on_univ : injective f ↔ inj_on f univ := ⟨λ h x hx y hy hxy, h hxy, λ h _ _ heq, h trivial trivial heq⟩ lemma inj_on_of_injective (h : injective f) (s : set α) : inj_on f s := λ x hx y hy hxy, h hxy alias inj_on_of_injective ← _root_.function.injective.inj_on theorem inj_on.comp (hg : inj_on g t) (hf: inj_on f s) (h : maps_to f s t) : inj_on (g ∘ f) s := λ x hx y hy heq, hf hx hy $ hg (h hx) (h hy) heq lemma inj_on_iff_injective : inj_on f s ↔ injective (s.restrict f) := ⟨λ H a b h, subtype.eq $ H a.2 b.2 h, λ H a as b bs h, congr_arg subtype.val $ @H ⟨a, as⟩ ⟨b, bs⟩ h⟩ alias inj_on_iff_injective ↔ inj_on.injective _ lemma exists_inj_on_iff_injective [nonempty β] : (∃ f : α → β, inj_on f s) ↔ ∃ f : s → β, injective f := ⟨λ ⟨f, hf⟩, ⟨_, hf.injective⟩, λ ⟨f, hf⟩, by { lift f to α → β using trivial, exact ⟨f, inj_on_iff_injective.2 hf⟩ }⟩ lemma inj_on_preimage {B : set (set β)} (hB : B ⊆ 𝒫 (range f)) : inj_on (preimage f) B := λ s hs t ht hst, (preimage_eq_preimage' (hB hs) (hB ht)).1 hst lemma inj_on.mem_of_mem_image {x} (hf : inj_on f s) (hs : s₁ ⊆ s) (h : x ∈ s) (h₁ : f x ∈ f '' s₁) : x ∈ s₁ := let ⟨x', h', eq⟩ := h₁ in hf (hs h') h eq ▸ h' lemma inj_on.mem_image_iff {x} (hf : inj_on f s) (hs : s₁ ⊆ s) (hx : x ∈ s) : f x ∈ f '' s₁ ↔ x ∈ s₁ := ⟨hf.mem_of_mem_image hs hx, mem_image_of_mem f⟩ lemma inj_on.preimage_image_inter (hf : inj_on f s) (hs : s₁ ⊆ s) : f ⁻¹' (f '' s₁) ∩ s = s₁ := ext $ λ x, ⟨λ ⟨h₁, h₂⟩, hf.mem_of_mem_image hs h₂ h₁, λ h, ⟨mem_image_of_mem _ h, hs h⟩⟩ lemma eq_on.cancel_left (h : s.eq_on (g ∘ f₁) (g ∘ f₂)) (hg : t.inj_on g) (hf₁ : s.maps_to f₁ t) (hf₂ : s.maps_to f₂ t) : s.eq_on f₁ f₂ := λ a ha, hg (hf₁ ha) (hf₂ ha) (h ha) lemma inj_on.cancel_left (hg : t.inj_on g) (hf₁ : s.maps_to f₁ t) (hf₂ : s.maps_to f₂ t) : s.eq_on (g ∘ f₁) (g ∘ f₂) ↔ s.eq_on f₁ f₂ := ⟨λ h, h.cancel_left hg hf₁ hf₂, eq_on.comp_left⟩ /-! ### Surjectivity on a set -/ /-- `f` is surjective from `a` to `b` if `b` is contained in the image of `a`. -/ def surj_on (f : α → β) (s : set α) (t : set β) : Prop := t ⊆ f '' s theorem surj_on.subset_range (h : surj_on f s t) : t ⊆ range f := subset.trans h $ image_subset_range f s lemma surj_on_iff_exists_map_subtype : surj_on f s t ↔ ∃ (t' : set β) (g : s → t'), t ⊆ t' ∧ surjective g ∧ ∀ x : s, f x = g x := ⟨λ h, ⟨_, (maps_to_image f s).restrict f s _, h, surjective_maps_to_image_restrict _ _, λ _, rfl⟩, λ ⟨t', g, htt', hg, hfg⟩ y hy, let ⟨x, hx⟩ := hg ⟨y, htt' hy⟩ in ⟨x, x.2, by rw [hfg, hx, subtype.coe_mk]⟩⟩ theorem surj_on_empty (f : α → β) (s : set α) : surj_on f s ∅ := empty_subset _ theorem surj_on_image (f : α → β) (s : set α) : surj_on f s (f '' s) := subset.rfl theorem surj_on.comap_nonempty (h : surj_on f s t) (ht : t.nonempty) : s.nonempty := (ht.mono h).of_image theorem surj_on.congr (h : surj_on f₁ s t) (H : eq_on f₁ f₂ s) : surj_on f₂ s t := by rwa [surj_on, ← H.image_eq] theorem eq_on.surj_on_iff (h : eq_on f₁ f₂ s) : surj_on f₁ s t ↔ surj_on f₂ s t := ⟨λ H, H.congr h, λ H, H.congr h.symm⟩ theorem surj_on.mono (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) (hf : surj_on f s₁ t₂) : surj_on f s₂ t₁ := subset.trans ht $ subset.trans hf $ image_subset _ hs theorem surj_on.union (h₁ : surj_on f s t₁) (h₂ : surj_on f s t₂) : surj_on f s (t₁ ∪ t₂) := λ x hx, hx.elim (λ hx, h₁ hx) (λ hx, h₂ hx) theorem surj_on.union_union (h₁ : surj_on f s₁ t₁) (h₂ : surj_on f s₂ t₂) : surj_on f (s₁ ∪ s₂) (t₁ ∪ t₂) := (h₁.mono (subset_union_left _ _) (subset.refl _)).union (h₂.mono (subset_union_right _ _) (subset.refl _)) theorem surj_on.inter_inter (h₁ : surj_on f s₁ t₁) (h₂ : surj_on f s₂ t₂) (h : inj_on f (s₁ ∪ s₂)) : surj_on f (s₁ ∩ s₂) (t₁ ∩ t₂) := begin intros y hy, rcases h₁ hy.1 with ⟨x₁, hx₁, rfl⟩, rcases h₂ hy.2 with ⟨x₂, hx₂, heq⟩, obtain rfl : x₁ = x₂ := h (or.inl hx₁) (or.inr hx₂) heq.symm, exact mem_image_of_mem f ⟨hx₁, hx₂⟩ end theorem surj_on.inter (h₁ : surj_on f s₁ t) (h₂ : surj_on f s₂ t) (h : inj_on f (s₁ ∪ s₂)) : surj_on f (s₁ ∩ s₂) t := inter_self t ▸ h₁.inter_inter h₂ h theorem surj_on.comp (hg : surj_on g t p) (hf : surj_on f s t) : surj_on (g ∘ f) s p := subset.trans hg $ subset.trans (image_subset g hf) $ (image_comp g f s) ▸ subset.refl _ lemma surjective_iff_surj_on_univ : surjective f ↔ surj_on f univ univ := by simp [surjective, surj_on, subset_def] lemma surj_on_iff_surjective : surj_on f s univ ↔ surjective (s.restrict f) := ⟨λ H b, let ⟨a, as, e⟩ := @H b trivial in ⟨⟨a, as⟩, e⟩, λ H b _, let ⟨⟨a, as⟩, e⟩ := H b in ⟨a, as, e⟩⟩ lemma surj_on.image_eq_of_maps_to (h₁ : surj_on f s t) (h₂ : maps_to f s t) : f '' s = t := eq_of_subset_of_subset h₂.image_subset h₁ lemma image_eq_iff_surj_on_maps_to : f '' s = t ↔ s.surj_on f t ∧ s.maps_to f t := begin refine ⟨_, λ h, h.1.image_eq_of_maps_to h.2⟩, rintro rfl, exact ⟨s.surj_on_image f, s.maps_to_image f⟩, end lemma surj_on.maps_to_compl (h : surj_on f s t) (h' : injective f) : maps_to f sᶜ tᶜ := λ x hs ht, let ⟨x', hx', heq⟩ := h ht in hs $ h' heq ▸ hx' lemma maps_to.surj_on_compl (h : maps_to f s t) (h' : surjective f) : surj_on f sᶜ tᶜ := h'.forall.2 $ λ x ht, mem_image_of_mem _ $ λ hs, ht (h hs) lemma eq_on.cancel_right (hf : s.eq_on (g₁ ∘ f) (g₂ ∘ f)) (hf' : s.surj_on f t) : t.eq_on g₁ g₂ := begin intros b hb, obtain ⟨a, ha, rfl⟩ := hf' hb, exact hf ha, end lemma surj_on.cancel_right (hf : s.surj_on f t) (hf' : s.maps_to f t) : s.eq_on (g₁ ∘ f) (g₂ ∘ f) ↔ t.eq_on g₁ g₂ := ⟨λ h, h.cancel_right hf, λ h, h.comp_right hf'⟩ lemma eq_on_comp_right_iff : s.eq_on (g₁ ∘ f) (g₂ ∘ f) ↔ (f '' s).eq_on g₁ g₂ := (s.surj_on_image f).cancel_right $ s.maps_to_image f /-! ### Bijectivity -/ /-- `f` is bijective from `s` to `t` if `f` is injective on `s` and `f '' s = t`. -/ def bij_on (f : α → β) (s : set α) (t : set β) : Prop := maps_to f s t ∧ inj_on f s ∧ surj_on f s t lemma bij_on.maps_to (h : bij_on f s t) : maps_to f s t := h.left lemma bij_on.inj_on (h : bij_on f s t) : inj_on f s := h.right.left lemma bij_on.surj_on (h : bij_on f s t) : surj_on f s t := h.right.right lemma bij_on.mk (h₁ : maps_to f s t) (h₂ : inj_on f s) (h₃ : surj_on f s t) : bij_on f s t := ⟨h₁, h₂, h₃⟩ lemma bij_on_empty (f : α → β) : bij_on f ∅ ∅ := ⟨maps_to_empty f ∅, inj_on_empty f, surj_on_empty f ∅⟩ lemma bij_on.inter (h₁ : bij_on f s₁ t₁) (h₂ : bij_on f s₂ t₂) (h : inj_on f (s₁ ∪ s₂)) : bij_on f (s₁ ∩ s₂) (t₁ ∩ t₂) := ⟨h₁.maps_to.inter_inter h₂.maps_to, h₁.inj_on.mono $ inter_subset_left _ _, h₁.surj_on.inter_inter h₂.surj_on h⟩ lemma bij_on.union (h₁ : bij_on f s₁ t₁) (h₂ : bij_on f s₂ t₂) (h : inj_on f (s₁ ∪ s₂)) : bij_on f (s₁ ∪ s₂) (t₁ ∪ t₂) := ⟨h₁.maps_to.union_union h₂.maps_to, h, h₁.surj_on.union_union h₂.surj_on⟩ theorem bij_on.subset_range (h : bij_on f s t) : t ⊆ range f := h.surj_on.subset_range lemma inj_on.bij_on_image (h : inj_on f s) : bij_on f s (f '' s) := bij_on.mk (maps_to_image f s) h (subset.refl _) theorem bij_on.congr (h₁ : bij_on f₁ s t) (h : eq_on f₁ f₂ s) : bij_on f₂ s t := bij_on.mk (h₁.maps_to.congr h) (h₁.inj_on.congr h) (h₁.surj_on.congr h) theorem eq_on.bij_on_iff (H : eq_on f₁ f₂ s) : bij_on f₁ s t ↔ bij_on f₂ s t := ⟨λ h, h.congr H, λ h, h.congr H.symm⟩ lemma bij_on.image_eq (h : bij_on f s t) : f '' s = t := h.surj_on.image_eq_of_maps_to h.maps_to theorem bij_on.comp (hg : bij_on g t p) (hf : bij_on f s t) : bij_on (g ∘ f) s p := bij_on.mk (hg.maps_to.comp hf.maps_to) (hg.inj_on.comp hf.inj_on hf.maps_to) (hg.surj_on.comp hf.surj_on) theorem bij_on.bijective (h : bij_on f s t) : bijective (t.cod_restrict (s.restrict f) $ λ x, h.maps_to x.val_prop) := ⟨λ x y h', subtype.ext $ h.inj_on x.2 y.2 $ subtype.ext_iff.1 h', λ ⟨y, hy⟩, let ⟨x, hx, hxy⟩ := h.surj_on hy in ⟨⟨x, hx⟩, subtype.eq hxy⟩⟩ lemma bijective_iff_bij_on_univ : bijective f ↔ bij_on f univ univ := iff.intro (λ h, let ⟨inj, surj⟩ := h in ⟨maps_to_univ f _, inj.inj_on _, iff.mp surjective_iff_surj_on_univ surj⟩) (λ h, let ⟨map, inj, surj⟩ := h in ⟨iff.mpr injective_iff_inj_on_univ inj, iff.mpr surjective_iff_surj_on_univ surj⟩) lemma bij_on.compl (hst : bij_on f s t) (hf : bijective f) : bij_on f sᶜ tᶜ := ⟨hst.surj_on.maps_to_compl hf.1, hf.1.inj_on _, hst.maps_to.surj_on_compl hf.2⟩ /-! ### left inverse -/ /-- `g` is a left inverse to `f` on `a` means that `g (f x) = x` for all `x ∈ a`. -/ def left_inv_on (f' : β → α) (f : α → β) (s : set α) : Prop := ∀ ⦃x⦄, x ∈ s → f' (f x) = x lemma left_inv_on.eq_on (h : left_inv_on f' f s) : eq_on (f' ∘ f) id s := h lemma left_inv_on.eq (h : left_inv_on f' f s) {x} (hx : x ∈ s) : f' (f x) = x := h hx lemma left_inv_on.congr_left (h₁ : left_inv_on f₁' f s) {t : set β} (h₁' : maps_to f s t) (heq : eq_on f₁' f₂' t) : left_inv_on f₂' f s := λ x hx, heq (h₁' hx) ▸ h₁ hx theorem left_inv_on.congr_right (h₁ : left_inv_on f₁' f₁ s) (heq : eq_on f₁ f₂ s) : left_inv_on f₁' f₂ s := λ x hx, heq hx ▸ h₁ hx theorem left_inv_on.inj_on (h : left_inv_on f₁' f s) : inj_on f s := λ x₁ h₁ x₂ h₂ heq, calc x₁ = f₁' (f x₁) : eq.symm $ h h₁ ... = f₁' (f x₂) : congr_arg f₁' heq ... = x₂ : h h₂ theorem left_inv_on.surj_on (h : left_inv_on f' f s) (hf : maps_to f s t) : surj_on f' t s := λ x hx, ⟨f x, hf hx, h hx⟩ theorem left_inv_on.maps_to (h : left_inv_on f' f s) (hf : surj_on f s t) : maps_to f' t s := λ y hy, let ⟨x, hs, hx⟩ := hf hy in by rwa [← hx, h hs] theorem left_inv_on.comp (hf' : left_inv_on f' f s) (hg' : left_inv_on g' g t) (hf : maps_to f s t) : left_inv_on (f' ∘ g') (g ∘ f) s := λ x h, calc (f' ∘ g') ((g ∘ f) x) = f' (f x) : congr_arg f' (hg' (hf h)) ... = x : hf' h theorem left_inv_on.mono (hf : left_inv_on f' f s) (ht : s₁ ⊆ s) : left_inv_on f' f s₁ := λ x hx, hf (ht hx) theorem left_inv_on.image_inter' (hf : left_inv_on f' f s) : f '' (s₁ ∩ s) = f' ⁻¹' s₁ ∩ f '' s := begin apply subset.antisymm, { rintro _ ⟨x, ⟨h₁, h⟩, rfl⟩, exact ⟨by rwa [mem_preimage, hf h], mem_image_of_mem _ h⟩ }, { rintro _ ⟨h₁, ⟨x, h, rfl⟩⟩, exact mem_image_of_mem _ ⟨by rwa ← hf h, h⟩ } end theorem left_inv_on.image_inter (hf : left_inv_on f' f s) : f '' (s₁ ∩ s) = f' ⁻¹' (s₁ ∩ s) ∩ f '' s := begin rw hf.image_inter', refine subset.antisymm _ (inter_subset_inter_left _ (preimage_mono $ inter_subset_left _ _)), rintro _ ⟨h₁, x, hx, rfl⟩, exact ⟨⟨h₁, by rwa hf hx⟩, mem_image_of_mem _ hx⟩ end theorem left_inv_on.image_image (hf : left_inv_on f' f s) : f' '' (f '' s) = s := by rw [image_image, image_congr hf, image_id'] theorem left_inv_on.image_image' (hf : left_inv_on f' f s) (hs : s₁ ⊆ s) : f' '' (f '' s₁) = s₁ := (hf.mono hs).image_image /-! ### Right inverse -/ /-- `g` is a right inverse to `f` on `b` if `f (g x) = x` for all `x ∈ b`. -/ @[reducible] def right_inv_on (f' : β → α) (f : α → β) (t : set β) : Prop := left_inv_on f f' t lemma right_inv_on.eq_on (h : right_inv_on f' f t) : eq_on (f ∘ f') id t := h lemma right_inv_on.eq (h : right_inv_on f' f t) {y} (hy : y ∈ t) : f (f' y) = y := h hy lemma left_inv_on.right_inv_on_image (h : left_inv_on f' f s) : right_inv_on f' f (f '' s) := λ y ⟨x, hx, eq⟩, eq ▸ congr_arg f $ h.eq hx theorem right_inv_on.congr_left (h₁ : right_inv_on f₁' f t) (heq : eq_on f₁' f₂' t) : right_inv_on f₂' f t := h₁.congr_right heq theorem right_inv_on.congr_right (h₁ : right_inv_on f' f₁ t) (hg : maps_to f' t s) (heq : eq_on f₁ f₂ s) : right_inv_on f' f₂ t := left_inv_on.congr_left h₁ hg heq theorem right_inv_on.surj_on (hf : right_inv_on f' f t) (hf' : maps_to f' t s) : surj_on f s t := hf.surj_on hf' theorem right_inv_on.maps_to (h : right_inv_on f' f t) (hf : surj_on f' t s) : maps_to f s t := h.maps_to hf theorem right_inv_on.comp (hf : right_inv_on f' f t) (hg : right_inv_on g' g p) (g'pt : maps_to g' p t) : right_inv_on (f' ∘ g') (g ∘ f) p := hg.comp hf g'pt theorem right_inv_on.mono (hf : right_inv_on f' f t) (ht : t₁ ⊆ t) : right_inv_on f' f t₁ := hf.mono ht theorem inj_on.right_inv_on_of_left_inv_on (hf : inj_on f s) (hf' : left_inv_on f f' t) (h₁ : maps_to f s t) (h₂ : maps_to f' t s) : right_inv_on f f' s := λ x h, hf (h₂ $ h₁ h) h (hf' (h₁ h)) theorem eq_on_of_left_inv_on_of_right_inv_on (h₁ : left_inv_on f₁' f s) (h₂ : right_inv_on f₂' f t) (h : maps_to f₂' t s) : eq_on f₁' f₂' t := λ y hy, calc f₁' y = (f₁' ∘ f ∘ f₂') y : congr_arg f₁' (h₂ hy).symm ... = f₂' y : h₁ (h hy) theorem surj_on.left_inv_on_of_right_inv_on (hf : surj_on f s t) (hf' : right_inv_on f f' s) : left_inv_on f f' t := λ y hy, let ⟨x, hx, heq⟩ := hf hy in by rw [← heq, hf' hx] /-! ### Two-side inverses -/ /-- `g` is an inverse to `f` viewed as a map from `a` to `b` -/ def inv_on (g : β → α) (f : α → β) (s : set α) (t : set β) : Prop := left_inv_on g f s ∧ right_inv_on g f t lemma inv_on.symm (h : inv_on f' f s t) : inv_on f f' t s := ⟨h.right, h.left⟩ lemma inv_on.mono (h : inv_on f' f s t) (hs : s₁ ⊆ s) (ht : t₁ ⊆ t) : inv_on f' f s₁ t₁ := ⟨h.1.mono hs, h.2.mono ht⟩ /-- If functions `f'` and `f` are inverse on `s` and `t`, `f` maps `s` into `t`, and `f'` maps `t` into `s`, then `f` is a bijection between `s` and `t`. The `maps_to` arguments can be deduced from `surj_on` statements using `left_inv_on.maps_to` and `right_inv_on.maps_to`. -/ theorem inv_on.bij_on (h : inv_on f' f s t) (hf : maps_to f s t) (hf' : maps_to f' t s) : bij_on f s t := ⟨hf, h.left.inj_on, h.right.surj_on hf'⟩ end set /-! ### `inv_fun_on` is a left/right inverse -/ namespace function variables [nonempty α] {s : set α} {f : α → β} {a : α} {b : β} local attribute [instance, priority 10] classical.prop_decidable /-- Construct the inverse for a function `f` on domain `s`. This function is a right inverse of `f` on `f '' s`. For a computable version, see `function.injective.inv_of_mem_range`. -/ noncomputable def inv_fun_on (f : α → β) (s : set α) (b : β) : α := if h : ∃a, a ∈ s ∧ f a = b then classical.some h else classical.choice ‹nonempty α› theorem inv_fun_on_pos (h : ∃a∈s, f a = b) : inv_fun_on f s b ∈ s ∧ f (inv_fun_on f s b) = b := by rw [bex_def] at h; rw [inv_fun_on, dif_pos h]; exact classical.some_spec h theorem inv_fun_on_mem (h : ∃a∈s, f a = b) : inv_fun_on f s b ∈ s := (inv_fun_on_pos h).left theorem inv_fun_on_eq (h : ∃a∈s, f a = b) : f (inv_fun_on f s b) = b := (inv_fun_on_pos h).right theorem inv_fun_on_neg (h : ¬ ∃a∈s, f a = b) : inv_fun_on f s b = classical.choice ‹nonempty α› := by rw [bex_def] at h; rw [inv_fun_on, dif_neg h] end function namespace set open function variables {s s₁ s₂ : set α} {t : set β} {f : α → β} theorem inj_on.left_inv_on_inv_fun_on [nonempty α] (h : inj_on f s) : left_inv_on (inv_fun_on f s) f s := λ a ha, have ∃a'∈s, f a' = f a, from ⟨a, ha, rfl⟩, h (inv_fun_on_mem this) ha (inv_fun_on_eq this) lemma inj_on.inv_fun_on_image [nonempty α] (h : inj_on f s₂) (ht : s₁ ⊆ s₂) : (inv_fun_on f s₂) '' (f '' s₁) = s₁ := h.left_inv_on_inv_fun_on.image_image' ht theorem surj_on.right_inv_on_inv_fun_on [nonempty α] (h : surj_on f s t) : right_inv_on (inv_fun_on f s) f t := λ y hy, inv_fun_on_eq $ mem_image_iff_bex.1 $ h hy theorem bij_on.inv_on_inv_fun_on [nonempty α] (h : bij_on f s t) : inv_on (inv_fun_on f s) f s t := ⟨h.inj_on.left_inv_on_inv_fun_on, h.surj_on.right_inv_on_inv_fun_on⟩ theorem surj_on.inv_on_inv_fun_on [nonempty α] (h : surj_on f s t) : inv_on (inv_fun_on f s) f (inv_fun_on f s '' t) t := begin refine ⟨_, h.right_inv_on_inv_fun_on⟩, rintros _ ⟨y, hy, rfl⟩, rw [h.right_inv_on_inv_fun_on hy] end theorem surj_on.maps_to_inv_fun_on [nonempty α] (h : surj_on f s t) : maps_to (inv_fun_on f s) t s := λ y hy, mem_preimage.2 $ inv_fun_on_mem $ mem_image_iff_bex.1 $ h hy theorem surj_on.bij_on_subset [nonempty α] (h : surj_on f s t) : bij_on f (inv_fun_on f s '' t) t := begin refine h.inv_on_inv_fun_on.bij_on _ (maps_to_image _ _), rintros _ ⟨y, hy, rfl⟩, rwa [h.right_inv_on_inv_fun_on hy] end theorem surj_on_iff_exists_bij_on_subset : surj_on f s t ↔ ∃ s' ⊆ s, bij_on f s' t := begin split, { rcases eq_empty_or_nonempty t with rfl|ht, { exact λ _, ⟨∅, empty_subset _, bij_on_empty f⟩ }, { assume h, haveI : nonempty α := ⟨classical.some (h.comap_nonempty ht)⟩, exact ⟨_, h.maps_to_inv_fun_on.image_subset, h.bij_on_subset⟩ }}, { rintros ⟨s', hs', hfs'⟩, exact hfs'.surj_on.mono hs' (subset.refl _) } end lemma preimage_inv_fun_of_mem [n : nonempty α] {f : α → β} (hf : injective f) {s : set α} (h : classical.choice n ∈ s) : inv_fun f ⁻¹' s = f '' s ∪ (range f)ᶜ := begin ext x, rcases em (x ∈ range f) with ⟨a, rfl⟩|hx, { simp [left_inverse_inv_fun hf _, hf.mem_set_image] }, { simp [mem_preimage, inv_fun_neg hx, h, hx] } end lemma preimage_inv_fun_of_not_mem [n : nonempty α] {f : α → β} (hf : injective f) {s : set α} (h : classical.choice n ∉ s) : inv_fun f ⁻¹' s = f '' s := begin ext x, rcases em (x ∈ range f) with ⟨a, rfl⟩|hx, { rw [mem_preimage, left_inverse_inv_fun hf, hf.mem_set_image] }, { have : x ∉ f '' s, from λ h', hx (image_subset_range _ _ h'), simp only [mem_preimage, inv_fun_neg hx, h, this] }, end end set /-! ### Monotone -/ namespace monotone variables [preorder α] [preorder β] {f : α → β} protected lemma restrict (h : monotone f) (s : set α) : monotone (s.restrict f) := λ x y hxy, h hxy protected lemma cod_restrict (h : monotone f) {s : set β} (hs : ∀ x, f x ∈ s) : monotone (s.cod_restrict f hs) := h protected lemma range_factorization (h : monotone f) : monotone (set.range_factorization f) := h end monotone /-! ### Piecewise defined function -/ namespace set variables {δ : α → Sort y} (s : set α) (f g : Πi, δ i) @[simp] lemma piecewise_empty [∀i : α, decidable (i ∈ (∅ : set α))] : piecewise ∅ f g = g := by { ext i, simp [piecewise] } @[simp] lemma piecewise_univ [∀i : α, decidable (i ∈ (set.univ : set α))] : piecewise set.univ f g = f := by { ext i, simp [piecewise] } @[simp] lemma piecewise_insert_self {j : α} [∀i, decidable (i ∈ insert j s)] : (insert j s).piecewise f g j = f j := by simp [piecewise] variable [∀j, decidable (j ∈ s)] instance compl.decidable_mem (j : α) : decidable (j ∈ sᶜ) := not.decidable lemma piecewise_insert [decidable_eq α] (j : α) [∀i, decidable (i ∈ insert j s)] : (insert j s).piecewise f g = function.update (s.piecewise f g) j (f j) := begin simp [piecewise], ext i, by_cases h : i = j, { rw h, simp }, { by_cases h' : i ∈ s; simp [h, h'] } end @[simp, priority 990] lemma piecewise_eq_of_mem {i : α} (hi : i ∈ s) : s.piecewise f g i = f i := if_pos hi @[simp, priority 990] lemma piecewise_eq_of_not_mem {i : α} (hi : i ∉ s) : s.piecewise f g i = g i := if_neg hi lemma piecewise_singleton (x : α) [Π y, decidable (y ∈ ({x} : set α))] [decidable_eq α] (f g : α → β) : piecewise {x} f g = function.update g x (f x) := by { ext y, by_cases hy : y = x, { subst y, simp }, { simp [hy] } } lemma piecewise_eq_on (f g : α → β) : eq_on (s.piecewise f g) f s := λ _, piecewise_eq_of_mem _ _ _ lemma piecewise_eq_on_compl (f g : α → β) : eq_on (s.piecewise f g) g sᶜ := λ _, piecewise_eq_of_not_mem _ _ _ lemma piecewise_le {δ : α → Type*} [Π i, preorder (δ i)] {s : set α} [Π j, decidable (j ∈ s)] {f₁ f₂ g : Π i, δ i} (h₁ : ∀ i ∈ s, f₁ i ≤ g i) (h₂ : ∀ i ∉ s, f₂ i ≤ g i) : s.piecewise f₁ f₂ ≤ g := λ i, if h : i ∈ s then by simp * else by simp * lemma le_piecewise {δ : α → Type*} [Π i, preorder (δ i)] {s : set α} [Π j, decidable (j ∈ s)] {f₁ f₂ g : Π i, δ i} (h₁ : ∀ i ∈ s, g i ≤ f₁ i) (h₂ : ∀ i ∉ s, g i ≤ f₂ i) : g ≤ s.piecewise f₁ f₂ := @piecewise_le α (λ i, (δ i)ᵒᵈ) _ s _ _ _ _ h₁ h₂ lemma piecewise_le_piecewise {δ : α → Type*} [Π i, preorder (δ i)] {s : set α} [Π j, decidable (j ∈ s)] {f₁ f₂ g₁ g₂ : Π i, δ i} (h₁ : ∀ i ∈ s, f₁ i ≤ g₁ i) (h₂ : ∀ i ∉ s, f₂ i ≤ g₂ i) : s.piecewise f₁ f₂ ≤ s.piecewise g₁ g₂ := by apply piecewise_le; intros; simp * @[simp, priority 990] lemma piecewise_insert_of_ne {i j : α} (h : i ≠ j) [∀i, decidable (i ∈ insert j s)] : (insert j s).piecewise f g i = s.piecewise f g i := by simp [piecewise, h] @[simp] lemma piecewise_compl [∀ i, decidable (i ∈ sᶜ)] : sᶜ.piecewise f g = s.piecewise g f := funext $ λ x, if hx : x ∈ s then by simp [hx] else by simp [hx] @[simp] lemma piecewise_range_comp {ι : Sort*} (f : ι → α) [Π j, decidable (j ∈ range f)] (g₁ g₂ : α → β) : (range f).piecewise g₁ g₂ ∘ f = g₁ ∘ f := comp_eq_of_eq_on_range $ piecewise_eq_on _ _ _ theorem maps_to.piecewise_ite {s s₁ s₂ : set α} {t t₁ t₂ : set β} {f₁ f₂ : α → β} [∀ i, decidable (i ∈ s)] (h₁ : maps_to f₁ (s₁ ∩ s) (t₁ ∩ t)) (h₂ : maps_to f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)) : maps_to (s.piecewise f₁ f₂) (s.ite s₁ s₂) (t.ite t₁ t₂) := begin refine (h₁.congr _).union_union (h₂.congr _), exacts [(piecewise_eq_on s f₁ f₂).symm.mono (inter_subset_right _ _), (piecewise_eq_on_compl s f₁ f₂).symm.mono (inter_subset_right _ _)] end theorem eq_on_piecewise {f f' g : α → β} {t} : eq_on (s.piecewise f f') g t ↔ eq_on f g (t ∩ s) ∧ eq_on f' g (t ∩ sᶜ) := begin simp only [eq_on, ← forall_and_distrib], refine forall_congr (λ a, _), by_cases a ∈ s; simp * end theorem eq_on.piecewise_ite' {f f' g : α → β} {t t'} (h : eq_on f g (t ∩ s)) (h' : eq_on f' g (t' ∩ sᶜ)) : eq_on (s.piecewise f f') g (s.ite t t') := by simp [eq_on_piecewise, *] theorem eq_on.piecewise_ite {f f' g : α → β} {t t'} (h : eq_on f g t) (h' : eq_on f' g t') : eq_on (s.piecewise f f') g (s.ite t t') := (h.mono (inter_subset_left _ _)).piecewise_ite' s (h'.mono (inter_subset_left _ _)) lemma piecewise_preimage (f g : α → β) (t) : s.piecewise f g ⁻¹' t = s.ite (f ⁻¹' t) (g ⁻¹' t) := ext $ λ x, by by_cases x ∈ s; simp [*, set.ite] lemma apply_piecewise {δ' : α → Sort*} (h : Π i, δ i → δ' i) {x : α} : h x (s.piecewise f g x) = s.piecewise (λ x, h x (f x)) (λ x, h x (g x)) x := by by_cases hx : x ∈ s; simp [hx] lemma apply_piecewise₂ {δ' δ'' : α → Sort*} (f' g' : Π i, δ' i) (h : Π i, δ i → δ' i → δ'' i) {x : α} : h x (s.piecewise f g x) (s.piecewise f' g' x) = s.piecewise (λ x, h x (f x) (f' x)) (λ x, h x (g x) (g' x)) x := by by_cases hx : x ∈ s; simp [hx] lemma piecewise_op {δ' : α → Sort*} (h : Π i, δ i → δ' i) : s.piecewise (λ x, h x (f x)) (λ x, h x (g x)) = λ x, h x (s.piecewise f g x) := funext $ λ x, (apply_piecewise _ _ _ _).symm lemma piecewise_op₂ {δ' δ'' : α → Sort*} (f' g' : Π i, δ' i) (h : Π i, δ i → δ' i → δ'' i) : s.piecewise (λ x, h x (f x) (f' x)) (λ x, h x (g x) (g' x)) = λ x, h x (s.piecewise f g x) (s.piecewise f' g' x) := funext $ λ x, (apply_piecewise₂ _ _ _ _ _ _).symm @[simp] lemma piecewise_same : s.piecewise f f = f := by { ext x, by_cases hx : x ∈ s; simp [hx] } lemma range_piecewise (f g : α → β) : range (s.piecewise f g) = f '' s ∪ g '' sᶜ := begin ext y, split, { rintro ⟨x, rfl⟩, by_cases h : x ∈ s;[left, right]; use x; simp [h] }, { rintro (⟨x, hx, rfl⟩|⟨x, hx, rfl⟩); use x; simp * at * } end lemma injective_piecewise_iff {f g : α → β} : injective (s.piecewise f g) ↔ inj_on f s ∧ inj_on g sᶜ ∧ (∀ (x ∈ s) (y ∉ s), f x ≠ g y) := begin rw [injective_iff_inj_on_univ, ← union_compl_self s, inj_on_union (@disjoint_compl_right _ s _), (piecewise_eq_on s f g).inj_on_iff, (piecewise_eq_on_compl s f g).inj_on_iff], refine and_congr iff.rfl (and_congr iff.rfl $ forall₄_congr $ λ x hx y hy, _), rw [piecewise_eq_of_mem s f g hx, piecewise_eq_of_not_mem s f g hy] end lemma piecewise_mem_pi {δ : α → Type*} {t : set α} {t' : Π i, set (δ i)} {f g} (hf : f ∈ pi t t') (hg : g ∈ pi t t') : s.piecewise f g ∈ pi t t' := by { intros i ht, by_cases hs : i ∈ s; simp [hf i ht, hg i ht, hs] } @[simp] lemma pi_piecewise {ι : Type*} {α : ι → Type*} (s s' : set ι) (t t' : Π i, set (α i)) [Π x, decidable (x ∈ s')] : pi s (s'.piecewise t t') = pi (s ∩ s') t ∩ pi (s \ s') t' := begin ext x, simp only [mem_pi, mem_inter_eq, ← forall_and_distrib], refine forall_congr (λ i, _), by_cases hi : i ∈ s'; simp * end lemma univ_pi_piecewise {ι : Type*} {α : ι → Type*} (s : set ι) (t : Π i, set (α i)) [Π x, decidable (x ∈ s)] : pi univ (s.piecewise t (λ _, univ)) = pi s t := by simp end set lemma strict_mono_on.inj_on [linear_order α] [preorder β] {f : α → β} {s : set α} (H : strict_mono_on f s) : s.inj_on f := λ x hx y hy hxy, show ordering.eq.compares x y, from (H.compares hx hy).1 hxy lemma strict_anti_on.inj_on [linear_order α] [preorder β] {f : α → β} {s : set α} (H : strict_anti_on f s) : s.inj_on f := @strict_mono_on.inj_on α βᵒᵈ _ _ f s H lemma strict_mono_on.comp [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} {t : set β} (hg : strict_mono_on g t) (hf : strict_mono_on f s) (hs : set.maps_to f s t) : strict_mono_on (g ∘ f) s := λ x hx y hy hxy, hg (hs hx) (hs hy) $ hf hx hy hxy lemma strict_mono_on.comp_strict_anti_on [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} {t : set β} (hg : strict_mono_on g t) (hf : strict_anti_on f s) (hs : set.maps_to f s t) : strict_anti_on (g ∘ f) s := λ x hx y hy hxy, hg (hs hy) (hs hx) $ hf hx hy hxy lemma strict_anti_on.comp [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} {t : set β} (hg : strict_anti_on g t) (hf : strict_anti_on f s) (hs : set.maps_to f s t) : strict_mono_on (g ∘ f) s := λ x hx y hy hxy, hg (hs hy) (hs hx) $ hf hx hy hxy lemma strict_anti_on.comp_strict_mono_on [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} {t : set β} (hg : strict_anti_on g t) (hf : strict_mono_on f s) (hs : set.maps_to f s t) : strict_anti_on (g ∘ f) s := λ x hx y hy hxy, hg (hs hx) (hs hy) $ hf hx hy hxy lemma strict_mono.cod_restrict [preorder α] [preorder β] {f : α → β} (hf : strict_mono f) {s : set β} (hs : ∀ x, f x ∈ s) : strict_mono (set.cod_restrict f s hs) := hf namespace function open set variables {fa : α → α} {fb : β → β} {f : α → β} {g : β → γ} {s t : set α} lemma injective.comp_inj_on (hg : injective g) (hf : s.inj_on f) : s.inj_on (g ∘ f) := (hg.inj_on univ).comp hf (maps_to_univ _ _) lemma surjective.surj_on (hf : surjective f) (s : set β) : surj_on f univ s := (surjective_iff_surj_on_univ.1 hf).mono (subset.refl _) (subset_univ _) lemma left_inverse.left_inv_on {g : β → α} (h : left_inverse f g) (s : set β) : left_inv_on f g s := λ x hx, h x lemma right_inverse.right_inv_on {g : β → α} (h : right_inverse f g) (s : set α) : right_inv_on f g s := λ x hx, h x lemma left_inverse.right_inv_on_range {g : β → α} (h : left_inverse f g) : right_inv_on f g (range g) := forall_range_iff.2 $ λ i, congr_arg g (h i) namespace semiconj lemma maps_to_image (h : semiconj f fa fb) (ha : maps_to fa s t) : maps_to fb (f '' s) (f '' t) := λ y ⟨x, hx, hy⟩, hy ▸ ⟨fa x, ha hx, h x⟩ lemma maps_to_range (h : semiconj f fa fb) : maps_to fb (range f) (range f) := λ y ⟨x, hy⟩, hy ▸ ⟨fa x, h x⟩ lemma surj_on_image (h : semiconj f fa fb) (ha : surj_on fa s t) : surj_on fb (f '' s) (f '' t) := begin rintros y ⟨x, hxt, rfl⟩, rcases ha hxt with ⟨x, hxs, rfl⟩, rw [h x], exact mem_image_of_mem _ (mem_image_of_mem _ hxs) end lemma surj_on_range (h : semiconj f fa fb) (ha : surjective fa) : surj_on fb (range f) (range f) := by { rw ← image_univ, exact h.surj_on_image (ha.surj_on univ) } lemma inj_on_image (h : semiconj f fa fb) (ha : inj_on fa s) (hf : inj_on f (fa '' s)) : inj_on fb (f '' s) := begin rintros _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ H, simp only [← h.eq] at H, exact congr_arg f (ha hx hy $ hf (mem_image_of_mem fa hx) (mem_image_of_mem fa hy) H) end lemma inj_on_range (h : semiconj f fa fb) (ha : injective fa) (hf : inj_on f (range fa)) : inj_on fb (range f) := by { rw ← image_univ at *, exact h.inj_on_image (ha.inj_on univ) hf } lemma bij_on_image (h : semiconj f fa fb) (ha : bij_on fa s t) (hf : inj_on f t) : bij_on fb (f '' s) (f '' t) := ⟨h.maps_to_image ha.maps_to, h.inj_on_image ha.inj_on (ha.image_eq.symm ▸ hf), h.surj_on_image ha.surj_on⟩ lemma bij_on_range (h : semiconj f fa fb) (ha : bijective fa) (hf : injective f) : bij_on fb (range f) (range f) := begin rw [← image_univ], exact h.bij_on_image (bijective_iff_bij_on_univ.1 ha) (hf.inj_on univ) end lemma maps_to_preimage (h : semiconj f fa fb) {s t : set β} (hb : maps_to fb s t) : maps_to fa (f ⁻¹' s) (f ⁻¹' t) := λ x hx, by simp only [mem_preimage, h x, hb hx] lemma inj_on_preimage (h : semiconj f fa fb) {s : set β} (hb : inj_on fb s) (hf : inj_on f (f ⁻¹' s)) : inj_on fa (f ⁻¹' s) := begin intros x hx y hy H, have := congr_arg f H, rw [h.eq, h.eq] at this, exact hf hx hy (hb hx hy this) end end semiconj lemma update_comp_eq_of_not_mem_range' {α β : Sort*} {γ : β → Sort*} [decidable_eq β] (g : Π b, γ b) {f : α → β} {i : β} (a : γ i) (h : i ∉ set.range f) : (λ j, (function.update g i a) (f j)) = (λ j, g (f j)) := update_comp_eq_of_forall_ne' _ _ $ λ x hx, h ⟨x, hx⟩ /-- Non-dependent version of `function.update_comp_eq_of_not_mem_range'` -/ lemma update_comp_eq_of_not_mem_range {α β γ : Sort*} [decidable_eq β] (g : β → γ) {f : α → β} {i : β} (a : γ) (h : i ∉ set.range f) : (function.update g i a) ∘ f = g ∘ f := update_comp_eq_of_not_mem_range' g a h lemma insert_inj_on (s : set α) : sᶜ.inj_on (λ a, insert a s) := λ a ha b _, (insert_inj ha).1 end function