152 lines
5 KiB
Text
152 lines
5 KiB
Text
/-
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Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.set.map
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Author: Jeremy Avigad, Andrew Zipperer
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Functions between subsets of finite types, bundled with the domain and range.
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-/
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import data.set.function
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open eq.ops set
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record map {X Y : Type} (a : set X) (b : set Y) := (func : X → Y) (mapsto : maps_to func a b)
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attribute map.func [coercion]
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namespace map
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variables {X Y Z: Type}
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variables {a : set X} {b : set Y} {c : set Z}
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/- the equivalence relation -/
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protected definition equiv [reducible] (f1 f2 : map a b) : Prop := eq_on f1 f2 a
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namespace equiv_notation
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infix `~` := map.equiv
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end equiv_notation
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open equiv_notation
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protected theorem equiv.refl (f : map a b) : f ~ f := take x, assume H, rfl
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protected theorem equiv.symm {f₁ f₂ : map a b} : f₁ ~ f₂ → f₂ ~ f₁ :=
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assume H : f₁ ~ f₂,
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take x, assume Ha : x ∈ a, eq.symm (H Ha)
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protected theorem equiv.trans {f₁ f₂ f₃ : map a b} : f₁ ~ f₂ → f₂ ~ f₃ → f₁ ~ f₃ :=
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assume H₁ : f₁ ~ f₂, assume H₂ : f₂ ~ f₃,
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take x, assume Ha : x ∈ a, eq.trans (H₁ Ha) (H₂ Ha)
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protected theorem equiv.is_equivalence {X Y : Type} (a : set X) (b : set Y) :
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equivalence (@equiv X Y a b) :=
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mk_equivalence (@equiv X Y a b) (@equiv.refl X Y a b) (@equiv.symm X Y a b) (@equiv.trans X Y a b)
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/- compose -/
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definition compose (g : map b c) (f : map a b) : map a c :=
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map.mk (#function g ∘ f) (maps_to_compose (mapsto g) (mapsto f))
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notation g ∘ f := compose g f
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/- range -/
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definition range (f : map a b) : set Y := image f a
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theorem range_eq_range_of_equiv {f1 f2 : map a b} (H : f1 ~ f2) : range f1 = range f2 :=
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image_eq_image_of_eq_on H
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/- injective -/
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definition injective (f : map a b) : Prop := inj_on f a
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theorem injective_of_equiv {f1 f2 : map a b} (H1 : f1 ~ f2) (H2 : injective f1) :
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injective f2 :=
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inj_on_of_eq_on H1 H2
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theorem injective_compose {g : map b c} {f : map a b} (Hg : injective g) (Hf: injective f) :
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injective (g ∘ f) :=
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inj_on_compose (mapsto f) Hg Hf
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/- surjective -/
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definition surjective (f : map a b) : Prop := surj_on f a b
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theorem surjective_of_equiv {f1 f2 : map a b} (H1 : f1 ~ f2) (H2 : surjective f1) :
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surjective f2 :=
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surj_on_of_eq_on H1 H2
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theorem surjective_compose {g : map b c} {f : map a b} (Hg : surjective g) (Hf: surjective f) :
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surjective (g ∘ f) :=
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surj_on_compose Hg Hf
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/- bijective -/
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definition bijective (f : map a b) : Prop := injective f ∧ surjective f
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theorem bijective_of_equiv {f1 f2 : map a b} (H1 : f1 ~ f2) (H2 : bijective f1) :
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bijective f2 :=
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and.intro (injective_of_equiv H1 (and.left H2)) (surjective_of_equiv H1 (and.right H2))
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theorem bijective_compose {g : map b c} {f : map a b} (Hg : bijective g) (Hf: bijective f) :
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bijective (g ∘ f) :=
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obtain Hg₁ Hg₂, from Hg,
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obtain Hf₁ Hf₂, from Hf,
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and.intro (injective_compose Hg₁ Hf₁) (surjective_compose Hg₂ Hf₂)
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/- left inverse -/
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-- g is a left inverse to f
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definition left_inverse (g : map b a) (f : map a b) : Prop := left_inv_on g f a
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theorem left_inverse_of_equiv_left {g1 g2 : map b a} {f : map a b} (eqg : g1 ~ g2)
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(H : left_inverse g1 f) : left_inverse g2 f :=
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left_inv_on_of_eq_on_left (mapsto f) eqg H
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theorem left_inverse_of_equiv_right {g : map b a} {f1 f2 : map a b} (eqf : f1 ~ f2)
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(H : left_inverse g f1) : left_inverse g f2 :=
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left_inv_on_of_eq_on_right eqf H
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theorem injective_of_left_inverse {g : map b a} {f : map a b} (H : left_inverse g f) :
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injective f :=
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inj_on_of_left_inv_on H
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theorem left_inverse_compose {f' : map b a} {g' : map c b} {g : map b c} {f : map a b}
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(Hf : left_inverse f' f) (Hg : left_inverse g' g) : left_inverse (f' ∘ g') (g ∘ f) :=
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left_inv_on_compose (mapsto f) Hf Hg
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/- right inverse -/
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-- g is a right inverse to f
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definition right_inverse (g : map b a) (f : map a b) : Prop := left_inverse f g
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theorem right_inverse_of_equiv_left {g1 g2 : map b a} {f : map a b} (eqg : g1 ~ g2)
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(H : right_inverse g1 f) : right_inverse g2 f :=
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left_inverse_of_equiv_right eqg H
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theorem right_inverse_of_equiv_right {g : map b a} {f1 f2 : map a b} (eqf : f1 ~ f2)
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(H : right_inverse g f1) : right_inverse g f2 :=
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left_inverse_of_equiv_left eqf H
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theorem surjective_of_right_inverse {g : map b a} {f : map a b} (H : right_inverse g f) :
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surjective f :=
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surj_on_of_right_inv_on (mapsto g) H
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theorem right_inverse_compose {f' : map b a} {g' : map c b} {g : map b c} {f : map a b}
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(Hf : right_inverse f' f) (Hg : right_inverse g' g) : right_inverse (f' ∘ g') (g ∘ f) :=
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left_inverse_compose Hg Hf
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theorem equiv_of_left_inverse_of_right_inverse {g1 g2 : map b a} {f : map a b}
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(H1 : left_inverse g1 f) (H2 : right_inverse g2 f) : g1 ~ g2 :=
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eq_on_of_left_inv_of_right_inv (mapsto g2) H1 H2
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/- inverse -/
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-- g is an inverse to f
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definition is_inverse (g : map b a) (f : map a b) : Prop := left_inverse g f ∧ right_inverse g f
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theorem bijective_of_is_inverse {g : map b a} {f : map a b} (H : is_inverse g f) : bijective f :=
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and.intro
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(injective_of_left_inverse (and.left H))
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(surjective_of_right_inverse (and.right H))
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end map
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