lean2/library/data/set/function.lean

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/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: data.set.function
Author: Jeremy Avigad, Andrew Zipperer, Haitao Zhang
Functions between subsets of finite types.
-/
import .basic
import algebra.function
open function eq.ops
namespace set
variables {X Y Z : Type}
abbreviation eq_on (f1 f2 : X → Y) (a : set X) : Prop :=
∀₀ x ∈ a, f1 x = f2 x
/- image -/
definition image (f : X → Y) (a : set X) : set Y := {y : Y | ∃x, x ∈ a ∧ f x = y}
notation f `'[`:max a `]` := image f a
theorem image_eq_image_of_eq_on {f1 f2 : X → Y} {a : set X} (H1 : eq_on f1 f2 a) :
f1 '[a] = f2 '[a] :=
setext (take y, iff.intro
(assume H2,
obtain x (H3 : x ∈ a ∧ f1 x = y), from H2,
have H4 : x ∈ a, from and.left H3,
have H5 : f2 x = y, from (H1 H4)⁻¹ ⬝ and.right H3,
exists.intro x (and.intro H4 H5))
(assume H2,
obtain x (H3 : x ∈ a ∧ f2 x = y), from H2,
have H4 : x ∈ a, from and.left H3,
have H5 : f1 x = y, from (H1 H4) ⬝ and.right H3,
exists.intro x (and.intro H4 H5)))
theorem in_image {f : X → Y} {a : set X} {x : X} {y : Y}
(H1 : x ∈ a) (H2 : f x = y) : y ∈ f '[a] :=
exists.intro x (and.intro H1 H2)
lemma image_compose (f : Y → Z) (g : X → Y) (a : set X) : (f ∘ g) '[a] = f '[g '[a]] :=
setext (take z,
iff.intro
(assume Hz : z ∈ (f ∘ g) '[a],
obtain x (Hx₁ : x ∈ a) (Hx₂ : f (g x) = z), from Hz,
have Hgx : g x ∈ g '[a], from in_image Hx₁ rfl,
show z ∈ f '[g '[a]], from in_image Hgx Hx₂)
(assume Hz : z ∈ f '[g '[a]],
obtain y (Hy₁ : y ∈ g '[a]) (Hy₂ : f y = z), from Hz,
obtain x (Hz₁ : x ∈ a) (Hz₂ : g x = y), from Hy₁,
show z ∈ (f ∘ g) '[a], from in_image Hz₁ (Hz₂⁻¹ ▸ Hy₂)))
/- maps to -/
definition maps_to [reducible] (f : X → Y) (a : set X) (b : set Y) : Prop := ∀⦃x⦄, x ∈ a → f x ∈ b
theorem maps_to_of_eq_on {f1 f2 : X → Y} {a : set X} {b : set Y} (eq_on_a : eq_on f1 f2 a)
(maps_to_f1 : maps_to f1 a b) :
maps_to f2 a b :=
take x,
assume xa : x ∈ a,
have H : f1 x ∈ b, from maps_to_f1 xa,
show f2 x ∈ b, from eq_on_a xa ▸ H
theorem maps_to_compose {g : Y → Z} {f : X → Y} {a : set X} {b : set Y} {c : set Z}
(H1 : maps_to g b c) (H2 : maps_to f a b) : maps_to (g ∘ f) a c :=
take x, assume H : x ∈ a, H1 (H2 H)
/- injectivity -/
definition inj_on [reducible] (f : X → Y) (a : set X) : Prop :=
∀⦃x1 x2 : X⦄, x1 ∈ a → x2 ∈ a → f x1 = f x2 → x1 = x2
theorem inj_on_of_eq_on {f1 f2 : X → Y} {a : set X} (eq_f1_f2 : eq_on f1 f2 a)
(inj_f1 : inj_on f1 a) :
inj_on f2 a :=
take x1 x2 : X,
assume ax1 : x1 ∈ a,
assume ax2 : x2 ∈ a,
assume H : f2 x1 = f2 x2,
have H' : f1 x1 = f1 x2, from eq_f1_f2 ax1 ⬝ H ⬝ (eq_f1_f2 ax2)⁻¹,
show x1 = x2, from inj_f1 ax1 ax2 H'
theorem inj_on_compose {g : Y → Z} {f : X → Y} {a : set X} {b : set Y}
(fab : maps_to f a b) (Hg : inj_on g b) (Hf: inj_on f a) :
inj_on (g ∘ f) a :=
take x1 x2 : X,
assume x1a : x1 ∈ a,
assume x2a : x2 ∈ a,
have fx1b : f x1 ∈ b, from fab x1a,
have fx2b : f x2 ∈ b, from fab x2a,
assume H1 : g (f x1) = g (f x2),
have H2 : f x1 = f x2, from Hg fx1b fx2b H1,
show x1 = x2, from Hf x1a x2a H2
/- surjectivity -/
definition surj_on [reducible] (f : X → Y) (a : set X) (b : set Y) : Prop := b ⊆ f '[a]
theorem surj_on_of_eq_on {f1 f2 : X → Y} {a : set X} {b : set Y} (eq_f1_f2 : eq_on f1 f2 a)
(surj_f1 : surj_on f1 a b) :
surj_on f2 a b :=
take y, assume H : y ∈ b,
obtain x (H1 : x ∈ a ∧ f1 x = y), from surj_f1 H,
have H2 : x ∈ a, from and.left H1,
have H3 : f2 x = y, from (eq_f1_f2 H2)⁻¹ ⬝ and.right H1,
exists.intro x (and.intro H2 H3)
theorem surj_on_compose {g : Y → Z} {f : X → Y} {a : set X} {b : set Y} {c : set Z}
(Hg : surj_on g b c) (Hf: surj_on f a b) :
surj_on (g ∘ f) a c :=
take z,
assume zc : z ∈ c,
obtain y (H1 : y ∈ b ∧ g y = z), from Hg zc,
obtain x (H2 : x ∈ a ∧ f x = y), from Hf (and.left H1),
show ∃x, x ∈ a ∧ g (f x) = z, from
exists.intro x
(and.intro
(and.left H2)
(calc
g (f x) = g y : {and.right H2}
... = z : and.right H1))
/- bijectivity -/
definition bij_on [reducible] (f : X → Y) (a : set X) (b : set Y) : Prop :=
maps_to f a b ∧ inj_on f a ∧ surj_on f a b
theorem bij_on_of_eq_on {f1 f2 : X → Y} {a : set X} {b : set Y} (eqf : eq_on f1 f2 a)
(H : bij_on f1 a b) : bij_on f2 a b :=
match H with and.intro Hmap (and.intro Hinj Hsurj) :=
and.intro
(maps_to_of_eq_on eqf Hmap)
(and.intro
(inj_on_of_eq_on eqf Hinj)
(surj_on_of_eq_on eqf Hsurj))
end
theorem bij_on_compose {g : Y → Z} {f : X → Y} {a : set X} {b : set Y} {c : set Z}
(Hg : bij_on g b c) (Hf: bij_on f a b) :
bij_on (g ∘ f) a c :=
match Hg with and.intro Hgmap (and.intro Hginj Hgsurj) :=
match Hf with and.intro Hfmap (and.intro Hfinj Hfsurj) :=
and.intro
(maps_to_compose Hgmap Hfmap)
(and.intro
(inj_on_compose Hfmap Hginj Hfinj)
(surj_on_compose Hgsurj Hfsurj))
end
end
/- left inverse -/
-- g is a left inverse to f on a
definition left_inv_on [reducible] (g : Y → X) (f : X → Y) (a : set X) : Prop :=
∀₀ x ∈ a, g (f x) = x
theorem left_inv_on_of_eq_on_left {g1 g2 : Y → X} {f : X → Y} {a : set X} {b : set Y}
(fab : maps_to f a b) (eqg : eq_on g1 g2 b) (H : left_inv_on g1 f a) : left_inv_on g2 f a :=
take x,
assume xa : x ∈ a,
calc
g2 (f x) = g1 (f x) : (eqg (fab xa))⁻¹
... = x : H xa
theorem left_inv_on_of_eq_on_right {g : Y → X} {f1 f2 : X → Y} {a : set X}
(eqf : eq_on f1 f2 a) (H : left_inv_on g f1 a) : left_inv_on g f2 a :=
take x,
assume xa : x ∈ a,
calc
g (f2 x) = g (f1 x) : {(eqf xa)⁻¹}
... = x : H xa
theorem inj_on_of_left_inv_on {g : Y → X} {f : X → Y} {a : set X} (H : left_inv_on g f a) :
inj_on f a :=
take x1 x2,
assume x1a : x1 ∈ a,
assume x2a : x2 ∈ a,
assume H1 : f x1 = f x2,
calc
x1 = g (f x1) : H x1a
... = g (f x2) : H1
... = x2 : H x2a
theorem left_inv_on_compose {f' : Y → X} {g' : Z → Y} {g : Y → Z} {f : X → Y}
{a : set X} {b : set Y} (fab : maps_to f a b)
(Hf : left_inv_on f' f a) (Hg : left_inv_on g' g b) : left_inv_on (f' ∘ g') (g ∘ f) a :=
take x : X,
assume xa : x ∈ a,
have fxb : f x ∈ b, from fab xa,
calc
f' (g' (g (f x))) = f' (f x) : Hg fxb
... = x : Hf xa
/- right inverse -/
-- g is a right inverse to f on a
definition right_inv_on [reducible] (g : Y → X) (f : X → Y) (b : set Y) : Prop :=
left_inv_on f g b
theorem right_inv_on_of_eq_on_left {g1 g2 : Y → X} {f : X → Y} {a : set X} {b : set Y}
(eqg : eq_on g1 g2 b) (H : right_inv_on g1 f b) : right_inv_on g2 f b :=
left_inv_on_of_eq_on_right eqg H
theorem right_inv_on_of_eq_on_right {g : Y → X} {f1 f2 : X → Y} {a : set X} {b : set Y}
(gba : maps_to g b a) (eqf : eq_on f1 f2 a) (H : right_inv_on g f1 b) : right_inv_on g f2 b :=
left_inv_on_of_eq_on_left gba eqf H
theorem surj_on_of_right_inv_on {g : Y → X} {f : X → Y} {a : set X} {b : set Y}
(gba : maps_to g b a) (H : right_inv_on g f b) :
surj_on f a b :=
take y,
assume yb : y ∈ b,
have gya : g y ∈ a, from gba yb,
have H1 : f (g y) = y, from H yb,
exists.intro (g y) (and.intro gya H1)
theorem right_inv_on_compose {f' : Y → X} {g' : Z → Y} {g : Y → Z} {f : X → Y}
{c : set Z} {b : set Y} (g'cb : maps_to g' c b)
(Hf : right_inv_on f' f b) (Hg : right_inv_on g' g c) : right_inv_on (f' ∘ g') (g ∘ f) c :=
left_inv_on_compose g'cb Hg Hf
theorem eq_on_of_left_inv_of_right_inv {g1 g2 : Y → X} {f : X → Y} {a : set X} {b : set Y}
(g2ba : maps_to g2 b a) (Hg1 : left_inv_on g1 f a) (Hg2 : right_inv_on g2 f b) : eq_on g1 g2 b :=
take y,
assume yb : y ∈ b,
calc
g1 y = g1 (f (g2 y)) : {(Hg2 yb)⁻¹}
... = g2 y : Hg1 (g2ba yb)
/- inverses -/
-- g is an inverse to f viewed as a map from a to b
definition inv_on [reducible] (g : Y → X) (f : X → Y) (a : set X) (b : set Y) : Prop :=
left_inv_on g f a ∧ right_inv_on g f b
theorem bij_on_of_inv_on {g : Y → X} {f : X → Y} {a : set X} {b : set Y} (fab : maps_to f a b)
(gba : maps_to g b a) (H : inv_on g f a b) : bij_on f a b :=
and.intro fab
(and.intro
(inj_on_of_left_inv_on (and.left H))
(surj_on_of_right_inv_on gba (and.right H)))
end set