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Coqで証明 関数に関する証明

2020/09/28に公開約2,200字

Coqには関数の型があります。述語や関係や論理結合子も関数の一種です。例えば、andはProp->Prop->Propの型を持っています。また、関数の型A->Bはforall _:A,Bの糖衣構文です。
今回はCoqの関数についていくつか証明してみました。証明は便利なタクティクを使って手抜きしました。

Require Import Utf8.

Set Implicit Arguments.

関数の相等、逆関数、合成、恒等関数、単射、全射を定義する。

Definition feq A B(f g:A->B):=
forall a,f a=g a.

Definition cancel A B(f:A->B)(g:B->A):=
forall a,g(f(a))=a.

Definition injection A B(f:A->B):=
forall a1 a2,f a1=f a2->a1=a2.

Definition surjection A B(f:A->B):=
forall b,exists a,f a=b.

Definition comp A B C(f:A->B)(g:B->C)(a:A):=g(f a).

Definition id {A}(a:A):=a.

証明本文。


Goal forall A B(f:A->B),
feq f f.
Proof.
split.
Qed.

Goal forall A B(f g:A->B),
feq f g->feq g f.
Proof.
easy.
Qed.

Goal forall A B(f g h:A->B),
feq f g->feq g h->feq f h.
Proof.
congruence.
Qed.

Goal forall A B(f:A->B),
feq(fun a=>f a)f.
Proof.
split.
Qed.

Goal forall A B C D(f:A->B)(g:B->C)(h:C->D),
comp(comp f g)h=comp f(comp g h).
Proof.
split.
Qed.

Goal forall A B(f:A->B),
feq(comp f id)f.
Proof.
split.
Qed.

Goal forall A B(f:A->B),
feq(comp id f)f.
Proof.
split.
Qed.

Goal forall A,injection(id:A->A).
Proof.
easy.
Qed.

Goal forall A,surjection(id:A->A).
Proof.
intros.
intro.
exists b.
split.
Qed.

Goal forall A B(f:A->B)(g:B->A),
cancel f g<->forall a b,f a=b->g b=a.
Proof.
firstorder.
congruence.
Qed.

Goal forall A B(f:A->B)(g:B->A),
cancel f g<->feq(comp f g)id.
Proof.
easy.
Qed.

Goal forall A(a:A),id a=a.
Proof.
split.
Qed.

Goal forall A B(f h:A->B)(g:B->A),
cancel f g->
cancel g h->
feq f h.
Proof.
congruence.
Qed.

Goal forall A B C(f:A->B)(g:B->C),
injection f->
injection g->
injection(comp f g).
Proof.
firstorder.
Qed.

Goal forall A B C(f:A->B)(g:B->C),
surjection f->
surjection g->
surjection(comp f g).
Proof.
intros.
intro.
destruct(H0 b).
destruct(H x).
exists x0.
unfold comp.
congruence.
Qed.

Goal forall A B C(f:A->B)(g:B->C),
injection(comp f g)->injection f.
Proof.
unfold injection,comp.
intros.
apply H.
congruence.
Qed.

Goal forall A B C(f:A->B)(g:B->C),
surjection(comp f g)->surjection g.
Proof.
firstorder.
Qed.

Goal forall A B(f:A->B)(g:B->A),
cancel f g->
injection f.
Proof.
congruence.
Qed.

Goal forall A B(f:A->B)(g:B->A),
cancel f g->
surjection g.
Proof.
firstorder.
Qed.

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