问题描述
我正在尝试证明Z_3类型的组公理：
Require Import Coq.Arith.PeanoNat.
Record Z_3 : Type := Z3
{
n :> nat;
proof : (Nat.ltb n 3) = true
}.
Proposition lt_0_3 : (0 <? 3) = true.
Proof.
simpl. reflexivity.
Qed.
Definition z3_0 : Z_3 := (Z3 0 lt_0_3).
Proposition lt_1_3 : (1 <? 3) = true.
Proof.
reflexivity.
Qed.
Definition z3_1 : Z_3 := (Z3 1 lt_1_3).
Proposition lt_2_3 : (2 <? 3) = true.
Proof.
reflexivity.
Qed.
Definition z3_2 : Z_3 := (Z3 2 lt_2_3).
Proposition three_ne_0 : 3 <> 0.
Proof.
discriminate.
Qed.
Lemma mod_upper_bound_bool : forall (a b : nat),(not (eq b O)) -> (Nat.ltb (a mod b) b) = true.
Proof.
intros a b H. apply (Nat.mod_upper_bound a b) in H. case Nat.ltb_spec0.
- reflexivity.
- intros Hcontr. contradiction.
Qed.
Definition Z3_op (x y: Z_3) : Z_3 :=
let a := (x + y) mod 3 in
Z3 a (mod_upper_bound_bool _ 3 three_ne_0).
Lemma Z3_eq n m p q : n = m -> Z3 n p = Z3 m q.
Proof.
intros H. revert p q. rewrite H. clear H. intros. apply f_equal.
我们快完成了：
1 subgoal (ID 41)
n,m : nat
p,q : (m <? 3) = true
============================
p = q
我应该使用什么定理证明p = q？
更新1
Theorem bool_dec :
(forall x y: bool,{x = y} + {x <> y}).
Proof.
intros x y. destruct x.
- destruct y.
+ left. reflexivity.
+ right. intro. discriminate H.
- destruct y.
+ right. intro. discriminate H.
+ left. reflexivity.
Qed.
Lemma Z3_eq n m p q : n = m -> Z3 n p = Z3 m q.
Proof.
intros H. revert p q. rewrite H. clear H. intros. apply f_equal. apply UIP_dec. apply bool_dec.
Qed.
解决方法
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