[英]How can I proove forall a b, a <=? b = true -> a <=? S b = true in Coq
Is it possible to prove this forall (ab: nat), a <=? b = true -> a <=? S b = true.
是否有可能证明这个forall (ab: nat), a <=? b = true -> a <=? S b = true.
forall (ab: nat), a <=? b = true -> a <=? S b = true.
in Coq?在考克?
I tried this so far到目前为止我试过这个
Lemma leb_0_r : forall x, x <=? 0 = true -> x = 0.
intros. induction x. reflexivity. discriminate H.
Qed.
Lemma leb_S : forall a b, a <=? b = true -> a <=? S b = true.
intros a b Hab. induction b. apply leb_0_r in Hab. now rewrite Hab.
But here I got stuck on the induction hypothesis但是在这里我陷入了归纳假设
1 subgoal
a, b : nat
Hab : (a <=? S b) = true
IHb : (a <=? b) = true -> (a <=? S b) = true
========================= (1 / 1)
(a <=? S (S b)) = true
I tried induction on a too我也试过感应
Lemma leb_S : forall a b, a <=? b = true -> a <=? S b = true.
intros a b Hab. induction a. reflexivity. simpl. destruct b.
discriminate Hab. simpl in Hab.
1 subgoal
a, b : nat
Hab : (a <=? b) = true
IHa : (a <=? S b) = true -> (a <=? S (S b)) = true
========================= (1 / 1)
(a <=? S b) = true
The problem is that I always get to S a <= b
or a <= S b
and I can't simplify that.问题是我总是得到S a <= b
或a <= S b
,我无法简化它。
After posting here I realized that conclusion of IHa is equal to the goal of second try and vice versa:thinking:在这里发帖后,我意识到 IHa 的结论等于第二次尝试的目标,反之亦然:思考:
You could try not to use induction, but transitivity of the <=
relation instead.您可以尝试不使用归纳,而是使用<=
关系的传递性。
I am nor sure if you are learning Coq and this is an exercise or if you are using Coq.我不确定您是否正在学习 Coq,这是一个练习,或者您是否正在使用 Coq。 In the latter case the answer is: I would have thought the lia tactic can do this, but it requires a bit of massaging:在后一种情况下,答案是:我原以为 lia 策略可以做到这一点,但它需要一点按摩:
Require Import PeanoNat.
Require Import Lia.
Lemma leb_0_r : forall x, x <=? 0 = true -> x = 0.
Proof.
intros.
Fail lia.
Search (_ <=? _ = true).
apply Nat.leb_le in H.
lia.
Qed.
In the former case, I would need to know what you are allowed to use.在前一种情况下,我需要知道您可以使用什么。 Eg this works:例如,这有效:
Require Import PeanoNat.
Lemma leb_0_r : forall x, x <=? 0 = true -> x = 0.
Proof.
intros.
apply Nat.leb_le in H.
inversion H.
reflexivity.
Qed.
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