如何在 Coq 中证明 0 <= a < b -> rem ab = a？

Question

I'm trying to prove this lemma

Require Import ZArith.
Require Import Lia.

Open Scope Z_scope.
Import Z.

Ltac Zify.zify_post_hook ::= Z.to_euclidean_division_equations.

Lemma rem_pos : forall {a b : Z},
  0 <= a < b ->
  rem a b = a.
 intros. lia.

At this point I receive Tactic failure: Cannot find witness. from lia .

Am I doing something wrong?

Answer 1

Search is your friend:

Require Import ZArith.

Open Scope Z_scope.
Import Z.

Lemma rem_pos : forall {a b : Z},
  0 <= a < b ->
  rem a b = a.
Proof.
  intros a b H.
  Search (rem ?a ?b = ?a).
  apply rem_small.
  assumption.
Qed.

Afaik there is no dedicated proof automation for modulo arithmetic. Coq-hammer is able to solve it with e-prover and Z3 backends (coming soon in Coq Platform):

Require Import ZArith.
From Hammer Require Import Hammer.

Open Scope Z_scope.
Import Z.

Lemma rem_pos : forall {a b : Z},
  0 <= a < b ->
  rem a b = a.
Proof.
  hammer.
Qed.

But this is trivial for an ATP, so not a reasonable test. The problem is that one can't easily combine ATPs with the power of lia.

In my experience your best bet is Search an adequate lemma and solve the preconditions with lia.