我有Z
证明的引理。所有变量都限制为大于或等于零。
nat
的引理,即使用该引理通过使用nat
的引理来证明Z
的类似引理?
示例:
Require Import ZArith.
Open Scope Z.
Lemma Z_lemma:
forall n n0 n1 n2 n3 n4 n5 n6 : Z,
n >= 0 -> n0 >= 0 -> n1 >= 0 ->
n2 >= 0 -> n3 >= 0 -> n4 >= 0 ->
n5 >= 0 -> n6 >= 0 ->
n5 + n4 = n6 + n3 ->
n1 + n0 = n2 + n ->
n5 * n1 + n6 * n2 + n3 * n0 + n * n4 =
n5 * n2 + n1 * n6 + n3 * n + n0 * n4.
Admitted.
Close Scope Z.
Lemma nat_lemma:
forall n n0 n1 n2 n3 n4 n5 n6 : nat,
n5 + n4 = n6 + n3 ->
n1 + n0 = n2 + n ->
n5 * n1 + n6 * n2 + n3 * n0 + n * n4 =
n5 * n2 + n1 * n6 + n3 * n + n0 * n4.
(* prove this using `Z_lemma` *)
答案 0 :(得分:4)
你可以通过定义一种利用Z.of_nat
是单射的并且分布在(+)
和(*)
上的事实的策略来对所有具有这种形状的引理进行一般性的处理:
Ltac solve_using_Z_and lemma :=
(* Apply Z.of_nat to both sides of the equation *)
apply Nat2Z.inj;
(* Push Z.of_nat through multiplications and additions *)
repeat (rewrite Nat2Z.inj_mul || rewrite Nat2Z.inj_add);
(* Apply the lemma passed as an argument*)
apply lemma;
(* Discharge all the goals with the shape Z.of_nat m >= 0 *)
try (apply Zle_ge, Nat2Z.is_nonneg);
(* Push the multiplications and additions back through Z.of_nat *)
repeat (rewrite <- Nat2Z.inj_mul || rewrite <- Nat2Z.inj_add);
(* Peal off Z.of_nat on each side of the equation *)
f_equal;
(* Look up the assumption in the environment*)
assumption.
nat_lemma
的证明现在变成了:
Lemma nat_lemma:
forall n n0 n1 n2 n3 n4 n5 n6 : nat,
n5 + n4 = n6 + n3 ->
n1 + n0 = n2 + n ->
n5 * n1 + n6 * n2 + n3 * n0 + n * n4 =
n5 * n2 + n1 * n6 + n3 * n + n0 * n4.
Proof.
intros; solve_using_Z_and Z_lemma.
Qed.