104 lines
2.8 KiB
Coq
Executable File
104 lines
2.8 KiB
Coq
Executable File
Require Import Coq.ZArith.ZArith.
|
|
Require Import Coq.Bool.Bool.
|
|
Require Import Coq.Lists.List.
|
|
Require Import Coq.Classes.RelationClasses.
|
|
Require Import Coq.Classes.Morphisms.
|
|
Require Import Coq.micromega.Psatz.
|
|
Require Import Permutation.
|
|
Require Import String.
|
|
From AUXLib Require Import int_auto Axioms Feq Idents List_lemma VMap.
|
|
Require Import SetsClass.SetsClass. Import SetsNotation.
|
|
From SimpleC.SL Require Import CommonAssertion Mem SeparationLogic IntLib.
|
|
Require Import Logic.LogicGenerator.demo932.Interface.
|
|
Local Open Scope Z_scope.
|
|
Local Open Scope sets.
|
|
Import ListNotations.
|
|
Local Open Scope list.
|
|
Require Import String.
|
|
Local Open Scope string.
|
|
Import naive_C_Rules.
|
|
Local Open Scope sac.
|
|
|
|
Module Aux.
|
|
|
|
Lemma Z_of_nat_succ: forall (n: nat),
|
|
Z.of_nat (S n) = Z.of_nat n + 1.
|
|
Proof. lia. Qed.
|
|
|
|
Lemma Zpow_add_1: forall (a b: Z),
|
|
a >= 0 -> b >= 0 ->
|
|
a ^ (b + 1) = a ^ b * a.
|
|
Proof.
|
|
intros.
|
|
rewrite (Z.pow_add_r a b 1); lia.
|
|
Qed.
|
|
|
|
Lemma Zmul_mod_cancel: forall (n a b: Z),
|
|
n >= 0 -> a > 0 -> b >= 0 ->
|
|
(n * a) mod (a ^ (b + 1)) = a * (n mod (a ^ b)).
|
|
Proof.
|
|
intros.
|
|
pose proof (Z_div_mod_eq_full n (a ^ b)).
|
|
pose proof (Z.mod_bound_pos n (a ^ b) ltac:(lia) ltac:(nia)).
|
|
remember (n / a ^ b) as q eqn:Hq.
|
|
remember (n mod a ^ b) as rem eqn:Hrem.
|
|
rewrite H2.
|
|
rewrite Z.mul_add_distr_r.
|
|
rewrite (Z.mul_comm (a ^ b) q); rewrite <-Z.mul_assoc.
|
|
rewrite <-Zpow_add_1; try lia.
|
|
assert (0 <= rem * a < a ^ (b + 1)). {
|
|
rewrite Zpow_add_1; try lia.
|
|
nia.
|
|
}
|
|
rewrite <-(Zmod_unique_full (q * a ^ (b + 1) + rem * a) (a ^ (b + 1)) q (rem * a)).
|
|
+ lia.
|
|
+ unfold Remainder.
|
|
lia.
|
|
+ lia.
|
|
Qed.
|
|
|
|
Lemma Zdiv_mod_pow: forall (n a b: Z),
|
|
a > 0 -> b >= 0 -> n >= 0 ->
|
|
(n / a) mod (a ^ b) = (n mod (a ^ (b + 1))) / a.
|
|
Proof.
|
|
intros.
|
|
pose proof (Z_div_mod_eq_full n (a ^ (b + 1))).
|
|
remember (n / (a ^ (b + 1))) as q eqn:Hq.
|
|
remember (n mod a ^ (b + 1)) as rem eqn:Hrem.
|
|
assert (n / a = a ^ b * q + rem / a). {
|
|
rewrite H2.
|
|
rewrite Zpow_add_1; try lia.
|
|
pose proof (Z_div_plus_full_l (a ^ b * q) a rem ltac:(lia)).
|
|
assert (a ^ b * q * a + rem = a ^ b * a * q + rem). { lia. }
|
|
rewrite H4 in H3.
|
|
tauto.
|
|
}
|
|
apply Znumtheory.Zdivide_mod_minus.
|
|
+ pose proof (Z.mod_bound_pos n (a ^ (b + 1)) ltac:(lia) (Z.pow_pos_nonneg a (b + 1) ltac:(lia) ltac:(lia))).
|
|
rewrite <-Hrem in H4.
|
|
rewrite Zpow_add_1 in H4; try lia.
|
|
pose proof (Z.div_lt_upper_bound rem a (a ^ b) ltac:(lia) ltac:(lia)).
|
|
split; try lia.
|
|
apply (Z_div_pos rem a ltac:(lia) ltac:(lia)).
|
|
+ unfold Z.divide.
|
|
exists q.
|
|
lia.
|
|
Qed.
|
|
|
|
Lemma list_app_cons: forall (l1 l2: list Z) (a: Z),
|
|
app l1 (a :: l2) = app (app l1 (a :: nil)) l2.
|
|
Proof.
|
|
intros.
|
|
revert a l2.
|
|
induction l1.
|
|
+ intros.
|
|
rewrite app_nil_l.
|
|
reflexivity.
|
|
+ intros.
|
|
simpl in *.
|
|
specialize (IHl1 a0 l2).
|
|
rewrite IHl1.
|
|
reflexivity.
|
|
Qed.
|
|
|
|
End Aux. |