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