Add annotations to gmp.c

projects/lib/GmpAux.v

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+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.