feat: Add definitions for basic structures.

projects/lib/gmp_aux.v

@@ -0,0 +1,63 @@
+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 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 Z_of_nat_succ: forall (n: nat),
+  Z.of_nat (S n) = Z.of_nat n + 1.
+
+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.
+      
+End Aux.