ZYM/MiniGmp-Verification
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Merge branch 'main' of https://github.com/xiaoh105/MiniGmp-Verification

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This commit is contained in:
ZhuangYumin
2025-06-13 12:50:34 +00:00
parent 596f66206e 257241df90
commit cce2450cd7
13 changed files with 3596 additions and 263 deletions
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25
Makefile
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@ -15,7 +15,9 @@ SETS_FLAG = -R $(SETS_DIR) SetsClass
COMPCERT_FLAG = -R $(COMPCERT_DIR) compcert.lib
DEP_FLAG = -R $(PV_DIR) PV -R $(SETS_DIR) SetsClass -R $(COMPCERT_DIR) compcert.lib
GMP_FLAG = $(shell grep -E '^-([RQ])' _CoqProject)
DEP_FLAG = $(GMP_FLAG)
SETS_FILE_NAMES = \
SetsClass.v SetsDomain.v SetElement.v RelsDomain.v SetElementProperties.v SetProd.v SetsClass_AxiomFree.v SetsDomain_Classic.v
@ -35,21 +37,20 @@ PV_FILE_NAMES = \
PV_FILES=$(PV_FILE_NAMES:%.v=$(PV_DIR)/%.v)
GMP_FILES = \
./projects/lib/GmpAux.v ./projects/lib/GmpNumber.v \
./projects/lib/gmp_goal.v
FILES = $(PV_FILES) \
$(SETS_FILES) \
$(COMPCERT_FILES)
$(SETS_FILES:%.v=%.vo): %.vo: %.v
@echo COQC $<
@$(COQC) $(SETS_FLAG) $<
$(COMPCERT_FILES:%.v=%.vo): %.vo: %.v
@echo COQC $<
@$(COQC) $(COMPCERT_FLAG) $<
$(GMP_FILES)
$(PV_FILES:%.v=%.vo): %.vo: %.v
@echo COQC $(<F)
@$(COQC) $(PV_FLAG) $<
@$(COQC) $(GMP_FLAG) $<
$(GMP_FILES:%.v=%.vo): %.vo: %.v
@echo COQC $(<F)
@$(COQC) $(GMP_FLAG) $<
all: $(FILES:%.v=%.vo)

126
projects/int_array.strategies Normal file
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@ -0,0 +1,126 @@
//test curent rules
//max rule id:30
#include "int_array_def.h"
#include "verification_list.h"
#include "verification_stdlib.h"
id : 1
priority : core(1)
left : store_int_array(?p, ?n, ?l) at 0
right : data_at(?p + (?i * sizeof(I32)), I32, ?v) at 1
check : infer(0 <= i);
infer(i < n);
action : right_erase(1);
left_erase(0);
left_add(store_int_array_missing_i_rec(p, i, 0, n, l));
right_add(v == l[i]);
id : 2
priority : post(1)
left : store_int_array_missing_i_rec(?p, ?i, 0, ?n, ?l) at 0
data_at(p + i * sizeof(I32), I32, l[i]) at 1
check : infer(0 <= i);
infer(i < n);
action : left_erase(1);
left_erase(0);
left_add(store_int_array(p, n, l));
id : 3
priority : post(3)
left : store_int_array_missing_i_rec(?p, ?i, 0, ?n, ?l) at 0
data_at(p + i * sizeof(I32), I32, ?v) at 1
check : infer(0 <= i);
infer(i < n);
action : left_erase(1);
left_erase(0);
left_add(store_int_array(p, n, replace_Znth{Z}(i, v, l)));
id : 4
priority : core(1)
left : store_int_array(?p, ?n, ?l1) at 0
right : store_int_array(p, n, ?l2) at 1
action : right_erase(1);
left_erase(0);
right_add(l1 == l2);
id : 5
priority : core(1)
left : store_int_array_missing_i_rec(?p, ?i, ?v, ?n, ?l) at 0
right : store_int_array_missing_i_rec(p, i, v, n, l) at 1
action : left_erase(0);
right_erase(1);
id : 6
priority : core(1)
left : store_int_array_rec(?p, ?x, ?y, ?l1) at 0
right : store_int_array_rec(?p, ?x, ?y, ?l2) at 1
action : right_erase(1);
left_erase(0);
right_add(l1 == l2);
id : 7
priority : core(1)
left : store_int_array_rec(?p, ?x, ?y, ?l) at 0
right : data_at(?p + (?i * sizeof(I32)), I32, ?v) at 1
check : infer(x <= i);
infer(i < y);
action : right_erase(1);
left_erase(0);
left_add(store_int_array_missing_i_rec(p, i, x, y, l));
right_add(v == l[i - x]);
id : 8
priority : core(1)
left : store_int_array_rec(?p, ?x, ?y, ?l1) at 0
store_int_array_rec(?p, ?y, ?z, ?l2) at 1
right : store_int_array_rec(?p, ?x, ?z, ?l3) at 2
check : infer(y <= z);
infer(x <= y);
action : right_erase(2);
left_erase(0);
left_erase(1);
right_add(l3 == app{Z}(l1, l2));
id : 9
priority : core(1)
left : store_int_array_rec(?p, ?x, ?z, ?l3) at 2
right : store_int_array_rec(?p, ?x, ?y, ?l1) at 0
store_int_array_rec(?p, ?y, ?z, ?l2) at 1
check : infer(y <= z);
infer(x <= y);
action : left_erase(2);
right_erase(0);
right_erase(1);
right_add(l3 == app{Z}(l1, l2));
right_add(Zlength{Z}(l1) == y - x);
id : 10
priority : core(1)
right : store_int_array_rec(?p, ?x, ?x, ?l) at 0
action : right_erase(0);
right_add(l == nil{Z});
id : 11
priority : post(1)
left : store_int_array_missing_i_rec(?p, ?i, ?x, ?y, ?l) at 0
data_at(p + ?i * sizeof(I32), I32, l[?i - ?x]) at 1
check : infer(x <= i);
infer(i < y);
action : left_erase(1);
left_erase(0);
left_add(store_int_array_rec(p, x, y, l));
id : 12
priority : post(4)
left : store_int_array_missing_i_rec(?p, ?i, ?x, ?y, ?l) at 0
data_at(p + ?i * sizeof(I32), I32, ?v) at 1
check : infer(x <= i);
infer(i < y);
action : left_erase(1);
left_erase(0);
left_add(store_int_array_rec(p, x, y, replace_Znth{Z}(i - x, v, l)));

19
projects/int_array_def.h Normal file
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@ -0,0 +1,19 @@
/*@ Extern Coq (sum : list Z -> Z)
(sublist : {A} -> Z -> Z -> list A -> list A)
(store_int_array : Z -> Z -> list Z -> Assertion)
(store_uint_array : Z -> Z -> list Z -> Assertion)
(store_undef_uint_array_rec : Z -> Z -> Z -> Assertion)
(store_int_array_missing_i_rec: Z -> Z -> Z -> Z -> list Z -> Assertion)
(store_uint_array_missing_i_rec: Z -> Z -> Z -> Z -> list Z -> Assertion)
(store_array_box: Z -> Z -> list Z -> Assertion)
(store_array_box_array: Z -> list Z -> Assertion)
(store_int_array_rec: Z -> Z -> Z -> list Z -> Assertion)
(store_uint_array_rec: Z -> Z -> Z -> list Z -> Assertion)
(Znth: {A} -> Z -> list A -> A -> A)
(replace_Znth: {A} -> Z -> A -> list A -> list A)
(zeros: Z -> list Z)
*/
/*@ include strategies "int_array.strategies" */
/*@ include strategies "uint_array.strategies" */

345
projects/lib/GmpAux.v Executable file
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@ -0,0 +1,345 @@
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.
Lemma list_app_single_l: forall (l: list Z) (a: Z),
([a] ++ l)%list = a :: l.
Proof.
intros.
induction l.
+ simpl; reflexivity.
+ rewrite list_app_cons.
simpl.
reflexivity.
Qed.
Lemma list_last_cons: forall (a: Z) (l: list Z),
l <> nil ->
last (a :: l) 1 = last l 1.
Proof.
intros.
destruct l.
+ contradiction.
+ simpl.
reflexivity.
Qed.
Lemma list_last_to_Znth: forall (l: list Z),
l <> nil ->
last l 1 = Znth ((Zlength l) - 1) l 0.
Proof.
intros.
induction l.
+ contradiction.
+ destruct l.
- simpl.
rewrite Znth0_cons.
lia.
- pose proof (@nil_cons Z z l).
specialize (IHl ltac:(auto)); clear H0.
rewrite list_last_cons; [ | pose proof (@nil_cons Z z l); auto ].
rewrite IHl.
pose proof (Zlength_cons a (z :: l)).
unfold Z.succ in H0; rewrite H0; clear H0.
pose proof (Zlength_nonneg l).
pose proof (Zlength_cons z l); unfold Z.succ in H1.
pose proof (Znth_cons (Zlength (z :: l)) a (z :: l) 0 ltac:(lia)).
assert (Zlength (z :: l) + 1 - 1 = Zlength (z :: l)). { lia. }
rewrite H3; clear H3.
rewrite H2.
reflexivity.
Qed.
Lemma Zlength_removelast: forall (l: list Z),
l <> [] ->
Zlength (removelast l) = Zlength l - 1.
Proof.
intros.
induction l.
+ contradiction.
+ destruct l.
- simpl.
rewrite Zlength_nil.
lia.
- pose proof (@nil_cons Z z l).
specialize (IHl ltac:(auto)).
assert (removelast (a :: z :: l) = a :: removelast(z :: l)). {
simpl.
reflexivity.
}
rewrite H1; clear H1.
repeat rewrite Zlength_cons; unfold Z.succ.
rewrite IHl.
rewrite Zlength_cons; unfold Z.succ.
lia.
Qed.
Lemma store_array_rec_false: forall x storeA lo hi (l: list Z),
lo > hi ->
store_array_rec storeA x lo hi l |-- [| False |].
Proof.
intros.
revert x storeA lo hi H.
induction l; intros.
+ simpl.
entailer!.
+ simpl.
specialize (IHl x storeA (lo + 1) hi ltac:(lia)).
sep_apply IHl.
entailer!.
Qed.
Lemma store_array_rec_empty: forall x storeA lo (l: list Z),
store_array_rec storeA x lo lo l |-- emp && [| l = nil |].
Proof.
intros.
destruct l.
+ simpl.
entailer!.
+ simpl.
sep_apply store_array_rec_false; [ entailer! | lia ].
Qed.
Lemma store_uint_array_rec_false: forall x lo hi l,
lo > hi ->
store_uint_array_rec x lo hi l |-- [| False |].
Proof.
intros.
unfold store_uint_array_rec.
sep_apply store_array_rec_false; [ entailer! | lia ].
Qed.
Lemma store_uint_array_rec_empty: forall x lo l,
store_uint_array_rec x lo lo l |-- emp && [| l = nil |].
Proof.
induction l.
+ unfold store_uint_array_rec.
simpl.
entailer!.
+ pose proof (store_uint_array_rec_false x (lo + 1) lo l ltac:(lia)).
unfold store_uint_array_rec in *.
simpl in *.
sep_apply H.
entailer!.
Qed.
Lemma store_uint_array_empty: forall x l,
store_uint_array x 0 l |-- emp && [| l = nil |].
Proof.
intros x l.
revert x.
induction l; intros.
+ unfold store_uint_array, store_array.
simpl.
entailer!.
+ unfold store_uint_array, store_array.
simpl.
sep_apply store_array_rec_false; [ entailer! | lia ].
Qed.
Lemma uint_array_rec_to_uint_array: forall x lo hi l,
lo <= hi ->
store_uint_array_rec x lo hi l --||--
store_uint_array (x + sizeof(UINT) * lo) (hi - lo) l.
Proof.
intros.
unfold store_uint_array_rec, store_uint_array, store_array.
pose proof (store_array_rec_base x 0 lo hi l (sizeof(UINT))
store_uint
(fun (x: addr) (lo a: Z) =>
(x + lo * sizeof(UINT)) # UInt |-> a) ltac:(reflexivity)).
assert (x + sizeof(UINT) * lo = x + lo * sizeof(UINT)). { lia. }
rewrite H1; clear H1.
assert (0 + lo = lo). { lia. }
repeat rewrite H1 in H0; clear H1.
destruct H0.
split.
+ sep_apply H0.
entailer!.
+ sep_apply H1.
entailer!.
Qed.
Lemma store_uint_array_single: forall x n a,
store_uint_array_rec x n (n + 1) [a] --||--
(x + n * sizeof(UINT)) # UInt |-> a.
Proof.
intros.
unfold store_uint_array_rec.
simpl.
split; entailer!.
Qed.
Lemma store_undef_uint_array_single: forall x n a,
(x + n * sizeof(UINT)) # UInt |-> a |--
store_undef_uint_array_rec x n (n + 1).
Proof.
intros.
unfold store_undef_uint_array_rec.
assert (Z.to_nat (n + 1 - n) = S O). { lia. }
rewrite H; clear H.
simpl.
entailer!.
sep_apply store_uint_undef_store_uint.
entailer!.
Qed.
Lemma store_uint_array_single_to_undef: forall x n a,
store_uint_array_rec x n (n + 1) [a] |--
store_undef_uint_array_rec x n (n + 1).
Proof.
intros.
pose proof (store_uint_array_single x n a).
destruct H as [H _].
sep_apply H.
sep_apply store_undef_uint_array_single.
entailer!.
Qed.
Lemma store_uint_array_divide_rec: forall x n l m,
0 <= m <= n ->
Zlength l = n ->
store_uint_array x n l --||--
store_uint_array x m (sublist 0 m l) **
store_uint_array_rec x m n (sublist m n l).
Proof.
intros.
pose proof (store_uint_array_divide x n l m H H0).
pose proof (uint_array_rec_to_uint_array x m n (sublist m n l) ltac:(lia)).
destruct H2 .
destruct H1.
rewrite Z.mul_comm in H2, H3.
rewrite H3 in H1.
rewrite <-H2 in H4.
clear H2 H3.
split; tauto.
Qed.
Lemma store_undef_uint_array_rec_divide: forall x l mid r,
0 <= l <= r ->
l <= mid <= r ->
store_undef_uint_array_rec x l r --||--
store_undef_uint_array_rec x l mid ** store_undef_uint_array_rec x mid r.
Proof.
intros.
unfold store_undef_uint_array_rec.
pose proof (store_undef_array_divide (x + l * sizeof(UINT)) (r - l) (mid - l) (sizeof(UINT))
(fun x => x # UInt |->_)
(fun x0 lo => (x0 + lo * sizeof(UINT)) # UInt |->_) ltac:(lia) ltac:(reflexivity)).
unfold store_undef_array in H1.
pose (fun (l: Z) (n: Z) (len: nat) => store_undef_array_rec_base x 0 l n len (sizeof(UINT))
(fun x => x # UInt |->_)
(fun x0 lo => (x0 + lo * sizeof(UINT)) # UInt |->_) ltac:(reflexivity)).
rewrite <-(l0 l r (Z.to_nat (r - l))) in H1.
rewrite <-(l0 l mid (Z.to_nat (mid - l))) in H1.
assert (r - l - (mid - l) = r - mid). { lia. }
rewrite H2 in H1; clear H2.
rewrite <-Z.add_assoc, <-Z.mul_add_distr_r in H1.
assert (l + (mid - l) = mid). { lia. }
rewrite H2 in H1; clear H2.
rewrite <-(l0 mid r (Z.to_nat (r - mid))) in H1.
repeat rewrite Z.add_0_l in H1.
clear l0.
rewrite H1.
split; entailer!.
Qed.
End Aux.

394
projects/lib/gmp_number.v → projects/lib/GmpNumber.v
View File

@ -9,6 +9,7 @@ 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 GmpLib.GmpAux.
Require Import Logic.LogicGenerator.demo932.Interface.
Local Open Scope Z_scope.
Local Open Scope sets.
@ -20,97 +21,14 @@ Import naive_C_Rules.
Local Open Scope sac.
Notation "'UINT_MOD'" := (4294967296).
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.
Notation "'LENGTH_MAX'" := (100000000).
Module Internal.
Definition mpd_store_list (ptr: addr) (data: list Z) (cap: Z): Assertion :=
[| Zlength data <= cap |] &&
store_uint_array ptr (Zlength data) data &&
store_undef_uint_array_rec ptr ((Zlength data) + 1) cap.
[| Zlength data <= cap |] && [| cap <= LENGTH_MAX |] &&
store_uint_array ptr (Zlength data) data **
store_undef_uint_array_rec ptr (Zlength data) cap.
Fixpoint list_to_Z (data: list Z): Z :=
match data with
@ -129,7 +47,47 @@ Definition list_store_Z (data: list Z) (n: Z): Prop :=
Definition mpd_store_Z (ptr: addr) (n: Z) (size: Z) (cap: Z): Assertion :=
EX data,
mpd_store_list ptr data cap && [| list_store_Z data n|] && [| size = Zlength data |].
[| list_store_Z data n |] && [| size = Zlength data |] && mpd_store_list ptr data cap.
Definition list_store_Z_compact (data: list Z) (n: Z): Prop :=
list_to_Z data = n /\ last data 1 >= 1 /\ list_within_bound data.
Definition mpd_store_Z_compact (ptr: addr) (n size cap: Z): Assertion :=
EX data,
[| list_store_Z_compact data n |] && [| size = Zlength data |] && mpd_store_list ptr data cap.
Lemma list_store_Z_normal_to_compact: forall (data: list Z) (n: Z),
list_store_Z data n ->
last data 1 >= 1 ->
list_store_Z_compact data n.
Proof.
unfold list_store_Z_compact, list_store_Z.
intros.
tauto.
Qed.
Lemma list_store_Z_compact_to_normal: forall data n,
list_store_Z_compact data n ->
list_store_Z data n.
Proof.
intros.
unfold list_store_Z_compact in H.
unfold list_store_Z.
tauto.
Qed.
Lemma list_store_Z_injection: forall l1 l2 n1 n2,
list_store_Z l1 n1 ->
list_store_Z l2 n2 ->
l1 = l2 -> n1 = n2.
Proof.
intros.
unfold list_store_Z in *.
destruct H, H0.
rewrite <-H, <-H0.
rewrite <-H1.
reflexivity.
Qed.
Lemma __list_within_bound_concat_r: forall (l1: list Z) (a: Z),
list_within_bound l1 ->
@ -164,6 +122,29 @@ Proof.
tauto.
Qed.
Lemma list_within_bound_Znth: forall (l: list Z) (i: Z),
0 <= i < Zlength l ->
list_within_bound l ->
0 <= Znth i l 0 < UINT_MOD.
Proof.
intros.
revert i H.
induction l; intros.
+ rewrite Zlength_nil in H.
lia.
+ assert (i = 0 \/ i > 0). { lia. }
destruct H1.
- rewrite H1.
rewrite (Znth0_cons a l 0).
simpl in H0.
lia.
- rewrite Znth_cons; try lia.
simpl in H0; destruct H0.
rewrite Zlength_cons in H; unfold Z.succ in H.
specialize (IHl H2 (i - 1) ltac:(lia)).
lia.
Qed.
Lemma __list_within_bound_split_r: forall (l1: list Z) (a: Z),
list_within_bound (l1 ++ [a]) ->
list_within_bound l1 /\ 0 <= a < UINT_MOD.
@ -371,6 +352,243 @@ Proof.
- pose proof (Zlength_nonneg l1); lia.
Qed.
Lemma list_store_Z_nth: forall (l: list Z) (n: Z) (i: Z),
0 <= i < Zlength l ->
list_store_Z l n ->
Znth i l 0 = (n / (UINT_MOD ^ i)) mod UINT_MOD.
Proof.
intros.
revert n i H H0.
induction l; intros.
+ rewrite Zlength_nil in H.
lia.
+ rewrite Zlength_cons in H; unfold Z.succ in H.
assert (i = 0 \/ i > 0). { lia. }
destruct H1.
- pose proof (list_store_Z_split [a] l n).
assert (a :: l = app [a] l). { auto. }
rewrite <-H3 in H2; clear H3.
specialize (H2 H0); destruct H2.
unfold list_store_Z, list_to_Z in H2.
unfold Zlength, Zlength_aux, Z.succ in H2.
destruct H2.
rewrite (Aux.Zpow_add_1 UINT_MOD 0) in H2; try lia.
rewrite Z.pow_0_r, Z.mul_1_l in H2.
simpl in H2.
rewrite H1; rewrite Znth0_cons.
rewrite Z.pow_0_r, Z.div_1_r.
lia.
- rewrite Znth_cons; [ | lia ].
unfold list_store_Z in H0; destruct H0.
simpl in H0.
simpl in H2.
unfold list_store_Z in IHl.
assert (list_to_Z l = (n - a) / UINT_MOD /\ list_within_bound l). {
rewrite (Z.div_unique_exact (n - a) UINT_MOD (list_to_Z l)); try lia; try tauto.
}
specialize (IHl ((n - a) / UINT_MOD) (i - 1) ltac:(lia) H3).
rewrite IHl.
assert ((n - a) / UINT_MOD / (UINT_MOD ^ (i - 1)) = (n - a) / (UINT_MOD ^ 1 * UINT_MOD ^ (i - 1))). {
rewrite Zdiv_Zdiv; try lia.
rewrite Z.pow_1_r.
reflexivity.
}
rewrite H4.
rewrite <-Z.pow_add_r; try lia.
assert (n / UINT_MOD ^ i = (n - a) / UINT_MOD ^ i). {
assert (i = 1 + (i - 1)). { lia. }
rewrite H5.
rewrite (Z.pow_add_r UINT_MOD 1 (i - 1)); try lia.
repeat rewrite <-Zdiv_Zdiv; try lia.
repeat rewrite Z.pow_1_r.
rewrite <-(Zdiv_unique_full n UINT_MOD (list_to_Z l) a); try lia.
+ destruct H3.
rewrite <-H3.
reflexivity.
+ unfold Remainder; lia.
}
rewrite H5.
assert (1 + (i - 1) = i). { lia. }
rewrite H6.
reflexivity.
Qed.
Lemma list_store_Z_cmp: forall l1 l2 n1 n2 i,
0 <= i < Zlength l1 ->
Zlength l1 = Zlength l2 ->
list_store_Z l1 n1 ->
list_store_Z l2 n2 ->
sublist (i + 1) (Zlength l1) l1 = sublist (i + 1) (Zlength l2) l2 ->
Znth i l1 0 < Znth i l2 0 ->
n1 < n2.
Proof.
intros.
assert (Zlength l1 = Z.of_nat (Datatypes.length l1)). { apply Zlength_correct. }
pose proof (sublist_split 0 (Zlength l1) i l1 ltac:(lia) ltac:(lia)).
assert (Zlength l2 = Z.of_nat (Datatypes.length l2)). { apply Zlength_correct. }
pose proof (sublist_split 0 (Zlength l2) i l2 ltac:(lia) ltac:(lia)).
clear H5 H7.
rewrite (sublist_self l1 (Zlength l1) ltac:(lia)) in H6.
rewrite (sublist_self l2 (Zlength l2) ltac:(lia)) in H8.
rewrite H6 in H1.
rewrite H8 in H2.
apply list_store_Z_split in H1, H2.
remember (Zlength l1) as n eqn:Hn.
assert (Zlength l2 = n). { lia. }
rewrite H5 in *.
rewrite Zlength_sublist0 in H1, H2; try lia.
destruct H1, H2.
remember (n1 mod UINT_MOD ^ i) as n1_lo eqn:Hn1_lo.
remember (n1 / UINT_MOD ^ i) as n1_hi eqn:Hn1_hi.
remember (n2 mod UINT_MOD ^ i) as n2_lo eqn:Hn2_lo.
remember (n2 / UINT_MOD ^ i) as n2_hi eqn:Hn2_hi.
remember (sublist 0 i l1) as l1_lo eqn:Hl1_lo.
remember (sublist i n l1) as l1_hi eqn:Hl1_hi.
remember (sublist 0 i l2) as l2_lo eqn:Hl2_lo.
remember (sublist i n l2) as l2_hi eqn:Hl2_hi.
assert (n1_lo - n2_lo < UINT_MOD ^ i). {
pose proof (list_store_Z_bound l1_lo n1_lo H1).
pose proof (list_store_Z_bound l2_lo n2_lo H2).
rewrite Hl1_lo, Zlength_sublist0 in H10; lia.
}
assert (n2_hi - n1_hi >= 1). {
assert (Zlength l1_hi >= 1 /\ Zlength l2_hi >= 1). {
pose proof (Zlength_sublist i n l1 ltac:(lia)).
pose proof (Zlength_sublist i n l2 ltac:(lia)).
rewrite <-Hl1_hi in H11.
rewrite <-Hl2_hi in H12.
lia.
}
destruct H11.
destruct l1_hi, l2_hi; try rewrite Zlength_nil in *; try lia.
unfold list_store_Z in H7, H9.
destruct H7, H9.
simpl in H7, H9.
assert (forall a (l0 l: list Z),
a :: l0 = sublist i n l ->
Zlength l = n ->
l0 = sublist (i + 1) n l /\ a = Znth i l 0
). {
intros.
assert (n = Z.of_nat(Datatypes.length l)). { rewrite <-Zlength_correct. lia. }
pose proof (sublist_split i n (i + 1) l ltac:(lia) ltac:(lia)).
rewrite (sublist_single i l 0) in H18; try lia.
rewrite <-H15 in H18.
rewrite Aux.list_app_single_l in H18.
injection H18; intros.
rewrite H19, H20.
split; reflexivity.
}
pose proof (H15 z0 l2_hi l2 Hl2_hi ltac:(lia)).
specialize (H15 z l1_hi l1 Hl1_hi ltac:(lia)).
destruct H15, H16.
rewrite H15, H17 in H7.
rewrite H16, H18 in H9.
rewrite <-H3 in H9.
lia.
}
pose proof (Zmod_eq_full n1 (UINT_MOD ^ i) ltac:(lia)).
pose proof (Zmod_eq_full n2 (UINT_MOD ^ i) ltac:(lia)).
rewrite <-Hn1_lo, <-Hn1_hi in H12.
rewrite <-Hn2_lo, <-Hn2_hi in H13.
assert (n2_hi * UINT_MOD ^ i - n1_hi * UINT_MOD ^ i >= UINT_MOD ^ i). {
rewrite <-Z.mul_sub_distr_r.
pose proof (Zmult_ge_compat_r (n2_hi - n1_hi) 1 (UINT_MOD ^ i) ltac:(lia) ltac:(lia)).
rewrite Z.mul_1_l in H14.
lia.
}
lia.
Qed.
Lemma list_store_Z_compact_cmp: forall l1 l2 n1 n2 i,
0 <= i < Zlength l1 ->
Zlength l1 = Zlength l2 ->
list_store_Z_compact l1 n1 ->
list_store_Z_compact l2 n2 ->
sublist (i + 1) (Zlength l1) l1 = sublist (i + 1) (Zlength l2) l2 ->
Znth i l1 0 < Znth i l2 0 ->
n1 < n2.
Proof.
intros.
apply list_store_Z_compact_to_normal in H1, H2.
apply (list_store_Z_cmp l1 l2 n1 n2 i); tauto.
Qed.
Lemma list_store_Z_cmp_length: forall l1 l2 n1 n2,
Zlength l1 < Zlength l2 ->
list_store_Z_compact l1 n1 ->
list_store_Z_compact l2 n2 ->
n1 < n2.
Proof.
intros.
unfold list_store_Z_compact in *.
destruct H0, H1, H2, H3.
assert (list_store_Z l1 n1 /\ list_store_Z l2 n2). {
unfold list_store_Z.
tauto.
}
clear H0 H1 H4 H5.
destruct H6.
assert (l2 <> []). {
pose proof (Zlength_nonneg l1).
assert (Zlength l2 >= 1). { lia. }
destruct l2.
+ rewrite Zlength_nil in H5.
lia.
+ pose proof (@nil_cons Z z l2).
auto.
}
pose proof (list_store_Z_bound l2 n2 H1) as Hn2.
pose proof (@app_removelast_last Z l2 1 H4).
rewrite H5 in H1.
apply list_store_Z_split in H1; destruct H1.
rewrite (Aux.Zlength_removelast l2) in H1, H6; try auto.
remember (Zlength l2 - 1) as n eqn:Hn.
unfold list_store_Z in H6; destruct H6.
simpl in H6.
assert (n2 >= UINT_MOD ^ n). {
assert (n2 / UINT_MOD ^ n >= 1). { lia. }
pose proof (Zmult_ge_compat_r (n2 / UINT_MOD ^ n) 1 (UINT_MOD ^ n) ltac:(lia) ltac:(lia)).
rewrite Z.mul_1_l in H9.
assert (n2 >= n2 / UINT_MOD ^ n * UINT_MOD ^ n). {
assert (n >= 0). {
destruct l2.
+ contradiction.
+ rewrite Zlength_cons in Hn; unfold Z.succ in Hn.
pose proof (Zlength_nonneg l2).
lia.
}
pose proof (Zmod_eq n2 (UINT_MOD ^ n) ltac:(lia)).
assert (n2 mod UINT_MOD ^ n >= 0). {
pose proof (Z.mod_bound_pos n2 (UINT_MOD ^ n) ltac:(lia) ltac:(lia)).
lia.
}
lia.
}
lia.
}
assert (Zlength l1 <= n). { lia. }
pose proof (list_store_Z_bound l1 n1 H0).
assert (UINT_MOD ^ n >= UINT_MOD ^ (Zlength l1)). {
assert (Zlength l1 = n \/ Zlength l1 < n). { lia. }
destruct H11.
+ rewrite H11.
lia.
+ assert (n >= 0). {
destruct l2.
+ contradiction.
+ rewrite Zlength_cons in Hn; unfold Z.succ in Hn.
pose proof (Zlength_nonneg l2).
lia.
}
pose proof (Z.pow_lt_mono_r_iff UINT_MOD (Zlength l1) n ltac:(lia) ltac:(lia)).
destruct H13 as [H13 _].
specialize (H13 H11).
lia.
}
lia.
Qed.
End Internal.
Record bigint_ent: Type := {

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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 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.

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From Require Import gmp_goal gmp_proof_auto gmp_proof_manual_tmp.
Module VC_Correctness : VC_Correct.
Include gmp_proof_auto.
Include gmp_proof_manual_tmp.
End VC_Correctness.

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Require Import Coq.ZArith.ZArith.
Require Import Coq.Bool.Bool.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
Require Import Coq.Classes.RelationClasses.
Require Import Coq.Classes.Morphisms.
Require Import Coq.micromega.Psatz.
Require Import Coq.Sorting.Permutation.
From AUXLib Require Import int_auto Axioms Feq Idents List_lemma VMap.
Require Import SetsClass.SetsClass. Import SetsNotation.
From SimpleC.SL Require Import Mem SeparationLogic.
From Require Import gmp_goal.
Require Import Logic.LogicGenerator.demo932.Interface.
Local Open Scope Z_scope.
Local Open Scope sets.
Local Open Scope string.
Local Open Scope list.
Import naive_C_Rules.
Local Open Scope sac.
Lemma proof_of_gmp_abs_safety_wit_1 : gmp_abs_safety_wit_1.
Proof. Admitted.
Lemma proof_of_gmp_cmp_safety_wit_1 : gmp_cmp_safety_wit_1.
Proof. Admitted.
Lemma proof_of_gmp_cmp_safety_wit_2 : gmp_cmp_safety_wit_2.
Proof. Admitted.
Lemma proof_of_gmp_cmp_safety_wit_3 : gmp_cmp_safety_wit_3.
Proof. Admitted.
Lemma proof_of_gmp_cmp_safety_wit_4 : gmp_cmp_safety_wit_4.
Proof. Admitted.
Lemma proof_of_gmp_cmp_return_wit_1_1 : gmp_cmp_return_wit_1_1.
Proof. Admitted.
Lemma proof_of_gmp_cmp_return_wit_1_3 : gmp_cmp_return_wit_1_3.
Proof. Admitted.
Lemma proof_of_mpn_copyi_safety_wit_1 : mpn_copyi_safety_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_copyi_safety_wit_2 : mpn_copyi_safety_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_copyi_partial_solve_wit_1 : mpn_copyi_partial_solve_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_copyi_partial_solve_wit_2_pure : mpn_copyi_partial_solve_wit_2_pure.
Proof. Admitted.
Lemma proof_of_mpn_copyi_partial_solve_wit_2 : mpn_copyi_partial_solve_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_copyi_partial_solve_wit_3 : mpn_copyi_partial_solve_wit_3.
Proof. Admitted.
Lemma proof_of_mpn_copyi_partial_solve_wit_4 : mpn_copyi_partial_solve_wit_4.
Proof. Admitted.
Lemma proof_of_mpn_copyi_partial_solve_wit_5 : mpn_copyi_partial_solve_wit_5.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_1 : mpn_cmp_safety_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_2 : mpn_cmp_safety_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_3 : mpn_cmp_safety_wit_3.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_4 : mpn_cmp_safety_wit_4.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_5 : mpn_cmp_safety_wit_5.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_6 : mpn_cmp_safety_wit_6.
Proof. Admitted.
Lemma proof_of_mpn_cmp_safety_wit_7 : mpn_cmp_safety_wit_7.
Proof. Admitted.
Lemma proof_of_mpn_cmp_partial_solve_wit_1 : mpn_cmp_partial_solve_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_cmp_partial_solve_wit_2 : mpn_cmp_partial_solve_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_cmp_partial_solve_wit_3 : mpn_cmp_partial_solve_wit_3.
Proof. Admitted.
Lemma proof_of_mpn_cmp4_safety_wit_1 : mpn_cmp4_safety_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_cmp4_safety_wit_2 : mpn_cmp4_safety_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_cmp4_safety_wit_3 : mpn_cmp4_safety_wit_3.
Proof. Admitted.
Lemma proof_of_mpn_cmp4_partial_solve_wit_1_pure : mpn_cmp4_partial_solve_wit_1_pure.
Proof. Admitted.
Lemma proof_of_mpn_cmp4_partial_solve_wit_1 : mpn_cmp4_partial_solve_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_cmp4_partial_solve_wit_2_pure : mpn_cmp4_partial_solve_wit_2_pure.
Proof. Admitted.
Lemma proof_of_mpn_cmp4_partial_solve_wit_2 : mpn_cmp4_partial_solve_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_cmp4_which_implies_wit_1 : mpn_cmp4_which_implies_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_normalized_size_safety_wit_1 : mpn_normalized_size_safety_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_normalized_size_safety_wit_2 : mpn_normalized_size_safety_wit_2.
Proof. Admitted.
Lemma proof_of_mpn_normalized_size_safety_wit_3 : mpn_normalized_size_safety_wit_3.
Proof. Admitted.
Lemma proof_of_mpn_normalized_size_safety_wit_4 : mpn_normalized_size_safety_wit_4.
Proof. Admitted.
Lemma proof_of_mpn_normalized_size_safety_wit_5 : mpn_normalized_size_safety_wit_5.
Proof. Admitted.
Lemma proof_of_mpn_normalized_size_entail_wit_1 : mpn_normalized_size_entail_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_normalized_size_partial_solve_wit_1 : mpn_normalized_size_partial_solve_wit_1.
Proof. Admitted.
Lemma proof_of_mpn_normalized_size_partial_solve_wit_2 : mpn_normalized_size_partial_solve_wit_2.
Proof. Admitted.

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Require Import Coq.ZArith.ZArith.
Require Import Coq.Bool.Bool.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
Require Import Coq.Classes.RelationClasses.
Require Import Coq.Classes.Morphisms.
Require Import Coq.micromega.Psatz.
Require Import Coq.Sorting.Permutation.
From AUXLib Require Import int_auto Axioms Feq Idents List_lemma VMap.
Require Import SetsClass.SetsClass. Import SetsNotation.
From SimpleC.SL Require Import Mem SeparationLogic.
From GmpLib Require Import gmp_goal.
Require Import GmpLib.GmpNumber. Import Internal.
Require Import GmpLib.GmpAux.
Require Import Logic.LogicGenerator.demo932.Interface.
Local Open Scope Z_scope.
Local Open Scope sets.
Local Open Scope string.
Local Open Scope list.
Import naive_C_Rules.
Local Open Scope sac.
Lemma proof_of_gmp_abs_return_wit_1_1 : gmp_abs_return_wit_1_1.
Proof. pre_process. Qed.
Lemma proof_of_gmp_abs_return_wit_1_2 : gmp_abs_return_wit_1_2.
Proof. pre_process. Qed.
Lemma proof_of_gmp_max_return_wit_1_1 : gmp_max_return_wit_1_1.
Proof.
pre_process.
entailer!.
unfold Zmax.
rewrite Z.max_r; lia.
Qed.
Lemma proof_of_gmp_max_return_wit_1_2 : gmp_max_return_wit_1_2.
Proof.
pre_process.
entailer!.
unfold Zmax.
rewrite Z.max_l; lia.
Qed.
Lemma proof_of_gmp_cmp_return_wit_1_2 : gmp_cmp_return_wit_1_2.
Proof.
pre_process.
repeat rewrite <-derivable1_orp_intros1.
entailer!.
Qed.
Lemma proof_of_mpn_copyi_entail_wit_1 : mpn_copyi_entail_wit_1.
Proof.
pre_process.
Exists l2 l_2.
entailer!.
pose proof (Zlength_nonneg l_2).
lia.
Qed.
Lemma proof_of_mpn_copyi_entail_wit_2 : mpn_copyi_entail_wit_2.
Proof.
pre_process.
Exists l2' l_3.
entailer!.
rewrite replace_Znth_app_r.
+ rewrite Zlength_sublist0; [ | lia ].
assert (i - i = 0). { lia. }
rewrite H15; clear H15.
assert (replace_Znth 0 (Znth i l_3 0) (a :: nil) = sublist i (i + 1) l_3). {
unfold replace_Znth, Z.to_nat, replace_nth.
rewrite (sublist_single i l_3 0); [ reflexivity | ].
rewrite <-Zlength_correct; lia.
}
rewrite H15; clear H15.
rewrite replace_Znth_nothing.
- rewrite <-sublist_split; try lia; try reflexivity.
rewrite <-Zlength_correct; lia.
- pose proof (Zlength_sublist0 i l_3 ltac:(lia)).
lia.
+ pose proof (Zlength_sublist0 i l_3); lia.
Qed.
Lemma proof_of_mpn_copyi_which_implies_wit_1 : mpn_copyi_which_implies_wit_1.
Proof.
pre_process.
unfold mpd_store_Z.
Intros l.
Exists l.
unfold mpd_store_list.
entailer!.
subst.
entailer!.
Qed.
Lemma proof_of_mpn_copyi_which_implies_wit_2 : mpn_copyi_which_implies_wit_2.
Proof.
pre_process.
pose proof (store_uint_array_divide d cap2 l2 0).
pose proof (Zlength_nonneg l2).
specialize (H0 ltac:(lia) ltac:(lia)).
destruct H0 as [H0 _].
simpl in H0.
entailer!.
rewrite (sublist_nil l2 0 0) in H0; [ | lia].
sep_apply H0.
entailer!.
unfold store_uint_array, store_uint_array_rec.
unfold store_array.
rewrite (sublist_self l2 cap2); [ | lia ].
assert (d + 0 = d). { lia. }
rewrite H2; clear H2.
assert (cap2 - 0 = cap2). { lia. }
rewrite H2; clear H2.
reflexivity.
Qed.
Lemma proof_of_mpn_copyi_which_implies_wit_3 : mpn_copyi_which_implies_wit_3.
Proof.
pre_process.
destruct l'. {
unfold store_uint_array_rec.
simpl.
entailer!.
}
pose proof (store_uint_array_rec_cons d i cap2 z l' ltac:(lia)).
sep_apply H2.
Exists z l'.
entailer!.
assert (i = 0 \/ i > 0). { lia. }
destruct H3.
+ subst.
unfold store_uint_array, store_array.
simpl.
entailer!.
+ pose proof (Aux.uint_array_rec_to_uint_array d 0 i (sublist 0 i l) ltac:(lia)).
destruct H4 as [_ H4].
assert (d + sizeof(UINT) * 0 = d). { lia. }
rewrite H5 in H4; clear H5.
assert (i - 0 = i). { lia. }
rewrite H5 in H4; clear H5.
sep_apply H4; clear H4.
pose proof (Aux.uint_array_rec_to_uint_array d 0 (i + 1) (sublist 0 i l ++ z :: nil) ltac:(lia)).
destruct H4 as [H4 _].
assert (i + 1 - 0 = i + 1). { lia. }
rewrite H5 in H4; clear H5.
assert (d + sizeof(UINT) * 0 = d). { lia. }
rewrite H5 in H4; clear H5.
rewrite <-H4.
sep_apply store_uint_array_rec_tail_merge; [ reflexivity | lia ].
Qed.
Lemma proof_of_mpn_cmp_entail_wit_1 : mpn_cmp_entail_wit_1.
Proof.
pre_process.
entailer!.
assert (n_pre - 1 + 1 = n_pre). { lia. }
rewrite H8; clear H8.
pose proof (Zlength_sublist n_pre n_pre l1 ltac:(lia)).
pose proof (Zlength_nil_inv (sublist n_pre n_pre l1) ltac:(lia)).
rewrite H9.
pose proof (Zlength_sublist n_pre n_pre l2 ltac:(lia)).
pose proof (Zlength_nil_inv (sublist n_pre n_pre l2) ltac:(lia)).
rewrite H11.
reflexivity.
Qed.
Lemma proof_of_mpn_cmp_entail_wit_2 : mpn_cmp_entail_wit_2.
Proof.
pre_process.
entailer!; try lia.
assert (n - 1 + 1 = n). { lia. }
rewrite H17; clear H17.
assert (n_pre <= Z.of_nat (Datatypes.length l1)). {
rewrite <-Zlength_correct.
lia.
}
assert (n_pre <= Z.of_nat (Datatypes.length l2)). {
rewrite <-Zlength_correct.
lia.
}
rewrite (sublist_split n n_pre (n + 1) l1 ltac:(lia) ltac:(lia)).
rewrite (sublist_split n n_pre (n + 1) l2 ltac:(lia) ltac:(lia)).
rewrite (sublist_single n l1 0 ltac:(lia)).
rewrite (sublist_single n l2 0 ltac:(lia)).
rewrite H.
rewrite H7.
reflexivity.
Qed.
Lemma proof_of_mpn_cmp_return_wit_1_1 : mpn_cmp_return_wit_1_1.
Proof.
pre_process.
entailer!.
Left; Left.
entailer!.
+ unfold mpd_store_Z_compact.
Exists l1 l2.
unfold mpd_store_list.
entailer!.
rewrite <-H6, <-H7.
entailer!.
+ assert (Znth n l1 0 < Znth n l2 0). { lia. }
clear H H0.
apply (list_store_Z_compact_cmp l1 l2 val1 val2 n ltac:(lia) ltac:(lia) H4 H5).
- rewrite <-H6, <-H7.
tauto.
- lia.
Qed.
Lemma proof_of_mpn_cmp_return_wit_1_2 : mpn_cmp_return_wit_1_2.
Proof.
pre_process.
Right.
entailer!.
+ unfold mpd_store_Z_compact, mpd_store_list.
Exists l1 l2.
rewrite <-H6, <-H7.
entailer!.
+ pose proof (list_store_Z_compact_cmp l2 l1 val2 val1 n ltac:(lia) ltac:(lia) H5 H4).
rewrite <-H6, <-H7 in H18.
rewrite H8 in H18.
specialize (H18 ltac:(reflexivity) ltac:(lia)).
lia.
Qed.
Lemma proof_of_mpn_cmp_which_implies_wit_1 : mpn_cmp_which_implies_wit_1.
Proof.
pre_process.
unfold mpd_store_Z_compact.
Intros l1 l2.
Exists l2 l1.
unfold mpd_store_list.
entailer!.
rewrite <-H0, <-H2.
entailer!.
Qed.
Lemma proof_of_mpn_cmp4_return_wit_1_1 : mpn_cmp4_return_wit_1_1.
Proof.
pre_process.
Right.
unfold mpd_store_Z_compact.
Intros l1 l2.
Exists l1 l2.
entailer!.
pose proof (list_store_Z_cmp_length l2 l1 val2 val1 ltac:(lia) H9 H7).
lia.
Qed.
Lemma proof_of_mpn_cmp4_return_wit_1_2 : mpn_cmp4_return_wit_1_2.
Proof.
pre_process.
Left; Left.
unfold mpd_store_Z_compact.
entailer!.
Intros l1 l2.
Exists l1 l2.
entailer!.
pose proof (list_store_Z_cmp_length l1 l2 val1 val2 ltac:(lia) H7 H9).
lia.
Qed.
Lemma proof_of_mpn_cmp4_return_wit_2_1 : mpn_cmp4_return_wit_2_1.
Proof.
pre_process.
Right.
unfold mpd_store_Z_compact.
Intros l1 l2.
Exists l1 l2.
entailer!.
Qed.
Lemma proof_of_mpn_cmp4_return_wit_2_2 : mpn_cmp4_return_wit_2_2.
Proof.
pre_process.
Left; Right.
unfold mpd_store_Z_compact.
Intros l1 l2.
Exists l1 l2.
entailer!.
Qed.
Lemma proof_of_mpn_cmp4_return_wit_2_3 : mpn_cmp4_return_wit_2_3.
Proof.
pre_process.
Left; Left.
unfold mpd_store_Z_compact.
Intros l1 l2.
Exists l1 l2.
entailer!.
Qed.
Lemma proof_of_mpn_normalized_size_entail_wit_2 : mpn_normalized_size_entail_wit_2.
Proof.
pre_process.
entailer!.
+ pose proof (store_uint_array_divide_rec
xp_pre n (sublist 0 n l) (n - 1) ltac:(lia)).
rewrite (Zlength_sublist0 n l ltac:(lia)) in H12.
specialize (H12 ltac:(lia)).
destruct H12 as [H12 _].
rewrite H12; clear H12.
rewrite (sublist_sublist00 (n - 1) n l ltac:(lia)).
rewrite (sublist_sublist0 n n (n - 1) l ltac:(lia) ltac:(lia)).
pose proof (Aux.uint_array_rec_to_uint_array xp_pre 0 (n - 1) (sublist 0 (n - 1) l) ltac:(lia)).
destruct H12 as [H12 _].
rewrite Z.mul_0_r, Z.add_0_r, Z.sub_0_r in H12.
rewrite H12; clear H12.
entailer!.
assert (n - 1 < Z.of_nat (Datatypes.length l)). {
rewrite <-Zlength_correct.
lia.
}
pose proof (sublist_single (n - 1) l 0 ltac:(lia)).
clear H12.
pose proof (Aux.store_uint_array_single_to_undef xp_pre (n - 1) (Znth (n - 1) l 0)).
assert (n - 1 + 1 = n). { lia. }
rewrite H14 in H12, H13; clear H14.
rewrite H13, H12; clear H13 H12.
pose proof (Aux.store_undef_uint_array_rec_divide xp_pre (n - 1) n cap ltac:(lia) ltac:(lia)).
rewrite <-H12.
entailer!.
+ assert (n <= Z.of_nat (Datatypes.length l)). {
rewrite <-Zlength_correct.
lia.
}
pose proof (sublist_split 0 n (n - 1) l ltac:(lia) ltac:(lia)).
clear H12.
rewrite H13 in H6.
apply (list_store_Z_split) in H6; destruct H6.
assert (Z.of_nat (Datatypes.length l) = Zlength l). {
rewrite (Zlength_correct l); reflexivity.
}
pose proof (sublist_single (n - 1) l 0 ltac:(lia)).
assert (n - 1 + 1 = n). { lia. }
rewrite H16 in H15; clear H16.
rewrite H15 in H12.
unfold list_store_Z in H12; destruct H12.
simpl in H12.
rewrite Znth_sublist0 in H; try lia.
rewrite H in H12.
rewrite (Zlength_sublist0 (n - 1) l) in *; try lia.
pose proof (Z_div_mod_eq_full val (UINT_MOD ^ (n - 1))).
rewrite <-H12, Z.mul_0_r, Z.add_0_l in H17.
rewrite <-H17 in H6.
tauto.
Qed.
Lemma proof_of_mpn_normalized_size_return_wit_1_1 : mpn_normalized_size_return_wit_1_1.
Proof.
pre_process.
assert (n = 0). { lia. }
clear H H0.
rewrite H11 in *.
unfold mpd_store_Z_compact, mpd_store_list.
Exists nil.
entailer!.
+ rewrite Zlength_nil.
lia.
+ unfold list_store_Z_compact.
simpl.
rewrite sublist_nil in H5; try lia.
unfold list_store_Z in H5; simpl in H5.
destruct H5.
lia.
Qed.
Lemma proof_of_mpn_normalized_size_return_wit_1_2 : mpn_normalized_size_return_wit_1_2.
Proof.
pre_process.
unfold mpd_store_Z_compact, mpd_store_list.
Exists (sublist 0 n l).
entailer!.
+ rewrite Zlength_sublist0; try lia.
entailer!.
+ rewrite Zlength_sublist0; lia.
+ rewrite Zlength_sublist0; lia.
+ unfold list_store_Z_compact.
unfold list_store_Z in H6.
destruct H6.
rewrite Aux.list_last_to_Znth.
- rewrite Zlength_sublist0; try lia.
repeat split; try tauto.
pose proof (list_within_bound_Znth (sublist 0 n l) (n - 1)).
rewrite Zlength_sublist0 in H13; try lia.
specialize (H13 ltac:(lia) H12).
lia.
- assert (sublist 0 n l = nil \/ sublist 0 n l <> nil). { tauto. }
destruct H13.
* pose proof (Zlength_sublist0 n l ltac:(lia)).
rewrite H13 in H14.
rewrite Zlength_nil in H14.
lia.
* tauto.
Qed.
Lemma proof_of_mpn_normalized_size_which_implies_wit_1 : mpn_normalized_size_which_implies_wit_1.
Proof.
pre_process.
unfold mpd_store_Z.
Intros l.
Exists l.
unfold mpd_store_list.
entailer!.
+ rewrite H0.
rewrite sublist_self; try lia.
entailer!.
+ rewrite sublist_self; try lia.
tauto.
Qed.

322
projects/mini-gmp.c
View File

@ -1,14 +1,36 @@
#include "mini-gmp.h"
#include "verification_stdlib.h"
#include "verification_list.h"
#include "int_array_def.h"
mp_size_t gmp_abs(mp_size_t x) {
int gmp_abs(int x)
/*@
Require INT_MIN < x && x <= INT_MAX
Ensure __return == Zabs(x)
*/
{
return x >= 0 ? x : -x;
}
mp_size_t gmp_max(mp_size_t a, mp_size_t b) {
int gmp_max(int a, int b)
/*@
Require emp
Ensure __return == Zmax(a, b)
*/
{
return a > b ? a : b;
}
int gmp_cmp(mp_size_t a, mp_size_t b) {
int gmp_cmp(int a, int b)
/*@
Require emp
Ensure
emp *
(a > b && __return == 1 ||
a == b && __return == 0 ||
a < b && __return == -1)
*/
{
return (a > b) - (a < b);
}
@ -16,73 +38,219 @@ int gmp_cmp(mp_size_t a, mp_size_t b) {
/* 从 低地址向高地址 顺序复制 */
void
mpn_copyi (mp_ptr d, mp_srcptr s, mp_size_t n)
mpn_copyi (unsigned int *d, unsigned int *s, int n)
/*@
With val l2 cap1 cap2
Require
mpd_store_Z(s, val, n, cap1) *
store_uint_array(d, cap2, l2) &&
Zlength(l2) == cap2 &&
cap2 >= n
Ensure
mpd_store_Z(s, val, n, cap1) *
mpd_store_Z(d, val, n, cap2)
*/
{
mp_size_t i;
for (i = 0; i < n; i++)
d[i] = s[i];
/*@
mpd_store_Z(s, val, n, cap1)
which implies
exists l,
n <= cap1 &&
Zlength(l) == n &&
cap1 <= 100000000 &&
store_uint_array(s, n, l) *
store_undef_uint_array_rec(s, n, cap1) &&
list_store_Z(l, val)
*/
/*@
store_uint_array(d, cap2, l2) && Zlength(l2) == cap2
which implies
store_uint_array_rec(d, 0, cap2, l2) * store_uint_array(d, 0, nil) &&
Zlength(l2) == cap2
*/
int i;
/*@ Inv
exists l l',
0 <= i && i <= n && Zlength(l) == n &&
list_store_Z(l, val) && n <= cap1 &&
store_uint_array(s, n, l) *
store_undef_uint_array_rec(s, n, cap1) *
store_uint_array(d, i, sublist(0, i, l)) *
store_uint_array_rec(d, i, cap2, l')
*/
for (i = 0; i < n; i++) {
/*@
Given l l'
*/
/*@
0 <= i && i < n && n <= cap2 &&
store_uint_array_rec(d, i, cap2, l') *
store_uint_array(d, i, sublist(0, i, l))
which implies
exists a l2',
l' == cons(a, l2') && i < n && n <= cap2 &&
store_uint_array_rec(d, i + 1, cap2, l2') *
store_uint_array(d, i + 1, app(sublist(0, i, l), cons(a, nil)))
*/
d[i] = s[i];
}
}
/* 大于返回1，小于返回-1，等于返回0 */
int
mpn_cmp (mp_srcptr ap, mp_srcptr bp, mp_size_t n)
mpn_cmp (unsigned int *ap, unsigned int *bp, int n)
/*@
With cap1 cap2 val1 val2
Require
mpd_store_Z_compact(ap, val1, n, cap1) *
mpd_store_Z_compact(bp, val2, n, cap2) &&
0 <= n && n <= cap1 && n <= cap2 &&
cap1 <= 100000000 && cap2 <= 100000000
Ensure
(val1 > val2 && __return == 1 ||
val1 == val2 && __return == 0 ||
val1 < val2 && __return == -1) &&
mpd_store_Z_compact(ap@pre, val1, n@pre, cap1) *
mpd_store_Z_compact(bp@pre, val2, n@pre, cap2)
*/
{
/*@
mpd_store_Z_compact(ap@pre, val1, n@pre, cap1) * mpd_store_Z_compact(bp@pre, val2, n@pre, cap2)
which implies
exists l1 l2,
store_uint_array(ap@pre, n@pre, l1) * store_uint_array(bp@pre, n@pre, l2) *
store_undef_uint_array_rec(ap@pre, n@pre, cap1) *
store_undef_uint_array_rec(bp@pre, n@pre, cap2) &&
list_store_Z_compact(l1, val1) && list_store_Z_compact(l2, val2) &&
n@pre == Zlength(l1) && n@pre == Zlength(l2)
*/
/*@
Given l1 l2
*/
--n;
/*@Inv
-1 <= n && n < n@pre &&
store_uint_array(ap@pre, n@pre, l1) * store_uint_array(bp@pre, n@pre, l2) *
store_undef_uint_array_rec(ap@pre, n@pre, cap1) *
store_undef_uint_array_rec(bp@pre, n@pre, cap2) &&
list_store_Z_compact(l1, val1) && list_store_Z_compact(l2, val2) &&
n@pre == Zlength(l1) && n@pre == Zlength(l2) &&
sublist(n + 1, n@pre, l1) == sublist(n + 1, n@pre, l2)
*/
while (n >= 0)
{
if (ap[n] != bp[n])
return ap[n] > bp[n] ? 1 : -1;
--n;
}
// Note: The parser cannot parse "--n" in loop condition so we paraphrased it.
/*
while (--n >= 0)
{
if (ap[n] != bp[n])
return ap[n] > bp[n] ? 1 : -1;
}
*/
return 0;
}
/*处理位数不同的情况*/
static int
mpn_cmp4 (mp_srcptr ap, mp_size_t an, mp_srcptr bp, mp_size_t bn)
mpn_cmp4 (unsigned int *ap, int an, unsigned int *bp, int bn)
/*@
With cap1 cap2 val1 val2
Require
mpd_store_Z_compact(ap, val1, an, cap1) *
mpd_store_Z_compact(bp, val2, bn, cap2) &&
an >= 0 && bn >= 0 && an <= cap1 && bn <= cap2 &&
cap1 <= 100000000 && cap2 <= 100000000
Ensure
(val1 > val2 && __return == 1 ||
val1 == val2 && __return == 0 ||
val1 < val2 && __return == -1) &&
mpd_store_Z_compact(ap@pre, val1, an@pre, cap1) *
mpd_store_Z_compact(bp@pre, val2, bn@pre, cap2)
*/
{
if (an != bn)
return an < bn ? -1 : 1;
else
else {
/*@
an@pre == bn@pre && bn@pre <= cap2
which implies
an@pre <= cap2
*/
return mpn_cmp (ap, bp, an);
}
}
/*返回非0的位数*/
static mp_size_t
mpn_normalized_size (mp_srcptr xp, mp_size_t n)
static int
mpn_normalized_size (unsigned int *xp, int n)
/*@
With cap val
Require
mpd_store_Z(xp, val, n, cap) &&
0 <= n && n <= cap && cap <= 100000000
Ensure
0 <= __return && __return <= cap &&
mpd_store_Z_compact(xp@pre, val, __return, cap)
*/
{
/*@
mpd_store_Z(xp@pre, val, n, cap)
which implies
exists l,
list_store_Z(sublist(0, n, l), val) &&
Zlength(l) == n &&
store_uint_array(xp@pre, n, sublist(0, n, l)) *
store_undef_uint_array_rec(xp@pre, n, cap)
*/
/*@
Given l
*/
/*@Inv
n >= 0 && n <= n@pre &&
n@pre >= 0 && n@pre <= cap && cap <= 100000000 &&
list_store_Z(sublist(0, n, l), val) &&
store_uint_array(xp@pre, n, sublist(0, n, l)) *
store_undef_uint_array_rec(xp@pre, n, cap)
*/
while (n > 0 && xp[n-1] == 0)
--n;
return n;
}
/* 多精度数ap 加上单精度数b，返回最后产生的进位 */
mp_limb_t
mpn_add_1 (mp_ptr rp, mp_srcptr ap, mp_size_t n, mp_limb_t b)
/*unsigned int
mpn_add_1 (unsigned int *rp, unsigned int *ap, int n, unsigned int b)
{
mp_size_t i;
int i;
//assert (n > 0);
i = 0;
do
{
mp_limb_t r = ap[i] + b;
/* Carry out */
unsigned int r = ap[i] + b;
// Carry out
b = (r < b);
rp[i] = r;
}
while (++i < n);
return b;
}
}*/
/* 位数相同的多精度数ap 加上多精度数bp，返回最后产生的进位 */
mp_limb_t
mpn_add_n (mp_ptr rp, mp_srcptr ap, mp_srcptr bp, mp_size_t n)
/*unsigned int
mpn_add_n (unsigned int *rp, unsigned int *ap, unsigned int *bp, int n)
{
mp_size_t i;
mp_limb_t cy;
int i;
unsigned int cy;
for (i = 0, cy = 0; i < n; i++)
{
mp_limb_t a, b, r;
unsigned int a, b, r;
a = ap[i]; b = bp[i];
r = a + cy;
cy = (r < cy);
@ -91,48 +259,48 @@ mpn_add_n (mp_ptr rp, mp_srcptr ap, mp_srcptr bp, mp_size_t n)
rp[i] = r;
}
return cy;
}
}*/
/*不同位数的多精度数相加，返回最后的进位*/
mp_limb_t
mpn_add (mp_ptr rp, mp_srcptr ap, mp_size_t an, mp_srcptr bp, mp_size_t bn)
/*unsigned int
mpn_add (unsigned int *rp, unsigned int *ap, int an, unsigned int *bp, int bn)
{
mp_limb_t cy;
unsigned int cy;
//assert (an >= bn);
cy = mpn_add_n (rp, ap, bp, bn);
if (an > bn)
cy = mpn_add_1 (rp + bn, ap + bn, an - bn, cy);
return cy;
}
}*/
mp_limb_t
mpn_sub_1 (mp_ptr rp, mp_srcptr ap, mp_size_t n, mp_limb_t b)
/*unsigned int
mpn_sub_1 (unsigned int *rp, unsigned int *ap, int n, unsigned int b)
{
mp_size_t i;
int i;
//assert (n > 0);
i = 0;
do
{
mp_limb_t a = ap[i];
/* Carry out */
mp_limb_t cy = a < b;
unsigned int a = ap[i];
// Carry out
unsigned int cy = a < b;
rp[i] = a - b;
b = cy;
}
while (++i < n);
return b;
}
}*/
mp_limb_t
mpn_sub_n (mp_ptr rp, mp_srcptr ap, mp_srcptr bp, mp_size_t n)
/*unsigned int
mpn_sub_n (unsigned int *rp, unsigned int *ap, unsigned int *bp, int n)
{
mp_size_t i;
mp_limb_t cy;
int i;
unsigned int cy;
for (i = 0, cy = 0; i < n; i++)
{
mp_limb_t a, b;
unsigned int a, b;
a = ap[i]; b = bp[i];
b += cy;
cy = (b < cy);
@ -140,30 +308,30 @@ mpn_sub_n (mp_ptr rp, mp_srcptr ap, mp_srcptr bp, mp_size_t n)
rp[i] = a - b;
}
return cy;
}
}*/
mp_limb_t
mpn_sub (mp_ptr rp, mp_srcptr ap, mp_size_t an, mp_srcptr bp, mp_size_t bn)
/*unsigned int
mpn_sub (unsigned int *rp, unsigned int *ap, int an, unsigned int *bp, int bn)
{
mp_limb_t cy;
unsigned int cy;
//assert (an >= bn);
cy = mpn_sub_n (rp, ap, bp, bn);
if (an > bn)
cy = mpn_sub_1 (rp + bn, ap + bn, an - bn, cy);
return cy;
}
}*/
/* MPZ interface */
void
/*void
mpz_clear (mpz_t r)
{
if (r->_mp_alloc)
gmp_free_limbs (r->_mp_d, r->_mp_alloc);
}
}*/
static mp_ptr
mpz_realloc (mpz_t r, mp_size_t size)
/*static unsigned int *
mpz_realloc (mpz_t r, int size)
{
size = gmp_max (size, 1);
@ -177,42 +345,42 @@ mpz_realloc (mpz_t r, mp_size_t size)
r->_mp_size = 0;
return r->_mp_d;
}
}*/
/* Realloc for an mpz_t WHAT if it has less than NEEDED limbs. */
mp_ptr mrz_realloc_if(mpz_t z,mp_size_t n) {
/*unsigned int *mrz_realloc_if(mpz_t z,int n) {
return n > z->_mp_alloc ? mpz_realloc(z, n) : z->_mp_d;
}
}*/
/* MPZ comparisons and the like. */
int
/*int
mpz_sgn (const mpz_t u)
{
return gmp_cmp (u->_mp_size, 0);
}
}*/
void
/*void
mpz_swap (mpz_t u, mpz_t v)
{
mp_size_t_swap (u->_mp_alloc, v->_mp_alloc);
mp_ptr_swap(u->_mp_d, v->_mp_d);
mp_size_t_swap (u->_mp_size, v->_mp_size);
}
int_swap (u->_mp_alloc, v->_mp_alloc);
unsigned int *_swap(u->_mp_d, v->_mp_d);
int_swap (u->_mp_size, v->_mp_size);
}*/
/* MPZ addition and subtraction */
static mp_size_t
/*static int
mpz_abs_add (mpz_t r, const mpz_t a, const mpz_t b)
{
mp_size_t an = gmp_abs (a->_mp_size);
mp_size_t bn = gmp_abs (b->_mp_size);
mp_ptr rp;
mp_limb_t cy;
int an = gmp_abs (a->_mp_size);
int bn = gmp_abs (b->_mp_size);
unsigned int *rp;
unsigned int cy;
if (an < bn)
{
mpz_srcptr_swap (a, b);
mp_size_t_swap (an, bn);
int_swap (an, bn);
}
rp = mrz_realloc_if (r, an + 1);
@ -221,15 +389,15 @@ mpz_abs_add (mpz_t r, const mpz_t a, const mpz_t b)
rp[an] = cy;
return an + cy;
}
}*/
static mp_size_t
/*static int
mpz_abs_sub (mpz_t r, const mpz_t a, const mpz_t b)
{
mp_size_t an = gmp_abs (a->_mp_size);
mp_size_t bn = gmp_abs (b->_mp_size);
int an = gmp_abs (a->_mp_size);
int bn = gmp_abs (b->_mp_size);
int cmp;
mp_ptr rp;
unsigned int *rp;
cmp = mpn_cmp4 (a->_mp_d, an, b->_mp_d, bn);
if (cmp > 0)
@ -246,12 +414,12 @@ mpz_abs_sub (mpz_t r, const mpz_t a, const mpz_t b)
}
else
return 0;
}
}*/
void
/*void
mpz_add (mpz_t r, const mpz_t a, const mpz_t b)
{
mp_size_t rn;
int rn;
if ( (a->_mp_size ^ b->_mp_size) >= 0)
rn = mpz_abs_add (r, a, b);
@ -259,12 +427,12 @@ mpz_add (mpz_t r, const mpz_t a, const mpz_t b)
rn = mpz_abs_sub (r, a, b);
r->_mp_size = a->_mp_size >= 0 ? rn : - rn;
}
}*/
void
/*void
mpz_sub (mpz_t r, const mpz_t a, const mpz_t b)
{
mp_size_t rn;
int rn;
if ( (a->_mp_size ^ b->_mp_size) >= 0)
rn = mpz_abs_sub (r, a, b);
@ -272,4 +440,4 @@ mpz_sub (mpz_t r, const mpz_t a, const mpz_t b)
rn = mpz_abs_add (r, a, b);
r->_mp_size = a->_mp_size >= 0 ? rn : - rn;
}
}*/

50
projects/mini-gmp.h
View File

@ -1,10 +1,3 @@
typedef unsigned long mp_limb_t;
typedef long mp_size_t;
typedef unsigned long mp_bitcnt_t;
typedef mp_limb_t *mp_ptr;
typedef const mp_limb_t *mp_srcptr;
typedef struct
{
int _mp_alloc; /* Number of *limbs* allocated and pointed
@ -12,7 +5,7 @@ typedef struct
int _mp_size; /* abs(_mp_size) is the number of limbs the
last field points to. If _mp_size is
negative this is a negative number. */
mp_limb_t *_mp_d; /* Pointer to the limbs. */
unsigned int *_mp_d; /* Pointer to the limbs. */
} __mpz_struct;
/* mpz_t is an array type that contains a single element of __mpz_struct, acting as a reference. */
@ -23,35 +16,35 @@ typedef const __mpz_struct *mpz_srcptr;
/* BEGIN Given Functions */
/* Swap functions. */
void mp_size_t_swap(mp_size_t x, mp_size_t y);
void int_swap(int x, int y);
void mp_ptr_swap(mp_ptr x, mp_ptr y);
void mp_ptr_swap(unsigned int *x, unsigned int *y);
void mpz_srcptr_swap(mpz_srcptr x, mpz_srcptr y);
/* Memory allocation functions. */
static mp_ptr
gmp_alloc_limbs (mp_size_t size);
static unsigned int *
gmp_alloc_limbs (int size);
static mp_ptr
gmp_realloc_limbs (mp_ptr old, mp_size_t old_size, mp_size_t size);
static unsigned int *
gmp_realloc_limbs (unsigned int *old, int old_size, int size);
static void
gmp_free_limbs (mp_ptr old, mp_size_t size);
gmp_free_limbs (unsigned int *old, int size);
/* END Given Functions */
void mpn_copyi (mp_ptr, mp_srcptr, mp_size_t);
void mpn_copyi (unsigned int *d, unsigned int *s, int n);
int mpn_cmp (mp_srcptr, mp_srcptr, mp_size_t);
int mpn_cmp (unsigned int *ap, unsigned int *bp, int n);
mp_limb_t mpn_add_1 (mp_ptr, mp_srcptr, mp_size_t, mp_limb_t);
mp_limb_t mpn_add_n (mp_ptr, mp_srcptr, mp_srcptr, mp_size_t);
mp_limb_t mpn_add (mp_ptr, mp_srcptr, mp_size_t, mp_srcptr, mp_size_t);
unsigned int mpn_add_1 (unsigned int *, unsigned int *, int, unsigned int);
unsigned int mpn_add_n (unsigned int *, unsigned int *, unsigned int *, int);
unsigned int mpn_add (unsigned int *, unsigned int *, int, unsigned int *, int);
mp_limb_t mpn_sub_1 (mp_ptr, mp_srcptr, mp_size_t, mp_limb_t);
mp_limb_t mpn_sub_n (mp_ptr, mp_srcptr, mp_srcptr, mp_size_t);
mp_limb_t mpn_sub (mp_ptr, mp_srcptr, mp_size_t, mp_srcptr, mp_size_t);
unsigned int mpn_sub_1 (unsigned int *, unsigned int *, int, unsigned int);
unsigned int mpn_sub_n (unsigned int *, unsigned int *, unsigned int *, int);
unsigned int mpn_sub (unsigned int *, unsigned int *, int, unsigned int *, int);
void mpz_clear (mpz_t);
@ -64,3 +57,14 @@ void mpz_add (mpz_t, const mpz_t, const mpz_t);
void mpz_sub (mpz_t, const mpz_t, const mpz_t);
void mpz_set (mpz_t, const mpz_t);
/*@
Extern Coq (Zabs : Z -> Z)
(Zmax : Z -> Z -> Z)
(mpd_store_Z : Z -> Z -> Z -> Z -> Assertion)
(mpd_store_Z_compact: Z -> Z -> Z -> Z -> Assertion)
(mpd_store_list : Z -> list Z -> Z -> Assertion)
(list_store_Z : list Z -> Z -> Prop)
(list_store_Z_compact: list Z -> Z -> Prop)
(last: list Z -> Z -> Z)
*/

126
projects/uint_array.strategies Normal file
View File

@ -0,0 +1,126 @@
//test curent rules
//max rule id:30
#include "int_array_def.h"
#include "verification_list.h"
#include "verification_stdlib.h"
id : 1
priority : core(1)
left : store_uint_array(?p, ?n, ?l) at 0
right : data_at(?p + (?i * sizeof(U32)), U32, ?v) at 1
check : infer(0 <= i);
infer(i < n);
action : right_erase(1);
left_erase(0);
left_add(store_uint_array_missing_i_rec(p, i, 0, n, l));
right_add(v == l[i]);
id : 2
priority : post(1)
left : store_uint_array_missing_i_rec(?p, ?i, 0, ?n, ?l) at 0
data_at(p + i * sizeof(U32), U32, l[i]) at 1
check : infer(0 <= i);
infer(i < n);
action : left_erase(1);
left_erase(0);
left_add(store_uint_array(p, n, l));
id : 3
priority : post(3)
left : store_uint_array_missing_i_rec(?p, ?i, 0, ?n, ?l) at 0
data_at(p + i * sizeof(U32), U32, ?v) at 1
check : infer(0 <= i);
infer(i < n);
action : left_erase(1);
left_erase(0);
left_add(store_uint_array(p, n, replace_Znth{Z}(i, v, l)));
id : 4
priority : core(1)
left : store_uint_array(?p, ?n, ?l1) at 0
right : store_uint_array(p, n, ?l2) at 1
action : right_erase(1);
left_erase(0);
right_add(l1 == l2);
id : 5
priority : core(1)
left : store_uint_array_missing_i_rec(?p, ?i, ?v, ?n, ?l) at 0
right : store_uint_array_missing_i_rec(p, i, v, n, l) at 1
action : left_erase(0);
right_erase(1);
id : 6
priority : core(1)
left : store_uint_array_rec(?p, ?x, ?y, ?l1) at 0
right : store_uint_array_rec(?p, ?x, ?y, ?l2) at 1
action : right_erase(1);
left_erase(0);
right_add(l1 == l2);
id : 7
priority : core(1)
left : store_uint_array_rec(?p, ?x, ?y, ?l) at 0
right : data_at(?p + (?i * sizeof(U32)), U32, ?v) at 1
check : infer(x <= i);
infer(i < y);
action : right_erase(1);
left_erase(0);
left_add(store_uint_array_missing_i_rec(p, i, x, y, l));
right_add(v == l[i - x]);
id : 8
priority : core(1)
left : store_uint_array_rec(?p, ?x, ?y, ?l1) at 0
store_uint_array_rec(?p, ?y, ?z, ?l2) at 1
right : store_uint_array_rec(?p, ?x, ?z, ?l3) at 2
check : infer(y <= z);
infer(x <= y);
action : right_erase(2);
left_erase(0);
left_erase(1);
right_add(l3 == app{Z}(l1, l2));
id : 9
priority : core(1)
left : store_uint_array_rec(?p, ?x, ?z, ?l3) at 2
right : store_uint_array_rec(?p, ?x, ?y, ?l1) at 0
store_uint_array_rec(?p, ?y, ?z, ?l2) at 1
check : infer(y <= z);
infer(x <= y);
action : left_erase(2);
right_erase(0);
right_erase(1);
right_add(l3 == app{Z}(l1, l2));
right_add(Zlength{Z}(l1) == y - x);
id : 10
priority : core(1)
right : store_uint_array_rec(?p, ?x, ?x, ?l) at 0
action : right_erase(0);
right_add(l == nil{Z});
id : 11
priority : post(1)
left : store_uint_array_missing_i_rec(?p, ?i, ?x, ?y, ?l) at 0
data_at(p + ?i * sizeof(U32), U32, l[?i - ?x]) at 1
check : infer(x <= i);
infer(i < y);
action : left_erase(1);
left_erase(0);
left_add(store_uint_array_rec(p, x, y, l));
id : 12
priority : post(4)
left : store_uint_array_missing_i_rec(?p, ?i, ?x, ?y, ?l) at 0
data_at(p + ?i * sizeof(U32), U32, ?v) at 1
check : infer(x <= i);
infer(i < y);
action : left_erase(1);
left_erase(0);
left_add(store_uint_array_rec(p, x, y, replace_Znth{Z}(i - x, v, l)));