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feat(cmp): Proved correctness of mpn_cmp.

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xiaoh105
2025-06-11 16:54:36 +08:00
parent 4c0b0e98fa
commit 36204b8877
7 changed files with 824 additions and 18 deletions
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projects/lib/gmp_proof_manual.v
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@ -149,4 +149,95 @@ Proof.
rewrite H5 in H4; clear H5.
rewrite <-H4.
sep_apply store_uint_array_rec_tail_merge; [ reflexivity | lia ].
Qed.
Lemma proof_of_mpn_cmp_safety_wit_1 : mpn_cmp_safety_wit_1.
Proof.
pre_process.
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!.
repeat rewrite <-derivable1_orp_intros1.
entailer!.
+ unfold mpd_store_Z.
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_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.
rewrite <-derivable1_orp_intros2.
entailer!.
+ unfold mpd_store_Z, mpd_store_list.
Exists l1 l2.
rewrite <-H6, <-H7.
entailer!.
+ pose proof (list_store_Z_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.
Intros l1 l2.
Exists l2 l1.
unfold mpd_store_list.
entailer!.
rewrite <-H4, <-H6.
entailer!.
Qed.