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