diff options
Diffstat (limited to 'src/spl/Operators.v')
| -rw-r--r-- | src/spl/Operators.v | 88 |
1 files changed, 44 insertions, 44 deletions
diff --git a/src/spl/Operators.v b/src/spl/Operators.v index 540de3f..95bbe8e 100644 --- a/src/spl/Operators.v +++ b/src/spl/Operators.v @@ -35,7 +35,7 @@ Section Operators. Fixpoint check_in x l := match l with | nil => false - | t::q => if x == t then true else check_in x q + | t::q => if x =? t then true else check_in x q end. @@ -43,7 +43,7 @@ Section Operators. Proof. intro x; induction l as [ |t q IHq]; simpl. split; intro H; try discriminate; elim H. - case_eq (x == t). + case_eq (x =? t). rewrite eqb_spec; intro; subst t; split; auto. intro H; rewrite IHq; split; auto; intros [H1|H1]; auto; rewrite H1, eqb_refl in H; discriminate. Qed. @@ -54,7 +54,7 @@ Section Operators. | nil => true | b::q => if aexistsbi (fun _ (x:option (int*int)) => match x with - | Some (a',b') => ((a == a') && (b == b')) || ((a == b') && (b == a')) + | Some (a',b') => ((a =? a') && (b =? b')) || ((a =? b') && (b =? a')) | None => false end ) t then check_diseqs_complete_aux a q t else false @@ -63,13 +63,13 @@ Section Operators. Lemma check_diseqs_complete_aux_spec : forall a dist t, check_diseqs_complete_aux a dist t = true <-> - forall y, In y dist -> exists i, i < length t /\ + forall y, In y dist -> exists i, i <? length t /\ (t.[i] = Some (a,y) \/ t.[i] = Some (y,a)). Proof. intros a dist t; induction dist as [ |b q IHq]; simpl; split; auto. intros _ y H; inversion H. - case_eq (aexistsbi (fun _ (x : option (int * int)) => match x with | Some (a', b') => (a == a') && (b == b') || (a == b') && (b == a') | None => false end) t); try discriminate; rewrite aexistsbi_spec; intros [i [H1 H2]]; rewrite IHq; clear IHq; intros H3 y [H4|H4]; auto; subst y; exists i; split; auto; generalize H2; case (t .[ i]); try discriminate; intros [a' b']; rewrite orb_true_iff, !andb_true_iff, !Int63.eqb_spec; intros [[H4 H5]|[H4 H5]]; subst a' b'; auto. - intro H1; case_eq (aexistsbi (fun _ (x : option (int * int)) => match x with | Some (a', b') => (a == a') && (b == b') || (a == b') && (b == a') | None => false end) t). + case_eq (aexistsbi (fun _ (x : option (int * int)) => match x with | Some (a', b') => (a =? a') && (b =? b') || (a =? b') && (b =? a') | None => false end) t); try discriminate; rewrite aexistsbi_spec; intros [i [H1 H2]]; rewrite IHq; clear IHq; intros H3 y [H4|H4]; auto; subst y; exists i; split; auto; generalize H2; case (t .[ i]); try discriminate; intros [a' b']; rewrite orb_true_iff, !andb_true_iff, !Int63.eqb_spec; intros [[H4 H5]|[H4 H5]]; subst a' b'; auto. + intro H1; case_eq (aexistsbi (fun _ (x : option (int * int)) => match x with | Some (a', b') => (a =? a') && (b =? b') || (a =? b') && (b =? a') | None => false end) t). intros _; rewrite IHq; clear IHq; intros y Hy; apply H1; auto. rewrite aexistsbi_false_spec; destruct (H1 b (or_introl (refl_equal b))) as [i [H2 H3]]; intro H; rewrite <- (H _ H2); destruct H3 as [H3|H3]; rewrite H3; rewrite !eqb_refl; auto; rewrite orb_true_r; auto. Qed. @@ -77,15 +77,15 @@ Section Operators. Lemma check_diseqs_complete_aux_false_spec : forall a dist t, check_diseqs_complete_aux a dist t = false <-> - exists y, In y dist /\ forall i, i < length t -> + exists y, In y dist /\ forall i, i <? length t -> (t.[i] <> Some (a,y) /\ t.[i] <> Some (y,a)). Proof. intros a dist t; induction dist as [ |b q IHq]; simpl; split; try discriminate. intros [y [H _]]; elim H. - case_eq (aexistsbi (fun _ (x : option (int * int)) => match x with | Some (a', b') => (a == a') && (b == b') || (a == b') && (b == a') | None => false end) t). + case_eq (aexistsbi (fun _ (x : option (int * int)) => match x with | Some (a', b') => (a =? a') && (b =? b') || (a =? b') && (b =? a') | None => false end) t). intros _; rewrite IHq; clear IHq; intros [y [H3 H4]]; exists y; auto. rewrite aexistsbi_false_spec; intros H _; exists b; split; auto; intros i Hi; split; intro H1; generalize (H _ Hi); rewrite H1, !eqb_refl; try discriminate; rewrite orb_true_r; discriminate. - intros [y [H1 H2]]; case_eq (aexistsbi (fun _ (x : option (int * int)) => match x with | Some (a', b') => (a == a') && (b == b') || (a == b') && (b == a') | None => false end) t); auto; rewrite aexistsbi_spec; intros [i [H3 H4]]; rewrite IHq; clear IHq; exists y; destruct H1 as [H1|H1]; auto; subst y; case_eq (t.[i]); [intros [a' b'] Heq|intro Heq]; rewrite Heq in H4; try discriminate; rewrite orb_true_iff, !andb_true_iff, !eqb_spec in H4; destruct H4 as [[H4 H5]|[H4 H5]]; subst a' b'; generalize (H2 _ H3); rewrite Heq; intros [H4 H5]; [elim H4|elim H5]; auto. + intros [y [H1 H2]]; case_eq (aexistsbi (fun _ (x : option (int * int)) => match x with | Some (a', b') => (a =? a') && (b =? b') || (a =? b') && (b =? a') | None => false end) t); auto; rewrite aexistsbi_spec; intros [i [H3 H4]]; rewrite IHq; clear IHq; exists y; destruct H1 as [H1|H1]; auto; subst y; case_eq (t.[i]); [intros [a' b'] Heq|intro Heq]; rewrite Heq in H4; try discriminate; rewrite orb_true_iff, !andb_true_iff, !eqb_spec in H4; destruct H4 as [[H4 H5]|[H4 H5]]; subst a' b'; generalize (H2 _ H3); rewrite Heq; intros [H4 H5]; [elim H4|elim H5]; auto. Qed. @@ -98,7 +98,7 @@ Section Operators. Lemma check_diseqs_complete_spec : forall dist t, check_diseqs_complete dist t = true <-> - forall x y, In2 x y dist -> exists i, i < length t /\ + forall x y, In2 x y dist -> exists i, i <? length t /\ (t.[i] = Some (x,y) \/ t.[i] = Some (y,x)). Proof. intros dist t; induction dist as [ |a q IHq]; simpl; split; auto. @@ -112,7 +112,7 @@ Section Operators. Lemma check_diseqs_complete_false_spec : forall dist t, check_diseqs_complete dist t = false <-> - exists x y, In2 x y dist /\ forall i, i < length t -> + exists x y, In2 x y dist /\ forall i, i <? length t -> (t.[i] <> Some (x,y) /\ t.[i] <> Some (y,x)). Proof. intros dist t; induction dist as [ |a q IHq]; simpl; split; try discriminate. @@ -135,7 +135,7 @@ Section Operators. | Fatom a => match get_atom a with | Atom.Abop (Atom.BO_eq A) h1 h2 => - if (Typ.eqb ty A) && (negb (h1 == h2)) && (check_in h1 dist) && (check_in h2 dist) then + if (Typ.eqb ty A) && (negb (h1 =? h2)) && (check_in h1 dist) && (check_in h2 dist) then Some (h1,h2) else None @@ -150,14 +150,14 @@ Section Operators. Lemma check_diseqs_spec : forall A dist diseq, check_diseqs A dist diseq = true <-> - ((forall i, i < length diseq -> + ((forall i, i <? length diseq -> let t := diseq.[i] in ~ Lit.is_pos t /\ exists a, get_form (Lit.blit t) = Fatom a /\ exists h1 h2, get_atom a = Atom.Abop (Atom.BO_eq A) h1 h2 /\ h1 <> h2 /\ (In2 h1 h2 dist \/ In2 h2 h1 dist)) /\ - (forall x y, In2 x y dist -> exists i, i < length diseq /\ + (forall x y, In2 x y dist -> exists i, i <? length diseq /\ let t := diseq.[i] in ~ Lit.is_pos t /\ exists a, get_form (Lit.blit t) = Fatom a /\ @@ -169,14 +169,14 @@ Section Operators. aforallbi_spec, check_diseqs_complete_spec, length_amap; split; intros [H1 H2]; split. clear H2; intros i Hi; generalize (H1 _ Hi); rewrite get_amap; auto; case_eq (Lit.is_pos (diseq .[ i])); try discriminate; intro Heq1; case_eq (get_form (Lit.blit (diseq .[ i]))); - try discriminate; intros a Heq2; case_eq (get_atom a); try discriminate; intros [ | | | | | | | B | | | | | | | | | | | | ]; try discriminate; intros h1 h2 Heq3; case_eq (Typ.eqb A B); try discriminate; change (Typ.eqb A B = true) with (is_true (Typ.eqb A B)); rewrite Typ.eqb_spec; intro; subst B; case_eq (h1 == h2); try discriminate; rewrite eqb_false_spec; intro H2; case_eq (check_in h1 dist); try discriminate; case_eq (check_in h2 dist); try discriminate; rewrite !check_in_spec; intros H3 H4 _; split; try discriminate; exists a; split; auto; exists h1, h2; repeat split; auto; rewrite <- In2_In; auto. + try discriminate; intros a Heq2; case_eq (get_atom a); try discriminate; intros [ | | | | | | | B | | | | | | | | | | | | ]; try discriminate; intros h1 h2 Heq3; case_eq (Typ.eqb A B); try discriminate; change (Typ.eqb A B = true) with (is_true (Typ.eqb A B)); rewrite Typ.eqb_spec; intro; subst B; case_eq (h1 =? h2); try discriminate; rewrite eqb_false_spec; intro H2; case_eq (check_in h1 dist); try discriminate; case_eq (check_in h2 dist); try discriminate; rewrite !check_in_spec; intros H3 H4 _; split; try discriminate; exists a; split; auto; exists h1, h2; repeat split; auto; rewrite <- In2_In; auto. clear H1; intros x y Hxy; destruct (H2 _ _ Hxy) as [i [H1 H4]]; clear H2; rewrite get_amap in H4; auto; exists i; split; auto; generalize H4; - case_eq (Lit.is_pos (diseq .[ i])); intro Heq; try (intros [H|H]; discriminate); case_eq (get_form (Lit.blit (diseq .[ i]))); [intros a| | |intros a1 a2|intros a1|intros a1|intros a1|intros a1 a2|intros a1 a2| intros a1 a2 a3|intros a ls]; intro Heq2; try (intros [H|H]; discriminate); case_eq (get_atom a); [intros a1|intros a1 a2|intros [ | | | | | | | B | | | | | | | | | | | | ] h1 h2|intros a1 a2|intros a1 a2 | intros a1 a2]; intro Heq3; try (intros [H|H]; discriminate); try (case_eq (Typ.eqb A B); try (intros _ [H|H]; discriminate); change (Typ.eqb A B = true) with (is_true (Typ.eqb A B)); rewrite Typ.eqb_spec; intro; subst B; case_eq (h1 == h2); try (intros _ [H|H]; discriminate); rewrite eqb_false_spec; intro H10; case (check_in h1 dist); try (intros [H|H]; discriminate); case (check_in h2 dist); try (intros [H|H]; discriminate); simpl; intro H3; split; try discriminate; exists a; split; auto; destruct H3 as [H3|H3]; inversion H3; subst; auto). + case_eq (Lit.is_pos (diseq .[ i])); intro Heq; try (intros [H|H]; discriminate); case_eq (get_form (Lit.blit (diseq .[ i]))); [intros a| | |intros a1 a2|intros a1|intros a1|intros a1|intros a1 a2|intros a1 a2| intros a1 a2 a3|intros a ls]; intro Heq2; try (intros [H|H]; discriminate); case_eq (get_atom a); [intros a1|intros a1 a2|intros [ | | | | | | | B | | | | | | | | | | | | ] h1 h2|intros a1 a2|intros a1 a2 | intros a1 a2]; intro Heq3; try (intros [H|H]; discriminate); try (case_eq (Typ.eqb A B); try (intros _ [H|H]; discriminate); change (Typ.eqb A B = true) with (is_true (Typ.eqb A B)); rewrite Typ.eqb_spec; intro; subst B; case_eq (h1 =? h2); try (intros _ [H|H]; discriminate); rewrite eqb_false_spec; intro H10; case (check_in h1 dist); try (intros [H|H]; discriminate); case (check_in h2 dist); try (intros [H|H]; discriminate); simpl; intro H3; split; try discriminate; exists a; split; auto; destruct H3 as [H3|H3]; inversion H3; subst; auto). intros. destruct H0; now contradict H0. - clear H2; intros i Hi; rewrite get_amap; auto; destruct (H1 _ Hi) as [H2 [a [H3 [h1 [h2 [H4 [H5 H6]]]]]]]; clear H1; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto); rewrite H3, H4, Typ.eqb_refl; simpl; replace (h1 == h2) with false by (case_eq (h1 == h2); auto; rewrite eqb_spec; intro H; elim H5; auto); simpl; rewrite <- In2_In, <- !check_in_spec in H6; auto; destruct H6 as [H6 H7]; rewrite H6, H7; auto. - clear H1; intros x y Hxy; destruct (H2 _ _ Hxy) as [i [H1 [H3 [a [H4 [H6 H5]]]]]]; clear H2; exists i; split; auto; rewrite get_amap; auto; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto); rewrite H4; assert (H7 := or_introl (In2 y x dist) Hxy); rewrite <- In2_In, <- !check_in_spec in H7; auto; destruct H7 as [H7 H8]; destruct H5 as [H5|H5]; rewrite H5, Typ.eqb_refl; [replace (x == y) with false by (case_eq (x == y); auto; rewrite eqb_spec; auto)|replace (y == x) with false by (case_eq (y == x); auto; rewrite eqb_spec; auto)]; simpl; rewrite H7, H8; auto. + clear H2; intros i Hi; rewrite get_amap; auto; destruct (H1 _ Hi) as [H2 [a [H3 [h1 [h2 [H4 [H5 H6]]]]]]]; clear H1; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto); rewrite H3, H4, Typ.eqb_refl; simpl; replace (h1 =? h2) with false by (case_eq (h1 =? h2); auto; rewrite eqb_spec; intro H; elim H5; auto); simpl; rewrite <- In2_In, <- !check_in_spec in H6; auto; destruct H6 as [H6 H7]; rewrite H6, H7; auto. + clear H1; intros x y Hxy; destruct (H2 _ _ Hxy) as [i [H1 [H3 [a [H4 [H6 H5]]]]]]; clear H2; exists i; split; auto; rewrite get_amap; auto; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto); rewrite H4; assert (H7 := or_introl (In2 y x dist) Hxy); rewrite <- In2_In, <- !check_in_spec in H7; auto; destruct H7 as [H7 H8]; destruct H5 as [H5|H5]; rewrite H5, Typ.eqb_refl; [replace (x =? y) with false by (case_eq (x =? y); auto; rewrite eqb_spec; auto)|replace (y =? x) with false by (case_eq (y =? x); auto; rewrite eqb_spec; auto)]; simpl; rewrite H7, H8; auto. Qed. @@ -187,7 +187,7 @@ intros. destruct H0; now contradict H0. (* | Fatom a => *) (* match get_atom a with *) (* | Atom.Abop (Atom.BO_eq A) h1 h2 => *) - (* (Typ.eqb ty A) && (negb (h1 == h2)) && (check_in h1 dist) && (check_in h2 dist) *) + (* (Typ.eqb ty A) && (negb (h1 =? h2)) && (check_in h1 dist) && (check_in h2 dist) *) (* | _ => false *) (* end *) (* | _ => false *) @@ -195,7 +195,7 @@ intros. destruct H0; now contradict H0. (* Lemma check_diseqs_spec : forall A dist diseq, *) - (* check_diseqs A dist diseq <-> forall i, i < length diseq -> *) + (* check_diseqs A dist diseq <-> forall i, i <? length diseq -> *) (* let t := diseq.[i] in *) (* ~ Lit.is_pos t /\ *) (* exists a, get_form (Lit.blit t) = Fatom a /\ *) @@ -204,7 +204,7 @@ intros. destruct H0; now contradict H0. (* Proof. *) (* intros A dist diseq; unfold check_diseqs; unfold is_true at 1; rewrite PArray.forallb_spec; split. *) (* intros H i Hi; generalize (H _ Hi); clear H; case (Lit.is_pos (diseq .[ i])); try discriminate; case (get_form (Lit.blit (diseq .[ i]))); try discriminate; intros a H1; split; try discriminate; exists a; split; auto; generalize H1; clear H1; case (get_atom a); try discriminate; intros [ | | | | | | |B] h1 h2; try discriminate; rewrite !andb_true_iff; change (check_in h1 dist = true) with (is_true (check_in h1 dist)); change (check_in h2 dist = true) with (is_true (check_in h2 dist)); rewrite !check_in_spec; intros [[[H1 H4] H2] H3]; change (is_true (Typ.eqb A B)) in H1; rewrite Typ.eqb_spec in H1; subst B; exists h1; exists h2; split; auto; assert (H5: h1 <> h2) by (intro H; rewrite H, eqb_refl in H4; discriminate); split; auto; rewrite <- In2_In; auto. *) - (* intros H i Hi; generalize (H _ Hi); clear H; case (Lit.is_pos (diseq .[ i])); try (intros [H _]; elim H; reflexivity); intros [_ [a [H1 [h1 [h2 [H2 [H3 H4]]]]]]]; rewrite H1, H2; rewrite !andb_true_iff; rewrite <- (In2_In H3) in H4; destruct H4 as [H4 H5]; change (check_in h1 dist = true) with (is_true (check_in h1 dist)); change (check_in h2 dist = true) with (is_true (check_in h2 dist)); rewrite !check_in_spec; repeat split; auto; case_eq (h1 == h2); auto; try (rewrite Typ.eqb_refl; auto); rewrite eqb_spec; intro; subst h1; elim H3; auto. *) + (* intros H i Hi; generalize (H _ Hi); clear H; case (Lit.is_pos (diseq .[ i])); try (intros [H _]; elim H; reflexivity); intros [_ [a [H1 [h1 [h2 [H2 [H3 H4]]]]]]]; rewrite H1, H2; rewrite !andb_true_iff; rewrite <- (In2_In H3) in H4; destruct H4 as [H4 H5]; change (check_in h1 dist = true) with (is_true (check_in h1 dist)); change (check_in h2 dist = true) with (is_true (check_in h2 dist)); rewrite !check_in_spec; repeat split; auto; case_eq (h1 =? h2); auto; try (rewrite Typ.eqb_refl; auto); rewrite eqb_spec; intro; subst h1; elim H3; auto. *) (* Qed. *) @@ -231,7 +231,7 @@ intros. destruct H0; now contradict H0. match get_form f1, get_form f2 with | Fatom ha, Fatom hb => match get_atom ha, get_atom hb with - | Atom.Anop (Atom.NO_distinct ty) (x::y::nil), Atom.Abop (Atom.BO_eq ty') x' y' => (Typ.eqb ty ty') && (((x == x') && (y == y')) || ((x == y') && (y == x'))) + | Atom.Anop (Atom.NO_distinct ty) (x::y::nil), Atom.Abop (Atom.BO_eq ty') x' y' => (Typ.eqb ty ty') && (((x =? x') && (y =? y')) || ((x =? y') && (y =? x'))) | _, _ => false end | _, _ => false @@ -287,12 +287,12 @@ intros. destruct H0; now contradict H0. Proof. intros ha diseq; rewrite check_distinct_spec; intros [A [dist [H1 H2]]]; rewrite check_diseqs_spec in H2; destruct H2 as [H2 H3]; unfold Atom.interp_form_hatom, Atom.interp_bool, Atom.interp_hatom; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op; rewrite H1; simpl; generalize (Atom.compute_interp_spec_rev t_i (get (Atom.t_interp t_i t_func t_atom)) A dist); case (Atom.compute_interp t_i (get (Atom.t_interp t_i t_func t_atom)) A nil); simpl. intros l H4; case_eq (distinct (Typ.i_eqb t_i A) (rev l)). - rewrite distinct_spec; intro H5; symmetry; apply afold_left_andb_true; rewrite length_amap; intros i Hi; rewrite get_amap by exact Hi; destruct (H2 _ Hi) as [H9 [a [H10 [h1 [h2 [H6 [H7 H8]]]]]]]; unfold Lit.interp; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto with smtcoq_spl_op smtcoq_core); unfold Var.interp; rewrite Form.wf_interp_form; auto with smtcoq_spl_op smtcoq_core; rewrite H10; simpl; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H6; simpl; unfold Atom.apply_binop; unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; assert (H11: a < length t_atom). - case_eq (a < length t_atom); auto with smtcoq_spl_op smtcoq_core; intro H11; rewrite (get_outofbound _ _ _ H11) in H6; rewrite default_t_atom in H6; inversion H6. + rewrite distinct_spec; intro H5; symmetry; apply afold_left_andb_true; rewrite length_amap; intros i Hi; rewrite get_amap by exact Hi; destruct (H2 _ Hi) as [H9 [a [H10 [h1 [h2 [H6 [H7 H8]]]]]]]; unfold Lit.interp; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto with smtcoq_spl_op smtcoq_core); unfold Var.interp; rewrite Form.wf_interp_form; auto with smtcoq_spl_op smtcoq_core; rewrite H10; simpl; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H6; simpl; unfold Atom.apply_binop; unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; assert (H11: a <? length t_atom). + case_eq (a <? length t_atom); auto with smtcoq_spl_op smtcoq_core; intro H11; rewrite (get_outofbound _ _ _ H11) in H6; rewrite default_t_atom in H6; inversion H6. generalize (wt_t_atom _ H11); rewrite H6; simpl; rewrite !andb_true_iff; change (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) h1) A = true) with (is_true (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) h1) A)); change (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) h2) A = true) with (is_true (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) h2) A)); rewrite !Typ.eqb_spec; intros [[_ H13] H12]; generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom h1); rewrite H13; intros [v1 HH1]; generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom h2); rewrite H12; intros [v2 HH2]; rewrite HH1, HH2; simpl; rewrite Typ.cast_refl; simpl; destruct H8 as [H8|H8]; [ |rewrite Typ.i_eqb_sym]; rewrite H5; auto with smtcoq_spl_op smtcoq_core; rewrite H4; [exists h2; exists h1|exists h1; exists h2]; auto with smtcoq_spl_op smtcoq_core. rewrite distinct_false_spec; intros [v2 [v1 [H5 H6]]]; rewrite H4 in H5; destruct H5 as [a [b [H5 [H7 H8]]]]; clear H4; change (Typ.i_eqb t_i A v2 v1 = true) with (is_true (Typ.i_eqb t_i A v2 v1)) in H6; rewrite Typ.i_eqb_spec in H6; subst v2; clear H2; destruct (H3 _ _ H5) as [i [H2 [H4 [hb [H6 [H9 H10]]]]]]; clear H3; symmetry; apply (afold_left_andb_false i); rewrite ?length_amap; auto with smtcoq_spl_op smtcoq_core; rewrite get_amap by assumption; unfold Lit.interp; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto with smtcoq_spl_op smtcoq_core); unfold Var.interp; rewrite Form.wf_interp_form; auto with smtcoq_spl_op smtcoq_core; rewrite H6; simpl; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; destruct H10 as [H10|H10]; rewrite H10; simpl; rewrite H7, H8; simpl; rewrite Typ.cast_refl; simpl; replace (Typ.i_eqb t_i A v1 v1) with true; auto with smtcoq_spl_op smtcoq_core; symmetry; change (is_true (Typ.i_eqb t_i A v1 v1)); rewrite Typ.i_eqb_spec; auto with smtcoq_spl_op smtcoq_core. - intros [a [H20 H21]]; assert (H4: ha < length t_atom). - case_eq (ha < length t_atom); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H1; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_atom; discriminate. + intros [a [H20 H21]]; assert (H4: ha <? length t_atom). + case_eq (ha <? length t_atom); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H1; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_atom; discriminate. unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; generalize (wt_t_atom _ H4); rewrite H1; simpl; rewrite andb_true_iff, forallb_forall; intros [_ H5]; assert (H6 := H5 _ H20); generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom a); intros [va Ha]; rewrite Ha in H21; simpl in H21; elim H21; apply Typ.eqb_spec; auto with smtcoq_spl_op smtcoq_core. Qed. @@ -300,7 +300,7 @@ intros. destruct H0; now contradict H0. check_distinct_two_args f1 f2 = true -> rho f1 = negb (rho f2). Proof. - intros f1 f2; rewrite check_distinct_two_args_spec; intros [ha [hb [A [x [y [H1 [H2 [H3 [H4|H4]]]]]]]]]; unfold Form.interp_state_var; assert (H5: f1 < length t_form) by (case_eq (f1 < length t_form); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H1; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_form; discriminate); assert (H6: f2 < length t_form) by (case_eq (f2 < length t_form); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H2; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_form; discriminate); rewrite !Form.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H1, H2; simpl; unfold Atom.interp_form_hatom, Atom.interp_hatom; rewrite !Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H3, H4; simpl; unfold Atom.wt,is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; assert (H7: hb < length t_atom) by (case_eq (hb < length t_atom); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H4; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_atom; discriminate); generalize (wt_t_atom _ H7); rewrite H4; simpl; case (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) hb); try discriminate; simpl; rewrite andb_true_iff; change (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) x) A = true) with (is_true (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) x) A)); change (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) y) A = true) with (is_true (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) y) A)); rewrite !Typ.eqb_spec; intros [H8 H9]; generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom x), (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom y); rewrite H8, H9; intros [v1 HH1] [v2 HH2]; rewrite HH1, HH2; simpl; rewrite Typ.cast_refl; auto with smtcoq_spl_op smtcoq_core; rewrite Typ.i_eqb_sym; auto with smtcoq_spl_op smtcoq_core. + intros f1 f2; rewrite check_distinct_two_args_spec; intros [ha [hb [A [x [y [H1 [H2 [H3 [H4|H4]]]]]]]]]; unfold Form.interp_state_var; assert (H5: f1 <? length t_form) by (case_eq (f1 <? length t_form); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H1; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_form; discriminate); assert (H6: f2 <? length t_form) by (case_eq (f2 <? length t_form); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H2; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_form; discriminate); rewrite !Form.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H1, H2; simpl; unfold Atom.interp_form_hatom, Atom.interp_hatom; rewrite !Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H3, H4; simpl; unfold Atom.wt,is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; assert (H7: hb <? length t_atom) by (case_eq (hb <? length t_atom); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H4; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_atom; discriminate); generalize (wt_t_atom _ H7); rewrite H4; simpl; case (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) hb); try discriminate; simpl; rewrite andb_true_iff; change (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) x) A = true) with (is_true (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) x) A)); change (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) y) A = true) with (is_true (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) y) A)); rewrite !Typ.eqb_spec; intros [H8 H9]; generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom x), (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom y); rewrite H8, H9; intros [v1 HH1] [v2 HH2]; rewrite HH1, HH2; simpl; rewrite Typ.cast_refl; auto with smtcoq_spl_op smtcoq_core; rewrite Typ.i_eqb_sym; auto with smtcoq_spl_op smtcoq_core. Qed. @@ -309,11 +309,11 @@ intros. destruct H0; now contradict H0. (* interp_form_hatom ha -> afold_left bool int true andb (Lit.interp rho) diseq. *) (* Proof. *) (* intros ha diseq; rewrite check_distinct_spec; intros [A [dist [H1 H]]]; rewrite check_diseqs_spec in H; unfold Atom.interp_form_hatom, Atom.interp_bool, Atom.interp_hatom; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H1; simpl; generalize (Atom.compute_interp_spec_rev t_i (get (Atom.t_interp t_i t_func t_atom)) A dist); case (Atom.compute_interp t_i (get (Atom.t_interp t_i t_func t_atom)) A nil); simpl. *) - (* intros l H2; unfold is_true; rewrite distinct_spec; intro H3; apply afold_left_andb_true; intros i Hi; destruct (H _ Hi) as [H4 [a [H5 [h1 [h2 [H6 [H7 H8]]]]]]]; unfold Lit.interp; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto with smtcoq_spl_op smtcoq_core); unfold Var.interp; rewrite Form.wf_interp_form; auto with smtcoq_spl_op smtcoq_core; rewrite H5; simpl; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H6; simpl; unfold Atom.apply_binop; unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; assert (H10: a < length t_atom). *) - (* case_eq (a < length t_atom); auto with smtcoq_spl_op smtcoq_core; intro H10; rewrite (get_outofbound _ _ _ H10) in H6; rewrite default_t_atom in H6; inversion H6. *) + (* intros l H2; unfold is_true; rewrite distinct_spec; intro H3; apply afold_left_andb_true; intros i Hi; destruct (H _ Hi) as [H4 [a [H5 [h1 [h2 [H6 [H7 H8]]]]]]]; unfold Lit.interp; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto with smtcoq_spl_op smtcoq_core); unfold Var.interp; rewrite Form.wf_interp_form; auto with smtcoq_spl_op smtcoq_core; rewrite H5; simpl; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H6; simpl; unfold Atom.apply_binop; unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; assert (H10: a <? length t_atom). *) + (* case_eq (a <? length t_atom); auto with smtcoq_spl_op smtcoq_core; intro H10; rewrite (get_outofbound _ _ _ H10) in H6; rewrite default_t_atom in H6; inversion H6. *) (* generalize (wt_t_atom _ H10); rewrite H6; simpl; rewrite !andb_true_iff. change (Typ.eqb (Atom.get_type t_i t_func t_atom h1) A = true) with (is_true (Typ.eqb (Atom.get_type t_i t_func t_atom h1) A)); change (Typ.eqb (Atom.get_type t_i t_func t_atom h2) A = true) with (is_true (Typ.eqb (Atom.get_type t_i t_func t_atom h2) A)); rewrite !Typ.eqb_spec; intros [[_ H11] H12]; generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom h1); rewrite H11; intros [v1 HH1]; generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom h2); rewrite H12; intros [v2 HH2]; rewrite HH1, HH2; simpl; rewrite Typ.cast_refl; simpl; destruct H8 as [H8|H8]; [ |rewrite Typ.i_eqb_sym]; rewrite H3; auto with smtcoq_spl_op smtcoq_core; rewrite H2; [exists h2; exists h1|exists h1; exists h2]; auto with smtcoq_spl_op smtcoq_core. *) - (* intros [a [H2 H3]] _; assert (H4: ha < length t_atom). *) - (* case_eq (ha < length t_atom); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H1; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_atom; discriminate. *) + (* intros [a [H2 H3]] _; assert (H4: ha <? length t_atom). *) + (* case_eq (ha <? length t_atom); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H1; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_atom; discriminate. *) (* unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; generalize (wt_t_atom _ H4); rewrite H1; simpl; rewrite andb_true_iff, forallb_forall; intros [_ H5]; assert (H6 := H5 _ H2); generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom a); intros [va Ha]; rewrite Ha in H3; simpl in H3; elim H3; apply Typ.eqb_spec; auto with smtcoq_spl_op smtcoq_core. *) (* Qed. *) @@ -325,22 +325,22 @@ intros. destruct H0; now contradict H0. Variable check_var : var -> var -> bool. Definition check_lit l1 l2 := - (l1 == l2) || ((Bool.eqb (Lit.is_pos l1) (Lit.is_pos l2)) && (check_var (Lit.blit l1) (Lit.blit l2))) || ((Bool.eqb (Lit.is_pos l1) (negb (Lit.is_pos l2))) && (check_distinct_two_args (Lit.blit l1) (Lit.blit l2))). + (l1 =? l2) || ((Bool.eqb (Lit.is_pos l1) (Lit.is_pos l2)) && (check_var (Lit.blit l1) (Lit.blit l2))) || ((Bool.eqb (Lit.is_pos l1) (negb (Lit.is_pos l2))) && (check_distinct_two_args (Lit.blit l1) (Lit.blit l2))). (* Definition check_lit l1 l2 := *) - (* (l1 == l2) || ((Lit.is_pos l1) && (Lit.is_pos l2) && (check_var (Lit.blit l1) (Lit.blit l2))) || ((negb (Lit.is_pos l1)) && (negb (Lit.is_pos l2)) && (check_var (Lit.blit l2) (Lit.blit l1))). *) + (* (l1 =? l2) || ((Lit.is_pos l1) && (Lit.is_pos l2) && (check_var (Lit.blit l1) (Lit.blit l2))) || ((negb (Lit.is_pos l1)) && (negb (Lit.is_pos l2)) && (check_var (Lit.blit l2) (Lit.blit l1))). *) Definition check_form_aux a b := match a, b with | Fatom ha, Fand diseq => check_distinct ha diseq - | Fatom a, Fatom b => a == b + | Fatom a, Fatom b => a =? b | Ftrue, Ftrue => true | Ffalse, Ffalse => true - | Fnot2 i1 l1, Fnot2 i2 l2 => (i1 == i2) && (check_lit l1 l2) - | Fand a1, Fand a2 => (length a1 == length a2) && (aforallbi (fun i l => check_lit l (a2.[i])) a1) - | For a1, For a2 => (length a1 == length a2) && (aforallbi (fun i l => check_lit l (a2.[i])) a1) - | Fimp a1, Fimp a2 => (length a1 == length a2) && (aforallbi (fun i l => check_lit l (a2.[i])) a1) - (* (length a1 == length a2) && (aforallbi (fun i l => if i < length a1 - 1 then check_lit (a2.[i]) l else check_lit l (a2.[i])) a1) *) + | Fnot2 i1 l1, Fnot2 i2 l2 => (i1 =? i2) && (check_lit l1 l2) + | Fand a1, Fand a2 => (length a1 =? length a2) && (aforallbi (fun i l => check_lit l (a2.[i])) a1) + | For a1, For a2 => (length a1 =? length a2) && (aforallbi (fun i l => check_lit l (a2.[i])) a1) + | Fimp a1, Fimp a2 => (length a1 =? length a2) && (aforallbi (fun i l => check_lit l (a2.[i])) a1) + (* (length a1 =? length a2) && (aforallbi (fun i l => if i <? length a1 - 1 then check_lit (a2.[i]) l else check_lit l (a2.[i])) a1) *) | Fxor l1 l2, Fxor j1 j2 => check_lit l1 j1 && check_lit l2 j2 (* check_lit l1 j1 && check_lit j1 l1 && check_lit l2 j2 && check_lit j2 l2 *) (* (* let a := check_lit l1 j1 in *) *) @@ -450,16 +450,16 @@ intros. destruct H0; now contradict H0. (* rewrite <- H1; eauto with smtcoq_core. *) (* eapply interp_check_lit; eauto with smtcoq_core. *) (* (* Implication *) *) - (* unfold is_true; rewrite andb_true_iff, Int63Properties.eqb_spec; intros [H1 H2]; rewrite aforallbi_spec in H2; intro H3; apply afold_right_implb_true; case_eq (length a1 == 0); intro Heq. *) + (* unfold is_true; rewrite andb_true_iff, Int63Properties.eqb_spec; intros [H1 H2]; rewrite aforallbi_spec in H2; intro H3; apply afold_right_implb_true; case_eq (length a1 =? 0); intro Heq. *) (* left; rewrite eqb_spec in Heq; rewrite <- H1; auto with smtcoq_core. *) (* destruct (afold_right_implb_true_inv _ _ _ H3) as [H4|[[i [H4 H5]]|H4]]. *) (* rewrite H4 in Heq; discriminate. *) - (* right; left; exists i; rewrite <- H1; split; auto with smtcoq_core; case_eq (Lit.interp rho (a2 .[ i])); auto with smtcoq_core; intro H6; assert (H7: i < length a1 = true). *) + (* right; left; exists i; rewrite <- H1; split; auto with smtcoq_core; case_eq (Lit.interp rho (a2 .[ i])); auto with smtcoq_core; intro H6; assert (H7: i <? length a1 = true). *) (* rewrite ltb_spec in *; rewrite eqb_false_spec in Heq; rewrite to_Z_sub_1_diff in H4; auto with smtcoq_core; omega. *) (* generalize (H2 _ H7); rewrite H4; intro H8; rewrite (interp_check_lit _ _ H8 H6) in H5; auto with smtcoq_core. *) - (* right; case_eq (aexistsbi (fun i l => (i < length a2 - 1) && (negb (Lit.interp rho l))) a2). *) + (* right; case_eq (aexistsbi (fun i l => (i <? length a2 - 1) && (negb (Lit.interp rho l))) a2). *) (* rewrite aexistsbi_spec; intros [i [_ H5]]; rewrite andb_true_iff in H5; destruct H5 as [H5 H6]; left; exists i; split; auto with smtcoq_core; generalize H6; case (Lit.interp rho (a2 .[ i])); auto with smtcoq_core; discriminate. *) - (* rewrite aexistsbi_false_spec; intro H; right; intros i Hi; assert (Hi' := Hi); rewrite <- H1 in Hi'; generalize (H2 _ Hi') (H _ Hi); rewrite <- H1; case (i < length a1 - 1); simpl. *) + (* rewrite aexistsbi_false_spec; intro H; right; intros i Hi; assert (Hi' := Hi); rewrite <- H1 in Hi'; generalize (H2 _ Hi') (H _ Hi); rewrite <- H1; case (i <? length a1 - 1); simpl. *) (* intros _; case (Lit.interp rho (a2 .[ i])); auto with smtcoq_core; discriminate. *) (* intros H5 _; apply (interp_check_lit _ _ H5); apply H4; auto with smtcoq_core. *) (* (* Xor *) *) @@ -474,7 +474,7 @@ intros. destruct H0; now contradict H0. Definition check_hform h1 h2 := foldi - (fun _ cont h1 h2 => (h1 == h2) || check_form_aux cont (get_form h1) (get_form h2)) + (fun _ cont h1 h2 => (h1 =? h2) || check_form_aux cont (get_form h1) (get_form h2)) 0 (PArray.length t_form) (fun h1 h2 => false) h1 h2. Definition check_form := check_form_aux check_hform. |