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Diffstat (limited to 'src/spl/Operators.v')
| -rw-r--r-- | src/spl/Operators.v | 549 |
1 files changed, 549 insertions, 0 deletions
diff --git a/src/spl/Operators.v b/src/spl/Operators.v new file mode 100644 index 0000000..90483db --- /dev/null +++ b/src/spl/Operators.v @@ -0,0 +1,549 @@ +(**************************************************************************) +(* *) +(* SMTCoq *) +(* Copyright (C) 2011 - 2015 *) +(* *) +(* Michaël Armand *) +(* Benjamin Grégoire *) +(* Chantal Keller *) +(* *) +(* Inria - École Polytechnique - MSR-Inria Joint Lab *) +(* *) +(* This file is distributed under the terms of the CeCILL-C licence *) +(* *) +(**************************************************************************) + +(*** Spl -- a small checker for simplifications ***) + +(* Add LoadPath ".." as SMTCoq. *) +(* Add LoadPath "../lia" as SMTCoq.lia. *) +Require Import List PArray Bool Int63 ZMicromega. +Require Import Misc State SMT_terms. +Require Lia. + +Local Open Scope array_scope. +Local Open Scope int63_scope. + + +(* Simplification of SMTLIB-2 operators *) + +Section Operators. + + Import Form. + + Variable t_form : PArray.array Form.form. + Variable t_atom : PArray.array Atom.atom. + + Local Notation get_form := (PArray.get t_form). + Local Notation get_atom := (PArray.get t_atom). + + + Fixpoint check_in x l := + match l with + | nil => false + | t::q => if x == t then true else check_in x q + end. + + + Lemma check_in_spec : forall x l, check_in x l = true <-> In x l. + Proof. + intro x; induction l as [ |t q IHq]; simpl. + split; intro H; try discriminate; elim H. + 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. + + + Fixpoint check_diseqs_complete_aux a dist t := + match dist with + | nil => true + | b::q => if PArray.existsb (fun (x:option (int*int)) => + match x with + | 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 + end. + + + 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 /\ + (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 (PArray.existsb (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 PArray.existsb_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, !Int63Properties.eqb_spec; intros [[H4 H5]|[H4 H5]]; subst a' b'; auto. + intro H1; case_eq (PArray.existsb (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 array_existsb_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. + + + 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 -> + (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 (PArray.existsb (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 array_existsb_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 (PArray.existsb (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 PArray.existsb_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. + + + Fixpoint check_diseqs_complete dist t := + match dist with + | nil => true + | a::q => if check_diseqs_complete_aux a q t then check_diseqs_complete q t else false + end. + + + 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 /\ + (t.[i] = Some (x,y) \/ t.[i] = Some (y,x)). + Proof. + intros dist t; induction dist as [ |a q IHq]; simpl; split; auto. + intros _ x y H; inversion H. + case_eq (check_diseqs_complete_aux a q t); try discriminate; rewrite check_diseqs_complete_aux_spec, IHq; clear IHq; intros H1 H2 x y H3; inversion H3; auto. + intro H; case_eq (check_diseqs_complete_aux a q t). + rewrite IHq; clear IHq; intros _ x y H1; apply H; constructor 2; auto. + rewrite check_diseqs_complete_aux_false_spec; clear IHq; intros [y [H1 H2]]; destruct (H _ _ (In2_hd _ a _ _ H1)) as [i [H3 [H4|H4]]]; elim (H2 _ H3); rewrite H4; intros H5 H6; [elim H5|elim H6]; auto. + Qed. + + + 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 -> + (t.[i] <> Some (x,y) /\ t.[i] <> Some (y,x)). + Proof. + intros dist t; induction dist as [ |a q IHq]; simpl; split; try discriminate. + intros [x [y [H _]]]; inversion H. + case_eq (check_diseqs_complete_aux a q t). + rewrite IHq; clear IHq; intros _ [x [y [H2 H3]]]; inversion H2; clear H2; subst q; exists x; exists y; split; auto; constructor 2. + constructor 1; auto. + constructor 2; auto. + rewrite check_diseqs_complete_aux_false_spec; intros [y [H1 H2]] _; clear IHq; exists a; exists y; split; auto; constructor; auto. + intros [x [y [H1 H2]]]; case_eq (check_diseqs_complete_aux a q t); auto; rewrite IHq; clear IHq; inversion H1; clear H1. + subst x q; rewrite check_diseqs_complete_aux_spec; intro H; destruct (H _ H0) as [i [H1 H3]]; elim (H2 _ H1); intros H4 H5; destruct H3; [elim H4|elim H5]; auto. + subst k q; intros _; exists x; exists y; split; auto. + Qed. + + + Definition check_diseqs ty dist diseq := + let t := PArray.mapi (fun _ t => + if Lit.is_pos t then None else + match get_form (Lit.blit t) with + | 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 + Some (h1,h2) + else + None + | _ => None + end + | _ => None + end + ) diseq in + PArray.forallb (fun x => match x with | None => false | _ => true end) t && + check_diseqs_complete dist t. + + + Lemma check_diseqs_spec : forall A dist diseq, + check_diseqs A dist diseq = true <-> + ((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 /\ + let t := diseq.[i] in + ~ Lit.is_pos t /\ + exists a, get_form (Lit.blit t) = Fatom a /\ + x <> y /\ + (get_atom a = Atom.Abop (Atom.BO_eq A) x y \/ + get_atom a = Atom.Abop (Atom.BO_eq A) y x))). + Proof. + intros A dist diseq; unfold check_diseqs; rewrite andb_true_iff, PArray.forallb_spec, check_diseqs_complete_spec, length_mapi; split; intros [H1 H2]; split. + clear H2; intros i Hi; generalize (H1 _ Hi); rewrite get_mapi; 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. + clear H1; intros x y Hxy; destruct (H2 _ _ Hxy) as [i [H1 H4]]; clear H2; rewrite get_mapi 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]; 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]; intro Heq3; try (intros [H|H]; discriminate); 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. + clear H2; intros i Hi; rewrite get_mapi; 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_mapi; 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. + + + (* Definition check_diseqs ty dist diseq := *) + (* PArray.forallb (fun t => *) + (* if Lit.is_pos t then false else *) + (* match get_form (Lit.blit t) with *) + (* | 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) *) + (* | _ => false *) + (* end *) + (* | _ => false *) + (* end) diseq. *) + + + (* Lemma check_diseqs_spec : forall A dist 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 /\ *) + (* exists h1 h2, get_atom a = Atom.Abop (Atom.BO_eq A) h1 h2 /\ *) + (* (h1 <> h2) /\ ((In2 h1 h2 dist) \/ (In2 h2 h1 dist)). *) + (* 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. *) + (* Qed. *) + + + Definition check_distinct ha diseq := + match get_atom ha with + | Atom.Anop (Atom.NO_distinct ty) dist => + check_diseqs ty dist diseq + | _ => false + end. + + + Lemma check_distinct_spec : forall ha diseq, + check_distinct ha diseq = true <-> exists A dist, + get_atom ha = Atom.Anop (Atom.NO_distinct A) dist /\ + check_diseqs A dist diseq = true. + Proof. + intros ha diseq; unfold check_distinct; split. + case (get_atom ha); try discriminate; intros [A] l H; exists A; exists l; auto. + intros [A [dist [H1 H2]]]; rewrite H1; auto. + Qed. + + + Definition check_distinct_two_args f1 f2 := + 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'))) + | _, _ => false + end + | _, _ => false + end. + + + Lemma check_distinct_two_args_spec : forall f1 f2, + check_distinct_two_args f1 f2 = true <-> exists ha hb ty x y, + get_form f1 = Fatom ha /\ + get_form f2 = Fatom hb /\ + get_atom ha = Atom.Anop (Atom.NO_distinct ty) (x::y::nil) /\ + (get_atom hb = Atom.Abop (Atom.BO_eq ty) x y \/ + get_atom hb = Atom.Abop (Atom.BO_eq ty) y x). + Proof. + intros f1 f2; unfold check_distinct_two_args; split. + case (get_form f1); try discriminate; intro ha; case (get_form f2); try discriminate; intro hb; case_eq (get_atom ha); try discriminate; intros [A] [ |x [ |y [ |l]]] Heq1; try discriminate; case_eq (get_atom hb); try discriminate; intros [ | | | | | | |B] x' y' Heq2; try discriminate; rewrite !andb_true_iff, orb_true_iff, !andb_true_iff; change (Typ.eqb A B = true) with (is_true (Typ.eqb A B)); rewrite Typ.eqb_spec, !Int63Properties.eqb_spec; intros [H1 [[H2 H3]|[H2 H3]]]; subst B x' y'; exists ha, hb, A, x, y; auto. + intros [ha [hb [A [x [y [H1 [H2 [H3 [H4|H4]]]]]]]]]; rewrite H1, H2, H3, H4, Typ.eqb_refl, !eqb_refl; auto; rewrite orb_true_r; auto. + Qed. + + + Section Valid1. + + Variables (t_i : array typ_eqb) + (t_func : array (Atom.tval t_i)) + (ch_atom : Atom.check_atom t_atom) + (ch_form : Form.check_form t_form) + (wt_t_atom : Atom.wt t_i t_func t_atom). + + Local Notation interp_form_hatom := + (Atom.interp_form_hatom t_i t_func t_atom). + Local Notation rho := + (Form.interp_state_var interp_form_hatom t_form). + + Let wf_t_atom : Atom.wf t_atom. + Proof. destruct (Atom.check_atom_correct _ ch_atom); auto. Qed. + + Let default_t_atom : default t_atom = Atom.Acop Atom.CO_xH. + Proof. destruct (Atom.check_atom_correct _ ch_atom); auto. Qed. + + Lemma default_t_form : default t_form = Ftrue. + Proof. destruct (Form.check_form_correct interp_form_hatom _ ch_form) as [[H _] _]; auto. Qed. + + Lemma wf_t_form : wf t_form. + Proof. destruct (Form.check_form_correct interp_form_hatom _ ch_form) as [[_ H] _]; auto. Qed. + + Local Hint Immediate wf_t_atom default_t_atom default_t_form wf_t_form. + + Lemma interp_check_distinct : forall ha diseq, + check_distinct ha diseq = true -> + 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 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; 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; intros i 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); unfold Var.interp; rewrite Form.wf_interp_form; auto; rewrite H10; simpl; rewrite Atom.t_interp_wf; auto; rewrite H6; simpl; unfold Atom.apply_binop; unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite forallbi_spec in wt_t_atom; assert (H11: a < length t_atom). + case_eq (a < length t_atom); auto; 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; rewrite H4; [exists h2; exists h1|exists h1; exists h2]; auto. + 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); auto; unfold Lit.interp; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto); unfold Var.interp; rewrite Form.wf_interp_form; auto; rewrite H6; simpl; rewrite Atom.t_interp_wf; auto; 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; symmetry; change (is_true (Typ.i_eqb t_i A v1 v1)); rewrite Typ.i_eqb_spec; auto. + intros [a [H20 H21]]; assert (H4: ha < length t_atom). + case_eq (ha < length t_atom); auto; intro Heq; generalize H1; rewrite get_outofbound; auto; rewrite default_t_atom; discriminate. + unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite forallbi_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. + Qed. + + Lemma interp_check_distinct_two_args : forall f1 f2, + 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; intro Heq; generalize H1; rewrite get_outofbound; auto; rewrite default_t_form; discriminate); assert (H6: f2 < length t_form) by (case_eq (f2 < length t_form); auto; intro Heq; generalize H2; rewrite get_outofbound; auto; rewrite default_t_form; discriminate); rewrite !Form.t_interp_wf; auto; rewrite H1, H2; simpl; unfold Atom.interp_form_hatom, Atom.interp_hatom; rewrite !Atom.t_interp_wf; auto; rewrite H3, H4; simpl; unfold Atom.wt,is_true in wt_t_atom; rewrite forallbi_spec in wt_t_atom; assert (H7: hb < length t_atom) by (case_eq (hb < length t_atom); auto; intro Heq; generalize H4; rewrite get_outofbound; auto; 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; rewrite Typ.i_eqb_sym; auto. + Qed. + + + (* Lemma interp_check_distinct : forall ha diseq, *) + (* check_distinct ha diseq -> *) + (* 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; 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); unfold Var.interp; rewrite Form.wf_interp_form; auto; rewrite H5; simpl; rewrite Atom.t_interp_wf; auto; rewrite H6; simpl; unfold Atom.apply_binop; unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite forallbi_spec in wt_t_atom; assert (H10: a < length t_atom). *) + (* case_eq (a < length t_atom); auto; 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; rewrite H2; [exists h2; exists h1|exists h1; exists h2]; auto. *) + (* intros [a [H2 H3]] _; assert (H4: ha < length t_atom). *) + (* case_eq (ha < length t_atom); auto; intro Heq; generalize H1; rewrite get_outofbound; auto; rewrite default_t_atom; discriminate. *) + (* unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite forallbi_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. *) + (* Qed. *) + + End Valid1. + + + Section AUX. + + 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))). + + (* 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))). *) + + Definition check_form_aux a b := + match a, b with + | Fatom ha, Fand diseq => check_distinct ha diseq + | 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) && (forallbi (fun i l => check_lit l (a2.[i])) a1) + | For a1, For a2 => (length a1 == length a2) && (forallbi (fun i l => check_lit l (a2.[i])) a1) + | Fimp a1, Fimp a2 => (length a1 == length a2) && (forallbi (fun i l => check_lit l (a2.[i])) a1) + (* (length a1 == length a2) && (forallbi (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 *) *) + (* (* let b := check_lit l2 j2 in *) *) + (* (* let c := check_lit l1 j2 in *) *) + (* (* let d := check_lit l2 j1 in *) *) + (* (* let e := check_lit j1 l1 in *) *) + (* (* let f := check_lit j1 l2 in *) *) + (* (* negb (((negb a) && b && (negb c)) || (c && e && (negb f)) || (b && (negb e) && f) || (a && (negb b) && (negb d))) *) *) + | Fiff l1 l2, Fiff 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 *) + | Fite l1 l2 l3, Fite j1 j2 j3 => check_lit l1 j1 && check_lit l2 j2 && check_lit l3 j3 + (* check_lit l1 j1 && check_lit j1 l1 && check_lit l2 j2 && check_lit l3 j3 *) + | _, _ => false + end. + + Variables (t_i : array typ_eqb) + (t_func : array (Atom.tval t_i)) + (ch_atom : Atom.check_atom t_atom) + (ch_form : Form.check_form t_form) + (wt_t_atom : Atom.wt t_i t_func t_atom). + + Local Notation interp_form_hatom := + (Atom.interp_form_hatom t_i t_func t_atom). + Local Notation rho := + (Form.interp_state_var interp_form_hatom t_form). + + Hypothesis interp_check_var : forall x y, + check_var x y -> Var.interp rho x = Var.interp rho y. + + (* Hypothesis interp_check_var : forall x y, *) + (* check_var x y -> Var.interp rho x -> Var.interp rho y. *) + + (* Local Hint Resolve interp_check_var. *) + + Lemma interp_check_lit : forall l1 l2, + check_lit l1 l2 -> Lit.interp rho l1 = Lit.interp rho l2. + Proof. + unfold check_lit; intros l1 l2; unfold is_true; rewrite !orb_true_iff, !andb_true_iff; intros [[H1|[H1 H2]]|[H1 H2]]. + rewrite eqb_spec in H1; rewrite H1; auto. + rewrite Bool.eqb_true_iff in H1; unfold Lit.interp; rewrite H1, (interp_check_var _ _ H2); auto. + generalize H1; unfold Lit.interp; case (Lit.is_pos l1); case (Lit.is_pos l2); try discriminate; intros _; unfold Var.interp; rewrite (interp_check_distinct_two_args _ t_func ch_atom ch_form wt_t_atom _ _ H2); auto; case (rho (Lit.blit l2)); auto. + Qed. + + (* Lemma interp_check_lit : forall l1 l2, *) + (* check_lit l1 l2 -> Lit.interp rho l1 -> Lit.interp rho l2 = true. *) + (* Proof. *) + (* unfold check_lit; intros l1 l2; unfold is_true; rewrite !orb_true_iff, !andb_true_iff; intros [[H1|[[H1 H2] H3]]|[[H1 H2] H3]]. *) + (* rewrite Int63Properties.eqb_spec in H1; subst l1; auto. *) + (* unfold Lit.interp; rewrite H1, H2; apply interp_check_var; auto. *) + (* unfold Lit.interp; case_eq (Lit.is_pos l1); intro Heq; rewrite Heq in H1; try discriminate; clear Heq H1; case_eq (Lit.is_pos l2); intro Heq; rewrite Heq in H2; try discriminate; clear Heq H2; case_eq (Var.interp rho (Lit.blit l1)); try discriminate; intros H4 _; case_eq (Var.interp rho (Lit.blit l2)); auto; intro H5; rewrite (interp_check_var _ _ H3 H5) in H4; discriminate. *) + (* Qed. *) + + (* Local Hint Resolve interp_check_lit. *) + + Lemma interp_check_form_aux : forall a b, + check_form_aux a b -> + Form.interp interp_form_hatom t_form a = Form.interp interp_form_hatom t_form b. + Proof. + intros [a| | |i1 l1|a1|a1|a1|l1 l2|l1 l2|l1 l2 l3] [b| | |j1 m1|a2|a2|a2|j1 j2|j1 j2|j1 j2 j3]; simpl; try discriminate;auto. + (* Atom *) + unfold is_true; rewrite Int63Properties.eqb_spec; intro; subst a; auto. + (* Interesting case *) + apply interp_check_distinct; auto. + (* Double negation *) + unfold is_true; rewrite andb_true_iff, Int63Properties.eqb_spec; intros [H1 H2]; subst j1. rewrite (interp_check_lit _ _ H2). auto. + (* Conjunction *) + unfold is_true; rewrite andb_true_iff, eqb_spec, forallbi_spec; intros [H1 H2]; apply afold_left_eq; auto; intros i Hi; apply interp_check_lit; auto. + (* Disjunction *) + unfold is_true; rewrite andb_true_iff, eqb_spec, forallbi_spec; intros [H1 H2]; apply afold_left_eq; auto; intros i Hi; apply interp_check_lit; auto. + (* Implication *) + unfold is_true; rewrite andb_true_iff, eqb_spec, forallbi_spec; intros [H1 H2]; apply afold_right_eq; auto; intros i Hi; apply interp_check_lit; auto. + (* Xor *) + unfold is_true; rewrite andb_true_iff; intros [H1 H2]; rewrite (interp_check_lit _ _ H1), (interp_check_lit _ _ H2); auto. + (* Iff *) + unfold is_true; rewrite andb_true_iff; intros [H1 H2]; rewrite (interp_check_lit _ _ H1), (interp_check_lit _ _ H2); auto. + (* Ite *) + unfold is_true; rewrite !andb_true_iff; intros [[H1 H2] H3]; rewrite (interp_check_lit _ _ H1), (interp_check_lit _ _ H2), (interp_check_lit _ _ H3); auto. + Qed. + + (* Lemma interp_check_lit_equiv : forall l1 l2, *) + (* check_lit l1 l2 -> check_lit l2 l1 -> *) + (* Lit.interp rho l1 = Lit.interp rho l2. *) + (* Proof. *) + (* intros l1 l2 H1 H2; generalize (interp_check_lit _ _ H1) (interp_check_lit _ _ H2); case (Lit.interp rho l1); case (Lit.interp rho l2); auto; symmetry; auto. *) + (* Qed. *) + + (* Lemma interp_check_form_aux : forall a b, *) + (* check_form_aux a b -> *) + (* Form.interp interp_form_hatom t_form a -> Form.interp interp_form_hatom t_form b. *) + (* Proof. *) + (* intros [a| | |i1 l1|a1|a1|a1|l1 l2|l1 l2|l1 l2 l3] [b| | |j1 m1|a2|a2|a2|j1 j2|j1 j2|j1 j2 j3]; simpl; try discriminate;auto. *) + (* (* Atom *) *) + (* unfold is_true; rewrite Int63Properties.eqb_spec; intro; subst a; auto. *) + (* (* Interesting case *) *) + (* apply interp_check_distinct; auto. *) + (* (* Double negation *) *) + (* unfold is_true; rewrite andb_true_iff, Int63Properties.eqb_spec; intros [H1 H2]; subst j1; apply (fold_ind2 _ _ (fun x y => x = true -> y = true)). *) + (* apply interp_check_lit; auto. *) + (* intros a b; case a; try discriminate; intros H _; rewrite H; auto. *) + (* (* Conjunction *) *) + (* unfold is_true; rewrite andb_true_iff, Int63Properties.eqb_spec; intros [H1 H2]; rewrite forallbi_spec in H2; intro H3; assert (H4 := afold_left_andb_true_inv _ _ _ H3); clear H3; apply afold_left_andb_true; rewrite <- H1; intros i Hi; eapply interp_check_lit; eauto. *) + (* (* Disjunction *) *) + (* unfold is_true; rewrite andb_true_iff, Int63Properties.eqb_spec; intros [H1 H2]; rewrite forallbi_spec in H2; intro H3; assert (H4 := afold_left_orb_true_inv _ _ _ H3); clear H3; destruct H4 as [i [H3 H4]]; eapply afold_left_orb_true. *) + (* rewrite <- H1; eauto. *) + (* eapply interp_check_lit; eauto. *) + (* (* Implication *) *) + (* unfold is_true; rewrite andb_true_iff, Int63Properties.eqb_spec; intros [H1 H2]; rewrite forallbi_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. *) + (* 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; case_eq (Lit.interp rho (a2 .[ i])); auto; 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; omega. *) + (* generalize (H2 _ H7); rewrite H4; intro H8; rewrite (interp_check_lit _ _ H8 H6) in H5; auto. *) + (* right; case_eq (existsbi (fun i l => (i < length a2 - 1) && (negb (Lit.interp rho l))) a2). *) + (* rewrite existsbi_spec; intros [i [_ H5]]; rewrite andb_true_iff in H5; destruct H5 as [H5 H6]; left; exists i; split; auto; generalize H6; case (Lit.interp rho (a2 .[ i])); auto; discriminate. *) + (* rewrite existsbi_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; discriminate. *) + (* intros H5 _; apply (interp_check_lit _ _ H5); apply H4; auto. *) + (* (* Xor *) *) + (* unfold is_true; rewrite !andb_true_iff; intros [[[H1 H2] H3] H4]; rewrite (interp_check_lit_equiv _ _ H1 H2), (interp_check_lit_equiv _ _ H3 H4); auto. *) + (* (* Iff *) *) + (* unfold is_true; rewrite !andb_true_iff; intros [[[H1 H2] H3] H4]; rewrite (interp_check_lit_equiv _ _ H1 H2), (interp_check_lit_equiv _ _ H3 H4); auto. *) + (* (* Ite *) *) + (* unfold is_true; rewrite !andb_true_iff; intros [[[H1 H2] H3] H4]; rewrite (interp_check_lit_equiv _ _ H1 H2); case (Lit.interp rho j1); apply interp_check_lit; auto. *) + (* Qed. *) + + End AUX. + + Definition check_hform h1 h2 := + foldi_down_cont + (fun _ cont h1 h2 => (h1 == h2) || check_form_aux cont (get_form h1) (get_form h2)) + (PArray.length t_form) 0 (fun h1 h2 => false) h1 h2. + + Definition check_form := check_form_aux check_hform. + + Definition check_lit' := check_lit check_hform. + + Fixpoint check_distinct_elim input res := + match input with + | nil => nil + | l::q => if check_lit' l res then res::q else l::(check_distinct_elim q res) + end. + + + Section Valid. + + Variables (t_i : array typ_eqb) + (t_func : array (Atom.tval t_i)) + (ch_atom : Atom.check_atom t_atom) + (ch_form : Form.check_form t_form) + (wt_t_atom : Atom.wt t_i t_func t_atom). + + Local Notation interp_form_hatom := + (Atom.interp_form_hatom t_i t_func t_atom). + Local Notation rho := + (Form.interp_state_var interp_form_hatom t_form). + + + Let wf_rho : Valuation.wf rho. + Proof. destruct (Form.check_form_correct interp_form_hatom _ ch_form); auto. Qed. + + Let default_t_form : default t_form = Ftrue. + Proof. destruct (Form.check_form_correct interp_form_hatom _ ch_form) as [[H _] _]; auto. Qed. + + Let wf_t_form : wf t_form. + Proof. destruct (Form.check_form_correct interp_form_hatom _ ch_form) as [[_ H] _]; auto. Qed. + + Local Hint Immediate wf_rho default_t_form wf_t_form. + + + Lemma interp_check_hform : forall h1 h2, + check_hform h1 h2 -> Var.interp rho h1 = Var.interp rho h2. + Proof. + unfold check_hform; apply foldi_down_cont_ind; try discriminate. intros i cont _ _ Hrec h1 h2. unfold is_true; rewrite orb_true_iff; intros [H|H]. + rewrite Int63Properties.eqb_spec in H; rewrite H; auto. + unfold Var.interp; rewrite !wf_interp_form; auto; eapply interp_check_form_aux; eauto. + Qed. + + Local Hint Resolve interp_check_hform. + + + Lemma interp_check_form : forall a b, + check_form a b -> + Form.interp interp_form_hatom t_form a = Form.interp interp_form_hatom t_form b. + Proof. apply interp_check_form_aux, interp_check_hform; auto. Qed. + + + Lemma interp_check_lit' : forall l res, + check_lit' l res -> Lit.interp rho l = Lit.interp rho res. + Proof. apply interp_check_lit, interp_check_hform; auto. Qed. + + + Lemma valid_check_distinct_elim : + forall input, C.valid rho input -> + forall res, C.valid rho (check_distinct_elim input res). + Proof. + induction input as [ |l c IHc]; auto; simpl; unfold C.valid; simpl; rewrite orb_true_iff; intros [H|H] res. + case_eq (check_lit' l res); intro Heq; simpl. + rewrite <- (interp_check_lit' _ _ Heq), H; auto. + rewrite H; auto. + case (check_lit' l res). + simpl; rewrite H, orb_true_r; auto. + simpl; rewrite (IHc H), orb_true_r; auto. + Qed. + + End Valid. + +End Operators. |