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diff --git a/src/spl/Syntactic.v b/src/spl/Syntactic.v new file mode 100644 index 0000000..d7d2594 --- /dev/null +++ b/src/spl/Syntactic.v @@ -0,0 +1,531 @@ +(**************************************************************************) +(* *) +(* 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. + + +(* Flattening and small arithmetic simplifications *) + +Section CheckAtom. + + Import Atom. + + Variable t_i : PArray.array typ_eqb. + Variable t_func : PArray.array (tval t_i). + Variable t_atom : PArray.array atom. + + Local Notation get_atom := (PArray.get t_atom). + + Section AUX. + + Variable check_hatom : hatom -> hatom -> bool. + + Definition check_atom_aux a b := + match a, b with + | Acop o1, Acop o2 => cop_eqb o1 o2 + + (* Two ways to define a negative integer *) + | Auop UO_Zopp p1, Auop UO_Zneg q => + match get_atom p1 with + | Auop UO_Zpos p => check_hatom p q + | _ => false + end + | Auop UO_Zneg p, Auop UO_Zopp q1 => + match get_atom q1 with + | Auop UO_Zpos q => check_hatom p q + | _ => false + end + + | Auop o1 a, Auop o2 b => uop_eqb o1 o2 && check_hatom a b + | Abop o1 a1 a2, Abop o2 b1 b2 => + match o1, o2 with + | BO_Zplus, BO_Zplus + | BO_Zmult, BO_Zmult => (check_hatom a1 b1 && check_hatom a2 b2) || (check_hatom a1 b2 && check_hatom a2 b1) + | BO_Zminus, BO_Zminus + | BO_Zlt, BO_Zlt + | BO_Zle, BO_Zle + | BO_Zge, BO_Zge + | BO_Zgt, BO_Zgt => check_hatom a1 b1 && check_hatom a2 b2 + | BO_Zge, BO_Zle + | BO_Zle, BO_Zge + | BO_Zgt, BO_Zlt + | BO_Zlt, BO_Zgt => check_hatom a1 b2 && check_hatom a2 b1 + | BO_eq t1, BO_eq t2 => + Typ.eqb t1 t2 && + ((check_hatom a1 b1 && check_hatom a2 b2) || + (check_hatom a1 b2 && check_hatom a2 b1)) + | _, _ => false + end + | Anop o1 l1, Anop o2 l2 => + match o1, o2 with + | NO_distinct t1, NO_distinct t2 => Typ.eqb t1 t2 && list_beq check_hatom l1 l2 + end + | Aapp f1 aargs, Aapp f2 bargs =>(f1 == f2) && list_beq check_hatom aargs bargs + + | _, _ => false + end. + + + Hypothesis check_hatom_correct : forall h1 h2, check_hatom h1 h2 -> + interp_hatom t_i t_func t_atom h1 = interp_hatom t_i t_func t_atom h2. + Hypothesis Hwf: wf t_atom. + Hypothesis Hd: default t_atom = Acop CO_xH. + + + Lemma list_beq_correct : forall l1 l2, + list_beq check_hatom l1 l2 = true -> + List.map (interp_hatom t_i t_func t_atom) l1 = + List.map (interp_hatom t_i t_func t_atom) l2. + Proof. + induction l1 as [ |h1 l1 IHl1]; intros [ |h2 l2]; simpl; try discriminate; auto; rewrite andb_true_iff; intros [H1 H2]; rewrite (IHl1 _ H2); rewrite (check_hatom_correct _ _ H1); auto. + Qed. + + + Lemma list_beq_compute_interp : forall t l1 l2, + list_beq check_hatom l1 l2 = true -> forall acc, + compute_interp t_i (interp_hatom t_i t_func t_atom) t acc l1 = + compute_interp t_i (interp_hatom t_i t_func t_atom) t acc l2. + Proof. + intro t; induction l1 as [ |h1 l1 IHl1]; intros [ |h2 l2]; simpl; try discriminate; auto; rewrite andb_true_iff; intros [H1 H2] acc; rewrite (check_hatom_correct _ _ H1); destruct (interp_hatom t_i t_func t_atom h2) as [ta va]; destruct (Typ.cast ta t) as [ka| ]; auto. + Qed. + + + Lemma check_atom_aux_correct : forall a1 a2, check_atom_aux a1 a2 -> + interp t_i t_func t_atom a1 = interp t_i t_func t_atom a2. + Proof. + intros [op1|op1 i1|op1 i1 j1|op1 li1|f1 args1]; simpl. + (* Constants *) + intros [op2|op2 i2|op2 i2 j2|op2 li2|f2 args2]; simpl; try discriminate; pose (H:=reflect_cop_eqb op1 op2); inversion H; try discriminate; subst op1; auto. + (* Unary operators *) + intros [op2|op2 i2|op2 i2 j2|op2 li2|f2 args2]; simpl; try discriminate; try (case op1; discriminate). + case op1; case op2; try discriminate; try (unfold is_true; rewrite andb_true_iff; intros [_ H]; rewrite (check_hatom_correct _ _ H); auto). + case_eq (get_atom i2); try discriminate; intros [ | | | | ] i Heq H; try discriminate; simpl; unfold apply_unop; rewrite (check_hatom_correct _ _ H); unfold interp_hatom; rewrite (t_interp_wf _ _ _ Hwf Hd i2), Heq; simpl; unfold apply_unop; destruct (t_interp t_i t_func t_atom .[ i]) as [A v]; destruct (Typ.cast A Typ.Tpositive) as [k| ]; auto. + case_eq (get_atom i1); try discriminate; intros [ | | | | ] i Heq H; try discriminate; simpl; unfold apply_unop; rewrite <- (check_hatom_correct _ _ H); unfold interp_hatom; rewrite (t_interp_wf _ _ _ Hwf Hd i1), Heq; simpl; unfold apply_unop; destruct (t_interp t_i t_func t_atom .[ i]) as [A v]; destruct (Typ.cast A Typ.Tpositive) as [k| ]; auto. + (* Binary operators *) + intros [op2|op2 i2|op2 i2 j2|op2 li2|f2 args2]; simpl; try discriminate; case op1; case op2; try discriminate; try (unfold is_true; rewrite andb_true_iff; intros [H1 H2]; rewrite (check_hatom_correct _ _ H1), (check_hatom_correct _ _ H2); auto). + unfold is_true, interp_bop, apply_binop. rewrite orb_true_iff, !andb_true_iff. intros [[H1 H2]|[H1 H2]]; rewrite (check_hatom_correct _ _ H1), (check_hatom_correct _ _ H2); destruct (interp_hatom t_i t_func t_atom i2) as [A v1]; destruct (interp_hatom t_i t_func t_atom j2) as [B v2]; destruct (Typ.cast B Typ.TZ) as [k2| ]; destruct (Typ.cast A Typ.TZ) as [k1| ]; auto; rewrite Z.add_comm; reflexivity. + unfold is_true, interp_bop, apply_binop. rewrite orb_true_iff, !andb_true_iff. intros [[H1 H2]|[H1 H2]]; rewrite (check_hatom_correct _ _ H1), (check_hatom_correct _ _ H2); destruct (interp_hatom t_i t_func t_atom i2) as [A v1]; destruct (interp_hatom t_i t_func t_atom j2) as [B v2]; destruct (Typ.cast B Typ.TZ) as [k2| ]; destruct (Typ.cast A Typ.TZ) as [k1| ]; auto; rewrite Z.mul_comm; reflexivity. + unfold interp_bop, apply_binop; destruct (interp_hatom t_i t_func t_atom j2) as [B v2]; destruct (interp_hatom t_i t_func t_atom i2) as [A v1]; destruct (Typ.cast B Typ.TZ) as [k2| ]; destruct (Typ.cast A Typ.TZ) as [k1| ]; auto; rewrite Z.gtb_ltb; auto. + unfold interp_bop, apply_binop; destruct (interp_hatom t_i t_func t_atom j2) as [B v2]; destruct (interp_hatom t_i t_func t_atom i2) as [A v1]; destruct (Typ.cast B Typ.TZ) as [k2| ]; destruct (Typ.cast A Typ.TZ) as [k1| ]; auto; rewrite Z.geb_leb; auto. + unfold interp_bop, apply_binop; destruct (interp_hatom t_i t_func t_atom j2) as [B v2]; destruct (interp_hatom t_i t_func t_atom i2) as [A v1]; destruct (Typ.cast B Typ.TZ) as [k2| ]; destruct (Typ.cast A Typ.TZ) as [k1| ]; auto; rewrite Z.geb_leb; auto. + unfold interp_bop, apply_binop; destruct (interp_hatom t_i t_func t_atom j2) as [B v2]; destruct (interp_hatom t_i t_func t_atom i2) as [A v1]; destruct (Typ.cast B Typ.TZ) as [k2| ]; destruct (Typ.cast A Typ.TZ) as [k1| ]; auto; rewrite Z.gtb_ltb; auto. + intros A B; unfold is_true; rewrite andb_true_iff, orb_true_iff; change (Typ.eqb B A = true) with (is_true (Typ.eqb B A)); rewrite Typ.eqb_spec; intros [H2 [H1|H1]]; subst B; rewrite andb_true_iff in H1; destruct H1 as [H1 H2]; rewrite (check_hatom_correct _ _ H1), (check_hatom_correct _ _ H2); auto; simpl; unfold apply_binop; destruct (interp_hatom t_i t_func t_atom j2) as [B v1]; destruct (interp_hatom t_i t_func t_atom i2) as [C v2]; destruct (Typ.cast B A) as [k1| ]; destruct (Typ.cast C A) as [k2| ]; auto; rewrite Typ.i_eqb_sym; auto. + (* N-ary operators *) + intros [op2|op2 i2|op2 i2 j2|op2 li2|f2 args2]; simpl; try discriminate; destruct op1 as [t1]; destruct op2 as [t2]; unfold is_true; rewrite andb_true_iff; change (Typ.eqb t1 t2 = true) with (is_true (Typ.eqb t1 t2)); rewrite Typ.eqb_spec; intros [H1 H2]; subst t2; rewrite (list_beq_compute_interp _ _ _ H2); auto. + (* Application *) + intros [op2|op2 i2|op2 i2 j2|op2 li2|f2 args2]; simpl; try discriminate; unfold is_true; rewrite andb_true_iff, Int63Properties.eqb_spec; intros [H2 H1]; subst f2; rewrite (list_beq_correct _ _ H1); auto. + Qed. + + End AUX. + + Definition check_hatom h1 h2 := + foldi_down_cont + (fun _ cont h1 h2 => (h1 == h2) || check_atom_aux cont (t_atom.[h1]) (t_atom.[h2])) + (PArray.length t_atom) 0 (fun h1 h2 => false) h1 h2. + + Definition check_atom := check_atom_aux check_hatom. + + Definition check_neg_hatom h1 h2 := + match get_atom h1, get_atom h2 with + | Abop op1 a1 a2, Abop op2 b1 b2 => + match op1, op2 with + | BO_Zlt, BO_Zle => check_hatom a1 b2 && check_hatom a2 b1 + | BO_Zlt, BO_Zge => check_hatom a1 b1 && check_hatom a2 b2 + | BO_Zle, BO_Zlt => check_hatom a1 b2 && check_hatom a2 b1 + | BO_Zle, BO_Zgt => check_hatom a1 b1 && check_hatom a2 b2 + | BO_Zge, BO_Zlt => check_hatom a1 b1 && check_hatom a2 b2 + | BO_Zge, BO_Zgt => check_hatom a1 b2 && check_hatom a2 b1 + | BO_Zgt, BO_Zle => check_hatom a1 b1 && check_hatom a2 b2 + | BO_Zgt, BO_Zge => check_hatom a1 b2 && check_hatom a2 b1 + | _, _ => false + end + | _, _ => false + end. + + (* TODO : move this *) + Lemma Zge_is_ge_bool : forall x y, (x >= y) <-> (Zge_bool x y = true). + Proof. + intros x y;assert (W:=Zge_cases x y);destruct (Zge_bool x y). + split;auto. + split;[intros;elimtype false;auto with zarith | discriminate]. + Qed. + + + (* Correctness of check_atom *) + + Lemma check_hatom_correct : wf t_atom -> + default t_atom = Acop CO_xH -> + forall h1 h2, check_hatom h1 h2 -> + interp_hatom t_i t_func t_atom h1 = interp_hatom t_i t_func t_atom h2. + Proof. + unfold check_hatom;intros Hwf Hdef. + 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; reflexivity. + unfold interp_hatom;rewrite !t_interp_wf;trivial. + apply check_atom_aux_correct with cont;trivial. + Qed. + + + Lemma check_atom_correct : wf t_atom -> + default t_atom = Acop CO_xH -> + forall a1 a2, check_atom a1 a2 -> + interp t_i t_func t_atom a1 = interp t_i t_func t_atom a2. + Proof. + intros Hwf Hdef;unfold check_atom;apply check_atom_aux_correct; auto. + apply check_hatom_correct;trivial. + Qed. + + + Lemma check_hatom_correct_bool : wf t_atom -> + default t_atom = Acop CO_xH -> + forall h1 h2, check_hatom h1 h2 -> + interp_form_hatom t_i t_func t_atom h1 = interp_form_hatom t_i t_func t_atom h2. + Proof. + unfold interp_form_hatom; intros H1 H2 h1 h2 H3; rewrite (check_hatom_correct H1 H2 h1 h2 H3); auto. + Qed. + + + (* Correctness of check_neg_atom *) + + Lemma check_neg_hatom_correct : wt t_i t_func t_atom -> + wf t_atom -> default t_atom = Acop CO_xH -> + forall h1 h2, check_neg_hatom h1 h2 -> + match interp_hatom t_i t_func t_atom h1, interp_hatom t_i t_func t_atom h2 with + | Val Typ.Tbool v1, Val Typ.Tbool v2 => v1 = negb v2 + | Val _ _, Val _ _ => False + end. + Proof. + unfold wt; unfold is_true at 1; rewrite forallbi_spec; intros Hwt Hwf Hdef h1 h2; unfold check_neg_hatom; case_eq (get_atom h1); try discriminate; intros b1 t11 t12 H1; case_eq (get_atom h2); try discriminate; intros b2 t21 t22 H2; assert (H7: h1 < length t_atom) by (apply PArray.get_not_default_lt; rewrite H1, Hdef; discriminate); generalize (Hwt _ H7); rewrite H1; simpl; generalize H1; case b1; try discriminate; clear H1 b1; simpl; intro H1; case (get_type' t_i (t_interp t_i t_func t_atom) h1); try discriminate; simpl; rewrite andb_true_iff; intros [H30 H31]; change (is_true (Typ.eqb (get_type' t_i (t_interp t_i t_func t_atom) t11) Typ.TZ)) in H30; change (is_true (Typ.eqb (get_type' t_i (t_interp t_i t_func t_atom) t12) Typ.TZ)) in H31; rewrite Typ.eqb_spec in H30, H31; generalize (check_aux_interp_hatom _ t_func _ Hwf t11), (check_aux_interp_hatom _ t_func _ Hwf t12); rewrite H30, H31; intros [v1 Hv1] [v2 Hv2]; generalize H2; case b2; try discriminate; clear H2 b2; intro H2; unfold is_true; rewrite andb_true_iff; intros [H3 H4]; generalize (check_hatom_correct Hwf Hdef _ _ H3), (check_hatom_correct Hwf Hdef _ _ H4); unfold interp_hatom; intros H5 H6; rewrite t_interp_wf; auto; rewrite H1; simpl; rewrite Hv1, Hv2; simpl; rewrite t_interp_wf; auto; rewrite H2; simpl; rewrite <- H5; rewrite <- H6, Hv1, Hv2; simpl. + rewrite Z.ltb_antisym; auto. + rewrite Z.geb_leb, Z.ltb_antisym; auto. + rewrite Z.leb_antisym; auto. + rewrite Z.gtb_ltb, Z.leb_antisym; auto. + rewrite Z.geb_leb, Z.leb_antisym; auto. + rewrite Z.geb_leb, Z.gtb_ltb, Z.leb_antisym; auto. + rewrite Z.gtb_ltb, Z.ltb_antisym; auto. + rewrite Z.geb_leb, Z.gtb_ltb, Z.ltb_antisym; auto. + Qed. + + + Lemma check_neg_hatom_correct_bool : wt t_i t_func t_atom -> + wf t_atom -> default t_atom = Acop CO_xH -> + forall h1 h2, check_neg_hatom h1 h2 -> + interp_form_hatom t_i t_func t_atom h1 = negb (interp_form_hatom t_i t_func t_atom h2). + Proof. + unfold interp_form_hatom. intros Hwt H1 H2 h1 h2 H3. unfold interp_bool. generalize (check_neg_hatom_correct Hwt H1 H2 _ _ H3). case (interp_hatom t_i t_func t_atom h1). case (interp_hatom t_i t_func t_atom h2). simpl. intros [i| | | ] v1 [j| | | ] v2; intro H; inversion H. rewrite Typ.cast_refl. auto. + Qed. + +End CheckAtom. + + +(* Flattening *) + +Section FLATTEN. + + Import Form. + + Variable t_form : PArray.array form. + + Local Notation get_form := (PArray.get t_form). + + Definition remove_not l := + match get_form (Lit.blit l) with + | Fnot2 _ l' => if Lit.is_pos l then l' else Lit.neg l' + | _ => l + end. + + Definition get_and l := + let l := remove_not l in + if Lit.is_pos l then + match get_form (Lit.blit l) with + | Fand args => Some args + | _ => None + end + else None. + + Definition get_or l := + let l := remove_not l in + if Lit.is_pos l then + match get_form (Lit.blit l) with + | For args => Some args + | _ => None + end + else None. + + Definition flatten_op_body (get_op:_lit -> option (array _lit)) + (frec : list _lit -> _lit -> list _lit) + (largs:list _lit) (l:_lit) : list _lit := + match get_op l with + | Some a => PArray.fold_left frec largs a + | None => l::largs + end. + Register flatten_op_body as PrimInline. + + + Definition flatten_op_lit (get_op:_lit -> option (array _lit)) max := + foldi_cont (fun _ => flatten_op_body get_op) 0 max (fun largs l => l::largs). + + Definition flatten_and t := + PArray.fold_left (flatten_op_lit get_and (PArray.length t_form)) nil t. + + Definition flatten_or t := + PArray.fold_left (flatten_op_lit get_or (PArray.length t_form)) nil t. + + + Variable check_atom check_neg_atom : atom -> atom -> bool. + + Definition check_flatten_body frec (l lf:_lit) := + let l := remove_not l in + let lf := remove_not lf in + if l == lf then true + else if 1 land (l lxor lf) == 0 then + match get_form (Lit.blit l), get_form (Lit.blit lf) with + | Fatom a1, Fatom a2 => check_atom a1 a2 + | Ftrue, Ftrue => true + | Ffalse, Ffalse => true + | Fand args1, Fand args2 => + let args1 := flatten_and args1 in + let args2 := flatten_and args2 in + forallb2 frec args1 args2 + | For args1, For args2 => + let args1 := flatten_or args1 in + let args2 := flatten_or args2 in + forallb2 frec args1 args2 + | Fxor l1 l2, Fxor lf1 lf2 => + frec l1 lf1 && frec l2 lf2 + | Fimp args1, Fimp args2 => + if PArray.length args1 == PArray.length args2 then + PArray.forallbi (fun i l => frec l (args2.[i])) args1 + else false + | Fiff l1 l2, Fiff lf1 lf2 => + frec l1 lf1 && frec l2 lf2 + | Fite l1 l2 l3, Fite lf1 lf2 lf3 => + frec l1 lf1 && frec l2 lf2 && frec l3 lf3 + | _, _ => false + end + else + match get_form (Lit.blit l), get_form (Lit.blit lf) with + | Fatom a1, Fatom a2 => check_neg_atom a1 a2 + | _, _ => false (* We maybe need to extend the rule here ... *) + end. + Register check_flatten_body as PrimInline. + + Definition check_flatten_aux l lf := + foldi_cont (fun _ => check_flatten_body) 0 (PArray.length t_form) (fun _ _ => false) l lf. + + Definition check_flatten s cid lf := + match S.get s cid with + | l :: nil => + if check_flatten_aux l lf then lf::nil else C._true + | _ => C._true + end. + + + (** Correctness proofs *) + Variable interp_atom : atom -> bool. + Hypothesis default_thf : default t_form = Ftrue. + Hypothesis wf_thf : wf t_form. + Hypothesis check_atom_correct : + forall a1 a2, check_atom a1 a2 -> interp_atom a1 = interp_atom a2. + Hypothesis check_neg_atom_correct : + forall a1 a2, check_neg_atom a1 a2 -> interp_atom a1 = negb (interp_atom a2). + + Local Notation interp_var := (interp_state_var interp_atom t_form). + Local Notation interp_lit := (Lit.interp interp_var). + + Lemma interp_Fnot2 : forall i l, interp interp_atom t_form (Fnot2 i l) = interp_lit l. + Proof. + intros i l;simpl;apply fold_ind;trivial. + intros a;rewrite negb_involutive;trivial. + Qed. + + Lemma remove_not_correct : + forall l, interp_lit (remove_not l) = interp_lit l. + Proof. + unfold remove_not;intros l. + case_eq (get_form (Lit.blit l));intros;trivial. + unfold Lit.interp, Var.interp. + rewrite (wf_interp_form interp_atom t_form default_thf wf_thf (Lit.blit l)), H, interp_Fnot2. + destruct(Lit.is_pos l);trivial. + rewrite Lit.is_pos_neg, Lit.blit_neg;unfold Lit.interp;destruct (Lit.is_pos i0);trivial. + rewrite negb_involutive;trivial. + Qed. + + Lemma get_and_correct : forall l args, get_and l = Some args -> + interp_lit l = interp interp_atom t_form (Fand args). + Proof. + unfold get_and;intros l args. + rewrite <- remove_not_correct;unfold Lit.interp;generalize (remove_not l). + intros l';unfold Var.interp. + destruct (Lit.is_pos l');[ | discriminate]. + rewrite wf_interp_form;trivial. + destruct (get_form (Lit.blit l'));intros Heq;inversion Heq;trivial. + Qed. + + Lemma get_or_correct : forall l args, get_or l = Some args -> + interp_lit l = interp interp_atom t_form (For args). + Proof. + unfold get_or;intros l args. + rewrite <- remove_not_correct;unfold Lit.interp;generalize (remove_not l). + intros l';unfold Var.interp. + destruct (Lit.is_pos l');[ | discriminate]. + rewrite wf_interp_form;trivial. + destruct (get_form (Lit.blit l'));intros Heq;inversion Heq;trivial. + Qed. + + Lemma flatten_and_correct : forall args, + List.fold_right (fun l res => andb res (interp_lit l)) true (flatten_and args) = + afold_left _ _ true andb interp_lit args. + Proof. + intros;rewrite afold_left_spec;auto;unfold flatten_and. + set (t:= true);unfold t at 2; + change true with + (List.fold_right (fun (l : int) (res : bool) => res && interp_lit l) true nil). + unfold t;clear t. + rewrite !fold_left_to_list. + generalize (@nil int);induction (to_list args);simpl;trivial. + intros l0;rewrite IHl. + clear IHl;f_equal; unfold flatten_op_lit. + clear l;revert a l0;apply foldi_cont_ind;simpl;trivial. + intros i cont _ Hle Hrec a l;unfold flatten_op_body. + case_eq (get_and a);intros;trivial. + rewrite get_and_correct with (1:= H);simpl. + rewrite afold_left_spec; auto; rewrite !fold_left_to_list. + rewrite <- !fold_left_rev_right. + clear H a;revert l;induction (List.rev (to_list a0));simpl. + intros l;rewrite andb_true_r;trivial. + intros;rewrite Hrec, IHl, andb_assoc;trivial. + Qed. + + Lemma flatten_or_correct : forall args, + List.fold_right (fun l res => orb res (interp_lit l)) false (flatten_or args) = + afold_left _ _ false orb interp_lit args. + Proof. + intros;rewrite afold_left_spec;auto;unfold flatten_or. + set (t:= false);unfold t at 2; + change false with + (List.fold_right (fun (l : int) (res : bool) => res || interp_lit l) false nil). + unfold t;clear t. + rewrite !fold_left_to_list. + generalize (@nil int);induction (to_list args);simpl;trivial. + intros l0;rewrite IHl. + clear IHl;f_equal; unfold flatten_op_lit. + clear l;revert a l0;apply foldi_cont_ind;simpl;trivial. + intros i cont _ Hle Hrec a l;unfold flatten_op_body. + case_eq (get_or a);intros;trivial. + rewrite get_or_correct with (1:= H);simpl. + rewrite afold_left_spec; auto; rewrite !fold_left_to_list. + rewrite <- !fold_left_rev_right. + clear H a;revert l;induction (List.rev (to_list a0));simpl. + intros l;rewrite orb_false_r;trivial. + intros;rewrite Hrec, IHl, orb_assoc;trivial. + Qed. + + Lemma check_flatten_aux_correct : forall l lf, + check_flatten_aux l lf = true -> + interp_lit l = interp_lit lf. + Proof. + unfold check_flatten_aux. + apply foldi_cont_ind. + discriminate. + intros i cont _ Hle Hrec l lf;unfold check_flatten_body. + rewrite <- (remove_not_correct l), <- (remove_not_correct lf). + generalize (remove_not l) (remove_not lf);clear l lf;intros l lf. + destruct (reflect_eqb l lf);[ intros;subst;trivial | ]. + destruct (reflect_eqb (1 land (l lxor lf)) 0). + unfold Lit.interp. + assert (Lit.is_pos l = Lit.is_pos lf). + unfold Lit.is_pos. + rewrite <- eqb_spec, land_comm in e. + change (is_true (is_even (l lxor lf))) in e. + rewrite is_even_xor in e. + destruct (is_even l);destruct (is_even lf);trivial;discriminate. + rewrite H;match goal with + |- ?P -> _ => + assert (W:P -> Var.interp interp_var (Lit.blit l) = Var.interp interp_var (Lit.blit lf)); + [ | intros;rewrite W;trivial] + end. + unfold Var.interp;rewrite !wf_interp_form;trivial. + clear e n H. + destruct (get_form (Lit.blit l)); + destruct (get_form (Lit.blit lf));intros;try discriminate;simpl;trivial. + (* atom *) + apply check_atom_correct;trivial. + (* and *) + rewrite <- !flatten_and_correct. + revert H;generalize (flatten_and a) (flatten_and a0);clear a a0. + induction l0;intros l1;destruct l1;simpl;trivial;try discriminate. + rewrite andb_true_iff;intros (H1, H2). + rewrite (Hrec _ _ H1), (IHl0 _ H2);trivial. + (* or *) + rewrite <- !flatten_or_correct. + revert H;generalize (flatten_or a) (flatten_or a0);clear a a0. + induction l0;intros l1;destruct l1;simpl;trivial;try discriminate. + rewrite andb_true_iff;intros (H1, H2). + rewrite (Hrec _ _ H1), (IHl0 _ H2);trivial. + (* implb *) + revert H;destruct (reflect_eqb (length a) (length a0));[intros|discriminate]. + apply afold_right_eq;trivial. + rewrite forallbi_spec in H;auto. + (* xorb *) + unfold is_true in H;rewrite andb_true_iff in H;destruct H as [H H0]. + rewrite (Hrec _ _ H), (Hrec _ _ H0);trivial. + (* eqb (i.e iff) *) + unfold is_true in H;rewrite andb_true_iff in H;destruct H as [H H0]. + rewrite (Hrec _ _ H), (Hrec _ _ H0);trivial. + (* ifb *) + unfold is_true in H;rewrite !andb_true_iff in H;destruct H as [[H H0] H1]. + rewrite (Hrec _ _ H), (Hrec _ _ H0), (Hrec _ _ H1);trivial. + (** opposite sign *) + assert (Lit.is_pos l = negb (Lit.is_pos lf)). + unfold Lit.is_pos. + rewrite <- eqb_spec, land_comm in n0. + change (~is_true (is_even (l lxor lf))) in n0. + rewrite is_even_xor in n0. + destruct (is_even l);destruct (is_even lf);trivial;elim n0;reflexivity. + unfold Lit.interp;rewrite H. match goal with + |- ?P -> _ => + assert (W:P -> Var.interp interp_var (Lit.blit l) = negb (Var.interp interp_var (Lit.blit lf))); + [ | intros;rewrite W;trivial] + end. + unfold Var.interp;rewrite !wf_interp_form;trivial. + destruct (get_form (Lit.blit l));try discriminate. + destruct (get_form (Lit.blit lf));try discriminate. + apply check_neg_atom_correct. + rewrite negb_involutive;destruct (Lit.is_pos lf);trivial. + Qed. + + Hypothesis Hwf: Valuation.wf interp_var. + + Lemma valid_check_flatten : forall s, S.valid interp_var s -> + forall cid lf, C.valid interp_var (check_flatten s cid lf). + Proof. + unfold check_flatten; intros s Hs cid lf; case_eq (S.get s cid). + intros; apply C.interp_true; auto. + intros i [ |l q] Heq; try apply C.interp_true; auto; case_eq (check_flatten_aux i lf); intro Heq2; try apply C.interp_true; auto; unfold C.valid; simpl; rewrite <- (check_flatten_aux_correct _ _ Heq2); unfold S.valid in Hs; generalize (Hs cid); rewrite Heq; auto. + Qed. + +End FLATTEN. |