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diff --git a/src/hls/GiblePargenproofEquiv.v b/src/hls/GiblePargenproofEquiv.v new file mode 100644 index 0000000..5f28d53 --- /dev/null +++ b/src/hls/GiblePargenproofEquiv.v @@ -0,0 +1,1736 @@ +(* + * Vericert: Verified high-level synthesis. + * Copyright (C) 2023 Yann Herklotz <git@yannherklotz.com> + * + * This program is free software: you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation, either version 3 of the License, or + * (at your option) any later version. + * + * This program is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program. If not, see <https://www.gnu.org/licenses/>. + *) + +Require Import Coq.Logic.Decidable. +Require Import Coq.Structures.Equalities. + +Require Import compcert.backend.Registers. +Require Import compcert.common.AST. +Require Import compcert.common.Globalenvs. +Require Import compcert.common.Memory. +Require Import compcert.common.Values. +Require Import compcert.lib.Floats. +Require Import compcert.lib.Integers. +Require Import compcert.lib.Maps. +Require compcert.verilog.Op. + +Require Import vericert.common.Vericertlib. +Require Import vericert.hls.GibleSeq. +Require Import vericert.hls.GiblePar. +Require Import vericert.hls.Gible. +Require Import vericert.hls.HashTree. +Require Import vericert.hls.Predicate. +Require Import vericert.common.DecEq. +Require Import vericert.hls.Abstr. +Require vericert.common.NonEmpty. +Module NE := NonEmpty. +Import NE.NonEmptyNotation. + +#[local] Open Scope non_empty_scope. +#[local] Open Scope positive. +#[local] Open Scope pred_op. +#[local] Open Scope forest. + +Fixpoint beq_expression (e1 e2: expression) {struct e1}: bool := + match e1, e2 with + | Ebase r1, Ebase r2 => if resource_eq r1 r2 then true else false + | Eop op1 el1, Eop op2 el2 => + if operation_eq op1 op2 then + beq_expression_list el1 el2 else false + | Eload chk1 addr1 el1 e1, Eload chk2 addr2 el2 e2 => + if memory_chunk_eq chk1 chk2 + then if addressing_eq addr1 addr2 + then if beq_expression_list el1 el2 + then beq_expression e1 e2 else false else false else false + | Estore e1 chk1 addr1 el1 m1, Estore e2 chk2 addr2 el2 m2 => + if memory_chunk_eq chk1 chk2 + then if addressing_eq addr1 addr2 + then if beq_expression_list el1 el2 + then if beq_expression m1 m2 + then beq_expression e1 e2 else false else false else false else false + | _, _ => false + end +with beq_expression_list (el1 el2: expression_list) {struct el1} : bool := + match el1, el2 with + | Enil, Enil => true + | Econs e1 t1, Econs e2 t2 => beq_expression e1 e2 && beq_expression_list t1 t2 + | _, _ => false + end +. + +Scheme expression_ind2 := Induction for expression Sort Prop + with expression_list_ind2 := Induction for expression_list Sort Prop. +Definition beq_pred_expression (e1 e2: pred_expression) : bool := + match e1, e2 with + | PEbase p1, PEbase p2 => if peq p1 p2 then true else false + | PEsetpred c1 el1, PEsetpred c2 el2 => + if condition_eq c1 c2 + then beq_expression_list el1 el2 else false + | _, _ => false + end. + +Definition beq_exit_expression (e1 e2: exit_expression) : bool := + match e1, e2 with + | EEbase, EEbase => true + | EEexit cf1, EEexit cf2 => if cf_instr_eq cf1 cf2 then true else false + | _, _ => false + end. + +Lemma beq_expression_correct: + forall e1 e2, beq_expression e1 e2 = true -> e1 = e2. +Proof. + intro e1; + apply expression_ind2 with + (P := fun (e1 : expression) => + forall e2, beq_expression e1 e2 = true -> e1 = e2) + (P0 := fun (e1 : expression_list) => + forall e2, beq_expression_list e1 e2 = true -> e1 = e2); simplify; + try solve [repeat match goal with + | [ H : context[match ?x with _ => _ end] |- _ ] => destruct x eqn:? + | [ H : context[if ?x then _ else _] |- _ ] => destruct x eqn:? + end; subst; f_equal; crush; eauto using Peqb_true_eq]. +Qed. + +Lemma beq_expression_refl: forall e, beq_expression e e = true. +Proof. + intros. + induction e using expression_ind2 with (P0 := fun el => beq_expression_list el el = true); + crush; repeat (destruct_match; crush); []. + crush. rewrite IHe. rewrite IHe0. auto. +Qed. + +Lemma beq_expression_list_refl: forall e, beq_expression_list e e = true. +Proof. induction e; auto. simplify. rewrite beq_expression_refl. auto. Qed. + +Lemma beq_expression_correct2: + forall e1 e2, beq_expression e1 e2 = false -> e1 <> e2. +Proof. + induction e1 using expression_ind2 + with (P0 := fun el1 => forall el2, beq_expression_list el1 el2 = false -> el1 <> el2). + - intros. simplify. repeat (destruct_match; crush). + - intros. simplify. repeat (destruct_match; crush). subst. apply IHe1 in H. + unfold not in *. intros. apply H. inv H0. auto. + - intros. simplify. repeat (destruct_match; crush); subst. + unfold not in *; intros. inv H0. rewrite beq_expression_refl in H. discriminate. + unfold not in *; intros. inv H. rewrite beq_expression_list_refl in Heqb. discriminate. + - simplify. repeat (destruct_match; crush); subst; + unfold not in *; intros. + inv H0. rewrite beq_expression_refl in H; crush. + inv H. rewrite beq_expression_refl in Heqb0; crush. + inv H. rewrite beq_expression_list_refl in Heqb; crush. + (* - simplify. repeat (destruct_match; crush); subst. *) + (* unfold not in *; intros. inv H0. rewrite beq_expression_list_refl in H; crush. *) + - simplify. repeat (destruct_match; crush); subst. + - simplify. repeat (destruct_match; crush); subst. + apply andb_false_iff in H. inv H. unfold not in *; intros. + inv H. rewrite beq_expression_refl in H0; discriminate. + unfold not in *; intros. inv H. rewrite beq_expression_list_refl in H0; discriminate. +Qed. + +Definition expression_dec: forall e1 e2: expression, {e1 = e2} + {e1 <> e2}. +Proof. + intros. + destruct (beq_expression e1 e2) eqn:?. apply beq_expression_correct in Heqb. auto. + apply beq_expression_correct2 in Heqb. auto. +Defined. + +Lemma beq_expression_list_correct: + forall e1 e2, beq_expression_list e1 e2 = true -> e1 = e2. +Proof. + induction e1; crush. + - destruct_match; crush. + - destruct_match; crush. + apply IHe1 in H1; subst. + apply beq_expression_correct in H0; subst. + trivial. +Qed. + +Lemma beq_expression_list_correct2: + forall e1 e2, beq_expression_list e1 e2 = false -> e1 <> e2. +Proof. + induction e1; crush. + - destruct_match; crush. + - destruct_match; crush. + apply andb_false_iff in H. inv H. apply beq_expression_correct2 in H0. + unfold not in *; intros. apply H0. inv H. auto. + apply IHe1 in H0. unfold not in *; intros. apply H0. inv H. + auto. +Qed. + +Lemma beq_pred_expression_correct: + forall e1 e2, beq_pred_expression e1 e2 = true -> e1 = e2. +Proof. + destruct e1, e2; crush. + - destruct_match; crush. + - destruct_match; subst; crush. + apply beq_expression_list_correct in H; subst. + trivial. +Qed. + +Lemma beq_pred_expression_refl: + forall e, beq_pred_expression e e = true. +Proof. + destruct e; crush; destruct_match; crush. + apply beq_expression_list_refl. +Qed. + +Lemma beq_pred_expression_correct2: + forall e1 e2, beq_pred_expression e1 e2 = false -> e1 <> e2. +Proof. + destruct e1, e2; unfold not; crush. + + destruct_match; crush. + + destruct_match; crush. inv H0. + now rewrite beq_expression_list_refl in H. +Qed. + +Lemma beq_exit_expression_correct: + forall e1 e2, beq_exit_expression e1 e2 = true <-> e1 = e2. +Proof. + destruct e1, e2; split; crush; + destruct_match; subst; crush. +Qed. + +Definition pred_expr_item_eq (pe1 pe2: pred_op * expression) : bool := + @equiv_dec _ SATSetoid _ (fst pe1) (fst pe2) && beq_expression (snd pe1) (snd pe2). + +Definition pred_eexpr_item_eq (pe1 pe2: pred_op * exit_expression) : bool := + @equiv_dec _ SATSetoid _ (fst pe1) (fst pe2) && beq_exit_expression (snd pe1) (snd pe2). + +Definition pred_expr_dec: forall (pe1 pe2: pred_op * expression), + {pred_expr_item_eq pe1 pe2 = true} + {pred_expr_item_eq pe1 pe2 = false}. +Proof. + intros; destruct (pred_expr_item_eq pe1 pe2) eqn:?; unfold not; [tauto | now right]. +Defined. + +Definition pred_expr_dec2: forall (pe1 pe2: pred_op * expression), + {pred_expr_item_eq pe1 pe2 = true} + {~ pred_expr_item_eq pe1 pe2 = true}. +Proof. + intros; destruct (pred_expr_item_eq pe1 pe2) eqn:?; unfold not; [tauto | now right]. +Defined. + +Definition pred_expression_dec: + forall e1 e2: pred_expression, {e1 = e2} + {e1 <> e2}. +Proof. + intros. destruct (beq_pred_expression e1 e2) eqn:?. + eauto using beq_pred_expression_correct. + eauto using beq_pred_expression_correct2. +Defined. + +Lemma exit_expression_refl: + forall e, beq_exit_expression e e = true. +Proof. destruct e; crush; destruct_match; crush. Qed. + +Definition exit_expression_dec: + forall e1 e2: exit_expression, {e1 = e2} + {e1 <> e2}. +Proof. + intros. destruct (beq_exit_expression e1 e2) eqn:?. + - left. eapply beq_exit_expression_correct; eauto. + - right. unfold not; intros. + assert (X: ~ (beq_exit_expression e1 e2 = true)) + by eauto with bool. + subst. apply X. apply exit_expression_refl. +Defined. + +Lemma pred_eexpression_dec: + forall (e1 e2: exit_expression) (p1 p2: pred_op), + {(p1, e1) = (p2, e2)} + {(p1, e1) <> (p2, e2)}. +Proof. + pose proof (Predicate.eq_dec peq). + pose proof (exit_expression_dec). + decide equality. +Defined. + +(*Fixpoint encode_expression_ne (max: predicate) (pe: pred_expr_ne) (h: hash_tree) + : (PTree.t pred_op) * hash_tree := + match pe with + | NE.singleton (p, e) => + let (p', h') := hash_value max e h in + (Por (Pnot p) (Pvar p'), h') + | (p, e) ::| pr => + let (p', h') := hash_value max e h in + let (p'', h'') := encode_expression_ne max pr h' in + (Pand (Por (Pnot p) (Pvar p')) p'', h'') + end.*) + +Fixpoint max_pred_expr (pe: pred_expr) : positive := + match pe with + | NE.singleton (p, e) => max_predicate p + | (p, e) ::| pe' => Pos.max (max_predicate p) (max_pred_expr pe') + end. + +Definition ge_preserved {A B C D: Type} (ge: Genv.t A B) (tge: Genv.t C D) : Prop := + (forall sp op vl m, Op.eval_operation ge sp op vl m = + Op.eval_operation tge sp op vl m) + /\ (forall sp addr vl, Op.eval_addressing ge sp addr vl = + Op.eval_addressing tge sp addr vl). + +Lemma ge_preserved_same: + forall A B ge, @ge_preserved A B A B ge ge. +Proof. unfold ge_preserved; auto. Qed. +#[local] Hint Resolve ge_preserved_same : core. + +Inductive match_states : instr_state -> instr_state -> Prop := +| match_states_intro: + forall ps ps' rs rs' m m', + (forall x, rs !! x = rs' !! x) -> + (forall x, ps !! x = ps' !! x) -> + m = m' -> + match_states (mk_instr_state rs ps m) (mk_instr_state rs' ps' m'). + +Lemma match_states_refl x : match_states x x. +Proof. destruct x; constructor; crush. Qed. + +Lemma match_states_commut x y : match_states x y -> match_states y x. +Proof. inversion 1; constructor; crush. Qed. + +Lemma match_states_trans x y z : + match_states x y -> match_states y z -> match_states x z. +Proof. repeat inversion 1; constructor; crush. Qed. + +#[global] Instance match_states_Equivalence : Equivalence match_states := + { Equivalence_Reflexive := match_states_refl ; + Equivalence_Symmetric := match_states_commut ; + Equivalence_Transitive := match_states_trans ; }. + +Inductive similar {A B} : @ctx A -> @ctx B -> Prop := +| similar_intro : + forall ist ist' sp ge tge, + ge_preserved ge tge -> + match_states ist ist' -> + similar (mk_ctx ist sp ge) (mk_ctx ist' sp tge). + +Lemma ge_preserved_refl: + forall A B (a: Genv.t A B), ge_preserved a a. +Proof. auto. Qed. + +Lemma similar_refl: + forall A (a: @Abstr.ctx A), similar a a. +Proof. intros; destruct a; constructor; auto. reflexivity. Qed. + +Lemma similar_commut: + forall A B (a: @Abstr.ctx A) (b: @Abstr.ctx B), similar a b -> similar b a. +Proof. + inversion 1; constructor; auto. + - unfold ge_preserved in *; inv H0; split; intros. + rewrite H4; auto. rewrite H5; auto. + - symmetry; auto. +Qed. + +Lemma similar_trans: + forall A B C (a: @Abstr.ctx A) (b: @Abstr.ctx B) (c: @Abstr.ctx C), + similar a b -> similar b c -> similar a c. +Proof. + repeat inversion 1; constructor. + - unfold ge_preserved in *; inv H0; inv H9; split; intros. + rewrite H11. rewrite H0; auto. + rewrite H12. rewrite H2. auto. + - transitivity ist'; auto. +Qed. + +Module HashExpr' <: MiniDecidableType. + Definition t := expression. + Definition eq_dec := expression_dec. +End HashExpr'. + +Module HashExpr := Make_UDT(HashExpr'). +Module Import HT := HashTree(HashExpr). + +Module PHashExpr' <: MiniDecidableType. + Definition t := pred_expression. + Definition eq_dec := pred_expression_dec. +End PHashExpr'. + +Module PHashExpr := Make_UDT(PHashExpr'). +Module PHT := HashTree(PHashExpr). + +Module EHashExpr' <: MiniDecidableType. + Definition t := exit_expression. + Definition eq_dec := exit_expression_dec. +End EHashExpr'. + +Module EHashExpr := Make_UDT(EHashExpr'). +Module EHT := HashTree(EHashExpr). + +Fixpoint hash_predicate (max: predicate) (ap: pred_pexpr) (h: PHT.hash_tree) + : pred_op * PHT.hash_tree := + match ap with + | Ptrue => (Ptrue, h) + | Pfalse => (Pfalse, h) + | Plit (b, ap') => + let (p', h') := PHT.hash_value max ap' h in + (Plit (b, p'), h') + | Pand p1 p2 => + let (p1', h') := hash_predicate max p1 h in + let (p2', h'') := hash_predicate max p2 h' in + (Pand p1' p2', h'') + | Por p1 p2 => + let (p1', h') := hash_predicate max p1 h in + let (p2', h'') := hash_predicate max p2 h' in + (Por p1' p2', h'') + end. + +Definition predicated_mutexcl {A: Type} (pe: predicated A): Prop := + (forall x y, NE.In x pe -> NE.In y pe -> x <> y -> fst x ⇒ ¬ fst y) + /\ NE.norepet pe. + +Lemma predicated_cons : + forall A (a: pred_op * A) x, + predicated_mutexcl (a ::| x) -> + predicated_mutexcl x. +Proof. + unfold predicated_mutexcl; intros. inv H. inv H1. split; auto. + intros. apply H0; auto; constructor; tauto. +Qed. + +Lemma predicated_singleton : + forall A (a: (pred_op * A)), predicated_mutexcl (NE.singleton a). +Proof. + unfold predicated_mutexcl; intros; split; intros. + { inv H. now inv H0. } + constructor. +Qed. + +(* + +Lemma norm_expr_constant : + forall x m h t h' e p, + HN.norm_expression m x h = (t, h') -> + h ! e = Some p -> + h' ! e = Some p. +Proof. Abort. + +Definition sat_aequiv ap1 ap2 := + exists h p1 p2, + sat_equiv p1 p2 + /\ hash_predicate 1 ap1 h = (p1, h) + /\ hash_predicate 1 ap2 h = (p2, h). + +Lemma aequiv_symm : forall a b, sat_aequiv a b -> sat_aequiv b a. +Proof. + unfold sat_aequiv; simplify; do 3 eexists; simplify; [symmetry | |]; eauto. +Qed. + +Lemma existsh : + forall ap1, + exists h p1, + hash_predicate 1 ap1 h = (p1, h). +Proof. Admitted. + +Lemma aequiv_refl : forall a, sat_aequiv a a. +Proof. + unfold sat_aequiv; intros. + pose proof (existsh a); simplify. + do 3 eexists; simplify; eauto. reflexivity. +Qed. + +Lemma aequiv_trans : + forall a b c, + sat_aequiv a b -> + sat_aequiv b c -> + sat_aequiv a c. +Proof. + unfold sat_aequiv; intros. + simplify. +Abort. + +Lemma norm_expr_mutexcl : + forall m pe h t h' e e' p p', + HN.norm_expression m pe h = (t, h') -> + predicated_mutexcl pe -> + t ! e = Some p -> + t ! e' = Some p' -> + e <> e' -> + p ⇒ ¬ p'. +Proof. Abort.*) + +Definition pred_expr_eqb: forall pe1 pe2: pred_expr, + {pe1 = pe2} + {pe1 <> pe2}. +Proof. + pose proof expression_dec. + pose proof NE.eq_dec. + pose proof (Predicate.eq_dec peq). + assert (forall a b: pred_op * expression, {a = b} + {a <> b}) + by decide equality. + decide equality. +Defined. + +Definition pred_pexpr_eqb: forall pe1 pe2: pred_pexpr, + {pe1 = pe2} + {pe1 <> pe2}. +Proof. + pose proof pred_expression_dec. + pose proof (Predicate.eq_dec pred_expression_dec). + apply X. +Defined. + +Definition beq_pred_pexpr (pe1 pe2: pred_pexpr): bool := + if pred_pexpr_eqb pe1 pe2 then true else + let (np1, h) := hash_predicate 1 pe1 (PTree.empty _) in + let (np2, h') := hash_predicate 1 pe2 h in + equiv_check np1 np2. + +Lemma inj_asgn_eg : forall a b, (a =? b)%positive = (a =? a)%positive -> a = b. +Proof. + intros. destruct (peq a b); subst. + auto. rewrite OrdersEx.Positive_as_OT.eqb_refl in H. + now apply Peqb_true_eq. +Qed. + +Lemma inj_asgn : + forall a b, (forall (f: positive -> bool), f a = f b) -> a = b. +Proof. intros. apply inj_asgn_eg. eauto. Qed. + +Lemma inj_asgn_false: + forall n1 n2 , ~ (forall c: positive -> bool, c n1 = negb (c n2)). +Proof. + unfold not; intros. specialize (H (fun x => true)). + simplify. discriminate. +Qed. + +Lemma negb_inj: + forall a b, + negb a = negb b -> a = b. +Proof. destruct a, b; crush. Qed. + +Lemma sat_predicate_Plit_inj : + forall p1 p2, + Plit p1 == Plit p2 -> p1 = p2. +Proof. + simplify. destruct p1, p2. + destruct b, b0. f_equal. unfold sat_equiv in H. cbn in H. now apply inj_asgn. + solve [exfalso; eapply inj_asgn_false; eauto]. + solve [exfalso; eapply inj_asgn_false; eauto]. + assert (p = p0). eapply inj_asgn. intros. specialize (H f). + apply negb_inj in H. auto. rewrite H0; auto. +Qed. + +Definition ind_preds t := + forall e1 e2 p1 p2 c, + e1 <> e2 -> + t ! e1 = Some p1 -> + t ! e2 = Some p2 -> + sat_predicate p1 c = true -> + sat_predicate p2 c = false. + +Definition ind_preds_l t := + forall (e1: predicate) e2 p1 p2 c, + e1 <> e2 -> + In (e1, p1) t -> + In (e2, p2) t -> + sat_predicate p1 c = true -> + sat_predicate p2 c = false. + +(*Lemma pred_equivalence_Some' : + forall ta tb e pe pe', + list_norepet (map fst ta) -> + list_norepet (map fst tb) -> + In (e, pe) ta -> + In (e, pe') tb -> + fold_right (fun p a => mk_pred_stmnt' (fst p) (snd p) ∧ a) T ta == + fold_right (fun p a => mk_pred_stmnt' (fst p) (snd p) ∧ a) T tb -> + pe == pe'. +Proof. + induction ta as [|hd tl Hta]; try solve [crush]. + - intros * NRP1 NRP2 IN1 IN2 FOLD. inv NRP1. inv IN1. + simpl in FOLD. + +Lemma pred_equivalence_Some : + forall (ta tb: PTree.t pred_op) e pe pe', + ta ! e = Some pe -> + tb ! e = Some pe' -> + mk_pred_stmnt ta == mk_pred_stmnt tb -> + pe == pe'. +Proof. + intros * SMEA SMEB EQ. unfold mk_pred_stmnt in *. + repeat rewrite PTree.fold_spec in EQ. + +Lemma pred_equivalence_None : + forall (ta tb: PTree.t pred_op) e pe, + (mk_pred_stmnt ta) == (mk_pred_stmnt tb) -> + ta ! e = Some pe -> + tb ! e = None -> + equiv pe ⟂. +Abort. + +Lemma pred_equivalence : + forall (ta tb: PTree.t pred_op) e pe, + equiv (mk_pred_stmnt ta) (mk_pred_stmnt tb) -> + ta ! e = Some pe -> + equiv pe ⟂ \/ (exists pe', tb ! e = Some pe' /\ equiv pe pe'). +Proof. + intros * EQ SME. destruct (tb ! e) eqn:HTB. + { right. econstructor. split; eauto. eapply pred_equivalence_Some; eauto. } + { left. eapply pred_equivalence_None; eauto. } +Qed.*) + +Section CORRECT. + + Context {FUN TFUN: Type}. + + Context (ictx: @ctx FUN) (octx: @ctx TFUN) (HSIM: similar ictx octx). + + Lemma sem_value_mem_det: + forall e v v' m m', + (sem_value ictx e v -> sem_value octx e v' -> v = v') + /\ (sem_mem ictx e m -> sem_mem octx e m' -> m = m'). + Proof using HSIM. + induction e using expression_ind2 + with (P0 := fun p => forall v v', + sem_val_list ictx p v -> sem_val_list octx p v' -> v = v'); + inv HSIM; match goal with H: context [match_states] |- _ => inv H end; repeat progress simplify; + try solve [match goal with + | H: sem_value _ _ _, H2: sem_value _ _ _ |- _ => inv H; inv H2; auto + | H: sem_mem _ _ _, H2: sem_mem _ _ _ |- _ => inv H; inv H2; auto + | H: sem_val_list _ _ _, H2: sem_val_list _ _ _ |- _ => inv H; inv H2; auto + end]. + - repeat match goal with + | H: sem_value _ _ _ |- _ => inv H + | H: sem_val_list {| ctx_ge := ge; |} ?e ?l1, + H2: sem_val_list {| ctx_ge := tge |} ?e ?l2, + IH: forall _ _, sem_val_list _ _ _ -> sem_val_list _ _ _ -> _ = _ |- _ => + assert (X: l1 = l2) by (apply IH; auto) + | H: ge_preserved _ _ |- _ => inv H + | |- context [ctx_rs] => unfold ctx_rs; cbn + | H: context [ctx_mem] |- _ => unfold ctx_mem in H; cbn + end; crush. + - repeat match goal with H: sem_value _ _ _ |- _ => inv H end; simplify; + assert (lv0 = lv) by (apply IHe; eauto); subst; + match goal with H: ge_preserved _ _ |- _ => inv H end; + match goal with H: context [Op.eval_addressing _ _ _ _ = Op.eval_addressing _ _ _ _] |- _ + => rewrite H in * end; + assert (a0 = a1) by crush; + assert (m'2 = m'1) by (apply IHe0; eauto); crush. + - inv H0; inv H3. simplify. + assert (lv = lv0) by ( apply IHe2; eauto). subst. + assert (a1 = a0). { inv H. rewrite H3 in *. crush. } + assert (v0 = v1). { apply IHe1; auto. } + assert (m'1 = m'2). { apply IHe3; auto. } crush. + - inv H0. inv H3. f_equal. apply IHe; auto. + apply IHe0; auto. + Qed. + + Lemma sem_value_mem_corr: + forall e v m, + (sem_value ictx e v -> sem_value octx e v) + /\ (sem_mem ictx e m -> sem_mem octx e m). + Proof using HSIM. + induction e using expression_ind2 + with (P0 := fun p => forall v, + sem_val_list ictx p v -> sem_val_list octx p v); + inv HSIM; match goal with H: context [match_states] |- _ => inv H end; repeat progress simplify. + - inv H0. unfold ctx_rs, ctx_ps, ctx_mem in *; cbn. rewrite H1. constructor. + - inv H0. unfold ctx_rs, ctx_ps, ctx_mem in *; cbn. constructor. + - inv H0. apply IHe in H6. econstructor; try eassumption. + unfold ctx_rs, ctx_ps, ctx_mem in *; cbn in *. inv H. crush. + - inv H0. + - inv H0. eapply IHe in H10. eapply IHe0 in H8; auto. + econstructor; try eassumption. + unfold ctx_rs, ctx_ps, ctx_mem in *; cbn in *. inv H; crush. + - inv H0. + - inv H0. + - inv H0. eapply IHe1 in H11; auto. eapply IHe2 in H12; auto. eapply IHe3 in H9; auto. + econstructor; try eassumption. + unfold ctx_rs, ctx_ps, ctx_mem in *; cbn in *. inv H; crush. + - inv H0. econstructor. + - inv H0. eapply IHe in H6; auto. eapply IHe0 in H8. + econstructor; eassumption. + Qed. + + Lemma sem_value_det: + forall e v v', sem_value ictx e v -> sem_value octx e v' -> v = v'. + Proof using HSIM. + intros. eapply sem_value_mem_det; eauto; apply Mem.empty. + Qed. + + Lemma sem_value_corr: + forall e v, sem_value ictx e v -> sem_value octx e v. + Proof using HSIM. + intros. eapply sem_value_mem_corr; eauto; apply Mem.empty. + Qed. + + Lemma sem_mem_det: + forall e v v', sem_mem ictx e v -> sem_mem octx e v' -> v = v'. + Proof using HSIM. + intros. eapply sem_value_mem_det; eauto; apply (Vint (Int.repr 0%Z)). + Qed. + + Lemma sem_mem_corr: + forall e v, sem_mem ictx e v -> sem_mem octx e v. + Proof using HSIM. + intros. eapply sem_value_mem_corr; eauto; apply (Vint (Int.repr 0%Z)). + Qed. + + Lemma sem_val_list_det: + forall e l l', sem_val_list ictx e l -> sem_val_list octx e l' -> l = l'. + Proof using HSIM. + induction e; simplify. + - inv H; inv H0; auto. + - inv H; inv H0. f_equal. eapply sem_value_det; eauto; try apply Mem.empty. + apply IHe; eauto. + Qed. + + Lemma sem_val_list_corr: + forall e l, sem_val_list ictx e l -> sem_val_list octx e l. + Proof using HSIM. + induction e; simplify. + - inv H; constructor. + - inv H. apply sem_value_corr in H3; auto; try apply Mem.empty; + apply IHe in H5; constructor; assumption. + Qed. + + Lemma sem_pred_det: + forall e v v', sem_pred ictx e v -> sem_pred octx e v' -> v = v'. + Proof using HSIM. + try solve [inversion 1]; pose proof sem_value_det; pose proof sem_val_list_det; inv HSIM; + match goal with H: match_states _ _ |- _ => inv H end; simplify. + inv H2; inv H5; auto. assert (lv = lv0) by (eapply H0; eauto). subst. unfold ctx_mem in *. + crush. + Qed. + + Lemma sem_pred_corr: + forall e v, sem_pred ictx e v -> sem_pred octx e v. + Proof using HSIM. + try solve [inversion 1]; pose proof sem_value_corr; pose proof sem_val_list_corr; inv HSIM; + match goal with H: match_states _ _ |- _ => inv H end; simplify. + inv H2; auto. apply H0 in H5. econstructor; eauto. + unfold ctx_ps; cbn. rewrite H4. constructor. + Qed. + + Lemma sem_exit_det: + forall e v v', sem_exit ictx e v -> sem_exit octx e v' -> v = v'. + Proof using HSIM. + try solve [inversion 1]; pose proof sem_value_det; pose proof sem_val_list_det; inv HSIM; + match goal with H: match_states _ _ |- _ => inv H end; simplify. + inv H2; inv H5; auto. + Qed. + + Lemma sem_exit_corr: + forall e v, sem_exit ictx e v -> sem_exit octx e v. + Proof using HSIM. + try solve [inversion 1]; pose proof sem_value_corr; pose proof sem_val_list_corr; inv HSIM; + match goal with H: match_states _ _ |- _ => inv H end; simplify. + inv H2; auto; constructor. + Qed. + + Lemma sem_pexpr_det : + forall p b1 b2, sem_pexpr ictx p b1 -> sem_pexpr octx p b2 -> b1 = b2. + Proof. + induction p; crush; inv H; inv H0; firstorder. + destruct b. + - apply sem_pred_det with (e:=p0); auto. + - apply negb_inj. apply sem_pred_det with (e:=p0); auto. + Qed. + + Lemma sem_pexpr_corr : + forall p b, sem_pexpr ictx p b -> sem_pexpr octx p b. + Proof. + induction p; crush; inv H; constructor; + try solve [try inv H3; firstorder]. + now apply sem_pred_corr. + Qed. + + Lemma sem_pred_exec_beq_correct2 : + forall A B (sem: forall G, @Abstr.ctx G -> A -> B -> Prop) a p r R, + (forall x y, + sem _ ictx x y -> + exists y', sem _ octx x y' /\ R y y') -> + sem_pred_expr a (sem _) ictx p r -> + exists r', sem_pred_expr a (sem _) octx p r' /\ R r r'. + Proof. + induction p; crush. + - inv H0. apply H in H4. simplify. + exists x; split; auto. + constructor; auto. + now apply sem_pexpr_corr. + - inv H0. + + apply H in H6; simplify. + exists x; split; auto. + constructor; auto. + now apply sem_pexpr_corr. + + exploit IHp; auto. exact H6. intros. simplify. + exists x; split; auto. + apply sem_pred_expr_cons_false; auto. + now apply sem_pexpr_corr. + Qed. + + Lemma sem_pred_expr_corr : + forall A B (sem: forall G, @Abstr.ctx G -> A -> B -> Prop) a p r, + (forall x y, sem _ ictx x y -> sem _ octx x y) -> + sem_pred_expr a (sem _) ictx p r -> + sem_pred_expr a (sem _) octx p r. + Proof. + intros. + assert + (forall x y, + sem _ ictx x y -> + exists y', sem _ octx x y' /\ eq y y') by firstorder. + pose proof (sem_pred_exec_beq_correct2 _ _ sem a p r _ H1 H0). + crush. + Qed. + + Lemma sem_correct: + forall f i cf, sem ictx f (i, cf) -> sem octx f (i, cf). + Proof. + intros. inv H. constructor. + - inv H2. constructor; intros. specialize (H x). + apply sem_pred_expr_corr; auto. exact sem_value_corr. + - inv H3; constructor; intros. specialize (H x). + now apply sem_pexpr_corr. + - apply sem_pred_expr_corr; auto. exact sem_mem_corr. + - apply sem_pred_expr_corr; auto. exact sem_exit_corr. + Qed. + +End CORRECT. + +Section SEM_PRED_EXEC. + + Context (A: Type). + Context (ctx: @Abstr.ctx A). + + Lemma sem_pexpr_negate : + forall p b, + sem_pexpr ctx p b -> + sem_pexpr ctx (¬ p) (negb b). + Proof. + induction p; crush. + - destruct_match. destruct b0; crush. inv Heqp0. + constructor. inv H. rewrite negb_involutive. auto. + constructor. inv H. auto. + - inv H. constructor. + - inv H. constructor. + - inv H. inv H3. + + apply IHp1 in H. solve [constructor; auto]. + + apply IHp2 in H. solve [constructor; auto]. + + apply IHp1 in H2. apply IHp2 in H4. solve [constructor; auto]. + - inv H. inv H3. + + apply IHp1 in H. solve [constructor; auto]. + + apply IHp2 in H. solve [constructor; auto]. + + apply IHp1 in H2. apply IHp2 in H4. solve [constructor; auto]. + Qed. + + Lemma sem_pexpr_negate2 : + forall p b, + sem_pexpr ctx (¬ p) (negb b) -> + sem_pexpr ctx p b. + Proof. + induction p; crush. + - destruct_match. destruct b0; crush. inv Heqp0. + constructor. inv H. rewrite negb_involutive in *. auto. + constructor. inv H. auto. + - inv H. destruct b; try discriminate. constructor. + - inv H. destruct b; try discriminate. constructor. + - inv H. destruct b; try discriminate. + + constructor. inv H1; eauto. + + destruct b; try discriminate. constructor; eauto. + - inv H. destruct b; try discriminate. + + constructor. inv H1; eauto. + + destruct b; try discriminate. constructor; eauto. + Qed. + + Lemma sem_pexpr_evaluable : + forall f_p ps, + (forall x, sem_pexpr ctx (get_forest_p' x f_p) ps !! x) -> + forall p, exists b, sem_pexpr ctx (from_pred_op f_p p) b. + Proof. + induction p; crush. + - destruct_match. inv Heqp0. destruct b. econstructor. eauto. + econstructor. eapply sem_pexpr_negate. eauto. + - econstructor. constructor. + - econstructor. constructor. + - destruct x0, x; solve [eexists; constructor; auto]. + - destruct x0, x; solve [eexists; constructor; auto]. + Qed. + + Lemma sem_pexpr_eval1 : + forall f_p ps, + (forall x, sem_pexpr ctx (get_forest_p' x f_p) ps !! x) -> + forall p, + eval_predf ps p = false -> + sem_pexpr ctx (from_pred_op f_p p) false. + Proof. + induction p; crush. + - destruct_match. inv Heqp0. + destruct b. + + cbn in H0. rewrite <- H0. eauto. + + replace false with (negb true) by auto. + apply sem_pexpr_negate. cbn in H0. + apply negb_true_iff in H0. rewrite negb_involutive in H0. + rewrite <- H0. eauto. + - constructor. + - rewrite eval_predf_Pand in H0. + apply andb_false_iff in H0. inv H0. eapply IHp1 in H1. + pose proof (sem_pexpr_evaluable _ _ H p2) as EVAL. + inversion_clear EVAL as [x EVAL2]. + replace false with (false && x) by auto. + constructor; auto. + eapply IHp2 in H1. + pose proof (sem_pexpr_evaluable _ _ H p1) as EVAL. + inversion_clear EVAL as [x EVAL2]. + replace false with (x && false) by eauto with bool. + apply sem_pexpr_Pand; auto. + - rewrite eval_predf_Por in H0. + apply orb_false_iff in H0. inv H0. + replace false with (false || false) by auto. + apply sem_pexpr_Por; auto. + Qed. + + Lemma sem_pexpr_eval2 : + forall f_p ps, + (forall x, sem_pexpr ctx (get_forest_p' x f_p) ps !! x) -> + forall p, + eval_predf ps p = true -> + sem_pexpr ctx (from_pred_op f_p p) true. + Proof. + induction p; crush. + - destruct_match. inv Heqp0. + destruct b. + + cbn in H0. rewrite <- H0. eauto. + + replace true with (negb false) by auto. + apply sem_pexpr_negate. cbn in H0. + apply negb_true_iff in H0. + rewrite <- H0. eauto. + - constructor. + - rewrite eval_predf_Pand in H0. + apply andb_true_iff in H0. inv H0. + replace true with (true && true) by auto. + constructor; auto. + - rewrite eval_predf_Por in H0. + apply orb_true_iff in H0. inv H0. eapply IHp1 in H1. + pose proof (sem_pexpr_evaluable _ _ H p2) as EVAL. + inversion_clear EVAL as [x EVAL2]. + replace true with (true || x) by auto. + apply sem_pexpr_Por; auto. + eapply IHp2 in H1. + pose proof (sem_pexpr_evaluable _ _ H p1) as EVAL. + inversion_clear EVAL as [x EVAL2]. + replace true with (x || true) by eauto with bool. + apply sem_pexpr_Por; auto. + Qed. + + Lemma sem_pexpr_eval : + forall f_p ps b, + (forall x, sem_pexpr ctx (get_forest_p' x f_p) ps !! x) -> + forall p, + eval_predf ps p = b -> + sem_pexpr ctx (from_pred_op f_p p) b. + Proof. + intros; destruct b; eauto using sem_pexpr_eval1, sem_pexpr_eval2. + Qed. + + Lemma sem_pexpr_eval_inv : + forall f_p ps b, + (forall x, sem_pexpr ctx (get_forest_p' x f_p) ps !! x) -> + forall p, + sem_pexpr ctx (from_pred_op f_p p) b -> + eval_predf ps p = b. + Proof. + induction p; intros. + - cbn in H0. destruct_match. destruct b0; cbn in *. + + specialize (H p0). eapply sem_pexpr_det; eauto. apply similar_refl. + + rewrite <- negb_involutive in H0. apply sem_pexpr_negate2 in H0. + symmetry; apply negb_sym. eapply sem_pexpr_det; eauto. + apply similar_refl. + - now inv H0. + - now inv H0. + - inv H0; try inv H4; rewrite eval_predf_Pand. + + apply IHp1 in H0. rewrite H0. auto. + + apply IHp2 in H0. rewrite H0. auto with bool. + + apply IHp2 in H5. apply IHp1 in H3. rewrite H3. rewrite H5. auto. + - inv H0; try inv H4; rewrite eval_predf_Por. + + apply IHp1 in H0. rewrite H0. auto. + + apply IHp2 in H0. rewrite H0. auto with bool. + + apply IHp2 in H5. apply IHp1 in H3. rewrite H3. rewrite H5. auto. + Qed. + + Context {C B: Type}. + Context (f: PTree.t pred_pexpr). + Context (ps: PMap.t bool). + Context (a_sem: @Abstr.ctx A -> C -> B -> Prop). + + Context (F_EVALULABLE: forall x, sem_pexpr ctx (get_forest_p' x f) ps !! x). + + Lemma sem_pexpr_equiv : + forall p1 p2 b, + p1 == p2 -> + sem_pexpr ctx (from_pred_op f p1) b -> + sem_pexpr ctx (from_pred_op f p2) b. + Proof. + intros. + eapply sem_pexpr_eval_inv in H0; eauto. + eapply sem_pexpr_eval; eauto. + Qed. + +End SEM_PRED_EXEC. + +Module HashExprNorm(HS: Hashable). + Module H := HashTree(HS). + + Definition norm_tree: Type := PTree.t pred_op * H.hash_tree. + + Fixpoint norm_expression (max: predicate) (pe: predicated HS.t) (h: H.hash_tree) + : norm_tree := + match pe with + | NE.singleton (p, e) => + let (hashed_e, h') := H.hash_value max e h in + (PTree.set hashed_e p (PTree.empty _), h') + | (p, e) ::| pr => + let (hashed_e, h') := H.hash_value max e h in + let (norm_pr, h'') := norm_expression max pr h' in + match norm_pr ! hashed_e with + | Some pr_op => + (PTree.set hashed_e (pr_op ∨ p) norm_pr, h'') + | None => + (PTree.set hashed_e p norm_pr, h'') + end + end. + + Definition mk_pred_stmnt' (e: predicate) p_e := ¬ p_e ∨ Plit (true, e). + + Definition mk_pred_stmnt t := PTree.fold (fun x a b => mk_pred_stmnt' a b ∧ x) t T. + + Definition mk_pred_stmnt_l (t: list (predicate * pred_op)) := + fold_left (fun x a => uncurry mk_pred_stmnt' a ∧ x) t T. + + Definition encode_expression max pe h := + let (tree, h) := norm_expression max pe h in + (mk_pred_stmnt tree, h). + + Definition pred_expr_dec: forall pe1 pe2: predicated HS.t, + {pe1 = pe2} + {pe1 <> pe2}. + Proof. + pose proof HS.eq_dec as X. + pose proof (Predicate.eq_dec peq). + pose proof (NE.eq_dec _ X). + assert (forall a b: pred_op * HS.t, {a = b} + {a <> b}) + by decide equality. + decide equality. + Defined. + + Definition beq_pred_expr' (pe1 pe2: predicated HS.t) : bool := + if pred_expr_dec pe1 pe2 then true else + let (p1, h) := encode_expression 1%positive pe1 (PTree.empty _) in + let (p2, h') := encode_expression 1%positive pe2 h in + equiv_check p1 p2. + + Lemma mk_pred_stmnt_equiv' : + forall l l' e p, + mk_pred_stmnt_l l == mk_pred_stmnt_l l' -> + In (e, p) l -> + list_norepet (map fst l) -> + (exists p', In (e, p') l' /\ p == p') + \/ ~ In e (map fst l') /\ p == ⟂. + Proof. Abort. + + Lemma mk_pred_stmnt_equiv : + forall tree tree', + mk_pred_stmnt tree == mk_pred_stmnt tree'. + Proof. Abort. + + Definition tree_equiv_check_el (np2: PTree.t pred_op) (n: positive) (p: pred_op): bool := + match np2 ! n with + | Some p' => equiv_check p p' + | None => equiv_check p ⟂ + end. + + Definition tree_equiv_check_None_el (np2: PTree.t pred_op) (n: positive) (p: pred_op): bool := + match np2 ! n with + | Some p' => true + | None => equiv_check p ⟂ + end. + + Definition beq_pred_expr (pe1 pe2: predicated HS.t) : bool := + if pred_expr_dec pe1 pe2 then true else + let (np1, h) := norm_expression 1 pe1 (PTree.empty _) in + let (np2, h') := norm_expression 1 pe2 h in + forall_ptree (tree_equiv_check_el np2) np1 + && forall_ptree (tree_equiv_check_None_el np1) np2. + + Lemma beq_pred_expr_correct: + forall np1 np2 e p p', + forall_ptree (tree_equiv_check_el np2) np1 = true -> + np1 ! e = Some p -> + np2 ! e = Some p' -> + p == p'. + Proof. + intros. + eapply forall_ptree_true in H; try eassumption. + unfold tree_equiv_check_el in H. + rewrite H1 in H. now apply equiv_check_correct. + Qed. + + Lemma beq_pred_expr_correct2: + forall np1 np2 e p, + forall_ptree (tree_equiv_check_el np2) np1 = true -> + np1 ! e = Some p -> + np2 ! e = None -> + p == ⟂. + Proof. + intros. + eapply forall_ptree_true in H; try eassumption. + unfold tree_equiv_check_el in H. + rewrite H1 in H. now apply equiv_check_correct. + Qed. + + Lemma beq_pred_expr_correct3: + forall np1 np2 e p, + forall_ptree (tree_equiv_check_None_el np1) np2 = true -> + np1 ! e = None -> + np2 ! e = Some p -> + p == ⟂. + Proof. + intros. + eapply forall_ptree_true in H; try eassumption. + unfold tree_equiv_check_None_el in H. + rewrite H0 in H. now apply equiv_check_correct. + Qed. + + Section PRED_PROOFS. + + Context {G B: Type}. + Context (f: PTree.t pred_pexpr). + Context (ps: PMap.t bool). + Context (a_sem: @Abstr.ctx G -> HS.t -> B -> Prop). + Context (ctx: @Abstr.ctx G). + + Context (F_EVALULABLE: forall x, sem_pexpr ctx (get_forest_p' x f) ps !! x). + + Variant sem_pred_tree: PTree.t HS.t -> PTree.t pred_op -> B -> Prop := + | sem_pred_tree_intro : + forall expr e v et pt pr, + sem_pexpr ctx (from_pred_op f pr) true -> + a_sem ctx expr v -> + pt ! e = Some pr -> + et ! e = Some expr -> + sem_pred_tree et pt v. + + Lemma norm_expression_in : + forall pe et pt h x y, + H.wf_hash_table h -> + norm_expression 1 pe h = (pt, et) -> + h ! x = Some y -> + et ! x = Some y. + Proof. + induction pe; crush; repeat (destruct_match; try discriminate; []). + - inv H0. eauto using H.hash_constant. + - destruct_match. + + inv H0. eapply IHpe. + eapply H.wf_hash_table_distr; eauto. eauto. + eauto using H.hash_constant. + + inv H0. eapply IHpe. + eapply H.wf_hash_table_distr; eauto. eauto. + eauto using H.hash_constant. + Qed. + + Lemma norm_expression_exists : + forall pe et pt h x y, + H.wf_hash_table h -> + norm_expression 1 pe h = (pt, et) -> + pt ! x = Some y -> + exists z, et ! x = Some z. + Proof. + induction pe; crush; repeat (destruct_match; try discriminate; []). + - inv H0. destruct (peq x h0); subst; inv H1. + + eexists. eauto using H.hash_value_in. + + rewrite PTree.gso in H2 by auto. now rewrite PTree.gempty in H2. + - assert (H.wf_hash_table h1) by eauto using H.wf_hash_table_distr. + destruct_match; inv H0. + + destruct (peq h0 x); subst; eauto. + rewrite PTree.gso in H1 by auto. eauto. + + destruct (peq h0 x); subst; eauto. + * pose proof Heqp0 as X. + eapply H.hash_value_in in Heqp0. + eapply norm_expression_in in Heqn; eauto. + * rewrite PTree.gso in H1 by auto. eauto. + Qed. + + Lemma norm_expression_wf : + forall pe et pt h, + H.wf_hash_table h -> + norm_expression 1 pe h = (pt, et) -> + H.wf_hash_table et. + Proof. + induction pe; crush; repeat (destruct_match; try discriminate; []). + - inv H0. eauto using H.wf_hash_table_distr. + - destruct_match. + + inv H0. eapply IHpe. + eapply H.wf_hash_table_distr; eauto. eauto. + + inv H0. eapply IHpe. + eapply H.wf_hash_table_distr; eauto. eauto. + Qed. + + Lemma sem_pred_expr_in_true : + forall pe v, + sem_pred_expr f a_sem ctx pe v -> + exists p e, NE.In (p, e) pe + /\ sem_pexpr ctx (from_pred_op f p) true + /\ a_sem ctx e v. + Proof. + induction pe; crush. + - inv H. do 2 eexists; split; try split; eauto. constructor. + - inv H. + + do 2 eexists; split; try split; eauto. constructor; tauto. + + exploit IHpe; eauto. simplify. + do 2 eexists; split; try split; eauto. constructor; tauto. + Qed. + + Definition pred_Ht_dec : + forall x y: pred_op * HS.t, {x = y} + {x <> y}. + Proof. + pose proof HS.eq_dec. + pose proof (@Predicate.eq_dec positive peq). + decide equality. + Defined. + + Lemma sem_pred_mutexcl : + forall pe p t v, + predicated_mutexcl ((p, t) ::| pe) -> + sem_pred_expr f a_sem ctx pe v -> + sem_pexpr ctx (from_pred_op f p) false. + Proof. + intros. unfold predicated_mutexcl in H. + exploit sem_pred_expr_in_true; eauto; simplify. + unfold "⇒" in *. inv H5. + destruct (pred_Ht_dec (x, x0) (p, t)); subst. + { inv e; exfalso; apply H7; auto. } + assert (NE.In (x, x0) ((p, t) ::| pe)) by (constructor; tauto). + assert (NE.In (p, t) ((p, t) ::| pe)) by (constructor; tauto). + pose proof (H3 _ _ H H5 n). + assert (forall c, eval_predf c x = true -> eval_predf c (¬ p) = true) + by eauto. + eapply sem_pexpr_eval_inv in H1; eauto. + eapply sem_pexpr_eval; eauto. apply H9 in H1. + unfold eval_predf in *. rewrite negate_correct in H1. + symmetry in H1. apply negb_sym in H1. auto. + Qed. + + Lemma exec_tree_exec_pe : + forall pe et pt v h + (MUTEXCL: predicated_mutexcl pe), + H.wf_hash_table h -> + norm_expression 1 pe h = (pt, et) -> + sem_pred_tree et pt v -> + sem_pred_expr f a_sem ctx pe v. + Proof. + induction pe; simplify; repeat (destruct_match; try discriminate; []). + - inv Heqp. inv H0. inv H1. + destruct (peq e h0); subst. + 2: { rewrite PTree.gso in H3 by auto. + rewrite PTree.gempty in H3. discriminate. } + assert (expr = t). + { apply H.hash_value_in in Heqp0. rewrite H4 in Heqp0. now inv Heqp0. } + subst. constructor; auto. rewrite PTree.gss in H3. inv H3; auto. + - inv Heqp. inv H1. destruct_match; inv H0; destruct (peq h0 e); subst. + + rewrite PTree.gss in H4. inv H4. inv H2. inv H1. + * exploit IHpe. eauto using predicated_cons. + eapply H.wf_hash_table_distr; eauto. eauto. + econstructor. eauto. eauto. eauto. eauto. intros. + assert (sem_pexpr ctx (from_pred_op f p) false) + by (eapply sem_pred_mutexcl; eauto). + eapply sem_pred_expr_cons_false; auto. + * assert (et ! e = Some t). + { eapply norm_expression_in. eapply H.wf_hash_table_distr; eauto. + eauto. apply H.hash_value_in in Heqp0. auto. } + rewrite H1 in H5. inv H5. + constructor; auto. + + exploit IHpe. eauto using predicated_cons. + eapply H.wf_hash_table_distr; eauto. eauto. + econstructor. eauto. eauto. rewrite PTree.gso in H4; eauto. auto. + intros. + assert (sem_pexpr ctx (from_pred_op f p) false) + by (eapply sem_pred_mutexcl; eauto). + eapply sem_pred_expr_cons_false; auto. + + rewrite PTree.gss in H4. inv H4. + econstructor; auto. + assert (et ! e = Some t). + { eapply norm_expression_in. eapply H.wf_hash_table_distr; eauto. + eauto. apply H.hash_value_in in Heqp0. auto. } + rewrite H0 in H5; inv H5. auto. + + rewrite PTree.gso in H4 by auto. + exploit IHpe. eauto using predicated_cons. + eapply H.wf_hash_table_distr; eauto. eauto. + econstructor. eauto. eauto. eauto. eauto. intros. + assert (sem_pexpr ctx (from_pred_op f p) false) + by (eapply sem_pred_mutexcl; eauto). + eapply sem_pred_expr_cons_false; auto. + Qed. + + Lemma exec_pe_exec_tree : + forall pe et pt v h + (MUTEXCL: predicated_mutexcl pe), + H.wf_hash_table h -> + norm_expression 1 pe h = (pt, et) -> + sem_pred_expr f a_sem ctx pe v -> + sem_pred_tree et pt v. + Proof. + induction pe; simplify; repeat (destruct_match; try discriminate; []). + - inv H0. inv H1. econstructor; eauto. apply PTree.gss. + eapply H.hash_value_in; eauto. + - inv H1. + + destruct_match. + * inv H0. econstructor. + 2: { eauto. } + 2: { apply PTree.gss. } + constructor; tauto. + eapply norm_expression_in. eapply H.wf_hash_table_distr; eauto. + eauto. eapply H.hash_value_in; eauto. + * inv H0. econstructor. eauto. eauto. apply PTree.gss. + eapply norm_expression_in. eapply H.wf_hash_table_distr; eauto. + eauto. eapply H.hash_value_in; eauto. + + destruct_match. + * inv H0. exploit IHpe. + eauto using predicated_cons. + eapply H.wf_hash_table_distr; eauto. + eauto. eauto. intros. inv H0. + destruct (peq e h0); subst. + -- rewrite H3 in Heqo. inv Heqo. + econstructor. + 3: { apply PTree.gss. } + constructor; tauto. eauto. auto. + -- econstructor. eauto. eauto. rewrite PTree.gso by eauto. auto. + auto. + * inv H0. exploit IHpe. + eauto using predicated_cons. + eapply H.wf_hash_table_distr; eauto. + eauto. eauto. intros. inv H0. + destruct (peq e h0); subst. + -- rewrite H3 in Heqo; discriminate. + -- econstructor; eauto. rewrite PTree.gso by auto. auto. + Qed. + + Lemma beq_pred_expr_correct_top: + forall p1 p2 v + (MUTEXCL1: predicated_mutexcl p1) + (MUTEXCL2: predicated_mutexcl p2), + beq_pred_expr p1 p2 = true -> + sem_pred_expr f a_sem ctx p1 v -> + sem_pred_expr f a_sem ctx p2 v. + Proof. + unfold beq_pred_expr; intros. + destruct_match; subst; auto. + repeat (destruct_match; []). + symmetry in H. apply andb_true_eq in H. inv H. + symmetry in H1. symmetry in H2. + pose proof Heqn0. eapply norm_expression_wf in H. + 2: { unfold H.wf_hash_table; intros. now rewrite PTree.gempty in H3. } + eapply exec_tree_exec_pe; eauto. + eapply exec_pe_exec_tree in H0; auto. + 3: { eauto. } + 2: { unfold H.wf_hash_table; intros. now rewrite PTree.gempty in H3. } + inv H0. destruct (t0 ! e) eqn:?. + - assert (pr == p) by eauto using beq_pred_expr_correct. + assert (sem_pexpr ctx (from_pred_op f p) true). + { eapply sem_pexpr_eval; eauto. eapply sem_pexpr_eval_inv in H3; eauto. } + econstructor. apply H7. eauto. eauto. + eapply norm_expression_in; eauto. + - assert (pr == ⟂) by eauto using beq_pred_expr_correct2. + eapply sem_pexpr_eval_inv in H3; eauto. now rewrite H0 in H3. + Qed. + + Lemma beq_pred_expr_correct_top2: + forall p1 p2 v + (MUTEXCL1: predicated_mutexcl p1) + (MUTEXCL2: predicated_mutexcl p2), + beq_pred_expr p1 p2 = true -> + sem_pred_expr f a_sem ctx p2 v -> + sem_pred_expr f a_sem ctx p1 v. + Proof. + unfold beq_pred_expr; intros. + destruct_match; subst; auto. + repeat (destruct_match; []). + symmetry in H. apply andb_true_eq in H. inv H. + symmetry in H1. symmetry in H2. + pose proof Heqn0. eapply norm_expression_wf in H. + 2: { unfold H.wf_hash_table; intros. now rewrite PTree.gempty in H3. } + eapply exec_tree_exec_pe; auto. + 2: { eauto. } + unfold H.wf_hash_table; intros. now rewrite PTree.gempty in H3. + eapply exec_pe_exec_tree in H0; auto. + 3: { eauto. } + 2: { auto. } + inv H0. destruct (t ! e) eqn:?. + - assert (p == pr) by eauto using beq_pred_expr_correct. + assert (sem_pexpr ctx (from_pred_op f p) true). + { eapply sem_pexpr_eval; eauto. eapply sem_pexpr_eval_inv in H3; eauto. } + econstructor. apply H7. eauto. eauto. + exploit norm_expression_exists. + 2: { eapply Heqn0. } unfold H.wf_hash_table; intros * YH. + now rewrite PTree.gempty in YH. eauto. simplify. + exploit norm_expression_in. 2: { eauto. } auto. eauto. intros. + crush. + - assert (pr == ⟂) by eauto using beq_pred_expr_correct3. + eapply sem_pexpr_eval_inv in H3; eauto. now rewrite H0 in H3. + Qed. + + End PRED_PROOFS. + +End HashExprNorm. + +Module HN := HashExprNorm(HashExpr). +Module EHN := HashExprNorm(EHashExpr). + +Definition check_mutexcl {A} (pe: predicated A) := + let preds := map fst (NE.to_list pe) in + let pairs := map (fun x => x → or_list (remove (Predicate.eq_dec peq) x preds)) preds in + match sat_pred_simple (simplify (negate (and_list pairs))) with + | None => true + | _ => false + end. + +Lemma check_mutexcl_correct: + forall A (pe: predicated A), + check_mutexcl pe = true -> + predicated_mutexcl pe. +Proof. Admitted. + +Definition check_mutexcl_tree {A} (f: PTree.t (predicated A)) := + forall_ptree (fun _ => check_mutexcl) f. + +Lemma check_mutexcl_tree_correct: + forall A (f: PTree.t (predicated A)) i pe, + check_mutexcl_tree f = true -> + f ! i = Some pe -> + predicated_mutexcl pe. +Proof. + unfold check_mutexcl_tree; intros. + eapply forall_ptree_true in H; eauto using check_mutexcl_correct. +Qed. + +Definition check f1 f2 := + RTree.beq HN.beq_pred_expr f1.(forest_regs) f2.(forest_regs) + && PTree.beq beq_pred_pexpr f1.(forest_preds) f2.(forest_preds) + && EHN.beq_pred_expr f1.(forest_exit) f2.(forest_exit) + && check_mutexcl_tree f1.(forest_regs) + && check_mutexcl_tree f2.(forest_regs) + && check_mutexcl f1.(forest_exit) + && check_mutexcl f2.(forest_exit). + +Lemma sem_pexpr_forward_eval1 : + forall A ctx f_p ps, + (forall x, sem_pexpr ctx (get_forest_p' x f_p) ps !! x) -> + forall p, + @sem_pexpr A ctx (from_pred_op f_p p) false -> + eval_predf ps p = false. +Proof. + induction p; crush. + - destruct_match. inv Heqp0. destruct b. + cbn. specialize (H p0). + eapply sem_pexpr_det; eauto. apply similar_refl. + specialize (H p0). + replace false with (negb true) in H0 by auto. + eapply sem_pexpr_negate2 in H0. cbn. + symmetry; apply negb_sym. cbn. + eapply sem_pexpr_det; eauto. apply similar_refl. + - inv H0. + - inv H0. inv H2. rewrite eval_predf_Pand. rewrite IHp1; eauto. + rewrite eval_predf_Pand. rewrite IHp2; eauto with bool. + - inv H0. rewrite eval_predf_Por. rewrite IHp1; eauto. +Qed. + +Lemma sem_pexpr_forward_eval2 : + forall A ctx f_p ps, + (forall x, sem_pexpr ctx (get_forest_p' x f_p) ps !! x) -> + forall p, + @sem_pexpr A ctx (from_pred_op f_p p) true -> + eval_predf ps p = true. +Proof. + induction p; crush. + - destruct_match. inv Heqp0. destruct b. + cbn. specialize (H p0). + eapply sem_pexpr_det; eauto. apply similar_refl. + cbn. symmetry. apply negb_sym; cbn. + replace true with (negb false) in H0 by auto. + eapply sem_pexpr_negate2 in H0. + eapply sem_pexpr_det; eauto. apply similar_refl. + - inv H0. + - inv H0. rewrite eval_predf_Pand. rewrite IHp1; eauto. + - inv H0. inv H2. rewrite eval_predf_Por. rewrite IHp1; eauto. + rewrite eval_predf_Por. rewrite IHp2; eauto with bool. +Qed. + +Lemma sem_pexpr_forward_eval : + forall A ctx f_p ps, + (forall x, sem_pexpr ctx (get_forest_p' x f_p) ps !! x) -> + forall p b, + @sem_pexpr A ctx (from_pred_op f_p p) b -> + eval_predf ps p = b. +Proof. + intros; destruct b; eauto using sem_pexpr_forward_eval1, sem_pexpr_forward_eval2. +Qed. + +Section BOOLEAN_EQUALITY. + + Context {A B: Type}. + Context (beqA: A -> B -> bool). + + Fixpoint beq2' (m1: PTree.tree' A) (m2: PTree.tree' B) {struct m1} : bool := + match m1, m2 with + | PTree.Node001 r1, PTree.Node001 r2 => beq2' r1 r2 + | PTree.Node010 x1, PTree.Node010 x2 => beqA x1 x2 + | PTree.Node011 x1 r1, PTree.Node011 x2 r2 => beqA x1 x2 && beq2' r1 r2 + | PTree.Node100 l1, PTree.Node100 l2 => beq2' l1 l2 + | PTree.Node101 l1 r1, PTree.Node101 l2 r2 => beq2' l1 l2 && beq2' r1 r2 + | PTree.Node110 l1 x1, PTree.Node110 l2 x2 => beqA x1 x2 && beq2' l1 l2 + | PTree.Node111 l1 x1 r1, PTree.Node111 l2 x2 r2 => beqA x1 x2 && beq2' l1 l2 && beq2' r1 r2 + | _, _ => false + end. + + Definition beq2 (m1: PTree.t A) (m2 : PTree.t B) : bool := + match m1, m2 with + | PTree.Empty, PTree.Empty => true + | PTree.Nodes m1', PTree.Nodes m2' => beq2' m1' m2' + | _, _ => false + end. + + Let beq2_optA (o1: option A) (o2: option B) : bool := + match o1, o2 with + | None, None => true + | Some a1, Some a2 => beqA a1 a2 + | _, _ => false + end. + + Lemma beq_correct_bool: + forall m1 m2, + beq2 m1 m2 = true <-> (forall x, beq2_optA (m1 ! x) (m2 ! x) = true). + Proof. + Local Transparent PTree.Node. + assert (beq_NN: forall l1 o1 r1 l2 o2 r2, + PTree.not_trivially_empty l1 o1 r1 -> + PTree.not_trivially_empty l2 o2 r2 -> + beq2 (PTree.Node l1 o1 r1) (PTree.Node l2 o2 r2) = + beq2 l1 l2 && beq2_optA o1 o2 && beq2 r1 r2). + { intros. + destruct l1, o1, r1; try contradiction; destruct l2, o2, r2; try contradiction; + simpl; rewrite ? andb_true_r, ? andb_false_r; auto. + rewrite andb_comm; auto. + f_equal; rewrite andb_comm; auto. } + induction m1 using PTree.tree_ind; [|induction m2 using PTree.tree_ind]. + - intros. simpl; split; intros. + + destruct m2; try discriminate. simpl; auto. + + replace m2 with (@PTree.Empty B); auto. apply PTree.extensionality; intros x. + specialize (H x). destruct (m2 ! x); simpl; easy. + - split; intros. + + destruct (PTree.Node l o r); try discriminate. simpl; auto. + + replace (PTree.Node l o r) with (@PTree.Empty A); auto. apply PTree.extensionality; intros x. + specialize (H0 x). destruct ((PTree.Node l o r) ! x); simpl in *; easy. + - rewrite beq_NN by auto. split; intros. + + InvBooleans. rewrite ! PTree.gNode. destruct x. + * apply IHm0; auto. + * apply IHm1; auto. + * auto. + + apply andb_true_intro; split; [apply andb_true_intro; split|]. + * apply IHm1. intros. specialize (H1 (xO x)); rewrite ! PTree.gNode in H1; auto. + * specialize (H1 xH); rewrite ! PTree.gNode in H1; auto. + * apply IHm0. intros. specialize (H1 (xI x)); rewrite ! PTree.gNode in H1; auto. + Qed. + + Theorem beq2_correct: + forall m1 m2, + beq2 m1 m2 = true <-> + (forall (x: PTree.elt), + match m1 ! x, m2 ! x with + | None, None => True + | Some y1, Some y2 => beqA y1 y2 = true + | _, _ => False + end). + Proof. + intros. rewrite beq_correct_bool. unfold beq2_optA. split; intros. + - specialize (H x). destruct (m1 ! x), (m2 ! x); intuition congruence. + - specialize (H x). destruct (m1 ! x), (m2 ! x); intuition auto. + Qed. + +End BOOLEAN_EQUALITY. + +Section GENERIC_CONTEXT. + +Context {A: Type}. +Context (ctx: @ctx A). + +(*| +Suitably restrict the predicate set so that one can evaluate a hashed predicate +using that predicate set. However, one issue might be that we do not know that +all the atoms of the original formula are actually evaluable. +|*) + +Definition match_pred_states + (ht: PHT.hash_tree) (p_out: pred_op) (pred_set: predset) := + forall (p: positive) (br: bool) (p_in: pred_expression), + PredIn p p_out -> + ht ! p = Some p_in -> + sem_pred ctx p_in (pred_set !! p). + +Lemma eval_hash_predicate : + forall max p_in ht p_out ht' br pred_set, + hash_predicate max p_in ht = (p_out, ht') -> + sem_pexpr ctx p_in br -> + match_pred_states ht' p_out pred_set -> + eval_predf pred_set p_out = br. +Proof. + induction p_in; simplify. + + repeat destruct_match. inv H. + unfold eval_predf. cbn. + inv H0. inv H4. unfold match_pred_states in H1. + specialize (H1 h br). +Abort. + +Lemma sem_pexpr_beq_correct : + forall p1 p2 b, + beq_pred_pexpr p1 p2 = true -> + sem_pexpr ctx p1 b -> + sem_pexpr ctx p2 b. +Proof. + unfold beq_pred_pexpr. + induction p1; intros; destruct_match; crush. + Admitted. + +(*| +This should only require a proof of sem_pexpr_beq_correct, the rest is +straightforward. +|*) + +Lemma pred_pexpr_beq_pred_pexpr : + forall pr a b br, + PTree.beq beq_pred_pexpr a b = true -> + sem_pexpr ctx (from_pred_op a pr) br -> + sem_pexpr ctx (from_pred_op b pr) br. +Proof. + induction pr; crush. + - destruct_match. inv Heqp0. destruct b0. + + unfold get_forest_p' in *. + apply PTree.beq_correct with (x := p0) in H. + destruct a ! p0; destruct b ! p0; try (exfalso; assumption); auto. + eapply sem_pexpr_beq_correct; eauto. + + replace br with (negb (negb br)) by (now apply negb_involutive). + replace br with (negb (negb br)) in H0 by (now apply negb_involutive). + apply sem_pexpr_negate. apply sem_pexpr_negate2 in H0. + unfold get_forest_p' in *. + apply PTree.beq_correct with (x := p0) in H. + destruct a ! p0; destruct b ! p0; try (exfalso; assumption); auto. + eapply sem_pexpr_beq_correct; eauto. + - inv H0; try inv H4. + + eapply IHpr1 in H0; eauto. apply sem_pexpr_Pand_false; tauto. + + eapply IHpr2 in H0; eauto. apply sem_pexpr_Pand_false; tauto. + + eapply IHpr1 in H3; eauto. eapply IHpr2 in H5; eauto. + apply sem_pexpr_Pand_true; auto. + - inv H0; try inv H4. + + eapply IHpr1 in H0; eauto. apply sem_pexpr_Por_true; tauto. + + eapply IHpr2 in H0; eauto. apply sem_pexpr_Por_true; tauto. + + eapply IHpr1 in H3; eauto. eapply IHpr2 in H5; eauto. + apply sem_pexpr_Por_false; auto. +Qed. + +(*| +This lemma requires a theorem that equivalence of symbolic predicates means they +will be. This further needs three-valued logic to be able to compare arbitrary +predicates with each other, that will also show that the predicate will also be +computable. +|*) + +Lemma sem_pred_exec_beq_correct : + forall A B (sem: Abstr.ctx -> A -> B -> Prop) p a b r, + PTree.beq beq_pred_pexpr a b = true -> + sem_pred_expr a sem ctx p r -> + sem_pred_expr b sem ctx p r. +Proof. + induction p; intros; inv H0. + - constructor; auto. eapply pred_pexpr_beq_pred_pexpr; eauto. + - constructor; auto. eapply pred_pexpr_beq_pred_pexpr; eauto. + - apply sem_pred_expr_cons_false; eauto. + eapply pred_pexpr_beq_pred_pexpr; eauto. +Qed. + +End GENERIC_CONTEXT. + +Lemma tree_beq_pred_pexpr : + forall a b x, + RTree.beq beq_pred_pexpr (forest_preds a) (forest_preds b) = true -> + beq_pred_pexpr a #p x b #p x = true. +Proof. + intros. unfold "#p". unfold get_forest_p'. + apply PTree.beq_correct with (x := x) in H. + destruct_match; destruct_match; auto. + unfold beq_pred_pexpr. destruct_match; auto. +Qed. + +Lemma tree_beq_pred_expr : + forall a b x, + RTree.beq HN.beq_pred_expr (forest_regs a) (forest_regs b) = true -> + HN.beq_pred_expr a #r x b #r x = true. +Proof. + intros. unfold "#r" in *. + apply PTree.beq_correct with (x := (R_indexed.index x)) in H. + unfold RTree.get in *. + unfold pred_expr in *. + destruct_match; destruct_match; auto. + unfold HN.beq_pred_expr. destruct_match; auto. +Qed. + +Section CORRECT. + +Context {FUN TFUN: Type}. +Context (ictx: @ctx FUN) (octx: @ctx TFUN) (HSIM: similar ictx octx). + +Lemma abstr_check_correct : + forall i' a b cf, + (exists ps, forall x, sem_pexpr ictx (get_forest_p' x (forest_preds a)) ps !! x) -> + check a b = true -> + sem ictx a (i', cf) -> + exists ti', sem octx b (ti', cf) /\ match_states i' ti'. +Proof. + intros * EVALUABLE **. unfold check in H. simplify. + inv H0. inv H10. inv H11. + eexists; split; constructor; auto. + - constructor. intros. + eapply sem_pred_exec_beq_correct; eauto. + eapply sem_pred_expr_corr; eauto. now apply sem_value_corr. + eapply HN.beq_pred_expr_correct_top; eauto. + { unfold "#r"; destruct_match; eauto using check_mutexcl_tree_correct, predicated_singleton. } + { unfold "#r"; destruct_match; eauto using check_mutexcl_tree_correct, predicated_singleton. } + eapply tree_beq_pred_expr; eauto. + - (* This is where three-valued logic would be needed. *) + constructor. intros. apply sem_pexpr_beq_correct with (p1 := a #p x0). + apply tree_beq_pred_pexpr; auto. + apply sem_pexpr_corr with (ictx:=ictx); auto. + - eapply sem_pred_exec_beq_correct; eauto. + eapply sem_pred_expr_corr; eauto. now apply sem_mem_corr. + eapply HN.beq_pred_expr_correct_top; eauto. + { unfold "#r"; destruct_match; eauto using check_mutexcl_tree_correct, predicated_singleton. } + { unfold "#r"; destruct_match; eauto using check_mutexcl_tree_correct, predicated_singleton. } + eapply tree_beq_pred_expr; eauto. + - eapply sem_pred_exec_beq_correct; eauto. + eapply sem_pred_expr_corr; eauto. now apply sem_exit_corr. + eapply EHN.beq_pred_expr_correct_top; eauto using check_mutexcl_correct. +Qed. + +(*| +Proof Sketch: + +Two abstract states can be equivalent, without it being obvious that they can +actually both be executed assuming one can be executed. One will therefore have +to add a few more assumptions to makes it possible to execute the other. + +It currently assumes that all the predicates in the predicate tree are +evaluable, which is actually something that can either be checked, or something +that can be proven constructively. I believe that it should already be possible +using the latter, so here it will only be assumed. + +Similarly, the current assumption is that mutual exclusivity of predicates is +being checked within the ``check`` function, which could possibly also be proven +constructively about the update function. This is a simpler short-term fix +though. +|*) + +End CORRECT. |