(* *********************************************************************) (* *) (* The Compcert verified compiler *) (* *) (* Xavier Leroy, INRIA Paris-Rocquencourt *) (* *) (* Copyright Institut National de Recherche en Informatique et en *) (* Automatique. All rights reserved. This file is distributed *) (* under the terms of the INRIA Non-Commercial License Agreement. *) (* *) (* *********************************************************************) (** Correctness proof for Cminor generation. *) Require Import Coq.Program.Equality. Require Import FSets. Require Import Coqlib. Require Intv. Require Import Errors. Require Import Maps. Require Import AST. Require Import Integers. Require Import Floats. Require Import Values. Require Import Memdata. Require Import Memory. Require Import Events. Require Import Globalenvs. Require Import Smallstep. Require Import Switch. Require Import Csharpminor. Require Import Cminor. Require Import Cminorgen. Open Local Scope error_monad_scope. Section TRANSLATION. Variable prog: Csharpminor.program. Variable tprog: program. Hypothesis TRANSL: transl_program prog = OK tprog. Let ge : Csharpminor.genv := Genv.globalenv prog. Let gvare : gvarenv := global_var_env prog. Let gve := (ge, gvare). Let gce : compilenv := build_global_compilenv prog. Let tge: genv := Genv.globalenv tprog. Lemma symbols_preserved: forall (s: ident), Genv.find_symbol tge s = Genv.find_symbol ge s. Proof (Genv.find_symbol_transf_partial2 (transl_fundef gce) transl_globvar _ TRANSL). Lemma function_ptr_translated: forall (b: block) (f: Csharpminor.fundef), Genv.find_funct_ptr ge b = Some f -> exists tf, Genv.find_funct_ptr tge b = Some tf /\ transl_fundef gce f = OK tf. Proof (Genv.find_funct_ptr_transf_partial2 (transl_fundef gce) transl_globvar _ TRANSL). Lemma functions_translated: forall (v: val) (f: Csharpminor.fundef), Genv.find_funct ge v = Some f -> exists tf, Genv.find_funct tge v = Some tf /\ transl_fundef gce f = OK tf. Proof (Genv.find_funct_transf_partial2 (transl_fundef gce) transl_globvar _ TRANSL). Lemma sig_preserved_body: forall f tf cenv size, transl_funbody cenv size f = OK tf -> tf.(fn_sig) = Csharpminor.fn_sig f. Proof. intros. monadInv H. reflexivity. Qed. Lemma sig_preserved: forall f tf, transl_fundef gce f = OK tf -> Cminor.funsig tf = Csharpminor.funsig f. Proof. intros until tf; destruct f; simpl. unfold transl_function. destruct (build_compilenv gce f). case (zle z Int.max_signed); simpl bind; try congruence. intros. monadInv H. simpl. eapply sig_preserved_body; eauto. intro. inv H. reflexivity. Qed. Definition global_compilenv_match (ce: compilenv) (gv: gvarenv) : Prop := forall id, match ce!!id with | Var_global_scalar chunk => gv!id = Some (Vscalar chunk) | Var_global_array => True | _ => False end. Lemma global_compilenv_charact: global_compilenv_match gce gvare. Proof. set (mkgve := fun gv (vars: list (ident * list init_data * var_kind)) => List.fold_left (fun gve x => match x with (id, init, k) => PTree.set id k gve end) vars gv). assert (forall vars gv ce, global_compilenv_match ce gv -> global_compilenv_match (List.fold_left assign_global_variable vars ce) (mkgve gv vars)). induction vars; simpl; intros. auto. apply IHvars. intro id1. unfold assign_global_variable. destruct a as [[id2 init2] lv2]. destruct lv2; simpl; rewrite PMap.gsspec; rewrite PTree.gsspec. case (peq id1 id2); intro. auto. apply H. case (peq id1 id2); intro. auto. apply H. change gvare with (mkgve (PTree.empty var_kind) prog.(prog_vars)). unfold gce, build_global_compilenv. apply H. intro. rewrite PMap.gi. auto. Qed. (** * Derived properties of memory operations *) Lemma load_freelist: forall fbl chunk m b ofs m', (forall b' lo hi, In (b', lo, hi) fbl -> b' <> b) -> Mem.free_list m fbl = Some m' -> Mem.load chunk m' b ofs = Mem.load chunk m b ofs. Proof. induction fbl; intros. simpl in H0. congruence. destruct a as [[b' lo] hi]. generalize H0. simpl. case_eq (Mem.free m b' lo hi); try congruence. intros m1 FR1 FRL. transitivity (Mem.load chunk m1 b ofs). eapply IHfbl; eauto. intros. eapply H. eauto with coqlib. eapply Mem.load_free; eauto. left. apply sym_not_equal. eapply H. auto with coqlib. Qed. Lemma perm_freelist: forall fbl m m' b ofs p, Mem.free_list m fbl = Some m' -> Mem.perm m' b ofs p -> Mem.perm m b ofs p. Proof. induction fbl; simpl; intros until p. congruence. destruct a as [[b' lo] hi]. case_eq (Mem.free m b' lo hi); try congruence. intros. eauto with mem. Qed. Lemma nextblock_freelist: forall fbl m m', Mem.free_list m fbl = Some m' -> Mem.nextblock m' = Mem.nextblock m. Proof. induction fbl; intros until m'; simpl. congruence. destruct a as [[b lo] hi]. case_eq (Mem.free m b lo hi); intros; try congruence. transitivity (Mem.nextblock m0). eauto. eapply Mem.nextblock_free; eauto. Qed. Lemma free_list_freeable: forall l m m', Mem.free_list m l = Some m' -> forall b lo hi, In (b, lo, hi) l -> Mem.range_perm m b lo hi Freeable. Proof. induction l; simpl; intros. contradiction. revert H. destruct a as [[b' lo'] hi']. caseEq (Mem.free m b' lo' hi'); try congruence. intros m1 FREE1 FREE2. destruct H0. inv H. eauto with mem. red; intros. eapply Mem.perm_free_3; eauto. exploit IHl; eauto. Qed. Lemma bounds_freelist: forall b l m m', Mem.free_list m l = Some m' -> Mem.bounds m' b = Mem.bounds m b. Proof. induction l; simpl; intros. inv H; auto. revert H. destruct a as [[b' lo'] hi']. caseEq (Mem.free m b' lo' hi'); try congruence. intros m1 FREE1 FREE2. transitivity (Mem.bounds m1 b). eauto. eapply Mem.bounds_free; eauto. Qed. Lemma nextblock_storev: forall chunk m addr v m', Mem.storev chunk m addr v = Some m' -> Mem.nextblock m' = Mem.nextblock m. Proof. unfold Mem.storev; intros. destruct addr; try discriminate. eapply Mem.nextblock_store; eauto. Qed. (** * Normalized values and operations over memory chunks *) (** A value is normalized with respect to a memory chunk if it is invariant under the cast (truncation, sign extension) corresponding to the chunk. *) Definition val_normalized (v: val) (chunk: memory_chunk) : Prop := Val.load_result chunk v = v. Lemma val_normalized_has_type: forall chunk v, val_normalized v chunk -> Val.has_type v (type_of_chunk chunk). Proof. intros until v; unfold val_normalized, Val.load_result. destruct chunk; destruct v; intro EQ; try (inv EQ); simpl; exact I. Qed. Lemma val_has_type_normalized: forall ty v, Val.has_type v ty -> val_normalized v (chunk_for_type ty). Proof. unfold Val.has_type, val_normalized; intros; destruct ty; destruct v; contradiction || reflexivity. Qed. Lemma chunktype_const_correct: forall c v, Csharpminor.eval_constant c = Some v -> val_normalized v (chunktype_const c). Proof. unfold Csharpminor.eval_constant; intros. destruct c; inv H; unfold val_normalized; auto. Qed. Lemma chunktype_unop_correct: forall op v1 v, Csharpminor.eval_unop op v1 = Some v -> val_normalized v (chunktype_unop op). Proof. intros; destruct op; simpl in *; unfold val_normalized. inv H. destruct v1; simpl; auto. rewrite Int.zero_ext_idem; auto. compute; auto. inv H. destruct v1; simpl; auto. rewrite Int.sign_ext_idem; auto. compute; auto. inv H. destruct v1; simpl; auto. rewrite Int.zero_ext_idem; auto. compute; auto. inv H. destruct v1; simpl; auto. rewrite Int.sign_ext_idem; auto. compute; auto. destruct v1; inv H; auto. destruct v1; inv H. destruct (Int.eq i Int.zero); auto. reflexivity. destruct v1; inv H; auto. destruct v1; inv H; auto. destruct v1; inv H; auto. inv H. destruct v1; simpl; auto. rewrite Float.singleoffloat_idem; auto. destruct v1; inv H; auto. destruct v1; inv H; auto. destruct v1; inv H; auto. destruct v1; inv H; auto. Qed. Lemma chunktype_binop_correct: forall op v1 v2 m v, Csharpminor.eval_binop op v1 v2 m = Some v -> val_normalized v (chunktype_binop op). Proof. intros; destruct op; simpl in *; unfold val_normalized; destruct v1; destruct v2; try (inv H; reflexivity). destruct (eq_block b b0); inv H; auto. destruct (Int.eq i0 Int.zero); inv H; auto. destruct (Int.eq i0 Int.zero); inv H; auto. destruct (Int.eq i0 Int.zero); inv H; auto. destruct (Int.eq i0 Int.zero); inv H; auto. destruct (Int.ltu i0 Int.iwordsize); inv H; auto. destruct (Int.ltu i0 Int.iwordsize); inv H; auto. destruct (Int.ltu i0 Int.iwordsize); inv H; auto. inv H; destruct (Int.cmp c i i0); reflexivity. unfold eval_compare_null in H. destruct (Int.eq i Int.zero). destruct c; inv H; auto. inv H. unfold eval_compare_null in H. destruct (Int.eq i0 Int.zero). destruct c; inv H; auto. inv H. destruct (Mem.valid_pointer m b (Int.signed i) && Mem.valid_pointer m b0 (Int.signed i0)). destruct (eq_block b b0); inv H. destruct (Int.cmp c i i0); auto. destruct c; inv H1; auto. inv H. inv H. destruct (Int.cmpu c i i0); auto. inv H. destruct (Float.cmp c f f0); auto. Qed. Lemma chunktype_compat_correct: forall src dst v, chunktype_compat src dst = true -> val_normalized v src -> val_normalized v dst. Proof. unfold val_normalized; intros. rewrite <- H0. destruct src; destruct dst; simpl in H; try discriminate; auto; destruct v; simpl; auto. Admitted. Lemma chunktype_merge_correct: forall c1 c2 c v, chunktype_merge c1 c2 = OK c -> val_normalized v c1 \/ val_normalized v c2 -> val_normalized v c. Proof. intros until v. unfold chunktype_merge. case_eq (chunktype_compat c1 c2). intros. inv H0. destruct H1. eapply chunktype_compat_correct; eauto. auto. case_eq (chunktype_compat c2 c1). intros. inv H1. destruct H2. auto. eapply chunktype_compat_correct; eauto. intros. destruct (typ_eq (type_of_chunk c1) (type_of_chunk c2)); inv H1. apply val_has_type_normalized. destruct H2. apply val_normalized_has_type. auto. rewrite e. apply val_normalized_has_type. auto. Qed. (** * Correspondence between Csharpminor's and Cminor's environments and memory states *) (** In Csharpminor, every variable is stored in a separate memory block. In the corresponding Cminor code, some of these variables reside in the local variable environment; others are sub-blocks of the stack data block. We capture these changes in memory via a memory injection [f]: - [f b = None] means that the Csharpminor block [b] no longer exist in the execution of the generated Cminor code. This corresponds to a Csharpminor local variable translated to a Cminor local variable. - [f b = Some(b', ofs)] means that Csharpminor block [b] corresponds to a sub-block of Cminor block [b] at offset [ofs]. A memory injection [f] defines a relation [val_inject f] between values and a relation [Mem.inject f] between memory states. These relations will be used intensively in our proof of simulation between Csharpminor and Cminor executions. In this section, we define the relation between Csharpminor and Cminor environments. *) (** Matching for a Csharpminor variable [id]. - If this variable is mapped to a Cminor local variable, the corresponding Csharpminor memory block [b] must no longer exist in Cminor ([f b = None]). Moreover, the content of block [b] must match the value of [id] found in the Cminor local environment [te]. - If this variable is mapped to a sub-block of the Cminor stack data at offset [ofs], the address of this variable in Csharpminor [Vptr b Int.zero] must match the address of the sub-block [Vptr sp (Int.repr ofs)]. *) Inductive match_var (f: meminj) (id: ident) (e: Csharpminor.env) (m: mem) (te: env) (sp: block) : var_info -> Prop := | match_var_local: forall chunk b v v', PTree.get id e = Some (b, Vscalar chunk) -> Mem.load chunk m b 0 = Some v -> f b = None -> PTree.get id te = Some v' -> val_inject f v v' -> match_var f id e m te sp (Var_local chunk) | match_var_stack_scalar: forall chunk ofs b, PTree.get id e = Some (b, Vscalar chunk) -> val_inject f (Vptr b Int.zero) (Vptr sp (Int.repr ofs)) -> match_var f id e m te sp (Var_stack_scalar chunk ofs) | match_var_stack_array: forall ofs sz b, PTree.get id e = Some (b, Varray sz) -> val_inject f (Vptr b Int.zero) (Vptr sp (Int.repr ofs)) -> match_var f id e m te sp (Var_stack_array ofs) | match_var_global_scalar: forall chunk, PTree.get id e = None -> PTree.get id gvare = Some (Vscalar chunk) -> match_var f id e m te sp (Var_global_scalar chunk) | match_var_global_array: PTree.get id e = None -> match_var f id e m te sp (Var_global_array). (** Matching between a Csharpminor environment [e] and a Cminor environment [te]. The [lo] and [hi] parameters delimit the range of addresses for the blocks referenced from [te]. *) Record match_env (f: meminj) (cenv: compilenv) (e: Csharpminor.env) (m: mem) (te: env) (sp: block) (lo hi: Z) : Prop := mk_match_env { (** Each variable mentioned in the compilation environment must match as defined above. *) me_vars: forall id, match_var f id e m te sp (PMap.get id cenv); (** [lo, hi] is a proper interval. *) me_low_high: lo <= hi; (** Every block appearing in the Csharpminor environment [e] must be in the range [lo, hi]. *) me_bounded: forall id b lv, PTree.get id e = Some(b, lv) -> lo <= b < hi; (** Distinct Csharpminor local variables must be mapped to distinct blocks. *) me_inj: forall id1 b1 lv1 id2 b2 lv2, PTree.get id1 e = Some(b1, lv1) -> PTree.get id2 e = Some(b2, lv2) -> id1 <> id2 -> b1 <> b2; (** All blocks mapped to sub-blocks of the Cminor stack data must be images of variables from the Csharpminor environment [e] *) me_inv: forall b delta, f b = Some(sp, delta) -> exists id, exists lv, PTree.get id e = Some(b, lv); (** All Csharpminor blocks below [lo] (i.e. allocated before the blocks referenced from [e]) must map to blocks that are below [sp] (i.e. allocated before the stack data for the current Cminor function). *) me_incr: forall b tb delta, f b = Some(tb, delta) -> b < lo -> tb < sp; (** The sizes of blocks appearing in [e] agree with their types *) me_bounds: forall id b lv, PTree.get id e = Some(b, lv) -> Mem.bounds m b = (0, sizeof lv) }. Hint Resolve me_low_high. (** Global environments match if the memory injection [f] leaves unchanged the references to global symbols and functions. *) Record match_globalenvs (f: meminj) : Prop := mk_match_globalenvs { mg_symbols: forall id b, Genv.find_symbol ge id = Some b -> f b = Some (b, 0) /\ Genv.find_symbol tge id = Some b; mg_functions: forall b, b < 0 -> f b = Some(b, 0) }. (** The remainder of this section is devoted to showing preservation of the [match_en] invariant under various assignments and memory stores. First: preservation by memory stores to ``mapped'' blocks (block that have a counterpart in the Cminor execution). *) Ltac geninv x := let H := fresh in (generalize x; intro H; inv H). Lemma match_env_store_mapped: forall f cenv e m1 m2 te sp lo hi chunk b ofs v, f b <> None -> Mem.store chunk m1 b ofs v = Some m2 -> match_env f cenv e m1 te sp lo hi -> match_env f cenv e m2 te sp lo hi. Proof. intros; inv H1; constructor; auto. (* vars *) intros. geninv (me_vars0 id); econstructor; eauto. rewrite <- H4. eapply Mem.load_store_other; eauto. left. congruence. (* bounds *) intros. rewrite (Mem.bounds_store _ _ _ _ _ _ H0). eauto. Qed. (** Preservation by assignment to a Csharpminor variable that is translated to a Cminor local variable. The value being assigned must be normalized with respect to the memory chunk of the variable, in the following sense. *) Lemma match_env_store_local: forall f cenv e m1 m2 te sp lo hi id b chunk v tv, e!id = Some(b, Vscalar chunk) -> Val.has_type v (type_of_chunk chunk) -> val_inject f (Val.load_result chunk v) tv -> Mem.store chunk m1 b 0 v = Some m2 -> match_env f cenv e m1 te sp lo hi -> match_env f cenv e m2 (PTree.set id tv te) sp lo hi. Proof. intros. inv H3. constructor; auto. (* vars *) intros. geninv (me_vars0 id0). (* var_local *) case (peq id id0); intro. (* the stored variable *) subst id0. assert (b0 = b) by congruence. subst. assert (chunk0 = chunk) by congruence. subst. econstructor. eauto. eapply Mem.load_store_same; eauto. auto. rewrite PTree.gss. reflexivity. auto. (* a different variable *) econstructor; eauto. rewrite <- H6. eapply Mem.load_store_other; eauto. rewrite PTree.gso; auto. (* var_stack_scalar *) econstructor; eauto. (* var_stack_array *) econstructor; eauto. (* var_global_scalar *) econstructor; eauto. (* var_global_array *) econstructor; eauto. (* bounds *) intros. rewrite (Mem.bounds_store _ _ _ _ _ _ H2). eauto. Qed. (** The [match_env] relation is preserved by any memory operation that preserves sizes and loads from blocks in the [lo, hi] range. *) Lemma match_env_invariant: forall f cenv e m1 m2 te sp lo hi, (forall b ofs chunk v, lo <= b < hi -> Mem.load chunk m1 b ofs = Some v -> Mem.load chunk m2 b ofs = Some v) -> (forall b, lo <= b < hi -> Mem.bounds m2 b = Mem.bounds m1 b) -> match_env f cenv e m1 te sp lo hi -> match_env f cenv e m2 te sp lo hi. Proof. intros. inv H1. constructor; eauto. (* vars *) intros. geninv (me_vars0 id); econstructor; eauto. (* bounds *) intros. rewrite H0. eauto. eauto. Qed. (** [match_env] is insensitive to the Cminor values of stack-allocated data. *) Lemma match_env_extensional: forall f cenv e m te1 sp lo hi te2, match_env f cenv e m te1 sp lo hi -> (forall id chunk, cenv!!id = Var_local chunk -> te2!id = te1!id) -> match_env f cenv e m te2 sp lo hi. Proof. intros. inv H; econstructor; eauto. intros. geninv (me_vars0 id); econstructor; eauto. rewrite <- H5. eauto. Qed. (** [match_env] and allocations *) Inductive alloc_condition: var_info -> var_kind -> block -> option (block * Z) -> Prop := | alloc_cond_local: forall chunk sp, alloc_condition (Var_local chunk) (Vscalar chunk) sp None | alloc_cond_stack_scalar: forall chunk pos sp, alloc_condition (Var_stack_scalar chunk pos) (Vscalar chunk) sp (Some(sp, pos)) | alloc_cond_stack_array: forall pos sz sp, alloc_condition (Var_stack_array pos) (Varray sz) sp (Some(sp, pos)). Lemma match_env_alloc_same: forall f1 cenv e m1 te sp lo lv m2 b f2 id info tv, match_env f1 cenv e m1 te sp lo (Mem.nextblock m1) -> Mem.alloc m1 0 (sizeof lv) = (m2, b) -> inject_incr f1 f2 -> alloc_condition info lv sp (f2 b) -> (forall b', b' <> b -> f2 b' = f1 b') -> te!id = Some tv -> e!id = None -> match_env f2 (PMap.set id info cenv) (PTree.set id (b, lv) e) m2 te sp lo (Mem.nextblock m2). Proof. intros until tv. intros ME ALLOC INCR ACOND OTHER TE E. (* assert (ALLOC_RES: b = Mem.nextblock m1) by eauto with mem. assert (ALLOC_NEXT: Mem.nextblock m2 = Zsucc(Mem.nextblock m1)) by eauto with mem. *) inv ME; constructor. (* vars *) intros. rewrite PMap.gsspec. destruct (peq id0 id). subst id0. (* the new var *) inv ACOND; econstructor. (* local *) rewrite PTree.gss. reflexivity. eapply Mem.load_alloc_same'; eauto. omega. simpl; omega. apply Zdivide_0. auto. eauto. constructor. (* stack scalar *) rewrite PTree.gss; reflexivity. econstructor; eauto. rewrite Int.add_commut; rewrite Int.add_zero; auto. (* stack array *) rewrite PTree.gss; reflexivity. econstructor; eauto. rewrite Int.add_commut; rewrite Int.add_zero; auto. (* the other vars *) geninv (me_vars0 id0); econstructor. (* local *) rewrite PTree.gso; eauto. eapply Mem.load_alloc_other; eauto. rewrite OTHER; auto. exploit me_bounded0; eauto. exploit Mem.alloc_result; eauto. unfold block; omega. eauto. eapply val_inject_incr; eauto. (* stack scalar *) rewrite PTree.gso; eauto. eapply val_inject_incr; eauto. (* stack array *) rewrite PTree.gso; eauto. eapply val_inject_incr; eauto. (* global scalar *) rewrite PTree.gso; auto. auto. (* global array *) rewrite PTree.gso; auto. (* low high *) exploit Mem.nextblock_alloc; eauto. unfold block in *; omega. (* bounded *) exploit Mem.alloc_result; eauto. intro RES. exploit Mem.nextblock_alloc; eauto. intro NEXT. intros until lv0. rewrite PTree.gsspec. destruct (peq id0 id); intro EQ. inv EQ. unfold block in *; omega. exploit me_bounded0; eauto. unfold block in *; omega. (* inj *) intros until lv2. repeat rewrite PTree.gsspec. exploit Mem.alloc_result; eauto. intro RES. destruct (peq id1 id); destruct (peq id2 id); subst; intros A1 A2 DIFF. congruence. inv A1. exploit me_bounded0; eauto. unfold block; omega. inv A2. exploit me_bounded0; eauto. unfold block; omega. eauto. (* inv *) intros. destruct (zeq b0 b). subst. exists id; exists lv. apply PTree.gss. exploit me_inv0; eauto. rewrite <- OTHER; eauto. intros [id' [lv' A]]. exists id'; exists lv'. rewrite PTree.gso; auto. congruence. (* incr *) intros. eapply me_incr0; eauto. rewrite <- OTHER; eauto. exploit Mem.alloc_result; eauto. unfold block; omega. (* bounds *) intros. rewrite PTree.gsspec in H. rewrite (Mem.bounds_alloc _ _ _ _ _ ALLOC). destruct (peq id0 id). inv H. apply dec_eq_true. rewrite dec_eq_false. eauto. apply Mem.valid_not_valid_diff with m1. exploit me_bounded0; eauto. intros [A B]. auto. eauto with mem. Qed. Lemma match_env_alloc_other: forall f1 cenv e m1 te sp lo hi sz m2 b f2, match_env f1 cenv e m1 te sp lo hi -> Mem.alloc m1 0 sz = (m2, b) -> inject_incr f1 f2 -> (forall b', b' <> b -> f2 b' = f1 b') -> hi <= b -> match f2 b with None => True | Some(b',ofs) => sp < b' end -> match_env f2 cenv e m2 te sp lo hi. Proof. intros until f2; intros ME ALLOC INCR OTHER BOUND TBOUND. inv ME. assert (BELOW: forall id b' lv, e!id = Some(b', lv) -> b' <> b). intros. exploit me_bounded0; eauto. exploit Mem.alloc_result; eauto. unfold block in *; omega. econstructor; eauto. (* vars *) intros. geninv (me_vars0 id); econstructor. (* local *) eauto. eapply Mem.load_alloc_other; eauto. rewrite OTHER; eauto. eauto. eapply val_inject_incr; eauto. (* stack scalar *) eauto. eapply val_inject_incr; eauto. (* stack array *) eauto. eapply val_inject_incr; eauto. (* global scalar *) auto. auto. (* global array *) auto. (* inv *) intros. rewrite OTHER in H. eauto. red; intro; subst b0. rewrite H in TBOUND. omegaContradiction. (* incr *) intros. eapply me_incr0; eauto. rewrite <- OTHER; eauto. exploit Mem.alloc_result; eauto. unfold block in *; omega. (* bounds *) intros. rewrite (Mem.bounds_alloc_other _ _ _ _ _ ALLOC). eauto. exploit me_bounded0; eauto. Qed. (** [match_env] and external calls *) Remark inject_incr_separated_same: forall f1 f2 m1 m1', inject_incr f1 f2 -> inject_separated f1 f2 m1 m1' -> forall b, Mem.valid_block m1 b -> f2 b = f1 b. Proof. intros. case_eq (f1 b). intros [b' delta] EQ. apply H; auto. intros EQ. case_eq (f2 b). intros [b'1 delta1] EQ1. exploit H0; eauto. intros [C D]. contradiction. auto. Qed. Remark inject_incr_separated_same': forall f1 f2 m1 m1', inject_incr f1 f2 -> inject_separated f1 f2 m1 m1' -> forall b b' delta, f2 b = Some(b', delta) -> Mem.valid_block m1' b' -> f1 b = Some(b', delta). Proof. intros. case_eq (f1 b). intros [b'1 delta1] EQ. exploit H; eauto. congruence. intros. exploit H0; eauto. intros [C D]. contradiction. Qed. Lemma match_env_external_call: forall f1 cenv e m1 te sp lo hi m2 f2 m1', match_env f1 cenv e m1 te sp lo hi -> mem_unchanged_on (loc_unmapped f1) m1 m2 -> inject_incr f1 f2 -> inject_separated f1 f2 m1 m1' -> (forall b, Mem.valid_block m1 b -> Mem.bounds m2 b = Mem.bounds m1 b) -> hi <= Mem.nextblock m1 -> sp < Mem.nextblock m1' -> match_env f2 cenv e m2 te sp lo hi. Proof. intros until m1'. intros ME UNCHANGED INCR SEPARATED BOUNDS VALID VALID'. destruct UNCHANGED as [UNCHANGED1 UNCHANGED2]. inversion ME. constructor; auto. (* vars *) intros. geninv (me_vars0 id); try (econstructor; eauto; fail). (* local *) econstructor. eauto. apply UNCHANGED2; eauto. rewrite <- H3. eapply inject_incr_separated_same; eauto. red. exploit me_bounded0; eauto. omega. eauto. eauto. (* inv *) intros. apply me_inv0 with delta. eapply inject_incr_separated_same'; eauto. (* incr *) intros. exploit inject_incr_separated_same; eauto. instantiate (1 := b). red; omega. intros. apply me_incr0 with b delta. congruence. auto. (* bounds *) intros. rewrite BOUNDS; eauto. red. exploit me_bounded0; eauto. omega. Qed. (** * Invariant on abstract call stacks *) (** Call stacks represent abstractly the execution state of the current Csharpminor and Cminor functions, as well as the states of the calling functions. A call stack is a list of frames, each frame collecting information on the current execution state of a Csharpminor function and its Cminor translation. *) Inductive frame : Type := Frame(cenv: compilenv) (tf: Cminor.function) (e: Csharpminor.env) (te: Cminor.env) (sp: block) (lo hi: Z). Definition callstack : Type := list frame. (** Matching of call stacks imply: - matching of environments for each of the frames - matching of the global environments - separation conditions over the memory blocks allocated for C#minor local variables; - separation conditions over the memory blocks allocated for Cminor stack data; - freeable permissions on the parts of the Cminor stack data blocks that are not images of C#minor local variable blocks. *) Definition padding_freeable (f: meminj) (m: mem) (tm: mem) (sp: block) (sz: Z) : Prop := forall ofs, 0 <= ofs < sz -> Mem.perm tm sp ofs Freeable \/ exists b, exists delta, f b = Some(sp, delta) /\ Mem.low_bound m b + delta <= ofs < Mem.high_bound m b + delta. Inductive match_callstack (f: meminj) (m: mem) (tm: mem): callstack -> Z -> Z -> Prop := | mcs_nil: forall bound tbound, match_globalenvs f -> match_callstack f m tm nil bound tbound | mcs_cons: forall cenv tf e te sp lo hi cs bound tbound (BOUND: hi <= bound) (TBOUND: sp < tbound) (MENV: match_env f cenv e m te sp lo hi) (PERM: padding_freeable f m tm sp tf.(fn_stackspace)) (MCS: match_callstack f m tm cs lo sp), match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) bound tbound. (** [match_callstack] implies [match_globalenvs]. *) Lemma match_callstack_match_globalenvs: forall f m tm cs bound tbound, match_callstack f m tm cs bound tbound -> match_globalenvs f. Proof. induction 1; eauto. Qed. (** We now show invariance properties for [match_callstack] that generalize those for [match_env]. *) Lemma padding_freeable_invariant: forall f1 m1 tm1 sp sz cenv e te lo hi f2 m2 tm2, padding_freeable f1 m1 tm1 sp sz -> match_env f1 cenv e m1 te sp lo hi -> (forall ofs, Mem.perm tm1 sp ofs Freeable -> Mem.perm tm2 sp ofs Freeable) -> (forall b, b < hi -> Mem.bounds m2 b = Mem.bounds m1 b) -> (forall b, b < hi -> f2 b = f1 b) -> padding_freeable f2 m2 tm2 sp sz. Proof. intros; red; intros. exploit H; eauto. intros [A | [b [delta [A B]]]]. left; auto. exploit me_inv; eauto. intros [id [lv C]]. exploit me_bounded; eauto. intros [D E]. right; exists b; exists delta. split. rewrite H3; auto. rewrite H2; auto. Qed. Lemma match_callstack_store_mapped: forall f m tm chunk b b' delta ofs ofs' v tv m' tm', f b = Some(b', delta) -> Mem.store chunk m b ofs v = Some m' -> Mem.store chunk tm b' ofs' tv = Some tm' -> forall cs lo hi, match_callstack f m tm cs lo hi -> match_callstack f m' tm' cs lo hi. Proof. induction 4. constructor; auto. constructor; auto. eapply match_env_store_mapped; eauto. congruence. eapply padding_freeable_invariant; eauto. intros; eauto with mem. intros. eapply Mem.bounds_store; eauto. Qed. Lemma match_callstack_storev_mapped: forall f m tm chunk a ta v tv m' tm', val_inject f a ta -> Mem.storev chunk m a v = Some m' -> Mem.storev chunk tm ta tv = Some tm' -> forall cs lo hi, match_callstack f m tm cs lo hi -> match_callstack f m' tm' cs lo hi. Proof. intros. destruct a; simpl in H0; try discriminate. inv H. simpl in H1. eapply match_callstack_store_mapped; eauto. Qed. Lemma match_callstack_invariant: forall f m tm cs bound tbound, match_callstack f m tm cs bound tbound -> forall m' tm', (forall cenv e te sp lo hi, hi <= bound -> match_env f cenv e m te sp lo hi -> match_env f cenv e m' te sp lo hi) -> (forall b, b < bound -> Mem.bounds m' b = Mem.bounds m b) -> (forall b ofs p, b < tbound -> Mem.perm tm b ofs p -> Mem.perm tm' b ofs p) -> match_callstack f m' tm' cs bound tbound. Proof. induction 1; intros. constructor; auto. constructor; auto. eapply padding_freeable_invariant; eauto. intros. apply H1. omega. eapply IHmatch_callstack; eauto. intros. eapply H0; eauto. inv MENV; omega. intros. apply H1; auto. inv MENV; omega. intros. apply H2; auto. omega. Qed. Lemma match_callstack_store_local: forall f cenv e te sp lo hi cs bound tbound m1 m2 tm tf id b chunk v tv, e!id = Some(b, Vscalar chunk) -> Val.has_type v (type_of_chunk chunk) -> val_inject f (Val.load_result chunk v) tv -> Mem.store chunk m1 b 0 v = Some m2 -> match_callstack f m1 tm (Frame cenv tf e te sp lo hi :: cs) bound tbound -> match_callstack f m2 tm (Frame cenv tf e (PTree.set id tv te) sp lo hi :: cs) bound tbound. Proof. intros. inv H3. constructor; auto. eapply match_env_store_local; eauto. eapply padding_freeable_invariant; eauto. intros. eapply Mem.bounds_store; eauto. eapply match_callstack_invariant; eauto. intros. apply match_env_invariant with m1; auto. intros. rewrite <- H6. eapply Mem.load_store_other; eauto. left. inv MENV. exploit me_bounded0; eauto. unfold block in *; omega. intros. eapply Mem.bounds_store; eauto. intros. eapply Mem.bounds_store; eauto. Qed. (** A variant of [match_callstack_store_local] where the Cminor environment [te] already associates to [id] a value that matches the assigned value. In this case, [match_callstack] is preserved even if no assignment takes place on the Cminor side. *) Lemma match_callstack_store_local_unchanged: forall f cenv e te sp lo hi cs bound tbound m1 m2 id b chunk v tv tf tm, e!id = Some(b, Vscalar chunk) -> Val.has_type v (type_of_chunk chunk) -> val_inject f (Val.load_result chunk v) tv -> Mem.store chunk m1 b 0 v = Some m2 -> te!id = Some tv -> match_callstack f m1 tm (Frame cenv tf e te sp lo hi :: cs) bound tbound -> match_callstack f m2 tm (Frame cenv tf e te sp lo hi :: cs) bound tbound. Proof. intros. exploit match_callstack_store_local; eauto. intro MCS. inv MCS. constructor; auto. eapply match_env_extensional; eauto. intros. rewrite PTree.gsspec. case (peq id0 id); intros. congruence. auto. Qed. Lemma match_callstack_incr_bound: forall f m tm cs bound tbound bound' tbound', match_callstack f m tm cs bound tbound -> bound <= bound' -> tbound <= tbound' -> match_callstack f m tm cs bound' tbound'. Proof. intros. inversion H; constructor; auto. omega. omega. Qed. (** Preservation of [match_callstack] by freeing all blocks allocated for local variables at function entry (on the Csharpminor side) and simultaneously freeing the Cminor stack data block. *) Lemma in_blocks_of_env: forall e id b lv, e!id = Some(b, lv) -> In (b, 0, sizeof lv) (blocks_of_env e). Proof. unfold blocks_of_env; intros. change (b, 0, sizeof lv) with (block_of_binding (id, (b, lv))). apply List.in_map. apply PTree.elements_correct. auto. Qed. Lemma in_blocks_of_env_inv: forall b lo hi e, In (b, lo, hi) (blocks_of_env e) -> exists id, exists lv, e!id = Some(b, lv) /\ lo = 0 /\ hi = sizeof lv. Proof. unfold blocks_of_env; intros. exploit list_in_map_inv; eauto. intros [[id [b' lv]] [A B]]. unfold block_of_binding in A. inv A. exists id; exists lv; intuition. apply PTree.elements_complete. auto. Qed. (* Lemma free_list_perm: forall l m m', Mem.free_list m l = Some m' -> forall b ofs p, Mem.perm m' b ofs p -> Mem.perm m b ofs p. Proof. induction l; simpl; intros. inv H; auto. revert H. destruct a as [[b' lo'] hi']. caseEq (Mem.free m b' lo' hi'); try congruence. intros m1 FREE1 FREE2. eauto with mem. Qed. *) Lemma match_callstack_freelist: forall f cenv tf e te sp lo hi cs m m' tm, Mem.inject f m tm -> Mem.free_list m (blocks_of_env e) = Some m' -> match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm) -> exists tm', Mem.free tm sp 0 tf.(fn_stackspace) = Some tm' /\ match_callstack f m' tm' cs (Mem.nextblock m') (Mem.nextblock tm') /\ Mem.inject f m' tm'. Proof. intros until tm; intros INJ FREELIST MCS. inv MCS. inv MENV. assert ({tm' | Mem.free tm sp 0 (fn_stackspace tf) = Some tm'}). apply Mem.range_perm_free. red; intros. exploit PERM; eauto. intros [A | [b [delta [A B]]]]. auto. exploit me_inv0; eauto. intros [id [lv C]]. exploit me_bounds0; eauto. intro D. rewrite D in B; simpl in B. assert (Mem.range_perm m b 0 (sizeof lv) Freeable). eapply free_list_freeable; eauto. eapply in_blocks_of_env; eauto. replace ofs with ((ofs - delta) + delta) by omega. eapply Mem.perm_inject; eauto. apply H0. omega. destruct X as [tm' FREE]. exploit nextblock_freelist; eauto. intro NEXT. exploit Mem.nextblock_free; eauto. intro NEXT'. exists tm'. split. auto. split. rewrite NEXT; rewrite NEXT'. apply match_callstack_incr_bound with lo sp; try omega. apply match_callstack_invariant with m tm; auto. intros. apply match_env_invariant with m; auto. intros. rewrite <- H2. eapply load_freelist; eauto. intros. exploit in_blocks_of_env_inv; eauto. intros [id [lv [A [B C]]]]. exploit me_bounded0; eauto. unfold block; omega. intros. eapply bounds_freelist; eauto. intros. eapply bounds_freelist; eauto. intros. eapply Mem.perm_free_1; eauto. left; unfold block; omega. eapply Mem.free_inject; eauto. intros. exploit me_inv0; eauto. intros [id [lv A]]. exists 0; exists (sizeof lv); split. eapply in_blocks_of_env; eauto. exploit me_bounds0; eauto. intro B. exploit Mem.perm_in_bounds; eauto. rewrite B; simpl. auto. Qed. (** Preservation of [match_callstack] by allocations. *) Lemma match_callstack_alloc_below: forall f1 m1 tm sz m2 b f2, Mem.alloc m1 0 sz = (m2, b) -> inject_incr f1 f2 -> (forall b', b' <> b -> f2 b' = f1 b') -> forall cs bound tbound, match_callstack f1 m1 tm cs bound tbound -> bound <= b -> match f2 b with None => True | Some(b',ofs) => tbound <= b' end -> match_callstack f2 m2 tm cs bound tbound. Proof. induction 4; intros. constructor. inv H2. constructor. intros. exploit mg_symbols0; eauto. intros [A B]. auto. intros. rewrite H1; auto. exploit Mem.alloc_result; eauto. generalize (Mem.nextblock_pos m1). unfold block; omega. constructor; auto. eapply match_env_alloc_other; eauto. omega. destruct (f2 b); auto. destruct p; omega. eapply padding_freeable_invariant; eauto. intros. eapply Mem.bounds_alloc_other; eauto. unfold block; omega. intros. apply H1. unfold block; omega. apply IHmatch_callstack. inv MENV; omega. destruct (f2 b); auto. destruct p; omega. Qed. Lemma match_callstack_alloc_left: forall f1 m1 tm cenv tf e te sp lo cs lv m2 b f2 info id tv, match_callstack f1 m1 tm (Frame cenv tf e te sp lo (Mem.nextblock m1) :: cs) (Mem.nextblock m1) (Mem.nextblock tm) -> Mem.alloc m1 0 (sizeof lv) = (m2, b) -> inject_incr f1 f2 -> alloc_condition info lv sp (f2 b) -> (forall b', b' <> b -> f2 b' = f1 b') -> te!id = Some tv -> e!id = None -> match_callstack f2 m2 tm (Frame (PMap.set id info cenv) tf (PTree.set id (b, lv) e) te sp lo (Mem.nextblock m2) :: cs) (Mem.nextblock m2) (Mem.nextblock tm). Proof. intros until tv; intros MCS ALLOC INCR ACOND OTHER TE E. inv MCS. exploit Mem.alloc_result; eauto. intro RESULT. exploit Mem.nextblock_alloc; eauto. intro NEXT. constructor. omega. auto. eapply match_env_alloc_same; eauto. eapply padding_freeable_invariant; eauto. intros. eapply Mem.bounds_alloc_other; eauto. unfold block in *; omega. intros. apply OTHER. unfold block in *; omega. eapply match_callstack_alloc_below; eauto. inv MENV. unfold block in *; omega. inv ACOND. auto. omega. omega. Qed. Lemma match_callstack_alloc_right: forall f m tm cs tf sp tm' te, match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm) -> Mem.alloc tm 0 tf.(fn_stackspace) = (tm', sp) -> Mem.inject f m tm -> match_callstack f m tm' (Frame gce tf empty_env te sp (Mem.nextblock m) (Mem.nextblock m) :: cs) (Mem.nextblock m) (Mem.nextblock tm'). Proof. intros. exploit Mem.alloc_result; eauto. intro RES. exploit Mem.nextblock_alloc; eauto. intro NEXT. constructor. omega. unfold block in *; omega. (* match env *) constructor. (* vars *) intros. generalize (global_compilenv_charact id); intro. destruct (gce!!id); try contradiction. constructor. apply PTree.gempty. auto. constructor. apply PTree.gempty. (* low high *) omega. (* bounded *) intros. rewrite PTree.gempty in H2. congruence. (* inj *) intros. rewrite PTree.gempty in H2. congruence. (* inv *) intros. assert (sp <> sp). apply Mem.valid_not_valid_diff with tm. eapply Mem.valid_block_inject_2; eauto. eauto with mem. tauto. (* incr *) intros. rewrite RES. change (Mem.valid_block tm tb). eapply Mem.valid_block_inject_2; eauto. (* bounds *) unfold empty_env; intros. rewrite PTree.gempty in H2. congruence. (* padding freeable *) red; intros. left. eapply Mem.perm_alloc_2; eauto. (* previous call stack *) rewrite RES. apply match_callstack_invariant with m tm; auto. intros. eapply Mem.perm_alloc_1; eauto. Qed. (** Decidability of the predicate "this is not a padding location" *) Definition is_reachable (f: meminj) (m: mem) (sp: block) (ofs: Z) : Prop := exists b, exists delta, f b = Some(sp, delta) /\ Mem.low_bound m b + delta <= ofs < Mem.high_bound m b + delta. Lemma is_reachable_dec: forall f cenv e m te sp lo hi ofs, match_env f cenv e m te sp lo hi -> {is_reachable f m sp ofs} + {~is_reachable f m sp ofs}. Proof. intros. set (P := fun (b: block) => match f b with | None => False | Some(b', delta) => b' = sp /\ Mem.low_bound m b + delta <= ofs < Mem.high_bound m b + delta end). assert ({forall b, Intv.In b (lo, hi) -> ~P b} + {exists b, Intv.In b (lo, hi) /\ P b}). apply Intv.forall_dec. intro b. unfold P. destruct (f b) as [[b' delta] | ]. destruct (eq_block b' sp). destruct (zle (Mem.low_bound m b + delta) ofs). destruct (zlt ofs (Mem.high_bound m b + delta)). right; auto. left; intuition. left; intuition. left; intuition. left; intuition. inv H. destruct H0. right; red; intros [b [delta [A [B C]]]]. elim (n b). exploit me_inv0; eauto. intros [id [lv D]]. exploit me_bounded0; eauto. red. rewrite A. auto. left. destruct e0 as [b [A B]]. red in B; revert B. case_eq (f b). intros [b' delta] EQ [C [D E]]. subst b'. exists b; exists delta. auto. tauto. Qed. (** Preservation of [match_callstack] by external calls. *) Lemma match_callstack_external_call: forall f1 f2 m1 m2 m1' m2', mem_unchanged_on (loc_unmapped f1) m1 m2 -> mem_unchanged_on (loc_out_of_reach f1 m1) m1' m2' -> inject_incr f1 f2 -> inject_separated f1 f2 m1 m1' -> (forall b, Mem.valid_block m1 b -> Mem.bounds m2 b = Mem.bounds m1 b) -> forall cs bound tbound, match_callstack f1 m1 m1' cs bound tbound -> bound <= Mem.nextblock m1 -> tbound <= Mem.nextblock m1' -> match_callstack f2 m2 m2' cs bound tbound. Proof. intros until m2'. intros UNMAPPED OUTOFREACH INCR SEPARATED BOUNDS. destruct OUTOFREACH as [OUTOFREACH1 OUTOFREACH2]. induction 1; intros; constructor. (* base case *) constructor; intros. exploit mg_symbols; eauto. intros [A B]. auto. replace (f2 b) with (f1 b). eapply mg_functions; eauto. symmetry. eapply inject_incr_separated_same; eauto. red. generalize (Mem.nextblock_pos m1); omega. (* inductive case *) auto. auto. eapply match_env_external_call; eauto. omega. omega. (* padding-freeable *) red; intros. destruct (is_reachable_dec _ _ _ _ _ _ _ _ ofs MENV). destruct i as [b [delta [A B]]]. right; exists b; exists delta; split. apply INCR; auto. rewrite BOUNDS. auto. exploit me_inv; eauto. intros [id [lv C]]. exploit me_bounded; eauto. intros. red; omega. exploit PERM; eauto. intros [A|A]; try contradiction. left. apply OUTOFREACH1; auto. red; intros. assert ((ofs < Mem.low_bound m1 b0 + delta \/ ofs >= Mem.high_bound m1 b0 + delta) \/ Mem.low_bound m1 b0 + delta <= ofs < Mem.high_bound m1 b0 + delta) by omega. destruct H4; auto. elim n. exists b0; exists delta; auto. (* induction *) eapply IHmatch_callstack; eauto. inv MENV; omega. omega. Qed. Remark external_call_nextblock_incr: forall ef vargs m1 t vres m2, external_call ef vargs m1 t vres m2 -> Mem.nextblock m1 <= Mem.nextblock m2. Proof. intros. generalize (external_call_valid_block _ _ _ _ _ _ (Mem.nextblock m1 - 1) H). unfold Mem.valid_block. omega. Qed. (** * Soundness of chunk and type inference. *) Lemma load_normalized: forall chunk m b ofs v, Mem.load chunk m b ofs = Some v -> val_normalized v chunk. Proof. intros. exploit Mem.load_type; eauto. intro TY. exploit Mem.load_cast; eauto. intro CST. red. destruct chunk; destruct v; simpl in *; auto; contradiction. Qed. Lemma chunktype_expr_correct: forall f m tm cenv tf e te sp lo hi cs bound tbound, match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) bound tbound -> forall a v, Csharpminor.eval_expr gve e m a v -> forall chunk (CTE: chunktype_expr cenv a = OK chunk), val_normalized v chunk. Proof. intros until tbound; intro MCS. induction 1; intros; try (monadInv CTE). (* var *) assert (chunk0 = chunk). unfold chunktype_expr in CTE. inv MCS. inv MENV. generalize (me_vars0 id); intro MV. inv MV; rewrite <- H1 in CTE; monadInv CTE; inv H; try congruence. unfold gve in H6. simpl in H6. congruence. subst chunk0. inv H; exploit load_normalized; eauto. unfold val_normalized; auto. (* const *) eapply chunktype_const_correct; eauto. (* unop *) eapply chunktype_unop_correct; eauto. (* binop *) eapply chunktype_binop_correct; eauto. (* load *) destruct v1; simpl in H0; try discriminate. eapply load_normalized; eauto. (* cond *) eapply chunktype_merge_correct; eauto. destruct vb1; eauto. Qed. Lemma type_expr_correct: forall f m tm cenv tf e te sp lo hi cs bound tbound, match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) bound tbound -> forall a v ty, Csharpminor.eval_expr gve e m a v -> type_expr cenv a = OK ty -> Val.has_type v ty. Proof. intros. monadInv H1. apply val_normalized_has_type. eapply chunktype_expr_correct; eauto. Qed. Lemma type_exprlist_correct: forall f m tm cenv tf e te sp lo hi cs bound tbound, match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) bound tbound -> forall al vl tyl, Csharpminor.eval_exprlist gve e m al vl -> type_exprlist cenv al = OK tyl -> Val.has_type_list vl tyl. Proof. intros. monadInv H1. generalize al vl H0 tyl H2. induction 1; intros. inv H3. simpl. auto. inv H5. simpl. split. eapply type_expr_correct; eauto. auto. Qed. (** * Correctness of Cminor construction functions *) Remark val_inject_val_of_bool: forall f b, val_inject f (Val.of_bool b) (Val.of_bool b). Proof. intros; destruct b; unfold Val.of_bool, Vtrue, Vfalse; constructor. Qed. Remark val_inject_eval_compare_mismatch: forall f c v, eval_compare_mismatch c = Some v -> val_inject f v v. Proof. unfold eval_compare_mismatch; intros. destruct c; inv H; unfold Vfalse, Vtrue; constructor. Qed. Remark val_inject_eval_compare_null: forall f i c v, eval_compare_null c i = Some v -> val_inject f v v. Proof. unfold eval_compare_null. intros. destruct (Int.eq i Int.zero). eapply val_inject_eval_compare_mismatch; eauto. discriminate. Qed. Hint Resolve eval_Econst eval_Eunop eval_Ebinop eval_Eload: evalexpr. Ltac TrivialOp := match goal with | [ |- exists y, _ /\ val_inject _ (Vint ?x) _ ] => exists (Vint x); split; [eauto with evalexpr | constructor] | [ |- exists y, _ /\ val_inject _ (Vfloat ?x) _ ] => exists (Vfloat x); split; [eauto with evalexpr | constructor] | [ |- exists y, _ /\ val_inject _ (Val.of_bool ?x) _ ] => exists (Val.of_bool x); split; [eauto with evalexpr | apply val_inject_val_of_bool] | [ |- exists y, Some ?x = Some y /\ val_inject _ _ _ ] => exists x; split; [auto | econstructor; eauto] | _ => idtac end. (** Correctness of [transl_constant]. *) Lemma transl_constant_correct: forall f sp cst v, Csharpminor.eval_constant cst = Some v -> exists tv, eval_constant tge sp (transl_constant cst) = Some tv /\ val_inject f v tv. Proof. destruct cst; simpl; intros; inv H; TrivialOp. Qed. (** Compatibility of [eval_unop] with respect to [val_inject]. *) Lemma eval_unop_compat: forall f op v1 tv1 v, eval_unop op v1 = Some v -> val_inject f v1 tv1 -> exists tv, eval_unop op tv1 = Some tv /\ val_inject f v tv. Proof. destruct op; simpl; intros. inv H; inv H0; simpl; TrivialOp. inv H; inv H0; simpl; TrivialOp. inv H; inv H0; simpl; TrivialOp. inv H; inv H0; simpl; TrivialOp. inv H0; inv H. TrivialOp. unfold Vfalse; TrivialOp. inv H0; inv H. TrivialOp. unfold Vfalse; TrivialOp. inv H0; inv H; TrivialOp. inv H0; inv H; TrivialOp. inv H0; inv H; TrivialOp. inv H0; inv H; TrivialOp. inv H0; inv H; TrivialOp. inv H0; inv H; TrivialOp. inv H0; inv H; TrivialOp. inv H0; inv H; TrivialOp. Qed. (** Compatibility of [eval_binop] with respect to [val_inject]. *) Lemma eval_binop_compat: forall f op v1 tv1 v2 tv2 v m tm, Csharpminor.eval_binop op v1 v2 m = Some v -> val_inject f v1 tv1 -> val_inject f v2 tv2 -> Mem.inject f m tm -> exists tv, Cminor.eval_binop op tv1 tv2 = Some tv /\ val_inject f v tv. Proof. destruct op; simpl; intros. inv H0; try discriminate; inv H1; inv H; TrivialOp. repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. repeat rewrite Int.add_assoc. decEq. apply Int.add_commut. inv H0; try discriminate; inv H1; inv H; TrivialOp. apply Int.sub_add_l. destruct (eq_block b1 b0); inv H4. assert (b3 = b2) by congruence. subst b3. unfold eq_block; rewrite zeq_true. TrivialOp. replace delta0 with delta by congruence. decEq. decEq. apply Int.sub_shifted. inv H0; try discriminate; inv H1; inv H; TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. destruct (Int.eq i0 Int.zero); inv H1. TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. destruct (Int.eq i0 Int.zero); inv H1. TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. destruct (Int.eq i0 Int.zero); inv H1. TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. destruct (Int.eq i0 Int.zero); inv H1. TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. destruct (Int.ltu i0 Int.iwordsize); inv H1. TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. destruct (Int.ltu i0 Int.iwordsize); inv H1. TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. destruct (Int.ltu i0 Int.iwordsize); inv H1. TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. inv H0; try discriminate; inv H1; inv H; TrivialOp. exists v; split; auto. eapply val_inject_eval_compare_null; eauto. exists v; split; auto. eapply val_inject_eval_compare_null; eauto. (* cmp ptr ptr *) caseEq (Mem.valid_pointer m b1 (Int.signed ofs1) && Mem.valid_pointer m b0 (Int.signed ofs0)); intro EQ; rewrite EQ in H4; try discriminate. elim (andb_prop _ _ EQ); intros. destruct (eq_block b1 b0); inv H4. (* same blocks in source *) assert (b3 = b2) by congruence. subst b3. assert (delta0 = delta) by congruence. subst delta0. exists (Val.of_bool (Int.cmp c ofs1 ofs0)); split. unfold eq_block; rewrite zeq_true; simpl. decEq. decEq. rewrite Int.translate_cmp. auto. eapply Mem.valid_pointer_inject_no_overflow; eauto. eapply Mem.valid_pointer_inject_no_overflow; eauto. apply val_inject_val_of_bool. (* different blocks in source *) simpl. exists v; split; [idtac | eapply val_inject_eval_compare_mismatch; eauto]. destruct (eq_block b2 b3); auto. exploit Mem.different_pointers_inject; eauto. intros [A|A]. congruence. decEq. destruct c; simpl in H6; inv H6; unfold Int.cmp. predSpec Int.eq Int.eq_spec (Int.add ofs1 (Int.repr delta)) (Int.add ofs0 (Int.repr delta0)). congruence. auto. predSpec Int.eq Int.eq_spec (Int.add ofs1 (Int.repr delta)) (Int.add ofs0 (Int.repr delta0)). congruence. auto. (* cmpu *) inv H0; try discriminate; inv H1; inv H; TrivialOp. (* cmpf *) inv H0; try discriminate; inv H1; inv H; TrivialOp. Qed. (** Correctness of [make_cast]. Note that the resulting Cminor value is normalized according to the given memory chunk. *) Lemma make_cast_correct: forall f sp te tm a v tv chunk, eval_expr tge sp te tm a tv -> val_inject f v tv -> exists tv', eval_expr tge sp te tm (make_cast chunk a) tv' /\ val_inject f (Val.load_result chunk v) tv'. Proof. intros. destruct chunk; simpl make_cast. exists (Val.sign_ext 8 tv). split. eauto with evalexpr. inversion H0; simpl; constructor. exists (Val.zero_ext 8 tv). split. eauto with evalexpr. inversion H0; simpl; constructor. exists (Val.sign_ext 16 tv). split. eauto with evalexpr. inversion H0; simpl; constructor. exists (Val.zero_ext 16 tv). split. eauto with evalexpr. inversion H0; simpl; constructor. exists tv. split. auto. inversion H0; simpl; econstructor; eauto. exists (Val.singleoffloat tv). split. eauto with evalexpr. inversion H0; simpl; constructor. exists tv. split. auto. inversion H0; simpl; econstructor; eauto. Qed. Lemma make_stackaddr_correct: forall sp te tm ofs, eval_expr tge (Vptr sp Int.zero) te tm (make_stackaddr ofs) (Vptr sp (Int.repr ofs)). Proof. intros; unfold make_stackaddr. eapply eval_Econst. simpl. decEq. decEq. rewrite Int.add_commut. apply Int.add_zero. Qed. Lemma make_globaladdr_correct: forall sp te tm id b, Genv.find_symbol tge id = Some b -> eval_expr tge (Vptr sp Int.zero) te tm (make_globaladdr id) (Vptr b Int.zero). Proof. intros; unfold make_globaladdr. eapply eval_Econst. simpl. rewrite H. auto. Qed. (** Correctness of [make_store]. *) Inductive val_content_inject (f: meminj): memory_chunk -> val -> val -> Prop := | val_content_inject_8_signed: forall n, val_content_inject f Mint8signed (Vint (Int.sign_ext 8 n)) (Vint n) | val_content_inject_8_unsigned: forall n, val_content_inject f Mint8unsigned (Vint (Int.zero_ext 8 n)) (Vint n) | val_content_inject_16_signed: forall n, val_content_inject f Mint16signed (Vint (Int.sign_ext 16 n)) (Vint n) | val_content_inject_16_unsigned: forall n, val_content_inject f Mint16unsigned (Vint (Int.zero_ext 16 n)) (Vint n) | val_content_inject_32: forall n, val_content_inject f Mfloat32 (Vfloat (Float.singleoffloat n)) (Vfloat n) | val_content_inject_base: forall chunk v1 v2, val_inject f v1 v2 -> val_content_inject f chunk v1 v2. Hint Resolve val_content_inject_base. Lemma store_arg_content_inject: forall f sp te tm a v va chunk, eval_expr tge sp te tm a va -> val_inject f v va -> exists vb, eval_expr tge sp te tm (store_arg chunk a) vb /\ val_content_inject f chunk v vb. Proof. intros. assert (exists vb, eval_expr tge sp te tm a vb /\ val_content_inject f chunk v vb). exists va; split. assumption. constructor. assumption. destruct a; simpl store_arg; trivial; destruct u; trivial; destruct chunk; trivial; inv H; simpl in H6; inv H6; econstructor; (split; [eauto|idtac]); destruct v1; simpl in H0; inv H0; constructor; constructor. Qed. Lemma storev_mapped_inject': forall f chunk m1 a1 v1 n1 m2 a2 v2, Mem.inject f m1 m2 -> Mem.storev chunk m1 a1 v1 = Some n1 -> val_inject f a1 a2 -> val_content_inject f chunk v1 v2 -> exists n2, Mem.storev chunk m2 a2 v2 = Some n2 /\ Mem.inject f n1 n2. Proof. intros. assert (forall v1', (forall b ofs, Mem.store chunk m1 b ofs v1 = Mem.store chunk m1 b ofs v1') -> Mem.storev chunk m1 a1 v1' = Some n1). intros. rewrite <- H0. destruct a1; simpl; auto. inv H2; (eapply Mem.storev_mapped_inject; [eauto|idtac|eauto|eauto]); auto; apply H3; intros. apply Mem.store_int8_sign_ext. apply Mem.store_int8_zero_ext. apply Mem.store_int16_sign_ext. apply Mem.store_int16_zero_ext. apply Mem.store_float32_truncate. Qed. Lemma make_store_correct: forall f sp te tm addr tvaddr rhs tvrhs chunk m vaddr vrhs m' fn k, eval_expr tge sp te tm addr tvaddr -> eval_expr tge sp te tm rhs tvrhs -> Mem.storev chunk m vaddr vrhs = Some m' -> Mem.inject f m tm -> val_inject f vaddr tvaddr -> val_inject f vrhs tvrhs -> exists tm', exists tvrhs', step tge (State fn (make_store chunk addr rhs) k sp te tm) E0 (State fn Sskip k sp te tm') /\ Mem.storev chunk tm tvaddr tvrhs' = Some tm' /\ Mem.inject f m' tm'. Proof. intros. unfold make_store. exploit store_arg_content_inject. eexact H0. eauto. intros [tv [EVAL VCINJ]]. exploit storev_mapped_inject'; eauto. intros [tm' [STORE MEMINJ]]. exists tm'; exists tv. split. eapply step_store; eauto. auto. Qed. (** Correctness of the variable accessors [var_get], [var_addr], and [var_set]. *) Lemma var_get_correct: forall cenv id a f tf e te sp lo hi m cs tm b chunk v, var_get cenv id = OK a -> match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm) -> Mem.inject f m tm -> eval_var_ref gve e id b chunk -> Mem.load chunk m b 0 = Some v -> exists tv, eval_expr tge (Vptr sp Int.zero) te tm a tv /\ val_inject f v tv. Proof. unfold var_get; intros. assert (match_var f id e m te sp cenv!!id). inv H0. inv MENV. auto. inv H4; rewrite <- H5 in H; inv H; inv H2; try congruence. (* var_local *) exists v'; split. apply eval_Evar. auto. congruence. (* var_stack_scalar *) assert (b0 = b). congruence. subst b0. assert (chunk0 = chunk). congruence. subst chunk0. exploit Mem.loadv_inject; eauto. unfold Mem.loadv. eexact H3. intros [tv [LOAD INJ]]. exists tv; split. eapply eval_Eload; eauto. eapply make_stackaddr_correct; eauto. auto. (* var_global_scalar *) simpl in *. assert (match_globalenvs f). eapply match_callstack_match_globalenvs; eauto. inv H2. exploit mg_symbols0; eauto. intros [A B]. assert (chunk0 = chunk). congruence. subst chunk0. assert (val_inject f (Vptr b Int.zero) (Vptr b Int.zero)). econstructor; eauto. exploit Mem.loadv_inject; eauto. simpl. eauto. intros [tv [LOAD INJ]]. exists tv; split. eapply eval_Eload; eauto. eapply make_globaladdr_correct; eauto. auto. Qed. Lemma var_addr_correct: forall cenv id a f tf e te sp lo hi m cs tm b, match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm) -> var_addr cenv id = OK a -> eval_var_addr gve e id b -> exists tv, eval_expr tge (Vptr sp Int.zero) te tm a tv /\ val_inject f (Vptr b Int.zero) tv. Proof. unfold var_addr; intros. assert (match_var f id e m te sp cenv!!id). inv H. inv MENV. auto. inv H2; rewrite <- H3 in H0; inv H0; inv H1; try congruence. (* var_stack_scalar *) exists (Vptr sp (Int.repr ofs)); split. eapply make_stackaddr_correct. congruence. (* var_stack_array *) exists (Vptr sp (Int.repr ofs)); split. eapply make_stackaddr_correct. congruence. (* var_global_scalar *) assert (match_globalenvs f). eapply match_callstack_match_globalenvs; eauto. inv H1. exploit mg_symbols0; eauto. intros [A B]. exists (Vptr b Int.zero); split. eapply make_globaladdr_correct. eauto. econstructor; eauto. (* var_global_array *) assert (match_globalenvs f). eapply match_callstack_match_globalenvs; eauto. inv H1. exploit mg_symbols0; eauto. intros [A B]. exists (Vptr b Int.zero); split. eapply make_globaladdr_correct. eauto. econstructor; eauto. Qed. Lemma var_set_correct: forall cenv id rhs rhs_chunk a f tf e te sp lo hi m cs tm tv v m' fn k, var_set cenv id rhs rhs_chunk = OK a -> match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm) -> eval_expr tge (Vptr sp Int.zero) te tm rhs tv -> val_inject f v tv -> Mem.inject f m tm -> exec_assign gve e m id v m' -> val_normalized v rhs_chunk -> exists te', exists tm', step tge (State fn a k (Vptr sp Int.zero) te tm) E0 (State fn Sskip k (Vptr sp Int.zero) te' tm') /\ Mem.inject f m' tm' /\ match_callstack f m' tm' (Frame cenv tf e te' sp lo hi :: cs) (Mem.nextblock m') (Mem.nextblock tm') /\ (forall id', id' <> id -> te'!id' = te!id'). Proof. intros until k. intros VS MCS EVAL VINJ MINJ ASG VNORM. unfold var_set in VS. inv ASG. assert (NEXTBLOCK: Mem.nextblock m' = Mem.nextblock m). eapply Mem.nextblock_store; eauto. assert (MV: match_var f id e m te sp cenv!!id). inv MCS. inv MENV. auto. inv MV; rewrite <- H1 in VS; inv VS; inv H; try congruence. (* var_local *) assert (b0 = b) by congruence. subst b0. assert (chunk0 = chunk) by congruence. subst chunk0. generalize H8; clear H8. case_eq (chunktype_compat rhs_chunk chunk). (* compatible chunks *) intros CCOMPAT EQ; inv EQ. exploit chunktype_compat_correct; eauto. intro VNORM'. exists (PTree.set id tv te); exists tm. split. eapply step_assign. eauto. split. eapply Mem.store_unmapped_inject; eauto. split. rewrite NEXTBLOCK. eapply match_callstack_store_local; eauto. eapply val_normalized_has_type; eauto. red in VNORM'. congruence. intros. apply PTree.gso; auto. (* incompatible chunks but same type *) intros. destruct (typ_eq (type_of_chunk chunk) (type_of_chunk rhs_chunk)); inv H8. exploit make_cast_correct; eauto. intros [tv' [EVAL' INJ']]. exists (PTree.set id tv' te); exists tm. split. eapply step_assign. eauto. split. eapply Mem.store_unmapped_inject; eauto. split. rewrite NEXTBLOCK. eapply match_callstack_store_local; eauto. rewrite e0. eapply val_normalized_has_type; eauto. intros. apply PTree.gso; auto. (* var_stack_scalar *) assert (b0 = b) by congruence. subst b0. assert (chunk0 = chunk) by congruence. subst chunk0. assert (Mem.storev chunk m (Vptr b Int.zero) v = Some m'). assumption. exploit make_store_correct. eapply make_stackaddr_correct. eauto. eauto. eauto. eauto. eauto. intros [tm' [tvrhs' [EVAL' [STORE' MEMINJ]]]]. exists te; exists tm'. split. eauto. split. auto. split. rewrite NEXTBLOCK. rewrite (nextblock_storev _ _ _ _ _ STORE'). eapply match_callstack_storev_mapped; eauto. auto. (* var_global_scalar *) simpl in *. assert (chunk0 = chunk) by congruence. subst chunk0. assert (Mem.storev chunk m (Vptr b Int.zero) v = Some m'). assumption. assert (match_globalenvs f). eapply match_callstack_match_globalenvs; eauto. exploit mg_symbols; eauto. intros [A B]. exploit make_store_correct. eapply make_globaladdr_correct; eauto. eauto. eauto. eauto. eauto. eauto. intros [tm' [tvrhs' [EVAL' [STORE' TNEXTBLOCK]]]]. exists te; exists tm'. split. eauto. split. auto. split. rewrite NEXTBLOCK. rewrite (nextblock_storev _ _ _ _ _ STORE'). eapply match_callstack_store_mapped; eauto. auto. Qed. Lemma match_callstack_extensional: forall f cenv tf e te1 te2 sp lo hi cs bound tbound m tm, (forall id chunk, cenv!!id = Var_local chunk -> te2!id = te1!id) -> match_callstack f m tm (Frame cenv tf e te1 sp lo hi :: cs) bound tbound -> match_callstack f m tm (Frame cenv tf e te2 sp lo hi :: cs) bound tbound. Proof. intros. inv H0. constructor; auto. apply match_env_extensional with te1; auto. Qed. Lemma var_set_self_correct: forall cenv id ty a f tf e te sp lo hi m cs tm tv te' v m' fn k, var_set_self cenv id ty = OK a -> match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm) -> val_inject f v tv -> Mem.inject f m tm -> exec_assign gve e m id v m' -> Val.has_type v ty -> te'!id = Some tv -> (forall i, i <> id -> te'!i = te!i) -> exists te'', exists tm', step tge (State fn a k (Vptr sp Int.zero) te' tm) E0 (State fn Sskip k (Vptr sp Int.zero) te'' tm') /\ Mem.inject f m' tm' /\ match_callstack f m' tm' (Frame cenv tf e te'' sp lo hi :: cs) (Mem.nextblock m') (Mem.nextblock tm') /\ (forall id', id' <> id -> te''!id' = te'!id'). Proof. intros until k. intros VS MCS VINJ MINJ ASG VTY VAL OTHERS. unfold var_set_self in VS. inv ASG. assert (NEXTBLOCK: Mem.nextblock m' = Mem.nextblock m). eapply Mem.nextblock_store; eauto. assert (MV: match_var f id e m te sp cenv!!id). inv MCS. inv MENV. auto. assert (EVAR: eval_expr tge (Vptr sp Int.zero) te' tm (Evar id) tv). constructor. auto. inv MV; rewrite <- H1 in VS; inv VS; inv H; try congruence. (* var_local *) assert (b0 = b) by congruence. subst b0. assert (chunk0 = chunk) by congruence. subst chunk0. destruct (typ_eq (type_of_chunk chunk) ty); inv H8. exploit make_cast_correct; eauto. intros [tv' [EVAL' INJ']]. exists (PTree.set id tv' te'); exists tm. split. eapply step_assign. eauto. split. eapply Mem.store_unmapped_inject; eauto. split. rewrite NEXTBLOCK. apply match_callstack_extensional with (PTree.set id tv' te). intros. repeat rewrite PTree.gsspec. destruct (peq id0 id); auto. eapply match_callstack_store_local; eauto. intros; apply PTree.gso; auto. (* var_stack_scalar *) assert (b0 = b) by congruence. subst b0. assert (chunk0 = chunk) by congruence. subst chunk0. assert (Mem.storev chunk m (Vptr b Int.zero) v = Some m'). assumption. exploit make_store_correct. eapply make_stackaddr_correct. eauto. eauto. eauto. eauto. eauto. intros [tm' [tvrhs' [EVAL' [STORE' MEMINJ]]]]. exists te'; exists tm'. split. eauto. split. auto. split. rewrite NEXTBLOCK. rewrite (nextblock_storev _ _ _ _ _ STORE'). apply match_callstack_extensional with te. intros. apply OTHERS. congruence. eapply match_callstack_storev_mapped; eauto. auto. (* var_global_scalar *) simpl in *. assert (chunk0 = chunk) by congruence. subst chunk0. assert (Mem.storev chunk m (Vptr b Int.zero) v = Some m'). assumption. assert (match_globalenvs f). eapply match_callstack_match_globalenvs; eauto. exploit mg_symbols; eauto. intros [A B]. exploit make_store_correct. eapply make_globaladdr_correct; eauto. eauto. eauto. eauto. eauto. eauto. intros [tm' [tvrhs' [EVAL' [STORE' MEMINJ]]]]. exists te'; exists tm'. split. eauto. split. auto. split. rewrite NEXTBLOCK. rewrite (nextblock_storev _ _ _ _ _ STORE'). apply match_callstack_extensional with te. intros. apply OTHERS. congruence. eapply match_callstack_store_mapped; eauto. auto. Qed. (** * Correctness of stack allocation of local variables *) (** This section shows the correctness of the translation of Csharpminor local variables, either as Cminor local variables or as sub-blocks of the Cminor stack data. This is the most difficult part of the proof. *) Remark array_alignment_pos: forall sz, array_alignment sz > 0. Proof. unfold array_alignment; intros. destruct (zlt sz 2). omega. destruct (zlt sz 4). omega. destruct (zlt sz 8); omega. Qed. Remark assign_variable_incr: forall atk id lv cenv sz cenv' sz', assign_variable atk (id, lv) (cenv, sz) = (cenv', sz') -> sz <= sz'. Proof. intros until sz'; simpl. destruct lv. case (Identset.mem id atk); intros. inv H. generalize (size_chunk_pos m). intro. generalize (align_le sz (size_chunk m) H). omega. inv H. omega. intros. inv H. generalize (align_le sz (array_alignment z) (array_alignment_pos z)). assert (0 <= Zmax 0 z). apply Zmax_bound_l. omega. omega. Qed. Remark assign_variables_incr: forall atk vars cenv sz cenv' sz', assign_variables atk vars (cenv, sz) = (cenv', sz') -> sz <= sz'. Proof. induction vars; intros until sz'. simpl; intros. replace sz' with sz. omega. congruence. Opaque assign_variable. destruct a as [id lv]. simpl. case_eq (assign_variable atk (id, lv) (cenv, sz)). intros cenv1 sz1 EQ1 EQ2. apply Zle_trans with sz1. eapply assign_variable_incr; eauto. eauto. Transparent assign_variable. Qed. Remark inj_offset_aligned_array: forall stacksize sz, Mem.inj_offset_aligned (align stacksize (array_alignment sz)) sz. Proof. intros; red; intros. apply Zdivides_trans with (array_alignment sz). unfold align_chunk. unfold array_alignment. generalize Zone_divide; intro. generalize Zdivide_refl; intro. assert (2 | 4). exists 2; auto. assert (2 | 8). exists 4; auto. assert (4 | 8). exists 2; auto. destruct (zlt sz 2). destruct chunk; simpl in *; auto; omegaContradiction. destruct (zlt sz 4). destruct chunk; simpl in *; auto; omegaContradiction. destruct (zlt sz 8). destruct chunk; simpl in *; auto; omegaContradiction. destruct chunk; simpl; auto. apply align_divides. apply array_alignment_pos. Qed. Remark inj_offset_aligned_array': forall stacksize sz, Mem.inj_offset_aligned (align stacksize (array_alignment sz)) (Zmax 0 sz). Proof. intros. replace (array_alignment sz) with (array_alignment (Zmax 0 sz)). apply inj_offset_aligned_array. rewrite Zmax_spec. destruct (zlt sz 0); auto. transitivity 1. reflexivity. unfold array_alignment. rewrite zlt_true. auto. omega. Qed. Remark inj_offset_aligned_var: forall stacksize chunk, Mem.inj_offset_aligned (align stacksize (size_chunk chunk)) (size_chunk chunk). Proof. intros. replace (align stacksize (size_chunk chunk)) with (align stacksize (array_alignment (size_chunk chunk))). apply inj_offset_aligned_array. decEq. destruct chunk; reflexivity. Qed. Lemma match_callstack_alloc_variable: forall atk id lv cenv sz cenv' sz' tm sp e tf m m' b te lo cs f tv, assign_variable atk (id, lv) (cenv, sz) = (cenv', sz') -> Mem.valid_block tm sp -> Mem.bounds tm sp = (0, tf.(fn_stackspace)) -> Mem.range_perm tm sp 0 tf.(fn_stackspace) Freeable -> tf.(fn_stackspace) <= Int.max_signed -> Mem.alloc m 0 (sizeof lv) = (m', b) -> match_callstack f m tm (Frame cenv tf e te sp lo (Mem.nextblock m) :: cs) (Mem.nextblock m) (Mem.nextblock tm) -> Mem.inject f m tm -> 0 <= sz -> sz' <= tf.(fn_stackspace) -> (forall b delta, f b = Some(sp, delta) -> Mem.high_bound m b + delta <= sz) -> e!id = None -> te!id = Some tv -> exists f', inject_incr f f' /\ Mem.inject f' m' tm /\ match_callstack f' m' tm (Frame cenv' tf (PTree.set id (b, lv) e) te sp lo (Mem.nextblock m') :: cs) (Mem.nextblock m') (Mem.nextblock tm) /\ (forall b delta, f' b = Some(sp, delta) -> Mem.high_bound m' b + delta <= sz'). Proof. intros until tv. intros ASV VALID BOUNDS PERMS NOOV ALLOC MCS INJ LO HI RANGE E TE. generalize ASV. unfold assign_variable. caseEq lv. (* 1. lv = LVscalar chunk *) intros chunk LV. case (Identset.mem id atk). (* 1.1 info = Var_stack_scalar chunk ofs *) set (ofs := align sz (size_chunk chunk)). intro EQ; injection EQ; intros; clear EQ. rewrite <- H0. generalize (size_chunk_pos chunk); intro SIZEPOS. generalize (align_le sz (size_chunk chunk) SIZEPOS). fold ofs. intro SZOFS. exploit Mem.alloc_left_mapped_inject. eauto. eauto. eauto. instantiate (1 := ofs). generalize Int.min_signed_neg. omega. right; rewrite BOUNDS; simpl. generalize Int.min_signed_neg. omega. intros. apply Mem.perm_implies with Freeable; auto with mem. apply PERMS. rewrite LV in H1. simpl in H1. omega. rewrite LV; simpl. rewrite Zminus_0_r. unfold ofs. apply inj_offset_aligned_var. intros. generalize (RANGE _ _ H1). omega. intros [f1 [MINJ1 [INCR1 [SAME OTHER]]]]. exists f1; split. auto. split. auto. split. eapply match_callstack_alloc_left; eauto. rewrite <- LV; auto. rewrite SAME; constructor. intros. rewrite (Mem.bounds_alloc _ _ _ _ _ ALLOC). destruct (eq_block b0 b); simpl. subst b0. assert (delta = ofs) by congruence. subst delta. rewrite LV. simpl. omega. rewrite OTHER in H1; eauto. generalize (RANGE _ _ H1). omega. (* 1.2 info = Var_local chunk *) intro EQ; injection EQ; intros; clear EQ. subst sz'. rewrite <- H0. exploit Mem.alloc_left_unmapped_inject; eauto. intros [f1 [MINJ1 [INCR1 [SAME OTHER]]]]. exists f1; split. auto. split. auto. split. eapply match_callstack_alloc_left; eauto. rewrite <- LV; auto. rewrite SAME; constructor. intros. rewrite (Mem.bounds_alloc _ _ _ _ _ ALLOC). destruct (eq_block b0 b); simpl. subst b0. congruence. rewrite OTHER in H; eauto. (* 2 info = Var_stack_array ofs *) intros dim LV EQ. injection EQ; clear EQ; intros. rewrite <- H0. assert (0 <= Zmax 0 dim). apply Zmax1. generalize (align_le sz (array_alignment dim) (array_alignment_pos dim)). intro. set (ofs := align sz (array_alignment dim)) in *. exploit Mem.alloc_left_mapped_inject. eauto. eauto. eauto. instantiate (1 := ofs). generalize Int.min_signed_neg. omega. right; rewrite BOUNDS; simpl. generalize Int.min_signed_neg. omega. intros. apply Mem.perm_implies with Freeable; auto with mem. apply PERMS. rewrite LV in H3. simpl in H3. omega. rewrite LV; simpl. rewrite Zminus_0_r. unfold ofs. apply inj_offset_aligned_array'. intros. generalize (RANGE _ _ H3). omega. intros [f1 [MINJ1 [INCR1 [SAME OTHER]]]]. exists f1; split. auto. split. auto. split. eapply match_callstack_alloc_left; eauto. rewrite <- LV; auto. rewrite SAME; constructor. intros. rewrite (Mem.bounds_alloc _ _ _ _ _ ALLOC). destruct (eq_block b0 b); simpl. subst b0. assert (delta = ofs) by congruence. subst delta. rewrite LV. simpl. omega. rewrite OTHER in H3; eauto. generalize (RANGE _ _ H3). omega. Qed. Lemma match_callstack_alloc_variables_rec: forall tm sp cenv' tf te lo cs atk, Mem.valid_block tm sp -> Mem.bounds tm sp = (0, tf.(fn_stackspace)) -> Mem.range_perm tm sp 0 tf.(fn_stackspace) Freeable -> tf.(fn_stackspace) <= Int.max_signed -> forall e m vars e' m', alloc_variables e m vars e' m' -> forall f cenv sz, assign_variables atk vars (cenv, sz) = (cenv', tf.(fn_stackspace)) -> match_callstack f m tm (Frame cenv tf e te sp lo (Mem.nextblock m) :: cs) (Mem.nextblock m) (Mem.nextblock tm) -> Mem.inject f m tm -> 0 <= sz -> (forall b delta, f b = Some(sp, delta) -> Mem.high_bound m b + delta <= sz) -> (forall id lv, In (id, lv) vars -> te!id <> None) -> list_norepet (List.map (@fst ident var_kind) vars) -> (forall id lv, In (id, lv) vars -> e!id = None) -> exists f', inject_incr f f' /\ Mem.inject f' m' tm /\ match_callstack f' m' tm (Frame cenv' tf e' te sp lo (Mem.nextblock m') :: cs) (Mem.nextblock m') (Mem.nextblock tm). Proof. intros until atk. intros VALID BOUNDS PERM NOOV. induction 1. (* base case *) intros. simpl in H. inversion H; subst cenv sz. exists f. split. apply inject_incr_refl. split. auto. auto. (* inductive case *) intros until sz. change (assign_variables atk ((id, lv) :: vars) (cenv, sz)) with (assign_variables atk vars (assign_variable atk (id, lv) (cenv, sz))). caseEq (assign_variable atk (id, lv) (cenv, sz)). intros cenv1 sz1 ASV1 ASVS MATCH MINJ SZPOS BOUND DEFINED NOREPET UNDEFINED. assert (DEFINED1: forall id0 lv0, In (id0, lv0) vars -> te!id0 <> None). intros. eapply DEFINED. simpl. right. eauto. assert (exists tv, te!id = Some tv). assert (te!id <> None). eapply DEFINED. simpl; left; auto. destruct (te!id). exists v; auto. congruence. destruct H1 as [tv TEID]. assert (sz1 <= fn_stackspace tf). eapply assign_variables_incr; eauto. exploit match_callstack_alloc_variable; eauto with coqlib. intros [f1 [INCR1 [INJ1 [MCS1 BOUND1]]]]. exploit IHalloc_variables; eauto. apply Zle_trans with sz; auto. eapply assign_variable_incr; eauto. inv NOREPET; auto. intros. rewrite PTree.gso. eapply UNDEFINED; eauto with coqlib. simpl in NOREPET. inversion NOREPET. red; intro; subst id0. elim H5. change id with (fst (id, lv0)). apply List.in_map. auto. intros [f2 [INCR2 [INJ2 MCS2]]]. exists f2; intuition. eapply inject_incr_trans; eauto. Qed. Lemma set_params_defined: forall params args id, In id params -> (set_params args params)!id <> None. Proof. induction params; simpl; intros. elim H. destruct args. rewrite PTree.gsspec. case (peq id a); intro. congruence. eapply IHparams. elim H; intro. congruence. auto. rewrite PTree.gsspec. case (peq id a); intro. congruence. eapply IHparams. elim H; intro. congruence. auto. Qed. Lemma set_locals_defined: forall e vars id, In id vars \/ e!id <> None -> (set_locals vars e)!id <> None. Proof. induction vars; simpl; intros. tauto. rewrite PTree.gsspec. case (peq id a); intro. congruence. apply IHvars. assert (a <> id). congruence. tauto. Qed. Lemma set_locals_params_defined: forall args params vars id, In id (params ++ vars) -> (set_locals vars (set_params args params))!id <> None. Proof. intros. apply set_locals_defined. elim (in_app_or _ _ _ H); intro. right. apply set_params_defined; auto. left; auto. Qed. (** Preservation of [match_callstack] by simultaneous allocation of Csharpminor local variables and of the Cminor stack data block. *) Lemma match_callstack_alloc_variables: forall fn cenv tf m e m' tm tm' sp f cs targs body, build_compilenv gce fn = (cenv, tf.(fn_stackspace)) -> tf.(fn_stackspace) <= Int.max_signed -> list_norepet (fn_params_names fn ++ fn_vars_names fn) -> alloc_variables Csharpminor.empty_env m (fn_variables fn) e m' -> Mem.alloc tm 0 tf.(fn_stackspace) = (tm', sp) -> match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm) -> Mem.inject f m tm -> let tvars := make_vars (fn_params_names fn) (fn_vars_names fn) body in let te := set_locals tvars (set_params targs (fn_params_names fn)) in exists f', inject_incr f f' /\ Mem.inject f' m' tm' /\ match_callstack f' m' tm' (Frame cenv tf e te sp (Mem.nextblock m) (Mem.nextblock m') :: cs) (Mem.nextblock m') (Mem.nextblock tm'). Proof. intros. unfold build_compilenv in H. eapply match_callstack_alloc_variables_rec; eauto with mem. eapply Mem.bounds_alloc_same; eauto. red; intros; eauto with mem. eapply match_callstack_alloc_right; eauto. eapply Mem.alloc_right_inject; eauto. omega. intros. elim (Mem.valid_not_valid_diff tm sp sp); eauto with mem. eapply Mem.valid_block_inject_2; eauto. intros. unfold te. apply set_locals_params_defined. elim (in_app_or _ _ _ H6); intros. elim (list_in_map_inv _ _ _ H7). intros x [A B]. apply in_or_app; left. inversion A. apply List.in_map. auto. apply in_or_app; right. unfold tvars, make_vars. apply in_or_app; left. change id with (fst (id, lv)). apply List.in_map; auto. (* norepet *) unfold fn_variables. rewrite List.map_app. rewrite list_map_compose. simpl. assumption. (* undef *) intros. unfold empty_env. apply PTree.gempty. Qed. (** Correctness of the code generated by [store_parameters] to store in memory the values of parameters that are stack-allocated. *) Inductive vars_vals_match (f:meminj): list (ident * memory_chunk) -> list val -> env -> Prop := | vars_vals_nil: forall te, vars_vals_match f nil nil te | vars_vals_cons: forall te id chunk vars v vals tv, te!id = Some tv -> val_inject f v tv -> Val.has_type v (type_of_chunk chunk) -> vars_vals_match f vars vals te -> vars_vals_match f ((id, chunk) :: vars) (v :: vals) te. Lemma vars_vals_match_extensional: forall f vars vals te, vars_vals_match f vars vals te -> forall te', (forall id lv, In (id, lv) vars -> te'!id = te!id) -> vars_vals_match f vars vals te'. Proof. induction 1; intros. constructor. econstructor; eauto. rewrite <- H. eauto with coqlib. apply IHvars_vals_match. intros. eapply H3; eauto with coqlib. Qed. Lemma store_parameters_correct: forall e m1 params vl m2, bind_parameters e m1 params vl m2 -> forall s f te1 cenv tf sp lo hi cs tm1 fn k, vars_vals_match f params vl te1 -> list_norepet (List.map param_name params) -> Mem.inject f m1 tm1 -> match_callstack f m1 tm1 (Frame cenv tf e te1 sp lo hi :: cs) (Mem.nextblock m1) (Mem.nextblock tm1) -> store_parameters cenv params = OK s -> exists te2, exists tm2, star step tge (State fn s k (Vptr sp Int.zero) te1 tm1) E0 (State fn Sskip k (Vptr sp Int.zero) te2 tm2) /\ Mem.inject f m2 tm2 /\ match_callstack f m2 tm2 (Frame cenv tf e te2 sp lo hi :: cs) (Mem.nextblock m2) (Mem.nextblock tm2). Proof. induction 1. (* base case *) intros; simpl. monadInv H3. exists te1; exists tm1. split. constructor. tauto. (* inductive case *) intros until k. intros VVM NOREPET MINJ MATCH STOREP. monadInv STOREP. inv VVM. inv NOREPET. exploit var_set_self_correct; eauto. econstructor; eauto. econstructor; eauto. intros [te2 [tm2 [EXEC1 [MINJ1 [MATCH1 UNCHANGED1]]]]]. assert (vars_vals_match f params vl te2). apply vars_vals_match_extensional with te1; auto. intros. apply UNCHANGED1. red; intro; subst id0. elim H4. change id with (param_name (id, lv)). apply List.in_map. auto. exploit IHbind_parameters; eauto. intros [te3 [tm3 [EXEC2 [MINJ2 MATCH2]]]]. exists te3; exists tm3. split. eapply star_left. constructor. eapply star_left. eexact EXEC1. eapply star_left. constructor. eexact EXEC2. reflexivity. reflexivity. reflexivity. auto. Qed. Lemma vars_vals_match_holds_1: forall f params args targs, list_norepet (List.map param_name params) -> val_list_inject f args targs -> Val.has_type_list args (List.map type_of_chunk (List.map param_chunk params)) -> vars_vals_match f params args (set_params targs (List.map (@fst ident memory_chunk) params)). Proof. induction params; simpl; intros. destruct args; simpl in H1; try contradiction. inv H0. constructor. destruct args; simpl in H1; try contradiction. destruct H1. inv H0. inv H. destruct a as [id chunk]; simpl in *. econstructor. rewrite PTree.gss. reflexivity. auto. auto. apply vars_vals_match_extensional with (set_params vl' (map param_name params)). eapply IHparams; eauto. intros. simpl. apply PTree.gso. red; intro; subst id0. elim H4. change id with (param_name (id, lv)). apply List.in_map; auto. Qed. Lemma vars_vals_match_holds: forall f params args targs, list_norepet (List.map param_name params) -> val_list_inject f args targs -> Val.has_type_list args (List.map type_of_chunk (List.map param_chunk params)) -> forall vars, list_norepet (vars ++ List.map param_name params) -> vars_vals_match f params args (set_locals vars (set_params targs (List.map param_name params))). Proof. induction vars; simpl; intros. eapply vars_vals_match_holds_1; eauto. inv H2. eapply vars_vals_match_extensional; eauto. intros. apply PTree.gso. red; intro; subst id; elim H5. apply in_or_app. right. change a with (param_name (a, lv)). apply List.in_map; auto. Qed. Remark identset_removelist_charact: forall l s x, Identset.In x (identset_removelist l s) <-> Identset.In x s /\ ~In x l. Proof. induction l; simpl; intros. tauto. split; intros. exploit Identset.remove_3; eauto. rewrite IHl. intros [P Q]. split. auto. intuition. elim (Identset.remove_1 H1 H). destruct H as [P Q]. apply Identset.remove_2. tauto. rewrite IHl. tauto. Qed. Remark InA_In: forall (A: Type) (x: A) (l: list A), InA (fun (x y: A) => x = y) x l <-> In x l. Proof. intros. rewrite InA_alt. split; intros. destruct H as [y [P Q]]. congruence. exists x; auto. Qed. Remark NoDupA_norepet: forall (A: Type) (l: list A), NoDupA (fun (x y: A) => x = y) l -> list_norepet l. Proof. induction 1. constructor. constructor; auto. red; intros; elim H. rewrite InA_In. auto. Qed. Lemma make_vars_norepet: forall fn body, list_norepet (fn_params_names fn ++ fn_vars_names fn) -> list_norepet (make_vars (fn_params_names fn) (fn_vars_names fn) body ++ fn_params_names fn). Proof. intros. rewrite list_norepet_app in H. destruct H as [A [B C]]. rewrite list_norepet_app. split. unfold make_vars. rewrite list_norepet_app. split. auto. split. apply NoDupA_norepet. apply Identset.elements_3w. red; intros. red; intros; subst y. rewrite <- InA_In in H0. exploit Identset.elements_2. eexact H0. rewrite identset_removelist_charact. intros [P Q]. elim Q. apply in_or_app. auto. split. auto. red; intros. unfold make_vars in H. destruct (in_app_or _ _ _ H). apply sym_not_equal. apply C; auto. rewrite <- InA_In in H1. exploit Identset.elements_2. eexact H1. rewrite identset_removelist_charact. intros [P Q]. red; intros; elim Q. apply in_or_app. left; congruence. Qed. (** The main result in this section: the behaviour of function entry in the generated Cminor code (allocate stack data block and store parameters whose address is taken) simulates what happens at function entry in the original Csharpminor (allocate one block per local variable and initialize the blocks corresponding to function parameters). *) Lemma function_entry_ok: forall fn m e m1 vargs m2 f cs tm cenv tf tm1 sp tvargs body s fn' k, list_norepet (fn_params_names fn ++ fn_vars_names fn) -> alloc_variables empty_env m (fn_variables fn) e m1 -> bind_parameters e m1 fn.(Csharpminor.fn_params) vargs m2 -> match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm) -> build_compilenv gce fn = (cenv, tf.(fn_stackspace)) -> tf.(fn_stackspace) <= Int.max_signed -> Mem.alloc tm 0 tf.(fn_stackspace) = (tm1, sp) -> let vars := make_vars (fn_params_names fn) (fn_vars_names fn) body in let te := set_locals vars (set_params tvargs (fn_params_names fn)) in val_list_inject f vargs tvargs -> Val.has_type_list vargs (Csharpminor.fn_sig fn).(sig_args) -> Mem.inject f m tm -> store_parameters cenv fn.(Csharpminor.fn_params) = OK s -> exists f2, exists te2, exists tm2, star step tge (State fn' s k (Vptr sp Int.zero) te tm1) E0 (State fn' Sskip k (Vptr sp Int.zero) te2 tm2) /\ Mem.inject f2 m2 tm2 /\ inject_incr f f2 /\ match_callstack f2 m2 tm2 (Frame cenv tf e te2 sp (Mem.nextblock m) (Mem.nextblock m1) :: cs) (Mem.nextblock m2) (Mem.nextblock tm2). Proof. intros. exploit match_callstack_alloc_variables; eauto. intros [f1 [INCR1 [MINJ1 MATCH1]]]. exploit vars_vals_match_holds. eapply list_norepet_append_left. eexact H. apply val_list_inject_incr with f. eauto. eauto. auto. eapply make_vars_norepet. auto. intro VVM. exploit store_parameters_correct. eauto. eauto. eapply list_norepet_append_left; eauto. eexact MINJ1. fold (fn_params_names fn). eexact MATCH1. eauto. intros [te2 [tm2 [EXEC [MINJ2 MATCH2]]]]. exists f1; exists te2; exists tm2. eauto. Qed. (** * Semantic preservation for the translation *) (** The proof of semantic preservation uses simulation diagrams of the following form: << e, m1, s ----------------- sp, te1, tm1, ts | | t| |t v v e, m2, out --------------- sp, te2, tm2, tout >> where [ts] is the Cminor statement obtained by translating the Csharpminor statement [s]. The left vertical arrow is an execution of a Csharpminor statement. The right vertical arrow is an execution of a Cminor statement. The precondition (top vertical bar) includes a [mem_inject] relation between the memory states [m1] and [tm1], and a [match_callstack] relation for any callstack having [e], [te1], [sp] as top frame. The postcondition (bottom vertical bar) is the existence of a memory injection [f2] that extends the injection [f1] we started with, preserves the [match_callstack] relation for the transformed callstack at the final state, and validates a [outcome_inject] relation between the outcomes [out] and [tout]. *) (** ** Semantic preservation for expressions *) Remark bool_of_val_inject: forall f v tv b, Val.bool_of_val v b -> val_inject f v tv -> Val.bool_of_val tv b. Proof. intros. inv H0; inv H; constructor; auto. Qed. Lemma transl_expr_correct: forall f m tm cenv tf e te sp lo hi cs (MINJ: Mem.inject f m tm) (MATCH: match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm)), forall a v, Csharpminor.eval_expr gve e m a v -> forall ta (TR: transl_expr cenv a = OK ta), exists tv, eval_expr tge (Vptr sp Int.zero) te tm ta tv /\ val_inject f v tv. Proof. induction 3; intros; simpl in TR; try (monadInv TR). (* Evar *) eapply var_get_correct; eauto. (* Eaddrof *) eapply var_addr_correct; eauto. (* Econst *) exploit transl_constant_correct; eauto. intros [tv [A B]]. exists tv; split. constructor; eauto. eauto. (* Eunop *) exploit IHeval_expr; eauto. intros [tv1 [EVAL1 INJ1]]. exploit eval_unop_compat; eauto. intros [tv [EVAL INJ]]. exists tv; split. econstructor; eauto. auto. (* Ebinop *) exploit IHeval_expr1; eauto. intros [tv1 [EVAL1 INJ1]]. exploit IHeval_expr2; eauto. intros [tv2 [EVAL2 INJ2]]. exploit eval_binop_compat; eauto. intros [tv [EVAL INJ]]. exists tv; split. econstructor; eauto. auto. (* Eload *) exploit IHeval_expr; eauto. intros [tv1 [EVAL1 INJ1]]. exploit Mem.loadv_inject; eauto. intros [tv [LOAD INJ]]. exists tv; split. econstructor; eauto. auto. (* Econdition *) exploit IHeval_expr1; eauto. intros [tv1 [EVAL1 INJ1]]. assert (transl_expr cenv (if vb1 then b else c) = OK (if vb1 then x0 else x1)). destruct vb1; auto. exploit IHeval_expr2; eauto. intros [tv2 [EVAL2 INJ2]]. exists tv2; split. eapply eval_Econdition; eauto. eapply bool_of_val_inject; eauto. auto. Qed. Lemma transl_exprlist_correct: forall f m tm cenv tf e te sp lo hi cs (MINJ: Mem.inject f m tm) (MATCH: match_callstack f m tm (Frame cenv tf e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm)), forall a v, Csharpminor.eval_exprlist gve e m a v -> forall ta (TR: transl_exprlist cenv a = OK ta), exists tv, eval_exprlist tge (Vptr sp Int.zero) te tm ta tv /\ val_list_inject f v tv. Proof. induction 3; intros; monadInv TR. exists (@nil val); split. constructor. constructor. exploit transl_expr_correct; eauto. intros [tv1 [EVAL1 VINJ1]]. exploit IHeval_exprlist; eauto. intros [tv2 [EVAL2 VINJ2]]. exists (tv1 :: tv2); split. constructor; auto. constructor; auto. Qed. (** ** Semantic preservation for statements and functions *) Inductive match_cont: Csharpminor.cont -> Cminor.cont -> option typ -> compilenv -> exit_env -> callstack -> Prop := | match_Kstop: forall ty cenv xenv, match_cont Csharpminor.Kstop Kstop ty cenv xenv nil | match_Kseq: forall s k ts tk ty cenv xenv cs, transl_stmt ty cenv xenv s = OK ts -> match_cont k tk ty cenv xenv cs -> match_cont (Csharpminor.Kseq s k) (Kseq ts tk) ty cenv xenv cs | match_Kseq2: forall s1 s2 k ts1 tk ty cenv xenv cs, transl_stmt ty cenv xenv s1 = OK ts1 -> match_cont (Csharpminor.Kseq s2 k) tk ty cenv xenv cs -> match_cont (Csharpminor.Kseq (Csharpminor.Sseq s1 s2) k) (Kseq ts1 tk) ty cenv xenv cs | match_Kblock: forall k tk ty cenv xenv cs, match_cont k tk ty cenv xenv cs -> match_cont (Csharpminor.Kblock k) (Kblock tk) ty cenv (true :: xenv) cs | match_Kblock2: forall k tk ty cenv xenv cs, match_cont k tk ty cenv xenv cs -> match_cont k (Kblock tk) ty cenv (false :: xenv) cs | match_Kcall_none: forall fn e k tfn sp te tk ty cenv xenv lo hi cs sz cenv', transl_funbody cenv sz fn = OK tfn -> match_cont k tk fn.(fn_return) cenv xenv cs -> match_cont (Csharpminor.Kcall None fn e k) (Kcall None tfn (Vptr sp Int.zero) te tk) ty cenv' nil (Frame cenv tfn e te sp lo hi :: cs) | match_Kcall_some: forall id fn e k tfn s sp te tk ty cenv xenv lo hi cs sz cenv', transl_funbody cenv sz fn = OK tfn -> var_set_self cenv id (typ_of_opttyp ty) = OK s -> match_cont k tk fn.(fn_return) cenv xenv cs -> match_cont (Csharpminor.Kcall (Some id) fn e k) (Kcall (Some id) tfn (Vptr sp Int.zero) te (Kseq s tk)) ty cenv' nil (Frame cenv tfn e te sp lo hi :: cs). Inductive match_states: Csharpminor.state -> Cminor.state -> Prop := | match_state: forall fn s k e m tfn ts tk sp te tm cenv xenv f lo hi cs sz (TRF: transl_funbody cenv sz fn = OK tfn) (TR: transl_stmt fn.(fn_return) cenv xenv s = OK ts) (MINJ: Mem.inject f m tm) (MCS: match_callstack f m tm (Frame cenv tfn e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm)) (MK: match_cont k tk fn.(fn_return) cenv xenv cs), match_states (Csharpminor.State fn s k e m) (State tfn ts tk (Vptr sp Int.zero) te tm) | match_state_seq: forall fn s1 s2 k e m tfn ts1 tk sp te tm cenv xenv f lo hi cs sz (TRF: transl_funbody cenv sz fn = OK tfn) (TR: transl_stmt fn.(fn_return) cenv xenv s1 = OK ts1) (MINJ: Mem.inject f m tm) (MCS: match_callstack f m tm (Frame cenv tfn e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm)) (MK: match_cont (Csharpminor.Kseq s2 k) tk fn.(fn_return) cenv xenv cs), match_states (Csharpminor.State fn (Csharpminor.Sseq s1 s2) k e m) (State tfn ts1 tk (Vptr sp Int.zero) te tm) | match_callstate: forall fd args k m tfd targs tk tm f cs cenv (TR: transl_fundef gce fd = OK tfd) (MINJ: Mem.inject f m tm) (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm)) (MK: match_cont k tk (Csharpminor.funsig fd).(sig_res) cenv nil cs) (ISCC: Csharpminor.is_call_cont k) (ARGSINJ: val_list_inject f args targs) (ARGSTY: Val.has_type_list args (Csharpminor.funsig fd).(sig_args)), match_states (Csharpminor.Callstate fd args k m) (Callstate tfd targs tk tm) | match_returnstate: forall v k m tv tk tm f cs ty cenv (MINJ: Mem.inject f m tm) (MCS: match_callstack f m tm cs (Mem.nextblock m) (Mem.nextblock tm)) (MK: match_cont k tk ty cenv nil cs) (RESINJ: val_inject f v tv) (RESTY: Val.has_type v (typ_of_opttyp ty)), match_states (Csharpminor.Returnstate v k m) (Returnstate tv tk tm). Remark val_inject_function_pointer: forall v fd f tv, Genv.find_funct tge v = Some fd -> match_globalenvs f -> val_inject f v tv -> tv = v. Proof. intros. exploit Genv.find_funct_inv; eauto. intros [b EQ]. subst v. rewrite Genv.find_funct_find_funct_ptr in H. assert (b < 0). unfold tge in H. eapply Genv.find_funct_ptr_negative; eauto. assert (f b = Some(b, 0)). eapply mg_functions; eauto. inv H1. rewrite H3 in H6; inv H6. reflexivity. Qed. Lemma match_call_cont: forall k tk ty cenv xenv cs, match_cont k tk ty cenv xenv cs -> match_cont (Csharpminor.call_cont k) (call_cont tk) ty cenv nil cs. Proof. induction 1; simpl; auto; econstructor; eauto. Qed. Lemma match_is_call_cont: forall tfn te sp tm k tk ty cenv xenv cs, match_cont k tk ty cenv xenv cs -> Csharpminor.is_call_cont k -> exists tk', star step tge (State tfn Sskip tk sp te tm) E0 (State tfn Sskip tk' sp te tm) /\ is_call_cont tk' /\ match_cont k tk' ty cenv nil cs. Proof. induction 1; simpl; intros; try contradiction. econstructor; split. apply star_refl. split. exact I. econstructor; eauto. exploit IHmatch_cont; eauto. intros [tk' [A B]]. exists tk'; split. eapply star_left; eauto. constructor. traceEq. auto. econstructor; split. apply star_refl. split. exact I. econstructor; eauto. econstructor; split. apply star_refl. split. exact I. econstructor; eauto. Qed. (** Properties of [switch] compilation *) Remark switch_table_shift: forall n sl base dfl, switch_target n (S dfl) (switch_table sl (S base)) = S (switch_target n dfl (switch_table sl base)). Proof. induction sl; intros; simpl. auto. destruct (Int.eq n i); auto. Qed. Remark length_switch_table: forall sl base1 base2, length (switch_table sl base1) = length (switch_table sl base2). Proof. induction sl; intros; simpl. auto. decEq; auto. Qed. Inductive transl_lblstmt_cont (ty: option typ) (cenv: compilenv) (xenv: exit_env): lbl_stmt -> cont -> cont -> Prop := | tlsc_default: forall s k ts, transl_stmt ty cenv (switch_env (LSdefault s) xenv) s = OK ts -> transl_lblstmt_cont ty cenv xenv (LSdefault s) k (Kblock (Kseq ts k)) | tlsc_case: forall i s ls k ts k', transl_stmt ty cenv (switch_env (LScase i s ls) xenv) s = OK ts -> transl_lblstmt_cont ty cenv xenv ls k k' -> transl_lblstmt_cont ty cenv xenv (LScase i s ls) k (Kblock (Kseq ts k')). Lemma switch_descent: forall ty cenv xenv k ls body s, transl_lblstmt ty cenv (switch_env ls xenv) ls body = OK s -> exists k', transl_lblstmt_cont ty cenv xenv ls k k' /\ (forall f sp e m, plus step tge (State f s k sp e m) E0 (State f body k' sp e m)). Proof. induction ls; intros. monadInv H. econstructor; split. econstructor; eauto. intros. eapply plus_left. constructor. apply star_one. constructor. traceEq. monadInv H. exploit IHls; eauto. intros [k' [A B]]. econstructor; split. econstructor; eauto. intros. eapply plus_star_trans. eauto. eapply star_left. constructor. apply star_one. constructor. reflexivity. traceEq. Qed. Lemma switch_ascent: forall f n sp e m ty cenv xenv k ls k1, let tbl := switch_table ls O in let ls' := select_switch n ls in transl_lblstmt_cont ty cenv xenv ls k k1 -> exists k2, star step tge (State f (Sexit (switch_target n (length tbl) tbl)) k1 sp e m) E0 (State f (Sexit O) k2 sp e m) /\ transl_lblstmt_cont ty cenv xenv ls' k k2. Proof. induction ls; intros; unfold tbl, ls'; simpl. inv H. econstructor; split. apply star_refl. econstructor; eauto. simpl in H. inv H. rewrite Int.eq_sym. destruct (Int.eq i n). econstructor; split. apply star_refl. econstructor; eauto. exploit IHls; eauto. intros [k2 [A B]]. rewrite (length_switch_table ls 1%nat 0%nat). rewrite switch_table_shift. econstructor; split. eapply star_left. constructor. eapply star_left. constructor. eexact A. reflexivity. traceEq. exact B. Qed. Lemma switch_match_cont: forall ty cenv xenv k cs tk ls tk', match_cont k tk ty cenv xenv cs -> transl_lblstmt_cont ty cenv xenv ls tk tk' -> match_cont (Csharpminor.Kseq (seq_of_lbl_stmt ls) k) tk' ty cenv (false :: switch_env ls xenv) cs. Proof. induction ls; intros; simpl. inv H0. apply match_Kblock2. econstructor; eauto. inv H0. apply match_Kblock2. eapply match_Kseq2. auto. eauto. Qed. Lemma transl_lblstmt_suffix: forall n ty cenv xenv ls body ts, transl_lblstmt ty cenv (switch_env ls xenv) ls body = OK ts -> let ls' := select_switch n ls in exists body', exists ts', transl_lblstmt ty cenv (switch_env ls' xenv) ls' body' = OK ts'. Proof. induction ls; simpl; intros. monadInv H. exists body; econstructor. rewrite EQ; eauto. simpl. reflexivity. monadInv H. destruct (Int.eq i n). exists body; econstructor. simpl. rewrite EQ; simpl. rewrite EQ0; simpl. reflexivity. eauto. Qed. Lemma switch_match_states: forall fn k e m tfn ts tk sp te tm cenv xenv f lo hi cs sz ls body tk' (TRF: transl_funbody cenv sz fn = OK tfn) (TR: transl_lblstmt (fn_return fn) cenv (switch_env ls xenv) ls body = OK ts) (MINJ: Mem.inject f m tm) (MCS: match_callstack f m tm (Frame cenv tfn e te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm)) (MK: match_cont k tk (fn_return fn) cenv xenv cs) (TK: transl_lblstmt_cont (fn_return fn) cenv xenv ls tk tk'), exists S, plus step tge (State tfn (Sexit O) tk' (Vptr sp Int.zero) te tm) E0 S /\ match_states (Csharpminor.State fn (seq_of_lbl_stmt ls) k e m) S. Proof. intros. destruct ls; simpl. inv TK. econstructor; split. eapply plus_left. constructor. apply star_one. constructor. traceEq. eapply match_state; eauto. inv TK. econstructor; split. eapply plus_left. constructor. apply star_one. constructor. traceEq. eapply match_state_seq; eauto. simpl. eapply switch_match_cont; eauto. Qed. (** Commutation between [find_label] and compilation *) Section FIND_LABEL. Variable lbl: label. Variable ty: option typ. Variable cenv: compilenv. Variable cs: callstack. Remark find_label_var_set: forall id e chunk s k, var_set cenv id e chunk = OK s -> find_label lbl s k = None. Proof. intros. unfold var_set in H. destruct (cenv!!id); try (monadInv H; reflexivity). destruct (chunktype_compat chunk m). inv H; auto. destruct (typ_eq (type_of_chunk m) (type_of_chunk chunk)); inv H; auto. Qed. Remark find_label_var_set_self: forall id ty s k, var_set_self cenv id ty = OK s -> find_label lbl s k = None. Proof. intros. unfold var_set_self in H. destruct (cenv!!id); try (monadInv H; reflexivity). destruct (typ_eq (type_of_chunk m) ty0); inv H; reflexivity. Qed. Lemma transl_lblstmt_find_label_context: forall xenv ls body ts tk1 tk2 ts' tk', transl_lblstmt ty cenv (switch_env ls xenv) ls body = OK ts -> transl_lblstmt_cont ty cenv xenv ls tk1 tk2 -> find_label lbl body tk2 = Some (ts', tk') -> find_label lbl ts tk1 = Some (ts', tk'). Proof. induction ls; intros. monadInv H. inv H0. simpl. simpl in H2. replace x with ts by congruence. rewrite H1. auto. monadInv H. inv H0. eapply IHls. eauto. eauto. simpl in H6. replace x with ts0 by congruence. simpl. rewrite H1. auto. Qed. Lemma transl_find_label: forall s k xenv ts tk, transl_stmt ty cenv xenv s = OK ts -> match_cont k tk ty cenv xenv cs -> match Csharpminor.find_label lbl s k with | None => find_label lbl ts tk = None | Some(s', k') => exists ts', exists tk', exists xenv', find_label lbl ts tk = Some(ts', tk') /\ transl_stmt ty cenv xenv' s' = OK ts' /\ match_cont k' tk' ty cenv xenv' cs end with transl_lblstmt_find_label: forall ls xenv body k ts tk tk1, transl_lblstmt ty cenv (switch_env ls xenv) ls body = OK ts -> match_cont k tk ty cenv xenv cs -> transl_lblstmt_cont ty cenv xenv ls tk tk1 -> find_label lbl body tk1 = None -> match Csharpminor.find_label_ls lbl ls k with | None => find_label lbl ts tk = None | Some(s', k') => exists ts', exists tk', exists xenv', find_label lbl ts tk = Some(ts', tk') /\ transl_stmt ty cenv xenv' s' = OK ts' /\ match_cont k' tk' ty cenv xenv' cs end. Proof. intros. destruct s; try (monadInv H); simpl; auto. (* assign *) eapply find_label_var_set; eauto. (* call *) destruct o; monadInv H; simpl; auto. destruct (list_eq_dec typ_eq x1 (sig_args s)); monadInv EQ4. simpl. eapply find_label_var_set_self; eauto. destruct (list_eq_dec typ_eq x1 (sig_args s)); monadInv EQ3. simpl; eauto. (* seq *) exploit (transl_find_label s1). eauto. eapply match_Kseq. eexact EQ1. eauto. destruct (Csharpminor.find_label lbl s1 (Csharpminor.Kseq s2 k)) as [[s' k'] | ]. intros [ts' [tk' [xenv' [A [B C]]]]]. exists ts'; exists tk'; exists xenv'. intuition. rewrite A; auto. intro. rewrite H. apply transl_find_label with xenv; auto. (* ifthenelse *) exploit (transl_find_label s1). eauto. eauto. destruct (Csharpminor.find_label lbl s1 k) as [[s' k'] | ]. intros [ts' [tk' [xenv' [A [B C]]]]]. exists ts'; exists tk'; exists xenv'. intuition. rewrite A; auto. intro. rewrite H. apply transl_find_label with xenv; auto. (* loop *) apply transl_find_label with xenv. auto. econstructor; eauto. simpl. rewrite EQ; auto. (* block *) apply transl_find_label with (true :: xenv). auto. constructor; auto. (* switch *) exploit switch_descent; eauto. intros [k' [A B]]. eapply transl_lblstmt_find_label. eauto. eauto. eauto. reflexivity. (* return *) destruct o; monadInv H; auto. destruct (typ_eq x0 (typ_of_opttyp ty)); monadInv EQ2; auto. (* label *) destruct (ident_eq lbl l). exists x; exists tk; exists xenv; auto. apply transl_find_label with xenv; auto. intros. destruct ls; monadInv H; simpl. (* default *) inv H1. simpl in H3. replace x with ts by congruence. rewrite H2. eapply transl_find_label; eauto. (* case *) inv H1. simpl in H7. exploit (transl_find_label s). eauto. eapply switch_match_cont; eauto. destruct (Csharpminor.find_label lbl s (Csharpminor.Kseq (seq_of_lbl_stmt ls) k)) as [[s' k''] | ]. intros [ts' [tk' [xenv' [A [B C]]]]]. exists ts'; exists tk'; exists xenv'; intuition. eapply transl_lblstmt_find_label_context; eauto. simpl. replace x with ts0 by congruence. rewrite H2. auto. intro. eapply transl_lblstmt_find_label. eauto. auto. eauto. simpl. replace x with ts0 by congruence. rewrite H2. auto. Qed. Remark find_label_store_parameters: forall vars s k, store_parameters cenv vars = OK s -> find_label lbl s k = None. Proof. induction vars; intros. monadInv H. auto. simpl in H. destruct a as [id lv]. monadInv H. simpl. rewrite (find_label_var_set_self id (type_of_chunk lv)); auto. Qed. End FIND_LABEL. Lemma transl_find_label_body: forall cenv xenv size f tf k tk cs lbl s' k', transl_funbody cenv size f = OK tf -> match_cont k tk (fn_return f) cenv xenv cs -> Csharpminor.find_label lbl f.(Csharpminor.fn_body) (Csharpminor.call_cont k) = Some (s', k') -> exists ts', exists tk', exists xenv', find_label lbl tf.(fn_body) (call_cont tk) = Some(ts', tk') /\ transl_stmt (fn_return f) cenv xenv' s' = OK ts' /\ match_cont k' tk' (fn_return f) cenv xenv' cs. Proof. intros. monadInv H. simpl. rewrite (find_label_store_parameters lbl cenv (Csharpminor.fn_params f)); auto. exploit transl_find_label. eexact EQ. eapply match_call_cont. eexact H0. instantiate (1 := lbl). rewrite H1. auto. Qed. (** The simulation diagram. *) Fixpoint seq_left_depth (s: Csharpminor.stmt) : nat := match s with | Csharpminor.Sseq s1 s2 => S (seq_left_depth s1) | _ => O end. Definition measure (S: Csharpminor.state) : nat := match S with | Csharpminor.State fn s k e m => seq_left_depth s | _ => O end. Lemma transl_step_correct: forall S1 t S2, Csharpminor.step gve S1 t S2 -> forall T1, match_states S1 T1 -> (exists T2, plus step tge T1 t T2 /\ match_states S2 T2) \/ (measure S2 < measure S1 /\ t = E0 /\ match_states S2 T1)%nat. Proof. induction 1; intros T1 MSTATE; inv MSTATE. (* skip seq *) monadInv TR. left. dependent induction MK. econstructor; split. apply plus_one. constructor. econstructor; eauto. econstructor; split. apply plus_one. constructor. eapply match_state_seq; eauto. exploit IHMK; eauto. intros [T2 [A B]]. exists T2; split. eapply plus_left. constructor. apply plus_star; eauto. traceEq. auto. (* skip block *) monadInv TR. left. dependent induction MK. econstructor; split. apply plus_one. constructor. econstructor; eauto. exploit IHMK; eauto. intros [T2 [A B]]. exists T2; split. eapply plus_left. constructor. apply plus_star; eauto. traceEq. auto. (* skip call *) monadInv TR. left. exploit match_is_call_cont; eauto. intros [tk' [A [B C]]]. exploit match_callstack_freelist; eauto. intros [tm' [P [Q R]]]. econstructor; split. eapply plus_right. eexact A. apply step_skip_call. auto. rewrite (sig_preserved_body _ _ _ _ TRF). auto. eauto. traceEq. econstructor; eauto. exact I. (* assign *) monadInv TR. exploit transl_expr_correct; eauto. intros [tv [EVAL VINJ]]. exploit var_set_correct; eauto. eapply chunktype_expr_correct; eauto. intros [te' [tm' [EXEC [MINJ' [MCS' OTHER]]]]]. left; econstructor; split. apply plus_one. eexact EXEC. econstructor; eauto. (* store *) monadInv TR. exploit transl_expr_correct. eauto. eauto. eexact H. eauto. intros [tv1 [EVAL1 VINJ1]]. exploit transl_expr_correct. eauto. eauto. eexact H0. eauto. intros [tv2 [EVAL2 VINJ2]]. exploit make_store_correct. eexact EVAL1. eexact EVAL2. eauto. eauto. auto. auto. intros [tm' [tv' [EXEC [STORE' MINJ']]]]. left; econstructor; split. apply plus_one. eexact EXEC. econstructor; eauto. eapply match_callstack_storev_mapped. eexact VINJ1. eauto. eauto. rewrite (nextblock_storev _ _ _ _ _ H1). rewrite (nextblock_storev _ _ _ _ _ STORE'). eauto. (* call *) simpl in H1. exploit functions_translated; eauto. intros [tfd [FIND TRANS]]. simpl in TR. destruct optid; monadInv TR. (* with return value *) destruct (list_eq_dec typ_eq x1 (sig_args (Csharpminor.funsig fd))); monadInv EQ4. exploit transl_expr_correct; eauto. intros [tvf [EVAL1 VINJ1]]. assert (tvf = vf). eapply val_inject_function_pointer; eauto. eapply match_callstack_match_globalenvs; eauto. subst tvf. exploit transl_exprlist_correct; eauto. intros [tvargs [EVAL2 VINJ2]]. left; econstructor; split. eapply plus_left. constructor. apply star_one. eapply step_call; eauto. apply sig_preserved; eauto. traceEq. econstructor; eauto. eapply match_Kcall_some with (cenv' := cenv); eauto. red; auto. eapply type_exprlist_correct; eauto. (* without return value *) destruct (list_eq_dec typ_eq x1 (sig_args (Csharpminor.funsig fd))); monadInv EQ3. exploit transl_expr_correct; eauto. intros [tvf [EVAL1 VINJ1]]. assert (tvf = vf). eapply val_inject_function_pointer; eauto. eapply match_callstack_match_globalenvs; eauto. subst tvf. exploit transl_exprlist_correct; eauto. intros [tvargs [EVAL2 VINJ2]]. left; econstructor; split. apply plus_one. eapply step_call; eauto. apply sig_preserved; eauto. econstructor; eauto. eapply match_Kcall_none with (cenv' := cenv); eauto. red; auto. eapply type_exprlist_correct; eauto. (* seq *) monadInv TR. left; econstructor; split. apply plus_one. constructor. econstructor; eauto. econstructor; eauto. (* seq 2 *) right. split. auto. split. auto. econstructor; eauto. (* ifthenelse *) monadInv TR. exploit transl_expr_correct; eauto. intros [tv [EVAL VINJ]]. left; exists (State tfn (if b then x0 else x1) tk (Vptr sp Int.zero) te tm); split. apply plus_one. eapply step_ifthenelse; eauto. eapply bool_of_val_inject; eauto. econstructor; eauto. destruct b; auto. (* loop *) monadInv TR. left; econstructor; split. apply plus_one. constructor. econstructor; eauto. econstructor; eauto. simpl. rewrite EQ; auto. (* block *) monadInv TR. left; econstructor; split. apply plus_one. constructor. econstructor; eauto. econstructor; eauto. (* exit seq *) monadInv TR. left. dependent induction MK. econstructor; split. apply plus_one. constructor. econstructor; eauto. simpl. auto. exploit IHMK; eauto. intros [T2 [A B]]. exists T2; split; auto. eapply plus_left. constructor. apply plus_star; eauto. traceEq. exploit IHMK; eauto. intros [T2 [A B]]. exists T2; split; auto. eapply plus_left. simpl. constructor. apply plus_star; eauto. traceEq. (* exit block 0 *) monadInv TR. left. dependent induction MK. econstructor; split. simpl. apply plus_one. constructor. econstructor; eauto. exploit IHMK; eauto. intros [T2 [A B]]. exists T2; split; auto. simpl. eapply plus_left. constructor. apply plus_star; eauto. traceEq. (* exit block n+1 *) monadInv TR. left. dependent induction MK. econstructor; split. simpl. apply plus_one. constructor. econstructor; eauto. auto. exploit IHMK; eauto. intros [T2 [A B]]. exists T2; split; auto. simpl. eapply plus_left. constructor. apply plus_star; eauto. traceEq. (* switch *) monadInv TR. left. exploit transl_expr_correct; eauto. intros [tv [EVAL VINJ]]. inv VINJ. exploit switch_descent; eauto. intros [k1 [A B]]. exploit switch_ascent; eauto. intros [k2 [C D]]. exploit transl_lblstmt_suffix; eauto. simpl. intros [body' [ts' E]]. exploit switch_match_states; eauto. intros [T2 [F G]]. exists T2; split. eapply plus_star_trans. eapply B. eapply star_left. econstructor; eauto. eapply star_trans. eexact C. apply plus_star. eexact F. reflexivity. reflexivity. traceEq. auto. (* return none *) monadInv TR. left. exploit match_callstack_freelist; eauto. intros [tm' [A [B C]]]. econstructor; split. apply plus_one. eapply step_return_0. eauto. econstructor; eauto. eapply match_call_cont; eauto. simpl; auto. (* return some *) monadInv TR. destruct (typ_eq x0 (typ_of_opttyp (fn_return f))); monadInv EQ2. left. exploit transl_expr_correct; eauto. intros [tv [EVAL VINJ]]. exploit match_callstack_freelist; eauto. intros [tm' [A [B C]]]. econstructor; split. apply plus_one. eapply step_return_1. eauto. eauto. econstructor; eauto. eapply match_call_cont; eauto. eapply type_expr_correct; eauto. (* label *) monadInv TR. left; econstructor; split. apply plus_one. constructor. econstructor; eauto. (* goto *) monadInv TR. exploit transl_find_label_body; eauto. intros [ts' [tk' [xenv' [A [B C]]]]]. left; econstructor; split. apply plus_one. apply step_goto. eexact A. econstructor; eauto. (* internal call *) monadInv TR. generalize EQ; clear EQ; unfold transl_function. caseEq (build_compilenv gce f). intros ce sz BC. destruct (zle sz Int.max_signed); try congruence. intro TRBODY. generalize TRBODY; intro TMP. monadInv TMP. set (tf := mkfunction (Csharpminor.fn_sig f) (fn_params_names f) (make_vars (fn_params_names f) (fn_vars_names f) (Sseq x1 x0)) sz (Sseq x1 x0)) in *. caseEq (Mem.alloc tm 0 (fn_stackspace tf)). intros tm' sp ALLOC'. exploit function_entry_ok; eauto; simpl; auto. intros [f2 [te2 [tm2 [EXEC [MINJ2 [IINCR MCS2]]]]]]. left; econstructor; split. eapply plus_left. constructor; simpl; eauto. simpl. eapply star_left. constructor. eapply star_right. eexact EXEC. constructor. reflexivity. reflexivity. traceEq. econstructor. eexact TRBODY. eauto. eexact MINJ2. eexact MCS2. inv MK; simpl in ISCC; contradiction || econstructor; eauto. (* external call *) monadInv TR. exploit external_call_mem_inject; eauto. intros [f' [vres' [tm' [EC [VINJ [MINJ' [UNMAPPED [OUTOFREACH [INCR SEPARATED]]]]]]]]]. left; econstructor; split. apply plus_one. econstructor; eauto. econstructor; eauto. apply match_callstack_incr_bound with (Mem.nextblock m) (Mem.nextblock tm). eapply match_callstack_external_call; eauto. intros. eapply external_call_bounds; eauto. omega. omega. eapply external_call_nextblock_incr; eauto. eapply external_call_nextblock_incr; eauto. simpl. change (Val.has_type vres (proj_sig_res (ef_sig ef))). eapply external_call_well_typed; eauto. (* return *) inv MK; inv H. (* no argument *) left; econstructor; split. apply plus_one. econstructor; eauto. simpl. econstructor; eauto. (* one argument *) exploit var_set_self_correct. eauto. eauto. eauto. eauto. eauto. eauto. instantiate (1 := PTree.set id tv te). apply PTree.gss. intros; apply PTree.gso; auto. intros [te' [tm' [A [B [C D]]]]]. left; econstructor; split. eapply plus_left. econstructor. simpl. eapply star_left. econstructor. eapply star_one. eexact A. reflexivity. traceEq. econstructor; eauto. Qed. Lemma match_globalenvs_init: forall m, Genv.init_mem prog = Some m -> match_globalenvs (Mem.flat_inj (Mem.nextblock m)). Proof. intros. constructor. intros. split. unfold Mem.flat_inj. rewrite zlt_true. auto. eapply Genv.find_symbol_not_fresh; eauto. rewrite <- H0. apply symbols_preserved. intros. unfold Mem.flat_inj. rewrite zlt_true. auto. generalize (Mem.nextblock_pos m). omega. Qed. Lemma transl_initial_states: forall S, Csharpminor.initial_state prog S -> exists R, Cminor.initial_state tprog R /\ match_states S R. Proof. induction 1. exploit function_ptr_translated; eauto. intros [tf [FIND TR]]. econstructor; split. econstructor. apply (Genv.init_mem_transf_partial2 _ _ _ TRANSL). eauto. simpl. fold tge. rewrite symbols_preserved. replace (prog_main tprog) with (prog_main prog). eexact H0. symmetry. unfold transl_program in TRANSL. eapply transform_partial_program2_main; eauto. eexact FIND. rewrite <- H2. apply sig_preserved; auto. eapply match_callstate with (f := Mem.flat_inj (Mem.nextblock m0)) (cs := @nil frame). auto. eapply Genv.initmem_inject; eauto. constructor. apply match_globalenvs_init. auto. instantiate (1 := gce). constructor. red; auto. constructor. rewrite H2; simpl; auto. Qed. Lemma transl_final_states: forall S R r, match_states S R -> Csharpminor.final_state S r -> Cminor.final_state R r. Proof. intros. inv H0. inv H. inv MK. inv RESINJ. constructor. Qed. Theorem transl_program_correct: forall (beh: program_behavior), not_wrong beh -> Csharpminor.exec_program prog beh -> Cminor.exec_program tprog beh. Proof. unfold Csharpminor.exec_program, Cminor.exec_program; intros. eapply simulation_star_preservation; eauto. eexact transl_initial_states. eexact transl_final_states. eexact transl_step_correct. Qed. End TRANSLATION.