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PPlusProperties.v
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Require Import Kami.Syntax.
Require Import Kami.Properties Kami.PProperties.
Import ListNotations.
Require Import Coq.Sorting.Permutation.
Require Import Coq.Sorting.PermutEq.
Require Import RelationClasses Setoid Morphisms.
Require Import ZArith Kami.Lib.EclecticLib.
Local Notation PPT_execs := (fun x => fst (snd x)).
Local Notation PPT_calls := (fun x => snd (snd x)).
Local Open Scope Z_scope.
(*TODO move somewhere else*)
Lemma existsb_nexists_str str l :
existsb (String.eqb str) l = false <->
~ In str l.
Proof.
split; repeat intro.
- assert (exists x, In x l /\ (String.eqb str) x = true) as P0.
{ exists str; split; auto. apply String.eqb_refl. }
rewrite <- existsb_exists in P0; rewrite P0 in *; discriminate.
- remember (existsb _ _) as exb; symmetry in Heqexb; destruct exb; auto.
exfalso; rewrite existsb_exists in Heqexb; dest.
rewrite String.eqb_eq in *; subst; auto.
Qed.
Section NeverCallAction.
Variable ty : Kind -> Type.
Inductive NeverCallActionT: forall k, ActionT ty k -> Prop :=
| NeverCallMCall meth s e lretT c: False -> @NeverCallActionT lretT (MCall meth s e c)
| NeverCallLetExpr k (e: Expr ty k) lretT c: (forall v, NeverCallActionT (c v)) -> @NeverCallActionT lretT (LetExpr e c)
| NeverCallLetAction k (a: ActionT ty k) lretT c: NeverCallActionT a -> (forall v, NeverCallActionT (c v)) -> @NeverCallActionT lretT (LetAction a c)
| NeverCallReadNondet k lretT c: (forall v, NeverCallActionT (c v)) -> @NeverCallActionT lretT (ReadNondet k c)
| NeverCallReadReg r k lretT c: (forall v, NeverCallActionT (c v)) -> @NeverCallActionT lretT (ReadReg r k c)
| NeverCallWriteReg r k (e: Expr ty k) lretT c: NeverCallActionT c -> @NeverCallActionT lretT (WriteReg r e c)
| NeverCallIfElse p k (atrue: ActionT ty k) afalse lretT c: (forall v, NeverCallActionT (c v)) -> NeverCallActionT atrue -> NeverCallActionT afalse -> @NeverCallActionT lretT (IfElse p atrue afalse c)
| NeverCallSys ls lretT c: NeverCallActionT c -> @NeverCallActionT lretT (Sys ls c)
| NeverCallReturn lretT e: @NeverCallActionT lretT (Return e).
End NeverCallAction.
Section NeverCallBaseModule.
Variable m : BaseModule.
Definition NeverCallBaseModule :=
(forall rule ty, In rule (getRules m) -> NeverCallActionT (snd rule ty)) /\
(forall meth ty, In meth (getMethods m) ->
forall v, NeverCallActionT (projT2 (snd meth) ty v)).
End NeverCallBaseModule.
Inductive NeverCallMod: Mod -> Prop :=
| BaseNeverCall m (HNCBaseModule: NeverCallBaseModule m): NeverCallMod (Base m)
| HideMethNeverCall m s (HNCModule: NeverCallMod m): NeverCallMod (HideMeth m s)
| ConcatModNeverCall m1 m2 (HNCModule1: NeverCallMod m1) (HNCModule2: NeverCallMod m2)
: NeverCallMod (ConcatMod m1 m2).
(*
Proves that the number of method calls returned
by [getNumCalls] is always greater than or
equal to 0.
*)
Lemma num_method_calls_positive
: forall (method : MethT) (labels : list FullLabel),
0 <= getNumCalls method labels.
Proof
fun method
=> list_ind _
(ltac:(discriminate) : 0 <= getNumCalls method [])
(fun (label : FullLabel) (labels : list FullLabel)
(H : 0 <= getNumFromCalls method (concat (map PPT_calls labels)))
=> list_ind _ H
(fun (method0 : MethT) (methods : MethsT)
(H0 : 0 <= getNumFromCalls method (methods ++ concat (map PPT_calls labels)))
=> sumbool_ind
(fun methods_eq
=> 0 <=
if methods_eq
then 1 + getNumFromCalls method (methods ++ concat (map PPT_calls labels))
else getNumFromCalls method (methods ++ concat (map PPT_calls labels)))
(fun _ => Z.add_nonneg_nonneg 1 _ (Zle_0_pos 1) H0)
(fun _ => H0)
(MethT_dec method method0))
(snd (snd label))).
(*
Proves that the number of method executions
counted by [getNumExecs] is always greater
than or equal to 0.
*)
Lemma num_method_execs_positive
: forall (method : MethT) (labels : list FullLabel),
0 <= getNumExecs method labels.
Proof.
induction labels; unfold getNumExecs in *; simpl; try lia.
destruct a; simpl; auto.
destruct p; simpl; auto.
destruct r0; simpl; auto.
destruct (MethT_dec method f); simpl; auto; subst.
destruct (getNumFromExecs f (map PPT_execs labels)); simpl; auto; try lia.
Defined.
Local Close Scope Z_scope.
Section PPlusTraceInclusion.
Definition getListFullLabel_diff_flat f (t : (RegsT *((list RuleOrMeth)*MethsT))) : Z:=
(getNumFromExecs f (PPT_execs t) - getNumFromCalls f (PPT_calls t))%Z.
Definition WeakInclusion_flat (t1 t2 : (RegsT *((list RuleOrMeth) * MethsT))) :=
(forall (f : MethT), (getListFullLabel_diff_flat f t1 = getListFullLabel_diff_flat f t2)%Z) /\
((exists rle, In (Rle rle) (PPT_execs t2)) ->
(exists rle, In (Rle rle) (PPT_execs t1))).
Inductive WeakInclusions_flat : list (RegsT * ((list RuleOrMeth) * MethsT)) -> list (RegsT *((list RuleOrMeth) * MethsT)) -> Prop :=
|WIf_Nil : WeakInclusions_flat nil nil
|WIf_Cons : forall (lt1 lt2 : list (RegsT *((list RuleOrMeth) * MethsT))) (t1 t2 : RegsT *((list RuleOrMeth) * MethsT)),
WeakInclusions_flat lt1 lt2 -> WeakInclusion_flat t1 t2 -> WeakInclusions_flat (t1::lt1) (t2::lt2).
Definition PPlusTraceList (m : BaseModule)(lt : list (RegsT * ((list RuleOrMeth) * MethsT))) :=
(exists (o : RegsT), PPlusTrace m o lt).
Definition PPlusTraceInclusion (m m' : BaseModule) :=
forall (o : RegsT)(tl : list (RegsT *((list RuleOrMeth) * MethsT))),
PPlusTrace m o tl -> exists (tl' : list (RegsT * ((list RuleOrMeth) * MethsT))), PPlusTraceList m' tl' /\ WeakInclusions_flat tl tl'.
Definition StrongPPlusTraceInclusion (m m' : BaseModule) :=
forall (o : RegsT)(tl : list (RegsT *((list RuleOrMeth) * MethsT))),
PPlusTrace m o tl -> exists (tl' : list (RegsT * ((list RuleOrMeth) * MethsT))), PPlusTrace m' o tl' /\ WeakInclusions_flat tl tl'.
End PPlusTraceInclusion.
Section BaseModule.
Variable m: BaseModule.
Variable o: RegsT.
Definition getLabelUpds (ls: list FullLabel) :=
concat (map (fun x => fst x) ls).
Definition getLabelExecs (ls: list FullLabel) :=
map (fun x => fst (snd x)) ls.
Definition getLabelCalls (ls: list FullLabel) :=
concat (map (fun x => (snd (snd x))) ls).
Lemma getLabelCalls_perm_rewrite l l' :
l [=] l' ->
getLabelCalls l [=] getLabelCalls l'.
Proof.
induction 1.
- reflexivity.
- unfold getLabelCalls; simpl; fold (getLabelCalls l); fold (getLabelCalls l').
rewrite IHPermutation; reflexivity.
- unfold getLabelCalls; simpl; fold (getLabelCalls l).
repeat rewrite app_assoc.
apply Permutation_app_tail, Permutation_app_comm.
- rewrite IHPermutation1, IHPermutation2.
reflexivity.
Qed.
Global Instance getLabelCalls_perm_rewrite' :
Proper (@Permutation (FullLabel) ==> @Permutation (MethT)) (@getLabelCalls) | 10.
Proof.
repeat red; intro; eauto using getLabelCalls_perm_rewrite.
Qed.
Lemma getLabelExecs_perm_rewrite l l' :
l [=] l' ->
getLabelExecs l [=] getLabelExecs l'.
Proof.
induction 1; auto.
- unfold getLabelExecs in *; simpl.
apply perm_skip; assumption.
- unfold getLabelExecs in *; simpl.
apply perm_swap.
- rewrite IHPermutation1, IHPermutation2; reflexivity.
Qed.
Lemma getLabelUpds_perm_rewrite l l' :
l [=] l' ->
getLabelUpds l [=] getLabelUpds l'.
Proof.
induction 1; auto; unfold getLabelUpds in *; simpl in *.
- apply Permutation_app_head; assumption.
- repeat rewrite app_assoc; apply Permutation_app_tail, Permutation_app_comm.
- rewrite IHPermutation1, IHPermutation2; reflexivity.
Qed.
Global Instance getLabelExecs_perm_rewrite' :
Proper (@Permutation (FullLabel) ==> @Permutation (RuleOrMeth)) (@getLabelExecs) | 10.
Proof.
repeat red; intro; eauto using getLabelExecs_perm_rewrite.
Qed.
Global Instance getLabelUpds_perm_rewrite' :
Proper (@Permutation (FullLabel) ==> @Permutation (string * {x : FullKind & fullType type x})) (@getLabelUpds) | 10.
Proof.
repeat red; intro; eauto using getLabelUpds_perm_rewrite.
Qed.
Lemma InCall_getLabelCalls f l:
InCall f l ->
In f (getLabelCalls l).
Proof.
induction l; unfold InCall,getLabelCalls in *; intros; simpl; dest; auto.
destruct H; subst;apply in_app_iff; [left; assumption|right; apply IHl].
exists x; auto.
Qed.
Lemma getLabelCalls_InCall f l:
In f (getLabelCalls l) ->
InCall f l.
Proof.
induction l; unfold InCall, getLabelCalls in *; intros; simpl in *;[contradiction|].
rewrite in_app_iff in H; destruct H;[exists a; auto|specialize (IHl H);dest].
exists x; auto.
Qed.
Corollary InCall_getLabelCalls_iff f l:
InCall f l <->
In f (getLabelCalls l).
Proof.
split; intro; eauto using InCall_getLabelCalls, getLabelCalls_InCall.
Qed.
Lemma PPlusSubsteps_PSubsteps:
forall upds execs calls,
PPlusSubsteps m o upds execs calls ->
exists l,
PSubsteps m o l /\
upds [=] getLabelUpds l /\
execs [=] getLabelExecs l /\
calls [=] getLabelCalls l.
Proof.
unfold getLabelUpds, getLabelExecs, getLabelCalls.
induction 1; dest.
- exists nil.
repeat split; auto; constructor; auto.
- exists ((u, (Rle rn, cs)) :: x).
repeat split; auto; try constructor; auto; simpl.
+ econstructor; eauto; intros.
* clear - H1 H4 HUpds HExecs HCalls HDisjRegs.
intro.
destruct (HDisjRegs k); auto.
left; intro.
clear - H1 H4 H H0.
rewrite H1 in H.
rewrite <- flat_map_concat_map in H.
rewrite in_map_iff in H.
setoid_rewrite in_flat_map in H.
rewrite in_map_iff in *; dest; subst.
firstorder fail.
* clear - H2 H4 HExecs HNoRle.
apply HNoRle.
rewrite H2.
rewrite in_map_iff.
firstorder fail.
+ rewrite H1 in HUpds; auto.
+ rewrite HExecs.
constructor; auto.
+ rewrite H3 in HCalls; auto.
- exists ((u, (Meth (fn, existT SignT (projT1 fb) (argV, retV)), cs)) :: x).
repeat split; auto; try constructor; auto; simpl.
+ econstructor 3; eauto; intros.
* clear - H1 H4 HUpds HExecs HCalls HDisjRegs.
intro.
destruct (HDisjRegs k); auto.
left; intro.
clear - H1 H4 H H0.
rewrite H1 in H.
rewrite <- flat_map_concat_map in H.
rewrite in_map_iff in H.
setoid_rewrite in_flat_map in H.
rewrite in_map_iff in *; dest; subst.
firstorder fail.
+ rewrite H1 in HUpds; auto.
+ rewrite HExecs.
constructor; auto.
+ rewrite H3 in HCalls; auto.
Qed.
End BaseModule.
Section PPlusSubsteps_rewrite.
Lemma PPlusSubsteps_rewrite_regs m o1 o2 upds execs calls:
(o1 [=] o2) ->
PPlusSubsteps m o1 upds execs calls ->
PPlusSubsteps m o2 upds execs calls.
Proof.
induction 2.
- econstructor 1.
rewrite <- H; assumption.
- econstructor 2;(rewrite <- H || apply (PSemAction_rewrite_state H) in HPAction); eauto.
- econstructor 3;(rewrite <- H || apply (PSemAction_rewrite_state H) in HPAction); eauto.
Qed.
Lemma PPlusSubsteps_rewrite_upds m o execs calls upds1 upds2:
(upds1 [=] upds2) ->
PPlusSubsteps m o upds1 execs calls ->
PPlusSubsteps m o upds2 execs calls.
Proof.
induction 2.
- apply Permutation_nil in H; rewrite H.
econstructor 1; assumption.
- econstructor 2; eauto.
rewrite H in HUpds.
assumption.
- econstructor 3; eauto.
rewrite H in HUpds.
assumption.
Qed.
Lemma PPlusSubsteps_rewrite_execs m o upds calls execs1 execs2:
(execs1 [=] execs2) ->
PPlusSubsteps m o upds execs1 calls ->
PPlusSubsteps m o upds execs2 calls.
Proof.
induction 2.
- apply Permutation_nil in H; rewrite H.
econstructor 1; assumption.
- econstructor 2; eauto.
rewrite H in HExecs.
assumption.
- econstructor 3; eauto.
rewrite H in HExecs.
assumption.
Qed.
Lemma PPlusSubsteps_rewrite_calls m o upds execs calls1 calls2:
(calls1 [=] calls2) ->
PPlusSubsteps m o upds execs calls1 ->
PPlusSubsteps m o upds execs calls2.
Proof.
induction 2.
- apply Permutation_nil in H; rewrite H.
econstructor 1; assumption.
- econstructor 2; eauto.
rewrite H in HCalls.
assumption.
- econstructor 3; eauto.
rewrite H in HCalls.
assumption.
Qed.
Lemma PPlusSubsteps_rewrite_all m o1 o2 upds1 execs1 calls1 upds2 execs2 calls2 :
o1 [=] o2 ->
upds1 [=] upds2 ->
execs1 [=] execs2 ->
calls1 [=] calls2 ->
PPlusSubsteps m o1 upds1 execs1 calls1 ->
PPlusSubsteps m o2 upds2 execs2 calls2.
Proof.
intros.
apply (PPlusSubsteps_rewrite_regs H) in H3;
apply (PPlusSubsteps_rewrite_upds H0) in H3;
apply (PPlusSubsteps_rewrite_execs H1) in H3;
apply (PPlusSubsteps_rewrite_calls H2) in H3;
assumption.
Qed.
Global Instance PPlusSubsteps_rewrite' :
Proper (Logic.eq ==>
@Permutation (string * {x : FullKind & fullType type x}) ==>
@Permutation (string * {x : FullKind & fullType type x}) ==>
@Permutation RuleOrMeth ==>
@Permutation MethT ==>
iff) (@PPlusSubsteps)| 10.
Proof.
repeat red; intros; split; intros; subst; eauto using Permutation_sym, PPlusSubsteps_rewrite_all.
symmetry in H0.
symmetry in H1.
symmetry in H2.
symmetry in H3.
eapply PPlusSubsteps_rewrite_all; eauto.
Qed.
End PPlusSubsteps_rewrite.
Lemma Permutation_flat_map_rewrite (A B : Type) (l1 l2 : list A) (f : A -> list B) :
l1 [=] l2 ->
flat_map f l1 [=] flat_map f l2.
Proof.
induction 1; simpl in *; auto.
- apply Permutation_app_head; assumption.
- repeat rewrite app_assoc; apply Permutation_app_tail.
rewrite Permutation_app_comm; reflexivity.
- rewrite IHPermutation1, IHPermutation2; reflexivity.
Qed.
Global Instance Permutation_flat_map_rewrite' (A B : Type)(f : A -> list B):
Proper (@Permutation A ==> @Permutation B) (@flat_map A B f) | 10.
repeat red; intros; intros; eauto using Permutation_flat_map_rewrite.
Qed.
Lemma PSubsteps_PPlusSubsteps:
forall m o l,
PSubsteps m o l ->
PPlusSubsteps m o (getLabelUpds l) (getLabelExecs l) (getLabelCalls l).
Proof.
induction 1; unfold getLabelUpds, getLabelExecs, getLabelCalls in *; try setoid_rewrite <- flat_map_concat_map.
- econstructor; eauto.
- rewrite HLabel; simpl; setoid_rewrite <-flat_map_concat_map in IHPSubsteps.
econstructor 2; intros; eauto.
+ clear - HDisjRegs.
induction ls.
* firstorder.
* intro; simpl in *; rewrite map_app, in_app_iff, DeM1.
assert (DisjKey (flat_map (fun x : FullLabel => fst x) ls) u);[eapply IHls; eauto|].
specialize (HDisjRegs a (or_introl _ eq_refl) k); specialize (H k).
firstorder fail.
+ rewrite in_map_iff in H0; dest; rewrite <- H0.
eapply HNoRle; eauto.
- rewrite HLabel; simpl; setoid_rewrite <- flat_map_concat_map in IHPSubsteps.
econstructor 3; intros; eauto.
+ clear - HDisjRegs.
induction ls.
* firstorder.
* intro; simpl in *; rewrite map_app, in_app_iff, DeM1.
assert (DisjKey (flat_map (fun x : FullLabel => fst x) ls) u);[eapply IHls; eauto|].
specialize (HDisjRegs a (or_introl _ eq_refl) k); specialize (H k).
firstorder fail.
Qed.
Section PPlusStep.
Variable m: BaseModule.
Variable o: RegsT.
Lemma PPlusStep_PStep:
forall upds execs calls,
PPlusStep m o upds execs calls ->
exists l,
PStep (Base m) o l /\
upds [=] getLabelUpds l /\
execs [=] getLabelExecs l /\
calls [=] getLabelCalls l.
Proof.
induction 1.
apply PPlusSubsteps_PSubsteps in H; dest.
exists x; repeat split; eauto.
econstructor 1; eauto.
intros f HInDef; specialize (H0 f HInDef).
unfold getNumCalls, getNumExecs, getLabelExecs, getLabelCalls in *.
rewrite <-H3, <-H2; assumption.
Qed.
Lemma PStep_PPlusStep :
forall l,
PStep (Base m) o l ->
PPlusStep m o (getLabelUpds l) (getLabelExecs l) (getLabelCalls l).
Proof.
intros; inv H; econstructor.
- apply PSubsteps_PPlusSubsteps in HPSubsteps; assumption.
- intros f HInDef; specialize (HMatching f).
apply HMatching; auto.
Qed.
End PPlusStep.
Section PPlusTrace.
Variable m: BaseModule.
Lemma PPlusTrace_PTrace o ls :
PPlusTrace m o ls ->
exists ls',
PTrace (Base m) o ls' /\
PermutationEquivLists (map fst ls) (map getLabelUpds ls') /\
PermutationEquivLists (map PPT_execs ls) (map getLabelExecs ls') /\
PermutationEquivLists (map PPT_calls ls) (map getLabelCalls ls').
Proof.
induction 1; subst; dest.
- exists nil; repeat split; econstructor; eauto.
- apply PPlusStep_PStep in HPPlusStep; dest.
exists (x0::x); repeat split; eauto; simpl in *; econstructor 2; eauto.
+ unfold PPlusUpdRegs in HUpdRegs; dest.
repeat split; eauto.
intros; destruct (H9 _ _ H10).
* rewrite H5 in H11; unfold getLabelUpds in H11.
rewrite <- flat_map_concat_map, in_flat_map in *; dest.
left; exists (fst x1); split; auto.
apply (in_map fst) in H11; assumption.
* destruct H11; right; split; auto.
intro; apply H11; dest.
unfold getLabelUpds in *.
rewrite H5, <- flat_map_concat_map, in_map_iff.
setoid_rewrite in_flat_map.
rewrite in_map_iff in H13,H14; dest.
exists x2; split; auto.
exists x3; subst; auto.
Qed.
Definition extractTriple (lfl : list FullLabel) : (RegsT * ((list RuleOrMeth) * MethsT)) :=
(getLabelUpds lfl, (getLabelExecs lfl, getLabelCalls lfl)).
Fixpoint extractTriples (llfl : list (list FullLabel)) : list (RegsT * ((list RuleOrMeth) * MethsT)) :=
match llfl with
|lfl::llfl' => (extractTriple lfl)::(extractTriples llfl')
|nil => nil
end.
Lemma extractTriples_nil l :
extractTriples l = nil -> l = nil.
Proof.
destruct l; intros; auto.
inv H.
Qed.
Lemma PTrace_PPlusTrace o ls:
PTrace (Base m) o ls ->
PPlusTrace m o (extractTriples ls).
Proof.
induction 1; subst; intros.
- econstructor; eauto.
- simpl; econstructor 2; eauto.
+ apply PStep_PPlusStep; apply HPStep.
+ unfold PUpdRegs,PPlusUpdRegs in *; dest; repeat split; eauto.
intros; destruct (H1 _ _ H2);[left|right]; unfold getLabelUpds; dest.
* rewrite <- flat_map_concat_map, in_flat_map.
rewrite (in_map_iff fst) in H3; dest; rewrite <- H3 in H4.
firstorder.
* split; auto; intro; apply H3.
rewrite <- flat_map_concat_map, in_map_iff in H5; dest.
rewrite in_flat_map in H6; dest.
exists (fst x0); split.
-- rewrite in_map_iff; exists x0; firstorder.
-- rewrite <- H5, in_map_iff; exists x; firstorder.
+ unfold extractTriple; reflexivity.
Qed.
End PPlusTrace.
Section PPlusTraceInclusion.
Lemma InExec_rewrite f l:
In (Meth f) (getLabelExecs l) <-> InExec f l.
Proof.
split; unfold InExec; induction l; simpl in *; intros; auto.
Qed.
Lemma InCall_rewrite f l :
In f (getLabelCalls l) <-> InCall f l.
Proof.
split; unfold InCall; induction l; simpl in *; intros; dest; try contradiction.
- unfold getLabelCalls in H. rewrite <- flat_map_concat_map, in_flat_map in H.
assumption.
- unfold getLabelCalls; rewrite <- flat_map_concat_map, in_flat_map.
firstorder fail.
Qed.
Lemma WeakInclusion_flat_WeakInclusion (l1 l2 : list FullLabel) :
WeakInclusion_flat (extractTriple l1) (extractTriple l2) <->
WeakInclusion l1 l2.
Proof.
split; auto.
Qed.
Lemma WeakInclusions_flat_WeakInclusions (ls1 ls2 : list (list FullLabel)) :
WeakInclusions_flat (extractTriples ls1) (extractTriples ls2) ->
WeakInclusions ls1 ls2.
Proof.
revert ls2; induction ls1; intros; simpl in *; inv H.
- symmetry in H0; apply extractTriples_nil in H0; subst; econstructor.
- destruct ls2; inv H2.
econstructor 2.
+ eapply IHls1; eauto.
+ apply WeakInclusion_flat_WeakInclusion; assumption.
Qed.
Lemma StrongPPlusTraceInclusion_PPlusTraceInclusion m m' :
StrongPPlusTraceInclusion m m' ->
PPlusTraceInclusion m m'.
Proof.
unfold StrongPPlusTraceInclusion, PPlusTraceInclusion, PPlusTraceList; intros.
specialize (H _ _ H0); dest.
exists x; split; auto.
exists o; auto.
Qed.
Lemma WeakInclusions_flat_PermutationEquivLists_r ls1:
forall l ls2,
WeakInclusions_flat (extractTriples ls1) l ->
PermutationEquivLists (map fst l) (map getLabelUpds ls2) ->
PermutationEquivLists (map PPT_execs l) (map getLabelExecs ls2) ->
PermutationEquivLists (map PPT_calls l) (map getLabelCalls ls2) ->
WeakInclusions_flat (extractTriples ls1) (extractTriples ls2).
Proof.
induction ls1; intros; inv H; simpl in *.
- destruct ls2; simpl in *.
+ econstructor.
+ inv H2.
- destruct ls2; inv H2; inv H1; inv H0; simpl in *.
econstructor.
+ eapply IHls1; eauto.
+ unfold WeakInclusion_flat in *; dest; simpl in *.
split; intros;
[unfold getListFullLabel_diff_flat in *; simpl in *; rewrite <-H9, <-H10; apply H|].
apply H0; setoid_rewrite H10; assumption.
Qed.
Lemma PPlusTraceInclusion_PTraceInclusion (m m' : BaseModule) :
PPlusTraceInclusion m m' ->
PTraceInclusion (Base m) (Base m').
Proof.
repeat intro.
apply (PTrace_PPlusTrace) in H0.
specialize (H o _ H0); dest.
destruct H.
apply (PPlusTrace_PTrace) in H; dest.
exists x1; split.
- exists x0; assumption.
- apply WeakInclusions_flat_WeakInclusions.
apply (WeakInclusions_flat_PermutationEquivLists_r _ _ H1 H2 H3 H4).
Qed.
Lemma PPlusTraceInclusion_TraceInclusion (m m' : BaseModule) (Wfm: WfMod type (Base m)) (Wfm': WfMod type (Base m')):
PPlusTraceInclusion m m' ->
TraceInclusion (Base m) (Base m').
Proof.
intros; apply PTraceInclusion_TraceInclusion, PPlusTraceInclusion_PTraceInclusion; auto.
Qed.
End PPlusTraceInclusion.
Lemma NoDup_app_iff (A : Type) (l1 l2 : list A) :
NoDup (l1++l2) <->
NoDup l1 /\
NoDup l2 /\
(forall a, In a l1 -> ~In a l2) /\
(forall a, In a l2 -> ~In a l1).
Proof.
repeat split; intros; dest.
- induction l1; econstructor; inv H; firstorder.
- induction l2; econstructor; apply NoDup_remove in H; dest; firstorder.
- induction l1; auto.
simpl in H; rewrite NoDup_cons_iff in H; dest; firstorder.
subst; firstorder.
- induction l2; auto.
apply NoDup_remove in H; dest; firstorder.
subst; firstorder.
- induction l1; simpl; eauto.
constructor.
+ rewrite in_app_iff, DeM1; split; firstorder.
inv H; assumption.
+ inv H; eapply IHl1; eauto; firstorder.
Qed.
Lemma NoDup_app_Disj (A : Type) (dec : forall (a1 a2 : A), {a1 = a2}+{a1 <> a2}) :
forall (l1 l2 : list A),
NoDup (l1++l2) ->
(forall a, ~In a l1 \/ ~In a l2).
Proof.
intros.
rewrite NoDup_app_iff in H; dest.
destruct (in_dec dec a l1); auto.
Qed.
Notation remove_calls := (fun x y => negb (getBool (string_dec (fst x) (fst y)))).
Notation keep_calls := (fun x y => (getBool (string_dec (fst x) (fst y)))).
Definition methcmp (m1 m2 : MethT) : bool := getBool (MethT_dec m1 m2).
Definition remove_execs (calls : MethsT) (exec : RuleOrMeth) : bool :=
match exec with
| Rle _ => false
| Meth f => existsb (methcmp f) calls
end.
Lemma key_not_In_filter (f : DefMethT) (calls : MethsT) :
key_not_In (fst f) calls ->
filter (remove_calls f) calls = calls.
Proof.
induction calls; unfold key_not_In in *; simpl in *; intros; auto.
destruct string_dec; pose proof (H (snd a)); simpl in *.
- apply False_ind; apply H0; left.
destruct a; simpl in *; rewrite e; reflexivity.
- rewrite IHcalls; auto.
repeat intro; specialize (H v); apply H; right; assumption.
Qed.
Lemma PSemAction_inline_notIn (f : DefMethT) o k (a : ActionT type k)
readRegs newRegs calls (fret : type k) :
PSemAction o a readRegs newRegs calls fret ->
~In (fst f) (map fst calls) ->
PSemAction o (inlineSingle a f) readRegs newRegs calls fret.
Proof.
induction 1; simpl; intros.
- destruct (fst f =? meth) eqn:G. rewrite <- (proj1 (String.eqb_eq _ _) G) in *.
+ apply False_ind; apply H0; rewrite HAcalls; simpl; left; reflexivity.
+ econstructor 1; eauto.
apply IHPSemAction; intro; apply H0; rewrite HAcalls; simpl; right; assumption.
- econstructor 2; eauto.
- econstructor 3; eauto;[eapply IHPSemAction1|eapply IHPSemAction2];
intro; apply H1; rewrite HUCalls, map_app, in_app_iff;[left|right]; assumption.
- econstructor 4; eauto.
- econstructor 5; eauto.
- econstructor 6; eauto.
- econstructor 7; eauto; [eapply IHPSemAction1|eapply IHPSemAction2];
intro; apply H1; rewrite HUCalls, map_app, in_app_iff;[left|right]; assumption.
- econstructor 8; eauto; [eapply IHPSemAction1|eapply IHPSemAction2];
intro; apply H1; rewrite HUCalls, map_app, in_app_iff;[left|right]; assumption.
- econstructor 9; eauto.
- econstructor 10; eauto.
Qed.
Inductive PSemAction_meth_collector (f : DefMethT) (o : RegsT) : RegsT -> RegsT -> MethsT -> MethsT -> Prop :=
|NilCalls : PSemAction_meth_collector f o nil nil nil nil
|ConsCalls reads1 reads2 reads upds1 upds2
upds calls1 calls2 calls calls'
calls'' argV retV:
PSemAction_meth_collector f o reads1 upds1 calls1 calls' ->
DisjKey upds1 upds2 ->
reads [=] reads1 ++ reads2 ->
upds [=] upds1 ++ upds2 ->
calls [=] calls1 ++ calls2 ->
calls'' [=] ((fst f, (existT _ (projT1 (snd f)) (argV, retV)))::calls') ->
PSemAction o (projT2 (snd f) type argV) reads2 upds2 calls2 retV ->
PSemAction_meth_collector f o reads upds calls calls''.
Lemma Produce_action_from_collector (f : DefMethT) (o : RegsT) reads upds calls calls' call:
In call calls' ->
PSemAction_meth_collector f o reads upds calls calls' ->
exists reads1 reads2 upds1 upds2 calls1 calls2 calls'' argV retV,
DisjKey upds1 upds2 /\
upds [=] upds2++upds1 /\
calls [=] calls2++calls1 /\
reads [=] reads2 ++ reads1 /\
call = (fst f, (existT _ (projT1 (snd f)) (argV, retV))) /\
calls' [=] call::calls'' /\
PSemAction o (projT2 (snd f) type argV) reads2 upds2 calls2 retV /\
PSemAction_meth_collector f o reads1 upds1 calls1 calls''.
Proof.
induction 2.
- contradiction.
- rewrite H5 in H.
destruct H.
+ exists reads1, reads2, upds1, upds2, calls1, calls2, calls', argV, retV; subst.
repeat split; auto; (rewrite H2 || rewrite H3 || rewrite H4 || rewrite H5); subst; auto; apply Permutation_app_comm.
+ specialize (IHPSemAction_meth_collector H); dest.
rewrite H8, H9, H10, H12 in *.
exists (reads2++x), x0, (upds2++x1), x2, (calls2++x3), x4, ((fst f, existT _ (projT1 (snd f)) (argV, retV))::x5), x6, x7.
repeat split; auto.
* intro; destruct (H7 k), (H1 k); rewrite map_app,in_app_iff in *; dest; firstorder fail.
* rewrite H3, <-app_assoc; apply Permutation_app_head, Permutation_app_comm.
* rewrite H4, <-app_assoc; apply Permutation_app_head, Permutation_app_comm.
* rewrite H2, <-app_assoc; apply Permutation_app_head, Permutation_app_comm.
* rewrite H5; apply perm_swap.
* econstructor.
-- apply H14.
-- assert (DisjKey x1 upds2).
++ intro; destruct (H1 k);[rewrite map_app,in_app_iff,DeM1 in *; dest; left; assumption|right; assumption].
++ apply H15.
-- apply Permutation_app_comm.
-- apply Permutation_app_comm.
-- apply Permutation_app_comm.
-- reflexivity.
-- assumption.
Qed.
Lemma collector_perm_rewrite f o reads upds calls calls1':
PSemAction_meth_collector f o reads upds calls calls1' ->
forall calls2',
calls1' [=] calls2' ->
PSemAction_meth_collector f o reads upds calls calls2'.
Proof.
induction 1; intros.
- apply Permutation_nil in H; subst.
constructor.
- rewrite H6 in H4.
econstructor; eauto.
Qed.
Global Instance collector_perm_rewrite' :
Proper (eq ==> eq ==> eq ==> eq ==> eq ==> (@Permutation MethT) ==> iff) (@PSemAction_meth_collector) | 10.
Proof.
repeat red; intro; split; intros; subst; eauto using collector_perm_rewrite, Permutation_sym.
Qed.
Lemma collector_split (f : DefMethT) o calls1' calls2' :
forall reads upds calls,
PSemAction_meth_collector f o reads upds calls (calls1'++calls2') ->
exists reads1 reads2 upds1 upds2 calls1 calls2,
reads [=] reads1 ++ reads2 /\
upds [=] upds1 ++ upds2 /\
calls [=] calls1 ++ calls2 /\
DisjKey upds1 upds2 /\
PSemAction_meth_collector f o reads1 upds1 calls1 calls1' /\
PSemAction_meth_collector f o reads2 upds2 calls2 calls2'.
Proof.
induction calls1'; simpl; intros.
- exists nil, reads, nil, upds, nil, calls.
repeat split; auto.
+ intro; auto.
+ constructor.
- specialize (Produce_action_from_collector _ (in_eq _ _) H) as TMP; dest.
apply Permutation_cons_inv in H5.
rewrite <- H5 in H7.
specialize (IHcalls1' _ _ _ H7) as TMP; dest.
exists (x0++x8), x9, (x2++x10), x11, (x4++x12), x13.
repeat split.
+ rewrite H8, app_assoc in H3; assumption.
+ rewrite H9, app_assoc in H1; assumption.
+ rewrite H10, app_assoc in H2; assumption.
+ intro; specialize (H11 k); specialize (H0 k); clear - H0 H11 H9; rewrite H9,map_app, in_app_iff in *; firstorder fail.
+ econstructor.
* apply H12.
* rewrite H9 in H0; assert (DisjKey x10 x2);[intro; specialize (H0 k);
rewrite map_app, in_app_iff in *;clear - H0; firstorder fail| apply H14].
* apply Permutation_app_comm.
* apply Permutation_app_comm.
* apply Permutation_app_comm.
* rewrite H4; reflexivity.
* assumption.
+ assumption.
Qed.
Lemma collector_correct_fst f o reads upds calls calls' call :
In call calls' ->
PSemAction_meth_collector f o reads upds calls calls' ->
fst call = fst f.
Proof.
intros.
specialize (Produce_action_from_collector _ H H0) as TMP; dest.
destruct call; inv H5; simpl; reflexivity.
Qed.
Lemma collector_correct_pair f o reads upds calls calls' call :
In call calls' ->
PSemAction_meth_collector f o reads upds calls calls' ->
(fst call, projT1 (snd call)) = (fst f, projT1 (snd f)).
Proof.
intros.
specialize (Produce_action_from_collector _ H H0) as TMP; dest.
destruct call; inv H5; simpl; reflexivity.
Qed.
Definition called_by (f : DefMethT) (call :MethT) : bool :=
(getBool (prod_dec string_dec Signature_dec (fst f, projT1 (snd f)) (fst call, projT1 (snd call)))).
Local Notation complement f := (fun x => negb (f x)).
Definition called_execs (calls : MethsT) (exec : RuleOrMeth) : bool :=
match exec with
| Rle _ => false
| Meth f => existsb (methcmp f) calls
end.
Lemma separate_calls_by_filter (A : Type) (l : list A) (f : A -> bool):
l [=] (filter f l) ++ (filter (complement f) l).
Proof.
induction l; auto; simpl.
destruct (f a); simpl; rewrite IHl at 1;[reflexivity|apply Permutation_middle].
Qed.
Lemma reduce_called_execs_list execs c :
~In (Meth c) execs ->
forall calls,
(filter (called_execs (c::calls)) execs) = (filter (called_execs calls) execs).
Proof.
induction execs; intros; auto.
unfold called_execs, methcmp in *; destruct a; simpl in *.
- eapply IHexecs.
intro; apply H.
right; assumption.
- destruct MethT_dec; simpl.
+ apply False_ind; apply H.
rewrite e; left; reflexivity.
+ rewrite IHexecs;[reflexivity|].
intro; apply H; right; assumption.
Qed.
Corollary collector_called_by_filter_irrel f o calls' :
forall reads upds calls,
PSemAction_meth_collector f o reads upds calls calls' ->
(filter (called_by f) calls') = calls'.
Proof.
induction calls'; intros; auto.
specialize (collector_correct_pair _ (in_eq _ _) H); intro.
Opaque prod_dec.
unfold called_by; simpl; destruct prod_dec;[simpl;fold (called_by f)|symmetry in H0; contradiction].
specialize (Produce_action_from_collector _ (in_eq _ _ ) H) as TMP; dest.
apply Permutation_cons_inv in H6.
rewrite <- H6 in H8.
erewrite IHcalls'; eauto.
Transparent prod_dec.
Qed.
Corollary collector_complement_called_by_filter_nil f o calls' :
forall reads upds calls,
PSemAction_meth_collector f o reads upds calls calls' ->
(filter (complement (called_by f)) calls') = nil.
Proof.
intros.
specialize (separate_calls_by_filter calls' (called_by f)) as parti.
rewrite (collector_called_by_filter_irrel H) in parti.
rewrite <- app_nil_r in parti at 1.
apply Permutation_app_inv_l in parti.
apply Permutation_nil in parti; assumption.
Qed.
Lemma collector_correct_snd f o reads upds calls calls' call :
In call calls' ->
PSemAction_meth_collector f o reads upds calls calls' ->
exists argV retV,
snd call = (existT _ (projT1 (snd f)) (argV, retV)).
Proof.
intros.
specialize (Produce_action_from_collector _ H H0) as TMP; dest.
destruct call; inv H5; simpl; exists x6, x7; reflexivity.
Qed.
Lemma notIn_filter_nil (f : DefMethT) (calls : MethsT) :
~In (fst f, projT1 (snd f)) (getKindAttr calls) ->
filter (called_by f) calls = nil.
Proof.
induction calls; auto; simpl.
intro; dest.
unfold called_by; destruct prod_dec; simpl; auto.
apply False_ind, H; auto.
Qed.
Lemma notIn_complement_filter_irrel (f : DefMethT) (calls : MethsT) :
~In (fst f, projT1 (snd f)) (getKindAttr calls) ->
filter (complement (called_by f)) calls = calls.
Proof.
induction calls; auto; simpl; intros.
unfold called_by in *; destruct prod_dec; simpl in *;
[apply False_ind, H|rewrite IHcalls];auto.
Qed.
Lemma filter_complement_disj (A : Type) (dec : forall (a1 a2 : A), {a1=a2}+{a1<>a2}) (f : A -> bool) (l : list A) :
forall x, ~In x (filter f l) \/ ~In x (filter (complement f) l).
Proof.
intros.
destruct (in_dec dec x (filter f l)).
- rewrite filter_In in i; dest.
right; intro.
rewrite filter_In in H1; dest.
rewrite H0 in H2; discriminate.
- left; assumption.
Qed.
Lemma PSemAction_NoDup_Key_Writes k o (a : ActionT type k) readRegs newRegs calls (fret : type k) :
PSemAction o a readRegs newRegs calls fret ->
NoDup (map fst newRegs).
Proof.
induction 1;
eauto;[|
rewrite HANewRegs; simpl; econstructor; eauto; intro;
specialize (fst_produce_snd _ _ H0) as TMP; dest; specialize (HDisjRegs x);
contradiction| | |subst; econstructor];
rewrite HUNewRegs; rewrite map_app,NoDup_app_iff; repeat split; eauto;
repeat intro; specialize (HDisjRegs a0); tauto.
Qed.
Corollary PSemAction_NoDup_Writes k o (a : ActionT type k) readRegs newRegs calls (fret : type k) :
PSemAction o a readRegs newRegs calls fret ->
NoDup newRegs.
Proof.
intros; apply PSemAction_NoDup_Key_Writes in H; apply NoDup_map_inv in H; assumption.
Qed.
Lemma collector_NoDupRegs1 (f : DefMethT) o reads upds calls calls' :
PSemAction_meth_collector f o reads upds calls calls' ->
NoDup (map fst upds).
Proof.
induction 1.
- constructor.
- rewrite H2, map_app, NoDup_app_iff.
specialize (PSemAction_NoDup_Key_Writes H5) as ND2.
repeat split; auto; repeat intro; specialize (H0 a); firstorder.
Qed.
Lemma PSemAction_inline_In (f : DefMethT) o:
forall {retK2} a calls1 calls2 readRegs newRegs (retV2 : type retK2),
PSemAction o a readRegs newRegs (calls1++calls2) retV2 ->
~In (fst f, projT1 (snd f)) (getKindAttr calls2) ->
forall readRegs' newRegs' calls',
DisjKey newRegs' newRegs ->
PSemAction_meth_collector f o readRegs' newRegs' calls' calls1 ->
PSemAction o (inlineSingle a f) (readRegs' ++ readRegs) (newRegs' ++ newRegs) (calls'++calls2) retV2.
Proof.
induction a; intros.
- simpl; destruct (fst f =? meth) eqn:G; [rewrite String.eqb_eq in G|rewrite eqb_neq in G]; [destruct Signature_dec | ]; subst.
+ inv H0; EqDep_subst.
assert (In (fst f, existT SignT (projT1 (snd f)) (evalExpr e, mret)) calls1).
{ case (in_app_or _ _ _ (Permutation_in _ (Permutation_sym (HAcalls)) (in_eq _ _)));
auto; intros TMP;apply (in_map (fun x => (fst x, projT1 (snd x)))) in TMP; contradiction. }
specialize (Produce_action_from_collector _ H0 H3) as TMP; destruct TMP as [creads1 [creads2 [cupds1 [cupds2 [ccalls1 [ccalls2 [ccalls'' [cargV [cretV decomp]]]]]]]]].
destruct decomp as [HDisju12 [HNr21 [HC21 [HRr21 [Hceq [HC1 [HSa HSac]]]]]]].
inv Hceq; EqDep_subst.
econstructor.
* rewrite HNr21 in H2.
assert (DisjKey cupds2 (cupds1++newRegs)) as HDjk21n;[|apply HDjk21n].
intro; specialize (H2 k); specialize (HDisju12 k); rewrite map_app,in_app_iff, DeM1 in *.
clear - H2 HDisju12; firstorder fail.
* econstructor; eauto.
* rewrite HRr21, <-app_assoc.
apply Permutation_app_head.
reflexivity.
* rewrite HNr21, <-app_assoc; reflexivity.
* rewrite HC21, <-app_assoc; reflexivity.
* eapply H; auto.