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1 changed files with 91 additions and 6 deletions
97
Connecting.v
97
Connecting.v
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@ -970,7 +970,7 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
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Inductive stmt :=
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| Skip
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| Assign (x : var) (e : exp)
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| Write (x : var) (e : exp)
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| Write (ae ve : exp)
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| Seq (s1 s2 : stmt)
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| IfThenElse (e : exp) (s1 s2 : stmt)
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| WhileLoop (e : exp) (s1 : stmt).
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@ -1008,11 +1008,11 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
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| StAssign : forall H V x e v,
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eval H V e v
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-> step (H, V, Assign x e) (H, V $+ (x, v), Skip)
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| StWrite : forall H V x e a v hv,
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V $? x = Some a
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-> eval H V e v
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| StWrite : forall H V ae ve a v hv,
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eval H V ae a
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-> eval H V ve v
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-> H $? a = Some hv
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-> step (H, V, Write x e) (H $+ (a, v), V, Skip)
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-> step (H, V, Write ae ve) (H $+ (a, v), V, Skip)
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| StSeq1 : forall H V s2,
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step (H, V, Seq Skip s2) (H, V, s2)
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| StSeq2 : forall H V s1 s2 H' V' s1',
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@ -1091,7 +1091,15 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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| TrRead : forall V B (a : wrd) (c : wrd -> cmd B) e x s1,
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translate_exp V a e
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-> (forall w, translate translate_return (V $+ (x, w)) (c w) s1)
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-> translate translate_return V (Bind (Read a) c) (Seq (Assign x (DE.Read e)) s1).
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-> translate translate_return V (Bind (Read a) c) (Seq (Assign x (DE.Read e)) s1)
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| TrWrite : forall V B a v (c : unit -> cmd B) ae ve s1,
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translate_exp V a ae
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-> translate_exp V v ve
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-> translate translate_return V (c tt) s1
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-> translate translate_return V (Bind (Write a v) c) (Seq (DE.Write ae ve) s1)
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| TrAssignedUnit : forall V B (c : unit -> cmd B) s1,
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translate translate_return V (c tt) s1
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-> translate translate_return V (Bind (Return tt) c) (Seq Skip s1).
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Ltac freshFor vm k :=
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let rec keepTrying x :=
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@ -1108,6 +1116,7 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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match V with
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| context[add _ ?y v] => apply TrVar with (x := y); simplify; equality
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end
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| [ |- translate_exp _ _ _ ] => eapply TrConst
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| [ |- translate _ _ (Return _) _ ] => apply TrDone; apply TrReturn
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| [ |- translate _ ?V (Bind (Return _) _) _ ] =>
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@ -1116,6 +1125,8 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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| [ |- translate _ ?V (Bind (Read _) _) _ ] =>
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freshFor V ltac:(fun y =>
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eapply TrRead with (x := y); [ | intro ])
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| [ |- translate _ ?V (Bind (Write _ _) _) _ ] =>
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eapply TrWrite
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end.
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Example adder (a b c : wrd) :=
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@ -1144,6 +1155,19 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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Definition reader_compiled := Eval simpl in proj1_sig translate_reader.
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Example incrementer (p : wrd) :=
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Bind (Read p) (fun v => Bind (Write p (v ^+ ^1)) (fun _ => Return v)).
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Lemma translate_incrementer : sig (fun s =>
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forall p, translate (translate_result "result") ($0 $+ ("p", p)) (incrementer p) s).
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Proof.
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eexists; simplify.
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unfold incrementer.
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repeat translate.
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Defined.
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Definition incrementer_compiled := Eval simpl in proj1_sig translate_incrementer.
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Inductive translated : forall {A}, DE.heap * valuation * stmt -> ME.heap * cmd A -> Prop :=
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| Translated : forall A H V s (c : cmd A),
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translate (translate_result "result") V c s
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@ -1224,6 +1248,21 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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econstructor.
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instantiate (1 := x); simplify; reflexivity.
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eauto.
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invert 1.
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invert H6.
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eapply eval_translate in H0; eauto; subst.
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eapply eval_translate in H; eauto; subst.
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do 2 eexists; propositional.
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eapply TrcFront.
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eauto.
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eauto.
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eapply TrAssignedUnit.
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assumption.
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invert 1.
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eauto.
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invert H4.
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Qed.
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Theorem translated_simulates : forall H V c s,
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@ -1292,6 +1331,20 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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econstructor.
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eauto.
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eauto.
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apply inj_pair2 in H8; subst.
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invert H6.
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eexists.
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econstructor.
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econstructor.
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eauto.
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eauto.
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eauto.
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apply inj_pair2 in H6; subst.
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invert H4.
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eauto.
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Qed.
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Theorem hoare_triple_sound : forall P (c : cmd wrd) Q V s H,
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@ -1398,6 +1451,38 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
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apply (proj2_sig translate_reader).
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simplify; equality.
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Qed.
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Theorem incrementer_ok : forall p v,
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{{p |-> v}}
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incrementer p
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{{r ~> [| r = v |] * p |-> (v ^+ ^1)}}.
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Proof.
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unfold incrementer.
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simplify.
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step.
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step.
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simp.
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step.
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step.
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simp.
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step.
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cancel.
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subst.
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cancel.
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Qed.
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Theorem incrementer_compiled_ok : forall p v,
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invariantFor (DE.trsys_of ($0 $+ (p, v)) ($0 $+ ("p", p)) incrementer_compiled)
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(fun p => snd p = Skip
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\/ exists p', DE.step p p').
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Proof.
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simplify.
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eapply hoare_triple_sound.
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apply incrementer_ok.
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constructor.
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apply (proj2_sig translate_incrementer).
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simplify; equality.
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Qed.
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End MixedToDeep.
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