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MessagesAndRefinement: refines_add2_with_tester
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Adam Chlipala
parent
137121dcdc
commit
9806321af1
1 changed files with 55 additions and 30 deletions
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@ -23,14 +23,13 @@ Inductive proc :=
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| Recv (ch : channel) {A : Set} (k : A -> proc)
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| Par (pr1 pr2 : proc)
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| Dup (pr : proc)
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| Done
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| Fail.
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| Done.
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Notation pid := nat (only parsing).
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Notation procs := (fmap pid proc).
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Notation channels := (set channel).
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Notation "'New' ls ( x ) ; k" := (NewChannel ls (fun x => k)) (right associativity, at level 45, ls at level 0).
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Notation "'New' ls ( x ) ; k" := (NewChannel ls (fun x => k)) (right associativity, at level 51, ls at level 0).
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Notation "!! ch ( v ) ; k" := (Send ch v k) (right associativity, at level 45, ch at level 0).
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Notation "?? ch ( x : T ) ; k" := (Recv ch (fun x : T => k)) (right associativity, at level 45, ch at level 0, x at level 0).
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Infix "||" := Par.
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@ -38,23 +37,25 @@ Infix "||" := Par.
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(** * Example *)
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Definition simple_addN (k : nat) (input output : channel) : proc :=
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Dup (??input(n : nat);
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!!output(n + k);
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Done).
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Definition addN (k : nat) (input output : channel) : proc :=
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??input(n : nat);
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!!output(n + k);
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Done.
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Definition addNs (k : nat) (input output : channel) : proc :=
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Dup (addN k input output).
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Definition add2 (input output : channel) : proc :=
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New[input;output](intermediate);
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(simple_addN 1 input intermediate
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|| simple_addN 1 intermediate output).
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Dup (New[input;output](intermediate);
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addN 1 input intermediate
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|| addN 1 intermediate output).
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Definition tester (k n : nat) (input output : channel) : proc :=
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!!input(n);
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??output(n' : nat);
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if n' ==n (n + k) then
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Done
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else
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Fail.
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Definition tester (metaInput input output metaOutput : channel) : proc :=
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??metaInput(n : nat);
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!!input(n * 2);
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??output(r : nat);
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!!metaOutput(r);
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Done.
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(** * Flat semantics *)
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@ -300,26 +301,21 @@ Ltac inferRead :=
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(** * Examples *)
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Module Example1.
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Definition simple_addN_once (k : nat) (input output : channel) : proc :=
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??input(n : nat);
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!!output(n + k);
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Done.
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Definition add2_once (input output : channel) : proc :=
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New[input;output](intermediate);
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(simple_addN_once 1 input intermediate
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|| simple_addN_once 1 intermediate output).
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(addN 1 input intermediate
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|| addN 1 intermediate output).
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Inductive R_add2 (input output : channel) : proc -> proc -> Prop :=
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| Initial : forall k,
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(forall ch, ~List.In ch [input; output]
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-> k ch = ??input(n : nat); !!ch(n + 1); Done || simple_addN_once 1 ch output)
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-> k ch = ??input(n : nat); !!ch(n + 1); Done || addN 1 ch output)
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-> R_add2 input output
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(NewChannel [input;output] k)
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(simple_addN_once 2 input output)
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(addN 2 input output)
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| GotInput : forall n k,
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(forall ch, ~List.In ch [input; output]
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-> k ch = !!ch(n + 1); Done || simple_addN_once 1 ch output)
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-> k ch = !!ch(n + 1); Done || addN 1 ch output)
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-> R_add2 input output
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(NewChannel [input;output] k)
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(!!output(n + 2); Done)
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@ -338,8 +334,8 @@ Module Example1.
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Hint Constructors R_add2.
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Theorem add2_once_refines_simple_addN_once : forall input output,
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add2_once input output <| simple_addN_once 2 input output.
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Theorem add2_once_refines_addN : forall input output,
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add2_once input output <| addN 2 input output.
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Proof.
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simplify.
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exists (R_add2 input output).
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@ -356,7 +352,7 @@ Module Example1.
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invert H; simplify; inverter.
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inferRead.
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inferProc.
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unfold simple_addN_once; eauto 10.
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unfold addN; eauto 10.
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impossible.
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inferAction.
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inferProc.
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@ -370,6 +366,14 @@ End Example1.
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(** * Compositional reasoning principles *)
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Theorem refines_refl : forall pr,
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pr <| pr.
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Proof.
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simplify.
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exists (fun pr1 pr2 => pr1 = pr2).
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first_order; subst; eauto.
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Qed.
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(** ** Dup *)
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Inductive RDup (R : proc -> proc -> Prop) : proc -> proc -> Prop :=
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@ -599,3 +603,24 @@ Proof.
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unfold simulates in *.
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propositional; eauto using refines_Par_Silent, refines_Par_Action.
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Qed.
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(** * Example again *)
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Theorem refines_add2 : forall input output,
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add2 input output <| addNs 2 input output.
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Proof.
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simplify.
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apply refines_Dup.
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apply Example1.add2_once_refines_addN.
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Qed.
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Theorem refines_add2_with_tester : forall metaInput input output metaOutput,
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add2 input output || tester metaInput input output metaOutput
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<| addNs 2 input output || tester metaInput input output metaOutput.
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Proof.
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simplify.
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apply refines_Par.
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apply refines_add2.
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apply refines_refl.
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Qed.
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