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MessagesAndRefinement: refines_add2_with_tester

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Adam Chlipala 2016-05-07 21:43:06 -04:00
parent 137121dcdc
commit 9806321af1
1 changed files with 55 additions and 30 deletions
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85
MessagesAndRefinement.v
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@ -23,14 +23,13 @@ Inductive proc :=
| Recv (ch : channel) {A : Set} (k : A -> proc) | Recv (ch : channel) {A : Set} (k : A -> proc)
| Par (pr1 pr2 : proc) | Par (pr1 pr2 : proc)
| Dup (pr : proc) | Dup (pr : proc)
| Done | Done.
| Fail.
Notation pid := nat (only parsing). Notation pid := nat (only parsing).
Notation procs := (fmap pid proc). Notation procs := (fmap pid proc).
Notation channels := (set channel). Notation channels := (set channel).
Notation "'New' ls ( x ) ; k" := (NewChannel ls (fun x => k)) (right associativity, at level 45, ls at level 0). Notation "'New' ls ( x ) ; k" := (NewChannel ls (fun x => k)) (right associativity, at level 51, ls at level 0).
Notation "!! ch ( v ) ; k" := (Send ch v k) (right associativity, at level 45, ch at level 0). Notation "!! ch ( v ) ; k" := (Send ch v k) (right associativity, at level 45, ch at level 0).
Notation "?? ch ( x : T ) ; k" := (Recv ch (fun x : T => k)) (right associativity, at level 45, ch at level 0, x at level 0). Notation "?? ch ( x : T ) ; k" := (Recv ch (fun x : T => k)) (right associativity, at level 45, ch at level 0, x at level 0).
Infix "||" := Par. Infix "||" := Par.
@ -38,23 +37,25 @@ Infix "||" := Par.
(** * Example *) (** * Example *)
Definition simple_addN (k : nat) (input output : channel) : proc := Definition addN (k : nat) (input output : channel) : proc :=
Dup (??input(n : nat); ??input(n : nat);
!!output(n + k); !!output(n + k);
Done). Done.
Definition addNs (k : nat) (input output : channel) : proc :=
Dup (addN k input output).
Definition add2 (input output : channel) : proc := Definition add2 (input output : channel) : proc :=
New[input;output](intermediate); Dup (New[input;output](intermediate);
(simple_addN 1 input intermediate addN 1 input intermediate
|| simple_addN 1 intermediate output). || addN 1 intermediate output).
Definition tester (k n : nat) (input output : channel) : proc := Definition tester (metaInput input output metaOutput : channel) : proc :=
!!input(n); ??metaInput(n : nat);
??output(n' : nat); !!input(n * 2);
if n' ==n (n + k) then ??output(r : nat);
Done !!metaOutput(r);
else Done.
Fail.
(** * Flat semantics *) (** * Flat semantics *)
@ -300,26 +301,21 @@ Ltac inferRead :=
(** * Examples *) (** * Examples *)
Module Example1. Module Example1.
Definition simple_addN_once (k : nat) (input output : channel) : proc :=
??input(n : nat);
!!output(n + k);
Done.
Definition add2_once (input output : channel) : proc := Definition add2_once (input output : channel) : proc :=
New[input;output](intermediate); New[input;output](intermediate);
(simple_addN_once 1 input intermediate (addN 1 input intermediate
|| simple_addN_once 1 intermediate output). || addN 1 intermediate output).
Inductive R_add2 (input output : channel) : proc -> proc -> Prop := Inductive R_add2 (input output : channel) : proc -> proc -> Prop :=
| Initial : forall k, | Initial : forall k,
(forall ch, ~List.In ch [input; output] (forall ch, ~List.In ch [input; output]
-> k ch = ??input(n : nat); !!ch(n + 1); Done || simple_addN_once 1 ch output) -> k ch = ??input(n : nat); !!ch(n + 1); Done || addN 1 ch output)
-> R_add2 input output -> R_add2 input output
(NewChannel [input;output] k) (NewChannel [input;output] k)
(simple_addN_once 2 input output) (addN 2 input output)
| GotInput : forall n k, | GotInput : forall n k,
(forall ch, ~List.In ch [input; output] (forall ch, ~List.In ch [input; output]
-> k ch = !!ch(n + 1); Done || simple_addN_once 1 ch output) -> k ch = !!ch(n + 1); Done || addN 1 ch output)
-> R_add2 input output -> R_add2 input output
(NewChannel [input;output] k) (NewChannel [input;output] k)
(!!output(n + 2); Done) (!!output(n + 2); Done)
@ -338,8 +334,8 @@ Module Example1.
Hint Constructors R_add2. Hint Constructors R_add2.
Theorem add2_once_refines_simple_addN_once : forall input output, Theorem add2_once_refines_addN : forall input output,
add2_once input output <| simple_addN_once 2 input output. add2_once input output <| addN 2 input output.
Proof. Proof.
simplify. simplify.
exists (R_add2 input output). exists (R_add2 input output).
@ -356,7 +352,7 @@ Module Example1.
invert H; simplify; inverter. invert H; simplify; inverter.
inferRead. inferRead.
inferProc. inferProc.
unfold simple_addN_once; eauto 10. unfold addN; eauto 10.
impossible. impossible.
inferAction. inferAction.
inferProc. inferProc.
@ -370,6 +366,14 @@ End Example1.
(** * Compositional reasoning principles *) (** * Compositional reasoning principles *)
Theorem refines_refl : forall pr,
pr <| pr.
Proof.
simplify.
exists (fun pr1 pr2 => pr1 = pr2).
first_order; subst; eauto.
Qed.
(** ** Dup *) (** ** Dup *)
Inductive RDup (R : proc -> proc -> Prop) : proc -> proc -> Prop := Inductive RDup (R : proc -> proc -> Prop) : proc -> proc -> Prop :=
@ -599,3 +603,24 @@ Proof.
unfold simulates in *. unfold simulates in *.
propositional; eauto using refines_Par_Silent, refines_Par_Action. propositional; eauto using refines_Par_Silent, refines_Par_Action.
Qed. Qed.
(** * Example again *)
Theorem refines_add2 : forall input output,
add2 input output <| addNs 2 input output.
Proof.
simplify.
apply refines_Dup.
apply Example1.add2_once_refines_addN.
Qed.
Theorem refines_add2_with_tester : forall metaInput input output metaOutput,
add2 input output || tester metaInput input output metaOutput
<| addNs 2 input output || tester metaInput input output metaOutput.
Proof.
simplify.
apply refines_Par.
apply refines_add2.
apply refines_refl.
Qed.