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MessagesAndRefinement: refines_Dup
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Adam Chlipala
parent
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commit
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1 changed files with 219 additions and 84 deletions
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@ -110,10 +110,6 @@ Inductive lstep : proc -> label -> proc -> Prop :=
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| LStepNewInside : forall chs k l k',
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(forall ch, ~List.In ch chs -> lstep (k ch) l (k' ch))
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-> lstep (NewChannel chs k) l (NewChannel chs k')
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| LStepNewUp1 : forall chs k pr,
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lstep (Par (NewChannel chs k) pr) Silent (NewChannel chs (fun ch => Par (k ch) pr))
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| LStepNewUp2 : forall chs k pr,
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lstep (Par pr (NewChannel chs k)) Silent (NewChannel chs (fun ch => Par pr (k ch)))
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| LStepPar1 : forall pr1 pr2 l pr1',
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lstep pr1 l pr1'
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-> lstep (Par pr1 pr2) l (Par pr1' pr2)
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@ -141,16 +137,28 @@ Inductive couldGenerate : proc -> list action -> Prop :=
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-> couldGenerate pr' tr
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-> couldGenerate pr (a :: tr).
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Definition lstepSilent (pr1 pr2 : proc) := lstep pr1 Silent pr2.
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Definition simulates (R : proc -> proc -> Prop) (pr1 pr2 : proc) :=
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(forall pr1 pr2, R pr1 pr2
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-> forall pr1', lstepSilent pr1 pr1'
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-> exists pr2', lstepSilent^* pr2 pr2'
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/\ R pr1' pr2')
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/\ (forall pr1 pr2, R pr1 pr2
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-> forall a pr1', lstep pr1 (Action a) pr1'
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-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
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/\ lstep pr2' (Action a) pr2''
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/\ R pr1' pr2'')
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/\ R pr1 pr2.
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Definition refines (pr1 pr2 : proc) :=
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forall tr, couldGenerate pr1 tr -> couldGenerate pr2 tr.
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exists R, simulates R pr1 pr2.
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Infix "<|" := refines (no associativity, at level 70).
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(** * Simulation: a proof method *)
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Definition lstepSilent (pr1 pr2 : proc) := lstep pr1 Silent pr2.
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Hint Constructors couldGenerate.
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Lemma lstepSilent_couldGenerate : forall pr1 pr2,
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@ -163,7 +171,7 @@ Qed.
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Hint Resolve lstepSilent_couldGenerate.
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Lemma simulates' : forall (R : proc -> proc -> Prop),
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Lemma simulates_couldGenerate' : forall (R : proc -> proc -> Prop),
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(forall pr1 pr2, R pr1 pr2
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-> forall pr1', lstepSilent pr1 pr1'
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-> exists pr2', lstepSilent^* pr2 pr2'
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@ -177,7 +185,6 @@ Lemma simulates' : forall (R : proc -> proc -> Prop),
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-> forall pr2, R pr1 pr2
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-> couldGenerate pr2 tr.
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Proof.
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unfold refines.
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induct 3; simplify; auto.
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eapply H in H1; eauto.
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@ -189,59 +196,15 @@ Proof.
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eauto.
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Qed.
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Theorem simulates : forall (R : proc -> proc -> Prop),
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(forall pr1 pr2, R pr1 pr2
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-> forall pr1', lstepSilent pr1 pr1'
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-> exists pr2', lstepSilent^* pr2 pr2'
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/\ R pr1' pr2')
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-> (forall pr1 pr2, R pr1 pr2
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-> forall a pr1', lstep pr1 (Action a) pr1'
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-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
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/\ lstep pr2' (Action a) pr2''
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/\ R pr1' pr2'')
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-> forall pr1 pr2, R pr1 pr2
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-> pr1 <| pr2.
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Theorem simulates_couldGenerate : forall pr1 pr2, pr1 <| pr2
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-> forall tr, couldGenerate pr1 tr
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-> couldGenerate pr2 tr.
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Proof.
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unfold refines; eauto using simulates'.
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unfold refines; first_order; eauto using simulates_couldGenerate'.
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Qed.
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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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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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-> 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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| 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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-> R_add2 input output
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(NewChannel [input;output] k)
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(!!output(n + 2); Done)
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| HandedOff : forall n k,
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(forall ch, ~List.In ch [input; output]
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-> k ch = Done || (!!output(n + 2); Done))
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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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| Finished : forall k,
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(forall ch, ~List.In ch [input; output]
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-> k ch = Done || Done)
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-> R_add2 input output
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(NewChannel [input;output] k)
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Done.
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Hint Constructors R_add2.
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(** * Tactics for automating refinement proofs *)
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Lemma invert_Recv : forall ch (A : Set) (k : A -> proc) (x : A) pr,
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lstep (Recv ch k) (Input {| Channel := ch; Value := x |}) pr
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@ -254,8 +217,6 @@ Ltac inverter :=
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repeat match goal with
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| [ H : lstep _ _ _ |- _ ] => (apply invert_Recv in H; try subst) || invert H
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| [ H : lstepSilent _ _ |- _ ] => invert H
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(*| [ H : forall ch : channel, lstep _ _ _ |- _ ] =>
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eapply anystep_somestep with (1 := 0); [ apply H | clear H; simplify ]*)
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end.
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Hint Constructors lstep.
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@ -335,29 +296,203 @@ Ltac inferRead :=
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first_order; subst
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end.
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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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(** * 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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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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-> 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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| 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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-> R_add2 input output
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(NewChannel [input;output] k)
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(!!output(n + 2); Done)
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| HandedOff : forall n k,
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(forall ch, ~List.In ch [input; output]
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-> k ch = Done || (!!output(n + 2); Done))
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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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| Finished : forall k,
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(forall ch, ~List.In ch [input; output]
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-> k ch = Done || Done)
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-> R_add2 input output
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(NewChannel [input;output] k)
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Done.
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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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Proof.
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simplify.
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exists (R_add2 input output).
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unfold simulates; propositional.
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invert H; simplify; inverter.
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impossible.
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inferProc.
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replace (n + 1 + 1) with (n + 2) in * by linear_arithmetic.
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eauto.
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impossible.
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impossible.
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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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impossible.
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inferAction.
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inferProc.
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eauto 10.
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impossible.
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unfold add2_once; eauto.
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Qed.
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End Example1.
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(** * Compositional reasoning principles *)
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(** ** Dup *)
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Inductive RDup (R : proc -> proc -> Prop) : proc -> proc -> Prop :=
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| RDupLeaf : forall pr1 pr2,
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R pr1 pr2
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-> RDup R pr1 pr2
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| RDupDup : forall pr1 pr2,
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RDup R pr1 pr2
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-> RDup R (Dup pr1) (Dup pr2)
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| RDupPar : forall pr1 pr2 pr1' pr2',
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RDup R pr1 pr1'
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-> RDup R pr2 pr2'
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-> RDup R (Par pr1 pr2) (Par pr1' pr2').
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Hint Constructors RDup.
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Hint Unfold lstepSilent.
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Lemma lstepSilent_Par1 : forall pr1 pr1' pr2,
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lstepSilent^* pr1 pr1'
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-> lstepSilent^* (Par pr1 pr2) (Par pr1' pr2).
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Proof.
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simplify.
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apply simulates with (R := R_add2 input output).
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invert 1; simplify; inverter.
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impossible.
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inferProc.
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replace (n + 1 + 1) with (n + 2) in * by linear_arithmetic.
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eauto.
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impossible.
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impossible.
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invert 1; 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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impossible.
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inferAction.
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inferProc.
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eauto 10.
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impossible.
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unfold add2_once; eauto.
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induct 1; eauto.
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Qed.
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Lemma lstepSilent_Par2 : forall pr2 pr2' pr1,
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lstepSilent^* pr2 pr2'
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-> lstepSilent^* (Par pr1 pr2) (Par pr1 pr2').
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Proof.
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induct 1; eauto.
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Qed.
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Hint Resolve lstepSilent_Par1 lstepSilent_Par2.
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Lemma refines_Dup_Action : forall R : _ -> _ -> Prop,
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(forall pr1 pr2, R pr1 pr2
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-> forall pr1' a, lstep pr1 (Action a) pr1'
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-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
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/\ lstep pr2' (Action a) pr2''
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/\ R pr1' pr2'')
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-> forall pr1 pr2, RDup R pr1 pr2
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-> forall pr1' a, lstep pr1 (Action a) pr1'
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-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
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/\ lstep pr2' (Action a) pr2''
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/\ RDup R pr1' pr2''.
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Proof.
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induct 2; simplify.
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eapply H in H1; eauto.
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first_order.
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eauto 6.
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invert H1.
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invert H0.
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apply IHRDup1 in H5.
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first_order.
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eauto 10.
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apply IHRDup2 in H5.
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first_order.
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eauto 10.
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Qed.
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Lemma refines_Dup_Silent : forall R : _ -> _ -> Prop,
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(forall pr1 pr2, R pr1 pr2
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-> forall pr1', lstepSilent pr1 pr1'
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-> exists pr2', lstepSilent^* pr2 pr2'
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/\ R pr1' pr2')
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-> (forall pr1 pr2, R pr1 pr2
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-> forall pr1' a, lstep pr1 (Action a) pr1'
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-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
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/\ lstep pr2' (Action a) pr2''
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/\ R pr1' pr2'')
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-> forall pr1 pr2, RDup R pr1 pr2
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-> forall pr1', lstepSilent pr1 pr1'
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-> exists pr2', lstepSilent^* pr2 pr2' /\ RDup R pr1' pr2'.
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Proof.
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induct 3; simplify.
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eapply H in H1; eauto.
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first_order.
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eauto.
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invert H2.
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eauto 10.
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invert H1.
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apply IHRDup1 in H6.
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first_order.
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eauto.
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apply IHRDup2 in H6.
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first_order.
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eauto.
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eapply refines_Dup_Action in H4; eauto.
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eapply refines_Dup_Action in H5; eauto.
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first_order.
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eexists; propositional.
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apply trc_trans with (x1 || pr2').
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eauto.
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apply trc_trans with (x1 || x).
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eauto.
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apply trc_one.
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eauto.
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eauto.
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eapply refines_Dup_Action in H4; eauto.
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eapply refines_Dup_Action in H5; eauto.
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first_order.
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eexists; propositional.
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apply trc_trans with (x1 || pr2').
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eauto.
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apply trc_trans with (x1 || x).
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eauto.
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apply trc_one.
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eauto.
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eauto.
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Qed.
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Theorem refines_Dup : forall pr1 pr2,
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pr1 <| pr2
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-> Dup pr1 <| Dup pr2.
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Proof.
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invert 1.
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exists (RDup x).
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unfold simulates in *.
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propositional; eauto using refines_Dup_Silent, refines_Dup_Action.
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Qed.
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