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frap/MessagesAndRefinement.v
Adam Chlipala 5455be7079 MessagesAndRefinement: Coq 8.4 compatibility
2016-05-09 10:42:13 -04:00

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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 15: Process Algebra and Behavioral Refinement
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Require Import Frap Eqdep FunctionalExtensionality.
Set Implicit Arguments.
Set Asymmetric Patterns.
(** * First, an unexplained tactic that will come in handy.... *)
Ltac invert H := (Frap.invert H || (inversion H; clear H));
repeat match goal with
| [ x : _ |- _ ] => subst x
| [ H : existT _ _ _ = existT _ _ _ |- _ ] => apply inj_pair2 in H; try subst
end.
(** * A process algebra: syntax and semantics *)
(* A process algebra defines a set of communicating *processes*, which might
* more commonly be called *threads*. Typically processes communicate not with
* locks and shared memory, as they have in the prior two chapters, but instead
* with *message passing*. Messages are passed over synchronous *channels*,
* which we will just represent as numbers. *)
Notation channel := nat (only parsing).
(* Here are the basic syntactic constructions of processes. *)
Inductive proc :=
| NewChannel (notThese : list channel) (k : channel -> proc)
(* Pick a new channel [ch] name not found in [notThese], and continue like
* [k ch]. *)
| BlockChannel (ch : channel) (pr : proc)
(* Act like [pr], but prevent interaction with other processes through channel
* [ch]. We effectively force [ch] to be *private*. *)
| Send (ch : channel) {A : Set} (v : A) (k : proc)
| Recv (ch : channel) {A : Set} (k : A -> proc)
(* When one process runs a [Send] and the other a [Recv] on the same channel
* simultaneously, the [Send] moves on to its [k], while the [Recv] moves on to
* its [k v], for [v] the value that was sent. *)
| Par (pr1 pr2 : proc)
(* This one, at least, is just like it was in the last chapter: parallel
* composition of threads. *)
| Dup (pr : proc)
(* An idiosyncratic convention of process algebra: [Dup pr] acts like an
* *infinite* number of *copies* of [pr]. It replaces conventional loops. *)
| Done
(* This process can't do anything *).
(* Some nicer notations: *)
Notation "'New' ls ( x ) ; k" := (NewChannel ls (fun x => k)) (right associativity, at level 51, ls at level 0).
Notation "'Block' ch ; k" := (BlockChannel ch k) (right associativity, at level 51).
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).
Infix "||" := Par.
(** * Example *)
(* Let's build highly exciting processes for adding constants to numbers. *)
(* This one accepts one number [n] on channel [input] and returns [n + k] as the
* result, by writing it to [output]. *)
Definition addN (k : nat) (input output : channel) : proc :=
??input(n : nat);
!!output(n + k);
Done.
(* We wrap that last one in a [Dup] to turn it into a kind of immortal server,
* happy to keep responding to "please add k to this" requests forever. *)
Definition addNs (k : nat) (input output : channel) : proc :=
Dup (addN k input output).
(* Chaining together two "+1" boxes is one way to build an alternative "+2"
* box! *)
Definition add2 (input output : channel) : proc :=
Dup (New[input;output](intermediate);
addN 1 input intermediate
|| addN 1 intermediate output).
(* However, we implement addition, we might compose with this tester process,
* which uses an adder as a subroutine in a larger protocol. [metaInput] and
* [metaOutput] are the input and output of the whole system, while [input] and
* [output] are used internally, say to communicate with an adder. *)
Definition tester (metaInput input output metaOutput : channel) : proc :=
??metaInput(n : nat);
!!input(n * 2);
??output(r : nat);
!!metaOutput(r);
Done.
(** * Labeled semantics *)
(* Let's explain how programs run. We'll give a flavor of operational semantics
* called a "labelled transition system," because each step will include a label
* that explains what happened. In this case, the only relevant happenings are
* sends or receives on channels. Crucially, we supress send/receive labels for
* operations blocked by [Block]s. *)
Record message := {
Channel : channel;
TypeOf : Set;
Value : TypeOf
}.
Inductive action :=
| Output (m : message)
| Input (m : message).
Inductive label :=
| Silent
| Action (a : action).
(* This command lets us use [action]s implicitly as [label]s. *)
Coercion Action : action >-> label.
(* This predicate captures when a label doesn't use a channel. *)
Definition notUse (ch : channel) (l : label) :=
match l with
| Action (Input r) => r.(Channel) <> ch
| Action (Output r) => r.(Channel) <> ch
| Silent => True
end.
(* Now, our labelled transition system: *)
Inductive lstep : proc -> label -> proc -> Prop :=
(* Sends and receives generate the expected labels. Note that, for a [Recv],
* the value being received is "pulled out of thin air"! However, it gets
* determined concretely by comparing against a matching [Send], in a rule that
* we get to shortly. *)
| LStepSend : forall ch {A : Set} (v : A) k,
lstep (Send ch v k)
(Output {| Channel := ch; Value := v |})
k
| LStepRecv : forall ch {A : Set} (k : A -> _) v,
lstep (Recv ch k)
(Input {| Channel := ch; Value := v |})
(k v)
(* A [Dup] always has the option of replicating itself further. *)
| LStepDup : forall pr,
lstep (Dup pr) Silent (Par (Dup pr) pr)
(* A channel allocation operation nondeterministically picks the new channel ID,
* only checking that it isn't in the provided blacklist. We install a [Block]
* node to keep this channel truly private. *)
| LStepNew : forall chs ch k,
~In ch chs
-> lstep (NewChannel chs k) Silent (BlockChannel ch (k ch))
(* [Block] nodes work as expected, disallowing labels that use the channel. *)
| LStepBlock : forall ch k l k',
lstep k l k'
-> notUse ch l
-> lstep (BlockChannel ch k) l (BlockChannel ch k')
(* When either side of a parallel composition can step, we may bubble that step
* up to the top. *)
| LStepPar1 : forall pr1 pr2 l pr1',
lstep pr1 l pr1'
-> lstep (Par pr1 pr2) l (Par pr1' pr2)
| LStepPar2 : forall pr1 pr2 l pr2',
lstep pr2 l pr2'
-> lstep (Par pr1 pr2) l (Par pr1 pr2')
(* These two symmetrical rules are the heart of how communication happens in our
* language. Namely, in a parallel composition, when one side is prepared to
* write a value to a channel, and the other side is prepared to read the same
* value from the same channel, the two sides *rendezvous*, and the value is
* exchanged. This is the only mechanism to let two transitions happen at
* once. *)
| LStepRendezvousLeft : forall pr1 ch {A : Set} (v : A) pr1' pr2 pr2',
lstep pr1 (Input {| Channel := ch; Value := v |}) pr1'
-> lstep pr2 (Output {| Channel := ch; Value := v |}) pr2'
-> lstep (Par pr1 pr2) Silent (Par pr1' pr2')
| LStepRendezvousRight : forall pr1 ch {A : Set} (v : A) pr1' pr2 pr2',
lstep pr1 (Output {| Channel := ch; Value := v |}) pr1'
-> lstep pr2 (Input {| Channel := ch; Value := v |}) pr2'
-> lstep (Par pr1 pr2) Silent (Par pr1' pr2').
(* Here's a shorthand for silent steps. *)
Definition lstepSilent (pr1 pr2 : proc) := lstep pr1 Silent pr2.
(* Our key proof task will be to prove that one process "acts like" another.
* We'll use *simulation* as the precise notion of "acts like." *)
(* We say that a relation [R] is a *simulation* when it satisfies the first two
* conditions below. The [simulates] predicate additionally asserts that a
* particular pair of processes belongs to [R]. *)
Definition simulates (R : proc -> proc -> Prop) (pr1 pr2 : proc) :=
(* First, consider a pair of processes related by [R]. When the lefthand
* process can take a silent step, the righthand process can take zero or more
* silent steps to "catch up," arriving at a new righthand process related to
* the new lefthand process. *)
(forall pr1 pr2, R pr1 pr2
-> forall pr1', lstepSilent pr1 pr1'
-> exists pr2', lstepSilent^* pr2 pr2'
/\ R pr1' pr2')
(* Now consider the same scenario where the lefthand process make a nonsilent
* step. We require that the righthand process can "catch up" in a way that
* generates the same label that the lefthand process generated. *)
/\ (forall pr1 pr2, R pr1 pr2
-> forall a pr1', lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ R pr1' pr2'')
(* Finally, the provided process pair is in the relation. *)
/\ R pr1 pr2.
(* One process *refines* another when there exists some simulation. *)
Definition refines (pr1 pr2 : proc) :=
exists R, simulates R pr1 pr2.
Infix "<|" := refines (no associativity, at level 70).
(* That's a somewhat fancy notion of compatibility! We can also relate it to
* more intuitive conditions that aren't strong enough for many of the proofs we
* want to do later. *)
(* This predicate captures all finite traces of actions that a process could
* generate. *)
Inductive couldGenerate : proc -> list action -> Prop :=
| CgNothing : forall pr,
couldGenerate pr []
| CgSilent : forall pr pr' tr,
lstep pr Silent pr'
-> couldGenerate pr' tr
-> couldGenerate pr tr
| CgAction : forall pr a pr' tr,
lstep pr (Action a) pr'
-> couldGenerate pr' tr
-> couldGenerate pr (a :: tr).
(* Skip ahead to [refines_couldGenerate] to see the top-level connection from
* [refines]. *)
Hint Constructors couldGenerate.
Lemma lstepSilent_couldGenerate : forall pr1 pr2,
lstepSilent^* pr1 pr2
-> forall tr, couldGenerate pr2 tr
-> couldGenerate pr1 tr.
Proof.
induct 1; eauto.
Qed.
Hint Resolve lstepSilent_couldGenerate.
Lemma simulates_couldGenerate' : forall (R : proc -> proc -> Prop),
(forall pr1 pr2, R pr1 pr2
-> forall pr1', lstepSilent pr1 pr1'
-> exists pr2', lstepSilent^* pr2 pr2'
/\ R pr1' pr2')
-> (forall pr1 pr2, R pr1 pr2
-> forall a pr1', lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ R pr1' pr2'')
-> forall pr1 tr, couldGenerate pr1 tr
-> forall pr2, R pr1 pr2
-> couldGenerate pr2 tr.
Proof.
induct 3; simplify; auto.
eapply H in H1; eauto.
first_order.
eauto.
eapply H0 in H1; eauto.
first_order.
eauto.
Qed.
(* This theorem says that refinement implies *trace inclusion*. *)
Theorem refines_couldGenerate : forall pr1 pr2, pr1 <| pr2
-> forall tr, couldGenerate pr1 tr
-> couldGenerate pr2 tr.
Proof.
unfold refines; first_order; eauto using simulates_couldGenerate'.
Qed.
(** * Tactics for automating refinement proofs *)
(* Well, you're used to unexplained automation tactics by now, so here are some
* more. ;-) *)
Lemma invert_Recv : forall ch (A : Set) (k : A -> proc) (x : A) pr,
lstep (Recv ch k) (Input {| Channel := ch; Value := x |}) pr
-> pr = k x.
Proof.
invert 1; auto.
Qed.
Ltac inverter :=
repeat match goal with
| [ H : lstep _ _ _ |- _ ] => (apply invert_Recv in H; try subst) || invert H
| [ H : lstepSilent _ _ |- _ ] => invert H
end.
Hint Constructors lstep.
Hint Unfold lstepSilent.
Ltac lists' :=
repeat match goal with
| [ H : NoDup _ |- _ ] => invert H
| [ |- NoDup _ ] => constructor
end; simplify; propositional; equality || linear_arithmetic.
Ltac lists := solve [ lists' ].
Hint Extern 1 (NoDup _) => lists.
(** * Examples *)
(* OK, let's verify a simplification of the example we started with. *)
Definition add2_once (input output : channel) : proc :=
New[input;output](intermediate);
(addN 1 input intermediate
|| addN 1 intermediate output).
(* Here's our first, painstakingly crafted simulation relation. It needs to
* identify all pairs of processes that should be considered compatible. Think
* of the first process as the fancy *implementation* and the second process as
* the simpler *specification*. *)
Inductive R_add2 : proc -> proc -> Prop :=
| Initial : forall input output,
input <> output
-> R_add2
(New[input;output](ch); ??input(n : nat); !!ch(n + 1); Done
|| ??ch(n : nat); !!output (n + 1); Done)
(??input(n : nat); !!output(n + 2); Done)
| ChoseIntermediate : forall input output intermediate,
NoDup [input; output; intermediate]
-> R_add2
(Block intermediate; ??input(n : nat); !!intermediate(n + 1); Done
|| ??intermediate(n : nat); !!output (n + 1); Done)
(??input(n : nat); !!output(n + 2); Done)
| GotInput : forall input output intermediate n,
NoDup [input; output; intermediate]
-> R_add2
(Block intermediate; !!intermediate(n + 1); Done
|| ??intermediate(n : nat); !!output (n + 1); Done)
(!!output(n + 2); Done)
| HandedOff : forall input output intermediate n,
NoDup [input; output; intermediate]
-> R_add2
(Block intermediate; Done || (!!output(n + 2); Done))
(!!output(n + 2); Done)
| Finished : forall input output intermediate,
NoDup [input; output; intermediate]
-> R_add2
(Block intermediate; Done || Done)
Done.
Hint Constructors R_add2.
Theorem add2_once_refines_addN : forall input output,
input <> output
-> add2_once input output <| addN 2 input output.
Proof.
simplify.
exists R_add2.
first_order.
invert H0; simplify; inverter; eauto.
replace (n + 1 + 1) with (n + 2) by linear_arithmetic.
eauto.
invert H0; simplify; inverter; eauto 10; simplify; equality.
unfold add2_once, addN; eauto.
Qed.
(* Well, good! The fancy version doesn't produce any traces that the simple
* version couldn't also produce. (It may, however, fail to produce some traces
* that the spec allows.) *)
(** * Compositional reasoning principles *)
(* It turns out that refinement has all sorts of convenient algebraic
* properties. To start with, it's a preorder. *)
Theorem refines_refl : forall pr,
pr <| pr.
Proof.
simplify.
exists (fun pr1 pr2 => pr1 = pr2).
first_order; subst; eauto.
Qed.
Lemma refines_trans' : forall R : _ -> _ -> Prop,
(forall pr1 pr2,
R pr1 pr2
-> forall pr1', lstepSilent pr1 pr1'
-> exists pr2', lstepSilent^* pr2 pr2' /\ R pr1' pr2')
-> forall pr1 pr1', lstepSilent^* pr1 pr1'
-> forall pr2, R pr1 pr2
-> exists pr2', R pr1' pr2' /\ lstepSilent^* pr2 pr2'.
Proof.
induct 2; simplify; eauto.
eapply H in H0; eauto.
first_order.
apply IHtrc in H3.
first_order.
eauto using trc_trans.
Qed.
Theorem refines_trans : forall pr1 pr2 pr3,
pr1 <| pr2
-> pr2 <| pr3
-> pr1 <| pr3.
Proof.
invert 1; invert 1.
exists (fun p q => exists r, x p r /\ x0 r q).
first_order.
match goal with
| [ H : _, H' : x _ _ |- _ ] => eapply H in H'; eauto; []
end.
first_order.
eapply refines_trans' with (R := x0) in H7; eauto.
first_order.
match goal with
| [ H : _, H' : x _ _ |- _ ] => eapply H in H'; eauto; []
end.
first_order.
eapply refines_trans' with (R := x0) in H7; eauto.
first_order.
match goal with
| [ H : _, H' : x0 _ _ |- _ ] => eapply H in H'; eauto; []
end.
first_order.
eauto 8 using trc_trans.
Qed.
(** ** Dup *)
(* Refinement can be "pushed inside" a [Dup] operation. *)
Inductive RDup (R : proc -> proc -> Prop) : proc -> proc -> Prop :=
| RDupLeaf : forall pr1 pr2,
R pr1 pr2
-> RDup R pr1 pr2
| RDupDup : forall pr1 pr2,
RDup R pr1 pr2
-> RDup R (Dup pr1) (Dup pr2)
| RDupPar : forall pr1 pr2 pr1' pr2',
RDup R pr1 pr1'
-> RDup R pr2 pr2'
-> RDup R (Par pr1 pr2) (Par pr1' pr2').
Hint Constructors RDup.
Hint Unfold lstepSilent.
Lemma lstepSilent_Par1 : forall pr1 pr1' pr2,
lstepSilent^* pr1 pr1'
-> lstepSilent^* (Par pr1 pr2) (Par pr1' pr2).
Proof.
induct 1; eauto.
Qed.
Lemma lstepSilent_Par2 : forall pr2 pr2' pr1,
lstepSilent^* pr2 pr2'
-> lstepSilent^* (Par pr1 pr2) (Par pr1 pr2').
Proof.
induct 1; eauto.
Qed.
Hint Resolve lstepSilent_Par1 lstepSilent_Par2.
Lemma refines_Dup_Action : forall R : _ -> _ -> Prop,
(forall pr1 pr2, R pr1 pr2
-> forall pr1' a, lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ R pr1' pr2'')
-> forall pr1 pr2, RDup R pr1 pr2
-> forall pr1' a, lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ RDup R pr1' pr2''.
Proof.
induct 2; simplify.
eapply H in H1; eauto.
first_order.
eauto 6.
invert H1.
invert H0.
apply IHRDup1 in H5.
first_order.
eauto 10.
apply IHRDup2 in H5.
first_order.
eauto 10.
Qed.
Lemma refines_Dup_Silent : forall R : _ -> _ -> Prop,
(forall pr1 pr2, R pr1 pr2
-> forall pr1', lstepSilent pr1 pr1'
-> exists pr2', lstepSilent^* pr2 pr2'
/\ R pr1' pr2')
-> (forall pr1 pr2, R pr1 pr2
-> forall pr1' a, lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ R pr1' pr2'')
-> forall pr1 pr2, RDup R pr1 pr2
-> forall pr1', lstepSilent pr1 pr1'
-> exists pr2', lstepSilent^* pr2 pr2' /\ RDup R pr1' pr2'.
Proof.
induct 3; simplify.
eapply H in H1; eauto.
first_order.
eauto.
invert H2.
eauto 10.
invert H1.
apply IHRDup1 in H6.
first_order.
eauto.
apply IHRDup2 in H6.
first_order.
eauto.
eapply refines_Dup_Action in H4; eauto.
eapply refines_Dup_Action in H5; eauto.
first_order.
eexists; propositional.
match goal with
| [ _ : lstepSilent^* pr1' ?x |- _ ] => apply trc_trans with (x || pr2')
end.
eauto.
match goal with
| [ _ : lstepSilent^* pr2' ?x' |- lstepSilent^* (?x || _) _ ] => eapply trc_trans with (x || x')
end.
eauto.
apply trc_one.
eauto.
eauto.
eapply refines_Dup_Action in H4; eauto.
eapply refines_Dup_Action in H5; eauto.
first_order.
eexists; propositional.
match goal with
| [ _ : lstepSilent^* pr1' ?x |- _ ] => apply trc_trans with (x || pr2')
end.
eauto.
match goal with
| [ _ : lstepSilent^* pr2' ?x' |- lstepSilent^* (?x || _) _ ] => eapply trc_trans with (x || x')
end.
eauto.
apply trc_one.
eauto.
eauto.
Qed.
Theorem refines_Dup : forall pr1 pr2,
pr1 <| pr2
-> Dup pr1 <| Dup pr2.
Proof.
invert 1.
exists (RDup x).
unfold simulates in *.
propositional; eauto using refines_Dup_Silent, refines_Dup_Action.
Qed.
(** ** Par *)
(* Refinement can also be "pushed inside" parallel composition. *)
Inductive RPar (R1 R2 : proc -> proc -> Prop) : proc -> proc -> Prop :=
| RPar1 : forall pr1 pr2 pr1' pr2',
R1 pr1 pr1'
-> R2 pr2 pr2'
-> RPar R1 R2 (pr1 || pr2) (pr1' || pr2').
Hint Constructors RPar.
Lemma refines_Par_Action : forall R1 R2 : _ -> _ -> Prop,
(forall pr1 pr2, R1 pr1 pr2
-> forall pr1' a, lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ R1 pr1' pr2'')
-> (forall pr1 pr2, R2 pr1 pr2
-> forall pr1' a, lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ R2 pr1' pr2'')
-> forall pr1 pr2, RPar R1 R2 pr1 pr2
-> forall pr1' a, lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ RPar R1 R2 pr1' pr2''.
Proof.
invert 3; simplify.
invert H1.
eapply H in H8; eauto.
first_order.
eauto 10.
eapply H0 in H8; eauto.
first_order.
eauto 10.
Qed.
Lemma refines_Par_Silent : forall R1 R2 : _ -> _ -> Prop,
(forall pr1 pr2, R1 pr1 pr2
-> forall pr1', lstepSilent pr1 pr1'
-> exists pr2', lstepSilent^* pr2 pr2'
/\ R1 pr1' pr2')
-> (forall pr1 pr2, R1 pr1 pr2
-> forall pr1' a, lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ R1 pr1' pr2'')
-> (forall pr1 pr2, R2 pr1 pr2
-> forall pr1', lstepSilent pr1 pr1'
-> exists pr2', lstepSilent^* pr2 pr2'
/\ R2 pr1' pr2')
-> (forall pr1 pr2, R2 pr1 pr2
-> forall pr1' a, lstep pr1 (Action a) pr1'
-> exists pr2' pr2'', lstepSilent^* pr2 pr2'
/\ lstep pr2' (Action a) pr2''
/\ R2 pr1' pr2'')
-> forall pr1 pr2, RPar R1 R2 pr1 pr2
-> forall pr1', lstepSilent pr1 pr1'
-> exists pr2', lstepSilent^* pr2 pr2' /\ RPar R1 R2 pr1' pr2'.
Proof.
invert 5; simplify.
invert H3.
eapply H in H10; eauto.
first_order; eauto.
eapply H1 in H10; eauto.
first_order; eauto.
eapply H0 in H8; eauto.
eapply H2 in H9; eauto.
first_order.
eexists; propositional.
match goal with
| [ _ : lstepSilent^* pr1' ?x |- _ ] => apply trc_trans with (x || pr2')
end.
eauto.
match goal with
| [ _ : lstepSilent^* pr2' ?x' |- lstepSilent^* (?x || _) _ ] => eapply trc_trans with (x || x')
end.
eauto.
apply trc_one.
eauto.
eauto.
eapply H0 in H8; eauto.
eapply H2 in H9; eauto.
first_order.
eexists; propositional.
match goal with
| [ _ : lstepSilent^* pr1' ?x |- _ ] => apply trc_trans with (x || pr2')
end.
eauto.
match goal with
| [ _ : lstepSilent^* pr2' ?x' |- lstepSilent^* (?x || _) _ ] => eapply trc_trans with (x || x')
end.
eauto.
apply trc_one.
eauto.
eauto.
Qed.
Theorem refines_Par : forall pr1 pr2 pr1' pr2',
pr1 <| pr1'
-> pr2 <| pr2'
-> pr1 || pr2 <| pr1' || pr2'.
Proof.
invert 1; invert 1.
exists (RPar x x0).
unfold simulates in *.
propositional; eauto using refines_Par_Silent, refines_Par_Action.
Qed.
(** ** Block *)
(* A few similar properties apply to [Block], too. *)
Inductive RBlock (R : proc -> proc -> Prop) : proc -> proc -> Prop :=
| RBlock1 : forall pr1 pr2 ch,
R pr1 pr2
-> RBlock R (Block ch; pr1) (Block ch; pr2).
Hint Constructors RBlock.
Hint Unfold notUse.
Lemma lstepSilent_Block : forall ch pr1 pr2,
lstepSilent^* pr1 pr2
-> lstepSilent^* (Block ch; pr1) (Block ch; pr2).
Proof.
induct 1; eauto.
Qed.
Hint Resolve lstepSilent_Block.
Theorem refines_Block : forall pr1 pr2 ch,
pr1 <| pr2
-> Block ch; pr1 <| Block ch; pr2.
Proof.
invert 1.
exists (RBlock x).
first_order; eauto.
invert H2.
invert H3.
eapply H in H6; eauto.
first_order.
eauto.
invert H2.
invert H3.
eapply H0 in H6; eauto.
first_order.
eauto 10.
Qed.
Inductive RBlock2 : proc -> proc -> Prop :=
| RBlock2_1 : forall ch1 ch2 pr,
RBlock2 (Block ch1; Block ch2; pr) (Block ch2; Block ch1; pr).
Hint Constructors RBlock2.
Theorem refines_Block2 : forall ch1 ch2 pr,
Block ch1; Block ch2; pr <| Block ch2; Block ch1; pr.
Proof.
exists RBlock2.
first_order; eauto.
invert H.
invert H0.
invert H2.
eauto 10.
invert H.
invert H0.
invert H2.
eauto 10.
Qed.
(* This predicate is handy for side conditions, to enforce that a process never
* uses a particular channel for anything. *)
Inductive neverUses (ch : channel) : proc -> Prop :=
| NuRecv : forall ch' (A : Set) (k : A -> _),
ch' <> ch
-> (forall v, neverUses ch (k v))
-> neverUses ch (Recv ch' k)
| NuSend : forall ch' (A : Set) (v : A) k,
ch' <> ch
-> neverUses ch k
-> neverUses ch (Send ch' v k)
| NuDup : forall pr,
neverUses ch pr
-> neverUses ch (Dup pr)
| NuPar : forall pr1 pr2,
neverUses ch pr1
-> neverUses ch pr2
-> neverUses ch (pr1 || pr2)
| NuDone :
neverUses ch Done.
Hint Constructors neverUses.
Lemma neverUses_step : forall ch pr1,
neverUses ch pr1
-> forall l pr2, lstep pr1 l pr2
-> neverUses ch pr2.
Proof.
induct 1; invert 1; eauto.
Qed.
Hint Resolve neverUses_step.
Inductive RBlockS : proc -> proc -> Prop :=
| RBlockS1 : forall ch pr1 pr2,
neverUses ch pr2
-> RBlockS (Block ch; pr1 || pr2) ((Block ch; pr1) || pr2).
Hint Constructors RBlockS.
Lemma neverUses_notUse : forall ch pr l,
neverUses ch pr
-> forall pr', lstep pr l pr'
-> notUse ch l.
Proof.
induct 1; invert 1; simplify; eauto.
Qed.
Lemma notUse_Input_Output : forall ch r,
notUse ch (Input r)
-> notUse ch (Output r).
Proof.
simplify; auto.
Qed.
Lemma notUse_Output_Input : forall ch r,
notUse ch (Output r)
-> notUse ch (Input r).
Proof.
simplify; auto.
Qed.
Hint Resolve neverUses_notUse.
Theorem refines_BlockS : forall ch pr1 pr2,
neverUses ch pr2
-> Block ch; pr1 || pr2 <| (Block ch; pr1) || pr2.
Proof.
exists RBlockS.
first_order; eauto.
invert H0.
invert H1.
invert H4; eauto 10.
eexists; propositional.
apply trc_one.
eapply LStepRendezvousLeft; eauto.
constructor; eauto.
apply notUse_Output_Input; eauto.
eauto.
eexists; propositional.
apply trc_one.
eapply LStepRendezvousRight; eauto.
constructor; eauto.
apply notUse_Input_Output; eauto.
eauto.
invert H0.
invert H1.
invert H4; eauto 10.
Qed.
(** * The first example again *)
(* Those tools will help us lift our earlier adder proof to the immortal-server
* case, without writing any new simulation relations ourselves. *)
Theorem refines_add2 : forall input output,
input <> output
-> add2 input output <| addNs 2 input output.
Proof.
simplify.
apply refines_Dup.
apply add2_once_refines_addN; auto.
Qed.
(* We can even check refinement of our different adders when run together with
* the tester, carefully marking internal channels as private with [Block]. *)
Theorem refines_add2_with_tester : forall metaInput input output metaOutput,
input <> output
-> Block input; Block output; add2 input output || tester metaInput input output metaOutput
<| Block input; Block output; addNs 2 input output || tester metaInput input output metaOutput.
Proof.
simplify.
do 2 apply refines_Block.
apply refines_Par.
apply refines_add2; auto.
apply refines_refl.
Qed.
(** * Tree membership *)
(* Here's one more example of a parallel program, which searches a binary tree
* in parallel, checking if a value is found at one of the leaves. *)
Inductive tree :=
| Leaf (n : nat)
| Node (l r : tree).
(* This function formalizes the membership property that we check. *)
Fixpoint mem (n : nat) (t : tree) : bool :=
match t with
| Leaf m => if m ==n n then true else false
| Node l r => mem n l || mem n r
end%bool.
(* Here's the lame (but straightforward) sequential implementation. Note that
* we do nothing if the value is not found, and we send exactly one [tt] value
* as output if the value is found. *)
Definition inTree_seq (t : tree) (input output : channel) :=
Dup (??input(n : nat);
if mem n t then
!!output(tt);
Done
else
Done).
(* Helper function for a fancier parallel version, creating many threads that
* are all allowed to send to a channel [output], if they find the value. *)
Fixpoint inTree_par' (n : nat) (t : tree) (output : channel) :=
match t with
| Leaf m =>
if m ==n n then
!!output(tt);
Done
else
Done
| Node l r =>
inTree_par' n l output || inTree_par' n r output
end.
(* Top-level wrapper for an immortal-server tree-searcher *)
Definition inTree_par (t : tree) (input output : channel) :=
Dup (??input(n : nat);
New[input;output](output');
inTree_par' n t output'
|| ??output'(_ : unit);
!!output(tt);
Done).
(* Note the second part of the parallel composition, which makes sure we send
* *at most one* notification to the outside world, though the internal threads
* may generate as many notifications as there are tree leaves. *)
(* OK, now we get into the complex part, to prove the simulation. We will let
* the relations and lemmas below "speak for themselves," though admittedly it's
* a pretty involved argument. *)
Inductive TreeThreads (output' : channel) : bool -> proc -> Prop :=
| TtDone : forall maySend,
TreeThreads output' maySend Done
| TtSend :
TreeThreads output' true (!!output'(tt); Done)
| TtPar : forall maySend pr1 pr2,
TreeThreads output' maySend pr1
-> TreeThreads output' maySend pr2
-> TreeThreads output' maySend (pr1 || pr2).
(* This is the main simulation relation. *)
Inductive RTree (t : tree) (input output : channel) : proc -> proc -> Prop :=
| TreeInit :
RTree t input output
(??input(n : nat);
New[input;output](output');
inTree_par' n t output'
|| ??output'(_ : unit);
!!output(tt);
Done)
(??input(n : nat);
if mem n t then
!!output(tt);
Done
else
Done)
| TreeGotInput : forall n,
RTree t input output
(New[input;output](output');
inTree_par' n t output'
|| ??output'(_ : unit);
!!output(tt);
Done)
(if mem n t then
!!output(tt);
Done
else
Done)
| TreeSearching : forall n output' threads,
~In output' [input; output]
-> TreeThreads output' (mem n t) threads
-> RTree t input output
(Block output';
threads
|| ??output'(_ : unit);
!!output(tt);
Done)
(if mem n t then
!!output(tt);
Done
else
Done)
| TreeFound : forall n output' threads,
mem n t = true
-> ~In output' [input; output]
-> TreeThreads output' true threads
-> RTree t input output
(Block output'; threads || !!output(tt); Done)
(!!output(tt); Done)
| TreeNotified : forall n output' threads,
mem n t = true
-> ~In output' [input; output]
-> TreeThreads output' true threads
-> RTree t input output
(Block output'; threads || Done)
Done.
Hint Constructors TreeThreads RTree.
Lemma TreeThreads_actionIs : forall ch maySend pr,
TreeThreads ch maySend pr
-> forall a pr', lstep pr (Action a) pr'
-> a = Output {| Channel := ch; Value := tt |}.
Proof.
induct 1; invert 1; eauto.
Qed.
Lemma TreeThreads_silent : forall ch maySend pr,
TreeThreads ch maySend pr
-> forall pr', lstep pr Silent pr'
-> False.
Proof.
induct 1; invert 1; simplify; eauto.
eapply TreeThreads_actionIs in H4; eauto.
equality.
eapply TreeThreads_actionIs in H5; eauto.
equality.
Qed.
Lemma TreeThreads_maySend : forall ch maySend pr,
TreeThreads ch maySend pr
-> forall a pr', lstep pr a pr'
-> maySend = true.
Proof.
induct 1; invert 1; eauto.
Qed.
Lemma TreeThreads_action : forall ch maySend pr,
TreeThreads ch maySend pr
-> forall a pr', lstep pr a pr'
-> TreeThreads ch maySend pr'.
Proof.
induct 1; invert 1; eauto.
Qed.
Lemma TreeThreads_weaken : forall ch maySend pr,
TreeThreads ch maySend pr
-> TreeThreads ch true pr.
Proof.
induct 1; eauto.
Qed.
Hint Resolve TreeThreads_silent TreeThreads_maySend TreeThreads_action TreeThreads_weaken.
Lemma TreeThreads_inTree_par' : forall n ch t,
TreeThreads ch (mem n t) (inTree_par' n t ch).
Proof.
induct t; simplify; eauto.
cases (n0 ==n n); eauto.
cases (mem n t1); simplify; eauto.
cases (mem n t2); simplify; eauto.
Qed.
Hint Resolve TreeThreads_inTree_par'.
(* Finally, the main theorem: *)
Theorem refines_inTree_par : forall t input output,
inTree_par t input output <| inTree_seq t input output.
Proof.
simplify.
apply refines_Dup.
exists (RTree t input output).
first_order; eauto.
invert H.
invert H0.
invert H0; eauto.
invert H0.
invert H4; eauto.
invert H6.
eapply TreeThreads_actionIs in H3; eauto; equality.
specialize (TreeThreads_actionIs H2 H3); invert 1.
invert H5.
assert (mem n t = true) by eauto.
rewrite H.
eauto 10.
invert H0.
invert H5.
eauto.
invert H7.
eapply TreeThreads_actionIs in H4; eauto; equality.
invert H6.
invert H0.
invert H5.
exfalso; eauto using TreeThreads_silent.
invert H7.
invert H6.
invert H6.
invert H.
invert H0; eauto.
invert H0.
invert H0.
invert H4.
eapply TreeThreads_actionIs in H6; eauto.
subst; simplify; equality.
invert H6.
simplify; equality.
invert H0.
invert H5.
eapply TreeThreads_actionIs in H7; eauto.
subst; simplify; equality.
invert H7.
eauto 10.
invert H0.
invert H5.
eapply TreeThreads_actionIs in H7; eauto.
subst; simplify; equality.
invert H7.
Qed.
(* Hey, let's reason about plugging together the adder and the tree-searcher,
* because we can! The adder produces a number that is fed into the
* tree-searcher as input. *)
Theorem gratuitous_composition : forall t ch1 ch2 ch3,
ch1 <> ch2
-> Block ch2; add2 ch1 ch2 || inTree_par t ch2 ch3
<| Block ch2; addNs 2 ch1 ch2 || inTree_seq t ch2 ch3.
Proof.
simplify.
apply refines_Block.
apply refines_Par.
apply refines_add2; auto.
apply refines_inTree_par.
Qed.
(* Note how we didn't need to revisit any details of the proofs of the
* individual components. Now that's modularity in action! *)
(** * One more example: handoff lemma *)
(* Let's prove an even simpler specification related to the last example proof.
* We define some relations and lemmas in service of the key handoff lemma, but
* feel free to search for [Theorem] to skip ahead to its (much simpler)
* statement. *)
Inductive manyOf (this : proc) : proc -> Prop :=
| MoOne : manyOf this this
| MoDup : manyOf this (Dup this)
| MoPar : forall pr1 pr2,
manyOf this pr1
-> manyOf this pr2
-> manyOf this (pr1 || pr2).
Inductive manyOfAndOneOf (common rare : proc) : proc -> Prop :=
| MooCommon : manyOfAndOneOf common rare common
| MooRare : manyOfAndOneOf common rare rare
| MooDup : manyOfAndOneOf common rare (Dup common)
| MooPar1 : forall pr1 pr2,
manyOfAndOneOf common rare pr1
-> manyOf common pr2
-> manyOfAndOneOf common rare (pr1 || pr2)
| MooPar2 : forall pr1 pr2,
manyOf common pr1
-> manyOfAndOneOf common rare pr2
-> manyOfAndOneOf common rare (pr1 || pr2).
Inductive Rhandoff (ch : channel) (A : Set) (v : A) (k : A -> proc) : proc -> proc -> Prop :=
| Rhandoff1 : forall recvs,
neverUses ch (k v)
-> manyOf (??ch(x : A); k x) recvs
-> Rhandoff ch v k (Block ch; !!ch(v); Done || recvs) (k v)
| Rhandoff2 : forall recvs rest,
neverUses ch rest
-> manyOfAndOneOf (??ch(x : A); k x) rest recvs
-> Rhandoff ch v k (Block ch; Done || recvs) rest
| Rhandoff3 : forall recvs rest,
neverUses ch rest
-> manyOf (??ch(x : A); k x) recvs
-> Rhandoff ch v k (Block ch; Done || recvs) rest.
Hint Constructors manyOf manyOfAndOneOf Rhandoff.
Lemma manyOf_action : forall this pr,
manyOf this pr
-> forall a pr', lstep pr (Action a) pr'
-> exists this', lstep this a this'.
Proof.
induct 1; simplify; eauto.
invert H.
invert H1; eauto.
Qed.
Lemma manyOf_silent : forall this, (forall this', lstepSilent this this' -> False)
-> (forall r this', lstep this (Output r) this' -> False)
-> forall pr, manyOf this pr
-> forall pr', lstep pr Silent pr'
-> manyOf this pr'.
Proof.
induct 1; simplify; eauto.
exfalso; eauto.
invert H1; eauto.
invert H1; eauto.
eapply manyOf_action in H5; eauto; first_order; exfalso; eauto.
eapply manyOf_action in H4; eauto; first_order; exfalso; eauto.
Qed.
Lemma manyOf_rendezvous : forall ch (A : Set) (v : A) (k : A -> _) pr,
manyOf (Recv ch k) pr
-> forall pr', lstep pr (Input {| Channel := ch; Value := v |}) pr'
-> manyOfAndOneOf (Recv ch k) (k v) pr'.
Proof.
induct 1; simplify; eauto.
invert H; eauto.
invert H.
invert H1; eauto.
Qed.
Hint Resolve manyOf_silent manyOf_rendezvous.
Lemma manyOfAndOneOf_output : forall ch (A : Set) (k : A -> _) rest ch0 (A0 : Set) (v0 : A0) pr,
manyOfAndOneOf (Recv ch k) rest pr
-> forall pr', lstep pr (Output {| Channel := ch0; Value := v0 |}) pr'
-> exists rest', lstep rest (Output {| Channel := ch0; Value := v0 |}) rest'
/\ manyOfAndOneOf (Recv ch k) rest' pr'.
Proof.
induct 1; simplify; eauto.
invert H.
invert H.
invert H1; eauto.
apply IHmanyOfAndOneOf in H6; first_order; eauto.
eapply manyOf_action in H0; eauto.
first_order.
invert H0.
invert H1; eauto.
eapply manyOf_action in H; eauto.
first_order.
invert H.
apply IHmanyOfAndOneOf in H6; first_order; eauto.
Qed.
Lemma manyOf_manyOfAndOneOf : forall this other pr,
manyOf this pr
-> manyOfAndOneOf this other pr.
Proof.
induct 1; simplify; eauto.
Qed.
Hint Resolve manyOf_manyOfAndOneOf.
Lemma no_rendezvous : forall ch0 (A0 : Set) (v : A0) pr1 rest (k : A0 -> _),
manyOfAndOneOf (??ch0 (x : _); k x) rest pr1
-> forall pr1', lstep pr1 (Output {| Channel := ch0; TypeOf := A0; Value := v |}) pr1'
-> neverUses ch0 rest
-> False.
Proof.
induct 1; simplify.
invert H.
invert H.
invert H0; equality.
invert H0.
invert H0.
eapply neverUses_notUse in H3; eauto.
simplify; equality.
invert H0.
eapply neverUses_notUse in H4; eauto.
simplify; equality.
invert H.
invert H1.
eauto.
eapply manyOf_action in H0; try eassumption.
first_order.
invert H0.
invert H1.
eapply manyOf_action in H7; try eassumption.
first_order.
invert H1.
eauto.
Qed.
Lemma manyOfAndOneOf_silent : forall ch (A : Set) (k : A -> _) rest pr,
manyOfAndOneOf (Recv ch k) rest pr
-> neverUses ch rest
-> forall pr', lstep pr Silent pr'
-> exists rest', manyOfAndOneOf (Recv ch k) rest' pr'
/\ (rest = rest' \/ lstep rest Silent rest').
Proof.
induct 1; simplify; eauto.
invert H0.
invert H0; eauto.
invert H2.
apply IHmanyOfAndOneOf in H7; auto.
first_order; eauto.
eexists; propositional.
apply MooPar1.
eauto.
eapply manyOf_silent; try eassumption; invert 1.
eapply manyOf_action in H6; eauto.
first_order.
invert H2.
eapply manyOf_action in H6; eauto.
first_order.
invert H2.
exfalso; eapply no_rendezvous; eassumption.
invert H2.
eexists; propositional.
apply MooPar2; auto.
eapply manyOf_silent; try eassumption; invert 1.
apply IHmanyOfAndOneOf in H7; first_order; eauto.
eapply manyOf_action in H5; eauto.
first_order.
invert H2.
exfalso; eapply no_rendezvous; eassumption.
eapply manyOf_action in H; eauto.
first_order.
invert H.
Qed.
Hint Resolve manyOfAndOneOf_silent manyOf_rendezvous.
Lemma manyOfAndOneOf_action : forall ch (A : Set) (k : A -> _) rest pr,
manyOfAndOneOf (Recv ch k) rest pr
-> forall a pr', lstep pr (Action a) pr'
-> (exists v : A, a = Input {| Channel := ch; Value := v |})
\/ exists rest', manyOfAndOneOf (Recv ch k) rest' pr'
/\ lstep rest a rest'.
Proof.
induct 1; simplify; eauto.
invert H; eauto.
invert H.
invert H1.
apply IHmanyOfAndOneOf in H6; first_order; subst; eauto.
eapply manyOf_action in H6; eauto.
first_order.
invert H1; eauto.
invert H1.
eapply manyOf_action in H6; eauto.
first_order.
invert H1; eauto.
apply IHmanyOfAndOneOf in H6; first_order; subst; eauto.
Qed.
(* When one thread is ready to send a message, and there is an immortal server
* ready to accept that message, the process is equivalent to one that just
* skips right to running a single server thread. It is crucial that the body
* of each server thread has nothing more to do with the channel we are using to
* send it requests! Otherwise, we would need to keep some [Dup] present
* explicitly in the spec (righthand argument of [<|]). *)
Theorem handoff : forall ch (A : Set) (v : A) k,
neverUses ch (k v)
-> Block ch; (!!ch(v); Done) || Dup (Recv ch k)
<| k v.
Proof.
simplify.
exists (Rhandoff ch v k).
first_order; eauto.
invert H0.
invert H1.
invert H5.
invert H7.
eexists; propositional; eauto.
apply Rhandoff1; auto.
eapply manyOf_silent; try eassumption; invert 1.
invert H4.
invert H4; eauto.
invert H1.
invert H5.
invert H7.
eapply manyOfAndOneOf_silent in H7; eauto.
first_order; subst; eauto.
eauto 6.
invert H4.
invert H4.
invert H1.
invert H5.
invert H7.
exists pr2; propositional; eauto.
apply Rhandoff3; auto.
eapply manyOf_silent; try eassumption; invert 1.
invert H4.
invert H4.
invert H0.
invert H1.
invert H5.
invert H7.
simplify; equality.
eapply manyOf_action in H7; eauto.
first_order.
invert H0.
simplify; equality.
invert H1.
invert H5.
invert H7.
eapply manyOfAndOneOf_action in H3; eauto.
first_order; subst; eauto 10.
simplify; equality.
invert H1.
invert H5.
invert H7.
eapply manyOf_action in H7; eauto.
first_order.
invert H0.
simplify; equality.
Qed.
Ltac neverUses :=
repeat match goal with
| [ |- context[if ?E then _ else _] ] => cases E
| _ => repeat (constructor; simplify)
end; lists.
(* OK, let's prove a final and satisfyingly simple spec for a system that
* combines an adder and a tree-searcher. We send some seed value to an adder,
* which forwards it to the tree-searcher. When the value we expect it to send
* is indeed present in the tree, the whole contraption is equivalent to just
* signalling a "yes" answer! In our setting, that means sending a message to
* the final output channel [ch3]. *)
Theorem gratuitous_composition_expanded : forall n t ch1 ch2 ch3,
mem (n + 2) t = true
-> NoDup [ch1; ch2; ch3]
-> Block ch1; Block ch2; !!ch1(n); Done || add2 ch1 ch2 || inTree_par t ch2 ch3
<| !!ch3(tt); Done.
Proof.
simplify.
eapply refines_trans.
do 2 eapply refines_Block.
apply refines_Par.
apply refines_Par.
apply refines_refl.
apply refines_add2; lists.
apply refines_inTree_par.
unfold addNs, addN, inTree_seq.
eapply refines_trans.
eapply refines_Block2.
eapply refines_trans.
eapply refines_Block.
eapply refines_trans.
apply refines_BlockS; neverUses.
apply refines_Par.
apply handoff; neverUses.
apply refines_refl.
eapply refines_trans.
apply handoff; neverUses.
rewrite H.
apply refines_refl.
Qed.
(* Note how, again, we used the correctness theorems for our components as black
* boxes, so that all that's left is algebraic reasoning over [<|]. *)
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