Revising for the final week of class

MessagesAndRefinement.v

@@ -29,11 +29,11 @@ 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 name [ch] not found in [notThese], and continue like
+(* Pick a new channel name [ch] 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
+(* Act like [pr] but prevent interaction with other processes through channel
  * [ch].  We effectively force [ch] to be *private*. *)
 
 | Send (ch : channel) {A : Type} (v : A) (k : proc)
@@ -146,7 +146,7 @@ Inductive lstep : proc -> label -> proc -> Prop :=
 | LStepDup : forall pr,
     lstep (Dup pr) Silent (Par (Dup pr) pr)
 
-(* A channel allocation operation nondeterministically picks the new channel ID,
+(* 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,
@@ -241,7 +241,7 @@ Inductive couldGenerate : proc -> list action -> Prop :=
 (* Skip ahead to [refines_couldGenerate] to see the top-level connection from
  * [refines]. *)
 
-Hint Constructors couldGenerate : core.
+Global Hint Constructors couldGenerate : core.
 
 Lemma lstepSilent_couldGenerate : forall pr1 pr2,
   lstepSilent^* pr1 pr2
@@ -251,7 +251,7 @@ Proof.
   induct 1; eauto.
 Qed.
 
-Hint Resolve lstepSilent_couldGenerate : core.
+Global Hint Resolve lstepSilent_couldGenerate : core.
 
 Lemma simulates_couldGenerate' : forall (R : proc -> proc -> Prop),
     (forall pr1 pr2, R pr1 pr2
@@ -305,8 +305,8 @@ Ltac inverter :=
          | [ H : lstepSilent _ _ |- _ ] => invert H
          end.
 
-Hint Constructors lstep : core.
-Hint Unfold lstepSilent : core.
+Global Hint Constructors lstep : core.
+Global Hint Unfold lstepSilent : core.
 
 Ltac lists' :=
   repeat match goal with
@@ -316,7 +316,7 @@ Ltac lists' :=
 
 Ltac lists := solve [ lists' ].
 
-Hint Extern 1 (NoDup _) => lists : core.
+Global Hint Extern 1 (NoDup _) => lists : core.
 
 
 (** * Examples *)
@@ -361,7 +361,7 @@ Inductive R_add2 : proc -> proc -> Prop :=
          (Block intermediate; Done || Done)
          Done.
 
-Hint Constructors R_add2 : core.
+Global Hint Constructors R_add2 : core.
 
 Theorem add2_once_refines_addN : forall input output,
     input <> output
@@ -462,9 +462,9 @@ Inductive RDup (R : proc -> proc -> Prop) : proc -> proc -> Prop :=
     -> RDup R pr2 pr2'
     -> RDup R (Par pr1 pr2) (Par pr1' pr2').
 
-Hint Constructors RDup : core.
+Global Hint Constructors RDup : core.
 
-Hint Unfold lstepSilent : core.
+Global Hint Unfold lstepSilent : core.
 
 Lemma lstepSilent_Par1 : forall pr1 pr1' pr2,
     lstepSilent^* pr1 pr1'
@@ -480,7 +480,7 @@ Proof.
   induct 1; eauto.
 Qed.
 
-Hint Resolve lstepSilent_Par1 lstepSilent_Par2 : core.
+Global Hint Resolve lstepSilent_Par1 lstepSilent_Par2 : core.
 
 Lemma refines_Dup_Action : forall R : _ -> _ -> Prop,
     (forall pr1 pr2, R pr1 pr2
@@ -593,7 +593,7 @@ Inductive RPar (R1 R2 : proc -> proc -> Prop) : proc -> proc -> Prop :=
     -> R2 pr2 pr2'
     -> RPar R1 R2 (pr1 || pr2) (pr1' || pr2').
 
-Hint Constructors RPar : core.
+Global Hint Constructors RPar : core.
 
 Lemma refines_Par_Action : forall R1 R2 : _ -> _ -> Prop,
     (forall pr1 pr2, R1 pr1 pr2
@@ -706,8 +706,8 @@ Inductive RBlock (R : proc -> proc -> Prop) : proc -> proc -> Prop :=
     R pr1 pr2
     -> RBlock R (Block ch; pr1) (Block ch; pr2).
 
-Hint Constructors RBlock : core.
-Hint Unfold notUse : core.
+Global Hint Constructors RBlock : core.
+Global Hint Unfold notUse : core.
 
 Lemma lstepSilent_Block : forall ch pr1 pr2,
     lstepSilent^* pr1 pr2
@@ -716,7 +716,7 @@ Proof.
   induct 1; eauto.
 Qed.
 
-Hint Resolve lstepSilent_Block : core.
+Global Hint Resolve lstepSilent_Block : core.
 
 Theorem refines_Block : forall pr1 pr2 ch,
     pr1 <| pr2
@@ -743,7 +743,7 @@ 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 : core.
+Global Hint Constructors RBlock2 : core.
 
 Theorem refines_Block2 : forall ch1 ch2 pr,
     Block ch1; Block ch2; pr <| Block ch2; Block ch1; pr.
@@ -783,7 +783,7 @@ Inductive neverUses (ch : channel) : proc -> Prop :=
 | NuDone :
     neverUses ch Done.
 
-Hint Constructors neverUses : core.
+Global Hint Constructors neverUses : core.
 
 Lemma neverUses_step : forall ch pr1,
     neverUses ch pr1
@@ -793,14 +793,14 @@ Proof.
   induct 1; invert 1; eauto.
 Qed.
 
-Hint Resolve neverUses_step : core.
+Global Hint Resolve neverUses_step : core.
 
 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 : core.
+Global Hint Constructors RBlockS : core.
 
 Lemma neverUses_notUse : forall ch pr l,
     neverUses ch pr
@@ -824,7 +824,7 @@ Proof.
   simplify; auto.
 Qed.
 
-Hint Resolve neverUses_notUse : core.
+Global Hint Resolve neverUses_notUse : core.
 
 Theorem refines_BlockS : forall ch pr1 pr2,
     neverUses ch pr2
@@ -1008,7 +1008,7 @@ Inductive RTree (t : tree) (input output : channel) : proc -> proc -> Prop :=
           (Block output'; threads || Done)
           Done.
 
-Hint Constructors TreeThreads RTree : core.
+Global Hint Constructors TreeThreads RTree : core.
 
 Lemma TreeThreads_actionIs : forall ch maySend pr,
     TreeThreads ch maySend pr
@@ -1055,7 +1055,7 @@ Proof.
   induct 1; eauto.
 Qed.
 
-Hint Resolve TreeThreads_silent TreeThreads_maySend TreeThreads_action TreeThreads_weaken : core.
+Global Hint Resolve TreeThreads_silent TreeThreads_maySend TreeThreads_action TreeThreads_weaken : core.
 
 Lemma TreeThreads_inTree_par' : forall n ch t,
     TreeThreads ch (mem n t) (inTree_par' n t ch).
@@ -1066,7 +1066,7 @@ Proof.
   cases (mem n t2); simplify; eauto.
 Qed.
 
-Hint Resolve TreeThreads_inTree_par' : core.
+Global Hint Resolve TreeThreads_inTree_par' : core.
 
 (* Finally, the main theorem: *)
 Theorem refines_inTree_par : forall t input output,
@@ -1186,7 +1186,7 @@ Inductive Rhandoff (ch : channel) (A : Type) (v : A) (k : A -> proc) : proc -> p
     -> manyOf (??ch(x : A); k x) recvs
     -> Rhandoff ch v k (Block ch; Done || recvs) rest.
 
-Hint Constructors manyOf manyOfAndOneOf Rhandoff : core.
+Global Hint Constructors manyOf manyOfAndOneOf Rhandoff : core.
 
 Lemma manyOf_action : forall this pr,
     manyOf this pr
@@ -1226,7 +1226,7 @@ Proof.
   invert H1; eauto.
 Qed.
 
-Hint Resolve manyOf_silent manyOf_rendezvous : core.
+Global Hint Resolve manyOf_silent manyOf_rendezvous : core.
 
 Lemma manyOfAndOneOf_output : forall ch (A : Type) (k : A -> _) rest ch0 (A0 : Type) (v0 : A0) pr,
     manyOfAndOneOf (Recv ch k) rest pr
@@ -1260,7 +1260,7 @@ Proof.
   induct 1; simplify; eauto.
 Qed.
 
-Hint Resolve manyOf_manyOfAndOneOf : core.
+Global Hint Resolve manyOf_manyOfAndOneOf : core.
 
 Lemma no_rendezvous : forall ch0 (A0 : Type) (v : A0) pr1 rest (k : A0 -> _),
     manyOfAndOneOf (??ch0 (x : _); k x) rest pr1
@@ -1335,7 +1335,7 @@ Proof.
   invert H.
 Qed.
 
-Hint Resolve manyOfAndOneOf_silent manyOf_rendezvous : core.
+Global Hint Resolve manyOfAndOneOf_silent manyOf_rendezvous : core.
 
 Lemma manyOfAndOneOf_action : forall ch (A : Type) (k : A -> _) rest pr,
     manyOfAndOneOf (Recv ch k) rest pr

SessionTypes.v

@@ -36,7 +36,7 @@ Inductive type :=
 (* This type applies to a process that begins by sending a value of type [A]
  * over channel [ch], then continuing according to type [t]. *)
 
-| TRecv (ch : channel) (A : Type) (t :  type)
+| TRecv (ch : channel) (A : Type) (t : type)
 (* This type is the dual of the last one: the process begins by receiving a
  * value of type [A] from channel [ch]. *)
 
@@ -128,7 +128,7 @@ Definition trsys_of pr := {|
 (* Note: here we force silent steps, so that all channel communication is
  * internal. *)
 
-Hint Constructors hasty : core.
+Global Hint Constructors hasty : core.
 
 (* The next two lemmas state some inversions that connect stepping and
  * typing. *)
@@ -347,7 +347,7 @@ Definition trsys_of pr := {|
   Step := lstepSilent
 |}.
 
-Hint Constructors hasty : core.
+Global Hint Constructors hasty : core.
 
 Lemma input_typed : forall pr ch A v pr',
     lstep pr (Action (Input {| Channel := ch; TypeOf := A; Value := v |})) pr'
@@ -515,7 +515,7 @@ Definition trsys_of pr := {|
   Step := lstepSilent
 |}.
 
-Hint Constructors hasty : core.
+Global Hint Constructors hasty : core.
 
 (* We prove that the type system rules out fancier constructs. *)
 
@@ -546,7 +546,7 @@ Proof.
   assumption.
 Qed.
 
-Hint Immediate hasty_not_Block hasty_not_Dup hasty_not_Par : core.
+Global Hint Immediate hasty_not_Block hasty_not_Dup hasty_not_Par : core.
 
 (* Next, we characterize how channels must be mapped, given typing of a
  * process. *)
@@ -605,7 +605,7 @@ Inductive typed_multistate party (channels : channel -> parties party) (t : type
     -> typed_multistate channels t ps pr2
     -> typed_multistate channels t (p :: ps) (pr1 || pr2).
 
-Hint Constructors typed_multistate : core.
+Global Hint Constructors typed_multistate : core.
 
 (* This fancier typing judgment gets a fancier tactic for type-checking. *)
 
@@ -652,7 +652,7 @@ Proof.
   assumption.
 Qed.
 
-Hint Immediate no_silent_steps : core.
+Global Hint Immediate no_silent_steps : core.
 
 Lemma complementarity_forever_done : forall party (channels : _ -> parties party) pr pr',
   lstep pr Silent pr'
@@ -676,7 +676,7 @@ Proof.
   assumption.
 Qed.
 
-Hint Immediate mayNotSend_really : core.
+Global Hint Immediate mayNotSend_really : core.
 
 Lemma may_not_output : forall (party : Type) pr pr' ch (A : Type) (v : A),
     lstep pr (Action (Output {| Channel := ch; Value := v |})) pr'
@@ -692,7 +692,7 @@ Proof.
   assumption.
 Qed.
 
-Hint Immediate may_not_output : core.
+Global Hint Immediate may_not_output : core.
 
 Lemma output_is_legit : forall (party : Type) pr pr' ch (A : Type) (v : A),
     lstep pr (Action (Output {| Channel := ch; Value := v |})) pr'
@@ -815,7 +815,7 @@ Proof.
   induct 1; eauto.
 Qed.
 
-Local Hint Immediate hasty_relax : core.
+Global Hint Immediate hasty_relax : core.
 
 Lemma complementarity_preserve_unused : forall party (channels : _ -> parties party)
                                                pr ch (A : Type) (t : A -> _) all_parties,
@@ -1058,7 +1058,7 @@ Inductive inert : proc -> Prop :=
     -> inert pr2
     -> inert (pr1 || pr2).
 
-Hint Constructors inert : core.
+Global Hint Constructors inert : core.
 
 (* Now a few more fiddly lemmas.  See you again at the [Theorem]. *)
 
@@ -1070,7 +1070,7 @@ Proof.
   invert H; eauto.
 Qed.
 
-Hint Immediate typed_multistate_inert : core.
+Global Hint Immediate typed_multistate_inert : core.
 
 Lemma deadlock_find_receiver : forall party (channels : _ -> parties party) all_parties
                                       ch (A : Type) (k : A -> _) pr,
@@ -1420,7 +1420,7 @@ Inductive mayTouch : proc -> channel -> Prop :=
     mayTouch pr1 ch
     -> mayTouch (Dup pr1) ch.
 
-Hint Constructors mayTouch : core.
+Global Hint Constructors mayTouch : core.
 
 Import BasicTwoParty Multiparty.
 
@@ -1435,7 +1435,7 @@ Proof.
     end.
 Qed.
 
-Hint Immediate lstep_mayTouch : core.
+Global Hint Immediate lstep_mayTouch : core.
 
 Lemma Input_mayTouch : forall pr ch (A : Type) (v : A) pr',
     lstep pr (Action (Input {| Channel := ch; Value := v |})) pr'
@@ -1451,7 +1451,7 @@ Proof.
   induct 1; eauto.
 Qed.
 
-Hint Immediate Input_mayTouch Output_mayTouch : core.
+Global Hint Immediate Input_mayTouch Output_mayTouch : core.
 
 Lemma independent_execution : forall pr1 pr2 pr,
   lstepSilent^* (pr1 || pr2) pr

frap_book.tex

@@ -5607,7 +5607,7 @@ In the \emph{synchronous}\index{synchronous message passing} or \emph{rendezvous
 The threads of the two complementary operations \emph{rendezvous} and pass the message in one atomic step.
 
 Packages of semantics and proof techniques for such languages are often called \emph{process algebras}\index{process algebra}, as they support an algebraic style of reasoning about the source code of message-passing programs.
-That is, we prove laws very similar to the familiar equations of algebra, and use those laws to ``rewrite'' inside larger processes, by replacing their subprocesses with others we have shown suitably equivalent.
+That is, we prove laws very similar to the familiar equations of algebra and use those laws to ``rewrite'' inside larger processes, by replacing their subprocesses with others we have shown suitably equivalent.
 It's a powerful technique for highly modular proofs, which we develop in the rest of this chapter for one concrete synchronous language.
 Well-known process algebras include the $\pi$-calculus\index{$\pi$-calculus} and the Calculus of Communicating Systems\index{Calculus of Communicating Systems}; the one we focus on is idiosyncratic and designed partly to make the Coq proofs manageable.
 
@@ -5722,7 +5722,6 @@ Instead, we formalize refinement via \emph{simulation}, very similarly to how we
 
 Intuitively, $R$ is a simulation when, starting in a pair of related processes, any step on the left can be matched by a step on the right, taking us back into $R$.
 The conditions are naturally illustrated with commuting diagrams\index{commuting diagram}.
-
 \[
 \begin{tikzcd}
 p_1 \arrow{r}{R} \arrow{d}{\forall \longrightarrow} & p_2 \arrow{d}{\exists \longrightarrow^*} \\