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SessionTypes: changed to make choices explicitly dependent on message contents
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Adam Chlipala
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0875f52b12
commit
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1 changed files with 31 additions and 158 deletions
189
SessionTypes.v
189
SessionTypes.v
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@ -3,7 +3,7 @@
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* Author: Adam Chlipala
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* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
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Require Import Frap MessagesAndRefinement.
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Require Import Frap FunctionalExtensionality MessagesAndRefinement.
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Set Implicit Arguments.
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Set Asymmetric Patterns.
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@ -12,40 +12,24 @@ Set Asymmetric Patterns.
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(** * Defining the Type System *)
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Inductive type :=
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| TSend (ch : channel) (A : Set) (t : type)
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| TRecv (ch : channel) (A : Set) (t : type)
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| TDone
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| InternalChoice (t1 t2 : type)
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| ExternalChoice (t1 t2 : type).
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| TSend (ch : channel) (A : Set) (t : A -> type)
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| TRecv (ch : channel) (A : Set) (t : A -> type)
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| TDone.
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Delimit Scope st_scope with st.
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Bind Scope st_scope with type.
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Notation "!!! ch ( A ) ; k" := (TSend ch A k%st) (right associativity, at level 45, ch at level 0) : st_scope.
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Notation "??? ch ( A ) ; k" := (TRecv ch A k%st) (right associativity, at level 45, ch at level 0) : st_scope.
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Infix "|?|" := InternalChoice (at level 40) : st_scope.
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Infix "?|?" := ExternalChoice (at level 40) : st_scope.
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Notation "!!! ch ( x : A ) ; k" := (TSend ch (fun x : A => k)%st) (right associativity, at level 45, ch at level 0, x at level 0) : st_scope.
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Notation "??? ch ( x : A ) ; k" := (TRecv ch (fun x : A => k)%st) (right associativity, at level 45, ch at level 0, x at level 0) : st_scope.
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Inductive hasty : proc -> type -> Prop :=
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| HtSend : forall ch (A : Set) (v : A) k t,
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hasty k t
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-> hasty (Send ch v k) (TSend ch A t)
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| HtRecv : forall ch (A : Set) (k : A -> _) t (v : A),
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(forall v, hasty (k v) t)
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-> hasty (Recv ch k) (TRecv ch A t)
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hasty k (t v)
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-> hasty (Send ch v k) (TSend ch t)
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| HtRecv : forall ch (A : Set) (k : A -> _) t,
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(forall v, hasty (k v) (t v))
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-> hasty (Recv ch k) (TRecv ch t)
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| HtDone :
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hasty Done TDone
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| HtInternalChoice1 : forall pr t1 t2,
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hasty pr t1
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-> hasty pr (InternalChoice t1 t2)
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| HtInternalChoice2 : forall pr t1 t2,
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hasty pr t2
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-> hasty pr (InternalChoice t1 t2)
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| HtExternalChoice : forall pr t1 t2,
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hasty pr t1
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-> hasty pr t2
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-> hasty pr (ExternalChoice t1 t2).
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hasty Done TDone.
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(** * Examples of Typed Processes *)
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@ -59,7 +43,7 @@ Definition addN (k : nat) (input output : channel) : proc :=
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Ltac hasty := simplify; repeat (constructor; simplify).
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Theorem addN_typed : forall k input output,
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hasty (addN k input output) (???input(nat); !!!output(nat); TDone).
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hasty (addN k input output) (???input(_ : nat); !!!output(_ : nat); TDone).
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Proof.
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hasty.
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Qed.
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@ -69,12 +53,9 @@ Qed.
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Fixpoint complement (t : type) : type :=
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match t with
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| TSend ch A t1 => TRecv ch A (complement t1)
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| TRecv ch A t1 => TSend ch A (complement t1)
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| TSend ch _ t1 => TRecv ch (fun v => complement (t1 v))
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| TRecv ch _ t1 => TSend ch (fun v => complement (t1 v))
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| TDone => TDone
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| InternalChoice t1 t2 => ExternalChoice (complement t1) (complement t2)
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| ExternalChoice t1 t2 => InternalChoice (complement t1) (complement t2)
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end.
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Definition add2_client (input output : channel) : proc :=
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@ -83,8 +64,7 @@ Definition add2_client (input output : channel) : proc :=
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Done.
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Theorem add2_client_typed : forall input output,
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input <> output
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-> hasty (add2_client input output) (complement (???input(nat); !!!output(nat); TDone)).
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hasty (add2_client input output) (complement (???input(_ : nat); !!!output(_ : nat); TDone)).
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Proof.
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hasty.
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Qed.
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@ -101,43 +81,12 @@ Definition trsys_of pr := {|
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Hint Constructors hasty.
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Lemma hasty_not_NewChannel : forall chs pr t,
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hasty (NewChannel chs pr) t
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-> False.
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Proof.
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induct 1; auto.
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Qed.
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Lemma hasty_not_BlockChannel : forall ch pr t,
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hasty (BlockChannel ch pr) t
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-> False.
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Proof.
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induct 1; auto.
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Qed.
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Lemma hasty_not_Dup : forall pr t,
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hasty (Dup pr) t
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-> False.
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Proof.
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induct 1; auto.
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Qed.
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Lemma hasty_not_Par : forall pr1 pr2 t,
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hasty (pr1 || pr2) t
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-> False.
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Proof.
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induct 1; auto.
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Qed.
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Hint Immediate hasty_not_NewChannel hasty_not_BlockChannel hasty_not_Dup hasty_not_Par.
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Lemma input_typed : forall pr ch A v pr',
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lstep pr (Input {| Channel := ch; TypeOf := A; Value := v |}) pr'
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-> forall t, hasty pr t
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-> exists k, pr = Recv ch k /\ pr' = k v.
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Proof.
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induct 1; simplify; try solve [ exfalso; eauto ].
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induct H; eauto.
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induct 1; invert 1; eauto.
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Qed.
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Lemma output_typed : forall pr ch A v pr',
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@ -145,22 +94,7 @@ Lemma output_typed : forall pr ch A v pr',
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-> forall t, hasty pr t
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-> exists k, pr = Send ch v k /\ pr' = k.
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Proof.
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induct 1; simplify; try solve [ exfalso; eauto ].
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induct H; eauto.
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Qed.
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Lemma complementarity_rendezvous : forall ch (A : Set) (k1 : A -> _) t,
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hasty (Recv ch k1) t
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-> forall (v : A) k2, hasty (Send ch v k2) (complement t)
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-> exists t', hasty (k1 v) t' /\ hasty k2 (complement t').
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Proof.
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induct 1; invert 1; simplify; eauto.
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Qed.
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Lemma complement_inverse : forall t,
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t = complement (complement t).
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Proof.
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induct t; simplify; equality.
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induct 1; invert 1; eauto.
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Qed.
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Lemma complementarity_forever : forall pr1 pr2 t,
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@ -182,73 +116,20 @@ Proof.
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first_order; subst.
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invert H2.
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invert H6; try solve [ exfalso; eauto ].
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invert H6; try solve [ exfalso; eauto ].
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invert H6; invert H0.
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invert H6; invert H1.
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eapply input_typed in H4; eauto.
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eapply output_typed in H5; eauto.
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first_order; subst.
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eapply complementarity_rendezvous in H0; eauto.
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first_order.
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invert H0.
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invert H1.
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eauto 7.
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eapply input_typed in H5; eauto.
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eapply output_typed in H4; eauto.
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first_order; subst.
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rewrite complement_inverse in H0.
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eapply complementarity_rendezvous in H1; eauto.
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first_order.
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rewrite complement_inverse in H.
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first_order.
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Qed.
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Lemma notstuck_send : forall pr1 t,
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hasty pr1 t
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-> forall pr2, hasty pr2 (complement t)
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-> forall ch (A : Set) (v : A) pr1', lstep pr1 (Output {| Channel := ch; Value := v |}) pr1'
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-> exists pr2', lstep pr2 (Input {| Channel := ch; Value := v |}) pr2'.
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Proof.
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induct 1; invert 1; simplify; eauto;
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match goal with
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| [ H : lstep _ _ _ |- _ ] => invert H; eauto
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end.
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Qed.
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Lemma notstuck_nosilent : forall pr1 t,
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hasty pr1 t
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-> forall pr1', lstep pr1 Silent pr1'
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-> False.
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Proof.
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induct 1; invert 1; simplify; eauto.
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Qed.
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Lemma notstuck_recv : forall pr1 t,
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hasty pr1 t
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-> forall pr2, hasty pr2 (complement t)
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-> forall ch (A : Set) (v : A) pr1', lstep pr1 (Input {| Channel := ch; Value := v |}) pr1'
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-> exists (v' : A) pr2', lstep pr2 (Output {| Channel := ch; Value := v' |}) pr2'.
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Proof.
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induct 1; invert 1; simplify; eauto;
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match goal with
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| [ H : lstep _ _ _ |- _ ] => invert H; eauto
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end.
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Qed.
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Lemma one_thread_progress : forall pr t,
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hasty pr t
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-> pr = Done \/ exists l pr', lstep pr l pr'.
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Proof.
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induct 1; first_order; subst; eauto.
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Unshelve.
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assumption.
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Qed.
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Lemma hasty_Done : forall t,
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hasty Done t
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-> forall pr, hasty pr (complement t)
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-> pr = Done.
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Proof.
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induct 1; invert 1; eauto.
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invert H0.
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invert H1.
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eauto 10.
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Qed.
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Theorem no_deadlock : forall pr1 pr2 t,
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@ -261,18 +142,10 @@ Proof.
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simplify.
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eapply invariant_weaken.
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eapply complementarity_forever; eauto.
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clear pr1 pr2 t H H0.
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simplify; first_order; subst.
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specialize (one_thread_progress H2); first_order; subst.
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eapply hasty_Done in H2; eauto.
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equality.
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cases x2.
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exfalso; eauto using notstuck_nosilent.
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right.
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cases a; cases m.
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eapply notstuck_send in H1; [ | eauto | eauto ].
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first_order; eauto.
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eapply notstuck_recv in H1; [ | eauto | eauto ].
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first_order; eauto.
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invert H0; invert H1; simplify; eauto.
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Unshelve.
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assumption.
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Qed.
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