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SessionTypes: simplified and proved a key invariant
This commit is contained in:
Adam Chlipala
parent
91fc06122d
commit
b9893a0e92
2 changed files with 120 additions and 142 deletions
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@ -335,7 +335,7 @@ Definition add2_once (input output : channel) : proc :=
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* of the first process as the fancy *implementation* and the second process as
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* the simpler *specification*. *)
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Inductive R_add2 : proc -> proc -> Prop :=
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| Initial : forall input output,
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| Starting : forall input output,
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input <> output
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-> R_add2
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(New[input;output](ch); ??input(n : nat); !!ch(n + 1); Done
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258
SessionTypes.v
258
SessionTypes.v
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@ -14,14 +14,11 @@ Set Asymmetric Patterns.
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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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| TPar (t1 t2 : type)
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| TDup (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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Infix "||" := Par : st_scope.
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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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@ -29,116 +26,13 @@ Notation "??? ch ( A ) ; k" := (TRecv ch A k%st) (right associativity, at level
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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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Inductive ignoresChannel (ch : channel) : type -> Prop :=
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| IcSend : forall ch' A t,
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ch' <> ch
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-> ignoresChannel ch t
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-> ignoresChannel ch (TSend ch' A t)
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| IcRecv : forall ch' A t,
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ch' <> ch
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-> ignoresChannel ch t
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-> ignoresChannel ch (TRecv ch' A t)
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| IcPar : forall t1 t2,
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ignoresChannel ch t1
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-> ignoresChannel ch t2
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-> ignoresChannel ch (TPar t1 t2)
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| IcDup : forall t,
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ignoresChannel ch t
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-> ignoresChannel ch (TDup t)
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| IcDone :
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ignoresChannel ch TDone
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| IcInternalChoice : forall t1 t2,
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ignoresChannel ch t1
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-> ignoresChannel ch t2
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-> ignoresChannel ch (InternalChoice t1 t2)
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| IcExternalChoice : forall t1 t2,
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ignoresChannel ch t1
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-> ignoresChannel ch t2
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-> ignoresChannel ch (ExternalChoice t1 t2).
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Inductive hideChannel (ch : channel) : type -> type -> Prop :=
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| HideIgnored : forall t,
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ignoresChannel ch t
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-> hideChannel ch t t
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| HideExtSend1 : forall ch' A t1 t2 t',
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ch' <> ch
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-> ignoresChannel ch' t2
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-> hideChannel ch (TPar t1 t2) t'
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-> hideChannel ch (TPar (TSend ch' A t1) t2) (TSend ch' A t')
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| HideExtRecv1 : forall ch' A t1 t2 t',
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ch' <> ch
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-> ignoresChannel ch' t2
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-> hideChannel ch (TPar t1 t2) t'
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-> hideChannel ch (TPar (TRecv ch' A t1) t2) (TRecv ch' A t')
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| HideExtSend2 : forall ch' A t1 t2 t',
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ch' <> ch
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-> ignoresChannel ch' t2
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-> hideChannel ch (TPar t1 t2) t'
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-> hideChannel ch (TPar t1 (TSend ch' A t2)) (TSend ch' A t')
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| HideExtRecv2 : forall ch' A t1 t2 t',
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ch' <> ch
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-> ignoresChannel ch' t2
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-> hideChannel ch (TPar t1 t2) t'
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-> hideChannel ch (TPar t1 (TRecv ch' A t2)) (TRecv ch' A t')
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| HideRendezvous1 : forall A t1 t2 t',
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hideChannel ch (TPar t1 t2) t'
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-> hideChannel ch (TPar (TSend ch A t1) (TRecv ch A t2)) t'
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| HideRendezvous2 : forall A t1 t2 t',
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hideChannel ch (TPar t1 t2) t'
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-> hideChannel ch (TPar (TRecv ch A t1) (TSend ch A t2)) t'.
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Fixpoint shrink (t : type) : type :=
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match t with
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| TSend ch A t1 => TSend ch A (shrink t1)
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| TRecv ch A t1 => TRecv ch A (shrink t1)
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| TPar t1 t2 =>
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let t1' := shrink t1 in
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let t2' := shrink t2 in
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match t1', t2' with
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| TDone, _ => t2'
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| _, TDone => t1'
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| _, _ => TPar t1' t2'
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end
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| TDup t1 =>
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let t1' := shrink t1 in
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match t1' with
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| TDone => TDone
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| _ => TDup t1'
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end
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| TDone => TDone
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| InternalChoice t1 t2 => InternalChoice (shrink t1) (shrink t2)
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| ExternalChoice t1 t2 => ExternalChoice (shrink t1) (shrink t2)
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end.
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Inductive hasty : proc -> type -> Prop :=
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| HtNewChannel : forall notThese k t tc tcs,
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(forall ch, ~In ch notThese -> hasty (k ch) (t ch))
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-> (forall ch, ~In ch notThese -> hideChannel ch (t ch) tc)
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-> shrink tc = tcs
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-> hasty (NewChannel notThese k) tcs
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| HtBlockChannel : forall ch pr t tc tcs,
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hasty pr t
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-> hideChannel ch t tc
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-> shrink tc = tcs
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-> hasty (BlockChannel ch pr) tcs
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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,
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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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| HtPar : forall pr1 pr2 t1 t2,
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hasty pr1 t1
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-> hasty pr2 t2
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-> hasty (Par pr1 pr2) (TPar t1 t2)
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| HtDup : forall pr t,
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hasty pr t
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-> hasty (Dup pr) (TDup t)
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| HtDone :
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hasty Done TDone
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@ -162,37 +56,10 @@ Definition addN (k : nat) (input output : channel) : proc :=
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!!output(n + k);
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Done.
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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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Proof.
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simplify.
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repeat (constructor; simplify).
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Qed.
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Definition add2 (input output : channel) : proc :=
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Dup (New[input;output](intermediate);
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addN 1 input intermediate
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|| addN 1 intermediate output).
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Ltac hide1 := apply HideRendezvous1 || apply HideRendezvous2
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|| (eapply HideIgnored; repeat constructor; equality)
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|| (eapply HideExtSend1; [ equality | repeat constructor; equality | ])
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|| (eapply HideExtRecv1; [ equality | repeat constructor; equality | ])
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|| (eapply HideExtSend2; [ equality | repeat constructor; equality | ])
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|| (eapply HideExtRecv2; [ equality | repeat constructor; equality | ]).
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Ltac hide := repeat hide1.
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Ltac hasty1 :=
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match goal with
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| [ |- hasty _ _ ] => econstructor; simplify
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end.
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Ltac hasty := simplify; repeat hasty1; simplify; hide; try equality.
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Theorem add2_typed : forall input output,
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input <> output
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-> hasty (add2 input output) (TDup (???input(nat); !!!output(nat); TDone)).
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Proof.
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hasty.
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Qed.
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@ -204,8 +71,6 @@ 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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| TPar t1 t2 => TPar (complement t1) (complement t2)
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| TDup t1 => TDup (complement t1)
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| TDone => TDone
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| InternalChoice t1 t2 => ExternalChoice (complement t1) (complement t2)
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@ -213,13 +78,126 @@ Fixpoint complement (t : type) : type :=
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end.
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Definition add2_client (input output : channel) : proc :=
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Dup (!!input(42);
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!!input(42);
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??output(_ : nat);
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Done).
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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 (TDup (???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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(** * Parallel execution preserves the existence of complementary session types. *)
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Definition trsys_of pr := {|
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Initial := {pr};
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Step := lstepSilent
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|}.
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(* Note: here we force silent steps, so that all channel communication is
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* internal. *)
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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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Qed.
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Lemma output_typed : forall pr ch A v pr',
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lstep pr (Output {| Channel := ch; TypeOf := A; Value := 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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Qed.
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Lemma complementarity_forever : forall pr1 pr2 t,
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hasty pr1 t
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-> hasty pr2 (complement t)
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-> invariantFor (trsys_of (pr1 || pr2))
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(fun pr => exists pr1' pr2' t',
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pr = pr1' || pr2'
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/\ hasty pr1' t'
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/\ hasty pr2' (complement t')).
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Proof.
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simplify.
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apply invariant_induction; simplify.
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propositional; subst.
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eauto 6.
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clear pr1 pr2 t H H0.
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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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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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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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