Start of SessionTypes

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Adam Chlipala 2018-05-12 14:53:37 -04:00
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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 20: Session Types
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Require Import Frap MessagesAndRefinement.
Set Implicit Arguments.
Set Asymmetric Patterns.
(** * Defining the Type System *)
Inductive type :=
| TSend (ch : channel) (A : Set) (t : type)
| TRecv (ch : channel) (A : Set) (t : type)
| TPar (t1 t2 : type)
| TDup (t : type)
| TDone
| InternalChoice (t1 t2 : type)
| ExternalChoice (t1 t2 : type).
Infix "||" := Par : st_scope.
Delimit Scope st_scope with st.
Bind Scope st_scope with type.
Notation "!!! ch ( A ) ; k" := (TSend ch A k%st) (right associativity, at level 45, ch at level 0) : st_scope.
Notation "??? ch ( A ) ; k" := (TRecv ch A k%st) (right associativity, at level 45, ch at level 0) : st_scope.
Infix "|?|" := InternalChoice (at level 40) : st_scope.
Infix "?|?" := ExternalChoice (at level 40) : st_scope.
Inductive ignoresChannel (ch : channel) : type -> Prop :=
| IcSend : forall ch' A t,
ch' <> ch
-> ignoresChannel ch t
-> ignoresChannel ch (TSend ch' A t)
| IcRecv : forall ch' A t,
ch' <> ch
-> ignoresChannel ch t
-> ignoresChannel ch (TRecv ch' A t)
| IcPar : forall t1 t2,
ignoresChannel ch t1
-> ignoresChannel ch t2
-> ignoresChannel ch (TPar t1 t2)
| IcDup : forall t,
ignoresChannel ch t
-> ignoresChannel ch (TDup t)
| IcDone :
ignoresChannel ch TDone
| IcInternalChoice : forall t1 t2,
ignoresChannel ch t1
-> ignoresChannel ch t2
-> ignoresChannel ch (InternalChoice t1 t2)
| IcExternalChoice : forall t1 t2,
ignoresChannel ch t1
-> ignoresChannel ch t2
-> ignoresChannel ch (ExternalChoice t1 t2).
Inductive hideChannel (ch : channel) : type -> type -> Prop :=
| HideIgnored : forall t,
ignoresChannel ch t
-> hideChannel ch t t
| HideExtSend1 : forall ch' A t1 t2 t',
ch' <> ch
-> ignoresChannel ch' t2
-> hideChannel ch (TPar t1 t2) t'
-> hideChannel ch (TPar (TSend ch' A t1) t2) (TSend ch' A t')
| HideExtRecv1 : forall ch' A t1 t2 t',
ch' <> ch
-> ignoresChannel ch' t2
-> hideChannel ch (TPar t1 t2) t'
-> hideChannel ch (TPar (TRecv ch' A t1) t2) (TRecv ch' A t')
| HideExtSend2 : forall ch' A t1 t2 t',
ch' <> ch
-> ignoresChannel ch' t2
-> hideChannel ch (TPar t1 t2) t'
-> hideChannel ch (TPar t1 (TSend ch' A t2)) (TSend ch' A t')
| HideExtRecv2 : forall ch' A t1 t2 t',
ch' <> ch
-> ignoresChannel ch' t2
-> hideChannel ch (TPar t1 t2) t'
-> hideChannel ch (TPar t1 (TRecv ch' A t2)) (TRecv ch' A t')
| HideRendezvous1 : forall A t1 t2 t',
hideChannel ch (TPar t1 t2) t'
-> hideChannel ch (TPar (TSend ch A t1) (TRecv ch A t2)) t'
| HideRendezvous2 : forall A t1 t2 t',
hideChannel ch (TPar t1 t2) t'
-> hideChannel ch (TPar (TRecv ch A t1) (TSend ch A t2)) t'.
Fixpoint shrink (t : type) : type :=
match t with
| TSend ch A t1 => TSend ch A (shrink t1)
| TRecv ch A t1 => TRecv ch A (shrink t1)
| TPar t1 t2 =>
let t1' := shrink t1 in
let t2' := shrink t2 in
match t1', t2' with
| TDone, _ => t2'
| _, TDone => t1'
| _, _ => TPar t1' t2'
end
| TDup t1 =>
let t1' := shrink t1 in
match t1' with
| TDone => TDone
| _ => TDup t1'
end
| TDone => TDone
| InternalChoice t1 t2 => InternalChoice (shrink t1) (shrink t2)
| ExternalChoice t1 t2 => ExternalChoice (shrink t1) (shrink t2)
end.
Inductive hasty : proc -> type -> Prop :=
| HtNewChannel : forall notThese k t tc tcs,
(forall ch, ~In ch notThese -> hasty (k ch) (t ch))
-> (forall ch, ~In ch notThese -> hideChannel ch (t ch) tc)
-> shrink tc = tcs
-> hasty (NewChannel notThese k) tcs
| HtBlockChannel : forall ch pr t tc tcs,
hasty pr t
-> hideChannel ch t tc
-> shrink tc = tcs
-> hasty (BlockChannel ch pr) tcs
| HtSend : forall ch (A : Set) (v : A) k t,
hasty k t
-> hasty (Send ch v k) (TSend ch A t)
| HtRecv : forall ch (A : Set) (k : A -> _) t,
(forall v, hasty (k v) t)
-> hasty (Recv ch k) (TRecv ch A t)
| HtPar : forall pr1 pr2 t1 t2,
hasty pr1 t1
-> hasty pr2 t2
-> hasty (Par pr1 pr2) (TPar t1 t2)
| HtDup : forall pr t,
hasty pr t
-> hasty (Dup pr) (TDup t)
| HtDone :
hasty Done TDone
| HtInternalChoice1 : forall pr t1 t2,
hasty pr t1
-> hasty pr (InternalChoice t1 t2)
| HtInternalChoice2 : forall pr t1 t2,
hasty pr t2
-> hasty pr (InternalChoice t1 t2)
| HtExternalChoice : forall pr t1 t2,
hasty pr t1
-> hasty pr t2
-> hasty pr (ExternalChoice t1 t2).
(** * Examples of Typed Processes *)
(* Recall our first example from last chapter. *)
Definition addN (k : nat) (input output : channel) : proc :=
??input(n : nat);
!!output(n + k);
Done.
Theorem addN_typed : forall k input output,
hasty (addN k input output) (???input(nat); !!!output(nat); TDone).
Proof.
simplify.
repeat (constructor; simplify).
Qed.
Definition add2 (input output : channel) : proc :=
Dup (New[input;output](intermediate);
addN 1 input intermediate
|| addN 1 intermediate output).
Ltac hide1 := apply HideRendezvous1 || apply HideRendezvous2
|| (eapply HideIgnored; repeat constructor; equality)
|| (eapply HideExtSend1; [ equality | repeat constructor; equality | ])
|| (eapply HideExtRecv1; [ equality | repeat constructor; equality | ])
|| (eapply HideExtSend2; [ equality | repeat constructor; equality | ])
|| (eapply HideExtRecv2; [ equality | repeat constructor; equality | ]).
Ltac hide := repeat hide1.
Ltac hasty1 :=
match goal with
| [ |- hasty _ _ ] => econstructor; simplify
end.
Ltac hasty := simplify; repeat hasty1; simplify; hide; try equality.
Theorem add2_typed : forall input output,
input <> output
-> hasty (add2 input output) (TDup (???input(nat); !!!output(nat); TDone)).
Proof.
hasty.
Qed.
(** * Complementing Types *)
Fixpoint complement (t : type) : type :=
match t with
| TSend ch A t1 => TRecv ch A (complement t1)
| TRecv ch A t1 => TSend ch A (complement t1)
| TPar t1 t2 => TPar (complement t1) (complement t2)
| TDup t1 => TDup (complement t1)
| TDone => TDone
| InternalChoice t1 t2 => ExternalChoice (complement t1) (complement t2)
| ExternalChoice t1 t2 => InternalChoice (complement t1) (complement t2)
end.
Definition add2_client (input output : channel) : proc :=
Dup (!!input(42);
??output(_ : nat);
Done).
Theorem add2_client_typed : forall input output,
input <> output
-> hasty (add2_client input output) (complement (TDup (???input(nat); !!!output(nat); TDone))).
Proof.
hasty.
Qed.