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SessionTypes: starting with a more basic version
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Adam Chlipala
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1 changed files with 166 additions and 34 deletions
200
SessionTypes.v
200
SessionTypes.v
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@ -9,6 +9,167 @@ Set Implicit Arguments.
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Set Asymmetric Patterns.
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(** * Basic Two-Party Session Types *)
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Module BasicTwoParty.
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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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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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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,
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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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| HtDone :
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hasty Done TDone.
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(** * Examples of typed processes *)
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(* Recall our first example from last chapter. *)
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Definition addN (k : nat) (input output : channel) : proc :=
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??input(n : nat);
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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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|| match goal with
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| [ |- hasty _ (match ?E with _ => _ end) ] => cases E
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| [ |- hasty (match ?E with _ => _ end) _ ] => cases E
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end).
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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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hasty.
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Qed.
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(** * Complementing types *)
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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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| TDone => TDone
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end.
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Definition add2_client (input output : channel) : proc :=
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!!input(42);
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??output(_ : nat);
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Done.
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Theorem add2_client_typed : forall input output,
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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 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; 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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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; invert 1; eauto.
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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; 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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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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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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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 => pr = (Done || Done)
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\/ exists pr', lstep pr Silent pr').
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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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invert H0; invert H1; simplify; eauto.
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Qed.
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Example adding_no_deadlock : forall k input output,
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input <> output
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-> invariantFor (trsys_of (addN k input output
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|| add2_client input output))
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(fun pr => pr = (Done || Done)
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\/ exists pr', lstep pr Silent pr').
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Proof.
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simplify.
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eapply no_deadlock with (t := (???input(nat); !!!output(nat); TDone)%st);
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hasty.
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Qed.
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End BasicTwoParty.
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(** * Two-Party Session Types *)
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Module TwoParty.
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@ -35,28 +196,12 @@ Inductive hasty : proc -> type -> Prop :=
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| HtDone :
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hasty Done TDone.
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(** * Examples of typed processes *)
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(* Recall our first example from last chapter. *)
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Definition addN (k : nat) (input output : channel) : proc :=
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??input(n : nat);
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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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|| match goal with
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| [ |- hasty _ (match ?E with _ => _ end) ] => cases E
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| [ |- hasty (match ?E with _ => _ end) _ ] => cases E
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end).
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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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hasty.
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Qed.
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(** * Complementing types *)
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Fixpoint complement (t : type) : type :=
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@ -66,17 +211,6 @@ Fixpoint complement (t : type) : type :=
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| TDone => TDone
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end.
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Definition add2_client (input output : channel) : proc :=
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!!input(42);
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??output(_ : nat);
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Done.
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Theorem add2_client_typed : forall input output,
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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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(** ** Example *)
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Section online_store.
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@ -207,7 +341,7 @@ Theorem no_deadlock : forall pr1 pr2 t,
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-> hasty pr2 (complement t)
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-> invariantFor (trsys_of (pr1 || pr2))
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(fun pr => pr = (Done || Done)
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\/ exists l' pr', lstep pr l' pr').
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\/ exists pr', lstep pr Silent pr').
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Proof.
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simplify.
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eapply invariant_weaken.
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@ -216,8 +350,6 @@ Proof.
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clear pr1 pr2 t H H0.
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simplify; first_order; subst.
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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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Example online_store_no_deadlock : forall request_product in_stock_or_not
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@ -230,7 +362,7 @@ Example online_store_no_deadlock : forall request_product in_stock_or_not
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send_payment_info payment_success add_review
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in_stock payment_checker))
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(fun pr => pr = (Done || Done)
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\/ exists l' pr', lstep pr l' pr').
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\/ exists pr', lstep pr Silent pr').
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Proof.
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simplify.
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eapply no_deadlock with (t := customer_type request_product in_stock_or_not
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@ -1145,12 +1277,12 @@ Section online_store_with_warehouse.
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{| Sender := Merchant';
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Receiver := Customer' |}.
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Example online_store_no_deadlock' : forall product payment_info in_stock good_infos,
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Example online_store_no_deadlock' : forall product payment_info payment_checker in_stock,
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NoDup [request_product; in_stock_or_not; send_payment_info; payment_success; add_review;
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merchant_to_warehouse; warehouse_to_merchant]
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-> invariantFor (trsys_of (customer' product payment_info
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|| (merchant' in_stock
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|| (warehouse good_infos || Done))))
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|| (merchant' payment_checker
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|| (warehouse in_stock || Done))))
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(fun pr => inert pr
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\/ exists pr', lstep pr Silent pr').
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Proof.
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