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Rename typing relations from hasty to has_ty
This commit is contained in:
Adam Chlipala
parent
04be648f88
commit
ac0a15e9f2
4 changed files with 549 additions and 549 deletions
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@ -80,25 +80,25 @@ Module Stlc.
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| Nat (* Numbers *)
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| Fun (dom ran : type) (* Functions *).
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Inductive hasty : fmap var type -> exp -> type -> Prop :=
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Inductive has_ty : fmap var type -> exp -> type -> Prop :=
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| HtVar : forall G x t,
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G $? x = Some t
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-> hasty G (Var x) t
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-> has_ty G (Var x) t
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| HtConst : forall G n,
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hasty G (Const n) Nat
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has_ty G (Const n) Nat
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| HtPlus : forall G e1 e2,
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hasty G e1 Nat
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-> hasty G e2 Nat
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-> hasty G (Plus e1 e2) Nat
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has_ty G e1 Nat
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-> has_ty G e2 Nat
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-> has_ty G (Plus e1 e2) Nat
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| HtAbs : forall G x e1 t1 t2,
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hasty (G $+ (x, t1)) e1 t2
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-> hasty G (Abs x e1) (Fun t1 t2)
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has_ty (G $+ (x, t1)) e1 t2
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-> has_ty G (Abs x e1) (Fun t1 t2)
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| HtApp : forall G e1 e2 t1 t2,
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hasty G e1 (Fun t1 t2)
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-> hasty G e2 t1
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-> hasty G (App e1 e2) t2.
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has_ty G e1 (Fun t1 t2)
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-> has_ty G e2 t1
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-> has_ty G (App e1 e2) t2.
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Local Hint Constructors value plug step0 step hasty : core.
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Local Hint Constructors value plug step0 step has_ty : core.
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(** * Now we adapt the automated proof of type soundness. *)
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@ -110,15 +110,15 @@ Module Stlc.
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| [ H : step _ _ |- _ ] => invert H
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| [ H : step0 _ _ |- _ ] => invert1 H
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| [ H : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H
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| [ H : hasty _ _ _ |- _ ] => invert1 H
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| [ H : has_ty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H
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| [ H : has_ty _ _ _ |- _ ] => invert1 H
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| [ H : plug _ _ _ |- _ ] => invert1 H
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end; subst.
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Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 6.
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Lemma progress : forall e t,
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hasty $0 e t
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has_ty $0 e t
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-> value e
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\/ (exists e' : exp, step e e').
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Proof.
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@ -137,29 +137,29 @@ Module Stlc.
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Local Hint Resolve weakening_override : core.
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Lemma weakening : forall G e t,
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hasty G e t
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has_ty G e t
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-> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
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-> hasty G' e t.
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-> has_ty G' e t.
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Proof.
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induct 1; t.
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Qed.
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Local Hint Resolve weakening : core.
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Lemma hasty_change : forall G e t,
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hasty G e t
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Lemma has_ty_change : forall G e t,
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has_ty G e t
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-> forall G', G' = G
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-> hasty G' e t.
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-> has_ty G' e t.
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Proof.
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t.
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Qed.
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Local Hint Resolve hasty_change : core.
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Local Hint Resolve has_ty_change : core.
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Lemma substitution : forall G x t' e t e',
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hasty (G $+ (x, t')) e t
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-> hasty $0 e' t'
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-> hasty G (subst e' x e) t.
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has_ty (G $+ (x, t')) e t
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-> has_ty $0 e' t'
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-> has_ty G (subst e' x e) t.
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Proof.
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induct 1; t.
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Qed.
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@ -168,8 +168,8 @@ Module Stlc.
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Lemma preservation0 : forall e1 e2,
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step0 e1 e2
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-> forall t, hasty $0 e1 t
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-> hasty $0 e2 t.
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-> forall t, has_ty $0 e1 t
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-> has_ty $0 e2 t.
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Proof.
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invert 1; t.
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Qed.
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@ -180,8 +180,8 @@ Module Stlc.
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plug C e1 e1'
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-> forall e2 e2' t, plug C e2 e2'
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-> step0 e1 e2
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-> hasty $0 e1' t
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-> hasty $0 e2' t.
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-> has_ty $0 e1' t
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-> has_ty $0 e2' t.
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Proof.
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induct 1; t.
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Qed.
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@ -190,21 +190,21 @@ Module Stlc.
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Lemma preservation : forall e1 e2,
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step e1 e2
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-> forall t, hasty $0 e1 t
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-> hasty $0 e2 t.
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-> forall t, has_ty $0 e1 t
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-> has_ty $0 e2 t.
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Proof.
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invert 1; t.
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Qed.
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Local Hint Resolve progress preservation : core.
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Theorem safety : forall e t, hasty $0 e t
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Theorem safety : forall e t, has_ty $0 e t
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-> invariantFor (trsys_of e)
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(fun e' => value e'
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\/ exists e'', step e' e'').
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Proof.
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simplify.
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apply invariant_weaken with (invariant1 := fun e' => hasty $0 e' t); eauto.
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apply invariant_weaken with (invariant1 := fun e' => has_ty $0 e' t); eauto.
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apply invariant_induction; simplify; eauto; equality.
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Qed.
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@ -390,33 +390,33 @@ Module StlcPairs.
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| Fun (dom ran : type)
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| Prod (t1 t2 : type) (* "Prod" for "product," as in Cartesian product *).
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Inductive hasty : fmap var type -> exp -> type -> Prop :=
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Inductive has_ty : fmap var type -> exp -> type -> Prop :=
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| HtVar : forall G x t,
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G $? x = Some t
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-> hasty G (Var x) t
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-> has_ty G (Var x) t
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| HtConst : forall G n,
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hasty G (Const n) Nat
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has_ty G (Const n) Nat
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| HtPlus : forall G e1 e2,
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hasty G e1 Nat
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-> hasty G e2 Nat
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-> hasty G (Plus e1 e2) Nat
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has_ty G e1 Nat
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-> has_ty G e2 Nat
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-> has_ty G (Plus e1 e2) Nat
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| HtAbs : forall G x e1 t1 t2,
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hasty (G $+ (x, t1)) e1 t2
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-> hasty G (Abs x e1) (Fun t1 t2)
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has_ty (G $+ (x, t1)) e1 t2
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-> has_ty G (Abs x e1) (Fun t1 t2)
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| HtApp : forall G e1 e2 t1 t2,
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hasty G e1 (Fun t1 t2)
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-> hasty G e2 t1
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-> hasty G (App e1 e2) t2
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has_ty G e1 (Fun t1 t2)
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-> has_ty G e2 t1
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-> has_ty G (App e1 e2) t2
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| HtPair : forall G e1 e2 t1 t2,
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hasty G e1 t1
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-> hasty G e2 t2
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-> hasty G (Pair e1 e2) (Prod t1 t2)
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has_ty G e1 t1
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-> has_ty G e2 t2
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-> has_ty G (Pair e1 e2) (Prod t1 t2)
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| HtFst : forall G e1 t1 t2,
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hasty G e1 (Prod t1 t2)
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-> hasty G (Fst e1) t1
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has_ty G e1 (Prod t1 t2)
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-> has_ty G (Fst e1) t1
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| HtSnd : forall G e1 t1 t2,
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hasty G e1 (Prod t1 t2)
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-> hasty G (Snd e1) t2.
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has_ty G e1 (Prod t1 t2)
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-> has_ty G (Snd e1) t2.
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(* Let's copy and paste the type-soundness proof from above and adapt it. *)
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End StlcPairs.
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@ -567,46 +567,46 @@ Module StlcSums.
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(* New case: *)
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| Sum (t1 t2 : type).
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Inductive hasty : fmap var type -> exp -> type -> Prop :=
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Inductive has_ty : fmap var type -> exp -> type -> Prop :=
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| HtVar : forall G x t,
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G $? x = Some t
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-> hasty G (Var x) t
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-> has_ty G (Var x) t
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| HtConst : forall G n,
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hasty G (Const n) Nat
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has_ty G (Const n) Nat
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| HtPlus : forall G e1 e2,
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hasty G e1 Nat
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-> hasty G e2 Nat
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-> hasty G (Plus e1 e2) Nat
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has_ty G e1 Nat
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-> has_ty G e2 Nat
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-> has_ty G (Plus e1 e2) Nat
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| HtAbs : forall G x e1 t1 t2,
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hasty (G $+ (x, t1)) e1 t2
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-> hasty G (Abs x e1) (Fun t1 t2)
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has_ty (G $+ (x, t1)) e1 t2
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-> has_ty G (Abs x e1) (Fun t1 t2)
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| HtApp : forall G e1 e2 t1 t2,
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hasty G e1 (Fun t1 t2)
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-> hasty G e2 t1
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-> hasty G (App e1 e2) t2
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has_ty G e1 (Fun t1 t2)
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-> has_ty G e2 t1
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-> has_ty G (App e1 e2) t2
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| HtPair : forall G e1 e2 t1 t2,
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hasty G e1 t1
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-> hasty G e2 t2
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-> hasty G (Pair e1 e2) (Prod t1 t2)
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has_ty G e1 t1
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-> has_ty G e2 t2
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-> has_ty G (Pair e1 e2) (Prod t1 t2)
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| HtFst : forall G e1 t1 t2,
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hasty G e1 (Prod t1 t2)
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-> hasty G (Fst e1) t1
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has_ty G e1 (Prod t1 t2)
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-> has_ty G (Fst e1) t1
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| HtSnd : forall G e1 t1 t2,
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hasty G e1 (Prod t1 t2)
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-> hasty G (Snd e1) t2
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has_ty G e1 (Prod t1 t2)
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-> has_ty G (Snd e1) t2
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(* New cases: *)
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| HtInl : forall G e1 t1 t2,
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hasty G e1 t1
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-> hasty G (Inl e1) (Sum t1 t2)
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has_ty G e1 t1
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-> has_ty G (Inl e1) (Sum t1 t2)
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| HtInr : forall G e1 t1 t2,
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hasty G e1 t2
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-> hasty G (Inr e1) (Sum t1 t2)
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has_ty G e1 t2
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-> has_ty G (Inr e1) (Sum t1 t2)
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| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
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hasty G e (Sum t1 t2)
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-> hasty (G $+ (x1, t1)) e1 t
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-> hasty (G $+ (x2, t2)) e2 t
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-> hasty G (Match e x1 e1 x2 e2) t.
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has_ty G e (Sum t1 t2)
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-> has_ty (G $+ (x1, t1)) e1 t
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-> has_ty (G $+ (x2, t2)) e2 t
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-> has_ty G (Match e x1 e1 x2 e2) t.
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(* Type-soundness proof here *)
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End StlcSums.
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@ -777,53 +777,53 @@ Module StlcExceptions.
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| Prod (t1 t2 : type)
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| Sum (t1 t2 : type).
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Inductive hasty : fmap var type -> exp -> type -> Prop :=
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Inductive has_ty : fmap var type -> exp -> type -> Prop :=
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| HtVar : forall G x t,
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G $? x = Some t
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-> hasty G (Var x) t
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-> has_ty G (Var x) t
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| HtConst : forall G n,
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hasty G (Const n) Nat
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has_ty G (Const n) Nat
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| HtPlus : forall G e1 e2,
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hasty G e1 Nat
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-> hasty G e2 Nat
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-> hasty G (Plus e1 e2) Nat
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has_ty G e1 Nat
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-> has_ty G e2 Nat
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-> has_ty G (Plus e1 e2) Nat
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| HtAbs : forall G x e1 t1 t2,
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hasty (G $+ (x, t1)) e1 t2
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-> hasty G (Abs x e1) (Fun t1 t2)
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has_ty (G $+ (x, t1)) e1 t2
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-> has_ty G (Abs x e1) (Fun t1 t2)
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| HtApp : forall G e1 e2 t1 t2,
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hasty G e1 (Fun t1 t2)
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-> hasty G e2 t1
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-> hasty G (App e1 e2) t2
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has_ty G e1 (Fun t1 t2)
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-> has_ty G e2 t1
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-> has_ty G (App e1 e2) t2
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| HtPair : forall G e1 e2 t1 t2,
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hasty G e1 t1
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-> hasty G e2 t2
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-> hasty G (Pair e1 e2) (Prod t1 t2)
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has_ty G e1 t1
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-> has_ty G e2 t2
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-> has_ty G (Pair e1 e2) (Prod t1 t2)
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| HtFst : forall G e1 t1 t2,
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hasty G e1 (Prod t1 t2)
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-> hasty G (Fst e1) t1
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has_ty G e1 (Prod t1 t2)
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-> has_ty G (Fst e1) t1
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| HtSnd : forall G e1 t1 t2,
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hasty G e1 (Prod t1 t2)
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-> hasty G (Snd e1) t2
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has_ty G e1 (Prod t1 t2)
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-> has_ty G (Snd e1) t2
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| HtInl : forall G e1 t1 t2,
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hasty G e1 t1
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-> hasty G (Inl e1) (Sum t1 t2)
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has_ty G e1 t1
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-> has_ty G (Inl e1) (Sum t1 t2)
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| HtInr : forall G e1 t1 t2,
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hasty G e1 t2
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-> hasty G (Inr e1) (Sum t1 t2)
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has_ty G e1 t2
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-> has_ty G (Inr e1) (Sum t1 t2)
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| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
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hasty G e (Sum t1 t2)
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-> hasty (G $+ (x1, t1)) e1 t
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-> hasty (G $+ (x2, t2)) e2 t
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-> hasty G (Match e x1 e1 x2 e2) t
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has_ty G e (Sum t1 t2)
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-> has_ty (G $+ (x1, t1)) e1 t
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-> has_ty (G $+ (x2, t2)) e2 t
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-> has_ty G (Match e x1 e1 x2 e2) t
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(* New cases: *)
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| HtThrow : forall G e1 t,
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hasty G e1 Nat
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-> hasty G (Throw e1) t
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has_ty G e1 Nat
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-> has_ty G (Throw e1) t
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| HtCatch : forall G e x1 e1 t,
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hasty G e t
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-> hasty (G $+ (x1, Nat)) e1 t
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-> hasty G (Catch e x1 e1) t.
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has_ty G e t
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-> has_ty (G $+ (x1, Nat)) e1 t
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-> has_ty G (Catch e x1 e1) t.
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End StlcExceptions.
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(** * Mutable Variables *)
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@ -991,53 +991,53 @@ Module StlcMutable.
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(* Now there will be two typing contexts, one for mutable variables and one
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* for immutable. *)
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Inductive hasty (M : fmap var type) : fmap var type -> exp -> type -> Prop :=
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Inductive has_ty (M : fmap var type) : fmap var type -> exp -> type -> Prop :=
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| HtVar : forall G x t,
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G $? x = Some t
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-> hasty M G (Var x) t
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-> has_ty M G (Var x) t
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| HtConst : forall G n,
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hasty M G (Const n) Nat
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has_ty M G (Const n) Nat
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| HtPlus : forall G e1 e2,
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hasty M G e1 Nat
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-> hasty M G e2 Nat
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-> hasty M G (Plus e1 e2) Nat
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has_ty M G e1 Nat
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-> has_ty M G e2 Nat
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-> has_ty M G (Plus e1 e2) Nat
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| HtAbs : forall G x e1 t1 t2,
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hasty M (G $+ (x, t1)) e1 t2
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-> hasty M G (Abs x e1) (Fun t1 t2)
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has_ty M (G $+ (x, t1)) e1 t2
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-> has_ty M G (Abs x e1) (Fun t1 t2)
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| HtApp : forall G e1 e2 t1 t2,
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hasty M G e1 (Fun t1 t2)
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-> hasty M G e2 t1
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-> hasty M G (App e1 e2) t2
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has_ty M G e1 (Fun t1 t2)
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-> has_ty M G e2 t1
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-> has_ty M G (App e1 e2) t2
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| HtPair : forall G e1 e2 t1 t2,
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hasty M G e1 t1
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-> hasty M G e2 t2
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-> hasty M G (Pair e1 e2) (Prod t1 t2)
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has_ty M G e1 t1
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-> has_ty M G e2 t2
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-> has_ty M G (Pair e1 e2) (Prod t1 t2)
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| HtFst : forall G e1 t1 t2,
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hasty M G e1 (Prod t1 t2)
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-> hasty M G (Fst e1) t1
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has_ty M G e1 (Prod t1 t2)
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-> has_ty M G (Fst e1) t1
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| HtSnd : forall G e1 t1 t2,
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||||
hasty M G e1 (Prod t1 t2)
|
||||
-> hasty M G (Snd e1) t2
|
||||
has_ty M G e1 (Prod t1 t2)
|
||||
-> has_ty M G (Snd e1) t2
|
||||
| HtInl : forall G e1 t1 t2,
|
||||
hasty M G e1 t1
|
||||
-> hasty M G (Inl e1) (Sum t1 t2)
|
||||
has_ty M G e1 t1
|
||||
-> has_ty M G (Inl e1) (Sum t1 t2)
|
||||
| HtInr : forall G e1 t1 t2,
|
||||
hasty M G e1 t2
|
||||
-> hasty M G (Inr e1) (Sum t1 t2)
|
||||
has_ty M G e1 t2
|
||||
-> has_ty M G (Inr e1) (Sum t1 t2)
|
||||
| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
|
||||
hasty M G e (Sum t1 t2)
|
||||
-> hasty M (G $+ (x1, t1)) e1 t
|
||||
-> hasty M (G $+ (x2, t2)) e2 t
|
||||
-> hasty M G (Match e x1 e1 x2 e2) t
|
||||
has_ty M G e (Sum t1 t2)
|
||||
-> has_ty M (G $+ (x1, t1)) e1 t
|
||||
-> has_ty M (G $+ (x2, t2)) e2 t
|
||||
-> has_ty M G (Match e x1 e1 x2 e2) t
|
||||
|
||||
(* New cases: *)
|
||||
| HtGetVar : forall G x t,
|
||||
M $? x = Some t
|
||||
-> hasty M G (GetVar x) t
|
||||
-> has_ty M G (GetVar x) t
|
||||
| HtSetVar : forall G x e t,
|
||||
M $? x = Some t
|
||||
-> hasty M G e t
|
||||
-> hasty M G (SetVar x e) t.
|
||||
-> has_ty M G e t
|
||||
-> has_ty M G (SetVar x e) t.
|
||||
End StlcMutable.
|
||||
|
||||
(** * Concurrency *)
|
||||
|
|
@ -1218,57 +1218,57 @@ Module StlcConcur.
|
|||
| Prod (t1 t2 : type)
|
||||
| Sum (t1 t2 : type).
|
||||
|
||||
Inductive hasty (M : fmap var type) : fmap var type -> exp -> type -> Prop :=
|
||||
Inductive has_ty (M : fmap var type) : fmap var type -> exp -> type -> Prop :=
|
||||
| HtVar : forall G x t,
|
||||
G $? x = Some t
|
||||
-> hasty M G (Var x) t
|
||||
-> has_ty M G (Var x) t
|
||||
| HtConst : forall G n,
|
||||
hasty M G (Const n) Nat
|
||||
has_ty M G (Const n) Nat
|
||||
| HtPlus : forall G e1 e2,
|
||||
hasty M G e1 Nat
|
||||
-> hasty M G e2 Nat
|
||||
-> hasty M G (Plus e1 e2) Nat
|
||||
has_ty M G e1 Nat
|
||||
-> has_ty M G e2 Nat
|
||||
-> has_ty M G (Plus e1 e2) Nat
|
||||
| HtAbs : forall G x e1 t1 t2,
|
||||
hasty M (G $+ (x, t1)) e1 t2
|
||||
-> hasty M G (Abs x e1) (Fun t1 t2)
|
||||
has_ty M (G $+ (x, t1)) e1 t2
|
||||
-> has_ty M G (Abs x e1) (Fun t1 t2)
|
||||
| HtApp : forall G e1 e2 t1 t2,
|
||||
hasty M G e1 (Fun t1 t2)
|
||||
-> hasty M G e2 t1
|
||||
-> hasty M G (App e1 e2) t2
|
||||
has_ty M G e1 (Fun t1 t2)
|
||||
-> has_ty M G e2 t1
|
||||
-> has_ty M G (App e1 e2) t2
|
||||
| HtPair : forall G e1 e2 t1 t2,
|
||||
hasty M G e1 t1
|
||||
-> hasty M G e2 t2
|
||||
-> hasty M G (Pair e1 e2) (Prod t1 t2)
|
||||
has_ty M G e1 t1
|
||||
-> has_ty M G e2 t2
|
||||
-> has_ty M G (Pair e1 e2) (Prod t1 t2)
|
||||
| HtFst : forall G e1 t1 t2,
|
||||
hasty M G e1 (Prod t1 t2)
|
||||
-> hasty M G (Fst e1) t1
|
||||
has_ty M G e1 (Prod t1 t2)
|
||||
-> has_ty M G (Fst e1) t1
|
||||
| HtSnd : forall G e1 t1 t2,
|
||||
hasty M G e1 (Prod t1 t2)
|
||||
-> hasty M G (Snd e1) t2
|
||||
has_ty M G e1 (Prod t1 t2)
|
||||
-> has_ty M G (Snd e1) t2
|
||||
| HtInl : forall G e1 t1 t2,
|
||||
hasty M G e1 t1
|
||||
-> hasty M G (Inl e1) (Sum t1 t2)
|
||||
has_ty M G e1 t1
|
||||
-> has_ty M G (Inl e1) (Sum t1 t2)
|
||||
| HtInr : forall G e1 t1 t2,
|
||||
hasty M G e1 t2
|
||||
-> hasty M G (Inr e1) (Sum t1 t2)
|
||||
has_ty M G e1 t2
|
||||
-> has_ty M G (Inr e1) (Sum t1 t2)
|
||||
| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
|
||||
hasty M G e (Sum t1 t2)
|
||||
-> hasty M (G $+ (x1, t1)) e1 t
|
||||
-> hasty M (G $+ (x2, t2)) e2 t
|
||||
-> hasty M G (Match e x1 e1 x2 e2) t
|
||||
has_ty M G e (Sum t1 t2)
|
||||
-> has_ty M (G $+ (x1, t1)) e1 t
|
||||
-> has_ty M (G $+ (x2, t2)) e2 t
|
||||
-> has_ty M G (Match e x1 e1 x2 e2) t
|
||||
| HtGetVar : forall G x t,
|
||||
M $? x = Some t
|
||||
-> hasty M G (GetVar x) t
|
||||
-> has_ty M G (GetVar x) t
|
||||
| HtSetVar : forall G x e t,
|
||||
M $? x = Some t
|
||||
-> hasty M G e t
|
||||
-> hasty M G (SetVar x e) t
|
||||
-> has_ty M G e t
|
||||
-> has_ty M G (SetVar x e) t
|
||||
|
||||
(* New cases: *)
|
||||
| HtPar : forall G e1 t1 e2 t2,
|
||||
hasty M G e1 t1
|
||||
-> hasty M G e2 t2
|
||||
-> hasty M G (Par e1 e2) (Prod t1 t2).
|
||||
has_ty M G e1 t1
|
||||
-> has_ty M G e2 t2
|
||||
-> has_ty M G (Par e1 e2) (Prod t1 t2).
|
||||
|
||||
(* Type-soundness proof *)
|
||||
End StlcConcur.
|
||||
|
|
|
|||
|
|
@ -477,25 +477,25 @@ Module Stlc.
|
|||
* (not to be confused with evaluation contexts) to map free variables to
|
||||
* their types. Free variables are those that don't refer to enclosing [Abs]
|
||||
* binders. *)
|
||||
Inductive hasty : fmap var type -> exp -> type -> Prop :=
|
||||
Inductive has_ty : fmap var type -> exp -> type -> Prop :=
|
||||
| HtVar : forall G x t,
|
||||
G $? x = Some t
|
||||
-> hasty G (Var x) t
|
||||
-> has_ty G (Var x) t
|
||||
| HtConst : forall G n,
|
||||
hasty G (Const n) Nat
|
||||
has_ty G (Const n) Nat
|
||||
| HtPlus : forall G e1 e2,
|
||||
hasty G e1 Nat
|
||||
-> hasty G e2 Nat
|
||||
-> hasty G (Plus e1 e2) Nat
|
||||
has_ty G e1 Nat
|
||||
-> has_ty G e2 Nat
|
||||
-> has_ty G (Plus e1 e2) Nat
|
||||
| HtAbs : forall G x e1 t1 t2,
|
||||
hasty (G $+ (x, t1)) e1 t2
|
||||
-> hasty G (Abs x e1) (Fun t1 t2)
|
||||
has_ty (G $+ (x, t1)) e1 t2
|
||||
-> has_ty G (Abs x e1) (Fun t1 t2)
|
||||
| HtApp : forall G e1 e2 t1 t2,
|
||||
hasty G e1 (Fun t1 t2)
|
||||
-> hasty G e2 t1
|
||||
-> hasty G (App e1 e2) t2.
|
||||
has_ty G e1 (Fun t1 t2)
|
||||
-> has_ty G e2 t1
|
||||
-> has_ty G (App e1 e2) t2.
|
||||
|
||||
Local Hint Constructors value step hasty : core.
|
||||
Local Hint Constructors value step has_ty : core.
|
||||
|
||||
(* Some notation to make it more pleasant to write programs *)
|
||||
Infix "-->" := Fun (at level 60, right associativity).
|
||||
|
|
@ -507,22 +507,22 @@ Module Stlc.
|
|||
|
||||
(* Some examples of typed programs *)
|
||||
|
||||
Example one_plus_one : hasty $0 (1 ^+^ 1) Nat.
|
||||
Example one_plus_one : has_ty $0 (1 ^+^ 1) Nat.
|
||||
Proof.
|
||||
repeat (econstructor; simplify).
|
||||
Qed.
|
||||
|
||||
Example add : hasty $0 (\"n", \"m", "n" ^+^ "m") (Nat --> Nat --> Nat).
|
||||
Example add : has_ty $0 (\"n", \"m", "n" ^+^ "m") (Nat --> Nat --> Nat).
|
||||
Proof.
|
||||
repeat (econstructor; simplify).
|
||||
Qed.
|
||||
|
||||
Example eleven : hasty $0 ((\"n", \"m", "n" ^+^ "m") @ 7 @ 4) Nat.
|
||||
Example eleven : has_ty $0 ((\"n", \"m", "n" ^+^ "m") @ 7 @ 4) Nat.
|
||||
Proof.
|
||||
repeat (econstructor; simplify).
|
||||
Qed.
|
||||
|
||||
Example seven_the_long_way : hasty $0 ((\"x", "x") @ (\"x", "x") @ 7) Nat.
|
||||
Example seven_the_long_way : has_ty $0 ((\"x", "x") @ (\"x", "x") @ 7) Nat.
|
||||
Proof.
|
||||
repeat (econstructor; simplify).
|
||||
Qed.
|
||||
|
|
@ -542,7 +542,7 @@ Module Stlc.
|
|||
(* Now we're ready for the first of the two key properties to establish that
|
||||
* invariant: well-typed programs are never stuck. *)
|
||||
Lemma progress : forall e t,
|
||||
hasty $0 e t
|
||||
has_ty $0 e t
|
||||
-> unstuck e.
|
||||
Proof.
|
||||
unfold unstuck; induct 1; simplify; try equality.
|
||||
|
|
@ -556,10 +556,10 @@ Module Stlc.
|
|||
(* Some automation is needed here to maintain compatibility with
|
||||
* name generation in different Coq versions. *)
|
||||
match goal with
|
||||
| [ H1 : value e1, H2 : hasty $0 e1 _ |- _ ] => invert H1; invert H2
|
||||
| [ H1 : value e1, H2 : has_ty $0 e1 _ |- _ ] => invert H1; invert H2
|
||||
end.
|
||||
match goal with
|
||||
| [ H1 : value e2, H2 : hasty $0 e2 _ |- _ ] => invert H1; invert H2
|
||||
| [ H1 : value e2, H2 : has_ty $0 e2 _ |- _ ] => invert H1; invert H2
|
||||
end.
|
||||
exists (Const (n + n0)).
|
||||
constructor.
|
||||
|
|
@ -596,7 +596,7 @@ Module Stlc.
|
|||
|
||||
right.
|
||||
match goal with
|
||||
| [ H1 : value e1, H2 : hasty $0 e1 _ |- _ ] => invert H1; invert H2
|
||||
| [ H1 : value e1, H2 : has_ty $0 e1 _ |- _ ] => invert H1; invert H2
|
||||
end.
|
||||
exists (subst e2 x e0).
|
||||
constructor.
|
||||
|
|
@ -643,9 +643,9 @@ Module Stlc.
|
|||
(* This lemma lets us transplant a typing derivation into a new context that
|
||||
* includes the old one. *)
|
||||
Lemma weakening : forall G e t,
|
||||
hasty G e t
|
||||
has_ty G e t
|
||||
-> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
|
||||
-> hasty G' e t.
|
||||
-> has_ty G' e t.
|
||||
Proof.
|
||||
induct 1; simplify.
|
||||
|
||||
|
|
@ -656,28 +656,28 @@ Module Stlc.
|
|||
constructor.
|
||||
|
||||
constructor.
|
||||
apply IHhasty1.
|
||||
apply IHhas_ty1.
|
||||
assumption.
|
||||
apply IHhasty2.
|
||||
apply IHhas_ty2.
|
||||
assumption.
|
||||
|
||||
constructor.
|
||||
apply IHhasty.
|
||||
apply IHhas_ty.
|
||||
apply weakening_override.
|
||||
assumption.
|
||||
|
||||
econstructor.
|
||||
apply IHhasty1.
|
||||
apply IHhas_ty1.
|
||||
assumption.
|
||||
apply IHhasty2.
|
||||
apply IHhas_ty2.
|
||||
assumption.
|
||||
Qed.
|
||||
|
||||
(* Replacing a variable with a properly typed term preserves typing. *)
|
||||
Lemma substitution : forall G x t' e t e',
|
||||
hasty (G $+ (x, t')) e t
|
||||
-> hasty $0 e' t'
|
||||
-> hasty G (subst e' x e) t.
|
||||
has_ty (G $+ (x, t')) e t
|
||||
-> has_ty $0 e' t'
|
||||
-> has_ty G (subst e' x e) t.
|
||||
Proof.
|
||||
induct 1; simplify.
|
||||
|
||||
|
|
@ -697,8 +697,8 @@ Module Stlc.
|
|||
constructor.
|
||||
|
||||
constructor.
|
||||
eapply IHhasty1; equality.
|
||||
eapply IHhasty2; equality.
|
||||
eapply IHhas_ty1; equality.
|
||||
eapply IHhas_ty2; equality.
|
||||
|
||||
cases (x0 ==v x).
|
||||
|
||||
|
|
@ -715,20 +715,20 @@ Module Stlc.
|
|||
assumption.
|
||||
|
||||
constructor.
|
||||
eapply IHhasty.
|
||||
eapply IHhas_ty.
|
||||
maps_equal.
|
||||
assumption.
|
||||
|
||||
econstructor.
|
||||
eapply IHhasty1; equality.
|
||||
eapply IHhasty2; equality.
|
||||
eapply IHhas_ty1; equality.
|
||||
eapply IHhas_ty2; equality.
|
||||
Qed.
|
||||
|
||||
(* OK, now we're almost done. Full steps really do preserve typing! *)
|
||||
Lemma preservation : forall e1 e2,
|
||||
step e1 e2
|
||||
-> forall t, hasty $0 e1 t
|
||||
-> hasty $0 e2 t.
|
||||
-> forall t, has_ty $0 e1 t
|
||||
-> has_ty $0 e2 t.
|
||||
Proof.
|
||||
induct 1; simplify.
|
||||
|
||||
|
|
@ -770,7 +770,7 @@ Module Stlc.
|
|||
* presented from scratch as a proof technique, it turns out that the two key
|
||||
* properties, progress and preservation, are just instances of the same methods
|
||||
* we've been applying all along with invariants of transition systems! *)
|
||||
Theorem safety : forall e t, hasty $0 e t
|
||||
Theorem safety : forall e t, has_ty $0 e t
|
||||
-> invariantFor (trsys_of e) unstuck.
|
||||
Proof.
|
||||
simplify.
|
||||
|
|
@ -778,7 +778,7 @@ Module Stlc.
|
|||
(* Step 1: strengthen the invariant. In particular, the typing relation is
|
||||
* exactly the right stronger invariant! Our progress theorem proves the
|
||||
* required invariant inclusion. *)
|
||||
apply invariant_weaken with (invariant1 := fun e' => hasty $0 e' t).
|
||||
apply invariant_weaken with (invariant1 := fun e' => has_ty $0 e' t).
|
||||
|
||||
(* Step 2: apply invariant induction, whose induction step turns out to match
|
||||
* our preservation theorem exactly! *)
|
||||
|
|
@ -805,14 +805,14 @@ Module Stlc.
|
|||
| [ H : Some _ = Some _ |- _ ] => invert H
|
||||
|
||||
| [ H : step _ _ |- _ ] => invert1 H
|
||||
| [ H : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H
|
||||
| [ H : hasty _ _ _ |- _ ] => invert1 H
|
||||
| [ H : has_ty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H
|
||||
| [ H : has_ty _ _ _ |- _ ] => invert1 H
|
||||
end; subst.
|
||||
|
||||
Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 6.
|
||||
|
||||
Lemma progress_snazzy : forall e t,
|
||||
hasty $0 e t
|
||||
has_ty $0 e t
|
||||
-> value e
|
||||
\/ (exists e' : exp, step e e').
|
||||
Proof.
|
||||
|
|
@ -822,9 +822,9 @@ Module Stlc.
|
|||
Local Hint Resolve weakening_override : core.
|
||||
|
||||
Lemma weakening_snazzy : forall G e t,
|
||||
hasty G e t
|
||||
has_ty G e t
|
||||
-> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
|
||||
-> hasty G' e t.
|
||||
-> has_ty G' e t.
|
||||
Proof.
|
||||
induct 1; t.
|
||||
Qed.
|
||||
|
|
@ -833,20 +833,20 @@ Module Stlc.
|
|||
|
||||
(* Replacing a typing context with an equal one has no effect (useful to guide
|
||||
* proof search as a hint). *)
|
||||
Lemma hasty_change : forall G e t,
|
||||
hasty G e t
|
||||
Lemma has_ty_change : forall G e t,
|
||||
has_ty G e t
|
||||
-> forall G', G' = G
|
||||
-> hasty G' e t.
|
||||
-> has_ty G' e t.
|
||||
Proof.
|
||||
t.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve hasty_change : core.
|
||||
Local Hint Resolve has_ty_change : core.
|
||||
|
||||
Lemma substitution_snazzy : forall G x t' e t e',
|
||||
hasty (G $+ (x, t')) e t
|
||||
-> hasty $0 e' t'
|
||||
-> hasty G (subst e' x e) t.
|
||||
has_ty (G $+ (x, t')) e t
|
||||
-> has_ty $0 e' t'
|
||||
-> has_ty G (subst e' x e) t.
|
||||
Proof.
|
||||
induct 1; t.
|
||||
Qed.
|
||||
|
|
@ -855,21 +855,21 @@ Module Stlc.
|
|||
|
||||
Lemma preservation_snazzy : forall e1 e2,
|
||||
step e1 e2
|
||||
-> forall t, hasty $0 e1 t
|
||||
-> hasty $0 e2 t.
|
||||
-> forall t, has_ty $0 e1 t
|
||||
-> has_ty $0 e2 t.
|
||||
Proof.
|
||||
induct 1; t.
|
||||
Qed.
|
||||
|
||||
Local Hint Resolve progress_snazzy preservation_snazzy : core.
|
||||
|
||||
Theorem safety_snazzy : forall e t, hasty $0 e t
|
||||
Theorem safety_snazzy : forall e t, has_ty $0 e t
|
||||
-> invariantFor (trsys_of e)
|
||||
(fun e' => value e'
|
||||
\/ exists e'', step e' e'').
|
||||
Proof.
|
||||
simplify.
|
||||
apply invariant_weaken with (invariant1 := fun e' => hasty $0 e' t); eauto.
|
||||
apply invariant_weaken with (invariant1 := fun e' => has_ty $0 e' t); eauto.
|
||||
apply invariant_induction; simplify; eauto; equality.
|
||||
Qed.
|
||||
End Stlc.
|
||||
|
|
|
|||
|
|
@ -419,25 +419,25 @@ Module Stlc.
|
|||
| Nat (* Numbers *)
|
||||
| Fun (dom ran : type) (* Functions *).
|
||||
|
||||
Inductive hasty : fmap var type -> exp -> type -> Prop :=
|
||||
Inductive has_ty : fmap var type -> exp -> type -> Prop :=
|
||||
| HtVar : forall G x t,
|
||||
G $? x = Some t
|
||||
-> hasty G (Var x) t
|
||||
-> has_ty G (Var x) t
|
||||
| HtConst : forall G n,
|
||||
hasty G (Const n) Nat
|
||||
has_ty G (Const n) Nat
|
||||
| HtPlus : forall G e1 e2,
|
||||
hasty G e1 Nat
|
||||
-> hasty G e2 Nat
|
||||
-> hasty G (Plus e1 e2) Nat
|
||||
has_ty G e1 Nat
|
||||
-> has_ty G e2 Nat
|
||||
-> has_ty G (Plus e1 e2) Nat
|
||||
| HtAbs : forall G x e1 t1 t2,
|
||||
hasty (G $+ (x, t1)) e1 t2
|
||||
-> hasty G (Abs x e1) (Fun t1 t2)
|
||||
has_ty (G $+ (x, t1)) e1 t2
|
||||
-> has_ty G (Abs x e1) (Fun t1 t2)
|
||||
| HtApp : forall G e1 e2 t1 t2,
|
||||
hasty G e1 (Fun t1 t2)
|
||||
-> hasty G e2 t1
|
||||
-> hasty G (App e1 e2) t2.
|
||||
has_ty G e1 (Fun t1 t2)
|
||||
-> has_ty G e2 t1
|
||||
-> has_ty G (App e1 e2) t2.
|
||||
|
||||
Local Hint Constructors value step hasty : core.
|
||||
Local Hint Constructors value step has_ty : core.
|
||||
|
||||
(* Some notation to make it more pleasant to write programs *)
|
||||
Infix "-->" := Fun (at level 60, right associativity).
|
||||
|
|
@ -449,22 +449,22 @@ Module Stlc.
|
|||
|
||||
(* Some examples of typed programs *)
|
||||
|
||||
Example one_plus_one : hasty $0 (1 ^+^ 1) Nat.
|
||||
Example one_plus_one : has_ty $0 (1 ^+^ 1) Nat.
|
||||
Proof.
|
||||
repeat (econstructor; simplify).
|
||||
Qed.
|
||||
|
||||
Example add : hasty $0 (\"n", \"m", "n" ^+^ "m") (Nat --> Nat --> Nat).
|
||||
Example add : has_ty $0 (\"n", \"m", "n" ^+^ "m") (Nat --> Nat --> Nat).
|
||||
Proof.
|
||||
repeat (econstructor; simplify).
|
||||
Qed.
|
||||
|
||||
Example eleven : hasty $0 ((\"n", \"m", "n" ^+^ "m") @ 7 @ 4) Nat.
|
||||
Example eleven : has_ty $0 ((\"n", \"m", "n" ^+^ "m") @ 7 @ 4) Nat.
|
||||
Proof.
|
||||
repeat (econstructor; simplify).
|
||||
Qed.
|
||||
|
||||
Example seven_the_long_way : hasty $0 ((\"x", "x") @ (\"x", "x") @ 7) Nat.
|
||||
Example seven_the_long_way : has_ty $0 ((\"x", "x") @ (\"x", "x") @ 7) Nat.
|
||||
Proof.
|
||||
repeat (econstructor; simplify).
|
||||
Qed.
|
||||
|
|
@ -476,7 +476,7 @@ Module Stlc.
|
|||
\/ (exists e' : exp, step e e').
|
||||
|
||||
Lemma progress : forall e t,
|
||||
hasty $0 e t
|
||||
has_ty $0 e t
|
||||
-> value e
|
||||
\/ (exists e' : exp, step e e').
|
||||
Proof.
|
||||
|
|
@ -484,23 +484,23 @@ Module Stlc.
|
|||
|
||||
(* Replacing a typing context with an equal one has no effect (useful to guide
|
||||
* proof search as a hint). *)
|
||||
Lemma hasty_change : forall G e t,
|
||||
hasty G e t
|
||||
Lemma has_ty_change : forall G e t,
|
||||
has_ty G e t
|
||||
-> forall G', G' = G
|
||||
-> hasty G' e t.
|
||||
-> has_ty G' e t.
|
||||
Proof.
|
||||
Admitted.
|
||||
|
||||
Local Hint Resolve hasty_change : core.
|
||||
Local Hint Resolve has_ty_change : core.
|
||||
|
||||
Lemma preservation : forall e1 e2,
|
||||
step e1 e2
|
||||
-> forall t, hasty $0 e1 t
|
||||
-> hasty $0 e2 t.
|
||||
-> forall t, has_ty $0 e1 t
|
||||
-> has_ty $0 e2 t.
|
||||
Proof.
|
||||
Admitted.
|
||||
|
||||
Theorem safety : forall e t, hasty $0 e t
|
||||
Theorem safety : forall e t, has_ty $0 e t
|
||||
-> invariantFor (trsys_of e) unstuck.
|
||||
Proof.
|
||||
simplify.
|
||||
|
|
@ -508,7 +508,7 @@ Module Stlc.
|
|||
(* Step 1: strengthen the invariant. In particular, the typing relation is
|
||||
* exactly the right stronger invariant! Our progress theorem proves the
|
||||
* required invariant inclusion. *)
|
||||
apply invariant_weaken with (invariant1 := fun e' => hasty $0 e' t).
|
||||
apply invariant_weaken with (invariant1 := fun e' => has_ty $0 e' t).
|
||||
|
||||
(* Step 2: apply invariant induction, whose induction step turns out to match
|
||||
* our preservation theorem exactly! *)
|
||||
|
|
|
|||
Loading…
Reference in a new issue