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EvaluationContexts: factored step0 into step0 and step1
This commit is contained in:
Adam Chlipala
parent
415aa99b88
commit
544e7fa500
1 changed files with 90 additions and 58 deletions
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@ -977,39 +977,46 @@ Module StlcMutable.
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-> plug (SetVar1 x C) e (SetVar x e').
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Definition valuation := fmap var exp.
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Inductive step0 : valuation * exp -> valuation * exp -> Prop :=
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| Beta : forall env x e v,
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Inductive step0 : exp -> exp -> Prop :=
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| Beta : forall x e v,
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value v
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-> step0 (env, App (Abs x e) v) (env, subst v x e)
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| Add : forall env n1 n2,
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step0 (env, Plus (Const n1) (Const n2)) (env, Const (n1 + n2))
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| FstPair : forall env v1 v2,
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-> step0 (App (Abs x e) v) (subst v x e)
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| Add : forall n1 n2,
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step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
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| FstPair : forall v1 v2,
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value v1
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-> value v2
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-> step0 (env, Fst (Pair v1 v2)) (env, v1)
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| SndPair : forall env v1 v2,
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-> step0 (Fst (Pair v1 v2)) v1
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| SndPair : forall v1 v2,
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value v1
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-> value v2
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-> step0 (env, Snd (Pair v1 v2)) (env, v2)
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| MatchInl : forall env v x1 e1 x2 e2,
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-> step0 (Snd (Pair v1 v2)) v2
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| MatchInl : forall v x1 e1 x2 e2,
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value v
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-> step0 (env, Match (Inl v) x1 e1 x2 e2) (env, subst v x1 e1)
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| MatchInr : forall env v x1 e1 x2 e2,
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-> step0 (Match (Inl v) x1 e1 x2 e2) (subst v x1 e1)
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| MatchInr : forall v x1 e1 x2 e2,
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value v
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-> step0 (env, Match (Inr v) x1 e1 x2 e2) (env, subst v x2 e2)
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-> step0 (Match (Inr v) x1 e1 x2 e2) (subst v x2 e2).
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(* We build an intermediate relation on top of [step0], allowing for
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* consultation of the mutable-variable valuation. *)
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Inductive step1 : valuation * exp -> valuation * exp -> Prop :=
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| Lift : forall env e1 e2,
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step0 e1 e2
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-> step1 (env, e1) (env, e2)
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| Read : forall env x v,
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env $? x = Some v
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-> step0 (env, GetVar x) (env, v)
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-> step1 (env, GetVar x) (env, v)
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| Overwrite : forall env x v,
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value v
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-> step0 (env, SetVar x v) (env $+ (x, v), v).
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-> step1 (env, SetVar x v) (env $+ (x, v), v).
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Inductive step : valuation * exp -> valuation * exp -> Prop :=
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| StepRule : forall C env1 e1 env2 e2 e1' e2',
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plug C e1 e1'
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-> plug C e2 e2'
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-> step0 (env1, e1) (env2, e2)
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-> step1 (env1, e1) (env2, e2)
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-> step (env1, e1') (env2, e2').
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Definition trsys_of (env : valuation) (e : exp) := {|
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@ -1074,7 +1081,7 @@ Module StlcMutable.
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-> hasty M G e t
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-> hasty M G (SetVar x e) t.
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Local Hint Constructors value plug step0 step hasty : core.
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Local Hint Constructors value plug step0 step1 step hasty : core.
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(* Let's characterize what makes a typing context and a variable valuation
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* compatible. *)
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@ -1187,9 +1194,19 @@ Module StlcMutable.
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Qed.
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Local Hint Immediate compatible_hasty : core.
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Lemma preservation0 : forall M e1 e2,
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step0 e1 e2
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-> forall t, hasty M $0 e1 t
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-> hasty M $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 preservation0 : core.
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Lemma preservation0_exp : forall M env1 e1 env2 e2,
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step0 (env1, e1) (env2, e2)
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Lemma preservation1_exp : forall M env1 e1 env2 e2,
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step1 (env1, e1) (env2, e2)
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-> forall t, hasty M $0 e1 t
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-> compatible M env1
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-> hasty M $0 e2 t.
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@ -1197,8 +1214,8 @@ Module StlcMutable.
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invert 1; t.
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Qed.
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Lemma preservation0_env : forall M env1 e1 env2 e2,
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step0 (env1, e1) (env2, e2)
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Lemma preservation1_env : forall M env1 e1 env2 e2,
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step1 (env1, e1) (env2, e2)
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-> forall t, hasty M $0 e1 t
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-> compatible M env1
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-> compatible M env2.
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@ -1206,12 +1223,12 @@ Module StlcMutable.
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invert 1; t.
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Qed.
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Local Hint Resolve preservation0_exp preservation0_env : core.
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Local Hint Resolve preservation1_exp preservation1_env : core.
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Lemma preservation'_exp : forall M C e1 e1',
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plug C e1 e1'
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-> forall e2 e2' t env1 env2, plug C e2 e2'
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-> step0 (env1, e1) (env2, e2)
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-> step1 (env1, e1) (env2, e2)
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-> hasty M $0 e1' t
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-> compatible M env1
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-> hasty M $0 e2' t.
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@ -1223,7 +1240,7 @@ Module StlcMutable.
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Lemma preservation'_env' : forall M C e1 e1',
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plug C e1 e1'
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-> forall t e2 env1 env2, step0 (env1, e1) (env2, e2)
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-> forall t e2 env1 env2, step1 (env1, e1) (env2, e2)
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-> hasty M $0 e1' t
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-> compatible M env1
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-> compatible M env2.
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@ -1232,7 +1249,7 @@ Module StlcMutable.
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Qed.
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Lemma preservation'_env : forall M C e1 e1' t e2 env1 env2,
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step0 (env1, e1) (env2, e2)
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step1 (env1, e1) (env2, e2)
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-> plug C e1 e1'
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-> hasty M $0 e1' t
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-> compatible M env1
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@ -1399,44 +1416,49 @@ Module StlcConcur.
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Definition valuation := fmap var exp.
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Inductive step0 : valuation * exp -> valuation * exp -> Prop :=
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| Beta : forall env x e v,
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Inductive step0 : exp -> exp -> Prop :=
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| Beta : forall x e v,
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value v
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-> step0 (env, App (Abs x e) v) (env, subst v x e)
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| Add : forall env n1 n2,
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step0 (env, Plus (Const n1) (Const n2)) (env, Const (n1 + n2))
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| FstPair : forall env v1 v2,
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-> step0 (App (Abs x e) v) (subst v x e)
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| Add : forall n1 n2,
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step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
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| FstPair : forall v1 v2,
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value v1
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-> value v2
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-> step0 (env, Fst (Pair v1 v2)) (env, v1)
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| SndPair : forall env v1 v2,
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-> step0 (Fst (Pair v1 v2)) v1
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| SndPair : forall v1 v2,
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value v1
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-> value v2
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-> step0 (env, Snd (Pair v1 v2)) (env, v2)
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| MatchInl : forall env v x1 e1 x2 e2,
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-> step0 (Snd (Pair v1 v2)) v2
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| MatchInl : forall v x1 e1 x2 e2,
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value v
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-> step0 (env, Match (Inl v) x1 e1 x2 e2) (env, subst v x1 e1)
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| MatchInr : forall env v x1 e1 x2 e2,
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-> step0 (Match (Inl v) x1 e1 x2 e2) (subst v x1 e1)
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| MatchInr : forall v x1 e1 x2 e2,
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value v
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-> step0 (env, Match (Inr v) x1 e1 x2 e2) (env, subst v x2 e2)
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| Read : forall env x v,
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env $? x = Some v
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-> step0 (env, GetVar x) (env, v)
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| Overwrite : forall env x v,
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value v
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-> step0 (env, SetVar x v) (env $+ (x, v), v)
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-> step0 (Match (Inr v) x1 e1 x2 e2) (subst v x2 e2)
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(* New case: *)
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| ParDone : forall env v1 v2,
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| ParDone : forall v1 v2,
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value v1
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-> value v2
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-> step0 (env, Par v1 v2) (env, Pair v1 v2).
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-> step0 (Par v1 v2) (Pair v1 v2).
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Inductive step1 : valuation * exp -> valuation * exp -> Prop :=
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| Lift : forall env e1 e2,
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step0 e1 e2
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-> step1 (env, e1) (env, e2)
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| Read : forall env x v,
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env $? x = Some v
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-> step1 (env, GetVar x) (env, v)
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| Overwrite : forall env x v,
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value v
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-> step1 (env, SetVar x v) (env $+ (x, v), v).
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Inductive step : valuation * exp -> valuation * exp -> Prop :=
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| StepRule : forall C env1 e1 env2 e2 e1' e2',
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plug C e1 e1'
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-> plug C e2 e2'
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-> step0 (env1, e1) (env2, e2)
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-> step1 (env1, e1) (env2, e2)
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-> step (env1, e1') (env2, e2').
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Definition trsys_of (env : valuation) (e : exp) := {|
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@ -1503,7 +1525,7 @@ Module StlcConcur.
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-> hasty M G e2 t2
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-> hasty M G (Par e1 e2) (Prod t1 t2).
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Local Hint Constructors value plug step0 step hasty : core.
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Local Hint Constructors value plug step0 step1 step hasty : core.
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Definition compatible (M : fmap var type) (env : valuation) :=
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forall x t, M $? x = Some t
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@ -1613,8 +1635,18 @@ Module StlcConcur.
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Local Hint Immediate compatible_hasty : core.
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Lemma preservation0_exp : forall M env1 e1 env2 e2,
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step0 (env1, e1) (env2, e2)
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Lemma preservation0 : forall M e1 e2,
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step0 e1 e2
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-> forall t, hasty M $0 e1 t
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-> hasty M $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 preservation0 : core.
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Lemma preservation1_exp : forall M env1 e1 env2 e2,
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step1 (env1, e1) (env2, e2)
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-> forall t, hasty M $0 e1 t
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-> compatible M env1
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-> hasty M $0 e2 t.
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@ -1622,8 +1654,8 @@ Module StlcConcur.
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invert 1; t.
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Qed.
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Lemma preservation0_env : forall M env1 e1 env2 e2,
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step0 (env1, e1) (env2, e2)
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Lemma preservation1_env : forall M env1 e1 env2 e2,
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step1 (env1, e1) (env2, e2)
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-> forall t, hasty M $0 e1 t
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-> compatible M env1
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-> compatible M env2.
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@ -1631,12 +1663,12 @@ Module StlcConcur.
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invert 1; t.
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Qed.
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Local Hint Resolve preservation0_exp preservation0_env : core.
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Local Hint Resolve preservation1_exp preservation1_env : core.
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Lemma preservation'_exp : forall M C e1 e1',
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plug C e1 e1'
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-> forall e2 e2' t env1 env2, plug C e2 e2'
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-> step0 (env1, e1) (env2, e2)
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-> step1 (env1, e1) (env2, e2)
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-> hasty M $0 e1' t
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-> compatible M env1
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-> hasty M $0 e2' t.
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@ -1648,7 +1680,7 @@ Module StlcConcur.
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Lemma preservation'_env' : forall M C e1 e1',
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plug C e1 e1'
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-> forall t e2 env1 env2, step0 (env1, e1) (env2, e2)
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-> forall t e2 env1 env2, step1 (env1, e1) (env2, e2)
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-> hasty M $0 e1' t
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-> compatible M env1
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-> compatible M env2.
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@ -1657,7 +1689,7 @@ Module StlcConcur.
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Qed.
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Lemma preservation'_env : forall M C e1 e1' t e2 env1 env2,
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step0 (env1, e1) (env2, e2)
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step1 (env1, e1) (env2, e2)
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-> plug C e1 e1'
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-> hasty M $0 e1' t
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-> compatible M env1
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