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EvaluationContexts: concurrency
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@ -1272,3 +1272,428 @@ Module StlcMutable.
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eauto.
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Qed.
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End StlcMutable.
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(** * Concurrency *)
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(* As we saw in Chapter 8, it's actually not all that much work to add
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* concurrency, once we have evaluation contexts. Let's replicate that result
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* for our functional language. *)
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Module StlcConcur.
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Inductive exp : Set :=
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| Var (x : var)
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| Const (n : nat)
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| Plus (e1 e2 : exp)
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| Abs (x : var) (e1 : exp)
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| App (e1 e2 : exp)
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| Pair (e1 e2 : exp)
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| Fst (e1 : exp)
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| Snd (e2 : exp)
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| Inl (e1 : exp)
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| Inr (e2 : exp)
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| Match (e' : exp) (x1 : var) (e1 : exp) (x2 : var) (e2 : exp)
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| GetVar (x : var)
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| SetVar (x : var) (e : exp)
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(* New cases: *)
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| Par (e1 e2 : exp).
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(* This form evaluates both expressions and forms a pair of their
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* results, if they terminate. *)
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Inductive value : exp -> Prop :=
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| VConst : forall n, value (Const n)
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| VAbs : forall x e1, value (Abs x e1)
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| VPair : forall v1 v2, value v1 -> value v2 -> value (Pair v1 v2)
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| VInl : forall v, value v -> value (Inl v)
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| VInr : forall v, value v -> value (Inr v).
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Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
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match e2 with
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| Var y => if y ==v x then e1 else Var y
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| Const n => Const n
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| Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
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| Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
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| App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
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| Pair e2' e2'' => Pair (subst e1 x e2') (subst e1 x e2'')
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| Fst e2' => Fst (subst e1 x e2')
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| Snd e2' => Snd (subst e1 x e2')
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| Inl e2' => Inl (subst e1 x e2')
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| Inr e2' => Inr (subst e1 x e2')
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| Match e2' x1 e21 x2 e22 => Match (subst e1 x e2')
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x1 (if x1 ==v x then e21 else subst e1 x e21)
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x2 (if x2 ==v x then e22 else subst e1 x e22)
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| GetVar y => GetVar y
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| SetVar y e2' => SetVar y (subst e1 x e2')
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| Par e2' e2'' => Par (subst e1 x e2') (subst e1 x e2'')
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end.
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Inductive context : Set :=
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| Hole : context
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| Plus1 : context -> exp -> context
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| Plus2 : exp -> context -> context
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| App1 : context -> exp -> context
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| App2 : exp -> context -> context
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| Pair1 : context -> exp -> context
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| Pair2 : exp -> context -> context
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| Fst1 : context -> context
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| Snd1 : context -> context
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| Inl1 : context -> context
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| Inr1 : context -> context
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| Match1 : context -> var -> exp -> var -> exp -> context
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| SetVar1 : var -> context -> context
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(* New cases: *)
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| Par1 : context -> exp -> context
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| Par2 : exp -> context -> context.
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Inductive plug : context -> exp -> exp -> Prop :=
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| PlugHole : forall e, plug Hole e e
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| PlugPlus1 : forall e e' C e2,
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plug C e e'
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-> plug (Plus1 C e2) e (Plus e' e2)
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| PlugPlus2 : forall e e' v1 C,
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value v1
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-> plug C e e'
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-> plug (Plus2 v1 C) e (Plus v1 e')
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| PlugApp1 : forall e e' C e2,
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plug C e e'
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-> plug (App1 C e2) e (App e' e2)
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| PlugApp2 : forall e e' v1 C,
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value v1
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-> plug C e e'
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-> plug (App2 v1 C) e (App v1 e')
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| PlugPair1 : forall e e' C e2,
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plug C e e'
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-> plug (Pair1 C e2) e (Pair e' e2)
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| PlugPair2 : forall e e' v1 C,
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value v1
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-> plug C e e'
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-> plug (Pair2 v1 C) e (Pair v1 e')
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| PlugFst1 : forall e e' C,
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plug C e e'
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-> plug (Fst1 C) e (Fst e')
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| PlugSnd1 : forall e e' C,
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plug C e e'
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-> plug (Snd1 C) e (Snd e')
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| PlugInl1 : forall e e' C,
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plug C e e'
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-> plug (Inl1 C) e (Inl e')
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| PlugInr1 : forall e e' C,
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plug C e e'
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-> plug (Inr1 C) e (Inr e')
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| PluMatch1 : forall e e' C x1 e1 x2 e2,
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plug C e e'
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-> plug (Match1 C x1 e1 x2 e2) e (Match e' x1 e1 x2 e2)
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| PluSetVar1 : forall x e e' C,
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plug C e e'
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-> plug (SetVar1 x C) e (SetVar x e')
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(* Our new plugging rules: *)
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| PlugPar1 : forall e e' C e2,
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plug C e e'
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-> plug (Par1 C e2) e (Par e' e2)
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| PlugPar2 : forall e e' e1 C,
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plug C e e'
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-> plug (Par2 e1 C) e (Par e1 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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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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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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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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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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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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(* New case: *)
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| ParDone : forall env 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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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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-> step (env1, e1') (env2, e2').
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Definition trsys_of (env : valuation) (e : exp) := {|
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Initial := {(env, e)};
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Step := step
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|}.
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Inductive type :=
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| Nat
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| Fun (dom ran : type)
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| Prod (t1 t2 : type)
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| Sum (t1 t2 : type).
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Inductive hasty (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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| HtConst : forall G n,
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hasty 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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| 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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| 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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| 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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| 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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| HtSnd : forall G e1 t1 t2,
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hasty M G e1 (Prod t1 t2)
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-> hasty M G (Snd e1) t2
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| HtInl : forall G e1 t1 t2,
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hasty M G e1 t1
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-> hasty M G (Inl e1) (Sum t1 t2)
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| HtInr : forall G e1 t1 t2,
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hasty M G e1 t2
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-> hasty M 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 M G e (Sum t1 t2)
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-> hasty M (G $+ (x1, t1)) e1 t
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-> hasty M (G $+ (x2, t2)) e2 t
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-> hasty M G (Match e x1 e1 x2 e2) t
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| HtGetVar : forall G x t,
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M $? x = Some t
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-> hasty M G (GetVar x) t
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| HtSetVar : forall G x e t,
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M $? x = Some t
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-> hasty M G e t
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-> hasty M G (SetVar x e) t
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(* New cases: *)
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| HtPar : forall G e1 t1 e2 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 (Par e1 e2) (Prod t1 t2).
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Local Hint Constructors value plug step0 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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-> exists v, env $? x = Some v
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/\ hasty M $0 v t.
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Ltac t0 := match goal with
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| [ H : ex _ |- _ ] => invert H
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| [ H : _ /\ _ |- _ ] => invert H
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| [ |- context[?x ==v ?y] ] => cases (x ==v y)
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| [ H : Some _ = Some _ |- _ ] => invert H
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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 _ _ ?e _, H' : value ?e |- _ ] => invert H'; invert H; [|]
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| [ H : hasty _ _ _ _ |- _ ] => invert1 H
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| [ H : plug _ _ _ |- _ ] => invert1 H
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| [ H1 : compatible ?M ?env, H2 : ?M $? ?x = Some _ |- _ ] =>
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(assert (exists v, env $? x = Some v) by eauto; fail 1)
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|| (pose proof (H1 _ _ H2); first_order)
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| [ H1 : forall env, compatible ?M env -> _, H2 : compatible ?M _ |- _ ] =>
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specialize (H1 _ H2); first_order
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end; subst.
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Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 7.
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Lemma progress : forall M e t,
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hasty M $0 e t
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-> forall env, compatible M env
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-> value e
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\/ exists st', step (env, e) st'.
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Proof.
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induct 1; t.
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Qed.
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Lemma weakening_override : forall (G G' : fmap var type) x t,
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(forall x' t', G $? x' = Some t' -> G' $? x' = Some t')
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-> (forall x' t', G $+ (x, t) $? x' = Some t'
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-> G' $+ (x, t) $? x' = Some t').
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Proof.
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simplify.
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cases (x ==v x'); simplify; eauto.
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Qed.
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Local Hint Resolve weakening_override : core.
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Lemma weakening : forall M G e t, hasty M 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 M 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 M G e t,
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hasty M G e t
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-> forall G', G' = G
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-> hasty M 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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Lemma substitution : forall M G x t' e t e',
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hasty M (G $+ (x, t')) e t
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-> hasty M $0 e' t'
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-> hasty M 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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Local Hint Resolve substitution : core.
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Lemma compatible_bind : forall M env x v t,
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compatible M env
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-> M $? x = Some t
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-> hasty M $0 v t
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-> compatible M (env $+ (x, v)).
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Proof.
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unfold compatible; first_order.
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cases (x ==v x0); subst; simplify.
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rewrite H0 in H2.
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invert H2.
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eauto.
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apply H in H0; first_order.
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Qed.
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Local Hint Resolve compatible_bind : core.
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Lemma compatible_hasty : forall M env x v t,
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compatible M env
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-> env $? x = Some v
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-> M $? x = Some t
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-> hasty M $0 v t.
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Proof.
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t.
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specialize (H _ _ H1); first_order.
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rewrite H0 in H.
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invert H.
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assumption.
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Qed.
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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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-> 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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Proof.
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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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-> 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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Proof.
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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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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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-> 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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Proof.
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induct 1; t.
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Qed.
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Local Hint Resolve preservation'_exp : core.
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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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-> hasty M $0 e1' t
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-> compatible M env1
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-> compatible M env2.
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Proof.
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induct 1; t.
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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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-> plug C e1 e1'
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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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Proof.
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simplify; eapply preservation'_env' with (e2 := e2); eauto.
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Qed.
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Local Hint Immediate preservation'_env : core.
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Lemma preservation : forall M env1 e1 env2 e2,
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step (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 /\ compatible M env2.
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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 M env e t, hasty M $0 e t
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-> compatible M env
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-> invariantFor (trsys_of env e)
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(fun st => value (snd st)
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\/ exists st', step st st').
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Proof.
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simplify.
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apply invariant_weaken with (invariant1 := fun st => hasty M $0 (snd st) t /\ compatible M (fst st)); eauto.
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apply invariant_induction; simplify.
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propositional; subst; auto.
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invert H1.
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cases s; cases s'; simplify.
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eauto.
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propositional.
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cases s; simplify.
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eauto.
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Qed.
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End StlcConcur.
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