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LambdaCalculusAndTypeSoundness_template update
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Adam Chlipala
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1 changed files with 74 additions and 147 deletions
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@ -37,7 +37,7 @@ Module Ulc.
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Inductive value : exp -> Prop :=
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| Value : forall x e, value (Abs x e).
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Hint Constructors eval value.
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Local Hint Constructors eval value : core.
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Theorem value_eval : forall v,
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value v
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@ -46,7 +46,7 @@ Module Ulc.
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invert 1; eauto.
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Qed.
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Hint Resolve value_eval.
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Local Hint Resolve value_eval : core.
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Theorem eval_value : forall e v,
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eval e v
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@ -55,7 +55,7 @@ Module Ulc.
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induct 1; eauto.
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Qed.
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Hint Resolve eval_value.
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Local Hint Resolve eval_value : core.
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(* Some notations, to let us write more normal-looking lambda terms *)
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Coercion Var : var >-> exp.
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@ -276,59 +276,51 @@ Module Ulc.
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Qed.
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(** * Small-step semantics with evaluation contexts *)
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Inductive context : Set :=
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| Hole : context
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| App1 : context -> exp -> context
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| App2 : exp -> context -> context.
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Inductive plug : context -> exp -> exp -> Prop :=
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| PlugHole : forall e,
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plug Hole e e
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| PlugApp1 : forall c e1 e2 e,
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plug c e1 e
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-> plug (App1 c e2) e1 (App e e2)
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| PlugApp2 : forall c e1 e2 e,
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value e1
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-> plug c e2 e
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-> plug (App2 e1 c) e2 (App e1 e).
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(** * Small-step semantics *)
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Inductive step : exp -> exp -> Prop :=
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| ContextBeta : forall c x e v e1 e2,
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| ContextBeta : forall x e v,
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value v
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-> plug c (App (Abs x e) v) e1
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-> plug c (subst v x e) e2
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-> step e1 e2.
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-> step (App (Abs x e) v) (subst v x e)
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Hint Constructors plug step.
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(* However, we also need bureaucractic rules for pushing evaluation inside
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* applications. *)
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| App1 : forall e1 e1' e2,
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step e1 e1'
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-> step (App e1 e2) (App e1' e2)
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| App2 : forall v e2 e2',
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value v
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-> step e2 e2'
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-> step (App v e2) (App v e2').
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Local Hint Constructors step : core.
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(* Here we now go through a proof of equivalence between big- and small-step
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* semantics, though we won't spend any further commentary on it. *)
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Lemma step_eval'' : forall v c x e e1 e2 v0,
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value v
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-> plug c (App (Abs x e) v) e1
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-> plug c (subst v x e) e2
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-> eval e2 v0
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-> eval e1 v0.
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Proof.
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induct c; invert 2; invert 1; simplify; eauto.
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invert H0; eauto.
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invert H0; eauto.
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Qed.
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Hint Resolve step_eval''.
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Lemma step_eval' : forall e1 e2,
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step e1 e2
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-> forall v, eval e2 v
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-> eval e1 v.
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Proof.
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invert 1; simplify; eauto.
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induct 1; simplify; eauto.
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invert H0.
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econstructor.
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apply IHstep.
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eassumption.
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eassumption.
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assumption.
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invert H1.
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econstructor.
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eassumption.
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apply IHstep.
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eassumption.
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assumption.
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Qed.
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Hint Resolve step_eval'.
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Local Hint Resolve step_eval' : core.
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Theorem step_eval : forall e v,
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step^* e v
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@ -338,76 +330,23 @@ Module Ulc.
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induct 1; eauto.
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Qed.
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Lemma plug_functional : forall C e e1,
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plug C e e1
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-> forall e2, plug C e e2
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-> e1 = e2.
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Local Hint Resolve eval_value : core.
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Theorem step_app1 : forall e1 e1' e2,
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step^* e1 e1'
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-> step^* (App e1 e2) (App e1' e2).
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Proof.
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induct 1; invert 1; simplify; try f_equal; eauto.
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induct 1; eauto.
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Qed.
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Lemma plug_mirror : forall C e e', plug C e e'
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-> forall e1, exists e1', plug C e1 e1'.
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Theorem step_app2 : forall e2 e2' v,
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value v
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-> step^* e2 e2'
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-> step^* (App v e2) (App v e2').
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Proof.
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induct 1; simplify; eauto.
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specialize (IHplug e0); first_order; eauto.
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specialize (IHplug e0); first_order; eauto.
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induct 2; eauto.
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Qed.
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Fixpoint compose (C1 C2 : context) : context :=
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match C2 with
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| Hole => C1
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| App1 C2' e => App1 (compose C1 C2') e
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| App2 v C2' => App2 v (compose C1 C2')
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end.
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Lemma compose_ok : forall C1 C2 e1 e2 e3,
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plug C1 e1 e2
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-> plug C2 e2 e3
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-> plug (compose C1 C2) e1 e3.
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Proof.
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induct 2; simplify; eauto.
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Qed.
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Hint Resolve compose_ok.
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Lemma step_plug : forall e1 e2,
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step e1 e2
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-> forall C e1' e2', plug C e1 e1'
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-> plug C e2 e2'
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-> step e1' e2'.
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Proof.
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invert 1; simplify; eauto.
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Qed.
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Lemma stepStar_plug : forall e1 e2,
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step^* e1 e2
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-> forall C e1' e2', plug C e1 e1'
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-> plug C e2 e2'
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-> step^* e1' e2'.
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Proof.
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induct 1; simplify.
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assert (e1' = e2') by (eapply plug_functional; eassumption).
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subst.
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constructor.
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assert (exists y', plug C y y') by eauto using plug_mirror.
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invert H3.
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eapply step_plug in H.
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econstructor.
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eassumption.
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eapply IHtrc.
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eassumption.
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assumption.
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eassumption.
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assumption.
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Qed.
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Hint Resolve stepStar_plug eval_value.
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Theorem eval_step : forall e v,
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eval e v
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-> step^* e v.
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@ -415,10 +354,16 @@ Module Ulc.
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induct 1; eauto.
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eapply trc_trans.
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eapply stepStar_plug with (e1 := e1) (e2 := Abs x e1') (C := App1 Hole e2); eauto.
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apply step_app1.
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eassumption.
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eapply trc_trans.
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eapply stepStar_plug with (e1 := e2) (e2 := v2) (C := App2 (Abs x e1') Hole); eauto.
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eapply step_app2.
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constructor.
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eassumption.
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econstructor.
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constructor.
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eauto.
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assumption.
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Qed.
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End Ulc.
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@ -444,50 +389,32 @@ Module Stlc.
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| App e2' e2'' => App (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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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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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 (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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Inductive step : exp -> exp -> Prop :=
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| StepRule : forall C e1 e2 e1' e2',
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plug C e1 e1'
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-> plug C e2 e2'
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-> step0 e1 e2
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-> step e1' e2'.
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| Beta : forall x e v,
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value v
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-> step (App (Abs x e) v) (subst v x e)
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| Add : forall n1 n2,
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step (Plus (Const n1) (Const n2)) (Const (n1 + n2))
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| App1 : forall e1 e1' e2,
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step e1 e1'
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-> step (App e1 e2) (App e1' e2)
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| App2 : forall v e2 e2',
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value v
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-> step e2 e2'
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-> step (App v e2) (App v e2')
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| Plus1 : forall e1 e1' e2,
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step e1 e1'
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-> step (Plus e1 e2) (Plus e1' e2)
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| Plus2 : forall v e2 e2',
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value v
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-> step e2 e2'
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-> step (Plus v e2) (Plus v e2').
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Definition trsys_of (e : exp) := {|
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Initial := {e};
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Step := step
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|}.
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Inductive type :=
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| Nat (* Numbers *)
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| Fun (dom ran : type) (* Functions *).
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@ -510,7 +437,7 @@ Module Stlc.
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-> hasty G e2 t1
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-> hasty G (App e1 e2) t2.
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Hint Constructors value plug step0 step hasty.
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Local Hint Constructors value step hasty : core.
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(* Some notation to make it more pleasant to write programs *)
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Infix "-->" := Fun (at level 60, right associativity).
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@ -564,7 +491,7 @@ Module Stlc.
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Proof.
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Admitted.
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Hint Resolve hasty_change.
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Local Hint Resolve hasty_change : core.
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Lemma preservation : forall e1 e2,
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step e1 e2
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