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Rename samples subdirectories
This commit is contained in:
Joshua Peek
733
samples/Coq/Stlc.v
Executable file
733
samples/Coq/Stlc.v
Executable file
@@ -0,0 +1,733 @@
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Require Export SfLib.
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Module STLC.
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Inductive ty : Type :=
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| ty_Bool : ty
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| ty_arrow : ty -> ty -> ty.
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Inductive tm : Type :=
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| tm_var : id -> tm
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| tm_app : tm -> tm -> tm
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| tm_abs : id -> ty -> tm -> tm
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| tm_true : tm
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| tm_false : tm
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| tm_if : tm -> tm -> tm -> tm.
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Tactic Notation "tm_cases" tactic(first) ident(c) :=
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first;
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[ Case_aux c "tm_var" | Case_aux c "tm_app"
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| Case_aux c "tm_abs" | Case_aux c "tm_true"
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| Case_aux c "tm_false" | Case_aux c "tm_if" ].
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Notation a := (Id 0).
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Notation b := (Id 1).
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Notation c := (Id 2).
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Notation idB :=
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(tm_abs a ty_Bool (tm_var a)).
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Notation idBB :=
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(tm_abs a (ty_arrow ty_Bool ty_Bool) (tm_var a)).
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Notation idBBBB :=
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(tm_abs a (ty_arrow (ty_arrow ty_Bool ty_Bool)
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(ty_arrow ty_Bool ty_Bool))
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(tm_var a)).
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Notation k := (tm_abs a ty_Bool (tm_abs b ty_Bool (tm_var a))).
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Inductive value : tm -> Prop :=
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| v_abs : forall x T t,
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value (tm_abs x T t)
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| t_true :
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value tm_true
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| t_false :
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value tm_false.
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Hint Constructors value.
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Fixpoint subst (s:tm) (x:id) (t:tm) : tm :=
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match t with
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| tm_var x' => if beq_id x x' then s else t
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| tm_abs x' T t1 => tm_abs x' T (if beq_id x x' then t1 else (subst s x t1))
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| tm_app t1 t2 => tm_app (subst s x t1) (subst s x t2)
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| tm_true => tm_true
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| tm_false => tm_false
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| tm_if t1 t2 t3 => tm_if (subst s x t1) (subst s x t2) (subst s x t3)
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end.
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Reserved Notation "t1 '==>' t2" (at level 40).
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Inductive step : tm -> tm -> Prop :=
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| ST_AppAbs : forall x T t12 v2,
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value v2 ->
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(tm_app (tm_abs x T t12) v2) ==> (subst v2 x t12)
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| ST_App1 : forall t1 t1' t2,
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t1 ==> t1' ->
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tm_app t1 t2 ==> tm_app t1' t2
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| ST_App2 : forall v1 t2 t2',
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value v1 ->
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t2 ==> t2' ->
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tm_app v1 t2 ==> tm_app v1 t2'
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| ST_IfTrue : forall t1 t2,
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(tm_if tm_true t1 t2) ==> t1
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| ST_IfFalse : forall t1 t2,
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(tm_if tm_false t1 t2) ==> t2
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| ST_If : forall t1 t1' t2 t3,
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t1 ==> t1' ->
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(tm_if t1 t2 t3) ==> (tm_if t1' t2 t3)
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where "t1 '==>' t2" := (step t1 t2).
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Tactic Notation "step_cases" tactic(first) ident(c) :=
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first;
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[ Case_aux c "ST_AppAbs" | Case_aux c "ST_App1"
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| Case_aux c "ST_App2" | Case_aux c "ST_IfTrue"
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| Case_aux c "ST_IfFalse" | Case_aux c "ST_If" ].
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Notation stepmany := (refl_step_closure step).
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Notation "t1 '==>*' t2" := (stepmany t1 t2) (at level 40).
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Hint Constructors step.
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Lemma step_example3 :
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(tm_app (tm_app idBBBB idBB) idB)
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==>* idB.
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Proof.
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eapply rsc_step.
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apply ST_App1.
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apply ST_AppAbs.
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apply v_abs.
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simpl.
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eapply rsc_step.
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apply ST_AppAbs.
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apply v_abs.
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simpl.
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apply rsc_refl.
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Qed.
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Definition context := partial_map ty.
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Module Context.
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Definition partial_map (A:Type) := id -> option A.
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Definition empty {A:Type} : partial_map A := (fun _ => None).
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Definition extend {A:Type} (Gamma : partial_map A) (x:id) (T : A) :=
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fun x' => if beq_id x x' then Some T else Gamma x'.
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Lemma extend_eq : forall A (ctxt: partial_map A) x T,
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(extend ctxt x T) x = Some T.
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Proof.
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intros. unfold extend. rewrite <- beq_id_refl. auto.
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Qed.
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Lemma extend_neq : forall A (ctxt: partial_map A) x1 T x2,
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beq_id x2 x1 = false ->
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(extend ctxt x2 T) x1 = ctxt x1.
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Proof.
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intros. unfold extend. rewrite H. auto.
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Qed.
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End Context.
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Inductive has_type : context -> tm -> ty -> Prop :=
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| T_Var : forall Gamma x T,
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Gamma x = Some T ->
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has_type Gamma (tm_var x) T
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| T_Abs : forall Gamma x T11 T12 t12,
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has_type (extend Gamma x T11) t12 T12 ->
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has_type Gamma (tm_abs x T11 t12) (ty_arrow T11 T12)
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| T_App : forall T11 T12 Gamma t1 t2,
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has_type Gamma t1 (ty_arrow T11 T12) ->
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has_type Gamma t2 T11 ->
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has_type Gamma (tm_app t1 t2) T12
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| T_True : forall Gamma,
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has_type Gamma tm_true ty_Bool
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| T_False : forall Gamma,
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has_type Gamma tm_false ty_Bool
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| T_If : forall t1 t2 t3 T Gamma,
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has_type Gamma t1 ty_Bool ->
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has_type Gamma t2 T ->
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has_type Gamma t3 T ->
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has_type Gamma (tm_if t1 t2 t3) T.
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Tactic Notation "has_type_cases" tactic(first) ident(c) :=
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first;
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[ Case_aux c "T_Var" | Case_aux c "T_Abs"
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| Case_aux c "T_App" | Case_aux c "T_True"
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| Case_aux c "T_False" | Case_aux c "T_If" ].
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Hint Constructors has_type.
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Hint Unfold beq_id beq_nat extend.
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Example typing_example_2_full :
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has_type empty
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(tm_abs a ty_Bool
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(tm_abs b (ty_arrow ty_Bool ty_Bool)
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(tm_app (tm_var b) (tm_app (tm_var b) (tm_var a)))))
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(ty_arrow ty_Bool (ty_arrow (ty_arrow ty_Bool ty_Bool) ty_Bool)).
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Proof.
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apply T_Abs.
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apply T_Abs.
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apply T_App with (T11 := ty_Bool).
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apply T_Var.
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unfold extend.
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simpl.
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reflexivity.
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apply T_App with (T11 := ty_Bool).
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apply T_Var.
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unfold extend.
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simpl.
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reflexivity.
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apply T_Var.
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unfold extend.
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simpl.
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reflexivity.
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Qed.
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Example typing_example_3 :
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exists T,
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has_type empty
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(tm_abs a (ty_arrow ty_Bool ty_Bool)
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(tm_abs b (ty_arrow ty_Bool ty_Bool)
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(tm_abs c ty_Bool
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(tm_app (tm_var b) (tm_app (tm_var a) (tm_var c))))))
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T.
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Proof with auto.
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exists
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(ty_arrow (ty_arrow ty_Bool ty_Bool)
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(ty_arrow (ty_arrow ty_Bool ty_Bool) (ty_arrow ty_Bool ty_Bool))).
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apply T_Abs.
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apply T_Abs.
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apply T_Abs.
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apply T_App with (T11 := ty_Bool).
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apply T_Var.
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unfold extend.
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simpl.
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reflexivity.
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apply T_App with (T11 := ty_Bool).
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auto.
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auto.
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Qed.
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Theorem coiso : forall a b e,
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a ==>* b ->
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tm_app a e ==>* tm_app b e.
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Proof.
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intros.
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induction H.
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apply rsc_refl.
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apply rsc_step with (tm_app y e).
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apply ST_App1.
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assumption.
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assumption.
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Qed.
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Theorem reptrans : forall a b c,
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a ==>* b ->
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b ==>* c ->
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a ==>* c.
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Proof.
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intros a b c H.
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induction H.
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intros.
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assumption.
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intros H1.
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apply IHrefl_step_closure in H1.
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apply rsc_step with y.
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assumption.
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assumption.
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Qed.
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(* TODO
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Example typing_nonexample_3 :
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~ (exists S, exists T,
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has_type empty
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(tm_abs a S
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(tm_app (tm_var a) (tm_var a)))
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T).
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Proof.
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*)
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Inductive appears_free_in : id -> tm -> Prop :=
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| afi_var : forall x,
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appears_free_in x (tm_var x)
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| afi_app1 : forall x t1 t2,
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appears_free_in x t1 -> appears_free_in x (tm_app t1 t2)
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| afi_app2 : forall x t1 t2,
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appears_free_in x t2 -> appears_free_in x (tm_app t1 t2)
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| afi_abs : forall x y T11 t12,
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y <> x ->
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appears_free_in x t12 ->
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appears_free_in x (tm_abs y T11 t12)
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| afi_if1 : forall x t1 t2 t3,
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appears_free_in x t1 ->
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appears_free_in x (tm_if t1 t2 t3)
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| afi_if2 : forall x t1 t2 t3,
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appears_free_in x t2 ->
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appears_free_in x (tm_if t1 t2 t3)
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| afi_if3 : forall x t1 t2 t3,
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appears_free_in x t3 ->
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appears_free_in x (tm_if t1 t2 t3).
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Tactic Notation "afi_cases" tactic(first) ident(c) :=
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first;
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[ Case_aux c "afi_var"
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| Case_aux c "afi_app1" | Case_aux c "afi_app2"
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| Case_aux c "afi_abs"
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| Case_aux c "afi_if1" | Case_aux c "afi_if2"
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| Case_aux c "afi_if3" ].
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Hint Constructors appears_free_in.
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Definition closed (t:tm) :=
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forall x, ~ appears_free_in x t.
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Lemma free_in_context : forall x t T Gamma,
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appears_free_in x t ->
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has_type Gamma t T ->
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exists T', Gamma x = Some T'.
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Proof.
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intros. generalize dependent Gamma. generalize dependent T.
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afi_cases (induction H) Case;
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intros; try solve [inversion H0; eauto].
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Case "afi_abs".
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inversion H1; subst.
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apply IHappears_free_in in H7.
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apply not_eq_beq_id_false in H.
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rewrite extend_neq in H7; assumption.
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Qed.
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Corollary typable_empty__closed : forall t T,
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has_type empty t T ->
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closed t.
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Proof.
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intros t T H x H1.
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remember (@empty ty) as Gamma.
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assert (exists t' : _, Gamma x = Some t').
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apply free_in_context with (t := t) (T := T).
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assumption.
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assumption.
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inversion H0.
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rewrite HeqGamma in H2.
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inversion H2.
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Qed.
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Lemma context_invariance : forall Gamma Gamma' t S,
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has_type Gamma t S ->
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(forall x, appears_free_in x t -> Gamma x = Gamma' x) ->
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has_type Gamma' t S.
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Proof with auto.
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intros.
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generalize dependent Gamma'.
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has_type_cases (induction H) Case; intros; auto.
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apply T_Var.
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rewrite <- H0...
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apply T_Abs.
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apply IHhas_type.
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intros x0 Hafi.
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unfold extend.
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remember (beq_id x x0) as e.
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destruct e.
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reflexivity.
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auto.
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apply H0.
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apply afi_abs.
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auto.
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eauto .
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apply beq_id_false_not_eq.
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rewrite Heqe.
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reflexivity.
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assumption.
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apply T_App with T11.
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auto.
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auto.
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Qed.
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Lemma substitution_preserves_typing : forall Gamma x U v t T,
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has_type (extend Gamma x U) t T ->
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has_type empty v U ->
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has_type Gamma (subst v x t) T.
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Proof with eauto.
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intros Gamma x U v t T Ht Hv.
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generalize dependent Gamma.
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generalize dependent T.
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tm_cases (induction t) Case; intros T Gamma H; inversion H; subst; simpl...
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Case "tm_var".
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rename i into y. remember (beq_id x y) as e. destruct e.
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SCase "x=y".
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apply beq_id_eq in Heqe. subst.
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rewrite extend_eq in H2.
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inversion H2; subst.
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clear H2.
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eapply context_invariance...
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intros x Hcontra.
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destruct (free_in_context _ _ T empty Hcontra) as (T', HT')...
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inversion HT'.
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apply T_Var.
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rewrite extend_neq in H2.
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assumption.
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rewrite Heqe.
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reflexivity.
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rename i into y.
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apply T_Abs.
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remember (beq_id x y) as e.
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destruct e.
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eapply context_invariance...
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apply beq_id_eq in Heqe.
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subst.
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intros x Hafi.
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unfold extend.
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destruct (beq_id y x).
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reflexivity.
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reflexivity.
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apply IHt.
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eapply context_invariance...
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intros x0 Hafi.
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unfold extend.
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remember (beq_id y x0) as Coiso1.
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remember (beq_id x x0) as Coiso2.
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destruct Coiso1.
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auto.
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eauto .
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destruct Coiso2.
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eauto .
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auto.
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apply beq_id_eq in HeqCoiso1.
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apply beq_id_eq in HeqCoiso2.
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subst.
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assert (x0 <> x0).
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apply beq_id_false_not_eq.
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rewrite Heqe.
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auto.
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apply ex_falso_quodlibet.
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apply H0.
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reflexivity.
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reflexivity.
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destruct Coiso2.
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auto.
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auto.
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Qed.
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Theorem preservation : forall t t' T,
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has_type empty t T ->
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t ==> t' ->
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has_type empty t' T.
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Proof.
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remember (@empty ty) as Gamma.
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intros t t' T HT.
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generalize dependent t'.
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induction HT.
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intros t' H1.
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inversion H1.
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intros t' H1.
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inversion H1.
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intros t' H1.
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inversion H1.
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apply substitution_preserves_typing with T11.
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subst.
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inversion HT1.
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subst.
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apply H2.
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subst.
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assumption.
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subst.
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apply T_App with T11.
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apply IHHT1.
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reflexivity.
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assumption.
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assumption.
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subst.
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apply T_App with T11.
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assumption.
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apply IHHT2.
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reflexivity.
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assumption.
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intros t' H.
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inversion H.
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intros t' H.
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inversion H.
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intros t' H.
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inversion H.
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subst.
|
||||
assumption.
|
||||
|
||||
subst.
|
||||
assumption.
|
||||
|
||||
subst.
|
||||
apply T_If.
|
||||
apply IHHT1.
|
||||
reflexivity.
|
||||
|
||||
assumption.
|
||||
|
||||
assumption.
|
||||
|
||||
assumption.
|
||||
Qed.
|
||||
|
||||
Theorem progress : forall t T,
|
||||
has_type empty t T ->
|
||||
value t \/ exists t', t ==> t'.
|
||||
Proof.
|
||||
intros t T.
|
||||
intros H.
|
||||
remember (@empty ty) as Gamma.
|
||||
induction H.
|
||||
rewrite HeqGamma in H.
|
||||
unfold empty in H.
|
||||
inversion H.
|
||||
|
||||
left.
|
||||
apply v_abs.
|
||||
|
||||
right.
|
||||
assert (value t1 \/ (exists t' : tm, t1 ==> t')).
|
||||
apply IHhas_type1.
|
||||
assumption.
|
||||
|
||||
assert (value t2 \/ (exists t' : tm, t2 ==> t')).
|
||||
apply IHhas_type2.
|
||||
assumption.
|
||||
|
||||
inversion H1.
|
||||
inversion H2.
|
||||
inversion H3.
|
||||
subst.
|
||||
exists (subst t2 x t).
|
||||
apply ST_AppAbs.
|
||||
assumption.
|
||||
|
||||
subst.
|
||||
inversion H.
|
||||
|
||||
subst.
|
||||
inversion H.
|
||||
|
||||
inversion H4.
|
||||
exists (tm_app t1 x).
|
||||
apply ST_App2.
|
||||
assumption.
|
||||
|
||||
assumption.
|
||||
|
||||
inversion H3.
|
||||
exists (tm_app x t2).
|
||||
apply ST_App1.
|
||||
assumption.
|
||||
|
||||
left.
|
||||
auto.
|
||||
|
||||
left.
|
||||
auto.
|
||||
|
||||
right.
|
||||
assert (value t1 \/ (exists t' : tm, t1 ==> t')).
|
||||
apply IHhas_type1.
|
||||
assumption.
|
||||
|
||||
inversion H2.
|
||||
inversion H3.
|
||||
subst.
|
||||
inversion H.
|
||||
|
||||
subst.
|
||||
exists t2.
|
||||
apply ST_IfTrue.
|
||||
|
||||
subst.
|
||||
exists t3.
|
||||
apply ST_IfFalse.
|
||||
|
||||
inversion H3.
|
||||
exists (tm_if x t2 t3).
|
||||
apply ST_If.
|
||||
assumption.
|
||||
Qed.
|
||||
|
||||
Theorem progress' : forall t T,
|
||||
has_type empty t T ->
|
||||
value t \/ exists t', t ==> t'.
|
||||
Proof.
|
||||
intros t.
|
||||
tm_cases (induction t) Case; intros T Ht; auto.
|
||||
inversion Ht.
|
||||
inversion H1.
|
||||
|
||||
right.
|
||||
inversion Ht.
|
||||
subst.
|
||||
assert (value t1 \/ (exists t' : tm, t1 ==> t')).
|
||||
apply IHt1 with (T := ty_arrow T11 T).
|
||||
assumption.
|
||||
|
||||
assert (value t2 \/ (exists t' : tm, t2 ==> t')).
|
||||
apply IHt2 with T11.
|
||||
assumption.
|
||||
|
||||
inversion H.
|
||||
inversion H1.
|
||||
subst.
|
||||
inversion H0.
|
||||
exists (subst t2 x t).
|
||||
apply ST_AppAbs.
|
||||
assumption.
|
||||
|
||||
inversion H3.
|
||||
exists (tm_app (tm_abs x T0 t) x0).
|
||||
apply ST_App2.
|
||||
assumption.
|
||||
|
||||
assumption.
|
||||
|
||||
subst.
|
||||
inversion H2.
|
||||
|
||||
subst.
|
||||
inversion H2.
|
||||
|
||||
inversion H1.
|
||||
exists (tm_app x t2).
|
||||
apply ST_App1.
|
||||
assumption.
|
||||
|
||||
right.
|
||||
inversion Ht.
|
||||
subst.
|
||||
assert (value t1 \/ (exists t' : tm, t1 ==> t')).
|
||||
apply IHt1 with ty_Bool.
|
||||
assumption.
|
||||
|
||||
assert (value t2 \/ (exists t' : tm, t2 ==> t')).
|
||||
apply IHt2 with T.
|
||||
assumption.
|
||||
|
||||
assert (value t3 \/ (exists t' : tm, t3 ==> t')).
|
||||
apply IHt3 with T.
|
||||
assumption.
|
||||
|
||||
inversion H.
|
||||
inversion H2.
|
||||
subst.
|
||||
inversion H3.
|
||||
|
||||
subst.
|
||||
subst.
|
||||
exists t2.
|
||||
apply ST_IfTrue.
|
||||
|
||||
exists t3.
|
||||
apply ST_IfFalse.
|
||||
|
||||
inversion H2.
|
||||
exists (tm_if x t2 t3).
|
||||
apply ST_If.
|
||||
assumption.
|
||||
Qed.
|
||||
|
||||
Theorem types_unique : forall t T Gamma,
|
||||
has_type Gamma t T ->
|
||||
(forall T', has_type Gamma t T' -> T' = T).
|
||||
Proof.
|
||||
intros t T Gamma H.
|
||||
induction H.
|
||||
intros T'.
|
||||
intros H1.
|
||||
inversion H1.
|
||||
subst.
|
||||
subst.
|
||||
auto.
|
||||
eauto .
|
||||
inversion H1.
|
||||
subst.
|
||||
assert (Some T = Some T').
|
||||
rewrite <- H3.
|
||||
rewrite <- H.
|
||||
reflexivity.
|
||||
|
||||
inversion H0.
|
||||
reflexivity.
|
||||
|
||||
intros T'.
|
||||
intros H1.
|
||||
inversion H1.
|
||||
subst.
|
||||
assert (T1 = T12).
|
||||
apply IHhas_type.
|
||||
assumption.
|
||||
|
||||
rewrite H0.
|
||||
reflexivity.
|
||||
|
||||
intros T' H1.
|
||||
subst.
|
||||
inversion H1.
|
||||
subst.
|
||||
assert (ty_arrow T0 T' = ty_arrow T11 T12).
|
||||
apply IHhas_type1.
|
||||
assumption.
|
||||
|
||||
inversion H2.
|
||||
reflexivity.
|
||||
|
||||
intros T'.
|
||||
intros H.
|
||||
inversion H.
|
||||
reflexivity.
|
||||
|
||||
intros.
|
||||
inversion H.
|
||||
reflexivity.
|
||||
|
||||
intros T'.
|
||||
intros H2.
|
||||
inversion H2.
|
||||
subst.
|
||||
apply IHhas_type2.
|
||||
assumption.
|
||||
Qed.
|
||||
Reference in New Issue
Block a user