Cleaned up example language modules.

NatLang.v

@@ -98,10 +98,6 @@ Module NatLang <: LocalLang.
       (e ⟨e| ξ1⟩) ⟨e| ξ2⟩ = e ⟨e| fun n => ξ2 (ξ1 n)⟩.
   Proof using. intro e; induction e; intros ξ1 ξ2; cbn; try rewrite IHe; reflexivity. Qed.
     
-  (* Theorem ExprSubstFusion : forall (e : Expr) (σ1 σ2 : nat -> Expr), *)
-  (*     (e [e| σ1]) [e|σ2] = e [e| fun n => σ1 n [e| σ2]]. *)
-  (* Proof using. intro e; induction e; intros σ1 σ2; cbn; try rewrite IHe; reflexivity. Qed. *)
-  
   Definition ExprUpSubst : (nat -> Expr) -> nat -> Expr :=
     fun σ n =>
       match n with

STLC.v

@@ -139,40 +139,5 @@ Module STLC <: (TypedLocalLang LambdaCalc).
   Proof.
     unfold ExprSubstTyping. intros Γ n. unfold ExprIdSubst. apply ExprVarTyping.
   Qed.                                                   
-
-  (* Lemma ExprSubstLWeakening : forall (Γ Δ1 Δ2 : nat -> ExprTyp) (σ : nat -> Expr) (ξ : nat -> nat), *)
-  (*     (forall n, Δ1 n = Δ2 (ξ n)) -> *)
-  (*     Γ ⊢es σ ⊣ Δ1 -> *)
-  (*     Γ ⊢es fun n => (σ n) ⟨e|ξ⟩ ⊣ Δ2. *)
-  (* Proof. *)
-  (*   intros Γ Δ1 Δ2 σ ξ subΔ typing. *)
-  (*   unfold ExprSubstTyping in *; intro n. *)
-  (*   eapply ExprWeakening; eauto. *)
-  (* Qed. *)
-
-  (* Lemma ExprSubstRWeakening : forall (Γ1 Γ2 Δ : nat -> ExprTyp) (σ : nat -> Expr) (ξ : nat -> nat), *)
-  (*     (forall n, Γ1 (ξ n) = Γ2 n) -> *)
-  (*     Γ1 ⊢es σ ⊣ Δ -> *)
-  (*     Γ2 ⊢es fun n => σ (ξ n) ⊣ Δ. *)
-  (* Proof. *)
-  (*   intros Γ1 Γ2 Δ σ ξ subΓ typing. *)
-  (*   unfold ExprSubstTyping in *; intro n. *)
-  (*   rewrite <- subΓ. apply typing. *)
-  (* Qed. *)
-  
-  (* Lemma ExprSubstTypeExpand : forall (Γ Δ : nat -> ExprTyp) (σ : nat -> Expr), *)
-  (*     Γ ⊢es σ ⊣ Δ -> *)
-  (*     forall τ : ExprTyp, ExprSubstTyping (fun n => match n with | 0 => τ | S n => Γ n end) *)
-  (*                                    (ExprUpSubst σ) *)
-  (*                                    (fun n => match n with |0 => τ | S n => Δ n end). *)
-  (* Proof. *)
-  (*   intros Γ Δ σ typing τ. *)
-  (*   unfold ExprUpSubst. *)
-  (*   unfold ExprSubstTyping in *. *)
-  (*   intro m. *)
-  (*   unfold ExprUpSubst. destruct m; simpl. *)
-  (*   apply ExprVarTyping. *)
-  (*   apply ExprWeakening with (Γ := Δ); auto. *)
-  (* Qed. *)
   
 End STLC.

TypedNatLang.v

@@ -104,40 +104,5 @@ Module TypedNatLang <: (TypedLocalLang NatLang).
   Proof.
     unfold ExprSubstTyping. intros Γ n. unfold ExprIdSubst. apply ExprVarTyping.
   Qed.                                                   
-
-  (* Lemma ExprSubstLWeakening : forall (Γ Δ1 Δ2 : nat -> ExprTyp) (σ : nat -> Expr) (ξ : nat -> nat), *)
-  (*     (forall n, Δ1 n = Δ2 (ξ n)) -> *)
-  (*     Γ ⊢es σ ⊣ Δ1 -> *)
-  (*     Γ ⊢es fun n => (σ n) ⟨e|ξ⟩ ⊣ Δ2. *)
-  (* Proof. *)
-  (*   intros Γ Δ1 Δ2 σ ξ subΔ typing. *)
-  (*   unfold ExprSubstTyping in *; intro n. *)
-  (*   eapply ExprWeakening; eauto. *)
-  (* Qed. *)
-
-  (* Lemma ExprSubstRWeakening : forall (Γ1 Γ2 Δ : nat -> ExprTyp) (σ : nat -> Expr) (ξ : nat -> nat), *)
-  (*     (forall n, Γ1 (ξ n) = Γ2 n) -> *)
-  (*     Γ1 ⊢es σ ⊣ Δ -> *)
-  (*     Γ2 ⊢es fun n => σ (ξ n) ⊣ Δ. *)
-  (* Proof. *)
-  (*   intros Γ1 Γ2 Δ σ ξ subΓ typing. *)
-  (*   unfold ExprSubstTyping in *; intro n. *)
-  (*   rewrite <- subΓ. apply typing. *)
-  (* Qed. *)
-  
-  (* Lemma ExprSubstTypeExpand : forall (Γ Δ : nat -> ExprTyp) (σ : nat -> Expr), *)
-  (*     Γ ⊢es σ ⊣ Δ -> *)
-  (*     forall τ : ExprTyp, ExprSubstTyping (fun n => match n with | 0 => τ | S n => Γ n end) *)
-  (*                                    (ExprUpSubst σ) *)
-  (*                                    (fun n => match n with |0 => τ | S n => Δ n end). *)
-  (* Proof. *)
-  (*   intros Γ Δ σ typing τ. *)
-  (*   unfold ExprUpSubst. *)
-  (*   unfold ExprSubstTyping in *. *)
-  (*   intro m. *)
-  (*   unfold ExprUpSubst. destruct m; simpl. *)
-  (*   apply ExprVarTyping. *)
-  (*   apply ExprWeakening with (Γ := Δ); auto. *)
-  (* Qed. *)
   
 End TypedNatLang.