diff --git a/coq/soundness.v b/coq/soundness.v
index e550a7d..5558dde 100644
--- a/coq/soundness.v
+++ b/coq/soundness.v
@@ -16,16 +16,21 @@ Proof.
   intros.
   induction H.
 
-  apply T_Var.
-  apply C_shuffle.
-  apply H.
+  - apply T_Var.
+    apply C_shuffle.
+    apply H.
   
-  apply T_Let with (σ:=σ0).
-  apply IHexpr_type1.
-  admit.
-
+  - apply T_Let with (σ:=σ0).
+    apply IHexpr_type1.
+    
+    admit.
+Admitted.
+Lemma map_type : forall Γ,
+  Γ |- [{ %"map"% }] \is [<
+       ∀"σ",∀"τ", (%"σ"% -> %"τ"%) -> [%"σ"%] -> [%"τ"%]
+  >].
+Proof.
 Admitted.
-
 
 Lemma morphism_path_solves_type : forall Γ τ τ' m,
   (translate_morphism_path Γ τ τ' m) ->
@@ -37,7 +42,7 @@ Proof.
 
   (* Sub *)
   apply T_MorphAbs.
-  apply T_DescendImplicit with (τ:=τ).
+  apply T_Descend with (τ:=τ).
   apply T_Var.
   apply C_take.
   apply H.
@@ -53,7 +58,7 @@ Proof.
   apply T_Var.
   apply C_take.
   apply TSubRepr_Ladder, TSubRepr_Refl, TEq_Refl.
-  apply M_Sub, TSubRepr_Refl, TEq_Refl.
+  apply id_morphism_path.
 
   (* Single *)
   apply T_Var.
@@ -65,7 +70,7 @@ Proof.
   apply T_MorphFun.
   apply typing_weakening.
   apply IHtranslate_morphism_path2.
-  
+
   apply T_App with (σ':=τ) (σ:=τ) (τ:=τ').
   apply T_MorphFun.
   apply typing_weakening.
@@ -73,15 +78,30 @@ Proof.
 
   apply T_Var.
   apply C_take.
-  apply M_Sub, TSubRepr_Refl, TEq_Refl.
-  apply M_Sub, TSubRepr_Refl, TEq_Refl.
+  apply id_morphism_path.
+  apply id_morphism_path.
 
   (* Map Sequence *)
   apply T_MorphAbs.
   apply T_App with (σ':=(type_spec (type_id "Seq") τ)) (σ:=(type_spec (type_id "Seq") τ)).
   apply T_App with (σ':=(type_fun τ τ')) (σ:=(type_fun τ τ')).
 
+  set (k:=[< (%"σ"% -> %"τ"%) -> <$"Seq"$ %"σ"%> -> <$"Seq"$ %"τ"%> >]).
+  set (k1:=[< (τ -> %"τ"%) -> <$"Seq"$ τ> -> <$"Seq"$ %"τ"%> >]).
+  set (k2:=[< (τ -> τ') -> <$"Seq"$ τ> -> <$"Seq"$ τ'> >]).
+
+  set (P:=(type_subst "τ" τ' k1) = k2).
+  
+(*  apply T_TypeApp with (α:="τ"%string) (τ:=k2).*)
+(*  apply T_TypeApp with (α:="τ"%string) (τ:=(type_subst "τ" τ' k1)).*)
+(*
+  apply map_type.
+
+  apply TSubst_UnivReplace.
   admit.
+  admit.
+
+  apply TSubst_UnivReplace.
 
   apply T_MorphFun.
   apply typing_weakening.
@@ -91,6 +111,7 @@ Proof.
   apply T_Var.
   apply C_take.
   apply M_Sub, TSubRepr_Refl, TEq_Refl.
+  *)
 Admitted.
 
 (* reduction step preserves well-typedness *)
@@ -185,7 +206,7 @@ Proof.
     apply T_App with (σ':=τ0) (σ:=τ0) (τ:=τ').
     apply T_MorphFun.
     apply T_MorphAbs.
-    apply T_DescendImplicit with (τ:=τ0).
+    apply T_Descend with (τ:=τ0).
     apply T_Var.
     apply C_take.
     apply H3.
@@ -303,7 +324,7 @@ Proof.
   apply H0.
 
   (* e is Desecension *)
-  apply T_DescendImplicit with (τ:=τ).
+  apply T_Descend with (τ:=τ).
   apply IHtranslate_typing.
   apply H0.
   apply H1.
diff --git a/coq/subst.v b/coq/subst.v
index 7cb13fc..5b8375f 100644
--- a/coq/subst.v
+++ b/coq/subst.v
@@ -1,60 +1,34 @@
- From Coq Require Import Strings.String.
+From Coq Require Import Strings.String.
+From Coq Require Import Lists.List.
+Import ListNotations.
 Require Import terms.
 
-(* Type Variable "x" is a free variable in type *)
-Inductive type_var_free (x:string) : type_term -> Prop :=
-  | TFree_Var :
-    (type_var_free x (type_var x))
-
-  | TFree_Ladder : forall τ1 τ2,
-    (type_var_free x τ1) ->
-    (type_var_free x τ2) ->
-    (type_var_free x (type_ladder τ1 τ2))
-
-  | TFree_Fun : forall τ1 τ2,
-    (type_var_free x τ1) ->
-    (type_var_free x τ2) ->
-    (type_var_free x (type_fun τ1 τ2))
-
-  | TFree_Morph : forall τ1 τ2,
-    (type_var_free x τ1) ->
-    (type_var_free x τ2) ->
-    (type_var_free x (type_morph τ1 τ2))
-
-  | TFree_Spec : forall τ1 τ2,
-    (type_var_free x τ1) ->
-    (type_var_free x τ2) ->
-    (type_var_free x (type_spec τ1 τ2))
-
-  | TFree_Univ : forall y τ,
-    ~(y = x) ->
-    (type_var_free x τ) ->
-    (type_var_free x (type_univ y τ))
-.
 
+Fixpoint type_fv (τ : type_term) {struct τ} : (list string) :=
+  match τ with
+  | type_id s => []
+  | type_var α => [α]
+  | type_univ α τ => (remove string_dec α (type_fv τ))
+  | type_spec σ τ => (type_fv σ) ++ (type_fv τ)
+  | type_fun σ τ => (type_fv σ) ++ (type_fv τ)
+  | type_morph σ τ => (type_fv σ) ++ (type_fv τ)
+  | type_ladder σ τ => (type_fv σ) ++ (type_fv τ)
+  end.
 
 Open Scope ladder_type_scope.
-Example ex_type_free_var1 :
-  (type_var_free "T" (type_univ "U" (type_var "T")))
+Example ex_type_fv1 :
+  (In "T"%string (type_fv [< ∀"U",%"T"% >]))
 .
-Proof.
-  apply TFree_Univ.
-  easy.
-  apply TFree_Var.
-Qed.
+Proof. simpl. left. auto. Qed.
 
 Open Scope ladder_type_scope.
-Example ex_type_free_var2 :
-  ~(type_var_free "T" (type_univ "T" (type_var "T")))
+Example ex_type_fv2 :
+  ~(In "T"%string (type_fv [< ∀"T",%"T"% >]))
 .
-Proof.
-  intro H.
-  inversion H.
-  contradiction.
-Qed.
+Proof. simpl. auto. Qed.
 
 (* scoped variable substitution in type terms $\label{coq:subst-type}$ *)
-Fixpoint type_subst (v:string) (n:type_term) (t0:type_term) :=
+Fixpoint type_subst (v:string) (n:type_term) (t0:type_term) : type_term :=
   match t0 with
   | type_var name => if (eqb v name) then n else t0
   | type_fun t1 t2 => (type_fun (type_subst v n t1) (type_subst v n t2))
@@ -64,12 +38,14 @@ Fixpoint type_subst (v:string) (n:type_term) (t0:type_term) :=
   | t => t
   end.
 
+(*
+
 Inductive type_subst1 (x:string) (σ:type_term) : type_term -> type_term -> Prop :=
   | TSubst_VarReplace :
     (type_subst1 x σ (type_var x) σ)
 
   | TSubst_VarKeep : forall y,
-    ~(x = y) ->
+    (x <> y) ->
     (type_subst1 x σ (type_var y) (type_var y))
 
   | TSubst_UnivReplace : forall y τ τ',
@@ -101,6 +77,56 @@ Inductive type_subst1 (x:string) (σ:type_term) : type_term -> type_term -> Prop
     (type_subst1 x σ τ2 τ2') ->
     (type_subst1 x σ (type_ladder τ1 τ2) (type_ladder τ1' τ2'))
 .
+*)
+
+Lemma type_subst_symm :
+  forall x y τ τ',
+  ((type_subst x (type_var y) τ) = τ') ->
+  ((type_subst y (type_var x) τ') = τ)
+.
+Proof.
+  intros.
+  induction H.
+
+  unfold type_subst.
+  induction τ.
+  reflexivity.
+
+  
+Admitted.
+
+Lemma type_subst_fresh :
+  forall α τ u,
+  ~ (In α (type_fv τ))
+  -> (type_subst α u τ) = τ
+.
+Proof.
+  intros.
+  unfold type_subst.
+  induction τ.
+  reflexivity.
+
+  
+  unfold eqb.
+
+  admit.
+(*
+  apply TSubst_Id.
+  apply TSubst_VarKeep.
+  contradict H.
+  rewrite H.
+  apply TFree_Var.
+
+  apply TSubst_Fun.
+  apply IHτ1.
+
+  contradict H.
+  apply TFree_Fun.
+  apply H.
+  apply 
+  *)
+Admitted.
+
 
 (* scoped variable substitution, replaces free occurences of v with n in expression e *)
 Fixpoint expr_subst (v:string) (n:expr_term) (e0:expr_term) :=
diff --git a/coq/typing.v b/coq/typing.v
index 1777111..8f1a845 100644
--- a/coq/typing.v
+++ b/coq/typing.v
@@ -11,6 +11,7 @@ Require Import morph.
 
 (** Typing Derivation *)
 
+Open Scope ladder_type_scope.
 Open Scope ladder_expr_scope.
 
 Reserved Notation "Gamma '|-' x '\is' X"  (at level 101, x at next level, X at level 0).
@@ -267,4 +268,116 @@ Proof.
   apply M_Sub. apply TSubRepr_Refl. apply TEq_Refl.
 Qed.
 
-End Typing.
+
+Ltac var_typing := auto using T_Var, C_shuffle, C_take.
+Ltac repr_subtype := auto using TSubRepr_Ladder, TSubRepr_Refl, TEq_Refl, TEq_LadderAssocLR.
+
+Example expand1 :
+  (translate_typing
+      (ctx_assign "60" [< $"ℝ"$ >]
+      (ctx_assign "360" [< $"ℝ"$ >]
+      (ctx_assign "/" [< $"ℝ"$ -> $"ℝ"$ -> $"ℝ"$ >]
+        ctx_empty)))
+  [{
+    let "deg2turns" :=
+       (λ"x",$"Angle"$~$"Degrees"$~$"ℝ"$
+                ↦morph ((%"/"% (%"x"% des $"ℝ"$) %"360"%) as $"Angle"$~$"Turns"$)) in
+    let "sin" := (λ"α",$"Angle"$~$"Turns"$~$"ℝ"$ ↦ (%"α"% des $"ℝ"$)) in
+    ( %"sin"% (%"60"% as $"Angle"$~$"Degrees"$) )
+  }]
+  [<
+     $"ℝ"$
+  >]
+  [{
+    let "deg2turns" :=
+       (λ"x",$"Angle"$~$"Degrees"$~$"ℝ"$
+                ↦morph ((%"/"% (%"x"% des $"ℝ"$) %"360"%) as $"Angle"$~$"Turns"$)) in
+    let "sin" := (λ"α",$"Angle"$~$"Turns"$~$"ℝ"$ ↦ (%"α"% des $"ℝ"$)) in
+    (%"sin"% (%"deg2turns"% (%"60"% as $"Angle"$~$"Degrees"$)))
+  }])
+.
+Proof.
+  apply Expand_Let with (σ:=[< ($"Angle"$~$"Degrees"$)~$"ℝ"$ ->morph ($"Angle"$~$"Turns"$)~$"ℝ"$ >]).
+
+  - apply T_DescendImplicit with (τ:=[< $"Angle"$~$"Degrees"$~$"ℝ"$ ->morph $"Angle"$~$"Turns"$ ~ $"ℝ"$ >]).
+    2: {
+      apply TSubRepr_Refl.
+      apply TEq_SubMorph.
+      apply TEq_LadderAssocRL.
+      apply TEq_LadderAssocRL.
+    }
+    apply T_MorphAbs.
+    apply T_DescendImplicit with (τ:=[< ($"Angle"$~$"Turns"$) ~ $"ℝ"$ >]).
+    2: {
+      apply TSubRepr_Refl.
+      apply TEq_LadderAssocLR.
+    }
+    apply T_Ascend with (τ:=[<$"ℝ"$>]) (τ':=[<$"Angle"$~$"Turns"$>]).
+    apply T_App with (σ':=[<$"ℝ"$>]) (σ:=[<$"ℝ"$>]).
+    apply T_App with (σ':=[<$"ℝ"$>]) (σ:=[<$"ℝ"$>]).
+    var_typing.
+    var_typing.
+    apply T_Descend with (τ:=[<$"Angle"$~$"Degrees"$~$"ℝ"$>]).
+    repr_subtype.
+    var_typing.
+    repr_subtype.
+    apply id_morphism_path.
+    var_typing.
+    apply id_morphism_path.
+
+  - apply T_Let with (σ:=[< $"Angle"$~$"Turns"$~$"ℝ"$ -> $"ℝ"$ >]).
+    apply T_Abs.
+    * apply T_Descend with (τ:=[<$"Angle"$~$"Turns"$~$"ℝ"$>]).
+      2: repr_subtype.
+      var_typing.
+
+    * apply T_App with (σ':=[<($"Angle"$~$"Degrees"$)~$"ℝ"$>])  (σ:=[<($"Angle"$~$"Turns"$)~$"ℝ"$>]) (τ:=[<$"ℝ"$>]).
+      apply T_DescendImplicit with (τ:=[< $"Angle"$~$"Turns"$~$"ℝ"$ -> $"ℝ"$ >]).
+      2: {
+        apply TSubRepr_Refl.
+        apply TEq_SubFun.
+        apply TEq_LadderAssocRL.
+        apply TEq_Refl.
+      }
+      var_typing.
+
+      apply T_Ascend with (τ':=[<$"Angle"$~$"Degrees"$>]) (τ:=[<$"ℝ"$>]).
+      var_typing.
+
+      apply M_Single with (h:="deg2turns"%string).
+      var_typing.
+
+  - admit.
+  (*
+  - apply Expand_MorphAbs.
+    * apply T_DescendImplicit with (τ:=[< ($"Angle"$~$"Turns"$) ~ $"ℝ"$ >]).
+      2: repr_subtype.
+      apply T_Ascend with (τ':=[<$"Angle"$~$"Turns"$>]) (τ:=[<$"ℝ"$>]).
+      apply T_App with (σ:=[<$"ℝ"$>]) (σ':=[<$"ℝ"$>]).
+      apply T_App with (σ:=[<$"ℝ"$>]) (σ':=[<$"ℝ"$>]).
+      auto using T_Var, C_shuffle, C_take.
+      apply T_Descend with (τ:=[<$"Angle"$~$"Degrees"$~$"ℝ"$>]).
+      2: repr_subtype.
+      var_typing.
+      apply id_morphism_path.
+      var_typing.
+      apply id_morphism_path.
+
+    * apply T_Abs.
+      apply T_DescendImplicit with (τ:=[< ($"Angle"$~$"Turns"$) ~ $"ℝ"$ >]).
+      2: repr_subtype.
+      apply T_Ascend with (τ':=[<$"Angle"$~$"Turns"$>]) (τ:=[<$"ℝ"$>]).
+      apply T_App with (σ':=[<$"ℝ"$>]) (σ:=[<$"ℝ"$>]).
+      apply T_App with (σ':=[<$"ℝ"$>]) (σ:=[<$"ℝ"$>]).
+      var_typing.
+      apply T_Descend with [<$"Angle"$~$"Degrees"$~$"ℝ"$>].
+      2: repr_subtype.
+      var_typing.
+      apply id_morphism_path.
+      var_typing.
+      apply id_morphism_path.
+
+    * apply Expand_Ascend.
+*)
+  - admit.
+Admitted.