From 6e5c832db7b155cd2cadb968d49cb33fcf928344 Mon Sep 17 00:00:00 2001
From: Michael Sippel <micha@fragmental.art>
Date: Wed, 18 Sep 2024 11:15:20 +0200
Subject: [PATCH] add notation for sequence types & use notations everywhere

---
 coq/bbencode.v  | 148 ++++++++++++++++++++++++++++--------------------
 coq/equiv.v     | 102 +++++++++++++++++----------------
 coq/morph.v     |  45 ++++++++-------
 coq/smallstep.v |  36 ++++++------
 coq/soundness.v |   1 -
 coq/terms.v     |  44 ++++++++------
 coq/typing.v    | 110 +++++++++++++++++------------------
 7 files changed, 260 insertions(+), 226 deletions(-)

diff --git a/coq/bbencode.v b/coq/bbencode.v
index 144680f..b8cd26c 100644
--- a/coq/bbencode.v
+++ b/coq/bbencode.v
@@ -4,77 +4,101 @@ Require Import terms.
 Require Import subst.
 Require Import smallstep.
 
-Include Terms.
-Include Subst.
-Include Smallstep.
-
 Open Scope ladder_type_scope.
 Open Scope ladder_expr_scope.
 
-(*  let bb_zero = Λα ↦ λs: α->α ↦ λz: α ↦ z
- *                ∀α.(α->α)->α->α
-*)
-Definition bb_zero : expr_term :=
-  (expr_ty_abs "α"
-    (expr_abs "s" (type_fun (type_var "α") (type_var "α"))
-      (expr_abs "z" (type_var "α")
-        (expr_var "z")))).
 
-(*  let bb_one = Λα ↦ λs: α->α ↦ λz: α ↦ s z
-*)
-Definition bb_one : expr_term :=
-  (expr_ty_abs "α"
-    (expr_abs "s" (type_fun (type_var "α") (type_var "α"))
-      (expr_abs "z" (type_var "α")
-        (expr_app (expr_var "s") (expr_var "z"))))).
+Definition bb_zero : expr_term := [{
+  (Λ"α" ↦
+  λ"s",(%"α"% -> %"α"%) ↦
+  λ"z",(%"α"%) ↦
+  %"z"%)
+  as $"ℕ"$~$"BBNat"$
+}].
 
-(*  let bb_two = Λα ↦ λs: α->α ↦ λz: α ↦ s (s z)
-*)
-Definition bb_two : expr_term :=
-  (expr_ty_abs "α"
-    (expr_abs "s" (type_fun (type_var "α") (type_var "α"))
-      (expr_abs "z" (type_var "α")
-        (expr_app (expr_var "s") (expr_app (expr_var "s") (expr_var "z")))))).
+Definition bb_one : expr_term := [{
+  (Λ"α" ↦
+  λ"s",(%"α"% -> %"α"%) ↦
+  λ"z",(%"α"%) ↦
+  (%"s"% %"z"%))
+  as $"ℕ"$~$"BBNat"$
+}].
 
+Definition bb_two : expr_term := [{
+  (Λ"α" ↦
+  λ"s",(%"α"% -> %"α"%) ↦
+  λ"z",(%"α"%) ↦
+  (%"s"% (%"s"% %"z"%)))
+  as $"ℕ"$~$"BBNat"$
+}].
 
-Definition bb_succ : expr_term :=
-  (expr_abs "n" (type_ladder (type_id "ℕ")
-                    (type_ladder (type_id "BBNat")
-                     (type_univ "α"
-                      (type_fun (type_fun (type_var "α") (type_var "α"))
-                         (type_fun (type_var "α") (type_var "α"))))))
+Definition bb_succ : expr_term := [{
+  λ"n",$"ℕ"$
+      ~$"BBNat"$
+      ~(∀"β",(%"β'"%->%"β"%)->%"β"%->%"β"%)
+      ↦
+    ((Λ"α" ↦
+      λ"s",((%"α"%->%"α"%)->%"α"%->%"α"%) ↦
+      λ"z",%"α"% ↦
+      (%"s"% ((%"n"% # %"α"%) %"s"% %"z"%)))
+      as $"ℕ"$~$"BBNat"$)
+}].
 
-       (expr_ascend (type_ladder (type_id "ℕ") (type_id "BBNat"))
-          (expr_ty_abs "α"
-            (expr_abs "s" (type_fun (type_var "α") (type_var "α"))
-              (expr_abs "z" (type_var "α")
-                (expr_app (expr_var "s")
-                  (expr_app (expr_app
-                      (expr_ty_app (expr_var "n") (type_var "α"))
-                      (expr_var "s"))
-                    (expr_var "z")))))))).
-
-Definition e1 : expr_term :=
-  (expr_let "bb-zero" bb_zero
-      (expr_app (expr_app (expr_var "+") (expr_var "bb-zero")) (expr_var "bb-zero"))
-  ).
-
-Definition t1 : expr_term := (expr_app (expr_var "x") (expr_var "x")).
-
-Compute (expr_subst "x"
-    (expr_ty_abs "α" (expr_abs "a" (type_var "α") (expr_var "a")))
-    bb_one
-).
-
-Example  example_let_reduction :
-    e1 -->β  (expr_app (expr_app (expr_var "+") bb_zero) bb_zero).
+Example  example_reduction :
+    (expr_app bb_succ bb_one) -->β* bb_two
+.
 Proof.
-    apply E_Let.
+
+  apply Multi_Step with (y:=[{
+    (λ"n",$"ℕ"$
+        ~$"BBNat"$
+        ~(∀"β",(%"β'"%->%"β"%)->%"β"%->%"β"%)
+        ↦
+      ((Λ"α" ↦
+        λ"s",((%"α"%->%"α"%)->%"α"%->%"α"%) ↦
+        λ"z",%"α"% ↦
+        (%"s"% ((%"n"% # %"α"%) %"s"% %"z"%)))
+        as $"ℕ"$~$"BBNat"$))
+
+    (Λ"β" ↦
+     λ"s",(%"β"% -> %"β"%) ↦
+     λ"z",(%"β"%) ↦
+      (%"s"% %"z"%))
+      as $"ℕ"$~$"BBNat"$
+  }]).
+Admitted.
+(*
+  apply E_Alpha.
+  apply E_App2.
+  apply EAlpha_Rename.
+
+  apply Multi_Step with (y:=[{
+      (Λ"α" ↦
+        λ"s",((%"α"%->%"α"%)->%"α"%->%"α"%) ↦
+        λ"z",%"α"% ↦
+        (%"s"% (
+        (((Λ"β" ↦
+          λ"s",(%"β"% -> %"β"%) ↦
+          λ"z",(%"β"%) ↦
+          (%"s"% %"z"%)) as $"ℕ"$~$"BBNat"$)
+          # %"α"%) %"s"% %"z"%)))
+    as $"ℕ"$~$"BBNat"$ }]).
+
+  apply E_AppLam.
+
+  apply Multi_Step with (y:=[{
+      (Λ"α" ↦
+        λ"s",((%"α"%->%"α"%)->%"α"%->%"α"%) ↦
+        λ"z",%"α"% ↦
+        (%"s"% (
+        (((Λ"α'" ↦
+          λ"s",(%"α'"% -> %"α'"%) ↦
+          λ"z",(%"α'"%) ↦
+          (%"s"% %"z"%)) as $"ℕ"$~$"BBNat"$)
+          # %"α"%) %"s"% %"z"%)))
+    as $"ℕ"$~$"BBNat"$ }]).
+  }].
 Qed.
+*)
 
 Compute     (expr_app bb_succ bb_zero) -->β bb_one.
-
-Example  example_succ :
-    (expr_app bb_succ bb_zero) -->β bb_one.
-Proof.
-Admitted.
diff --git a/coq/equiv.v b/coq/equiv.v
index 7ed4272..630783f 100644
--- a/coq/equiv.v
+++ b/coq/equiv.v
@@ -38,46 +38,44 @@ From Coq Require Import Strings.String.
 
 Reserved Notation "S '--->α' T" (at level 40).
 Inductive type_conv_alpha : type_term -> type_term -> Prop :=
-  | TAlpha_Rename : forall x y t t',
-
-  (type_subst1 x (type_var y) t t') ->
-  (type_univ x t) --->α (type_univ y t')
+  | TAlpha_Rename : forall x y t,
+  (type_univ x t) --->α (type_univ y (type_subst x (type_var y) t))
 
   | TAlpha_SubUniv : forall x τ τ',
     (τ --->α τ') ->
-    (type_univ x τ) --->α (type_univ x τ')
+    [< ∀x,τ >] --->α [< ∀x,τ' >]
 
   | TAlpha_SubSpec1 : forall τ1 τ1' τ2,
     (τ1 --->α τ1') ->
-    (type_spec τ1 τ2) --->α (type_spec τ1' τ2)
+    [< <τ1 τ2> >] --->α [< <τ1' τ2> >]
 
   | TAlpha_SubSpec2 : forall τ1 τ2 τ2',
     (τ2 --->α τ2') ->
-    (type_spec τ1 τ2) --->α (type_spec τ1 τ2')
+    [< <τ1 τ2> >] --->α [< <τ1 τ2'> >]
 
   | TAlpha_SubFun1 : forall τ1 τ1' τ2,
     (τ1 --->α τ1') ->
-    (type_fun τ1 τ2) --->α (type_fun τ1' τ2)
+    [< τ1 -> τ2 >] --->α [< τ1' -> τ2 >]
 
   | TAlpha_SubFun2 : forall τ1 τ2 τ2',
     (τ2 --->α τ2') ->
-    (type_fun τ1 τ2) --->α (type_fun τ1 τ2')
+    [< τ1 -> τ2 >] --->α [< τ1 -> τ2' >]
 
   | TAlpha_SubMorph1 : forall τ1 τ1' τ2,
     (τ1 --->α τ1') ->
-    (type_morph τ1 τ2) --->α (type_morph τ1' τ2)
+    [< τ1 ->morph τ2 >] --->α [< τ1' ->morph τ2 >]
 
   | TAlpha_SubMorph2 : forall τ1 τ2 τ2',
     (τ2 --->α τ2') ->
-    (type_morph τ1 τ2) --->α (type_morph τ1 τ2')
+    [< τ1 ->morph τ2 >] --->α [< τ1 ->morph τ2' >]
 
   | TAlpha_SubLadder1 : forall τ1 τ1' τ2,
     (τ1 --->α τ1') ->
-    (type_ladder τ1 τ2) --->α (type_ladder τ1' τ2)
+    [< τ1 ~ τ2 >] --->α [< τ1' ~ τ2 >]
 
   | TAlpha_SubLadder2 : forall τ1 τ2 τ2',
     (τ2 --->α τ2') ->
-    (type_ladder τ1 τ2) --->α (type_ladder τ1 τ2')
+    [< τ1 ~ τ2 >] --->α [< τ1 ~ τ2' >]
 
 where "S '--->α' T" := (type_conv_alpha S T).
 
@@ -86,38 +84,48 @@ Lemma type_alpha_symm :
   forall σ τ,
   (σ --->α τ) -> (τ --->α σ).
 Proof.
- (* TODO *)
+  intros.
+  induction H.
+  - admit.
+  - auto using TAlpha_Rename,type_subst,type_subst_symm, TAlpha_SubUniv.
+  - auto using TAlpha_SubSpec1.
+  - auto using TAlpha_SubSpec2.
+  - auto using TAlpha_SubFun1.
+  - auto using TAlpha_SubFun2.
+  - auto using TAlpha_SubMorph1.
+  - auto using TAlpha_SubMorph2.
+  - auto using TAlpha_SubLadder1.
+  - auto using TAlpha_SubLadder2.
 Admitted.
 
-
 (** Define all rewrite steps $\label{coq:type-dist}$ *)
 
 Reserved Notation "S '-->distribute-ladder' T" (at level 40).
 Inductive type_distribute_ladder : type_term -> type_term -> Prop :=
   | L_DistributeOverSpec1 : forall x x' y,
-    (type_spec (type_ladder x x') y)
+    [<  <x~x' y>  >]
     -->distribute-ladder
-    (type_ladder (type_spec x y) (type_spec x' y))
+    [<  <x y>~<x' y>  >]
 
   | L_DistributeOverSpec2 : forall x y y',
-    (type_spec x (type_ladder y y'))
+    [<  <x y~y'>  >]
     -->distribute-ladder
-    (type_ladder (type_spec x y) (type_spec x y'))
+    [<  <x y>~<x y'>  >]
 
   | L_DistributeOverFun1 : forall x x' y,
-    (type_fun (type_ladder x x') y)
+    [<  (x~x' -> y)  >]
     -->distribute-ladder
-    (type_ladder (type_fun x y) (type_fun x' y))
+    [<  (x -> y) ~ (x' -> y)  >]
 
   | L_DistributeOverFun2 : forall x y y',
-    (type_fun x (type_ladder y y'))
+    [<  (x -> y~y')  >]
     -->distribute-ladder
-    (type_ladder (type_fun x y) (type_fun x y'))
+    [<  (x -> y) ~ (x -> y')  >]
 
   | L_DistributeOverMorph1 : forall x x' y,
-    (type_morph (type_ladder x x') y)
+    [<  (x~x' ->morph y)  >]
     -->distribute-ladder
-    (type_ladder (type_morph x y) (type_morph x' y))
+    [<  (x ->morph y) ~ (x' ->morph y)  >]
 
   | L_DistributeOverMorph2 : forall x y y',
     (type_morph x (type_ladder y y'))
@@ -131,34 +139,34 @@ where "S '-->distribute-ladder' T" := (type_distribute_ladder S T).
 Reserved Notation "S '-->condense-ladder' T" (at level 40).
 Inductive type_condense_ladder : type_term -> type_term -> Prop :=
   | L_CondenseOverSpec1 : forall x x' y,
-    (type_ladder (type_spec x y) (type_spec x' y))
+    [< <x y>~<x' y> >]
     -->condense-ladder
-    (type_spec (type_ladder x x') y)
+    [< <x~x' y> >]
 
   | L_CondenseOverSpec2 : forall x y y',
-    (type_ladder (type_spec x y) (type_spec x y'))
+    [< <x y>~<x y'> >]
     -->condense-ladder
-    (type_spec x (type_ladder y y'))
+    [< <x y~y'> >]
 
   | L_CondenseOverFun1 : forall x x' y,
-    (type_ladder (type_fun x y) (type_fun x' y))
+    [< (x -> y) ~ (x' -> y) >]
     -->condense-ladder
-    (type_fun (type_ladder x x') y)
+    [< (x~x') -> y >]
 
   | L_CondenseOverFun2 : forall x y y',
-    (type_ladder (type_fun x y) (type_fun x y'))
+    [< (x -> y) ~ (x -> y') >]
     -->condense-ladder
-    (type_fun x (type_ladder y y'))
+    [< (x -> y~y') >]
 
   | L_CondenseOverMorph1 : forall x x' y,
-    (type_ladder (type_morph x y) (type_morph x' y))
+    [< (x ->morph y) ~ (x' ->morph y) >]
     -->condense-ladder
-    (type_morph (type_ladder x x') y)
+    [< (x~x' ->morph y) >]
 
   | L_CondenseOverMorph2 : forall x y y',
-    (type_ladder (type_morph x y) (type_morph x y'))
+    [< (x ->morph y) ~ (x ->morph y') >]
     -->condense-ladder
-    (type_morph x (type_ladder y y'))
+    [< (x ->morph y~y') >]
 
 where "S '-->condense-ladder' T" := (type_condense_ladder S T).
 
@@ -213,14 +221,14 @@ Inductive type_eq : type_term -> type_term -> Prop :=
     x === z
 
   | TEq_LadderAssocLR : forall x y z,
-    (type_ladder (type_ladder x y) z)
+    [< (x~y)~z >]
     ===
-    (type_ladder x (type_ladder y z))
+    [< x~(y~z) >]
 
   | TEq_LadderAssocRL : forall x y z,
-    (type_ladder x (type_ladder y z))
+    [< x~(y~z) >]
     ===
-    (type_ladder (type_ladder x y) z)
+    [< (x~y)~z >]
 
   | TEq_Alpha : forall x y,
     x --->α y ->
@@ -378,7 +386,7 @@ Admitted.
  *)
 
 Example example_flat_type :
-    (type_is_flat (type_spec (type_id "PosInt") (type_id "10"))).
+    type_is_flat [< <$"PosInt"$ $"10"$> >].
 Proof.
     apply FlatApp.
     apply FlatId.
@@ -386,17 +394,15 @@ Proof.
 Qed.
 
 Example example_lnf_type :
-    (type_is_lnf
-    (type_ladder (type_spec (type_id "Seq") (type_id "Char"))
-               (type_spec (type_id "Seq") (type_id "Byte")))).
+    type_is_lnf [< [$"Char"$] ~ [$"Byte"$] >].
 Proof.
 Admitted.
 
 Example example_type_eq :
-    (type_spec (type_id "Seq") (type_ladder (type_id "Char") (type_id "Byte")))
+    [< [$"Char"$ ~ $"Byte"$] >]
     ===
-    (type_ladder (type_spec (type_id "Seq") (type_id "Char"))
-               (type_spec (type_id "Seq") (type_id "Byte"))).
+    [< [$"Char"$] ~ [$"Byte"$] >]
+.
 Proof.
   apply TEq_Distribute.
   apply L_DistributeOverSpec2.
diff --git a/coq/morph.v b/coq/morph.v
index a686102..4202d4a 100644
--- a/coq/morph.v
+++ b/coq/morph.v
@@ -8,13 +8,15 @@ Require Import context.
 (* Given a context, there is a morphism path from τ to τ' *)
 Reserved Notation "Γ '|-' σ '~>' τ" (at level 101, σ at next level, τ at next level).
 
+Open Scope ladder_expr_scope.
+
 Inductive morphism_path : context -> type_term -> type_term -> Prop :=
   | M_Sub : forall Γ τ τ',
     (τ :<= τ') ->
     (Γ |- τ ~> τ')
 
   | M_Single : forall Γ h τ τ',
-    (context_contains Γ h (type_morph τ τ')) ->
+    (context_contains Γ h [< τ ->morph τ' >]) ->
     (Γ |- τ ~> τ')
 
   | M_Chain : forall Γ τ τ' τ'',
@@ -24,58 +26,57 @@ Inductive morphism_path : context -> type_term -> type_term -> Prop :=
 
   | M_Lift : forall Γ σ τ τ',
     (Γ |- τ ~> τ') ->
-    (Γ |- (type_ladder σ τ) ~> (type_ladder σ τ'))
+    (Γ |- [< σ ~ τ >] ~> [< σ ~ τ' >])
 
   | M_MapSeq : forall Γ τ τ',
     (Γ |- τ ~> τ') ->
-    (Γ |- (type_spec (type_id "Seq") τ) ~> (type_spec (type_id "Seq") τ'))
+    (Γ |- [< [τ] >] ~> [< [τ'] >])
 
 where "Γ '|-' s '~>' t" := (morphism_path Γ s t).
 
+Lemma id_morphism_path : forall Γ τ, Γ |- τ ~> τ.
+Proof.
+  intros.
+  apply M_Sub, TSubRepr_Refl, TEq_Refl.
+Qed.
 
 Inductive translate_morphism_path : context -> type_term -> type_term -> expr_term -> Prop :=
-  | Translate_Subtype : forall Γ τ τ',
+  | Translate_Descend : forall Γ τ τ',
     (τ :<= τ') ->
     (translate_morphism_path Γ τ τ'
-        (expr_morph "x" τ (expr_var "x")))
+      (expr_morph "x" τ [{ %"x"% des τ' }]))
 
   | Translate_Lift : forall Γ σ τ τ' m,
     (Γ |- τ ~> τ') ->
     (translate_morphism_path Γ τ τ' m) ->
-    (translate_morphism_path Γ (type_ladder σ τ) (type_ladder σ τ')
-        (expr_morph "x" (type_ladder σ τ)
-          (expr_ascend σ (expr_app m (expr_descend τ (expr_var "x"))))))
+    (translate_morphism_path Γ [< σ ~ τ >] [< σ ~ τ' >]
+      (expr_morph "x" [< σ ~ τ >] [{ (m (%"x"% des τ)) as σ }]))
 
   | Translate_Single : forall Γ h τ τ',
-    (context_contains Γ h (type_morph τ τ')) ->
-    (translate_morphism_path Γ τ τ' (expr_var h))
+    (context_contains Γ h [< τ ->morph τ' >]) ->
+    (translate_morphism_path Γ τ τ' [{ %h% }])
 
   | Translate_Chain : forall Γ τ τ' τ'' m1 m2,
     (translate_morphism_path Γ τ τ' m1) -> 
     (translate_morphism_path Γ τ' τ'' m2) ->
     (translate_morphism_path Γ τ τ''
-      (expr_morph "x" τ (expr_app m2 (expr_app m1 (expr_var "x")))))
+      (expr_morph "x" τ [{ m2 (m1 %"x"%) }]))
 
   | Translate_MapSeq : forall Γ τ τ' m,
     (translate_morphism_path Γ τ τ' m) ->
-    (translate_morphism_path Γ
-      (type_spec (type_id "Seq") τ)
-      (type_spec (type_id "Seq") τ')
-      (expr_morph "xs"
-        (type_spec (type_id "Seq") τ)
-        (expr_app (expr_app (expr_ty_app (expr_ty_app
-          (expr_var "map") τ) τ') m)
-          (expr_var "xs"))))
+    (translate_morphism_path Γ [< [τ] >] [< [τ'] >]
+      [{
+        λ"xs",[τ] ↦morph (%"map"% # τ # τ' m %"xs"%)
+      }])
 .
 
-
 Example morphism_paths :
   (ctx_assign "degrees-to-turns" [< $"Angle"$~$"Degrees"$~$"ℝ"$ ->morph $"Angle"$~$"Turns"$~$"ℝ"$ >]
     (ctx_assign "turns-to-radians" [< $"Angle"$~$"Turns"$~$"ℝ"$ ->morph $"Angle"$~$"Radians"$~$"ℝ"$ >]
       ctx_empty))
 
-  |- [< <$"Seq"$ $"Hue"$~$"Angle"$~$"Degrees"$~$"ℝ"$> >]
-  ~> [< <$"Seq"$ $"Hue"$~$"Angle"$~$"Radians"$~$"ℝ"$> >]
+  |- [< [ $"Hue"$~$"Angle"$~$"Degrees"$~$"ℝ"$ ] >]
+  ~> [< [ $"Hue"$~$"Angle"$~$"Radians"$~$"ℝ"$ ] >]
 .
 Proof.
   intros.
diff --git a/coq/smallstep.v b/coq/smallstep.v
index d5cb87b..ab6bfe4 100644
--- a/coq/smallstep.v
+++ b/coq/smallstep.v
@@ -16,19 +16,19 @@ Inductive expr_alpha : expr_term -> expr_term -> Prop :=
 
   | EAlpha_SubAbs : forall x τ e e',
     (e -->α e') ->
-    (expr_abs x τ e) -->α (expr_abs x τ e')
+    [{ λ x , τ ↦ e }] -->α [{ λ x , τ ↦ e' }]
 
   | EAlpha_SubTyAbs : forall α e e',
     (e -->α e') ->
-    (expr_ty_abs α e) -->α (expr_ty_abs α e')
+    [{ Λ α ↦ e }] -->α [{ Λ α ↦ e' }]
 
   | EAlpha_SubApp1 : forall e1 e1' e2,
     (e1 -->α e1') ->
-    (expr_app e1 e2) -->α (expr_app e1' e2)
+    [{ e1 e2 }] -->α [{ e1' e2 }]
 
   | EAlpha_SubApp2 : forall e1 e2 e2',
     (e2 -->α e2') ->
-    (expr_app e1 e2) -->α (expr_app e1 e2')
+    [{ e1 e2 }] -->α [{ e1 e2' }]
 
 where "s '-->α' t" := (expr_alpha s t).
 
@@ -45,45 +45,45 @@ Qed.
 Inductive beta_step : expr_term -> expr_term -> Prop :=
   | E_App1 : forall e1 e1' e2,
     e1 -->β e1' ->
-    (expr_app e1 e2) -->β (expr_app e1' e2)
+    [{ e1 e2 }] -->β [{ e1' e2 }]
 
   | E_App2 : forall v1 e2 e2',
     (is_value v1) ->
     e2 -->β e2' ->
-    (expr_app v1 e2) -->β (expr_app v1 e2')
+    [{ v1 e2 }] -->β [{ v1 e2' }]
 
   | E_TypApp : forall e e' τ,
     e -->β e' ->
-    (expr_ty_app e τ) -->β (expr_ty_app e' τ)
+    [{ Λ τ ↦ e }] -->β [{ Λ τ ↦ e' }]
 
   | E_TypAppLam : forall α e τ,
-    (expr_ty_app (expr_ty_abs α e) τ) -->β (expr_specialize α τ e)
+    [{ (Λ α ↦ e) # τ }] -->β (expr_specialize α τ e)
 
   | E_AppLam : forall x τ e a,
-    (expr_app (expr_abs x τ e) a) -->β (expr_subst x a e)
+    [{ (λ x , τ ↦ e) a }] -->β (expr_subst x a e)
 
   | E_AppMorph : forall x τ e a,
-    (expr_app (expr_morph x τ e) a) -->β (expr_subst x a e)
+    [{ (λ x , τ ↦morph e) a }] -->β (expr_subst x a e)
 
   | E_Let : forall x e a,
-    (expr_let x a e) -->β (expr_subst x a e)
+    [{ let x := a in e }] -->β (expr_subst x a e)
 
   | E_StripAscend : forall τ e,
-    (expr_ascend τ e) -->β e
+    [{ e as τ }] -->β e
 
   | E_StripDescend : forall τ e,
-    (expr_descend τ e) -->β e
+    [{ e des τ }] -->β e
 
   | E_Ascend : forall τ e e',
     (e -->β e') ->
-    (expr_ascend τ e) -->β (expr_ascend τ e')
+    [{ e as τ }] -->β [{ e' as τ }]
 
   | E_AscendCollapse : forall τ' τ e,
-    (expr_ascend τ' (expr_ascend τ e)) -->β (expr_ascend (type_ladder τ' τ) e)
+    [{ (e as τ) as τ' }] -->β [{ e as (τ'~τ) }]
 
   | E_DescendCollapse : forall τ' τ e,
     (τ':<=τ) ->
-    (expr_descend τ (expr_descend τ' e)) -->β (expr_descend τ e)
+    [{ (e des τ') des τ }] -->β [{ e des τ }]
 
 where "s '-->β' t" := (beta_step s t).
 
@@ -102,7 +102,7 @@ Notation " s -->β* t " := (multi beta_step s t) (at level 40).
 Example reduce1 :
   [{
     let "deg2turns" :=
-       (λ"x" $"Angle"$~$"Degrees"$~$"ℝ"$
+       (λ"x",$"Angle"$~$"Degrees"$~$"ℝ"$
                 ↦morph ((%"/"% (%"x"% des $"ℝ"$) %"360"%) as $"Angle"$~$"Turns"$))
     in ( %"deg2turns"% (%"60"% as $"Angle"$~$"Degrees"$) )
   }]
@@ -111,7 +111,7 @@ Example reduce1 :
      ((%"/"% %"60"%) %"360"%) as $"Angle"$~$"Turns"$
   }].
 Proof.
-  apply Multi_Step with (y:=[{ (λ"x" $"Angle"$~$"Degrees"$~$"ℝ"$
+  apply Multi_Step with (y:=[{ (λ"x",$"Angle"$~$"Degrees"$~$"ℝ"$
                 ↦morph (((%"/"% (%"x"% des $"ℝ"$)) %"360"%) as $"Angle"$~$"Turns"$)) (%"60"% as $"Angle"$~$"Degrees"$) }]).
   apply E_Let.
 
diff --git a/coq/soundness.v b/coq/soundness.v
index d121044..e550a7d 100644
--- a/coq/soundness.v
+++ b/coq/soundness.v
@@ -8,7 +8,6 @@ Require Import morph.
 Require Import smallstep.
 Require Import typing.
 
-
 Lemma typing_weakening : forall Γ e τ x σ,
   (Γ |- e \is τ) ->
   ((ctx_assign x σ Γ) |- e \is τ)
diff --git a/coq/terms.v b/coq/terms.v
index 2ae8d33..5b7e657 100644
--- a/coq/terms.v
+++ b/coq/terms.v
@@ -64,58 +64,66 @@ Declare Custom Entry ladder_type.
 Declare Custom Entry ladder_expr.
 
 Notation "[< t >]" := t
-  (t custom ladder_type at level 80) : ladder_type_scope.
+  (t custom ladder_type at level 99) : ladder_type_scope.
 Notation "t" := t
   (in custom ladder_type at level 0, t ident) : ladder_type_scope.
 Notation "'∀' x ',' t" := (type_univ x t)
   (t custom ladder_type at level 80, in custom ladder_type at level 80, x constr).
 Notation "'<' σ τ '>'" := (type_spec σ τ)
   (in custom ladder_type at level 80, left associativity) : ladder_type_scope.
-Notation "'(' τ ')'" := τ
+Notation "'[' τ ']'" := (type_spec (type_id "Seq") τ)
   (in custom ladder_type at level 70) : ladder_type_scope.
+Notation "'(' τ ')'" := τ
+  (in custom ladder_type at level 5) : ladder_type_scope.
 Notation "σ '->' τ" := (type_fun σ τ)
   (in custom ladder_type at level 75, right associativity) : ladder_type_scope.
 Notation "σ '->morph' τ" := (type_morph σ τ)
   (in custom ladder_type at level 75, right associativity, τ at level 80) : ladder_type_scope.
 Notation "σ '~' τ" := (type_ladder σ τ)
-  (in custom ladder_type at level 70, right associativity) : ladder_type_scope.
+  (in custom ladder_type at level 20, right associativity) : ladder_type_scope.
 Notation "'$' x '$'" := (type_id x%string)
-  (in custom ladder_type at level 0, x constr) : ladder_type_scope.
+  (in custom ladder_type at level 10, x constr) : ladder_type_scope.
 Notation "'%' x '%'" := (type_var x%string)
-  (in custom ladder_type at level 0, x constr) : ladder_type_scope.
+  (in custom ladder_type at level 10, x constr) : ladder_type_scope.
 
 Notation "[{ e }]" := e
-  (e custom ladder_expr at level 80) : ladder_expr_scope.
+  (e custom ladder_expr at level 99) : ladder_expr_scope.
 Notation "e" := e
   (in custom ladder_expr at level 0, e ident) : ladder_expr_scope.
 Notation "'%' x '%'" := (expr_var x%string)
-  (in custom ladder_expr at level 0, x constr) : ladder_expr_scope.
+  (in custom ladder_expr at level 10, x constr) : ladder_expr_scope.
 Notation "'Λ' t '↦' e" := (expr_ty_abs t e)
-  (in custom ladder_expr at level 10, t constr, e custom ladder_expr at level 80) : ladder_expr_scope.
-Notation "'λ' x τ '↦' e" := (expr_abs x τ e)
-  (in custom ladder_expr at level 10, x constr, τ custom ladder_type at level 99, e custom ladder_expr at level 99) :ladder_expr_scope.
-Notation "'λ' x τ '↦morph' e" := (expr_morph x τ e)
-  (in custom ladder_expr at level 10, x constr, τ custom ladder_type at level 99, e custom ladder_expr at level 99) :ladder_expr_scope.
+  (in custom ladder_expr at level 10, t constr, e custom ladder_expr at level 80, right associativity) : ladder_expr_scope.
+Notation "'λ' x ',' τ '↦' e" := (expr_abs x τ e)
+  (in custom ladder_expr at level 70, x constr, τ custom ladder_type at level 90, e custom ladder_expr at level 80, right associativity) :ladder_expr_scope.
+Notation "'λ' x ',' τ '↦morph' e" := (expr_morph x τ e)
+  (in custom ladder_expr at level 70, x constr, τ custom ladder_type at level 90, e custom ladder_expr at level 80, right associativity) :ladder_expr_scope.
 Notation "'let' x ':=' e 'in' t" := (expr_let x e t)
-  (in custom ladder_expr at level 20, x constr, e custom ladder_expr at level 99, t custom ladder_expr at level 99) : ladder_expr_scope.
+  (in custom ladder_expr at level 60, x constr, e custom ladder_expr at level 80, t custom ladder_expr at level 80, right associativity) : ladder_expr_scope.
 Notation "e 'as' τ" := (expr_ascend τ e)
   (in custom ladder_expr at level 30, e custom ladder_expr, τ custom ladder_type at level 99) : ladder_expr_scope.
 Notation "e 'des' τ" := (expr_descend τ e)
   (in custom ladder_expr at level 30, e custom ladder_expr, τ custom ladder_type at level 99) : ladder_expr_scope.
 Notation "e1 e2" := (expr_app e1 e2)
-  (in custom ladder_expr at level 50) : ladder_expr_scope.
+  (in custom ladder_expr at level 90, e2 custom ladder_expr at next level) : ladder_expr_scope.
+Notation "e '#' τ" := (expr_ty_app e τ)
+  (in custom ladder_expr at level 80, τ custom ladder_type at level 101, left associativity) : ladder_expr_scope.
 Notation "'(' e ')'" := e
-  (in custom ladder_expr at level 0) : ladder_expr_scope.
+  (in custom ladder_expr, e custom ladder_expr at next level, left associativity) : ladder_expr_scope.
+
+
+Print Grammar constr.
 
 (* EXAMPLES *)
 
 Open Scope ladder_type_scope.
 Open Scope ladder_expr_scope.
 
-Check [< ∀"α", (< $"Seq"$ %"α"% > ~ < $"List"$ %"α"% >) >].
+Check [< ∀"α", [%"α"%] ~ <$"List"$ %"α"%> >].
 
-Definition polymorphic_identity1 : expr_term := [{ Λ"T" ↦ λ"x"%"T"% ↦ (%"x"%) }].
-Definition polymorphic_identity2 : expr_term := [{ Λ"T" ↦ λ"y"%"T"% ↦ %"y"% }].
+
+Definition polymorphic_identity1 : expr_term := [{ Λ"T" ↦ λ"x",%"T"% ↦ (%"x"%) }].
+Definition polymorphic_identity2 : expr_term := [{ Λ"T" ↦ λ"y",%"T"% ↦ %"y"% }].
 
 Compute polymorphic_identity1.
 
diff --git a/coq/typing.v b/coq/typing.v
index 34710c7..1777111 100644
--- a/coq/typing.v
+++ b/coq/typing.v
@@ -11,40 +11,41 @@ Require Import morph.
 
 (** Typing Derivation *)
 
+Open Scope ladder_expr_scope.
+
 Reserved Notation "Gamma '|-' x '\is' X"  (at level 101, x at next level, X at level 0).
 
 Inductive expr_type : context -> expr_term -> type_term -> Prop :=
   | T_Var : forall Γ x τ,
-    (context_contains Γ x τ) ->
-    (Γ |- (expr_var x) \is τ)
+    context_contains Γ x τ ->
+    Γ |- [{ %x% }] \is τ
 
   | T_Let : forall Γ s σ t τ x,
-    (Γ |- s \is σ) ->
-    ((ctx_assign x σ Γ) |- t \is τ) ->
-    (Γ |- (expr_let x s t) \is τ)
+    Γ |- s \is σ ->
+    (ctx_assign x σ Γ) |- t \is τ ->
+    Γ |- [{ let x := s in t }] \is τ
 
   | T_TypeAbs : forall Γ e τ α,
     Γ |- e \is τ ->
-    Γ |- (expr_ty_abs α e) \is (type_univ α τ)
+    Γ |- [{ Λ α ↦ e }] \is [< ∀α,τ >]
 
-  | T_TypeApp : forall Γ α e σ τ τ',
-    Γ |- e \is (type_univ α τ) ->
-    (type_subst1 α σ τ τ') ->
-    Γ |- (expr_ty_app e σ) \is τ'
+  | T_TypeApp : forall Γ α e σ τ,
+    Γ |- e \is [< ∀α, τ >] ->
+    Γ |- [{ e # σ }] \is (type_subst α σ τ)
 
   | T_Abs : forall Γ x σ t τ,
-    ((ctx_assign x σ Γ) |- t \is τ) ->
-    (Γ |- (expr_abs x σ t) \is (type_fun σ τ))
+    (ctx_assign x σ Γ) |- t \is τ ->
+    Γ |- [{ λ x , σ ↦ t }] \is [< σ -> τ >]
 
-  | T_MorphAbs : forall Γ x σ e τ,
-    ((ctx_assign x σ Γ) |- e \is τ) ->
-    Γ |- (expr_morph x σ e) \is (type_morph σ τ)
+  | T_MorphAbs : forall Γ x σ t τ,
+    (ctx_assign x σ Γ) |- t \is τ ->
+    Γ |- [{ λ x , σ ↦morph t }] \is [< σ ->morph τ >]
 
   | T_App : forall Γ f a σ' σ τ,
-    (Γ |- f \is (type_fun σ τ)) ->
+    (Γ |- f \is [< σ -> τ >]) ->
     (Γ |- a \is σ') ->
     (Γ |- σ' ~> σ) ->
-    (Γ |- (expr_app f a) \is τ)
+    (Γ |- [{ (f a) }] \is τ)
 
   | T_MorphFun : forall Γ f σ τ,
     Γ |- f \is (type_morph σ τ) ->
@@ -52,7 +53,7 @@ Inductive expr_type : context -> expr_term -> type_term -> Prop :=
 
   | T_Ascend : forall Γ e τ τ',
     (Γ |- e \is τ) ->
-    (Γ |- (expr_ascend τ' e) \is (type_ladder τ' τ))
+    (Γ |- [{ e as τ' }] \is [< τ' ~ τ >])
 
   | T_DescendImplicit : forall Γ x τ τ',
     Γ |- x \is τ ->
@@ -62,7 +63,7 @@ Inductive expr_type : context -> expr_term -> type_term -> Prop :=
   | T_Descend : forall Γ x τ τ',
     Γ |- x \is τ ->
     (τ :<= τ') ->
-    Γ |- (expr_descend τ' x) \is τ'
+    Γ |- [{ x des τ' }] \is τ'
 
 where "Γ '|-' x '\is' τ" := (expr_type Γ x τ).
 
@@ -75,73 +76,85 @@ Definition is_well_typed (e:expr_term) : Prop :=
 Inductive translate_typing : context -> expr_term -> type_term -> expr_term -> Prop :=
   | Expand_Var : forall Γ x τ,
     (Γ |- (expr_var x) \is τ) ->
-    (translate_typing Γ (expr_var x) τ (expr_var x))
+    (translate_typing Γ [{ %x% }] τ [{ %x% }])
 
   | Expand_Let : forall Γ x e e' t t' σ τ,
     (Γ |- e \is σ) ->
     ((ctx_assign x σ Γ) |- t \is τ) ->
     (translate_typing Γ e σ e') ->
     (translate_typing (ctx_assign x σ Γ) t τ t') ->
-    (translate_typing Γ (expr_let x e t) τ (expr_let x e' t'))
+    (translate_typing Γ
+      [{ let x := e in t }]
+      τ
+      [{ let x := e' in t' }])
 
   | Expand_TypeAbs : forall Γ α e e' τ,
     (Γ |- e \is τ) ->
     (translate_typing Γ e τ e') ->
     (translate_typing Γ
-      (expr_ty_abs α e)
-      (type_univ α τ)
-      (expr_ty_abs α e'))
+      [{ Λ α ↦ e }]
+      [< ∀ α,τ >]
+      [{ Λ α ↦ e' }])
 
   | Expand_TypeApp : forall Γ e e' σ α τ,
     (Γ |- e \is (type_univ α τ)) ->
     (translate_typing Γ e τ e') ->
-    (translate_typing Γ (expr_ty_app e τ) (type_subst α σ τ) (expr_ty_app e' τ))
+    (translate_typing Γ
+      [{ e # τ }]
+      (type_subst α σ τ)
+      [{ e' # τ }])
 
   | Expand_Abs : forall Γ x σ e e' τ,
     ((ctx_assign x σ Γ) |- e \is τ) ->
     (Γ |- (expr_abs x σ e) \is (type_fun σ τ)) ->
     (translate_typing (ctx_assign x σ Γ) e τ e') ->
-    (translate_typing Γ (expr_abs x σ e) (type_fun σ τ) (expr_abs x σ e'))
+    (translate_typing Γ
+      [{ λ x , σ ↦ e }]
+      [< σ -> τ >]
+      [{ λ x , σ ↦ e' }])
 
   | Expand_MorphAbs : forall Γ x σ e e' τ,
     ((ctx_assign x σ Γ) |- e \is τ) ->
     (Γ |- (expr_abs x σ e) \is (type_fun σ τ)) ->
     (translate_typing (ctx_assign x σ Γ) e τ e') ->
-    (translate_typing Γ (expr_morph x σ e) (type_morph σ τ) (expr_morph x σ e'))
+    (translate_typing Γ
+      [{ λ x , σ ↦morph e }]
+      [< σ ->morph τ >]
+      [{ λ x , σ ↦morph e' }])
 
   | Expand_App : forall Γ f f' a a' m σ τ σ',
     (Γ |- f \is (type_fun σ τ)) ->
     (Γ |- a \is σ') ->
     (Γ |- σ' ~> σ) ->
-    (translate_typing Γ f (type_fun σ τ) f') ->
+    (translate_typing Γ f [< σ -> τ >] f') ->
     (translate_typing Γ a σ' a') ->
     (translate_morphism_path Γ σ' σ m) ->
-    (translate_typing Γ (expr_app f a) τ (expr_app f' (expr_app m a')))
+    (translate_typing Γ [{ f a }] τ [{ f' (m a') }])
 
   | Expand_MorphFun : forall Γ f f' σ τ,
     (Γ |- f \is (type_morph σ τ)) ->
     (Γ |- f \is (type_fun σ τ)) ->
-    (translate_typing Γ f (type_morph σ τ) f') ->
-    (translate_typing Γ f (type_fun σ τ) f')
+    (translate_typing Γ f [< σ ->morph τ >] f') ->
+    (translate_typing Γ f [< σ -> τ >] f')
 
   | Expand_Ascend : forall Γ e e' τ τ',
     (Γ |- e \is τ) ->
-    (Γ |- (expr_ascend τ' e) \is (type_ladder τ' τ)) ->
+    (Γ |- [{ e as τ' }] \is [< τ' ~ τ >]) ->
     (translate_typing Γ e τ e') ->
-    (translate_typing Γ (expr_ascend τ' e) (type_ladder τ' τ) (expr_ascend τ' e'))
+    (translate_typing Γ [{ e as τ' }] [< τ' ~ τ >] [{ e' as τ' }])
 
   | Expand_Descend : forall Γ e e' τ τ',
     (Γ |- e \is τ) ->
     (τ :<= τ') ->
-    (Γ |- e \is τ') ->
+    (Γ |- [{ e des τ' }] \is τ') ->
     (translate_typing Γ e τ e') ->
-    (translate_typing Γ e τ' e')
+    (translate_typing Γ e τ' [{ e' des τ' }])
 .
 
 (* Examples *)
 
 Example typing1 :
-  ctx_empty |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"T", %"T"% -> %"T"% >].
+  ctx_empty |- [{ Λ"T" ↦ λ "x",%"T"% ↦ %"x"% }] \is [< ∀"T", %"T"% -> %"T"% >].
 Proof.
   intros.
   apply T_TypeAbs.
@@ -151,7 +164,7 @@ Proof.
 Qed.
 
 Example typing2 :
-  ctx_empty |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"U", %"U"% -> %"U"% >].
+  ctx_empty |- [{ Λ"T" ↦ λ "x",%"T"% ↦ %"x"% }] \is [< ∀"U", %"U"% -> %"U"% >].
 Proof.
   intros.
   apply T_DescendImplicit with (τ:=[< ∀"T",(%"T"% -> %"T"%) >]).
@@ -163,14 +176,11 @@ Proof.
   apply TSubRepr_Refl.
   apply TEq_Alpha.
   apply TAlpha_Rename.
-  apply TSubst_Fun.
-  apply TSubst_VarReplace.
-  apply TSubst_VarReplace.
 Qed.
 
 Example typing3 :
   ctx_empty |- [{
-    Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"y"%
+    Λ"T" ↦ Λ"U" ↦ λ"x",%"T"% ↦ λ"y",%"U"% ↦ %"y"%
   }] \is [<
        ∀"S",∀"T",(%"S"%->%"T"%->%"T"%)
   >].
@@ -186,26 +196,14 @@ Proof.
   apply TEq_Trans with (y:= [< ∀"S",∀"U",(%"S"%->%"U"%->%"U"%) >] ).
   apply TEq_Alpha.
   apply TAlpha_Rename.
-  apply TSubst_UnivReplace. discriminate.
-  easy.
-  apply TSubst_Fun.
-  apply TSubst_VarReplace.
-  apply TSubst_Fun.
-  apply TSubst_VarKeep. discriminate.
-  apply TSubst_VarKeep. discriminate.
 
   apply TEq_Alpha.
   apply TAlpha_SubUniv.
   apply TAlpha_Rename.
-  apply TSubst_Fun.
-  apply TSubst_VarKeep. discriminate.
-  apply TSubst_Fun.
-  apply TSubst_VarReplace.
-  apply TSubst_VarReplace.
 Qed.
 
 Example typing4 : (is_well_typed
-  [{ Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"x"% }]
+  [{ Λ"T" ↦ Λ"U" ↦ λ"x",%"T"% ↦ λ"y",%"U"% ↦ %"x"% }]
 ).
 Proof.
   unfold is_well_typed.
@@ -218,8 +216,6 @@ Proof.
   apply C_shuffle, C_take.
 Qed.
 
-Open Scope ladder_expr_scope.
-
 Example typing5 :
   (ctx_assign "60" [< $"ℝ"$ >]
   (ctx_assign "360" [< $"ℝ"$ >]
@@ -228,7 +224,7 @@ Example typing5 :
   |-
   [{
     let "deg2turns" :=
-       (λ"x" $"Angle"$~$"Degrees"$~$"ℝ"$
+       (λ"x",$"Angle"$~$"Degrees"$~$"ℝ"$
                 ↦morph (((%"/"% %"x"%) %"360"%) as $"Angle"$~$"Turns"$))
     in ( %"deg2turns"% (%"60"% as $"Angle"$~$"Degrees"$) )
   }]