From 2e9baff1b908068f13bc542e916110f21e7f0baa Mon Sep 17 00:00:00 2001
From: Junyoung/Clare Jang <jjc9310@gmail.com>
Date: Wed, 13 Dec 2023 05:12:47 -0500
Subject: [PATCH] Add some syntax

---
 theories/Syntax.v | 44 ++++++++++++++++++++------------------------
 theories/System.v | 11 +++++------
 2 files changed, 25 insertions(+), 30 deletions(-)

diff --git a/theories/Syntax.v b/theories/Syntax.v
index 78494661..50ed4d7a 100644
--- a/theories/Syntax.v
+++ b/theories/Syntax.v
@@ -39,9 +39,6 @@ with subst : Set :=
 Infix "∙" := a_compose (at level 70).
 Infix ",," := a_extend (at level 80).
 
-Declare Custom Entry ast.
-Declare Scope ast_scope.
-
 Fixpoint nat_to_exp n : exp :=
   match n with
   | 0 => a_zero
@@ -65,28 +62,27 @@ Definition exp_to_num e :=
   | None => None
   end.
 
-Notation "{{ e }}" := e (at level 0, e custom ast at level 99) : ast_scope.
-Notation "( x )" := x (in custom ast at level 0, x custom ast at level 99) : ast_scope.
-Notation "~{ x }" := x (in custom ast at level 0, x constr at level 0) : ast_scope.
-Notation "x" := x (in custom ast at level 0, x constr at level 0) : ast_scope.
-Notation "e [ s ]" := (a_sub e s) (in custom ast at level 0) : ast_scope.
-Notation "'λ' T -> e" := (a_fn T e) (in custom ast at level 0, e custom ast at level 99) : ast_scope.
-Notation "f x .. y" := (a_app .. (a_app f x) .. y) (in custom ast at level 20, f custom ast, x custom ast at level 9, y custom ast at level 9) : ast_scope.
-Notation "'ℕ'" := a_nat (in custom ast) : ast_scope.
-Notation "'Type<' n >" := (a_typ n) (in custom ast at level 0, n constr at level 0) : ast_scope.
-Notation "'Π' T -> S" := (a_pi T S) (in custom ast at level 99, T custom ast at level 0, S custom ast at level 0) : ast_scope.
-Number Notation exp num_to_exp exp_to_num : ast_scope.
-Notation "0" := a_zero (in custom ast) : ast_scope.
-Notation "1" := (a_succ a_zero) (in custom ast) : ast_scope.
-Notation "2" := (a_succ (a_succ a_zero)) (in custom ast) : ast_scope.
-Notation "'S' e" := (a_succ e) (in custom ast at level 20, e custom ast at level 10) : ast_scope.
-Notation "'#' n" := (a_var n) (in custom ast at level 70, n constr at level 0) : ast_scope.
-Notation "!" := a_id (in custom ast at level 0) : ast_scope.
+Declare Custom Entry exp.
+Declare Scope exp_scope.
+Delimit Scope exp_scope with exp.
 
-Open Scope ast_scope.
-Example example := {{ (λ Type<0> -> S S # 0 Type<0>) Type<0> ℕ }}.
-Set Printing All.
-Print example.
+Notation "{{ e }}" := e (at level 0, e custom exp at level 99) : exp_scope.
+Notation "( x )" := x (in custom exp at level 0, x custom exp at level 99) : exp_scope.
+Notation "~{ x }" := x (in custom exp at level 0, x constr at level 0) : exp_scope.
+Notation "x" := x (in custom exp at level 0, x constr at level 0) : exp_scope.
+Notation "e [ s ]" := (a_sub e s) (in custom exp at level 0, s custom exp at level 99) : exp_scope.
+Notation "'λ' T e" := (a_fn T e) (in custom exp at level 0, T custom exp at level 30, e custom exp at level 30) : exp_scope.
+Notation "f x .. y" := (a_app .. (a_app f x) .. y) (in custom exp at level 40, f custom exp, x custom exp at next level, y custom exp at next level) : exp_scope.
+Notation "'ℕ'" := a_nat (in custom exp) : exp_scope.
+Notation "'Type(' n ')'" := (a_typ n) (in custom exp at level 0, n constr at level 99) : exp_scope.
+Notation "'Π' T S" := (a_pi T S) (in custom exp at level 0, T custom exp at level 30, S custom exp at level 30) : exp_scope.
+Number Notation exp num_to_exp exp_to_num : exp_scope.
+Notation "'suc' e" := (a_succ e) (in custom exp at level 30, e custom exp at level 30) : exp_scope.
+Notation "'#' n" := (a_var n) (in custom exp at level 0, n constr at level 0) : exp_scope.
+Notation "'$id'" := a_id (in custom exp at level 0) : exp_scope.
+Notation "'$wk'" := a_weaken (in custom exp at level 0) : exp_scope.
+Infix "∙" := a_compose (in custom exp at level 70) : exp_scope.
+Infix "," := a_extend (in custom exp at level 80) : exp_scope.
 
 Notation Ctx := (list exp).
 Notation Sb := subst.
diff --git a/theories/System.v b/theories/System.v
index de228d13..89aa7a92 100644
--- a/theories/System.v
+++ b/theories/System.v
@@ -11,14 +11,13 @@ Reserved Notation "Γ ⊢ A ≈ B : T" (at level 80, A at next level, B at next
 Reserved Notation "Γ ⊢ t : T" (no associativity, at level 80, t at next level).
 Reserved Notation "Γ ⊢s σ : Δ" (no associativity, at level 80, σ at next level).
 Reserved Notation "Γ ⊢s S1 ≈ S2 : Δ" (no associativity, at level 80, S1 at next level, S2 at next level).
-Reserved Notation "x : T ∈! Γ" (no associativity, at level 80). 
+Reserved Notation "x : T ∈! Γ" (no associativity, at level 80).
 
 Generalizable All Variables.
 
-
 Inductive ctx_lookup : nat -> Typ -> Ctx -> Prop :=
-  | here : `( 0 :  (a_sub T a_weaken) ∈! (T :: Γ))
-  | there : `( n : T ∈! Γ -> (S n) : (a_sub T a_weaken) ∈! (T' :: Γ))              
+  | here : `( 0 : ( {{ T[$wk] }}%exp ) ∈! (T :: Γ) )
+  | there : `( n : T ∈! Γ -> (S n) : {{ T[$wk] }}%exp ∈! (T' :: Γ) )
 where "x : T ∈! Γ" := (ctx_lookup x T Γ).
 
 
@@ -59,14 +58,14 @@ with wf_term : Ctx -> exp -> Typ -> Prop :=
   | wf_vlookup : `(
       ⊢ Γ ->
       x : T ∈! Γ ->
-      Γ ⊢ a_var x : T
+      Γ ⊢ {{ #x }}%exp : T
     )
 | wf_fun_e: `(
       Γ ⊢ A : typ i ->          
       A :: Γ ⊢ B : typ i ->            
       Γ ⊢ M : Π A B ->
       Γ ⊢ N : A ->
-      Γ ⊢ a_app M N : a_sub B (a_id ,, N)
+      Γ ⊢ {{ M N }}%exp : {{ B[$id,N] }}%exp
     )
   | wf_fun_i : `(
       Γ ⊢ A : typ i ->