Split neutrals from canonical values so stuck eliminators arent encoded via constructor overloading

BidirTT/Value.lean

@@ -3,11 +3,18 @@ import BidirTT.Syntax
 namespace BidirTT
 
 mutual
+  inductive Neutral where
+    | var  : Nat → Neutral
+    | app  : Neutral → Val → Neutral
+    | fst  : Neutral → Neutral
+    | snd  : Neutral → Neutral
+    | natElim : Val → Val → Val → Neutral → Neutral
+    | unitElim : Val → Val → Neutral → Neutral
+    | emptyElim : Val → Neutral → Neutral
+    | idElim : Val → Val → Val → Neutral → Neutral
+
   inductive Val where
-    | var  : Nat → Val
-    | app  : Val → Val → Val
-    | fst  : Val → Val
-    | snd  : Val → Val
+    | neu  : Neutral → Val
     | lam  : Closure → Val
     | pi   : Val → Closure → Val
     | sig  : Val → Closure → Val
@@ -15,15 +22,11 @@ mutual
     | nat  : Val
     | zero : Val
     | succ : Val → Val
-    | natElim : Val → Val → Val → Val → Val
     | unit : Val
     | triv : Val
-    | unitElim : Val → Val → Val → Val
     | empty : Val
-    | emptyElim : Val → Val → Val
     | id   : Val → Val → Val → Val
     | refl : Val
-    | idElim : Val → Val → Val → Val → Val
     | univ : Nat → Val
 
   inductive Closure where
@@ -33,6 +36,7 @@ end
 abbrev Env := List Val
 abbrev Lvl := Nat
 
+instance : Inhabited Neutral := ⟨.var 0⟩
 instance : Inhabited Val := ⟨.univ 0⟩
 instance : Inhabited Closure := ⟨.mk [] (.univ 0)⟩