Write Nat/Unit/Empty/Id Eliminators Through NbE and Bidir Elaboration

BidirTT/Examples.lean

@@ -46,6 +46,32 @@ def fstDepPair : Raw := .fst depPairAnn
 
 def sndDepPair : Raw := .snd depPairAnn
 
+def natTwo : Raw :=
+  .succ (.succ .zero)
+
+def natFoldId : Raw :=
+  .ann
+    (.natElim "n" .nat .zero "k" "ih" (.succ (.var "ih")) natTwo)
+    .nat
+
+def unitToNat : Raw :=
+  .ann
+    (.unitElim "u" .nat natTwo .triv)
+    .nat
+
+def absurdNat : Raw :=
+  .ann
+    (.lam "e" (.emptyElim "x" .nat (.var "e")))
+    (.pi "e" .empty .nat)
+
+def reflZero : Raw :=
+  .ann .refl (.id .nat .zero .zero)
+
+def idElimNat : Raw :=
+  .ann
+    (.idElim "y" "p" .nat .zero .zero reflZero)
+    .nat
+
 def omegaTy : Raw :=
   .pi "A" (.univ 0) (.var "A")
 
@@ -67,6 +93,11 @@ def letUniverse : Raw :=
     (.letE "A" (.univ 1) (.pi "_" (.univ 0) (.univ 0)) (.var "A"))
     (.univ 1)
 
+def badSucc : Raw := .succ (.univ 0)
+
+def badRefl : Raw :=
+  .ann .refl (.id .nat .zero (.succ .zero))
+
 def univ0 : Raw := .univ 0
 
 end BidirTT.Examples