Add definitional transport and congruence combinators derived from Id so equality programming isnt all raw idElim

BidirTT/Examples.lean

@@ -74,6 +74,44 @@ def idElimNat : Raw :=
     (.idElim "y" "p" .nat .zero .zero reflZero)
     .nat
 
+def transportNatTy : Raw :=
+  .pi "x" .nat
+    (.pi "y" .nat
+      (.pi "p" (.id .nat (.var "x") (.var "y"))
+        (.pi "z" .nat .nat)))
+
+def transportNatTm : Raw :=
+  .lam "x" (.lam "y" (.lam "p" (.lam "z"
+    (.idElim "y'" "p'" .nat (.var "z") (.var "y") (.var "p")))))
+
+def transportNat : Raw := .ann transportNatTm transportNatTy
+
+def transportNatRefl : Raw :=
+  .ann
+    (.app (.app (.app (.app transportNat .zero) .zero) reflZero) natTwo)
+    .nat
+
+def congSuccTy : Raw :=
+  .pi "x" .nat
+    (.pi "y" .nat
+      (.pi "p" (.id .nat (.var "x") (.var "y"))
+        (.id .nat (.succ (.var "x")) (.succ (.var "y")))))
+
+def congSuccTm : Raw :=
+  .lam "x" (.lam "y" (.lam "p"
+    (.idElim "y'" "p'"
+      (.id .nat (.succ (.var "x")) (.succ (.var "y'")))
+      .refl
+      (.var "y")
+      (.var "p"))))
+
+def congSucc : Raw := .ann congSuccTm congSuccTy
+
+def congSuccRefl : Raw :=
+  .ann
+    (.app (.app (.app congSucc .zero) .zero) reflZero)
+    (.id .nat (.succ .zero) (.succ .zero))
+
 def omegaTy : Raw :=
   .pi "A" (.univ 0) (.var "A")