test(tests/lean/run/nateq): add example that triggered previous modification

tests/lean/run/nateq.lean

@@ -0,0 +1,33 @@
+import data.nat.basic data.bool
+open bool nat eq.ops
+reducible nat.rec_on
+
+definition is_eq (a b : nat) : bool :=
+nat.rec_on a
+  (λ b, nat.cases_on b tt (λb₁, ff))
+  (λ a₁ r₁ b, nat.cases_on b ff (λb₁, r₁ b₁))
+  b
+
+example : is_eq 3 3 = tt :=
+rfl
+
+example : is_eq 3 5 = ff :=
+rfl
+
+theorem eq.to_is_eq (a b : nat) (H : a = b) : is_eq a b = tt :=
+have aux : is_eq a a = tt, from
+  nat.induction_on a
+    rfl
+    (λ (a₁ : nat) (ih : is_eq a₁ a₁ = tt), ih),
+H ▸ aux
+
+theorem is_eq.to_eq (a b : nat) : is_eq a b = tt → a = b :=
+nat.induction_on a
+  (λb, nat.cases_on b (λh, rfl) (λb₁ H, absurd H !ff_ne_tt))
+  (λa₁ (ih : ∀b, is_eq a₁ b = tt → a₁ = b) (b : nat),
+    nat.cases_on b
+      (λ (H  : is_eq (succ a₁) zero = tt), absurd H !ff_ne_tt)
+      (λb₁ (H : is_eq (succ a₁) (succ b₁) = tt),
+         have aux : a₁ = b₁, from ih b₁ H,
+         aux ▸ rfl))
+  b