test(tests/lean/run): another test for the unifier

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

tests/lean/run/nat_bug5.lean

@@ -0,0 +1,34 @@
+import logic num
+using num eq_proofs
+
+inductive nat : Type :=
+| zero : nat
+| succ : nat → nat
+
+definition add (x y : nat) : nat := nat_rec x (λn r, succ r) y
+infixl `+`:65 := add
+definition mul (n m : nat) := nat_rec zero (fun m x, x + n) m
+infixl `*`:75 := mul
+
+axiom add_one (n:nat) : n + (succ zero) = succ n
+axiom mul_zero_right (n : nat) : n * zero = zero
+axiom add_zero_right (n : nat) : n + zero = n
+axiom mul_succ_right (n m : nat) : n * succ m = n * m + n
+axiom add_assoc (n m k : nat) : (n + m) + k = n + (m + k)
+axiom add_right_comm (n m k : nat) : n + m + k = n + k + m
+axiom induction_on {P : nat → Prop} (a : nat) (H1 : P zero) (H2 : ∀ (n : nat) (IH : P n), P (succ n)) : P a
+
+theorem mul_add_distr_left (n m k : nat) : (n + m) * k = n * k + m * k
+:= induction_on k
+    (calc
+      (n + m) * zero = zero : refl _
+        ...          = n * zero + m * zero : refl _)
+    (take l IH,
+        calc
+        (n + m) * succ l = (n + m) * l + (n + m) : mul_succ_right _ _
+          ... = n * l + m * l + (n + m) : {IH}
+          ... = n * l + m * l + n + m : symm (add_assoc _ _ _)
+          ... = n * l + n + m * l + m : {add_right_comm _ _ _}
+          ... = n * l + n + (m * l + m) : add_assoc _ _ _
+          ... = n * succ l + (m * l + m) : {symm (mul_succ_right _ _)}
+          ... = n * succ l + m * succ l : {symm (mul_succ_right _ _)})