diff --git a/nat.ml b/nat.ml
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--- /dev/null
+++ b/nat.ml
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+type nat =
+  | Z
+  | S of nat
+
+type proof_plus_is =
+  | PZero
+  | PSucc of proof_plus_is
+
+type proof_times_is =
+  | TZero
+  | TSucc of proof_times_is * proof_plus_is * nat
+
+let rec check_plus a b c proof = match proof with
+  | PZero -> a = Z && b = c
+  | PSucc proof_0 -> begin match a, c with
+    | S a', S c' -> check_plus a' b c' proof_0
+    | _, _ -> false
+  end
+
+let rec check_times a b c proof = match proof with
+  | TZero -> a = Z && c = Z
+  | TSucc (proof_0, proof_1, nat_0) -> begin match a, b, c with
+    | S a', b, c -> check_times a' b nat_0 proof_0 && check_plus b nat_0 c proof_1
+    | _, _, _ -> false
+  end
+
+let rec find_plus_proof = fun a b c -> match a, c with
+  | Z, _ -> if b = c then Some(PZero) else None
+  | S a', S c' -> begin match find_plus_proof a' b c' with
+    | None -> None
+    | Some proof_0 -> Some(PSucc proof_0)
+  end
+  | _, _ -> None
+
+(* option option int *)
+(* string | null | null *)
+
+
+(* Z plus S(Z) is S(Z) *)
+let proof = find_plus_proof Z (S Z) Z;;
+(* printf proof *)
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