feat(builtin/Nat): add basic axioms and theorems

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

src/builtin/Nat.lean

@@ -4,7 +4,6 @@ Variable Nat : Type.
 Alias ℕ : Nat.
 
 Namespace Nat.
-
 Builtin numeral.
 
 Builtin add : Nat → Nat → Nat.
@@ -30,9 +29,44 @@ Infix 50 >  : gt.
 Definition id (a : Nat) := a.
 Notation 55 | _ | : id.
 
+Axiom PlusZero (a : Nat)   : a + 0 = a.
+Axiom PlusSucc (a b : Nat) : a + (b + 1) = (a + b) + 1.
+Axiom Induction {P : Nat -> Bool} (Hb : P 0) (Hi : Pi (n : Nat) (H : P n), P (n + 1)) (a : Nat) : P a.
+
+Theorem ZeroPlus (a : Nat) : 0 + a = a
+:= Induction (show 0 + 0 = 0, Trivial)
+             (fun (n : Nat) (H : 0 + n = n),
+                  (show 0 + (n + 1) = n + 1,
+                       let L1 : 0 + (n + 1) = (0 + n) + 1 := PlusSucc 0 n
+                       in Subst L1 H))
+             a.
+
+Theorem SuccPlus (a b : Nat) : (a + 1) + b = (a + b) + 1
+:= Induction (show (a + 1) + 0 = (a + 0) + 1,
+                   (Subst (PlusZero (a + 1)) (Symm (PlusZero a))))
+             (fun (n : Nat) (H : (a + 1) + n = (a + n) + 1),
+                  (show (a + 1) + (n + 1) = (a + (n + 1)) + 1,
+                        let L1 : (a + 1) + (n + 1) = ((a + 1) + n) + 1 := PlusSucc (a + 1) n,
+                            L2 : (a + 1) + (n + 1) = ((a + n) + 1) + 1 := Subst L1 H,
+                            L3 : (a + n) + 1 = a + (n + 1)             := Symm (PlusSucc a n)
+                         in Subst L2 L3))
+             b.
+
+Theorem PlusComm (a b : Nat) : a + b = b + a
+:= Induction (show a + 0 = 0 + a,
+                   let L1 : a + 0 = a  := PlusZero a,
+                       L2 : a = 0 + a  := Symm (ZeroPlus a)
+                   in Trans L1 L2)
+             (fun (n : Nat) (H : a + n = n + a),
+                  (show a + (n + 1) = (n + 1) + a,
+                        let L1 : a + (n + 1) = (a + n) + 1 := PlusSucc a n,
+                            L2 : a + (n + 1) = (n + a) + 1 := Subst L1 H,
+                            L3 : (n + a) + 1 = (n + 1) + a := Symm (SuccPlus n a)
+                        in Trans L2 L3))
+             b.
+
 SetOpaque ge true.
 SetOpaque lt true.
 SetOpaque gt true.
 SetOpaque id true.
-
 EndNamespace.