feat(library/data/nat/wf): add nat.lt is well founded theorem

library/data/nat/default.lean

@@ -2,4 +2,4 @@
 -- Released under Apache 2.0 license as described in the file LICENSE.
 -- Author: Jeremy Avigad
 
-import .basic .order .sub .div
+import .basic .order .sub .div .wf

library/data/nat/wf.lean

@@ -0,0 +1,17 @@
+-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+-- Released under Apache 2.0 license as described in the file LICENSE.
+-- Author: Leonardo de Moura
+import data.nat.order logic.wf
+open nat eq.ops
+
+theorem lt.wf [instance] : well_founded lt :=
+well_founded.intro
+ (take n, nat.induction_on n
+    (acc.intro zero (λ (y : nat) (H : y < 0),
+      absurd H !not_lt_zero))
+    (λ (n : nat) (iH : acc lt n),
+      acc.intro (succ n) (λ (m : nat) (H : m < succ n),
+        have H₁ : m < n ∨ m = n, from le_imp_lt_or_eq (succ_le_cancel (lt_imp_le_succ H)),
+        or.elim H₁
+          (assume Hlt : m < n, acc.inv iH Hlt)
+          (assume Heq : m = n, Heq⁻¹ ▸ iH))))