doc(library/data/nat/examples): add tail recursive fib example

library/data/nat/examples/fib2.lean

@@ -0,0 +1,41 @@
+/-
+Copyright (c) 2015 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Leonardo de Moura
+
+Show that tail recursive fib is equal to standard one.
+-/
+import data.nat
+open nat
+
+definition fib : nat → nat
+| 0     := 1
+| 1     := 1
+| (n+2) := fib (n+1) + fib n
+
+private definition fib_fast_aux : nat → nat → nat → nat
+| 0     i j := j
+| (n+1) i j := fib_fast_aux n j (j+i)
+
+lemma fib_fast_aux_succ : ∀ n i j, fib_fast_aux (succ n) i j = fib_fast_aux n j (j+i) :=
+λ n i j, rfl
+
+lemma fib_fast_aux_lemma : ∀ n m, fib_fast_aux n (fib m) (fib (succ m)) = fib (succ (n + m))
+| 0        m := by rewrite zero_add
+| (succ n) m :=
+  begin
+    have ih : fib_fast_aux n (fib (succ m)) (fib (succ (succ m))) = fib (succ (n + succ m)), from fib_fast_aux_lemma n (succ m),
+    have h₁ : fib (succ m) + fib m = fib (succ (succ m)), from rfl,
+    rewrite [fib_fast_aux_succ, h₁, ih, succ_add, add_succ],
+  end
+
+definition fib_fast (n: nat) :=
+fib_fast_aux n 0 1
+
+lemma fib_fast_eq_fib : ∀ n, fib_fast n = fib n
+| 0        := rfl
+| (succ n) :=
+  begin
+    have h₁ : fib_fast_aux n (fib 0) (fib 1) = fib (succ n), from !fib_fast_aux_lemma,
+    unfold fib_fast, krewrite [fib_fast_aux_succ, h₁]
+  end