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