53 lines
1.4 KiB
Text
53 lines
1.4 KiB
Text
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/-
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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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-/
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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)
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| 0 := (0, 1)
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| 1 := (1, 1)
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| (n+2) :=
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match fib_fast_aux (n+1) with
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| (fn, fn1) := (fn1, fn1 + fn)
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end
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open prod.ops -- Get .1 .2 notation for pairs
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definition fib_fast (n : nat) := (fib_fast_aux n).2
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-- We now prove that fib_fast and fib are equal
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lemma fib_fast_aux_succ_succ : ∀ n, fib_fast_aux (succ (succ n)) = match fib_fast_aux (succ n) with | (fn, fn1) := (fn1, fn1 + fn) end :=
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λ n, rfl
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lemma fib_fast_aux_lemma : ∀ n, (fib_fast_aux (succ n)).1 = (fib_fast_aux n).2
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| 0 := rfl
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| 1 := rfl
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| (succ (succ n)) :=
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begin
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rewrite [fib_fast_aux_succ_succ],
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rewrite [-prod.eta (fib_fast_aux (succ (succ n)))],
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end
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theorem fib_eq_fib_fast : ∀ n, fib n = fib_fast n
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| 0 := rfl
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| 1 := rfl
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| (n+2) :=
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begin
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have feq : fib n = fib_fast n, from fib_eq_fib_fast n,
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have f1eq : fib (succ n) = fib_fast (succ n), from fib_eq_fib_fast (succ n),
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unfold [fib, fib_fast, fib_fast_aux],
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rewrite [-prod.eta (fib_fast_aux (succ n))], esimp,
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fold fib_fast (succ n), rewrite -f1eq,
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rewrite fib_fast_aux_lemma,
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fold fib_fast n, rewrite -feq,
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end
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