34 lines
1.1 KiB
Text
34 lines
1.1 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 partial_sum : nat → nat
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| 0 := 0
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| (succ n) := succ n + partial_sum n
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example : partial_sum 5 = 15 :=
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rfl
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example : partial_sum 6 = 21 :=
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rfl
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lemma two_mul_partial_sum_eq : ∀ n, 2 * partial_sum n = (succ n) * n
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| 0 := by reflexivity
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| (succ n) := calc
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2 * (succ n + partial_sum n) = 2 * succ n + 2 * partial_sum n : mul.left_distrib
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... = 2 * succ n + succ n * n : two_mul_partial_sum_eq
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... = 2 * succ n + n * succ n : mul.comm
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... = (2 + n) * succ n : mul.right_distrib
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... = (n + 2) * succ n : add.comm
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... = (succ (succ n)) * succ n : rfl
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theorem partial_sum_eq : ∀ n, partial_sum n = ((n + 1) * n) div 2 :=
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take n,
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assert h₁ : (2 * partial_sum n) div 2 = ((succ n) * n) div 2, by rewrite two_mul_partial_sum_eq,
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assert h₂ : 2 > 0, from dec_trivial,
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by rewrite [mul_div_cancel_left _ h₂ at h₁]; exact h₁
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