59 lines
2.2 KiB
Text
59 lines
2.2 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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Module: data.nat.sqrt
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Authors: Leonardo de Moura
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Very simple (sqrt n) function that returns s s.t.
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s*s ≤ n ≤ s*s + s + s
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-/
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import data.nat.order
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namespace nat
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open decidable
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-- This is the simplest possible function that just performs a linear search
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definition sqrt_aux : nat → nat → nat
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| 0 n := 0
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| (succ s) n := if (succ s)*(succ s) ≤ n then succ s else sqrt_aux s n
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theorem sqrt_aux_suc_of_pos {s n} : (succ s)*(succ s) ≤ n → sqrt_aux (succ s) n = (succ s) :=
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assume h, if_pos h
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theorem sqrt_aux_suc_of_neg {s n} : ¬ (succ s)*(succ s) ≤ n → sqrt_aux (succ s) n = sqrt_aux s n :=
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assume h, if_neg h
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definition sqrt (n : nat) : nat :=
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sqrt_aux n n
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theorem sqrt_aux_lower : ∀ {s n : nat}, s ≤ n → sqrt_aux s n * sqrt_aux s n ≤ n
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| 0 n h := h
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| (succ s) n h := by_cases
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(λ h₁ : (succ s)*(succ s) ≤ n, by rewrite [sqrt_aux_suc_of_pos h₁]; exact h₁)
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(λ h₂ : ¬ (succ s)*(succ s) ≤ n,
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assert aux : s ≤ n, from lt.step (lt_of_succ_le h),
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by rewrite [sqrt_aux_suc_of_neg h₂]; exact (sqrt_aux_lower aux))
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theorem sqrt_lower (n : nat) : sqrt n * sqrt n ≤ n :=
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sqrt_aux_lower (le.refl n)
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theorem succ_squared (n : nat) : succ n * succ n = n*n + n + n + 1 :=
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calc succ n * succ n = (n+1)*(n+1) : by rewrite [add_one]
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... = n*n + n + n + 1 : by rewrite [mul.right_distrib, mul.left_distrib, one_mul, mul_one]
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theorem sqrt_aux_upper : ∀ {s n : nat}, n ≤ s*s + s + s → n ≤ sqrt_aux s n * sqrt_aux s n + sqrt_aux s n + sqrt_aux s n
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| 0 n h := h
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| (succ s) n h := by_cases
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(λ h₁ : (succ s)*(succ s) ≤ n,
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by rewrite [sqrt_aux_suc_of_pos h₁]; exact h)
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(λ h₂ : ¬ (succ s)*(succ s) ≤ n,
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assert h₃ : n < (succ s) * (succ s), from lt_of_not_le h₂,
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assert h₄ : n ≤ s * s + s + s, by rewrite [succ_squared at h₃]; exact h₃,
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by rewrite [sqrt_aux_suc_of_neg h₂]; exact (sqrt_aux_upper h₄))
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theorem sqrt_upper (n : nat) : n ≤ sqrt n * sqrt n + sqrt n + sqrt n :=
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have aux : n ≤ n*n + n + n, from le_add_of_le_right (le_add_of_le_left (le.refl n)),
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sqrt_aux_upper aux
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end nat
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