97 lines
3.4 KiB
Text
97 lines
3.4 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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Authors: Leonardo de Moura
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Elegant pairing function.
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-/
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import data.nat.sqrt data.nat.div
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open prod decidable
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open algebra
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namespace nat
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definition mkpair (a b : nat) : nat :=
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if a < b then b*b + a else a*a + a + b
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definition unpair (n : nat) : nat × nat :=
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let s := sqrt n in
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if n - s*s < s then (n - s*s, s) else (s, n - s*s - s)
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theorem mkpair_unpair (n : nat) : mkpair (pr1 (unpair n)) (pr2 (unpair n)) = n :=
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let s := sqrt n in
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by_cases
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(suppose n - s*s < s,
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begin
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esimp [unpair],
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rewrite [if_pos this],
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esimp [mkpair],
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rewrite [if_pos this, add_sub_of_le (sqrt_lower n)]
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end)
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(suppose h₁ : ¬ n - s*s < s,
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have s ≤ n - s*s, from le_of_not_gt h₁,
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assert s + s*s ≤ n - s*s + s*s, from add_le_add_right this (s*s),
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assert s*s + s ≤ n, by rewrite [nat.sub_add_cancel (sqrt_lower n) at this,
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add.comm at this]; assumption,
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have n ≤ s*s + s + s, from sqrt_upper n,
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have n - s*s ≤ s + s, from calc
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n - s*s ≤ (s*s + s + s) - s*s : nat.sub_le_sub_right this (s*s)
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... = (s*s + (s+s)) - s*s : by rewrite add.assoc
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... = s + s : by rewrite nat.add_sub_cancel_left,
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have n - s*s - s ≤ s, from calc
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n - s*s - s ≤ (s + s) - s : nat.sub_le_sub_right this s
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... = s : by rewrite nat.add_sub_cancel_left,
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assert h₂ : ¬ s < n - s*s - s, from not_lt_of_ge this,
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begin
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esimp [unpair],
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rewrite [if_neg h₁], esimp,
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esimp [mkpair],
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rewrite [if_neg h₂, nat.sub_sub, add_sub_of_le `s*s + s ≤ n`],
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end)
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theorem unpair_mkpair (a b : nat) : unpair (mkpair a b) = (a, b) :=
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by_cases
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(suppose a < b,
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assert a ≤ b + b, from calc
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a ≤ b : le_of_lt this
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... ≤ b+b : !le_add_right,
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begin
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esimp [mkpair],
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rewrite [if_pos `a < b`],
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esimp [unpair],
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rewrite [sqrt_offset_eq `a ≤ b + b`, nat.add_sub_cancel_left, if_pos `a < b`]
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end)
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(suppose ¬ a < b,
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have b ≤ a, from le_of_not_gt this,
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assert a + b ≤ a + a, from add_le_add_left this a,
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have a + b ≥ a, from !le_add_right,
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assert ¬ a + b < a, from not_lt_of_ge this,
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begin
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esimp [mkpair],
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rewrite [if_neg `¬ a < b`],
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esimp [unpair],
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rewrite [add.assoc (a * a) a b, sqrt_offset_eq `a + b ≤ a + a`, *nat.add_sub_cancel_left,
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if_neg `¬ a + b < a`]
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end)
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open prod.ops
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theorem unpair_lt_aux {n : nat} : n ≥ 1 → (unpair n).1 < n :=
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suppose n ≥ 1,
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or.elim (eq_or_lt_of_le this)
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(suppose 1 = n, by subst n; exact dec_trivial)
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(suppose n > 1,
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let s := sqrt n in
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by_cases
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(suppose h : n - s*s < s,
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assert n > 0, from lt_of_succ_lt `n > 1`,
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assert sqrt n > 0, from sqrt_pos_of_pos this,
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assert sqrt n * sqrt n > 0, from mul_pos this this,
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begin unfold unpair, rewrite [if_pos h], esimp, exact sub_lt `n > 0` `sqrt n * sqrt n > 0` end)
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(suppose ¬ n - s*s < s, begin unfold unpair, rewrite [if_neg this], esimp, apply sqrt_lt `n > 1` end))
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theorem unpair_lt : ∀ (n : nat), (unpair n).1 < succ n
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| 0 := dec_trivial
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| (succ n) :=
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have (unpair (succ n)).1 < succ n, from unpair_lt_aux dec_trivial,
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lt.step this
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end nat
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