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feat(data/nat/pairing): ported data/nat/pairing.lean from Lean2
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/- | ||
Copyright (c) 2015 Microsoft Corporation. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Leonardo de Moura | ||
Elegant pairing function. | ||
-/ | ||
import data.nat.sqrt | ||
open prod decidable | ||
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namespace nat | ||
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def mkpair (a b : nat) : nat := | ||
if a < b then b*b + a else a*a + a + b | ||
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def unpair (n : nat) : nat × nat := | ||
let s := sqrt n in | ||
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 (unpair n).1 (unpair n).2 = n := | ||
let s := sqrt n in | ||
by_cases | ||
(assume : n - s*s < s, by simp [unpair, mkpair, this, nat.add_sub_of_le sqrt_lower]) | ||
(assume h₁ : ¬ n - s*s < s, | ||
have s ≤ n - s*s, from le_of_not_gt h₁, | ||
have s + s*s ≤ n - s*s + s*s, from add_le_add_right this (s*s), | ||
have h₂ : s*s + s ≤ n, by rwa [nat.sub_add_cancel sqrt_lower, add_comm] at this, | ||
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have n < (s + 1) * (s + 1), from sqrt_upper, | ||
have n < (s * s + s + s) + 1, by simp [mul_add, add_mul] at this; simp [this], | ||
have n - s*s ≤ s + s, from calc | ||
n - s*s ≤ (s*s + s + s) - s*s : nat.sub_le_sub_right (le_of_succ_le_succ this) (s*s) | ||
... = (s*s + (s+s)) - s*s : by rewrite add_assoc | ||
... = s + s : by rewrite nat.add_sub_cancel_left, | ||
have n - s*s - s ≤ s, from calc | ||
n - s*s - s ≤ (s + s) - s : nat.sub_le_sub_right this s | ||
... = s : by rewrite nat.add_sub_cancel_left, | ||
have h₃ : ¬ s < n - s*s - s, from not_lt_of_ge this, | ||
begin | ||
simp [h₁, h₃, unpair, mkpair, -add_comm, -add_assoc], | ||
rewrite [nat.sub_sub, add_sub_of_le h₂] | ||
end) | ||
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theorem unpair_mkpair (a b : nat) : unpair (mkpair a b) = (a, b) := | ||
by_cases | ||
(assume : a < b, | ||
have a + b * b < (b + 1) * (b + 1), | ||
from calc a + b * b < b + b * b : add_lt_add_right this _ | ||
... ≤ b + b * b + (b + 1) : le_add_right _ _ | ||
... = (b + 1) * (b + 1) : by simp [mul_add, add_mul], | ||
have sqrt_mkpair : sqrt (mkpair a b) = b, | ||
by simp [mkpair, *]; exact sqrt_eq (le_add_left _ _) this, | ||
have mkpair a b - b * b = a, by simp [mkpair, ‹a < b›, -add_comm, nat.add_sub_cancel_left], | ||
by simp [unpair, sqrt_mkpair, this, ‹a < b›]) | ||
(assume : ¬ a < b, | ||
have a + (b + a * a) < (a + 1) * (a + 1), | ||
from calc a + (b + a * a) ≤ a + (a + a * a) : | ||
add_le_add_left (add_le_add_right (le_of_not_gt this) _) _ | ||
... < (a + 1) * (a + 1) : lt_of_succ_le $ by simp [mul_add, add_mul, succ_eq_add_one], | ||
have sqrt_mkpair : sqrt (mkpair a b) = a, | ||
by simp [mkpair, *]; exact sqrt_eq (le_add_of_nonneg_of_le (zero_le _) (le_add_left _ _)) this, | ||
have mkpair_sub : mkpair a b - a * a = a + b, | ||
by simp [mkpair, ‹¬ a < b›]; rw [←add_assoc, nat.add_sub_cancel], | ||
have ¬ a + b < a, from not_lt_of_ge $ le_add_right _ _, | ||
by simp [unpair, ‹¬ a < b›, sqrt_mkpair, mkpair_sub, this, nat.add_sub_cancel_left]) | ||
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theorem unpair_lt_aux {n : nat} : n ≥ 1 → (unpair n).1 < n := | ||
assume : n ≥ 1, | ||
or.elim (nat.eq_or_lt_of_le this) | ||
(assume : 1 = n, by subst n; exact dec_trivial) | ||
(assume : n > 1, | ||
let s := sqrt n in | ||
by_cases | ||
(assume h : n - s*s < s, | ||
have n > 0, from lt_of_succ_lt ‹n > 1›, | ||
have sqrt n > 0, from sqrt_pos_of_pos this, | ||
have sqrt n * sqrt n > 0, from mul_pos this this, | ||
by simp [unpair, h]; exact sub_lt ‹n > 0› ‹sqrt n * sqrt n > 0›) | ||
(assume : ¬ n - s*s < s, by simp [unpair, this]; exact sqrt_lt ‹n > 1›)) | ||
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theorem unpair_lt : ∀ (n : nat), (unpair n).1 < succ n | ||
| 0 := dec_trivial | ||
| (succ n) := | ||
have (unpair (succ n)).1 < succ n, from unpair_lt_aux dec_trivial, | ||
lt.step this | ||
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end nat |