lean2/library/data/nat/pairing.lean
Leonardo de Moura a618bd7d6c refactor(library): use type classes for encoding all arithmetic operations
Before this commit we were using overloading for concrete structures and
type classes for abstract ones.

This is the first of series of commits that implement this modification
2015-11-08 14:04:54 -08:00

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