464 lines
19 KiB
Text
464 lines
19 KiB
Text
/-
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Copyright (c) 2014 Jeremy Avigad. 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.div
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Authors: Jeremy Avigad, Leonardo de Moura
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Definitions of div, mod, and gcd on the natural numbers.
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-/
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import data.nat.sub logic
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import algebra.relation
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import tools.fake_simplifier
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open eq.ops well_founded decidable fake_simplifier prod
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open relation relation.iff_ops
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namespace nat
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-- Auxiliary lemma used to justify div
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private definition div_rec_lemma {x y : nat} (H : 0 < y ∧ y ≤ x) : x - y < x :=
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and.rec_on H (λ ypos ylex,
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sub_lt (lt_of_lt_of_le ypos ylex) ypos)
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private definition div.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y : nat) : nat :=
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if H : 0 < y ∧ y ≤ x then f (x - y) (div_rec_lemma H) y + 1 else zero
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definition divide (x y : nat) :=
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fix div.F x y
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theorem divide_def (x y : nat) : divide x y = if 0 < y ∧ y ≤ x then divide (x - y) y + 1 else 0 :=
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congr_fun (fix_eq div.F x) y
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notation a div b := divide a b
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theorem div_zero (a : ℕ) : a div 0 = 0 :=
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divide_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0))
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theorem div_less {a b : ℕ} (h : a < b) : a div b = 0 :=
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divide_def a b ⬝ if_neg (!not_and_of_not_right (not_le_of_lt h))
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theorem zero_div (b : ℕ) : 0 div b = 0 :=
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divide_def 0 b ⬝ if_neg (λ h, and.rec_on h (λ l r, absurd (lt_of_lt_of_le l r) (lt.irrefl 0)))
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theorem div_rec {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a div b = succ ((a - b) div b) :=
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divide_def a b ⬝ if_pos (and.intro h₁ h₂)
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theorem div_add_self_right {x z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) :=
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calc (x + z) div z
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= if 0 < z ∧ z ≤ x + z then divide (x + z - z) z + 1 else 0 : !divide_def
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... = divide (x + z - z) z + 1 : if_pos (and.intro H (le_add_left z x))
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... = succ (x div z) : sub_add_left
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theorem div_add_mul_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y :=
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induction_on y
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(calc (x + zero * z) div z = (x + zero) div z : zero_mul
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... = x div z : add_zero
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... = x div z + zero : add_zero)
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(take y,
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assume IH : (x + y * z) div z = x div z + y, calc
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(x + succ y * z) div z = (x + y * z + z) div z : by simp
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... = succ ((x + y * z) div z) : div_add_self_right H
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... = x div z + succ y : by simp)
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private definition mod.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y : nat) : nat :=
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if H : 0 < y ∧ y ≤ x then f (x - y) (div_rec_lemma H) y else x
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definition modulo (x y : nat) :=
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fix mod.F x y
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notation a mod b := modulo a b
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theorem modulo_def (x y : nat) : modulo x y = if 0 < y ∧ y ≤ x then modulo (x - y) y else x :=
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congr_fun (fix_eq mod.F x) y
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theorem mod_zero (a : ℕ) : a mod 0 = a :=
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modulo_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0))
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theorem mod_less {a b : ℕ} (h : a < b) : a mod b = a :=
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modulo_def a b ⬝ if_neg (!not_and_of_not_right (not_le_of_lt h))
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theorem zero_mod (b : ℕ) : 0 mod b = 0 :=
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modulo_def 0 b ⬝ if_neg (λ h, and.rec_on h (λ l r, absurd (lt_of_lt_of_le l r) (lt.irrefl 0)))
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theorem mod_rec {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a mod b = (a - b) mod b :=
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modulo_def a b ⬝ if_pos (and.intro h₁ h₂)
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theorem mod_add_self_right {x z : ℕ} (H : z > 0) : (x + z) mod z = x mod z :=
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calc (x + z) mod z
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= if 0 < z ∧ z ≤ x + z then (x + z - z) mod z else _ : modulo_def
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... = (x + z - z) mod z : if_pos (and.intro H (le_add_left z x))
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... = x mod z : sub_add_left
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theorem mod_add_mul_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z :=
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induction_on y
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(calc (x + zero * z) mod z = (x + zero) mod z : zero_mul
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... = x mod z : add_zero)
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(take y,
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assume IH : (x + y * z) mod z = x mod z,
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calc
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(x + succ y * z) mod z = (x + (y * z + z)) mod z : succ_mul
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... = (x + y * z + z) mod z : add.assoc
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... = (x + y * z) mod z : mod_add_self_right H
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... = x mod z : IH)
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theorem mod_mul_self_right {m n : ℕ} : (m * n) mod n = 0 :=
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by_cases_zero_pos n (by simp)
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(take n,
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assume npos : n > 0,
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(by simp) ▸ (@mod_add_mul_self_right 0 m _ npos))
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-- add_rewrite mod_mul_self_right
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theorem mod_mul_self_left {m n : ℕ} : (m * n) mod m = 0 :=
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!mul.comm ▸ mod_mul_self_right
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-- add_rewrite mod_mul_self_left
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-- ### properties of div and mod together
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theorem mod_lt {x y : ℕ} (H : y > 0) : x mod y < y :=
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case_strong_induction_on x
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(show 0 mod y < y, from !zero_mod⁻¹ ▸ H)
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(take x,
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assume IH : ∀x', x' ≤ x → x' mod y < y,
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show succ x mod y < y, from
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by_cases -- (succ x < y)
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(assume H1 : succ x < y,
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have H2 : succ x mod y = succ x, from mod_less H1,
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show succ x mod y < y, from H2⁻¹ ▸ H1)
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(assume H1 : ¬ succ x < y,
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have H2 : y ≤ succ x, from le_of_not_lt H1,
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have H3 : succ x mod y = (succ x - y) mod y, from mod_rec H H2,
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have H4 : succ x - y < succ x, from sub_lt !succ_pos H,
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have H5 : succ x - y ≤ x, from le_of_lt_succ H4,
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show succ x mod y < y, from H3⁻¹ ▸ IH _ H5))
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theorem div_mod_eq {x y : ℕ} : x = x div y * y + x mod y :=
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by_cases_zero_pos y
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(show x = x div 0 * 0 + x mod 0, from
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(calc
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x div 0 * 0 + x mod 0 = 0 + x mod 0 : mul_zero
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... = x mod 0 : zero_add
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... = x : mod_zero)⁻¹)
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(take y,
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assume H : y > 0,
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show x = x div y * y + x mod y, from
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case_strong_induction_on x
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(show 0 = (0 div y) * y + 0 mod y, by simp)
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(take x,
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assume IH : ∀x', x' ≤ x → x' = x' div y * y + x' mod y,
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show succ x = succ x div y * y + succ x mod y, from
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by_cases -- (succ x < y)
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(assume H1 : succ x < y,
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have H2 : succ x div y = 0, from div_less H1,
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have H3 : succ x mod y = succ x, from mod_less H1,
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by simp)
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(assume H1 : ¬ succ x < y,
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have H2 : y ≤ succ x, from le_of_not_lt H1,
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have H3 : succ x div y = succ ((succ x - y) div y), from div_rec H H2,
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have H4 : succ x mod y = (succ x - y) mod y, from mod_rec H H2,
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have H5 : succ x - y < succ x, from sub_lt !succ_pos H,
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have H6 : succ x - y ≤ x, from le_of_lt_succ H5,
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(calc
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succ x div y * y + succ x mod y = succ ((succ x - y) div y) * y + succ x mod y : H3
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... = ((succ x - y) div y) * y + y + succ x mod y : succ_mul
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... = ((succ x - y) div y) * y + y + (succ x - y) mod y : H4
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... = ((succ x - y) div y) * y + (succ x - y) mod y + y : add.right_comm
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... = succ x - y + y : {!(IH _ H6)⁻¹}
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... = succ x : add_sub_ge_left H2)⁻¹)))
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theorem mod_le {x y : ℕ} : x mod y ≤ x :=
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div_mod_eq⁻¹ ▸ !le_add_left
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--- a good example where simplifying using the context causes problems
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theorem remainder_unique {y : ℕ} (H : y > 0) {q1 r1 q2 r2 : ℕ} (H1 : r1 < y) (H2 : r2 < y)
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(H3 : q1 * y + r1 = q2 * y + r2) : r1 = r2 :=
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calc
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r1 = r1 mod y : by simp
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... = (r1 + q1 * y) mod y : (mod_add_mul_self_right H)⁻¹
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... = (q1 * y + r1) mod y : add.comm
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... = (r2 + q2 * y) mod y : by simp
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... = r2 mod y : mod_add_mul_self_right H
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... = r2 : by simp
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theorem quotient_unique {y : ℕ} (H : y > 0) {q1 r1 q2 r2 : ℕ} (H1 : r1 < y) (H2 : r2 < y)
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(H3 : q1 * y + r1 = q2 * y + r2) : q1 = q2 :=
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have H4 : q1 * y + r2 = q2 * y + r2, from (remainder_unique H H1 H2 H3) ▸ H3,
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have H5 : q1 * y = q2 * y, from add.cancel_right H4,
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have H6 : y > 0, from lt_of_le_of_lt !zero_le H1,
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show q1 = q2, from eq_of_mul_eq_mul_right H6 H5
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theorem div_mul_mul {z x y : ℕ} (zpos : z > 0) : (z * x) div (z * y) = x div y :=
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by_cases -- (y = 0)
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(assume H : y = 0, by simp)
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(assume H : y ≠ 0,
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have ypos : y > 0, from pos_of_ne_zero H,
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have zypos : z * y > 0, from mul_pos zpos ypos,
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have H1 : (z * x) mod (z * y) < z * y, from mod_lt zypos,
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have H2 : z * (x mod y) < z * y, from mul_lt_mul_of_pos_left (mod_lt ypos) zpos,
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quotient_unique zypos H1 H2
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(calc
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((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : div_mod_eq
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... = z * (x div y * y + x mod y) : div_mod_eq
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... = z * (x div y * y) + z * (x mod y) : mul.left_distrib
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... = (x div y) * (z * y) + z * (x mod y) : mul.left_comm))
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--- something wrong with the term order
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--- ... = (x div y) * (z * y) + z * (x mod y) : by simp))
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theorem mod_mul_mul {z x y : ℕ} (zpos : z > 0) : (z * x) mod (z * y) = z * (x mod y) :=
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by_cases -- (y = 0)
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(assume H : y = 0, by simp)
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(assume H : y ≠ 0,
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have ypos : y > 0, from pos_of_ne_zero H,
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have zypos : z * y > 0, from mul_pos zpos ypos,
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have H1 : (z * x) mod (z * y) < z * y, from mod_lt zypos,
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have H2 : z * (x mod y) < z * y, from mul_lt_mul_of_pos_left (mod_lt ypos) zpos,
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remainder_unique zypos H1 H2
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(calc
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((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : div_mod_eq
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... = z * (x div y * y + x mod y) : div_mod_eq
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... = z * (x div y * y) + z * (x mod y) : mul.left_distrib
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... = (x div y) * (z * y) + z * (x mod y) : mul.left_comm))
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theorem mod_one (x : ℕ) : x mod 1 = 0 :=
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have H1 : x mod 1 < 1, from mod_lt !succ_pos,
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eq_zero_of_le_zero (le_of_lt_succ H1)
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-- add_rewrite mod_one
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theorem mod_self (n : ℕ) : n mod n = 0 :=
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cases_on n (by simp)
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(take n,
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have H : (succ n * 1) mod (succ n * 1) = succ n * (1 mod 1),
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from mod_mul_mul !succ_pos,
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(by simp) ▸ H)
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-- add_rewrite mod_self
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theorem div_one (n : ℕ) : n div 1 = n :=
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have H : n div 1 * 1 + n mod 1 = n, from div_mod_eq⁻¹,
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(by simp) ▸ H
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-- add_rewrite div_one
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theorem pos_div_self {n : ℕ} (H : n > 0) : n div n = 1 :=
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have H1 : (n * 1) div (n * 1) = 1 div 1, from div_mul_mul H,
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(by simp) ▸ H1
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-- add_rewrite pos_div_self
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-- Divides
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-- -------
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-- definition dvd (x y : ℕ) : Prop := y mod x = 0
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-- infix `|` := dvd
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-- theorem dvd_iff_mod_eq_zero {x y : ℕ} : x | y ↔ y mod x = 0 :=
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-- iff.of_eq rfl
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theorem mod_eq_zero_imp_div_mul_eq {x y : ℕ} (H : x mod y = 0) : x div y * y = x :=
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(calc
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x = x div y * y + x mod y : div_mod_eq
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... = x div y * y + 0 : H
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... = x div y * y : !add_zero)⁻¹
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-- add_rewrite dvd_imp_div_mul_eq
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theorem mul_eq_imp_mod_eq_zero {z x y : ℕ} (H : z * y = x) : x mod y = 0 :=
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have H1 : z * y = x mod y + x div y * y, from
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H ⬝ div_mod_eq ⬝ !add.comm,
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have H2 : (z - x div y) * y = x mod y, from
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calc
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(z - x div y) * y = z * y - x div y * y : mul_sub_distr_right
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... = x mod y + x div y * y - x div y * y : H1
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... = x mod y : sub_add_left,
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show x mod y = 0, from
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by_cases
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(assume yz : y = 0,
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have xz : x = 0, from
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calc
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x = z * y : H
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... = z * 0 : yz
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... = 0 : mul_zero,
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calc
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x mod y = x mod 0 : yz
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... = x : mod_zero
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... = 0 : xz)
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(assume ynz : y ≠ 0,
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have ypos : y > 0, from pos_of_ne_zero ynz,
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have H3 : (z - x div y) * y < y, from H2⁻¹ ▸ mod_lt ypos,
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have H4 : (z - x div y) * y < 1 * y, from !one_mul⁻¹ ▸ H3,
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have H5 : z - x div y < 1, from lt_of_mul_lt_mul_right H4,
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have H6 : z - x div y = 0, from eq_zero_of_le_zero (le_of_lt_succ H5),
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calc
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x mod y = (z - x div y) * y : H2
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... = 0 * y : H6
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... = 0 : zero_mul)
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theorem dvd_of_mod_eq_zero {x y : ℕ} (H : y mod x = 0) : x | y :=
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dvd.intro (!mul.comm ▸ mod_eq_zero_imp_div_mul_eq H)
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theorem mod_eq_zero_of_dvd {x y : ℕ} (H : x | y) : y mod x = 0 :=
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dvd.elim H (
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take z,
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assume H1 : x * z = y,
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mul_eq_imp_mod_eq_zero (!mul.comm ▸ H1))
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theorem dvd_iff_mod_eq_zero (x y : ℕ) : x | y ↔ y mod x = 0 :=
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iff.intro mod_eq_zero_of_dvd dvd_of_mod_eq_zero
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definition dvd.decidable_rel [instance] : decidable_rel dvd :=
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take m n, decidable_of_decidable_of_iff _ (!dvd_iff_mod_eq_zero⁻¹)
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theorem dvd_imp_div_mul_eq {x y : ℕ} (H : y | x) : x div y * y = x :=
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mod_eq_zero_imp_div_mul_eq (mod_eq_zero_of_dvd H)
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theorem dvd_of_dvd_add_left {m n1 n2 : ℕ} : m | (n1 + n2) → m | n1 → m | n2 :=
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by_cases_zero_pos m
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(assume (H1 : 0 | n1 + n2) (H2 : 0 | n1),
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have H3 : n1 + n2 = 0, from eq_zero_of_zero_dvd H1,
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have H4 : n1 = 0, from eq_zero_of_zero_dvd H2,
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have H5 : n2 = 0, from calc
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n2 = 0 + n2 : zero_add
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... = n1 + n2 : H4
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... = 0 : H3,
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show 0 | n2, from H5 ▸ dvd.refl n2)
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(take m,
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assume mpos : m > 0,
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assume H1 : m | (n1 + n2),
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assume H2 : m | n1,
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have H3 : n1 + n2 = n1 + n2 div m * m, from calc
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n1 + n2 = (n1 + n2) div m * m : dvd_imp_div_mul_eq H1
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... = (n1 div m * m + n2) div m * m : dvd_imp_div_mul_eq H2
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... = (n2 + n1 div m * m) div m * m : add.comm
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... = (n2 div m + n1 div m) * m : div_add_mul_self_right mpos
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... = n2 div m * m + n1 div m * m : mul.right_distrib
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... = n1 div m * m + n2 div m * m : add.comm
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... = n1 + n2 div m * m : dvd_imp_div_mul_eq H2,
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have H4 : n2 = n2 div m * m, from add.cancel_left H3,
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have H5 : m * (n2 div m) = n2, from !mul.comm ▸ H4⁻¹,
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dvd.intro H5)
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theorem dvd_add_cancel_right {m n1 n2 : ℕ} (H : m | (n1 + n2)) : m | n2 → m | n1 :=
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dvd_of_dvd_add_left (!add.comm ▸ H)
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theorem dvd_sub {m n1 n2 : ℕ} (H1 : m | n1) (H2 : m | n2) : m | (n1 - n2) :=
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by_cases
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(assume H3 : n1 ≥ n2,
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have H4 : n1 = n1 - n2 + n2, from (add_sub_ge_left H3)⁻¹,
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show m | n1 - n2, from dvd_add_cancel_right (H4 ▸ H1) H2)
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(assume H3 : ¬ (n1 ≥ n2),
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have H4 : n1 - n2 = 0, from le_imp_sub_eq_zero (le_of_lt (lt_of_not_le H3)),
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show m | n1 - n2, from H4⁻¹ ▸ dvd_zero _)
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-- Gcd and lcm
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-- -----------
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private definition pair_nat.lt : nat × nat → nat × nat → Prop := measure pr₂
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private definition pair_nat.lt.wf : well_founded pair_nat.lt :=
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intro_k (measure.wf pr₂) 20 -- Remark: we use intro_k to be able to execute gcd efficiently in the kernel
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instance pair_nat.lt.wf -- Remark: instance will not be saved in .olean
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infixl [local] `≺`:50 := pair_nat.lt
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private definition gcd.lt.dec (x y₁ : nat) : (succ y₁, x mod succ y₁) ≺ (x, succ y₁) :=
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mod_lt (succ_pos y₁)
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definition gcd.F (p₁ : nat × nat) : (Π p₂ : nat × nat, p₂ ≺ p₁ → nat) → nat :=
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prod.cases_on p₁ (λx y, cases_on y
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(λ f, x)
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(λ y₁ (f : Πp₂, p₂ ≺ (x, succ y₁) → nat), f (succ y₁, x mod succ y₁) !gcd.lt.dec))
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definition gcd (x y : nat) :=
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fix gcd.F (pair x y)
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theorem gcd_zero (x : nat) : gcd x 0 = x :=
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well_founded.fix_eq gcd.F (x, 0)
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theorem gcd_succ (x y : nat) : gcd x (succ y) = gcd (succ y) (x mod succ y) :=
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well_founded.fix_eq gcd.F (x, succ y)
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theorem gcd_one (n : ℕ) : gcd n 1 = 1 :=
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calc gcd n 1 = gcd 1 (n mod 1) : gcd_succ n zero
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... = gcd 1 0 : mod_one
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... = 1 : gcd_zero
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|
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theorem gcd_def (x y : ℕ) : gcd x y = if y = 0 then x else gcd y (x mod y) :=
|
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cases_on y
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(calc gcd x 0 = x : gcd_zero x
|
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... = if 0 = 0 then x else gcd zero (x mod zero) : (if_pos rfl)⁻¹)
|
||
(λy₁, calc
|
||
gcd x (succ y₁) = gcd (succ y₁) (x mod succ y₁) : gcd_succ x y₁
|
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... = if succ y₁ = 0 then x else gcd (succ y₁) (x mod succ y₁) : (if_neg (succ_ne_zero y₁))⁻¹)
|
||
|
||
theorem gcd_pos (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m mod n) :=
|
||
gcd_def m n ⬝ if_neg (ne_zero_of_pos H)
|
||
|
||
theorem gcd_self (n : ℕ) : gcd n n = n :=
|
||
cases_on n
|
||
rfl
|
||
(λn₁, calc
|
||
gcd (succ n₁) (succ n₁) = gcd (succ n₁) (succ n₁ mod succ n₁) : gcd_succ (succ n₁) n₁
|
||
... = gcd (succ n₁) 0 : mod_self (succ n₁)
|
||
... = succ n₁ : gcd_zero)
|
||
|
||
theorem gcd_zero_left (n : nat) : gcd 0 n = n :=
|
||
cases_on n
|
||
rfl
|
||
(λ n₁, calc
|
||
gcd 0 (succ n₁) = gcd (succ n₁) (0 mod succ n₁) : gcd_succ
|
||
... = gcd (succ n₁) 0 : zero_mod
|
||
... = (succ n₁) : gcd_zero)
|
||
|
||
theorem gcd_induct {P : ℕ → ℕ → Prop}
|
||
(m n : ℕ)
|
||
(H0 : ∀m, P m 0)
|
||
(H1 : ∀m n, 0 < n → P n (m mod n) → P m n)
|
||
: P m n :=
|
||
let Q : nat × nat → Prop := λ p : nat × nat, P (pr₁ p) (pr₂ p) in
|
||
have aux : Q (m, n), from
|
||
well_founded.induction (m, n) (λp, prod.cases_on p
|
||
(λm n, cases_on n
|
||
(λ ih, show P (pr₁ (m, 0)) (pr₂ (m, 0)), from H0 m)
|
||
(λ n₁ (ih : ∀p₂, p₂ ≺ (m, succ n₁) → P (pr₁ p₂) (pr₂ p₂)),
|
||
have hlt₁ : 0 < succ n₁, from succ_pos n₁,
|
||
have hlt₂ : (succ n₁, m mod succ n₁) ≺ (m, succ n₁), from gcd.lt.dec _ _,
|
||
have hp : P (succ n₁) (m mod succ n₁), from ih _ hlt₂,
|
||
show P m (succ n₁), from
|
||
H1 m (succ n₁) hlt₁ hp))),
|
||
aux
|
||
|
||
theorem gcd_dvd (m n : ℕ) : (gcd m n | m) ∧ (gcd m n | n) :=
|
||
gcd_induct m n
|
||
(take m,
|
||
show (gcd m 0 | m) ∧ (gcd m 0 | 0), by simp)
|
||
(take m n,
|
||
assume npos : 0 < n,
|
||
assume IH : (gcd n (m mod n) | n) ∧ (gcd n (m mod n) | (m mod n)),
|
||
have H : gcd n (m mod n) | (m div n * n + m mod n), from
|
||
dvd_add (dvd.trans (and.elim_left IH) !dvd_mul_left) (and.elim_right IH),
|
||
have H1 : gcd n (m mod n) | m, from div_mod_eq⁻¹ ▸ H,
|
||
have gcd_eq : gcd n (m mod n) = gcd m n, from (gcd_pos _ npos)⁻¹,
|
||
show (gcd m n | m) ∧ (gcd m n | n), from gcd_eq ▸ (and.intro H1 (and.elim_left IH)))
|
||
|
||
theorem gcd_dvd_left (m n : ℕ) : (gcd m n | m) := and.elim_left !gcd_dvd
|
||
|
||
theorem gcd_dvd_right (m n : ℕ) : (gcd m n | n) := and.elim_right !gcd_dvd
|
||
|
||
theorem gcd_greatest {m n k : ℕ} : k | m → k | n → k | (gcd m n) :=
|
||
gcd_induct m n
|
||
(take m, assume (h₁ : k | m) (h₂ : k | 0),
|
||
show k | gcd m 0, from !gcd_zero⁻¹ ▸ h₁)
|
||
(take m n,
|
||
assume npos : n > 0,
|
||
assume IH : k | n → k | (m mod n) → k | gcd n (m mod n),
|
||
assume H1 : k | m,
|
||
assume H2 : k | n,
|
||
have H3 : k | m div n * n + m mod n, from div_mod_eq ▸ H1,
|
||
have H4 : k | m mod n, from nat.dvd_of_dvd_add_left H3 (dvd.trans H2 (by simp)),
|
||
have gcd_eq : gcd n (m mod n) = gcd m n, from (gcd_pos _ npos)⁻¹,
|
||
show k | gcd m n, from gcd_eq ▸ IH H2 H4)
|
||
|
||
end nat
|