refactor/feat(library/data/nat): fix up sub and div, rename theorems, add theorems
This commit is contained in:
Jeremy Avigad
parent
0dcf06000b
commit
b68ce77650
3 changed files with 152 additions and 160 deletions
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@ -2,10 +2,10 @@
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Copyright (c) 2014 Floris van Doorn. 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.algebra
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Authors: Jeremy Avigad
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Module: data.nat.comm_semiring
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Author: Jeremy Avigad
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nat is a comm_semiring
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ℕ is a comm_semiring.
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-/
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import data.nat.basic algebra.binary algebra.ring
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open binary
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@ -1,4 +1,4 @@
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/-
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/-
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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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@ -8,25 +8,21 @@ 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 data.nat.comm_semiring logic
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import algebra.relation
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import tools.fake_simplifier
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import data.nat.sub data.nat.comm_semiring 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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/- div and mod -/
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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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and.rec_on H (λ ypos ylex, 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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definition divide (x y : nat) := 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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@ -36,22 +32,25 @@ 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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theorem div_eq_zero_of_lt {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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theorem div_eq_succ_sub_div {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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theorem add_div_left {x z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) :=
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calc
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(x + z) div z = if 0 < z ∧ z ≤ x + z then (x + z - z) div z + 1 else 0 : !divide_def
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... = (x + z - z) div z + 1 : if_pos (and.intro H (le_add_left z x))
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... = succ (x div z) : add_sub_cancel
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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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theorem add_div_right {x z : ℕ} (H : x > 0) : (x + z) div x = succ (z div x) :=
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!add.comm ▸ add_div_left H
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theorem add_mul_div_left {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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@ -59,14 +58,16 @@ induction_on y
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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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... = succ ((x + y * z) div z) : add_div_left H
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... = x div z + succ y : by simp)
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theorem add_mul_div_right {x y z : ℕ} (H : y > 0) : (x + y * z) div y = x div y + z :=
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!mul.comm ▸ add_mul_div_left H
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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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definition modulo (x y : nat) := fix mod.F x y
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notation a mod b := modulo a b
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@ -76,22 +77,25 @@ 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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theorem mod_eq_of_lt {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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theorem mod_eq_sub_mod {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 : add_sub_cancel
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theorem add_mod_left {x z : ℕ} (H : z > 0) : (x + z) mod z = x mod z :=
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calc
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(x + z) mod z = 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 : add_sub_cancel
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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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theorem add_mod_right {x z : ℕ} (H : x > 0) : (x + z) mod x = z mod x :=
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!add.comm ▸ add_mod_left H
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theorem add_mul_mod_left {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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@ -100,23 +104,20 @@ induction_on y
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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 + y * z) mod z : add_mod_left 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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theorem add_mul_mod_right {x y z : ℕ} (H : y > 0) : (x + y * z) mod y = x mod y :=
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!mul.comm ▸ add_mul_mod_left H
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theorem mul_mod_left {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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(by simp) ▸ (@add_mul_mod_left 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 mul_mod_right {m n : ℕ} : (m * n) mod m = 0 :=
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!mul.comm ▸ !mul_mod_left
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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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@ -126,16 +127,19 @@ case_strong_induction_on x
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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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have H2 : succ x mod y = succ x, from mod_eq_of_lt 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 H3 : succ x mod y = (succ x - y) mod y, from mod_eq_sub_mod 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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/- properties of div and mod together -/
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-- the quotient / remainder theorem
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theorem eq_div_mul_add_mod {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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@ -152,45 +156,45 @@ by_cases_zero_pos 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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have H2 : succ x div y = 0, from div_eq_zero_of_lt H1,
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have H3 : succ x mod y = succ x, from mod_eq_of_lt 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 H3 : succ x div y = succ ((succ x - y) div y), from div_eq_succ_sub_div H H2,
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have H4 : succ x mod y = (succ x - y) mod y, from mod_eq_sub_mod 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 div y * y + succ x mod y =
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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 : sub_add_cancel H2)⁻¹)))
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... = succ x : sub_add_cancel 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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eq_div_mul_add_mod⁻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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theorem eq_remainder {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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... = (r1 + q1 * y) mod y : (add_mul_mod_left 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 mod y : add_mul_mod_left 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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theorem eq_quotient {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 H4 : q1 * y + r2 = q2 * y + r2, from (eq_remainder 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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theorem mul_div_mul_left {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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@ -198,16 +202,17 @@ by_cases -- (y = 0)
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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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eq_quotient 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 (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : eq_div_mul_add_mod
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... = z * (x div y * y + x mod y) : eq_div_mul_add_mod
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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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theorem mul_div_mul_right {x z y : ℕ} (zpos : z > 0) : (x * z) div (y * z) = x div y :=
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!mul.comm ▸ !mul.comm ▸ mul_div_mul_left zpos
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theorem mul_mod_mul_left {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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@ -215,106 +220,60 @@ by_cases -- (y = 0)
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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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eq_remainder 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 (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : eq_div_mul_add_mod
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... = z * (x div y * y + x mod y) : eq_div_mul_add_mod
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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 mul_mod_mul_right {x z y : ℕ} (zpos : z > 0) : (x * z) mod (y * z) = (x mod y) * z :=
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mul.comm z x ▸ mul.comm z y ▸ !mul.comm ▸ mul_mod_mul_left zpos
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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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from mul_mod_mul_left !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,
|
||||
have H : n div 1 * 1 + n mod 1 = n, from eq_div_mul_add_mod⁻¹,
|
||||
(by simp) ▸ H
|
||||
|
||||
-- add_rewrite div_one
|
||||
|
||||
theorem pos_div_self {n : ℕ} (H : n > 0) : n div n = 1 :=
|
||||
have H1 : (n * 1) div (n * 1) = 1 div 1, from div_mul_mul H,
|
||||
theorem div_self {n : ℕ} (H : n > 0) : n div n = 1 :=
|
||||
have H1 : (n * 1) div (n * 1) = 1 div 1, from mul_div_mul_left H,
|
||||
(by simp) ▸ H1
|
||||
|
||||
-- add_rewrite pos_div_self
|
||||
|
||||
-- Divides
|
||||
-- -------
|
||||
|
||||
-- definition dvd (x y : ℕ) : Prop := y mod x = 0
|
||||
|
||||
-- infix `|` := dvd
|
||||
|
||||
-- theorem dvd_iff_mod_eq_zero {x y : ℕ} : x | y ↔ y mod x = 0 :=
|
||||
-- iff.of_eq rfl
|
||||
|
||||
theorem mod_eq_zero_imp_div_mul_eq {x y : ℕ} (H : x mod y = 0) : x div y * y = x :=
|
||||
theorem div_mul_eq_of_mod_eq_zero {x y : ℕ} (H : x mod y = 0) : x div y * y = x :=
|
||||
(calc
|
||||
x = x div y * y + x mod y : div_mod_eq
|
||||
x = x div y * y + x mod y : eq_div_mul_add_mod
|
||||
... = x div y * y + 0 : H
|
||||
... = x div y * y : !add_zero)⁻¹
|
||||
|
||||
-- add_rewrite dvd_imp_div_mul_eq
|
||||
|
||||
theorem mul_eq_imp_mod_eq_zero {z x y : ℕ} (H : z * y = x) : x mod y = 0 :=
|
||||
have H1 : z * y = x mod y + x div y * y, from
|
||||
H ⬝ div_mod_eq ⬝ !add.comm,
|
||||
have H2 : (z - x div y) * y = x mod y, from
|
||||
calc
|
||||
(z - x div y) * y = z * y - x div y * y : mul_sub_right_distrib
|
||||
... = x mod y + x div y * y - x div y * y : H1
|
||||
... = x mod y : add_sub_cancel,
|
||||
show x mod y = 0, from
|
||||
by_cases
|
||||
(assume yz : y = 0,
|
||||
have xz : x = 0, from
|
||||
calc
|
||||
x = z * y : H
|
||||
... = z * 0 : yz
|
||||
... = 0 : mul_zero,
|
||||
calc
|
||||
x mod y = x mod 0 : yz
|
||||
... = x : mod_zero
|
||||
... = 0 : xz)
|
||||
(assume ynz : y ≠ 0,
|
||||
have ypos : y > 0, from pos_of_ne_zero ynz,
|
||||
have H3 : (z - x div y) * y < y, from H2⁻¹ ▸ mod_lt ypos,
|
||||
have H4 : (z - x div y) * y < 1 * y, from !one_mul⁻¹ ▸ H3,
|
||||
have H5 : z - x div y < 1, from lt_of_mul_lt_mul_right H4,
|
||||
have H6 : z - x div y = 0, from eq_zero_of_le_zero (le_of_lt_succ H5),
|
||||
calc
|
||||
x mod y = (z - x div y) * y : H2
|
||||
... = 0 * y : H6
|
||||
... = 0 : zero_mul)
|
||||
/- divides -/
|
||||
|
||||
theorem dvd_of_mod_eq_zero {x y : ℕ} (H : y mod x = 0) : x | y :=
|
||||
dvd.intro (!mul.comm ▸ mod_eq_zero_imp_div_mul_eq H)
|
||||
dvd.intro (!mul.comm ▸ div_mul_eq_of_mod_eq_zero H)
|
||||
|
||||
theorem mod_eq_zero_of_dvd {x y : ℕ} (H : x | y) : y mod x = 0 :=
|
||||
dvd.elim H (
|
||||
take z,
|
||||
dvd.elim H
|
||||
(take z,
|
||||
assume H1 : x * z = y,
|
||||
mul_eq_imp_mod_eq_zero (!mul.comm ▸ H1))
|
||||
H1 ▸ !mul_mod_right)
|
||||
|
||||
theorem dvd_iff_mod_eq_zero (x y : ℕ) : x | y ↔ y mod x = 0 :=
|
||||
iff.intro mod_eq_zero_of_dvd dvd_of_mod_eq_zero
|
||||
|
||||
definition dvd.decidable_rel [instance] : decidable_rel dvd :=
|
||||
take m n, decidable_of_decidable_of_iff _ (!dvd_iff_mod_eq_zero⁻¹)
|
||||
take m n, decidable_of_decidable_of_iff _ (iff.symm !dvd_iff_mod_eq_zero)
|
||||
|
||||
theorem dvd_imp_div_mul_eq {x y : ℕ} (H : y | x) : x div y * y = x :=
|
||||
mod_eq_zero_imp_div_mul_eq (mod_eq_zero_of_dvd H)
|
||||
theorem div_mul_eq_of_dvd {x y : ℕ} (H : y | x) : x div y * y = x :=
|
||||
div_mul_eq_of_mod_eq_zero (mod_eq_zero_of_dvd H)
|
||||
|
||||
theorem dvd_of_dvd_add_left {m n1 n2 : ℕ} : m | (n1 + n2) → m | n1 → m | n2 :=
|
||||
by_cases_zero_pos m
|
||||
|
|
@ -331,36 +290,50 @@ by_cases_zero_pos m
|
|||
assume H1 : m | (n1 + n2),
|
||||
assume H2 : m | n1,
|
||||
have H3 : n1 + n2 = n1 + n2 div m * m, from calc
|
||||
n1 + n2 = (n1 + n2) div m * m : dvd_imp_div_mul_eq H1
|
||||
... = (n1 div m * m + n2) div m * m : dvd_imp_div_mul_eq H2
|
||||
n1 + n2 = (n1 + n2) div m * m : div_mul_eq_of_dvd H1
|
||||
... = (n1 div m * m + n2) div m * m : div_mul_eq_of_dvd H2
|
||||
... = (n2 + n1 div m * m) div m * m : add.comm
|
||||
... = (n2 div m + n1 div m) * m : div_add_mul_self_right mpos
|
||||
... = (n2 div m + n1 div m) * m : add_mul_div_left mpos
|
||||
... = n2 div m * m + n1 div m * m : mul.right_distrib
|
||||
... = n1 div m * m + n2 div m * m : add.comm
|
||||
... = n1 + n2 div m * m : dvd_imp_div_mul_eq H2,
|
||||
... = n1 + n2 div m * m : div_mul_eq_of_dvd H2,
|
||||
have H4 : n2 = n2 div m * m, from add.cancel_left H3,
|
||||
have H5 : m * (n2 div m) = n2, from !mul.comm ▸ H4⁻¹,
|
||||
dvd.intro H5)
|
||||
|
||||
theorem dvd_add_cancel_right {m n1 n2 : ℕ} (H : m | (n1 + n2)) : m | n2 → m | n1 :=
|
||||
theorem dvd_of_dvd_add_right {m n1 n2 : ℕ} (H : m | (n1 + n2)) : m | n2 → m | n1 :=
|
||||
dvd_of_dvd_add_left (!add.comm ▸ H)
|
||||
|
||||
theorem dvd_sub {m n1 n2 : ℕ} (H1 : m | n1) (H2 : m | n2) : m | (n1 - n2) :=
|
||||
by_cases
|
||||
(assume H3 : n1 ≥ n2,
|
||||
have H4 : n1 = n1 - n2 + n2, from (sub_add_cancel H3)⁻¹,
|
||||
show m | n1 - n2, from dvd_add_cancel_right (H4 ▸ H1) H2)
|
||||
show m | n1 - n2, from dvd_of_dvd_add_right (H4 ▸ H1) H2)
|
||||
(assume H3 : ¬ (n1 ≥ n2),
|
||||
have H4 : n1 - n2 = 0, from sub_eq_zero_of_le (le_of_lt (lt_of_not_le H3)),
|
||||
show m | n1 - n2, from H4⁻¹ ▸ dvd_zero _)
|
||||
|
||||
-- Gcd and lcm
|
||||
-- -----------
|
||||
theorem dvd.antisymm {m n : ℕ} : m | n → n | m → m = n :=
|
||||
by_cases_zero_pos n
|
||||
(assume H1, assume H2 : 0 | m, eq_zero_of_zero_dvd H2)
|
||||
(take n,
|
||||
assume Hpos : n > 0,
|
||||
assume H1 : m | n,
|
||||
assume H2 : n | m,
|
||||
obtain k (Hk : m * k = n), from dvd.ex H1,
|
||||
obtain l (Hl : n * l = m), from dvd.ex H2,
|
||||
have H3 : n * (l * k) = n, from !mul.assoc ▸ Hl⁻¹ ▸ Hk,
|
||||
have H4 : l * k = 1, from eq_one_of_mul_eq_self_right Hpos H3,
|
||||
have H5 : k = 1, from eq_one_of_mul_eq_one_left H4,
|
||||
show m = n, from (mul_one m)⁻¹ ⬝ (H5 ▸ Hk))
|
||||
|
||||
/- gcd and lcm -/
|
||||
|
||||
private definition pair_nat.lt : nat × nat → nat × nat → Prop := measure pr₂
|
||||
private definition pair_nat.lt.wf : well_founded pair_nat.lt :=
|
||||
intro_k (measure.wf pr₂) 20 -- Remark: we use intro_k to be able to execute gcd efficiently in the kernel
|
||||
instance pair_nat.lt.wf -- Remark: instance will not be saved in .olean
|
||||
intro_k (measure.wf pr₂) 20 -- we use intro_k to be able to execute gcd efficiently in the kernel
|
||||
|
||||
instance pair_nat.lt.wf -- instance will not be saved in .olean
|
||||
infixl [local] `≺`:50 := pair_nat.lt
|
||||
|
||||
private definition gcd.lt.dec (x y₁ : nat) : (succ y₁, x mod succ y₁) ≺ (x, succ y₁) :=
|
||||
|
|
@ -371,8 +344,7 @@ prod.cases_on p₁ (λx y, cases_on y
|
|||
(λ f, x)
|
||||
(λ y₁ (f : Πp₂, p₂ ≺ (x, succ y₁) → nat), f (succ y₁, x mod succ y₁) !gcd.lt.dec))
|
||||
|
||||
definition gcd (x y : nat) :=
|
||||
fix gcd.F (pair x y)
|
||||
definition gcd (x y : nat) := fix gcd.F (pair x y)
|
||||
|
||||
theorem gcd_zero (x : nat) : gcd x 0 = x :=
|
||||
well_founded.fix_eq gcd.F (x, 0)
|
||||
|
|
@ -390,10 +362,10 @@ cases_on y
|
|||
(calc gcd x 0 = x : gcd_zero x
|
||||
... = 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
|
||||
... = if succ y₁ = 0 then x else gcd (succ y₁) (x mod succ y₁) : (if_neg (succ_ne_zero y₁))⁻¹)
|
||||
gcd x (succ y₁) = gcd (succ y₁) (x mod succ y₁) : gcd_succ x y₁
|
||||
... = 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) :=
|
||||
theorem gcd_rec (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 :=
|
||||
|
|
@ -412,11 +384,11 @@ cases_on n
|
|||
... = gcd (succ n₁) 0 : zero_mod
|
||||
... = (succ n₁) : gcd_zero)
|
||||
|
||||
theorem gcd_induct {P : ℕ → ℕ → Prop}
|
||||
theorem gcd.induction {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 :=
|
||||
(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
|
||||
|
|
@ -431,7 +403,7 @@ have aux : Q (m, n), from
|
|||
aux
|
||||
|
||||
theorem gcd_dvd (m n : ℕ) : (gcd m n | m) ∧ (gcd m n | n) :=
|
||||
gcd_induct m n
|
||||
gcd.induction m n
|
||||
(take m,
|
||||
show (gcd m 0 | m) ∧ (gcd m 0 | 0), by simp)
|
||||
(take m n,
|
||||
|
|
@ -439,16 +411,16 @@ gcd_induct m 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,
|
||||
have H1 : gcd n (m mod n) | m, from eq_div_mul_add_mod⁻¹ ▸ H,
|
||||
have gcd_eq : gcd n (m mod n) = gcd m n, from (gcd_rec _ 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
|
||||
theorem dvd_gcd {m n k : ℕ} : k | m → k | n → k | (gcd m n) :=
|
||||
gcd.induction m n
|
||||
(take m, assume (h₁ : k | m) (h₂ : k | 0),
|
||||
show k | gcd m 0, from !gcd_zero⁻¹ ▸ h₁)
|
||||
(take m n,
|
||||
|
|
@ -456,9 +428,23 @@ gcd_induct m n
|
|||
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 H3 : k | m div n * n + m mod n, from eq_div_mul_add_mod ▸ 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,
|
||||
have gcd_eq : gcd n (m mod n) = gcd m n, from (gcd_rec _ npos)⁻¹,
|
||||
show k | gcd m n, from gcd_eq ▸ IH H2 H4)
|
||||
|
||||
theorem gcd.comm (m n : ℕ) : gcd m n = gcd n m :=
|
||||
dvd.antisymm
|
||||
(dvd_gcd !gcd_dvd_right !gcd_dvd_left)
|
||||
(dvd_gcd !gcd_dvd_right !gcd_dvd_left)
|
||||
|
||||
theorem gcd.assoc (m n k : ℕ) : gcd (gcd m n) k = gcd m (gcd n k) :=
|
||||
dvd.antisymm
|
||||
(dvd_gcd
|
||||
(dvd.trans !gcd_dvd_left !gcd_dvd_left)
|
||||
(dvd_gcd (dvd.trans !gcd_dvd_left !gcd_dvd_right) !gcd_dvd_right))
|
||||
(dvd_gcd
|
||||
(dvd_gcd !gcd_dvd_left (dvd.trans !gcd_dvd_right !gcd_dvd_left))
|
||||
(dvd.trans !gcd_dvd_right !gcd_dvd_right))
|
||||
|
||||
end nat
|
||||
|
|
|
|||
|
|
@ -411,17 +411,17 @@ have H4 : m ≤ k, from le_of_mul_le_mul_left H2 Hn,
|
|||
have H5 : k ≤ m, from le_of_mul_le_mul_left H3 Hn,
|
||||
le.antisymm H4 H5
|
||||
|
||||
theorem eq_of_mul_eq_mul_right {n m k : ℕ} (Hm : m > 0) (H : n * m = k * m) : n = k :=
|
||||
eq_of_mul_eq_mul_left Hm (!mul.comm ▸ !mul.comm ▸ H)
|
||||
|
||||
theorem eq_zero_or_eq_of_mul_eq_mul_left {n m k : ℕ} (H : n * m = n * k) : n = 0 ∨ m = k :=
|
||||
or_of_or_of_imp_right zero_or_pos
|
||||
(assume Hn : n > 0, eq_of_mul_eq_mul_left Hn H)
|
||||
|
||||
theorem eq_of_mul_eq_mul_right {n m k : ℕ} (Hm : m > 0) (H : n * m = k * m) : n = k :=
|
||||
eq_of_mul_eq_mul_left Hm (!mul.comm ▸ !mul.comm ▸ H)
|
||||
|
||||
theorem eq_zero_or_eq_of_mul_eq_mul_right {n m k : ℕ} (H : n * m = k * m) : m = 0 ∨ n = k :=
|
||||
eq_zero_or_eq_of_mul_eq_mul_left (!mul.comm ▸ !mul.comm ▸ H)
|
||||
|
||||
theorem eq_one_of_mul_eq_one_left {n m : ℕ} (H : n * m = 1) : n = 1 :=
|
||||
theorem eq_one_of_mul_eq_one_right {n m : ℕ} (H : n * m = 1) : n = 1 :=
|
||||
have H2 : n * m > 0, from H⁻¹ ▸ !succ_pos,
|
||||
have H3 : n > 0, from pos_of_mul_pos_right H2,
|
||||
have H4 : m > 0, from pos_of_mul_pos_left H2,
|
||||
|
|
@ -433,7 +433,13 @@ or.elim (le_or_gt n 1)
|
|||
have H7 : 1 ≥ 2, from !mul_one ▸ H ▸ H6,
|
||||
absurd !self_lt_succ (not_lt_of_le H7))
|
||||
|
||||
theorem eq_one_of_mul_eq_one_right {n m : ℕ} (H : n * m = 1) : m = 1 :=
|
||||
eq_one_of_mul_eq_one_left (!mul.comm ▸ H)
|
||||
theorem eq_one_of_mul_eq_one_left {n m : ℕ} (H : n * m = 1) : m = 1 :=
|
||||
eq_one_of_mul_eq_one_right (!mul.comm ▸ H)
|
||||
|
||||
theorem eq_one_of_mul_eq_self_left {n m : ℕ} (Hpos : n > 0) (H : m * n = n) : m = 1 :=
|
||||
eq_of_mul_eq_mul_right Hpos (H ⬝ !one_mul⁻¹)
|
||||
|
||||
theorem eq_one_of_mul_eq_self_right {n m : ℕ} (Hpos : m > 0) (H : m * n = m) : n = 1 :=
|
||||
eq_one_of_mul_eq_self_left Hpos (!mul.comm ▸ H)
|
||||
|
||||
end nat
|
||||
|
|
|
|||
Loading…
Reference in a new issue