# Theory Squarefree

theory Squarefree
imports Primes
```(*
File:      HOL/Computational_Algebra/Squarefree.thy
Author:    Manuel Eberl <eberlm@in.tum.de>

Squarefreeness and decomposition of ring elements into square part and squarefree part
*)
section ‹Squarefreeness›
theory Squarefree
imports Primes
begin

(* TODO: Generalise to n-th powers *)

definition squarefree :: "'a :: comm_monoid_mult ⇒ bool" where
"squarefree n ⟷ (∀x. x ^ 2 dvd n ⟶ x dvd 1)"

lemma squarefreeI: "(⋀x. x ^ 2 dvd n ⟹ x dvd 1) ⟹ squarefree n"
by (auto simp: squarefree_def)

lemma squarefreeD: "squarefree n ⟹ x ^ 2 dvd n ⟹ x dvd 1"
by (auto simp: squarefree_def)

lemma not_squarefreeI: "x ^ 2 dvd n ⟹ ¬x dvd 1 ⟹ ¬squarefree n"
by (auto simp: squarefree_def)

lemma not_squarefreeE [case_names square_dvd]:
"¬squarefree n ⟹ (⋀x. x ^ 2 dvd n ⟹ ¬x dvd 1 ⟹ P) ⟹ P"
by (auto simp: squarefree_def)

lemma not_squarefree_0 [simp]: "¬squarefree (0 :: 'a :: comm_semiring_1)"
by (rule not_squarefreeI[of 0]) auto

lemma squarefree_factorial_semiring:
assumes "n ≠ 0"
shows   "squarefree (n :: 'a :: factorial_semiring) ⟷ (∀p. prime p ⟶ ¬p ^ 2 dvd n)"
unfolding squarefree_def
proof safe
assume *: "∀p. prime p ⟶ ¬p ^ 2 dvd n"
fix x :: 'a assume x: "x ^ 2 dvd n"
{
assume "¬is_unit x"
moreover from assms and x have "x ≠ 0" by auto
ultimately obtain p where "p dvd x" "prime p"
using prime_divisor_exists by blast
with * have "¬p ^ 2 dvd n" by blast
moreover from ‹p dvd x› have "p ^ 2 dvd x ^ 2" by (rule dvd_power_same)
ultimately have "¬x ^ 2 dvd n" by (blast dest: dvd_trans)
with x have False by contradiction
}
thus "is_unit x" by blast
qed auto

lemma squarefree_factorial_semiring':
assumes "n ≠ 0"
shows   "squarefree (n :: 'a :: factorial_semiring) ⟷
(∀p∈prime_factors n. multiplicity p n = 1)"
proof (subst squarefree_factorial_semiring [OF assms], safe)
fix p assume "∀p∈#prime_factorization n. multiplicity p n = 1" "prime p" "p^2 dvd n"
with assms show False
by (cases "p dvd n")
(auto simp: prime_factors_dvd power_dvd_iff_le_multiplicity not_dvd_imp_multiplicity_0)
qed (auto intro!: multiplicity_eqI simp: power2_eq_square [symmetric])

lemma squarefree_factorial_semiring'':
assumes "n ≠ 0"
shows   "squarefree (n :: 'a :: factorial_semiring) ⟷
(∀p. prime p ⟶ multiplicity p n ≤ 1)"
by (subst squarefree_factorial_semiring'[OF assms]) (auto simp: prime_factors_multiplicity)

lemma squarefree_unit [simp]: "is_unit n ⟹ squarefree n"
proof (rule squarefreeI)
fix x assume "x^2 dvd n" "n dvd 1"
hence "is_unit (x^2)" by (rule dvd_unit_imp_unit)
thus "is_unit x" by (simp add: is_unit_power_iff)
qed

lemma squarefree_1 [simp]: "squarefree (1 :: 'a :: algebraic_semidom)"
by simp

lemma squarefree_minus [simp]: "squarefree (-n :: 'a :: comm_ring_1) ⟷ squarefree n"

lemma squarefree_mono: "a dvd b ⟹ squarefree b ⟹ squarefree a"
by (auto simp: squarefree_def intro: dvd_trans)

lemma squarefree_multD:
assumes "squarefree (a * b)"
shows   "squarefree a" "squarefree b"
by (rule squarefree_mono[OF _ assms], simp)+

lemma squarefree_prime_elem:
assumes "prime_elem (p :: 'a :: factorial_semiring)"
shows   "squarefree p"
proof -
from assms have "p ≠ 0" by auto
show ?thesis
proof (subst squarefree_factorial_semiring [OF ‹p ≠ 0›]; safe)
fix q assume *: "prime q" "q^2 dvd p"
with assms have "multiplicity q p ≥ 2" by (intro multiplicity_geI) auto
thus False using assms ‹prime q› prime_multiplicity_other[of q "normalize p"]
by (cases "q = normalize p") simp_all
qed
qed

lemma squarefree_prime:
assumes "prime (p :: 'a :: factorial_semiring)"
shows   "squarefree p"
using assms by (intro squarefree_prime_elem) auto

lemma squarefree_mult_coprime:
fixes a b :: "'a :: factorial_semiring_gcd"
assumes "coprime a b" "squarefree a" "squarefree b"
shows   "squarefree (a * b)"
proof -
from assms have nz: "a * b ≠ 0" by auto
show ?thesis unfolding squarefree_factorial_semiring'[OF nz]
proof
fix p assume p: "p ∈ prime_factors (a * b)"
with nz have "prime p"
have "¬ (p dvd a ∧ p dvd b)"
proof
assume "p dvd a ∧ p dvd b"
with ‹coprime a b› have "is_unit p"
by (auto intro: coprime_common_divisor)
with ‹prime p› show False
by simp
qed
moreover from p have "p dvd a ∨ p dvd b" using nz
by (auto simp: prime_factors_dvd prime_dvd_mult_iff)
ultimately show "multiplicity p (a * b) = 1" using nz p assms(2,3)
by (auto simp: prime_elem_multiplicity_mult_distrib prime_factors_multiplicity
not_dvd_imp_multiplicity_0 squarefree_factorial_semiring')
qed
qed

lemma squarefree_prod_coprime:
fixes f :: "'a ⇒ 'b :: factorial_semiring_gcd"
assumes "⋀a b. a ∈ A ⟹ b ∈ A ⟹ a ≠ b ⟹ coprime (f a) (f b)"
assumes "⋀a. a ∈ A ⟹ squarefree (f a)"
shows   "squarefree (prod f A)"
using assms
by (induction A rule: infinite_finite_induct)
(auto intro!: squarefree_mult_coprime prod_coprime_right)

lemma squarefree_powerD: "m > 0 ⟹ squarefree (n ^ m) ⟹ squarefree n"
by (cases m) (auto dest: squarefree_multD)

lemma squarefree_power_iff:
"squarefree (n ^ m) ⟷ m = 0 ∨ is_unit n ∨ (squarefree n ∧ m = 1)"
proof safe
assume "squarefree (n ^ m)" "m > 0" "¬is_unit n"
show "m = 1"
proof (rule ccontr)
assume "m ≠ 1"
with ‹m > 0› have "n ^ 2 dvd n ^ m" by (intro le_imp_power_dvd) auto
from this and ‹¬is_unit n› have "¬squarefree (n ^ m)" by (rule not_squarefreeI)
with ‹squarefree (n ^ m)› show False by contradiction
qed
qed (auto simp: is_unit_power_iff dest: squarefree_powerD)

definition squarefree_nat :: "nat ⇒ bool" where
[code_abbrev]: "squarefree_nat = squarefree"

lemma squarefree_nat_code_naive [code]:
"squarefree_nat n ⟷ n ≠ 0 ∧ (∀k∈{2..n}. ¬k ^ 2 dvd n)"
proof safe
assume *: "∀k∈{2..n}. ¬ k⇧2 dvd n" and n: "n > 0"
show "squarefree_nat n" unfolding squarefree_nat_def
proof (rule squarefreeI)
fix k assume k: "k ^ 2 dvd n"
have "k dvd n" by (rule dvd_trans[OF _ k]) auto
with n have "k ≤ n" by (intro dvd_imp_le)
with bspec[OF *, of k] k have "¬k > 1" by (intro notI) auto
moreover from k and n have "k ≠ 0" by (intro notI) auto
ultimately have "k = 1" by presburger
thus "is_unit k" by simp
qed
qed (auto simp: squarefree_nat_def squarefree_def intro!: Nat.gr0I)

definition square_part :: "'a :: factorial_semiring ⇒ 'a" where
"square_part n = (if n = 0 then 0 else
normalize (∏p∈prime_factors n. p ^ (multiplicity p n div 2)))"

lemma square_part_nonzero:
"n ≠ 0 ⟹ square_part n = normalize (∏p∈prime_factors n. p ^ (multiplicity p n div 2))"

lemma square_part_0 [simp]: "square_part 0 = 0"

lemma square_part_unit [simp]: "is_unit x ⟹ square_part x = 1"
by (auto simp: square_part_def prime_factorization_unit)

lemma square_part_1 [simp]: "square_part 1 = 1"
by simp

lemma square_part_0_iff [simp]: "square_part n = 0 ⟷ n = 0"

lemma normalize_uminus [simp]:
"normalize (-x :: 'a :: {normalization_semidom, comm_ring_1}) = normalize x"
by (rule associatedI) auto

lemma multiplicity_uminus_right [simp]:
"multiplicity (x :: 'a :: {factorial_semiring, comm_ring_1}) (-y) = multiplicity x y"
proof -
have "multiplicity x (-y) = multiplicity x (normalize (-y))"
by (rule multiplicity_normalize_right [symmetric])
also have "… = multiplicity x y" by simp
finally show ?thesis .
qed

lemma multiplicity_uminus_left [simp]:
"multiplicity (-x :: 'a :: {factorial_semiring, comm_ring_1}) y = multiplicity x y"
proof -
have "multiplicity (-x) y = multiplicity (normalize (-x)) y"
by (rule multiplicity_normalize_left [symmetric])
also have "… = multiplicity x y" by simp
finally show ?thesis .
qed

lemma prime_factorization_uminus [simp]:
"prime_factorization (-x :: 'a :: {factorial_semiring, comm_ring_1}) = prime_factorization x"
by (rule prime_factorization_cong) simp_all

lemma square_part_uminus [simp]:
"square_part (-x :: 'a :: {factorial_semiring, comm_ring_1}) = square_part x"

lemma prime_multiplicity_square_part:
assumes "prime p"
shows   "multiplicity p (square_part n) = multiplicity p n div 2"
proof (cases "n = 0")
case False
thus ?thesis unfolding square_part_nonzero[OF False] multiplicity_normalize_right
using finite_prime_divisors[of n] assms
by (subst multiplicity_prod_prime_powers)
(auto simp: not_dvd_imp_multiplicity_0 prime_factors_dvd multiplicity_prod_prime_powers)
qed auto

lemma square_part_square_dvd [simp, intro]: "square_part n ^ 2 dvd n"
proof (cases "n = 0")
case False
thus ?thesis
by (intro multiplicity_le_imp_dvd)
(auto simp: prime_multiplicity_square_part prime_elem_multiplicity_power_distrib)
qed auto

lemma prime_multiplicity_le_imp_dvd:
assumes "x ≠ 0" "y ≠ 0"
shows   "x dvd y ⟷ (∀p. prime p ⟶ multiplicity p x ≤ multiplicity p y)"
using assms by (auto intro: multiplicity_le_imp_dvd dvd_imp_multiplicity_le)

lemma dvd_square_part_iff: "x dvd square_part n ⟷ x ^ 2 dvd n"
proof (cases "x = 0"; cases "n = 0")
assume nz: "x ≠ 0" "n ≠ 0"
thus ?thesis
by (subst (1 2) prime_multiplicity_le_imp_dvd)
(auto simp: prime_multiplicity_square_part prime_elem_multiplicity_power_distrib)
qed auto

definition squarefree_part :: "'a :: factorial_semiring ⇒ 'a" where
"squarefree_part n = (if n = 0 then 1 else n div square_part n ^ 2)"

lemma squarefree_part_0 [simp]: "squarefree_part 0 = 1"

lemma squarefree_part_unit [simp]: "is_unit n ⟹ squarefree_part n = n"

lemma squarefree_part_1 [simp]: "squarefree_part 1 = 1"
by simp

lemma squarefree_decompose: "n = squarefree_part n * square_part n ^ 2"

lemma squarefree_part_uminus [simp]:
assumes "x ≠ 0"
shows   "squarefree_part (-x :: 'a :: {factorial_semiring, comm_ring_1}) = -squarefree_part x"
proof -
have "-(squarefree_part x * square_part x ^ 2) = -x"
by (subst squarefree_decompose [symmetric]) auto
also have "… = squarefree_part (-x) * square_part (-x) ^ 2" by (rule squarefree_decompose)
finally have "(- squarefree_part x) * square_part x ^ 2 =
squarefree_part (-x) * square_part x ^ 2" by simp
thus ?thesis using assms by (subst (asm) mult_right_cancel) auto
qed

lemma squarefree_part_nonzero [simp]: "squarefree_part n ≠ 0"
using squarefree_decompose[of n] by (cases "n ≠ 0") auto

lemma prime_multiplicity_squarefree_part:
assumes "prime p"
shows   "multiplicity p (squarefree_part n) = multiplicity p n mod 2"
proof (cases "n = 0")
case False
hence n: "n ≠ 0" by auto
have "multiplicity p n mod 2 + 2 * (multiplicity p n div 2) = multiplicity p n" by simp
also have "… = multiplicity p (squarefree_part n * square_part n ^ 2)"
by (subst squarefree_decompose[of n]) simp
also from assms n have "… = multiplicity p (squarefree_part n) + 2 * (multiplicity p n div 2)"
by (subst prime_elem_multiplicity_mult_distrib)
(auto simp: prime_elem_multiplicity_power_distrib prime_multiplicity_square_part)
finally show ?thesis by (subst (asm) add_right_cancel) simp
qed auto

lemma prime_multiplicity_squarefree_part_le_Suc_0 [intro]:
assumes "prime p"
shows   "multiplicity p (squarefree_part n) ≤ Suc 0"

lemma squarefree_squarefree_part [simp, intro]: "squarefree (squarefree_part n)"
by (subst squarefree_factorial_semiring'')
(auto simp: prime_multiplicity_squarefree_part_le_Suc_0)

lemma squarefree_decomposition_unique:
assumes "square_part m = square_part n"
assumes "squarefree_part m = squarefree_part n"
shows   "m = n"
by (subst (1 2) squarefree_decompose) (simp_all add: assms)

lemma normalize_square_part [simp]: "normalize (square_part x) = square_part x"

lemma square_part_even_power': "square_part (x ^ (2 * n)) = normalize (x ^ n)"
proof (cases "x = 0")
case False
have "normalize (square_part (x ^ (2 * n))) = normalize (x ^ n)" using False
by (intro multiplicity_eq_imp_eq)
(auto simp: prime_multiplicity_square_part prime_elem_multiplicity_power_distrib)
thus ?thesis by simp
qed (auto simp: power_0_left)

lemma square_part_even_power: "even n ⟹ square_part (x ^ n) = normalize (x ^ (n div 2))"
by (subst square_part_even_power' [symmetric]) auto

lemma square_part_odd_power': "square_part (x ^ (Suc (2 * n))) = normalize (x ^ n * square_part x)"
proof (cases "x = 0")
case False
have "normalize (square_part (x ^ (Suc (2 * n)))) = normalize (square_part x * x ^ n)"
proof (rule multiplicity_eq_imp_eq, goal_cases)
case (3 p)
hence "multiplicity p (square_part (x ^ Suc (2 * n))) =
(2 * (n * multiplicity p x) + multiplicity p x) div 2"
by (subst prime_multiplicity_square_part)
(auto simp: False prime_elem_multiplicity_power_distrib algebra_simps simp del: power_Suc)
also from 3 False have "… = multiplicity p (square_part x * x ^ n)"
by (subst div_mult_self4) (auto simp: prime_multiplicity_square_part
prime_elem_multiplicity_mult_distrib prime_elem_multiplicity_power_distrib)
finally show ?case .
qed (insert False, auto)
thus ?thesis by (simp add: mult_ac)
qed auto

lemma square_part_odd_power:
"odd n ⟹ square_part (x ^ n) = normalize (x ^ (n div 2) * square_part x)"
by (subst square_part_odd_power' [symmetric]) auto

end
```