src/HOL/Computational_Algebra/Squarefree.thy

+(*
+  File:      HOL/Computational_Algebra/Squarefree.thy
+  Author:    Manuel Eberl <[e-mail]>
+
+  Squarefreeness and decomposition of ring elements into square part and squarefree part
+*)
+section \<open>Squarefreeness\<close>
+theory Squarefree
+imports Primes
+begin
+  
+(* TODO: Generalise to n-th powers *)
+
+definition squarefree :: "'a :: comm_monoid_mult \<Rightarrow> bool" where
+  "squarefree n \<longleftrightarrow> (\<forall>x. x ^ 2 dvd n \<longrightarrow> x dvd 1)"
+  
+lemma squarefreeI: "(\<And>x. x ^ 2 dvd n \<Longrightarrow> x dvd 1) \<Longrightarrow> squarefree n"
+  by (auto simp: squarefree_def)
+
+lemma squarefreeD: "squarefree n \<Longrightarrow> x ^ 2 dvd n \<Longrightarrow> x dvd 1"
+  by (auto simp: squarefree_def)
+
+lemma not_squarefreeI: "x ^ 2 dvd n \<Longrightarrow> \<not>x dvd 1 \<Longrightarrow> \<not>squarefree n"
+  by (auto simp: squarefree_def)
+
+lemma not_squarefreeE [case_names square_dvd]: 
+  "\<not>squarefree n \<Longrightarrow> (\<And>x. x ^ 2 dvd n \<Longrightarrow> \<not>x dvd 1 \<Longrightarrow> P) \<Longrightarrow> P"
+  by (auto simp: squarefree_def)
+
+lemma not_squarefree_0 [simp]: "\<not>squarefree (0 :: 'a :: comm_semiring_1)"
+  by (rule not_squarefreeI[of 0]) auto
+
+lemma squarefree_factorial_semiring:
+  assumes "n \<noteq> 0"
+  shows   "squarefree (n :: 'a :: factorial_semiring) \<longleftrightarrow> (\<forall>p. prime p \<longrightarrow> \<not>p ^ 2 dvd n)"
+  unfolding squarefree_def
+proof safe
+  assume *: "\<forall>p. prime p \<longrightarrow> \<not>p ^ 2 dvd n"
+  fix x :: 'a assume x: "x ^ 2 dvd n"
+  {
+    assume "\<not>is_unit x"
+    moreover from assms and x have "x \<noteq> 0" by auto
+    ultimately obtain p where "p dvd x" "prime p"
+      using prime_divisor_exists by blast
+    with * have "\<not>p ^ 2 dvd n" by blast
+    moreover from \<open>p dvd x\<close> have "p ^ 2 dvd x ^ 2" by (rule dvd_power_same)
+    ultimately have "\<not>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 \<noteq> 0"
+  shows   "squarefree (n :: 'a :: factorial_semiring) \<longleftrightarrow> 
+             (\<forall>p\<in>prime_factors n. multiplicity p n = 1)"
+proof (subst squarefree_factorial_semiring [OF assms], safe)
+  fix p assume "\<forall>p\<in>#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 \<noteq> 0"
+  shows   "squarefree (n :: 'a :: factorial_semiring) \<longleftrightarrow> 
+             (\<forall>p. prime p \<longrightarrow> multiplicity p n \<le> 1)"
+  by (subst squarefree_factorial_semiring'[OF assms]) (auto simp: prime_factors_multiplicity)
+
+lemma squarefree_unit [simp]: "is_unit n \<Longrightarrow> 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) \<longleftrightarrow> squarefree n"
+  by (simp add: squarefree_def)
+
+lemma squarefree_mono: "a dvd b \<Longrightarrow> squarefree b \<Longrightarrow> 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 \<noteq> 0" by auto
+  show ?thesis
+  proof (subst squarefree_factorial_semiring [OF \<open>p \<noteq> 0\<close>]; safe)
+    fix q assume *: "prime q" "q^2 dvd p"
+    with assms have "multiplicity q p \<ge> 2" by (intro multiplicity_geI) auto
+    thus False using assms \<open>prime q\<close> 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 \<noteq> 0" by auto
+  show ?thesis unfolding squarefree_factorial_semiring'[OF nz]
+  proof
+    fix p assume p: "p \<in> prime_factors (a * b)"
+    {
+      assume "p dvd a \<and> p dvd b"
+      hence "p dvd gcd a b" by simp
+      also have "gcd a b = 1" by fact
+      finally have False using nz using p by (auto simp: prime_factors_dvd)
+    }
+    hence "\<not>(p dvd a \<and> p dvd b)" by blast
+    moreover from p have "p dvd a \<or> 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 \<Rightarrow> 'b :: factorial_semiring_gcd"
+  assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> coprime (f a) (f b)"
+  assumes "\<And>a. a \<in> A \<Longrightarrow> squarefree (f a)"
+  shows   "squarefree (prod f A)"
+  using assms 
+  by (induction A rule: infinite_finite_induct) 
+     (auto intro!: squarefree_mult_coprime prod_coprime')
+
+lemma squarefree_powerD: "m > 0 \<Longrightarrow> squarefree (n ^ m) \<Longrightarrow> squarefree n"
+  by (cases m) (auto dest: squarefree_multD)
+
+lemma squarefree_power_iff: 
+  "squarefree (n ^ m) \<longleftrightarrow> m = 0 \<or> is_unit n \<or> (squarefree n \<and> m = 1)"
+proof safe
+  assume "squarefree (n ^ m)" "m > 0" "\<not>is_unit n"
+  show "m = 1"
+  proof (rule ccontr)
+    assume "m \<noteq> 1"
+    with \<open>m > 0\<close> have "n ^ 2 dvd n ^ m" by (intro le_imp_power_dvd) auto
+    from this and \<open>\<not>is_unit n\<close> have "\<not>squarefree (n ^ m)" by (rule not_squarefreeI)
+    with \<open>squarefree (n ^ m)\<close> show False by contradiction
+  qed
+qed (auto simp: is_unit_power_iff dest: squarefree_powerD)
+
+definition squarefree_nat :: "nat \<Rightarrow> bool" where
+  [code_abbrev]: "squarefree_nat = squarefree"
+  
+lemma squarefree_nat_code_naive [code]: 
+  "squarefree_nat n \<longleftrightarrow> n \<noteq> 0 \<and> (\<forall>k\<in>{2..n}. \<not>k ^ 2 dvd n)"
+proof safe
+  assume *: "\<forall>k\<in>{2..n}. \<not> k\<^sup>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 \<le> n" by (intro dvd_imp_le)
+    with bspec[OF *, of k] k have "\<not>k > 1" by (intro notI) auto
+    moreover from k and n have "k \<noteq> 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 \<Rightarrow> 'a" where
+  "square_part n = (if n = 0 then 0 else 
+     normalize (\<Prod>p\<in>prime_factors n. p ^ (multiplicity p n div 2)))"
+  
+lemma square_part_nonzero: 
+  "n \<noteq> 0 \<Longrightarrow> square_part n = normalize (\<Prod>p\<in>prime_factors n. p ^ (multiplicity p n div 2))"
+  by (simp add: square_part_def)
+  
+lemma square_part_0 [simp]: "square_part 0 = 0"
+  by (simp add: square_part_def)
+
+lemma square_part_unit [simp]: "is_unit x \<Longrightarrow> 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 \<longleftrightarrow> n = 0"
+  by (simp add: square_part_def)
+
+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 "\<dots> = 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 "\<dots> = 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"
+  by (simp add: square_part_def)
+  
+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 \<noteq> 0" "y \<noteq> 0"
+  shows   "x dvd y \<longleftrightarrow> (\<forall>p. prime p \<longrightarrow> multiplicity p x \<le> 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 \<longleftrightarrow> x ^ 2 dvd n"
+proof (cases "x = 0"; cases "n = 0")
+  assume nz: "x \<noteq> 0" "n \<noteq> 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 \<Rightarrow> '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"
+  by (simp add: squarefree_part_def)
+
+lemma squarefree_part_unit [simp]: "is_unit n \<Longrightarrow> squarefree_part n = n"
+  by (auto simp add: squarefree_part_def)
+  
+lemma squarefree_part_1 [simp]: "squarefree_part 1 = 1"
+  by simp
+    
+lemma squarefree_decompose: "n = squarefree_part n * square_part n ^ 2"
+  by (simp add: squarefree_part_def)
+
+lemma squarefree_part_uminus [simp]: 
+  assumes "x \<noteq> 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 "\<dots> = 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 \<noteq> 0"
+  using squarefree_decompose[of n] by (cases "n \<noteq> 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 \<noteq> 0" by auto
+  have "multiplicity p n mod 2 + 2 * (multiplicity p n div 2) = multiplicity p n" by simp
+  also have "\<dots> = multiplicity p (squarefree_part n * square_part n ^ 2)"
+    by (subst squarefree_decompose[of n]) simp
+  also from assms n have "\<dots> = 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) \<le> Suc 0"
+  by (simp add: assms prime_multiplicity_squarefree_part)
+
+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"
+  by (simp add: square_part_def)
+
+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 \<Longrightarrow> 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 "\<dots> = 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 \<Longrightarrow> square_part (x ^ n) = normalize (x ^ (n div 2) * square_part x)"
+  by (subst square_part_odd_power' [symmetric]) auto
+
+end