(* Author: René Thiemann Akihisa Yamada Jose Divason License: BSD *) section ‹Missing Unsorted› text ‹This theory contains several lemmas which might be of interest to the Isabelle distribution. For instance, we prove that $b^n \cdot n^k$ is bounded by a constant whenever $0 < b < 1$.› theory Missing_Unsorted imports HOL.Complex "HOL-Computational_Algebra.Factorial_Ring" begin lemma bernoulli_inequality: assumes x: "-1 ≤ (x :: 'a :: linordered_field)" shows "1 + of_nat n * x ≤ (1 + x) ^ n" proof (induct n) case (Suc n) have "1 + of_nat (Suc n) * x = 1 + x + of_nat n * x" by (simp add: field_simps) also have "… ≤ … + of_nat n * x ^ 2" by simp also have "… = (1 + of_nat n * x) * (1 + x)" by (simp add: field_simps power2_eq_square) also have "… ≤ (1 + x) ^ n * (1 + x)" by (rule mult_right_mono[OF Suc], insert x, auto) also have "… = (1 + x) ^ (Suc n)" by simp finally show ?case . qed simp context fixes b :: "'a :: archimedean_field" assumes b: "0 < b" "b < 1" begin private lemma pow_one: "b ^ x ≤ 1" using power_Suc_less_one[OF b, of "x - 1"] by (cases x, auto) private lemma pow_zero: "0 < b ^ x" using b(1) by simp lemma exp_tends_to_zero: assumes c: "c > 0" shows "∃ x. b ^ x ≤ c" proof (rule ccontr) assume not: "¬ ?thesis" define bb where "bb = inverse b" define cc where "cc = inverse c" from b have bb: "bb > 1" unfolding bb_def by (rule one_less_inverse) from c have cc: "cc > 0" unfolding cc_def by simp define bbb where "bbb = bb - 1" have id: "bb = 1 + bbb" and bbb: "bbb > 0" and bm1: "bbb ≥ -1" unfolding bbb_def using bb by auto have "∃ n. cc / bbb < of_nat n" by (rule reals_Archimedean2) then obtain n where lt: "cc / bbb < of_nat n" by auto from not have "¬ b ^ n ≤ c" by auto hence bnc: "b ^ n > c" by simp have "bb ^ n = inverse (b ^ n)" unfolding bb_def by (rule power_inverse) also have "… < cc" unfolding cc_def by (rule less_imp_inverse_less[OF bnc c]) also have "… < bbb * of_nat n" using lt bbb by (metis mult.commute pos_divide_less_eq) also have "… ≤ bb ^ n" using bernoulli_inequality[OF bm1, folded id, of n] by (simp add: ac_simps) finally show False by simp qed lemma linear_exp_bound: "∃ p. ∀ x. b ^ x * of_nat x ≤ p" proof - from b have "1 - b > 0" by simp from exp_tends_to_zero[OF this] obtain x0 where x0: "b ^ x0 ≤ 1 - b" .. { fix x assume "x ≥ x0" hence "∃ y. x = x0 + y" by arith then obtain y where x: "x = x0 + y" by auto have "b ^ x = b ^ x0 * b ^ y" unfolding x by (simp add: power_add) also have "… ≤ b ^ x0" using pow_one[of y] pow_zero[of x0] by auto also have "… ≤ 1 - b" by (rule x0) finally have "b ^ x ≤ 1 - b" . } note x0 = this define bs where "bs = insert 1 { b ^ Suc x * of_nat (Suc x) | x . x ≤ x0}" have bs: "finite bs" unfolding bs_def by auto define p where "p = Max bs" have bs: "⋀ b. b ∈ bs ⟹ b ≤ p" unfolding p_def using bs by simp hence p1: "p ≥ 1" unfolding bs_def by auto show ?thesis proof (rule exI[of _ p], intro allI) fix x show "b ^ x * of_nat x ≤ p" proof (induct x) case (Suc x) show ?case proof (cases "x ≤ x0") case True show ?thesis by (rule bs, unfold bs_def, insert True, auto) next case False let ?x = "of_nat x :: 'a" have "b ^ (Suc x) * of_nat (Suc x) = b * (b ^ x * ?x) + b ^ Suc x" by (simp add: field_simps) also have "… ≤ b * p + b ^ Suc x" by (rule add_right_mono[OF mult_left_mono[OF Suc]], insert b, auto) also have "… = p - ((1 - b) * p - b ^ (Suc x))" by (simp add: field_simps) also have "… ≤ p - 0" proof - have "b ^ Suc x ≤ 1 - b" using x0[of "Suc x"] False by auto also have "… ≤ (1 - b) * p" using b p1 by auto finally show ?thesis by (intro diff_left_mono, simp) qed finally show ?thesis by simp qed qed (insert p1, auto) qed qed lemma poly_exp_bound: "∃ p. ∀ x. b ^ x * of_nat x ^ deg ≤ p" proof - show ?thesis proof (induct deg) case 0 show ?case by (rule exI[of _ 1], intro allI, insert pow_one, auto) next case (Suc deg) then obtain q where IH: "⋀ x. b ^ x * (of_nat x) ^ deg ≤ q" by auto define p where "p = max 0 q" from IH have IH: "⋀ x. b ^ x * (of_nat x) ^ deg ≤ p" unfolding p_def using le_max_iff_disj by blast have p: "p ≥ 0" unfolding p_def by simp show ?case proof (cases "deg = 0") case True thus ?thesis using linear_exp_bound by simp next case False note deg = this define p' where "p' = p*p * 2 ^ Suc deg * inverse b" let ?f = "λ x. b ^ x * (of_nat x) ^ Suc deg" define f where "f = ?f" { fix x let ?x = "of_nat x :: 'a" have "f (2 * x) ≤ (2 ^ Suc deg) * (p * p)" proof (cases "x = 0") case False hence x1: "?x ≥ 1" by (cases x, auto) from x1 have x: "?x ^ (deg - 1) ≥ 1" by simp from x1 have xx: "?x ^ Suc deg ≥ 1" by (rule one_le_power) define c where "c = b ^ x * b ^ x * (2 ^ Suc deg)" have c: "c > 0" unfolding c_def using b by auto have "f (2 * x) = ?f (2 * x)" unfolding f_def by simp also have "b ^ (2 * x) = (b ^ x) * (b ^ x)" by (simp add: power2_eq_square power_even_eq) also have "of_nat (2 * x) = 2 * ?x" by simp also have "(2 * ?x) ^ Suc deg = 2 ^ Suc deg * ?x ^ Suc deg" by simp finally have "f (2 * x) = c * ?x ^ Suc deg" unfolding c_def by (simp add: ac_simps) also have "… ≤ c * ?x ^ Suc deg * ?x ^ (deg - 1)" proof - have "c * ?x ^ Suc deg > 0" using c xx by simp thus ?thesis unfolding mult_le_cancel_left1 using x by simp qed also have "… = c * ?x ^ (Suc deg + (deg - 1))" by (simp add: power_add) also have "Suc deg + (deg - 1) = deg + deg" using deg by simp also have "?x ^ (deg + deg) = (?x ^ deg) * (?x ^ deg)" by (simp add: power_add) also have "c * … = (2 ^ Suc deg) * ((b ^ x * ?x ^ deg) * (b ^ x * ?x ^ deg))" unfolding c_def by (simp add: ac_simps) also have "… ≤ (2 ^ Suc deg) * (p * p)" by (rule mult_left_mono[OF mult_mono[OF IH IH p]], insert pow_zero[of x], auto) finally show "f (2 * x) ≤ (2 ^ Suc deg) * (p * p)" . qed (auto simp: f_def) hence "?f (2 * x) ≤ (2 ^ Suc deg) * (p * p)" unfolding f_def . } note even = this show ?thesis proof (rule exI[of _ p'], intro allI) fix y show "?f y ≤ p'" proof (cases "even y") case True define x where "x = y div 2" have "y = 2 * x" unfolding x_def using True by simp from even[of x, folded this] have "?f y ≤ 2 ^ Suc deg * (p * p)" . also have "… ≤ … * inverse b" unfolding mult_le_cancel_left1 using b p by (simp add: algebra_split_simps one_le_inverse) also have "… = p'" unfolding p'_def by (simp add: ac_simps) finally show "?f y ≤ p'" . next case False define x where "x = y div 2" have "y = 2 * x + 1" unfolding x_def using False by simp hence "?f y = ?f (2 * x + 1)" by simp also have "… ≤ b ^ (2 * x + 1) * of_nat (2 * x + 2) ^ Suc deg" by (rule mult_left_mono[OF power_mono], insert b, auto) also have "b ^ (2 * x + 1) = b ^ (2 * x + 2) * inverse b" using b by auto also have "b ^ (2 * x + 2) * inverse b * of_nat (2 * x + 2) ^ Suc deg = inverse b * ?f (2 * (x + 1))" by (simp add: ac_simps) also have "… ≤ inverse b * ((2 ^ Suc deg) * (p * p))" by (rule mult_left_mono[OF even], insert b, auto) also have "… = p'" unfolding p'_def by (simp add: ac_simps) finally show "?f y ≤ p'" . qed qed qed qed qed end lemma prod_list_replicate[simp]: "prod_list (replicate n a) = a ^ n" by (induct n, auto) lemma prod_list_power: fixes xs :: "'a :: comm_monoid_mult list" shows "prod_list xs ^ n = (∏x←xs. x ^ n)" by (induct xs, auto simp: power_mult_distrib) lemma set_upt_Suc: "{0 ..< Suc i} = insert i {0 ..< i}" by (fact atLeast0_lessThan_Suc) lemma prod_pow[simp]: "(∏i = 0..<n. p) = (p :: 'a :: comm_monoid_mult) ^ n" by (induct n, auto simp: set_upt_Suc) (* GCD and LCM part *) lemma dvd_abs_mult_left_int [simp]: "¦a¦ * y dvd x ⟷ a * y dvd x" for x y a :: int using abs_dvd_iff [of "a * y"] abs_dvd_iff [of "¦a¦ * y"] by (simp add: abs_mult) lemma gcd_abs_mult_right_int [simp]: "gcd x (¦a¦ * y) = gcd x (a * y)" for x y a :: int using gcd_abs2_int [of _ "a * y"] gcd_abs2_int [of _ "¦a¦ * y"] by (simp add: abs_mult) lemma lcm_abs_mult_right_int [simp]: "lcm x (¦a¦ * y) = lcm x (a * y)" for x y a :: int using lcm_abs2_int [of _ "a * y"] lcm_abs2_int [of _ "¦a¦ * y"] by (simp add: abs_mult) lemma gcd_abs_mult_left_int [simp]: "gcd x (a * ¦y¦) = gcd x (a * y)" for x y a :: int using gcd_abs2_int [of _ "a * ¦y¦"] gcd_abs2_int [of _ "a * y"] by (simp add: abs_mult) lemma lcm_abs_mult_left_int [simp]: "lcm x (a * ¦y¦) = lcm x (a * y)" for x y a :: int using lcm_abs2_int [of _ "a * ¦y¦"] lcm_abs2_int [of _ "a * y"] by (simp add: abs_mult) abbreviation (input) list_gcd :: "'a :: semiring_gcd list ⇒ 'a" where "list_gcd ≡ gcd_list" abbreviation (input) list_lcm :: "'a :: semiring_gcd list ⇒ 'a" where "list_lcm ≡ lcm_list" lemma list_gcd_simps: "list_gcd [] = 0" "list_gcd (x # xs) = gcd x (list_gcd xs)" by simp_all lemma list_gcd: "x ∈ set xs ⟹ list_gcd xs dvd x" by (fact Gcd_fin_dvd) lemma list_gcd_greatest: "(⋀ x. x ∈ set xs ⟹ y dvd x) ⟹ y dvd (list_gcd xs)" by (fact gcd_list_greatest) lemma list_gcd_mult_int [simp]: fixes xs :: "int list" shows "list_gcd (map (times a) xs) = ¦a¦ * list_gcd xs" by (simp add: Gcd_mult abs_mult) lemma list_lcm_simps: "list_lcm [] = 1" "list_lcm (x # xs) = lcm x (list_lcm xs)" by simp_all lemma list_lcm: "x ∈ set xs ⟹ x dvd list_lcm xs" by (fact dvd_Lcm_fin) lemma list_lcm_least: "(⋀ x. x ∈ set xs ⟹ x dvd y) ⟹ list_lcm xs dvd y" by (fact lcm_list_least) lemma lcm_mult_distrib_nat: "(k :: nat) * lcm m n = lcm (k * m) (k * n)" by (simp add: lcm_mult_left) lemma lcm_mult_distrib_int: "abs (k::int) * lcm m n = lcm (k * m) (k * n)" by (simp add: lcm_mult_left abs_mult) lemma list_lcm_mult_int [simp]: fixes xs :: "int list" shows "list_lcm (map (times a) xs) = (if xs = [] then 1 else ¦a¦ * list_lcm xs)" by (simp add: Lcm_mult abs_mult) lemma list_lcm_pos: "list_lcm xs ≥ (0 :: int)" "0 ∉ set xs ⟹ list_lcm xs ≠ 0" "0 ∉ set xs ⟹ list_lcm xs > 0" proof - have "0 ≤ ¦Lcm (set xs)¦" by (simp only: abs_ge_zero) then have "0 ≤ Lcm (set xs)" by simp then show "list_lcm xs ≥ 0" by simp assume "0 ∉ set xs" then show "list_lcm xs ≠ 0" by (simp add: Lcm_0_iff) with ‹list_lcm xs ≥ 0› show "list_lcm xs > 0" by (simp add: le_less) qed lemma quotient_of_nonzero: "snd (quotient_of r) > 0" "snd (quotient_of r) ≠ 0" using quotient_of_denom_pos' [of r] by simp_all lemma quotient_of_int_div: assumes q: "quotient_of (of_int x / of_int y) = (a, b)" and y: "y ≠ 0" shows "∃ z. z ≠ 0 ∧ x = a * z ∧ y = b * z" proof - let ?r = "rat_of_int" define z where "z = gcd x y" define x' where "x' = x div z" define y' where "y' = y div z" have id: "x = z * x'" "y = z * y'" unfolding x'_def y'_def z_def by auto from y have y': "y' ≠ 0" unfolding id by auto have z: "z ≠ 0" unfolding z_def using y by auto have cop: "coprime x' y'" unfolding x'_def y'_def z_def using div_gcd_coprime y by blast have "?r x / ?r y = ?r x' / ?r y'" unfolding id using z y y' by (auto simp: field_simps) from assms[unfolded this] have quot: "quotient_of (?r x' / ?r y') = (a, b)" by auto from quotient_of_coprime[OF quot] have cop': "coprime a b" . hence cop: "coprime b a" by (simp add: ac_simps) from quotient_of_denom_pos[OF quot] have b: "b > 0" "b ≠ 0" by auto from quotient_of_div[OF quot] quotient_of_denom_pos[OF quot] y' have "?r x' * ?r b = ?r a * ?r y'" by (auto simp: field_simps) hence id': "x' * b = a * y'" unfolding of_int_mult[symmetric] by linarith from id'[symmetric] have "b dvd y' * a" unfolding mult.commute[of y'] by auto with cop y' have "b dvd y'" by (simp add: coprime_dvd_mult_left_iff) then obtain z' where ybz: "y' = b * z'" unfolding dvd_def by auto from id[unfolded y' this] have y: "y = b * (z * z')" by auto with ‹y ≠ 0› have zz: "z * z' ≠ 0" by auto from quotient_of_div[OF q] ‹y ≠ 0› ‹b ≠ 0› have "?r x * ?r b = ?r y * ?r a" by (auto simp: field_simps) hence id': "x * b = y * a" unfolding of_int_mult[symmetric] by linarith from this[unfolded y] b have x: "x = a * (z * z')" by auto show ?thesis unfolding x y using zz by blast qed fun max_list_non_empty :: "('a :: linorder) list ⇒ 'a" where "max_list_non_empty [x] = x" | "max_list_non_empty (x # xs) = max x (max_list_non_empty xs)" lemma max_list_non_empty: "x ∈ set xs ⟹ x ≤ max_list_non_empty xs" proof (induct xs) case (Cons y ys) note oCons = this show ?case proof (cases ys) case (Cons z zs) hence id: "max_list_non_empty (y # ys) = max y (max_list_non_empty ys)" by simp from oCons show ?thesis unfolding id by (auto simp: max.coboundedI2) qed (insert oCons, auto) qed simp lemma cnj_reals[simp]: "(cnj c ∈ ℝ) = (c ∈ ℝ)" using Reals_cnj_iff by fastforce lemma sgn_real_mono: "x ≤ y ⟹ sgn x ≤ sgn (y :: real)" unfolding sgn_real_def by (auto split: if_splits) lemma sgn_minus_rat: "sgn (- (x :: rat)) = - sgn x" by (fact Rings.sgn_minus) lemma real_of_rat_sgn: "sgn (of_rat x) = real_of_rat (sgn x)" unfolding sgn_real_def sgn_rat_def by auto lemma inverse_le_iff_sgn: assumes sgn: "sgn x = sgn y" shows "(inverse (x :: real) ≤ inverse y) = (y ≤ x)" proof (cases "x = 0") case True with sgn have "sgn y = 0" by simp hence "y = 0" unfolding sgn_real_def by (cases "y = 0"; cases "y < 0"; auto) thus ?thesis using True by simp next case False note x = this show ?thesis proof (cases "x < 0") case True with x sgn have "sgn y = -1" by simp hence "y < 0" unfolding sgn_real_def by (cases "y = 0"; cases "y < 0", auto) show ?thesis by (rule inverse_le_iff_le_neg[OF True ‹y < 0›]) next case False with x have x: "x > 0" by auto with sgn have "sgn y = 1" by auto hence "y > 0" unfolding sgn_real_def by (cases "y = 0"; cases "y < 0", auto) show ?thesis by (rule inverse_le_iff_le[OF x ‹y > 0›]) qed qed lemma inverse_le_sgn: assumes sgn: "sgn x = sgn y" and xy: "x ≤ (y :: real)" shows "inverse y ≤ inverse x" using xy inverse_le_iff_sgn[OF sgn] by auto lemma set_list_update: "set (xs [i := k]) = (if i < length xs then insert k (set (take i xs) ∪ set (drop (Suc i) xs)) else set xs)" proof (induct xs arbitrary: i) case (Cons x xs i) thus ?case by (cases i, auto) qed simp lemma prod_list_dvd: assumes "(x :: 'a :: comm_monoid_mult) ∈ set xs" shows "x dvd prod_list xs" proof - from assms[unfolded in_set_conv_decomp] obtain ys zs where xs: "xs = ys @ x # zs" by auto show ?thesis unfolding xs dvd_def by (intro exI[of _ "prod_list (ys @ zs)"], simp add: ac_simps) qed lemma dvd_prod: fixes A::"'b set" assumes "∃b∈A. a dvd f b" "finite A" shows "a dvd prod f A" using assms(2,1) proof (induct A) case (insert x A) thus ?case using comm_monoid_mult_class.dvd_mult dvd_mult2 insert_iff prod.insert by auto qed auto context fixes xs :: "'a :: comm_monoid_mult list" begin lemma prod_list_filter: "prod_list (filter f xs) * prod_list (filter (λ x. ¬ f x) xs) = prod_list xs" by (induct xs, auto simp: ac_simps) lemma prod_list_partition: assumes "partition f xs = (ys, zs)" shows "prod_list xs = prod_list ys * prod_list zs" using assms by (subst prod_list_filter[symmetric, of f], auto simp: o_def) end lemma dvd_imp_mult_div_cancel_left[simp]: assumes "(a :: 'a :: semidom_divide) dvd b" shows "a * (b div a) = b" proof(cases "b = 0") case True then show ?thesis by auto next case False with dvdE[OF assms] obtain c where *: "b = a * c" by auto also with False have "a ≠ 0" by auto then have "a * c div a = c" by auto also note *[symmetric] finally show ?thesis. qed lemma (in semidom) prod_list_zero_iff[simp]: "prod_list xs = 0 ⟷ 0 ∈ set xs" by (induction xs, auto) context comm_monoid_mult begin lemma unit_prod [intro]: shows "a dvd 1 ⟹ b dvd 1 ⟹ (a * b) dvd 1" by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono) lemma is_unit_mult_iff[simp]: shows "(a * b) dvd 1 ⟷ a dvd 1 ∧ b dvd 1" by (auto dest: dvd_mult_left dvd_mult_right) end context comm_semiring_1 begin lemma irreducibleE[elim]: assumes "irreducible p" and "p ≠ 0 ⟹ ¬ p dvd 1 ⟹ (⋀a b. p = a * b ⟹ a dvd 1 ∨ b dvd 1) ⟹ thesis" shows thesis using assms by (auto simp: irreducible_def) lemma not_irreducibleE: assumes "¬ irreducible x" and "x = 0 ⟹ thesis" and "x dvd 1 ⟹ thesis" and "⋀a b. x = a * b ⟹ ¬ a dvd 1 ⟹ ¬ b dvd 1 ⟹ thesis" shows thesis using assms unfolding irreducible_def by auto lemma prime_elem_dvd_prod_list: assumes p: "prime_elem p" and pA: "p dvd prod_list A" shows "∃a ∈ set A. p dvd a" proof(insert pA, induct A) case Nil with p show ?case by (simp add: prime_elem_not_unit) next case (Cons a A) then show ?case by (auto simp: prime_elem_dvd_mult_iff[OF p]) qed lemma prime_elem_dvd_prod_mset: assumes p: "prime_elem p" and pA: "p dvd prod_mset A" shows "∃a ∈# A. p dvd a" proof(insert pA, induct A) case empty with p show ?case by (simp add: prime_elem_not_unit) next case (add a A) then show ?case by (auto simp: prime_elem_dvd_mult_iff[OF p]) qed lemma mult_unit_dvd_iff[simp]: assumes "b dvd 1" shows "a * b dvd c ⟷ a dvd c" proof assume "a * b dvd c" with assms show "a dvd c" using dvd_mult_left[of a b c] by simp next assume "a dvd c" with assms mult_dvd_mono show "a * b dvd c" by fastforce qed lemma mult_unit_dvd_iff'[simp]: "a dvd 1 ⟹ (a * b) dvd c ⟷ b dvd c" using mult_unit_dvd_iff [of a b c] by (simp add: ac_simps) lemma irreducibleD': assumes "irreducible a" "b dvd a" shows "a dvd b ∨ b dvd 1" proof - from assms obtain c where c: "a = b * c" by (elim dvdE) from irreducibleD[OF assms(1) this] have "b dvd 1 ∨ c dvd 1" . thus ?thesis by (auto simp: c) qed end context idom begin text ‹ Following lemmas are adapted and generalized so that they don't use "algebraic" classes. › lemma dvd_times_left_cancel_iff [simp]: assumes "a ≠ 0" shows "a * b dvd a * c ⟷ b dvd c" (is "?lhs ⟷ ?rhs") proof assume ?lhs then obtain d where "a * c = a * b * d" .. with assms have "c = b * d" by (auto simp add: ac_simps) then show ?rhs .. next assume ?rhs then obtain d where "c = b * d" .. then have "a * c = a * b * d" by (simp add: ac_simps) then show ?lhs .. qed lemma dvd_times_right_cancel_iff [simp]: assumes "a ≠ 0" shows "b * a dvd c * a ⟷ b dvd c" using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps) lemma irreducibleI': assumes "a ≠ 0" "¬ a dvd 1" "⋀b. b dvd a ⟹ a dvd b ∨ b dvd 1" shows "irreducible a" proof (rule irreducibleI) fix b c assume a_eq: "a = b * c" hence "a dvd b ∨ b dvd 1" by (intro assms) simp_all thus "b dvd 1 ∨ c dvd 1" proof assume "a dvd b" hence "b * c dvd b * 1" by (simp add: a_eq) moreover from ‹a ≠ 0› a_eq have "b ≠ 0" by auto ultimately show ?thesis using dvd_times_left_cancel_iff by fastforce qed blast qed (simp_all add: assms(1,2)) lemma irreducible_altdef: shows "irreducible x ⟷ x ≠ 0 ∧ ¬ x dvd 1 ∧ (∀b. b dvd x ⟶ x dvd b ∨ b dvd 1)" using irreducibleI'[of x] irreducibleD'[of x] irreducible_not_unit[of x] by auto lemma dvd_mult_unit_iff: assumes b: "b dvd 1" shows "a dvd c * b ⟷ a dvd c" proof- from b obtain b' where 1: "b * b' = 1" by (elim dvdE, auto) then have b0: "b ≠ 0" by auto from 1 have "a = (a * b') * b" by (simp add: ac_simps) also have "… dvd c * b ⟷ a * b' dvd c" using b0 by auto finally show ?thesis by (auto intro: dvd_mult_left) qed lemma dvd_mult_unit_iff': "b dvd 1 ⟹ a dvd b * c ⟷ a dvd c" using dvd_mult_unit_iff [of b a c] by (simp add: ac_simps) lemma irreducible_mult_unit_left: shows "a dvd 1 ⟹ irreducible (a * p) ⟷ irreducible p" by (auto simp: irreducible_altdef mult.commute[of a] dvd_mult_unit_iff) lemma irreducible_mult_unit_right: shows "a dvd 1 ⟹ irreducible (p * a) ⟷ irreducible p" by (auto simp: irreducible_altdef mult.commute[of a] dvd_mult_unit_iff) lemma prime_elem_imp_irreducible: assumes "prime_elem p" shows "irreducible p" proof (rule irreducibleI) fix a b assume p_eq: "p = a * b" with assms have nz: "a ≠ 0" "b ≠ 0" by auto from p_eq have "p dvd a * b" by simp with ‹prime_elem p› have "p dvd a ∨ p dvd b" by (rule prime_elem_dvd_multD) with ‹p = a * b› have "a * b dvd 1 * b ∨ a * b dvd a * 1" by auto thus "a dvd 1 ∨ b dvd 1" by (simp only: dvd_times_left_cancel_iff[OF nz(1)] dvd_times_right_cancel_iff[OF nz(2)]) qed (insert assms, simp_all add: prime_elem_def) lemma unit_imp_dvd [dest]: "b dvd 1 ⟹ b dvd a" by (rule dvd_trans [of _ 1]) simp_all lemma unit_mult_left_cancel: "a dvd 1 ⟹ a * b = a * c ⟷ b = c" using mult_cancel_left [of a b c] by auto lemma unit_mult_right_cancel: "a dvd 1 ⟹ b * a = c * a ⟷ b = c" using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps) text ‹New parts from here› lemma irreducible_multD: assumes l: "irreducible (a*b)" shows "a dvd 1 ∧ irreducible b ∨ b dvd 1 ∧ irreducible a" proof- from l have "a dvd 1 ∨ b dvd 1" using irreducibleD by auto then show ?thesis proof(elim disjE) assume a: "a dvd 1" with l have "irreducible b" unfolding irreducible_def by (metis is_unit_mult_iff mult.left_commute mult_not_zero) with a show ?thesis by auto next assume a: "b dvd 1" with l have "irreducible a" unfolding irreducible_def by (meson is_unit_mult_iff mult_not_zero semiring_normalization_rules(16)) with a show ?thesis by auto qed qed end lemma (in field) irreducible_field[simp]: "irreducible x ⟷ False" by (auto simp: dvd_field_iff irreducible_def) lemma (in idom) irreducible_mult: shows "irreducible (a*b) ⟷ a dvd 1 ∧ irreducible b ∨ b dvd 1 ∧ irreducible a" by (auto dest: irreducible_multD simp: irreducible_mult_unit_left irreducible_mult_unit_right) end