added parallel_proofs flag (default true, cf. usedir option -Q), which can be disabled in low-memory situations;
(* Title: HOL/Rational.thy Author: Markus Wenzel, TU Muenchen*)header {* Rational numbers *}theory Rationalimports Nat_Int_Bij GCDuses ("Tools/rat_arith.ML")beginsubsection {* Rational numbers as quotient *}subsubsection {* Construction of the type of rational numbers *}definition ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where "ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"lemma ratrel_iff [simp]: "(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x" by (simp add: ratrel_def)lemma refl_ratrel: "refl {x. snd x \<noteq> 0} ratrel" by (auto simp add: refl_def ratrel_def)lemma sym_ratrel: "sym ratrel" by (simp add: ratrel_def sym_def)lemma trans_ratrel: "trans ratrel"proof (rule transI, unfold split_paired_all) fix a b a' b' a'' b'' :: int assume A: "((a, b), (a', b')) \<in> ratrel" assume B: "((a', b'), (a'', b'')) \<in> ratrel" have "b' * (a * b'') = b'' * (a * b')" by simp also from A have "a * b' = a' * b" by auto also have "b'' * (a' * b) = b * (a' * b'')" by simp also from B have "a' * b'' = a'' * b'" by auto also have "b * (a'' * b') = b' * (a'' * b)" by simp finally have "b' * (a * b'') = b' * (a'' * b)" . moreover from B have "b' \<noteq> 0" by auto ultimately have "a * b'' = a'' * b" by simp with A B show "((a, b), (a'', b'')) \<in> ratrel" by autoqedlemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel" by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]lemma equiv_ratrel_iff [iff]: assumes "snd x \<noteq> 0" and "snd y \<noteq> 0" shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel" by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"proof have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI)qedlemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat" by (simp add: Rat_def quotientI)declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]subsubsection {* Representation and basic operations *}definition Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where [code del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"code_datatype Fractlemma Rat_cases [case_names Fract, cases type: rat]: assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C" shows C using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)lemma Rat_induct [case_names Fract, induct type: rat]: assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)" shows "P q" using assms by (cases q) simplemma eq_rat: shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b" and "\<And>a. Fract a 0 = Fract 0 1" and "\<And>a c. Fract 0 a = Fract 0 c" by (simp_all add: Fract_def)instantiation rat :: "{comm_ring_1, recpower}"begindefinition Zero_rat_def [code, code unfold]: "0 = Fract 0 1"definition One_rat_def [code, code unfold]: "1 = Fract 1 1"definition add_rat_def [code del]: "q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"lemma add_rat [simp]: assumes "b \<noteq> 0" and "d \<noteq> 0" shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"proof - have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)}) respects2 ratrel" by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib) with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)qeddefinition minus_rat_def [code del]: "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"lemma minus_rat [simp, code]: "- Fract a b = Fract (- a) b"proof - have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel" by (simp add: congruent_def) then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)qedlemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b" by (cases "b = 0") (simp_all add: eq_rat)definition diff_rat_def [code del]: "q - r = q + - (r::rat)"lemma diff_rat [simp]: assumes "b \<noteq> 0" and "d \<noteq> 0" shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)" using assms by (simp add: diff_rat_def)definition mult_rat_def [code del]: "q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. ratrel``{(fst x * fst y, snd x * snd y)})"lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"proof - have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel" by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)qedlemma mult_rat_cancel: assumes "c \<noteq> 0" shows "Fract (c * a) (c * b) = Fract a b"proof - from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def) then show ?thesis by (simp add: mult_rat [symmetric])qedprimrec power_ratwhere rat_power_0: "q ^ 0 = (1\<Colon>rat)" | rat_power_Suc: "q ^ Suc n = (q\<Colon>rat) * (q ^ n)"instance proof fix q r s :: rat show "(q * r) * s = q * (r * s)" by (cases q, cases r, cases s) (simp add: eq_rat)next fix q r :: rat show "q * r = r * q" by (cases q, cases r) (simp add: eq_rat)next fix q :: rat show "1 * q = q" by (cases q) (simp add: One_rat_def eq_rat)next fix q r s :: rat show "(q + r) + s = q + (r + s)" by (cases q, cases r, cases s) (simp add: eq_rat ring_simps)next fix q r :: rat show "q + r = r + q" by (cases q, cases r) (simp add: eq_rat)next fix q :: rat show "0 + q = q" by (cases q) (simp add: Zero_rat_def eq_rat)next fix q :: rat show "- q + q = 0" by (cases q) (simp add: Zero_rat_def eq_rat)next fix q r :: rat show "q - r = q + - r" by (cases q, cases r) (simp add: eq_rat)next fix q r s :: rat show "(q + r) * s = q * s + r * s" by (cases q, cases r, cases s) (simp add: eq_rat ring_simps)next show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)next fix q :: rat show "q * 1 = q" by (cases q) (simp add: One_rat_def eq_rat)next fix q :: rat fix n :: nat show "q ^ 0 = 1" by simp show "q ^ (Suc n) = q * (q ^ n)" by simpqedendlemma of_nat_rat: "of_nat k = Fract (of_nat k) 1" by (induct k) (simp_all add: Zero_rat_def One_rat_def)lemma of_int_rat: "of_int k = Fract k 1" by (cases k rule: int_diff_cases) (simp add: of_nat_rat)lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" by (rule of_nat_rat [symmetric])lemma Fract_of_int_eq: "Fract k 1 = of_int k" by (rule of_int_rat [symmetric])instantiation rat :: number_ringbegindefinition rat_number_of_def [code del]: "number_of w = Fract w 1"instance by intro_classes (simp add: rat_number_of_def of_int_rat)endlemma rat_number_collapse [code post]: "Fract 0 k = 0" "Fract 1 1 = 1" "Fract (number_of k) 1 = number_of k" "Fract k 0 = 0" by (cases "k = 0") (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)lemma rat_number_expand [code unfold]: "0 = Fract 0 1" "1 = Fract 1 1" "number_of k = Fract (number_of k) 1" by (simp_all add: rat_number_collapse)lemma iszero_rat [simp]: "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)" by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)lemma Rat_cases_nonzero [case_names Fract 0]: assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C" assumes 0: "q = 0 \<Longrightarrow> C" shows Cproof (cases "q = 0") case True then show C using 0 by autonext case False then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto moreover with False have "0 \<noteq> Fract a b" by simp with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat) with Fract `q = Fract a b` `b \<noteq> 0` show C by autoqedsubsubsection {* The field of rational numbers *}instantiation rat :: "{field, division_by_zero}"begindefinition inverse_rat_def [code del]: "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"proof - have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel" by (auto simp add: congruent_def mult_commute) then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)qeddefinition divide_rat_def [code del]: "q / r = q * inverse (r::rat)"lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)" by (simp add: divide_rat_def)instance proof show "inverse 0 = (0::rat)" by (simp add: rat_number_expand) (simp add: rat_number_collapse)next fix q :: rat assume "q \<noteq> 0" then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero) (simp_all add: mult_rat inverse_rat rat_number_expand eq_rat)next fix q r :: rat show "q / r = q * inverse r" by (simp add: divide_rat_def)qedendsubsubsection {* Various *}lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1" by (simp add: rat_number_expand)lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l" by (simp add: Fract_of_int_eq [symmetric])lemma Fract_number_of_quotient [code post]: "Fract (number_of k) (number_of l) = number_of k / number_of l" unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..lemma Fract_1_number_of [code post]: "Fract 1 (number_of k) = 1 / number_of k" unfolding Fract_of_int_quotient number_of_eq by simpsubsubsection {* The ordered field of rational numbers *}instantiation rat :: linorderbegindefinition le_rat_def [code del]: "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"lemma le_rat [simp]: assumes "b \<noteq> 0" and "d \<noteq> 0" shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"proof - have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)}) respects2 ratrel" proof (clarsimp simp add: congruent2_def) fix a b a' b' c d c' d'::int assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" assume eq1: "a * b' = a' * b" assume eq2: "c * d' = c' * d" let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))" { fix a b c d x :: int assume x: "x \<noteq> 0" have "?le a b c d = ?le (a * x) (b * x) c d" proof - from x have "0 < x * x" by (auto simp add: zero_less_mult_iff) hence "?le a b c d = ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))" by (simp add: mult_le_cancel_right) also have "... = ?le (a * x) (b * x) c d" by (simp add: mult_ac) finally show ?thesis . qed } note le_factor = this let ?D = "b * d" and ?D' = "b' * d'" from neq have D: "?D \<noteq> 0" by simp from neq have "?D' \<noteq> 0" by simp hence "?le a b c d = ?le (a * ?D') (b * ?D') c d" by (rule le_factor) also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" by (simp add: mult_ac) also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')" by (simp only: eq1 eq2) also have "... = ?le (a' * ?D) (b' * ?D) c' d'" by (simp add: mult_ac) also from D have "... = ?le a' b' c' d'" by (rule le_factor [symmetric]) finally show "?le a b c d = ?le a' b' c' d'" . qed with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)qeddefinition less_rat_def [code del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"lemma less_rat [simp]: assumes "b \<noteq> 0" and "d \<noteq> 0" shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)" using assms by (simp add: less_rat_def eq_rat order_less_le)instance proof fix q r s :: rat { assume "q \<le> r" and "r \<le> s" show "q \<le> s" proof (insert prems, induct q, induct r, induct s) fix a b c d e f :: int assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f" show "Fract a b \<le> Fract e f" proof - from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f" by (auto simp add: zero_less_mult_iff linorder_neq_iff) have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)" proof - from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" by simp with ff show ?thesis by (simp add: mult_le_cancel_right) qed also have "... = (c * f) * (d * f) * (b * b)" by algebra also have "... \<le> (e * d) * (d * f) * (b * b)" proof - from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)" by simp with bb show ?thesis by (simp add: mult_le_cancel_right) qed finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)" by (simp only: mult_ac) with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)" by (simp add: mult_le_cancel_right) with neq show ?thesis by simp qed qed next assume "q \<le> r" and "r \<le> q" show "q = r" proof (insert prems, induct q, induct r) fix a b c d :: int assume neq: "b \<noteq> 0" "d \<noteq> 0" assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b" show "Fract a b = Fract c d" proof - from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" by simp also have "... \<le> (a * d) * (b * d)" proof - from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)" by simp thus ?thesis by (simp only: mult_ac) qed finally have "(a * d) * (b * d) = (c * b) * (b * d)" . moreover from neq have "b * d \<noteq> 0" by simp ultimately have "a * d = c * b" by simp with neq show ?thesis by (simp add: eq_rat) qed qed next show "q \<le> q" by (induct q) simp show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)" by (induct q, induct r) (auto simp add: le_less mult_commute) show "q \<le> r \<or> r \<le> q" by (induct q, induct r) (simp add: mult_commute, rule linorder_linear) }qedendinstantiation rat :: "{distrib_lattice, abs_if, sgn_if}"begindefinition abs_rat_def [code del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" by (auto simp add: abs_rat_def zabs_def Zero_rat_def less_rat not_less le_less minus_rat eq_rat zero_compare_simps)definition sgn_rat_def [code del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)" unfolding Fract_of_int_eq by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat) (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)definition "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"definition "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"instance by intro_classes (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)endinstance rat :: ordered_fieldproof fix q r s :: rat show "q \<le> r ==> s + q \<le> s + r" proof (induct q, induct r, induct s) fix a b c d e f :: int assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" assume le: "Fract a b \<le> Fract c d" show "Fract e f + Fract a b \<le> Fract e f + Fract c d" proof - let ?F = "f * f" from neq have F: "0 < ?F" by (auto simp add: zero_less_mult_iff) from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)" by simp with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F" by (simp add: mult_le_cancel_right) with neq show ?thesis by (simp add: mult_ac int_distrib) qed qed show "q < r ==> 0 < s ==> s * q < s * r" proof (induct q, induct r, induct s) fix a b c d e f :: int assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" assume le: "Fract a b < Fract c d" assume gt: "0 < Fract e f" show "Fract e f * Fract a b < Fract e f * Fract c d" proof - let ?E = "e * f" and ?F = "f * f" from neq gt have "0 < ?E" by (auto simp add: Zero_rat_def order_less_le eq_rat) moreover from neq have "0 < ?F" by (auto simp add: zero_less_mult_iff) moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)" by simp ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F" by (simp add: mult_less_cancel_right) with neq show ?thesis by (simp add: mult_ac) qed qedqed autolemma Rat_induct_pos [case_names Fract, induct type: rat]: assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)" shows "P q"proof (cases q) have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)" proof - fix a::int and b::int assume b: "b < 0" hence "0 < -b" by simp hence "P (Fract (-a) (-b))" by (rule step) thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b]) qed case (Fract a b) thus "P q" by (force simp add: linorder_neq_iff step step')qedlemma zero_less_Fract_iff: "0 < b ==> (0 < Fract a b) = (0 < a)"by (simp add: Zero_rat_def order_less_imp_not_eq2 zero_less_mult_iff)subsection {* Arithmetic setup *}use "Tools/rat_arith.ML"declaration {* K rat_arith_setup *}subsection {* Embedding from Rationals to other Fields *}class field_char_0 = field + ring_char_0subclass (in ordered_field) field_char_0 ..context field_char_0begindefinition of_rat :: "rat \<Rightarrow> 'a" where [code del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"endlemma of_rat_congruent: "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"apply (rule congruent.intro)apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)apply (simp only: of_int_mult [symmetric])donelemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b" unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)lemma of_rat_0 [simp]: "of_rat 0 = 0"by (simp add: Zero_rat_def of_rat_rat)lemma of_rat_1 [simp]: "of_rat 1 = 1"by (simp add: One_rat_def of_rat_rat)lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"by (induct a, induct b, simp add: of_rat_rat add_frac_eq)lemma of_rat_minus: "of_rat (- a) = - of_rat a"by (induct a, simp add: of_rat_rat)lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"by (simp only: diff_minus of_rat_add of_rat_minus)lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"apply (induct a, induct b, simp add: of_rat_rat)apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)donelemma nonzero_of_rat_inverse: "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"apply (rule inverse_unique [symmetric])apply (simp add: of_rat_mult [symmetric])donelemma of_rat_inverse: "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) = inverse (of_rat a)"by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)lemma nonzero_of_rat_divide: "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)lemma of_rat_divide: "(of_rat (a / b)::'a::{field_char_0,division_by_zero}) = of_rat a / of_rat b"by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)lemma of_rat_power: "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"by (induct n) (simp_all add: of_rat_mult power_Suc)lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"apply (induct a, induct b)apply (simp add: of_rat_rat eq_rat)apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)apply (simp only: of_int_mult [symmetric] of_int_eq_iff)donelemma of_rat_less: "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"proof (induct r, induct s) fix a b c d :: int assume not_zero: "b > 0" "d > 0" then have "b * d > 0" by (rule mult_pos_pos) have of_int_divide_less_eq: "(of_int a :: 'a) / of_int b < of_int c / of_int d \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b" using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq) show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d) \<longleftrightarrow> Fract a b < Fract c d" using not_zero `b * d > 0` by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult) (auto intro: mult_strict_right_mono mult_right_less_imp_less)qedlemma of_rat_less_eq: "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s" unfolding le_less by (auto simp add: of_rat_less)lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]lemma of_rat_eq_id [simp]: "of_rat = id"proof fix a show "of_rat a = id a" by (induct a) (simp add: of_rat_rat Fract_of_int_eq [symmetric])qedtext{*Collapse nested embeddings*}lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"by (induct n) (simp_all add: of_rat_add)lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"by (cases z rule: int_diff_cases) (simp add: of_rat_diff)lemma of_rat_number_of_eq [simp]: "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"by (simp add: number_of_eq)lemmas zero_rat = Zero_rat_deflemmas one_rat = One_rat_defabbreviation rat_of_nat :: "nat \<Rightarrow> rat"where "rat_of_nat \<equiv> of_nat"abbreviation rat_of_int :: "int \<Rightarrow> rat"where "rat_of_int \<equiv> of_int"subsection {* The Set of Rational Numbers *}context field_char_0begindefinition Rats :: "'a set" where [code del]: "Rats = range of_rat"notation (xsymbols) Rats ("\<rat>")endlemma Rats_of_rat [simp]: "of_rat r \<in> Rats"by (simp add: Rats_def)lemma Rats_of_int [simp]: "of_int z \<in> Rats"by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)lemma Rats_number_of [simp]: "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)lemma Rats_0 [simp]: "0 \<in> Rats"apply (unfold Rats_def)apply (rule range_eqI)apply (rule of_rat_0 [symmetric])donelemma Rats_1 [simp]: "1 \<in> Rats"apply (unfold Rats_def)apply (rule range_eqI)apply (rule of_rat_1 [symmetric])donelemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"apply (auto simp add: Rats_def)apply (rule range_eqI)apply (rule of_rat_add [symmetric])donelemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"apply (auto simp add: Rats_def)apply (rule range_eqI)apply (rule of_rat_minus [symmetric])donelemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"apply (auto simp add: Rats_def)apply (rule range_eqI)apply (rule of_rat_diff [symmetric])donelemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"apply (auto simp add: Rats_def)apply (rule range_eqI)apply (rule of_rat_mult [symmetric])donelemma nonzero_Rats_inverse: fixes a :: "'a::field_char_0" shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"apply (auto simp add: Rats_def)apply (rule range_eqI)apply (erule nonzero_of_rat_inverse [symmetric])donelemma Rats_inverse [simp]: fixes a :: "'a::{field_char_0,division_by_zero}" shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"apply (auto simp add: Rats_def)apply (rule range_eqI)apply (rule of_rat_inverse [symmetric])donelemma nonzero_Rats_divide: fixes a b :: "'a::field_char_0" shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"apply (auto simp add: Rats_def)apply (rule range_eqI)apply (erule nonzero_of_rat_divide [symmetric])donelemma Rats_divide [simp]: fixes a b :: "'a::{field_char_0,division_by_zero}" shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"apply (auto simp add: Rats_def)apply (rule range_eqI)apply (rule of_rat_divide [symmetric])donelemma Rats_power [simp]: fixes a :: "'a::{field_char_0,recpower}" shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"apply (auto simp add: Rats_def)apply (rule range_eqI)apply (rule of_rat_power [symmetric])donelemma Rats_cases [cases set: Rats]: assumes "q \<in> \<rat>" obtains (of_rat) r where "q = of_rat r" unfolding Rats_defproof - from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def . then obtain r where "q = of_rat r" .. then show thesis ..qedlemma Rats_induct [case_names of_rat, induct set: Rats]: "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q" by (rule Rats_cases) autosubsection {* The Rationals are Countably Infinite *}definition nat_to_rat_surj :: "nat \<Rightarrow> rat" where"nat_to_rat_surj n = (let (a,b) = nat_to_nat2 n in Fract (nat_to_int_bij a) (nat_to_int_bij b))"lemma surj_nat_to_rat_surj: "surj nat_to_rat_surj"unfolding surj_defproof fix r::rat show "\<exists>n. r = nat_to_rat_surj n" proof(cases r) fix i j assume [simp]: "r = Fract i j" and "j \<noteq> 0" have "r = (let m = inv nat_to_int_bij i; n = inv nat_to_int_bij j in nat_to_rat_surj(nat2_to_nat (m,n)))" using nat2_to_nat_inj surj_f_inv_f[OF surj_nat_to_int_bij] by(simp add:Let_def nat_to_rat_surj_def nat_to_nat2_def) thus "\<exists>n. r = nat_to_rat_surj n" by(auto simp:Let_def) qedqedlemma Rats_eq_range_nat_to_rat_surj: "\<rat> = range nat_to_rat_surj"by (simp add: Rats_def surj_nat_to_rat_surj surj_range)context field_char_0beginlemma Rats_eq_range_of_rat_o_nat_to_rat_surj: "\<rat> = range (of_rat o nat_to_rat_surj)"using surj_nat_to_rat_surjby (auto simp: Rats_def image_def surj_def) (blast intro: arg_cong[where f = of_rat])lemma surj_of_rat_nat_to_rat_surj: "r\<in>\<rat> \<Longrightarrow> \<exists>n. r = of_rat(nat_to_rat_surj n)"by(simp add: Rats_eq_range_of_rat_o_nat_to_rat_surj image_def)endsubsection {* Implementation of rational numbers as pairs of integers *}lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"proof (cases "a = 0 \<or> b = 0") case True then show ?thesis by (auto simp add: eq_rat)next let ?c = "zgcd a b" case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto then have "?c \<noteq> 0" by simp then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat) moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b" by (simp add: semiring_div_class.mod_div_equality) moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric]) moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric]) ultimately show ?thesis by (simp add: mult_rat [symmetric])qeddefinition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where [simp, code del]: "Fract_norm a b = Fract a b"lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))" by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)lemma [code]: "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)" by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)instantiation rat :: eqbegindefinition [code del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"instance by default (simp add: eq_rat_def)lemma rat_eq_code [code]: "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0 then c = 0 \<or> d = 0 else if d = 0 then a = 0 \<or> b = 0 else a * d = b * c)" by (auto simp add: eq eq_rat)lemma rat_eq_refl [code nbe]: "eq_class.eq (r::rat) r \<longleftrightarrow> True" by (rule HOL.eq_refl)endlemma le_rat': assumes "b \<noteq> 0" and "d \<noteq> 0" shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"proof - have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)" proof (cases "b * d > 0") case True moreover from True have "sgn b * sgn d = 1" by (simp add: sgn_times [symmetric] sgn_1_pos) ultimately show ?thesis by (simp add: mult_le_cancel_right) next case False with assms have "b * d < 0" by (simp add: less_le) moreover from this have "sgn b * sgn d = - 1" by (simp only: sgn_times [symmetric] sgn_1_neg) ultimately show ?thesis by (simp add: mult_le_cancel_right) qed also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d" by (simp add: abs_sgn mult_ac) finally show ?thesis using assms by simpqedlemma less_rat': assumes "b \<noteq> 0" and "d \<noteq> 0" shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"proof - have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)" proof (cases "b * d > 0") case True moreover from True have "sgn b * sgn d = 1" by (simp add: sgn_times [symmetric] sgn_1_pos) ultimately show ?thesis by (simp add: mult_less_cancel_right) next case False with assms have "b * d < 0" by (simp add: less_le) moreover from this have "sgn b * sgn d = - 1" by (simp only: sgn_times [symmetric] sgn_1_neg) ultimately show ?thesis by (simp add: mult_less_cancel_right) qed also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d" by (simp add: abs_sgn mult_ac) finally show ?thesis using assms by simpqedlemma rat_less_eq_code [code]: "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0 then sgn c * sgn d \<ge> 0 else if d = 0 then sgn a * sgn b \<le> 0 else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat) (auto simp add: sgn_times sgn_0_0 le_less sgn_1_pos [symmetric] sgn_1_neg [symmetric])lemma rat_le_eq_code [code]: "Fract a b < Fract c d \<longleftrightarrow> (if b = 0 then sgn c * sgn d > 0 else if d = 0 then sgn a * sgn b < 0 else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat) (auto simp add: sgn_times sgn_0_0 sgn_1_pos [symmetric] sgn_1_neg [symmetric], auto simp add: sgn_1_pos)lemma rat_plus_code [code]: "Fract a b + Fract c d = (if b = 0 then Fract c d else if d = 0 then Fract a b else Fract_norm (a * d + c * b) (b * d))" by (simp add: eq_rat, simp add: Zero_rat_def)lemma rat_times_code [code]: "Fract a b * Fract c d = Fract_norm (a * c) (b * d)" by simplemma rat_minus_code [code]: "Fract a b - Fract c d = (if b = 0 then Fract (- c) d else if d = 0 then Fract a b else Fract_norm (a * d - c * b) (b * d))" by (simp add: eq_rat, simp add: Zero_rat_def)lemma rat_inverse_code [code]: "inverse (Fract a b) = (if b = 0 then Fract 1 0 else if a < 0 then Fract (- b) (- a) else Fract b a)" by (simp add: eq_rat)lemma rat_divide_code [code]: "Fract a b / Fract c d = Fract_norm (a * d) (b * c)" by simphide (open) const Fract_normtext {* Setup for SML code generator *}types_code rat ("(int */ int)")attach (term_of) {*fun term_of_rat (p, q) = let val rT = Type ("Rational.rat", []) in if q = 1 orelse p = 0 then HOLogic.mk_number rT p else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $ HOLogic.mk_number rT p $ HOLogic.mk_number rT q end;*}attach (test) {*fun gen_rat i = let val p = random_range 0 i; val q = random_range 1 (i + 1); val g = Integer.gcd p q; val p' = p div g; val q' = q div g; val r = (if one_of [true, false] then p' else ~ p', if p' = 0 then 0 else q') in (r, fn () => term_of_rat r) end;*}consts_code Fract ("(_,/ _)")consts_code "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")attach {*fun rat_of_int 0 = (0, 0) | rat_of_int i = (i, 1);*}end