src/HOL/Library/Fraction_Field.thy
author wenzelm
Wed Jun 17 11:03:05 2015 +0200 (2015-06-17)
changeset 60500 903bb1495239
parent 60429 d3d1e185cd63
child 61076 bdc1e2f0a86a
child 61106 5bafa612ede4
permissions -rw-r--r--
isabelle update_cartouches;
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(*  Title:      HOL/Library/Fraction_Field.thy
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    Author:     Amine Chaieb, University of Cambridge
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*)
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section\<open>A formalization of the fraction field of any integral domain;
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         generalization of theory Rat from int to any integral domain\<close>
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theory Fraction_Field
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imports Main
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begin
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subsection \<open>General fractions construction\<close>
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subsubsection \<open>Construction of the type of fractions\<close>
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definition fractrel :: "(('a::idom * 'a ) * ('a * 'a)) set" where
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  "fractrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
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lemma fractrel_iff [simp]:
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  "(x, y) \<in> fractrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
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  by (simp add: fractrel_def)
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lemma refl_fractrel: "refl_on {x. snd x \<noteq> 0} fractrel"
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  by (auto simp add: refl_on_def fractrel_def)
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lemma sym_fractrel: "sym fractrel"
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  by (simp add: fractrel_def sym_def)
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lemma trans_fractrel: "trans fractrel"
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proof (rule transI, unfold split_paired_all)
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  fix a b a' b' a'' b'' :: 'a
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  assume A: "((a, b), (a', b')) \<in> fractrel"
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  assume B: "((a', b'), (a'', b'')) \<in> fractrel"
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  have "b' * (a * b'') = b'' * (a * b')" by (simp add: ac_simps)
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  also from A have "a * b' = a' * b" by auto
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  also have "b'' * (a' * b) = b * (a' * b'')" by (simp add: ac_simps)
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  also from B have "a' * b'' = a'' * b'" by auto
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  also have "b * (a'' * b') = b' * (a'' * b)" by (simp add: ac_simps)
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  finally have "b' * (a * b'') = b' * (a'' * b)" .
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  moreover from B have "b' \<noteq> 0" by auto
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  ultimately have "a * b'' = a'' * b" by simp
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  with A B show "((a, b), (a'', b'')) \<in> fractrel" by auto
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qed
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lemma equiv_fractrel: "equiv {x. snd x \<noteq> 0} fractrel"
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  by (rule equivI [OF refl_fractrel sym_fractrel trans_fractrel])
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lemmas UN_fractrel = UN_equiv_class [OF equiv_fractrel]
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lemmas UN_fractrel2 = UN_equiv_class2 [OF equiv_fractrel equiv_fractrel]
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lemma equiv_fractrel_iff [iff]:
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  assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
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  shows "fractrel `` {x} = fractrel `` {y} \<longleftrightarrow> (x, y) \<in> fractrel"
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  by (rule eq_equiv_class_iff, rule equiv_fractrel) (auto simp add: assms)
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definition "fract = {(x::'a\<times>'a). snd x \<noteq> (0::'a::idom)} // fractrel"
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typedef 'a fract = "fract :: ('a * 'a::idom) set set"
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  unfolding fract_def
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proof
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  have "(0::'a, 1::'a) \<in> {x. snd x \<noteq> 0}" by simp
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  then show "fractrel `` {(0::'a, 1)} \<in> {x. snd x \<noteq> 0} // fractrel"
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    by (rule quotientI)
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qed
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lemma fractrel_in_fract [simp]: "snd x \<noteq> 0 \<Longrightarrow> fractrel `` {x} \<in> fract"
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  by (simp add: fract_def quotientI)
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declare Abs_fract_inject [simp] Abs_fract_inverse [simp]
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subsubsection \<open>Representation and basic operations\<close>
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definition Fract :: "'a::idom \<Rightarrow> 'a \<Rightarrow> 'a fract"
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  where "Fract a b = Abs_fract (fractrel `` {if b = 0 then (0, 1) else (a, b)})"
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code_datatype Fract
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lemma Fract_cases [cases type: fract]:
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  obtains (Fract) a b where "q = Fract a b" "b \<noteq> 0"
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  by (cases q) (clarsimp simp add: Fract_def fract_def quotient_def)
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lemma Fract_induct [case_names Fract, induct type: fract]:
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  "(\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)) \<Longrightarrow> P q"
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  by (cases q) simp
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lemma eq_fract:
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  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"
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    and "\<And>a. Fract a 0 = Fract 0 1"
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    and "\<And>a c. Fract 0 a = Fract 0 c"
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  by (simp_all add: Fract_def)
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instantiation fract :: (idom) "{comm_ring_1,power}"
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begin
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definition Zero_fract_def [code_unfold]: "0 = Fract 0 1"
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definition One_fract_def [code_unfold]: "1 = Fract 1 1"
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definition add_fract_def:
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  "q + r = Abs_fract (\<Union>x \<in> Rep_fract q. \<Union>y \<in> Rep_fract r.
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    fractrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
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lemma add_fract [simp]:
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  assumes "b \<noteq> (0::'a::idom)"
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    and "d \<noteq> 0"
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  shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
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proof -
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  have "(\<lambda>x y. fractrel``{(fst x * snd y + fst y * snd x, snd x * snd y :: 'a)}) respects2 fractrel"
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    by (rule equiv_fractrel [THEN congruent2_commuteI]) (simp_all add: algebra_simps)
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  with assms show ?thesis by (simp add: Fract_def add_fract_def UN_fractrel2)
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qed
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definition minus_fract_def:
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  "- q = Abs_fract (\<Union>x \<in> Rep_fract q. fractrel `` {(- fst x, snd x)})"
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lemma minus_fract [simp, code]:
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  fixes a b :: "'a::idom"
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  shows "- Fract a b = Fract (- a) b"
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proof -
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  have "(\<lambda>x. fractrel `` {(- fst x, snd x :: 'a)}) respects fractrel"
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    by (simp add: congruent_def split_paired_all)
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  then show ?thesis by (simp add: Fract_def minus_fract_def UN_fractrel)
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qed
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lemma minus_fract_cancel [simp]: "Fract (- a) (- b) = Fract a b"
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  by (cases "b = 0") (simp_all add: eq_fract)
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definition diff_fract_def: "q - r = q + - (r::'a fract)"
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lemma diff_fract [simp]:
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  assumes "b \<noteq> 0"
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    and "d \<noteq> 0"
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  shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
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  using assms by (simp add: diff_fract_def)
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definition mult_fract_def:
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  "q * r = Abs_fract (\<Union>x \<in> Rep_fract q. \<Union>y \<in> Rep_fract r.
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    fractrel``{(fst x * fst y, snd x * snd y)})"
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lemma mult_fract [simp]: "Fract (a::'a::idom) b * Fract c d = Fract (a * c) (b * d)"
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proof -
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  have "(\<lambda>x y. fractrel `` {(fst x * fst y, snd x * snd y :: 'a)}) respects2 fractrel"
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    by (rule equiv_fractrel [THEN congruent2_commuteI]) (simp_all add: algebra_simps)
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  then show ?thesis by (simp add: Fract_def mult_fract_def UN_fractrel2)
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qed
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lemma mult_fract_cancel:
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  assumes "c \<noteq> (0::'a)"
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  shows "Fract (c * a) (c * b) = Fract a b"
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proof -
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  from assms have "Fract c c = Fract 1 1"
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    by (simp add: Fract_def)
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  then show ?thesis
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    by (simp add: mult_fract [symmetric])
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qed
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instance
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proof
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  fix q r s :: "'a fract"
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  show "(q * r) * s = q * (r * s)"
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    by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
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  show "q * r = r * q"
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    by (cases q, cases r) (simp add: eq_fract algebra_simps)
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  show "1 * q = q"
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    by (cases q) (simp add: One_fract_def eq_fract)
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  show "(q + r) + s = q + (r + s)"
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    by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
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  show "q + r = r + q"
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    by (cases q, cases r) (simp add: eq_fract algebra_simps)
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  show "0 + q = q"
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    by (cases q) (simp add: Zero_fract_def eq_fract)
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  show "- q + q = 0"
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    by (cases q) (simp add: Zero_fract_def eq_fract)
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  show "q - r = q + - r"
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    by (cases q, cases r) (simp add: eq_fract)
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  show "(q + r) * s = q * s + r * s"
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    by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
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  show "(0::'a fract) \<noteq> 1"
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    by (simp add: Zero_fract_def One_fract_def eq_fract)
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qed
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end
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lemma of_nat_fract: "of_nat k = Fract (of_nat k) 1"
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  by (induct k) (simp_all add: Zero_fract_def One_fract_def)
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lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
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  by (rule of_nat_fract [symmetric])
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lemma fract_collapse [code_post]:
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  "Fract 0 k = 0"
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  "Fract 1 1 = 1"
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  "Fract k 0 = 0"
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  by (cases "k = 0")
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    (simp_all add: Zero_fract_def One_fract_def eq_fract Fract_def)
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lemma fract_expand [code_unfold]:
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  "0 = Fract 0 1"
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  "1 = Fract 1 1"
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  by (simp_all add: fract_collapse)
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lemma Fract_cases_nonzero:
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  obtains (Fract) a b where "q = Fract a b" and "b \<noteq> 0" and "a \<noteq> 0"
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    | (0) "q = 0"
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proof (cases "q = 0")
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  case True
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  then show thesis using 0 by auto
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next
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  case False
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  then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
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  with False have "0 \<noteq> Fract a b" by simp
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  with \<open>b \<noteq> 0\<close> have "a \<noteq> 0" by (simp add: Zero_fract_def eq_fract)
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  with Fract \<open>q = Fract a b\<close> \<open>b \<noteq> 0\<close> show thesis by auto
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qed
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subsubsection \<open>The field of rational numbers\<close>
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context idom
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begin
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subclass ring_no_zero_divisors ..
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end
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instantiation fract :: (idom) field
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begin
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definition inverse_fract_def:
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  "inverse q = Abs_fract (\<Union>x \<in> Rep_fract q.
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     fractrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
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lemma inverse_fract [simp]: "inverse (Fract a b) = Fract (b::'a::idom) a"
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proof -
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  have *: "\<And>x. (0::'a) = x \<longleftrightarrow> x = 0"
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    by auto
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  have "(\<lambda>x. fractrel `` {if fst x = 0 then (0, 1) else (snd x, fst x :: 'a)}) respects fractrel"
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    by (auto simp add: congruent_def * algebra_simps)
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  then show ?thesis
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    by (simp add: Fract_def inverse_fract_def UN_fractrel)
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qed
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definition divide_fract_def: "q div r = q * inverse (r:: 'a fract)"
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lemma divide_fract [simp]: "Fract a b div Fract c d = Fract (a * d) (b * c)"
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  by (simp add: divide_fract_def)
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instance
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proof
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  fix q :: "'a fract"
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  assume "q \<noteq> 0"
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  then show "inverse q * q = 1"
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    by (cases q rule: Fract_cases_nonzero)
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      (simp_all add: fract_expand eq_fract mult.commute)
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next
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  fix q r :: "'a fract"
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  show "q div r = q * inverse r" by (simp add: divide_fract_def)
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next
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  show "inverse 0 = (0:: 'a fract)"
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    by (simp add: fract_expand) (simp add: fract_collapse)
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qed
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end
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subsubsection \<open>The ordered field of fractions over an ordered idom\<close>
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lemma le_congruent2:
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  "(\<lambda>x y::'a \<times> 'a::linordered_idom.
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    {(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)})
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    respects2 fractrel"
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proof (clarsimp simp add: congruent2_def)
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  fix a b a' b' c d c' d' :: 'a
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  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
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  assume eq1: "a * b' = a' * b"
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  assume eq2: "c * d' = c' * d"
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  let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
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  {
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    fix a b c d x :: 'a
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    assume x: "x \<noteq> 0"
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    have "?le a b c d = ?le (a * x) (b * x) c d"
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    proof -
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      from x have "0 < x * x"
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        by (auto simp add: zero_less_mult_iff)
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      then have "?le a b c d =
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          ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
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        by (simp add: mult_le_cancel_right)
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      also have "... = ?le (a * x) (b * x) c d"
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        by (simp add: ac_simps)
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      finally show ?thesis .
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    qed
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  } note le_factor = this
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  let ?D = "b * d" and ?D' = "b' * d'"
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  from neq have D: "?D \<noteq> 0" by simp
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  from neq have "?D' \<noteq> 0" by simp
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  then have "?le a b c d = ?le (a * ?D') (b * ?D') c d"
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    by (rule le_factor)
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   301
  also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
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   302
    by (simp add: ac_simps)
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   303
  also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
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   304
    by (simp only: eq1 eq2)
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   305
  also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
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   306
    by (simp add: ac_simps)
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   307
  also from D have "... = ?le a' b' c' d'"
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   308
    by (rule le_factor [symmetric])
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   309
  finally show "?le a b c d = ?le a' b' c' d'" .
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   310
qed
huffman@36331
   311
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   312
instantiation fract :: (linordered_idom) linorder
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   313
begin
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   314
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   315
definition le_fract_def:
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   316
  "q \<le> r \<longleftrightarrow> the_elem (\<Union>x \<in> Rep_fract q. \<Union>y \<in> Rep_fract r.
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   317
    {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
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   318
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   319
definition less_fract_def: "z < (w::'a fract) \<longleftrightarrow> z \<le> w \<and> \<not> w \<le> z"
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   320
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   321
lemma le_fract [simp]:
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   322
  assumes "b \<noteq> 0"
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   323
    and "d \<noteq> 0"
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   324
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
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   325
  by (simp add: Fract_def le_fract_def le_congruent2 UN_fractrel2 assms)
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   326
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   327
lemma less_fract [simp]:
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   328
  assumes "b \<noteq> 0"
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   329
    and "d \<noteq> 0"
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   330
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
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   331
  by (simp add: less_fract_def less_le_not_le ac_simps assms)
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   332
wenzelm@47252
   333
instance
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   334
proof
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   335
  fix q r s :: "'a fract"
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  assume "q \<le> r" and "r \<le> s"
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   337
  then show "q \<le> s"
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   338
  proof (induct q, induct r, induct s)
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   339
    fix a b c d e f :: 'a
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   340
    assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
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   341
    assume 1: "Fract a b \<le> Fract c d"
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   342
    assume 2: "Fract c d \<le> Fract e f"
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   343
    show "Fract a b \<le> Fract e f"
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   344
    proof -
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   345
      from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
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   346
        by (auto simp add: zero_less_mult_iff linorder_neq_iff)
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   347
      have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
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   348
      proof -
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   349
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
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   350
          by simp
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   351
        with ff show ?thesis by (simp add: mult_le_cancel_right)
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   352
      qed
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   353
      also have "... = (c * f) * (d * f) * (b * b)"
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   354
        by (simp only: ac_simps)
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   355
      also have "... \<le> (e * d) * (d * f) * (b * b)"
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   356
      proof -
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   357
        from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
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   358
          by simp
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   359
        with bb show ?thesis by (simp add: mult_le_cancel_right)
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   360
      qed
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   361
      finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
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   362
        by (simp only: ac_simps)
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   363
      with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
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   364
        by (simp add: mult_le_cancel_right)
huffman@36331
   365
      with neq show ?thesis by simp
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   366
    qed
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   367
  qed
huffman@36331
   368
next
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   369
  fix q r :: "'a fract"
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   370
  assume "q \<le> r" and "r \<le> q"
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   371
  then show "q = r"
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   372
  proof (induct q, induct r)
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   373
    fix a b c d :: 'a
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   374
    assume neq: "b \<noteq> 0" "d \<noteq> 0"
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   375
    assume 1: "Fract a b \<le> Fract c d"
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   376
    assume 2: "Fract c d \<le> Fract a b"
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   377
    show "Fract a b = Fract c d"
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   378
    proof -
huffman@36331
   379
      from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
huffman@36331
   380
        by simp
huffman@36331
   381
      also have "... \<le> (a * d) * (b * d)"
huffman@36331
   382
      proof -
huffman@36331
   383
        from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
huffman@36331
   384
          by simp
haftmann@57514
   385
        then show ?thesis by (simp only: ac_simps)
huffman@36331
   386
      qed
huffman@36331
   387
      finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
huffman@36331
   388
      moreover from neq have "b * d \<noteq> 0" by simp
huffman@36331
   389
      ultimately have "a * d = c * b" by simp
huffman@36331
   390
      with neq show ?thesis by (simp add: eq_fract)
huffman@36331
   391
    qed
huffman@36331
   392
  qed
huffman@36331
   393
next
huffman@36331
   394
  fix q r :: "'a fract"
huffman@36331
   395
  show "q \<le> q"
huffman@36331
   396
    by (induct q) simp
huffman@36331
   397
  show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
huffman@36331
   398
    by (simp only: less_fract_def)
huffman@36331
   399
  show "q \<le> r \<or> r \<le> q"
huffman@36331
   400
    by (induct q, induct r)
haftmann@57512
   401
       (simp add: mult.commute, rule linorder_linear)
huffman@36331
   402
qed
huffman@36331
   403
huffman@36331
   404
end
huffman@36331
   405
wenzelm@54463
   406
instantiation fract :: (linordered_idom) "{distrib_lattice,abs_if,sgn_if}"
huffman@36331
   407
begin
huffman@36331
   408
wenzelm@46573
   409
definition abs_fract_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::'a fract))"
huffman@36331
   410
wenzelm@46573
   411
definition sgn_fract_def:
wenzelm@54463
   412
  "sgn (q::'a fract) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
huffman@36331
   413
huffman@36331
   414
theorem abs_fract [simp]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
huffman@36331
   415
  by (auto simp add: abs_fract_def Zero_fract_def le_less
huffman@36331
   416
      eq_fract zero_less_mult_iff mult_less_0_iff split: abs_split)
huffman@36331
   417
wenzelm@46573
   418
definition inf_fract_def:
wenzelm@46573
   419
  "(inf \<Colon> 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = min"
huffman@36331
   420
wenzelm@46573
   421
definition sup_fract_def:
wenzelm@46573
   422
  "(sup \<Colon> 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = max"
huffman@36331
   423
wenzelm@46573
   424
instance
wenzelm@46573
   425
  by intro_classes
wenzelm@46573
   426
    (auto simp add: abs_fract_def sgn_fract_def
haftmann@54863
   427
      max_min_distrib2 inf_fract_def sup_fract_def)
huffman@36331
   428
huffman@36331
   429
end
huffman@36331
   430
haftmann@59867
   431
instance fract :: (linordered_idom) linordered_field
huffman@36331
   432
proof
huffman@36331
   433
  fix q r s :: "'a fract"
wenzelm@53196
   434
  assume "q \<le> r"
wenzelm@53196
   435
  then show "s + q \<le> s + r"
huffman@36331
   436
  proof (induct q, induct r, induct s)
huffman@36331
   437
    fix a b c d e f :: 'a
wenzelm@53196
   438
    assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
huffman@36331
   439
    assume le: "Fract a b \<le> Fract c d"
huffman@36331
   440
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
huffman@36331
   441
    proof -
huffman@36331
   442
      let ?F = "f * f" from neq have F: "0 < ?F"
huffman@36331
   443
        by (auto simp add: zero_less_mult_iff)
huffman@36331
   444
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
huffman@36331
   445
        by simp
huffman@36331
   446
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
huffman@36331
   447
        by (simp add: mult_le_cancel_right)
haftmann@36348
   448
      with neq show ?thesis by (simp add: field_simps)
huffman@36331
   449
    qed
huffman@36331
   450
  qed
wenzelm@53196
   451
next
wenzelm@53196
   452
  fix q r s :: "'a fract"
wenzelm@53196
   453
  assume "q < r" and "0 < s"
wenzelm@53196
   454
  then show "s * q < s * r"
huffman@36331
   455
  proof (induct q, induct r, induct s)
huffman@36331
   456
    fix a b c d e f :: 'a
wenzelm@54463
   457
    assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
huffman@36331
   458
    assume le: "Fract a b < Fract c d"
huffman@36331
   459
    assume gt: "0 < Fract e f"
huffman@36331
   460
    show "Fract e f * Fract a b < Fract e f * Fract c d"
huffman@36331
   461
    proof -
huffman@36331
   462
      let ?E = "e * f" and ?F = "f * f"
huffman@36331
   463
      from neq gt have "0 < ?E"
huffman@36331
   464
        by (auto simp add: Zero_fract_def order_less_le eq_fract)
huffman@36331
   465
      moreover from neq have "0 < ?F"
huffman@36331
   466
        by (auto simp add: zero_less_mult_iff)
huffman@36331
   467
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
huffman@36331
   468
        by simp
huffman@36331
   469
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
huffman@36331
   470
        by (simp add: mult_less_cancel_right)
huffman@36331
   471
      with neq show ?thesis
haftmann@57514
   472
        by (simp add: ac_simps)
huffman@36331
   473
    qed
huffman@36331
   474
  qed
huffman@36331
   475
qed
huffman@36331
   476
huffman@36331
   477
lemma fract_induct_pos [case_names Fract]:
huffman@36331
   478
  fixes P :: "'a::linordered_idom fract \<Rightarrow> bool"
huffman@36331
   479
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
huffman@36331
   480
  shows "P q"
huffman@36331
   481
proof (cases q)
wenzelm@54463
   482
  case (Fract a b)
wenzelm@54463
   483
  {
wenzelm@54463
   484
    fix a b :: 'a
huffman@36331
   485
    assume b: "b < 0"
wenzelm@54463
   486
    have "P (Fract a b)"
wenzelm@54463
   487
    proof -
wenzelm@54463
   488
      from b have "0 < - b" by simp
wenzelm@54463
   489
      then have "P (Fract (- a) (- b))"
wenzelm@54463
   490
        by (rule step)
wenzelm@54463
   491
      then show "P (Fract a b)"
wenzelm@54463
   492
        by (simp add: order_less_imp_not_eq [OF b])
wenzelm@54463
   493
    qed
wenzelm@54463
   494
  }
wenzelm@54463
   495
  with Fract show "P q"
wenzelm@54463
   496
    by (auto simp add: linorder_neq_iff step)
huffman@36331
   497
qed
huffman@36331
   498
wenzelm@53196
   499
lemma zero_less_Fract_iff: "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
huffman@36331
   500
  by (auto simp add: Zero_fract_def zero_less_mult_iff)
huffman@36331
   501
wenzelm@53196
   502
lemma Fract_less_zero_iff: "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
huffman@36331
   503
  by (auto simp add: Zero_fract_def mult_less_0_iff)
huffman@36331
   504
wenzelm@53196
   505
lemma zero_le_Fract_iff: "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
huffman@36331
   506
  by (auto simp add: Zero_fract_def zero_le_mult_iff)
huffman@36331
   507
wenzelm@53196
   508
lemma Fract_le_zero_iff: "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
huffman@36331
   509
  by (auto simp add: Zero_fract_def mult_le_0_iff)
huffman@36331
   510
wenzelm@53196
   511
lemma one_less_Fract_iff: "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
huffman@36331
   512
  by (auto simp add: One_fract_def mult_less_cancel_right_disj)
huffman@36331
   513
wenzelm@53196
   514
lemma Fract_less_one_iff: "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
huffman@36331
   515
  by (auto simp add: One_fract_def mult_less_cancel_right_disj)
huffman@36331
   516
wenzelm@53196
   517
lemma one_le_Fract_iff: "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
huffman@36331
   518
  by (auto simp add: One_fract_def mult_le_cancel_right)
huffman@36331
   519
wenzelm@53196
   520
lemma Fract_le_one_iff: "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
huffman@36331
   521
  by (auto simp add: One_fract_def mult_le_cancel_right)
huffman@36331
   522
huffman@36331
   523
end