haftmann@28952: (*  Title:  HOL/Rational.thy
paulson@14365:     Author: Markus Wenzel, TU Muenchen
paulson@14365: *)
paulson@14365: 
wenzelm@14691: header {* Rational numbers *}
paulson@14365: 
nipkow@15131: theory Rational
huffman@29880: imports GCD
haftmann@28952: uses ("Tools/rat_arith.ML")
nipkow@15131: begin
paulson@14365: 
haftmann@27551: subsection {* Rational numbers as quotient *}
paulson@14365: 
haftmann@27551: subsubsection {* Construction of the type of rational numbers *}
huffman@18913: 
wenzelm@21404: definition
wenzelm@21404:   ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where
haftmann@27551:   "ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
paulson@14365: 
huffman@18913: lemma ratrel_iff [simp]:
haftmann@27551:   "(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
haftmann@27551:   by (simp add: ratrel_def)
paulson@14365: 
haftmann@27551: lemma refl_ratrel: "refl {x. snd x \<noteq> 0} ratrel"
haftmann@27551:   by (auto simp add: refl_def ratrel_def)
huffman@18913: 
huffman@18913: lemma sym_ratrel: "sym ratrel"
haftmann@27551:   by (simp add: ratrel_def sym_def)
paulson@14365: 
huffman@18913: lemma trans_ratrel: "trans ratrel"
haftmann@27551: proof (rule transI, unfold split_paired_all)
haftmann@27551:   fix a b a' b' a'' b'' :: int
haftmann@27551:   assume A: "((a, b), (a', b')) \<in> ratrel"
haftmann@27551:   assume B: "((a', b'), (a'', b'')) \<in> ratrel"
haftmann@27551:   have "b' * (a * b'') = b'' * (a * b')" by simp
haftmann@27551:   also from A have "a * b' = a' * b" by auto
haftmann@27551:   also have "b'' * (a' * b) = b * (a' * b'')" by simp
haftmann@27551:   also from B have "a' * b'' = a'' * b'" by auto
haftmann@27551:   also have "b * (a'' * b') = b' * (a'' * b)" by simp
haftmann@27551:   finally have "b' * (a * b'') = b' * (a'' * b)" .
haftmann@27551:   moreover from B have "b' \<noteq> 0" by auto
haftmann@27551:   ultimately have "a * b'' = a'' * b" by simp
haftmann@27551:   with A B show "((a, b), (a'', b'')) \<in> ratrel" by auto
paulson@14365: qed
haftmann@27551:   
haftmann@27551: lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"
haftmann@27551:   by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])
paulson@14365: 
huffman@18913: lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
huffman@18913: lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
paulson@14365: 
haftmann@27551: lemma equiv_ratrel_iff [iff]: 
haftmann@27551:   assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
haftmann@27551:   shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel"
haftmann@27551:   by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)
paulson@14365: 
haftmann@27551: typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"
haftmann@27551: proof
haftmann@27551:   have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp
haftmann@27551:   then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI)
haftmann@27551: qed
haftmann@27551: 
haftmann@27551: lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat"
haftmann@27551:   by (simp add: Rat_def quotientI)
haftmann@27551: 
haftmann@27551: declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]
haftmann@27551: 
haftmann@27551: 
haftmann@27551: subsubsection {* Representation and basic operations *}
haftmann@27551: 
haftmann@27551: definition
haftmann@27551:   Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
haftmann@28562:   [code del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
paulson@14365: 
haftmann@27551: code_datatype Fract
haftmann@27551: 
haftmann@27551: lemma Rat_cases [case_names Fract, cases type: rat]:
haftmann@27551:   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C"
haftmann@27551:   shows C
haftmann@27551:   using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
haftmann@27551: 
haftmann@27551: lemma Rat_induct [case_names Fract, induct type: rat]:
haftmann@27551:   assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)"
haftmann@27551:   shows "P q"
haftmann@27551:   using assms by (cases q) simp
haftmann@27551: 
haftmann@27551: lemma eq_rat:
haftmann@27551:   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"
haftmann@27652:   and "\<And>a. Fract a 0 = Fract 0 1"
haftmann@27652:   and "\<And>a c. Fract 0 a = Fract 0 c"
haftmann@27551:   by (simp_all add: Fract_def)
haftmann@27551: 
haftmann@27551: instantiation rat :: "{comm_ring_1, recpower}"
haftmann@25571: begin
haftmann@25571: 
haftmann@25571: definition
haftmann@27551:   Zero_rat_def [code, code unfold]: "0 = Fract 0 1"
paulson@14365: 
haftmann@25571: definition
haftmann@27551:   One_rat_def [code, code unfold]: "1 = Fract 1 1"
huffman@18913: 
haftmann@25571: definition
haftmann@28562:   add_rat_def [code del]:
haftmann@27551:   "q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
haftmann@27551:     ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
haftmann@27551: 
haftmann@27652: lemma add_rat [simp]:
haftmann@27551:   assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551:   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
haftmann@27551: proof -
haftmann@27551:   have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
haftmann@27551:     respects2 ratrel"
haftmann@27551:   by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)
haftmann@27551:   with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)
haftmann@27551: qed
huffman@18913: 
haftmann@25571: definition
haftmann@28562:   minus_rat_def [code del]:
haftmann@27551:   "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
haftmann@27551: 
haftmann@27652: lemma minus_rat [simp, code]: "- Fract a b = Fract (- a) b"
haftmann@27551: proof -
haftmann@27551:   have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
haftmann@27551:     by (simp add: congruent_def)
haftmann@27551:   then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
haftmann@27551: qed
haftmann@27551: 
haftmann@27652: lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
haftmann@27551:   by (cases "b = 0") (simp_all add: eq_rat)
haftmann@25571: 
haftmann@25571: definition
haftmann@28562:   diff_rat_def [code del]: "q - r = q + - (r::rat)"
huffman@18913: 
haftmann@27652: lemma diff_rat [simp]:
haftmann@27551:   assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551:   shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
haftmann@27652:   using assms by (simp add: diff_rat_def)
haftmann@25571: 
haftmann@25571: definition
haftmann@28562:   mult_rat_def [code del]:
haftmann@27551:   "q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
haftmann@27551:     ratrel``{(fst x * fst y, snd x * snd y)})"
paulson@14365: 
haftmann@27652: lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
haftmann@27551: proof -
haftmann@27551:   have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel"
haftmann@27551:     by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all
haftmann@27551:   then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)
paulson@14365: qed
paulson@14365: 
haftmann@27652: lemma mult_rat_cancel:
haftmann@27551:   assumes "c \<noteq> 0"
haftmann@27551:   shows "Fract (c * a) (c * b) = Fract a b"
haftmann@27551: proof -
haftmann@27551:   from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
haftmann@27652:   then show ?thesis by (simp add: mult_rat [symmetric])
haftmann@27551: qed
huffman@27509: 
huffman@27509: primrec power_rat
huffman@27509: where
haftmann@27551:   rat_power_0:     "q ^ 0 = (1\<Colon>rat)"
haftmann@27551:   | rat_power_Suc: "q ^ Suc n = (q\<Colon>rat) * (q ^ n)"
huffman@27509: 
huffman@27509: instance proof
chaieb@27668:   fix q r s :: rat show "(q * r) * s = q * (r * s)" 
haftmann@27652:     by (cases q, cases r, cases s) (simp add: eq_rat)
haftmann@27551: next
haftmann@27551:   fix q r :: rat show "q * r = r * q"
haftmann@27652:     by (cases q, cases r) (simp add: eq_rat)
haftmann@27551: next
haftmann@27551:   fix q :: rat show "1 * q = q"
haftmann@27652:     by (cases q) (simp add: One_rat_def eq_rat)
haftmann@27551: next
haftmann@27551:   fix q r s :: rat show "(q + r) + s = q + (r + s)"
nipkow@29667:     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
haftmann@27551: next
haftmann@27551:   fix q r :: rat show "q + r = r + q"
haftmann@27652:     by (cases q, cases r) (simp add: eq_rat)
haftmann@27551: next
haftmann@27551:   fix q :: rat show "0 + q = q"
haftmann@27652:     by (cases q) (simp add: Zero_rat_def eq_rat)
haftmann@27551: next
haftmann@27551:   fix q :: rat show "- q + q = 0"
haftmann@27652:     by (cases q) (simp add: Zero_rat_def eq_rat)
haftmann@27551: next
haftmann@27551:   fix q r :: rat show "q - r = q + - r"
haftmann@27652:     by (cases q, cases r) (simp add: eq_rat)
haftmann@27551: next
haftmann@27551:   fix q r s :: rat show "(q + r) * s = q * s + r * s"
nipkow@29667:     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
haftmann@27551: next
haftmann@27551:   show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
haftmann@27551: next
haftmann@27551:   fix q :: rat show "q * 1 = q"
haftmann@27652:     by (cases q) (simp add: One_rat_def eq_rat)
haftmann@27551: next
huffman@27509:   fix q :: rat
huffman@27509:   fix n :: nat
huffman@27509:   show "q ^ 0 = 1" by simp
huffman@27509:   show "q ^ (Suc n) = q * (q ^ n)" by simp
huffman@27509: qed
huffman@27509: 
huffman@27509: end
huffman@27509: 
haftmann@27551: lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
haftmann@27652:   by (induct k) (simp_all add: Zero_rat_def One_rat_def)
haftmann@27551: 
haftmann@27551: lemma of_int_rat: "of_int k = Fract k 1"
haftmann@27652:   by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
haftmann@27551: 
haftmann@27551: lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
haftmann@27551:   by (rule of_nat_rat [symmetric])
haftmann@27551: 
haftmann@27551: lemma Fract_of_int_eq: "Fract k 1 = of_int k"
haftmann@27551:   by (rule of_int_rat [symmetric])
haftmann@27551: 
haftmann@27551: instantiation rat :: number_ring
haftmann@27551: begin
haftmann@27551: 
haftmann@27551: definition
haftmann@28562:   rat_number_of_def [code del]: "number_of w = Fract w 1"
haftmann@27551: 
haftmann@27551: instance by intro_classes (simp add: rat_number_of_def of_int_rat)
haftmann@27551: 
haftmann@27551: end
haftmann@27551: 
haftmann@27551: lemma rat_number_collapse [code post]:
haftmann@27551:   "Fract 0 k = 0"
haftmann@27551:   "Fract 1 1 = 1"
haftmann@27551:   "Fract (number_of k) 1 = number_of k"
haftmann@27551:   "Fract k 0 = 0"
haftmann@27551:   by (cases "k = 0")
haftmann@27551:     (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)
haftmann@27551: 
haftmann@27551: lemma rat_number_expand [code unfold]:
haftmann@27551:   "0 = Fract 0 1"
haftmann@27551:   "1 = Fract 1 1"
haftmann@27551:   "number_of k = Fract (number_of k) 1"
haftmann@27551:   by (simp_all add: rat_number_collapse)
haftmann@27551: 
haftmann@27551: lemma iszero_rat [simp]:
haftmann@27551:   "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
haftmann@27551:   by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
haftmann@27551: 
haftmann@27551: lemma Rat_cases_nonzero [case_names Fract 0]:
haftmann@27551:   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
haftmann@27551:   assumes 0: "q = 0 \<Longrightarrow> C"
haftmann@27551:   shows C
haftmann@27551: proof (cases "q = 0")
haftmann@27551:   case True then show C using 0 by auto
haftmann@27551: next
haftmann@27551:   case False
haftmann@27551:   then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
haftmann@27551:   moreover with False have "0 \<noteq> Fract a b" by simp
haftmann@27551:   with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
haftmann@27551:   with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
haftmann@27551: qed
haftmann@27551:   
haftmann@27551: 
haftmann@27551: 
haftmann@27551: subsubsection {* The field of rational numbers *}
haftmann@27551: 
haftmann@27551: instantiation rat :: "{field, division_by_zero}"
haftmann@27551: begin
haftmann@27551: 
haftmann@27551: definition
haftmann@28562:   inverse_rat_def [code del]:
haftmann@27551:   "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
haftmann@27551:      ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
haftmann@27551: 
haftmann@27652: lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
haftmann@27551: proof -
haftmann@27551:   have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
haftmann@27551:     by (auto simp add: congruent_def mult_commute)
haftmann@27551:   then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
huffman@27509: qed
huffman@27509: 
haftmann@27551: definition
haftmann@28562:   divide_rat_def [code del]: "q / r = q * inverse (r::rat)"
haftmann@27551: 
haftmann@27652: lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
haftmann@27652:   by (simp add: divide_rat_def)
haftmann@27551: 
haftmann@27551: instance proof
haftmann@27652:   show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
haftmann@27551:     (simp add: rat_number_collapse)
haftmann@27551: next
haftmann@27551:   fix q :: rat
haftmann@27551:   assume "q \<noteq> 0"
haftmann@27551:   then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
haftmann@27551:    (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
haftmann@27551: next
haftmann@27551:   fix q r :: rat
haftmann@27551:   show "q / r = q * inverse r" by (simp add: divide_rat_def)
haftmann@27551: qed
haftmann@27551: 
haftmann@27551: end
haftmann@27551: 
haftmann@27551: 
haftmann@27551: subsubsection {* Various *}
haftmann@27551: 
haftmann@27551: lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
haftmann@27652:   by (simp add: rat_number_expand)
haftmann@27551: 
haftmann@27551: lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
haftmann@27652:   by (simp add: Fract_of_int_eq [symmetric])
haftmann@27551: 
haftmann@27551: lemma Fract_number_of_quotient [code post]:
haftmann@27551:   "Fract (number_of k) (number_of l) = number_of k / number_of l"
haftmann@27551:   unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
haftmann@27551: 
haftmann@27652: lemma Fract_1_number_of [code post]:
haftmann@27652:   "Fract 1 (number_of k) = 1 / number_of k"
haftmann@27652:   unfolding Fract_of_int_quotient number_of_eq by simp
haftmann@27551: 
haftmann@27551: subsubsection {* The ordered field of rational numbers *}
huffman@27509: 
huffman@27509: instantiation rat :: linorder
huffman@27509: begin
huffman@27509: 
huffman@27509: definition
haftmann@28562:   le_rat_def [code del]:
huffman@27509:    "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
haftmann@27551:       {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
haftmann@27551: 
haftmann@27652: lemma le_rat [simp]:
haftmann@27551:   assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551:   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27551: proof -
haftmann@27551:   have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
haftmann@27551:     respects2 ratrel"
haftmann@27551:   proof (clarsimp simp add: congruent2_def)
haftmann@27551:     fix a b a' b' c d c' d'::int
haftmann@27551:     assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
haftmann@27551:     assume eq1: "a * b' = a' * b"
haftmann@27551:     assume eq2: "c * d' = c' * d"
haftmann@27551: 
haftmann@27551:     let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
haftmann@27551:     {
haftmann@27551:       fix a b c d x :: int assume x: "x \<noteq> 0"
haftmann@27551:       have "?le a b c d = ?le (a * x) (b * x) c d"
haftmann@27551:       proof -
haftmann@27551:         from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
haftmann@27551:         hence "?le a b c d =
haftmann@27551:             ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
haftmann@27551:           by (simp add: mult_le_cancel_right)
haftmann@27551:         also have "... = ?le (a * x) (b * x) c d"
haftmann@27551:           by (simp add: mult_ac)
haftmann@27551:         finally show ?thesis .
haftmann@27551:       qed
haftmann@27551:     } note le_factor = this
haftmann@27551: 
haftmann@27551:     let ?D = "b * d" and ?D' = "b' * d'"
haftmann@27551:     from neq have D: "?D \<noteq> 0" by simp
haftmann@27551:     from neq have "?D' \<noteq> 0" by simp
haftmann@27551:     hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
haftmann@27551:       by (rule le_factor)
chaieb@27668:     also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" 
haftmann@27551:       by (simp add: mult_ac)
haftmann@27551:     also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
haftmann@27551:       by (simp only: eq1 eq2)
haftmann@27551:     also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
haftmann@27551:       by (simp add: mult_ac)
haftmann@27551:     also from D have "... = ?le a' b' c' d'"
haftmann@27551:       by (rule le_factor [symmetric])
haftmann@27551:     finally show "?le a b c d = ?le a' b' c' d'" .
haftmann@27551:   qed
haftmann@27551:   with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
haftmann@27551: qed
huffman@27509: 
huffman@27509: definition
haftmann@28562:   less_rat_def [code del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
huffman@27509: 
haftmann@27652: lemma less_rat [simp]:
haftmann@27551:   assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551:   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652:   using assms by (simp add: less_rat_def eq_rat order_less_le)
huffman@27509: 
huffman@27509: instance proof
paulson@14365:   fix q r s :: rat
paulson@14365:   {
paulson@14365:     assume "q \<le> r" and "r \<le> s"
paulson@14365:     show "q \<le> s"
paulson@14365:     proof (insert prems, induct q, induct r, induct s)
paulson@14365:       fix a b c d e f :: int
paulson@14365:       assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365:       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
paulson@14365:       show "Fract a b \<le> Fract e f"
paulson@14365:       proof -
paulson@14365:         from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
paulson@14365:           by (auto simp add: zero_less_mult_iff linorder_neq_iff)
paulson@14365:         have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
paulson@14365:         proof -
paulson@14365:           from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652:             by simp
paulson@14365:           with ff show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365:         qed
chaieb@27668:         also have "... = (c * f) * (d * f) * (b * b)" by algebra
paulson@14365:         also have "... \<le> (e * d) * (d * f) * (b * b)"
paulson@14365:         proof -
paulson@14365:           from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
haftmann@27652:             by simp
paulson@14365:           with bb show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365:         qed
paulson@14365:         finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
paulson@14365:           by (simp only: mult_ac)
paulson@14365:         with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
paulson@14365:           by (simp add: mult_le_cancel_right)
haftmann@27652:         with neq show ?thesis by simp
paulson@14365:       qed
paulson@14365:     qed
paulson@14365:   next
paulson@14365:     assume "q \<le> r" and "r \<le> q"
paulson@14365:     show "q = r"
paulson@14365:     proof (insert prems, induct q, induct r)
paulson@14365:       fix a b c d :: int
paulson@14365:       assume neq: "b \<noteq> 0"  "d \<noteq> 0"
paulson@14365:       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
paulson@14365:       show "Fract a b = Fract c d"
paulson@14365:       proof -
paulson@14365:         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652:           by simp
paulson@14365:         also have "... \<le> (a * d) * (b * d)"
paulson@14365:         proof -
paulson@14365:           from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
haftmann@27652:             by simp
paulson@14365:           thus ?thesis by (simp only: mult_ac)
paulson@14365:         qed
paulson@14365:         finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
paulson@14365:         moreover from neq have "b * d \<noteq> 0" by simp
paulson@14365:         ultimately have "a * d = c * b" by simp
paulson@14365:         with neq show ?thesis by (simp add: eq_rat)
paulson@14365:       qed
paulson@14365:     qed
paulson@14365:   next
paulson@14365:     show "q \<le> q"
haftmann@27652:       by (induct q) simp
haftmann@27682:     show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
haftmann@27682:       by (induct q, induct r) (auto simp add: le_less mult_commute)
paulson@14365:     show "q \<le> r \<or> r \<le> q"
huffman@18913:       by (induct q, induct r)
haftmann@27652:          (simp add: mult_commute, rule linorder_linear)
paulson@14365:   }
paulson@14365: qed
paulson@14365: 
huffman@27509: end
huffman@27509: 
haftmann@27551: instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
haftmann@25571: begin
haftmann@25571: 
haftmann@25571: definition
haftmann@28562:   abs_rat_def [code del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
haftmann@27551: 
haftmann@27652: lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
haftmann@27551:   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)
haftmann@27551: 
haftmann@27551: definition
haftmann@28562:   sgn_rat_def [code del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
haftmann@27551: 
haftmann@27652: lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
haftmann@27551:   unfolding Fract_of_int_eq
haftmann@27652:   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
haftmann@27551:     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
haftmann@27551: 
haftmann@27551: definition
haftmann@25571:   "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
haftmann@25571: 
haftmann@25571: definition
haftmann@25571:   "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
haftmann@25571: 
haftmann@27551: instance by intro_classes
haftmann@27551:   (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
haftmann@22456: 
haftmann@25571: end
haftmann@25571: 
haftmann@27551: instance rat :: ordered_field
haftmann@27551: proof
paulson@14365:   fix q r s :: rat
paulson@14365:   show "q \<le> r ==> s + q \<le> s + r"
paulson@14365:   proof (induct q, induct r, induct s)
paulson@14365:     fix a b c d e f :: int
paulson@14365:     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365:     assume le: "Fract a b \<le> Fract c d"
paulson@14365:     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
paulson@14365:     proof -
paulson@14365:       let ?F = "f * f" from neq have F: "0 < ?F"
paulson@14365:         by (auto simp add: zero_less_mult_iff)
paulson@14365:       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652:         by simp
paulson@14365:       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
paulson@14365:         by (simp add: mult_le_cancel_right)
haftmann@27652:       with neq show ?thesis by (simp add: mult_ac int_distrib)
paulson@14365:     qed
paulson@14365:   qed
paulson@14365:   show "q < r ==> 0 < s ==> s * q < s * r"
paulson@14365:   proof (induct q, induct r, induct s)
paulson@14365:     fix a b c d e f :: int
paulson@14365:     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365:     assume le: "Fract a b < Fract c d"
paulson@14365:     assume gt: "0 < Fract e f"
paulson@14365:     show "Fract e f * Fract a b < Fract e f * Fract c d"
paulson@14365:     proof -
paulson@14365:       let ?E = "e * f" and ?F = "f * f"
paulson@14365:       from neq gt have "0 < ?E"
haftmann@27652:         by (auto simp add: Zero_rat_def order_less_le eq_rat)
paulson@14365:       moreover from neq have "0 < ?F"
paulson@14365:         by (auto simp add: zero_less_mult_iff)
paulson@14365:       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652:         by simp
paulson@14365:       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
paulson@14365:         by (simp add: mult_less_cancel_right)
paulson@14365:       with neq show ?thesis
haftmann@27652:         by (simp add: mult_ac)
paulson@14365:     qed
paulson@14365:   qed
haftmann@27551: qed auto
paulson@14365: 
haftmann@27551: lemma Rat_induct_pos [case_names Fract, induct type: rat]:
haftmann@27551:   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
haftmann@27551:   shows "P q"
paulson@14365: proof (cases q)
haftmann@27551:   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
paulson@14365:   proof -
paulson@14365:     fix a::int and b::int
paulson@14365:     assume b: "b < 0"
paulson@14365:     hence "0 < -b" by simp
paulson@14365:     hence "P (Fract (-a) (-b))" by (rule step)
paulson@14365:     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
paulson@14365:   qed
paulson@14365:   case (Fract a b)
paulson@14365:   thus "P q" by (force simp add: linorder_neq_iff step step')
paulson@14365: qed
paulson@14365: 
paulson@14365: lemma zero_less_Fract_iff:
haftmann@27652:   "0 < b ==> (0 < Fract a b) = (0 < a)"
haftmann@27652: by (simp add: Zero_rat_def order_less_imp_not_eq2 zero_less_mult_iff)
paulson@14365: 
paulson@14378: 
haftmann@27551: subsection {* Arithmetic setup *}
paulson@14387: 
haftmann@28952: use "Tools/rat_arith.ML"
wenzelm@24075: declaration {* K rat_arith_setup *}
paulson@14387: 
huffman@23342: 
huffman@23342: subsection {* Embedding from Rationals to other Fields *}
huffman@23342: 
haftmann@24198: class field_char_0 = field + ring_char_0
huffman@23342: 
haftmann@27551: subclass (in ordered_field) field_char_0 ..
huffman@23342: 
haftmann@27551: context field_char_0
haftmann@27551: begin
haftmann@27551: 
haftmann@27551: definition of_rat :: "rat \<Rightarrow> 'a" where
haftmann@28562:   [code del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
huffman@23342: 
haftmann@27551: end
haftmann@27551: 
huffman@23342: lemma of_rat_congruent:
haftmann@27551:   "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
huffman@23342: apply (rule congruent.intro)
huffman@23342: apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23342: apply (simp only: of_int_mult [symmetric])
huffman@23342: done
huffman@23342: 
haftmann@27551: lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
haftmann@27551:   unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
huffman@23342: 
huffman@23342: lemma of_rat_0 [simp]: "of_rat 0 = 0"
huffman@23342: by (simp add: Zero_rat_def of_rat_rat)
huffman@23342: 
huffman@23342: lemma of_rat_1 [simp]: "of_rat 1 = 1"
huffman@23342: by (simp add: One_rat_def of_rat_rat)
huffman@23342: 
huffman@23342: lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
haftmann@27652: by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
huffman@23342: 
huffman@23343: lemma of_rat_minus: "of_rat (- a) = - of_rat a"
haftmann@27652: by (induct a, simp add: of_rat_rat)
huffman@23343: 
huffman@23343: lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
huffman@23343: by (simp only: diff_minus of_rat_add of_rat_minus)
huffman@23343: 
huffman@23342: lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
haftmann@27652: apply (induct a, induct b, simp add: of_rat_rat)
huffman@23342: apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
huffman@23342: done
huffman@23342: 
huffman@23342: lemma nonzero_of_rat_inverse:
huffman@23342:   "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
huffman@23343: apply (rule inverse_unique [symmetric])
huffman@23343: apply (simp add: of_rat_mult [symmetric])
huffman@23342: done
huffman@23342: 
huffman@23342: lemma of_rat_inverse:
huffman@23342:   "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
huffman@23342:    inverse (of_rat a)"
huffman@23342: by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
huffman@23342: 
huffman@23342: lemma nonzero_of_rat_divide:
huffman@23342:   "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
huffman@23342: by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
huffman@23342: 
huffman@23342: lemma of_rat_divide:
huffman@23342:   "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
huffman@23342:    = of_rat a / of_rat b"
haftmann@27652: by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
huffman@23342: 
huffman@23343: lemma of_rat_power:
huffman@23343:   "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
huffman@23343: by (induct n) (simp_all add: of_rat_mult power_Suc)
huffman@23343: 
huffman@23343: lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
huffman@23343: apply (induct a, induct b)
huffman@23343: apply (simp add: of_rat_rat eq_rat)
huffman@23343: apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23343: apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
huffman@23343: done
huffman@23343: 
haftmann@27652: lemma of_rat_less:
haftmann@27652:   "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"
haftmann@27652: proof (induct r, induct s)
haftmann@27652:   fix a b c d :: int
haftmann@27652:   assume not_zero: "b > 0" "d > 0"
haftmann@27652:   then have "b * d > 0" by (rule mult_pos_pos)
haftmann@27652:   have of_int_divide_less_eq:
haftmann@27652:     "(of_int a :: 'a) / of_int b < of_int c / of_int d
haftmann@27652:       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
haftmann@27652:     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
haftmann@27652:   show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)
haftmann@27652:     \<longleftrightarrow> Fract a b < Fract c d"
haftmann@27652:     using not_zero `b * d > 0`
haftmann@27652:     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
haftmann@27652:       (auto intro: mult_strict_right_mono mult_right_less_imp_less)
haftmann@27652: qed
haftmann@27652: 
haftmann@27652: lemma of_rat_less_eq:
haftmann@27652:   "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
haftmann@27652:   unfolding le_less by (auto simp add: of_rat_less)
haftmann@27652: 
huffman@23343: lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
huffman@23343: 
haftmann@27652: lemma of_rat_eq_id [simp]: "of_rat = id"
huffman@23343: proof
huffman@23343:   fix a
huffman@23343:   show "of_rat a = id a"
huffman@23343:   by (induct a)
haftmann@27652:      (simp add: of_rat_rat Fract_of_int_eq [symmetric])
huffman@23343: qed
huffman@23343: 
huffman@23343: text{*Collapse nested embeddings*}
huffman@23343: lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
huffman@23343: by (induct n) (simp_all add: of_rat_add)
huffman@23343: 
huffman@23343: lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
haftmann@27652: by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
huffman@23343: 
huffman@23343: lemma of_rat_number_of_eq [simp]:
huffman@23343:   "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
huffman@23343: by (simp add: number_of_eq)
huffman@23343: 
haftmann@23879: lemmas zero_rat = Zero_rat_def
haftmann@23879: lemmas one_rat = One_rat_def
haftmann@23879: 
haftmann@24198: abbreviation
haftmann@24198:   rat_of_nat :: "nat \<Rightarrow> rat"
haftmann@24198: where
haftmann@24198:   "rat_of_nat \<equiv> of_nat"
haftmann@24198: 
haftmann@24198: abbreviation
haftmann@24198:   rat_of_int :: "int \<Rightarrow> rat"
haftmann@24198: where
haftmann@24198:   "rat_of_int \<equiv> of_int"
haftmann@24198: 
huffman@28010: subsection {* The Set of Rational Numbers *}
berghofe@24533: 
nipkow@28001: context field_char_0
nipkow@28001: begin
nipkow@28001: 
nipkow@28001: definition
nipkow@28001:   Rats  :: "'a set" where
haftmann@28562:   [code del]: "Rats = range of_rat"
nipkow@28001: 
nipkow@28001: notation (xsymbols)
nipkow@28001:   Rats  ("\<rat>")
nipkow@28001: 
nipkow@28001: end
nipkow@28001: 
huffman@28010: lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
huffman@28010: by (simp add: Rats_def)
huffman@28010: 
huffman@28010: lemma Rats_of_int [simp]: "of_int z \<in> Rats"
huffman@28010: by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
huffman@28010: 
huffman@28010: lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
huffman@28010: by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
huffman@28010: 
huffman@28010: lemma Rats_number_of [simp]:
huffman@28010:   "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
huffman@28010: by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
huffman@28010: 
huffman@28010: lemma Rats_0 [simp]: "0 \<in> Rats"
huffman@28010: apply (unfold Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (rule of_rat_0 [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma Rats_1 [simp]: "1 \<in> Rats"
huffman@28010: apply (unfold Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (rule of_rat_1 [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
huffman@28010: apply (auto simp add: Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (rule of_rat_add [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
huffman@28010: apply (auto simp add: Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (rule of_rat_minus [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
huffman@28010: apply (auto simp add: Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (rule of_rat_diff [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
huffman@28010: apply (auto simp add: Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (rule of_rat_mult [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma nonzero_Rats_inverse:
huffman@28010:   fixes a :: "'a::field_char_0"
huffman@28010:   shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
huffman@28010: apply (auto simp add: Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (erule nonzero_of_rat_inverse [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma Rats_inverse [simp]:
huffman@28010:   fixes a :: "'a::{field_char_0,division_by_zero}"
huffman@28010:   shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
huffman@28010: apply (auto simp add: Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (rule of_rat_inverse [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma nonzero_Rats_divide:
huffman@28010:   fixes a b :: "'a::field_char_0"
huffman@28010:   shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010: apply (auto simp add: Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (erule nonzero_of_rat_divide [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma Rats_divide [simp]:
huffman@28010:   fixes a b :: "'a::{field_char_0,division_by_zero}"
huffman@28010:   shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010: apply (auto simp add: Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (rule of_rat_divide [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma Rats_power [simp]:
huffman@28010:   fixes a :: "'a::{field_char_0,recpower}"
huffman@28010:   shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
huffman@28010: apply (auto simp add: Rats_def)
huffman@28010: apply (rule range_eqI)
huffman@28010: apply (rule of_rat_power [symmetric])
huffman@28010: done
huffman@28010: 
huffman@28010: lemma Rats_cases [cases set: Rats]:
huffman@28010:   assumes "q \<in> \<rat>"
huffman@28010:   obtains (of_rat) r where "q = of_rat r"
huffman@28010:   unfolding Rats_def
huffman@28010: proof -
huffman@28010:   from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
huffman@28010:   then obtain r where "q = of_rat r" ..
huffman@28010:   then show thesis ..
huffman@28010: qed
huffman@28010: 
huffman@28010: lemma Rats_induct [case_names of_rat, induct set: Rats]:
huffman@28010:   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
huffman@28010:   by (rule Rats_cases) auto
huffman@28010: 
nipkow@28001: 
berghofe@24533: subsection {* Implementation of rational numbers as pairs of integers *}
berghofe@24533: 
haftmann@27652: lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"
haftmann@27652: proof (cases "a = 0 \<or> b = 0")
haftmann@27652:   case True then show ?thesis by (auto simp add: eq_rat)
haftmann@27652: next
haftmann@27652:   let ?c = "zgcd a b"
haftmann@27652:   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@27652:   then have "?c \<noteq> 0" by simp
haftmann@27652:   then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
haftmann@27652:   moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
nipkow@29925:     by (simp add: semiring_div_class.mod_div_equality)
haftmann@27652:   moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
haftmann@27652:   moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
haftmann@27652:   ultimately show ?thesis
haftmann@27652:     by (simp add: mult_rat [symmetric])
haftmann@27652: qed
berghofe@24533: 
haftmann@27652: definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
haftmann@28562:   [simp, code del]: "Fract_norm a b = Fract a b"
haftmann@27652: 
haftmann@29332: lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in
haftmann@27652:   if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
haftmann@27652:   by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
berghofe@24533: 
berghofe@24533: lemma [code]:
haftmann@27652:   "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
haftmann@27652:   by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
berghofe@24533: 
haftmann@26513: instantiation rat :: eq
haftmann@26513: begin
haftmann@26513: 
haftmann@28562: definition [code del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
berghofe@24533: 
haftmann@26513: instance by default (simp add: eq_rat_def)
haftmann@26513: 
haftmann@27652: lemma rat_eq_code [code]:
haftmann@27652:   "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
haftmann@27652:        then c = 0 \<or> d = 0
haftmann@27652:      else if d = 0
haftmann@27652:        then a = 0 \<or> b = 0
haftmann@29332:      else a * d = b * c)"
haftmann@27652:   by (auto simp add: eq eq_rat)
haftmann@26513: 
haftmann@28351: lemma rat_eq_refl [code nbe]:
haftmann@28351:   "eq_class.eq (r::rat) r \<longleftrightarrow> True"
haftmann@28351:   by (rule HOL.eq_refl)
haftmann@28351: 
haftmann@26513: end
berghofe@24533: 
haftmann@27652: lemma le_rat':
haftmann@27652:   assumes "b \<noteq> 0"
haftmann@27652:     and "d \<noteq> 0"
haftmann@27652:   shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
berghofe@24533: proof -
haftmann@27652:   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
haftmann@27652:   have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"
haftmann@27652:   proof (cases "b * d > 0")
haftmann@27652:     case True
haftmann@27652:     moreover from True have "sgn b * sgn d = 1"
haftmann@27652:       by (simp add: sgn_times [symmetric] sgn_1_pos)
haftmann@27652:     ultimately show ?thesis by (simp add: mult_le_cancel_right)
haftmann@27652:   next
haftmann@27652:     case False with assms have "b * d < 0" by (simp add: less_le)
haftmann@27652:     moreover from this have "sgn b * sgn d = - 1"
haftmann@27652:       by (simp only: sgn_times [symmetric] sgn_1_neg)
haftmann@27652:     ultimately show ?thesis by (simp add: mult_le_cancel_right)
haftmann@27652:   qed
haftmann@27652:   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
haftmann@27652:     by (simp add: abs_sgn mult_ac)
haftmann@27652:   finally show ?thesis using assms by simp
berghofe@24533: qed
berghofe@24533: 
haftmann@27652: lemma less_rat': 
haftmann@27652:   assumes "b \<noteq> 0"
haftmann@27652:     and "d \<noteq> 0"
haftmann@27652:   shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
berghofe@24533: proof -
haftmann@27652:   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
haftmann@27652:   have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
haftmann@27652:   proof (cases "b * d > 0")
haftmann@27652:     case True
haftmann@27652:     moreover from True have "sgn b * sgn d = 1"
haftmann@27652:       by (simp add: sgn_times [symmetric] sgn_1_pos)
haftmann@27652:     ultimately show ?thesis by (simp add: mult_less_cancel_right)
haftmann@27652:   next
haftmann@27652:     case False with assms have "b * d < 0" by (simp add: less_le)
haftmann@27652:     moreover from this have "sgn b * sgn d = - 1"
haftmann@27652:       by (simp only: sgn_times [symmetric] sgn_1_neg)
haftmann@27652:     ultimately show ?thesis by (simp add: mult_less_cancel_right)
haftmann@27652:   qed
haftmann@27652:   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
haftmann@27652:     by (simp add: abs_sgn mult_ac)
haftmann@27652:   finally show ?thesis using assms by simp
berghofe@24533: qed
berghofe@24533: 
haftmann@29940: lemma (in ordered_idom) sgn_greater [simp]:
haftmann@29940:   "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940:   unfolding sgn_if by auto
haftmann@29940: 
haftmann@29940: lemma (in ordered_idom) sgn_less [simp]:
haftmann@29940:   "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940:   unfolding sgn_if by auto
berghofe@24533: 
haftmann@27652: lemma rat_le_eq_code [code]:
haftmann@27652:   "Fract a b < Fract c d \<longleftrightarrow> (if b = 0
haftmann@27652:        then sgn c * sgn d > 0
haftmann@27652:      else if d = 0
haftmann@27652:        then sgn a * sgn b < 0
haftmann@27652:      else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
haftmann@29940:   by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
haftmann@29940: 
haftmann@29940: lemma rat_less_eq_code [code]:
haftmann@29940:   "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
haftmann@29940:        then sgn c * sgn d \<ge> 0
haftmann@29940:      else if d = 0
haftmann@29940:        then sgn a * sgn b \<le> 0
haftmann@29940:      else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
haftmann@29940:   by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
haftmann@29940:     (auto simp add: le_less not_less sgn_0_0)
haftmann@29940: 
berghofe@24533: 
haftmann@27652: lemma rat_plus_code [code]:
haftmann@27652:   "Fract a b + Fract c d = (if b = 0
haftmann@27652:      then Fract c d
haftmann@27652:    else if d = 0
haftmann@27652:      then Fract a b
haftmann@27652:    else Fract_norm (a * d + c * b) (b * d))"
haftmann@27652:   by (simp add: eq_rat, simp add: Zero_rat_def)
haftmann@27652: 
haftmann@27652: lemma rat_times_code [code]:
haftmann@27652:   "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
haftmann@27652:   by simp
berghofe@24533: 
haftmann@27652: lemma rat_minus_code [code]:
haftmann@27652:   "Fract a b - Fract c d = (if b = 0
haftmann@27652:      then Fract (- c) d
haftmann@27652:    else if d = 0
haftmann@27652:      then Fract a b
haftmann@27652:    else Fract_norm (a * d - c * b) (b * d))"
haftmann@27652:   by (simp add: eq_rat, simp add: Zero_rat_def)
berghofe@24533: 
haftmann@27652: lemma rat_inverse_code [code]:
haftmann@27652:   "inverse (Fract a b) = (if b = 0 then Fract 1 0
haftmann@27652:     else if a < 0 then Fract (- b) (- a)
haftmann@27652:     else Fract b a)"
haftmann@27652:   by (simp add: eq_rat)
haftmann@27652: 
haftmann@27652: lemma rat_divide_code [code]:
haftmann@27652:   "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
haftmann@27652:   by simp
haftmann@27652: 
haftmann@27652: hide (open) const Fract_norm
berghofe@24533: 
haftmann@24622: text {* Setup for SML code generator *}
berghofe@24533: 
berghofe@24533: types_code
berghofe@24533:   rat ("(int */ int)")
berghofe@24533: attach (term_of) {*
berghofe@24533: fun term_of_rat (p, q) =
haftmann@24622:   let
haftmann@24661:     val rT = Type ("Rational.rat", [])
berghofe@24533:   in
berghofe@24533:     if q = 1 orelse p = 0 then HOLogic.mk_number rT p
berghofe@25885:     else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
berghofe@24533:       HOLogic.mk_number rT p $ HOLogic.mk_number rT q
berghofe@24533:   end;
berghofe@24533: *}
berghofe@24533: attach (test) {*
berghofe@24533: fun gen_rat i =
berghofe@24533:   let
berghofe@24533:     val p = random_range 0 i;
berghofe@24533:     val q = random_range 1 (i + 1);
berghofe@24533:     val g = Integer.gcd p q;
wenzelm@24630:     val p' = p div g;
wenzelm@24630:     val q' = q div g;
berghofe@25885:     val r = (if one_of [true, false] then p' else ~ p',
berghofe@25885:       if p' = 0 then 0 else q')
berghofe@24533:   in
berghofe@25885:     (r, fn () => term_of_rat r)
berghofe@24533:   end;
berghofe@24533: *}
berghofe@24533: 
berghofe@24533: consts_code
haftmann@27551:   Fract ("(_,/ _)")
berghofe@24533: 
berghofe@24533: consts_code
berghofe@24533:   "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
berghofe@24533: attach {*
berghofe@24533: fun rat_of_int 0 = (0, 0)
berghofe@24533:   | rat_of_int i = (i, 1);
berghofe@24533: *}
berghofe@24533: 
huffman@29880: end