haftmann@35372: (*  Title:  HOL/Rat.thy
paulson@14365:     Author: Markus Wenzel, TU Muenchen
paulson@14365: *)
paulson@14365: 
wenzelm@14691: header {* Rational numbers *}
paulson@14365: 
haftmann@35372: theory Rat
huffman@30097: imports GCD Archimedean_Field
huffman@35343: uses ("Tools/float_syntax.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: 
nipkow@30198: lemma refl_on_ratrel: "refl_on {x. snd x \<noteq> 0} ratrel"
nipkow@30198:   by (auto simp add: refl_on_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@40815:   by (rule equivI [OF refl_on_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: 
wenzelm@45694: definition "Rat = {x. snd x \<noteq> 0} // ratrel"
wenzelm@45694: 
wenzelm@45694: typedef (open) rat = Rat
wenzelm@45694:   morphisms Rep_Rat Abs_Rat
wenzelm@45694:   unfolding Rat_def
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@35369:   "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
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@35369: lemma Rat_cases [case_names Fract, cases type: rat]:
haftmann@35369:   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
haftmann@35369:   shows C
haftmann@35369: proof -
haftmann@35369:   obtain a b :: int where "q = Fract a b" and "b \<noteq> 0"
haftmann@35369:     by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
haftmann@35369:   let ?a = "a div gcd a b"
haftmann@35369:   let ?b = "b div gcd a b"
haftmann@35369:   from `b \<noteq> 0` have "?b * gcd a b = b"
haftmann@35369:     by (simp add: dvd_div_mult_self)
haftmann@35369:   with `b \<noteq> 0` have "?b \<noteq> 0" by auto
haftmann@35369:   from `q = Fract a b` `b \<noteq> 0` `?b \<noteq> 0` have q: "q = Fract ?a ?b"
haftmann@35369:     by (simp add: eq_rat dvd_div_mult mult_commute [of a])
haftmann@35369:   from `b \<noteq> 0` have coprime: "coprime ?a ?b"
haftmann@35369:     by (auto intro: div_gcd_coprime_int)
haftmann@35369:   show C proof (cases "b > 0")
haftmann@35369:     case True
haftmann@35369:     note assms
haftmann@35369:     moreover note q
haftmann@35369:     moreover from True have "?b > 0" by (simp add: nonneg1_imp_zdiv_pos_iff)
haftmann@35369:     moreover note coprime
haftmann@35369:     ultimately show C .
haftmann@35369:   next
haftmann@35369:     case False
haftmann@35369:     note assms
haftmann@35369:     moreover from q have "q = Fract (- ?a) (- ?b)" by (simp add: Fract_def)
haftmann@35369:     moreover from False `b \<noteq> 0` have "- ?b > 0" by (simp add: pos_imp_zdiv_neg_iff)
haftmann@35369:     moreover from coprime have "coprime (- ?a) (- ?b)" by simp
haftmann@35369:     ultimately show C .
haftmann@35369:   qed
haftmann@35369: qed
haftmann@35369: 
haftmann@35369: lemma Rat_induct [case_names Fract, induct type: rat]:
haftmann@35369:   assumes "\<And>a b. b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> P (Fract a b)"
haftmann@35369:   shows "P q"
haftmann@35369:   using assms by (cases q) simp
haftmann@35369: 
haftmann@31017: instantiation rat :: comm_ring_1
haftmann@25571: begin
haftmann@25571: 
haftmann@25571: definition
haftmann@35369:   Zero_rat_def: "0 = Fract 0 1"
paulson@14365: 
haftmann@25571: definition
haftmann@35369:   One_rat_def: "1 = Fract 1 1"
huffman@18913: 
haftmann@25571: definition
haftmann@35369:   add_rat_def:
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@35369:   minus_rat_def:
haftmann@27551:   "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
haftmann@27551: 
haftmann@35369: lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
haftmann@27551: proof -
haftmann@27551:   have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
haftmann@40819:     by (simp add: congruent_def split_paired_all)
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@35369:   diff_rat_def: "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@35369:   mult_rat_def:
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: 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)
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@35369: lemma rat_number_collapse:
haftmann@27551:   "Fract 0 k = 0"
haftmann@27551:   "Fract 1 1 = 1"
huffman@47108:   "Fract (numeral w) 1 = numeral w"
huffman@47108:   "Fract (neg_numeral w) 1 = neg_numeral w"
haftmann@27551:   "Fract k 0 = 0"
huffman@47108:   using Fract_of_int_eq [of "numeral w"]
huffman@47108:   using Fract_of_int_eq [of "neg_numeral w"]
huffman@47108:   by (simp_all add: Zero_rat_def One_rat_def eq_rat)
haftmann@27551: 
huffman@47108: lemma rat_number_expand:
haftmann@27551:   "0 = Fract 0 1"
haftmann@27551:   "1 = Fract 1 1"
huffman@47108:   "numeral k = Fract (numeral k) 1"
huffman@47108:   "neg_numeral k = Fract (neg_numeral k) 1"
haftmann@27551:   by (simp_all add: rat_number_collapse)
haftmann@27551: 
haftmann@27551: lemma Rat_cases_nonzero [case_names Fract 0]:
haftmann@35369:   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> coprime a b \<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@35369:   then obtain a b where "q = Fract a b" and "b > 0" and "coprime a b" by (cases q) auto
haftmann@27551:   moreover with False have "0 \<noteq> Fract a b" by simp
haftmann@35369:   with `b > 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
haftmann@35369:   with Fract `q = Fract a b` `b > 0` `coprime a b` show C by blast
haftmann@27551: qed
haftmann@27551: 
nipkow@33805: subsubsection {* Function @{text normalize} *}
nipkow@33805: 
haftmann@35369: lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
haftmann@35369: proof (cases "b = 0")
haftmann@35369:   case True then show ?thesis by (simp add: eq_rat)
haftmann@35369: next
haftmann@35369:   case False
haftmann@35369:   moreover have "b div gcd a b * gcd a b = b"
haftmann@35369:     by (rule dvd_div_mult_self) simp
haftmann@35369:   ultimately have "b div gcd a b \<noteq> 0" by auto
haftmann@35369:   with False show ?thesis by (simp add: eq_rat dvd_div_mult mult_commute [of a])
haftmann@35369: qed
nipkow@33805: 
haftmann@35369: definition normalize :: "int \<times> int \<Rightarrow> int \<times> int" where
haftmann@35369:   "normalize p = (if snd p > 0 then (let a = gcd (fst p) (snd p) in (fst p div a, snd p div a))
haftmann@35369:     else if snd p = 0 then (0, 1)
haftmann@35369:     else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"
haftmann@35369: 
haftmann@35369: lemma normalize_crossproduct:
haftmann@35369:   assumes "q \<noteq> 0" "s \<noteq> 0"
haftmann@35369:   assumes "normalize (p, q) = normalize (r, s)"
haftmann@35369:   shows "p * s = r * q"
haftmann@35369: proof -
haftmann@35369:   have aux: "p * gcd r s = sgn (q * s) * r * gcd p q \<Longrightarrow> q * gcd r s = sgn (q * s) * s * gcd p q \<Longrightarrow> p * s = q * r"
haftmann@35369:   proof -
haftmann@35369:     assume "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
haftmann@35369:     then have "(p * gcd r s) * (sgn (q * s) * s * gcd p q) = (q * gcd r s) * (sgn (q * s) * r * gcd p q)" by simp
haftmann@35369:     with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
haftmann@35369:   qed
haftmann@35369:   from assms show ?thesis
haftmann@35369:     by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
nipkow@33805: qed
nipkow@33805: 
haftmann@35369: lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369:   by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
haftmann@35369:     split:split_if_asm)
haftmann@35369: 
haftmann@35369: lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369:   by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
haftmann@35369:     split:split_if_asm)
haftmann@35369: 
haftmann@35369: lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369:   by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
haftmann@35369:     split:split_if_asm)
haftmann@35369: 
haftmann@35369: lemma normalize_stable [simp]:
haftmann@35369:   "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
haftmann@35369:   by (simp add: normalize_def)
haftmann@35369: 
haftmann@35369: lemma normalize_denom_zero [simp]:
haftmann@35369:   "normalize (p, 0) = (0, 1)"
haftmann@35369:   by (simp add: normalize_def)
haftmann@35369: 
haftmann@35369: lemma normalize_negative [simp]:
haftmann@35369:   "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
haftmann@35369:   by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
haftmann@35369: 
haftmann@35369: text{*
haftmann@35369:   Decompose a fraction into normalized, i.e. coprime numerator and denominator:
haftmann@35369: *}
haftmann@35369: 
haftmann@35369: definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
haftmann@35369:   "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
haftmann@35369:                    snd pair > 0 & coprime (fst pair) (snd pair))"
haftmann@35369: 
haftmann@35369: lemma quotient_of_unique:
haftmann@35369:   "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369: proof (cases r)
haftmann@35369:   case (Fract a b)
haftmann@35369:   then have "r = Fract (fst (a, b)) (snd (a, b)) \<and> snd (a, b) > 0 \<and> coprime (fst (a, b)) (snd (a, b))" by auto
haftmann@35369:   then show ?thesis proof (rule ex1I)
haftmann@35369:     fix p
haftmann@35369:     obtain c d :: int where p: "p = (c, d)" by (cases p)
haftmann@35369:     assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369:     with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
haftmann@35369:     have "c = a \<and> d = b"
haftmann@35369:     proof (cases "a = 0")
haftmann@35369:       case True with Fract Fract' show ?thesis by (simp add: eq_rat)
haftmann@35369:     next
haftmann@35369:       case False
haftmann@35369:       with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
haftmann@35369:       then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
haftmann@35369:       with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
haftmann@35369:       with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
haftmann@35369:       from `coprime a b` `coprime c d` have "\<bar>a\<bar> * \<bar>d\<bar> = \<bar>c\<bar> * \<bar>b\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> \<bar>d\<bar> = \<bar>b\<bar>"
haftmann@35369:         by (simp add: coprime_crossproduct_int)
haftmann@35369:       with `b > 0` `d > 0` have "\<bar>a\<bar> * d = \<bar>c\<bar> * b \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> d = b" by simp
haftmann@35369:       then have "a * sgn a * d = c * sgn c * b \<longleftrightarrow> a * sgn a = c * sgn c \<and> d = b" by (simp add: abs_sgn)
haftmann@35369:       with sgn * show ?thesis by (auto simp add: sgn_0_0)
nipkow@33805:     qed
haftmann@35369:     with p show "p = (a, b)" by simp
nipkow@33805:   qed
nipkow@33805: qed
nipkow@33805: 
haftmann@35369: lemma quotient_of_Fract [code]:
haftmann@35369:   "quotient_of (Fract a b) = normalize (a, b)"
haftmann@35369: proof -
haftmann@35369:   have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
haftmann@35369:     by (rule sym) (auto intro: normalize_eq)
haftmann@35369:   moreover have "0 < snd (normalize (a, b))" (is ?denom_pos) 
haftmann@35369:     by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
haftmann@35369:   moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
haftmann@35369:     by (rule normalize_coprime) simp
haftmann@35369:   ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
haftmann@35369:   with quotient_of_unique have
haftmann@35369:     "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)) = normalize (a, b)"
haftmann@35369:     by (rule the1_equality)
haftmann@35369:   then show ?thesis by (simp add: quotient_of_def)
haftmann@35369: qed
haftmann@35369: 
haftmann@35369: lemma quotient_of_number [simp]:
haftmann@35369:   "quotient_of 0 = (0, 1)"
haftmann@35369:   "quotient_of 1 = (1, 1)"
huffman@47108:   "quotient_of (numeral k) = (numeral k, 1)"
huffman@47108:   "quotient_of (neg_numeral k) = (neg_numeral k, 1)"
haftmann@35369:   by (simp_all add: rat_number_expand quotient_of_Fract)
nipkow@33805: 
haftmann@35369: lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369:   by (simp add: quotient_of_Fract normalize_eq)
haftmann@35369: 
haftmann@35369: lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369:   by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
haftmann@35369: 
haftmann@35369: lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369:   by (cases r) (simp add: quotient_of_Fract normalize_coprime)
nipkow@33805: 
haftmann@35369: lemma quotient_of_inject:
haftmann@35369:   assumes "quotient_of a = quotient_of b"
haftmann@35369:   shows "a = b"
haftmann@35369: proof -
haftmann@35369:   obtain p q r s where a: "a = Fract p q"
haftmann@35369:     and b: "b = Fract r s"
haftmann@35369:     and "q > 0" and "s > 0" by (cases a, cases b)
haftmann@35369:   with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
haftmann@35369: qed
haftmann@35369: 
haftmann@35369: lemma quotient_of_inject_eq:
haftmann@35369:   "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
haftmann@35369:   by (auto simp add: quotient_of_inject)
nipkow@33805: 
haftmann@27551: 
haftmann@27551: subsubsection {* The field of rational numbers *}
haftmann@27551: 
haftmann@36409: instantiation rat :: field_inverse_zero
haftmann@27551: begin
haftmann@27551: 
haftmann@27551: definition
haftmann@35369:   inverse_rat_def:
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@35369:   divide_rat_def: "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@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)
huffman@35216:    (simp_all add: 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@36415: next
haftmann@36415:   show "inverse 0 = (0::rat)" by (simp add: rat_number_expand, simp add: rat_number_collapse)
haftmann@36415: qed
haftmann@27551: 
haftmann@27551: end
haftmann@27551: 
haftmann@27551: 
haftmann@27551: subsubsection {* Various *}
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: 
huffman@47108: lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
huffman@47108:   by (simp add: rat_number_expand)
haftmann@27551: 
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@35369:   le_rat_def:
haftmann@39910:    "q \<le> r \<longleftrightarrow> the_elem (\<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@35369:   less_rat_def: "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"
haftmann@35369:     then show "q \<le> s" 
haftmann@35369:     proof (induct q, induct r, induct s)
paulson@14365:       fix a b c d e f :: int
haftmann@35369:       assume neq: "b > 0"  "d > 0"  "f > 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"
haftmann@35369:     then show "q = r"
haftmann@35369:     proof (induct q, induct r)
paulson@14365:       fix a b c d :: int
haftmann@35369:       assume neq: "b > 0"  "d > 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@35369:   abs_rat_def: "\<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>"
huffman@35216:   by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
haftmann@27551: 
haftmann@27551: definition
haftmann@35369:   sgn_rat_def: "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@36409: instance rat :: linordered_field_inverse_zero
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
haftmann@35369:     assume neq: "b > 0"  "d > 0"  "f > 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
haftmann@35369:     assume neq: "b > 0"  "d > 0"  "f > 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:
huffman@30095:   "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
huffman@30095:   by (simp add: Zero_rat_def zero_less_mult_iff)
huffman@30095: 
huffman@30095: lemma Fract_less_zero_iff:
huffman@30095:   "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
huffman@30095:   by (simp add: Zero_rat_def mult_less_0_iff)
huffman@30095: 
huffman@30095: lemma zero_le_Fract_iff:
huffman@30095:   "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
huffman@30095:   by (simp add: Zero_rat_def zero_le_mult_iff)
huffman@30095: 
huffman@30095: lemma Fract_le_zero_iff:
huffman@30095:   "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
huffman@30095:   by (simp add: Zero_rat_def mult_le_0_iff)
huffman@30095: 
huffman@30095: lemma one_less_Fract_iff:
huffman@30095:   "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
huffman@30095:   by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095: 
huffman@30095: lemma Fract_less_one_iff:
huffman@30095:   "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
huffman@30095:   by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095: 
huffman@30095: lemma one_le_Fract_iff:
huffman@30095:   "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
huffman@30095:   by (simp add: One_rat_def mult_le_cancel_right)
huffman@30095: 
huffman@30095: lemma Fract_le_one_iff:
huffman@30095:   "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
huffman@30095:   by (simp add: One_rat_def mult_le_cancel_right)
paulson@14365: 
paulson@14378: 
huffman@30097: subsubsection {* Rationals are an Archimedean field *}
huffman@30097: 
huffman@30097: lemma rat_floor_lemma:
huffman@30097:   shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
huffman@30097: proof -
huffman@30097:   have "Fract a b = of_int (a div b) + Fract (a mod b) b"
huffman@35293:     by (cases "b = 0", simp, simp add: of_int_rat)
huffman@30097:   moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
huffman@35293:     unfolding Fract_of_int_quotient
haftmann@36409:     by (rule linorder_cases [of b 0]) (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
huffman@30097:   ultimately show ?thesis by simp
huffman@30097: qed
huffman@30097: 
huffman@30097: instance rat :: archimedean_field
huffman@30097: proof
huffman@30097:   fix r :: rat
huffman@30097:   show "\<exists>z. r \<le> of_int z"
huffman@30097:   proof (induct r)
huffman@30097:     case (Fract a b)
huffman@35293:     have "Fract a b \<le> of_int (a div b + 1)"
huffman@35293:       using rat_floor_lemma [of a b] by simp
huffman@30097:     then show "\<exists>z. Fract a b \<le> of_int z" ..
huffman@30097:   qed
huffman@30097: qed
huffman@30097: 
bulwahn@43732: instantiation rat :: floor_ceiling
bulwahn@43732: begin
bulwahn@43732: 
bulwahn@43732: definition [code del]:
bulwahn@43732:   "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
bulwahn@43732: 
bulwahn@43732: instance proof
bulwahn@43732:   fix x :: rat
bulwahn@43732:   show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
bulwahn@43732:     unfolding floor_rat_def using floor_exists1 by (rule theI')
bulwahn@43732: qed
bulwahn@43732: 
bulwahn@43732: end
bulwahn@43732: 
huffman@35293: lemma floor_Fract: "floor (Fract a b) = a div b"
huffman@35293:   using rat_floor_lemma [of a b]
huffman@30097:   by (simp add: floor_unique)
huffman@30097: 
huffman@30097: 
haftmann@31100: subsection {* Linear arithmetic setup *}
paulson@14387: 
haftmann@31100: declaration {*
haftmann@31100:   K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
haftmann@31100:     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
haftmann@31100:   #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
haftmann@31100:     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
haftmann@31100:   #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
haftmann@31100:       @{thm True_implies_equals},
huffman@47108:       read_instantiate @{context} [(("a", 0), "(numeral ?v)")] @{thm right_distrib},
huffman@47108:       read_instantiate @{context} [(("a", 0), "(neg_numeral ?v)")] @{thm right_distrib},
haftmann@31100:       @{thm divide_1}, @{thm divide_zero_left},
haftmann@31100:       @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
haftmann@31100:       @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
haftmann@31100:       @{thm of_int_minus}, @{thm of_int_diff},
haftmann@31100:       @{thm of_int_of_nat_eq}]
haftmann@31100:   #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
haftmann@31100:   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
haftmann@31100:   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
haftmann@31100: *}
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@35028: subclass (in linordered_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@39910:   "of_rat q = the_elem (\<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"
haftmann@40816: apply (rule congruentI)
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:
haftmann@36409:   "(of_rat (inverse a)::'a::{field_char_0, field_inverse_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:
haftmann@36409:   "(of_rat (a / b)::'a::{field_char_0, field_inverse_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:
haftmann@31017:   "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
huffman@30273: by (induct n) (simp_all add: of_rat_mult)
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@35028:   "(of_rat r :: 'a::linordered_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@35028:   show "(of_rat (Fract a b) :: 'a::linordered_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: qed
haftmann@27652: 
haftmann@27652: lemma of_rat_less_eq:
haftmann@35028:   "(of_rat r :: 'a::linordered_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@47108: lemma of_rat_numeral_eq [simp]:
huffman@47108:   "of_rat (numeral w) = numeral w"
huffman@47108: using of_rat_of_int_eq [of "numeral w"] by simp
huffman@47108: 
huffman@47108: lemma of_rat_neg_numeral_eq [simp]:
huffman@47108:   "of_rat (neg_numeral w) = neg_numeral w"
huffman@47108: using of_rat_of_int_eq [of "neg_numeral w"] by simp
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@35369:   "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@47108: lemma Rats_number_of [simp]: "numeral w \<in> Rats"
huffman@47108: by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
huffman@47108: 
huffman@47108: lemma Rats_neg_number_of [simp]: "neg_numeral w \<in> Rats"
huffman@47108: by (subst of_rat_neg_numeral_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]:
haftmann@36409:   fixes a :: "'a::{field_char_0, field_inverse_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]:
haftmann@36409:   fixes a b :: "'a::{field_char_0, field_inverse_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]:
haftmann@31017:   fixes a :: "'a::field_char_0"
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: 
huffman@47108: text {* Formal constructor *}
huffman@47108: 
haftmann@35369: definition Frct :: "int \<times> int \<Rightarrow> rat" where
haftmann@35369:   [simp]: "Frct p = Fract (fst p) (snd p)"
haftmann@35369: 
haftmann@36112: lemma [code abstype]:
haftmann@36112:   "Frct (quotient_of q) = q"
haftmann@36112:   by (cases q) (auto intro: quotient_of_eq)
haftmann@35369: 
huffman@47108: 
huffman@47108: text {* Numerals *}
haftmann@35369: 
haftmann@35369: declare quotient_of_Fract [code abstract]
haftmann@35369: 
huffman@47108: definition of_int :: "int \<Rightarrow> rat"
huffman@47108: where
huffman@47108:   [code_abbrev]: "of_int = Int.of_int"
huffman@47108: hide_const (open) of_int
huffman@47108: 
huffman@47108: lemma quotient_of_int [code abstract]:
huffman@47108:   "quotient_of (Rat.of_int a) = (a, 1)"
huffman@47108:   by (simp add: of_int_def of_int_rat quotient_of_Fract)
huffman@47108: 
huffman@47108: lemma [code_unfold]:
huffman@47108:   "numeral k = Rat.of_int (numeral k)"
huffman@47108:   by (simp add: Rat.of_int_def)
huffman@47108: 
huffman@47108: lemma [code_unfold]:
huffman@47108:   "neg_numeral k = Rat.of_int (neg_numeral k)"
huffman@47108:   by (simp add: Rat.of_int_def)
huffman@47108: 
huffman@47108: lemma Frct_code_post [code_post]:
huffman@47108:   "Frct (0, a) = 0"
huffman@47108:   "Frct (a, 0) = 0"
huffman@47108:   "Frct (1, 1) = 1"
huffman@47108:   "Frct (numeral k, 1) = numeral k"
huffman@47108:   "Frct (neg_numeral k, 1) = neg_numeral k"
huffman@47108:   "Frct (1, numeral k) = 1 / numeral k"
huffman@47108:   "Frct (1, neg_numeral k) = 1 / neg_numeral k"
huffman@47108:   "Frct (numeral k, numeral l) = numeral k / numeral l"
huffman@47108:   "Frct (numeral k, neg_numeral l) = numeral k / neg_numeral l"
huffman@47108:   "Frct (neg_numeral k, numeral l) = neg_numeral k / numeral l"
huffman@47108:   "Frct (neg_numeral k, neg_numeral l) = neg_numeral k / neg_numeral l"
huffman@47108:   by (simp_all add: Fract_of_int_quotient)
huffman@47108: 
huffman@47108: 
huffman@47108: text {* Operations *}
huffman@47108: 
haftmann@35369: lemma rat_zero_code [code abstract]:
haftmann@35369:   "quotient_of 0 = (0, 1)"
haftmann@35369:   by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
haftmann@35369: 
haftmann@35369: lemma rat_one_code [code abstract]:
haftmann@35369:   "quotient_of 1 = (1, 1)"
haftmann@35369:   by (simp add: One_rat_def quotient_of_Fract normalize_def)
haftmann@35369: 
haftmann@35369: lemma rat_plus_code [code abstract]:
haftmann@35369:   "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369:      in normalize (a * d + b * c, c * d))"
haftmann@35369:   by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@27652: 
haftmann@35369: lemma rat_uminus_code [code abstract]:
haftmann@35369:   "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
haftmann@35369:   by (cases p) (simp add: quotient_of_Fract)
haftmann@35369: 
haftmann@35369: lemma rat_minus_code [code abstract]:
haftmann@35369:   "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369:      in normalize (a * d - b * c, c * d))"
haftmann@35369:   by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369: 
haftmann@35369: lemma rat_times_code [code abstract]:
haftmann@35369:   "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369:      in normalize (a * b, c * d))"
haftmann@35369:   by (cases p, cases q) (simp add: quotient_of_Fract)
berghofe@24533: 
haftmann@35369: lemma rat_inverse_code [code abstract]:
haftmann@35369:   "quotient_of (inverse p) = (let (a, b) = quotient_of p
haftmann@35369:     in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
haftmann@35369: proof (cases p)
haftmann@35369:   case (Fract a b) then show ?thesis
haftmann@35369:     by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
haftmann@35369: qed
haftmann@35369: 
haftmann@35369: lemma rat_divide_code [code abstract]:
haftmann@35369:   "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369:      in normalize (a * d, c * b))"
haftmann@35369:   by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369: 
haftmann@35369: lemma rat_abs_code [code abstract]:
haftmann@35369:   "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
haftmann@35369:   by (cases p) (simp add: quotient_of_Fract)
haftmann@35369: 
haftmann@35369: lemma rat_sgn_code [code abstract]:
haftmann@35369:   "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
haftmann@35369: proof (cases p)
haftmann@35369:   case (Fract a b) then show ?thesis
haftmann@35369:   by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
haftmann@35369: qed
berghofe@24533: 
bulwahn@43733: lemma rat_floor_code [code]:
bulwahn@43733:   "floor p = (let (a, b) = quotient_of p in a div b)"
bulwahn@43733: by (cases p) (simp add: quotient_of_Fract floor_Fract)
bulwahn@43733: 
haftmann@38857: instantiation rat :: equal
haftmann@26513: begin
haftmann@26513: 
haftmann@35369: definition [code]:
haftmann@38857:   "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
haftmann@26513: 
haftmann@35369: instance proof
haftmann@38857: qed (simp add: equal_rat_def quotient_of_inject_eq)
haftmann@26513: 
haftmann@28351: lemma rat_eq_refl [code nbe]:
haftmann@38857:   "HOL.equal (r::rat) r \<longleftrightarrow> True"
haftmann@38857:   by (rule equal_refl)
haftmann@28351: 
haftmann@26513: end
berghofe@24533: 
haftmann@35369: lemma rat_less_eq_code [code]:
haftmann@35369:   "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
haftmann@35726:   by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533: 
haftmann@35369: lemma rat_less_code [code]:
haftmann@35369:   "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
haftmann@35726:   by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533: 
haftmann@35369: lemma [code]:
haftmann@35369:   "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
haftmann@35369:   by (cases p) (simp add: quotient_of_Fract of_rat_rat)
haftmann@27652: 
huffman@47108: 
huffman@47108: text {* Quickcheck *}
huffman@47108: 
haftmann@31203: definition (in term_syntax)
haftmann@32657:   valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
haftmann@32657:   [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
haftmann@31203: 
haftmann@37751: notation fcomp (infixl "\<circ>>" 60)
haftmann@37751: notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203: 
haftmann@31203: instantiation rat :: random
haftmann@31203: begin
haftmann@31203: 
haftmann@31203: definition
haftmann@37751:   "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
haftmann@31205:      let j = Code_Numeral.int_of (denom + 1)
haftmann@32657:      in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
haftmann@31203: 
haftmann@31203: instance ..
haftmann@31203: 
haftmann@31203: end
haftmann@31203: 
haftmann@37751: no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751: no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203: 
bulwahn@41920: instantiation rat :: exhaustive
bulwahn@41231: begin
bulwahn@41231: 
bulwahn@41231: definition
bulwahn@45818:   "exhaustive_rat f d = Quickcheck_Exhaustive.exhaustive (%l. Quickcheck_Exhaustive.exhaustive (%k. f (Fract k (Code_Numeral.int_of l + 1))) d) d"
bulwahn@42311: 
bulwahn@42311: instance ..
bulwahn@42311: 
bulwahn@42311: end
bulwahn@42311: 
bulwahn@42311: instantiation rat :: full_exhaustive
bulwahn@42311: begin
bulwahn@42311: 
bulwahn@42311: definition
bulwahn@45818:   "full_exhaustive_rat f d = Quickcheck_Exhaustive.full_exhaustive (%(l, _). Quickcheck_Exhaustive.full_exhaustive (%k.
bulwahn@45507:      f (let j = Code_Numeral.int_of l + 1
bulwahn@45507:         in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
bulwahn@41231: 
bulwahn@41231: instance ..
bulwahn@41231: 
bulwahn@41231: end
bulwahn@41231: 
bulwahn@43889: instantiation rat :: partial_term_of
bulwahn@43889: begin
bulwahn@43889: 
bulwahn@43889: instance ..
bulwahn@43889: 
bulwahn@43889: end
bulwahn@43889: 
bulwahn@43889: lemma [code]:
bulwahn@46758:   "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
bulwahn@46758:   "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ==
bulwahn@45507:      Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
bulwahn@45507:      (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
bulwahn@45507:         Typerep.Typerep (STR ''Rat.rat'') []])) (Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.Const (STR ''Product_Type.Pair'') (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []]]])) (partial_term_of (TYPE(int)) l)) (partial_term_of (TYPE(int)) k))"
bulwahn@43889: by (rule partial_term_of_anything)+
bulwahn@43889: 
bulwahn@43887: instantiation rat :: narrowing
bulwahn@43887: begin
bulwahn@43887: 
bulwahn@43887: definition
bulwahn@45507:   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
bulwahn@45507:     (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
bulwahn@43887: 
bulwahn@43887: instance ..
bulwahn@43887: 
bulwahn@43887: end
bulwahn@43887: 
bulwahn@43887: 
bulwahn@45183: subsection {* Setup for Nitpick *}
berghofe@24533: 
blanchet@38287: declaration {*
blanchet@38287:   Nitpick_HOL.register_frac_type @{type_name rat}
wenzelm@33209:    [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
wenzelm@33209:     (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
wenzelm@33209:     (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
wenzelm@33209:     (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
wenzelm@33209:     (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
wenzelm@33209:     (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
blanchet@37397:     (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
wenzelm@33209:     (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
blanchet@45478:     (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
blanchet@33197: *}
blanchet@33197: 
blanchet@41792: lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
huffman@47108:   one_rat_inst.one_rat ord_rat_inst.less_rat
blanchet@37397:   ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
blanchet@37397:   uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
blanchet@33197: 
huffman@35343: subsection{* Float syntax *}
huffman@35343: 
huffman@35343: syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
huffman@35343: 
huffman@35343: use "Tools/float_syntax.ML"
huffman@35343: setup Float_Syntax.setup
huffman@35343: 
huffman@35343: text{* Test: *}
huffman@35343: lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
huffman@35343: by simp
huffman@35343: 
wenzelm@37143: 
wenzelm@37143: hide_const (open) normalize
wenzelm@37143: 
huffman@29880: end