Theory Rat

```(*  Title:      HOL/Rat.thy
Author:     Markus Wenzel, TU Muenchen
*)

section ‹Rational numbers›

theory Rat
imports Archimedean_Field
begin

subsection ‹Rational numbers as quotient›

subsubsection ‹Construction of the type of rational numbers›

definition ratrel :: "(int × int) ⇒ (int × int) ⇒ bool"
where "ratrel = (λx y. snd x ≠ 0 ∧ snd y ≠ 0 ∧ fst x * snd y = fst y * snd x)"

lemma ratrel_iff [simp]: "ratrel x y ⟷ snd x ≠ 0 ∧ snd y ≠ 0 ∧ fst x * snd y = fst y * snd x"

lemma exists_ratrel_refl: "∃x. ratrel x x"
by (auto intro!: one_neq_zero)

lemma symp_ratrel: "symp ratrel"

lemma transp_ratrel: "transp ratrel"
proof (rule transpI, unfold split_paired_all)
fix a b a' b' a'' b'' :: int
assume *: "ratrel (a, b) (a', b')"
assume **: "ratrel (a', b') (a'', b'')"
have "b' * (a * b'') = b'' * (a * b')" by simp
also from * have "a * b' = a' * b" by auto
also have "b'' * (a' * b) = b * (a' * b'')" by simp
also from ** have "a' * b'' = a'' * b'" by auto
also have "b * (a'' * b') = b' * (a'' * b)" by simp
finally have "b' * (a * b'') = b' * (a'' * b)" .
moreover from ** have "b' ≠ 0" by auto
ultimately have "a * b'' = a'' * b" by simp
with * ** show "ratrel (a, b) (a'', b'')" by auto
qed

lemma part_equivp_ratrel: "part_equivp ratrel"
by (rule part_equivpI [OF exists_ratrel_refl symp_ratrel transp_ratrel])

quotient_type rat = "int × int" / partial: "ratrel"
morphisms Rep_Rat Abs_Rat
by (rule part_equivp_ratrel)

lemma Domainp_cr_rat [transfer_domain_rule]: "Domainp pcr_rat = (λx. snd x ≠ 0)"

subsubsection ‹Representation and basic operations›

lift_definition Fract :: "int ⇒ int ⇒ rat"
is "λa b. if b = 0 then (0, 1) else (a, b)"
by simp

lemma eq_rat:
"⋀a b c d. b ≠ 0 ⟹ d ≠ 0 ⟹ Fract a b = Fract c d ⟷ a * d = c * b"
"⋀a. Fract a 0 = Fract 0 1"
"⋀a c. Fract 0 a = Fract 0 c"
by (transfer, simp)+

lemma Rat_cases [case_names Fract, cases type: rat]:
assumes that: "⋀a b. q = Fract a b ⟹ b > 0 ⟹ coprime a b ⟹ C"
shows C
proof -
obtain a b :: int where q: "q = Fract a b" and b: "b ≠ 0"
by transfer simp
let ?a = "a div gcd a b"
let ?b = "b div gcd a b"
from b have "?b * gcd a b = b"
by simp
with b have "?b ≠ 0"
by fastforce
with q b have q2: "q = Fract ?a ?b"
by (simp add: eq_rat dvd_div_mult mult.commute [of a])
from b have coprime: "coprime ?a ?b"
by (auto intro: div_gcd_coprime)
show C
proof (cases "b > 0")
case True
then have "?b > 0"
from q2 this coprime show C by (rule that)
next
case False
have "q = Fract (- ?a) (- ?b)"
unfolding q2 by transfer simp
moreover from False b have "- ?b > 0"
moreover from coprime have "coprime (- ?a) (- ?b)"
by simp
ultimately show C
by (rule that)
qed
qed

lemma Rat_induct [case_names Fract, induct type: rat]:
assumes "⋀a b. b > 0 ⟹ coprime a b ⟹ P (Fract a b)"
shows "P q"
using assms by (cases q) simp

instantiation rat :: field
begin

lift_definition zero_rat :: "rat" is "(0, 1)"
by simp

lift_definition one_rat :: "rat" is "(1, 1)"
by simp

lemma Zero_rat_def: "0 = Fract 0 1"
by transfer simp

lemma One_rat_def: "1 = Fract 1 1"
by transfer simp

lift_definition plus_rat :: "rat ⇒ rat ⇒ rat"
is "λx y. (fst x * snd y + fst y * snd x, snd x * snd y)"
by (auto simp: distrib_right) (simp add: ac_simps)

assumes "b ≠ 0" and "d ≠ 0"
shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
using assms by transfer simp

lift_definition uminus_rat :: "rat ⇒ rat" is "λx. (- fst x, snd x)"
by simp

lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
by transfer simp

lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
by (cases "b = 0") (simp_all add: eq_rat)

definition diff_rat_def: "q - r = q + - r" for q r :: rat

lemma diff_rat [simp]:
"b ≠ 0 ⟹ d ≠ 0 ⟹ Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"

lift_definition times_rat :: "rat ⇒ rat ⇒ rat"
is "λx y. (fst x * fst y, snd x * snd y)"

lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
by transfer simp

lemma mult_rat_cancel: "c ≠ 0 ⟹ Fract (c * a) (c * b) = Fract a b"
by transfer simp

lift_definition inverse_rat :: "rat ⇒ rat"
is "λx. if fst x = 0 then (0, 1) else (snd x, fst x)"

lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
by transfer simp

definition divide_rat_def: "q div r = q * inverse r" for q r :: rat

lemma divide_rat [simp]: "Fract a b div Fract c d = Fract (a * d) (b * c)"

instance
proof
fix q r s :: rat
show "(q * r) * s = q * (r * s)"
by transfer simp
show "q * r = r * q"
by transfer simp
show "1 * q = q"
by transfer simp
show "(q + r) + s = q + (r + s)"
show "q + r = r + q"
by transfer simp
show "0 + q = q"
by transfer simp
show "- q + q = 0"
by transfer simp
show "q - r = q + - r"
by (fact diff_rat_def)
show "(q + r) * s = q * s + r * s"
show "(0::rat) ≠ 1"
by transfer simp
show "inverse q * q = 1" if "q ≠ 0"
using that by transfer simp
show "q div r = q * inverse r"
by (fact divide_rat_def)
show "inverse 0 = (0::rat)"
by transfer simp
qed

end

(* We cannot state these two rules earlier because of pending sort hypotheses *)
assumes "NO_MATCH (x :: 'b :: field) b" "(b :: 'a :: euclidean_semiring_cancel) ≠ 0"
shows "(b + a) div b = a div b + 1"

assumes "NO_MATCH (x :: 'b :: field) b" "(b :: 'a :: euclidean_semiring_cancel) ≠ 0"
shows "(a + b) div b = a div b + 1"

lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
by (induct k) (simp_all add: Zero_rat_def One_rat_def)

lemma of_int_rat: "of_int k = Fract k 1"
by (cases k rule: int_diff_cases) (simp add: of_nat_rat)

lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
by (rule of_nat_rat [symmetric])

lemma Fract_of_int_eq: "Fract k 1 = of_int k"
by (rule of_int_rat [symmetric])

lemma rat_number_collapse:
"Fract 0 k = 0"
"Fract 1 1 = 1"
"Fract (numeral w) 1 = numeral w"
"Fract (- numeral w) 1 = - numeral w"
"Fract (- 1) 1 = - 1"
"Fract k 0 = 0"
using Fract_of_int_eq [of "numeral w"]
and Fract_of_int_eq [of "- numeral w"]
by (simp_all add: Zero_rat_def One_rat_def eq_rat)

lemma rat_number_expand:
"0 = Fract 0 1"
"1 = Fract 1 1"
"numeral k = Fract (numeral k) 1"
"- 1 = Fract (- 1) 1"
"- numeral k = Fract (- numeral k) 1"

lemma Rat_cases_nonzero [case_names Fract 0]:
assumes Fract: "⋀a b. q = Fract a b ⟹ b > 0 ⟹ a ≠ 0 ⟹ coprime a b ⟹ C"
and 0: "q = 0 ⟹ C"
shows C
proof (cases "q = 0")
case True
then show C using 0 by auto
next
case False
then obtain a b where *: "q = Fract a b" "b > 0" "coprime a b"
by (cases q) auto
with False have "0 ≠ Fract a b"
by simp
with ‹b > 0› have "a ≠ 0"
with Fract * show C by blast
qed

subsubsection ‹Function ‹normalize››

lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
proof (cases "b = 0")
case True
then show ?thesis
next
case False
moreover have "b div gcd a b * gcd a b = b"
by (rule dvd_div_mult_self) simp
ultimately have "b div gcd a b * gcd a b ≠ 0"
by simp
then have "b div gcd a b ≠ 0"
by fastforce
with False show ?thesis
by (simp add: eq_rat dvd_div_mult mult.commute [of a])
qed

definition normalize :: "int × int ⇒ int × int"
where "normalize p =
(if snd p > 0 then (let a = gcd (fst p) (snd p) in (fst p div a, snd p div a))
else if snd p = 0 then (0, 1)
else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"

lemma normalize_crossproduct:
assumes "q ≠ 0" "s ≠ 0"
assumes "normalize (p, q) = normalize (r, s)"
shows "p * s = r * q"
proof -
have *: "p * s = q * r"
if "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
proof -
from that 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
with assms show ?thesis
by (auto simp add: ac_simps sgn_mult sgn_0_0)
qed
from assms show ?thesis
by (auto simp: normalize_def Let_def dvd_div_div_eq_mult mult.commute sgn_mult
split: if_splits intro: *)
qed

lemma normalize_eq: "normalize (a, b) = (p, q) ⟹ Fract p q = Fract a b"
by (auto simp: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
split: if_split_asm)

lemma normalize_denom_pos: "normalize r = (p, q) ⟹ q > 0"
by (auto simp: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
split: if_split_asm)

lemma normalize_coprime: "normalize r = (p, q) ⟹ coprime p q"
by (auto simp: normalize_def Let_def dvd_div_neg div_gcd_coprime split: if_split_asm)

lemma normalize_stable [simp]: "q > 0 ⟹ coprime p q ⟹ normalize (p, q) = (p, q)"

lemma normalize_denom_zero [simp]: "normalize (p, 0) = (0, 1)"

lemma normalize_negative [simp]: "q < 0 ⟹ normalize (p, q) = normalize (- p, - q)"
by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)

text‹
Decompose a fraction into normalized, i.e. coprime numerator and denominator:
›

definition quotient_of :: "rat ⇒ int × int"
where "quotient_of x =
(THE pair. x = Fract (fst pair) (snd pair) ∧ snd pair > 0 ∧ coprime (fst pair) (snd pair))"

lemma quotient_of_unique: "∃!p. r = Fract (fst p) (snd p) ∧ snd p > 0 ∧ coprime (fst p) (snd p)"
proof (cases r)
case (Fract a b)
then have "r = Fract (fst (a, b)) (snd (a, b)) ∧
snd (a, b) > 0 ∧ coprime (fst (a, b)) (snd (a, b))"
by auto
then show ?thesis
proof (rule ex1I)
fix p
assume r: "r = Fract (fst p) (snd p) ∧ snd p > 0 ∧ coprime (fst p) (snd p)"
obtain c d where p: "p = (c, d)" by (cases p)
with r have Fract': "r = Fract c d" "d > 0" "coprime c d"
by simp_all
have "(c, d) = (a, b)"
proof (cases "a = 0")
case True
with Fract Fract' show ?thesis
next
case False
with Fract Fract' have *: "c * b = a * d" and "c ≠ 0"
then have "c * b > 0 ⟷ a * d > 0"
by auto
with ‹b > 0› ‹d > 0› have "a > 0 ⟷ c > 0"
with ‹a ≠ 0› ‹c ≠ 0› have sgn: "sgn a = sgn c"
from ‹coprime a b› ‹coprime c d› have "¦a¦ * ¦d¦ = ¦c¦ * ¦b¦ ⟷ ¦a¦ = ¦c¦ ∧ ¦d¦ = ¦b¦"
with ‹b > 0› ‹d > 0› have "¦a¦ * d = ¦c¦ * b ⟷ ¦a¦ = ¦c¦ ∧ d = b"
by simp
then have "a * sgn a * d = c * sgn c * b ⟷ a * sgn a = c * sgn c ∧ d = b"
with sgn * show ?thesis
qed
with p show "p = (a, b)"
by simp
qed
qed

lemma quotient_of_Fract [code]: "quotient_of (Fract a b) = normalize (a, b)"
proof -
have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
by (rule sym) (auto intro: normalize_eq)
moreover have "0 < snd (normalize (a, b))" (is ?denom_pos)
by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
by (rule normalize_coprime) simp
ultimately have "?Fract ∧ ?denom_pos ∧ ?coprime" by blast
then have "(THE p. Fract a b = Fract (fst p) (snd p) ∧ 0 < snd p ∧
coprime (fst p) (snd p)) = normalize (a, b)"
by (rule the1_equality [OF quotient_of_unique])
then show ?thesis by (simp add: quotient_of_def)
qed

lemma quotient_of_number [simp]:
"quotient_of 0 = (0, 1)"
"quotient_of 1 = (1, 1)"
"quotient_of (numeral k) = (numeral k, 1)"
"quotient_of (- 1) = (- 1, 1)"
"quotient_of (- numeral k) = (- numeral k, 1)"

lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) ⟹ Fract p q = Fract a b"

lemma quotient_of_denom_pos: "quotient_of r = (p, q) ⟹ q > 0"
by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)

lemma quotient_of_denom_pos': "snd (quotient_of r) > 0"
using quotient_of_denom_pos [of r] by (simp add: prod_eq_iff)

lemma quotient_of_coprime: "quotient_of r = (p, q) ⟹ coprime p q"
by (cases r) (simp add: quotient_of_Fract normalize_coprime)

lemma quotient_of_inject:
assumes "quotient_of a = quotient_of b"
shows "a = b"
proof -
obtain p q r s where a: "a = Fract p q" and b: "b = Fract r s" and "q > 0" and "s > 0"
by (cases a, cases b)
with assms show ?thesis
by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
qed

lemma quotient_of_inject_eq: "quotient_of a = quotient_of b ⟷ a = b"

subsubsection ‹Various›

lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"

lemma Fract_add_one: "n ≠ 0 ⟹ Fract (m + n) n = Fract m n + 1"

lemma quotient_of_div:
assumes r: "quotient_of r = (n,d)"
shows "r = of_int n / of_int d"
proof -
from theI'[OF quotient_of_unique[of r], unfolded r[unfolded quotient_of_def]]
have "r = Fract n d" by simp
then show ?thesis using Fract_of_int_quotient
by simp
qed

subsubsection ‹The ordered field of rational numbers›

lift_definition positive :: "rat ⇒ bool"
is "λx. 0 < fst x * snd x"
proof clarsimp
fix a b c d :: int
assume "b ≠ 0" and "d ≠ 0" and "a * d = c * b"
then have "a * d * b * d = c * b * b * d"
by simp
then have "a * b * d⇧2 = c * d * b⇧2"
unfolding power2_eq_square by (simp add: ac_simps)
then have "0 < a * b * d⇧2 ⟷ 0 < c * d * b⇧2"
by simp
then show "0 < a * b ⟷ 0 < c * d"
using ‹b ≠ 0› and ‹d ≠ 0›
qed

lemma positive_zero: "¬ positive 0"
by transfer simp

lemma positive_add: "positive x ⟹ positive y ⟹ positive (x + y)"
apply transfer
done

lemma positive_mult: "positive x ⟹ positive y ⟹ positive (x * y)"
apply transfer
by (metis fst_conv mult.commute mult_pos_neg2 snd_conv zero_less_mult_iff)

lemma positive_minus: "¬ positive x ⟹ x ≠ 0 ⟹ positive (- x)"
by transfer (auto simp: neq_iff zero_less_mult_iff mult_less_0_iff)

instantiation rat :: linordered_field
begin

definition "x < y ⟷ positive (y - x)"

definition "x ≤ y ⟷ x < y ∨ x = y" for x y :: rat

definition "¦a¦ = (if a < 0 then - a else a)" for a :: rat

definition "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)" for a :: rat

instance
proof
fix a b c :: rat
show "¦a¦ = (if a < 0 then - a else a)"
by (rule abs_rat_def)
show "a < b ⟷ a ≤ b ∧ ¬ b ≤ a"
unfolding less_eq_rat_def less_rat_def
show "a ≤ a"
unfolding less_eq_rat_def by simp
show "a ≤ b ⟹ b ≤ c ⟹ a ≤ c"
unfolding less_eq_rat_def less_rat_def
show "a ≤ b ⟹ b ≤ a ⟹ a = b"
unfolding less_eq_rat_def less_rat_def
show "a ≤ b ⟹ c + a ≤ c + b"
unfolding less_eq_rat_def less_rat_def by auto
show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
by (rule sgn_rat_def)
show "a ≤ b ∨ b ≤ a"
unfolding less_eq_rat_def less_rat_def
by (auto dest!: positive_minus)
show "a < b ⟹ 0 < c ⟹ c * a < c * b"
unfolding less_rat_def
by (metis diff_zero positive_mult right_diff_distrib')
qed

end

instantiation rat :: distrib_lattice
begin

definition "(inf :: rat ⇒ rat ⇒ rat) = min"

definition "(sup :: rat ⇒ rat ⇒ rat) = max"

instance
by standard (auto simp add: inf_rat_def sup_rat_def max_min_distrib2)

end

lemma positive_rat: "positive (Fract a b) ⟷ 0 < a * b"
by transfer simp

lemma less_rat [simp]:
"b ≠ 0 ⟹ d ≠ 0 ⟹ Fract a b < Fract c d ⟷ (a * d) * (b * d) < (c * b) * (b * d)"
by (simp add: less_rat_def positive_rat algebra_simps)

lemma le_rat [simp]:
"b ≠ 0 ⟹ d ≠ 0 ⟹ Fract a b ≤ Fract c d ⟷ (a * d) * (b * d) ≤ (c * b) * (b * d)"

lemma abs_rat [simp, code]: "¦Fract a b¦ = Fract ¦a¦ ¦b¦"
by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)

lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
unfolding Fract_of_int_eq
by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
(auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)

lemma Rat_induct_pos [case_names Fract, induct type: rat]:
assumes step: "⋀a b. 0 < b ⟹ P (Fract a b)"
shows "P q"
proof (cases q)
case (Fract a b)
have step': "P (Fract a b)" if b: "b < 0" for a b :: int
proof -
from b have "0 < - b"
by simp
then have "P (Fract (- a) (- b))"
by (rule step)
then show "P (Fract a b)"
by (simp add: order_less_imp_not_eq [OF b])
qed
from Fract show "P q"
by (auto simp add: linorder_neq_iff step step')
qed

lemma zero_less_Fract_iff: "0 < b ⟹ 0 < Fract a b ⟷ 0 < a"

lemma Fract_less_zero_iff: "0 < b ⟹ Fract a b < 0 ⟷ a < 0"

lemma zero_le_Fract_iff: "0 < b ⟹ 0 ≤ Fract a b ⟷ 0 ≤ a"

lemma Fract_le_zero_iff: "0 < b ⟹ Fract a b ≤ 0 ⟷ a ≤ 0"

lemma one_less_Fract_iff: "0 < b ⟹ 1 < Fract a b ⟷ b < a"

lemma Fract_less_one_iff: "0 < b ⟹ Fract a b < 1 ⟷ a < b"

lemma one_le_Fract_iff: "0 < b ⟹ 1 ≤ Fract a b ⟷ b ≤ a"

lemma Fract_le_one_iff: "0 < b ⟹ Fract a b ≤ 1 ⟷ a ≤ b"

subsubsection ‹Rationals are an Archimedean field›

lemma rat_floor_lemma: "of_int (a div b) ≤ Fract a b ∧ Fract a b < of_int (a div b + 1)"
proof -
have "Fract a b = of_int (a div b) + Fract (a mod b) b"
by (cases "b = 0") (simp, simp add: of_int_rat)
moreover have "0 ≤ Fract (a mod b) b ∧ Fract (a mod b) b < 1"
unfolding Fract_of_int_quotient
by (rule linorder_cases [of b 0]) (simp_all add: divide_nonpos_neg)
ultimately show ?thesis by simp
qed

instance rat :: archimedean_field
proof
show "∃z. r ≤ of_int z" for r :: rat
proof (induct r)
case (Fract a b)
have "Fract a b ≤ of_int (a div b + 1)"
using rat_floor_lemma [of a b] by simp
then show "∃z. Fract a b ≤ of_int z" ..
qed
qed

instantiation rat :: floor_ceiling
begin

definition floor_rat :: "rat ⇒ int"
where"⌊x⌋ = (THE z. of_int z ≤ x ∧ x < of_int (z + 1))" for x :: rat

instance
proof
show "of_int ⌊x⌋ ≤ x ∧ x < of_int (⌊x⌋ + 1)" for x :: rat
unfolding floor_rat_def using floor_exists1 by (rule theI')
qed

end

lemma floor_Fract [simp]: "⌊Fract a b⌋ = a div b"

subsection ‹Linear arithmetic setup›

declaration ‹
K (Lin_Arith.add_inj_thms @{thms of_int_le_iff [THEN iffD2] of_int_eq_iff [THEN iffD2]}
(* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
#> Lin_Arith.add_inj_const (\<^const_name>‹of_nat›, \<^typ>‹nat ⇒ rat›)
#> Lin_Arith.add_inj_const (\<^const_name>‹of_int›, \<^typ>‹int ⇒ rat›))
›

subsection ‹Embedding from Rationals to other Fields›

context field_char_0
begin

lift_definition of_rat :: "rat ⇒ 'a"
is "λx. of_int (fst x) / of_int (snd x)"
by (auto simp: nonzero_divide_eq_eq nonzero_eq_divide_eq) (simp only: of_int_mult [symmetric])

end

lemma of_rat_rat: "b ≠ 0 ⟹ of_rat (Fract a b) = of_int a / of_int b"
by transfer simp

lemma of_rat_0 [simp]: "of_rat 0 = 0"
by transfer simp

lemma of_rat_1 [simp]: "of_rat 1 = 1"
by transfer simp

lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"

lemma of_rat_minus: "of_rat (- a) = - of_rat a"
by transfer simp

lemma of_rat_neg_one [simp]: "of_rat (- 1) = - 1"

lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"

lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
by transfer (simp add: divide_inverse nonzero_inverse_mult_distrib ac_simps)

lemma of_rat_sum: "of_rat (∑a∈A. f a) = (∑a∈A. of_rat (f a))"
by (induct rule: infinite_finite_induct) (auto simp: of_rat_add)

lemma of_rat_prod: "of_rat (∏a∈A. f a) = (∏a∈A. of_rat (f a))"
by (induct rule: infinite_finite_induct) (auto simp: of_rat_mult)

lemma nonzero_of_rat_inverse: "a ≠ 0 ⟹ of_rat (inverse a) = inverse (of_rat a)"
by (rule inverse_unique [symmetric]) (simp add: of_rat_mult [symmetric])

lemma of_rat_inverse: "(of_rat (inverse a) :: 'a::field_char_0) = inverse (of_rat a)"
by (cases "a = 0") (simp_all add: nonzero_of_rat_inverse)

lemma nonzero_of_rat_divide: "b ≠ 0 ⟹ of_rat (a / b) = of_rat a / of_rat b"
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)

lemma of_rat_divide: "(of_rat (a / b) :: 'a::field_char_0) = of_rat a / of_rat b"
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)

lemma of_rat_power: "(of_rat (a ^ n) :: 'a::field_char_0) = of_rat a ^ n"
by (induct n) (simp_all add: of_rat_mult)

lemma of_rat_eq_iff [simp]: "of_rat a = of_rat b ⟷ a = b"
apply transfer
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq flip: of_int_mult)
done

lemma of_rat_eq_0_iff [simp]: "of_rat a = 0 ⟷ a = 0"
using of_rat_eq_iff [of _ 0] by simp

lemma zero_eq_of_rat_iff [simp]: "0 = of_rat a ⟷ 0 = a"
by simp

lemma of_rat_eq_1_iff [simp]: "of_rat a = 1 ⟷ a = 1"
using of_rat_eq_iff [of _ 1] by simp

lemma one_eq_of_rat_iff [simp]: "1 = of_rat a ⟷ 1 = a"
by simp

lemma of_rat_less: "(of_rat r :: 'a::linordered_field) < of_rat s ⟷ r < s"
proof (induct r, induct s)
fix a b c d :: int
assume not_zero: "b > 0" "d > 0"
then have "b * d > 0" by simp
have of_int_divide_less_eq:
"(of_int a :: 'a) / of_int b < of_int c / of_int d ⟷
(of_int a :: 'a) * of_int d < of_int c * of_int b"
using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d) ⟷
Fract a b < Fract c d"
using not_zero ‹b * d > 0›
by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
qed

lemma of_rat_less_eq: "(of_rat r :: 'a::linordered_field) ≤ of_rat s ⟷ r ≤ s"
unfolding le_less by (auto simp add: of_rat_less)

lemma of_rat_le_0_iff [simp]: "(of_rat r :: 'a::linordered_field) ≤ 0 ⟷ r ≤ 0"
using of_rat_less_eq [of r 0, where 'a = 'a] by simp

lemma zero_le_of_rat_iff [simp]: "0 ≤ (of_rat r :: 'a::linordered_field) ⟷ 0 ≤ r"
using of_rat_less_eq [of 0 r, where 'a = 'a] by simp

lemma of_rat_le_1_iff [simp]: "(of_rat r :: 'a::linordered_field) ≤ 1 ⟷ r ≤ 1"
using of_rat_less_eq [of r 1] by simp

lemma one_le_of_rat_iff [simp]: "1 ≤ (of_rat r :: 'a::linordered_field) ⟷ 1 ≤ r"
using of_rat_less_eq [of 1 r] by simp

lemma of_rat_less_0_iff [simp]: "(of_rat r :: 'a::linordered_field) < 0 ⟷ r < 0"
using of_rat_less [of r 0, where 'a = 'a] by simp

lemma zero_less_of_rat_iff [simp]: "0 < (of_rat r :: 'a::linordered_field) ⟷ 0 < r"
using of_rat_less [of 0 r, where 'a = 'a] by simp

lemma of_rat_less_1_iff [simp]: "(of_rat r :: 'a::linordered_field) < 1 ⟷ r < 1"
using of_rat_less [of r 1] by simp

lemma one_less_of_rat_iff [simp]: "1 < (of_rat r :: 'a::linordered_field) ⟷ 1 < r"
using of_rat_less [of 1 r] by simp

lemma of_rat_eq_id [simp]: "of_rat = id"
proof
show "of_rat a = id a" for a
by (induct a) (simp add: of_rat_rat Fract_of_int_eq [symmetric])
qed

lemma abs_of_rat [simp]:
"¦of_rat r¦ = (of_rat ¦r¦ :: 'a :: linordered_field)"
by (cases "r ≥ 0") (simp_all add: not_le of_rat_minus)

text ‹Collapse nested embeddings.›
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"

lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)

lemma of_rat_numeral_eq [simp]: "of_rat (numeral w) = numeral w"
using of_rat_of_int_eq [of "numeral w"] by simp

lemma of_rat_neg_numeral_eq [simp]: "of_rat (- numeral w) = - numeral w"
using of_rat_of_int_eq [of "- numeral w"] by simp

lemma of_rat_floor [simp]:
"⌊of_rat r⌋ = ⌊r⌋"
by (cases r) (simp add: Fract_of_int_quotient of_rat_divide floor_divide_of_int_eq)

lemma of_rat_ceiling [simp]:
"⌈of_rat r⌉ = ⌈r⌉"
using of_rat_floor [of "- r"] by (simp add: of_rat_minus ceiling_def)

lemmas zero_rat = Zero_rat_def
lemmas one_rat = One_rat_def

abbreviation rat_of_nat :: "nat ⇒ rat"
where "rat_of_nat ≡ of_nat"

abbreviation rat_of_int :: "int ⇒ rat"
where "rat_of_int ≡ of_int"

subsection ‹The Set of Rational Numbers›

context field_char_0
begin

definition Rats :: "'a set" ("ℚ")
where "ℚ = range of_rat"

end

lemma Rats_cases [cases set: Rats]:
assumes "q ∈ ℚ"
obtains (of_rat) r where "q = of_rat r"
proof -
from ‹q ∈ ℚ› have "q ∈ range of_rat"
by (simp only: Rats_def)
then obtain r where "q = of_rat r" ..
then show thesis ..
qed

lemma Rats_cases':
assumes "(x :: 'a :: field_char_0) ∈ ℚ"
obtains a b where "b > 0" "coprime a b" "x = of_int a / of_int b"
proof -
from assms obtain r where "x = of_rat r"
by (auto simp: Rats_def)
obtain a b where quot: "quotient_of r = (a,b)" by force
have "b > 0" using quotient_of_denom_pos[OF quot] .
moreover have "coprime a b" using quotient_of_coprime[OF quot] .
moreover have "x = of_int a / of_int b" unfolding ‹x = of_rat r›
quotient_of_div[OF quot] by (simp add: of_rat_divide)
ultimately show ?thesis using that by blast
qed

lemma Rats_of_rat [simp]: "of_rat r ∈ ℚ"

lemma Rats_of_int [simp]: "of_int z ∈ ℚ"
by (subst of_rat_of_int_eq [symmetric]) (rule Rats_of_rat)

lemma Ints_subset_Rats: "ℤ ⊆ ℚ"
using Ints_cases Rats_of_int by blast

lemma Rats_of_nat [simp]: "of_nat n ∈ ℚ"
by (subst of_rat_of_nat_eq [symmetric]) (rule Rats_of_rat)

lemma Nats_subset_Rats: "ℕ ⊆ ℚ"
using Ints_subset_Rats Nats_subset_Ints by blast

lemma Rats_number_of [simp]: "numeral w ∈ ℚ"
by (subst of_rat_numeral_eq [symmetric]) (rule Rats_of_rat)

lemma Rats_0 [simp]: "0 ∈ ℚ"
unfolding Rats_def by (rule range_eqI) (rule of_rat_0 [symmetric])

lemma Rats_1 [simp]: "1 ∈ ℚ"
unfolding Rats_def by (rule range_eqI) (rule of_rat_1 [symmetric])

lemma Rats_add [simp]: "a ∈ ℚ ⟹ b ∈ ℚ ⟹ a + b ∈ ℚ"

lemma Rats_minus_iff [simp]: "- a ∈ ℚ ⟷ a ∈ ℚ"
by (metis Rats_cases Rats_of_rat add.inverse_inverse of_rat_minus)

lemma Rats_diff [simp]: "a ∈ ℚ ⟹ b ∈ ℚ ⟹ a - b ∈ ℚ"

lemma Rats_mult [simp]: "a ∈ ℚ ⟹ b ∈ ℚ ⟹ a * b ∈ ℚ"
by (metis Rats_cases Rats_of_rat of_rat_mult)

lemma Rats_inverse [simp]: "a ∈ ℚ ⟹ inverse a ∈ ℚ"
for a :: "'a::field_char_0"
by (metis Rats_cases Rats_of_rat of_rat_inverse)

lemma Rats_divide [simp]: "a ∈ ℚ ⟹ b ∈ ℚ ⟹ a / b ∈ ℚ"
for a b :: "'a::field_char_0"

lemma Rats_power [simp]: "a ∈ ℚ ⟹ a ^ n ∈ ℚ"
for a :: "'a::field_char_0"
by (metis Rats_cases Rats_of_rat of_rat_power)

lemma Rats_sum [intro]: "(⋀x. x ∈ A ⟹ f x ∈ ℚ) ⟹ sum f A ∈ ℚ"
by (induction A rule: infinite_finite_induct) auto

lemma Rats_prod [intro]: "(⋀x. x ∈ A ⟹ f x ∈ ℚ) ⟹ prod f A ∈ ℚ"
by (induction A rule: infinite_finite_induct) auto

lemma Rats_induct [case_names of_rat, induct set: Rats]: "q ∈ ℚ ⟹ (⋀r. P (of_rat r)) ⟹ P q"
by (rule Rats_cases) auto

lemma Rats_infinite: "¬ finite ℚ"
by (auto dest!: finite_imageD simp: inj_on_def infinite_UNIV_char_0 Rats_def)

lemma Rats_add_iff: "a ∈ ℚ ∨ b ∈ ℚ ⟹ a+b ∈ ℚ ⟷ a ∈ ℚ ∧ b ∈ ℚ"

lemma Rats_diff_iff: "a ∈ ℚ ∨ b ∈ ℚ ⟹ a-b ∈ ℚ ⟷ a ∈ ℚ ∧ b ∈ ℚ"

lemma Rats_mult_iff: "a ∈ ℚ-{0} ∨ b ∈ ℚ-{0} ⟹ a*b ∈ ℚ ⟷ a ∈ ℚ ∧ b ∈ ℚ"
by (metis Diff_iff Rats_divide Rats_mult insertI1 mult.commute nonzero_divide_eq_eq)

lemma Rats_inverse_iff [simp]: "inverse a ∈ ℚ ⟷ a ∈ ℚ"
using Rats_inverse by force

lemma Rats_divide_iff: "a ∈ ℚ-{0} ∨ b ∈ ℚ-{0} ⟹ a/b ∈ ℚ ⟷ a ∈ ℚ ∧ b ∈ ℚ"
by (metis Rats_divide Rats_mult_iff divide_eq_0_iff divide_inverse nonzero_mult_div_cancel_right)

subsection ‹Implementation of rational numbers as pairs of integers›

text ‹Formal constructor›

definition Frct :: "int × int ⇒ rat"
where [simp]: "Frct p = Fract (fst p) (snd p)"

lemma [code abstype]: "Frct (quotient_of q) = q"
by (cases q) (auto intro: quotient_of_eq)

text ‹Numerals›

declare quotient_of_Fract [code abstract]

definition of_int :: "int ⇒ rat"
where [code_abbrev]: "of_int = Int.of_int"

hide_const (open) of_int

lemma quotient_of_int [code abstract]: "quotient_of (Rat.of_int a) = (a, 1)"
by (simp add: of_int_def of_int_rat quotient_of_Fract)

lemma [code_unfold]: "numeral k = Rat.of_int (numeral k)"

lemma [code_unfold]: "- numeral k = Rat.of_int (- numeral k)"

lemma Frct_code_post [code_post]:
"Frct (0, a) = 0"
"Frct (a, 0) = 0"
"Frct (1, 1) = 1"
"Frct (numeral k, 1) = numeral k"
"Frct (1, numeral k) = 1 / numeral k"
"Frct (numeral k, numeral l) = numeral k / numeral l"
"Frct (- a, b) = - Frct (a, b)"
"Frct (a, - b) = - Frct (a, b)"
"- (- Frct q) = Frct q"

text ‹Operations›

lemma rat_zero_code [code abstract]: "quotient_of 0 = (0, 1)"
by (simp add: Zero_rat_def quotient_of_Fract normalize_def)

lemma rat_one_code [code abstract]: "quotient_of 1 = (1, 1)"
by (simp add: One_rat_def quotient_of_Fract normalize_def)

lemma rat_plus_code [code abstract]:
"quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
in normalize (a * d + b * c, c * d))"
by (cases p, cases q) (simp add: quotient_of_Fract)

lemma rat_uminus_code [code abstract]:
"quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
by (cases p) (simp add: quotient_of_Fract)

lemma rat_minus_code [code abstract]:
"quotient_of (p - q) =
(let (a, c) = quotient_of p; (b, d) = quotient_of q
in normalize (a * d - b * c, c * d))"
by (cases p, cases q) (simp add: quotient_of_Fract)

lemma rat_times_code [code abstract]:
"quotient_of (p * q) =
(let (a, c) = quotient_of p; (b, d) = quotient_of q
in normalize (a * b, c * d))"
by (cases p, cases q) (simp add: quotient_of_Fract)

lemma rat_inverse_code [code abstract]:
"quotient_of (inverse p) =
(let (a, b) = quotient_of p
in if a = 0 then (0, 1) else (sgn a * b, ¦a¦))"
proof (cases p)
case (Fract a b)
then show ?thesis
by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract ac_simps)
qed

lemma rat_divide_code [code abstract]:
"quotient_of (p / q) =
(let (a, c) = quotient_of p; (b, d) = quotient_of q
in normalize (a * d, c * b))"
by (cases p, cases q) (simp add: quotient_of_Fract)

lemma rat_abs_code [code abstract]:
"quotient_of ¦p¦ = (let (a, b) = quotient_of p in (¦a¦, b))"
by (cases p) (simp add: quotient_of_Fract)

lemma rat_sgn_code [code abstract]: "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
proof (cases p)
case (Fract a b)
then show ?thesis
by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
qed

lemma rat_floor_code [code]: "⌊p⌋ = (let (a, b) = quotient_of p in a div b)"
by (cases p) (simp add: quotient_of_Fract floor_Fract)

instantiation rat :: equal
begin

definition [code]: "HOL.equal a b ⟷ quotient_of a = quotient_of b"

instance
by standard (simp add: equal_rat_def quotient_of_inject_eq)

lemma rat_eq_refl [code nbe]: "HOL.equal (r::rat) r ⟷ True"
by (rule equal_refl)

end

lemma rat_less_eq_code [code]:
"p ≤ q ⟷ (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d ≤ c * b)"
by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)

lemma rat_less_code [code]:
"p < q ⟷ (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)

lemma [code]: "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
by (cases p) (simp add: quotient_of_Fract of_rat_rat)

text ‹Quickcheck›

context
includes term_syntax
begin

definition
valterm_fract :: "int × (unit ⇒ Code_Evaluation.term) ⇒
int × (unit ⇒ Code_Evaluation.term) ⇒
rat × (unit ⇒ Code_Evaluation.term)"
where [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {⋅} k {⋅} l"

end

instantiation rat :: random
begin

context
includes state_combinator_syntax
begin

definition
"Quickcheck_Random.random i =
Quickcheck_Random.random i ∘→ (λnum. Random.range i ∘→ (λdenom. Pair
(let j = int_of_integer (integer_of_natural (denom + 1))
in valterm_fract num (j, λu. Code_Evaluation.term_of j))))"

instance ..

end

end

instantiation rat :: exhaustive
begin

definition
"exhaustive_rat f d =
Quickcheck_Exhaustive.exhaustive
(λl. Quickcheck_Exhaustive.exhaustive
(λk. f (Fract k (int_of_integer (integer_of_natural l) + 1))) d) d"

instance ..

end

instantiation rat :: full_exhaustive
begin

definition
"full_exhaustive_rat f d =
Quickcheck_Exhaustive.full_exhaustive
(λ(l, _). Quickcheck_Exhaustive.full_exhaustive
(λk. f
(let j = int_of_integer (integer_of_natural l) + 1
in valterm_fract k (j, λ_. Code_Evaluation.term_of j))) d) d"

instance ..

end

instance rat :: partial_term_of ..

lemma [code]:
"partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) ≡
Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
"partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ≡
Code_Evaluation.App
(Code_Evaluation.Const (STR ''Rat.Frct'')
(Typerep.Typerep (STR ''fun'')
[Typerep.Typerep (STR ''Product_Type.prod'')
[Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
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))"
by (rule partial_term_of_anything)+

instantiation rat :: narrowing
begin

definition
"narrowing =
Quickcheck_Narrowing.apply
(Quickcheck_Narrowing.apply
(Quickcheck_Narrowing.cons (λnom denom. Fract nom denom)) narrowing) narrowing"

instance ..

end

subsection ‹Setup for Nitpick›

declaration ‹
Nitpick_HOL.register_frac_type \<^type_name>‹rat›
[(\<^const_name>‹Abs_Rat›, \<^const_name>‹Nitpick.Abs_Frac›),
(\<^const_name>‹zero_rat_inst.zero_rat›, \<^const_name>‹Nitpick.zero_frac›),
(\<^const_name>‹one_rat_inst.one_rat›, \<^const_name>‹Nitpick.one_frac›),
(\<^const_name>‹plus_rat_inst.plus_rat›, \<^const_name>‹Nitpick.plus_frac›),
(\<^const_name>‹times_rat_inst.times_rat›, \<^const_name>‹Nitpick.times_frac›),
(\<^const_name>‹uminus_rat_inst.uminus_rat›, \<^const_name>‹Nitpick.uminus_frac›),
(\<^const_name>‹inverse_rat_inst.inverse_rat›, \<^const_name>‹Nitpick.inverse_frac›),
(\<^const_name>‹ord_rat_inst.less_rat›, \<^const_name>‹Nitpick.less_frac›),
(\<^const_name>‹ord_rat_inst.less_eq_rat›, \<^const_name>‹Nitpick.less_eq_frac›),
(\<^const_name>‹field_char_0_class.of_rat›, \<^const_name>‹Nitpick.of_frac›)]
›

lemmas [nitpick_unfold] =
inverse_rat_inst.inverse_rat
one_rat_inst.one_rat ord_rat_inst.less_rat
ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat

subsection ‹Float syntax›

syntax "_Float" :: "float_const ⇒ 'a"    ("_")

parse_translation ‹
let
fun mk_frac str =
let
val {mant = i, exp = n} = Lexicon.read_float str;
val exp = Syntax.const \<^const_syntax>‹Power.power›;
val ten = Numeral.mk_number_syntax 10;
val exp10 = if n = 1 then ten else exp \$ ten \$ Numeral.mk_number_syntax n;
in Syntax.const \<^const_syntax>‹Fields.inverse_divide› \$ Numeral.mk_number_syntax i \$ exp10 end;

fun float_tr [(c as Const (\<^syntax_const>‹_constrain›, _)) \$ t \$ u] = c \$ float_tr [t] \$ u
| float_tr [t as Const (str, _)] = mk_frac str
| float_tr ts = raise TERM ("float_tr", ts);
in [(\<^syntax_const>‹_Float›, K float_tr)] end
›

text‹Test:›
lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
by simp

subsection ‹Hiding implementation details›

hide_const (open) normalize positive

lifting_update rat.lifting
lifting_forget rat.lifting

end
```