(* Title: HOL/Rat.thy
Author: Markus Wenzel, TU Muenchen
*)
section \<open>Rational numbers\<close>
theory Rat
imports Archimedean_Field
begin
subsection \<open>Rational numbers as quotient\<close>
subsubsection \<open>Construction of the type of rational numbers\<close>
definition ratrel :: "(int \ int) \ (int \ int) \ bool"
where "ratrel = (\x 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"
by (simp add: ratrel_def)
lemma exists_ratrel_refl: "\x. ratrel x x"
by (auto intro!: one_neq_zero)
lemma symp_ratrel: "symp ratrel"
by (simp add: ratrel_def symp_def)
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 = (\x. snd x \ 0)"
by (simp add: rat.domain_eq)
subsubsection \<open>Representation and basic operations\<close>
lift_definition Fract :: "int \ int \ rat"
is "\a 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"
by (simp add: nonneg1_imp_zdiv_pos_iff)
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"
by (simp add: pos_imp_zdiv_neg_iff)
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 "\x y. (fst x * snd y + fst y * snd x, snd x * snd y)"
by (auto simp: distrib_right) (simp add: ac_simps)
lemma add_rat [simp]:
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 "\x. (- 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)"
by (simp add: diff_rat_def)
lift_definition times_rat :: "rat \ rat \ rat"
is "\x y. (fst x * fst y, snd x * snd y)"
by (simp add: ac_simps)
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 "\x. if fst x = 0 then (0, 1) else (snd x, fst x)"
by (auto simp add: mult.commute)
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)"
by (simp add: divide_rat_def)
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)"
by transfer (simp add: algebra_simps)
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"
by transfer (simp add: algebra_simps)
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 *)
lemma div_add_self1_no_field [simp]:
assumes "NO_MATCH (x :: 'b :: field) b" "(b :: 'a :: euclidean_semiring_cancel) \ 0"
shows "(b + a) div b = a div b + 1"
using assms(2) by (fact div_add_self1)
lemma div_add_self2_no_field [simp]:
assumes "NO_MATCH (x :: 'b :: field) b" "(b :: 'a :: euclidean_semiring_cancel) \ 0"
shows "(a + b) div b = a div b + 1"
using assms(2) by (fact div_add_self2)
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"
by (simp_all add: rat_number_collapse)
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 \<open>b > 0\<close> have "a \<noteq> 0"
by (simp add: Zero_rat_def eq_rat)
with Fract * show C by blast
qed
subsubsection \<open>Function \<open>normalize\<close>\<close>
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
by (simp add: eq_rat)
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)"
by (simp add: normalize_def)
lemma normalize_denom_zero [simp]: "normalize (p, 0) = (0, 1)"
by (simp add: normalize_def)
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\<open>
Decompose a fraction into normalized, i.e. coprime numerator and denominator:
\<close>
definition quotient_of :: "rat \ int \ int"
where "quotient_of x =
(THE pair. x = Fract (fst pair) (snd pair) \<and> snd pair > 0 \<and> 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 \<and> 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
by (simp add: eq_rat)
next
case False
with Fract Fract' have *: "c * b = a * d" and "c \ 0"
by (auto simp add: eq_rat)
then have "c * b > 0 \ a * d > 0"
by auto
with \<open>b > 0\<close> \<open>d > 0\<close> have "a > 0 \<longleftrightarrow> c > 0"
by (simp add: zero_less_mult_iff)
with \<open>a \<noteq> 0\<close> \<open>c \<noteq> 0\<close> have sgn: "sgn a = sgn c"
by (auto simp add: not_less)
from \<open>coprime a b\<close> \<open>coprime c d\<close> 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>"
by (simp add: coprime_crossproduct_int)
with \<open>b > 0\<close> \<open>d > 0\<close> have "\<bar>a\<bar> * d = \<bar>c\<bar> * b \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> d = b"
by simp
then have "a * sgn a * d = c * sgn c * b \ a * sgn a = c * sgn c \ d = b"
by (simp add: abs_sgn)
with sgn * show ?thesis
by (auto simp add: sgn_0_0)
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)"
by (simp_all add: rat_number_expand quotient_of_Fract)
lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \ Fract p q = Fract a b"
by (simp add: quotient_of_Fract normalize_eq)
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"
by (auto simp add: quotient_of_inject)
subsubsection \<open>Various\<close>
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
by (simp add: Fract_of_int_eq [symmetric])
lemma Fract_add_one: "n \ 0 \ Fract (m + n) n = Fract m n + 1"
by (simp add: rat_number_expand)
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 \<open>The ordered field of rational numbers\<close>
lift_definition positive :: "rat \ bool"
is "\x. 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\<^sup>2 = c * d * b\<^sup>2"
unfolding power2_eq_square by (simp add: ac_simps)
then have "0 < a * b * d\<^sup>2 \ 0 < c * d * b\<^sup>2"
by simp
then show "0 < a * b \ 0 < c * d"
using \<open>b \<noteq> 0\<close> and \<open>d \<noteq> 0\<close>
by (simp add: zero_less_mult_iff)
qed
lemma positive_zero: "\ positive 0"
by transfer simp
lemma positive_add: "positive x \ positive y \ positive (x + y)"
apply transfer
apply (auto simp add: zero_less_mult_iff add_pos_pos add_neg_neg mult_pos_neg mult_neg_pos mult_neg_neg)
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
using positive_add positive_zero by force
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
using positive_add by fastforce
show "a \ b \ b \ a \ a = b"
unfolding less_eq_rat_def less_rat_def
using positive_add positive_zero by fastforce
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)"
by (simp add: le_less eq_rat)
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"
by (simp add: Zero_rat_def zero_less_mult_iff)
lemma Fract_less_zero_iff: "0 < b \ Fract a b < 0 \ a < 0"
by (simp add: Zero_rat_def mult_less_0_iff)
lemma zero_le_Fract_iff: "0 < b \ 0 \ Fract a b \ 0 \ a"
by (simp add: Zero_rat_def zero_le_mult_iff)
lemma Fract_le_zero_iff: "0 < b \ Fract a b \ 0 \ a \ 0"
by (simp add: Zero_rat_def mult_le_0_iff)
lemma one_less_Fract_iff: "0 < b \ 1 < Fract a b \ b < a"
by (simp add: One_rat_def mult_less_cancel_right_disj)
lemma Fract_less_one_iff: "0 < b \ Fract a b < 1 \ a < b"
by (simp add: One_rat_def mult_less_cancel_right_disj)
lemma one_le_Fract_iff: "0 < b \ 1 \ Fract a b \ b \ a"
by (simp add: One_rat_def mult_le_cancel_right)
lemma Fract_le_one_iff: "0 < b \ Fract a b \ 1 \ a \ b"
by (simp add: One_rat_def mult_le_cancel_right)
subsubsection \<open>Rationals are an Archimedean field\<close>
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"
by (simp add: Fract_of_int_quotient floor_divide_of_int_eq)
subsection \<open>Linear arithmetic setup\<close>
declaration \<open>
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>\<open>of_nat\<close>, \<^typ>\<open>nat \<Rightarrow> rat\<close>)
#> Lin_Arith.add_inj_const (\<^const_name>\<open>of_int\<close>, \<^typ>\<open>int \<Rightarrow> rat\<close>))
\<close>
subsection \<open>Embedding from Rationals to other Fields\<close>
context field_char_0
begin
lift_definition of_rat :: "rat \ 'a"
is "\x. 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"
by transfer (simp add: add_frac_eq)
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
by transfer simp
lemma of_rat_neg_one [simp]: "of_rat (- 1) = - 1"
by (simp add: of_rat_minus)
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
using of_rat_add [of a "- b"] by (simp add: of_rat_minus)
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 \<open>b * d > 0\<close>
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 \<open>Collapse nested embeddings.\<close>
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
by (induct n) (simp_all add: of_rat_add)
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
lemma of_rat_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 \<open>The Set of Rational Numbers\<close>
context field_char_0
begin
definition Rats :: "'a set" ("\")
where "\ = range of_rat"
end
lemma Rats_cases [cases set: Rats]:
assumes "q \ \"
obtains (of_rat) r where "q = of_rat r"
proof -
from \<open>q \<in> \<rat>\<close> have "q \<in> 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) \ \"
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 \<open>x = of_rat r\<close>
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 \ \"
by (simp add: Rats_def)
lemma Rats_of_int [simp]: "of_int z \ \"
by (subst of_rat_of_int_eq [symmetric]) (rule Rats_of_rat)
lemma Ints_subset_Rats: "\ \ \"
using Ints_cases Rats_of_int by blast
lemma Rats_of_nat [simp]: "of_nat n \ \"
by (subst of_rat_of_nat_eq [symmetric]) (rule Rats_of_rat)
lemma Nats_subset_Rats: "\ \ \"
using Ints_subset_Rats Nats_subset_Ints by blast
lemma Rats_number_of [simp]: "numeral w \ \"
by (subst of_rat_numeral_eq [symmetric]) (rule Rats_of_rat)
lemma Rats_0 [simp]: "0 \ \"
unfolding Rats_def by (rule range_eqI) (rule of_rat_0 [symmetric])
lemma Rats_1 [simp]: "1 \ \"
unfolding Rats_def by (rule range_eqI) (rule of_rat_1 [symmetric])
lemma Rats_add [simp]: "a \ \ \ b \ \ \ a + b \ \"
by (metis Rats_cases Rats_of_rat of_rat_add)
lemma Rats_minus_iff [simp]: "- a \ \ \ a \ \"
by (metis Rats_cases Rats_of_rat add.inverse_inverse of_rat_minus)
lemma Rats_diff [simp]: "a \ \ \ b \ \ \ a - b \ \"
by (metis Rats_add Rats_minus_iff diff_conv_add_uminus)
lemma Rats_mult [simp]: "a \ \ \ b \ \ \ a * b \ \"
by (metis Rats_cases Rats_of_rat of_rat_mult)
lemma Rats_inverse [simp]: "a \ \ \ inverse a \ \"
for a :: "'a::field_char_0"
by (metis Rats_cases Rats_of_rat of_rat_inverse)
lemma Rats_divide [simp]: "a \ \ \ b \ \ \ a / b \ \"
for a b :: "'a::field_char_0"
by (simp add: divide_inverse)
lemma Rats_power [simp]: "a \ \ \ a ^ n \ \"
for a :: "'a::field_char_0"
by (metis Rats_cases Rats_of_rat of_rat_power)
lemma Rats_induct [case_names of_rat, induct set: Rats]: "q \ \ \ (\r. P (of_rat r)) \ P q"
by (rule Rats_cases) auto
lemma Rats_infinite: "\ finite \"
by (auto dest!: finite_imageD simp: inj_on_def infinite_UNIV_char_0 Rats_def)
subsection \<open>Implementation of rational numbers as pairs of integers\<close>
text \<open>Formal constructor\<close>
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 \<open>Numerals\<close>
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)"
by (simp add: Rat.of_int_def)
lemma [code_unfold]: "- numeral k = Rat.of_int (- numeral k)"
by (simp add: Rat.of_int_def)
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"
by (simp_all add: Fract_of_int_quotient)
text \<open>Operations\<close>
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, \<bar>a\<bar>))"
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 \<open>Quickcheck\<close>
context
includes term_syntax
begin
definition
valterm_fract :: "int \ (unit \ Code_Evaluation.term) \
int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>
rat \<times> (unit \<Rightarrow> 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 \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair
(let j = int_of_integer (integer_of_natural (denom + 1))
in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
instance ..
end
end
instantiation rat :: exhaustive
begin
definition
"exhaustive_rat f d =
Quickcheck_Exhaustive.exhaustive
(\<lambda>l. Quickcheck_Exhaustive.exhaustive
(\<lambda>k. 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
(\<lambda>(l, _). Quickcheck_Exhaustive.full_exhaustive
(\<lambda>k. f
(let j = int_of_integer (integer_of_natural l) + 1
in valterm_fract k (j, \<lambda>_. 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 (\<lambda>nom denom. Fract nom denom)) narrowing) narrowing"
instance ..
end
subsection \<open>Setup for Nitpick\<close>
declaration \<open>
Nitpick_HOL.register_frac_type \<^type_name>\<open>rat\<close>
[(\<^const_name>\<open>Abs_Rat\<close>, \<^const_name>\<open>Nitpick.Abs_Frac\<close>),
(\<^const_name>\<open>zero_rat_inst.zero_rat\<close>, \<^const_name>\<open>Nitpick.zero_frac\<close>),
(\<^const_name>\<open>one_rat_inst.one_rat\<close>, \<^const_name>\<open>Nitpick.one_frac\<close>),
(\<^const_name>\<open>plus_rat_inst.plus_rat\<close>, \<^const_name>\<open>Nitpick.plus_frac\<close>),
(\<^const_name>\<open>times_rat_inst.times_rat\<close>, \<^const_name>\<open>Nitpick.times_frac\<close>),
(\<^const_name>\<open>uminus_rat_inst.uminus_rat\<close>, \<^const_name>\<open>Nitpick.uminus_frac\<close>),
(\<^const_name>\<open>inverse_rat_inst.inverse_rat\<close>, \<^const_name>\<open>Nitpick.inverse_frac\<close>),
(\<^const_name>\<open>ord_rat_inst.less_rat\<close>, \<^const_name>\<open>Nitpick.less_frac\<close>),
(\<^const_name>\<open>ord_rat_inst.less_eq_rat\<close>, \<^const_name>\<open>Nitpick.less_eq_frac\<close>),
(\<^const_name>\<open>field_char_0_class.of_rat\<close>, \<^const_name>\<open>Nitpick.of_frac\<close>)]
\<close>
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 \<open>Float syntax\<close>
syntax "_Float" :: "float_const \ 'a" ("_")
parse_translation \<open>
let
fun mk_frac str =
let
val {mant = i, exp = n} = Lexicon.read_float str;
val exp = Syntax.const \<^const_syntax>\<open>Power.power\<close>;
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>\<open>Fields.inverse_divide\<close> $ Numeral.mk_number_syntax i $ exp10 end;
fun float_tr [(c as Const (\<^syntax_const>\<open>_constrain\<close>, _)) $ 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>\<open>_Float\<close>, K float_tr)] end
\<close>
text\<open>Test:\<close>
lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
by simp
subsection \<open>Hiding implementation details\<close>
hide_const (open) normalize positive
lifting_update rat.lifting
lifting_forget rat.lifting
end
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