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