(* Title: HOL/Library/Rational.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen
*)
header {* Rational numbers *}
theory Rational
imports Main
uses ("rat_arith.ML")
begin
subsection {* Rational numbers *}
subsubsection {* Equivalence of fractions *}
constdefs
fraction :: "(int \<times> int) set"
"fraction \<equiv> {x. snd x \<noteq> 0}"
ratrel :: "((int \<times> int) \<times> (int \<times> int)) set"
"ratrel \<equiv> {(x,y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
lemma fraction_iff [simp]: "(x \<in> fraction) = (snd x \<noteq> 0)"
by (simp add: fraction_def)
lemma ratrel_iff [simp]:
"((x,y) \<in> ratrel) =
(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 fraction ratrel"
by (auto simp add: refl_def fraction_def ratrel_def)
lemma sym_ratrel: "sym ratrel"
by (simp add: ratrel_def sym_def)
lemma trans_ratrel_lemma:
assumes 1: "a * b' = a' * b"
assumes 2: "a' * b'' = a'' * b'"
assumes 3: "b' \<noteq> (0::int)"
shows "a * b'' = a'' * b"
proof -
have "b' * (a * b'') = b'' * (a * b')" by simp
also note 1
also have "b'' * (a' * b) = b * (a' * b'')" by simp
also note 2
also have "b * (a'' * b') = b' * (a'' * b)" by simp
finally have "b' * (a * b'') = b' * (a'' * b)" .
with 3 show "a * b'' = a'' * b" by simp
qed
lemma trans_ratrel: "trans ratrel"
by (auto simp add: trans_def elim: trans_ratrel_lemma)
lemma equiv_ratrel: "equiv fraction ratrel"
by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])
lemmas equiv_ratrel_iff [iff] = eq_equiv_class_iff [OF equiv_ratrel]
lemma equiv_ratrel_iff2:
"\<lbrakk>snd x \<noteq> 0; snd y \<noteq> 0\<rbrakk>
\<Longrightarrow> (ratrel `` {x} = ratrel `` {y}) = ((x,y) \<in> ratrel)"
by (rule eq_equiv_class_iff [OF equiv_ratrel], simp_all)
subsubsection {* The type of rational numbers *}
typedef (Rat) rat = "fraction//ratrel"
proof
have "(0,1) \<in> fraction" by (simp add: fraction_def)
thus "ratrel``{(0,1)} \<in> fraction//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]
constdefs
Fract :: "int \<Rightarrow> int \<Rightarrow> rat"
"Fract a b \<equiv> Abs_Rat (ratrel``{(a,b)})"
theorem Rat_cases [case_names Fract, cases type: rat]:
"(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"
by (cases q, clarsimp simp add: Fract_def Rat_def fraction_def quotient_def)
theorem Rat_induct [case_names Fract, induct type: rat]:
"(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q"
by (cases q) simp
subsubsection {* Congruence lemmas *}
lemma add_congruent2:
"(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
respects2 ratrel"
apply (rule equiv_ratrel [THEN congruent2_commuteI])
apply (simp_all add: left_distrib)
done
lemma minus_congruent:
"(\<lambda>x. ratrel``{(- fst x, snd x)}) respects ratrel"
by (simp add: congruent_def)
lemma mult_congruent2:
"(\<lambda>x y. ratrel``{(fst x * fst y, snd x * snd y)}) respects2 ratrel"
by (rule equiv_ratrel [THEN congruent2_commuteI], simp_all)
lemma inverse_congruent:
"(\<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)
lemma le_congruent2:
"(\<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
lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
subsubsection {* Standard operations on rational numbers *}
instance rat :: "{ord, zero, one, plus, times, minus, inverse}" ..
defs (overloaded)
Zero_rat_def: "0 == Fract 0 1"
One_rat_def: "1 == Fract 1 1"
add_rat_def:
"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)})"
minus_rat_def:
"- q == Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel``{(- fst x, snd x)})"
diff_rat_def: "q - r == q + - (r::rat)"
mult_rat_def:
"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)})"
inverse_rat_def:
"inverse q ==
Abs_Rat (\<Union>x \<in> Rep_Rat q.
ratrel``{if fst x=0 then (0,1) else (snd x, fst x)})"
divide_rat_def: "q / r == q * inverse (r::rat)"
le_rat_def:
"q \<le> r == 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)})"
less_rat_def: "(z < (w::rat)) == (z \<le> w & z \<noteq> w)"
abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)"
lemma zero_rat: "0 = Fract 0 1"
by (simp add: Zero_rat_def)
lemma one_rat: "1 = Fract 1 1"
by (simp add: One_rat_def)
theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
(Fract a b = Fract c d) = (a * d = c * b)"
by (simp add: Fract_def)
theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
by (simp add: Fract_def add_rat_def add_congruent2 UN_ratrel2)
theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b"
by (simp add: Fract_def minus_rat_def minus_congruent UN_ratrel)
theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
by (simp add: diff_rat_def add_rat minus_rat)
theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
Fract a b * Fract c d = Fract (a * c) (b * d)"
by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2)
theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==>
inverse (Fract a b) = Fract b a"
by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel)
theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>
Fract a b / Fract c d = Fract (a * d) (b * c)"
by (simp add: divide_rat_def inverse_rat mult_rat)
theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
(Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
by (simp add: Fract_def le_rat_def le_congruent2 UN_ratrel2)
theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
(Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))"
by (simp add: less_rat_def le_rat eq_rat order_less_le)
theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat)
(auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less
split: abs_split)
subsubsection {* The ordered field of rational numbers *}
lemma rat_add_assoc: "(q + r) + s = q + (r + (s::rat))"
by (induct q, induct r, induct s)
(simp add: add_rat add_ac mult_ac int_distrib)
lemma rat_add_0: "0 + q = (q::rat)"
by (induct q) (simp add: zero_rat add_rat)
lemma rat_left_minus: "(-q) + q = (0::rat)"
by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat)
instance rat :: field
proof
fix q r s :: rat
show "(q + r) + s = q + (r + s)"
by (induct q, induct r, induct s)
(simp add: add_rat add_ac mult_ac int_distrib)
show "q + r = r + q"
by (induct q, induct r) (simp add: add_rat add_ac mult_ac)
show "0 + q = q"
by (induct q) (simp add: zero_rat add_rat)
show "(-q) + q = 0"
by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat)
show "q - r = q + (-r)"
by (induct q, induct r) (simp add: add_rat minus_rat diff_rat)
show "(q * r) * s = q * (r * s)"
by (induct q, induct r, induct s) (simp add: mult_rat mult_ac)
show "q * r = r * q"
by (induct q, induct r) (simp add: mult_rat mult_ac)
show "1 * q = q"
by (induct q) (simp add: one_rat mult_rat)
show "(q + r) * s = q * s + r * s"
by (induct q, induct r, induct s)
(simp add: add_rat mult_rat eq_rat int_distrib)
show "q \<noteq> 0 ==> inverse q * q = 1"
by (induct q) (simp add: inverse_rat mult_rat one_rat zero_rat eq_rat)
show "q / r = q * inverse r"
by (simp add: divide_rat_def)
show "0 \<noteq> (1::rat)"
by (simp add: zero_rat one_rat eq_rat)
qed
instance rat :: linorder
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 add: le_rat)
with ff show ?thesis by (simp add: mult_le_cancel_right)
qed
also have "... = (c * f) * (d * f) * (b * b)"
by (simp only: mult_ac)
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 add: le_rat)
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 add: le_rat)
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 add: le_rat)
also have "... \<le> (a * d) * (b * d)"
proof -
from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
by (simp add: le_rat)
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 add: le_rat)
show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
by (simp only: less_rat_def)
show "q \<le> r \<or> r \<le> q"
by (induct q, induct r)
(simp add: le_rat mult_commute, rule linorder_linear)
}
qed
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 add: le_rat)
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: add_rat le_rat 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 less_rat le_rat 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 add: less_rat)
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: less_rat mult_rat mult_ac)
qed
qed
show "\<bar>q\<bar> = (if q < 0 then -q else q)"
by (simp only: abs_rat_def)
qed
instance rat :: division_by_zero
proof
show "inverse 0 = (0::rat)"
by (simp add: zero_rat Fract_def inverse_rat_def
inverse_congruent UN_ratrel)
qed
subsection {* Various Other Results *}
lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b"
by (simp add: eq_rat)
theorem 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)
have step': "!!a b. b < 0 ==> 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 less_rat order_less_imp_not_eq2 zero_less_mult_iff)
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
apply (insert add_rat [of concl: m n 1 1])
apply (simp add: one_rat [symmetric])
done
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
apply (induct k)
apply (simp add: zero_rat)
apply (simp add: Fract_add_one)
done
lemma Fract_of_int_eq: "Fract k 1 = of_int k"
proof (cases k rule: int_cases)
case (nonneg n)
thus ?thesis by (simp add: int_eq_of_nat Fract_of_nat_eq)
next
case (neg n)
hence "Fract k 1 = - (Fract (of_nat (Suc n)) 1)"
by (simp only: minus_rat int_eq_of_nat)
also have "... = - (of_nat (Suc n))"
by (simp only: Fract_of_nat_eq)
finally show ?thesis
by (simp add: only: prems int_eq_of_nat of_int_minus of_int_of_nat_eq)
qed
subsection {* Numerals and Arithmetic *}
instance rat :: number ..
defs (overloaded)
rat_number_of_def: "(number_of w :: rat) == of_int (Rep_Bin w)"
--{*the type constraint is essential!*}
instance rat :: number_ring
by (intro_classes, simp add: rat_number_of_def)
declare diff_rat_def [symmetric]
use "rat_arith.ML"
setup rat_arith_setup
end