--- a/src/HOL/Real/Rational.thy Fri Jul 18 18:25:53 2008 +0200
+++ b/src/HOL/Real/Rational.thy Fri Jul 18 18:25:56 2008 +0200
@@ -6,7 +6,7 @@
header {* Rational numbers *}
theory Rational
-imports "../Presburger" GCD Abstract_Rat
+imports "../Presburger" GCD
uses ("rat_arith.ML")
begin
@@ -87,7 +87,8 @@
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 c. Fract a 0 = Fract c 0"
+ 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}"
@@ -104,7 +105,7 @@
"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:
+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 -
@@ -118,43 +119,42 @@
minus_rat_def [code func del]:
"- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
-lemma minus_rat: "- Fract a b = Fract (- a) b"
+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"
+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:
+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 add_rat minus_rat)
+ 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: "Fract a b * Fract c d = Fract (a * c) (b * d)"
+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 [simp]:
+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] mult_rat)
+ then show ?thesis by (simp add: mult_rat [symmetric])
qed
primrec power_rat
@@ -164,36 +164,36 @@
instance proof
fix q r s :: rat show "(q * r) * s = q * (r * s)"
- by (cases q, cases r, cases s) (simp add: mult_rat eq_rat)
+ 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: mult_rat eq_rat)
+ 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 mult_rat eq_rat)
+ 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: add_rat eq_rat ring_simps)
+ 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: add_rat eq_rat)
+ 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 add_rat eq_rat)
+ 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 add_rat minus_rat eq_rat)
+ 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: diff_rat add_rat minus_rat eq_rat)
+ 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: add_rat mult_rat eq_rat ring_simps)
+ 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 mult_rat eq_rat)
+ by (cases q) (simp add: One_rat_def eq_rat)
next
fix q :: rat
fix n :: nat
@@ -204,10 +204,10 @@
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 add_rat)
+ 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 diff_rat)
+ 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])
@@ -269,7 +269,7 @@
"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: "inverse (Fract a b) = Fract b a"
+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)
@@ -279,11 +279,11 @@
definition
divide_rat_def [code func del]: "q / r = q * inverse (r::rat)"
-lemma divide_rat: "Fract a b / Fract c d = Fract (a * d) (b * c)"
- by (simp add: divide_rat_def inverse_rat mult_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 inverse_rat)
+ show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
(simp add: rat_number_collapse)
next
fix q :: rat
@@ -301,15 +301,18 @@
subsubsection {* Various *}
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
- by (simp add: rat_number_expand add_rat)
+ 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] divide_rat)
+ 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 *}
@@ -321,7 +324,7 @@
"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:
+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 -
@@ -369,10 +372,10 @@
definition
less_rat_def [code func del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
-lemma less_rat:
+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 le_rat eq_rat order_less_le)
+ using assms by (simp add: less_rat_def eq_rat order_less_le)
instance proof
fix q r s :: rat
@@ -390,7 +393,7 @@
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)
+ by simp
with ff show ?thesis by (simp add: mult_le_cancel_right)
qed
also have "... = (c * f) * (d * f) * (b * b)"
@@ -398,14 +401,14 @@
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)
+ 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 add: le_rat)
+ with neq show ?thesis by simp
qed
qed
next
@@ -418,11 +421,11 @@
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)
+ 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 add: le_rat)
+ by simp
thus ?thesis by (simp only: mult_ac)
qed
finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
@@ -433,12 +436,12 @@
qed
next
show "q \<le> q"
- by (induct q) (simp add: le_rat)
+ by (induct q) simp
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)
+ (simp add: mult_commute, rule linorder_linear)
}
qed
@@ -448,17 +451,17 @@
begin
definition
- abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
+ abs_rat_def [code func del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
-lemma abs_rat: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
+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: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
+ 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: "sgn (Fract a b) = Fract (sgn a * sgn b) 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 less_rat)
+ 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
@@ -485,10 +488,10 @@
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)
+ 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: add_rat le_rat mult_ac int_distrib)
+ with neq show ?thesis by (simp add: mult_ac int_distrib)
qed
qed
show "q < r ==> 0 < s ==> s * q < s * r"
@@ -501,15 +504,15 @@
proof -
let ?E = "e * f" and ?F = "f * f"
from neq gt have "0 < ?E"
- by (auto simp add: Zero_rat_def less_rat le_rat order_less_le eq_rat)
+ 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 add: less_rat)
+ 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: less_rat mult_rat mult_ac)
+ by (simp add: mult_ac)
qed
qed
qed auto
@@ -531,8 +534,8 @@
qed
lemma zero_less_Fract_iff:
- "0 < b ==> (0 < Fract a b) = (0 < a)"
-by (simp add: Zero_rat_def less_rat order_less_imp_not_eq2 zero_less_mult_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 *}
@@ -572,16 +575,16 @@
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: add_rat of_rat_rat add_frac_eq)
+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: minus_rat of_rat_rat)
+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: mult_rat of_rat_rat)
+apply (induct a, induct b, simp add: of_rat_rat)
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
done
@@ -603,7 +606,7 @@
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)
+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"
@@ -616,14 +619,35 @@
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 :: rat \<Rightarrow> rat)"
+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 divide_rat Fract_of_int_eq [symmetric])
+ (simp add: of_rat_rat Fract_of_int_eq [symmetric])
qed
text{*Collapse nested embeddings*}
@@ -631,7 +655,7 @@
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)
+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})"
@@ -653,61 +677,146 @@
subsection {* Implementation of rational numbers as pairs of integers *}
-lemma INum_Fract [simp]: "INum = split Fract"
- by (auto simp add: expand_fun_eq INum_def Fract_of_int_quotient)
+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: times_div_mod_plus_zero_one.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
-lemma split_Fract_normNum [simp]: "split Fract (normNum (k, l)) = Fract k l"
- unfolding INum_Fract [symmetric] normNum by simp
+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 k l) = (if l \<noteq> 0 then of_int k / of_int l else 0)"
- by (cases "l = 0") (simp_all add: rat_number_collapse of_rat_rat)
+ "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 (r\<Colon>rat) s \<longleftrightarrow> r - s = 0"
+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 k l) (Fract r s) \<longleftrightarrow> eq_class.eq (normNum (k, l)) (normNum (r, s))"
- by (simp add: eq INum_normNum_iff [where ?'a = rat, symmetric])
+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 rat_less_eq_code [code]: "Fract k l \<le> Fract r s \<longleftrightarrow> normNum (k, l) \<le>\<^sub>N normNum (r, s)"
+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 "normNum (k, l) \<le>\<^sub>N normNum (r, s) \<longleftrightarrow> split Fract (normNum (k, l)) \<le> split Fract (normNum (r, s))"
- by (simp add: INum_Fract [symmetric] del: INum_Fract normNum)
- also have "\<dots> = (Fract k l \<le> Fract r s)" by simp
- finally show ?thesis by simp
+ 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 rat_less_code [code]: "Fract k l < Fract r s \<longleftrightarrow> normNum (k, l) <\<^sub>N normNum (r, s)"
+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 "normNum (k, l) <\<^sub>N normNum (r, s) \<longleftrightarrow> split Fract (normNum (k, l)) < split Fract (normNum (r, s))"
- by (simp add: INum_Fract [symmetric] del: INum_Fract normNum)
- also have "\<dots> = (Fract k l < Fract r s)" by simp
- finally show ?thesis by simp
+ 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_add_code [code]: "Fract k l + Fract r s = split Fract ((k, l) +\<^sub>N (r, s))"
- by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
+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_mul_code [code]: "Fract k l * Fract r s = split Fract ((k, l) *\<^sub>N (r, s))"
- by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
-
-lemma rat_neg_code [code]: "- Fract k l = split Fract (~\<^sub>N (k, l))"
- by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
+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_sub_code [code]: "Fract k l - Fract r s = split Fract ((k, l) -\<^sub>N (r, s))"
- by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
+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_inv_code [code]: "inverse (Fract k l) = split Fract (Ninv (k, l))"
- by (simp add: INum_Fract [symmetric] del: INum_Fract, simp add: divide_rat_def)
+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_div_code [code]: "Fract k l / Fract r s = split Fract ((k, l) \<div>\<^sub>N (r, s))"
- by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
+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 *}
@@ -748,4 +857,4 @@
| rat_of_int i = (i, 1);
*}
-end
+end
\ No newline at end of file