author | paulson |
Tue, 27 Jan 2004 15:39:51 +0100 | |
changeset 14365 | 3d4df8c166ae |
child 14378 | 69c4d5997669 |
permissions | -rw-r--r-- |
14365
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(* Title: HOL/Library/Rational.thy |
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ID: $Id$ |
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Author: Markus Wenzel, TU Muenchen |
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License: GPL (GNU GENERAL PUBLIC LICENSE) |
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*) |
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|
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header {* |
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\title{Rational numbers} |
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\author{Markus Wenzel} |
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*} |
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|
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theory Rational = Quotient + Ring_and_Field: |
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|
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subsection {* Fractions *} |
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|
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subsubsection {* The type of fractions *} |
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|
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typedef fraction = "{(a, b) :: int \<times> int | a b. b \<noteq> 0}" |
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proof |
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show "(0, 1) \<in> ?fraction" by simp |
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qed |
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|
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constdefs |
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fract :: "int => int => fraction" |
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"fract a b == Abs_fraction (a, b)" |
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num :: "fraction => int" |
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"num Q == fst (Rep_fraction Q)" |
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den :: "fraction => int" |
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"den Q == snd (Rep_fraction Q)" |
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|
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lemma fract_num [simp]: "b \<noteq> 0 ==> num (fract a b) = a" |
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by (simp add: fract_def num_def fraction_def Abs_fraction_inverse) |
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|
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lemma fract_den [simp]: "b \<noteq> 0 ==> den (fract a b) = b" |
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by (simp add: fract_def den_def fraction_def Abs_fraction_inverse) |
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|
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lemma fraction_cases [case_names fract, cases type: fraction]: |
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"(!!a b. Q = fract a b ==> b \<noteq> 0 ==> C) ==> C" |
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proof - |
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assume r: "!!a b. Q = fract a b ==> b \<noteq> 0 ==> C" |
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obtain a b where "Q = fract a b" and "b \<noteq> 0" |
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by (cases Q) (auto simp add: fract_def fraction_def) |
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thus C by (rule r) |
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qed |
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|
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lemma fraction_induct [case_names fract, induct type: fraction]: |
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"(!!a b. b \<noteq> 0 ==> P (fract a b)) ==> P Q" |
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by (cases Q) simp |
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|
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|
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subsubsection {* Equivalence of fractions *} |
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instance fraction :: eqv .. |
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|
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defs (overloaded) |
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equiv_fraction_def: "Q \<sim> R == num Q * den R = num R * den Q" |
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|
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lemma equiv_fraction_iff [iff]: |
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"b \<noteq> 0 ==> b' \<noteq> 0 ==> (fract a b \<sim> fract a' b') = (a * b' = a' * b)" |
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by (simp add: equiv_fraction_def) |
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|
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instance fraction :: equiv |
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proof |
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fix Q R S :: fraction |
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{ |
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show "Q \<sim> Q" |
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proof (induct Q) |
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fix a b :: int |
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assume "b \<noteq> 0" and "b \<noteq> 0" |
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with refl show "fract a b \<sim> fract a b" .. |
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qed |
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next |
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assume "Q \<sim> R" and "R \<sim> S" |
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show "Q \<sim> S" |
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proof (insert prems, induct Q, induct R, induct S) |
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fix a b a' b' a'' b'' :: int |
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assume b: "b \<noteq> 0" and b': "b' \<noteq> 0" and b'': "b'' \<noteq> 0" |
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assume "fract a b \<sim> fract a' b'" hence eq1: "a * b' = a' * b" .. |
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assume "fract a' b' \<sim> fract a'' b''" hence eq2: "a' * b'' = a'' * b'" .. |
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have "a * b'' = a'' * b" |
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proof cases |
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assume "a' = 0" |
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with b' eq1 eq2 have "a = 0 \<and> a'' = 0" by auto |
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thus ?thesis by simp |
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85 |
next |
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86 |
assume a': "a' \<noteq> 0" |
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from eq1 eq2 have "(a * b') * (a' * b'') = (a' * b) * (a'' * b')" by simp |
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hence "(a * b'') * (a' * b') = (a'' * b) * (a' * b')" by (simp only: mult_ac) |
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with a' b' show ?thesis by simp |
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90 |
qed |
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91 |
thus "fract a b \<sim> fract a'' b''" .. |
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92 |
qed |
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93 |
next |
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94 |
show "Q \<sim> R ==> R \<sim> Q" |
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proof (induct Q, induct R) |
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96 |
fix a b a' b' :: int |
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97 |
assume b: "b \<noteq> 0" and b': "b' \<noteq> 0" |
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98 |
assume "fract a b \<sim> fract a' b'" |
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99 |
hence "a * b' = a' * b" .. |
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100 |
hence "a' * b = a * b'" .. |
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101 |
thus "fract a' b' \<sim> fract a b" .. |
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102 |
qed |
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103 |
} |
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104 |
qed |
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105 |
|
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106 |
lemma eq_fraction_iff [iff]: |
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107 |
"b \<noteq> 0 ==> b' \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>) = (a * b' = a' * b)" |
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108 |
by (simp add: equiv_fraction_iff quot_equality) |
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109 |
|
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110 |
|
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111 |
subsubsection {* Operations on fractions *} |
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112 |
|
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113 |
text {* |
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114 |
We define the basic arithmetic operations on fractions and |
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115 |
demonstrate their ``well-definedness'', i.e.\ congruence with respect |
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116 |
to equivalence of fractions. |
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117 |
*} |
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118 |
|
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119 |
instance fraction :: zero .. |
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120 |
instance fraction :: one .. |
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121 |
instance fraction :: plus .. |
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122 |
instance fraction :: minus .. |
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123 |
instance fraction :: times .. |
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124 |
instance fraction :: inverse .. |
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|
125 |
instance fraction :: ord .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
126 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
127 |
defs (overloaded) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
128 |
zero_fraction_def: "0 == fract 0 1" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
129 |
one_fraction_def: "1 == fract 1 1" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
130 |
add_fraction_def: "Q + R == |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
131 |
fract (num Q * den R + num R * den Q) (den Q * den R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
132 |
minus_fraction_def: "-Q == fract (-(num Q)) (den Q)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
133 |
mult_fraction_def: "Q * R == fract (num Q * num R) (den Q * den R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
134 |
inverse_fraction_def: "inverse Q == fract (den Q) (num Q)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
135 |
le_fraction_def: "Q \<le> R == |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
136 |
(num Q * den R) * (den Q * den R) \<le> (num R * den Q) * (den Q * den R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
137 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
138 |
lemma is_zero_fraction_iff: "b \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>0\<rfloor>) = (a = 0)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
139 |
by (simp add: zero_fraction_def eq_fraction_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
140 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
141 |
theorem add_fraction_cong: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
142 |
"\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
143 |
==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
144 |
==> \<lfloor>fract a b + fract c d\<rfloor> = \<lfloor>fract a' b' + fract c' d'\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
145 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
146 |
assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
147 |
assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
148 |
assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
149 |
have "\<lfloor>fract (a * d + c * b) (b * d)\<rfloor> = \<lfloor>fract (a' * d' + c' * b') (b' * d')\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
150 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
151 |
show "(a * d + c * b) * (b' * d') = (a' * d' + c' * b') * (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
152 |
(is "?lhs = ?rhs") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
153 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
154 |
have "?lhs = (a * b') * (d * d') + (c * d') * (b * b')" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
155 |
by (simp add: int_distrib mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
156 |
also have "... = (a' * b) * (d * d') + (c' * d) * (b * b')" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
157 |
by (simp only: eq1 eq2) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
158 |
also have "... = ?rhs" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
159 |
by (simp add: int_distrib mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
160 |
finally show "?lhs = ?rhs" . |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
161 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
162 |
from neq show "b * d \<noteq> 0" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
163 |
from neq show "b' * d' \<noteq> 0" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
164 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
165 |
with neq show ?thesis by (simp add: add_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
166 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
167 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
168 |
theorem minus_fraction_cong: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
169 |
"\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
170 |
==> \<lfloor>-(fract a b)\<rfloor> = \<lfloor>-(fract a' b')\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
171 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
172 |
assume neq: "b \<noteq> 0" "b' \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
173 |
assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
174 |
hence "a * b' = a' * b" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
175 |
hence "-a * b' = -a' * b" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
176 |
hence "\<lfloor>fract (-a) b\<rfloor> = \<lfloor>fract (-a') b'\<rfloor>" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
177 |
with neq show ?thesis by (simp add: minus_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
178 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
179 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
180 |
theorem mult_fraction_cong: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
181 |
"\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
182 |
==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
183 |
==> \<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
184 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
185 |
assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
186 |
assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
187 |
assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
188 |
have "\<lfloor>fract (a * c) (b * d)\<rfloor> = \<lfloor>fract (a' * c') (b' * d')\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
189 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
190 |
from eq1 eq2 have "(a * b') * (c * d') = (a' * b) * (c' * d)" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
191 |
thus "(a * c) * (b' * d') = (a' * c') * (b * d)" by (simp add: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
192 |
from neq show "b * d \<noteq> 0" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
193 |
from neq show "b' * d' \<noteq> 0" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
194 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
195 |
with neq show "\<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
196 |
by (simp add: mult_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
197 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
198 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
199 |
theorem inverse_fraction_cong: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
200 |
"\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor> ==> \<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
201 |
==> b \<noteq> 0 ==> b' \<noteq> 0 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
202 |
==> \<lfloor>inverse (fract a b)\<rfloor> = \<lfloor>inverse (fract a' b')\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
203 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
204 |
assume neq: "b \<noteq> 0" "b' \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
205 |
assume "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>" and "\<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
206 |
with neq obtain "a \<noteq> 0" and "a' \<noteq> 0" by (simp add: is_zero_fraction_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
207 |
assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
208 |
hence "a * b' = a' * b" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
209 |
hence "b * a' = b' * a" by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
210 |
hence "\<lfloor>fract b a\<rfloor> = \<lfloor>fract b' a'\<rfloor>" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
211 |
with neq show ?thesis by (simp add: inverse_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
212 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
213 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
214 |
theorem le_fraction_cong: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
215 |
"\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
216 |
==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
217 |
==> (fract a b \<le> fract c d) = (fract a' b' \<le> fract c' d')" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
218 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
219 |
assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
220 |
assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
221 |
assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
222 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
223 |
let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
224 |
{ |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
225 |
fix a b c d x :: int assume x: "x \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
226 |
have "?le a b c d = ?le (a * x) (b * x) c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
227 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
228 |
from x have "0 < x * x" by (auto simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
229 |
hence "?le a b c d = |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
230 |
((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
231 |
by (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
232 |
also have "... = ?le (a * x) (b * x) c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
233 |
by (simp add: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
234 |
finally show ?thesis . |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
235 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
236 |
} note le_factor = this |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
237 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
238 |
let ?D = "b * d" and ?D' = "b' * d'" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
239 |
from neq have D: "?D \<noteq> 0" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
240 |
from neq have "?D' \<noteq> 0" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
241 |
hence "?le a b c d = ?le (a * ?D') (b * ?D') c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
242 |
by (rule le_factor) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
243 |
also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
244 |
by (simp add: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
245 |
also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
246 |
by (simp only: eq1 eq2) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
247 |
also have "... = ?le (a' * ?D) (b' * ?D) c' d'" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
248 |
by (simp add: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
249 |
also from D have "... = ?le a' b' c' d'" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
250 |
by (rule le_factor [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
251 |
finally have "?le a b c d = ?le a' b' c' d'" . |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
252 |
with neq show ?thesis by (simp add: le_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
253 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
254 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
255 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
256 |
subsection {* Rational numbers *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
257 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
258 |
subsubsection {* The type of rational numbers *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
259 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
260 |
typedef (Rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
261 |
rat = "UNIV :: fraction quot set" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
262 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
263 |
lemma RatI [intro, simp]: "Q \<in> Rat" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
264 |
by (simp add: Rat_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
265 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
266 |
constdefs |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
267 |
fraction_of :: "rat => fraction" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
268 |
"fraction_of q == pick (Rep_Rat q)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
269 |
rat_of :: "fraction => rat" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
270 |
"rat_of Q == Abs_Rat \<lfloor>Q\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
271 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
272 |
theorem rat_of_equality [iff?]: "(rat_of Q = rat_of Q') = (\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor>)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
273 |
by (simp add: rat_of_def Abs_Rat_inject) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
274 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
275 |
lemma rat_of: "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> rat_of Q = rat_of Q'" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
276 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
277 |
constdefs |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
278 |
Fract :: "int => int => rat" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
279 |
"Fract a b == rat_of (fract a b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
280 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
281 |
theorem Fract_inverse: "\<lfloor>fraction_of (Fract a b)\<rfloor> = \<lfloor>fract a b\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
282 |
by (simp add: fraction_of_def rat_of_def Fract_def Abs_Rat_inverse pick_inverse) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
283 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
284 |
theorem Fract_equality [iff?]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
285 |
"(Fract a b = Fract c d) = (\<lfloor>fract a b\<rfloor> = \<lfloor>fract c d\<rfloor>)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
286 |
by (simp add: Fract_def rat_of_equality) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
287 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
288 |
theorem eq_rat: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
289 |
"b \<noteq> 0 ==> d \<noteq> 0 ==> (Fract a b = Fract c d) = (a * d = c * b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
290 |
by (simp add: Fract_equality eq_fraction_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
291 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
292 |
theorem Rat_cases [case_names Fract, cases type: rat]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
293 |
"(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
294 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
295 |
assume r: "!!a b. q = Fract a b ==> b \<noteq> 0 ==> C" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
296 |
obtain x where "q = Abs_Rat x" by (cases q) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
297 |
moreover obtain Q where "x = \<lfloor>Q\<rfloor>" by (cases x) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
298 |
moreover obtain a b where "Q = fract a b" and "b \<noteq> 0" by (cases Q) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
299 |
ultimately have "q = Fract a b" by (simp only: Fract_def rat_of_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
300 |
thus ?thesis by (rule r) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
301 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
302 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
303 |
theorem Rat_induct [case_names Fract, induct type: rat]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
304 |
"(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
305 |
by (cases q) simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
306 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
307 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
308 |
subsubsection {* Canonical function definitions *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
309 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
310 |
text {* |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
311 |
Note that the unconditional version below is much easier to read. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
312 |
*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
313 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
314 |
theorem rat_cond_function: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
315 |
"(!!q r. P \<lfloor>fraction_of q\<rfloor> \<lfloor>fraction_of r\<rfloor> ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
316 |
f q r == g (fraction_of q) (fraction_of r)) ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
317 |
(!!a b a' b' c d c' d'. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
318 |
\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
319 |
P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==> P \<lfloor>fract a' b'\<rfloor> \<lfloor>fract c' d'\<rfloor> ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
320 |
b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
321 |
g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
322 |
P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
323 |
f (Fract a b) (Fract c d) = g (fract a b) (fract c d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
324 |
(is "PROP ?eq ==> PROP ?cong ==> ?P ==> _") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
325 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
326 |
assume eq: "PROP ?eq" and cong: "PROP ?cong" and P: ?P |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
327 |
have "f (Abs_Rat \<lfloor>fract a b\<rfloor>) (Abs_Rat \<lfloor>fract c d\<rfloor>) = g (fract a b) (fract c d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
328 |
proof (rule quot_cond_function) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
329 |
fix X Y assume "P X Y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
330 |
with eq show "f (Abs_Rat X) (Abs_Rat Y) == g (pick X) (pick Y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
331 |
by (simp add: fraction_of_def pick_inverse Abs_Rat_inverse) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
332 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
333 |
fix Q Q' R R' :: fraction |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
334 |
show "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> \<lfloor>R\<rfloor> = \<lfloor>R'\<rfloor> ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
335 |
P \<lfloor>Q\<rfloor> \<lfloor>R\<rfloor> ==> P \<lfloor>Q'\<rfloor> \<lfloor>R'\<rfloor> ==> g Q R = g Q' R'" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
336 |
by (induct Q, induct Q', induct R, induct R') (rule cong) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
337 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
338 |
thus ?thesis by (unfold Fract_def rat_of_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
339 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
340 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
341 |
theorem rat_function: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
342 |
"(!!q r. f q r == g (fraction_of q) (fraction_of r)) ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
343 |
(!!a b a' b' c d c' d'. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
344 |
\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
345 |
b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
346 |
g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
347 |
f (Fract a b) (Fract c d) = g (fract a b) (fract c d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
348 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
349 |
case rule_context from this TrueI |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
350 |
show ?thesis by (rule rat_cond_function) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
351 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
352 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
353 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
354 |
subsubsection {* Standard operations on rational numbers *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
355 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
356 |
instance rat :: zero .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
357 |
instance rat :: one .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
358 |
instance rat :: plus .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
359 |
instance rat :: minus .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
360 |
instance rat :: times .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
361 |
instance rat :: inverse .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
362 |
instance rat :: ord .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
363 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
364 |
defs (overloaded) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
365 |
zero_rat_def: "0 == rat_of 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
366 |
one_rat_def: "1 == rat_of 1" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
367 |
add_rat_def: "q + r == rat_of (fraction_of q + fraction_of r)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
368 |
minus_rat_def: "-q == rat_of (-(fraction_of q))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
369 |
diff_rat_def: "q - r == q + (-(r::rat))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
370 |
mult_rat_def: "q * r == rat_of (fraction_of q * fraction_of r)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
371 |
inverse_rat_def: "inverse q == |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
372 |
if q=0 then 0 else rat_of (inverse (fraction_of q))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
373 |
divide_rat_def: "q / r == q * inverse (r::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
374 |
le_rat_def: "q \<le> r == fraction_of q \<le> fraction_of r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
375 |
less_rat_def: "q < r == q \<le> r \<and> q \<noteq> (r::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
376 |
abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
377 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
378 |
theorem zero_rat: "0 = Fract 0 1" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
379 |
by (simp add: zero_rat_def zero_fraction_def rat_of_def Fract_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
380 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
381 |
theorem one_rat: "1 = Fract 1 1" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
382 |
by (simp add: one_rat_def one_fraction_def rat_of_def Fract_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
383 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
384 |
theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
385 |
Fract a b + Fract c d = Fract (a * d + c * b) (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
386 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
387 |
have "Fract a b + Fract c d = rat_of (fract a b + fract c d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
388 |
by (rule rat_function, rule add_rat_def, rule rat_of, rule add_fraction_cong) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
389 |
also |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
390 |
assume "b \<noteq> 0" "d \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
391 |
hence "fract a b + fract c d = fract (a * d + c * b) (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
392 |
by (simp add: add_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
393 |
finally show ?thesis by (unfold Fract_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
394 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
395 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
396 |
theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
397 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
398 |
have "-(Fract a b) = rat_of (-(fract a b))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
399 |
by (rule rat_function, rule minus_rat_def, rule rat_of, rule minus_fraction_cong) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
400 |
also assume "b \<noteq> 0" hence "-(fract a b) = fract (-a) b" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
401 |
by (simp add: minus_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
402 |
finally show ?thesis by (unfold Fract_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
403 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
404 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
405 |
theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
406 |
Fract a b - Fract c d = Fract (a * d - c * b) (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
407 |
by (simp add: diff_rat_def add_rat minus_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
408 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
409 |
theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
410 |
Fract a b * Fract c d = Fract (a * c) (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
411 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
412 |
have "Fract a b * Fract c d = rat_of (fract a b * fract c d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
413 |
by (rule rat_function, rule mult_rat_def, rule rat_of, rule mult_fraction_cong) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
414 |
also |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
415 |
assume "b \<noteq> 0" "d \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
416 |
hence "fract a b * fract c d = fract (a * c) (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
417 |
by (simp add: mult_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
418 |
finally show ?thesis by (unfold Fract_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
419 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
420 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
421 |
theorem inverse_rat: "Fract a b \<noteq> 0 ==> b \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
422 |
inverse (Fract a b) = Fract b a" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
423 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
424 |
assume neq: "b \<noteq> 0" and nonzero: "Fract a b \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
425 |
hence "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
426 |
by (simp add: zero_rat eq_rat is_zero_fraction_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
427 |
with _ inverse_fraction_cong [THEN rat_of] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
428 |
have "inverse (Fract a b) = rat_of (inverse (fract a b))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
429 |
proof (rule rat_cond_function) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
430 |
fix q assume cond: "\<lfloor>fraction_of q\<rfloor> \<noteq> \<lfloor>0\<rfloor>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
431 |
have "q \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
432 |
proof (cases q) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
433 |
fix a b assume "b \<noteq> 0" and "q = Fract a b" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
434 |
from this cond show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
435 |
by (simp add: Fract_inverse is_zero_fraction_iff zero_rat eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
436 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
437 |
thus "inverse q == rat_of (inverse (fraction_of q))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
438 |
by (simp add: inverse_rat_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
439 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
440 |
also from neq nonzero have "inverse (fract a b) = fract b a" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
441 |
by (simp add: inverse_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
442 |
finally show ?thesis by (unfold Fract_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
443 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
444 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
445 |
theorem divide_rat: "Fract c d \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
446 |
Fract a b / Fract c d = Fract (a * d) (b * c)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
447 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
448 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" and nonzero: "Fract c d \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
449 |
hence "c \<noteq> 0" by (simp add: zero_rat eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
450 |
with neq nonzero show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
451 |
by (simp add: divide_rat_def inverse_rat mult_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
452 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
453 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
454 |
theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
455 |
(Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
456 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
457 |
have "(Fract a b \<le> Fract c d) = (fract a b \<le> fract c d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
458 |
by (rule rat_function, rule le_rat_def, rule le_fraction_cong) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
459 |
also |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
460 |
assume "b \<noteq> 0" "d \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
461 |
hence "(fract a b \<le> fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
462 |
by (simp add: le_fraction_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
463 |
finally show ?thesis . |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
464 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
465 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
466 |
theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
467 |
(Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
468 |
by (simp add: less_rat_def le_rat eq_rat int_less_le) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
469 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
470 |
theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
471 |
by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
472 |
(auto simp add: mult_less_0_iff zero_less_mult_iff int_le_less |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
473 |
split: abs_split) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
474 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
475 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
476 |
subsubsection {* The ordered field of rational numbers *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
477 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
478 |
lemma rat_add_assoc: "(q + r) + s = q + (r + (s::rat))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
479 |
by (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
480 |
(simp add: add_rat add_ac mult_ac int_distrib) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
481 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
482 |
lemma rat_add_0: "0 + q = (q::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
483 |
by (induct q) (simp add: zero_rat add_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
484 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
485 |
lemma rat_left_minus: "(-q) + q = (0::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
486 |
by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
487 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
488 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
489 |
instance rat :: field |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
490 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
491 |
fix q r s :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
492 |
show "(q + r) + s = q + (r + s)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
493 |
by (rule rat_add_assoc) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
494 |
show "q + r = r + q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
495 |
by (induct q, induct r) (simp add: add_rat add_ac mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
496 |
show "0 + q = q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
497 |
by (induct q) (simp add: zero_rat add_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
498 |
show "(-q) + q = 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
499 |
by (rule rat_left_minus) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
500 |
show "q - r = q + (-r)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
501 |
by (induct q, induct r) (simp add: add_rat minus_rat diff_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
502 |
show "(q * r) * s = q * (r * s)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
503 |
by (induct q, induct r, induct s) (simp add: mult_rat mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
504 |
show "q * r = r * q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
505 |
by (induct q, induct r) (simp add: mult_rat mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
506 |
show "1 * q = q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
507 |
by (induct q) (simp add: one_rat mult_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
508 |
show "(q + r) * s = q * s + r * s" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
509 |
by (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
510 |
(simp add: add_rat mult_rat eq_rat int_distrib) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
511 |
show "q \<noteq> 0 ==> inverse q * q = 1" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
512 |
by (induct q) (simp add: inverse_rat mult_rat one_rat zero_rat eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
513 |
show "r \<noteq> 0 ==> q / r = q * inverse r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
514 |
by (induct q, induct r) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
515 |
(simp add: mult_rat divide_rat inverse_rat zero_rat eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
516 |
show "0 \<noteq> (1::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
517 |
by (simp add: zero_rat one_rat eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
518 |
assume eq: "s+q = s+r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
519 |
hence "(-s + s) + q = (-s + s) + r" by (simp only: eq rat_add_assoc) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
520 |
thus "q = r" by (simp add: rat_left_minus rat_add_0) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
521 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
522 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
523 |
instance rat :: linorder |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
524 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
525 |
fix q r s :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
526 |
{ |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
527 |
assume "q \<le> r" and "r \<le> s" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
528 |
show "q \<le> s" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
529 |
proof (insert prems, induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
530 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
531 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
532 |
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
533 |
show "Fract a b \<le> Fract e f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
534 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
535 |
from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
536 |
by (auto simp add: zero_less_mult_iff linorder_neq_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
537 |
have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
538 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
539 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
540 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
541 |
with ff show ?thesis by (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
542 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
543 |
also have "... = (c * f) * (d * f) * (b * b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
544 |
by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
545 |
also have "... \<le> (e * d) * (d * f) * (b * b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
546 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
547 |
from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
548 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
549 |
with bb show ?thesis by (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
550 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
551 |
finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
552 |
by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
553 |
with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
554 |
by (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
555 |
with neq show ?thesis by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
556 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
557 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
558 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
559 |
assume "q \<le> r" and "r \<le> q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
560 |
show "q = r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
561 |
proof (insert prems, induct q, induct r) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
562 |
fix a b c d :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
563 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
564 |
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
565 |
show "Fract a b = Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
566 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
567 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
568 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
569 |
also have "... \<le> (a * d) * (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
570 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
571 |
from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
572 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
573 |
thus ?thesis by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
574 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
575 |
finally have "(a * d) * (b * d) = (c * b) * (b * d)" . |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
576 |
moreover from neq have "b * d \<noteq> 0" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
577 |
ultimately have "a * d = c * b" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
578 |
with neq show ?thesis by (simp add: eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
579 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
580 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
581 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
582 |
show "q \<le> q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
583 |
by (induct q) (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
584 |
show "(q < r) = (q \<le> r \<and> q \<noteq> r)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
585 |
by (simp only: less_rat_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
586 |
show "q \<le> r \<or> r \<le> q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
587 |
by (induct q, induct r) (simp add: le_rat mult_ac, arith) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
588 |
} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
589 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
590 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
591 |
instance rat :: ordered_field |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
592 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
593 |
fix q r s :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
594 |
show "0 < (1::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
595 |
by (simp add: zero_rat one_rat less_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
596 |
show "q \<le> r ==> s + q \<le> s + r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
597 |
proof (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
598 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
599 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
600 |
assume le: "Fract a b \<le> Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
601 |
show "Fract e f + Fract a b \<le> Fract e f + Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
602 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
603 |
let ?F = "f * f" from neq have F: "0 < ?F" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
604 |
by (auto simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
605 |
from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
606 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
607 |
with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
608 |
by (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
609 |
with neq show ?thesis by (simp add: add_rat le_rat mult_ac int_distrib) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
610 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
611 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
612 |
show "q < r ==> 0 < s ==> s * q < s * r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
613 |
proof (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
614 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
615 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
616 |
assume le: "Fract a b < Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
617 |
assume gt: "0 < Fract e f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
618 |
show "Fract e f * Fract a b < Fract e f * Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
619 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
620 |
let ?E = "e * f" and ?F = "f * f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
621 |
from neq gt have "0 < ?E" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
622 |
by (auto simp add: zero_rat less_rat le_rat int_less_le eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
623 |
moreover from neq have "0 < ?F" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
624 |
by (auto simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
625 |
moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
626 |
by (simp add: less_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
627 |
ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
628 |
by (simp add: mult_less_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
629 |
with neq show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
630 |
by (simp add: less_rat mult_rat mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
631 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
632 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
633 |
show "\<bar>q\<bar> = (if q < 0 then -q else q)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
634 |
by (simp only: abs_rat_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
635 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
636 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
637 |
instance rat :: division_by_zero |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
638 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
639 |
fix x :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
640 |
show "inverse 0 = (0::rat)" by (simp add: inverse_rat_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
641 |
show "x/0 = 0" by (simp add: divide_rat_def inverse_rat_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
642 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
643 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
644 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
645 |
subsection {* Embedding integers: The Injection @{term rat} *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
646 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
647 |
constdefs |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
648 |
rat :: "int => rat" (* FIXME generalize int to any numeric subtype (?) *) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
649 |
"rat z == Fract z 1" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
650 |
int_set :: "rat set" ("\<int>") (* FIXME generalize rat to any numeric supertype (?) *) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
651 |
"\<int> == range rat" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
652 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
653 |
lemma int_set_cases [case_names rat, cases set: int_set]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
654 |
"q \<in> \<int> ==> (!!z. q = rat z ==> C) ==> C" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
655 |
proof (unfold int_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
656 |
assume "!!z. q = rat z ==> C" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
657 |
assume "q \<in> range rat" thus C .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
658 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
659 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
660 |
lemma int_set_induct [case_names rat, induct set: int_set]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
661 |
"q \<in> \<int> ==> (!!z. P (rat z)) ==> P q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
662 |
by (rule int_set_cases) auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
663 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
664 |
lemma rat_of_int_zero [simp]: "rat (0::int) = (0::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
665 |
by (simp add: rat_def zero_rat [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
666 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
667 |
lemma rat_of_int_one [simp]: "rat (1::int) = (1::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
668 |
by (simp add: rat_def one_rat [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
669 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
670 |
lemma rat_of_int_add_distrib [simp]: "rat (x + y) = rat (x::int) + rat y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
671 |
by (simp add: rat_def add_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
672 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
673 |
lemma rat_of_int_minus_distrib [simp]: "rat (-x) = -rat (x::int)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
674 |
by (simp add: rat_def minus_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
675 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
676 |
lemma rat_of_int_diff_distrib [simp]: "rat (x - y) = rat (x::int) - rat y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
677 |
by (simp add: rat_def diff_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
678 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
679 |
lemma rat_of_int_mult_distrib [simp]: "rat (x * y) = rat (x::int) * rat y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
680 |
by (simp add: rat_def mult_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
681 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
682 |
lemma rat_inject [simp]: "(rat z = rat w) = (z = w)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
683 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
684 |
assume "rat z = rat w" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
685 |
hence "Fract z 1 = Fract w 1" by (unfold rat_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
686 |
hence "\<lfloor>fract z 1\<rfloor> = \<lfloor>fract w 1\<rfloor>" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
687 |
thus "z = w" by auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
688 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
689 |
assume "z = w" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
690 |
thus "rat z = rat w" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
691 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
692 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
693 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
694 |
lemma rat_of_int_zero_cancel [simp]: "(rat x = 0) = (x = 0)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
695 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
696 |
have "(rat x = 0) = (rat x = rat 0)" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
697 |
also have "... = (x = 0)" by (rule rat_inject) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
698 |
finally show ?thesis . |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
699 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
700 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
701 |
lemma rat_of_int_less_iff [simp]: "rat (x::int) < rat y = (x < y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
702 |
by (simp add: rat_def less_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
703 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
704 |
lemma rat_of_int_le_iff [simp]: "(rat (x::int) \<le> rat y) = (x \<le> y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
705 |
by (simp add: linorder_not_less [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
706 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
707 |
lemma zero_less_rat_of_int_iff [simp]: "(0 < rat y) = (0 < y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
708 |
by (insert rat_of_int_less_iff [of 0 y], simp) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
709 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
710 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
711 |
subsection {* Various Other Results *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
712 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
713 |
lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
714 |
by (simp add: Fract_equality eq_fraction_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
715 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
716 |
theorem Rat_induct_pos [case_names Fract, induct type: rat]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
717 |
assumes step: "!!a b. 0 < b ==> P (Fract a b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
718 |
shows "P q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
719 |
proof (cases q) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
720 |
have step': "!!a b. b < 0 ==> P (Fract a b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
721 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
722 |
fix a::int and b::int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
723 |
assume b: "b < 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
724 |
hence "0 < -b" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
725 |
hence "P (Fract (-a) (-b))" by (rule step) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
726 |
thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
727 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
728 |
case (Fract a b) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
729 |
thus "P q" by (force simp add: linorder_neq_iff step step') |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
730 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
731 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
732 |
lemma zero_less_Fract_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
733 |
"0 < b ==> (0 < Fract a b) = (0 < a)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
734 |
by (simp add: zero_rat less_rat order_less_imp_not_eq2 zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
735 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
736 |
end |