author | ballarin |
Thu, 19 Feb 2004 15:57:34 +0100 | |
changeset 14398 | c5c47703f763 |
parent 14387 | e96d5c42c4b0 |
child 14421 | ee97b6463cb4 |
permissions | -rw-r--r-- |
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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|
1 |
(* Title: HOL/Ring_and_Field.thy |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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|
2 |
ID: $Id$ |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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|
3 |
Author: Gertrud Bauer and Markus Wenzel, TU Muenchen |
14269 | 4 |
Lawrence C Paulson, University of Cambridge |
14265
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
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5 |
License: GPL (GNU GENERAL PUBLIC LICENSE) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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6 |
*) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
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7 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
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8 |
header {* |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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9 |
\title{Ring and field structures} |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
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10 |
\author{Gertrud Bauer, L. C. Paulson and Markus Wenzel} |
14265
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
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11 |
*} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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12 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
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13 |
theory Ring_and_Field = Inductive: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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14 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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15 |
subsection {* Abstract algebraic structures *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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16 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
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17 |
axclass semiring \<subseteq> zero, one, plus, times |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
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18 |
add_assoc: "(a + b) + c = a + (b + c)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
19 |
add_commute: "a + b = b + a" |
14288 | 20 |
add_0 [simp]: "0 + a = a" |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
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21 |
add_left_imp_eq: "a + b = a + c ==> b=c" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
22 |
--{*This axiom is needed for semirings only: for rings, etc., it is |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
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23 |
redundant. Including it allows many more of the following results |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
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24 |
to be proved for semirings too. The drawback is that this redundant |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
25 |
axiom must be proved for instances of rings.*} |
14265
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
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26 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
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27 |
mult_assoc: "(a * b) * c = a * (b * c)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
28 |
mult_commute: "a * b = b * a" |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
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29 |
mult_1 [simp]: "1 * a = a" |
14265
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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30 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
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31 |
left_distrib: "(a + b) * c = a * c + b * c" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
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32 |
zero_neq_one [simp]: "0 \<noteq> 1" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
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33 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
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34 |
axclass ring \<subseteq> semiring, minus |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
35 |
left_minus [simp]: "- a + a = 0" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
36 |
diff_minus: "a - b = a + (-b)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
37 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
38 |
axclass ordered_semiring \<subseteq> semiring, linorder |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
39 |
zero_less_one [simp]: "0 < 1" --{*This too is needed for semirings only.*} |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
40 |
add_left_mono: "a \<le> b ==> c + a \<le> c + b" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
41 |
mult_strict_left_mono: "a < b ==> 0 < c ==> c * a < c * b" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
42 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
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43 |
axclass ordered_ring \<subseteq> ordered_semiring, ring |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
44 |
abs_if: "\<bar>a\<bar> = (if a < 0 then -a else a)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
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45 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
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46 |
axclass field \<subseteq> ring, inverse |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
47 |
left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
48 |
divide_inverse: "b \<noteq> 0 ==> a / b = a * inverse b" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
49 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
50 |
axclass ordered_field \<subseteq> ordered_ring, field |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
51 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
52 |
axclass division_by_zero \<subseteq> zero, inverse |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
53 |
inverse_zero [simp]: "inverse 0 = 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
54 |
divide_zero [simp]: "a / 0 = 0" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
55 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
56 |
|
14270 | 57 |
subsection {* Derived Rules for Addition *} |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
58 |
|
14288 | 59 |
lemma add_0_right [simp]: "a + 0 = (a::'a::semiring)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
60 |
proof - |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
61 |
have "a + 0 = 0 + a" by (simp only: add_commute) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
62 |
also have "... = a" by simp |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
63 |
finally show ?thesis . |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
64 |
qed |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
65 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
66 |
lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::semiring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
67 |
by (rule mk_left_commute [of "op +", OF add_assoc add_commute]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
68 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
69 |
theorems add_ac = add_assoc add_commute add_left_commute |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
70 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
71 |
lemma right_minus [simp]: "a + -(a::'a::ring) = 0" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
72 |
proof - |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
73 |
have "a + -a = -a + a" by (simp add: add_ac) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
74 |
also have "... = 0" by simp |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
75 |
finally show ?thesis . |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
76 |
qed |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
77 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
78 |
lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::ring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
79 |
proof |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
80 |
have "a = a - b + b" by (simp add: diff_minus add_ac) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
81 |
also assume "a - b = 0" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
82 |
finally show "a = b" by simp |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
83 |
next |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
84 |
assume "a = b" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
85 |
thus "a - b = 0" by (simp add: diff_minus) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
86 |
qed |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
87 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
88 |
lemma add_left_cancel [simp]: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
89 |
"(a + b = a + c) = (b = (c::'a::semiring))" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
90 |
by (blast dest: add_left_imp_eq) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
91 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
92 |
lemma add_right_cancel [simp]: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
93 |
"(b + a = c + a) = (b = (c::'a::semiring))" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
94 |
by (simp add: add_commute) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
95 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
96 |
lemma minus_minus [simp]: "- (- (a::'a::ring)) = a" |
14377 | 97 |
proof (rule add_left_cancel [of "-a", THEN iffD1]) |
98 |
show "(-a + -(-a) = -a + a)" |
|
99 |
by simp |
|
100 |
qed |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
101 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
102 |
lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::ring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
103 |
apply (rule right_minus_eq [THEN iffD1, symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
104 |
apply (simp add: diff_minus add_commute) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
105 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
106 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
107 |
lemma minus_zero [simp]: "- 0 = (0::'a::ring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
108 |
by (simp add: equals_zero_I) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
109 |
|
14270 | 110 |
lemma diff_self [simp]: "a - (a::'a::ring) = 0" |
111 |
by (simp add: diff_minus) |
|
112 |
||
113 |
lemma diff_0 [simp]: "(0::'a::ring) - a = -a" |
|
114 |
by (simp add: diff_minus) |
|
115 |
||
116 |
lemma diff_0_right [simp]: "a - (0::'a::ring) = a" |
|
117 |
by (simp add: diff_minus) |
|
118 |
||
14288 | 119 |
lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::ring)" |
120 |
by (simp add: diff_minus) |
|
121 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
122 |
lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ring))" |
14377 | 123 |
proof |
124 |
assume "- a = - b" |
|
125 |
hence "- (- a) = - (- b)" |
|
126 |
by simp |
|
127 |
thus "a=b" by simp |
|
128 |
next |
|
129 |
assume "a=b" |
|
130 |
thus "-a = -b" by simp |
|
131 |
qed |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
132 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
133 |
lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
134 |
by (subst neg_equal_iff_equal [symmetric], simp) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
135 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
136 |
lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
137 |
by (subst neg_equal_iff_equal [symmetric], simp) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
138 |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
139 |
text{*The next two equations can make the simplifier loop!*} |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
140 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
141 |
lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ring))" |
14377 | 142 |
proof - |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
143 |
have "(- (-a) = - b) = (- a = b)" by (rule neg_equal_iff_equal) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
144 |
thus ?thesis by (simp add: eq_commute) |
14377 | 145 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
146 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
147 |
lemma minus_equation_iff: "(- a = b) = (- (b::'a::ring) = a)" |
14377 | 148 |
proof - |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
149 |
have "(- a = - (-b)) = (a = -b)" by (rule neg_equal_iff_equal) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
150 |
thus ?thesis by (simp add: eq_commute) |
14377 | 151 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
152 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
153 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
154 |
subsection {* Derived rules for multiplication *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
155 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
156 |
lemma mult_1_right [simp]: "a * (1::'a::semiring) = a" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
157 |
proof - |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
158 |
have "a * 1 = 1 * a" by (simp add: mult_commute) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
159 |
also have "... = a" by simp |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
160 |
finally show ?thesis . |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
161 |
qed |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
162 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
163 |
lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::semiring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
164 |
by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
165 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
166 |
theorems mult_ac = mult_assoc mult_commute mult_left_commute |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
167 |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
168 |
lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
169 |
proof - |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
170 |
have "0*a + 0*a = 0*a + 0" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
171 |
by (simp add: left_distrib [symmetric]) |
14266 | 172 |
thus ?thesis by (simp only: add_left_cancel) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
173 |
qed |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
174 |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
175 |
lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
176 |
by (simp add: mult_commute) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
177 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
178 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
179 |
subsection {* Distribution rules *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
180 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
181 |
lemma right_distrib: "a * (b + c) = a * b + a * (c::'a::semiring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
182 |
proof - |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
183 |
have "a * (b + c) = (b + c) * a" by (simp add: mult_ac) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
184 |
also have "... = b * a + c * a" by (simp only: left_distrib) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
185 |
also have "... = a * b + a * c" by (simp add: mult_ac) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
186 |
finally show ?thesis . |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
187 |
qed |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
188 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
189 |
theorems ring_distrib = right_distrib left_distrib |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
190 |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
191 |
text{*For the @{text combine_numerals} simproc*} |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
192 |
lemma combine_common_factor: "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
193 |
by (simp add: left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
194 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
195 |
lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
196 |
apply (rule equals_zero_I) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
197 |
apply (simp add: add_ac) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
198 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
199 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
200 |
lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
201 |
apply (rule equals_zero_I) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
202 |
apply (simp add: left_distrib [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
203 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
204 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
205 |
lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
206 |
apply (rule equals_zero_I) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
207 |
apply (simp add: right_distrib [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
208 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
209 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
210 |
lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
211 |
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
212 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
213 |
lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
214 |
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
215 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
216 |
lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
217 |
by (simp add: right_distrib diff_minus |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
218 |
minus_mult_left [symmetric] minus_mult_right [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
219 |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
220 |
lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
221 |
by (simp add: mult_commute [of _ c] right_diff_distrib) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
222 |
|
14270 | 223 |
lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ring)" |
224 |
by (simp add: diff_minus add_commute) |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
225 |
|
14270 | 226 |
|
227 |
subsection {* Ordering Rules for Addition *} |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
228 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
229 |
lemma add_right_mono: "a \<le> (b::'a::ordered_semiring) ==> a + c \<le> b + c" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
230 |
by (simp add: add_commute [of _ c] add_left_mono) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
231 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
232 |
text {* non-strict, in both arguments *} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
233 |
lemma add_mono: "[|a \<le> b; c \<le> d|] ==> a + c \<le> b + (d::'a::ordered_semiring)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
234 |
apply (erule add_right_mono [THEN order_trans]) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
235 |
apply (simp add: add_commute add_left_mono) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
236 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
237 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
238 |
lemma add_strict_left_mono: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
239 |
"a < b ==> c + a < c + (b::'a::ordered_semiring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
240 |
by (simp add: order_less_le add_left_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
241 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
242 |
lemma add_strict_right_mono: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
243 |
"a < b ==> a + c < b + (c::'a::ordered_semiring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
244 |
by (simp add: add_commute [of _ c] add_strict_left_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
245 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
246 |
text{*Strict monotonicity in both arguments*} |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
247 |
lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::ordered_semiring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
248 |
apply (erule add_strict_right_mono [THEN order_less_trans]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
249 |
apply (erule add_strict_left_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
250 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
251 |
|
14370 | 252 |
lemma add_less_le_mono: |
253 |
"[| a<b; c\<le>d |] ==> a + c < b + (d::'a::ordered_semiring)" |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
254 |
apply (erule add_strict_right_mono [THEN order_less_le_trans]) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
255 |
apply (erule add_left_mono) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
256 |
done |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
257 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
258 |
lemma add_le_less_mono: |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
259 |
"[| a\<le>b; c<d |] ==> a + c < b + (d::'a::ordered_semiring)" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
260 |
apply (erule add_right_mono [THEN order_le_less_trans]) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
261 |
apply (erule add_strict_left_mono) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
262 |
done |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
263 |
|
14270 | 264 |
lemma add_less_imp_less_left: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
265 |
assumes less: "c + a < c + b" shows "a < (b::'a::ordered_semiring)" |
14377 | 266 |
proof (rule ccontr) |
267 |
assume "~ a < b" |
|
268 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
269 |
hence "c+b \<le> c+a" by (rule add_left_mono) |
|
270 |
with this and less show False |
|
271 |
by (simp add: linorder_not_less [symmetric]) |
|
272 |
qed |
|
14270 | 273 |
|
274 |
lemma add_less_imp_less_right: |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
275 |
"a + c < b + c ==> a < (b::'a::ordered_semiring)" |
14270 | 276 |
apply (rule add_less_imp_less_left [of c]) |
277 |
apply (simp add: add_commute) |
|
278 |
done |
|
279 |
||
280 |
lemma add_less_cancel_left [simp]: |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
281 |
"(c+a < c+b) = (a < (b::'a::ordered_semiring))" |
14270 | 282 |
by (blast intro: add_less_imp_less_left add_strict_left_mono) |
283 |
||
284 |
lemma add_less_cancel_right [simp]: |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
285 |
"(a+c < b+c) = (a < (b::'a::ordered_semiring))" |
14270 | 286 |
by (blast intro: add_less_imp_less_right add_strict_right_mono) |
287 |
||
288 |
lemma add_le_cancel_left [simp]: |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
289 |
"(c+a \<le> c+b) = (a \<le> (b::'a::ordered_semiring))" |
14270 | 290 |
by (simp add: linorder_not_less [symmetric]) |
291 |
||
292 |
lemma add_le_cancel_right [simp]: |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
293 |
"(a+c \<le> b+c) = (a \<le> (b::'a::ordered_semiring))" |
14270 | 294 |
by (simp add: linorder_not_less [symmetric]) |
295 |
||
296 |
lemma add_le_imp_le_left: |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
297 |
"c + a \<le> c + b ==> a \<le> (b::'a::ordered_semiring)" |
14270 | 298 |
by simp |
299 |
||
300 |
lemma add_le_imp_le_right: |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
301 |
"a + c \<le> b + c ==> a \<le> (b::'a::ordered_semiring)" |
14270 | 302 |
by simp |
303 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
304 |
lemma add_increasing: "[|0\<le>a; b\<le>c|] ==> b \<le> a + (c::'a::ordered_semiring)" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
305 |
by (insert add_mono [of 0 a b c], simp) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
306 |
|
14270 | 307 |
|
308 |
subsection {* Ordering Rules for Unary Minus *} |
|
309 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
310 |
lemma le_imp_neg_le: |
14269 | 311 |
assumes "a \<le> (b::'a::ordered_ring)" shows "-b \<le> -a" |
14377 | 312 |
proof - |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
313 |
have "-a+a \<le> -a+b" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
314 |
by (rule add_left_mono) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
315 |
hence "0 \<le> -a+b" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
316 |
by simp |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
317 |
hence "0 + (-b) \<le> (-a + b) + (-b)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
318 |
by (rule add_right_mono) |
14266 | 319 |
thus ?thesis |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
320 |
by (simp add: add_assoc) |
14377 | 321 |
qed |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
322 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
323 |
lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::ordered_ring))" |
14377 | 324 |
proof |
325 |
assume "- b \<le> - a" |
|
326 |
hence "- (- a) \<le> - (- b)" |
|
327 |
by (rule le_imp_neg_le) |
|
328 |
thus "a\<le>b" by simp |
|
329 |
next |
|
330 |
assume "a\<le>b" |
|
331 |
thus "-b \<le> -a" by (rule le_imp_neg_le) |
|
332 |
qed |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
333 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
334 |
lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::ordered_ring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
335 |
by (subst neg_le_iff_le [symmetric], simp) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
336 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
337 |
lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::ordered_ring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
338 |
by (subst neg_le_iff_le [symmetric], simp) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
339 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
340 |
lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::ordered_ring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
341 |
by (force simp add: order_less_le) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
342 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
343 |
lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::ordered_ring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
344 |
by (subst neg_less_iff_less [symmetric], simp) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
345 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
346 |
lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::ordered_ring))" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
347 |
by (subst neg_less_iff_less [symmetric], simp) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
348 |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
349 |
text{*The next several equations can make the simplifier loop!*} |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
350 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
351 |
lemma less_minus_iff: "(a < - b) = (b < - (a::'a::ordered_ring))" |
14377 | 352 |
proof - |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
353 |
have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
354 |
thus ?thesis by simp |
14377 | 355 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
356 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
357 |
lemma minus_less_iff: "(- a < b) = (- b < (a::'a::ordered_ring))" |
14377 | 358 |
proof - |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
359 |
have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
360 |
thus ?thesis by simp |
14377 | 361 |
qed |
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
362 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
363 |
lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::ordered_ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
364 |
apply (simp add: linorder_not_less [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
365 |
apply (rule minus_less_iff) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
366 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
367 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
368 |
lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::ordered_ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
369 |
apply (simp add: linorder_not_less [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
370 |
apply (rule less_minus_iff) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
371 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
372 |
|
14270 | 373 |
|
374 |
subsection{*Subtraction Laws*} |
|
375 |
||
376 |
lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ring)" |
|
377 |
by (simp add: diff_minus add_ac) |
|
378 |
||
379 |
lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ring)" |
|
380 |
by (simp add: diff_minus add_ac) |
|
381 |
||
382 |
lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ring))" |
|
383 |
by (auto simp add: diff_minus add_assoc) |
|
384 |
||
385 |
lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ring) = c)" |
|
386 |
by (auto simp add: diff_minus add_assoc) |
|
387 |
||
388 |
lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ring))" |
|
389 |
by (simp add: diff_minus add_ac) |
|
390 |
||
391 |
lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ring)" |
|
392 |
by (simp add: diff_minus add_ac) |
|
393 |
||
394 |
text{*Further subtraction laws for ordered rings*} |
|
395 |
||
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
396 |
lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::ordered_ring))" |
14270 | 397 |
proof - |
398 |
have "(a < b) = (a + (- b) < b + (-b))" |
|
399 |
by (simp only: add_less_cancel_right) |
|
400 |
also have "... = (a - b < 0)" by (simp add: diff_minus) |
|
401 |
finally show ?thesis . |
|
402 |
qed |
|
403 |
||
404 |
lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::ordered_ring))" |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
405 |
apply (subst less_iff_diff_less_0) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
406 |
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) |
14270 | 407 |
apply (simp add: diff_minus add_ac) |
408 |
done |
|
409 |
||
410 |
lemma less_diff_eq: "(a < c-b) = (a + (b::'a::ordered_ring) < c)" |
|
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
411 |
apply (subst less_iff_diff_less_0) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
412 |
apply (rule less_iff_diff_less_0 [of _ "c-b", THEN ssubst]) |
14270 | 413 |
apply (simp add: diff_minus add_ac) |
414 |
done |
|
415 |
||
416 |
lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::ordered_ring))" |
|
417 |
by (simp add: linorder_not_less [symmetric] less_diff_eq) |
|
418 |
||
419 |
lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::ordered_ring) \<le> c)" |
|
420 |
by (simp add: linorder_not_less [symmetric] diff_less_eq) |
|
421 |
||
422 |
text{*This list of rewrites simplifies (in)equalities by bringing subtractions |
|
423 |
to the top and then moving negative terms to the other side. |
|
424 |
Use with @{text add_ac}*} |
|
425 |
lemmas compare_rls = |
|
426 |
diff_minus [symmetric] |
|
427 |
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 |
|
428 |
diff_less_eq less_diff_eq diff_le_eq le_diff_eq |
|
429 |
diff_eq_eq eq_diff_eq |
|
430 |
||
431 |
||
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
432 |
subsection{*Lemmas for the @{text cancel_numerals} simproc*} |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
433 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
434 |
lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
435 |
by (simp add: compare_rls) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
436 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
437 |
lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::ordered_ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
438 |
by (simp add: compare_rls) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
439 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
440 |
lemma eq_add_iff1: |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
441 |
"(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
442 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
443 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
444 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
445 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
446 |
lemma eq_add_iff2: |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
447 |
"(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
448 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
449 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
450 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
451 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
452 |
lemma less_add_iff1: |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
453 |
"(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::ordered_ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
454 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
455 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
456 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
457 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
458 |
lemma less_add_iff2: |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
459 |
"(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::ordered_ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
460 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
461 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
462 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
463 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
464 |
lemma le_add_iff1: |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
465 |
"(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::ordered_ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
466 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
467 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
468 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
469 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
470 |
lemma le_add_iff2: |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
471 |
"(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::ordered_ring))" |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
472 |
apply (simp add: diff_minus left_distrib add_ac) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
473 |
apply (simp add: compare_rls minus_mult_left [symmetric]) |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
474 |
done |
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
475 |
|
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset
|
476 |
|
14270 | 477 |
subsection {* Ordering Rules for Multiplication *} |
478 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
479 |
lemma mult_strict_right_mono: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
480 |
"[|a < b; 0 < c|] ==> a * c < b * (c::'a::ordered_semiring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
481 |
by (simp add: mult_commute [of _ c] mult_strict_left_mono) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
482 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
483 |
lemma mult_left_mono: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
484 |
"[|a \<le> b; 0 \<le> c|] ==> c * a \<le> c * (b::'a::ordered_semiring)" |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
485 |
apply (case_tac "c=0", simp) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
486 |
apply (force simp add: mult_strict_left_mono order_le_less) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
487 |
done |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
488 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
489 |
lemma mult_right_mono: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
490 |
"[|a \<le> b; 0 \<le> c|] ==> a*c \<le> b * (c::'a::ordered_semiring)" |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
491 |
by (simp add: mult_left_mono mult_commute [of _ c]) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
492 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
493 |
lemma mult_left_le_imp_le: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
494 |
"[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring)" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
495 |
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric]) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
496 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
497 |
lemma mult_right_le_imp_le: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
498 |
"[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring)" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
499 |
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric]) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
500 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
501 |
lemma mult_left_less_imp_less: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
502 |
"[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
503 |
by (force simp add: mult_left_mono linorder_not_le [symmetric]) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
504 |
|
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
505 |
lemma mult_right_less_imp_less: |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
506 |
"[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
507 |
by (force simp add: mult_right_mono linorder_not_le [symmetric]) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
508 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
509 |
lemma mult_strict_left_mono_neg: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
510 |
"[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
511 |
apply (drule mult_strict_left_mono [of _ _ "-c"]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
512 |
apply (simp_all add: minus_mult_left [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
513 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
514 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
515 |
lemma mult_strict_right_mono_neg: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
516 |
"[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
517 |
apply (drule mult_strict_right_mono [of _ _ "-c"]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
518 |
apply (simp_all add: minus_mult_right [symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
519 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
520 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
521 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
522 |
subsection{* Products of Signs *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
523 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
524 |
lemma mult_pos: "[| (0::'a::ordered_semiring) < a; 0 < b |] ==> 0 < a*b" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
525 |
by (drule mult_strict_left_mono [of 0 b], auto) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
526 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
527 |
lemma mult_pos_neg: "[| (0::'a::ordered_semiring) < a; b < 0 |] ==> a*b < 0" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
528 |
by (drule mult_strict_left_mono [of b 0], auto) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
529 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
530 |
lemma mult_neg: "[| a < (0::'a::ordered_ring); b < 0 |] ==> 0 < a*b" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
531 |
by (drule mult_strict_right_mono_neg, auto) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
532 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
533 |
lemma zero_less_mult_pos: |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
534 |
"[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
535 |
apply (case_tac "b\<le>0") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
536 |
apply (auto simp add: order_le_less linorder_not_less) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
537 |
apply (drule_tac mult_pos_neg [of a b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
538 |
apply (auto dest: order_less_not_sym) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
539 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
540 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
541 |
lemma zero_less_mult_iff: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
542 |
"((0::'a::ordered_ring) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
543 |
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
544 |
apply (blast dest: zero_less_mult_pos) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
545 |
apply (simp add: mult_commute [of a b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
546 |
apply (blast dest: zero_less_mult_pos) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
547 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
548 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
549 |
text{*A field has no "zero divisors", and this theorem holds without the |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
550 |
assumption of an ordering. See @{text field_mult_eq_0_iff} below.*} |
14266 | 551 |
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring)) = (a = 0 | b = 0)" |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
552 |
apply (case_tac "a < 0") |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
553 |
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
554 |
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+ |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
555 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
556 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
557 |
lemma zero_le_mult_iff: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
558 |
"((0::'a::ordered_ring) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
559 |
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
560 |
zero_less_mult_iff) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
561 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
562 |
lemma mult_less_0_iff: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
563 |
"(a*b < (0::'a::ordered_ring)) = (0 < a & b < 0 | a < 0 & 0 < b)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
564 |
apply (insert zero_less_mult_iff [of "-a" b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
565 |
apply (force simp add: minus_mult_left[symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
566 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
567 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
568 |
lemma mult_le_0_iff: |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
569 |
"(a*b \<le> (0::'a::ordered_ring)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
570 |
apply (insert zero_le_mult_iff [of "-a" b]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
571 |
apply (force simp add: minus_mult_left[symmetric]) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
572 |
done |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
573 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
574 |
lemma zero_le_square: "(0::'a::ordered_ring) \<le> a*a" |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
575 |
by (simp add: zero_le_mult_iff linorder_linear) |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
576 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
577 |
text{*All three types of comparision involving 0 and 1 are covered.*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
578 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
579 |
declare zero_neq_one [THEN not_sym, simp] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
580 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
581 |
lemma zero_le_one [simp]: "(0::'a::ordered_semiring) \<le> 1" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
582 |
by (rule zero_less_one [THEN order_less_imp_le]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
583 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
584 |
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semiring) \<le> 0" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
585 |
by (simp add: linorder_not_le zero_less_one) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
586 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
587 |
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semiring) < 0" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
588 |
by (simp add: linorder_not_less zero_le_one) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
589 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
590 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
591 |
subsection{*More Monotonicity*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
592 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
593 |
lemma mult_left_mono_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
594 |
"[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::ordered_ring)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
595 |
apply (drule mult_left_mono [of _ _ "-c"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
596 |
apply (simp_all add: minus_mult_left [symmetric]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
597 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
598 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
599 |
lemma mult_right_mono_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
600 |
"[|b \<le> a; c \<le> 0|] ==> a * c \<le> b * (c::'a::ordered_ring)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
601 |
by (simp add: mult_left_mono_neg mult_commute [of _ c]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
602 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
603 |
text{*Strict monotonicity in both arguments*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
604 |
lemma mult_strict_mono: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
605 |
"[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
606 |
apply (case_tac "c=0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
607 |
apply (simp add: mult_pos) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
608 |
apply (erule mult_strict_right_mono [THEN order_less_trans]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
609 |
apply (force simp add: order_le_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
610 |
apply (erule mult_strict_left_mono, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
611 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
612 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
613 |
text{*This weaker variant has more natural premises*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
614 |
lemma mult_strict_mono': |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
615 |
"[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
616 |
apply (rule mult_strict_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
617 |
apply (blast intro: order_le_less_trans)+ |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
618 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
619 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
620 |
lemma mult_mono: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
621 |
"[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
622 |
==> a * c \<le> b * (d::'a::ordered_semiring)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
623 |
apply (erule mult_right_mono [THEN order_trans], assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
624 |
apply (erule mult_left_mono, assumption) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
625 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
626 |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
627 |
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semiring)" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
628 |
apply (insert mult_strict_mono [of 1 m 1 n]) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
629 |
apply (simp add: order_less_trans [OF zero_less_one]); |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
630 |
done |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
631 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
632 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
633 |
subsection{*Cancellation Laws for Relationships With a Common Factor*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
634 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
635 |
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"}, |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
636 |
also with the relations @{text "\<le>"} and equality.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
637 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
638 |
lemma mult_less_cancel_right: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
639 |
"(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring)))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
640 |
apply (case_tac "c = 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
641 |
apply (auto simp add: linorder_neq_iff mult_strict_right_mono |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
642 |
mult_strict_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
643 |
apply (auto simp add: linorder_not_less |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
644 |
linorder_not_le [symmetric, of "a*c"] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
645 |
linorder_not_le [symmetric, of a]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
646 |
apply (erule_tac [!] notE) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
647 |
apply (auto simp add: order_less_imp_le mult_right_mono |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
648 |
mult_right_mono_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
649 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
650 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
651 |
lemma mult_less_cancel_left: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
652 |
"(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring)))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
653 |
by (simp add: mult_commute [of c] mult_less_cancel_right) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
654 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
655 |
lemma mult_le_cancel_right: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
656 |
"(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring)))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
657 |
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
658 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
659 |
lemma mult_le_cancel_left: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
660 |
"(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring)))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
661 |
by (simp add: mult_commute [of c] mult_le_cancel_right) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
662 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
663 |
lemma mult_less_imp_less_left: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
664 |
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
665 |
shows "a < (b::'a::ordered_semiring)" |
14377 | 666 |
proof (rule ccontr) |
667 |
assume "~ a < b" |
|
668 |
hence "b \<le> a" by (simp add: linorder_not_less) |
|
669 |
hence "c*b \<le> c*a" by (rule mult_left_mono) |
|
670 |
with this and less show False |
|
671 |
by (simp add: linorder_not_less [symmetric]) |
|
672 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
673 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
674 |
lemma mult_less_imp_less_right: |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
675 |
"[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
676 |
by (rule mult_less_imp_less_left, simp add: mult_commute) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
677 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
678 |
text{*Cancellation of equalities with a common factor*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
679 |
lemma mult_cancel_right [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
680 |
"(a*c = b*c) = (c = (0::'a::ordered_ring) | a=b)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
681 |
apply (cut_tac linorder_less_linear [of 0 c]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
682 |
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
683 |
simp add: linorder_neq_iff) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
684 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
685 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
686 |
text{*These cancellation theorems require an ordering. Versions are proved |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
687 |
below that work for fields without an ordering.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
688 |
lemma mult_cancel_left [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
689 |
"(c*a = c*b) = (c = (0::'a::ordered_ring) | a=b)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
690 |
by (simp add: mult_commute [of c] mult_cancel_right) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
691 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
692 |
|
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
693 |
subsection {* Fields *} |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
694 |
|
14288 | 695 |
lemma right_inverse [simp]: |
696 |
assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1" |
|
697 |
proof - |
|
698 |
have "a * inverse a = inverse a * a" by (simp add: mult_ac) |
|
699 |
also have "... = 1" using not0 by simp |
|
700 |
finally show ?thesis . |
|
701 |
qed |
|
702 |
||
703 |
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" |
|
704 |
proof |
|
705 |
assume neq: "b \<noteq> 0" |
|
706 |
{ |
|
707 |
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) |
|
708 |
also assume "a / b = 1" |
|
709 |
finally show "a = b" by simp |
|
710 |
next |
|
711 |
assume "a = b" |
|
712 |
with neq show "a / b = 1" by (simp add: divide_inverse) |
|
713 |
} |
|
714 |
qed |
|
715 |
||
716 |
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" |
|
717 |
by (simp add: divide_inverse) |
|
718 |
||
719 |
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" |
|
720 |
by (simp add: divide_inverse) |
|
721 |
||
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
722 |
lemma divide_inverse_zero: "a/b = a * inverse(b::'a::{field,division_by_zero})" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
723 |
apply (case_tac "b = 0") |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
724 |
apply (simp_all add: divide_inverse) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
725 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
726 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
727 |
lemma divide_zero_left [simp]: "0/a = (0::'a::{field,division_by_zero})" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
728 |
by (simp add: divide_inverse_zero) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
729 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
730 |
lemma inverse_eq_divide: "inverse (a::'a::{field,division_by_zero}) = 1/a" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
731 |
by (simp add: divide_inverse_zero) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
732 |
|
14293 | 733 |
lemma nonzero_add_divide_distrib: "c \<noteq> 0 ==> (a+b)/(c::'a::field) = a/c + b/c" |
734 |
by (simp add: divide_inverse left_distrib) |
|
735 |
||
736 |
lemma add_divide_distrib: "(a+b)/(c::'a::{field,division_by_zero}) = a/c + b/c" |
|
737 |
apply (case_tac "c=0", simp) |
|
738 |
apply (simp add: nonzero_add_divide_distrib) |
|
739 |
done |
|
740 |
||
14270 | 741 |
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement |
742 |
of an ordering.*} |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
743 |
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)" |
14377 | 744 |
proof cases |
745 |
assume "a=0" thus ?thesis by simp |
|
746 |
next |
|
747 |
assume anz [simp]: "a\<noteq>0" |
|
748 |
{ assume "a * b = 0" |
|
749 |
hence "inverse a * (a * b) = 0" by simp |
|
750 |
hence "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric])} |
|
751 |
thus ?thesis by force |
|
752 |
qed |
|
14270 | 753 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
754 |
text{*Cancellation of equalities with a common factor*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
755 |
lemma field_mult_cancel_right_lemma: |
14269 | 756 |
assumes cnz: "c \<noteq> (0::'a::field)" |
757 |
and eq: "a*c = b*c" |
|
758 |
shows "a=b" |
|
14377 | 759 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
760 |
have "(a * c) * inverse c = (b * c) * inverse c" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
761 |
by (simp add: eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
762 |
thus "a=b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
763 |
by (simp add: mult_assoc cnz) |
14377 | 764 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
765 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
766 |
lemma field_mult_cancel_right [simp]: |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
767 |
"(a*c = b*c) = (c = (0::'a::field) | a=b)" |
14377 | 768 |
proof cases |
769 |
assume "c=0" thus ?thesis by simp |
|
770 |
next |
|
771 |
assume "c\<noteq>0" |
|
772 |
thus ?thesis by (force dest: field_mult_cancel_right_lemma) |
|
773 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
774 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
775 |
lemma field_mult_cancel_left [simp]: |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
776 |
"(c*a = c*b) = (c = (0::'a::field) | a=b)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
777 |
by (simp add: mult_commute [of c] field_mult_cancel_right) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
778 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
779 |
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)" |
14377 | 780 |
proof |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
781 |
assume ianz: "inverse a = 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
782 |
assume "a \<noteq> 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
783 |
hence "1 = a * inverse a" by simp |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
784 |
also have "... = 0" by (simp add: ianz) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
785 |
finally have "1 = (0::'a::field)" . |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
786 |
thus False by (simp add: eq_commute) |
14377 | 787 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
788 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
789 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
790 |
subsection{*Basic Properties of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
791 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
792 |
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
793 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
794 |
apply (blast dest: nonzero_imp_inverse_nonzero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
795 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
796 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
797 |
lemma inverse_nonzero_imp_nonzero: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
798 |
"inverse a = 0 ==> a = (0::'a::field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
799 |
apply (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
800 |
apply (blast dest: nonzero_imp_inverse_nonzero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
801 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
802 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
803 |
lemma inverse_nonzero_iff_nonzero [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
804 |
"(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
805 |
by (force dest: inverse_nonzero_imp_nonzero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
806 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
807 |
lemma nonzero_inverse_minus_eq: |
14269 | 808 |
assumes [simp]: "a\<noteq>0" shows "inverse(-a) = -inverse(a::'a::field)" |
14377 | 809 |
proof - |
810 |
have "-a * inverse (- a) = -a * - inverse a" |
|
811 |
by simp |
|
812 |
thus ?thesis |
|
813 |
by (simp only: field_mult_cancel_left, simp) |
|
814 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
815 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
816 |
lemma inverse_minus_eq [simp]: |
14377 | 817 |
"inverse(-a) = -inverse(a::'a::{field,division_by_zero})"; |
818 |
proof cases |
|
819 |
assume "a=0" thus ?thesis by (simp add: inverse_zero) |
|
820 |
next |
|
821 |
assume "a\<noteq>0" |
|
822 |
thus ?thesis by (simp add: nonzero_inverse_minus_eq) |
|
823 |
qed |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
824 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
825 |
lemma nonzero_inverse_eq_imp_eq: |
14269 | 826 |
assumes inveq: "inverse a = inverse b" |
827 |
and anz: "a \<noteq> 0" |
|
828 |
and bnz: "b \<noteq> 0" |
|
829 |
shows "a = (b::'a::field)" |
|
14377 | 830 |
proof - |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
831 |
have "a * inverse b = a * inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
832 |
by (simp add: inveq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
833 |
hence "(a * inverse b) * b = (a * inverse a) * b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
834 |
by simp |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
835 |
thus "a = b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
836 |
by (simp add: mult_assoc anz bnz) |
14377 | 837 |
qed |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
838 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
839 |
lemma inverse_eq_imp_eq: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
840 |
"inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
841 |
apply (case_tac "a=0 | b=0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
842 |
apply (force dest!: inverse_zero_imp_zero |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
843 |
simp add: eq_commute [of "0::'a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
844 |
apply (force dest!: nonzero_inverse_eq_imp_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
845 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
846 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
847 |
lemma inverse_eq_iff_eq [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
848 |
"(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
849 |
by (force dest!: inverse_eq_imp_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
850 |
|
14270 | 851 |
lemma nonzero_inverse_inverse_eq: |
852 |
assumes [simp]: "a \<noteq> 0" shows "inverse(inverse (a::'a::field)) = a" |
|
853 |
proof - |
|
854 |
have "(inverse (inverse a) * inverse a) * a = a" |
|
855 |
by (simp add: nonzero_imp_inverse_nonzero) |
|
856 |
thus ?thesis |
|
857 |
by (simp add: mult_assoc) |
|
858 |
qed |
|
859 |
||
860 |
lemma inverse_inverse_eq [simp]: |
|
861 |
"inverse(inverse (a::'a::{field,division_by_zero})) = a" |
|
862 |
proof cases |
|
863 |
assume "a=0" thus ?thesis by simp |
|
864 |
next |
|
865 |
assume "a\<noteq>0" |
|
866 |
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) |
|
867 |
qed |
|
868 |
||
869 |
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)" |
|
870 |
proof - |
|
871 |
have "inverse 1 * 1 = (1::'a::field)" |
|
872 |
by (rule left_inverse [OF zero_neq_one [symmetric]]) |
|
873 |
thus ?thesis by simp |
|
874 |
qed |
|
875 |
||
876 |
lemma nonzero_inverse_mult_distrib: |
|
877 |
assumes anz: "a \<noteq> 0" |
|
878 |
and bnz: "b \<noteq> 0" |
|
879 |
shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)" |
|
880 |
proof - |
|
881 |
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" |
|
882 |
by (simp add: field_mult_eq_0_iff anz bnz) |
|
883 |
hence "inverse(a*b) * a = inverse(b)" |
|
884 |
by (simp add: mult_assoc bnz) |
|
885 |
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" |
|
886 |
by simp |
|
887 |
thus ?thesis |
|
888 |
by (simp add: mult_assoc anz) |
|
889 |
qed |
|
890 |
||
891 |
text{*This version builds in division by zero while also re-orienting |
|
892 |
the right-hand side.*} |
|
893 |
lemma inverse_mult_distrib [simp]: |
|
894 |
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" |
|
895 |
proof cases |
|
896 |
assume "a \<noteq> 0 & b \<noteq> 0" |
|
897 |
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) |
|
898 |
next |
|
899 |
assume "~ (a \<noteq> 0 & b \<noteq> 0)" |
|
900 |
thus ?thesis by force |
|
901 |
qed |
|
902 |
||
903 |
text{*There is no slick version using division by zero.*} |
|
904 |
lemma inverse_add: |
|
905 |
"[|a \<noteq> 0; b \<noteq> 0|] |
|
906 |
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" |
|
907 |
apply (simp add: left_distrib mult_assoc) |
|
908 |
apply (simp add: mult_commute [of "inverse a"]) |
|
909 |
apply (simp add: mult_assoc [symmetric] add_commute) |
|
910 |
done |
|
911 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
912 |
lemma inverse_divide [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
913 |
"inverse (a/b) = b / (a::'a::{field,division_by_zero})" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
914 |
by (simp add: divide_inverse_zero mult_commute) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
915 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
916 |
lemma nonzero_mult_divide_cancel_left: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
917 |
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
918 |
shows "(c*a)/(c*b) = a/(b::'a::field)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
919 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
920 |
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
921 |
by (simp add: field_mult_eq_0_iff divide_inverse |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
922 |
nonzero_inverse_mult_distrib) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
923 |
also have "... = a * inverse b * (inverse c * c)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
924 |
by (simp only: mult_ac) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
925 |
also have "... = a * inverse b" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
926 |
by simp |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
927 |
finally show ?thesis |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
928 |
by (simp add: divide_inverse) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
929 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
930 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
931 |
lemma mult_divide_cancel_left: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
932 |
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
933 |
apply (case_tac "b = 0") |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
934 |
apply (simp_all add: nonzero_mult_divide_cancel_left) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
935 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
936 |
|
14321 | 937 |
lemma nonzero_mult_divide_cancel_right: |
938 |
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)" |
|
939 |
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) |
|
940 |
||
941 |
lemma mult_divide_cancel_right: |
|
942 |
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" |
|
943 |
apply (case_tac "b = 0") |
|
944 |
apply (simp_all add: nonzero_mult_divide_cancel_right) |
|
945 |
done |
|
946 |
||
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
947 |
(*For ExtractCommonTerm*) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
948 |
lemma mult_divide_cancel_eq_if: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
949 |
"(c*a) / (c*b) = |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
950 |
(if c=0 then 0 else a / (b::'a::{field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
951 |
by (simp add: mult_divide_cancel_left) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
952 |
|
14284
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
953 |
lemma divide_1 [simp]: "a/1 = (a::'a::field)" |
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
954 |
by (simp add: divide_inverse [OF not_sym]) |
f1abe67c448a
re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset
|
955 |
|
14288 | 956 |
lemma times_divide_eq_right [simp]: |
957 |
"a * (b/c) = (a*b) / (c::'a::{field,division_by_zero})" |
|
958 |
by (simp add: divide_inverse_zero mult_assoc) |
|
959 |
||
960 |
lemma times_divide_eq_left [simp]: |
|
961 |
"(b/c) * a = (b*a) / (c::'a::{field,division_by_zero})" |
|
962 |
by (simp add: divide_inverse_zero mult_ac) |
|
963 |
||
964 |
lemma divide_divide_eq_right [simp]: |
|
965 |
"a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})" |
|
966 |
by (simp add: divide_inverse_zero mult_ac) |
|
967 |
||
968 |
lemma divide_divide_eq_left [simp]: |
|
969 |
"(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)" |
|
970 |
by (simp add: divide_inverse_zero mult_assoc) |
|
971 |
||
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
972 |
|
14293 | 973 |
subsection {* Division and Unary Minus *} |
974 |
||
975 |
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)" |
|
976 |
by (simp add: divide_inverse minus_mult_left) |
|
977 |
||
978 |
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)" |
|
979 |
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) |
|
980 |
||
981 |
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)" |
|
982 |
by (simp add: divide_inverse nonzero_inverse_minus_eq) |
|
983 |
||
984 |
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::{field,division_by_zero})" |
|
985 |
apply (case_tac "b=0", simp) |
|
986 |
apply (simp add: nonzero_minus_divide_left) |
|
987 |
done |
|
988 |
||
989 |
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})" |
|
990 |
apply (case_tac "b=0", simp) |
|
991 |
by (rule nonzero_minus_divide_right) |
|
992 |
||
993 |
text{*The effect is to extract signs from divisions*} |
|
994 |
declare minus_divide_left [symmetric, simp] |
|
995 |
declare minus_divide_right [symmetric, simp] |
|
996 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
997 |
text{*Also, extract signs from products*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
998 |
declare minus_mult_left [symmetric, simp] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
999 |
declare minus_mult_right [symmetric, simp] |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1000 |
|
14293 | 1001 |
lemma minus_divide_divide [simp]: |
1002 |
"(-a)/(-b) = a / (b::'a::{field,division_by_zero})" |
|
1003 |
apply (case_tac "b=0", simp) |
|
1004 |
apply (simp add: nonzero_minus_divide_divide) |
|
1005 |
done |
|
1006 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1007 |
lemma diff_divide_distrib: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1008 |
"(a-b)/(c::'a::{field,division_by_zero}) = a/c - b/c" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1009 |
by (simp add: diff_minus add_divide_distrib) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1010 |
|
14293 | 1011 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1012 |
subsection {* Ordered Fields *} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1013 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1014 |
lemma positive_imp_inverse_positive: |
14269 | 1015 |
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)" |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1016 |
proof - |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1017 |
have "0 < a * inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1018 |
by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1019 |
thus "0 < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1020 |
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1021 |
qed |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1022 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1023 |
lemma negative_imp_inverse_negative: |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1024 |
"a < 0 ==> inverse a < (0::'a::ordered_field)" |
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1025 |
by (insert positive_imp_inverse_positive [of "-a"], |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1026 |
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1027 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1028 |
lemma inverse_le_imp_le: |
14269 | 1029 |
assumes invle: "inverse a \<le> inverse b" |
1030 |
and apos: "0 < a" |
|
1031 |
shows "b \<le> (a::'a::ordered_field)" |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1032 |
proof (rule classical) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1033 |
assume "~ b \<le> a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1034 |
hence "a < b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1035 |
by (simp add: linorder_not_le) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1036 |
hence bpos: "0 < b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1037 |
by (blast intro: apos order_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1038 |
hence "a * inverse a \<le> a * inverse b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1039 |
by (simp add: apos invle order_less_imp_le mult_left_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1040 |
hence "(a * inverse a) * b \<le> (a * inverse b) * b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1041 |
by (simp add: bpos order_less_imp_le mult_right_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1042 |
thus "b \<le> a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1043 |
by (simp add: mult_assoc apos bpos order_less_imp_not_eq2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1044 |
qed |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1045 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1046 |
lemma inverse_positive_imp_positive: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1047 |
assumes inv_gt_0: "0 < inverse a" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1048 |
and [simp]: "a \<noteq> 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1049 |
shows "0 < (a::'a::ordered_field)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1050 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1051 |
have "0 < inverse (inverse a)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1052 |
by (rule positive_imp_inverse_positive) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1053 |
thus "0 < a" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1054 |
by (simp add: nonzero_inverse_inverse_eq) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1055 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1056 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1057 |
lemma inverse_positive_iff_positive [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1058 |
"(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1059 |
apply (case_tac "a = 0", simp) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1060 |
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1061 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1062 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1063 |
lemma inverse_negative_imp_negative: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1064 |
assumes inv_less_0: "inverse a < 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1065 |
and [simp]: "a \<noteq> 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1066 |
shows "a < (0::'a::ordered_field)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1067 |
proof - |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1068 |
have "inverse (inverse a) < 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1069 |
by (rule negative_imp_inverse_negative) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1070 |
thus "a < 0" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1071 |
by (simp add: nonzero_inverse_inverse_eq) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1072 |
qed |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1073 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1074 |
lemma inverse_negative_iff_negative [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1075 |
"(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1076 |
apply (case_tac "a = 0", simp) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1077 |
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1078 |
done |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1079 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1080 |
lemma inverse_nonnegative_iff_nonnegative [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1081 |
"(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1082 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1083 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1084 |
lemma inverse_nonpositive_iff_nonpositive [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1085 |
"(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1086 |
by (simp add: linorder_not_less [symmetric]) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1087 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1088 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1089 |
subsection{*Anti-Monotonicity of @{term inverse}*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1090 |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1091 |
lemma less_imp_inverse_less: |
14269 | 1092 |
assumes less: "a < b" |
1093 |
and apos: "0 < a" |
|
1094 |
shows "inverse b < inverse (a::'a::ordered_field)" |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1095 |
proof (rule ccontr) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1096 |
assume "~ inverse b < inverse a" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1097 |
hence "inverse a \<le> inverse b" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1098 |
by (simp add: linorder_not_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1099 |
hence "~ (a < b)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1100 |
by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1101 |
thus False |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1102 |
by (rule notE [OF _ less]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1103 |
qed |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1104 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1105 |
lemma inverse_less_imp_less: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1106 |
"[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1107 |
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1108 |
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1109 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1110 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1111 |
text{*Both premises are essential. Consider -1 and 1.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1112 |
lemma inverse_less_iff_less [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1113 |
"[|0 < a; 0 < b|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1114 |
==> (inverse a < inverse b) = (b < (a::'a::ordered_field))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1115 |
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1116 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1117 |
lemma le_imp_inverse_le: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1118 |
"[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1119 |
by (force simp add: order_le_less less_imp_inverse_less) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1120 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1121 |
lemma inverse_le_iff_le [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1122 |
"[|0 < a; 0 < b|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1123 |
==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1124 |
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1125 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1126 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1127 |
text{*These results refer to both operands being negative. The opposite-sign |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1128 |
case is trivial, since inverse preserves signs.*} |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1129 |
lemma inverse_le_imp_le_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1130 |
"[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1131 |
apply (rule classical) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1132 |
apply (subgoal_tac "a < 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1133 |
prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1134 |
apply (insert inverse_le_imp_le [of "-b" "-a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1135 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1136 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1137 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1138 |
lemma less_imp_inverse_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1139 |
"[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1140 |
apply (subgoal_tac "a < 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1141 |
prefer 2 apply (blast intro: order_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1142 |
apply (insert less_imp_inverse_less [of "-b" "-a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1143 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1144 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1145 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1146 |
lemma inverse_less_imp_less_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1147 |
"[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1148 |
apply (rule classical) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1149 |
apply (subgoal_tac "a < 0") |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1150 |
prefer 2 |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1151 |
apply (force simp add: linorder_not_less intro: order_le_less_trans) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1152 |
apply (insert inverse_less_imp_less [of "-b" "-a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1153 |
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1154 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1155 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1156 |
lemma inverse_less_iff_less_neg [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1157 |
"[|a < 0; b < 0|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1158 |
==> (inverse a < inverse b) = (b < (a::'a::ordered_field))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1159 |
apply (insert inverse_less_iff_less [of "-b" "-a"]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1160 |
apply (simp del: inverse_less_iff_less |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1161 |
add: order_less_imp_not_eq nonzero_inverse_minus_eq) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1162 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1163 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1164 |
lemma le_imp_inverse_le_neg: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1165 |
"[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1166 |
by (force simp add: order_le_less less_imp_inverse_less_neg) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1167 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1168 |
lemma inverse_le_iff_le_neg [simp]: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1169 |
"[|a < 0; b < 0|] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1170 |
==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset
|
1171 |
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1172 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1173 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1174 |
subsection{*Inverses and the Number One*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1175 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1176 |
lemma one_less_inverse_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1177 |
"(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1178 |
assume "0 < x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1179 |
with inverse_less_iff_less [OF zero_less_one, of x] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1180 |
show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1181 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1182 |
assume notless: "~ (0 < x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1183 |
have "~ (1 < inverse x)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1184 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1185 |
assume "1 < inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1186 |
also with notless have "... \<le> 0" by (simp add: linorder_not_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1187 |
also have "... < 1" by (rule zero_less_one) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1188 |
finally show False by auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1189 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1190 |
with notless show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1191 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1192 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1193 |
lemma inverse_eq_1_iff [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1194 |
"(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1195 |
by (insert inverse_eq_iff_eq [of x 1], simp) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1196 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1197 |
lemma one_le_inverse_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1198 |
"(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1199 |
by (force simp add: order_le_less one_less_inverse_iff zero_less_one |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1200 |
eq_commute [of 1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1201 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1202 |
lemma inverse_less_1_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1203 |
"(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1204 |
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1205 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1206 |
lemma inverse_le_1_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1207 |
"(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1208 |
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1209 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1210 |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1211 |
subsection{*Division and Signs*} |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1212 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1213 |
lemma zero_less_divide_iff: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1214 |
"((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1215 |
by (simp add: divide_inverse_zero zero_less_mult_iff) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1216 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1217 |
lemma divide_less_0_iff: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1218 |
"(a/b < (0::'a::{ordered_field,division_by_zero})) = |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1219 |
(0 < a & b < 0 | a < 0 & 0 < b)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1220 |
by (simp add: divide_inverse_zero mult_less_0_iff) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1221 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1222 |
lemma zero_le_divide_iff: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1223 |
"((0::'a::{ordered_field,division_by_zero}) \<le> a/b) = |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1224 |
(0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1225 |
by (simp add: divide_inverse_zero zero_le_mult_iff) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1226 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1227 |
lemma divide_le_0_iff: |
14288 | 1228 |
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) = |
1229 |
(0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)" |
|
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1230 |
by (simp add: divide_inverse_zero mult_le_0_iff) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1231 |
|
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1232 |
lemma divide_eq_0_iff [simp]: |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1233 |
"(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))" |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1234 |
by (simp add: divide_inverse_zero field_mult_eq_0_iff) |
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset
|
1235 |
|
14288 | 1236 |
|
1237 |
subsection{*Simplification of Inequalities Involving Literal Divisors*} |
|
1238 |
||
1239 |
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" |
|
1240 |
proof - |
|
1241 |
assume less: "0<c" |
|
1242 |
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" |
|
1243 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1244 |
also have "... = (a*c \<le> b)" |
|
1245 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1246 |
finally show ?thesis . |
|
1247 |
qed |
|
1248 |
||
1249 |
||
1250 |
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" |
|
1251 |
proof - |
|
1252 |
assume less: "c<0" |
|
1253 |
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" |
|
1254 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1255 |
also have "... = (b \<le> a*c)" |
|
1256 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1257 |
finally show ?thesis . |
|
1258 |
qed |
|
1259 |
||
1260 |
lemma le_divide_eq: |
|
1261 |
"(a \<le> b/c) = |
|
1262 |
(if 0 < c then a*c \<le> b |
|
1263 |
else if c < 0 then b \<le> a*c |
|
1264 |
else a \<le> (0::'a::{ordered_field,division_by_zero}))" |
|
1265 |
apply (case_tac "c=0", simp) |
|
1266 |
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) |
|
1267 |
done |
|
1268 |
||
1269 |
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" |
|
1270 |
proof - |
|
1271 |
assume less: "0<c" |
|
1272 |
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" |
|
1273 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1274 |
also have "... = (b \<le> a*c)" |
|
1275 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1276 |
finally show ?thesis . |
|
1277 |
qed |
|
1278 |
||
1279 |
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" |
|
1280 |
proof - |
|
1281 |
assume less: "c<0" |
|
1282 |
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" |
|
1283 |
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) |
|
1284 |
also have "... = (a*c \<le> b)" |
|
1285 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1286 |
finally show ?thesis . |
|
1287 |
qed |
|
1288 |
||
1289 |
lemma divide_le_eq: |
|
1290 |
"(b/c \<le> a) = |
|
1291 |
(if 0 < c then b \<le> a*c |
|
1292 |
else if c < 0 then a*c \<le> b |
|
1293 |
else 0 \<le> (a::'a::{ordered_field,division_by_zero}))" |
|
1294 |
apply (case_tac "c=0", simp) |
|
1295 |
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) |
|
1296 |
done |
|
1297 |
||
1298 |
||
1299 |
lemma pos_less_divide_eq: |
|
1300 |
"0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)" |
|
1301 |
proof - |
|
1302 |
assume less: "0<c" |
|
1303 |
hence "(a < b/c) = (a*c < (b/c)*c)" |
|
1304 |
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) |
|
1305 |
also have "... = (a*c < b)" |
|
1306 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1307 |
finally show ?thesis . |
|
1308 |
qed |
|
1309 |
||
1310 |
lemma neg_less_divide_eq: |
|
1311 |
"c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)" |
|
1312 |
proof - |
|
1313 |
assume less: "c<0" |
|
1314 |
hence "(a < b/c) = ((b/c)*c < a*c)" |
|
1315 |
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) |
|
1316 |
also have "... = (b < a*c)" |
|
1317 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1318 |
finally show ?thesis . |
|
1319 |
qed |
|
1320 |
||
1321 |
lemma less_divide_eq: |
|
1322 |
"(a < b/c) = |
|
1323 |
(if 0 < c then a*c < b |
|
1324 |
else if c < 0 then b < a*c |
|
1325 |
else a < (0::'a::{ordered_field,division_by_zero}))" |
|
1326 |
apply (case_tac "c=0", simp) |
|
1327 |
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) |
|
1328 |
done |
|
1329 |
||
1330 |
lemma pos_divide_less_eq: |
|
1331 |
"0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)" |
|
1332 |
proof - |
|
1333 |
assume less: "0<c" |
|
1334 |
hence "(b/c < a) = ((b/c)*c < a*c)" |
|
1335 |
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) |
|
1336 |
also have "... = (b < a*c)" |
|
1337 |
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) |
|
1338 |
finally show ?thesis . |
|
1339 |
qed |
|
1340 |
||
1341 |
lemma neg_divide_less_eq: |
|
1342 |
"c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)" |
|
1343 |
proof - |
|
1344 |
assume less: "c<0" |
|
1345 |
hence "(b/c < a) = (a*c < (b/c)*c)" |
|
1346 |
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) |
|
1347 |
also have "... = (a*c < b)" |
|
1348 |
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) |
|
1349 |
finally show ?thesis . |
|
1350 |
qed |
|
1351 |
||
1352 |
lemma divide_less_eq: |
|
1353 |
"(b/c < a) = |
|
1354 |
(if 0 < c then b < a*c |
|
1355 |
else if c < 0 then a*c < b |
|
1356 |
else 0 < (a::'a::{ordered_field,division_by_zero}))" |
|
1357 |
apply (case_tac "c=0", simp) |
|
1358 |
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) |
|
1359 |
done |
|
1360 |
||
1361 |
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)" |
|
1362 |
proof - |
|
1363 |
assume [simp]: "c\<noteq>0" |
|
1364 |
have "(a = b/c) = (a*c = (b/c)*c)" |
|
1365 |
by (simp add: field_mult_cancel_right) |
|
1366 |
also have "... = (a*c = b)" |
|
1367 |
by (simp add: divide_inverse mult_assoc) |
|
1368 |
finally show ?thesis . |
|
1369 |
qed |
|
1370 |
||
1371 |
lemma eq_divide_eq: |
|
1372 |
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" |
|
1373 |
by (simp add: nonzero_eq_divide_eq) |
|
1374 |
||
1375 |
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)" |
|
1376 |
proof - |
|
1377 |
assume [simp]: "c\<noteq>0" |
|
1378 |
have "(b/c = a) = ((b/c)*c = a*c)" |
|
1379 |
by (simp add: field_mult_cancel_right) |
|
1380 |
also have "... = (b = a*c)" |
|
1381 |
by (simp add: divide_inverse mult_assoc) |
|
1382 |
finally show ?thesis . |
|
1383 |
qed |
|
1384 |
||
1385 |
lemma divide_eq_eq: |
|
1386 |
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" |
|
1387 |
by (force simp add: nonzero_divide_eq_eq) |
|
1388 |
||
1389 |
subsection{*Cancellation Laws for Division*} |
|
1390 |
||
1391 |
lemma divide_cancel_right [simp]: |
|
1392 |
"(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
|
1393 |
apply (case_tac "c=0", simp) |
|
1394 |
apply (simp add: divide_inverse_zero field_mult_cancel_right) |
|
1395 |
done |
|
1396 |
||
1397 |
lemma divide_cancel_left [simp]: |
|
1398 |
"(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" |
|
1399 |
apply (case_tac "c=0", simp) |
|
1400 |
apply (simp add: divide_inverse_zero field_mult_cancel_left) |
|
1401 |
done |
|
1402 |
||
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1403 |
subsection {* Division and the Number One *} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1404 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1405 |
text{*Simplify expressions equated with 1*} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1406 |
lemma divide_eq_1_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1407 |
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1408 |
apply (case_tac "b=0", simp) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1409 |
apply (simp add: right_inverse_eq) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1410 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1411 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1412 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1413 |
lemma one_eq_divide_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1414 |
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1415 |
by (simp add: eq_commute [of 1]) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1416 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1417 |
lemma zero_eq_1_divide_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1418 |
"((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1419 |
apply (case_tac "a=0", simp) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1420 |
apply (auto simp add: nonzero_eq_divide_eq) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1421 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1422 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1423 |
lemma one_divide_eq_0_iff [simp]: |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1424 |
"(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)" |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1425 |
apply (case_tac "a=0", simp) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1426 |
apply (insert zero_neq_one [THEN not_sym]) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1427 |
apply (auto simp add: nonzero_divide_eq_eq) |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1428 |
done |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1429 |
|
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1430 |
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1431 |
declare zero_less_divide_iff [of "1", simp] |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1432 |
declare divide_less_0_iff [of "1", simp] |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1433 |
declare zero_le_divide_iff [of "1", simp] |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1434 |
declare divide_le_0_iff [of "1", simp] |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1435 |
|
14288 | 1436 |
|
14293 | 1437 |
subsection {* Ordering Rules for Division *} |
1438 |
||
1439 |
lemma divide_strict_right_mono: |
|
1440 |
"[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)" |
|
1441 |
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono |
|
1442 |
positive_imp_inverse_positive) |
|
1443 |
||
1444 |
lemma divide_right_mono: |
|
1445 |
"[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})" |
|
1446 |
by (force simp add: divide_strict_right_mono order_le_less) |
|
1447 |
||
1448 |
||
1449 |
text{*The last premise ensures that @{term a} and @{term b} |
|
1450 |
have the same sign*} |
|
1451 |
lemma divide_strict_left_mono: |
|
1452 |
"[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
|
1453 |
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono |
|
1454 |
order_less_imp_not_eq order_less_imp_not_eq2 |
|
1455 |
less_imp_inverse_less less_imp_inverse_less_neg) |
|
1456 |
||
1457 |
lemma divide_left_mono: |
|
1458 |
"[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)" |
|
1459 |
apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") |
|
1460 |
prefer 2 |
|
1461 |
apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) |
|
1462 |
apply (case_tac "c=0", simp add: divide_inverse) |
|
1463 |
apply (force simp add: divide_strict_left_mono order_le_less) |
|
1464 |
done |
|
1465 |
||
1466 |
lemma divide_strict_left_mono_neg: |
|
1467 |
"[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)" |
|
1468 |
apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") |
|
1469 |
prefer 2 |
|
1470 |
apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) |
|
1471 |
apply (drule divide_strict_left_mono [of _ _ "-c"]) |
|
1472 |
apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) |
|
1473 |
done |
|
1474 |
||
1475 |
lemma divide_strict_right_mono_neg: |
|
1476 |
"[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)" |
|
1477 |
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) |
|
1478 |
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) |
|
1479 |
done |
|
1480 |
||
1481 |
||
1482 |
subsection {* Ordered Fields are Dense *} |
|
1483 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1484 |
lemma less_add_one: "a < (a+1::'a::ordered_semiring)" |
14293 | 1485 |
proof - |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1486 |
have "a+0 < (a+1::'a::ordered_semiring)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1487 |
by (blast intro: zero_less_one add_strict_left_mono) |
14293 | 1488 |
thus ?thesis by simp |
1489 |
qed |
|
1490 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1491 |
lemma zero_less_two: "0 < (1+1::'a::ordered_semiring)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1492 |
by (blast intro: order_less_trans zero_less_one less_add_one) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1493 |
|
14293 | 1494 |
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)" |
1495 |
by (simp add: zero_less_two pos_less_divide_eq right_distrib) |
|
1496 |
||
1497 |
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b" |
|
1498 |
by (simp add: zero_less_two pos_divide_less_eq right_distrib) |
|
1499 |
||
1500 |
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b" |
|
1501 |
by (blast intro!: less_half_sum gt_half_sum) |
|
1502 |
||
1503 |
||
1504 |
subsection {* Absolute Value *} |
|
1505 |
||
1506 |
lemma abs_zero [simp]: "abs 0 = (0::'a::ordered_ring)" |
|
1507 |
by (simp add: abs_if) |
|
1508 |
||
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1509 |
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_ring)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1510 |
by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1511 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1512 |
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_ring)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1513 |
apply (case_tac "a=0 | b=0", force) |
14293 | 1514 |
apply (auto elim: order_less_asym |
1515 |
simp add: abs_if mult_less_0_iff linorder_neq_iff |
|
1516 |
minus_mult_left [symmetric] minus_mult_right [symmetric]) |
|
1517 |
done |
|
1518 |
||
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
1519 |
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_ring)" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
1520 |
by (simp add: abs_if) |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
1521 |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1522 |
lemma abs_eq_0 [simp]: "(abs a = 0) = (a = (0::'a::ordered_ring))" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1523 |
by (simp add: abs_if) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1524 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1525 |
lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::ordered_ring))" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1526 |
by (simp add: abs_if linorder_neq_iff) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1527 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1528 |
lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::ordered_ring)" |
14398
c5c47703f763
Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents:
14387
diff
changeset
|
1529 |
apply (simp add: abs_if) |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1530 |
by (simp add: abs_if order_less_not_sym [of a 0]) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1531 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1532 |
lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::ordered_ring)) = (a = 0)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1533 |
by (simp add: order_le_less) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1534 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1535 |
lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::ordered_ring)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1536 |
apply (auto simp add: abs_if linorder_not_less order_less_not_sym [of 0 a]) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1537 |
apply (drule order_antisym, assumption, simp) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1538 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1539 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1540 |
lemma abs_ge_zero [simp]: "(0::'a::ordered_ring) \<le> abs a" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1541 |
apply (simp add: abs_if order_less_imp_le) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1542 |
apply (simp add: linorder_not_less) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1543 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1544 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1545 |
lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::ordered_ring)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1546 |
by (force elim: order_less_asym simp add: abs_if) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1547 |
|
14305
f17ca9f6dc8c
tidying first part of HyperArith0.ML, using generic lemmas
paulson
parents:
14295
diff
changeset
|
1548 |
lemma abs_zero_iff [simp]: "(abs a = 0) = (a = (0::'a::ordered_ring))" |
14293 | 1549 |
by (simp add: abs_if) |
1550 |
||
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1551 |
lemma abs_ge_self: "a \<le> abs (a::'a::ordered_ring)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1552 |
apply (simp add: abs_if) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1553 |
apply (simp add: order_less_imp_le order_trans [of _ 0]) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1554 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1555 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1556 |
lemma abs_ge_minus_self: "-a \<le> abs (a::'a::ordered_ring)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1557 |
by (insert abs_ge_self [of "-a"], simp) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1558 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1559 |
lemma nonzero_abs_inverse: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1560 |
"a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1561 |
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1562 |
negative_imp_inverse_negative) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1563 |
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1564 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1565 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1566 |
lemma abs_inverse [simp]: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1567 |
"abs (inverse (a::'a::{ordered_field,division_by_zero})) = |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1568 |
inverse (abs a)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1569 |
apply (case_tac "a=0", simp) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1570 |
apply (simp add: nonzero_abs_inverse) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1571 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1572 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1573 |
lemma nonzero_abs_divide: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1574 |
"b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1575 |
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1576 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1577 |
lemma abs_divide: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1578 |
"abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1579 |
apply (case_tac "b=0", simp) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1580 |
apply (simp add: nonzero_abs_divide) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1581 |
done |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1582 |
|
14295 | 1583 |
lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::ordered_ring)" |
1584 |
by (simp add: abs_if) |
|
1585 |
||
1586 |
lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::ordered_ring))" |
|
1587 |
proof |
|
1588 |
assume ale: "a \<le> -a" |
|
1589 |
show "a\<le>0" |
|
1590 |
apply (rule classical) |
|
1591 |
apply (simp add: linorder_not_le) |
|
1592 |
apply (blast intro: ale order_trans order_less_imp_le |
|
1593 |
neg_0_le_iff_le [THEN iffD1]) |
|
1594 |
done |
|
1595 |
next |
|
1596 |
assume "a\<le>0" |
|
1597 |
hence "0 \<le> -a" by (simp only: neg_0_le_iff_le) |
|
1598 |
thus "a \<le> -a" by (blast intro: prems order_trans) |
|
1599 |
qed |
|
1600 |
||
1601 |
lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::ordered_ring))" |
|
1602 |
by (insert le_minus_self_iff [of "-a"], simp) |
|
1603 |
||
1604 |
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_ring))" |
|
1605 |
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff) |
|
1606 |
||
1607 |
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_ring))" |
|
1608 |
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff) |
|
1609 |
||
1610 |
lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::ordered_ring)" |
|
1611 |
apply (simp add: abs_if split: split_if_asm) |
|
1612 |
apply (rule order_trans [of _ "-a"]) |
|
1613 |
apply (simp add: less_minus_self_iff order_less_imp_le, assumption) |
|
1614 |
done |
|
1615 |
||
1616 |
lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::ordered_ring)" |
|
1617 |
by (insert abs_le_D1 [of "-a"], simp) |
|
1618 |
||
1619 |
lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::ordered_ring))" |
|
1620 |
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2) |
|
1621 |
||
1622 |
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_ring))" |
|
1623 |
apply (simp add: order_less_le abs_le_iff) |
|
14398
c5c47703f763
Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents:
14387
diff
changeset
|
1624 |
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff) |
c5c47703f763
Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents:
14387
diff
changeset
|
1625 |
apply (simp add: le_minus_self_iff linorder_neq_iff) |
c5c47703f763
Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents:
14387
diff
changeset
|
1626 |
done |
c5c47703f763
Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents:
14387
diff
changeset
|
1627 |
(* |
c5c47703f763
Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents:
14387
diff
changeset
|
1628 |
apply (simp add: order_less_le abs_le_iff) |
14295 | 1629 |
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff) |
14398
c5c47703f763
Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents:
14387
diff
changeset
|
1630 |
apply (simp add: linorder_not_less [symmetric]) |
14295 | 1631 |
apply (simp add: le_minus_self_iff linorder_neq_iff) |
1632 |
apply (simp add: linorder_not_less [symmetric]) |
|
1633 |
done |
|
14398
c5c47703f763
Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents:
14387
diff
changeset
|
1634 |
*) |
14295 | 1635 |
|
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1636 |
lemma abs_triangle_ineq: "abs (a+b) \<le> abs a + abs (b::'a::ordered_ring)" |
14295 | 1637 |
by (force simp add: abs_le_iff abs_ge_self abs_ge_minus_self add_mono) |
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1638 |
|
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1639 |
lemma abs_mult_less: |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1640 |
"[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_ring)" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1641 |
proof - |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1642 |
assume ac: "abs a < c" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1643 |
hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1644 |
assume "abs b < d" |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1645 |
thus ?thesis by (simp add: ac cpos mult_strict_mono) |
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset
|
1646 |
qed |
14293 | 1647 |
|
14331 | 1648 |
ML |
1649 |
{* |
|
14334 | 1650 |
val add_assoc = thm"add_assoc"; |
1651 |
val add_commute = thm"add_commute"; |
|
1652 |
val mult_assoc = thm"mult_assoc"; |
|
1653 |
val mult_commute = thm"mult_commute"; |
|
1654 |
val zero_neq_one = thm"zero_neq_one"; |
|
1655 |
val diff_minus = thm"diff_minus"; |
|
1656 |
val abs_if = thm"abs_if"; |
|
1657 |
val divide_inverse = thm"divide_inverse"; |
|
1658 |
val inverse_zero = thm"inverse_zero"; |
|
1659 |
val divide_zero = thm"divide_zero"; |
|
14368
2763da611ad9
converted Real/Lubs to Isar script. Converting arithmetic setup
paulson
parents:
14365
diff
changeset
|
1660 |
|
14334 | 1661 |
val add_0 = thm"add_0"; |
14331 | 1662 |
val add_0_right = thm"add_0_right"; |
14368
2763da611ad9
converted Real/Lubs to Isar script. Converting arithmetic setup
paulson
parents:
14365
diff
changeset
|
1663 |
val add_zero_left = thm"add_0"; |
2763da611ad9
converted Real/Lubs to Isar script. Converting arithmetic setup
paulson
parents:
14365
diff
changeset
|
1664 |
val add_zero_right = thm"add_0_right"; |
2763da611ad9
converted Real/Lubs to Isar script. Converting arithmetic setup
paulson
parents:
14365
diff
changeset
|
1665 |
|
14331 | 1666 |
val add_left_commute = thm"add_left_commute"; |
14334 | 1667 |
val left_minus = thm"left_minus"; |
14331 | 1668 |
val right_minus = thm"right_minus"; |
1669 |
val right_minus_eq = thm"right_minus_eq"; |
|
1670 |
val add_left_cancel = thm"add_left_cancel"; |
|
1671 |
val add_right_cancel = thm"add_right_cancel"; |
|
1672 |
val minus_minus = thm"minus_minus"; |
|
1673 |
val equals_zero_I = thm"equals_zero_I"; |
|
1674 |
val minus_zero = thm"minus_zero"; |
|
1675 |
val diff_self = thm"diff_self"; |
|
1676 |
val diff_0 = thm"diff_0"; |
|
1677 |
val diff_0_right = thm"diff_0_right"; |
|
1678 |
val diff_minus_eq_add = thm"diff_minus_eq_add"; |
|
1679 |
val neg_equal_iff_equal = thm"neg_equal_iff_equal"; |
|
1680 |
val neg_equal_0_iff_equal = thm"neg_equal_0_iff_equal"; |
|
1681 |
val neg_0_equal_iff_equal = thm"neg_0_equal_iff_equal"; |
|
1682 |
val equation_minus_iff = thm"equation_minus_iff"; |
|
1683 |
val minus_equation_iff = thm"minus_equation_iff"; |
|
14334 | 1684 |
val mult_1 = thm"mult_1"; |
14331 | 1685 |
val mult_1_right = thm"mult_1_right"; |
1686 |
val mult_left_commute = thm"mult_left_commute"; |
|
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1687 |
val mult_zero_left = thm"mult_zero_left"; |
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
1688 |
val mult_zero_right = thm"mult_zero_right"; |
14334 | 1689 |
val left_distrib = thm "left_distrib"; |
14331 | 1690 |
val right_distrib = thm"right_distrib"; |
1691 |
val combine_common_factor = thm"combine_common_factor"; |
|
1692 |
val minus_add_distrib = thm"minus_add_distrib"; |
|
1693 |
val minus_mult_left = thm"minus_mult_left"; |
|
1694 |
val minus_mult_right = thm"minus_mult_right"; |
|
1695 |
val minus_mult_minus = thm"minus_mult_minus"; |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1696 |
val minus_mult_commute = thm"minus_mult_commute"; |
14331 | 1697 |
val right_diff_distrib = thm"right_diff_distrib"; |
1698 |
val left_diff_distrib = thm"left_diff_distrib"; |
|
1699 |
val minus_diff_eq = thm"minus_diff_eq"; |
|
14334 | 1700 |
val add_left_mono = thm"add_left_mono"; |
14331 | 1701 |
val add_right_mono = thm"add_right_mono"; |
1702 |
val add_mono = thm"add_mono"; |
|
1703 |
val add_strict_left_mono = thm"add_strict_left_mono"; |
|
1704 |
val add_strict_right_mono = thm"add_strict_right_mono"; |
|
1705 |
val add_strict_mono = thm"add_strict_mono"; |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
1706 |
val add_less_le_mono = thm"add_less_le_mono"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset
|
1707 |
val add_le_less_mono = thm"add_le_less_mono"; |
14331 | 1708 |
val add_less_imp_less_left = thm"add_less_imp_less_left"; |
1709 |
val add_less_imp_less_right = thm"add_less_imp_less_right"; |
|
1710 |
val add_less_cancel_left = thm"add_less_cancel_left"; |
|
1711 |
val add_less_cancel_right = thm"add_less_cancel_right"; |
|
1712 |
val add_le_cancel_left = thm"add_le_cancel_left"; |
|
1713 |
val add_le_cancel_right = thm"add_le_cancel_right"; |
|
1714 |
val add_le_imp_le_left = thm"add_le_imp_le_left"; |
|
1715 |
val add_le_imp_le_right = thm"add_le_imp_le_right"; |
|
1716 |
val le_imp_neg_le = thm"le_imp_neg_le"; |
|
1717 |
val neg_le_iff_le = thm"neg_le_iff_le"; |
|
1718 |
val neg_le_0_iff_le = thm"neg_le_0_iff_le"; |
|
1719 |
val neg_0_le_iff_le = thm"neg_0_le_iff_le"; |
|
1720 |
val neg_less_iff_less = thm"neg_less_iff_less"; |
|
1721 |
val neg_less_0_iff_less = thm"neg_less_0_iff_less"; |
|
1722 |
val neg_0_less_iff_less = thm"neg_0_less_iff_less"; |
|
1723 |
val less_minus_iff = thm"less_minus_iff"; |
|
1724 |
val minus_less_iff = thm"minus_less_iff"; |
|
1725 |
val le_minus_iff = thm"le_minus_iff"; |
|
1726 |
val minus_le_iff = thm"minus_le_iff"; |
|
1727 |
val add_diff_eq = thm"add_diff_eq"; |
|
1728 |
val diff_add_eq = thm"diff_add_eq"; |
|
1729 |
val diff_eq_eq = thm"diff_eq_eq"; |
|
1730 |
val eq_diff_eq = thm"eq_diff_eq"; |
|
1731 |
val diff_diff_eq = thm"diff_diff_eq"; |
|
1732 |
val diff_diff_eq2 = thm"diff_diff_eq2"; |
|
1733 |
val less_iff_diff_less_0 = thm"less_iff_diff_less_0"; |
|
1734 |
val diff_less_eq = thm"diff_less_eq"; |
|
1735 |
val less_diff_eq = thm"less_diff_eq"; |
|
1736 |
val diff_le_eq = thm"diff_le_eq"; |
|
1737 |
val le_diff_eq = thm"le_diff_eq"; |
|
1738 |
val eq_iff_diff_eq_0 = thm"eq_iff_diff_eq_0"; |
|
1739 |
val le_iff_diff_le_0 = thm"le_iff_diff_le_0"; |
|
1740 |
val eq_add_iff1 = thm"eq_add_iff1"; |
|
1741 |
val eq_add_iff2 = thm"eq_add_iff2"; |
|
1742 |
val less_add_iff1 = thm"less_add_iff1"; |
|
1743 |
val less_add_iff2 = thm"less_add_iff2"; |
|
1744 |
val le_add_iff1 = thm"le_add_iff1"; |
|
1745 |
val le_add_iff2 = thm"le_add_iff2"; |
|
14334 | 1746 |
val mult_strict_left_mono = thm"mult_strict_left_mono"; |
14331 | 1747 |
val mult_strict_right_mono = thm"mult_strict_right_mono"; |
1748 |
val mult_left_mono = thm"mult_left_mono"; |
|
1749 |
val mult_right_mono = thm"mult_right_mono"; |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
1750 |
val mult_left_le_imp_le = thm"mult_left_le_imp_le"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
1751 |
val mult_right_le_imp_le = thm"mult_right_le_imp_le"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
1752 |
val mult_left_less_imp_less = thm"mult_left_less_imp_less"; |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
1753 |
val mult_right_less_imp_less = thm"mult_right_less_imp_less"; |
14331 | 1754 |
val mult_strict_left_mono_neg = thm"mult_strict_left_mono_neg"; |
1755 |
val mult_strict_right_mono_neg = thm"mult_strict_right_mono_neg"; |
|
1756 |
val mult_pos = thm"mult_pos"; |
|
1757 |
val mult_pos_neg = thm"mult_pos_neg"; |
|
1758 |
val mult_neg = thm"mult_neg"; |
|
1759 |
val zero_less_mult_pos = thm"zero_less_mult_pos"; |
|
1760 |
val zero_less_mult_iff = thm"zero_less_mult_iff"; |
|
1761 |
val mult_eq_0_iff = thm"mult_eq_0_iff"; |
|
1762 |
val zero_le_mult_iff = thm"zero_le_mult_iff"; |
|
1763 |
val mult_less_0_iff = thm"mult_less_0_iff"; |
|
1764 |
val mult_le_0_iff = thm"mult_le_0_iff"; |
|
1765 |
val zero_le_square = thm"zero_le_square"; |
|
1766 |
val zero_less_one = thm"zero_less_one"; |
|
1767 |
val zero_le_one = thm"zero_le_one"; |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1768 |
val not_one_less_zero = thm"not_one_less_zero"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
1769 |
val not_one_le_zero = thm"not_one_le_zero"; |
14331 | 1770 |
val mult_left_mono_neg = thm"mult_left_mono_neg"; |
1771 |
val mult_right_mono_neg = thm"mult_right_mono_neg"; |
|
1772 |
val mult_strict_mono = thm"mult_strict_mono"; |
|
1773 |
val mult_strict_mono' = thm"mult_strict_mono'"; |
|
1774 |
val mult_mono = thm"mult_mono"; |
|
1775 |
val mult_less_cancel_right = thm"mult_less_cancel_right"; |
|
1776 |
val mult_less_cancel_left = thm"mult_less_cancel_left"; |
|
1777 |
val mult_le_cancel_right = thm"mult_le_cancel_right"; |
|
1778 |
val mult_le_cancel_left = thm"mult_le_cancel_left"; |
|
1779 |
val mult_less_imp_less_left = thm"mult_less_imp_less_left"; |
|
1780 |
val mult_less_imp_less_right = thm"mult_less_imp_less_right"; |
|
1781 |
val mult_cancel_right = thm"mult_cancel_right"; |
|
1782 |
val mult_cancel_left = thm"mult_cancel_left"; |
|
1783 |
val left_inverse = thm "left_inverse"; |
|
1784 |
val right_inverse = thm"right_inverse"; |
|
1785 |
val right_inverse_eq = thm"right_inverse_eq"; |
|
1786 |
val nonzero_inverse_eq_divide = thm"nonzero_inverse_eq_divide"; |
|
1787 |
val divide_self = thm"divide_self"; |
|
1788 |
val divide_inverse_zero = thm"divide_inverse_zero"; |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset
|
1789 |
val inverse_divide = thm"inverse_divide"; |
14331 | 1790 |
val divide_zero_left = thm"divide_zero_left"; |
1791 |
val inverse_eq_divide = thm"inverse_eq_divide"; |
|
1792 |
val nonzero_add_divide_distrib = thm"nonzero_add_divide_distrib"; |
|
1793 |
val add_divide_distrib = thm"add_divide_distrib"; |
|
1794 |
val field_mult_eq_0_iff = thm"field_mult_eq_0_iff"; |
|
1795 |
val field_mult_cancel_right = thm"field_mult_cancel_right"; |
|
1796 |
val field_mult_cancel_left = thm"field_mult_cancel_left"; |
|
1797 |
val nonzero_imp_inverse_nonzero = thm"nonzero_imp_inverse_nonzero"; |
|
1798 |
val inverse_zero_imp_zero = thm"inverse_zero_imp_zero"; |
|
1799 |
val inverse_nonzero_imp_nonzero = thm"inverse_nonzero_imp_nonzero"; |
|
1800 |
val inverse_nonzero_iff_nonzero = thm"inverse_nonzero_iff_nonzero"; |
|
1801 |
val nonzero_inverse_minus_eq = thm"nonzero_inverse_minus_eq"; |
|
1802 |
val inverse_minus_eq = thm"inverse_minus_eq"; |
|
1803 |
val nonzero_inverse_eq_imp_eq = thm"nonzero_inverse_eq_imp_eq"; |
|
1804 |
val inverse_eq_imp_eq = thm"inverse_eq_imp_eq"; |
|
1805 |
val inverse_eq_iff_eq = thm"inverse_eq_iff_eq"; |
|
1806 |
val nonzero_inverse_inverse_eq = thm"nonzero_inverse_inverse_eq"; |
|
1807 |
val inverse_inverse_eq = thm"inverse_inverse_eq"; |
|
1808 |
val inverse_1 = thm"inverse_1"; |
|
1809 |
val nonzero_inverse_mult_distrib = thm"nonzero_inverse_mult_distrib"; |
|
1810 |
val inverse_mult_distrib = thm"inverse_mult_distrib"; |
|
1811 |
val inverse_add = thm"inverse_add"; |
|
1812 |
val nonzero_mult_divide_cancel_left = thm"nonzero_mult_divide_cancel_left"; |
|
1813 |
val mult_divide_cancel_left = thm"mult_divide_cancel_left"; |
|
1814 |
val nonzero_mult_divide_cancel_right = thm"nonzero_mult_divide_cancel_right"; |
|
1815 |
val mult_divide_cancel_right = thm"mult_divide_cancel_right"; |
|
1816 |
val mult_divide_cancel_eq_if = thm"mult_divide_cancel_eq_if"; |
|
1817 |
val divide_1 = thm"divide_1"; |
|
1818 |
val times_divide_eq_right = thm"times_divide_eq_right"; |
|
1819 |
val times_divide_eq_left = thm"times_divide_eq_left"; |
|
1820 |
val divide_divide_eq_right = thm"divide_divide_eq_right"; |
|
1821 |
val divide_divide_eq_left = thm"divide_divide_eq_left"; |
|
1822 |
val nonzero_minus_divide_left = thm"nonzero_minus_divide_left"; |
|
1823 |
val nonzero_minus_divide_right = thm"nonzero_minus_divide_right"; |
|
1824 |
val nonzero_minus_divide_divide = thm"nonzero_minus_divide_divide"; |
|
1825 |
val minus_divide_left = thm"minus_divide_left"; |
|
1826 |
val minus_divide_right = thm"minus_divide_right"; |
|
1827 |
val minus_divide_divide = thm"minus_divide_divide"; |
|
1828 |
val positive_imp_inverse_positive = thm"positive_imp_inverse_positive"; |
|
1829 |
val negative_imp_inverse_negative = thm"negative_imp_inverse_negative"; |
|
1830 |
val inverse_le_imp_le = thm"inverse_le_imp_le"; |
|
1831 |
val inverse_positive_imp_positive = thm"inverse_positive_imp_positive"; |
|
1832 |
val inverse_positive_iff_positive = thm"inverse_positive_iff_positive"; |
|
1833 |
val inverse_negative_imp_negative = thm"inverse_negative_imp_negative"; |
|
1834 |
val inverse_negative_iff_negative = thm"inverse_negative_iff_negative"; |
|
1835 |
val inverse_nonnegative_iff_nonnegative = thm"inverse_nonnegative_iff_nonnegative"; |
|
1836 |
val inverse_nonpositive_iff_nonpositive = thm"inverse_nonpositive_iff_nonpositive"; |
|
1837 |
val less_imp_inverse_less = thm"less_imp_inverse_less"; |
|
1838 |
val inverse_less_imp_less = thm"inverse_less_imp_less"; |
|
1839 |
val inverse_less_iff_less = thm"inverse_less_iff_less"; |
|
1840 |
val le_imp_inverse_le = thm"le_imp_inverse_le"; |
|
1841 |
val inverse_le_iff_le = thm"inverse_le_iff_le"; |
|
1842 |
val inverse_le_imp_le_neg = thm"inverse_le_imp_le_neg"; |
|
1843 |
val less_imp_inverse_less_neg = thm"less_imp_inverse_less_neg"; |
|
1844 |
val inverse_less_imp_less_neg = thm"inverse_less_imp_less_neg"; |
|
1845 |
val inverse_less_iff_less_neg = thm"inverse_less_iff_less_neg"; |
|
1846 |
val le_imp_inverse_le_neg = thm"le_imp_inverse_le_neg"; |
|
1847 |
val inverse_le_iff_le_neg = thm"inverse_le_iff_le_neg"; |
|
1848 |
val zero_less_divide_iff = thm"zero_less_divide_iff"; |
|
1849 |
val divide_less_0_iff = thm"divide_less_0_iff"; |
|
1850 |
val zero_le_divide_iff = thm"zero_le_divide_iff"; |
|
1851 |
val divide_le_0_iff = thm"divide_le_0_iff"; |
|
1852 |
val divide_eq_0_iff = thm"divide_eq_0_iff"; |
|
1853 |
val pos_le_divide_eq = thm"pos_le_divide_eq"; |
|
1854 |
val neg_le_divide_eq = thm"neg_le_divide_eq"; |
|
1855 |
val le_divide_eq = thm"le_divide_eq"; |
|
1856 |
val pos_divide_le_eq = thm"pos_divide_le_eq"; |
|
1857 |
val neg_divide_le_eq = thm"neg_divide_le_eq"; |
|
1858 |
val divide_le_eq = thm"divide_le_eq"; |
|
1859 |
val pos_less_divide_eq = thm"pos_less_divide_eq"; |
|
1860 |
val neg_less_divide_eq = thm"neg_less_divide_eq"; |
|
1861 |
val less_divide_eq = thm"less_divide_eq"; |
|
1862 |
val pos_divide_less_eq = thm"pos_divide_less_eq"; |
|
1863 |
val neg_divide_less_eq = thm"neg_divide_less_eq"; |
|
1864 |
val divide_less_eq = thm"divide_less_eq"; |
|
1865 |
val nonzero_eq_divide_eq = thm"nonzero_eq_divide_eq"; |
|
1866 |
val eq_divide_eq = thm"eq_divide_eq"; |
|
1867 |
val nonzero_divide_eq_eq = thm"nonzero_divide_eq_eq"; |
|
1868 |
val divide_eq_eq = thm"divide_eq_eq"; |
|
1869 |
val divide_cancel_right = thm"divide_cancel_right"; |
|
1870 |
val divide_cancel_left = thm"divide_cancel_left"; |
|
1871 |
val divide_strict_right_mono = thm"divide_strict_right_mono"; |
|
1872 |
val divide_right_mono = thm"divide_right_mono"; |
|
1873 |
val divide_strict_left_mono = thm"divide_strict_left_mono"; |
|
1874 |
val divide_left_mono = thm"divide_left_mono"; |
|
1875 |
val divide_strict_left_mono_neg = thm"divide_strict_left_mono_neg"; |
|
1876 |
val divide_strict_right_mono_neg = thm"divide_strict_right_mono_neg"; |
|
1877 |
val zero_less_two = thm"zero_less_two"; |
|
1878 |
val less_half_sum = thm"less_half_sum"; |
|
1879 |
val gt_half_sum = thm"gt_half_sum"; |
|
1880 |
val dense = thm"dense"; |
|
1881 |
val abs_zero = thm"abs_zero"; |
|
1882 |
val abs_one = thm"abs_one"; |
|
1883 |
val abs_mult = thm"abs_mult"; |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
1884 |
val abs_mult_self = thm"abs_mult_self"; |
14331 | 1885 |
val abs_eq_0 = thm"abs_eq_0"; |
1886 |
val zero_less_abs_iff = thm"zero_less_abs_iff"; |
|
1887 |
val abs_not_less_zero = thm"abs_not_less_zero"; |
|
1888 |
val abs_le_zero_iff = thm"abs_le_zero_iff"; |
|
1889 |
val abs_minus_cancel = thm"abs_minus_cancel"; |
|
1890 |
val abs_ge_zero = thm"abs_ge_zero"; |
|
1891 |
val abs_idempotent = thm"abs_idempotent"; |
|
1892 |
val abs_zero_iff = thm"abs_zero_iff"; |
|
1893 |
val abs_ge_self = thm"abs_ge_self"; |
|
1894 |
val abs_ge_minus_self = thm"abs_ge_minus_self"; |
|
1895 |
val nonzero_abs_inverse = thm"nonzero_abs_inverse"; |
|
1896 |
val abs_inverse = thm"abs_inverse"; |
|
1897 |
val nonzero_abs_divide = thm"nonzero_abs_divide"; |
|
1898 |
val abs_divide = thm"abs_divide"; |
|
1899 |
val abs_leI = thm"abs_leI"; |
|
1900 |
val le_minus_self_iff = thm"le_minus_self_iff"; |
|
1901 |
val minus_le_self_iff = thm"minus_le_self_iff"; |
|
1902 |
val eq_minus_self_iff = thm"eq_minus_self_iff"; |
|
1903 |
val less_minus_self_iff = thm"less_minus_self_iff"; |
|
1904 |
val abs_le_D1 = thm"abs_le_D1"; |
|
1905 |
val abs_le_D2 = thm"abs_le_D2"; |
|
1906 |
val abs_le_iff = thm"abs_le_iff"; |
|
1907 |
val abs_less_iff = thm"abs_less_iff"; |
|
1908 |
val abs_triangle_ineq = thm"abs_triangle_ineq"; |
|
1909 |
val abs_mult_less = thm"abs_mult_less"; |
|
1910 |
||
1911 |
val compare_rls = thms"compare_rls"; |
|
1912 |
*} |
|
1913 |
||
14293 | 1914 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset
|
1915 |
end |