author  paulson 
Tue, 06 Jan 2004 10:40:15 +0100  
changeset 14341  a09441bd4f1e 
parent 14334  6137d24eef79 
child 14348  744c868ee0b7 
permissions  rwrr 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

1 
(* Title: HOL/Ring_and_Field.thy 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

2 
ID: $Id$ 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

3 
Author: Gertrud Bauer and Markus Wenzel, TU Muenchen 
14269  4 
Lawrence C Paulson, University of Cambridge 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

5 
License: GPL (GNU GENERAL PUBLIC LICENSE) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

6 
*) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

7 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

8 
header {* 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

9 
\title{Ring and field structures} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

10 
\author{Gertrud Bauer and Markus Wenzel} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

11 
*} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

12 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

13 
theory Ring_and_Field = Inductive: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

14 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

15 
text{*Lemmas and extension to semirings by L. C. Paulson*} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

16 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

17 
subsection {* Abstract algebraic structures *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

18 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

19 
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

20 
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

21 
add_commute: "a + b = b + a" 
14288  22 
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

23 
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

24 
{*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

25 
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

26 
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

27 
axiom must be proved for instances of rings.*} 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

28 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

29 
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

30 
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

31 
mult_1 [simp]: "1 * a = a" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

32 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

33 
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

34 
zero_neq_one [simp]: "0 \<noteq> 1" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

35 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

36 
axclass ring \<subseteq> semiring, minus 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

37 
left_minus [simp]: " a + a = 0" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

38 
diff_minus: "a  b = a + (b)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

39 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

40 
axclass ordered_semiring \<subseteq> semiring, linorder 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

41 
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

42 
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

43 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

44 
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

45 
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

46 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

47 
axclass field \<subseteq> ring, inverse 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

48 
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

49 
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

50 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

51 
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

52 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

53 
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

54 
inverse_zero [simp]: "inverse 0 = 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

55 
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

56 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

57 

14270  58 
subsection {* Derived Rules for Addition *} 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

59 

14288  60 
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

61 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

62 
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

63 
also have "... = a" by simp 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

64 
finally show ?thesis . 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

65 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

66 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

67 
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

68 
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

69 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

70 
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

71 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

72 
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

73 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

74 
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

75 
also have "... = 0" by simp 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

76 
finally show ?thesis . 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

77 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

78 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

79 
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

80 
proof 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

81 
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

82 
also assume "a  b = 0" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

83 
finally show "a = b" by simp 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

84 
next 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

85 
assume "a = b" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

86 
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

87 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

88 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

89 
lemma add_left_cancel [simp]: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

90 
"(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

91 
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

92 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

93 
lemma add_right_cancel [simp]: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

94 
"(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

95 
by (simp add: add_commute) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

96 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

97 
lemma minus_minus [simp]: " ( (a::'a::ring)) = a" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

98 
proof (rule add_left_cancel [of "a", THEN iffD1]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

99 
show "(a + (a) = a + a)" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

100 
by simp 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

101 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

102 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

103 
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

104 
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

105 
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

106 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

107 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

108 
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

109 
by (simp add: equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

110 

14270  111 
lemma diff_self [simp]: "a  (a::'a::ring) = 0" 
112 
by (simp add: diff_minus) 

113 

114 
lemma diff_0 [simp]: "(0::'a::ring)  a = a" 

115 
by (simp add: diff_minus) 

116 

117 
lemma diff_0_right [simp]: "a  (0::'a::ring) = a" 

118 
by (simp add: diff_minus) 

119 

14288  120 
lemma diff_minus_eq_add [simp]: "a   b = a + (b::'a::ring)" 
121 
by (simp add: diff_minus) 

122 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

123 
lemma neg_equal_iff_equal [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

124 
proof 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

125 
assume " a =  b" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

126 
hence " ( a) =  ( b)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

127 
by simp 
14266  128 
thus "a=b" by simp 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

129 
next 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

130 
assume "a=b" 
14266  131 
thus "a = b" by simp 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

132 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

133 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

134 
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

135 
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

136 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

137 
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

138 
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

139 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

140 
text{*The next two equations can make the simplifier loop!*} 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

141 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

142 
lemma equation_minus_iff: "(a =  b) = (b =  (a::'a::ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

143 
proof  
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

144 
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

145 
thus ?thesis by (simp add: eq_commute) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

146 
qed 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

147 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

148 
lemma minus_equation_iff: "( a = b) = ( (b::'a::ring) = a)" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

149 
proof  
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

150 
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

151 
thus ?thesis by (simp add: eq_commute) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

152 
qed 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

153 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

154 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

155 
subsection {* Derived rules for multiplication *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

156 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

157 
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

158 
proof  
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

159 
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

160 
also have "... = a" by simp 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

161 
finally show ?thesis . 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

162 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

163 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

164 
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

165 
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

166 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

167 
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

168 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

169 
lemma mult_left_zero [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

170 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

171 
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

172 
by (simp add: left_distrib [symmetric]) 
14266  173 
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

174 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

175 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

176 
lemma mult_right_zero [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

177 
by (simp add: mult_commute) 
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 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

180 
subsection {* Distribution rules *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

181 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

182 
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

183 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

184 
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

185 
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

186 
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

187 
finally show ?thesis . 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

188 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

189 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

190 
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

191 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

192 
text{*For the @{text combine_numerals} simproc*} 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

193 
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

194 
by (simp add: left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

195 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

196 
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

197 
apply (rule equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

198 
apply (simp add: add_ac) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

199 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

200 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

201 
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

202 
apply (rule equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

203 
apply (simp add: left_distrib [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

204 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

205 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

206 
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

207 
apply (rule equals_zero_I) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

208 
apply (simp add: right_distrib [symmetric]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

209 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

210 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

211 
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

212 
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

213 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

214 
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

215 
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

216 
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

217 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

218 
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

219 
by (simp add: mult_commute [of _ c] right_diff_distrib) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

220 

14270  221 
lemma minus_diff_eq [simp]: " (a  b) = b  (a::'a::ring)" 
222 
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

223 

14270  224 

225 
subsection {* Ordering Rules for Addition *} 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

226 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

227 
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

228 
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

229 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

230 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

231 
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

232 
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

233 
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

234 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

235 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

236 
lemma add_strict_left_mono: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

237 
"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

238 
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

239 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

240 
lemma add_strict_right_mono: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

241 
"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

242 
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

243 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

244 
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

245 
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

246 
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

247 
apply (erule add_strict_left_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

248 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

249 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

250 
lemma add_less_le_mono: "[ a<b; c\<le>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

251 
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

252 
apply (erule add_left_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

253 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

254 

a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

255 
lemma add_le_less_mono: 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

256 
"[ 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

257 
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

258 
apply (erule add_strict_left_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

259 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

260 

14270  261 
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

262 
assumes less: "c + a < c + b" shows "a < (b::'a::ordered_semiring)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

263 
proof (rule ccontr) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

264 
assume "~ a < b" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

265 
hence "b \<le> a" by (simp add: linorder_not_less) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

266 
hence "c+b \<le> c+a" by (rule add_left_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

267 
with this and less show False 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

268 
by (simp add: linorder_not_less [symmetric]) 
14270  269 
qed 
270 

271 
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

272 
"a + c < b + c ==> a < (b::'a::ordered_semiring)" 
14270  273 
apply (rule add_less_imp_less_left [of c]) 
274 
apply (simp add: add_commute) 

275 
done 

276 

277 
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

278 
"(c+a < c+b) = (a < (b::'a::ordered_semiring))" 
14270  279 
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
280 

281 
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

282 
"(a+c < b+c) = (a < (b::'a::ordered_semiring))" 
14270  283 
by (blast intro: add_less_imp_less_right add_strict_right_mono) 
284 

285 
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

286 
"(c+a \<le> c+b) = (a \<le> (b::'a::ordered_semiring))" 
14270  287 
by (simp add: linorder_not_less [symmetric]) 
288 

289 
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

290 
"(a+c \<le> b+c) = (a \<le> (b::'a::ordered_semiring))" 
14270  291 
by (simp add: linorder_not_less [symmetric]) 
292 

293 
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

294 
"c + a \<le> c + b ==> a \<le> (b::'a::ordered_semiring)" 
14270  295 
by simp 
296 

297 
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

298 
"a + c \<le> b + c ==> a \<le> (b::'a::ordered_semiring)" 
14270  299 
by simp 
300 

301 

302 
subsection {* Ordering Rules for Unary Minus *} 

303 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

304 
lemma le_imp_neg_le: 
14269  305 
assumes "a \<le> (b::'a::ordered_ring)" shows "b \<le> a" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

306 
proof  
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

307 
have "a+a \<le> a+b" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

308 
by (rule add_left_mono) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

309 
hence "0 \<le> a+b" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

310 
by simp 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

311 
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

312 
by (rule add_right_mono) 
14266  313 
thus ?thesis 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

314 
by (simp add: add_assoc) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

315 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

316 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

317 
lemma neg_le_iff_le [simp]: "(b \<le> a) = (a \<le> (b::'a::ordered_ring))" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

318 
proof 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

319 
assume " b \<le>  a" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

320 
hence " ( a) \<le>  ( b)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

321 
by (rule le_imp_neg_le) 
14266  322 
thus "a\<le>b" by simp 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

323 
next 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

324 
assume "a\<le>b" 
14266  325 
thus "b \<le> a" by (rule le_imp_neg_le) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

326 
qed 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

327 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

328 
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

329 
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

330 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

331 
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

332 
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

333 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

334 
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

335 
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

336 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

337 
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

338 
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

339 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

340 
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

341 
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

342 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

343 
text{*The next several equations can make the simplifier loop!*} 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

344 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

345 
lemma less_minus_iff: "(a <  b) = (b <  (a::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

346 
proof  
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

347 
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

348 
thus ?thesis by simp 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

349 
qed 
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 minus_less_iff: "( a < b) = ( b < (a::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

352 
proof  
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 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

355 
qed 
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 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

358 
apply (simp add: linorder_not_less [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

359 
apply (rule minus_less_iff) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

360 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

361 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

362 
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

363 
apply (simp add: linorder_not_less [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

364 
apply (rule less_minus_iff) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

365 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

366 

14270  367 

368 
subsection{*Subtraction Laws*} 

369 

370 
lemma add_diff_eq: "a + (b  c) = (a + b)  (c::'a::ring)" 

371 
by (simp add: diff_minus add_ac) 

372 

373 
lemma diff_add_eq: "(a  b) + c = (a + c)  (b::'a::ring)" 

374 
by (simp add: diff_minus add_ac) 

375 

376 
lemma diff_eq_eq: "(ab = c) = (a = c + (b::'a::ring))" 

377 
by (auto simp add: diff_minus add_assoc) 

378 

379 
lemma eq_diff_eq: "(a = cb) = (a + (b::'a::ring) = c)" 

380 
by (auto simp add: diff_minus add_assoc) 

381 

382 
lemma diff_diff_eq: "(a  b)  c = a  (b + (c::'a::ring))" 

383 
by (simp add: diff_minus add_ac) 

384 

385 
lemma diff_diff_eq2: "a  (b  c) = (a + c)  (b::'a::ring)" 

386 
by (simp add: diff_minus add_ac) 

387 

388 
text{*Further subtraction laws for ordered rings*} 

389 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

390 
lemma less_iff_diff_less_0: "(a < b) = (a  b < (0::'a::ordered_ring))" 
14270  391 
proof  
392 
have "(a < b) = (a + ( b) < b + (b))" 

393 
by (simp only: add_less_cancel_right) 

394 
also have "... = (a  b < 0)" by (simp add: diff_minus) 

395 
finally show ?thesis . 

396 
qed 

397 

398 
lemma diff_less_eq: "(ab < c) = (a < c + (b::'a::ordered_ring))" 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

399 
apply (subst less_iff_diff_less_0) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

400 
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) 
14270  401 
apply (simp add: diff_minus add_ac) 
402 
done 

403 

404 
lemma less_diff_eq: "(a < cb) = (a + (b::'a::ordered_ring) < c)" 

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 _ "cb", THEN ssubst]) 
14270  407 
apply (simp add: diff_minus add_ac) 
408 
done 

409 

410 
lemma diff_le_eq: "(ab \<le> c) = (a \<le> c + (b::'a::ordered_ring))" 

411 
by (simp add: linorder_not_less [symmetric] less_diff_eq) 

412 

413 
lemma le_diff_eq: "(a \<le> cb) = (a + (b::'a::ordered_ring) \<le> c)" 

414 
by (simp add: linorder_not_less [symmetric] diff_less_eq) 

415 

416 
text{*This list of rewrites simplifies (in)equalities by bringing subtractions 

417 
to the top and then moving negative terms to the other side. 

418 
Use with @{text add_ac}*} 

419 
lemmas compare_rls = 

420 
diff_minus [symmetric] 

421 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 

422 
diff_less_eq less_diff_eq diff_le_eq le_diff_eq 

423 
diff_eq_eq eq_diff_eq 

424 

425 

14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

426 
subsection{*Lemmas for the @{text cancel_numerals} simproc*} 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

427 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

428 
lemma eq_iff_diff_eq_0: "(a = b) = (ab = (0::'a::ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

429 
by (simp add: compare_rls) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

430 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

431 
lemma le_iff_diff_le_0: "(a \<le> b) = (ab \<le> (0::'a::ordered_ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

432 
by (simp add: compare_rls) 
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_add_iff1: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

435 
"(a*e + c = b*e + d) = ((ab)*e + c = (d::'a::ring))" 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

436 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

437 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

438 
done 
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_iff2: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

441 
"(a*e + c = b*e + d) = (c = (ba)*e + (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 less_add_iff1: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

447 
"(a*e + c < b*e + d) = ((ab)*e + c < (d::'a::ordered_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_iff2: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

453 
"(a*e + c < b*e + d) = (c < (ba)*e + (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 le_add_iff1: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

459 
"(a*e + c \<le> b*e + d) = ((ab)*e + c \<le> (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_iff2: 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

465 
"(a*e + c \<le> b*e + d) = (c \<le> (ba)*e + (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 

14270  471 
subsection {* Ordering Rules for Multiplication *} 
472 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

473 
lemma mult_strict_right_mono: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

474 
"[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

475 
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

476 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

477 
lemma mult_left_mono: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

478 
"[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

479 
apply (case_tac "c=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

480 
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

481 
done 
14265
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_right_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] ==> 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

485 
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

486 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

487 
lemma mult_strict_left_mono_neg: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

488 
"[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

489 
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

490 
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

491 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

492 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

493 
lemma mult_strict_right_mono_neg: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

494 
"[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

495 
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

496 
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

497 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

498 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

499 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

500 
subsection{* Products of Signs *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

501 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

502 
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

503 
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

504 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

505 
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

506 
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

507 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

508 
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

509 
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

510 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

511 
lemma zero_less_mult_pos: 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

512 
"[ 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

513 
apply (case_tac "b\<le>0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

514 
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

515 
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

516 
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

517 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

518 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

519 
lemma zero_less_mult_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

520 
"((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

521 
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

522 
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

523 
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

524 
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

525 
done 
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 
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

528 
assumption of an ordering. See @{text field_mult_eq_0_iff} below.*} 
14266  529 
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

530 
apply (case_tac "a < 0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

531 
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

532 
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

533 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

534 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

535 
lemma zero_le_mult_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

536 
"((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

537 
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

538 
zero_less_mult_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

539 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

540 
lemma mult_less_0_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

541 
"(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

542 
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

543 
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

544 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

545 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

546 
lemma mult_le_0_iff: 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

547 
"(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

548 
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

549 
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

550 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

551 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

552 
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

553 
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

554 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

555 
lemma zero_less_one: "(0::'a::ordered_ring) < 1" 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

556 
apply (insert zero_le_square [of 1]) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

557 
apply (simp add: order_less_le) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

558 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

559 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

560 
lemma zero_le_one: "(0::'a::ordered_ring) \<le> 1" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

561 
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

562 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

563 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

564 
subsection{*More Monotonicity*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

565 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

566 
lemma mult_left_mono_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

567 
"[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

568 
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

569 
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

570 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

571 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

572 
lemma mult_right_mono_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

573 
"[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

574 
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

575 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

576 
text{*Strict monotonicity in both arguments*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

577 
lemma mult_strict_mono: 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

578 
"[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

579 
apply (case_tac "c=0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

580 
apply (simp add: mult_pos) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

581 
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

582 
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

583 
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

584 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

585 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

586 
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

587 
lemma mult_strict_mono': 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

588 
"[ 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

589 
apply (rule mult_strict_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

590 
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

591 
done 
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_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

594 
"[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

595 
==> 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

596 
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

597 
apply (erule mult_left_mono, assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

598 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

599 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

600 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

601 
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

602 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

603 
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

604 
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

605 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

606 
lemma mult_less_cancel_right: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

607 
"(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

608 
apply (case_tac "c = 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

609 
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

610 
mult_strict_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

611 
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

612 
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

613 
linorder_not_le [symmetric, of a]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

614 
apply (erule_tac [!] notE) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

615 
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

616 
mult_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

617 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

618 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

619 
lemma mult_less_cancel_left: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

620 
"(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

621 
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

622 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

623 
lemma mult_le_cancel_right: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

624 
"(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

625 
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

626 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

627 
lemma mult_le_cancel_left: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

628 
"(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

629 
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

630 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

631 
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

632 
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

633 
shows "a < (b::'a::ordered_semiring)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

634 
proof (rule ccontr) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

635 
assume "~ a < b" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

636 
hence "b \<le> a" by (simp add: linorder_not_less) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

637 
hence "c*b \<le> c*a" by (rule mult_left_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

638 
with this and less show False 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

639 
by (simp add: linorder_not_less [symmetric]) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

640 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

641 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

642 
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

643 
"[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

644 
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

645 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

646 
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

647 
lemma mult_cancel_right [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

648 
"(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

649 
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

650 
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

651 
simp add: linorder_neq_iff) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

652 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

653 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

654 
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

655 
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

656 
lemma mult_cancel_left [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

657 
"(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

658 
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

659 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

660 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

661 
subsection {* Fields *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

662 

14288  663 
lemma right_inverse [simp]: 
664 
assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1" 

665 
proof  

666 
have "a * inverse a = inverse a * a" by (simp add: mult_ac) 

667 
also have "... = 1" using not0 by simp 

668 
finally show ?thesis . 

669 
qed 

670 

671 
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" 

672 
proof 

673 
assume neq: "b \<noteq> 0" 

674 
{ 

675 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) 

676 
also assume "a / b = 1" 

677 
finally show "a = b" by simp 

678 
next 

679 
assume "a = b" 

680 
with neq show "a / b = 1" by (simp add: divide_inverse) 

681 
} 

682 
qed 

683 

684 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" 

685 
by (simp add: divide_inverse) 

686 

687 
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" 

688 
by (simp add: divide_inverse) 

689 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

690 
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

691 
apply (case_tac "b = 0") 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

692 
apply (simp_all add: divide_inverse) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

693 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

694 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

695 
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

696 
by (simp add: divide_inverse_zero) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

697 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

698 
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

699 
by (simp add: divide_inverse_zero) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

700 

14293  701 
lemma nonzero_add_divide_distrib: "c \<noteq> 0 ==> (a+b)/(c::'a::field) = a/c + b/c" 
702 
by (simp add: divide_inverse left_distrib) 

703 

704 
lemma add_divide_distrib: "(a+b)/(c::'a::{field,division_by_zero}) = a/c + b/c" 

705 
apply (case_tac "c=0", simp) 

706 
apply (simp add: nonzero_add_divide_distrib) 

707 
done 

708 

14270  709 
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement 
710 
of an ordering.*} 

711 
lemma field_mult_eq_0_iff: "(a*b = (0::'a::field)) = (a = 0  b = 0)" 

712 
proof cases 

713 
assume "a=0" thus ?thesis by simp 

714 
next 

715 
assume anz [simp]: "a\<noteq>0" 

716 
thus ?thesis 

717 
proof auto 

718 
assume "a * b = 0" 

719 
hence "inverse a * (a * b) = 0" by simp 

720 
thus "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric]) 

721 
qed 

722 
qed 

723 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

724 
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

725 
lemma field_mult_cancel_right_lemma: 
14269  726 
assumes cnz: "c \<noteq> (0::'a::field)" 
727 
and eq: "a*c = b*c" 

728 
shows "a=b" 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

729 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

730 
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

731 
by (simp add: eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

732 
thus "a=b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

733 
by (simp add: mult_assoc cnz) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

734 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

735 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

736 
lemma field_mult_cancel_right: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

737 
"(a*c = b*c) = (c = (0::'a::field)  a=b)" 
14269  738 
proof cases 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

739 
assume "c=0" thus ?thesis by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

740 
next 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

741 
assume "c\<noteq>0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

742 
thus ?thesis by (force dest: field_mult_cancel_right_lemma) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

743 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

744 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

745 
lemma field_mult_cancel_left: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

746 
"(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

747 
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

748 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

749 
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

750 
proof 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

751 
assume ianz: "inverse a = 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

752 
assume "a \<noteq> 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

753 
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

754 
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

755 
finally have "1 = (0::'a::field)" . 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

756 
thus False by (simp add: eq_commute) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

757 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

758 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

759 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

760 
subsection{*Basic Properties of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

761 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

762 
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

763 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

764 
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

765 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

766 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

767 
lemma inverse_nonzero_imp_nonzero: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

768 
"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

769 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

770 
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

771 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

772 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

773 
lemma inverse_nonzero_iff_nonzero [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

774 
"(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

775 
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

776 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

777 
lemma nonzero_inverse_minus_eq: 
14269  778 
assumes [simp]: "a\<noteq>0" shows "inverse(a) = inverse(a::'a::field)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

779 
proof  
14269  780 
have "a * inverse ( a) = a *  inverse a" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

781 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

782 
thus ?thesis 
14269  783 
by (simp only: field_mult_cancel_left, simp) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

784 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

785 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

786 
lemma inverse_minus_eq [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

787 
"inverse(a) = inverse(a::'a::{field,division_by_zero})"; 
14269  788 
proof cases 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

789 
assume "a=0" thus ?thesis by (simp add: inverse_zero) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

790 
next 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

791 
assume "a\<noteq>0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

792 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

793 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

794 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

795 
lemma nonzero_inverse_eq_imp_eq: 
14269  796 
assumes inveq: "inverse a = inverse b" 
797 
and anz: "a \<noteq> 0" 

798 
and bnz: "b \<noteq> 0" 

799 
shows "a = (b::'a::field)" 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

800 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

801 
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

802 
by (simp add: inveq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

803 
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

804 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

805 
thus "a = b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

806 
by (simp add: mult_assoc anz bnz) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

807 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

808 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

809 
lemma inverse_eq_imp_eq: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

810 
"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

811 
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

812 
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

813 
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

814 
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

815 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

816 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

817 
lemma inverse_eq_iff_eq [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

818 
"(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

819 
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

820 

14270  821 
lemma nonzero_inverse_inverse_eq: 
822 
assumes [simp]: "a \<noteq> 0" shows "inverse(inverse (a::'a::field)) = a" 

823 
proof  

824 
have "(inverse (inverse a) * inverse a) * a = a" 

825 
by (simp add: nonzero_imp_inverse_nonzero) 

826 
thus ?thesis 

827 
by (simp add: mult_assoc) 

828 
qed 

829 

830 
lemma inverse_inverse_eq [simp]: 

831 
"inverse(inverse (a::'a::{field,division_by_zero})) = a" 

832 
proof cases 

833 
assume "a=0" thus ?thesis by simp 

834 
next 

835 
assume "a\<noteq>0" 

836 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 

837 
qed 

838 

839 
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)" 

840 
proof  

841 
have "inverse 1 * 1 = (1::'a::field)" 

842 
by (rule left_inverse [OF zero_neq_one [symmetric]]) 

843 
thus ?thesis by simp 

844 
qed 

845 

846 
lemma nonzero_inverse_mult_distrib: 

847 
assumes anz: "a \<noteq> 0" 

848 
and bnz: "b \<noteq> 0" 

849 
shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)" 

850 
proof  

851 
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 

852 
by (simp add: field_mult_eq_0_iff anz bnz) 

853 
hence "inverse(a*b) * a = inverse(b)" 

854 
by (simp add: mult_assoc bnz) 

855 
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 

856 
by simp 

857 
thus ?thesis 

858 
by (simp add: mult_assoc anz) 

859 
qed 

860 

861 
text{*This version builds in division by zero while also reorienting 

862 
the righthand side.*} 

863 
lemma inverse_mult_distrib [simp]: 

864 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

865 
proof cases 

866 
assume "a \<noteq> 0 & b \<noteq> 0" 

867 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) 

868 
next 

869 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

870 
thus ?thesis by force 

871 
qed 

872 

873 
text{*There is no slick version using division by zero.*} 

874 
lemma inverse_add: 

875 
"[a \<noteq> 0; b \<noteq> 0] 

876 
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" 

877 
apply (simp add: left_distrib mult_assoc) 

878 
apply (simp add: mult_commute [of "inverse a"]) 

879 
apply (simp add: mult_assoc [symmetric] add_commute) 

880 
done 

881 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

882 
lemma nonzero_mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

883 
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

884 
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

885 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

886 
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

887 
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

888 
nonzero_inverse_mult_distrib) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

889 
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

890 
by (simp only: mult_ac) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

891 
also have "... = a * inverse b" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

892 
by simp 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

893 
finally show ?thesis 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

894 
by (simp add: divide_inverse) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

895 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

896 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

897 
lemma mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

898 
"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

899 
apply (case_tac "b = 0") 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

900 
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

901 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

902 

14321  903 
lemma nonzero_mult_divide_cancel_right: 
904 
"[b\<noteq>0; c\<noteq>0] ==> (a*c) / (b*c) = a/(b::'a::field)" 

905 
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 

906 

907 
lemma mult_divide_cancel_right: 

908 
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" 

909 
apply (case_tac "b = 0") 

910 
apply (simp_all add: nonzero_mult_divide_cancel_right) 

911 
done 

912 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

913 
(*For ExtractCommonTerm*) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

914 
lemma mult_divide_cancel_eq_if: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

915 
"(c*a) / (c*b) = 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

916 
(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

917 
by (simp add: mult_divide_cancel_left) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

918 

14284
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

919 
lemma divide_1 [simp]: "a/1 = (a::'a::field)" 
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

920 
by (simp add: divide_inverse [OF not_sym]) 
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

921 

14288  922 
lemma times_divide_eq_right [simp]: 
923 
"a * (b/c) = (a*b) / (c::'a::{field,division_by_zero})" 

924 
by (simp add: divide_inverse_zero mult_assoc) 

925 

926 
lemma times_divide_eq_left [simp]: 

927 
"(b/c) * a = (b*a) / (c::'a::{field,division_by_zero})" 

928 
by (simp add: divide_inverse_zero mult_ac) 

929 

930 
lemma divide_divide_eq_right [simp]: 

931 
"a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})" 

932 
by (simp add: divide_inverse_zero mult_ac) 

933 

934 
lemma divide_divide_eq_left [simp]: 

935 
"(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)" 

936 
by (simp add: divide_inverse_zero mult_assoc) 

937 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

938 

14293  939 
subsection {* Division and Unary Minus *} 
940 

941 
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==>  (a/b) = (a) / (b::'a::field)" 

942 
by (simp add: divide_inverse minus_mult_left) 

943 

944 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a/b) = a / (b::'a::field)" 

945 
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) 

946 

947 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a)/(b) = a / (b::'a::field)" 

948 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 

949 

950 
lemma minus_divide_left: " (a/b) = (a) / (b::'a::{field,division_by_zero})" 

951 
apply (case_tac "b=0", simp) 

952 
apply (simp add: nonzero_minus_divide_left) 

953 
done 

954 

955 
lemma minus_divide_right: " (a/b) = a / (b::'a::{field,division_by_zero})" 

956 
apply (case_tac "b=0", simp) 

957 
by (rule nonzero_minus_divide_right) 

958 

959 
text{*The effect is to extract signs from divisions*} 

960 
declare minus_divide_left [symmetric, simp] 

961 
declare minus_divide_right [symmetric, simp] 

962 

963 
lemma minus_divide_divide [simp]: 

964 
"(a)/(b) = a / (b::'a::{field,division_by_zero})" 

965 
apply (case_tac "b=0", simp) 

966 
apply (simp add: nonzero_minus_divide_divide) 

967 
done 

968 

969 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

970 
subsection {* Ordered Fields *} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

971 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

972 
lemma positive_imp_inverse_positive: 
14269  973 
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

974 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

975 
have "0 < a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

976 
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

977 
thus "0 < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

978 
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

979 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

980 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

981 
lemma negative_imp_inverse_negative: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

982 
"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

983 
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

984 
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

985 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

986 
lemma inverse_le_imp_le: 
14269  987 
assumes invle: "inverse a \<le> inverse b" 
988 
and apos: "0 < a" 

989 
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

990 
proof (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

991 
assume "~ b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

992 
hence "a < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

993 
by (simp add: linorder_not_le) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

994 
hence bpos: "0 < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

995 
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

996 
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

997 
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

998 
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

999 
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

1000 
thus "b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1001 
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

1002 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1003 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1004 
lemma inverse_positive_imp_positive: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1005 
assumes inv_gt_0: "0 < inverse a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1006 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1007 
shows "0 < (a::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1008 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1009 
have "0 < inverse (inverse a)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1010 
by (rule positive_imp_inverse_positive) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1011 
thus "0 < a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1012 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1013 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1014 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1015 
lemma inverse_positive_iff_positive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1016 
"(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

1017 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1018 
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

1019 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1020 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1021 
lemma inverse_negative_imp_negative: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1022 
assumes inv_less_0: "inverse a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1023 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1024 
shows "a < (0::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1025 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1026 
have "inverse (inverse a) < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1027 
by (rule negative_imp_inverse_negative) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1028 
thus "a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1029 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1030 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1031 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1032 
lemma inverse_negative_iff_negative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1033 
"(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

1034 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1035 
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

1036 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1037 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1038 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1039 
"(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

1040 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1041 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1042 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1043 
"(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

1044 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1045 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1046 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1047 
subsection{*AntiMonotonicity of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1048 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1049 
lemma less_imp_inverse_less: 
14269  1050 
assumes less: "a < b" 
1051 
and apos: "0 < a" 

1052 
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

1053 
proof (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1054 
assume "~ inverse b < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1055 
hence "inverse a \<le> inverse b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1056 
by (simp add: linorder_not_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1057 
hence "~ (a < b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1058 
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

1059 
thus False 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1060 
by (rule notE [OF _ less]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1061 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1062 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1063 
lemma inverse_less_imp_less: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1064 
"[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

1065 
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

1066 
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

1067 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1068 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1069 
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

1070 
lemma inverse_less_iff_less [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1071 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1072 
==> (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

1073 
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

1074 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1075 
lemma le_imp_inverse_le: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1076 
"[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

1077 
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

1078 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1079 
lemma inverse_le_iff_le [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1080 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1081 
==> (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

1082 
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

1083 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1084 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1085 
text{*These results refer to both operands being negative. The oppositesign 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1086 
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

1087 
lemma inverse_le_imp_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1088 
"[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

1089 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1090 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1091 
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

1092 
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

1093 
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

1094 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1095 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1096 
lemma less_imp_inverse_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1097 
"[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

1098 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1099 
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

1100 
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

1101 
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

1102 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1103 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1104 
lemma inverse_less_imp_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1105 
"[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

1106 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1107 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1108 
prefer 2 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1109 
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

1110 
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

1111 
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

1112 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1113 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1114 
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

1115 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1116 
==> (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

1117 
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

1118 
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

1119 
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

1120 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1121 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1122 
lemma le_imp_inverse_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1123 
"[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

1124 
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

1125 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1126 
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

1127 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1128 
==> (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

1129 
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

1130 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1131 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1132 
subsection{*Division and Signs*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1133 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1134 
lemma zero_less_divide_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1135 
"((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

1136 
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

1137 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1138 
lemma divide_less_0_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1139 
"(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

1140 
(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

1141 
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

1142 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1143 
lemma zero_le_divide_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1144 
"((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

1145 
(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

1146 
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

1147 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1148 
lemma divide_le_0_iff: 
14288  1149 
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) = 
1150 
(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

1151 
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

1152 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1153 
lemma divide_eq_0_iff [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1154 
"(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

1155 
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

1156 

14288  1157 

1158 
subsection{*Simplification of Inequalities Involving Literal Divisors*} 

1159 

1160 
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" 

1161 
proof  

1162 
assume less: "0<c" 

1163 
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" 

1164 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1165 
also have "... = (a*c \<le> b)" 

1166 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1167 
finally show ?thesis . 

1168 
qed 

1169 

1170 

1171 
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" 

1172 
proof  

1173 
assume less: "c<0" 

1174 
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" 

1175 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1176 
also have "... = (b \<le> a*c)" 

1177 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1178 
finally show ?thesis . 

1179 
qed 

1180 

1181 
lemma le_divide_eq: 

1182 
"(a \<le> b/c) = 

1183 
(if 0 < c then a*c \<le> b 

1184 
else if c < 0 then b \<le> a*c 

1185 
else a \<le> (0::'a::{ordered_field,division_by_zero}))" 

1186 
apply (case_tac "c=0", simp) 

1187 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 

1188 
done 

1189 

1190 
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" 

1191 
proof  

1192 
assume less: "0<c" 

1193 
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" 

1194 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1195 
also have "... = (b \<le> a*c)" 

1196 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1197 
finally show ?thesis . 

1198 
qed 

1199 

1200 
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" 

1201 
proof  

1202 
assume less: "c<0" 

1203 
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" 

1204 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1205 
also have "... = (a*c \<le> b)" 

1206 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1207 
finally show ?thesis . 

1208 
qed 

1209 

1210 
lemma divide_le_eq: 

1211 
"(b/c \<le> a) = 

1212 
(if 0 < c then b \<le> a*c 

1213 
else if c < 0 then a*c \<le> b 

1214 
else 0 \<le> (a::'a::{ordered_field,division_by_zero}))" 

1215 
apply (case_tac "c=0", simp) 

1216 
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 

1217 
done 

1218 

1219 

1220 
lemma pos_less_divide_eq: 

1221 
"0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)" 

1222 
proof  

1223 
assume less: "0<c" 

1224 
hence "(a < b/c) = (a*c < (b/c)*c)" 

1225 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1226 
also have "... = (a*c < b)" 

1227 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1228 
finally show ?thesis . 

1229 
qed 

1230 

1231 
lemma neg_less_divide_eq: 

1232 
"c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)" 

1233 
proof  

1234 
assume less: "c<0" 

1235 
hence "(a < b/c) = ((b/c)*c < a*c)" 

1236 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1237 
also have "... = (b < a*c)" 

1238 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1239 
finally show ?thesis . 

1240 
qed 

1241 

1242 
lemma less_divide_eq: 

1243 
"(a < b/c) = 

1244 
(if 0 < c then a*c < b 

1245 
else if c < 0 then b < a*c 

1246 
else a < (0::'a::{ordered_field,division_by_zero}))" 

1247 
apply (case_tac "c=0", simp) 

1248 
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 

1249 
done 

1250 

1251 
lemma pos_divide_less_eq: 

1252 
"0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)" 

1253 
proof  

1254 
assume less: "0<c" 

1255 
hence "(b/c < a) = ((b/c)*c < a*c)" 

1256 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1257 
also have "... = (b < a*c)" 

1258 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1259 
finally show ?thesis . 

1260 
qed 

1261 

1262 
lemma neg_divide_less_eq: 

1263 
"c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)" 

1264 
proof  

1265 
assume less: "c<0" 

1266 
hence "(b/c < a) = (a*c < (b/c)*c)" 

1267 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1268 
also have "... = (a*c < b)" 

1269 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1270 
finally show ?thesis . 

1271 
qed 

1272 

1273 
lemma divide_less_eq: 

1274 
"(b/c < a) = 

1275 
(if 0 < c then b < a*c 

1276 
else if c < 0 then a*c < b 

1277 
else 0 < (a::'a::{ordered_field,division_by_zero}))" 

1278 
apply (case_tac "c=0", simp) 

1279 
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 

1280 
done 

1281 

1282 
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)" 

1283 
proof  

1284 
assume [simp]: "c\<noteq>0" 

1285 
have "(a = b/c) = (a*c = (b/c)*c)" 

1286 
by (simp add: field_mult_cancel_right) 

1287 
also have "... = (a*c = b)" 

1288 
by (simp add: divide_inverse mult_assoc) 

1289 
finally show ?thesis . 

1290 
qed 

1291 

1292 
lemma eq_divide_eq: 

1293 
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" 

1294 
by (simp add: nonzero_eq_divide_eq) 

1295 

1296 
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)" 

1297 
proof  

1298 
assume [simp]: "c\<noteq>0" 

1299 
have "(b/c = a) = ((b/c)*c = a*c)" 

1300 
by (simp add: field_mult_cancel_right) 

1301 
also have "... = (b = a*c)" 

1302 
by (simp add: divide_inverse mult_assoc) 

1303 
finally show ?thesis . 

1304 
qed 

1305 

1306 
lemma divide_eq_eq: 

1307 
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" 

1308 
by (force simp add: nonzero_divide_eq_eq) 

1309 

d149e3cbdb39
Moving some theorems from Real/RealArith0.ML
