author | haftmann |
Fri, 01 Nov 2013 18:51:14 +0100 | |
changeset 54230 | b1d955791529 |
parent 54148 | c8cc5ab4a863 |
child 54250 | 7d2544dd3988 |
permissions | -rw-r--r-- |
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1 |
(* Title: HOL/Groups.thy |
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Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson, Markus Wenzel, Jeremy Avigad |
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*) |
4 |
||
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header {* Groups, also combined with orderings *} |
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theory Groups |
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8 |
imports Orderings |
15131 | 9 |
begin |
14738 | 10 |
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subsection {* Fact collections *} |
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ML {* |
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structure Ac_Simps = Named_Thms |
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( |
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val name = @{binding ac_simps} |
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val description = "associativity and commutativity simplification rules" |
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) |
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*} |
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||
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setup Ac_Simps.setup |
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||
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text{* The rewrites accumulated in @{text algebra_simps} deal with the |
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classical algebraic structures of groups, rings and family. They simplify |
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terms by multiplying everything out (in case of a ring) and bringing sums and |
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products into a canonical form (by ordered rewriting). As a result it decides |
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group and ring equalities but also helps with inequalities. |
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|
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Of course it also works for fields, but it knows nothing about multiplicative |
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inverses or division. This is catered for by @{text field_simps}. *} |
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ML {* |
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structure Algebra_Simps = Named_Thms |
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( |
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val name = @{binding algebra_simps} |
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val description = "algebra simplification rules" |
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) |
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*} |
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39 |
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setup Algebra_Simps.setup |
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41 |
|
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text{* Lemmas @{text field_simps} multiply with denominators in (in)equations |
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if they can be proved to be non-zero (for equations) or positive/negative |
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(for inequations). Can be too aggressive and is therefore separate from the |
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more benign @{text algebra_simps}. *} |
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46 |
|
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ML {* |
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structure Field_Simps = Named_Thms |
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( |
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val name = @{binding field_simps} |
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val description = "algebra simplification rules for fields" |
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) |
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*} |
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54 |
|
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setup Field_Simps.setup |
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subsection {* Abstract structures *} |
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59 |
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text {* |
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These locales provide basic structures for interpretation into |
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bigger structures; extensions require careful thinking, otherwise |
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undesired effects may occur due to interpretation. |
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*} |
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65 |
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locale semigroup = |
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fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70) |
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assumes assoc [ac_simps]: "a * b * c = a * (b * c)" |
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69 |
|
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locale abel_semigroup = semigroup + |
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assumes commute [ac_simps]: "a * b = b * a" |
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begin |
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|
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lemma left_commute [ac_simps]: |
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"b * (a * c) = a * (b * c)" |
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proof - |
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have "(b * a) * c = (a * b) * c" |
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by (simp only: commute) |
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then show ?thesis |
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by (simp only: assoc) |
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qed |
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82 |
|
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end |
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84 |
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locale monoid = semigroup + |
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fixes z :: 'a ("1") |
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assumes left_neutral [simp]: "1 * a = a" |
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assumes right_neutral [simp]: "a * 1 = a" |
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locale comm_monoid = abel_semigroup + |
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fixes z :: 'a ("1") |
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assumes comm_neutral: "a * 1 = a" |
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sublocale comm_monoid < monoid |
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by default (simp_all add: commute comm_neutral) |
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97 |
|
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subsection {* Generic operations *} |
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99 |
|
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class zero = |
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fixes zero :: 'a ("0") |
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102 |
|
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class one = |
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fixes one :: 'a ("1") |
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105 |
|
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hide_const (open) zero one |
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|
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lemma Let_0 [simp]: "Let 0 f = f 0" |
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unfolding Let_def .. |
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lemma Let_1 [simp]: "Let 1 f = f 1" |
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unfolding Let_def .. |
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113 |
|
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setup {* |
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115 |
Reorient_Proc.add |
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(fn Const(@{const_name Groups.zero}, _) => true |
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| Const(@{const_name Groups.one}, _) => true |
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| _ => false) |
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*} |
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120 |
|
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simproc_setup reorient_zero ("0 = x") = Reorient_Proc.proc |
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simproc_setup reorient_one ("1 = x") = Reorient_Proc.proc |
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123 |
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typed_print_translation {* |
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125 |
let |
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126 |
fun tr' c = (c, fn ctxt => fn T => fn ts => |
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if null ts andalso Printer.type_emphasis ctxt T then |
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Syntax.const @{syntax_const "_constrain"} $ Syntax.const c $ |
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129 |
Syntax_Phases.term_of_typ ctxt T |
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130 |
else raise Match); |
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in map tr' [@{const_syntax Groups.one}, @{const_syntax Groups.zero}] end; |
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*} -- {* show types that are presumably too general *} |
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133 |
|
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134 |
class plus = |
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135 |
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65) |
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136 |
|
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137 |
class minus = |
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138 |
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65) |
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139 |
|
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class uminus = |
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fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80) |
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142 |
|
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143 |
class times = |
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fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70) |
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|
145 |
|
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146 |
|
23085 | 147 |
subsection {* Semigroups and Monoids *} |
14738 | 148 |
|
22390 | 149 |
class semigroup_add = plus + |
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150 |
assumes add_assoc [algebra_simps, field_simps]: "(a + b) + c = a + (b + c)" |
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151 |
|
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152 |
sublocale semigroup_add < add!: semigroup plus |
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by default (fact add_assoc) |
22390 | 154 |
|
155 |
class ab_semigroup_add = semigroup_add + |
|
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156 |
assumes add_commute [algebra_simps, field_simps]: "a + b = b + a" |
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157 |
|
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158 |
sublocale ab_semigroup_add < add!: abel_semigroup plus |
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159 |
by default (fact add_commute) |
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160 |
|
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161 |
context ab_semigroup_add |
25062 | 162 |
begin |
14738 | 163 |
|
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164 |
lemmas add_left_commute [algebra_simps, field_simps] = add.left_commute |
25062 | 165 |
|
166 |
theorems add_ac = add_assoc add_commute add_left_commute |
|
167 |
||
168 |
end |
|
14738 | 169 |
|
170 |
theorems add_ac = add_assoc add_commute add_left_commute |
|
171 |
||
22390 | 172 |
class semigroup_mult = times + |
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173 |
assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)" |
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174 |
|
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175 |
sublocale semigroup_mult < mult!: semigroup times |
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176 |
by default (fact mult_assoc) |
14738 | 177 |
|
22390 | 178 |
class ab_semigroup_mult = semigroup_mult + |
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179 |
assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a" |
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180 |
|
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181 |
sublocale ab_semigroup_mult < mult!: abel_semigroup times |
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182 |
by default (fact mult_commute) |
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|
183 |
|
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|
184 |
context ab_semigroup_mult |
23181 | 185 |
begin |
14738 | 186 |
|
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|
187 |
lemmas mult_left_commute [algebra_simps, field_simps] = mult.left_commute |
25062 | 188 |
|
189 |
theorems mult_ac = mult_assoc mult_commute mult_left_commute |
|
23181 | 190 |
|
191 |
end |
|
14738 | 192 |
|
193 |
theorems mult_ac = mult_assoc mult_commute mult_left_commute |
|
194 |
||
23085 | 195 |
class monoid_add = zero + semigroup_add + |
35720 | 196 |
assumes add_0_left: "0 + a = a" |
197 |
and add_0_right: "a + 0 = a" |
|
198 |
||
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199 |
sublocale monoid_add < add!: monoid plus 0 |
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200 |
by default (fact add_0_left add_0_right)+ |
23085 | 201 |
|
26071 | 202 |
lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0" |
29667 | 203 |
by (rule eq_commute) |
26071 | 204 |
|
22390 | 205 |
class comm_monoid_add = zero + ab_semigroup_add + |
25062 | 206 |
assumes add_0: "0 + a = a" |
23085 | 207 |
|
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208 |
sublocale comm_monoid_add < add!: comm_monoid plus 0 |
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209 |
by default (insert add_0, simp add: ac_simps) |
25062 | 210 |
|
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211 |
subclass (in comm_monoid_add) monoid_add |
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212 |
by default (fact add.left_neutral add.right_neutral)+ |
14738 | 213 |
|
49388 | 214 |
class comm_monoid_diff = comm_monoid_add + minus + |
215 |
assumes diff_zero [simp]: "a - 0 = a" |
|
216 |
and zero_diff [simp]: "0 - a = 0" |
|
217 |
and add_diff_cancel_left [simp]: "(c + a) - (c + b) = a - b" |
|
218 |
and diff_diff_add: "a - b - c = a - (b + c)" |
|
219 |
begin |
|
220 |
||
221 |
lemma add_diff_cancel_right [simp]: |
|
222 |
"(a + c) - (b + c) = a - b" |
|
223 |
using add_diff_cancel_left [symmetric] by (simp add: add.commute) |
|
224 |
||
225 |
lemma add_diff_cancel_left' [simp]: |
|
226 |
"(b + a) - b = a" |
|
227 |
proof - |
|
228 |
have "(b + a) - (b + 0) = a" by (simp only: add_diff_cancel_left diff_zero) |
|
229 |
then show ?thesis by simp |
|
230 |
qed |
|
231 |
||
232 |
lemma add_diff_cancel_right' [simp]: |
|
233 |
"(a + b) - b = a" |
|
234 |
using add_diff_cancel_left' [symmetric] by (simp add: add.commute) |
|
235 |
||
236 |
lemma diff_add_zero [simp]: |
|
237 |
"a - (a + b) = 0" |
|
238 |
proof - |
|
239 |
have "a - (a + b) = (a + 0) - (a + b)" by simp |
|
240 |
also have "\<dots> = 0" by (simp only: add_diff_cancel_left zero_diff) |
|
241 |
finally show ?thesis . |
|
242 |
qed |
|
243 |
||
244 |
lemma diff_cancel [simp]: |
|
245 |
"a - a = 0" |
|
246 |
proof - |
|
247 |
have "(a + 0) - (a + 0) = 0" by (simp only: add_diff_cancel_left diff_zero) |
|
248 |
then show ?thesis by simp |
|
249 |
qed |
|
250 |
||
251 |
lemma diff_right_commute: |
|
252 |
"a - c - b = a - b - c" |
|
253 |
by (simp add: diff_diff_add add.commute) |
|
254 |
||
255 |
lemma add_implies_diff: |
|
256 |
assumes "c + b = a" |
|
257 |
shows "c = a - b" |
|
258 |
proof - |
|
259 |
from assms have "(b + c) - (b + 0) = a - b" by (simp add: add.commute) |
|
260 |
then show "c = a - b" by simp |
|
261 |
qed |
|
262 |
||
263 |
end |
|
264 |
||
22390 | 265 |
class monoid_mult = one + semigroup_mult + |
35720 | 266 |
assumes mult_1_left: "1 * a = a" |
267 |
and mult_1_right: "a * 1 = a" |
|
268 |
||
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269 |
sublocale monoid_mult < mult!: monoid times 1 |
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|
270 |
by default (fact mult_1_left mult_1_right)+ |
14738 | 271 |
|
26071 | 272 |
lemma one_reorient: "1 = x \<longleftrightarrow> x = 1" |
29667 | 273 |
by (rule eq_commute) |
26071 | 274 |
|
22390 | 275 |
class comm_monoid_mult = one + ab_semigroup_mult + |
25062 | 276 |
assumes mult_1: "1 * a = a" |
14738 | 277 |
|
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278 |
sublocale comm_monoid_mult < mult!: comm_monoid times 1 |
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|
279 |
by default (insert mult_1, simp add: ac_simps) |
25062 | 280 |
|
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|
281 |
subclass (in comm_monoid_mult) monoid_mult |
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|
282 |
by default (fact mult.left_neutral mult.right_neutral)+ |
14738 | 283 |
|
22390 | 284 |
class cancel_semigroup_add = semigroup_add + |
25062 | 285 |
assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c" |
286 |
assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c" |
|
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|
287 |
begin |
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|
288 |
|
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|
289 |
lemma add_left_cancel [simp]: |
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|
290 |
"a + b = a + c \<longleftrightarrow> b = c" |
29667 | 291 |
by (blast dest: add_left_imp_eq) |
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|
292 |
|
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|
293 |
lemma add_right_cancel [simp]: |
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|
294 |
"b + a = c + a \<longleftrightarrow> b = c" |
29667 | 295 |
by (blast dest: add_right_imp_eq) |
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|
296 |
|
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|
297 |
end |
14738 | 298 |
|
22390 | 299 |
class cancel_ab_semigroup_add = ab_semigroup_add + |
25062 | 300 |
assumes add_imp_eq: "a + b = a + c \<Longrightarrow> b = c" |
25267 | 301 |
begin |
14738 | 302 |
|
25267 | 303 |
subclass cancel_semigroup_add |
28823 | 304 |
proof |
22390 | 305 |
fix a b c :: 'a |
306 |
assume "a + b = a + c" |
|
307 |
then show "b = c" by (rule add_imp_eq) |
|
308 |
next |
|
14738 | 309 |
fix a b c :: 'a |
310 |
assume "b + a = c + a" |
|
22390 | 311 |
then have "a + b = a + c" by (simp only: add_commute) |
312 |
then show "b = c" by (rule add_imp_eq) |
|
14738 | 313 |
qed |
314 |
||
25267 | 315 |
end |
316 |
||
29904 | 317 |
class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add |
318 |
||
319 |
||
23085 | 320 |
subsection {* Groups *} |
321 |
||
25762 | 322 |
class group_add = minus + uminus + monoid_add + |
25062 | 323 |
assumes left_minus [simp]: "- a + a = 0" |
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|
324 |
assumes add_uminus_conv_diff [simp]: "a + (- b) = a - b" |
25062 | 325 |
begin |
23085 | 326 |
|
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|
327 |
lemma diff_conv_add_uminus: |
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|
328 |
"a - b = a + (- b)" |
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|
329 |
by simp |
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|
330 |
|
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|
331 |
lemma minus_unique: |
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|
332 |
assumes "a + b = 0" shows "- a = b" |
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|
333 |
proof - |
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|
334 |
have "- a = - a + (a + b)" using assms by simp |
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|
335 |
also have "\<dots> = b" by (simp add: add_assoc [symmetric]) |
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|
336 |
finally show ?thesis . |
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|
337 |
qed |
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|
338 |
|
25062 | 339 |
lemma minus_zero [simp]: "- 0 = 0" |
14738 | 340 |
proof - |
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|
341 |
have "0 + 0 = 0" by (rule add_0_right) |
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|
342 |
thus "- 0 = 0" by (rule minus_unique) |
14738 | 343 |
qed |
344 |
||
25062 | 345 |
lemma minus_minus [simp]: "- (- a) = a" |
23085 | 346 |
proof - |
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|
347 |
have "- a + a = 0" by (rule left_minus) |
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|
348 |
thus "- (- a) = a" by (rule minus_unique) |
23085 | 349 |
qed |
14738 | 350 |
|
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|
351 |
lemma right_minus: "a + - a = 0" |
14738 | 352 |
proof - |
25062 | 353 |
have "a + - a = - (- a) + - a" by simp |
354 |
also have "\<dots> = 0" by (rule left_minus) |
|
14738 | 355 |
finally show ?thesis . |
356 |
qed |
|
357 |
||
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|
358 |
lemma diff_self [simp]: |
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|
359 |
"a - a = 0" |
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changeset
|
360 |
using right_minus [of a] by simp |
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changeset
|
361 |
|
40368
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changeset
|
362 |
subclass cancel_semigroup_add |
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changeset
|
363 |
proof |
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changeset
|
364 |
fix a b c :: 'a |
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added class relation group_add < cancel_semigroup_add
haftmann
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changeset
|
365 |
assume "a + b = a + c" |
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changeset
|
366 |
then have "- a + a + b = - a + a + c" |
47c186c8577d
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changeset
|
367 |
unfolding add_assoc by simp |
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added class relation group_add < cancel_semigroup_add
haftmann
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changeset
|
368 |
then show "b = c" by simp |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
369 |
next |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
370 |
fix a b c :: 'a |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
371 |
assume "b + a = c + a" |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
372 |
then have "b + a + - a = c + a + - a" by simp |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
373 |
then show "b = c" unfolding add_assoc by simp |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
374 |
qed |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
375 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
376 |
lemma minus_add_cancel [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
377 |
"- a + (a + b) = b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
378 |
by (simp add: add_assoc [symmetric]) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
379 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
380 |
lemma add_minus_cancel [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
381 |
"a + (- a + b) = b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
382 |
by (simp add: add_assoc [symmetric]) |
34147
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
383 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
384 |
lemma diff_add_cancel [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
385 |
"a - b + b = a" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
386 |
by (simp only: diff_conv_add_uminus add_assoc) simp |
34147
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
387 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
388 |
lemma add_diff_cancel [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
389 |
"a + b - b = a" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
390 |
by (simp only: diff_conv_add_uminus add_assoc) simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
391 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
392 |
lemma minus_add: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
393 |
"- (a + b) = - b + - a" |
34147
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
394 |
proof - |
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
395 |
have "(a + b) + (- b + - a) = 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
396 |
by (simp only: add_assoc add_minus_cancel) simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
397 |
then show "- (a + b) = - b + - a" |
34147
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
398 |
by (rule minus_unique) |
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
399 |
qed |
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
400 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
401 |
lemma right_minus_eq [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
402 |
"a - b = 0 \<longleftrightarrow> a = b" |
14738 | 403 |
proof |
23085 | 404 |
assume "a - b = 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
405 |
have "a = (a - b) + b" by (simp add: add_assoc) |
23085 | 406 |
also have "\<dots> = b" using `a - b = 0` by simp |
407 |
finally show "a = b" . |
|
14738 | 408 |
next |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
409 |
assume "a = b" thus "a - b = 0" by simp |
14738 | 410 |
qed |
411 |
||
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
412 |
lemma eq_iff_diff_eq_0: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
413 |
"a = b \<longleftrightarrow> a - b = 0" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
414 |
by (fact right_minus_eq [symmetric]) |
14738 | 415 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
416 |
lemma diff_0 [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
417 |
"0 - a = - a" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
418 |
by (simp only: diff_conv_add_uminus add_0_left) |
14738 | 419 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
420 |
lemma diff_0_right [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
421 |
"a - 0 = a" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
422 |
by (simp only: diff_conv_add_uminus minus_zero add_0_right) |
14738 | 423 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
424 |
lemma diff_minus_eq_add [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
425 |
"a - - b = a + b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
426 |
by (simp only: diff_conv_add_uminus minus_minus) |
14738 | 427 |
|
25062 | 428 |
lemma neg_equal_iff_equal [simp]: |
429 |
"- a = - b \<longleftrightarrow> a = b" |
|
14738 | 430 |
proof |
431 |
assume "- a = - b" |
|
29667 | 432 |
hence "- (- a) = - (- b)" by simp |
25062 | 433 |
thus "a = b" by simp |
14738 | 434 |
next |
25062 | 435 |
assume "a = b" |
436 |
thus "- a = - b" by simp |
|
14738 | 437 |
qed |
438 |
||
25062 | 439 |
lemma neg_equal_0_iff_equal [simp]: |
440 |
"- a = 0 \<longleftrightarrow> a = 0" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
441 |
by (subst neg_equal_iff_equal [symmetric]) simp |
14738 | 442 |
|
25062 | 443 |
lemma neg_0_equal_iff_equal [simp]: |
444 |
"0 = - a \<longleftrightarrow> 0 = a" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
445 |
by (subst neg_equal_iff_equal [symmetric]) simp |
14738 | 446 |
|
447 |
text{*The next two equations can make the simplifier loop!*} |
|
448 |
||
25062 | 449 |
lemma equation_minus_iff: |
450 |
"a = - b \<longleftrightarrow> b = - a" |
|
14738 | 451 |
proof - |
25062 | 452 |
have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal) |
453 |
thus ?thesis by (simp add: eq_commute) |
|
454 |
qed |
|
455 |
||
456 |
lemma minus_equation_iff: |
|
457 |
"- a = b \<longleftrightarrow> - b = a" |
|
458 |
proof - |
|
459 |
have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal) |
|
14738 | 460 |
thus ?thesis by (simp add: eq_commute) |
461 |
qed |
|
462 |
||
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
463 |
lemma eq_neg_iff_add_eq_0: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
464 |
"a = - b \<longleftrightarrow> a + b = 0" |
29914
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
465 |
proof |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
466 |
assume "a = - b" then show "a + b = 0" by simp |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
467 |
next |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
468 |
assume "a + b = 0" |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
469 |
moreover have "a + (b + - b) = (a + b) + - b" |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
470 |
by (simp only: add_assoc) |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
471 |
ultimately show "a = - b" by simp |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
472 |
qed |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
473 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
474 |
lemma add_eq_0_iff2: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
475 |
"a + b = 0 \<longleftrightarrow> a = - b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
476 |
by (fact eq_neg_iff_add_eq_0 [symmetric]) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
477 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
478 |
lemma neg_eq_iff_add_eq_0: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
479 |
"- a = b \<longleftrightarrow> a + b = 0" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
480 |
by (auto simp add: add_eq_0_iff2) |
44348 | 481 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
482 |
lemma add_eq_0_iff: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
483 |
"a + b = 0 \<longleftrightarrow> b = - a" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
484 |
by (auto simp add: neg_eq_iff_add_eq_0 [symmetric]) |
45548
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
485 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
486 |
lemma minus_diff_eq [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
487 |
"- (a - b) = b - a" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
488 |
by (simp only: neg_eq_iff_add_eq_0 diff_conv_add_uminus add_assoc minus_add_cancel) simp |
45548
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
489 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
490 |
lemma add_diff_eq [algebra_simps, field_simps]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
491 |
"a + (b - c) = (a + b) - c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
492 |
by (simp only: diff_conv_add_uminus add_assoc) |
45548
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
493 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
494 |
lemma diff_add_eq_diff_diff_swap: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
495 |
"a - (b + c) = a - c - b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
496 |
by (simp only: diff_conv_add_uminus add_assoc minus_add) |
45548
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
497 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
498 |
lemma diff_eq_eq [algebra_simps, field_simps]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
499 |
"a - b = c \<longleftrightarrow> a = c + b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
500 |
by auto |
45548
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
501 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
502 |
lemma eq_diff_eq [algebra_simps, field_simps]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
503 |
"a = c - b \<longleftrightarrow> a + b = c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
504 |
by auto |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
505 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
506 |
lemma diff_diff_eq2 [algebra_simps, field_simps]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
507 |
"a - (b - c) = (a + c) - b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
508 |
by (simp only: diff_conv_add_uminus add_assoc) simp |
45548
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
509 |
|
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
510 |
lemma diff_eq_diff_eq: |
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
511 |
"a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
512 |
by (simp only: eq_iff_diff_eq_0 [of a b] eq_iff_diff_eq_0 [of c d]) |
45548
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
513 |
|
25062 | 514 |
end |
515 |
||
25762 | 516 |
class ab_group_add = minus + uminus + comm_monoid_add + |
25062 | 517 |
assumes ab_left_minus: "- a + a = 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
518 |
assumes ab_add_uminus_conv_diff: "a - b = a + (- b)" |
25267 | 519 |
begin |
25062 | 520 |
|
25267 | 521 |
subclass group_add |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
522 |
proof qed (simp_all add: ab_left_minus ab_add_uminus_conv_diff) |
25062 | 523 |
|
29904 | 524 |
subclass cancel_comm_monoid_add |
28823 | 525 |
proof |
25062 | 526 |
fix a b c :: 'a |
527 |
assume "a + b = a + c" |
|
528 |
then have "- a + a + b = - a + a + c" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
529 |
by (simp only: add_assoc) |
25062 | 530 |
then show "b = c" by simp |
531 |
qed |
|
532 |
||
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
533 |
lemma uminus_add_conv_diff [simp]: |
25062 | 534 |
"- a + b = b - a" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
535 |
by (simp add: add_commute) |
25062 | 536 |
|
537 |
lemma minus_add_distrib [simp]: |
|
538 |
"- (a + b) = - a + - b" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
539 |
by (simp add: algebra_simps) |
25062 | 540 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
541 |
lemma diff_add_eq [algebra_simps, field_simps]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
542 |
"(a - b) + c = (a + c) - b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
543 |
by (simp add: algebra_simps) |
25077 | 544 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
545 |
lemma diff_diff_eq [algebra_simps, field_simps]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
546 |
"(a - b) - c = a - (b + c)" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
547 |
by (simp add: algebra_simps) |
30629 | 548 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
549 |
lemma diff_add_eq_diff_diff: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
550 |
"a - (b + c) = a - b - c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
551 |
using diff_add_eq_diff_diff_swap [of a c b] by (simp add: add.commute) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
552 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
553 |
lemma add_diff_cancel_left [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
554 |
"(c + a) - (c + b) = a - b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
555 |
by (simp add: algebra_simps) |
48556
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
556 |
|
25062 | 557 |
end |
14738 | 558 |
|
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
559 |
|
14738 | 560 |
subsection {* (Partially) Ordered Groups *} |
561 |
||
35301
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
562 |
text {* |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
563 |
The theory of partially ordered groups is taken from the books: |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
564 |
\begin{itemize} |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
565 |
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
566 |
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
567 |
\end{itemize} |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
568 |
Most of the used notions can also be looked up in |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
569 |
\begin{itemize} |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
570 |
\item \url{http://www.mathworld.com} by Eric Weisstein et. al. |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
571 |
\item \emph{Algebra I} by van der Waerden, Springer. |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
572 |
\end{itemize} |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
573 |
*} |
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
574 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
575 |
class ordered_ab_semigroup_add = order + ab_semigroup_add + |
25062 | 576 |
assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b" |
577 |
begin |
|
24380
c215e256beca
moved ordered_ab_semigroup_add to OrderedGroup.thy
haftmann
parents:
24286
diff
changeset
|
578 |
|
25062 | 579 |
lemma add_right_mono: |
580 |
"a \<le> b \<Longrightarrow> a + c \<le> b + c" |
|
29667 | 581 |
by (simp add: add_commute [of _ c] add_left_mono) |
14738 | 582 |
|
583 |
text {* non-strict, in both arguments *} |
|
584 |
lemma add_mono: |
|
25062 | 585 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d" |
14738 | 586 |
apply (erule add_right_mono [THEN order_trans]) |
587 |
apply (simp add: add_commute add_left_mono) |
|
588 |
done |
|
589 |
||
25062 | 590 |
end |
591 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
592 |
class ordered_cancel_ab_semigroup_add = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
593 |
ordered_ab_semigroup_add + cancel_ab_semigroup_add |
25062 | 594 |
begin |
595 |
||
14738 | 596 |
lemma add_strict_left_mono: |
25062 | 597 |
"a < b \<Longrightarrow> c + a < c + b" |
29667 | 598 |
by (auto simp add: less_le add_left_mono) |
14738 | 599 |
|
600 |
lemma add_strict_right_mono: |
|
25062 | 601 |
"a < b \<Longrightarrow> a + c < b + c" |
29667 | 602 |
by (simp add: add_commute [of _ c] add_strict_left_mono) |
14738 | 603 |
|
604 |
text{*Strict monotonicity in both arguments*} |
|
25062 | 605 |
lemma add_strict_mono: |
606 |
"a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" |
|
607 |
apply (erule add_strict_right_mono [THEN less_trans]) |
|
14738 | 608 |
apply (erule add_strict_left_mono) |
609 |
done |
|
610 |
||
611 |
lemma add_less_le_mono: |
|
25062 | 612 |
"a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d" |
613 |
apply (erule add_strict_right_mono [THEN less_le_trans]) |
|
614 |
apply (erule add_left_mono) |
|
14738 | 615 |
done |
616 |
||
617 |
lemma add_le_less_mono: |
|
25062 | 618 |
"a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" |
619 |
apply (erule add_right_mono [THEN le_less_trans]) |
|
14738 | 620 |
apply (erule add_strict_left_mono) |
621 |
done |
|
622 |
||
25062 | 623 |
end |
624 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
625 |
class ordered_ab_semigroup_add_imp_le = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
626 |
ordered_cancel_ab_semigroup_add + |
25062 | 627 |
assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b" |
628 |
begin |
|
629 |
||
14738 | 630 |
lemma add_less_imp_less_left: |
29667 | 631 |
assumes less: "c + a < c + b" shows "a < b" |
14738 | 632 |
proof - |
633 |
from less have le: "c + a <= c + b" by (simp add: order_le_less) |
|
634 |
have "a <= b" |
|
635 |
apply (insert le) |
|
636 |
apply (drule add_le_imp_le_left) |
|
637 |
by (insert le, drule add_le_imp_le_left, assumption) |
|
638 |
moreover have "a \<noteq> b" |
|
639 |
proof (rule ccontr) |
|
640 |
assume "~(a \<noteq> b)" |
|
641 |
then have "a = b" by simp |
|
642 |
then have "c + a = c + b" by simp |
|
643 |
with less show "False"by simp |
|
644 |
qed |
|
645 |
ultimately show "a < b" by (simp add: order_le_less) |
|
646 |
qed |
|
647 |
||
648 |
lemma add_less_imp_less_right: |
|
25062 | 649 |
"a + c < b + c \<Longrightarrow> a < b" |
14738 | 650 |
apply (rule add_less_imp_less_left [of c]) |
651 |
apply (simp add: add_commute) |
|
652 |
done |
|
653 |
||
654 |
lemma add_less_cancel_left [simp]: |
|
25062 | 655 |
"c + a < c + b \<longleftrightarrow> a < b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
656 |
by (blast intro: add_less_imp_less_left add_strict_left_mono) |
14738 | 657 |
|
658 |
lemma add_less_cancel_right [simp]: |
|
25062 | 659 |
"a + c < b + c \<longleftrightarrow> a < b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
660 |
by (blast intro: add_less_imp_less_right add_strict_right_mono) |
14738 | 661 |
|
662 |
lemma add_le_cancel_left [simp]: |
|
25062 | 663 |
"c + a \<le> c + b \<longleftrightarrow> a \<le> b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
664 |
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) |
14738 | 665 |
|
666 |
lemma add_le_cancel_right [simp]: |
|
25062 | 667 |
"a + c \<le> b + c \<longleftrightarrow> a \<le> b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
668 |
by (simp add: add_commute [of a c] add_commute [of b c]) |
14738 | 669 |
|
670 |
lemma add_le_imp_le_right: |
|
25062 | 671 |
"a + c \<le> b + c \<Longrightarrow> a \<le> b" |
29667 | 672 |
by simp |
25062 | 673 |
|
25077 | 674 |
lemma max_add_distrib_left: |
675 |
"max x y + z = max (x + z) (y + z)" |
|
676 |
unfolding max_def by auto |
|
677 |
||
678 |
lemma min_add_distrib_left: |
|
679 |
"min x y + z = min (x + z) (y + z)" |
|
680 |
unfolding min_def by auto |
|
681 |
||
44848
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
682 |
lemma max_add_distrib_right: |
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
683 |
"x + max y z = max (x + y) (x + z)" |
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
684 |
unfolding max_def by auto |
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
685 |
|
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
686 |
lemma min_add_distrib_right: |
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
687 |
"x + min y z = min (x + y) (x + z)" |
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
688 |
unfolding min_def by auto |
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
689 |
|
25062 | 690 |
end |
691 |
||
52289 | 692 |
class ordered_cancel_comm_monoid_diff = comm_monoid_diff + ordered_ab_semigroup_add_imp_le + |
693 |
assumes le_iff_add: "a \<le> b \<longleftrightarrow> (\<exists>c. b = a + c)" |
|
694 |
begin |
|
695 |
||
696 |
context |
|
697 |
fixes a b |
|
698 |
assumes "a \<le> b" |
|
699 |
begin |
|
700 |
||
701 |
lemma add_diff_inverse: |
|
702 |
"a + (b - a) = b" |
|
703 |
using `a \<le> b` by (auto simp add: le_iff_add) |
|
704 |
||
705 |
lemma add_diff_assoc: |
|
706 |
"c + (b - a) = c + b - a" |
|
707 |
using `a \<le> b` by (auto simp add: le_iff_add add_left_commute [of c]) |
|
708 |
||
709 |
lemma add_diff_assoc2: |
|
710 |
"b - a + c = b + c - a" |
|
711 |
using `a \<le> b` by (auto simp add: le_iff_add add_assoc) |
|
712 |
||
713 |
lemma diff_add_assoc: |
|
714 |
"c + b - a = c + (b - a)" |
|
715 |
using `a \<le> b` by (simp add: add_commute add_diff_assoc) |
|
716 |
||
717 |
lemma diff_add_assoc2: |
|
718 |
"b + c - a = b - a + c" |
|
719 |
using `a \<le> b`by (simp add: add_commute add_diff_assoc) |
|
720 |
||
721 |
lemma diff_diff_right: |
|
722 |
"c - (b - a) = c + a - b" |
|
723 |
by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add_commute) |
|
724 |
||
725 |
lemma diff_add: |
|
726 |
"b - a + a = b" |
|
727 |
by (simp add: add_commute add_diff_inverse) |
|
728 |
||
729 |
lemma le_add_diff: |
|
730 |
"c \<le> b + c - a" |
|
731 |
by (auto simp add: add_commute diff_add_assoc2 le_iff_add) |
|
732 |
||
733 |
lemma le_imp_diff_is_add: |
|
734 |
"a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a" |
|
735 |
by (auto simp add: add_commute add_diff_inverse) |
|
736 |
||
737 |
lemma le_diff_conv2: |
|
738 |
"c \<le> b - a \<longleftrightarrow> c + a \<le> b" (is "?P \<longleftrightarrow> ?Q") |
|
739 |
proof |
|
740 |
assume ?P |
|
741 |
then have "c + a \<le> b - a + a" by (rule add_right_mono) |
|
742 |
then show ?Q by (simp add: add_diff_inverse add_commute) |
|
743 |
next |
|
744 |
assume ?Q |
|
745 |
then have "a + c \<le> a + (b - a)" by (simp add: add_diff_inverse add_commute) |
|
746 |
then show ?P by simp |
|
747 |
qed |
|
748 |
||
749 |
end |
|
750 |
||
751 |
end |
|
752 |
||
753 |
||
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
754 |
subsection {* Support for reasoning about signs *} |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
755 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
756 |
class ordered_comm_monoid_add = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
757 |
ordered_cancel_ab_semigroup_add + comm_monoid_add |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
758 |
begin |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
759 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
760 |
lemma add_pos_nonneg: |
29667 | 761 |
assumes "0 < a" and "0 \<le> b" shows "0 < a + b" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
762 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
763 |
have "0 + 0 < a + b" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
764 |
using assms by (rule add_less_le_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
765 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
766 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
767 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
768 |
lemma add_pos_pos: |
29667 | 769 |
assumes "0 < a" and "0 < b" shows "0 < a + b" |
770 |
by (rule add_pos_nonneg) (insert assms, auto) |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
771 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
772 |
lemma add_nonneg_pos: |
29667 | 773 |
assumes "0 \<le> a" and "0 < b" shows "0 < a + b" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
774 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
775 |
have "0 + 0 < a + b" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
776 |
using assms by (rule add_le_less_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
777 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
778 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
779 |
|
36977
71c8973a604b
declare add_nonneg_nonneg [simp]; remove now-redundant lemmas realpow_two_le_order(2)
huffman
parents:
36348
diff
changeset
|
780 |
lemma add_nonneg_nonneg [simp]: |
29667 | 781 |
assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
782 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
783 |
have "0 + 0 \<le> a + b" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
784 |
using assms by (rule add_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
785 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
786 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
787 |
|
30691 | 788 |
lemma add_neg_nonpos: |
29667 | 789 |
assumes "a < 0" and "b \<le> 0" shows "a + b < 0" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
790 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
791 |
have "a + b < 0 + 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
792 |
using assms by (rule add_less_le_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
793 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
794 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
795 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
796 |
lemma add_neg_neg: |
29667 | 797 |
assumes "a < 0" and "b < 0" shows "a + b < 0" |
798 |
by (rule add_neg_nonpos) (insert assms, auto) |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
799 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
800 |
lemma add_nonpos_neg: |
29667 | 801 |
assumes "a \<le> 0" and "b < 0" shows "a + b < 0" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
802 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
803 |
have "a + b < 0 + 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
804 |
using assms by (rule add_le_less_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
805 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
806 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
807 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
808 |
lemma add_nonpos_nonpos: |
29667 | 809 |
assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
810 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
811 |
have "a + b \<le> 0 + 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
812 |
using assms by (rule add_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
813 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
814 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
815 |
|
30691 | 816 |
lemmas add_sign_intros = |
817 |
add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg |
|
818 |
add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos |
|
819 |
||
29886 | 820 |
lemma add_nonneg_eq_0_iff: |
821 |
assumes x: "0 \<le> x" and y: "0 \<le> y" |
|
822 |
shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0" |
|
823 |
proof (intro iffI conjI) |
|
824 |
have "x = x + 0" by simp |
|
825 |
also have "x + 0 \<le> x + y" using y by (rule add_left_mono) |
|
826 |
also assume "x + y = 0" |
|
827 |
also have "0 \<le> x" using x . |
|
828 |
finally show "x = 0" . |
|
829 |
next |
|
830 |
have "y = 0 + y" by simp |
|
831 |
also have "0 + y \<le> x + y" using x by (rule add_right_mono) |
|
832 |
also assume "x + y = 0" |
|
833 |
also have "0 \<le> y" using y . |
|
834 |
finally show "y = 0" . |
|
835 |
next |
|
836 |
assume "x = 0 \<and> y = 0" |
|
837 |
then show "x + y = 0" by simp |
|
838 |
qed |
|
839 |
||
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
840 |
lemma add_increasing: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
841 |
"0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
842 |
by (insert add_mono [of 0 a b c], simp) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
843 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
844 |
lemma add_increasing2: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
845 |
"0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
846 |
by (simp add: add_increasing add_commute [of a]) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
847 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
848 |
lemma add_strict_increasing: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
849 |
"0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
850 |
by (insert add_less_le_mono [of 0 a b c], simp) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
851 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
852 |
lemma add_strict_increasing2: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
853 |
"0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
854 |
by (insert add_le_less_mono [of 0 a b c], simp) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
855 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
856 |
end |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
857 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
858 |
class ordered_ab_group_add = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
859 |
ab_group_add + ordered_ab_semigroup_add |
25062 | 860 |
begin |
861 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
862 |
subclass ordered_cancel_ab_semigroup_add .. |
25062 | 863 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
864 |
subclass ordered_ab_semigroup_add_imp_le |
28823 | 865 |
proof |
25062 | 866 |
fix a b c :: 'a |
867 |
assume "c + a \<le> c + b" |
|
868 |
hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono) |
|
869 |
hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc) |
|
870 |
thus "a \<le> b" by simp |
|
871 |
qed |
|
872 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
873 |
subclass ordered_comm_monoid_add .. |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
874 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
875 |
lemma add_less_same_cancel1 [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
876 |
"b + a < b \<longleftrightarrow> a < 0" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
877 |
using add_less_cancel_left [of _ _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
878 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
879 |
lemma add_less_same_cancel2 [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
880 |
"a + b < b \<longleftrightarrow> a < 0" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
881 |
using add_less_cancel_right [of _ _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
882 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
883 |
lemma less_add_same_cancel1 [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
884 |
"a < a + b \<longleftrightarrow> 0 < b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
885 |
using add_less_cancel_left [of _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
886 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
887 |
lemma less_add_same_cancel2 [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
888 |
"a < b + a \<longleftrightarrow> 0 < b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
889 |
using add_less_cancel_right [of 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
890 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
891 |
lemma add_le_same_cancel1 [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
892 |
"b + a \<le> b \<longleftrightarrow> a \<le> 0" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
893 |
using add_le_cancel_left [of _ _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
894 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
895 |
lemma add_le_same_cancel2 [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
896 |
"a + b \<le> b \<longleftrightarrow> a \<le> 0" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
897 |
using add_le_cancel_right [of _ _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
898 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
899 |
lemma le_add_same_cancel1 [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
900 |
"a \<le> a + b \<longleftrightarrow> 0 \<le> b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
901 |
using add_le_cancel_left [of _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
902 |
|
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
903 |
lemma le_add_same_cancel2 [simp]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
904 |
"a \<le> b + a \<longleftrightarrow> 0 \<le> b" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
905 |
using add_le_cancel_right [of 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
906 |
|
25077 | 907 |
lemma max_diff_distrib_left: |
908 |
shows "max x y - z = max (x - z) (y - z)" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
909 |
using max_add_distrib_left [of x y "- z"] by simp |
25077 | 910 |
|
911 |
lemma min_diff_distrib_left: |
|
912 |
shows "min x y - z = min (x - z) (y - z)" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
913 |
using min_add_distrib_left [of x y "- z"] by simp |
25077 | 914 |
|
915 |
lemma le_imp_neg_le: |
|
29667 | 916 |
assumes "a \<le> b" shows "-b \<le> -a" |
25077 | 917 |
proof - |
29667 | 918 |
have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
919 |
then have "0 \<le> -a+b" by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
920 |
then have "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
921 |
then show ?thesis by (simp add: algebra_simps) |
25077 | 922 |
qed |
923 |
||
924 |
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b" |
|
925 |
proof |
|
926 |
assume "- b \<le> - a" |
|
29667 | 927 |
hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le) |
25077 | 928 |
thus "a\<le>b" by simp |
929 |
next |
|
930 |
assume "a\<le>b" |
|
931 |
thus "-b \<le> -a" by (rule le_imp_neg_le) |
|
932 |
qed |
|
933 |
||
934 |
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a" |
|
29667 | 935 |
by (subst neg_le_iff_le [symmetric], simp) |
25077 | 936 |
|
937 |
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0" |
|
29667 | 938 |
by (subst neg_le_iff_le [symmetric], simp) |
25077 | 939 |
|
940 |
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b" |
|
29667 | 941 |
by (force simp add: less_le) |
25077 | 942 |
|
943 |
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a" |
|
29667 | 944 |
by (subst neg_less_iff_less [symmetric], simp) |
25077 | 945 |
|
946 |
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0" |
|
29667 | 947 |
by (subst neg_less_iff_less [symmetric], simp) |
25077 | 948 |
|
949 |
text{*The next several equations can make the simplifier loop!*} |
|
950 |
||
951 |
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a" |
|
952 |
proof - |
|
953 |
have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less) |
|
954 |
thus ?thesis by simp |
|
955 |
qed |
|
956 |
||
957 |
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a" |
|
958 |
proof - |
|
959 |
have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less) |
|
960 |
thus ?thesis by simp |
|
961 |
qed |
|
962 |
||
963 |
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a" |
|
964 |
proof - |
|
965 |
have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff) |
|
966 |
have "(- (- a) <= -b) = (b <= - a)" |
|
967 |
apply (auto simp only: le_less) |
|
968 |
apply (drule mm) |
|
969 |
apply (simp_all) |
|
970 |
apply (drule mm[simplified], assumption) |
|
971 |
done |
|
972 |
then show ?thesis by simp |
|
973 |
qed |
|
974 |
||
975 |
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a" |
|
29667 | 976 |
by (auto simp add: le_less minus_less_iff) |
25077 | 977 |
|
54148 | 978 |
lemma diff_less_0_iff_less [simp]: |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
979 |
"a - b < 0 \<longleftrightarrow> a < b" |
25077 | 980 |
proof - |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
981 |
have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by simp |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
982 |
also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right) |
25077 | 983 |
finally show ?thesis . |
984 |
qed |
|
985 |
||
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
986 |
lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric] |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
987 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
988 |
lemma diff_less_eq [algebra_simps, field_simps]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
989 |
"a - b < c \<longleftrightarrow> a < c + b" |
25077 | 990 |
apply (subst less_iff_diff_less_0 [of a]) |
991 |
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
992 |
apply (simp add: algebra_simps) |
25077 | 993 |
done |
994 |
||
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
995 |
lemma less_diff_eq[algebra_simps, field_simps]: |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
996 |
"a < c - b \<longleftrightarrow> a + b < c" |
36302 | 997 |
apply (subst less_iff_diff_less_0 [of "a + b"]) |
25077 | 998 |
apply (subst less_iff_diff_less_0 [of a]) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
999 |
apply (simp add: algebra_simps) |
25077 | 1000 |
done |
1001 |
||
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset
|
1002 |
lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
1003 |
by (auto simp add: le_less diff_less_eq ) |
25077 | 1004 |
|
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset
|
1005 |
lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
1006 |
by (auto simp add: le_less less_diff_eq) |
25077 | 1007 |
|
54148 | 1008 |
lemma diff_le_0_iff_le [simp]: |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1009 |
"a - b \<le> 0 \<longleftrightarrow> a \<le> b" |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1010 |
by (simp add: algebra_simps) |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1011 |
|
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1012 |
lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric] |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1013 |
|
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1014 |
lemma diff_eq_diff_less: |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1015 |
"a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d" |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1016 |
by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d]) |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1017 |
|
37889
0d8058e0c270
keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents:
37884
diff
changeset
|
1018 |
lemma diff_eq_diff_less_eq: |
0d8058e0c270
keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents:
37884
diff
changeset
|
1019 |
"a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d" |
0d8058e0c270
keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents:
37884
diff
changeset
|
1020 |
by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d]) |
25077 | 1021 |
|
1022 |
end |
|
1023 |
||
48891 | 1024 |
ML_file "Tools/group_cancel.ML" |
48556
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1025 |
|
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1026 |
simproc_setup group_cancel_add ("a + b::'a::ab_group_add") = |
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1027 |
{* fn phi => fn ss => try Group_Cancel.cancel_add_conv *} |
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1028 |
|
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1029 |
simproc_setup group_cancel_diff ("a - b::'a::ab_group_add") = |
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1030 |
{* fn phi => fn ss => try Group_Cancel.cancel_diff_conv *} |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1031 |
|
48556
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1032 |
simproc_setup group_cancel_eq ("a = (b::'a::ab_group_add)") = |
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1033 |
{* fn phi => fn ss => try Group_Cancel.cancel_eq_conv *} |
37889
0d8058e0c270
keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents:
37884
diff
changeset
|
1034 |
|
48556
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1035 |
simproc_setup group_cancel_le ("a \<le> (b::'a::ordered_ab_group_add)") = |
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1036 |
{* fn phi => fn ss => try Group_Cancel.cancel_le_conv *} |
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1037 |
|
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1038 |
simproc_setup group_cancel_less ("a < (b::'a::ordered_ab_group_add)") = |
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
1039 |
{* fn phi => fn ss => try Group_Cancel.cancel_less_conv *} |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
1040 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1041 |
class linordered_ab_semigroup_add = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1042 |
linorder + ordered_ab_semigroup_add |
25062 | 1043 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1044 |
class linordered_cancel_ab_semigroup_add = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1045 |
linorder + ordered_cancel_ab_semigroup_add |
25267 | 1046 |
begin |
25062 | 1047 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1048 |
subclass linordered_ab_semigroup_add .. |
25062 | 1049 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1050 |
subclass ordered_ab_semigroup_add_imp_le |
28823 | 1051 |
proof |
25062 | 1052 |
fix a b c :: 'a |
1053 |
assume le: "c + a <= c + b" |
|
1054 |
show "a <= b" |
|
1055 |
proof (rule ccontr) |
|
1056 |
assume w: "~ a \<le> b" |
|
1057 |
hence "b <= a" by (simp add: linorder_not_le) |
|
1058 |
hence le2: "c + b <= c + a" by (rule add_left_mono) |
|
1059 |
have "a = b" |
|
1060 |
apply (insert le) |
|
1061 |
apply (insert le2) |
|
1062 |
apply (drule antisym, simp_all) |
|
1063 |
done |
|
1064 |
with w show False |
|
1065 |
by (simp add: linorder_not_le [symmetric]) |
|
1066 |
qed |
|
1067 |
qed |
|
1068 |
||
25267 | 1069 |
end |
1070 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1071 |
class linordered_ab_group_add = linorder + ordered_ab_group_add |
25267 | 1072 |
begin |
25230 | 1073 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1074 |
subclass linordered_cancel_ab_semigroup_add .. |
25230 | 1075 |
|
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1076 |
lemma neg_less_eq_nonneg [simp]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1077 |
"- a \<le> a \<longleftrightarrow> 0 \<le> a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1078 |
proof |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1079 |
assume A: "- a \<le> a" show "0 \<le> a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1080 |
proof (rule classical) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1081 |
assume "\<not> 0 \<le> a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1082 |
then have "a < 0" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1083 |
with A have "- a < 0" by (rule le_less_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1084 |
then show ?thesis by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1085 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1086 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1087 |
assume A: "0 \<le> a" show "- a \<le> a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1088 |
proof (rule order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1089 |
show "- a \<le> 0" using A by (simp add: minus_le_iff) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1090 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1091 |
show "0 \<le> a" using A . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1092 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1093 |
qed |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1094 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1095 |
lemma neg_less_nonneg [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1096 |
"- a < a \<longleftrightarrow> 0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1097 |
proof |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1098 |
assume A: "- a < a" show "0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1099 |
proof (rule classical) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1100 |
assume "\<not> 0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1101 |
then have "a \<le> 0" by auto |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1102 |
with A have "- a < 0" by (rule less_le_trans) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1103 |
then show ?thesis by auto |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1104 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1105 |
next |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1106 |
assume A: "0 < a" show "- a < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1107 |
proof (rule less_trans) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1108 |
show "- a < 0" using A by (simp add: minus_le_iff) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1109 |
next |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1110 |
show "0 < a" using A . |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1111 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1112 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1113 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1114 |
lemma less_eq_neg_nonpos [simp]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1115 |
"a \<le> - a \<longleftrightarrow> a \<le> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1116 |
proof |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1117 |
assume A: "a \<le> - a" show "a \<le> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1118 |
proof (rule classical) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1119 |
assume "\<not> a \<le> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1120 |
then have "0 < a" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1121 |
then have "0 < - a" using A by (rule less_le_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1122 |
then show ?thesis by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1123 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1124 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1125 |
assume A: "a \<le> 0" show "a \<le> - a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1126 |
proof (rule order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1127 |
show "0 \<le> - a" using A by (simp add: minus_le_iff) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1128 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1129 |
show "a \<le> 0" using A . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1130 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1131 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1132 |
|
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1133 |
lemma equal_neg_zero [simp]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1134 |
"a = - a \<longleftrightarrow> a = 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1135 |
proof |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1136 |
assume "a = 0" then show "a = - a" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1137 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1138 |
assume A: "a = - a" show "a = 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1139 |
proof (cases "0 \<le> a") |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1140 |
case True with A have "0 \<le> - a" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1141 |
with le_minus_iff have "a \<le> 0" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1142 |
with True show ?thesis by (auto intro: order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1143 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1144 |
case False then have B: "a \<le> 0" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1145 |
with A have "- a \<le> 0" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1146 |
with B show ?thesis by (auto intro: order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1147 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1148 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1149 |
|
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1150 |
lemma neg_equal_zero [simp]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1151 |
"- a = a \<longleftrightarrow> a = 0" |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1152 |
by (auto dest: sym) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1153 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1154 |
lemma double_zero [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1155 |
"a + a = 0 \<longleftrightarrow> a = 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1156 |
proof |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1157 |
assume assm: "a + a = 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1158 |
then have a: "- a = a" by (rule minus_unique) |
35216 | 1159 |
then show "a = 0" by (simp only: neg_equal_zero) |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1160 |
qed simp |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1161 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1162 |
lemma double_zero_sym [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1163 |
"0 = a + a \<longleftrightarrow> a = 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1164 |
by (rule, drule sym) simp_all |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1165 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1166 |
lemma zero_less_double_add_iff_zero_less_single_add [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1167 |
"0 < a + a \<longleftrightarrow> 0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1168 |
proof |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1169 |
assume "0 < a + a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1170 |
then have "0 - a < a" by (simp only: diff_less_eq) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1171 |
then have "- a < a" by simp |
35216 | 1172 |
then show "0 < a" by (simp only: neg_less_nonneg) |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1173 |
next |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1174 |
assume "0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1175 |
with this have "0 + 0 < a + a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1176 |
by (rule add_strict_mono) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1177 |
then show "0 < a + a" by simp |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1178 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1179 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1180 |
lemma zero_le_double_add_iff_zero_le_single_add [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1181 |
"0 \<le> a + a \<longleftrightarrow> 0 \<le> a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1182 |
by (auto simp add: le_less) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1183 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1184 |
lemma double_add_less_zero_iff_single_add_less_zero [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1185 |
"a + a < 0 \<longleftrightarrow> a < 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1186 |
proof - |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1187 |
have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1188 |
by (simp add: not_less) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1189 |
then show ?thesis by simp |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1190 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1191 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1192 |
lemma double_add_le_zero_iff_single_add_le_zero [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1193 |
"a + a \<le> 0 \<longleftrightarrow> a \<le> 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1194 |
proof - |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1195 |
have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1196 |
by (simp add: not_le) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1197 |
then show ?thesis by simp |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1198 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1199 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1200 |
lemma le_minus_self_iff: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1201 |
"a \<le> - a \<longleftrightarrow> a \<le> 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
1202 |
by (fact less_eq_neg_nonpos) |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1203 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1204 |
lemma minus_le_self_iff: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1205 |
"- a \<le> a \<longleftrightarrow> 0 \<le> a" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
1206 |
by (fact neg_less_eq_nonneg) |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1207 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1208 |
lemma minus_max_eq_min: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1209 |
"- max x y = min (-x) (-y)" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1210 |
by (auto simp add: max_def min_def) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1211 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1212 |
lemma minus_min_eq_max: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1213 |
"- min x y = max (-x) (-y)" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1214 |
by (auto simp add: max_def min_def) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1215 |
|
25267 | 1216 |
end |
1217 |
||
35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1218 |
class abs = |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1219 |
fixes abs :: "'a \<Rightarrow> 'a" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1220 |
begin |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1221 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1222 |
notation (xsymbols) |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1223 |
abs ("\<bar>_\<bar>") |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1224 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1225 |
notation (HTML output) |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1226 |
abs ("\<bar>_\<bar>") |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1227 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1228 |
end |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1229 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1230 |
class sgn = |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1231 |
fixes sgn :: "'a \<Rightarrow> 'a" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1232 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1233 |
class abs_if = minus + uminus + ord + zero + abs + |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1234 |
assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1235 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1236 |
class sgn_if = minus + uminus + zero + one + ord + sgn + |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1237 |
assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1238 |
begin |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1239 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1240 |
lemma sgn0 [simp]: "sgn 0 = 0" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1241 |
by (simp add:sgn_if) |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1242 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1243 |
end |
14738 | 1244 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1245 |
class ordered_ab_group_add_abs = ordered_ab_group_add + abs + |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1246 |
assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1247 |
and abs_ge_self: "a \<le> \<bar>a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1248 |
and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1249 |
and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1250 |
and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1251 |
begin |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1252 |
|
25307 | 1253 |
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0" |
1254 |
unfolding neg_le_0_iff_le by simp |
|
1255 |
||
1256 |
lemma abs_of_nonneg [simp]: |
|
29667 | 1257 |
assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a" |
25307 | 1258 |
proof (rule antisym) |
1259 |
from nonneg le_imp_neg_le have "- a \<le> 0" by simp |
|
1260 |
from this nonneg have "- a \<le> a" by (rule order_trans) |
|
1261 |
then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI) |
|
1262 |
qed (rule abs_ge_self) |
|
1263 |
||
1264 |
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>" |
|
29667 | 1265 |
by (rule antisym) |
36302 | 1266 |
(auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"]) |
25307 | 1267 |
|
1268 |
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0" |
|
1269 |
proof - |
|
1270 |
have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0" |
|
1271 |
proof (rule antisym) |
|
1272 |
assume zero: "\<bar>a\<bar> = 0" |
|
1273 |
with abs_ge_self show "a \<le> 0" by auto |
|
1274 |
from zero have "\<bar>-a\<bar> = 0" by simp |
|
36302 | 1275 |
with abs_ge_self [of "- a"] have "- a \<le> 0" by auto |
25307 | 1276 |
with neg_le_0_iff_le show "0 \<le> a" by auto |
1277 |
qed |
|
1278 |
then show ?thesis by auto |
|
1279 |
qed |
|
1280 |
||
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1281 |
lemma abs_zero [simp]: "\<bar>0\<bar> = 0" |
29667 | 1282 |
by simp |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1283 |
|
54148 | 1284 |
lemma abs_0_eq [simp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1285 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1286 |
have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1287 |
thus ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1288 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1289 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1290 |
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1291 |
proof |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1292 |
assume "\<bar>a\<bar> \<le> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1293 |
then have "\<bar>a\<bar> = 0" by (rule antisym) simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1294 |
thus "a = 0" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1295 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1296 |
assume "a = 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1297 |
thus "\<bar>a\<bar> \<le> 0" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1298 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1299 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1300 |
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0" |
29667 | 1301 |
by (simp add: less_le) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1302 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1303 |
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1304 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1305 |
have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1306 |
show ?thesis by (simp add: a) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1307 |
qed |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1308 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1309 |
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1310 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1311 |
have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1312 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1313 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1314 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1315 |
lemma abs_minus_commute: |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1316 |
"\<bar>a - b\<bar> = \<bar>b - a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1317 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1318 |
have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1319 |
also have "... = \<bar>b - a\<bar>" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1320 |
finally show ?thesis . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1321 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1322 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1323 |
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a" |
29667 | 1324 |
by (rule abs_of_nonneg, rule less_imp_le) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1325 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1326 |
lemma abs_of_nonpos [simp]: |
29667 | 1327 |
assumes "a \<le> 0" shows "\<bar>a\<bar> = - a" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1328 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1329 |
let ?b = "- a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1330 |
have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1331 |
unfolding abs_minus_cancel [of "?b"] |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1332 |
unfolding neg_le_0_iff_le [of "?b"] |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1333 |
unfolding minus_minus by (erule abs_of_nonneg) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1334 |
then show ?thesis using assms by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1335 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1336 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1337 |
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a" |
29667 | 1338 |
by (rule abs_of_nonpos, rule less_imp_le) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1339 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1340 |
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b" |
29667 | 1341 |
by (insert abs_ge_self, blast intro: order_trans) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1342 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1343 |
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b" |
36302 | 1344 |
by (insert abs_le_D1 [of "- a"], simp) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1345 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1346 |
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b" |
29667 | 1347 |
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1348 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1349 |
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>" |
36302 | 1350 |
proof - |
1351 |
have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
1352 |
by (simp add: algebra_simps) |
36302 | 1353 |
then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>" |
1354 |
by (simp add: abs_triangle_ineq) |
|
1355 |
then show ?thesis |
|
1356 |
by (simp add: algebra_simps) |
|
1357 |
qed |
|
1358 |
||
1359 |
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>" |
|
1360 |
by (simp only: abs_minus_commute [of b] abs_triangle_ineq2) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1361 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1362 |
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>" |
36302 | 1363 |
by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1364 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1365 |
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1366 |
proof - |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
1367 |
have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (simp add: algebra_simps) |
36302 | 1368 |
also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq) |
29667 | 1369 |
finally show ?thesis by simp |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1370 |
qed |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1371 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1372 |
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1373 |
proof - |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
1374 |
have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: algebra_simps) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1375 |
also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1376 |
finally show ?thesis . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1377 |
qed |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1378 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1379 |
lemma abs_add_abs [simp]: |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1380 |
"\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R") |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1381 |
proof (rule antisym) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1382 |
show "?L \<ge> ?R" by(rule abs_ge_self) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1383 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1384 |
have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1385 |
also have "\<dots> = ?R" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1386 |
finally show "?L \<le> ?R" . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1387 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1388 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1389 |
end |
14738 | 1390 |
|
15178 | 1391 |
|
25090 | 1392 |
subsection {* Tools setup *} |
1393 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
1394 |
lemma add_mono_thms_linordered_semiring: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1395 |
fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add" |
25077 | 1396 |
shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
1397 |
and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
|
1398 |
and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l" |
|
1399 |
and "i = j \<and> k = l \<Longrightarrow> i + k = j + l" |
|
1400 |
by (rule add_mono, clarify+)+ |
|
1401 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
1402 |
lemma add_mono_thms_linordered_field: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1403 |
fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add" |
25077 | 1404 |
shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l" |
1405 |
and "i = j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
1406 |
and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l" |
|
1407 |
and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
1408 |
and "i < j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
1409 |
by (auto intro: add_strict_right_mono add_strict_left_mono |
|
1410 |
add_less_le_mono add_le_less_mono add_strict_mono) |
|
1411 |
||
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
52289
diff
changeset
|
1412 |
code_identifier |
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
52289
diff
changeset
|
1413 |
code_module Groups \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith |
33364 | 1414 |
|
14738 | 1415 |
end |
49388 | 1416 |