author | wenzelm |
Mon, 20 Jun 2016 21:40:48 +0200 | |
changeset 63325 | 1086d56cde86 |
parent 63290 | 9ac558ab0906 |
child 63364 | 4fa441c2f20c |
permissions | -rw-r--r-- |
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(* Title: HOL/Groups.thy |
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Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson, Markus Wenzel, Jeremy Avigad |
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*) |
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section \<open>Groups, also combined with orderings\<close> |
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theory Groups |
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imports Orderings |
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begin |
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subsection \<open>Dynamic facts\<close> |
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named_theorems ac_simps "associativity and commutativity simplification rules" |
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text \<open> |
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The rewrites accumulated in \<open>algebra_simps\<close> 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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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 \<open>field_simps\<close>. |
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\<close> |
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named_theorems algebra_simps "algebra simplification rules" |
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text \<open> |
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Lemmas \<open>field_simps\<close> 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 \<open>algebra_simps\<close>. |
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\<close> |
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named_theorems field_simps "algebra simplification rules for fields" |
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subsection \<open>Abstract structures\<close> |
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text \<open> |
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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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\<close> |
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locale semigroup = |
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fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^bold>*" 70) |
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assumes assoc [ac_simps]: "a \<^bold>* b \<^bold>* c = a \<^bold>* (b \<^bold>* c)" |
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locale abel_semigroup = semigroup + |
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assumes commute [ac_simps]: "a \<^bold>* b = b \<^bold>* a" |
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begin |
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lemma left_commute [ac_simps]: "b \<^bold>* (a \<^bold>* c) = a \<^bold>* (b \<^bold>* c)" |
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proof - |
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have "(b \<^bold>* a) \<^bold>* c = (a \<^bold>* b) \<^bold>* 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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end |
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locale monoid = semigroup + |
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fixes z :: 'a ("\<^bold>1") |
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assumes left_neutral [simp]: "\<^bold>1 \<^bold>* a = a" |
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assumes right_neutral [simp]: "a \<^bold>* \<^bold>1 = a" |
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locale comm_monoid = abel_semigroup + |
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fixes z :: 'a ("\<^bold>1") |
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assumes comm_neutral: "a \<^bold>* \<^bold>1 = a" |
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begin |
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sublocale monoid |
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by standard (simp_all add: commute comm_neutral) |
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end |
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subsection \<open>Generic operations\<close> |
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class zero = |
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fixes zero :: 'a ("0") |
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class one = |
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fixes one :: 'a ("1") |
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hide_const (open) zero one |
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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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setup \<open> |
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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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\<close> |
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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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typed_print_translation \<open> |
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let |
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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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Syntax_Phases.term_of_typ ctxt T |
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else raise Match); |
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in map tr' [@{const_syntax Groups.one}, @{const_syntax Groups.zero}] end; |
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\<close> \<comment> \<open>show types that are presumably too general\<close> |
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class plus = |
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fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65) |
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class minus = |
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fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65) |
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class uminus = |
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fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80) |
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class times = |
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fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70) |
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subsection \<open>Semigroups and Monoids\<close> |
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class semigroup_add = plus + |
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assumes add_assoc [algebra_simps, field_simps]: "(a + b) + c = a + (b + c)" |
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begin |
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sublocale add: semigroup plus |
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by standard (fact add_assoc) |
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end |
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hide_fact add_assoc |
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class ab_semigroup_add = semigroup_add + |
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assumes add_commute [algebra_simps, field_simps]: "a + b = b + a" |
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begin |
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sublocale add: abel_semigroup plus |
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by standard (fact add_commute) |
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declare add.left_commute [algebra_simps, field_simps] |
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lemmas add_ac = add.assoc add.commute add.left_commute |
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end |
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hide_fact add_commute |
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lemmas add_ac = add.assoc add.commute add.left_commute |
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class semigroup_mult = times + |
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assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)" |
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begin |
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sublocale mult: semigroup times |
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by standard (fact mult_assoc) |
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54868 | 168 |
end |
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hide_fact mult_assoc |
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class ab_semigroup_mult = semigroup_mult + |
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|
173 |
assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a" |
54868 | 174 |
begin |
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|
175 |
|
61605 | 176 |
sublocale mult: abel_semigroup times |
61169 | 177 |
by standard (fact mult_commute) |
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|
178 |
|
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|
179 |
declare mult.left_commute [algebra_simps, field_simps] |
25062 | 180 |
|
61337 | 181 |
lemmas mult_ac = mult.assoc mult.commute mult.left_commute |
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|
182 |
|
23181 | 183 |
end |
14738 | 184 |
|
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185 |
hide_fact mult_commute |
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|
186 |
|
61337 | 187 |
lemmas mult_ac = mult.assoc mult.commute mult.left_commute |
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|
188 |
|
23085 | 189 |
class monoid_add = zero + semigroup_add + |
35720 | 190 |
assumes add_0_left: "0 + a = a" |
191 |
and add_0_right: "a + 0 = a" |
|
54868 | 192 |
begin |
35720 | 193 |
|
61605 | 194 |
sublocale add: monoid plus 0 |
61169 | 195 |
by standard (fact add_0_left add_0_right)+ |
23085 | 196 |
|
54868 | 197 |
end |
198 |
||
26071 | 199 |
lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0" |
54868 | 200 |
by (fact eq_commute) |
26071 | 201 |
|
22390 | 202 |
class comm_monoid_add = zero + ab_semigroup_add + |
25062 | 203 |
assumes add_0: "0 + a = a" |
54868 | 204 |
begin |
23085 | 205 |
|
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|
206 |
subclass monoid_add |
61169 | 207 |
by standard (simp_all add: add_0 add.commute [of _ 0]) |
25062 | 208 |
|
61605 | 209 |
sublocale add: comm_monoid plus 0 |
61169 | 210 |
by standard (simp add: ac_simps) |
14738 | 211 |
|
54868 | 212 |
end |
213 |
||
22390 | 214 |
class monoid_mult = one + semigroup_mult + |
35720 | 215 |
assumes mult_1_left: "1 * a = a" |
216 |
and mult_1_right: "a * 1 = a" |
|
54868 | 217 |
begin |
35720 | 218 |
|
61605 | 219 |
sublocale mult: monoid times 1 |
61169 | 220 |
by standard (fact mult_1_left mult_1_right)+ |
14738 | 221 |
|
54868 | 222 |
end |
223 |
||
26071 | 224 |
lemma one_reorient: "1 = x \<longleftrightarrow> x = 1" |
54868 | 225 |
by (fact eq_commute) |
26071 | 226 |
|
22390 | 227 |
class comm_monoid_mult = one + ab_semigroup_mult + |
25062 | 228 |
assumes mult_1: "1 * a = a" |
54868 | 229 |
begin |
14738 | 230 |
|
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|
231 |
subclass monoid_mult |
61169 | 232 |
by standard (simp_all add: mult_1 mult.commute [of _ 1]) |
25062 | 233 |
|
61605 | 234 |
sublocale mult: comm_monoid times 1 |
61169 | 235 |
by standard (simp add: ac_simps) |
14738 | 236 |
|
54868 | 237 |
end |
238 |
||
22390 | 239 |
class cancel_semigroup_add = semigroup_add + |
25062 | 240 |
assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c" |
241 |
assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c" |
|
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|
242 |
begin |
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|
243 |
|
63325 | 244 |
lemma add_left_cancel [simp]: "a + b = a + c \<longleftrightarrow> b = c" |
245 |
by (blast dest: add_left_imp_eq) |
|
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|
246 |
|
63325 | 247 |
lemma add_right_cancel [simp]: "b + a = c + a \<longleftrightarrow> b = c" |
248 |
by (blast dest: add_right_imp_eq) |
|
27474
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|
249 |
|
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|
250 |
end |
14738 | 251 |
|
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252 |
class cancel_ab_semigroup_add = ab_semigroup_add + minus + |
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|
253 |
assumes add_diff_cancel_left' [simp]: "(a + b) - a = b" |
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|
254 |
assumes diff_diff_add [algebra_simps, field_simps]: "a - b - c = a - (b + c)" |
25267 | 255 |
begin |
14738 | 256 |
|
63325 | 257 |
lemma add_diff_cancel_right' [simp]: "(a + b) - b = a" |
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|
258 |
using add_diff_cancel_left' [of b a] by (simp add: ac_simps) |
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|
259 |
|
25267 | 260 |
subclass cancel_semigroup_add |
28823 | 261 |
proof |
22390 | 262 |
fix a b c :: 'a |
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|
263 |
assume "a + b = a + c" |
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explicit commutative additive inverse operation;
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|
264 |
then have "a + b - a = a + c - a" |
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explicit commutative additive inverse operation;
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parents:
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|
265 |
by simp |
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|
266 |
then show "b = c" |
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|
267 |
by simp |
22390 | 268 |
next |
14738 | 269 |
fix a b c :: 'a |
270 |
assume "b + a = c + a" |
|
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|
271 |
then have "b + a - a = c + a - a" |
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|
272 |
by simp |
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|
273 |
then show "b = c" |
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|
274 |
by simp |
14738 | 275 |
qed |
276 |
||
63325 | 277 |
lemma add_diff_cancel_left [simp]: "(c + a) - (c + b) = a - b" |
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|
278 |
unfolding diff_diff_add [symmetric] by simp |
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|
279 |
|
63325 | 280 |
lemma add_diff_cancel_right [simp]: "(a + c) - (b + c) = a - b" |
59815
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|
281 |
using add_diff_cancel_left [symmetric] by (simp add: ac_simps) |
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|
282 |
|
63325 | 283 |
lemma diff_right_commute: "a - c - b = a - b - c" |
59815
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|
284 |
by (simp add: diff_diff_add add.commute) |
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|
285 |
|
25267 | 286 |
end |
287 |
||
29904 | 288 |
class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add |
59322 | 289 |
begin |
290 |
||
63325 | 291 |
lemma diff_zero [simp]: "a - 0 = a" |
59815
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|
292 |
using add_diff_cancel_right' [of a 0] by simp |
59322 | 293 |
|
63325 | 294 |
lemma diff_cancel [simp]: "a - a = 0" |
59322 | 295 |
proof - |
63325 | 296 |
have "(a + 0) - (a + 0) = 0" |
297 |
by (simp only: add_diff_cancel_left diff_zero) |
|
59322 | 298 |
then show ?thesis by simp |
299 |
qed |
|
300 |
||
301 |
lemma add_implies_diff: |
|
302 |
assumes "c + b = a" |
|
303 |
shows "c = a - b" |
|
304 |
proof - |
|
63325 | 305 |
from assms have "(b + c) - (b + 0) = a - b" |
306 |
by (simp add: add.commute) |
|
59322 | 307 |
then show "c = a - b" by simp |
308 |
qed |
|
309 |
||
63325 | 310 |
lemma add_cancel_right_right [simp]: "a = a + b \<longleftrightarrow> b = 0" |
311 |
(is "?P \<longleftrightarrow> ?Q") |
|
62608 | 312 |
proof |
63325 | 313 |
assume ?Q |
314 |
then show ?P by simp |
|
62608 | 315 |
next |
63325 | 316 |
assume ?P |
317 |
then have "a - a = a + b - a" by simp |
|
62608 | 318 |
then show ?Q by simp |
319 |
qed |
|
320 |
||
63325 | 321 |
lemma add_cancel_right_left [simp]: "a = b + a \<longleftrightarrow> b = 0" |
62608 | 322 |
using add_cancel_right_right [of a b] by (simp add: ac_simps) |
323 |
||
63325 | 324 |
lemma add_cancel_left_right [simp]: "a + b = a \<longleftrightarrow> b = 0" |
62608 | 325 |
by (auto dest: sym) |
326 |
||
63325 | 327 |
lemma add_cancel_left_left [simp]: "b + a = a \<longleftrightarrow> b = 0" |
62608 | 328 |
by (auto dest: sym) |
329 |
||
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
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parents:
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diff
changeset
|
330 |
end |
59815
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|
331 |
|
cce82e360c2f
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|
332 |
class comm_monoid_diff = cancel_comm_monoid_add + |
cce82e360c2f
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|
333 |
assumes zero_diff [simp]: "0 - a = 0" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
334 |
begin |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
335 |
|
63325 | 336 |
lemma diff_add_zero [simp]: "a - (a + b) = 0" |
59815
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haftmann
parents:
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diff
changeset
|
337 |
proof - |
63325 | 338 |
have "a - (a + b) = (a + 0) - (a + b)" |
339 |
by simp |
|
340 |
also have "\<dots> = 0" |
|
341 |
by (simp only: add_diff_cancel_left zero_diff) |
|
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
342 |
finally show ?thesis . |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
343 |
qed |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
344 |
|
59322 | 345 |
end |
346 |
||
29904 | 347 |
|
60758 | 348 |
subsection \<open>Groups\<close> |
23085 | 349 |
|
25762 | 350 |
class group_add = minus + uminus + monoid_add + |
25062 | 351 |
assumes left_minus [simp]: "- a + a = 0" |
54230
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54148
diff
changeset
|
352 |
assumes add_uminus_conv_diff [simp]: "a + (- b) = a - b" |
25062 | 353 |
begin |
23085 | 354 |
|
63325 | 355 |
lemma diff_conv_add_uminus: "a - b = a + (- b)" |
54230
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more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
356 |
by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
357 |
|
34147
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
358 |
lemma minus_unique: |
63325 | 359 |
assumes "a + b = 0" |
360 |
shows "- a = b" |
|
34147
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
361 |
proof - |
63325 | 362 |
from assms have "- a = - a + (a + b)" by simp |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
363 |
also have "\<dots> = b" 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
|
364 |
finally show ?thesis . |
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
365 |
qed |
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
366 |
|
25062 | 367 |
lemma minus_zero [simp]: "- 0 = 0" |
14738 | 368 |
proof - |
34147
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
369 |
have "0 + 0 = 0" by (rule add_0_right) |
63325 | 370 |
then show "- 0 = 0" by (rule minus_unique) |
14738 | 371 |
qed |
372 |
||
25062 | 373 |
lemma minus_minus [simp]: "- (- a) = a" |
23085 | 374 |
proof - |
34147
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
375 |
have "- a + a = 0" by (rule left_minus) |
63325 | 376 |
then show "- (- a) = a" by (rule minus_unique) |
23085 | 377 |
qed |
14738 | 378 |
|
54230
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more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
379 |
lemma right_minus: "a + - a = 0" |
14738 | 380 |
proof - |
25062 | 381 |
have "a + - a = - (- a) + - a" by simp |
382 |
also have "\<dots> = 0" by (rule left_minus) |
|
14738 | 383 |
finally show ?thesis . |
384 |
qed |
|
385 |
||
63325 | 386 |
lemma diff_self [simp]: "a - a = 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
387 |
using right_minus [of a] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
388 |
|
40368
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
389 |
subclass cancel_semigroup_add |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
390 |
proof |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
391 |
fix a b c :: 'a |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
392 |
assume "a + b = a + c" |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
393 |
then have "- a + a + b = - a + a + c" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
394 |
unfolding add.assoc by simp |
40368
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
395 |
then show "b = c" by simp |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
396 |
next |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
397 |
fix a b c :: 'a |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
398 |
assume "b + a = c + a" |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
399 |
then have "b + a + - a = c + a + - a" by simp |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
400 |
then show "b = c" unfolding add.assoc by simp |
40368
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
401 |
qed |
47c186c8577d
added class relation group_add < cancel_semigroup_add
haftmann
parents:
39134
diff
changeset
|
402 |
|
63325 | 403 |
lemma minus_add_cancel [simp]: "- a + (a + b) = b" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
404 |
by (simp add: add.assoc [symmetric]) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
405 |
|
63325 | 406 |
lemma add_minus_cancel [simp]: "a + (- a + b) = b" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
407 |
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
|
408 |
|
63325 | 409 |
lemma diff_add_cancel [simp]: "a - b + b = a" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
410 |
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
|
411 |
|
63325 | 412 |
lemma add_diff_cancel [simp]: "a + b - b = a" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
413 |
by (simp only: diff_conv_add_uminus add.assoc) simp |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
414 |
|
63325 | 415 |
lemma minus_add: "- (a + b) = - b + - a" |
34147
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
416 |
proof - |
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
417 |
have "(a + b) + (- b + - a) = 0" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
418 |
by (simp only: add.assoc add_minus_cancel) simp |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
419 |
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
|
420 |
by (rule minus_unique) |
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
421 |
qed |
319616f4eecf
generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents:
34146
diff
changeset
|
422 |
|
63325 | 423 |
lemma right_minus_eq [simp]: "a - b = 0 \<longleftrightarrow> a = b" |
14738 | 424 |
proof |
23085 | 425 |
assume "a - b = 0" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
426 |
have "a = (a - b) + b" by (simp add: add.assoc) |
60758 | 427 |
also have "\<dots> = b" using \<open>a - b = 0\<close> by simp |
23085 | 428 |
finally show "a = b" . |
14738 | 429 |
next |
63325 | 430 |
assume "a = b" |
431 |
then show "a - b = 0" by simp |
|
14738 | 432 |
qed |
433 |
||
63325 | 434 |
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
435 |
by (fact right_minus_eq [symmetric]) |
14738 | 436 |
|
63325 | 437 |
lemma diff_0 [simp]: "0 - a = - a" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
438 |
by (simp only: diff_conv_add_uminus add_0_left) |
14738 | 439 |
|
63325 | 440 |
lemma diff_0_right [simp]: "a - 0 = a" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
441 |
by (simp only: diff_conv_add_uminus minus_zero add_0_right) |
14738 | 442 |
|
63325 | 443 |
lemma diff_minus_eq_add [simp]: "a - - b = a + b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
444 |
by (simp only: diff_conv_add_uminus minus_minus) |
14738 | 445 |
|
63325 | 446 |
lemma neg_equal_iff_equal [simp]: "- a = - b \<longleftrightarrow> a = b" |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
447 |
proof |
14738 | 448 |
assume "- a = - b" |
63325 | 449 |
then have "- (- a) = - (- b)" by simp |
450 |
then show "a = b" by simp |
|
14738 | 451 |
next |
25062 | 452 |
assume "a = b" |
63325 | 453 |
then show "- a = - b" by simp |
14738 | 454 |
qed |
455 |
||
63325 | 456 |
lemma neg_equal_0_iff_equal [simp]: "- a = 0 \<longleftrightarrow> a = 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
457 |
by (subst neg_equal_iff_equal [symmetric]) simp |
14738 | 458 |
|
63325 | 459 |
lemma neg_0_equal_iff_equal [simp]: "0 = - a \<longleftrightarrow> 0 = a" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
460 |
by (subst neg_equal_iff_equal [symmetric]) simp |
14738 | 461 |
|
63325 | 462 |
text \<open>The next two equations can make the simplifier loop!\<close> |
14738 | 463 |
|
63325 | 464 |
lemma equation_minus_iff: "a = - b \<longleftrightarrow> b = - a" |
14738 | 465 |
proof - |
63325 | 466 |
have "- (- a) = - b \<longleftrightarrow> - a = b" |
467 |
by (rule neg_equal_iff_equal) |
|
468 |
then show ?thesis |
|
469 |
by (simp add: eq_commute) |
|
25062 | 470 |
qed |
471 |
||
63325 | 472 |
lemma minus_equation_iff: "- a = b \<longleftrightarrow> - b = a" |
25062 | 473 |
proof - |
63325 | 474 |
have "- a = - (- b) \<longleftrightarrow> a = -b" |
475 |
by (rule neg_equal_iff_equal) |
|
476 |
then show ?thesis |
|
477 |
by (simp add: eq_commute) |
|
14738 | 478 |
qed |
479 |
||
63325 | 480 |
lemma eq_neg_iff_add_eq_0: "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
|
481 |
proof |
63325 | 482 |
assume "a = - b" |
483 |
then show "a + b = 0" by simp |
|
29914
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
484 |
next |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
485 |
assume "a + b = 0" |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
486 |
moreover have "a + (b + - b) = (a + b) + - b" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
487 |
by (simp only: add.assoc) |
63325 | 488 |
ultimately show "a = - b" |
489 |
by simp |
|
29914
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
490 |
qed |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
491 |
|
63325 | 492 |
lemma add_eq_0_iff2: "a + b = 0 \<longleftrightarrow> a = - b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
493 |
by (fact eq_neg_iff_add_eq_0 [symmetric]) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
494 |
|
63325 | 495 |
lemma neg_eq_iff_add_eq_0: "- a = b \<longleftrightarrow> a + b = 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
496 |
by (auto simp add: add_eq_0_iff2) |
44348 | 497 |
|
63325 | 498 |
lemma add_eq_0_iff: "a + b = 0 \<longleftrightarrow> b = - a" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
499 |
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
|
500 |
|
63325 | 501 |
lemma minus_diff_eq [simp]: "- (a - b) = b - a" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
502 |
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
|
503 |
|
63325 | 504 |
lemma add_diff_eq [algebra_simps, field_simps]: "a + (b - c) = (a + b) - c" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
505 |
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
|
506 |
|
63325 | 507 |
lemma diff_add_eq_diff_diff_swap: "a - (b + c) = a - c - b" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
508 |
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
|
509 |
|
63325 | 510 |
lemma diff_eq_eq [algebra_simps, field_simps]: "a - b = c \<longleftrightarrow> a = c + b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
511 |
by auto |
45548
3e2722d66169
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents:
45294
diff
changeset
|
512 |
|
63325 | 513 |
lemma eq_diff_eq [algebra_simps, field_simps]: "a = c - b \<longleftrightarrow> a + b = c" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
514 |
by auto |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
515 |
|
63325 | 516 |
lemma diff_diff_eq2 [algebra_simps, field_simps]: "a - (b - c) = (a + c) - b" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
517 |
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
|
518 |
|
63325 | 519 |
lemma diff_eq_diff_eq: "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
|
520 |
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
|
521 |
|
25062 | 522 |
end |
523 |
||
25762 | 524 |
class ab_group_add = minus + uminus + comm_monoid_add + |
25062 | 525 |
assumes ab_left_minus: "- a + a = 0" |
59557 | 526 |
assumes ab_diff_conv_add_uminus: "a - b = a + (- b)" |
25267 | 527 |
begin |
25062 | 528 |
|
25267 | 529 |
subclass group_add |
63325 | 530 |
by standard (simp_all add: ab_left_minus ab_diff_conv_add_uminus) |
25062 | 531 |
|
29904 | 532 |
subclass cancel_comm_monoid_add |
28823 | 533 |
proof |
25062 | 534 |
fix a b c :: 'a |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
535 |
have "b + a - a = b" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
536 |
by simp |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
537 |
then show "a + b - a = b" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
538 |
by (simp add: ac_simps) |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
539 |
show "a - b - c = a - (b + c)" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
540 |
by (simp add: algebra_simps) |
25062 | 541 |
qed |
542 |
||
63325 | 543 |
lemma uminus_add_conv_diff [simp]: "- a + b = b - a" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
544 |
by (simp add: add.commute) |
25062 | 545 |
|
63325 | 546 |
lemma minus_add_distrib [simp]: "- (a + b) = - a + - b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
547 |
by (simp add: algebra_simps) |
25062 | 548 |
|
63325 | 549 |
lemma diff_add_eq [algebra_simps, field_simps]: "(a - b) + c = (a + c) - b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
550 |
by (simp add: algebra_simps) |
25077 | 551 |
|
25062 | 552 |
end |
14738 | 553 |
|
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
554 |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
555 |
subsection \<open>(Partially) Ordered Groups\<close> |
14738 | 556 |
|
60758 | 557 |
text \<open> |
35301
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
558 |
The theory of partially ordered groups is taken from the books: |
63325 | 559 |
|
560 |
\<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979 |
|
561 |
\<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963 |
|
562 |
||
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
563 |
Most of the used notions can also be looked up in |
63325 | 564 |
\<^item> @{url "http://www.mathworld.com"} by Eric Weisstein et. al. |
565 |
\<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer |
|
60758 | 566 |
\<close> |
35301
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35267
diff
changeset
|
567 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
568 |
class ordered_ab_semigroup_add = order + ab_semigroup_add + |
25062 | 569 |
assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b" |
570 |
begin |
|
24380
c215e256beca
moved ordered_ab_semigroup_add to OrderedGroup.thy
haftmann
parents:
24286
diff
changeset
|
571 |
|
63325 | 572 |
lemma add_right_mono: "a \<le> b \<Longrightarrow> a + c \<le> b + c" |
573 |
by (simp add: add.commute [of _ c] add_left_mono) |
|
14738 | 574 |
|
60758 | 575 |
text \<open>non-strict, in both arguments\<close> |
63325 | 576 |
lemma add_mono: "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d" |
14738 | 577 |
apply (erule add_right_mono [THEN order_trans]) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
578 |
apply (simp add: add.commute add_left_mono) |
14738 | 579 |
done |
580 |
||
25062 | 581 |
end |
582 |
||
63325 | 583 |
text \<open>Strict monotonicity in both arguments\<close> |
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
584 |
class strict_ordered_ab_semigroup_add = ordered_ab_semigroup_add + |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
585 |
assumes add_strict_mono: "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
586 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
587 |
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
|
588 |
ordered_ab_semigroup_add + cancel_ab_semigroup_add |
25062 | 589 |
begin |
590 |
||
63325 | 591 |
lemma add_strict_left_mono: "a < b \<Longrightarrow> c + a < c + b" |
592 |
by (auto simp add: less_le add_left_mono) |
|
14738 | 593 |
|
63325 | 594 |
lemma add_strict_right_mono: "a < b \<Longrightarrow> a + c < b + c" |
595 |
by (simp add: add.commute [of _ c] add_strict_left_mono) |
|
14738 | 596 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
597 |
subclass strict_ordered_ab_semigroup_add |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
598 |
apply standard |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
599 |
apply (erule add_strict_right_mono [THEN less_trans]) |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
600 |
apply (erule add_strict_left_mono) |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
601 |
done |
14738 | 602 |
|
63325 | 603 |
lemma add_less_le_mono: "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d" |
604 |
apply (erule add_strict_right_mono [THEN less_le_trans]) |
|
605 |
apply (erule add_left_mono) |
|
606 |
done |
|
14738 | 607 |
|
63325 | 608 |
lemma add_le_less_mono: "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" |
609 |
apply (erule add_right_mono [THEN le_less_trans]) |
|
610 |
apply (erule add_strict_left_mono) |
|
611 |
done |
|
14738 | 612 |
|
25062 | 613 |
end |
614 |
||
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
615 |
class ordered_ab_semigroup_add_imp_le = ordered_cancel_ab_semigroup_add + |
25062 | 616 |
assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b" |
617 |
begin |
|
618 |
||
14738 | 619 |
lemma add_less_imp_less_left: |
63325 | 620 |
assumes less: "c + a < c + b" |
621 |
shows "a < b" |
|
14738 | 622 |
proof - |
63325 | 623 |
from less have le: "c + a \<le> c + b" |
624 |
by (simp add: order_le_less) |
|
625 |
have "a \<le> b" |
|
14738 | 626 |
apply (insert le) |
627 |
apply (drule add_le_imp_le_left) |
|
63325 | 628 |
apply (insert le) |
629 |
apply (drule add_le_imp_le_left) |
|
630 |
apply assumption |
|
631 |
done |
|
14738 | 632 |
moreover have "a \<noteq> b" |
633 |
proof (rule ccontr) |
|
63325 | 634 |
assume "\<not> ?thesis" |
14738 | 635 |
then have "a = b" by simp |
636 |
then have "c + a = c + b" by simp |
|
63325 | 637 |
with less show "False" by simp |
14738 | 638 |
qed |
63325 | 639 |
ultimately show "a < b" |
640 |
by (simp add: order_le_less) |
|
14738 | 641 |
qed |
642 |
||
63325 | 643 |
lemma add_less_imp_less_right: "a + c < b + c \<Longrightarrow> a < b" |
644 |
by (rule add_less_imp_less_left [of c]) (simp add: add.commute) |
|
14738 | 645 |
|
63325 | 646 |
lemma add_less_cancel_left [simp]: "c + a < c + b \<longleftrightarrow> a < b" |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
647 |
by (blast intro: add_less_imp_less_left add_strict_left_mono) |
14738 | 648 |
|
63325 | 649 |
lemma add_less_cancel_right [simp]: "a + c < b + c \<longleftrightarrow> a < b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
650 |
by (blast intro: add_less_imp_less_right add_strict_right_mono) |
14738 | 651 |
|
63325 | 652 |
lemma add_le_cancel_left [simp]: "c + a \<le> c + b \<longleftrightarrow> a \<le> b" |
653 |
apply auto |
|
654 |
apply (drule add_le_imp_le_left) |
|
655 |
apply (simp_all add: add_left_mono) |
|
656 |
done |
|
14738 | 657 |
|
63325 | 658 |
lemma add_le_cancel_right [simp]: "a + c \<le> b + c \<longleftrightarrow> a \<le> b" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56950
diff
changeset
|
659 |
by (simp add: add.commute [of a c] add.commute [of b c]) |
14738 | 660 |
|
63325 | 661 |
lemma add_le_imp_le_right: "a + c \<le> b + c \<Longrightarrow> a \<le> b" |
662 |
by simp |
|
25062 | 663 |
|
63325 | 664 |
lemma max_add_distrib_left: "max x y + z = max (x + z) (y + z)" |
25077 | 665 |
unfolding max_def by auto |
666 |
||
63325 | 667 |
lemma min_add_distrib_left: "min x y + z = min (x + z) (y + z)" |
25077 | 668 |
unfolding min_def by auto |
669 |
||
63325 | 670 |
lemma max_add_distrib_right: "x + max y z = max (x + y) (x + z)" |
44848
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
671 |
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
|
672 |
|
63325 | 673 |
lemma min_add_distrib_right: "x + min y z = min (x + y) (x + z)" |
44848
f4d0b060c7ca
remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents:
44433
diff
changeset
|
674 |
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
|
675 |
|
25062 | 676 |
end |
677 |
||
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
678 |
subsection \<open>Support for reasoning about signs\<close> |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
679 |
|
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
680 |
class ordered_comm_monoid_add = comm_monoid_add + ordered_ab_semigroup_add |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
681 |
begin |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
682 |
|
63325 | 683 |
lemma add_nonneg_nonneg [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b" |
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
684 |
using add_mono[of 0 a 0 b] by simp |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
685 |
|
63325 | 686 |
lemma add_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> a + b \<le> 0" |
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
687 |
using add_mono[of a 0 b 0] by simp |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
688 |
|
63325 | 689 |
lemma add_nonneg_eq_0_iff: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0" |
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
690 |
using add_left_mono[of 0 y x] add_right_mono[of 0 x y] by auto |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
691 |
|
63325 | 692 |
lemma add_nonpos_eq_0_iff: "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0" |
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
693 |
using add_left_mono[of y 0 x] add_right_mono[of x 0 y] by auto |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
694 |
|
63325 | 695 |
lemma add_increasing: "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c" |
696 |
using add_mono [of 0 a b c] by simp |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
697 |
|
63325 | 698 |
lemma add_increasing2: "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c" |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
699 |
by (simp add: add_increasing add.commute [of a]) |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
700 |
|
63325 | 701 |
lemma add_decreasing: "a \<le> 0 \<Longrightarrow> c \<le> b \<Longrightarrow> a + c \<le> b" |
702 |
using add_mono [of a 0 c b] by simp |
|
52289 | 703 |
|
63325 | 704 |
lemma add_decreasing2: "c \<le> 0 \<Longrightarrow> a \<le> b \<Longrightarrow> a + c \<le> b" |
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
705 |
using add_mono[of a b c 0] by simp |
52289 | 706 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
707 |
lemma add_pos_nonneg: "0 < a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < a + b" |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
708 |
using less_le_trans[of 0 a "a + b"] by (simp add: add_increasing2) |
52289 | 709 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
710 |
lemma add_pos_pos: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a + b" |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
711 |
by (intro add_pos_nonneg less_imp_le) |
52289 | 712 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
713 |
lemma add_nonneg_pos: "0 \<le> a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a + b" |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
714 |
using add_pos_nonneg[of b a] by (simp add: add_commute) |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
715 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
716 |
lemma add_neg_nonpos: "a < 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> a + b < 0" |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
717 |
using le_less_trans[of "a + b" a 0] by (simp add: add_decreasing2) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
718 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
719 |
lemma add_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> a + b < 0" |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
720 |
by (intro add_neg_nonpos less_imp_le) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
721 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
722 |
lemma add_nonpos_neg: "a \<le> 0 \<Longrightarrow> b < 0 \<Longrightarrow> a + b < 0" |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
723 |
using add_neg_nonpos[of b a] by (simp add: add_commute) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
724 |
|
30691 | 725 |
lemmas add_sign_intros = |
726 |
add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg |
|
727 |
add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos |
|
728 |
||
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
729 |
end |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
730 |
|
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
731 |
class strict_ordered_comm_monoid_add = comm_monoid_add + strict_ordered_ab_semigroup_add |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
732 |
begin |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
733 |
|
63325 | 734 |
lemma pos_add_strict: "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c" |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
735 |
using add_strict_mono [of 0 a b c] by simp |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
736 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
737 |
end |
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
738 |
|
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
739 |
class ordered_cancel_comm_monoid_add = ordered_comm_monoid_add + cancel_ab_semigroup_add |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
740 |
begin |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
741 |
|
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
742 |
subclass ordered_cancel_ab_semigroup_add .. |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
743 |
subclass strict_ordered_comm_monoid_add .. |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
744 |
|
63325 | 745 |
lemma add_strict_increasing: "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c" |
746 |
using add_less_le_mono [of 0 a b c] by simp |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
747 |
|
63325 | 748 |
lemma add_strict_increasing2: "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c" |
749 |
using add_le_less_mono [of 0 a b c] by simp |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
750 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
751 |
end |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
752 |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
753 |
class ordered_ab_group_add = ab_group_add + ordered_ab_semigroup_add |
25062 | 754 |
begin |
755 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
756 |
subclass ordered_cancel_ab_semigroup_add .. |
25062 | 757 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
758 |
subclass ordered_ab_semigroup_add_imp_le |
28823 | 759 |
proof |
25062 | 760 |
fix a b c :: 'a |
761 |
assume "c + a \<le> c + b" |
|
63325 | 762 |
then have "(-c) + (c + a) \<le> (-c) + (c + b)" |
763 |
by (rule add_left_mono) |
|
764 |
then have "((-c) + c) + a \<le> ((-c) + c) + b" |
|
765 |
by (simp only: add.assoc) |
|
766 |
then show "a \<le> b" by simp |
|
25062 | 767 |
qed |
768 |
||
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
769 |
subclass ordered_cancel_comm_monoid_add .. |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
770 |
|
63325 | 771 |
lemma add_less_same_cancel1 [simp]: "b + a < b \<longleftrightarrow> a < 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
772 |
using add_less_cancel_left [of _ _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
773 |
|
63325 | 774 |
lemma add_less_same_cancel2 [simp]: "a + b < b \<longleftrightarrow> a < 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
775 |
using add_less_cancel_right [of _ _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
776 |
|
63325 | 777 |
lemma less_add_same_cancel1 [simp]: "a < a + b \<longleftrightarrow> 0 < b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
778 |
using add_less_cancel_left [of _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
779 |
|
63325 | 780 |
lemma less_add_same_cancel2 [simp]: "a < b + a \<longleftrightarrow> 0 < b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
781 |
using add_less_cancel_right [of 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
782 |
|
63325 | 783 |
lemma add_le_same_cancel1 [simp]: "b + a \<le> b \<longleftrightarrow> a \<le> 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
784 |
using add_le_cancel_left [of _ _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
785 |
|
63325 | 786 |
lemma add_le_same_cancel2 [simp]: "a + b \<le> b \<longleftrightarrow> a \<le> 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
787 |
using add_le_cancel_right [of _ _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
788 |
|
63325 | 789 |
lemma le_add_same_cancel1 [simp]: "a \<le> a + b \<longleftrightarrow> 0 \<le> b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
790 |
using add_le_cancel_left [of _ 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
791 |
|
63325 | 792 |
lemma le_add_same_cancel2 [simp]: "a \<le> b + a \<longleftrightarrow> 0 \<le> b" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
793 |
using add_le_cancel_right [of 0] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
794 |
|
63325 | 795 |
lemma max_diff_distrib_left: "max x y - z = max (x - z) (y - z)" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
796 |
using max_add_distrib_left [of x y "- z"] by simp |
25077 | 797 |
|
63325 | 798 |
lemma min_diff_distrib_left: "min x y - z = min (x - z) (y - z)" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
799 |
using min_add_distrib_left [of x y "- z"] by simp |
25077 | 800 |
|
801 |
lemma le_imp_neg_le: |
|
63325 | 802 |
assumes "a \<le> b" |
803 |
shows "- b \<le> - a" |
|
25077 | 804 |
proof - |
63325 | 805 |
from assms have "- a + a \<le> - a + b" |
806 |
by (rule add_left_mono) |
|
807 |
then have "0 \<le> - a + b" |
|
808 |
by simp |
|
809 |
then have "0 + (- b) \<le> (- a + b) + (- b)" |
|
810 |
by (rule add_right_mono) |
|
811 |
then show ?thesis |
|
812 |
by (simp add: algebra_simps) |
|
25077 | 813 |
qed |
814 |
||
815 |
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b" |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
816 |
proof |
25077 | 817 |
assume "- b \<le> - a" |
63325 | 818 |
then have "- (- a) \<le> - (- b)" |
819 |
by (rule le_imp_neg_le) |
|
820 |
then show "a \<le> b" |
|
821 |
by simp |
|
25077 | 822 |
next |
63325 | 823 |
assume "a \<le> b" |
824 |
then show "- b \<le> - a" |
|
825 |
by (rule le_imp_neg_le) |
|
25077 | 826 |
qed |
827 |
||
828 |
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a" |
|
63325 | 829 |
by (subst neg_le_iff_le [symmetric]) simp |
25077 | 830 |
|
831 |
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0" |
|
63325 | 832 |
by (subst neg_le_iff_le [symmetric]) simp |
25077 | 833 |
|
834 |
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b" |
|
63325 | 835 |
by (auto simp add: less_le) |
25077 | 836 |
|
837 |
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a" |
|
63325 | 838 |
by (subst neg_less_iff_less [symmetric]) simp |
25077 | 839 |
|
840 |
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0" |
|
63325 | 841 |
by (subst neg_less_iff_less [symmetric]) simp |
25077 | 842 |
|
63325 | 843 |
text \<open>The next several equations can make the simplifier loop!\<close> |
25077 | 844 |
|
845 |
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a" |
|
846 |
proof - |
|
63325 | 847 |
have "- (-a) < - b \<longleftrightarrow> b < - a" |
848 |
by (rule neg_less_iff_less) |
|
849 |
then show ?thesis by simp |
|
25077 | 850 |
qed |
851 |
||
852 |
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a" |
|
853 |
proof - |
|
63325 | 854 |
have "- a < - (- b) \<longleftrightarrow> - b < a" |
855 |
by (rule neg_less_iff_less) |
|
856 |
then show ?thesis by simp |
|
25077 | 857 |
qed |
858 |
||
859 |
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a" |
|
860 |
proof - |
|
63325 | 861 |
have mm: "- (- a) < -b \<Longrightarrow> - (- b) < -a" for a b :: 'a |
862 |
by (simp only: minus_less_iff) |
|
863 |
have "- (- a) \<le> -b \<longleftrightarrow> b \<le> - a" |
|
25077 | 864 |
apply (auto simp only: le_less) |
865 |
apply (drule mm) |
|
866 |
apply (simp_all) |
|
867 |
apply (drule mm[simplified], assumption) |
|
868 |
done |
|
869 |
then show ?thesis by simp |
|
870 |
qed |
|
871 |
||
872 |
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a" |
|
63325 | 873 |
by (auto simp add: le_less minus_less_iff) |
25077 | 874 |
|
63325 | 875 |
lemma diff_less_0_iff_less [simp]: "a - b < 0 \<longleftrightarrow> a < b" |
25077 | 876 |
proof - |
63325 | 877 |
have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" |
878 |
by simp |
|
879 |
also have "\<dots> \<longleftrightarrow> a < b" |
|
880 |
by (simp only: add_less_cancel_right) |
|
25077 | 881 |
finally show ?thesis . |
882 |
qed |
|
883 |
||
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
884 |
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
|
885 |
|
63325 | 886 |
lemma diff_less_eq [algebra_simps, field_simps]: "a - b < c \<longleftrightarrow> a < c + b" |
887 |
apply (subst less_iff_diff_less_0 [of a]) |
|
888 |
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) |
|
889 |
apply (simp add: algebra_simps) |
|
890 |
done |
|
25077 | 891 |
|
63325 | 892 |
lemma less_diff_eq[algebra_simps, field_simps]: "a < c - b \<longleftrightarrow> a + b < c" |
893 |
apply (subst less_iff_diff_less_0 [of "a + b"]) |
|
894 |
apply (subst less_iff_diff_less_0 [of a]) |
|
895 |
apply (simp add: algebra_simps) |
|
896 |
done |
|
25077 | 897 |
|
63325 | 898 |
lemma diff_gt_0_iff_gt [simp]: "a - b > 0 \<longleftrightarrow> a > b" |
62348 | 899 |
by (simp add: less_diff_eq) |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61605
diff
changeset
|
900 |
|
63325 | 901 |
lemma diff_le_eq [algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b" |
62348 | 902 |
by (auto simp add: le_less diff_less_eq ) |
25077 | 903 |
|
63325 | 904 |
lemma le_diff_eq [algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c" |
62348 | 905 |
by (auto simp add: le_less less_diff_eq) |
25077 | 906 |
|
63325 | 907 |
lemma diff_le_0_iff_le [simp]: "a - b \<le> 0 \<longleftrightarrow> a \<le> b" |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
908 |
by (simp add: algebra_simps) |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
909 |
|
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
910 |
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
|
911 |
|
63325 | 912 |
lemma diff_ge_0_iff_ge [simp]: "a - b \<ge> 0 \<longleftrightarrow> a \<ge> b" |
62348 | 913 |
by (simp add: le_diff_eq) |
914 |
||
63325 | 915 |
lemma diff_eq_diff_less: "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d" |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
916 |
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
|
917 |
|
63325 | 918 |
lemma diff_eq_diff_less_eq: "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d" |
37889
0d8058e0c270
keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents:
37884
diff
changeset
|
919 |
by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d]) |
25077 | 920 |
|
56950 | 921 |
lemma diff_mono: "a \<le> b \<Longrightarrow> d \<le> c \<Longrightarrow> a - c \<le> b - d" |
922 |
by (simp add: field_simps add_mono) |
|
923 |
||
924 |
lemma diff_left_mono: "b \<le> a \<Longrightarrow> c - a \<le> c - b" |
|
925 |
by (simp add: field_simps) |
|
926 |
||
927 |
lemma diff_right_mono: "a \<le> b \<Longrightarrow> a - c \<le> b - c" |
|
928 |
by (simp add: field_simps) |
|
929 |
||
930 |
lemma diff_strict_mono: "a < b \<Longrightarrow> d < c \<Longrightarrow> a - c < b - d" |
|
931 |
by (simp add: field_simps add_strict_mono) |
|
932 |
||
933 |
lemma diff_strict_left_mono: "b < a \<Longrightarrow> c - a < c - b" |
|
934 |
by (simp add: field_simps) |
|
935 |
||
936 |
lemma diff_strict_right_mono: "a < b \<Longrightarrow> a - c < b - c" |
|
937 |
by (simp add: field_simps) |
|
938 |
||
25077 | 939 |
end |
940 |
||
48891 | 941 |
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
|
942 |
|
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
943 |
simproc_setup group_cancel_add ("a + b::'a::ab_group_add") = |
60758 | 944 |
\<open>fn phi => fn ss => try Group_Cancel.cancel_add_conv\<close> |
48556
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
945 |
|
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
946 |
simproc_setup group_cancel_diff ("a - b::'a::ab_group_add") = |
60758 | 947 |
\<open>fn phi => fn ss => try Group_Cancel.cancel_diff_conv\<close> |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
948 |
|
48556
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
949 |
simproc_setup group_cancel_eq ("a = (b::'a::ab_group_add)") = |
60758 | 950 |
\<open>fn phi => fn ss => try Group_Cancel.cancel_eq_conv\<close> |
37889
0d8058e0c270
keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents:
37884
diff
changeset
|
951 |
|
48556
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
952 |
simproc_setup group_cancel_le ("a \<le> (b::'a::ordered_ab_group_add)") = |
60758 | 953 |
\<open>fn phi => fn ss => try Group_Cancel.cancel_le_conv\<close> |
48556
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
954 |
|
62a3fbf9d35b
replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents:
45548
diff
changeset
|
955 |
simproc_setup group_cancel_less ("a < (b::'a::ordered_ab_group_add)") = |
60758 | 956 |
\<open>fn phi => fn ss => try Group_Cancel.cancel_less_conv\<close> |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36977
diff
changeset
|
957 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
958 |
class linordered_ab_semigroup_add = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
959 |
linorder + ordered_ab_semigroup_add |
25062 | 960 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
961 |
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
|
962 |
linorder + ordered_cancel_ab_semigroup_add |
25267 | 963 |
begin |
25062 | 964 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
965 |
subclass linordered_ab_semigroup_add .. |
25062 | 966 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
967 |
subclass ordered_ab_semigroup_add_imp_le |
28823 | 968 |
proof |
25062 | 969 |
fix a b c :: 'a |
63325 | 970 |
assume le1: "c + a \<le> c + b" |
971 |
show "a \<le> b" |
|
25062 | 972 |
proof (rule ccontr) |
63325 | 973 |
assume *: "\<not> ?thesis" |
974 |
then have "b \<le> a" by (simp add: linorder_not_le) |
|
975 |
then have le2: "c + b \<le> c + a" by (rule add_left_mono) |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
976 |
have "a = b" |
63325 | 977 |
apply (insert le1 le2) |
978 |
apply (drule antisym) |
|
979 |
apply simp_all |
|
25062 | 980 |
done |
63325 | 981 |
with * show False |
25062 | 982 |
by (simp add: linorder_not_le [symmetric]) |
983 |
qed |
|
984 |
qed |
|
985 |
||
25267 | 986 |
end |
987 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
988 |
class linordered_ab_group_add = linorder + ordered_ab_group_add |
25267 | 989 |
begin |
25230 | 990 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
991 |
subclass linordered_cancel_ab_semigroup_add .. |
25230 | 992 |
|
63325 | 993 |
lemma equal_neg_zero [simp]: "a = - a \<longleftrightarrow> a = 0" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
994 |
proof |
63325 | 995 |
assume "a = 0" |
996 |
then show "a = - a" by simp |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
997 |
next |
63325 | 998 |
assume A: "a = - a" |
999 |
show "a = 0" |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1000 |
proof (cases "0 \<le> a") |
63325 | 1001 |
case True |
1002 |
with A have "0 \<le> - a" by auto |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1003 |
with le_minus_iff have "a \<le> 0" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1004 |
with True show ?thesis by (auto intro: order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1005 |
next |
63325 | 1006 |
case False |
1007 |
then have B: "a \<le> 0" by auto |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1008 |
with A have "- a \<le> 0" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1009 |
with B show ?thesis by (auto intro: order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1010 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1011 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1012 |
|
63325 | 1013 |
lemma neg_equal_zero [simp]: "- 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
|
1014 |
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
|
1015 |
|
63325 | 1016 |
lemma neg_less_eq_nonneg [simp]: "- a \<le> a \<longleftrightarrow> 0 \<le> a" |
54250 | 1017 |
proof |
63325 | 1018 |
assume *: "- a \<le> a" |
1019 |
show "0 \<le> a" |
|
54250 | 1020 |
proof (rule classical) |
63325 | 1021 |
assume "\<not> ?thesis" |
54250 | 1022 |
then have "a < 0" by auto |
63325 | 1023 |
with * have "- a < 0" by (rule le_less_trans) |
54250 | 1024 |
then show ?thesis by auto |
1025 |
qed |
|
1026 |
next |
|
63325 | 1027 |
assume *: "0 \<le> a" |
1028 |
then have "- a \<le> 0" by (simp add: minus_le_iff) |
|
1029 |
from this * show "- a \<le> a" by (rule order_trans) |
|
54250 | 1030 |
qed |
1031 |
||
63325 | 1032 |
lemma neg_less_pos [simp]: "- a < a \<longleftrightarrow> 0 < a" |
54250 | 1033 |
by (auto simp add: less_le) |
1034 |
||
63325 | 1035 |
lemma less_eq_neg_nonpos [simp]: "a \<le> - a \<longleftrightarrow> a \<le> 0" |
54250 | 1036 |
using neg_less_eq_nonneg [of "- a"] by simp |
1037 |
||
63325 | 1038 |
lemma less_neg_neg [simp]: "a < - a \<longleftrightarrow> a < 0" |
54250 | 1039 |
using neg_less_pos [of "- a"] by simp |
1040 |
||
63325 | 1041 |
lemma double_zero [simp]: "a + a = 0 \<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
|
1042 |
proof |
63325 | 1043 |
assume "a + a = 0" |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1044 |
then have a: "- a = a" by (rule minus_unique) |
35216 | 1045 |
then show "a = 0" by (simp only: neg_equal_zero) |
63325 | 1046 |
next |
1047 |
assume "a = 0" |
|
1048 |
then show "a + a = 0" by simp |
|
1049 |
qed |
|
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1050 |
|
63325 | 1051 |
lemma double_zero_sym [simp]: "0 = a + a \<longleftrightarrow> a = 0" |
1052 |
apply (rule iffI) |
|
1053 |
apply (drule sym) |
|
1054 |
apply simp_all |
|
1055 |
done |
|
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1056 |
|
63325 | 1057 |
lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a" |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1058 |
proof |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1059 |
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
|
1060 |
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
|
1061 |
then have "- a < a" by simp |
54250 | 1062 |
then show "0 < a" by simp |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1063 |
next |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1064 |
assume "0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1065 |
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
|
1066 |
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
|
1067 |
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
|
1068 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1069 |
|
63325 | 1070 |
lemma zero_le_double_add_iff_zero_le_single_add [simp]: "0 \<le> a + a \<longleftrightarrow> 0 \<le> a" |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1071 |
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
|
1072 |
|
63325 | 1073 |
lemma double_add_less_zero_iff_single_add_less_zero [simp]: "a + a < 0 \<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
|
1074 |
proof - |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1075 |
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
|
1076 |
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
|
1077 |
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
|
1078 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1079 |
|
63325 | 1080 |
lemma double_add_le_zero_iff_single_add_le_zero [simp]: "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1081 |
proof - |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1082 |
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
|
1083 |
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
|
1084 |
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
|
1085 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1086 |
|
63325 | 1087 |
lemma minus_max_eq_min: "- max x y = min (- x) (- y)" |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1088 |
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
|
1089 |
|
63325 | 1090 |
lemma minus_min_eq_max: "- min x y = max (- x) (- y)" |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
1091 |
by (auto simp add: max_def min_def) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1092 |
|
25267 | 1093 |
end |
1094 |
||
35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1095 |
class abs = |
61944 | 1096 |
fixes abs :: "'a \<Rightarrow> 'a" ("\<bar>_\<bar>") |
35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1097 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1098 |
class sgn = |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1099 |
fixes sgn :: "'a \<Rightarrow> 'a" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1100 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1101 |
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
|
1102 |
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
|
1103 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1104 |
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
|
1105 |
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
|
1106 |
begin |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1107 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1108 |
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
|
1109 |
by (simp add:sgn_if) |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1110 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
1111 |
end |
14738 | 1112 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1113 |
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
|
1114 |
assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1115 |
and abs_ge_self: "a \<le> \<bar>a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1116 |
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
|
1117 |
and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1118 |
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
|
1119 |
begin |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1120 |
|
25307 | 1121 |
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0" |
1122 |
unfolding neg_le_0_iff_le by simp |
|
1123 |
||
1124 |
lemma abs_of_nonneg [simp]: |
|
63325 | 1125 |
assumes nonneg: "0 \<le> a" |
1126 |
shows "\<bar>a\<bar> = a" |
|
25307 | 1127 |
proof (rule antisym) |
63325 | 1128 |
show "a \<le> \<bar>a\<bar>" by (rule abs_ge_self) |
25307 | 1129 |
from nonneg le_imp_neg_le have "- a \<le> 0" by simp |
1130 |
from this nonneg have "- a \<le> a" by (rule order_trans) |
|
1131 |
then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI) |
|
63325 | 1132 |
qed |
25307 | 1133 |
|
1134 |
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>" |
|
63325 | 1135 |
by (rule antisym) (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"]) |
25307 | 1136 |
|
1137 |
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0" |
|
1138 |
proof - |
|
1139 |
have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0" |
|
1140 |
proof (rule antisym) |
|
1141 |
assume zero: "\<bar>a\<bar> = 0" |
|
1142 |
with abs_ge_self show "a \<le> 0" by auto |
|
1143 |
from zero have "\<bar>-a\<bar> = 0" by simp |
|
36302 | 1144 |
with abs_ge_self [of "- a"] have "- a \<le> 0" by auto |
25307 | 1145 |
with neg_le_0_iff_le show "0 \<le> a" by auto |
1146 |
qed |
|
1147 |
then show ?thesis by auto |
|
1148 |
qed |
|
1149 |
||
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1150 |
lemma abs_zero [simp]: "\<bar>0\<bar> = 0" |
63325 | 1151 |
by simp |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1152 |
|
54148 | 1153 |
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
|
1154 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1155 |
have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac) |
63325 | 1156 |
then show ?thesis by simp |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1157 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1158 |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
1159 |
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1160 |
proof |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1161 |
assume "\<bar>a\<bar> \<le> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1162 |
then have "\<bar>a\<bar> = 0" by (rule antisym) simp |
63325 | 1163 |
then show "a = 0" by simp |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1164 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1165 |
assume "a = 0" |
63325 | 1166 |
then show "\<bar>a\<bar> \<le> 0" by simp |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1167 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1168 |
|
62379
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1169 |
lemma abs_le_self_iff [simp]: "\<bar>a\<bar> \<le> a \<longleftrightarrow> 0 \<le> a" |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1170 |
proof - |
63325 | 1171 |
have "0 \<le> \<bar>a\<bar>" |
62379
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1172 |
using abs_ge_zero by blast |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1173 |
then have "\<bar>a\<bar> \<le> a \<Longrightarrow> 0 \<le> a" |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1174 |
using order.trans by blast |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1175 |
then show ?thesis |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1176 |
using abs_of_nonneg eq_refl by blast |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1177 |
qed |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62378
diff
changeset
|
1178 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1179 |
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0" |
63325 | 1180 |
by (simp add: less_le) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1181 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1182 |
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1183 |
proof - |
63325 | 1184 |
have "x \<le> y \<Longrightarrow> \<not> y < x" for x y by auto |
1185 |
then show ?thesis by simp |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1186 |
qed |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1187 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1188 |
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1189 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1190 |
have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1191 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1192 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1193 |
|
63325 | 1194 |
lemma abs_minus_commute: "\<bar>a - b\<bar> = \<bar>b - a\<bar>" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1195 |
proof - |
63325 | 1196 |
have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" |
1197 |
by (simp only: abs_minus_cancel) |
|
1198 |
also have "\<dots> = \<bar>b - a\<bar>" by simp |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1199 |
finally show ?thesis . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1200 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1201 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1202 |
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a" |
63325 | 1203 |
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
|
1204 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1205 |
lemma abs_of_nonpos [simp]: |
63325 | 1206 |
assumes "a \<le> 0" |
1207 |
shows "\<bar>a\<bar> = - a" |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1208 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1209 |
let ?b = "- a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1210 |
have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)" |
63325 | 1211 |
unfolding abs_minus_cancel [of ?b] |
1212 |
unfolding neg_le_0_iff_le [of ?b] |
|
1213 |
unfolding minus_minus by (erule abs_of_nonneg) |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1214 |
then show ?thesis using assms by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1215 |
qed |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62348
diff
changeset
|
1216 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1217 |
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a" |
63325 | 1218 |
by (rule abs_of_nonpos) (rule less_imp_le) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1219 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1220 |
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b" |
63325 | 1221 |
using abs_ge_self by (blast intro: order_trans) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1222 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1223 |
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b" |
63325 | 1224 |
using abs_le_D1 [of "- a"] by simp |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1225 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1226 |
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b" |
63325 | 1227 |
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
|
1228 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1229 |
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>" |
36302 | 1230 |
proof - |
1231 |
have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54148
diff
changeset
|
1232 |
by (simp add: algebra_simps) |
36302 | 1233 |
then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>" |
1234 |
by (simp add: abs_triangle_ineq) |
|
1235 |
then show ?thesis |
|
1236 |
by (simp add: algebra_simps) |
|
1237 |
qed |
|
1238 |
||
1239 |
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>" |
|
1240 |
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
|
1241 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1242 |
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>" |
36302 | 1243 |
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
|
1244 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1245 |
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
|
1246 |
proof - |
63325 | 1247 |
have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" |
1248 |
by (simp add: algebra_simps) |
|
1249 |
also have "\<dots> \<le> \<bar>a\<bar> + \<bar>- b\<bar>" |
|
1250 |
by (rule abs_triangle_ineq) |
|
29667 | 1251 |
finally show ?thesis by simp |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1252 |
qed |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1253 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1254 |
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
|
1255 |
proof - |
63325 | 1256 |
have "\<bar>a + b - (c + d)\<bar> = \<bar>(a - c) + (b - d)\<bar>" |
1257 |
by (simp add: algebra_simps) |
|
1258 |
also have "\<dots> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>" |
|
1259 |
by (rule abs_triangle_ineq) |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1260 |
finally show ?thesis . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1261 |
qed |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1262 |
|
63325 | 1263 |
lemma abs_add_abs [simp]: "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" |
1264 |
(is "?L = ?R") |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1265 |
proof (rule antisym) |
63325 | 1266 |
show "?L \<ge> ?R" by (rule abs_ge_self) |
1267 |
have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by (rule abs_triangle_ineq) |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1268 |
also have "\<dots> = ?R" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1269 |
finally show "?L \<le> ?R" . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1270 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1271 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1272 |
end |
14738 | 1273 |
|
60762 | 1274 |
lemma dense_eq0_I: |
1275 |
fixes x::"'a::{dense_linorder,ordered_ab_group_add_abs}" |
|
63325 | 1276 |
shows "(\<And>e. 0 < e \<Longrightarrow> \<bar>x\<bar> \<le> e) \<Longrightarrow> x = 0" |
1277 |
apply (cases "\<bar>x\<bar> = 0") |
|
1278 |
apply simp |
|
60762 | 1279 |
apply (simp only: zero_less_abs_iff [symmetric]) |
1280 |
apply (drule dense) |
|
1281 |
apply (auto simp add: not_less [symmetric]) |
|
1282 |
done |
|
1283 |
||
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
1284 |
hide_fact (open) ab_diff_conv_add_uminus add_0 mult_1 ab_left_minus |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
1285 |
|
63325 | 1286 |
lemmas add_0 = add_0_left (* FIXME duplicate *) |
1287 |
lemmas mult_1 = mult_1_left (* FIXME duplicate *) |
|
1288 |
lemmas ab_left_minus = left_minus (* FIXME duplicate *) |
|
1289 |
lemmas diff_diff_eq = diff_diff_add (* FIXME duplicate *) |
|
1290 |
||
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59559
diff
changeset
|
1291 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1292 |
subsection \<open>Canonically ordered monoids\<close> |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1293 |
|
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1294 |
text \<open>Canonically ordered monoids are never groups.\<close> |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1295 |
|
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1296 |
class canonically_ordered_monoid_add = comm_monoid_add + order + |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1297 |
assumes le_iff_add: "a \<le> b \<longleftrightarrow> (\<exists>c. b = a + c)" |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1298 |
begin |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1299 |
|
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1300 |
lemma zero_le[simp]: "0 \<le> x" |
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1301 |
by (auto simp: le_iff_add) |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1302 |
|
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1303 |
lemma le_zero_eq[simp]: "n \<le> 0 \<longleftrightarrow> n = 0" |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1304 |
by (auto intro: antisym) |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1305 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1306 |
lemma not_less_zero[simp]: "\<not> n < 0" |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1307 |
by (auto simp: less_le) |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1308 |
|
63325 | 1309 |
lemma zero_less_iff_neq_zero: "0 < n \<longleftrightarrow> n \<noteq> 0" |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1310 |
by (auto simp: less_le) |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1311 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1312 |
text \<open>This theorem is useful with \<open>blast\<close>\<close> |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1313 |
lemma gr_zeroI: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n" |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1314 |
by (rule zero_less_iff_neq_zero[THEN iffD2]) iprover |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1315 |
|
63325 | 1316 |
lemma not_gr_zero[simp]: "\<not> 0 < n \<longleftrightarrow> n = 0" |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1317 |
by (simp add: zero_less_iff_neq_zero) |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1318 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1319 |
subclass ordered_comm_monoid_add |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1320 |
proof qed (auto simp: le_iff_add add_ac) |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1321 |
|
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1322 |
lemma add_eq_0_iff_both_eq_0: "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0" |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1323 |
by (intro add_nonneg_eq_0_iff zero_le) |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1324 |
|
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1325 |
lemma gr_implies_not_zero: "m < n \<Longrightarrow> n \<noteq> 0" |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1326 |
using add_eq_0_iff_both_eq_0[of m] by (auto simp: le_iff_add less_le) |
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1327 |
|
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1328 |
lemmas zero_order = zero_le le_zero_eq not_less_zero zero_less_iff_neq_zero not_gr_zero |
63145 | 1329 |
\<comment> \<open>This should be attributed with \<open>[iff]\<close>, but then \<open>blast\<close> fails in \<open>Set\<close>.\<close> |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62377
diff
changeset
|
1330 |
|
62377
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1331 |
end |
ace69956d018
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents:
62376
diff
changeset
|
1332 |
|
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|
1333 |
class ordered_cancel_comm_monoid_diff = |
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changeset
|
1334 |
canonically_ordered_monoid_add + comm_monoid_diff + ordered_ab_semigroup_add_imp_le |
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|
1335 |
begin |
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diff
changeset
|
1336 |
|
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diff
changeset
|
1337 |
context |
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diff
changeset
|
1338 |
fixes a b |
63325 | 1339 |
assumes le: "a \<le> b" |
62377
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diff
changeset
|
1340 |
begin |
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changeset
|
1341 |
|
63325 | 1342 |
lemma add_diff_inverse: "a + (b - a) = b" |
1343 |
using le by (auto simp add: le_iff_add) |
|
62377
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parents:
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diff
changeset
|
1344 |
|
63325 | 1345 |
lemma add_diff_assoc: "c + (b - a) = c + b - a" |
1346 |
using le by (auto simp add: le_iff_add add.left_commute [of c]) |
|
62377
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diff
changeset
|
1347 |
|
63325 | 1348 |
lemma add_diff_assoc2: "b - a + c = b + c - a" |
1349 |
using le by (auto simp add: le_iff_add add.assoc) |
|
62377
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diff
changeset
|
1350 |
|
63325 | 1351 |
lemma diff_add_assoc: "c + b - a = c + (b - a)" |
1352 |
using le by (simp add: add.commute add_diff_assoc) |
|
62377
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diff
changeset
|
1353 |
|
63325 | 1354 |
lemma diff_add_assoc2: "b + c - a = b - a + c" |
1355 |
using le by (simp add: add.commute add_diff_assoc) |
|
62377
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diff
changeset
|
1356 |
|
63325 | 1357 |
lemma diff_diff_right: "c - (b - a) = c + a - b" |
62377
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diff
changeset
|
1358 |
by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add.commute) |
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changeset
|
1359 |
|
63325 | 1360 |
lemma diff_add: "b - a + a = b" |
62377
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changeset
|
1361 |
by (simp add: add.commute add_diff_inverse) |
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changeset
|
1362 |
|
63325 | 1363 |
lemma le_add_diff: "c \<le> b + c - a" |
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changeset
|
1364 |
by (auto simp add: add.commute diff_add_assoc2 le_iff_add) |
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changeset
|
1365 |
|
63325 | 1366 |
lemma le_imp_diff_is_add: "a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a" |
62377
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changeset
|
1367 |
by (auto simp add: add.commute add_diff_inverse) |
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changeset
|
1368 |
|
63325 | 1369 |
lemma le_diff_conv2: "c \<le> b - a \<longleftrightarrow> c + a \<le> b" |
1370 |
(is "?P \<longleftrightarrow> ?Q") |
|
62377
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changeset
|
1371 |
proof |
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changeset
|
1372 |
assume ?P |
63325 | 1373 |
then have "c + a \<le> b - a + a" |
1374 |
by (rule add_right_mono) |
|
1375 |
then show ?Q |
|
1376 |
by (simp add: add_diff_inverse add.commute) |
|
62377
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changeset
|
1377 |
next |
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changeset
|
1378 |
assume ?Q |
63325 | 1379 |
then have "a + c \<le> a + (b - a)" |
1380 |
by (simp add: add_diff_inverse add.commute) |
|
62377
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changeset
|
1381 |
then show ?P by simp |
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changeset
|
1382 |
qed |
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changeset
|
1383 |
|
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parents:
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changeset
|
1384 |
end |
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parents:
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changeset
|
1385 |
|
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parents:
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changeset
|
1386 |
end |
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parents:
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changeset
|
1387 |
|
63325 | 1388 |
|
60758 | 1389 |
subsection \<open>Tools setup\<close> |
25090 | 1390 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
1391 |
lemma add_mono_thms_linordered_semiring: |
61076 | 1392 |
fixes i j k :: "'a::ordered_ab_semigroup_add" |
25077 | 1393 |
shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
1394 |
and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
|
1395 |
and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l" |
|
1396 |
and "i = j \<and> k = l \<Longrightarrow> i + k = j + l" |
|
63325 | 1397 |
by (rule add_mono, clarify+)+ |
25077 | 1398 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
1399 |
lemma add_mono_thms_linordered_field: |
61076 | 1400 |
fixes i j k :: "'a::ordered_cancel_ab_semigroup_add" |
25077 | 1401 |
shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l" |
1402 |
and "i = j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
1403 |
and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l" |
|
1404 |
and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
1405 |
and "i < j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
63325 | 1406 |
by (auto intro: add_strict_right_mono add_strict_left_mono |
1407 |
add_less_le_mono add_le_less_mono add_strict_mono) |
|
25077 | 1408 |
|
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
52289
diff
changeset
|
1409 |
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
|
1410 |
code_module Groups \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith |
33364 | 1411 |
|
14738 | 1412 |
end |