author | blanchet |
Thu, 18 Feb 2010 18:48:07 +0100 | |
changeset 35220 | 2bcdae5f4fdb |
parent 35092 | cfe605c54e50 |
child 35216 | 7641e8d831d2 |
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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header {* Groups, also combined with orderings *} |
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theory Groups |
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imports Orderings |
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uses "~~/src/Provers/Arith/abel_cancel.ML" |
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begin |
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text {* |
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The theory of partially ordered groups is taken from the books: |
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\begin{itemize} |
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\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
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\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
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\end{itemize} |
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Most of the used notions can also be looked up in |
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\begin{itemize} |
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\item \url{http://www.mathworld.com} by Eric Weisstein et. al. |
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\item \emph{Algebra I} by van der Waerden, Springer. |
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\end{itemize} |
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*} |
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ML {* |
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structure Algebra_Simps = Named_Thms( |
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val name = "algebra_simps" |
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val description = "algebra simplification rules" |
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) |
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*} |
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setup Algebra_Simps.setup |
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text{* The rewrites accumulated in @{text algebra_simps} deal with the |
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classical algebraic structures of groups, rings and family. They simplify |
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terms by multiplying everything out (in case of a ring) and bringing sums and |
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products into a canonical form (by ordered rewriting). As a result it decides |
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group and ring equalities but also helps with inequalities. |
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Of course it also works for fields, but it knows nothing about multiplicative |
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inverses or division. This is catered for by @{text field_simps}. *} |
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subsection {* Semigroups and Monoids *} |
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class semigroup_add = plus + |
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assumes add_assoc [algebra_simps]: "(a + b) + c = a + (b + c)" |
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|
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sublocale semigroup_add < plus!: semigroup plus proof |
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qed (fact add_assoc) |
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class ab_semigroup_add = semigroup_add + |
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assumes add_commute [algebra_simps]: "a + b = b + a" |
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sublocale ab_semigroup_add < plus!: abel_semigroup plus proof |
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qed (fact add_commute) |
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context ab_semigroup_add |
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begin |
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|
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lemmas add_left_commute [algebra_simps] = plus.left_commute |
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theorems add_ac = add_assoc add_commute add_left_commute |
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65 |
end |
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theorems 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]: "(a * b) * c = a * (b * c)" |
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sublocale semigroup_mult < times!: semigroup times proof |
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qed (fact mult_assoc) |
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class ab_semigroup_mult = semigroup_mult + |
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assumes mult_commute [algebra_simps]: "a * b = b * a" |
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|
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sublocale ab_semigroup_mult < times!: abel_semigroup times proof |
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qed (fact mult_commute) |
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context ab_semigroup_mult |
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begin |
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lemmas mult_left_commute [algebra_simps] = times.left_commute |
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theorems mult_ac = mult_assoc mult_commute mult_left_commute |
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end |
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14738 | 89 |
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theorems mult_ac = mult_assoc mult_commute mult_left_commute |
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class ab_semigroup_idem_mult = ab_semigroup_mult + |
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assumes mult_idem: "x * x = x" |
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|
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sublocale ab_semigroup_idem_mult < times!: semilattice times proof |
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qed (fact mult_idem) |
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context ab_semigroup_idem_mult |
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begin |
100 |
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lemmas mult_left_idem = times.left_idem |
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end |
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class monoid_add = zero + semigroup_add + |
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assumes add_0_left [simp]: "0 + a = a" |
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and add_0_right [simp]: "a + 0 = a" |
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lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0" |
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by (rule eq_commute) |
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class comm_monoid_add = zero + ab_semigroup_add + |
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assumes add_0: "0 + a = a" |
114 |
begin |
|
23085 | 115 |
|
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subclass monoid_add |
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proof qed (insert add_0, simp_all add: add_commute) |
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|
119 |
end |
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class monoid_mult = one + semigroup_mult + |
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assumes mult_1_left [simp]: "1 * a = a" |
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assumes mult_1_right [simp]: "a * 1 = a" |
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lemma one_reorient: "1 = x \<longleftrightarrow> x = 1" |
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by (rule eq_commute) |
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class comm_monoid_mult = one + ab_semigroup_mult + |
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assumes mult_1: "1 * a = a" |
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begin |
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|
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subclass monoid_mult |
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proof qed (insert mult_1, simp_all add: mult_commute) |
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end |
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class cancel_semigroup_add = semigroup_add + |
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assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c" |
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assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c" |
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begin |
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|
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lemma add_left_cancel [simp]: |
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"a + b = a + c \<longleftrightarrow> b = c" |
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by (blast dest: add_left_imp_eq) |
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|
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lemma add_right_cancel [simp]: |
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"b + a = c + a \<longleftrightarrow> b = c" |
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by (blast dest: add_right_imp_eq) |
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|
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end |
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class cancel_ab_semigroup_add = ab_semigroup_add + |
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assumes add_imp_eq: "a + b = a + c \<Longrightarrow> b = c" |
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begin |
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|
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subclass cancel_semigroup_add |
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proof |
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fix a b c :: 'a |
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assume "a + b = a + c" |
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then show "b = c" by (rule add_imp_eq) |
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next |
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fix a b c :: 'a |
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assume "b + a = c + a" |
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then have "a + b = a + c" by (simp only: add_commute) |
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then show "b = c" by (rule add_imp_eq) |
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qed |
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25267 | 168 |
end |
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class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add |
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||
172 |
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subsection {* Groups *} |
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class group_add = minus + uminus + monoid_add + |
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assumes left_minus [simp]: "- a + a = 0" |
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assumes diff_minus: "a - b = a + (- b)" |
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begin |
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23085 | 179 |
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lemma minus_unique: |
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assumes "a + b = 0" shows "- a = b" |
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proof - |
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have "- a = - a + (a + b)" using assms by simp |
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also have "\<dots> = b" by (simp add: add_assoc [symmetric]) |
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finally show ?thesis . |
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qed |
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|
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lemmas equals_zero_I = minus_unique (* legacy name *) |
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lemma minus_zero [simp]: "- 0 = 0" |
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proof - |
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have "0 + 0 = 0" by (rule add_0_right) |
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thus "- 0 = 0" by (rule minus_unique) |
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qed |
195 |
||
25062 | 196 |
lemma minus_minus [simp]: "- (- a) = a" |
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proof - |
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have "- a + a = 0" by (rule left_minus) |
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thus "- (- a) = a" by (rule minus_unique) |
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qed |
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|
25062 | 202 |
lemma right_minus [simp]: "a + - a = 0" |
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proof - |
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have "a + - a = - (- a) + - a" by simp |
205 |
also have "\<dots> = 0" by (rule left_minus) |
|
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finally show ?thesis . |
207 |
qed |
|
208 |
||
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lemma minus_add_cancel: "- a + (a + b) = b" |
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by (simp add: add_assoc [symmetric]) |
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|
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lemma add_minus_cancel: "a + (- a + b) = b" |
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by (simp add: add_assoc [symmetric]) |
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|
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lemma minus_add: "- (a + b) = - b + - a" |
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proof - |
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have "(a + b) + (- b + - a) = 0" |
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by (simp add: add_assoc add_minus_cancel) |
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thus "- (a + b) = - b + - a" |
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by (rule minus_unique) |
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qed |
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222 |
|
25062 | 223 |
lemma right_minus_eq: "a - b = 0 \<longleftrightarrow> a = b" |
14738 | 224 |
proof |
23085 | 225 |
assume "a - b = 0" |
226 |
have "a = (a - b) + b" by (simp add:diff_minus add_assoc) |
|
227 |
also have "\<dots> = b" using `a - b = 0` by simp |
|
228 |
finally show "a = b" . |
|
14738 | 229 |
next |
23085 | 230 |
assume "a = b" thus "a - b = 0" by (simp add: diff_minus) |
14738 | 231 |
qed |
232 |
||
25062 | 233 |
lemma diff_self [simp]: "a - a = 0" |
29667 | 234 |
by (simp add: diff_minus) |
14738 | 235 |
|
25062 | 236 |
lemma diff_0 [simp]: "0 - a = - a" |
29667 | 237 |
by (simp add: diff_minus) |
14738 | 238 |
|
25062 | 239 |
lemma diff_0_right [simp]: "a - 0 = a" |
29667 | 240 |
by (simp add: diff_minus) |
14738 | 241 |
|
25062 | 242 |
lemma diff_minus_eq_add [simp]: "a - - b = a + b" |
29667 | 243 |
by (simp add: diff_minus) |
14738 | 244 |
|
25062 | 245 |
lemma neg_equal_iff_equal [simp]: |
246 |
"- a = - b \<longleftrightarrow> a = b" |
|
14738 | 247 |
proof |
248 |
assume "- a = - b" |
|
29667 | 249 |
hence "- (- a) = - (- b)" by simp |
25062 | 250 |
thus "a = b" by simp |
14738 | 251 |
next |
25062 | 252 |
assume "a = b" |
253 |
thus "- a = - b" by simp |
|
14738 | 254 |
qed |
255 |
||
25062 | 256 |
lemma neg_equal_0_iff_equal [simp]: |
257 |
"- a = 0 \<longleftrightarrow> a = 0" |
|
29667 | 258 |
by (subst neg_equal_iff_equal [symmetric], simp) |
14738 | 259 |
|
25062 | 260 |
lemma neg_0_equal_iff_equal [simp]: |
261 |
"0 = - a \<longleftrightarrow> 0 = a" |
|
29667 | 262 |
by (subst neg_equal_iff_equal [symmetric], simp) |
14738 | 263 |
|
264 |
text{*The next two equations can make the simplifier loop!*} |
|
265 |
||
25062 | 266 |
lemma equation_minus_iff: |
267 |
"a = - b \<longleftrightarrow> b = - a" |
|
14738 | 268 |
proof - |
25062 | 269 |
have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal) |
270 |
thus ?thesis by (simp add: eq_commute) |
|
271 |
qed |
|
272 |
||
273 |
lemma minus_equation_iff: |
|
274 |
"- a = b \<longleftrightarrow> - b = a" |
|
275 |
proof - |
|
276 |
have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal) |
|
14738 | 277 |
thus ?thesis by (simp add: eq_commute) |
278 |
qed |
|
279 |
||
28130
32b4185bfdc7
move diff_add_cancel, add_diff_cancel from class ab_group_add to group_add
huffman
parents:
27516
diff
changeset
|
280 |
lemma diff_add_cancel: "a - b + b = a" |
29667 | 281 |
by (simp add: diff_minus add_assoc) |
28130
32b4185bfdc7
move diff_add_cancel, add_diff_cancel from class ab_group_add to group_add
huffman
parents:
27516
diff
changeset
|
282 |
|
32b4185bfdc7
move diff_add_cancel, add_diff_cancel from class ab_group_add to group_add
huffman
parents:
27516
diff
changeset
|
283 |
lemma add_diff_cancel: "a + b - b = a" |
29667 | 284 |
by (simp add: diff_minus add_assoc) |
285 |
||
286 |
declare diff_minus[symmetric, algebra_simps] |
|
28130
32b4185bfdc7
move diff_add_cancel, add_diff_cancel from class ab_group_add to group_add
huffman
parents:
27516
diff
changeset
|
287 |
|
29914
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
288 |
lemma eq_neg_iff_add_eq_0: "a = - b \<longleftrightarrow> a + b = 0" |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
289 |
proof |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
290 |
assume "a = - b" then show "a + b = 0" by simp |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
291 |
next |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
292 |
assume "a + b = 0" |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
293 |
moreover have "a + (b + - b) = (a + b) + - b" |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
294 |
by (simp only: add_assoc) |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
295 |
ultimately show "a = - b" by simp |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
296 |
qed |
c9ced4f54e82
generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents:
29904
diff
changeset
|
297 |
|
25062 | 298 |
end |
299 |
||
25762 | 300 |
class ab_group_add = minus + uminus + comm_monoid_add + |
25062 | 301 |
assumes ab_left_minus: "- a + a = 0" |
302 |
assumes ab_diff_minus: "a - b = a + (- b)" |
|
25267 | 303 |
begin |
25062 | 304 |
|
25267 | 305 |
subclass group_add |
28823 | 306 |
proof qed (simp_all add: ab_left_minus ab_diff_minus) |
25062 | 307 |
|
29904 | 308 |
subclass cancel_comm_monoid_add |
28823 | 309 |
proof |
25062 | 310 |
fix a b c :: 'a |
311 |
assume "a + b = a + c" |
|
312 |
then have "- a + a + b = - a + a + c" |
|
313 |
unfolding add_assoc by simp |
|
314 |
then show "b = c" by simp |
|
315 |
qed |
|
316 |
||
29667 | 317 |
lemma uminus_add_conv_diff[algebra_simps]: |
25062 | 318 |
"- a + b = b - a" |
29667 | 319 |
by (simp add:diff_minus add_commute) |
25062 | 320 |
|
321 |
lemma minus_add_distrib [simp]: |
|
322 |
"- (a + b) = - a + - b" |
|
34146
14595e0c27e8
rename equals_zero_I to minus_unique (keep old name too)
huffman
parents:
33364
diff
changeset
|
323 |
by (rule minus_unique) (simp add: add_ac) |
25062 | 324 |
|
325 |
lemma minus_diff_eq [simp]: |
|
326 |
"- (a - b) = b - a" |
|
29667 | 327 |
by (simp add: diff_minus add_commute) |
25077 | 328 |
|
29667 | 329 |
lemma add_diff_eq[algebra_simps]: "a + (b - c) = (a + b) - c" |
330 |
by (simp add: diff_minus add_ac) |
|
25077 | 331 |
|
29667 | 332 |
lemma diff_add_eq[algebra_simps]: "(a - b) + c = (a + c) - b" |
333 |
by (simp add: diff_minus add_ac) |
|
25077 | 334 |
|
29667 | 335 |
lemma diff_eq_eq[algebra_simps]: "a - b = c \<longleftrightarrow> a = c + b" |
336 |
by (auto simp add: diff_minus add_assoc) |
|
25077 | 337 |
|
29667 | 338 |
lemma eq_diff_eq[algebra_simps]: "a = c - b \<longleftrightarrow> a + b = c" |
339 |
by (auto simp add: diff_minus add_assoc) |
|
25077 | 340 |
|
29667 | 341 |
lemma diff_diff_eq[algebra_simps]: "(a - b) - c = a - (b + c)" |
342 |
by (simp add: diff_minus add_ac) |
|
25077 | 343 |
|
29667 | 344 |
lemma diff_diff_eq2[algebra_simps]: "a - (b - c) = (a + c) - b" |
345 |
by (simp add: diff_minus add_ac) |
|
25077 | 346 |
|
347 |
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0" |
|
29667 | 348 |
by (simp add: algebra_simps) |
25077 | 349 |
|
30629 | 350 |
lemma diff_eq_0_iff_eq [simp, noatp]: "a - b = 0 \<longleftrightarrow> a = b" |
351 |
by (simp add: algebra_simps) |
|
352 |
||
25062 | 353 |
end |
14738 | 354 |
|
355 |
subsection {* (Partially) Ordered Groups *} |
|
356 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
357 |
class ordered_ab_semigroup_add = order + ab_semigroup_add + |
25062 | 358 |
assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b" |
359 |
begin |
|
24380
c215e256beca
moved ordered_ab_semigroup_add to OrderedGroup.thy
haftmann
parents:
24286
diff
changeset
|
360 |
|
25062 | 361 |
lemma add_right_mono: |
362 |
"a \<le> b \<Longrightarrow> a + c \<le> b + c" |
|
29667 | 363 |
by (simp add: add_commute [of _ c] add_left_mono) |
14738 | 364 |
|
365 |
text {* non-strict, in both arguments *} |
|
366 |
lemma add_mono: |
|
25062 | 367 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d" |
14738 | 368 |
apply (erule add_right_mono [THEN order_trans]) |
369 |
apply (simp add: add_commute add_left_mono) |
|
370 |
done |
|
371 |
||
25062 | 372 |
end |
373 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
374 |
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
|
375 |
ordered_ab_semigroup_add + cancel_ab_semigroup_add |
25062 | 376 |
begin |
377 |
||
14738 | 378 |
lemma add_strict_left_mono: |
25062 | 379 |
"a < b \<Longrightarrow> c + a < c + b" |
29667 | 380 |
by (auto simp add: less_le add_left_mono) |
14738 | 381 |
|
382 |
lemma add_strict_right_mono: |
|
25062 | 383 |
"a < b \<Longrightarrow> a + c < b + c" |
29667 | 384 |
by (simp add: add_commute [of _ c] add_strict_left_mono) |
14738 | 385 |
|
386 |
text{*Strict monotonicity in both arguments*} |
|
25062 | 387 |
lemma add_strict_mono: |
388 |
"a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" |
|
389 |
apply (erule add_strict_right_mono [THEN less_trans]) |
|
14738 | 390 |
apply (erule add_strict_left_mono) |
391 |
done |
|
392 |
||
393 |
lemma add_less_le_mono: |
|
25062 | 394 |
"a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d" |
395 |
apply (erule add_strict_right_mono [THEN less_le_trans]) |
|
396 |
apply (erule add_left_mono) |
|
14738 | 397 |
done |
398 |
||
399 |
lemma add_le_less_mono: |
|
25062 | 400 |
"a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" |
401 |
apply (erule add_right_mono [THEN le_less_trans]) |
|
14738 | 402 |
apply (erule add_strict_left_mono) |
403 |
done |
|
404 |
||
25062 | 405 |
end |
406 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
407 |
class ordered_ab_semigroup_add_imp_le = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
408 |
ordered_cancel_ab_semigroup_add + |
25062 | 409 |
assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b" |
410 |
begin |
|
411 |
||
14738 | 412 |
lemma add_less_imp_less_left: |
29667 | 413 |
assumes less: "c + a < c + b" shows "a < b" |
14738 | 414 |
proof - |
415 |
from less have le: "c + a <= c + b" by (simp add: order_le_less) |
|
416 |
have "a <= b" |
|
417 |
apply (insert le) |
|
418 |
apply (drule add_le_imp_le_left) |
|
419 |
by (insert le, drule add_le_imp_le_left, assumption) |
|
420 |
moreover have "a \<noteq> b" |
|
421 |
proof (rule ccontr) |
|
422 |
assume "~(a \<noteq> b)" |
|
423 |
then have "a = b" by simp |
|
424 |
then have "c + a = c + b" by simp |
|
425 |
with less show "False"by simp |
|
426 |
qed |
|
427 |
ultimately show "a < b" by (simp add: order_le_less) |
|
428 |
qed |
|
429 |
||
430 |
lemma add_less_imp_less_right: |
|
25062 | 431 |
"a + c < b + c \<Longrightarrow> a < b" |
14738 | 432 |
apply (rule add_less_imp_less_left [of c]) |
433 |
apply (simp add: add_commute) |
|
434 |
done |
|
435 |
||
436 |
lemma add_less_cancel_left [simp]: |
|
25062 | 437 |
"c + a < c + b \<longleftrightarrow> a < b" |
29667 | 438 |
by (blast intro: add_less_imp_less_left add_strict_left_mono) |
14738 | 439 |
|
440 |
lemma add_less_cancel_right [simp]: |
|
25062 | 441 |
"a + c < b + c \<longleftrightarrow> a < b" |
29667 | 442 |
by (blast intro: add_less_imp_less_right add_strict_right_mono) |
14738 | 443 |
|
444 |
lemma add_le_cancel_left [simp]: |
|
25062 | 445 |
"c + a \<le> c + b \<longleftrightarrow> a \<le> b" |
29667 | 446 |
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) |
14738 | 447 |
|
448 |
lemma add_le_cancel_right [simp]: |
|
25062 | 449 |
"a + c \<le> b + c \<longleftrightarrow> a \<le> b" |
29667 | 450 |
by (simp add: add_commute [of a c] add_commute [of b c]) |
14738 | 451 |
|
452 |
lemma add_le_imp_le_right: |
|
25062 | 453 |
"a + c \<le> b + c \<Longrightarrow> a \<le> b" |
29667 | 454 |
by simp |
25062 | 455 |
|
25077 | 456 |
lemma max_add_distrib_left: |
457 |
"max x y + z = max (x + z) (y + z)" |
|
458 |
unfolding max_def by auto |
|
459 |
||
460 |
lemma min_add_distrib_left: |
|
461 |
"min x y + z = min (x + z) (y + z)" |
|
462 |
unfolding min_def by auto |
|
463 |
||
25062 | 464 |
end |
465 |
||
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
466 |
subsection {* Support for reasoning about signs *} |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
467 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
468 |
class ordered_comm_monoid_add = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
469 |
ordered_cancel_ab_semigroup_add + comm_monoid_add |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
470 |
begin |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
471 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
472 |
lemma add_pos_nonneg: |
29667 | 473 |
assumes "0 < a" and "0 \<le> b" shows "0 < a + b" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
474 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
475 |
have "0 + 0 < a + b" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
476 |
using assms by (rule add_less_le_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
477 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
478 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
479 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
480 |
lemma add_pos_pos: |
29667 | 481 |
assumes "0 < a" and "0 < b" shows "0 < a + b" |
482 |
by (rule add_pos_nonneg) (insert assms, auto) |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
483 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
484 |
lemma add_nonneg_pos: |
29667 | 485 |
assumes "0 \<le> a" and "0 < b" shows "0 < a + b" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
486 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
487 |
have "0 + 0 < a + b" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
488 |
using assms by (rule add_le_less_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
489 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
490 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
491 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
492 |
lemma add_nonneg_nonneg: |
29667 | 493 |
assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
494 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
495 |
have "0 + 0 \<le> a + b" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
496 |
using assms by (rule add_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
497 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
498 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
499 |
|
30691 | 500 |
lemma add_neg_nonpos: |
29667 | 501 |
assumes "a < 0" and "b \<le> 0" shows "a + b < 0" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
502 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
503 |
have "a + b < 0 + 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
504 |
using assms by (rule add_less_le_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
505 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
506 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
507 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
508 |
lemma add_neg_neg: |
29667 | 509 |
assumes "a < 0" and "b < 0" shows "a + b < 0" |
510 |
by (rule add_neg_nonpos) (insert assms, auto) |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
511 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
512 |
lemma add_nonpos_neg: |
29667 | 513 |
assumes "a \<le> 0" and "b < 0" shows "a + b < 0" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
514 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
515 |
have "a + b < 0 + 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
516 |
using assms by (rule add_le_less_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
517 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
518 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
519 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
520 |
lemma add_nonpos_nonpos: |
29667 | 521 |
assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
522 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
523 |
have "a + b \<le> 0 + 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
524 |
using assms by (rule add_mono) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
525 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
526 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
527 |
|
30691 | 528 |
lemmas add_sign_intros = |
529 |
add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg |
|
530 |
add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos |
|
531 |
||
29886 | 532 |
lemma add_nonneg_eq_0_iff: |
533 |
assumes x: "0 \<le> x" and y: "0 \<le> y" |
|
534 |
shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0" |
|
535 |
proof (intro iffI conjI) |
|
536 |
have "x = x + 0" by simp |
|
537 |
also have "x + 0 \<le> x + y" using y by (rule add_left_mono) |
|
538 |
also assume "x + y = 0" |
|
539 |
also have "0 \<le> x" using x . |
|
540 |
finally show "x = 0" . |
|
541 |
next |
|
542 |
have "y = 0 + y" by simp |
|
543 |
also have "0 + y \<le> x + y" using x by (rule add_right_mono) |
|
544 |
also assume "x + y = 0" |
|
545 |
also have "0 \<le> y" using y . |
|
546 |
finally show "y = 0" . |
|
547 |
next |
|
548 |
assume "x = 0 \<and> y = 0" |
|
549 |
then show "x + y = 0" by simp |
|
550 |
qed |
|
551 |
||
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
552 |
end |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
553 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
554 |
class ordered_ab_group_add = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
555 |
ab_group_add + ordered_ab_semigroup_add |
25062 | 556 |
begin |
557 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
558 |
subclass ordered_cancel_ab_semigroup_add .. |
25062 | 559 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
560 |
subclass ordered_ab_semigroup_add_imp_le |
28823 | 561 |
proof |
25062 | 562 |
fix a b c :: 'a |
563 |
assume "c + a \<le> c + b" |
|
564 |
hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono) |
|
565 |
hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc) |
|
566 |
thus "a \<le> b" by simp |
|
567 |
qed |
|
568 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
569 |
subclass ordered_comm_monoid_add .. |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
570 |
|
25077 | 571 |
lemma max_diff_distrib_left: |
572 |
shows "max x y - z = max (x - z) (y - z)" |
|
29667 | 573 |
by (simp add: diff_minus, rule max_add_distrib_left) |
25077 | 574 |
|
575 |
lemma min_diff_distrib_left: |
|
576 |
shows "min x y - z = min (x - z) (y - z)" |
|
29667 | 577 |
by (simp add: diff_minus, rule min_add_distrib_left) |
25077 | 578 |
|
579 |
lemma le_imp_neg_le: |
|
29667 | 580 |
assumes "a \<le> b" shows "-b \<le> -a" |
25077 | 581 |
proof - |
29667 | 582 |
have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) |
583 |
hence "0 \<le> -a+b" by simp |
|
584 |
hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) |
|
585 |
thus ?thesis by (simp add: add_assoc) |
|
25077 | 586 |
qed |
587 |
||
588 |
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b" |
|
589 |
proof |
|
590 |
assume "- b \<le> - a" |
|
29667 | 591 |
hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le) |
25077 | 592 |
thus "a\<le>b" by simp |
593 |
next |
|
594 |
assume "a\<le>b" |
|
595 |
thus "-b \<le> -a" by (rule le_imp_neg_le) |
|
596 |
qed |
|
597 |
||
598 |
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a" |
|
29667 | 599 |
by (subst neg_le_iff_le [symmetric], simp) |
25077 | 600 |
|
601 |
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0" |
|
29667 | 602 |
by (subst neg_le_iff_le [symmetric], simp) |
25077 | 603 |
|
604 |
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b" |
|
29667 | 605 |
by (force simp add: less_le) |
25077 | 606 |
|
607 |
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a" |
|
29667 | 608 |
by (subst neg_less_iff_less [symmetric], simp) |
25077 | 609 |
|
610 |
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0" |
|
29667 | 611 |
by (subst neg_less_iff_less [symmetric], simp) |
25077 | 612 |
|
613 |
text{*The next several equations can make the simplifier loop!*} |
|
614 |
||
615 |
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a" |
|
616 |
proof - |
|
617 |
have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less) |
|
618 |
thus ?thesis by simp |
|
619 |
qed |
|
620 |
||
621 |
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a" |
|
622 |
proof - |
|
623 |
have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less) |
|
624 |
thus ?thesis by simp |
|
625 |
qed |
|
626 |
||
627 |
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a" |
|
628 |
proof - |
|
629 |
have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff) |
|
630 |
have "(- (- a) <= -b) = (b <= - a)" |
|
631 |
apply (auto simp only: le_less) |
|
632 |
apply (drule mm) |
|
633 |
apply (simp_all) |
|
634 |
apply (drule mm[simplified], assumption) |
|
635 |
done |
|
636 |
then show ?thesis by simp |
|
637 |
qed |
|
638 |
||
639 |
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a" |
|
29667 | 640 |
by (auto simp add: le_less minus_less_iff) |
25077 | 641 |
|
642 |
lemma less_iff_diff_less_0: "a < b \<longleftrightarrow> a - b < 0" |
|
643 |
proof - |
|
644 |
have "(a < b) = (a + (- b) < b + (-b))" |
|
645 |
by (simp only: add_less_cancel_right) |
|
646 |
also have "... = (a - b < 0)" by (simp add: diff_minus) |
|
647 |
finally show ?thesis . |
|
648 |
qed |
|
649 |
||
29667 | 650 |
lemma diff_less_eq[algebra_simps]: "a - b < c \<longleftrightarrow> a < c + b" |
25077 | 651 |
apply (subst less_iff_diff_less_0 [of a]) |
652 |
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) |
|
653 |
apply (simp add: diff_minus add_ac) |
|
654 |
done |
|
655 |
||
29667 | 656 |
lemma less_diff_eq[algebra_simps]: "a < c - b \<longleftrightarrow> a + b < c" |
25077 | 657 |
apply (subst less_iff_diff_less_0 [of "plus a b"]) |
658 |
apply (subst less_iff_diff_less_0 [of a]) |
|
659 |
apply (simp add: diff_minus add_ac) |
|
660 |
done |
|
661 |
||
29667 | 662 |
lemma diff_le_eq[algebra_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b" |
663 |
by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel) |
|
25077 | 664 |
|
29667 | 665 |
lemma le_diff_eq[algebra_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c" |
666 |
by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel) |
|
25077 | 667 |
|
668 |
lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0" |
|
29667 | 669 |
by (simp add: algebra_simps) |
25077 | 670 |
|
29667 | 671 |
text{*Legacy - use @{text algebra_simps} *} |
29833 | 672 |
lemmas group_simps[noatp] = algebra_simps |
25230 | 673 |
|
25077 | 674 |
end |
675 |
||
29667 | 676 |
text{*Legacy - use @{text algebra_simps} *} |
29833 | 677 |
lemmas group_simps[noatp] = algebra_simps |
25230 | 678 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
679 |
class linordered_ab_semigroup_add = |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
680 |
linorder + ordered_ab_semigroup_add |
25062 | 681 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
682 |
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
|
683 |
linorder + ordered_cancel_ab_semigroup_add |
25267 | 684 |
begin |
25062 | 685 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
686 |
subclass linordered_ab_semigroup_add .. |
25062 | 687 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
688 |
subclass ordered_ab_semigroup_add_imp_le |
28823 | 689 |
proof |
25062 | 690 |
fix a b c :: 'a |
691 |
assume le: "c + a <= c + b" |
|
692 |
show "a <= b" |
|
693 |
proof (rule ccontr) |
|
694 |
assume w: "~ a \<le> b" |
|
695 |
hence "b <= a" by (simp add: linorder_not_le) |
|
696 |
hence le2: "c + b <= c + a" by (rule add_left_mono) |
|
697 |
have "a = b" |
|
698 |
apply (insert le) |
|
699 |
apply (insert le2) |
|
700 |
apply (drule antisym, simp_all) |
|
701 |
done |
|
702 |
with w show False |
|
703 |
by (simp add: linorder_not_le [symmetric]) |
|
704 |
qed |
|
705 |
qed |
|
706 |
||
25267 | 707 |
end |
708 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
709 |
class linordered_ab_group_add = linorder + ordered_ab_group_add |
25267 | 710 |
begin |
25230 | 711 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
712 |
subclass linordered_cancel_ab_semigroup_add .. |
25230 | 713 |
|
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
714 |
lemma neg_less_eq_nonneg [simp]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
715 |
"- a \<le> a \<longleftrightarrow> 0 \<le> a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
716 |
proof |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
717 |
assume A: "- a \<le> a" show "0 \<le> a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
718 |
proof (rule classical) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
719 |
assume "\<not> 0 \<le> a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
720 |
then have "a < 0" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
721 |
with A have "- a < 0" by (rule le_less_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
722 |
then show ?thesis by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
723 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
724 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
725 |
assume A: "0 \<le> a" show "- a \<le> a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
726 |
proof (rule order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
727 |
show "- a \<le> 0" using A by (simp add: minus_le_iff) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
728 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
729 |
show "0 \<le> a" using A . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
730 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
731 |
qed |
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
732 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
733 |
lemma neg_less_nonneg [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
734 |
"- a < a \<longleftrightarrow> 0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
735 |
proof |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
736 |
assume A: "- a < a" show "0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
737 |
proof (rule classical) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
738 |
assume "\<not> 0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
739 |
then have "a \<le> 0" by auto |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
740 |
with A have "- a < 0" by (rule less_le_trans) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
741 |
then show ?thesis by auto |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
742 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
743 |
next |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
744 |
assume A: "0 < a" show "- a < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
745 |
proof (rule less_trans) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
746 |
show "- a < 0" using A by (simp add: minus_le_iff) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
747 |
next |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
748 |
show "0 < a" using A . |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
749 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
750 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
751 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
752 |
lemma less_eq_neg_nonpos [simp]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
753 |
"a \<le> - a \<longleftrightarrow> a \<le> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
754 |
proof |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
755 |
assume A: "a \<le> - a" show "a \<le> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
756 |
proof (rule classical) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
757 |
assume "\<not> a \<le> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
758 |
then have "0 < a" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
759 |
then have "0 < - a" using A by (rule less_le_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
760 |
then show ?thesis by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
761 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
762 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
763 |
assume A: "a \<le> 0" show "a \<le> - a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
764 |
proof (rule order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
765 |
show "0 \<le> - a" using A by (simp add: minus_le_iff) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
766 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
767 |
show "a \<le> 0" using A . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
768 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
769 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
770 |
|
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
771 |
lemma equal_neg_zero [simp]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
772 |
"a = - a \<longleftrightarrow> a = 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
773 |
proof |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
774 |
assume "a = 0" then show "a = - a" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
775 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
776 |
assume A: "a = - a" show "a = 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
777 |
proof (cases "0 \<le> a") |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
778 |
case True with A have "0 \<le> - a" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
779 |
with le_minus_iff have "a \<le> 0" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
780 |
with True show ?thesis by (auto intro: order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
781 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
782 |
case False then have B: "a \<le> 0" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
783 |
with A have "- a \<le> 0" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
784 |
with B show ?thesis by (auto intro: order_trans) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
785 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
786 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
787 |
|
35036
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
788 |
lemma neg_equal_zero [simp]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
789 |
"- 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
|
790 |
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
|
791 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
792 |
lemma double_zero [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
793 |
"a + a = 0 \<longleftrightarrow> a = 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
794 |
proof |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
795 |
assume assm: "a + a = 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
796 |
then have a: "- a = a" by (rule minus_unique) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
797 |
then show "a = 0" by (simp add: neg_equal_zero) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
798 |
qed simp |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
799 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
800 |
lemma double_zero_sym [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
801 |
"0 = a + a \<longleftrightarrow> a = 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
802 |
by (rule, drule sym) simp_all |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
803 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
804 |
lemma zero_less_double_add_iff_zero_less_single_add [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
805 |
"0 < a + a \<longleftrightarrow> 0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
806 |
proof |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
807 |
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
|
808 |
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
|
809 |
then have "- 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
|
810 |
then show "0 < a" by (simp add: neg_less_nonneg) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
811 |
next |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
812 |
assume "0 < a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
813 |
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
|
814 |
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
|
815 |
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
|
816 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
817 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
818 |
lemma zero_le_double_add_iff_zero_le_single_add [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
819 |
"0 \<le> a + a \<longleftrightarrow> 0 \<le> a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
820 |
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
|
821 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
822 |
lemma double_add_less_zero_iff_single_add_less_zero [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
823 |
"a + a < 0 \<longleftrightarrow> a < 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
824 |
proof - |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
825 |
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
|
826 |
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
|
827 |
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
|
828 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
829 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
830 |
lemma double_add_le_zero_iff_single_add_le_zero [simp]: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
831 |
"a + a \<le> 0 \<longleftrightarrow> a \<le> 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
832 |
proof - |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
833 |
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
|
834 |
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
|
835 |
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
|
836 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
837 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
838 |
lemma le_minus_self_iff: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
839 |
"a \<le> - a \<longleftrightarrow> a \<le> 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
840 |
proof - |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
841 |
from add_le_cancel_left [of "- a" "a + a" 0] |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
842 |
have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
843 |
by (simp add: add_assoc [symmetric]) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
844 |
thus ?thesis by simp |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
845 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
846 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
847 |
lemma minus_le_self_iff: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
848 |
"- a \<le> a \<longleftrightarrow> 0 \<le> a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
849 |
proof - |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
850 |
from add_le_cancel_left [of "- a" 0 "a + a"] |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
851 |
have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
852 |
by (simp add: add_assoc [symmetric]) |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
853 |
thus ?thesis by simp |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
854 |
qed |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
855 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
856 |
lemma minus_max_eq_min: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
857 |
"- max x y = min (-x) (-y)" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
858 |
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
|
859 |
|
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
860 |
lemma minus_min_eq_max: |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
861 |
"- min x y = max (-x) (-y)" |
b8c8d01cc20d
separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents:
35028
diff
changeset
|
862 |
by (auto simp add: max_def min_def) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
863 |
|
25267 | 864 |
end |
865 |
||
25077 | 866 |
-- {* FIXME localize the following *} |
14738 | 867 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
868 |
lemma add_increasing: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
869 |
fixes c :: "'a::{ordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
870 |
shows "[|0\<le>a; b\<le>c|] ==> b \<le> a + c" |
14738 | 871 |
by (insert add_mono [of 0 a b c], simp) |
872 |
||
15539 | 873 |
lemma add_increasing2: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
874 |
fixes c :: "'a::{ordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
15539 | 875 |
shows "[|0\<le>c; b\<le>a|] ==> b \<le> a + c" |
876 |
by (simp add:add_increasing add_commute[of a]) |
|
877 |
||
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
878 |
lemma add_strict_increasing: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
879 |
fixes c :: "'a::{ordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
880 |
shows "[|0<a; b\<le>c|] ==> b < a + c" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
881 |
by (insert add_less_le_mono [of 0 a b c], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
882 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
883 |
lemma add_strict_increasing2: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
884 |
fixes c :: "'a::{ordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
885 |
shows "[|0\<le>a; b<c|] ==> b < a + c" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
886 |
by (insert add_le_less_mono [of 0 a b c], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
887 |
|
35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
888 |
class abs = |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
889 |
fixes abs :: "'a \<Rightarrow> 'a" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
890 |
begin |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
891 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
892 |
notation (xsymbols) |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
893 |
abs ("\<bar>_\<bar>") |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
894 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
895 |
notation (HTML output) |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
896 |
abs ("\<bar>_\<bar>") |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
897 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
898 |
end |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
899 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
900 |
class sgn = |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
901 |
fixes sgn :: "'a \<Rightarrow> 'a" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
902 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
903 |
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
|
904 |
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
|
905 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
906 |
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
|
907 |
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
|
908 |
begin |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
909 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
910 |
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
|
911 |
by (simp add:sgn_if) |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
912 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35050
diff
changeset
|
913 |
end |
14738 | 914 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
915 |
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
|
916 |
assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
917 |
and abs_ge_self: "a \<le> \<bar>a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
918 |
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
|
919 |
and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
920 |
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
|
921 |
begin |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
922 |
|
25307 | 923 |
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0" |
924 |
unfolding neg_le_0_iff_le by simp |
|
925 |
||
926 |
lemma abs_of_nonneg [simp]: |
|
29667 | 927 |
assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a" |
25307 | 928 |
proof (rule antisym) |
929 |
from nonneg le_imp_neg_le have "- a \<le> 0" by simp |
|
930 |
from this nonneg have "- a \<le> a" by (rule order_trans) |
|
931 |
then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI) |
|
932 |
qed (rule abs_ge_self) |
|
933 |
||
934 |
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>" |
|
29667 | 935 |
by (rule antisym) |
936 |
(auto intro!: abs_ge_self abs_leI order_trans [of "uminus (abs a)" zero "abs a"]) |
|
25307 | 937 |
|
938 |
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0" |
|
939 |
proof - |
|
940 |
have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0" |
|
941 |
proof (rule antisym) |
|
942 |
assume zero: "\<bar>a\<bar> = 0" |
|
943 |
with abs_ge_self show "a \<le> 0" by auto |
|
944 |
from zero have "\<bar>-a\<bar> = 0" by simp |
|
945 |
with abs_ge_self [of "uminus a"] have "- a \<le> 0" by auto |
|
946 |
with neg_le_0_iff_le show "0 \<le> a" by auto |
|
947 |
qed |
|
948 |
then show ?thesis by auto |
|
949 |
qed |
|
950 |
||
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
951 |
lemma abs_zero [simp]: "\<bar>0\<bar> = 0" |
29667 | 952 |
by simp |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
953 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
954 |
lemma abs_0_eq [simp, noatp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
955 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
956 |
have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
957 |
thus ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
958 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
959 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
960 |
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
961 |
proof |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
962 |
assume "\<bar>a\<bar> \<le> 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
963 |
then have "\<bar>a\<bar> = 0" by (rule antisym) simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
964 |
thus "a = 0" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
965 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
966 |
assume "a = 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
967 |
thus "\<bar>a\<bar> \<le> 0" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
968 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
969 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
970 |
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0" |
29667 | 971 |
by (simp add: less_le) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
972 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
973 |
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
974 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
975 |
have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
976 |
show ?thesis by (simp add: a) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
977 |
qed |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
978 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
979 |
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
980 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
981 |
have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
982 |
then show ?thesis by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
983 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
984 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
985 |
lemma abs_minus_commute: |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
986 |
"\<bar>a - b\<bar> = \<bar>b - a\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
987 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
988 |
have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
989 |
also have "... = \<bar>b - a\<bar>" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
990 |
finally show ?thesis . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
991 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
992 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
993 |
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a" |
29667 | 994 |
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
|
995 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
996 |
lemma abs_of_nonpos [simp]: |
29667 | 997 |
assumes "a \<le> 0" shows "\<bar>a\<bar> = - a" |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
998 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
999 |
let ?b = "- a" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1000 |
have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1001 |
unfolding abs_minus_cancel [of "?b"] |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1002 |
unfolding neg_le_0_iff_le [of "?b"] |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1003 |
unfolding minus_minus by (erule abs_of_nonneg) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1004 |
then show ?thesis using assms by auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1005 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1006 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1007 |
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a" |
29667 | 1008 |
by (rule abs_of_nonpos, rule less_imp_le) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1009 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1010 |
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b" |
29667 | 1011 |
by (insert abs_ge_self, blast intro: order_trans) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1012 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1013 |
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b" |
29667 | 1014 |
by (insert abs_le_D1 [of "uminus a"], simp) |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1015 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1016 |
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b" |
29667 | 1017 |
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
|
1018 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1019 |
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>" |
29667 | 1020 |
apply (simp add: algebra_simps) |
1021 |
apply (subgoal_tac "abs a = abs (plus b (minus a b))") |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1022 |
apply (erule ssubst) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1023 |
apply (rule abs_triangle_ineq) |
29667 | 1024 |
apply (rule arg_cong[of _ _ abs]) |
1025 |
apply (simp add: algebra_simps) |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1026 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1027 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1028 |
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>" |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1029 |
apply (subst abs_le_iff) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1030 |
apply auto |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1031 |
apply (rule abs_triangle_ineq2) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1032 |
apply (subst abs_minus_commute) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1033 |
apply (rule abs_triangle_ineq2) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1034 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1035 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1036 |
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
|
1037 |
proof - |
29667 | 1038 |
have "abs(a - b) = abs(a + - b)" by (subst diff_minus, rule refl) |
1039 |
also have "... <= abs a + abs (- b)" by (rule abs_triangle_ineq) |
|
1040 |
finally show ?thesis by simp |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1041 |
qed |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1042 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1043 |
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
|
1044 |
proof - |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1045 |
have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1046 |
also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1047 |
finally show ?thesis . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1048 |
qed |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1049 |
|
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1050 |
lemma abs_add_abs [simp]: |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1051 |
"\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R") |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1052 |
proof (rule antisym) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1053 |
show "?L \<ge> ?R" by(rule abs_ge_self) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1054 |
next |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1055 |
have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq) |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1056 |
also have "\<dots> = ?R" by simp |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1057 |
finally show "?L \<le> ?R" . |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1058 |
qed |
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1059 |
|
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25267
diff
changeset
|
1060 |
end |
14738 | 1061 |
|
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1062 |
text {* Needed for abelian cancellation simprocs: *} |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1063 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1064 |
lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1065 |
apply (subst add_left_commute) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1066 |
apply (subst add_left_cancel) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1067 |
apply simp |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1068 |
done |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1069 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1070 |
lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1071 |
apply (subst add_cancel_21[of _ _ _ 0, simplified]) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1072 |
apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified]) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1073 |
done |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1074 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1075 |
lemma less_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')" |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1076 |
by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y']) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1077 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1078 |
lemma le_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')" |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1079 |
apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of y' x']) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1080 |
apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0]) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1081 |
done |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1082 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1083 |
lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')" |
30629 | 1084 |
by (simp only: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y']) |
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1085 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1086 |
lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1087 |
by (simp add: diff_minus) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1088 |
|
25090 | 1089 |
lemma le_add_right_mono: |
15178 | 1090 |
assumes |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1091 |
"a <= b + (c::'a::ordered_ab_group_add)" |
15178 | 1092 |
"c <= d" |
1093 |
shows "a <= b + d" |
|
1094 |
apply (rule_tac order_trans[where y = "b+c"]) |
|
1095 |
apply (simp_all add: prems) |
|
1096 |
done |
|
1097 |
||
1098 |
||
25090 | 1099 |
subsection {* Tools setup *} |
1100 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1101 |
lemma add_mono_thms_linordered_semiring [noatp]: |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1102 |
fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add" |
25077 | 1103 |
shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
1104 |
and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
|
1105 |
and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l" |
|
1106 |
and "i = j \<and> k = l \<Longrightarrow> i + k = j + l" |
|
1107 |
by (rule add_mono, clarify+)+ |
|
1108 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1109 |
lemma add_mono_thms_linordered_field [noatp]: |
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1110 |
fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add" |
25077 | 1111 |
shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l" |
1112 |
and "i = j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
1113 |
and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l" |
|
1114 |
and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
1115 |
and "i < j \<and> k < l \<Longrightarrow> i + k < j + l" |
|
1116 |
by (auto intro: add_strict_right_mono add_strict_left_mono |
|
1117 |
add_less_le_mono add_le_less_mono add_strict_mono) |
|
1118 |
||
17085 | 1119 |
text{*Simplification of @{term "x-y < 0"}, etc.*} |
29833 | 1120 |
lemmas diff_less_0_iff_less [simp, noatp] = less_iff_diff_less_0 [symmetric] |
1121 |
lemmas diff_le_0_iff_le [simp, noatp] = le_iff_diff_le_0 [symmetric] |
|
17085 | 1122 |
|
22482 | 1123 |
ML {* |
27250 | 1124 |
structure ab_group_add_cancel = Abel_Cancel |
1125 |
( |
|
22482 | 1126 |
|
1127 |
(* term order for abelian groups *) |
|
1128 |
||
1129 |
fun agrp_ord (Const (a, _)) = find_index (fn a' => a = a') |
|
34973
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
34147
diff
changeset
|
1130 |
[@{const_name Algebras.zero}, @{const_name Algebras.plus}, |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
34147
diff
changeset
|
1131 |
@{const_name Algebras.uminus}, @{const_name Algebras.minus}] |
22482 | 1132 |
| agrp_ord _ = ~1; |
1133 |
||
29269
5c25a2012975
moved term order operations to structure TermOrd (cf. Pure/term_ord.ML);
wenzelm
parents:
28823
diff
changeset
|
1134 |
fun termless_agrp (a, b) = (TermOrd.term_lpo agrp_ord (a, b) = LESS); |
22482 | 1135 |
|
1136 |
local |
|
1137 |
val ac1 = mk_meta_eq @{thm add_assoc}; |
|
1138 |
val ac2 = mk_meta_eq @{thm add_commute}; |
|
1139 |
val ac3 = mk_meta_eq @{thm add_left_commute}; |
|
34973
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
34147
diff
changeset
|
1140 |
fun solve_add_ac thy _ (_ $ (Const (@{const_name Algebras.plus},_) $ _ $ _) $ _) = |
22482 | 1141 |
SOME ac1 |
34973
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
34147
diff
changeset
|
1142 |
| solve_add_ac thy _ (_ $ x $ (Const (@{const_name Algebras.plus},_) $ y $ z)) = |
22482 | 1143 |
if termless_agrp (y, x) then SOME ac3 else NONE |
1144 |
| solve_add_ac thy _ (_ $ x $ y) = |
|
1145 |
if termless_agrp (y, x) then SOME ac2 else NONE |
|
1146 |
| solve_add_ac thy _ _ = NONE |
|
1147 |
in |
|
32010 | 1148 |
val add_ac_proc = Simplifier.simproc @{theory} |
22482 | 1149 |
"add_ac_proc" ["x + y::'a::ab_semigroup_add"] solve_add_ac; |
1150 |
end; |
|
1151 |
||
27250 | 1152 |
val eq_reflection = @{thm eq_reflection}; |
1153 |
||
1154 |
val T = @{typ "'a::ab_group_add"}; |
|
1155 |
||
22482 | 1156 |
val cancel_ss = HOL_basic_ss settermless termless_agrp |
1157 |
addsimprocs [add_ac_proc] addsimps |
|
23085 | 1158 |
[@{thm add_0_left}, @{thm add_0_right}, @{thm diff_def}, |
22482 | 1159 |
@{thm minus_add_distrib}, @{thm minus_minus}, @{thm minus_zero}, |
1160 |
@{thm right_minus}, @{thm left_minus}, @{thm add_minus_cancel}, |
|
1161 |
@{thm minus_add_cancel}]; |
|
27250 | 1162 |
|
1163 |
val sum_pats = [@{cterm "x + y::'a::ab_group_add"}, @{cterm "x - y::'a::ab_group_add"}]; |
|
22482 | 1164 |
|
22548 | 1165 |
val eqI_rules = [@{thm less_eqI}, @{thm le_eqI}, @{thm eq_eqI}]; |
22482 | 1166 |
|
1167 |
val dest_eqI = |
|
1168 |
fst o HOLogic.dest_bin "op =" HOLogic.boolT o HOLogic.dest_Trueprop o concl_of; |
|
1169 |
||
27250 | 1170 |
); |
22482 | 1171 |
*} |
1172 |
||
26480 | 1173 |
ML {* |
22482 | 1174 |
Addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]; |
1175 |
*} |
|
17085 | 1176 |
|
33364 | 1177 |
code_modulename SML |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35036
diff
changeset
|
1178 |
Groups Arith |
33364 | 1179 |
|
1180 |
code_modulename OCaml |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35036
diff
changeset
|
1181 |
Groups Arith |
33364 | 1182 |
|
1183 |
code_modulename Haskell |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35036
diff
changeset
|
1184 |
Groups Arith |
33364 | 1185 |
|
14738 | 1186 |
end |