src/HOL/OrderedGroup.thy
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(*  Title:   HOL/OrderedGroup.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 {* Ordered Groups *}
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theory OrderedGroup
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imports Lattices
19798
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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 AlgebraSimps =
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  NamedThmsFun(val name = "algebra_simps"
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               val description = "algebra simplification rules");
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*}
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setup AlgebraSimps.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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class ab_semigroup_add = semigroup_add +
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  assumes add_commute[algebra_simps]: "a + b = b + a"
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begin
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lemma add_left_commute[algebra_simps]: "a + (b + c) = b + (a + c)"
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by (rule mk_left_commute [of "plus", OF add_assoc add_commute])
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theorems add_ac = add_assoc add_commute add_left_commute
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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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class ab_semigroup_mult = semigroup_mult +
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  assumes mult_commute[algebra_simps]: "a * b = b * a"
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begin
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lemma mult_left_commute[algebra_simps]: "a * (b * c) = b * (a * c)"
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by (rule mk_left_commute [of "times", OF mult_assoc mult_commute])
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theorems mult_ac = mult_assoc mult_commute mult_left_commute
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end
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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[simp]: "x * x = x"
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begin
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lemma mult_left_idem[simp]: "x * (x * y) = x * y"
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  unfolding mult_assoc [symmetric, of x] mult_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"
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begin
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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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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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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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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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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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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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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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end
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class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add
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856f16a3b436 add class cancel_comm_monoid_add
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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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lemma minus_add_cancel: "- a + (a + b) = b"
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by (simp add: add_assoc[symmetric])
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parents:
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   162
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lemma minus_zero [simp]: "- 0 = 0"
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parents:
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   164
proof -
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   165
  have "- 0 = - 0 + (0 + 0)" by (simp only: add_0_right)
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parents: 24748
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   166
  also have "\<dots> = 0" by (rule minus_add_cancel)
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parents:
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  finally show ?thesis .
83f1a514dcb4 changes made due to new Ring_and_Field theory
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parents:
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qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
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parents:
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   169
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lemma minus_minus [simp]: "- (- a) = a"
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parents: 22997
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   171
proof -
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   172
  have "- (- a) = - (- a) + (- a + a)" by simp
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  also have "\<dots> = a" by (rule minus_add_cancel)
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parents: 22997
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   174
  finally show ?thesis .
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
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   175
qed
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parents:
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lemma right_minus [simp]: "a + - a = 0"
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parents:
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   178
proof -
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   179
  have "a + - a = - (- a) + - a" by simp
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  also have "\<dots> = 0" by (rule left_minus)
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  finally show ?thesis .
83f1a514dcb4 changes made due to new Ring_and_Field theory
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parents:
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qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
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parents:
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   183
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   184
lemma right_minus_eq: "a - b = 0 \<longleftrightarrow> a = b"
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   185
proof
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   186
  assume "a - b = 0"
fd30d75a6614 Introduced new classes monoid_add and group_add
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parents: 22997
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   187
  have "a = (a - b) + b" by (simp add:diff_minus add_assoc)
fd30d75a6614 Introduced new classes monoid_add and group_add
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parents: 22997
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   188
  also have "\<dots> = b" using `a - b = 0` by simp
fd30d75a6614 Introduced new classes monoid_add and group_add
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parents: 22997
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   189
  finally show "a = b" .
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   190
next
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parents: 22997
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   191
  assume "a = b" thus "a - b = 0" by (simp add: diff_minus)
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   192
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
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parents:
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   193
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   194
lemma equals_zero_I:
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  assumes "a + b = 0" shows "- a = b"
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parents: 22997
diff changeset
   196
proof -
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   197
  have "- a = - a + (a + b)" using assms by simp
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   198
  also have "\<dots> = b" by (simp add: add_assoc[symmetric])
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parents: 22997
diff changeset
   199
  finally show ?thesis .
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
   200
qed
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parents:
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   201
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   202
lemma diff_self [simp]: "a - a = 0"
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by (simp add: diff_minus)
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parents:
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   204
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   205
lemma diff_0 [simp]: "0 - a = - a"
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by (simp add: diff_minus)
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parents:
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   207
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   208
lemma diff_0_right [simp]: "a - 0 = a" 
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   209
by (simp add: diff_minus)
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obua
parents:
diff changeset
   210
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   211
lemma diff_minus_eq_add [simp]: "a - - b = a + b"
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   212
by (simp add: diff_minus)
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obua
parents:
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   213
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   214
lemma neg_equal_iff_equal [simp]:
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   215
  "- a = - b \<longleftrightarrow> a = b" 
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obua
parents:
diff changeset
   216
proof 
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parents:
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   217
  assume "- a = - b"
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   218
  hence "- (- a) = - (- b)" by simp
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   219
  thus "a = b" by simp
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parents:
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   220
next
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   221
  assume "a = b"
af5ef0d4d655 global class syntax
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   222
  thus "- a = - b" by simp
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obua
parents:
diff changeset
   223
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   224
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   225
lemma neg_equal_0_iff_equal [simp]:
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parents: 24748
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   226
  "- a = 0 \<longleftrightarrow> a = 0"
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   227
by (subst neg_equal_iff_equal [symmetric], simp)
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obua
parents:
diff changeset
   228
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   229
lemma neg_0_equal_iff_equal [simp]:
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parents: 24748
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   230
  "0 = - a \<longleftrightarrow> 0 = a"
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diff changeset
   231
by (subst neg_equal_iff_equal [symmetric], simp)
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   232
83f1a514dcb4 changes made due to new Ring_and_Field theory
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parents:
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   233
text{*The next two equations can make the simplifier loop!*}
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parents:
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   234
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   235
lemma equation_minus_iff:
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parents: 24748
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   236
  "a = - b \<longleftrightarrow> b = - a"
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parents:
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   237
proof -
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parents: 24748
diff changeset
   238
  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   239
  thus ?thesis by (simp add: eq_commute)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   240
qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   241
af5ef0d4d655 global class syntax
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diff changeset
   242
lemma minus_equation_iff:
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   243
  "- a = b \<longleftrightarrow> - b = a"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   244
proof -
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   245
  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
14738
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obua
parents:
diff changeset
   246
  thus ?thesis by (simp add: eq_commute)
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obua
parents:
diff changeset
   247
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   248
28130
32b4185bfdc7 move diff_add_cancel, add_diff_cancel from class ab_group_add to group_add
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   249
lemma diff_add_cancel: "a - b + b = a"
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   250
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
   251
32b4185bfdc7 move diff_add_cancel, add_diff_cancel from class ab_group_add to group_add
huffman
parents: 27516
diff changeset
   252
lemma add_diff_cancel: "a + b - b = a"
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   253
by (simp add: diff_minus add_assoc)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
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parents: 29269
diff changeset
   254
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   255
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
   256
29914
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   257
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
   258
proof
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   259
  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
   260
next
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   261
  assume "a + b = 0"
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   262
  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
   263
    by (simp only: add_assoc)
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   264
  ultimately show "a = - b" by simp
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   265
qed
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   266
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diff changeset
   267
end
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   268
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c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25613
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   269
class ab_group_add = minus + uminus + comm_monoid_add +
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   270
  assumes ab_left_minus: "- a + a = 0"
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parents: 24748
diff changeset
   271
  assumes ab_diff_minus: "a - b = a + (- b)"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   272
begin
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parents: 24748
diff changeset
   273
25267
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haftmann
parents: 25230
diff changeset
   274
subclass group_add
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28262
diff changeset
   275
  proof qed (simp_all add: ab_left_minus ab_diff_minus)
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parents: 24748
diff changeset
   276
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29886
diff changeset
   277
subclass cancel_comm_monoid_add
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28262
diff changeset
   278
proof
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diff changeset
   279
  fix a b c :: 'a
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   280
  assume "a + b = a + c"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   281
  then have "- a + a + b = - a + a + c"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   282
    unfolding add_assoc by simp
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   283
  then show "b = c" by simp
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   284
qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   285
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   286
lemma uminus_add_conv_diff[algebra_simps]:
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   287
  "- a + b = b - a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   288
by (simp add:diff_minus add_commute)
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haftmann
parents: 24748
diff changeset
   289
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   290
lemma minus_add_distrib [simp]:
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parents: 24748
diff changeset
   291
  "- (a + b) = - a + - b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   292
by (rule equals_zero_I) (simp add: add_ac)
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parents: 24748
diff changeset
   293
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   294
lemma minus_diff_eq [simp]:
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parents: 24748
diff changeset
   295
  "- (a - b) = b - a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   296
by (simp add: diff_minus add_commute)
25077
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haftmann
parents: 25062
diff changeset
   297
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   298
lemma add_diff_eq[algebra_simps]: "a + (b - c) = (a + b) - c"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   299
by (simp add: diff_minus add_ac)
25077
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haftmann
parents: 25062
diff changeset
   300
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   301
lemma diff_add_eq[algebra_simps]: "(a - b) + c = (a + c) - b"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   302
by (simp add: diff_minus add_ac)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   303
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   304
lemma diff_eq_eq[algebra_simps]: "a - b = c \<longleftrightarrow> a = c + b"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   305
by (auto simp add: diff_minus add_assoc)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   306
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   307
lemma eq_diff_eq[algebra_simps]: "a = c - b \<longleftrightarrow> a + b = c"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   308
by (auto simp add: diff_minus add_assoc)
25077
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haftmann
parents: 25062
diff changeset
   309
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   310
lemma diff_diff_eq[algebra_simps]: "(a - b) - c = a - (b + c)"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   311
by (simp add: diff_minus add_ac)
25077
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haftmann
parents: 25062
diff changeset
   312
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   313
lemma diff_diff_eq2[algebra_simps]: "a - (b - c) = (a + c) - b"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   314
by (simp add: diff_minus add_ac)
25077
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haftmann
parents: 25062
diff changeset
   315
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   316
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
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   317
by (simp add: algebra_simps)
25077
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   318
30629
5cd9b19edef3 move diff_eq_0_iff_eq into class locale context
huffman
parents: 29914
diff changeset
   319
lemma diff_eq_0_iff_eq [simp, noatp]: "a - b = 0 \<longleftrightarrow> a = b"
5cd9b19edef3 move diff_eq_0_iff_eq into class locale context
huffman
parents: 29914
diff changeset
   320
by (simp add: algebra_simps)
5cd9b19edef3 move diff_eq_0_iff_eq into class locale context
huffman
parents: 29914
diff changeset
   321
25062
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   322
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
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   323
83f1a514dcb4 changes made due to new Ring_and_Field theory
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   324
subsection {* (Partially) Ordered Groups *} 
83f1a514dcb4 changes made due to new Ring_and_Field theory
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parents:
diff changeset
   325
22390
378f34b1e380 now using "class"
haftmann
parents: 21382
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   326
class pordered_ab_semigroup_add = order + ab_semigroup_add +
25062
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parents: 24748
diff changeset
   327
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   328
begin
24380
c215e256beca moved ordered_ab_semigroup_add to OrderedGroup.thy
haftmann
parents: 24286
diff changeset
   329
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af5ef0d4d655 global class syntax
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parents: 24748
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   330
lemma add_right_mono:
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   331
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   332
by (simp add: add_commute [of _ c] add_left_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   333
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   334
text {* non-strict, in both arguments *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   335
lemma add_mono:
25062
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   336
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   337
  apply (erule add_right_mono [THEN order_trans])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   338
  apply (simp add: add_commute add_left_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   339
  done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   340
25062
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parents: 24748
diff changeset
   341
end
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   342
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   343
class pordered_cancel_ab_semigroup_add =
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   344
  pordered_ab_semigroup_add + cancel_ab_semigroup_add
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   345
begin
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   346
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   347
lemma add_strict_left_mono:
25062
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   348
  "a < b \<Longrightarrow> c + a < c + b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   349
by (auto simp add: less_le add_left_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   350
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   351
lemma add_strict_right_mono:
25062
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   352
  "a < b \<Longrightarrow> a + c < b + c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   353
by (simp add: add_commute [of _ c] add_strict_left_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   354
83f1a514dcb4 changes made due to new Ring_and_Field theory
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parents:
diff changeset
   355
text{*Strict monotonicity in both arguments*}
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   356
lemma add_strict_mono:
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   357
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   358
apply (erule add_strict_right_mono [THEN less_trans])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   359
apply (erule add_strict_left_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   360
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   361
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   362
lemma add_less_le_mono:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   363
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   364
apply (erule add_strict_right_mono [THEN less_le_trans])
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   365
apply (erule add_left_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   366
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   367
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   368
lemma add_le_less_mono:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   369
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   370
apply (erule add_right_mono [THEN le_less_trans])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   371
apply (erule add_strict_left_mono) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   372
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   373
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   374
end
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   375
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   376
class pordered_ab_semigroup_add_imp_le =
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   377
  pordered_cancel_ab_semigroup_add +
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   378
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   379
begin
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   380
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   381
lemma add_less_imp_less_left:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   382
  assumes less: "c + a < c + b" shows "a < b"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   383
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   384
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   385
  have "a <= b" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   386
    apply (insert le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   387
    apply (drule add_le_imp_le_left)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   388
    by (insert le, drule add_le_imp_le_left, assumption)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   389
  moreover have "a \<noteq> b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   390
  proof (rule ccontr)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   391
    assume "~(a \<noteq> b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   392
    then have "a = b" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   393
    then have "c + a = c + b" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   394
    with less show "False"by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   395
  qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   396
  ultimately show "a < b" by (simp add: order_le_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   397
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   398
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   399
lemma add_less_imp_less_right:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   400
  "a + c < b + c \<Longrightarrow> a < b"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   401
apply (rule add_less_imp_less_left [of c])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   402
apply (simp add: add_commute)  
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   403
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   404
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   405
lemma add_less_cancel_left [simp]:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   406
  "c + a < c + b \<longleftrightarrow> a < b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   407
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   408
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   409
lemma add_less_cancel_right [simp]:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   410
  "a + c < b + c \<longleftrightarrow> a < b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   411
by (blast intro: add_less_imp_less_right add_strict_right_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   412
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   413
lemma add_le_cancel_left [simp]:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   414
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   415
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   416
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   417
lemma add_le_cancel_right [simp]:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   418
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   419
by (simp add: add_commute [of a c] add_commute [of b c])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   420
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   421
lemma add_le_imp_le_right:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   422
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   423
by simp
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   424
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   425
lemma max_add_distrib_left:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   426
  "max x y + z = max (x + z) (y + z)"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   427
  unfolding max_def by auto
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   428
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   429
lemma min_add_distrib_left:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   430
  "min x y + z = min (x + z) (y + z)"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   431
  unfolding min_def by auto
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   432
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   433
end
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   434
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   435
subsection {* Support for reasoning about signs *}
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   436
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   437
class pordered_comm_monoid_add =
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   438
  pordered_cancel_ab_semigroup_add + comm_monoid_add
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   439
begin
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   440
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   441
lemma add_pos_nonneg:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   442
  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
   443
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   444
  have "0 + 0 < a + b" 
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   445
    using assms by (rule add_less_le_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   446
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   447
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   448
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   449
lemma add_pos_pos:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   450
  assumes "0 < a" and "0 < b" shows "0 < a + b"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   451
by (rule add_pos_nonneg) (insert assms, auto)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   452
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   453
lemma add_nonneg_pos:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   454
  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
   455
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   456
  have "0 + 0 < a + b" 
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   457
    using assms by (rule add_le_less_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   458
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   459
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   460
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   461
lemma add_nonneg_nonneg:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   462
  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
   463
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   464
  have "0 + 0 \<le> a + b" 
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   465
    using assms by (rule add_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   466
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   467
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   468
30691
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   469
lemma add_neg_nonpos:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   470
  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
   471
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   472
  have "a + b < 0 + 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   473
    using assms by (rule add_less_le_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   474
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   475
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   476
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   477
lemma add_neg_neg: 
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   478
  assumes "a < 0" and "b < 0" shows "a + b < 0"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   479
by (rule add_neg_nonpos) (insert assms, auto)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   480
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   481
lemma add_nonpos_neg:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   482
  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
   483
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   484
  have "a + b < 0 + 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   485
    using assms by (rule add_le_less_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   486
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   487
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   488
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   489
lemma add_nonpos_nonpos:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   490
  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
   491
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   492
  have "a + b \<le> 0 + 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   493
    using assms by (rule add_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   494
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   495
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   496
30691
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   497
lemmas add_sign_intros =
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   498
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   499
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   500
29886
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   501
lemma add_nonneg_eq_0_iff:
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   502
  assumes x: "0 \<le> x" and y: "0 \<le> y"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   503
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   504
proof (intro iffI conjI)
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   505
  have "x = x + 0" by simp
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   506
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   507
  also assume "x + y = 0"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   508
  also have "0 \<le> x" using x .
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   509
  finally show "x = 0" .
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   510
next
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   511
  have "y = 0 + y" by simp
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   512
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   513
  also assume "x + y = 0"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   514
  also have "0 \<le> y" using y .
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   515
  finally show "y = 0" .
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   516
next
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   517
  assume "x = 0 \<and> y = 0"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   518
  then show "x + y = 0" by simp
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   519
qed
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   520
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   521
end
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   522
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   523
class pordered_ab_group_add =
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   524
  ab_group_add + pordered_ab_semigroup_add
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   525
begin
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   526
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 27474
diff changeset
   527
subclass pordered_cancel_ab_semigroup_add ..
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   528
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   529
subclass pordered_ab_semigroup_add_imp_le
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28262
diff changeset
   530
proof
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   531
  fix a b c :: 'a
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   532
  assume "c + a \<le> c + b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   533
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   534
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   535
  thus "a \<le> b" by simp
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   536
qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   537
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 27474
diff changeset
   538
subclass pordered_comm_monoid_add ..
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   539
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   540
lemma max_diff_distrib_left:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   541
  shows "max x y - z = max (x - z) (y - z)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   542
by (simp add: diff_minus, rule max_add_distrib_left) 
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   543
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   544
lemma min_diff_distrib_left:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   545
  shows "min x y - z = min (x - z) (y - z)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   546
by (simp add: diff_minus, rule min_add_distrib_left) 
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   547
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   548
lemma le_imp_neg_le:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   549
  assumes "a \<le> b" shows "-b \<le> -a"
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   550
proof -
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   551
  have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   552
  hence "0 \<le> -a+b" by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   553
  hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   554
  thus ?thesis by (simp add: add_assoc)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   555
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   556
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   557
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   558
proof 
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   559
  assume "- b \<le> - a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   560
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   561
  thus "a\<le>b" by simp
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   562
next
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   563
  assume "a\<le>b"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   564
  thus "-b \<le> -a" by (rule le_imp_neg_le)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   565
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   566
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   567
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   568
by (subst neg_le_iff_le [symmetric], simp)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   569
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   570
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   571
by (subst neg_le_iff_le [symmetric], simp)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   572
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   573
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   574
by (force simp add: less_le) 
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   575
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   576
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   577
by (subst neg_less_iff_less [symmetric], simp)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   578
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   579
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   580
by (subst neg_less_iff_less [symmetric], simp)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   581
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   582
text{*The next several equations can make the simplifier loop!*}
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   583
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   584
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   585
proof -
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   586
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   587
  thus ?thesis by simp
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   588
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   589
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   590
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   591
proof -
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   592
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   593
  thus ?thesis by simp
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   594
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   595
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   596
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   597
proof -
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   598
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   599
  have "(- (- a) <= -b) = (b <= - a)" 
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   600
    apply (auto simp only: le_less)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   601
    apply (drule mm)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   602
    apply (simp_all)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   603
    apply (drule mm[simplified], assumption)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   604
    done
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   605
  then show ?thesis by simp
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   606
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   607
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   608
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   609
by (auto simp add: le_less minus_less_iff)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   610
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   611
lemma less_iff_diff_less_0: "a < b \<longleftrightarrow> a - b < 0"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   612
proof -
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   613
  have  "(a < b) = (a + (- b) < b + (-b))"  
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   614
    by (simp only: add_less_cancel_right)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   615
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   616
  finally show ?thesis .
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   617
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   618
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   619
lemma diff_less_eq[algebra_simps]: "a - b < c \<longleftrightarrow> a < c + b"
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   620
apply (subst less_iff_diff_less_0 [of a])
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   621
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   622
apply (simp add: diff_minus add_ac)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   623
done
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   624
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   625
lemma less_diff_eq[algebra_simps]: "a < c - b \<longleftrightarrow> a + b < c"
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   626
apply (subst less_iff_diff_less_0 [of "plus a b"])
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   627
apply (subst less_iff_diff_less_0 [of a])
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   628
apply (simp add: diff_minus add_ac)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   629
done
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   630
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   631
lemma diff_le_eq[algebra_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   632
by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   633
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   634
lemma le_diff_eq[algebra_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   635
by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   636
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   637
lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   638
by (simp add: algebra_simps)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   639
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   640
text{*Legacy - use @{text algebra_simps} *}
29833
409138c4de12 added noatps
nipkow
parents: 29670
diff changeset
   641
lemmas group_simps[noatp] = algebra_simps
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
   642
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   643
end
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   644
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   645
text{*Legacy - use @{text algebra_simps} *}
29833
409138c4de12 added noatps
nipkow
parents: 29670
diff changeset
   646
lemmas group_simps[noatp] = algebra_simps
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
   647
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   648
class ordered_ab_semigroup_add =
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   649
  linorder + pordered_ab_semigroup_add
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   650
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   651
class ordered_cancel_ab_semigroup_add =
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   652
  linorder + pordered_cancel_ab_semigroup_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   653
begin
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   654
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 27474
diff changeset
   655
subclass ordered_ab_semigroup_add ..
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   656
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   657
subclass pordered_ab_semigroup_add_imp_le
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28262
diff changeset
   658
proof
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   659
  fix a b c :: 'a
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   660
  assume le: "c + a <= c + b"  
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   661
  show "a <= b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   662
  proof (rule ccontr)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   663
    assume w: "~ a \<le> b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   664
    hence "b <= a" by (simp add: linorder_not_le)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   665
    hence le2: "c + b <= c + a" by (rule add_left_mono)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   666
    have "a = b" 
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   667
      apply (insert le)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   668
      apply (insert le2)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   669
      apply (drule antisym, simp_all)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   670
      done
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   671
    with w show False 
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   672
      by (simp add: linorder_not_le [symmetric])
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   673
  qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   674
qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   675
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   676
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   677
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
   678
class ordered_ab_group_add =
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
   679
  linorder + pordered_ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   680
begin
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
   681
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 27474
diff changeset
   682
subclass ordered_cancel_ab_semigroup_add ..
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
   683
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   684
lemma neg_less_eq_nonneg:
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   685
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   686
proof
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   687
  assume A: "- a \<le> a" show "0 \<le> a"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   688
  proof (rule classical)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   689
    assume "\<not> 0 \<le> a"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   690
    then have "a < 0" by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   691
    with A have "- a < 0" by (rule le_less_trans)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   692
    then show ?thesis by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   693
  qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   694
next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   695
  assume A: "0 \<le> a" show "- a \<le> a"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   696
  proof (rule order_trans)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   697
    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
   698
  next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   699
    show "0 \<le> a" using A .
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   700
  qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   701
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   702
  
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   703
lemma less_eq_neg_nonpos:
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   704
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   705
proof
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   706
  assume A: "a \<le> - a" show "a \<le> 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   707
  proof (rule classical)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   708
    assume "\<not> a \<le> 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   709
    then have "0 < a" by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   710
    then have "0 < - a" using A by (rule less_le_trans)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   711
    then show ?thesis by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   712
  qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   713
next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   714
  assume A: "a \<le> 0" show "a \<le> - a"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   715
  proof (rule order_trans)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   716
    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
   717
  next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   718
    show "a \<le> 0" using A .
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   719
  qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   720
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   721
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   722
lemma equal_neg_zero:
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   723
  "a = - a \<longleftrightarrow> a = 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   724
proof
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   725
  assume "a = 0" then show "a = - a" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   726
next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   727
  assume A: "a = - a" show "a = 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   728
  proof (cases "0 \<le> a")
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   729
    case True with A have "0 \<le> - a" by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   730
    with le_minus_iff have "a \<le> 0" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   731
    with True show ?thesis by (auto intro: order_trans)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   732
  next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   733
    case False then have B: "a \<le> 0" by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   734
    with A have "- a \<le> 0" by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   735
    with B show ?thesis by (auto intro: order_trans)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   736
  qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   737
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   738
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   739
lemma neg_equal_zero:
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   740
  "- a = a \<longleftrightarrow> a = 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   741
  unfolding equal_neg_zero [symmetric] by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   742
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   743
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   744
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   745
-- {* FIXME localize the following *}
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   746
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   747
lemma add_increasing:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   748
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   749
  shows  "[|0\<le>a; b\<le>c|] ==> b \<le> a + c"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   750
by (insert add_mono [of 0 a b c], simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   751
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   752
lemma add_increasing2:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   753
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   754
  shows  "[|0\<le>c; b\<le>a|] ==> b \<le> a + c"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   755
by (simp add:add_increasing add_commute[of a])
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   756
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   757
lemma add_strict_increasing:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   758
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   759
  shows "[|0<a; b\<le>c|] ==> b < a + c"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   760
by (insert add_less_le_mono [of 0 a b c], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   761
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   762
lemma add_strict_increasing2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   763
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   764
  shows "[|0\<le>a; b<c|] ==> b < a + c"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   765
by (insert add_le_less_mono [of 0 a b c], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   766
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   767
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   768
class pordered_ab_group_add_abs = pordered_ab_group_add + abs +
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   769
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   770
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   771
    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
   772
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   773
    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
   774
begin
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   775
25307
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   776
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   777
  unfolding neg_le_0_iff_le by simp
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   778
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   779
lemma abs_of_nonneg [simp]:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   780
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
25307
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   781
proof (rule antisym)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   782
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   783
  from this nonneg have "- a \<le> a" by (rule order_trans)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   784
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   785
qed (rule abs_ge_self)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   786
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   787
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   788
by (rule antisym)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   789
   (auto intro!: abs_ge_self abs_leI order_trans [of "uminus (abs a)" zero "abs a"])
25307
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   790
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   791
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   792
proof -
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   793
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   794
  proof (rule antisym)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   795
    assume zero: "\<bar>a\<bar> = 0"
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   796
    with abs_ge_self show "a \<le> 0" by auto
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   797
    from zero have "\<bar>-a\<bar> = 0" by simp
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   798
    with abs_ge_self [of "uminus a"] have "- a \<le> 0" by auto
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   799
    with neg_le_0_iff_le show "0 \<le> a" by auto
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   800
  qed
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   801
  then show ?thesis by auto
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   802
qed
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
   803
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   804
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   805
by simp
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   806
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   807
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
   808
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   809
  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
   810
  thus ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   811
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   812
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   813
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
   814
proof
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   815
  assume "\<bar>a\<bar> \<le> 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   816
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   817
  thus "a = 0" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   818
next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   819
  assume "a = 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   820
  thus "\<bar>a\<bar> \<le> 0" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   821
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   822
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   823
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   824
by (simp add: less_le)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   825
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   826
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   827
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   828
  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
   829
  show ?thesis by (simp add: a)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   830
qed
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   831
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   832
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   833
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   834
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   835
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   836
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   837
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   838
lemma abs_minus_commute: 
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   839
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   840
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   841
  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
   842
  also have "... = \<bar>b - a\<bar>" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   843
  finally show ?thesis .
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   844
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   845
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   846
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   847
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
   848
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   849
lemma abs_of_nonpos [simp]:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   850
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   851
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   852
  let ?b = "- a"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   853
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   854
  unfolding abs_minus_cancel [of "?b"]
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   855
  unfolding neg_le_0_iff_le [of "?b"]
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   856
  unfolding minus_minus by (erule abs_of_nonneg)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   857
  then show ?thesis using assms by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   858
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   859
  
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   860
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   861
by (rule abs_of_nonpos, rule less_imp_le)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   862
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   863
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   864
by (insert abs_ge_self, blast intro: order_trans)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   865
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   866
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   867
by (insert abs_le_D1 [of "uminus a"], simp)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   868
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   869
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   870
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
   871
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   872
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   873
  apply (simp add: algebra_simps)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   874
  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
   875
  apply (erule ssubst)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   876
  apply (rule abs_triangle_ineq)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   877
  apply (rule arg_cong[of _ _ abs])
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   878
  apply (simp add: algebra_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   879
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   880
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   881
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
   882
  apply (subst abs_le_iff)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   883
  apply auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   884
  apply (rule abs_triangle_ineq2)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   885
  apply (subst abs_minus_commute)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   886
  apply (rule abs_triangle_ineq2)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   887
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   888
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   889
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
   890
proof -
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   891
  have "abs(a - b) = abs(a + - b)" by (subst diff_minus, rule refl)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   892
  also have "... <= abs a + abs (- b)" by (rule abs_triangle_ineq)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   893
  finally show ?thesis by simp
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   894
qed
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   895
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   896
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
   897
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   898
  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
   899
  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
   900
  finally show ?thesis .
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   901
qed
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   902
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   903
lemma abs_add_abs [simp]:
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   904
  "\<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
   905
proof (rule antisym)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   906
  show "?L \<ge> ?R" by(rule abs_ge_self)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   907
next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   908
  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
   909
  also have "\<dots> = ?R" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   910
  finally show "?L \<le> ?R" .
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   911
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   912
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   913
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   914
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   915
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   916
subsection {* Lattice Ordered (Abelian) Groups *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   917
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   918
class lordered_ab_group_add_meet = pordered_ab_group_add + lower_semilattice
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   919
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   920
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   921
lemma add_inf_distrib_left:
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   922
  "a + inf b c = inf (a + b) (a + c)"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   923
apply (rule antisym)
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
   924
apply (simp_all add: le_infI)
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   925
apply (rule add_le_imp_le_left [of "uminus a"])
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   926
apply (simp only: add_assoc [symmetric], simp)
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 21245
diff changeset
   927
apply rule
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 21245
diff changeset
   928
apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   929
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   930
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   931
lemma add_inf_distrib_right:
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   932
  "inf a b + c = inf (a + c) (b + c)"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   933
proof -
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   934
  have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   935
  thus ?thesis by (simp add: add_commute)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   936
qed
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   937
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   938
end
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   939
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   940
class lordered_ab_group_add_join = pordered_ab_group_add + upper_semilattice
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   941
begin
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   942
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   943
lemma add_sup_distrib_left:
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   944
  "a + sup b c = sup (a + b) (a + c)" 
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   945
apply (rule antisym)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   946
apply (rule add_le_imp_le_left [of "uminus a"])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   947
apply (simp only: add_assoc[symmetric], simp)
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 21245
diff changeset
   948
apply rule
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 21245
diff changeset
   949
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
   950
apply (rule le_supI)
21312
1d39091a3208 started reorgnization of lattice theories
nipkow
parents: 21245
diff changeset
   951
apply (simp_all)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   952
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   953
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   954
lemma add_sup_distrib_right:
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   955
  "sup a b + c = sup (a+c) (b+c)"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   956
proof -
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   957
  have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   958
  thus ?thesis by (simp add: add_commute)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   959
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   960
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   961
end
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   962
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   963
class lordered_ab_group_add = pordered_ab_group_add + lattice
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   964
begin
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   965
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 27474
diff changeset
   966
subclass lordered_ab_group_add_meet ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 27474
diff changeset
   967
subclass lordered_ab_group_add_join ..
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   968
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
   969
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   970
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   971
lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   972
proof (rule inf_unique)
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   973
  fix a b :: 'a
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   974
  show "- sup (-a) (-b) \<le> a"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   975
    by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   976
      (simp, simp add: add_sup_distrib_left)
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   977
next
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   978
  fix a b :: 'a
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   979
  show "- sup (-a) (-b) \<le> b"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   980
    by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   981
      (simp, simp add: add_sup_distrib_left)
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   982
next
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   983
  fix a b c :: 'a
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   984
  assume "a \<le> b" "a \<le> c"
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   985
  then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric])
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   986
    (simp add: le_supI)
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   987
qed
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   988
  
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   989
lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   990
proof (rule sup_unique)
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   991
  fix a b :: 'a
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   992
  show "a \<le> - inf (-a) (-b)"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   993
    by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   994
      (simp, simp add: add_inf_distrib_left)
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   995
next
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   996
  fix a b :: 'a
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   997
  show "b \<le> - inf (-a) (-b)"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   998
    by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
   999
      (simp, simp add: add_inf_distrib_left)
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
  1000
next
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
  1001
  fix a b c :: 'a
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
  1002
  assume "a \<le> c" "b \<le> c"
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
  1003
  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric])
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
  1004
    (simp add: le_infI)
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
  1005
qed
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1006
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1007
lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1008
by (simp add: inf_eq_neg_sup)
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1009
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1010
lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1011
by (simp add: sup_eq_neg_inf)
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1012
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1013
lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1014
proof -
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1015
  have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1016
  hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1017
  hence "0 = (-a + sup a b) + (inf a b + (-b))"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1018
    by (simp add: add_sup_distrib_left add_inf_distrib_right)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1019
       (simp add: algebra_simps)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1020
  thus ?thesis by (simp add: algebra_simps)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1021
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1022
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1023
subsection {* Positive Part, Negative Part, Absolute Value *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1024
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1025
definition
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1026
  nprt :: "'a \<Rightarrow> 'a" where
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1027
  "nprt x = inf x 0"
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1028
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1029
definition
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1030
  pprt :: "'a \<Rightarrow> 'a" where
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1031
  "pprt x = sup x 0"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1032
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1033
lemma pprt_neg: "pprt (- x) = - nprt x"
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1034
proof -
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1035
  have "sup (- x) 0 = sup (- x) (- 0)" unfolding minus_zero ..
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1036
  also have "\<dots> = - inf x 0" unfolding neg_inf_eq_sup ..
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1037
  finally have "sup (- x) 0 = - inf x 0" .
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1038
  then show ?thesis unfolding pprt_def nprt_def .
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1039
qed
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1040
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1041
lemma nprt_neg: "nprt (- x) = - pprt x"
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1042
proof -
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1043
  from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1044
  then have "pprt x = - nprt (- x)" by simp
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1045
  then show ?thesis by simp
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1046
qed
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1047
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1048
lemma prts: "a = pprt a + nprt a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1049
by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1050
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1051
lemma zero_le_pprt[simp]: "0 \<le> pprt a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1052
by (simp add: pprt_def)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1053
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1054
lemma nprt_le_zero[simp]: "nprt a \<le> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1055
by (simp add: nprt_def)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1056
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1057
lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1058
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1059
  have a: "?l \<longrightarrow> ?r"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1060
    apply (auto)
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1061
    apply (rule add_le_imp_le_right[of _ "uminus b" _])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1062
    apply (simp add: add_assoc)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1063
    done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1064
  have b: "?r \<longrightarrow> ?l"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1065
    apply (auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1066
    apply (rule add_le_imp_le_right[of _ "b" _])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1067
    apply (simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1068
    done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1069
  from a b show ?thesis by blast
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1070
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1071
15580
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
  1072
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
  1073
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
  1074
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1075
lemma pprt_eq_id [simp, noatp]: "0 \<le> x \<Longrightarrow> pprt x = x"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1076
by (simp add: pprt_def le_iff_sup sup_ACI)
15580
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
  1077
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1078
lemma nprt_eq_id [simp, noatp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1079
by (simp add: nprt_def le_iff_inf inf_ACI)
15580
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
  1080
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1081
lemma pprt_eq_0 [simp, noatp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1082
by (simp add: pprt_def le_iff_sup sup_ACI)
15580
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
  1083
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1084
lemma nprt_eq_0 [simp, noatp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1085
by (simp add: nprt_def le_iff_inf inf_ACI)
15580
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
  1086
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1087
lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1088
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1089
  {
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1090
    fix a::'a
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1091
    assume hyp: "sup a (-a) = 0"
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1092
    hence "sup a (-a) + a = a" by (simp)
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1093
    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right) 
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1094
    hence "sup (a+a) 0 <= a" by (simp)
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1095
    hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1096
  }
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1097
  note p = this
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1098
  assume hyp:"sup a (-a) = 0"
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1099
  hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1100
  from p[OF hyp] p[OF hyp2] show "a = 0" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1101
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1102
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1103
lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1104
apply (simp add: inf_eq_neg_sup)
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1105
apply (simp add: sup_commute)
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1106
apply (erule sup_0_imp_0)
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15234
diff changeset
  1107
done
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1108
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1109
lemma inf_0_eq_0 [simp, noatp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1110
by (rule, erule inf_0_imp_0) simp
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1111
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1112
lemma sup_0_eq_0 [simp, noatp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1113
by (rule, erule sup_0_imp_0) simp
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1114
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1115
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1116
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1117
proof
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1118
  assume "0 <= a + a"
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1119
  hence a:"inf (a+a) 0 = 0" by (simp add: le_iff_inf inf_commute)
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1120
  have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_")
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1121
    by (simp add: add_sup_inf_distribs inf_ACI)
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1122
  hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1123
  hence "inf a 0 = 0" by (simp only: add_right_cancel)
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22390
diff changeset
  1124
  then show "0 <= a" by (simp add: le_iff_inf inf_commute)    
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1125
next  
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1126
  assume a: "0 <= a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1127
  show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1128
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1129
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1130
lemma double_zero: "a + a = 0 \<longleftrightarrow> a = 0"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1131
proof
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1132
  assume assm: "a + a = 0"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1133
  then have "a + a + - a = - a" by simp
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1134
  then have "a + (a + - a) = - a" by (simp only: add_assoc)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1135
  then have a: "- a = a" by simp (*FIXME tune proof*)
25102
db3e412c4cb1 antisymmetry not a default intro rule any longer
haftmann
parents: 25090
diff changeset
  1136
  show "a = 0" apply (rule antisym)
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1137
  apply (unfold neg_le_iff_le [symmetric, of a])
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1138
  unfolding a apply simp
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1139
  unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1140
  unfolding assm unfolding le_less apply simp_all done
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1141
next
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1142
  assume "a = 0" then show "a + a = 0" by simp
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1143
qed
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1144
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1145
lemma zero_less_double_add_iff_zero_less_single_add:
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1146
  "0 < a + a \<longleftrightarrow> 0 < a"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1147
proof (cases "a = 0")
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1148
  case True then show ?thesis by auto
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1149
next
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1150
  case False then show ?thesis (*FIXME tune proof*)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1151
  unfolding less_le apply simp apply rule
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1152
  apply clarify
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1153
  apply rule
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1154
  apply assumption
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1155
  apply (rule notI)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1156
  unfolding double_zero [symmetric, of a] apply simp
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1157
  done
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1158
qed
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1159
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1160
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1161
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1162
proof -
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1163
  have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1164
  moreover have "\<dots> \<longleftrightarrow> a \<le> 0" by (simp add: zero_le_double_add_iff_zero_le_single_add)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1165
  ultimately show ?thesis by blast
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1166
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1167
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1168
lemma double_add_less_zero_iff_single_less_zero [simp]:
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1169
  "a + a < 0 \<longleftrightarrow> a < 0"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1170
proof -
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1171
  have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1172
  moreover have "\<dots> \<longleftrightarrow> a < 0" by (simp add: zero_less_double_add_iff_zero_less_single_add)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1173
  ultimately show ?thesis by blast
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1174
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1175
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1176
declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp]
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1177
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1178
lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1179
proof -
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1180
  from add_le_cancel_left [of "uminus a" "plus a a" zero]
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1181
  have "(a <= -a) = (a+a <= 0)" 
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1182
    by (simp add: add_assoc[symmetric])
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1183
  thus ?thesis by simp
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1184
qed
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1185
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1186
lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1187
proof -
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1188
  from add_le_cancel_left [of "uminus a" zero "plus a a"]
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1189
  have "(-a <= a) = (0 <= a+a)" 
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1190
    by (simp add: add_assoc[symmetric])
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1191
  thus ?thesis by simp
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1192
qed
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1193
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1194
lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1195
by (simp add: le_iff_inf nprt_def inf_commute)
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1196
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1197
lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1198
by (simp add: le_iff_sup pprt_def sup_commute)
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1199
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1200
lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1201
by (simp add: le_iff_sup pprt_def sup_commute)
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1202
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1203
lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1204
by (simp add: le_iff_inf nprt_def inf_commute)
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1205
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1206
lemma pprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1207
by (simp add: le_iff_sup pprt_def sup_ACI sup_assoc [symmetric, of a])
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1208
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1209
lemma nprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1210
by (simp add: le_iff_inf nprt_def inf_ACI inf_assoc [symmetric, of a])
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1211
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1212
end
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1213
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1214
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1215
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1216
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1217
class lordered_ab_group_add_abs = lordered_ab_group_add + abs +
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1218
  assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1219
begin
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1220
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1221
lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1222
proof -
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1223
  have "0 \<le> \<bar>a\<bar>"
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1224
  proof -
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1225
    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1226
    show ?thesis by (rule add_mono [OF a b, simplified])
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1227
  qed
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1228
  then have "0 \<le> sup a (- a)" unfolding abs_lattice .
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1229
  then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1230
  then show ?thesis
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1231
    by (simp add: add_sup_inf_distribs sup_ACI
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1232
      pprt_def nprt_def diff_minus abs_lattice)
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1233
qed
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1234
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1235
subclass pordered_ab_group_add_abs
29557
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1236
proof
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1237
  have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1238
  proof -
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1239
    fix a b
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1240
    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1241
    show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified])
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1242
  qed
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1243
  have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1244
    by (simp add: abs_lattice le_supI)
29557
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1245
  fix a b
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1246
  show "0 \<le> \<bar>a\<bar>" by simp
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1247
  show "a \<le> \<bar>a\<bar>"
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1248
    by (auto simp add: abs_lattice)
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1249
  show "\<bar>-a\<bar> = \<bar>a\<bar>"
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1250
    by (simp add: abs_lattice sup_commute)
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1251
  show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (fact abs_leI)
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1252
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1253
  proof -
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1254
    have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1255
      by (simp add: abs_lattice add_sup_inf_distribs sup_ACI diff_minus)
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1256
    have a:"a+b <= sup ?m ?n" by (simp)
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1257
    have b:"-a-b <= ?n" by (simp) 
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1258
    have c:"?n <= sup ?m ?n" by (simp)
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1259
    from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1260
    have e:"-a-b = -(a+b)" by (simp add: diff_minus)
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1261
    from a d e have "abs(a+b) <= sup ?m ?n" 
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1262
      by (drule_tac abs_leI, auto)
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1263
    with g[symmetric] show ?thesis by simp
5362cc5ee3a8 tuned proof
haftmann
parents: 29269
diff changeset
  1264
  qed
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1265
qed
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1266
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1267
end
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1268
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1269
lemma sup_eq_if:
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1270
  fixes a :: "'a\<Colon>{lordered_ab_group_add, linorder}"
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1271
  shows "sup a (- a) = (if a < 0 then - a else a)"
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1272
proof -
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1273
  note add_le_cancel_right [of a a "- a", symmetric, simplified]
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1274
  moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified]
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1275
  then show ?thesis by (auto simp: sup_max max_def)
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1276
qed
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1277
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1278
lemma abs_if_lattice:
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1279
  fixes a :: "'a\<Colon>{lordered_ab_group_add_abs, linorder}"
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1280
  shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1281
by auto
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1282
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1283
14754
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1284
text {* Needed for abelian cancellation simprocs: *}
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1285
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1286
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
  1287
apply (subst add_left_commute)
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1288
apply (subst add_left_cancel)
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1289
apply simp
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1290
done
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1291
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1292
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
  1293
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
  1294
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
  1295
done
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1296
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1297
lemma less_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1298
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
  1299
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1300
lemma le_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1301
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
  1302
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
  1303
done
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1304
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1305
lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')"
30629
5cd9b19edef3 move diff_eq_0_iff_eq into class locale context
huffman
parents: 29914
diff changeset
  1306
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
  1307
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1308
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
  1309
by (simp add: diff_minus)
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1310
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1311
lemma add_minus_cancel: "(a::'a::ab_group_add) + (-a + b) = b"
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1312
by (simp add: add_assoc[symmetric])
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1313
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1314
lemma le_add_right_mono: 
15178
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1315
  assumes 
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1316
  "a <= b + (c::'a::pordered_ab_group_add)"
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1317
  "c <= d"    
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1318
  shows "a <= b + d"
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1319
  apply (rule_tac order_trans[where y = "b+c"])
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1320
  apply (simp_all add: prems)
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1321
  done
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1322
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1323
lemma estimate_by_abs:
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1324
  "a + b <= (c::'a::lordered_ab_group_add_abs) \<Longrightarrow> a <= c + abs b" 
15178
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1325
proof -
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23389
diff changeset
  1326
  assume "a+b <= c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1327
  hence 2: "a <= c+(-b)" by (simp add: algebra_simps)
15178
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1328
  have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1329
  show ?thesis by (rule le_add_right_mono[OF 2 3])
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1330
qed
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1331
25090
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1332
subsection {* Tools setup *}
4a50b958391a 98% localized
haftmann
parents: 25077
diff changeset
  1333
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1334
lemma add_mono_thms_ordered_semiring [noatp]:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1335
  fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1336
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1337
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1338
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1339
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1340
by (rule add_mono, clarify+)+
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1341
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1342
lemma add_mono_thms_ordered_field [noatp]:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1343
  fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1344
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1345
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1346
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1347
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1348
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1349
by (auto intro: add_strict_right_mono add_strict_left_mono
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1350
  add_less_le_mono add_le_less_mono add_strict_mono)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1351
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1352
text{*Simplification of @{term "x-y < 0"}, etc.*}
29833
409138c4de12 added noatps
nipkow
parents: 29670
diff changeset
  1353
lemmas diff_less_0_iff_less [simp, noatp] = less_iff_diff_less_0 [symmetric]
409138c4de12 added noatps
nipkow
parents: 29670
diff changeset
  1354
lemmas diff_le_0_iff_le [simp, noatp] = le_iff_diff_le_0 [symmetric]
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1355
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1356
ML {*
27250
7eef2b183032 simplified Abel_Cancel setup;
wenzelm
parents: 26480
diff changeset
  1357
structure ab_group_add_cancel = Abel_Cancel
7eef2b183032 simplified Abel_Cancel setup;
wenzelm
parents: 26480
diff changeset
  1358
(
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1359
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1360
(* term order for abelian groups *)
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1361
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1362
fun agrp_ord (Const (a, _)) = find_index (fn a' => a = a')
22997
d4f3b015b50b canonical prefixing of class constants
haftmann
parents: 22986
diff changeset
  1363
      [@{const_name HOL.zero}, @{const_name HOL.plus},
d4f3b015b50b canonical prefixing of class constants
haftmann
parents: 22986
diff changeset
  1364
        @{const_name HOL.uminus}, @{const_name HOL.minus}]
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1365
  | agrp_ord _ = ~1;
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1366
29269
5c25a2012975 moved term order operations to structure TermOrd (cf. Pure/term_ord.ML);
wenzelm
parents: 28823
diff changeset
  1367
fun termless_agrp (a, b) = (TermOrd.term_lpo agrp_ord (a, b) = LESS);
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1368
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1369
local
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1370
  val ac1 = mk_meta_eq @{thm add_assoc};
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1371
  val ac2 = mk_meta_eq @{thm add_commute};
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1372
  val ac3 = mk_meta_eq @{thm add_left_commute};
22997
d4f3b015b50b canonical prefixing of class constants
haftmann
parents: 22986
diff changeset
  1373
  fun solve_add_ac thy _ (_ $ (Const (@{const_name HOL.plus},_) $ _ $ _) $ _) =
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1374
        SOME ac1
22997
d4f3b015b50b canonical prefixing of class constants
haftmann
parents: 22986
diff changeset
  1375
    | solve_add_ac thy _ (_ $ x $ (Const (@{const_name HOL.plus},_) $ y $ z)) =
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1376
        if termless_agrp (y, x) then SOME ac3 else NONE
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1377
    | solve_add_ac thy _ (_ $ x $ y) =
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1378
        if termless_agrp (y, x) then SOME ac2 else NONE
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1379
    | solve_add_ac thy _ _ = NONE
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1380
in
28262
aa7ca36d67fd back to dynamic the_context(), because static @{theory} is invalidated if ML environment changes within the same code block;
wenzelm
parents: 28130
diff changeset
  1381
  val add_ac_proc = Simplifier.simproc (the_context ())
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1382
    "add_ac_proc" ["x + y::'a::ab_semigroup_add"] solve_add_ac;
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1383
end;
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1384
27250
7eef2b183032 simplified Abel_Cancel setup;
wenzelm
parents: 26480
diff changeset
  1385
val eq_reflection = @{thm eq_reflection};
7eef2b183032 simplified Abel_Cancel setup;
wenzelm
parents: 26480
diff changeset
  1386
  
7eef2b183032 simplified Abel_Cancel setup;
wenzelm
parents: 26480
diff changeset
  1387
val T = @{typ "'a::ab_group_add"};
7eef2b183032 simplified Abel_Cancel setup;
wenzelm
parents: 26480
diff changeset
  1388
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1389
val cancel_ss = HOL_basic_ss settermless termless_agrp
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1390
  addsimprocs [add_ac_proc] addsimps
23085
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
  1391
  [@{thm add_0_left}, @{thm add_0_right}, @{thm diff_def},
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1392
   @{thm minus_add_distrib}, @{thm minus_minus}, @{thm minus_zero},
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1393
   @{thm right_minus}, @{thm left_minus}, @{thm add_minus_cancel},
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1394
   @{thm minus_add_cancel}];
27250
7eef2b183032 simplified Abel_Cancel setup;
wenzelm
parents: 26480
diff changeset
  1395
7eef2b183032 simplified Abel_Cancel setup;
wenzelm
parents: 26480
diff changeset
  1396
val sum_pats = [@{cterm "x + y::'a::ab_group_add"}, @{cterm "x - y::'a::ab_group_add"}];
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1397
  
22548
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22482
diff changeset
  1398
val eqI_rules = [@{thm less_eqI}, @{thm le_eqI}, @{thm eq_eqI}];
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1399
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1400
val dest_eqI = 
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1401
  fst o HOLogic.dest_bin "op =" HOLogic.boolT o HOLogic.dest_Trueprop o concl_of;
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1402
27250
7eef2b183032 simplified Abel_Cancel setup;
wenzelm
parents: 26480
diff changeset
  1403
);
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1404
*}
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1405
26480
544cef16045b replaced 'ML_setup' by 'ML';
wenzelm
parents: 26071
diff changeset
  1406
ML {*
22482
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1407
  Addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv];
8fc3d7237e03 dropped OrderedGroup.ML
haftmann
parents: 22452
diff changeset
  1408
*}
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1409
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1410
end