src/HOL/OrderedGroup.thy
author wenzelm
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proper definition of add_zero_left/right;
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(*  Title:   HOL/OrderedGroup.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel,
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             with contributions by Jeremy Avigad
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*)
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header {* Ordered Groups *}
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theory OrderedGroup
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imports Inductive LOrder
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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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subsection {* Semigroups, Groups *}
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axclass semigroup_add \<subseteq> plus
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  add_assoc: "(a + b) + c = a + (b + c)"
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axclass ab_semigroup_add \<subseteq> semigroup_add
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  add_commute: "a + b = b + a"
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lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::ab_semigroup_add))"
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  by (rule mk_left_commute [of "op +", OF add_assoc add_commute])
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theorems add_ac = add_assoc add_commute add_left_commute
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axclass semigroup_mult \<subseteq> times
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  mult_assoc: "(a * b) * c = a * (b * c)"
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axclass ab_semigroup_mult \<subseteq> semigroup_mult
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  mult_commute: "a * b = b * a"
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lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::ab_semigroup_mult))"
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  by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute])
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theorems mult_ac = mult_assoc mult_commute mult_left_commute
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axclass comm_monoid_add \<subseteq> zero, ab_semigroup_add
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  add_0[simp]: "0 + a = a"
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axclass monoid_mult \<subseteq> one, semigroup_mult
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  mult_1_left[simp]: "1 * a  = a"
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  mult_1_right[simp]: "a * 1 = a"
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axclass comm_monoid_mult \<subseteq> one, ab_semigroup_mult
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  mult_1: "1 * a = a"
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instance comm_monoid_mult \<subseteq> monoid_mult
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by (intro_classes, insert mult_1, simp_all add: mult_commute, auto)
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axclass cancel_semigroup_add \<subseteq> semigroup_add
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  add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
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  add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
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axclass cancel_ab_semigroup_add \<subseteq> ab_semigroup_add
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  add_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
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instance cancel_ab_semigroup_add \<subseteq> cancel_semigroup_add
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proof
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  {
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    fix a b c :: 'a
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    assume "a + b = a + c"
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    thus "b = c" by (rule add_imp_eq)
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  }
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  note f = this
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  fix a b c :: 'a
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  assume "b + a = c + a"
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  hence "a + b = a + c" by (simp only: add_commute)
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  thus "b = c" by (rule f)
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qed
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axclass ab_group_add \<subseteq> minus, comm_monoid_add
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  left_minus[simp]: " - a + a = 0"
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  diff_minus: "a - b = a + (-b)"
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instance ab_group_add \<subseteq> cancel_ab_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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  hence "-a + a + b = -a + a + c" by (simp only: add_assoc)
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  thus "b = c" by simp 
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qed
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lemma add_0_right [simp]: "a + 0 = (a::'a::comm_monoid_add)"
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proof -
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  have "a + 0 = 0 + a" by (simp only: add_commute)
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  also have "... = a" by simp
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  finally show ?thesis .
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qed
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lemmas add_zero_left = add_0
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  and add_zero_right = add_0_right
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lemma add_left_cancel [simp]:
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     "(a + b = a + c) = (b = (c::'a::cancel_semigroup_add))"
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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) = (b = (c::'a::cancel_semigroup_add))"
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  by (blast dest: add_right_imp_eq)
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lemma right_minus [simp]: "a + -(a::'a::ab_group_add) = 0"
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proof -
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  have "a + -a = -a + a" by (simp add: add_ac)
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  also have "... = 0" by simp
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  finally show ?thesis .
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qed
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lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::ab_group_add))"
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proof
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  have "a = a - b + b" by (simp add: diff_minus add_ac)
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  also assume "a - b = 0"
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  finally show "a = b" by simp
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next
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  assume "a = b"
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  thus "a - b = 0" by (simp add: diff_minus)
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qed
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lemma minus_minus [simp]: "- (- (a::'a::ab_group_add)) = a"
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proof (rule add_left_cancel [of "-a", THEN iffD1])
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  show "(-a + -(-a) = -a + a)"
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  by simp
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qed
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lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::ab_group_add)"
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apply (rule right_minus_eq [THEN iffD1, symmetric])
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apply (simp add: diff_minus add_commute) 
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done
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lemma minus_zero [simp]: "- 0 = (0::'a::ab_group_add)"
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by (simp add: equals_zero_I)
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lemma diff_self [simp]: "a - (a::'a::ab_group_add) = 0"
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  by (simp add: diff_minus)
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lemma diff_0 [simp]: "(0::'a::ab_group_add) - a = -a"
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by (simp add: diff_minus)
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lemma diff_0_right [simp]: "a - (0::'a::ab_group_add) = a" 
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by (simp add: diff_minus)
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lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::ab_group_add)"
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by (simp add: diff_minus)
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lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ab_group_add))" 
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obua
parents:
diff changeset
   159
proof 
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obua
parents:
diff changeset
   160
  assume "- a = - b"
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obua
parents:
diff changeset
   161
  hence "- (- a) = - (- b)"
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obua
parents:
diff changeset
   162
    by simp
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obua
parents:
diff changeset
   163
  thus "a=b" by simp
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obua
parents:
diff changeset
   164
next
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parents:
diff changeset
   165
  assume "a=b"
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obua
parents:
diff changeset
   166
  thus "-a = -b" by simp
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obua
parents:
diff changeset
   167
qed
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obua
parents:
diff changeset
   168
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   169
lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ab_group_add))"
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parents:
diff changeset
   170
by (subst neg_equal_iff_equal [symmetric], simp)
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obua
parents:
diff changeset
   171
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obua
parents:
diff changeset
   172
lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ab_group_add))"
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obua
parents:
diff changeset
   173
by (subst neg_equal_iff_equal [symmetric], simp)
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obua
parents:
diff changeset
   174
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obua
parents:
diff changeset
   175
text{*The next two equations can make the simplifier loop!*}
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obua
parents:
diff changeset
   176
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obua
parents:
diff changeset
   177
lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ab_group_add))"
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obua
parents:
diff changeset
   178
proof -
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obua
parents:
diff changeset
   179
  have "(- (-a) = - b) = (- a = b)" by (rule neg_equal_iff_equal)
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obua
parents:
diff changeset
   180
  thus ?thesis by (simp add: eq_commute)
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obua
parents:
diff changeset
   181
qed
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obua
parents:
diff changeset
   182
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   183
lemma minus_equation_iff: "(- a = b) = (- (b::'a::ab_group_add) = a)"
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obua
parents:
diff changeset
   184
proof -
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obua
parents:
diff changeset
   185
  have "(- a = - (-b)) = (a = -b)" by (rule neg_equal_iff_equal)
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obua
parents:
diff changeset
   186
  thus ?thesis by (simp add: eq_commute)
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obua
parents:
diff changeset
   187
qed
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obua
parents:
diff changeset
   188
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   189
lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ab_group_add)"
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obua
parents:
diff changeset
   190
apply (rule equals_zero_I)
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obua
parents:
diff changeset
   191
apply (simp add: add_ac) 
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obua
parents:
diff changeset
   192
done
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obua
parents:
diff changeset
   193
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   194
lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ab_group_add)"
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obua
parents:
diff changeset
   195
by (simp add: diff_minus add_commute)
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obua
parents:
diff changeset
   196
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   197
subsection {* (Partially) Ordered Groups *} 
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obua
parents:
diff changeset
   198
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parents:
diff changeset
   199
axclass pordered_ab_semigroup_add \<subseteq> order, ab_semigroup_add
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parents:
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   200
  add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
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obua
parents:
diff changeset
   201
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   202
axclass pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add, cancel_ab_semigroup_add
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obua
parents:
diff changeset
   203
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   204
instance pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add ..
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obua
parents:
diff changeset
   205
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obua
parents:
diff changeset
   206
axclass pordered_ab_semigroup_add_imp_le \<subseteq> pordered_cancel_ab_semigroup_add
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obua
parents:
diff changeset
   207
  add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
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obua
parents:
diff changeset
   208
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   209
axclass pordered_ab_group_add \<subseteq> ab_group_add, pordered_ab_semigroup_add
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obua
parents:
diff changeset
   210
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   211
instance pordered_ab_group_add \<subseteq> pordered_ab_semigroup_add_imp_le
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obua
parents:
diff changeset
   212
proof
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obua
parents:
diff changeset
   213
  fix a b c :: 'a
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obua
parents:
diff changeset
   214
  assume "c + a \<le> c + b"
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obua
parents:
diff changeset
   215
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   216
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
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obua
parents:
diff changeset
   217
  thus "a \<le> b" by simp
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obua
parents:
diff changeset
   218
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   219
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   220
axclass ordered_cancel_ab_semigroup_add \<subseteq> pordered_cancel_ab_semigroup_add, linorder
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obua
parents:
diff changeset
   221
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   222
instance ordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add_imp_le
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obua
parents:
diff changeset
   223
proof
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   224
  fix a b c :: 'a
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obua
parents:
diff changeset
   225
  assume le: "c + a <= c + b"  
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obua
parents:
diff changeset
   226
  show "a <= b"
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obua
parents:
diff changeset
   227
  proof (rule ccontr)
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obua
parents:
diff changeset
   228
    assume w: "~ a \<le> b"
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obua
parents:
diff changeset
   229
    hence "b <= a" by (simp add: linorder_not_le)
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obua
parents:
diff changeset
   230
    hence le2: "c+b <= c+a" by (rule add_left_mono)
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obua
parents:
diff changeset
   231
    have "a = b" 
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obua
parents:
diff changeset
   232
      apply (insert le)
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obua
parents:
diff changeset
   233
      apply (insert le2)
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obua
parents:
diff changeset
   234
      apply (drule order_antisym, simp_all)
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obua
parents:
diff changeset
   235
      done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   236
    with w  show False 
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obua
parents:
diff changeset
   237
      by (simp add: linorder_not_le [symmetric])
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obua
parents:
diff changeset
   238
  qed
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obua
parents:
diff changeset
   239
qed
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obua
parents:
diff changeset
   240
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   241
lemma add_right_mono: "a \<le> (b::'a::pordered_ab_semigroup_add) ==> a + c \<le> b + c"
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obua
parents:
diff changeset
   242
by (simp add: add_commute[of _ c] add_left_mono)
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obua
parents:
diff changeset
   243
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   244
text {* non-strict, in both arguments *}
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obua
parents:
diff changeset
   245
lemma add_mono:
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obua
parents:
diff changeset
   246
     "[|a \<le> b;  c \<le> d|] ==> a + c \<le> b + (d::'a::pordered_ab_semigroup_add)"
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obua
parents:
diff changeset
   247
  apply (erule add_right_mono [THEN order_trans])
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obua
parents:
diff changeset
   248
  apply (simp add: add_commute add_left_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   249
  done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   250
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   251
lemma add_strict_left_mono:
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obua
parents:
diff changeset
   252
     "a < b ==> c + a < c + (b::'a::pordered_cancel_ab_semigroup_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   253
 by (simp add: order_less_le add_left_mono) 
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obua
parents:
diff changeset
   254
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   255
lemma add_strict_right_mono:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   256
     "a < b ==> a + c < b + (c::'a::pordered_cancel_ab_semigroup_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   257
 by (simp add: add_commute [of _ c] add_strict_left_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   258
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   259
text{*Strict monotonicity in both arguments*}
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obua
parents:
diff changeset
   260
lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   261
apply (erule add_strict_right_mono [THEN order_less_trans])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   262
apply (erule add_strict_left_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   263
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   264
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   265
lemma add_less_le_mono:
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obua
parents:
diff changeset
   266
     "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   267
apply (erule add_strict_right_mono [THEN order_less_le_trans])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   268
apply (erule add_left_mono) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   269
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   270
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   271
lemma add_le_less_mono:
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obua
parents:
diff changeset
   272
     "[| a\<le>b; c<d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   273
apply (erule add_right_mono [THEN order_le_less_trans])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   274
apply (erule add_strict_left_mono) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   275
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   276
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   277
lemma add_less_imp_less_left:
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obua
parents:
diff changeset
   278
      assumes less: "c + a < c + b"  shows "a < (b::'a::pordered_ab_semigroup_add_imp_le)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   279
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   280
  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
   281
  have "a <= b" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   282
    apply (insert le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   283
    apply (drule add_le_imp_le_left)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   284
    by (insert le, drule add_le_imp_le_left, assumption)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   285
  moreover have "a \<noteq> b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   286
  proof (rule ccontr)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   287
    assume "~(a \<noteq> b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   288
    then have "a = b" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   289
    then have "c + a = c + b" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   290
    with less show "False"by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   291
  qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   292
  ultimately show "a < b" by (simp add: order_le_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   293
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   294
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   295
lemma add_less_imp_less_right:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   296
      "a + c < b + c ==> a < (b::'a::pordered_ab_semigroup_add_imp_le)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   297
apply (rule add_less_imp_less_left [of c])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   298
apply (simp add: add_commute)  
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   299
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   300
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   301
lemma add_less_cancel_left [simp]:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   302
    "(c+a < c+b) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   303
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   304
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   305
lemma add_less_cancel_right [simp]:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   306
    "(a+c < b+c) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   307
by (blast intro: add_less_imp_less_right add_strict_right_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   308
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   309
lemma add_le_cancel_left [simp]:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   310
    "(c+a \<le> c+b) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   311
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   312
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   313
lemma add_le_cancel_right [simp]:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   314
    "(a+c \<le> b+c) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   315
by (simp add: add_commute[of a c] add_commute[of b c])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   316
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   317
lemma add_le_imp_le_right:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   318
      "a + c \<le> b + c ==> a \<le> (b::'a::pordered_ab_semigroup_add_imp_le)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   319
by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   320
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   321
lemma add_increasing:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   322
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   323
  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
   324
by (insert add_mono [of 0 a b c], simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   325
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   326
lemma add_increasing2:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   327
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   328
  shows  "[|0\<le>c; b\<le>a|] ==> b \<le> a + c"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   329
by (simp add:add_increasing add_commute[of a])
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   330
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   331
lemma add_strict_increasing:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   332
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   333
  shows "[|0<a; b\<le>c|] ==> b < a + c"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   334
by (insert add_less_le_mono [of 0 a b c], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   335
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   336
lemma add_strict_increasing2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   337
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   338
  shows "[|0\<le>a; b<c|] ==> b < a + c"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   339
by (insert add_le_less_mono [of 0 a b c], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   340
19527
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   341
lemma max_add_distrib_left:
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   342
  fixes z :: "'a::pordered_ab_semigroup_add_imp_le"
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   343
  shows  "(max x y) + z = max (x+z) (y+z)"
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   344
by (rule max_of_mono [THEN sym], rule add_le_cancel_right)
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   345
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   346
lemma min_add_distrib_left:
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   347
  fixes z :: "'a::pordered_ab_semigroup_add_imp_le"
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   348
  shows  "(min x y) + z = min (x+z) (y+z)"
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   349
by (rule min_of_mono [THEN sym], rule add_le_cancel_right)
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   350
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   351
lemma max_diff_distrib_left:
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   352
  fixes z :: "'a::pordered_ab_group_add"
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   353
  shows  "(max x y) - z = max (x-z) (y-z)"
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   354
by (simp add: diff_minus, rule max_add_distrib_left) 
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   355
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   356
lemma min_diff_distrib_left:
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   357
  fixes z :: "'a::pordered_ab_group_add"
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   358
  shows  "(min x y) - z = min (x-z) (y-z)"
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   359
by (simp add: diff_minus, rule min_add_distrib_left) 
9b5c38e8e780 some facts about min, max and add, diff
paulson
parents: 19404
diff changeset
   360
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   361
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   362
subsection {* Ordering Rules for Unary Minus *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   363
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   364
lemma le_imp_neg_le:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   365
      assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "-b \<le> -a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   366
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   367
  have "-a+a \<le> -a+b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   368
    by (rule add_left_mono) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   369
  hence "0 \<le> -a+b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   370
    by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   371
  hence "0 + (-b) \<le> (-a + b) + (-b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   372
    by (rule add_right_mono) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   373
  thus ?thesis
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   374
    by (simp add: add_assoc)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   375
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   376
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   377
lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   378
proof 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   379
  assume "- b \<le> - a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   380
  hence "- (- a) \<le> - (- b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   381
    by (rule le_imp_neg_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   382
  thus "a\<le>b" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   383
next
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   384
  assume "a\<le>b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   385
  thus "-b \<le> -a" by (rule le_imp_neg_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   386
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   387
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   388
lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   389
by (subst neg_le_iff_le [symmetric], simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   390
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   391
lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   392
by (subst neg_le_iff_le [symmetric], simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   393
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   394
lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   395
by (force simp add: order_less_le) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   396
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   397
lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   398
by (subst neg_less_iff_less [symmetric], simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   399
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   400
lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   401
by (subst neg_less_iff_less [symmetric], simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   402
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   403
text{*The next several equations can make the simplifier loop!*}
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 less_minus_iff: "(a < - b) = (b < - (a::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   406
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   407
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   408
  thus ?thesis by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   409
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   410
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   411
lemma minus_less_iff: "(- a < b) = (- b < (a::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   412
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   413
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   414
  thus ?thesis by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   415
qed
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 le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   418
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   419
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   420
  have "(- (- a) <= -b) = (b <= - a)" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   421
    apply (auto simp only: order_le_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   422
    apply (drule mm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   423
    apply (simp_all)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   424
    apply (drule mm[simplified], assumption)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   425
    done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   426
  then show ?thesis by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   427
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   428
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   429
lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   430
by (auto simp add: order_le_less minus_less_iff)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   431
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   432
lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ab_group_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   433
by (simp add: diff_minus add_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   434
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   435
lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ab_group_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   436
by (simp add: diff_minus add_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   437
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   438
lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   439
by (auto simp add: diff_minus add_assoc)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   440
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   441
lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ab_group_add) = c)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   442
by (auto simp add: diff_minus add_assoc)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   443
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   444
lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   445
by (simp add: diff_minus add_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   446
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   447
lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ab_group_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   448
by (simp add: diff_minus add_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   449
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   450
lemma diff_add_cancel: "a - b + b = (a::'a::ab_group_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   451
by (simp add: diff_minus add_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   452
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   453
lemma add_diff_cancel: "a + b - b = (a::'a::ab_group_add)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   454
by (simp add: diff_minus add_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   455
14754
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   456
text{*Further subtraction laws*}
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   457
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   458
lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   459
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   460
  have  "(a < b) = (a + (- b) < b + (-b))"  
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   461
    by (simp only: add_less_cancel_right)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   462
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   463
  finally show ?thesis .
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   464
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   465
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   466
lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::pordered_ab_group_add))"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15234
diff changeset
   467
apply (subst less_iff_diff_less_0 [of a])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   468
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   469
apply (simp add: diff_minus add_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   470
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   471
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   472
lemma less_diff_eq: "(a < c-b) = (a + (b::'a::pordered_ab_group_add) < c)"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15234
diff changeset
   473
apply (subst less_iff_diff_less_0 [of "a+b"])
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15234
diff changeset
   474
apply (subst less_iff_diff_less_0 [of a])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   475
apply (simp add: diff_minus add_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   476
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   477
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   478
lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   479
by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   480
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   481
lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::pordered_ab_group_add) \<le> c)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   482
by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   483
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   484
text{*This list of rewrites simplifies (in)equalities by bringing subtractions
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   485
  to the top and then moving negative terms to the other side.
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   486
  Use with @{text add_ac}*}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   487
lemmas compare_rls =
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   488
       diff_minus [symmetric]
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   489
       add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   490
       diff_less_eq less_diff_eq diff_le_eq le_diff_eq
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   491
       diff_eq_eq eq_diff_eq
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   492
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   493
subsection {* Support for reasoning about signs *}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   494
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   495
lemma add_pos_pos: "0 < 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   496
    (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   497
      ==> 0 < y ==> 0 < x + y"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   498
apply (subgoal_tac "0 + 0 < x + y")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   499
apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   500
apply (erule add_less_le_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   501
apply (erule order_less_imp_le)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   502
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   503
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   504
lemma add_pos_nonneg: "0 < 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   505
    (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   506
      ==> 0 <= y ==> 0 < x + y"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   507
apply (subgoal_tac "0 + 0 < x + y")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   508
apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   509
apply (erule add_less_le_mono, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   510
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   511
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   512
lemma add_nonneg_pos: "0 <= 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   513
    (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   514
      ==> 0 < y ==> 0 < x + y"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   515
apply (subgoal_tac "0 + 0 < x + y")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   516
apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   517
apply (erule add_le_less_mono, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   518
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   519
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   520
lemma add_nonneg_nonneg: "0 <= 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   521
    (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   522
      ==> 0 <= y ==> 0 <= x + y"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   523
apply (subgoal_tac "0 + 0 <= x + y")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   524
apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   525
apply (erule add_mono, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   526
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   527
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   528
lemma add_neg_neg: "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add})
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   529
    < 0 ==> y < 0 ==> x + y < 0"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   530
apply (subgoal_tac "x + y < 0 + 0")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   531
apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   532
apply (erule add_less_le_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   533
apply (erule order_less_imp_le)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   534
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   535
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   536
lemma add_neg_nonpos: 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   537
    "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) < 0 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   538
      ==> y <= 0 ==> x + y < 0"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   539
apply (subgoal_tac "x + y < 0 + 0")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   540
apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   541
apply (erule add_less_le_mono, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   542
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   543
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   544
lemma add_nonpos_neg: 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   545
    "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   546
      ==> y < 0 ==> x + y < 0"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   547
apply (subgoal_tac "x + y < 0 + 0")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   548
apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   549
apply (erule add_le_less_mono, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   550
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   551
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   552
lemma add_nonpos_nonpos: 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   553
    "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   554
      ==> y <= 0 ==> x + y <= 0"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   555
apply (subgoal_tac "x + y <= 0 + 0")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   556
apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   557
apply (erule add_mono, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   558
done
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   559
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   560
subsection{*Lemmas for the @{text cancel_numerals} simproc*}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   561
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   562
lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   563
by (simp add: compare_rls)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   564
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   565
lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::pordered_ab_group_add))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   566
by (simp add: compare_rls)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   567
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   568
subsection {* Lattice Ordered (Abelian) Groups *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   569
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   570
axclass lordered_ab_group_meet < pordered_ab_group_add, meet_semilorder
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   571
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   572
axclass lordered_ab_group_join < pordered_ab_group_add, join_semilorder
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   573
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   574
lemma add_meet_distrib_left: "a + (meet b c) = meet (a + b) (a + (c::'a::{pordered_ab_group_add, meet_semilorder}))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   575
apply (rule order_antisym)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   576
apply (rule meet_imp_le, simp_all add: meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   577
apply (rule add_le_imp_le_left [of "-a"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   578
apply (simp only: add_assoc[symmetric], simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   579
apply (rule meet_imp_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   580
apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   581
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   582
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   583
lemma add_join_distrib_left: "a + (join b c) = join (a + b) (a+ (c::'a::{pordered_ab_group_add, join_semilorder}))" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   584
apply (rule order_antisym)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   585
apply (rule add_le_imp_le_left [of "-a"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   586
apply (simp only: add_assoc[symmetric], simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   587
apply (rule join_imp_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   588
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   589
apply (rule join_imp_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   590
apply (simp_all add: meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   591
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   592
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   593
lemma is_join_neg_meet: "is_join (% (a::'a::{pordered_ab_group_add, meet_semilorder}) b. - (meet (-a) (-b)))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   594
apply (auto simp add: is_join_def)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   595
apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   596
apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   597
apply (subst neg_le_iff_le[symmetric]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   598
apply (simp add: meet_imp_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   599
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   600
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   601
lemma is_meet_neg_join: "is_meet (% (a::'a::{pordered_ab_group_add, join_semilorder}) b. - (join (-a) (-b)))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   602
apply (auto simp add: is_meet_def)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   603
apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   604
apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   605
apply (subst neg_le_iff_le[symmetric]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   606
apply (simp add: join_imp_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   607
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   608
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   609
axclass lordered_ab_group \<subseteq> pordered_ab_group_add, lorder
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   610
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   611
instance lordered_ab_group_meet \<subseteq> lordered_ab_group
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   612
proof 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   613
  show "? j. is_join (j::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_meet))" by (blast intro: is_join_neg_meet)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   614
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   615
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   616
instance lordered_ab_group_join \<subseteq> lordered_ab_group
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   617
proof
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   618
  show "? m. is_meet (m::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_join))" by (blast intro: is_meet_neg_join)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   619
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   620
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   621
lemma add_join_distrib_right: "(join a b) + (c::'a::lordered_ab_group) = join (a+c) (b+c)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   622
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   623
  have "c + (join a b) = join (c+a) (c+b)" by (simp add: add_join_distrib_left)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   624
  thus ?thesis by (simp add: add_commute)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   625
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   626
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   627
lemma add_meet_distrib_right: "(meet a b) + (c::'a::lordered_ab_group) = meet (a+c) (b+c)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   628
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   629
  have "c + (meet a b) = meet (c+a) (c+b)" by (simp add: add_meet_distrib_left)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   630
  thus ?thesis by (simp add: add_commute)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   631
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   632
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   633
lemmas add_meet_join_distribs = add_meet_distrib_right add_meet_distrib_left add_join_distrib_right add_join_distrib_left
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   634
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   635
lemma join_eq_neg_meet: "join a (b::'a::lordered_ab_group) = - meet (-a) (-b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   636
by (simp add: is_join_unique[OF is_join_join is_join_neg_meet])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   637
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   638
lemma meet_eq_neg_join: "meet a (b::'a::lordered_ab_group) = - join (-a) (-b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   639
by (simp add: is_meet_unique[OF is_meet_meet is_meet_neg_join])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   640
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   641
lemma add_eq_meet_join: "a + b = (join a b) + (meet a (b::'a::lordered_ab_group))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   642
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   643
  have "0 = - meet 0 (a-b) + meet (a-b) 0" by (simp add: meet_comm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   644
  hence "0 = join 0 (b-a) + meet (a-b) 0" by (simp add: meet_eq_neg_join)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   645
  hence "0 = (-a + join a b) + (meet a b + (-b))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   646
    apply (simp add: add_join_distrib_left add_meet_distrib_right)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   647
    by (simp add: diff_minus add_commute)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   648
  thus ?thesis
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   649
    apply (simp add: compare_rls)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   650
    apply (subst add_left_cancel[symmetric, of "a+b" "join a b + meet a b" "-a"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   651
    apply (simp only: add_assoc, simp add: add_assoc[symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   652
    done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   653
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   654
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   655
subsection {* Positive Part, Negative Part, Absolute Value *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   656
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   657
constdefs
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   658
  pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   659
  "pprt x == join x 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   660
  nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   661
  "nprt x == meet x 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   662
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   663
lemma prts: "a = pprt a + nprt a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   664
by (simp add: pprt_def nprt_def add_eq_meet_join[symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   665
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   666
lemma zero_le_pprt[simp]: "0 \<le> pprt a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   667
by (simp add: pprt_def meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   668
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   669
lemma nprt_le_zero[simp]: "nprt a \<le> 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   670
by (simp add: nprt_def meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   671
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   672
lemma le_eq_neg: "(a \<le> -b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r")
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   673
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   674
  have a: "?l \<longrightarrow> ?r"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   675
    apply (auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   676
    apply (rule add_le_imp_le_right[of _ "-b" _])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   677
    apply (simp add: add_assoc)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   678
    done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   679
  have b: "?r \<longrightarrow> ?l"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   680
    apply (auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   681
    apply (rule add_le_imp_le_right[of _ "b" _])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   682
    apply (simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   683
    done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   684
  from a b show ?thesis by blast
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   685
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   686
15580
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   687
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   688
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   689
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   690
lemma pprt_eq_id[simp]: "0 <= x \<Longrightarrow> pprt x = x"
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   691
  by (simp add: pprt_def le_def_join join_aci)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   692
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   693
lemma nprt_eq_id[simp]: "x <= 0 \<Longrightarrow> nprt x = x"
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   694
  by (simp add: nprt_def le_def_meet meet_aci)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   695
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   696
lemma pprt_eq_0[simp]: "x <= 0 \<Longrightarrow> pprt x = 0"
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   697
  by (simp add: pprt_def le_def_join join_aci)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   698
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   699
lemma nprt_eq_0[simp]: "0 <= x \<Longrightarrow> nprt x = 0"
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   700
  by (simp add: nprt_def le_def_meet meet_aci)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   701
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   702
lemma join_0_imp_0: "join a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   703
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   704
  {
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   705
    fix a::'a
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   706
    assume hyp: "join a (-a) = 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   707
    hence "join a (-a) + a = a" by (simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   708
    hence "join (a+a) 0 = a" by (simp add: add_join_distrib_right) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   709
    hence "join (a+a) 0 <= a" by (simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   710
    hence "0 <= a" by (blast intro: order_trans meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   711
  }
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   712
  note p = this
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   713
  assume hyp:"join a (-a) = 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   714
  hence hyp2:"join (-a) (-(-a)) = 0" by (simp add: join_comm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   715
  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
   716
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   717
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   718
lemma meet_0_imp_0: "meet a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   719
apply (simp add: meet_eq_neg_join)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   720
apply (simp add: join_comm)
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15234
diff changeset
   721
apply (erule join_0_imp_0)
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15234
diff changeset
   722
done
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   723
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   724
lemma join_0_eq_0[simp]: "(join a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   725
by (auto, erule join_0_imp_0)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   726
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   727
lemma meet_0_eq_0[simp]: "(meet a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   728
by (auto, erule meet_0_imp_0)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   729
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   730
lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   731
proof
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   732
  assume "0 <= a + a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   733
  hence a:"meet (a+a) 0 = 0" by (simp add: le_def_meet meet_comm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   734
  have "(meet a 0)+(meet a 0) = meet (meet (a+a) 0) a" (is "?l=_") by (simp add: add_meet_join_distribs meet_aci)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   735
  hence "?l = 0 + meet a 0" by (simp add: a, simp add: meet_comm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   736
  hence "meet a 0 = 0" by (simp only: add_right_cancel)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   737
  then show "0 <= a" by (simp add: le_def_meet meet_comm)    
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   738
next  
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   739
  assume a: "0 <= a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   740
  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
   741
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   742
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   743
lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   744
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   745
  have "(a + a <= 0) = (0 <= -(a+a))" by (subst le_minus_iff, simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   746
  moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   747
  ultimately show ?thesis by blast
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   748
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   749
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   750
lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   751
proof cases
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   752
  assume a: "a < 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   753
  thus ?s by (simp add:  add_strict_mono[OF a a, simplified])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   754
next
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   755
  assume "~(a < 0)" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   756
  hence a:"0 <= a" by (simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   757
  hence "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
   758
  hence "~(a+a < 0)" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   759
  with a show ?thesis by simp 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   760
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   761
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   762
axclass lordered_ab_group_abs \<subseteq> lordered_ab_group
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   763
  abs_lattice: "abs x = join x (-x)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   764
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   765
lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   766
by (simp add: abs_lattice)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   767
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   768
lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   769
by (simp add: abs_lattice)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   770
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   771
lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   772
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   773
  have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   774
  thus ?thesis by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   775
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   776
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   777
lemma neg_meet_eq_join[simp]: "- meet a (b::_::lordered_ab_group) = join (-a) (-b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   778
by (simp add: meet_eq_neg_join)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   779
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   780
lemma neg_join_eq_meet[simp]: "- join a (b::_::lordered_ab_group) = meet (-a) (-b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   781
by (simp del: neg_meet_eq_join add: join_eq_neg_meet)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   782
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   783
lemma join_eq_if: "join a (-a) = (if a < 0 then -a else (a::'a::{lordered_ab_group, linorder}))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   784
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   785
  note b = add_le_cancel_right[of a a "-a",symmetric,simplified]
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   786
  have c: "a + a = 0 \<Longrightarrow> -a = a" by (rule add_right_imp_eq[of _ a], simp)
15197
19e735596e51 Added antisymmetry simproc
nipkow
parents: 15178
diff changeset
   787
  show ?thesis by (auto simp add: join_max max_def b linorder_not_less)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   788
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   789
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   790
lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then -a else (a::'a::{lordered_ab_group_abs, linorder}))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   791
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   792
  show ?thesis by (simp add: abs_lattice join_eq_if)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   793
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   794
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   795
lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   796
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   797
  have a:"a <= abs a" and b:"-a <= abs a" by (auto simp add: abs_lattice meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   798
  show ?thesis by (rule add_mono[OF a b, simplified])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   799
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   800
  
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   801
lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   802
proof
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   803
  assume "abs a <= 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   804
  hence "abs a = 0" by (auto dest: order_antisym)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   805
  thus "a = 0" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   806
next
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   807
  assume "a = 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   808
  thus "abs a <= 0" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   809
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   810
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   811
lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   812
by (simp add: order_less_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   813
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   814
lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   815
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   816
  have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   817
  show ?thesis by (simp add: a)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   818
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   819
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   820
lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   821
by (simp add: abs_lattice meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   822
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   823
lemma abs_ge_minus_self: "-a \<le> abs (a::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   824
by (simp add: abs_lattice meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   825
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   826
lemma le_imp_join_eq: "a \<le> b \<Longrightarrow> join a b = b" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   827
by (simp add: le_def_join)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   828
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   829
lemma ge_imp_join_eq: "b \<le> a \<Longrightarrow> join a b = a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   830
by (simp add: le_def_join join_aci)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   831
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   832
lemma le_imp_meet_eq: "a \<le> b \<Longrightarrow> meet a b = a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   833
by (simp add: le_def_meet)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   834
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   835
lemma ge_imp_meet_eq: "b \<le> a \<Longrightarrow> meet a b = b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   836
by (simp add: le_def_meet meet_aci)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   837
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   838
lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a - nprt a"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   839
apply (simp add: pprt_def nprt_def diff_minus)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   840
apply (simp add: add_meet_join_distribs join_aci abs_lattice[symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   841
apply (subst le_imp_join_eq, auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   842
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   843
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   844
lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   845
by (simp add: abs_lattice join_comm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   846
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   847
lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   848
apply (simp add: abs_lattice[of "abs a"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   849
apply (subst ge_imp_join_eq)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   850
apply (rule order_trans[of _ 0])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   851
by auto
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   852
15093
49ede01e9ee6 conversion of Integration and NSPrimes to Isar scripts
paulson
parents: 15010
diff changeset
   853
lemma abs_minus_commute: 
49ede01e9ee6 conversion of Integration and NSPrimes to Isar scripts
paulson
parents: 15010
diff changeset
   854
  fixes a :: "'a::lordered_ab_group_abs"
49ede01e9ee6 conversion of Integration and NSPrimes to Isar scripts
paulson
parents: 15010
diff changeset
   855
  shows "abs (a-b) = abs(b-a)"
49ede01e9ee6 conversion of Integration and NSPrimes to Isar scripts
paulson
parents: 15010
diff changeset
   856
proof -
49ede01e9ee6 conversion of Integration and NSPrimes to Isar scripts
paulson
parents: 15010
diff changeset
   857
  have "abs (a-b) = abs (- (a-b))" by (simp only: abs_minus_cancel)
49ede01e9ee6 conversion of Integration and NSPrimes to Isar scripts
paulson
parents: 15010
diff changeset
   858
  also have "... = abs(b-a)" by simp
49ede01e9ee6 conversion of Integration and NSPrimes to Isar scripts
paulson
parents: 15010
diff changeset
   859
  finally show ?thesis .
49ede01e9ee6 conversion of Integration and NSPrimes to Isar scripts
paulson
parents: 15010
diff changeset
   860
qed
49ede01e9ee6 conversion of Integration and NSPrimes to Isar scripts
paulson
parents: 15010
diff changeset
   861
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   862
lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   863
by (simp add: le_def_meet nprt_def meet_comm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   864
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   865
lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   866
by (simp add: le_def_join pprt_def join_comm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   867
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   868
lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   869
by (simp add: le_def_join pprt_def join_comm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   870
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   871
lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   872
by (simp add: le_def_meet nprt_def meet_comm)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   873
15580
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   874
lemma pprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> pprt a <= pprt b"
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   875
  by (simp add: le_def_join pprt_def join_aci)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   876
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   877
lemma nprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> nprt a <= nprt b"
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   878
  by (simp add: le_def_meet nprt_def meet_aci)
900291ee0af8 Cleaning up HOL/Matrix
obua
parents: 15539
diff changeset
   879
19404
9bf2cdc9e8e8 Moved stuff from Ring_and_Field to Matrix
obua
parents: 19233
diff changeset
   880
lemma pprt_neg: "pprt (-x) = - nprt x"
9bf2cdc9e8e8 Moved stuff from Ring_and_Field to Matrix
obua
parents: 19233
diff changeset
   881
  by (simp add: pprt_def nprt_def)
9bf2cdc9e8e8 Moved stuff from Ring_and_Field to Matrix
obua
parents: 19233
diff changeset
   882
9bf2cdc9e8e8 Moved stuff from Ring_and_Field to Matrix
obua
parents: 19233
diff changeset
   883
lemma nprt_neg: "nprt (-x) = - pprt x"
9bf2cdc9e8e8 Moved stuff from Ring_and_Field to Matrix
obua
parents: 19233
diff changeset
   884
  by (simp add: pprt_def nprt_def)
9bf2cdc9e8e8 Moved stuff from Ring_and_Field to Matrix
obua
parents: 19233
diff changeset
   885
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   886
lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   887
by (simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   888
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   889
lemma abs_of_nonneg [simp]: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   890
by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   891
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   892
lemma abs_of_pos: "0 < (x::'a::lordered_ab_group_abs) ==> abs x = x";
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   893
by (rule abs_of_nonneg, rule order_less_imp_le);
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   894
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   895
lemma abs_of_nonpos [simp]: "a \<le> 0 \<Longrightarrow> abs a = -(a::'a::lordered_ab_group_abs)"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   896
by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   897
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   898
lemma abs_of_neg: "(x::'a::lordered_ab_group_abs) <  0 ==> 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   899
  abs x = - x"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   900
by (rule abs_of_nonpos, rule order_less_imp_le)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   901
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   902
lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   903
by (simp add: abs_lattice join_imp_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   904
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   905
lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::lordered_ab_group))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   906
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   907
  from add_le_cancel_left[of "-a" "a+a" "0"] have "(a <= -a) = (a+a <= 0)" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   908
    by (simp add: add_assoc[symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   909
  thus ?thesis by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   910
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   911
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   912
lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   913
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   914
  from add_le_cancel_left[of "-a" "0" "a+a"] have "(-a <= a) = (0 <= a+a)" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   915
    by (simp add: add_assoc[symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   916
  thus ?thesis by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   917
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   918
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   919
lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   920
by (insert abs_ge_self, blast intro: order_trans)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   921
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   922
lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::lordered_ab_group_abs)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   923
by (insert abs_le_D1 [of "-a"], simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   924
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   925
lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::lordered_ab_group_abs))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   926
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   927
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   928
lemma abs_triangle_ineq: "abs(a+b) \<le> abs a + abs(b::'a::lordered_ab_group_abs)"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   929
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   930
  have g:"abs a + abs b = join (a+b) (join (-a-b) (join (-a+b) (a + (-b))))" (is "_=join ?m ?n")
19233
77ca20b0ed77 renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents: 17085
diff changeset
   931
    by (simp add: abs_lattice add_meet_join_distribs join_aci diff_minus)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   932
  have a:"a+b <= join ?m ?n" by (simp add: meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   933
  have b:"-a-b <= ?n" by (simp add: meet_join_le) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   934
  have c:"?n <= join ?m ?n" by (simp add: meet_join_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   935
  from b c have d: "-a-b <= join ?m ?n" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   936
  have e:"-a-b = -(a+b)" by (simp add: diff_minus)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   937
  from a d e have "abs(a+b) <= join ?m ?n" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   938
    by (drule_tac abs_leI, auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   939
  with g[symmetric] show ?thesis by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   940
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   941
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   942
lemma abs_triangle_ineq2: "abs (a::'a::lordered_ab_group_abs) - 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   943
    abs b <= abs (a - b)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   944
  apply (simp add: compare_rls)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   945
  apply (subgoal_tac "abs a = abs (a - b + b)")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   946
  apply (erule ssubst)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   947
  apply (rule abs_triangle_ineq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   948
  apply (rule arg_cong);back;
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   949
  apply (simp add: compare_rls)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   950
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   951
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   952
lemma abs_triangle_ineq3: 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   953
    "abs(abs (a::'a::lordered_ab_group_abs) - abs b) <= abs (a - b)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   954
  apply (subst abs_le_iff)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   955
  apply auto
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   956
  apply (rule abs_triangle_ineq2)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   957
  apply (subst abs_minus_commute)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   958
  apply (rule abs_triangle_ineq2)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   959
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   960
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   961
lemma abs_triangle_ineq4: "abs ((a::'a::lordered_ab_group_abs) - b) <= 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   962
    abs a + abs b"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   963
proof -;
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   964
  have "abs(a - b) = abs(a + - b)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   965
    by (subst diff_minus, rule refl)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   966
  also have "... <= abs a + abs (- b)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   967
    by (rule abs_triangle_ineq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   968
  finally show ?thesis
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   969
    by simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   970
qed
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
   971
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   972
lemma abs_diff_triangle_ineq:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   973
     "\<bar>(a::'a::lordered_ab_group_abs) + b - (c+d)\<bar> \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   974
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   975
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   976
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   977
  finally show ?thesis .
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   978
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   979
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   980
lemma abs_add_abs[simp]:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   981
fixes a:: "'a::{lordered_ab_group_abs}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   982
shows "abs(abs a + abs b) = abs a + abs b" (is "?L = ?R")
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   983
proof (rule order_antisym)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   984
  show "?L \<ge> ?R" by(rule abs_ge_self)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   985
next
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   986
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   987
  also have "\<dots> = ?R" by simp
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   988
  finally show "?L \<le> ?R" .
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   989
qed
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15481
diff changeset
   990
14754
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   991
text {* Needed for abelian cancellation simprocs: *}
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   992
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   993
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
   994
apply (subst add_left_commute)
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   995
apply (subst add_left_cancel)
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   996
apply simp
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   997
done
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   998
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   999
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
  1000
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
  1001
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
  1002
done
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1003
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1004
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
  1005
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
  1006
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1007
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
  1008
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
  1009
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
  1010
done
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1011
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1012
lemma eq_eqI: "(x::'a::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
  1013
by (simp add: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y'])
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1014
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1015
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
  1016
by (simp add: diff_minus)
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1017
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1018
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
  1019
by (simp add: add_assoc[symmetric])
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1020
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1021
lemma minus_add_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
  1022
by (simp add: add_assoc[symmetric])
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
  1023
15178
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1024
lemma  le_add_right_mono: 
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1025
  assumes 
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1026
  "a <= b + (c::'a::pordered_ab_group_add)"
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1027
  "c <= d"    
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1028
  shows "a <= b + d"
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1029
  apply (rule_tac order_trans[where y = "b+c"])
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1030
  apply (simp_all add: prems)
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1031
  done
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1032
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1033
lemmas group_eq_simps =
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1034
  mult_ac
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1035
  add_ac
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1036
  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1037
  diff_eq_eq eq_diff_eq
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1038
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1039
lemma estimate_by_abs:
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1040
"a + b <= (c::'a::lordered_ab_group_abs) \<Longrightarrow> a <= c + abs b" 
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1041
proof -
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1042
  assume 1: "a+b <= c"
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1043
  have 2: "a <= c+(-b)"
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1044
    apply (insert 1)
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1045
    apply (drule_tac add_right_mono[where c="-b"])
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1046
    apply (simp add: group_eq_simps)
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1047
    done
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1048
  have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1049
  show ?thesis by (rule le_add_right_mono[OF 2 3])
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1050
qed
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
  1051
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1052
text{*Simplification of @{term "x-y < 0"}, etc.*}
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1053
lemmas diff_less_0_iff_less = less_iff_diff_less_0 [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1054
lemmas diff_eq_0_iff_eq = eq_iff_diff_eq_0 [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1055
lemmas diff_le_0_iff_le = le_iff_diff_le_0 [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1056
declare diff_less_0_iff_less [simp]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1057
declare diff_eq_0_iff_eq [simp]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1058
declare diff_le_0_iff_le [simp]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1059
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1060
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1061
ML {*
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1062
val add_assoc = thm "add_assoc";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1063
val add_commute = thm "add_commute";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1064
val add_left_commute = thm "add_left_commute";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1065
val add_ac = thms "add_ac";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1066
val mult_assoc = thm "mult_assoc";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1067
val mult_commute = thm "mult_commute";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1068
val mult_left_commute = thm "mult_left_commute";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1069
val mult_ac = thms "mult_ac";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1070
val add_0 = thm "add_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1071
val mult_1_left = thm "mult_1_left";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1072
val mult_1_right = thm "mult_1_right";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1073
val mult_1 = thm "mult_1";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1074
val add_left_imp_eq = thm "add_left_imp_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1075
val add_right_imp_eq = thm "add_right_imp_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1076
val add_imp_eq = thm "add_imp_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1077
val left_minus = thm "left_minus";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1078
val diff_minus = thm "diff_minus";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1079
val add_0_right = thm "add_0_right";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1080
val add_left_cancel = thm "add_left_cancel";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1081
val add_right_cancel = thm "add_right_cancel";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1082
val right_minus = thm "right_minus";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1083
val right_minus_eq = thm "right_minus_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1084
val minus_minus = thm "minus_minus";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1085
val equals_zero_I = thm "equals_zero_I";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1086
val minus_zero = thm "minus_zero";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1087
val diff_self = thm "diff_self";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1088
val diff_0 = thm "diff_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1089
val diff_0_right = thm "diff_0_right";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1090
val diff_minus_eq_add = thm "diff_minus_eq_add";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1091
val neg_equal_iff_equal = thm "neg_equal_iff_equal";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1092
val neg_equal_0_iff_equal = thm "neg_equal_0_iff_equal";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1093
val neg_0_equal_iff_equal = thm "neg_0_equal_iff_equal";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1094
val equation_minus_iff = thm "equation_minus_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1095
val minus_equation_iff = thm "minus_equation_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1096
val minus_add_distrib = thm "minus_add_distrib";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1097
val minus_diff_eq = thm "minus_diff_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1098
val add_left_mono = thm "add_left_mono";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1099
val add_le_imp_le_left = thm "add_le_imp_le_left";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1100
val add_right_mono = thm "add_right_mono";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1101
val add_mono = thm "add_mono";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1102
val add_strict_left_mono = thm "add_strict_left_mono";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1103
val add_strict_right_mono = thm "add_strict_right_mono";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1104
val add_strict_mono = thm "add_strict_mono";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1105
val add_less_le_mono = thm "add_less_le_mono";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1106
val add_le_less_mono = thm "add_le_less_mono";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1107
val add_less_imp_less_left = thm "add_less_imp_less_left";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1108
val add_less_imp_less_right = thm "add_less_imp_less_right";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1109
val add_less_cancel_left = thm "add_less_cancel_left";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1110
val add_less_cancel_right = thm "add_less_cancel_right";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1111
val add_le_cancel_left = thm "add_le_cancel_left";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1112
val add_le_cancel_right = thm "add_le_cancel_right";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1113
val add_le_imp_le_right = thm "add_le_imp_le_right";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1114
val add_increasing = thm "add_increasing";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1115
val le_imp_neg_le = thm "le_imp_neg_le";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1116
val neg_le_iff_le = thm "neg_le_iff_le";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1117
val neg_le_0_iff_le = thm "neg_le_0_iff_le";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1118
val neg_0_le_iff_le = thm "neg_0_le_iff_le";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1119
val neg_less_iff_less = thm "neg_less_iff_less";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1120
val neg_less_0_iff_less = thm "neg_less_0_iff_less";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1121
val neg_0_less_iff_less = thm "neg_0_less_iff_less";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1122
val less_minus_iff = thm "less_minus_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1123
val minus_less_iff = thm "minus_less_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1124
val le_minus_iff = thm "le_minus_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1125
val minus_le_iff = thm "minus_le_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1126
val add_diff_eq = thm "add_diff_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1127
val diff_add_eq = thm "diff_add_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1128
val diff_eq_eq = thm "diff_eq_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1129
val eq_diff_eq = thm "eq_diff_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1130
val diff_diff_eq = thm "diff_diff_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1131
val diff_diff_eq2 = thm "diff_diff_eq2";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1132
val diff_add_cancel = thm "diff_add_cancel";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1133
val add_diff_cancel = thm "add_diff_cancel";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1134
val less_iff_diff_less_0 = thm "less_iff_diff_less_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1135
val diff_less_eq = thm "diff_less_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1136
val less_diff_eq = thm "less_diff_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1137
val diff_le_eq = thm "diff_le_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1138
val le_diff_eq = thm "le_diff_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1139
val compare_rls = thms "compare_rls";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1140
val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1141
val le_iff_diff_le_0 = thm "le_iff_diff_le_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1142
val add_meet_distrib_left = thm "add_meet_distrib_left";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1143
val add_join_distrib_left = thm "add_join_distrib_left";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1144
val is_join_neg_meet = thm "is_join_neg_meet";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1145
val is_meet_neg_join = thm "is_meet_neg_join";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1146
val add_join_distrib_right = thm "add_join_distrib_right";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1147
val add_meet_distrib_right = thm "add_meet_distrib_right";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1148
val add_meet_join_distribs = thms "add_meet_join_distribs";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1149
val join_eq_neg_meet = thm "join_eq_neg_meet";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1150
val meet_eq_neg_join = thm "meet_eq_neg_join";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1151
val add_eq_meet_join = thm "add_eq_meet_join";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1152
val prts = thm "prts";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1153
val zero_le_pprt = thm "zero_le_pprt";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1154
val nprt_le_zero = thm "nprt_le_zero";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1155
val le_eq_neg = thm "le_eq_neg";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1156
val join_0_imp_0 = thm "join_0_imp_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1157
val meet_0_imp_0 = thm "meet_0_imp_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1158
val join_0_eq_0 = thm "join_0_eq_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1159
val meet_0_eq_0 = thm "meet_0_eq_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1160
val zero_le_double_add_iff_zero_le_single_add = thm "zero_le_double_add_iff_zero_le_single_add";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1161
val double_add_le_zero_iff_single_add_le_zero = thm "double_add_le_zero_iff_single_add_le_zero";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1162
val double_add_less_zero_iff_single_less_zero = thm "double_add_less_zero_iff_single_less_zero";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1163
val abs_lattice = thm "abs_lattice";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1164
val abs_zero = thm "abs_zero";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1165
val abs_eq_0 = thm "abs_eq_0";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1166
val abs_0_eq = thm "abs_0_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1167
val neg_meet_eq_join = thm "neg_meet_eq_join";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1168
val neg_join_eq_meet = thm "neg_join_eq_meet";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1169
val join_eq_if = thm "join_eq_if";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1170
val abs_if_lattice = thm "abs_if_lattice";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1171
val abs_ge_zero = thm "abs_ge_zero";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1172
val abs_le_zero_iff = thm "abs_le_zero_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1173
val zero_less_abs_iff = thm "zero_less_abs_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1174
val abs_not_less_zero = thm "abs_not_less_zero";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1175
val abs_ge_self = thm "abs_ge_self";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1176
val abs_ge_minus_self = thm "abs_ge_minus_self";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1177
val le_imp_join_eq = thm "le_imp_join_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1178
val ge_imp_join_eq = thm "ge_imp_join_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1179
val le_imp_meet_eq = thm "le_imp_meet_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1180
val ge_imp_meet_eq = thm "ge_imp_meet_eq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1181
val abs_prts = thm "abs_prts";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1182
val abs_minus_cancel = thm "abs_minus_cancel";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1183
val abs_idempotent = thm "abs_idempotent";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1184
val zero_le_iff_zero_nprt = thm "zero_le_iff_zero_nprt";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1185
val le_zero_iff_zero_pprt = thm "le_zero_iff_zero_pprt";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1186
val le_zero_iff_pprt_id = thm "le_zero_iff_pprt_id";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1187
val zero_le_iff_nprt_id = thm "zero_le_iff_nprt_id";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1188
val iff2imp = thm "iff2imp";
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
  1189
(* val imp_abs_id = thm "imp_abs_id";
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
  1190
val imp_abs_neg_id = thm "imp_abs_neg_id"; *)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1191
val abs_leI = thm "abs_leI";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1192
val le_minus_self_iff = thm "le_minus_self_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1193
val minus_le_self_iff = thm "minus_le_self_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1194
val abs_le_D1 = thm "abs_le_D1";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1195
val abs_le_D2 = thm "abs_le_D2";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1196
val abs_le_iff = thm "abs_le_iff";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1197
val abs_triangle_ineq = thm "abs_triangle_ineq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1198
val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq";
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1199
*}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1200
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1201
end