author | wenzelm |
Wed, 08 Nov 2006 13:48:29 +0100 | |
changeset 21243 | afffe1f72143 |
parent 19798 | 94f12468bbba |
child 21245 | 23e6eb4d0975 |
permissions | -rw-r--r-- |
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(* Title: HOL/OrderedGroup.thy |
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ID: $Id$ |
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added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
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Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel, |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
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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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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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proof |
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assume "- a = - b" |
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hence "- (- a) = - (- b)" |
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by simp |
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thus "a=b" by simp |
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next |
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assume "a=b" |
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thus "-a = -b" by simp |
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qed |
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lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ab_group_add))" |
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by (subst neg_equal_iff_equal [symmetric], simp) |
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lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ab_group_add))" |
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by (subst neg_equal_iff_equal [symmetric], simp) |
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text{*The next two equations can make the simplifier loop!*} |
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lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ab_group_add))" |
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proof - |
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have "(- (-a) = - b) = (- a = b)" by (rule neg_equal_iff_equal) |
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thus ?thesis by (simp add: eq_commute) |
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qed |
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lemma minus_equation_iff: "(- a = b) = (- (b::'a::ab_group_add) = a)" |
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proof - |
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have "(- a = - (-b)) = (a = -b)" by (rule neg_equal_iff_equal) |
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thus ?thesis by (simp add: eq_commute) |
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qed |
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lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ab_group_add)" |
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apply (rule equals_zero_I) |
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apply (simp add: add_ac) |
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done |
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lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ab_group_add)" |
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by (simp add: diff_minus add_commute) |
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subsection {* (Partially) Ordered Groups *} |
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axclass pordered_ab_semigroup_add \<subseteq> order, ab_semigroup_add |
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add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b" |
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axclass pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add, cancel_ab_semigroup_add |
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instance pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add .. |
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axclass pordered_ab_semigroup_add_imp_le \<subseteq> pordered_cancel_ab_semigroup_add |
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add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b" |
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axclass pordered_ab_group_add \<subseteq> ab_group_add, pordered_ab_semigroup_add |
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instance pordered_ab_group_add \<subseteq> pordered_ab_semigroup_add_imp_le |
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proof |
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fix a b c :: 'a |
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assume "c + a \<le> c + b" |
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hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono) |
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hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc) |
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thus "a \<le> b" by simp |
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qed |
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axclass ordered_cancel_ab_semigroup_add \<subseteq> pordered_cancel_ab_semigroup_add, linorder |
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instance ordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add_imp_le |
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proof |
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fix a b c :: 'a |
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assume le: "c + a <= c + b" |
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show "a <= b" |
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proof (rule ccontr) |
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assume w: "~ a \<le> b" |
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hence "b <= a" by (simp add: linorder_not_le) |
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hence le2: "c+b <= c+a" by (rule add_left_mono) |
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have "a = b" |
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apply (insert le) |
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apply (insert le2) |
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apply (drule order_antisym, simp_all) |
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done |
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with w show False |
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by (simp add: linorder_not_le [symmetric]) |
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qed |
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qed |
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lemma add_right_mono: "a \<le> (b::'a::pordered_ab_semigroup_add) ==> a + c \<le> b + c" |
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by (simp add: add_commute[of _ c] add_left_mono) |
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text {* non-strict, in both arguments *} |
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lemma add_mono: |
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"[|a \<le> b; c \<le> d|] ==> a + c \<le> b + (d::'a::pordered_ab_semigroup_add)" |
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apply (erule add_right_mono [THEN order_trans]) |
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apply (simp add: add_commute add_left_mono) |
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done |
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lemma add_strict_left_mono: |
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"a < b ==> c + a < c + (b::'a::pordered_cancel_ab_semigroup_add)" |
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by (simp add: order_less_le add_left_mono) |
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lemma add_strict_right_mono: |
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"a < b ==> a + c < b + (c::'a::pordered_cancel_ab_semigroup_add)" |
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by (simp add: add_commute [of _ c] add_strict_left_mono) |
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text{*Strict monotonicity in both arguments*} |
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lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" |
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apply (erule add_strict_right_mono [THEN order_less_trans]) |
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apply (erule add_strict_left_mono) |
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done |
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lemma add_less_le_mono: |
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"[| a<b; c\<le>d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" |
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apply (erule add_strict_right_mono [THEN order_less_le_trans]) |
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apply (erule add_left_mono) |
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done |
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lemma add_le_less_mono: |
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"[| a\<le>b; c<d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" |
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apply (erule add_right_mono [THEN order_le_less_trans]) |
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apply (erule add_strict_left_mono) |
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done |
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lemma add_less_imp_less_left: |
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assumes less: "c + a < c + b" shows "a < (b::'a::pordered_ab_semigroup_add_imp_le)" |
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proof - |
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from less have le: "c + a <= c + b" by (simp add: order_le_less) |
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have "a <= b" |
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apply (insert le) |
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apply (drule add_le_imp_le_left) |
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by (insert le, drule add_le_imp_le_left, assumption) |
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moreover have "a \<noteq> b" |
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proof (rule ccontr) |
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assume "~(a \<noteq> b)" |
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then have "a = b" by simp |
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then have "c + a = c + b" by simp |
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with less show "False"by simp |
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qed |
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ultimately show "a < b" by (simp add: order_le_less) |
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qed |
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lemma add_less_imp_less_right: |
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"a + c < b + c ==> a < (b::'a::pordered_ab_semigroup_add_imp_le)" |
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apply (rule add_less_imp_less_left [of c]) |
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apply (simp add: add_commute) |
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done |
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lemma add_less_cancel_left [simp]: |
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"(c+a < c+b) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))" |
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by (blast intro: add_less_imp_less_left add_strict_left_mono) |
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lemma add_less_cancel_right [simp]: |
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"(a+c < b+c) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))" |
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by (blast intro: add_less_imp_less_right add_strict_right_mono) |
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lemma add_le_cancel_left [simp]: |
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"(c+a \<le> c+b) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))" |
|
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by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) |
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310 |
lemma add_le_cancel_right [simp]: |
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311 |
"(a+c \<le> b+c) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))" |
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by (simp add: add_commute[of a c] add_commute[of b c]) |
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314 |
lemma add_le_imp_le_right: |
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315 |
"a + c \<le> b + c ==> a \<le> (b::'a::pordered_ab_semigroup_add_imp_le)" |
|
316 |
by simp |
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317 |
||
15234
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318 |
lemma add_increasing: |
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319 |
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
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|
320 |
shows "[|0\<le>a; b\<le>c|] ==> b \<le> a + c" |
14738 | 321 |
by (insert add_mono [of 0 a b c], simp) |
322 |
||
15539 | 323 |
lemma add_increasing2: |
324 |
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
|
325 |
shows "[|0\<le>c; b\<le>a|] ==> b \<le> a + c" |
|
326 |
by (simp add:add_increasing add_commute[of a]) |
|
327 |
||
15234
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simplification tweaks for better arithmetic reasoning
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15229
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|
328 |
lemma add_strict_increasing: |
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diff
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|
329 |
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
330 |
shows "[|0<a; b\<le>c|] ==> b < a + c" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
331 |
by (insert add_less_le_mono [of 0 a b c], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
332 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
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diff
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|
333 |
lemma add_strict_increasing2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
334 |
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
335 |
shows "[|0\<le>a; b<c|] ==> b < a + c" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
336 |
by (insert add_le_less_mono [of 0 a b c], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
337 |
|
19527 | 338 |
lemma max_add_distrib_left: |
339 |
fixes z :: "'a::pordered_ab_semigroup_add_imp_le" |
|
340 |
shows "(max x y) + z = max (x+z) (y+z)" |
|
341 |
by (rule max_of_mono [THEN sym], rule add_le_cancel_right) |
|
342 |
||
343 |
lemma min_add_distrib_left: |
|
344 |
fixes z :: "'a::pordered_ab_semigroup_add_imp_le" |
|
345 |
shows "(min x y) + z = min (x+z) (y+z)" |
|
346 |
by (rule min_of_mono [THEN sym], rule add_le_cancel_right) |
|
347 |
||
348 |
lemma max_diff_distrib_left: |
|
349 |
fixes z :: "'a::pordered_ab_group_add" |
|
350 |
shows "(max x y) - z = max (x-z) (y-z)" |
|
351 |
by (simp add: diff_minus, rule max_add_distrib_left) |
|
352 |
||
353 |
lemma min_diff_distrib_left: |
|
354 |
fixes z :: "'a::pordered_ab_group_add" |
|
355 |
shows "(min x y) - z = min (x-z) (y-z)" |
|
356 |
by (simp add: diff_minus, rule min_add_distrib_left) |
|
357 |
||
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
358 |
|
14738 | 359 |
subsection {* Ordering Rules for Unary Minus *} |
360 |
||
361 |
lemma le_imp_neg_le: |
|
362 |
assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "-b \<le> -a" |
|
363 |
proof - |
|
364 |
have "-a+a \<le> -a+b" |
|
365 |
by (rule add_left_mono) |
|
366 |
hence "0 \<le> -a+b" |
|
367 |
by simp |
|
368 |
hence "0 + (-b) \<le> (-a + b) + (-b)" |
|
369 |
by (rule add_right_mono) |
|
370 |
thus ?thesis |
|
371 |
by (simp add: add_assoc) |
|
372 |
qed |
|
373 |
||
374 |
lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::pordered_ab_group_add))" |
|
375 |
proof |
|
376 |
assume "- b \<le> - a" |
|
377 |
hence "- (- a) \<le> - (- b)" |
|
378 |
by (rule le_imp_neg_le) |
|
379 |
thus "a\<le>b" by simp |
|
380 |
next |
|
381 |
assume "a\<le>b" |
|
382 |
thus "-b \<le> -a" by (rule le_imp_neg_le) |
|
383 |
qed |
|
384 |
||
385 |
lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))" |
|
386 |
by (subst neg_le_iff_le [symmetric], simp) |
|
387 |
||
388 |
lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::pordered_ab_group_add))" |
|
389 |
by (subst neg_le_iff_le [symmetric], simp) |
|
390 |
||
391 |
lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::pordered_ab_group_add))" |
|
392 |
by (force simp add: order_less_le) |
|
393 |
||
394 |
lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::pordered_ab_group_add))" |
|
395 |
by (subst neg_less_iff_less [symmetric], simp) |
|
396 |
||
397 |
lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::pordered_ab_group_add))" |
|
398 |
by (subst neg_less_iff_less [symmetric], simp) |
|
399 |
||
400 |
text{*The next several equations can make the simplifier loop!*} |
|
401 |
||
402 |
lemma less_minus_iff: "(a < - b) = (b < - (a::'a::pordered_ab_group_add))" |
|
403 |
proof - |
|
404 |
have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less) |
|
405 |
thus ?thesis by simp |
|
406 |
qed |
|
407 |
||
408 |
lemma minus_less_iff: "(- a < b) = (- b < (a::'a::pordered_ab_group_add))" |
|
409 |
proof - |
|
410 |
have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less) |
|
411 |
thus ?thesis by simp |
|
412 |
qed |
|
413 |
||
414 |
lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::pordered_ab_group_add))" |
|
415 |
proof - |
|
416 |
have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff) |
|
417 |
have "(- (- a) <= -b) = (b <= - a)" |
|
418 |
apply (auto simp only: order_le_less) |
|
419 |
apply (drule mm) |
|
420 |
apply (simp_all) |
|
421 |
apply (drule mm[simplified], assumption) |
|
422 |
done |
|
423 |
then show ?thesis by simp |
|
424 |
qed |
|
425 |
||
426 |
lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::pordered_ab_group_add))" |
|
427 |
by (auto simp add: order_le_less minus_less_iff) |
|
428 |
||
429 |
lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ab_group_add)" |
|
430 |
by (simp add: diff_minus add_ac) |
|
431 |
||
432 |
lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ab_group_add)" |
|
433 |
by (simp add: diff_minus add_ac) |
|
434 |
||
435 |
lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ab_group_add))" |
|
436 |
by (auto simp add: diff_minus add_assoc) |
|
437 |
||
438 |
lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ab_group_add) = c)" |
|
439 |
by (auto simp add: diff_minus add_assoc) |
|
440 |
||
441 |
lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ab_group_add))" |
|
442 |
by (simp add: diff_minus add_ac) |
|
443 |
||
444 |
lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ab_group_add)" |
|
445 |
by (simp add: diff_minus add_ac) |
|
446 |
||
447 |
lemma diff_add_cancel: "a - b + b = (a::'a::ab_group_add)" |
|
448 |
by (simp add: diff_minus add_ac) |
|
449 |
||
450 |
lemma add_diff_cancel: "a + b - b = (a::'a::ab_group_add)" |
|
451 |
by (simp add: diff_minus add_ac) |
|
452 |
||
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
453 |
text{*Further subtraction laws*} |
14738 | 454 |
|
455 |
lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::pordered_ab_group_add))" |
|
456 |
proof - |
|
457 |
have "(a < b) = (a + (- b) < b + (-b))" |
|
458 |
by (simp only: add_less_cancel_right) |
|
459 |
also have "... = (a - b < 0)" by (simp add: diff_minus) |
|
460 |
finally show ?thesis . |
|
461 |
qed |
|
462 |
||
463 |
lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::pordered_ab_group_add))" |
|
15481 | 464 |
apply (subst less_iff_diff_less_0 [of a]) |
14738 | 465 |
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) |
466 |
apply (simp add: diff_minus add_ac) |
|
467 |
done |
|
468 |
||
469 |
lemma less_diff_eq: "(a < c-b) = (a + (b::'a::pordered_ab_group_add) < c)" |
|
15481 | 470 |
apply (subst less_iff_diff_less_0 [of "a+b"]) |
471 |
apply (subst less_iff_diff_less_0 [of a]) |
|
14738 | 472 |
apply (simp add: diff_minus add_ac) |
473 |
done |
|
474 |
||
475 |
lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))" |
|
476 |
by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel) |
|
477 |
||
478 |
lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::pordered_ab_group_add) \<le> c)" |
|
479 |
by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel) |
|
480 |
||
481 |
text{*This list of rewrites simplifies (in)equalities by bringing subtractions |
|
482 |
to the top and then moving negative terms to the other side. |
|
483 |
Use with @{text add_ac}*} |
|
484 |
lemmas compare_rls = |
|
485 |
diff_minus [symmetric] |
|
486 |
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 |
|
487 |
diff_less_eq less_diff_eq diff_le_eq le_diff_eq |
|
488 |
diff_eq_eq eq_diff_eq |
|
489 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
490 |
subsection {* Support for reasoning about signs *} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
491 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
492 |
lemma add_pos_pos: "0 < |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
493 |
(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
|
494 |
==> 0 < y ==> 0 < x + y" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
495 |
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
|
496 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
497 |
apply (erule add_less_le_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
498 |
apply (erule order_less_imp_le) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
499 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
500 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
501 |
lemma add_pos_nonneg: "0 < |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
502 |
(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
|
503 |
==> 0 <= y ==> 0 < x + y" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
504 |
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
|
505 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
506 |
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
|
507 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
508 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
509 |
lemma add_nonneg_pos: "0 <= |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
510 |
(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
|
511 |
==> 0 < y ==> 0 < x + y" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
512 |
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
|
513 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
514 |
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
|
515 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
516 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
517 |
lemma add_nonneg_nonneg: "0 <= |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
518 |
(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
|
519 |
==> 0 <= y ==> 0 <= x + y" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
520 |
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
|
521 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
522 |
apply (erule add_mono, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
523 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
524 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
525 |
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
|
526 |
< 0 ==> y < 0 ==> x + y < 0" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
527 |
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
|
528 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
529 |
apply (erule add_less_le_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
530 |
apply (erule order_less_imp_le) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
531 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
532 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
533 |
lemma add_neg_nonpos: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
534 |
"(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
|
535 |
==> y <= 0 ==> x + y < 0" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
536 |
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
|
537 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
538 |
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
|
539 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
540 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
541 |
lemma add_nonpos_neg: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
542 |
"(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
|
543 |
==> y < 0 ==> x + y < 0" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
544 |
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
|
545 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
546 |
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
|
547 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
548 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
549 |
lemma add_nonpos_nonpos: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
550 |
"(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
|
551 |
==> y <= 0 ==> x + y <= 0" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
552 |
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
|
553 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
554 |
apply (erule add_mono, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
555 |
done |
14738 | 556 |
|
557 |
subsection{*Lemmas for the @{text cancel_numerals} simproc*} |
|
558 |
||
559 |
lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ab_group_add))" |
|
560 |
by (simp add: compare_rls) |
|
561 |
||
562 |
lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::pordered_ab_group_add))" |
|
563 |
by (simp add: compare_rls) |
|
564 |
||
565 |
subsection {* Lattice Ordered (Abelian) Groups *} |
|
566 |
||
567 |
axclass lordered_ab_group_meet < pordered_ab_group_add, meet_semilorder |
|
568 |
||
569 |
axclass lordered_ab_group_join < pordered_ab_group_add, join_semilorder |
|
570 |
||
571 |
lemma add_meet_distrib_left: "a + (meet b c) = meet (a + b) (a + (c::'a::{pordered_ab_group_add, meet_semilorder}))" |
|
572 |
apply (rule order_antisym) |
|
573 |
apply (rule meet_imp_le, simp_all add: meet_join_le) |
|
574 |
apply (rule add_le_imp_le_left [of "-a"]) |
|
575 |
apply (simp only: add_assoc[symmetric], simp) |
|
576 |
apply (rule meet_imp_le) |
|
577 |
apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+ |
|
578 |
done |
|
579 |
||
580 |
lemma add_join_distrib_left: "a + (join b c) = join (a + b) (a+ (c::'a::{pordered_ab_group_add, join_semilorder}))" |
|
581 |
apply (rule order_antisym) |
|
582 |
apply (rule add_le_imp_le_left [of "-a"]) |
|
583 |
apply (simp only: add_assoc[symmetric], simp) |
|
584 |
apply (rule join_imp_le) |
|
585 |
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+ |
|
586 |
apply (rule join_imp_le) |
|
587 |
apply (simp_all add: meet_join_le) |
|
588 |
done |
|
589 |
||
590 |
lemma is_join_neg_meet: "is_join (% (a::'a::{pordered_ab_group_add, meet_semilorder}) b. - (meet (-a) (-b)))" |
|
591 |
apply (auto simp add: is_join_def) |
|
592 |
apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le) |
|
593 |
apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le) |
|
594 |
apply (subst neg_le_iff_le[symmetric]) |
|
595 |
apply (simp add: meet_imp_le) |
|
596 |
done |
|
597 |
||
598 |
lemma is_meet_neg_join: "is_meet (% (a::'a::{pordered_ab_group_add, join_semilorder}) b. - (join (-a) (-b)))" |
|
599 |
apply (auto simp add: is_meet_def) |
|
600 |
apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le) |
|
601 |
apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le) |
|
602 |
apply (subst neg_le_iff_le[symmetric]) |
|
603 |
apply (simp add: join_imp_le) |
|
604 |
done |
|
605 |
||
606 |
axclass lordered_ab_group \<subseteq> pordered_ab_group_add, lorder |
|
607 |
||
608 |
instance lordered_ab_group_meet \<subseteq> lordered_ab_group |
|
609 |
proof |
|
610 |
show "? j. is_join (j::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_meet))" by (blast intro: is_join_neg_meet) |
|
611 |
qed |
|
612 |
||
613 |
instance lordered_ab_group_join \<subseteq> lordered_ab_group |
|
614 |
proof |
|
615 |
show "? m. is_meet (m::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_join))" by (blast intro: is_meet_neg_join) |
|
616 |
qed |
|
617 |
||
618 |
lemma add_join_distrib_right: "(join a b) + (c::'a::lordered_ab_group) = join (a+c) (b+c)" |
|
619 |
proof - |
|
620 |
have "c + (join a b) = join (c+a) (c+b)" by (simp add: add_join_distrib_left) |
|
621 |
thus ?thesis by (simp add: add_commute) |
|
622 |
qed |
|
623 |
||
624 |
lemma add_meet_distrib_right: "(meet a b) + (c::'a::lordered_ab_group) = meet (a+c) (b+c)" |
|
625 |
proof - |
|
626 |
have "c + (meet a b) = meet (c+a) (c+b)" by (simp add: add_meet_distrib_left) |
|
627 |
thus ?thesis by (simp add: add_commute) |
|
628 |
qed |
|
629 |
||
630 |
lemmas add_meet_join_distribs = add_meet_distrib_right add_meet_distrib_left add_join_distrib_right add_join_distrib_left |
|
631 |
||
632 |
lemma join_eq_neg_meet: "join a (b::'a::lordered_ab_group) = - meet (-a) (-b)" |
|
633 |
by (simp add: is_join_unique[OF is_join_join is_join_neg_meet]) |
|
634 |
||
635 |
lemma meet_eq_neg_join: "meet a (b::'a::lordered_ab_group) = - join (-a) (-b)" |
|
636 |
by (simp add: is_meet_unique[OF is_meet_meet is_meet_neg_join]) |
|
637 |
||
638 |
lemma add_eq_meet_join: "a + b = (join a b) + (meet a (b::'a::lordered_ab_group))" |
|
639 |
proof - |
|
640 |
have "0 = - meet 0 (a-b) + meet (a-b) 0" by (simp add: meet_comm) |
|
641 |
hence "0 = join 0 (b-a) + meet (a-b) 0" by (simp add: meet_eq_neg_join) |
|
642 |
hence "0 = (-a + join a b) + (meet a b + (-b))" |
|
643 |
apply (simp add: add_join_distrib_left add_meet_distrib_right) |
|
644 |
by (simp add: diff_minus add_commute) |
|
645 |
thus ?thesis |
|
646 |
apply (simp add: compare_rls) |
|
647 |
apply (subst add_left_cancel[symmetric, of "a+b" "join a b + meet a b" "-a"]) |
|
648 |
apply (simp only: add_assoc, simp add: add_assoc[symmetric]) |
|
649 |
done |
|
650 |
qed |
|
651 |
||
652 |
subsection {* Positive Part, Negative Part, Absolute Value *} |
|
653 |
||
654 |
constdefs |
|
655 |
pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" |
|
656 |
"pprt x == join x 0" |
|
657 |
nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" |
|
658 |
"nprt x == meet x 0" |
|
659 |
||
660 |
lemma prts: "a = pprt a + nprt a" |
|
661 |
by (simp add: pprt_def nprt_def add_eq_meet_join[symmetric]) |
|
662 |
||
663 |
lemma zero_le_pprt[simp]: "0 \<le> pprt a" |
|
664 |
by (simp add: pprt_def meet_join_le) |
|
665 |
||
666 |
lemma nprt_le_zero[simp]: "nprt a \<le> 0" |
|
667 |
by (simp add: nprt_def meet_join_le) |
|
668 |
||
669 |
lemma le_eq_neg: "(a \<le> -b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r") |
|
670 |
proof - |
|
671 |
have a: "?l \<longrightarrow> ?r" |
|
672 |
apply (auto) |
|
673 |
apply (rule add_le_imp_le_right[of _ "-b" _]) |
|
674 |
apply (simp add: add_assoc) |
|
675 |
done |
|
676 |
have b: "?r \<longrightarrow> ?l" |
|
677 |
apply (auto) |
|
678 |
apply (rule add_le_imp_le_right[of _ "b" _]) |
|
679 |
apply (simp) |
|
680 |
done |
|
681 |
from a b show ?thesis by blast |
|
682 |
qed |
|
683 |
||
15580 | 684 |
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def) |
685 |
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def) |
|
686 |
||
687 |
lemma pprt_eq_id[simp]: "0 <= x \<Longrightarrow> pprt x = x" |
|
688 |
by (simp add: pprt_def le_def_join join_aci) |
|
689 |
||
690 |
lemma nprt_eq_id[simp]: "x <= 0 \<Longrightarrow> nprt x = x" |
|
691 |
by (simp add: nprt_def le_def_meet meet_aci) |
|
692 |
||
693 |
lemma pprt_eq_0[simp]: "x <= 0 \<Longrightarrow> pprt x = 0" |
|
694 |
by (simp add: pprt_def le_def_join join_aci) |
|
695 |
||
696 |
lemma nprt_eq_0[simp]: "0 <= x \<Longrightarrow> nprt x = 0" |
|
697 |
by (simp add: nprt_def le_def_meet meet_aci) |
|
698 |
||
14738 | 699 |
lemma join_0_imp_0: "join a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" |
700 |
proof - |
|
701 |
{ |
|
702 |
fix a::'a |
|
703 |
assume hyp: "join a (-a) = 0" |
|
704 |
hence "join a (-a) + a = a" by (simp) |
|
705 |
hence "join (a+a) 0 = a" by (simp add: add_join_distrib_right) |
|
706 |
hence "join (a+a) 0 <= a" by (simp) |
|
707 |
hence "0 <= a" by (blast intro: order_trans meet_join_le) |
|
708 |
} |
|
709 |
note p = this |
|
710 |
assume hyp:"join a (-a) = 0" |
|
711 |
hence hyp2:"join (-a) (-(-a)) = 0" by (simp add: join_comm) |
|
712 |
from p[OF hyp] p[OF hyp2] show "a = 0" by simp |
|
713 |
qed |
|
714 |
||
715 |
lemma meet_0_imp_0: "meet a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" |
|
716 |
apply (simp add: meet_eq_neg_join) |
|
717 |
apply (simp add: join_comm) |
|
15481 | 718 |
apply (erule join_0_imp_0) |
719 |
done |
|
14738 | 720 |
|
721 |
lemma join_0_eq_0[simp]: "(join a (-a) = 0) = (a = (0::'a::lordered_ab_group))" |
|
722 |
by (auto, erule join_0_imp_0) |
|
723 |
||
724 |
lemma meet_0_eq_0[simp]: "(meet a (-a) = 0) = (a = (0::'a::lordered_ab_group))" |
|
725 |
by (auto, erule meet_0_imp_0) |
|
726 |
||
727 |
lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))" |
|
728 |
proof |
|
729 |
assume "0 <= a + a" |
|
730 |
hence a:"meet (a+a) 0 = 0" by (simp add: le_def_meet meet_comm) |
|
731 |
have "(meet a 0)+(meet a 0) = meet (meet (a+a) 0) a" (is "?l=_") by (simp add: add_meet_join_distribs meet_aci) |
|
732 |
hence "?l = 0 + meet a 0" by (simp add: a, simp add: meet_comm) |
|
733 |
hence "meet a 0 = 0" by (simp only: add_right_cancel) |
|
734 |
then show "0 <= a" by (simp add: le_def_meet meet_comm) |
|
735 |
next |
|
736 |
assume a: "0 <= a" |
|
737 |
show "0 <= a + a" by (simp add: add_mono[OF a a, simplified]) |
|
738 |
qed |
|
739 |
||
740 |
lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)" |
|
741 |
proof - |
|
742 |
have "(a + a <= 0) = (0 <= -(a+a))" by (subst le_minus_iff, simp) |
|
743 |
moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add) |
|
744 |
ultimately show ?thesis by blast |
|
745 |
qed |
|
746 |
||
747 |
lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s) |
|
748 |
proof cases |
|
749 |
assume a: "a < 0" |
|
750 |
thus ?s by (simp add: add_strict_mono[OF a a, simplified]) |
|
751 |
next |
|
752 |
assume "~(a < 0)" |
|
753 |
hence a:"0 <= a" by (simp) |
|
754 |
hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified]) |
|
755 |
hence "~(a+a < 0)" by simp |
|
756 |
with a show ?thesis by simp |
|
757 |
qed |
|
758 |
||
759 |
axclass lordered_ab_group_abs \<subseteq> lordered_ab_group |
|
760 |
abs_lattice: "abs x = join x (-x)" |
|
761 |
||
762 |
lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)" |
|
763 |
by (simp add: abs_lattice) |
|
764 |
||
765 |
lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))" |
|
766 |
by (simp add: abs_lattice) |
|
767 |
||
768 |
lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))" |
|
769 |
proof - |
|
770 |
have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac) |
|
771 |
thus ?thesis by simp |
|
772 |
qed |
|
773 |
||
774 |
lemma neg_meet_eq_join[simp]: "- meet a (b::_::lordered_ab_group) = join (-a) (-b)" |
|
775 |
by (simp add: meet_eq_neg_join) |
|
776 |
||
777 |
lemma neg_join_eq_meet[simp]: "- join a (b::_::lordered_ab_group) = meet (-a) (-b)" |
|
778 |
by (simp del: neg_meet_eq_join add: join_eq_neg_meet) |
|
779 |
||
780 |
lemma join_eq_if: "join a (-a) = (if a < 0 then -a else (a::'a::{lordered_ab_group, linorder}))" |
|
781 |
proof - |
|
782 |
note b = add_le_cancel_right[of a a "-a",symmetric,simplified] |
|
783 |
have c: "a + a = 0 \<Longrightarrow> -a = a" by (rule add_right_imp_eq[of _ a], simp) |
|
15197 | 784 |
show ?thesis by (auto simp add: join_max max_def b linorder_not_less) |
14738 | 785 |
qed |
786 |
||
787 |
lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then -a else (a::'a::{lordered_ab_group_abs, linorder}))" |
|
788 |
proof - |
|
789 |
show ?thesis by (simp add: abs_lattice join_eq_if) |
|
790 |
qed |
|
791 |
||
792 |
lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)" |
|
793 |
proof - |
|
794 |
have a:"a <= abs a" and b:"-a <= abs a" by (auto simp add: abs_lattice meet_join_le) |
|
795 |
show ?thesis by (rule add_mono[OF a b, simplified]) |
|
796 |
qed |
|
797 |
||
798 |
lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)" |
|
799 |
proof |
|
800 |
assume "abs a <= 0" |
|
801 |
hence "abs a = 0" by (auto dest: order_antisym) |
|
802 |
thus "a = 0" by simp |
|
803 |
next |
|
804 |
assume "a = 0" |
|
805 |
thus "abs a <= 0" by simp |
|
806 |
qed |
|
807 |
||
808 |
lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))" |
|
809 |
by (simp add: order_less_le) |
|
810 |
||
811 |
lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)" |
|
812 |
proof - |
|
813 |
have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto |
|
814 |
show ?thesis by (simp add: a) |
|
815 |
qed |
|
816 |
||
817 |
lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)" |
|
818 |
by (simp add: abs_lattice meet_join_le) |
|
819 |
||
820 |
lemma abs_ge_minus_self: "-a \<le> abs (a::'a::lordered_ab_group_abs)" |
|
821 |
by (simp add: abs_lattice meet_join_le) |
|
822 |
||
823 |
lemma le_imp_join_eq: "a \<le> b \<Longrightarrow> join a b = b" |
|
824 |
by (simp add: le_def_join) |
|
825 |
||
826 |
lemma ge_imp_join_eq: "b \<le> a \<Longrightarrow> join a b = a" |
|
827 |
by (simp add: le_def_join join_aci) |
|
828 |
||
829 |
lemma le_imp_meet_eq: "a \<le> b \<Longrightarrow> meet a b = a" |
|
830 |
by (simp add: le_def_meet) |
|
831 |
||
832 |
lemma ge_imp_meet_eq: "b \<le> a \<Longrightarrow> meet a b = b" |
|
833 |
by (simp add: le_def_meet meet_aci) |
|
834 |
||
835 |
lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a - nprt a" |
|
836 |
apply (simp add: pprt_def nprt_def diff_minus) |
|
837 |
apply (simp add: add_meet_join_distribs join_aci abs_lattice[symmetric]) |
|
838 |
apply (subst le_imp_join_eq, auto) |
|
839 |
done |
|
840 |
||
841 |
lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::lordered_ab_group_abs)" |
|
842 |
by (simp add: abs_lattice join_comm) |
|
843 |
||
844 |
lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)" |
|
845 |
apply (simp add: abs_lattice[of "abs a"]) |
|
846 |
apply (subst ge_imp_join_eq) |
|
847 |
apply (rule order_trans[of _ 0]) |
|
848 |
by auto |
|
849 |
||
15093
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
850 |
lemma abs_minus_commute: |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
851 |
fixes a :: "'a::lordered_ab_group_abs" |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
852 |
shows "abs (a-b) = abs(b-a)" |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
853 |
proof - |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
854 |
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
|
855 |
also have "... = abs(b-a)" by simp |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
856 |
finally show ?thesis . |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
857 |
qed |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
858 |
|
14738 | 859 |
lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)" |
860 |
by (simp add: le_def_meet nprt_def meet_comm) |
|
861 |
||
862 |
lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)" |
|
863 |
by (simp add: le_def_join pprt_def join_comm) |
|
864 |
||
865 |
lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)" |
|
866 |
by (simp add: le_def_join pprt_def join_comm) |
|
867 |
||
868 |
lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)" |
|
869 |
by (simp add: le_def_meet nprt_def meet_comm) |
|
870 |
||
15580 | 871 |
lemma pprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> pprt a <= pprt b" |
872 |
by (simp add: le_def_join pprt_def join_aci) |
|
873 |
||
874 |
lemma nprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> nprt a <= nprt b" |
|
875 |
by (simp add: le_def_meet nprt_def meet_aci) |
|
876 |
||
19404 | 877 |
lemma pprt_neg: "pprt (-x) = - nprt x" |
878 |
by (simp add: pprt_def nprt_def) |
|
879 |
||
880 |
lemma nprt_neg: "nprt (-x) = - pprt x" |
|
881 |
by (simp add: pprt_def nprt_def) |
|
882 |
||
14738 | 883 |
lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)" |
884 |
by (simp) |
|
885 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
886 |
lemma abs_of_nonneg [simp]: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)" |
14738 | 887 |
by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts) |
888 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
889 |
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
|
890 |
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
|
891 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
892 |
lemma abs_of_nonpos [simp]: "a \<le> 0 \<Longrightarrow> abs a = -(a::'a::lordered_ab_group_abs)" |
14738 | 893 |
by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts) |
894 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
895 |
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
|
896 |
abs x = - x" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
897 |
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
|
898 |
|
14738 | 899 |
lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::lordered_ab_group_abs)" |
900 |
by (simp add: abs_lattice join_imp_le) |
|
901 |
||
902 |
lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::lordered_ab_group))" |
|
903 |
proof - |
|
904 |
from add_le_cancel_left[of "-a" "a+a" "0"] have "(a <= -a) = (a+a <= 0)" |
|
905 |
by (simp add: add_assoc[symmetric]) |
|
906 |
thus ?thesis by simp |
|
907 |
qed |
|
908 |
||
909 |
lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))" |
|
910 |
proof - |
|
911 |
from add_le_cancel_left[of "-a" "0" "a+a"] have "(-a <= a) = (0 <= a+a)" |
|
912 |
by (simp add: add_assoc[symmetric]) |
|
913 |
thus ?thesis by simp |
|
914 |
qed |
|
915 |
||
916 |
lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)" |
|
917 |
by (insert abs_ge_self, blast intro: order_trans) |
|
918 |
||
919 |
lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::lordered_ab_group_abs)" |
|
920 |
by (insert abs_le_D1 [of "-a"], simp) |
|
921 |
||
922 |
lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::lordered_ab_group_abs))" |
|
923 |
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2) |
|
924 |
||
15539 | 925 |
lemma abs_triangle_ineq: "abs(a+b) \<le> abs a + abs(b::'a::lordered_ab_group_abs)" |
14738 | 926 |
proof - |
927 |
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
|
928 |
by (simp add: abs_lattice add_meet_join_distribs join_aci diff_minus) |
14738 | 929 |
have a:"a+b <= join ?m ?n" by (simp add: meet_join_le) |
930 |
have b:"-a-b <= ?n" by (simp add: meet_join_le) |
|
931 |
have c:"?n <= join ?m ?n" by (simp add: meet_join_le) |
|
932 |
from b c have d: "-a-b <= join ?m ?n" by simp |
|
933 |
have e:"-a-b = -(a+b)" by (simp add: diff_minus) |
|
934 |
from a d e have "abs(a+b) <= join ?m ?n" |
|
935 |
by (drule_tac abs_leI, auto) |
|
936 |
with g[symmetric] show ?thesis by simp |
|
937 |
qed |
|
938 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
939 |
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
|
940 |
abs b <= abs (a - b)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
941 |
apply (simp add: compare_rls) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
942 |
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
|
943 |
apply (erule ssubst) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
944 |
apply (rule abs_triangle_ineq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
945 |
apply (rule arg_cong);back; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
946 |
apply (simp add: compare_rls) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
947 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
948 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
949 |
lemma abs_triangle_ineq3: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
950 |
"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
|
951 |
apply (subst abs_le_iff) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
952 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
953 |
apply (rule abs_triangle_ineq2) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
954 |
apply (subst abs_minus_commute) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
955 |
apply (rule abs_triangle_ineq2) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
956 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
957 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
958 |
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
|
959 |
abs a + abs b" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
960 |
proof -; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
961 |
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
|
962 |
by (subst diff_minus, rule refl) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
963 |
also have "... <= abs a + abs (- b)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
964 |
by (rule abs_triangle_ineq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
965 |
finally show ?thesis |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
966 |
by simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
967 |
qed |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
968 |
|
14738 | 969 |
lemma abs_diff_triangle_ineq: |
970 |
"\<bar>(a::'a::lordered_ab_group_abs) + b - (c+d)\<bar> \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" |
|
971 |
proof - |
|
972 |
have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac) |
|
973 |
also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq) |
|
974 |
finally show ?thesis . |
|
975 |
qed |
|
976 |
||
15539 | 977 |
lemma abs_add_abs[simp]: |
978 |
fixes a:: "'a::{lordered_ab_group_abs}" |
|
979 |
shows "abs(abs a + abs b) = abs a + abs b" (is "?L = ?R") |
|
980 |
proof (rule order_antisym) |
|
981 |
show "?L \<ge> ?R" by(rule abs_ge_self) |
|
982 |
next |
|
983 |
have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq) |
|
984 |
also have "\<dots> = ?R" by simp |
|
985 |
finally show "?L \<le> ?R" . |
|
986 |
qed |
|
987 |
||
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
988 |
text {* Needed for abelian cancellation simprocs: *} |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
989 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
990 |
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
|
991 |
apply (subst add_left_commute) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
992 |
apply (subst add_left_cancel) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
993 |
apply simp |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
994 |
done |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
995 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
996 |
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
|
997 |
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
|
998 |
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
|
999 |
done |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1000 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1001 |
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
|
1002 |
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
|
1003 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1004 |
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
|
1005 |
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
|
1006 |
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
|
1007 |
done |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1008 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1009 |
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
|
1010 |
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
|
1011 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1012 |
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
|
1013 |
by (simp add: diff_minus) |
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 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
|
1016 |
by (simp add: add_assoc[symmetric]) |
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 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
|
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 |
|
15178 | 1021 |
lemma le_add_right_mono: |
1022 |
assumes |
|
1023 |
"a <= b + (c::'a::pordered_ab_group_add)" |
|
1024 |
"c <= d" |
|
1025 |
shows "a <= b + d" |
|
1026 |
apply (rule_tac order_trans[where y = "b+c"]) |
|
1027 |
apply (simp_all add: prems) |
|
1028 |
done |
|
1029 |
||
1030 |
lemmas group_eq_simps = |
|
1031 |
mult_ac |
|
1032 |
add_ac |
|
1033 |
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 |
|
1034 |
diff_eq_eq eq_diff_eq |
|
1035 |
||
1036 |
lemma estimate_by_abs: |
|
1037 |
"a + b <= (c::'a::lordered_ab_group_abs) \<Longrightarrow> a <= c + abs b" |
|
1038 |
proof - |
|
1039 |
assume 1: "a+b <= c" |
|
1040 |
have 2: "a <= c+(-b)" |
|
1041 |
apply (insert 1) |
|
1042 |
apply (drule_tac add_right_mono[where c="-b"]) |
|
1043 |
apply (simp add: group_eq_simps) |
|
1044 |
done |
|
1045 |
have 3: "(-b) <= abs b" by (rule abs_ge_minus_self) |
|
1046 |
show ?thesis by (rule le_add_right_mono[OF 2 3]) |
|
1047 |
qed |
|
1048 |
||
17085 | 1049 |
text{*Simplification of @{term "x-y < 0"}, etc.*} |
1050 |
lemmas diff_less_0_iff_less = less_iff_diff_less_0 [symmetric] |
|
1051 |
lemmas diff_eq_0_iff_eq = eq_iff_diff_eq_0 [symmetric] |
|
1052 |
lemmas diff_le_0_iff_le = le_iff_diff_le_0 [symmetric] |
|
1053 |
declare diff_less_0_iff_less [simp] |
|
1054 |
declare diff_eq_0_iff_eq [simp] |
|
1055 |
declare diff_le_0_iff_le [simp] |
|
1056 |
||
1057 |
||
19404 | 1058 |
|
1059 |
||
14738 | 1060 |
ML {* |
1061 |
val add_zero_left = thm"add_0"; |
|
1062 |
val add_zero_right = thm"add_0_right"; |
|
1063 |
*} |
|
1064 |
||
1065 |
ML {* |
|
1066 |
val add_assoc = thm "add_assoc"; |
|
1067 |
val add_commute = thm "add_commute"; |
|
1068 |
val add_left_commute = thm "add_left_commute"; |
|
1069 |
val add_ac = thms "add_ac"; |
|
1070 |
val mult_assoc = thm "mult_assoc"; |
|
1071 |
val mult_commute = thm "mult_commute"; |
|
1072 |
val mult_left_commute = thm "mult_left_commute"; |
|
1073 |
val mult_ac = thms "mult_ac"; |
|
1074 |
val add_0 = thm "add_0"; |
|
1075 |
val mult_1_left = thm "mult_1_left"; |
|
1076 |
val mult_1_right = thm "mult_1_right"; |
|
1077 |
val mult_1 = thm "mult_1"; |
|
1078 |
val add_left_imp_eq = thm "add_left_imp_eq"; |
|
1079 |
val add_right_imp_eq = thm "add_right_imp_eq"; |
|
1080 |
val add_imp_eq = thm "add_imp_eq"; |
|
1081 |
val left_minus = thm "left_minus"; |
|
1082 |
val diff_minus = thm "diff_minus"; |
|
1083 |
val add_0_right = thm "add_0_right"; |
|
1084 |
val add_left_cancel = thm "add_left_cancel"; |
|
1085 |
val add_right_cancel = thm "add_right_cancel"; |
|
1086 |
val right_minus = thm "right_minus"; |
|
1087 |
val right_minus_eq = thm "right_minus_eq"; |
|
1088 |
val minus_minus = thm "minus_minus"; |
|
1089 |
val equals_zero_I = thm "equals_zero_I"; |
|
1090 |
val minus_zero = thm "minus_zero"; |
|
1091 |
val diff_self = thm "diff_self"; |
|
1092 |
val diff_0 = thm "diff_0"; |
|
1093 |
val diff_0_right = thm "diff_0_right"; |
|
1094 |
val diff_minus_eq_add = thm "diff_minus_eq_add"; |
|
1095 |
val neg_equal_iff_equal = thm "neg_equal_iff_equal"; |
|
1096 |
val neg_equal_0_iff_equal = thm "neg_equal_0_iff_equal"; |
|
1097 |
val neg_0_equal_iff_equal = thm "neg_0_equal_iff_equal"; |
|
1098 |
val equation_minus_iff = thm "equation_minus_iff"; |
|
1099 |
val minus_equation_iff = thm "minus_equation_iff"; |
|
1100 |
val minus_add_distrib = thm "minus_add_distrib"; |
|
1101 |
val minus_diff_eq = thm "minus_diff_eq"; |
|
1102 |
val add_left_mono = thm "add_left_mono"; |
|
1103 |
val add_le_imp_le_left = thm "add_le_imp_le_left"; |
|
1104 |
val add_right_mono = thm "add_right_mono"; |
|
1105 |
val add_mono = thm "add_mono"; |
|
1106 |
val add_strict_left_mono = thm "add_strict_left_mono"; |
|
1107 |
val add_strict_right_mono = thm "add_strict_right_mono"; |
|
1108 |
val add_strict_mono = thm "add_strict_mono"; |
|
1109 |
val add_less_le_mono = thm "add_less_le_mono"; |
|
1110 |
val add_le_less_mono = thm "add_le_less_mono"; |
|
1111 |
val add_less_imp_less_left = thm "add_less_imp_less_left"; |
|
1112 |
val add_less_imp_less_right = thm "add_less_imp_less_right"; |
|
1113 |
val add_less_cancel_left = thm "add_less_cancel_left"; |
|
1114 |
val add_less_cancel_right = thm "add_less_cancel_right"; |
|
1115 |
val add_le_cancel_left = thm "add_le_cancel_left"; |
|
1116 |
val add_le_cancel_right = thm "add_le_cancel_right"; |
|
1117 |
val add_le_imp_le_right = thm "add_le_imp_le_right"; |
|
1118 |
val add_increasing = thm "add_increasing"; |
|
1119 |
val le_imp_neg_le = thm "le_imp_neg_le"; |
|
1120 |
val neg_le_iff_le = thm "neg_le_iff_le"; |
|
1121 |
val neg_le_0_iff_le = thm "neg_le_0_iff_le"; |
|
1122 |
val neg_0_le_iff_le = thm "neg_0_le_iff_le"; |
|
1123 |
val neg_less_iff_less = thm "neg_less_iff_less"; |
|
1124 |
val neg_less_0_iff_less = thm "neg_less_0_iff_less"; |
|
1125 |
val neg_0_less_iff_less = thm "neg_0_less_iff_less"; |
|
1126 |
val less_minus_iff = thm "less_minus_iff"; |
|
1127 |
val minus_less_iff = thm "minus_less_iff"; |
|
1128 |
val le_minus_iff = thm "le_minus_iff"; |
|
1129 |
val minus_le_iff = thm "minus_le_iff"; |
|
1130 |
val add_diff_eq = thm "add_diff_eq"; |
|
1131 |
val diff_add_eq = thm "diff_add_eq"; |
|
1132 |
val diff_eq_eq = thm "diff_eq_eq"; |
|
1133 |
val eq_diff_eq = thm "eq_diff_eq"; |
|
1134 |
val diff_diff_eq = thm "diff_diff_eq"; |
|
1135 |
val diff_diff_eq2 = thm "diff_diff_eq2"; |
|
1136 |
val diff_add_cancel = thm "diff_add_cancel"; |
|
1137 |
val add_diff_cancel = thm "add_diff_cancel"; |
|
1138 |
val less_iff_diff_less_0 = thm "less_iff_diff_less_0"; |
|
1139 |
val diff_less_eq = thm "diff_less_eq"; |
|
1140 |
val less_diff_eq = thm "less_diff_eq"; |
|
1141 |
val diff_le_eq = thm "diff_le_eq"; |
|
1142 |
val le_diff_eq = thm "le_diff_eq"; |
|
1143 |
val compare_rls = thms "compare_rls"; |
|
1144 |
val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0"; |
|
1145 |
val le_iff_diff_le_0 = thm "le_iff_diff_le_0"; |
|
1146 |
val add_meet_distrib_left = thm "add_meet_distrib_left"; |
|
1147 |
val add_join_distrib_left = thm "add_join_distrib_left"; |
|
1148 |
val is_join_neg_meet = thm "is_join_neg_meet"; |
|
1149 |
val is_meet_neg_join = thm "is_meet_neg_join"; |
|
1150 |
val add_join_distrib_right = thm "add_join_distrib_right"; |
|
1151 |
val add_meet_distrib_right = thm "add_meet_distrib_right"; |
|
1152 |
val add_meet_join_distribs = thms "add_meet_join_distribs"; |
|
1153 |
val join_eq_neg_meet = thm "join_eq_neg_meet"; |
|
1154 |
val meet_eq_neg_join = thm "meet_eq_neg_join"; |
|
1155 |
val add_eq_meet_join = thm "add_eq_meet_join"; |
|
1156 |
val prts = thm "prts"; |
|
1157 |
val zero_le_pprt = thm "zero_le_pprt"; |
|
1158 |
val nprt_le_zero = thm "nprt_le_zero"; |
|
1159 |
val le_eq_neg = thm "le_eq_neg"; |
|
1160 |
val join_0_imp_0 = thm "join_0_imp_0"; |
|
1161 |
val meet_0_imp_0 = thm "meet_0_imp_0"; |
|
1162 |
val join_0_eq_0 = thm "join_0_eq_0"; |
|
1163 |
val meet_0_eq_0 = thm "meet_0_eq_0"; |
|
1164 |
val zero_le_double_add_iff_zero_le_single_add = thm "zero_le_double_add_iff_zero_le_single_add"; |
|
1165 |
val double_add_le_zero_iff_single_add_le_zero = thm "double_add_le_zero_iff_single_add_le_zero"; |
|
1166 |
val double_add_less_zero_iff_single_less_zero = thm "double_add_less_zero_iff_single_less_zero"; |
|
1167 |
val abs_lattice = thm "abs_lattice"; |
|
1168 |
val abs_zero = thm "abs_zero"; |
|
1169 |
val abs_eq_0 = thm "abs_eq_0"; |
|
1170 |
val abs_0_eq = thm "abs_0_eq"; |
|
1171 |
val neg_meet_eq_join = thm "neg_meet_eq_join"; |
|
1172 |
val neg_join_eq_meet = thm "neg_join_eq_meet"; |
|
1173 |
val join_eq_if = thm "join_eq_if"; |
|
1174 |
val abs_if_lattice = thm "abs_if_lattice"; |
|
1175 |
val abs_ge_zero = thm "abs_ge_zero"; |
|
1176 |
val abs_le_zero_iff = thm "abs_le_zero_iff"; |
|
1177 |
val zero_less_abs_iff = thm "zero_less_abs_iff"; |
|
1178 |
val abs_not_less_zero = thm "abs_not_less_zero"; |
|
1179 |
val abs_ge_self = thm "abs_ge_self"; |
|
1180 |
val abs_ge_minus_self = thm "abs_ge_minus_self"; |
|
1181 |
val le_imp_join_eq = thm "le_imp_join_eq"; |
|
1182 |
val ge_imp_join_eq = thm "ge_imp_join_eq"; |
|
1183 |
val le_imp_meet_eq = thm "le_imp_meet_eq"; |
|
1184 |
val ge_imp_meet_eq = thm "ge_imp_meet_eq"; |
|
1185 |
val abs_prts = thm "abs_prts"; |
|
1186 |
val abs_minus_cancel = thm "abs_minus_cancel"; |
|
1187 |
val abs_idempotent = thm "abs_idempotent"; |
|
1188 |
val zero_le_iff_zero_nprt = thm "zero_le_iff_zero_nprt"; |
|
1189 |
val le_zero_iff_zero_pprt = thm "le_zero_iff_zero_pprt"; |
|
1190 |
val le_zero_iff_pprt_id = thm "le_zero_iff_pprt_id"; |
|
1191 |
val zero_le_iff_nprt_id = thm "zero_le_iff_nprt_id"; |
|
1192 |
val iff2imp = thm "iff2imp"; |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1193 |
(* 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
|
1194 |
val imp_abs_neg_id = thm "imp_abs_neg_id"; *) |
14738 | 1195 |
val abs_leI = thm "abs_leI"; |
1196 |
val le_minus_self_iff = thm "le_minus_self_iff"; |
|
1197 |
val minus_le_self_iff = thm "minus_le_self_iff"; |
|
1198 |
val abs_le_D1 = thm "abs_le_D1"; |
|
1199 |
val abs_le_D2 = thm "abs_le_D2"; |
|
1200 |
val abs_le_iff = thm "abs_le_iff"; |
|
1201 |
val abs_triangle_ineq = thm "abs_triangle_ineq"; |
|
1202 |
val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq"; |
|
1203 |
*} |
|
1204 |
||
1205 |
end |