src/HOL/Ring_and_Field.thy
author paulson
Sun Feb 15 10:46:37 2004 +0100 (2004-02-15)
changeset 14387 e96d5c42c4b0
parent 14377 f454b3004f8f
child 14398 c5c47703f763
permissions -rw-r--r--
Polymorphic treatment of binary arithmetic using axclasses
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer and Markus Wenzel, TU Muenchen
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             Lawrence C Paulson, University of Cambridge
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    License: GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {*
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  \title{Ring and field structures}
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  \author{Gertrud Bauer, L. C. Paulson and Markus Wenzel}
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*}
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theory Ring_and_Field = Inductive:
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subsection {* Abstract algebraic structures *}
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axclass semiring \<subseteq> zero, one, plus, times
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  add_assoc: "(a + b) + c = a + (b + c)"
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  add_commute: "a + b = b + a"
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  add_0 [simp]: "0 + a = a"
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  add_left_imp_eq: "a + b = a + c ==> b=c"
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    --{*This axiom is needed for semirings only: for rings, etc., it is
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        redundant. Including it allows many more of the following results
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        to be proved for semirings too. The drawback is that this redundant
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        axiom must be proved for instances of rings.*}
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  mult_assoc: "(a * b) * c = a * (b * c)"
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  mult_commute: "a * b = b * a"
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  mult_1 [simp]: "1 * a = a"
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  left_distrib: "(a + b) * c = a * c + b * c"
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  zero_neq_one [simp]: "0 \<noteq> 1"
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axclass ring \<subseteq> semiring, minus
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  left_minus [simp]: "- a + a = 0"
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  diff_minus: "a - b = a + (-b)"
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axclass ordered_semiring \<subseteq> semiring, linorder
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  zero_less_one [simp]: "0 < 1" --{*This too is needed for semirings only.*}
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  add_left_mono: "a \<le> b ==> c + a \<le> c + b"
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  mult_strict_left_mono: "a < b ==> 0 < c ==> c * a < c * b"
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axclass ordered_ring \<subseteq> ordered_semiring, ring
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  abs_if: "\<bar>a\<bar> = (if a < 0 then -a else a)"
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axclass field \<subseteq> ring, inverse
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  left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"
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  divide_inverse:      "b \<noteq> 0 ==> a / b = a * inverse b"
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axclass ordered_field \<subseteq> ordered_ring, field
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axclass division_by_zero \<subseteq> zero, inverse
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  inverse_zero [simp]: "inverse 0 = 0"
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  divide_zero [simp]: "a / 0 = 0"
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subsection {* Derived Rules for Addition *}
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lemma add_0_right [simp]: "a + 0 = (a::'a::semiring)"
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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_commute: "a + (b + c) = b + (a + (c::'a::semiring))"
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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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lemma right_minus [simp]: "a + -(a::'a::ring) = 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::ring))"
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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 add_left_cancel [simp]:
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     "(a + b = a + c) = (b = (c::'a::semiring))"
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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::semiring))"
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  by (simp add: add_commute)
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lemma minus_minus [simp]: "- (- (a::'a::ring)) = 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::ring)"
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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::ring)"
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by (simp add: equals_zero_I)
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lemma diff_self [simp]: "a - (a::'a::ring) = 0"
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  by (simp add: diff_minus)
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lemma diff_0 [simp]: "(0::'a::ring) - a = -a"
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by (simp add: diff_minus)
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lemma diff_0_right [simp]: "a - (0::'a::ring) = 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::ring)"
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by (simp add: diff_minus)
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lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ring))" 
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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::ring))"
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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::ring))"
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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::ring))"
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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::ring) = 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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subsection {* Derived rules for multiplication *}
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lemma mult_1_right [simp]: "a * (1::'a::semiring) = a"
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proof -
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  have "a * 1 = 1 * a" by (simp add: mult_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 mult_left_commute: "a * (b * c) = b * (a * (c::'a::semiring))"
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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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lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring)"
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proof -
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  have "0*a + 0*a = 0*a + 0"
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    by (simp add: left_distrib [symmetric])
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  thus ?thesis by (simp only: add_left_cancel)
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qed
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lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring)"
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  by (simp add: mult_commute)
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subsection {* Distribution rules *}
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lemma right_distrib: "a * (b + c) = a * b + a * (c::'a::semiring)"
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proof -
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  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
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  also have "... = b * a + c * a" by (simp only: left_distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show ?thesis .
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qed
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theorems ring_distrib = right_distrib left_distrib
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor: "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
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by (simp add: left_distrib add_ac)
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lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ring)"
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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_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: left_distrib [symmetric]) 
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done
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lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: right_distrib [symmetric]) 
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done
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lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
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by (simp add: right_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
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by (simp add: mult_commute [of _ c] right_diff_distrib) 
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lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ring)"
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by (simp add: diff_minus add_commute) 
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subsection {* Ordering Rules for Addition *}
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lemma add_right_mono: "a \<le> (b::'a::ordered_semiring) ==> 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: "[|a \<le> b;  c \<le> d|] ==> a + c \<le> b + (d::'a::ordered_semiring)"
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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::ordered_semiring)"
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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::ordered_semiring)"
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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::ordered_semiring)"
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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::ordered_semiring)"
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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::ordered_semiring)"
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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::ordered_semiring)"
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proof (rule ccontr)
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  assume "~ a < b"
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  hence "b \<le> a" by (simp add: linorder_not_less)
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  hence "c+b \<le> c+a" by (rule add_left_mono)
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  with this and less show False 
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    by (simp add: linorder_not_less [symmetric])
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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::ordered_semiring)"
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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::ordered_semiring))"
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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::ordered_semiring))"
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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::ordered_semiring))"
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by (simp add: linorder_not_less [symmetric]) 
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lemma add_le_cancel_right [simp]:
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    "(a+c \<le> b+c) = (a \<le> (b::'a::ordered_semiring))"
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by (simp add: linorder_not_less [symmetric]) 
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lemma add_le_imp_le_left:
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      "c + a \<le> c + b ==> a \<le> (b::'a::ordered_semiring)"
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by simp
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lemma add_le_imp_le_right:
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      "a + c \<le> b + c ==> a \<le> (b::'a::ordered_semiring)"
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by simp
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lemma add_increasing: "[|0\<le>a; b\<le>c|] ==> b \<le> a + (c::'a::ordered_semiring)"
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by (insert add_mono [of 0 a b c], simp)
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subsection {* Ordering Rules for Unary Minus *}
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lemma le_imp_neg_le:
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      assumes "a \<le> (b::'a::ordered_ring)" shows "-b \<le> -a"
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proof -
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  have "-a+a \<le> -a+b"
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    by (rule add_left_mono) 
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  hence "0 \<le> -a+b"
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    by simp
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  hence "0 + (-b) \<le> (-a + b) + (-b)"
paulson@14265
   318
    by (rule add_right_mono) 
paulson@14266
   319
  thus ?thesis
paulson@14265
   320
    by (simp add: add_assoc)
paulson@14377
   321
qed
paulson@14265
   322
paulson@14265
   323
lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::ordered_ring))"
paulson@14377
   324
proof 
paulson@14377
   325
  assume "- b \<le> - a"
paulson@14377
   326
  hence "- (- a) \<le> - (- b)"
paulson@14377
   327
    by (rule le_imp_neg_le)
paulson@14377
   328
  thus "a\<le>b" by simp
paulson@14377
   329
next
paulson@14377
   330
  assume "a\<le>b"
paulson@14377
   331
  thus "-b \<le> -a" by (rule le_imp_neg_le)
paulson@14377
   332
qed
paulson@14265
   333
paulson@14265
   334
lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::ordered_ring))"
paulson@14265
   335
by (subst neg_le_iff_le [symmetric], simp)
paulson@14265
   336
paulson@14265
   337
lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::ordered_ring))"
paulson@14265
   338
by (subst neg_le_iff_le [symmetric], simp)
paulson@14265
   339
paulson@14265
   340
lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::ordered_ring))"
paulson@14265
   341
by (force simp add: order_less_le) 
paulson@14265
   342
paulson@14265
   343
lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::ordered_ring))"
paulson@14265
   344
by (subst neg_less_iff_less [symmetric], simp)
paulson@14265
   345
paulson@14265
   346
lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::ordered_ring))"
paulson@14265
   347
by (subst neg_less_iff_less [symmetric], simp)
paulson@14265
   348
paulson@14272
   349
text{*The next several equations can make the simplifier loop!*}
paulson@14272
   350
paulson@14272
   351
lemma less_minus_iff: "(a < - b) = (b < - (a::'a::ordered_ring))"
paulson@14377
   352
proof -
paulson@14272
   353
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
paulson@14272
   354
  thus ?thesis by simp
paulson@14377
   355
qed
paulson@14272
   356
paulson@14272
   357
lemma minus_less_iff: "(- a < b) = (- b < (a::'a::ordered_ring))"
paulson@14377
   358
proof -
paulson@14272
   359
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
paulson@14272
   360
  thus ?thesis by simp
paulson@14377
   361
qed
paulson@14272
   362
paulson@14272
   363
lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::ordered_ring))"
paulson@14272
   364
apply (simp add: linorder_not_less [symmetric])
paulson@14272
   365
apply (rule minus_less_iff) 
paulson@14272
   366
done
paulson@14272
   367
paulson@14272
   368
lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::ordered_ring))"
paulson@14272
   369
apply (simp add: linorder_not_less [symmetric])
paulson@14272
   370
apply (rule less_minus_iff) 
paulson@14272
   371
done
paulson@14272
   372
paulson@14270
   373
paulson@14270
   374
subsection{*Subtraction Laws*}
paulson@14270
   375
paulson@14270
   376
lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ring)"
paulson@14270
   377
by (simp add: diff_minus add_ac)
paulson@14270
   378
paulson@14270
   379
lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ring)"
paulson@14270
   380
by (simp add: diff_minus add_ac)
paulson@14270
   381
paulson@14270
   382
lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ring))"
paulson@14270
   383
by (auto simp add: diff_minus add_assoc)
paulson@14270
   384
paulson@14270
   385
lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ring) = c)"
paulson@14270
   386
by (auto simp add: diff_minus add_assoc)
paulson@14270
   387
paulson@14270
   388
lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ring))"
paulson@14270
   389
by (simp add: diff_minus add_ac)
paulson@14270
   390
paulson@14270
   391
lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ring)"
paulson@14270
   392
by (simp add: diff_minus add_ac)
paulson@14270
   393
paulson@14270
   394
text{*Further subtraction laws for ordered rings*}
paulson@14270
   395
paulson@14272
   396
lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::ordered_ring))"
paulson@14270
   397
proof -
paulson@14270
   398
  have  "(a < b) = (a + (- b) < b + (-b))"  
paulson@14270
   399
    by (simp only: add_less_cancel_right)
paulson@14270
   400
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
paulson@14270
   401
  finally show ?thesis .
paulson@14270
   402
qed
paulson@14270
   403
paulson@14270
   404
lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::ordered_ring))"
paulson@14272
   405
apply (subst less_iff_diff_less_0)
paulson@14272
   406
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
paulson@14270
   407
apply (simp add: diff_minus add_ac)
paulson@14270
   408
done
paulson@14270
   409
paulson@14270
   410
lemma less_diff_eq: "(a < c-b) = (a + (b::'a::ordered_ring) < c)"
paulson@14272
   411
apply (subst less_iff_diff_less_0)
paulson@14272
   412
apply (rule less_iff_diff_less_0 [of _ "c-b", THEN ssubst])
paulson@14270
   413
apply (simp add: diff_minus add_ac)
paulson@14270
   414
done
paulson@14270
   415
paulson@14270
   416
lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::ordered_ring))"
paulson@14270
   417
by (simp add: linorder_not_less [symmetric] less_diff_eq)
paulson@14270
   418
paulson@14270
   419
lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::ordered_ring) \<le> c)"
paulson@14270
   420
by (simp add: linorder_not_less [symmetric] diff_less_eq)
paulson@14270
   421
paulson@14270
   422
text{*This list of rewrites simplifies (in)equalities by bringing subtractions
paulson@14270
   423
  to the top and then moving negative terms to the other side.
paulson@14270
   424
  Use with @{text add_ac}*}
paulson@14270
   425
lemmas compare_rls =
paulson@14270
   426
       diff_minus [symmetric]
paulson@14270
   427
       add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
paulson@14270
   428
       diff_less_eq less_diff_eq diff_le_eq le_diff_eq
paulson@14270
   429
       diff_eq_eq eq_diff_eq
paulson@14270
   430
paulson@14270
   431
paulson@14272
   432
subsection{*Lemmas for the @{text cancel_numerals} simproc*}
paulson@14272
   433
paulson@14272
   434
lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ring))"
paulson@14272
   435
by (simp add: compare_rls)
paulson@14272
   436
paulson@14272
   437
lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::ordered_ring))"
paulson@14272
   438
by (simp add: compare_rls)
paulson@14272
   439
paulson@14272
   440
lemma eq_add_iff1:
paulson@14272
   441
     "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
paulson@14272
   442
apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   443
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   444
done
paulson@14272
   445
paulson@14272
   446
lemma eq_add_iff2:
paulson@14272
   447
     "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
paulson@14272
   448
apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   449
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   450
done
paulson@14272
   451
paulson@14272
   452
lemma less_add_iff1:
paulson@14272
   453
     "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::ordered_ring))"
paulson@14272
   454
apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   455
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   456
done
paulson@14272
   457
paulson@14272
   458
lemma less_add_iff2:
paulson@14272
   459
     "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::ordered_ring))"
paulson@14272
   460
apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   461
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   462
done
paulson@14272
   463
paulson@14272
   464
lemma le_add_iff1:
paulson@14272
   465
     "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::ordered_ring))"
paulson@14272
   466
apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   467
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   468
done
paulson@14272
   469
paulson@14272
   470
lemma le_add_iff2:
paulson@14272
   471
     "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::ordered_ring))"
paulson@14272
   472
apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   473
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   474
done
paulson@14272
   475
paulson@14272
   476
paulson@14270
   477
subsection {* Ordering Rules for Multiplication *}
paulson@14270
   478
paulson@14265
   479
lemma mult_strict_right_mono:
paulson@14265
   480
     "[|a < b; 0 < c|] ==> a * c < b * (c::'a::ordered_semiring)"
paulson@14265
   481
by (simp add: mult_commute [of _ c] mult_strict_left_mono)
paulson@14265
   482
paulson@14265
   483
lemma mult_left_mono:
paulson@14341
   484
     "[|a \<le> b; 0 \<le> c|] ==> c * a \<le> c * (b::'a::ordered_semiring)"
paulson@14267
   485
  apply (case_tac "c=0", simp)
paulson@14267
   486
  apply (force simp add: mult_strict_left_mono order_le_less) 
paulson@14267
   487
  done
paulson@14265
   488
paulson@14265
   489
lemma mult_right_mono:
paulson@14341
   490
     "[|a \<le> b; 0 \<le> c|] ==> a*c \<le> b * (c::'a::ordered_semiring)"
paulson@14267
   491
  by (simp add: mult_left_mono mult_commute [of _ c]) 
paulson@14265
   492
paulson@14348
   493
lemma mult_left_le_imp_le:
paulson@14348
   494
     "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring)"
paulson@14348
   495
  by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
paulson@14348
   496
 
paulson@14348
   497
lemma mult_right_le_imp_le:
paulson@14348
   498
     "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring)"
paulson@14348
   499
  by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
paulson@14348
   500
paulson@14348
   501
lemma mult_left_less_imp_less:
paulson@14348
   502
     "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
paulson@14348
   503
  by (force simp add: mult_left_mono linorder_not_le [symmetric])
paulson@14348
   504
 
paulson@14348
   505
lemma mult_right_less_imp_less:
paulson@14348
   506
     "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
paulson@14348
   507
  by (force simp add: mult_right_mono linorder_not_le [symmetric])
paulson@14348
   508
paulson@14265
   509
lemma mult_strict_left_mono_neg:
paulson@14265
   510
     "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring)"
paulson@14265
   511
apply (drule mult_strict_left_mono [of _ _ "-c"])
paulson@14265
   512
apply (simp_all add: minus_mult_left [symmetric]) 
paulson@14265
   513
done
paulson@14265
   514
paulson@14265
   515
lemma mult_strict_right_mono_neg:
paulson@14265
   516
     "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring)"
paulson@14265
   517
apply (drule mult_strict_right_mono [of _ _ "-c"])
paulson@14265
   518
apply (simp_all add: minus_mult_right [symmetric]) 
paulson@14265
   519
done
paulson@14265
   520
paulson@14265
   521
paulson@14265
   522
subsection{* Products of Signs *}
paulson@14265
   523
paulson@14341
   524
lemma mult_pos: "[| (0::'a::ordered_semiring) < a; 0 < b |] ==> 0 < a*b"
paulson@14265
   525
by (drule mult_strict_left_mono [of 0 b], auto)
paulson@14265
   526
paulson@14341
   527
lemma mult_pos_neg: "[| (0::'a::ordered_semiring) < a; b < 0 |] ==> a*b < 0"
paulson@14265
   528
by (drule mult_strict_left_mono [of b 0], auto)
paulson@14265
   529
paulson@14265
   530
lemma mult_neg: "[| a < (0::'a::ordered_ring); b < 0 |] ==> 0 < a*b"
paulson@14265
   531
by (drule mult_strict_right_mono_neg, auto)
paulson@14265
   532
paulson@14341
   533
lemma zero_less_mult_pos:
paulson@14341
   534
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring)"
paulson@14265
   535
apply (case_tac "b\<le>0") 
paulson@14265
   536
 apply (auto simp add: order_le_less linorder_not_less)
paulson@14265
   537
apply (drule_tac mult_pos_neg [of a b]) 
paulson@14265
   538
 apply (auto dest: order_less_not_sym)
paulson@14265
   539
done
paulson@14265
   540
paulson@14265
   541
lemma zero_less_mult_iff:
paulson@14265
   542
     "((0::'a::ordered_ring) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
paulson@14265
   543
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
paulson@14265
   544
apply (blast dest: zero_less_mult_pos) 
paulson@14265
   545
apply (simp add: mult_commute [of a b]) 
paulson@14265
   546
apply (blast dest: zero_less_mult_pos) 
paulson@14265
   547
done
paulson@14265
   548
paulson@14341
   549
text{*A field has no "zero divisors", and this theorem holds without the
paulson@14277
   550
      assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}
paulson@14266
   551
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring)) = (a = 0 | b = 0)"
paulson@14265
   552
apply (case_tac "a < 0")
paulson@14265
   553
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
paulson@14265
   554
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
paulson@14265
   555
done
paulson@14265
   556
paulson@14265
   557
lemma zero_le_mult_iff:
paulson@14265
   558
     "((0::'a::ordered_ring) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14265
   559
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
paulson@14265
   560
                   zero_less_mult_iff)
paulson@14265
   561
paulson@14265
   562
lemma mult_less_0_iff:
paulson@14265
   563
     "(a*b < (0::'a::ordered_ring)) = (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14265
   564
apply (insert zero_less_mult_iff [of "-a" b]) 
paulson@14265
   565
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   566
done
paulson@14265
   567
paulson@14265
   568
lemma mult_le_0_iff:
paulson@14265
   569
     "(a*b \<le> (0::'a::ordered_ring)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14265
   570
apply (insert zero_le_mult_iff [of "-a" b]) 
paulson@14265
   571
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   572
done
paulson@14265
   573
paulson@14265
   574
lemma zero_le_square: "(0::'a::ordered_ring) \<le> a*a"
paulson@14265
   575
by (simp add: zero_le_mult_iff linorder_linear) 
paulson@14265
   576
paulson@14387
   577
text{*All three types of comparision involving 0 and 1 are covered.*}
paulson@14387
   578
paulson@14387
   579
declare zero_neq_one [THEN not_sym, simp]
paulson@14387
   580
paulson@14387
   581
lemma zero_le_one [simp]: "(0::'a::ordered_semiring) \<le> 1"
paulson@14268
   582
  by (rule zero_less_one [THEN order_less_imp_le]) 
paulson@14268
   583
paulson@14387
   584
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semiring) \<le> 0"
paulson@14387
   585
by (simp add: linorder_not_le zero_less_one) 
paulson@14387
   586
paulson@14387
   587
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semiring) < 0"
paulson@14387
   588
by (simp add: linorder_not_less zero_le_one) 
paulson@14387
   589
paulson@14268
   590
paulson@14268
   591
subsection{*More Monotonicity*}
paulson@14268
   592
paulson@14268
   593
lemma mult_left_mono_neg:
paulson@14268
   594
     "[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::ordered_ring)"
paulson@14268
   595
apply (drule mult_left_mono [of _ _ "-c"]) 
paulson@14268
   596
apply (simp_all add: minus_mult_left [symmetric]) 
paulson@14268
   597
done
paulson@14268
   598
paulson@14268
   599
lemma mult_right_mono_neg:
paulson@14268
   600
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le> b * (c::'a::ordered_ring)"
paulson@14268
   601
  by (simp add: mult_left_mono_neg mult_commute [of _ c]) 
paulson@14268
   602
paulson@14268
   603
text{*Strict monotonicity in both arguments*}
paulson@14268
   604
lemma mult_strict_mono:
paulson@14341
   605
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring)"
paulson@14268
   606
apply (case_tac "c=0")
paulson@14268
   607
 apply (simp add: mult_pos) 
paulson@14268
   608
apply (erule mult_strict_right_mono [THEN order_less_trans])
paulson@14268
   609
 apply (force simp add: order_le_less) 
paulson@14268
   610
apply (erule mult_strict_left_mono, assumption)
paulson@14268
   611
done
paulson@14268
   612
paulson@14268
   613
text{*This weaker variant has more natural premises*}
paulson@14268
   614
lemma mult_strict_mono':
paulson@14341
   615
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring)"
paulson@14268
   616
apply (rule mult_strict_mono)
paulson@14268
   617
apply (blast intro: order_le_less_trans)+
paulson@14268
   618
done
paulson@14268
   619
paulson@14268
   620
lemma mult_mono:
paulson@14268
   621
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
paulson@14341
   622
      ==> a * c  \<le>  b * (d::'a::ordered_semiring)"
paulson@14268
   623
apply (erule mult_right_mono [THEN order_trans], assumption)
paulson@14268
   624
apply (erule mult_left_mono, assumption)
paulson@14268
   625
done
paulson@14268
   626
paulson@14387
   627
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semiring)"
paulson@14387
   628
apply (insert mult_strict_mono [of 1 m 1 n]) 
paulson@14387
   629
apply (simp add:  order_less_trans [OF zero_less_one]); 
paulson@14387
   630
done
paulson@14387
   631
paulson@14268
   632
paulson@14268
   633
subsection{*Cancellation Laws for Relationships With a Common Factor*}
paulson@14268
   634
paulson@14268
   635
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
paulson@14268
   636
   also with the relations @{text "\<le>"} and equality.*}
paulson@14268
   637
paulson@14268
   638
lemma mult_less_cancel_right:
paulson@14268
   639
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring)))"
paulson@14268
   640
apply (case_tac "c = 0")
paulson@14268
   641
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
paulson@14268
   642
                      mult_strict_right_mono_neg)
paulson@14268
   643
apply (auto simp add: linorder_not_less 
paulson@14268
   644
                      linorder_not_le [symmetric, of "a*c"]
paulson@14268
   645
                      linorder_not_le [symmetric, of a])
paulson@14268
   646
apply (erule_tac [!] notE)
paulson@14268
   647
apply (auto simp add: order_less_imp_le mult_right_mono 
paulson@14268
   648
                      mult_right_mono_neg)
paulson@14268
   649
done
paulson@14268
   650
paulson@14268
   651
lemma mult_less_cancel_left:
paulson@14268
   652
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring)))"
paulson@14268
   653
by (simp add: mult_commute [of c] mult_less_cancel_right)
paulson@14268
   654
paulson@14268
   655
lemma mult_le_cancel_right:
paulson@14268
   656
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring)))"
paulson@14268
   657
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right)
paulson@14268
   658
paulson@14268
   659
lemma mult_le_cancel_left:
paulson@14268
   660
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring)))"
paulson@14268
   661
by (simp add: mult_commute [of c] mult_le_cancel_right)
paulson@14268
   662
paulson@14268
   663
lemma mult_less_imp_less_left:
paulson@14341
   664
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
paulson@14341
   665
      shows "a < (b::'a::ordered_semiring)"
paulson@14377
   666
proof (rule ccontr)
paulson@14377
   667
  assume "~ a < b"
paulson@14377
   668
  hence "b \<le> a" by (simp add: linorder_not_less)
paulson@14377
   669
  hence "c*b \<le> c*a" by (rule mult_left_mono)
paulson@14377
   670
  with this and less show False 
paulson@14377
   671
    by (simp add: linorder_not_less [symmetric])
paulson@14377
   672
qed
paulson@14268
   673
paulson@14268
   674
lemma mult_less_imp_less_right:
paulson@14341
   675
    "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
paulson@14341
   676
  by (rule mult_less_imp_less_left, simp add: mult_commute)
paulson@14268
   677
paulson@14268
   678
text{*Cancellation of equalities with a common factor*}
paulson@14268
   679
lemma mult_cancel_right [simp]:
paulson@14268
   680
     "(a*c = b*c) = (c = (0::'a::ordered_ring) | a=b)"
paulson@14268
   681
apply (cut_tac linorder_less_linear [of 0 c])
paulson@14268
   682
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono
paulson@14268
   683
             simp add: linorder_neq_iff)
paulson@14268
   684
done
paulson@14268
   685
paulson@14268
   686
text{*These cancellation theorems require an ordering. Versions are proved
paulson@14268
   687
      below that work for fields without an ordering.*}
paulson@14268
   688
lemma mult_cancel_left [simp]:
paulson@14268
   689
     "(c*a = c*b) = (c = (0::'a::ordered_ring) | a=b)"
paulson@14268
   690
by (simp add: mult_commute [of c] mult_cancel_right)
paulson@14268
   691
paulson@14265
   692
paulson@14265
   693
subsection {* Fields *}
paulson@14265
   694
paulson@14288
   695
lemma right_inverse [simp]:
paulson@14288
   696
      assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"
paulson@14288
   697
proof -
paulson@14288
   698
  have "a * inverse a = inverse a * a" by (simp add: mult_ac)
paulson@14288
   699
  also have "... = 1" using not0 by simp
paulson@14288
   700
  finally show ?thesis .
paulson@14288
   701
qed
paulson@14288
   702
paulson@14288
   703
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
paulson@14288
   704
proof
paulson@14288
   705
  assume neq: "b \<noteq> 0"
paulson@14288
   706
  {
paulson@14288
   707
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
paulson@14288
   708
    also assume "a / b = 1"
paulson@14288
   709
    finally show "a = b" by simp
paulson@14288
   710
  next
paulson@14288
   711
    assume "a = b"
paulson@14288
   712
    with neq show "a / b = 1" by (simp add: divide_inverse)
paulson@14288
   713
  }
paulson@14288
   714
qed
paulson@14288
   715
paulson@14288
   716
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
paulson@14288
   717
by (simp add: divide_inverse)
paulson@14288
   718
paulson@14288
   719
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
paulson@14288
   720
  by (simp add: divide_inverse)
paulson@14288
   721
paulson@14277
   722
lemma divide_inverse_zero: "a/b = a * inverse(b::'a::{field,division_by_zero})"
paulson@14277
   723
apply (case_tac "b = 0")
paulson@14277
   724
apply (simp_all add: divide_inverse)
paulson@14277
   725
done
paulson@14277
   726
paulson@14277
   727
lemma divide_zero_left [simp]: "0/a = (0::'a::{field,division_by_zero})"
paulson@14277
   728
by (simp add: divide_inverse_zero)
paulson@14277
   729
paulson@14277
   730
lemma inverse_eq_divide: "inverse (a::'a::{field,division_by_zero}) = 1/a"
paulson@14277
   731
by (simp add: divide_inverse_zero)
paulson@14277
   732
paulson@14293
   733
lemma nonzero_add_divide_distrib: "c \<noteq> 0 ==> (a+b)/(c::'a::field) = a/c + b/c"
paulson@14293
   734
by (simp add: divide_inverse left_distrib) 
paulson@14293
   735
paulson@14293
   736
lemma add_divide_distrib: "(a+b)/(c::'a::{field,division_by_zero}) = a/c + b/c"
paulson@14293
   737
apply (case_tac "c=0", simp) 
paulson@14293
   738
apply (simp add: nonzero_add_divide_distrib) 
paulson@14293
   739
done
paulson@14293
   740
paulson@14270
   741
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
paulson@14270
   742
      of an ordering.*}
paulson@14348
   743
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
paulson@14377
   744
proof cases
paulson@14377
   745
  assume "a=0" thus ?thesis by simp
paulson@14377
   746
next
paulson@14377
   747
  assume anz [simp]: "a\<noteq>0"
paulson@14377
   748
  { assume "a * b = 0"
paulson@14377
   749
    hence "inverse a * (a * b) = 0" by simp
paulson@14377
   750
    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
paulson@14377
   751
  thus ?thesis by force
paulson@14377
   752
qed
paulson@14270
   753
paulson@14268
   754
text{*Cancellation of equalities with a common factor*}
paulson@14268
   755
lemma field_mult_cancel_right_lemma:
paulson@14269
   756
      assumes cnz: "c \<noteq> (0::'a::field)"
paulson@14269
   757
	  and eq:  "a*c = b*c"
paulson@14269
   758
	 shows "a=b"
paulson@14377
   759
proof -
paulson@14268
   760
  have "(a * c) * inverse c = (b * c) * inverse c"
paulson@14268
   761
    by (simp add: eq)
paulson@14268
   762
  thus "a=b"
paulson@14268
   763
    by (simp add: mult_assoc cnz)
paulson@14377
   764
qed
paulson@14268
   765
paulson@14348
   766
lemma field_mult_cancel_right [simp]:
paulson@14268
   767
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
paulson@14377
   768
proof cases
paulson@14377
   769
  assume "c=0" thus ?thesis by simp
paulson@14377
   770
next
paulson@14377
   771
  assume "c\<noteq>0" 
paulson@14377
   772
  thus ?thesis by (force dest: field_mult_cancel_right_lemma)
paulson@14377
   773
qed
paulson@14268
   774
paulson@14348
   775
lemma field_mult_cancel_left [simp]:
paulson@14268
   776
     "(c*a = c*b) = (c = (0::'a::field) | a=b)"
paulson@14268
   777
  by (simp add: mult_commute [of c] field_mult_cancel_right) 
paulson@14268
   778
paulson@14268
   779
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
paulson@14377
   780
proof
paulson@14268
   781
  assume ianz: "inverse a = 0"
paulson@14268
   782
  assume "a \<noteq> 0"
paulson@14268
   783
  hence "1 = a * inverse a" by simp
paulson@14268
   784
  also have "... = 0" by (simp add: ianz)
paulson@14268
   785
  finally have "1 = (0::'a::field)" .
paulson@14268
   786
  thus False by (simp add: eq_commute)
paulson@14377
   787
qed
paulson@14268
   788
paulson@14277
   789
paulson@14277
   790
subsection{*Basic Properties of @{term inverse}*}
paulson@14277
   791
paulson@14268
   792
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"
paulson@14268
   793
apply (rule ccontr) 
paulson@14268
   794
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   795
done
paulson@14268
   796
paulson@14268
   797
lemma inverse_nonzero_imp_nonzero:
paulson@14268
   798
   "inverse a = 0 ==> a = (0::'a::field)"
paulson@14268
   799
apply (rule ccontr) 
paulson@14268
   800
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   801
done
paulson@14268
   802
paulson@14268
   803
lemma inverse_nonzero_iff_nonzero [simp]:
paulson@14268
   804
   "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"
paulson@14268
   805
by (force dest: inverse_nonzero_imp_nonzero) 
paulson@14268
   806
paulson@14268
   807
lemma nonzero_inverse_minus_eq:
paulson@14269
   808
      assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
paulson@14377
   809
proof -
paulson@14377
   810
  have "-a * inverse (- a) = -a * - inverse a"
paulson@14377
   811
    by simp
paulson@14377
   812
  thus ?thesis 
paulson@14377
   813
    by (simp only: field_mult_cancel_left, simp)
paulson@14377
   814
qed
paulson@14268
   815
paulson@14268
   816
lemma inverse_minus_eq [simp]:
paulson@14377
   817
   "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
paulson@14377
   818
proof cases
paulson@14377
   819
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
   820
next
paulson@14377
   821
  assume "a\<noteq>0" 
paulson@14377
   822
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
   823
qed
paulson@14268
   824
paulson@14268
   825
lemma nonzero_inverse_eq_imp_eq:
paulson@14269
   826
      assumes inveq: "inverse a = inverse b"
paulson@14269
   827
	  and anz:  "a \<noteq> 0"
paulson@14269
   828
	  and bnz:  "b \<noteq> 0"
paulson@14269
   829
	 shows "a = (b::'a::field)"
paulson@14377
   830
proof -
paulson@14268
   831
  have "a * inverse b = a * inverse a"
paulson@14268
   832
    by (simp add: inveq)
paulson@14268
   833
  hence "(a * inverse b) * b = (a * inverse a) * b"
paulson@14268
   834
    by simp
paulson@14268
   835
  thus "a = b"
paulson@14268
   836
    by (simp add: mult_assoc anz bnz)
paulson@14377
   837
qed
paulson@14268
   838
paulson@14268
   839
lemma inverse_eq_imp_eq:
paulson@14268
   840
     "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"
paulson@14268
   841
apply (case_tac "a=0 | b=0") 
paulson@14268
   842
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
   843
              simp add: eq_commute [of "0::'a"])
paulson@14268
   844
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
   845
done
paulson@14268
   846
paulson@14268
   847
lemma inverse_eq_iff_eq [simp]:
paulson@14268
   848
     "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"
paulson@14268
   849
by (force dest!: inverse_eq_imp_eq) 
paulson@14268
   850
paulson@14270
   851
lemma nonzero_inverse_inverse_eq:
paulson@14270
   852
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
paulson@14270
   853
  proof -
paulson@14270
   854
  have "(inverse (inverse a) * inverse a) * a = a" 
paulson@14270
   855
    by (simp add: nonzero_imp_inverse_nonzero)
paulson@14270
   856
  thus ?thesis
paulson@14270
   857
    by (simp add: mult_assoc)
paulson@14270
   858
  qed
paulson@14270
   859
paulson@14270
   860
lemma inverse_inverse_eq [simp]:
paulson@14270
   861
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
paulson@14270
   862
  proof cases
paulson@14270
   863
    assume "a=0" thus ?thesis by simp
paulson@14270
   864
  next
paulson@14270
   865
    assume "a\<noteq>0" 
paulson@14270
   866
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
   867
  qed
paulson@14270
   868
paulson@14270
   869
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
paulson@14270
   870
  proof -
paulson@14270
   871
  have "inverse 1 * 1 = (1::'a::field)" 
paulson@14270
   872
    by (rule left_inverse [OF zero_neq_one [symmetric]])
paulson@14270
   873
  thus ?thesis  by simp
paulson@14270
   874
  qed
paulson@14270
   875
paulson@14270
   876
lemma nonzero_inverse_mult_distrib: 
paulson@14270
   877
      assumes anz: "a \<noteq> 0"
paulson@14270
   878
          and bnz: "b \<noteq> 0"
paulson@14270
   879
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
paulson@14270
   880
  proof -
paulson@14270
   881
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
paulson@14270
   882
    by (simp add: field_mult_eq_0_iff anz bnz)
paulson@14270
   883
  hence "inverse(a*b) * a = inverse(b)" 
paulson@14270
   884
    by (simp add: mult_assoc bnz)
paulson@14270
   885
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
paulson@14270
   886
    by simp
paulson@14270
   887
  thus ?thesis
paulson@14270
   888
    by (simp add: mult_assoc anz)
paulson@14270
   889
  qed
paulson@14270
   890
paulson@14270
   891
text{*This version builds in division by zero while also re-orienting
paulson@14270
   892
      the right-hand side.*}
paulson@14270
   893
lemma inverse_mult_distrib [simp]:
paulson@14270
   894
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
   895
  proof cases
paulson@14270
   896
    assume "a \<noteq> 0 & b \<noteq> 0" 
paulson@14270
   897
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
   898
  next
paulson@14270
   899
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
paulson@14270
   900
    thus ?thesis  by force
paulson@14270
   901
  qed
paulson@14270
   902
paulson@14270
   903
text{*There is no slick version using division by zero.*}
paulson@14270
   904
lemma inverse_add:
paulson@14270
   905
     "[|a \<noteq> 0;  b \<noteq> 0|]
paulson@14270
   906
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
paulson@14270
   907
apply (simp add: left_distrib mult_assoc)
paulson@14270
   908
apply (simp add: mult_commute [of "inverse a"]) 
paulson@14270
   909
apply (simp add: mult_assoc [symmetric] add_commute)
paulson@14270
   910
done
paulson@14270
   911
paulson@14365
   912
lemma inverse_divide [simp]:
paulson@14365
   913
      "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
paulson@14365
   914
  by (simp add: divide_inverse_zero mult_commute)
paulson@14365
   915
paulson@14277
   916
lemma nonzero_mult_divide_cancel_left:
paulson@14277
   917
  assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
paulson@14277
   918
    shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
   919
proof -
paulson@14277
   920
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
paulson@14277
   921
    by (simp add: field_mult_eq_0_iff divide_inverse 
paulson@14277
   922
                  nonzero_inverse_mult_distrib)
paulson@14277
   923
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
   924
    by (simp only: mult_ac)
paulson@14277
   925
  also have "... =  a * inverse b"
paulson@14277
   926
    by simp
paulson@14277
   927
    finally show ?thesis 
paulson@14277
   928
    by (simp add: divide_inverse)
paulson@14277
   929
qed
paulson@14277
   930
paulson@14277
   931
lemma mult_divide_cancel_left:
paulson@14277
   932
     "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
paulson@14277
   933
apply (case_tac "b = 0")
paulson@14277
   934
apply (simp_all add: nonzero_mult_divide_cancel_left)
paulson@14277
   935
done
paulson@14277
   936
paulson@14321
   937
lemma nonzero_mult_divide_cancel_right:
paulson@14321
   938
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
paulson@14321
   939
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
paulson@14321
   940
paulson@14321
   941
lemma mult_divide_cancel_right:
paulson@14321
   942
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
paulson@14321
   943
apply (case_tac "b = 0")
paulson@14321
   944
apply (simp_all add: nonzero_mult_divide_cancel_right)
paulson@14321
   945
done
paulson@14321
   946
paulson@14277
   947
(*For ExtractCommonTerm*)
paulson@14277
   948
lemma mult_divide_cancel_eq_if:
paulson@14277
   949
     "(c*a) / (c*b) = 
paulson@14277
   950
      (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
paulson@14277
   951
  by (simp add: mult_divide_cancel_left)
paulson@14277
   952
paulson@14284
   953
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
paulson@14284
   954
  by (simp add: divide_inverse [OF not_sym])
paulson@14284
   955
paulson@14288
   956
lemma times_divide_eq_right [simp]:
paulson@14288
   957
     "a * (b/c) = (a*b) / (c::'a::{field,division_by_zero})"
paulson@14288
   958
by (simp add: divide_inverse_zero mult_assoc)
paulson@14288
   959
paulson@14288
   960
lemma times_divide_eq_left [simp]:
paulson@14288
   961
     "(b/c) * a = (b*a) / (c::'a::{field,division_by_zero})"
paulson@14288
   962
by (simp add: divide_inverse_zero mult_ac)
paulson@14288
   963
paulson@14288
   964
lemma divide_divide_eq_right [simp]:
paulson@14288
   965
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14288
   966
by (simp add: divide_inverse_zero mult_ac)
paulson@14288
   967
paulson@14288
   968
lemma divide_divide_eq_left [simp]:
paulson@14288
   969
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14288
   970
by (simp add: divide_inverse_zero mult_assoc)
paulson@14288
   971
paulson@14268
   972
paulson@14293
   973
subsection {* Division and Unary Minus *}
paulson@14293
   974
paulson@14293
   975
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
   976
by (simp add: divide_inverse minus_mult_left)
paulson@14293
   977
paulson@14293
   978
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
   979
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
   980
paulson@14293
   981
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
   982
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
   983
paulson@14293
   984
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::{field,division_by_zero})"
paulson@14293
   985
apply (case_tac "b=0", simp) 
paulson@14293
   986
apply (simp add: nonzero_minus_divide_left) 
paulson@14293
   987
done
paulson@14293
   988
paulson@14293
   989
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14293
   990
apply (case_tac "b=0", simp) 
paulson@14293
   991
by (rule nonzero_minus_divide_right) 
paulson@14293
   992
paulson@14293
   993
text{*The effect is to extract signs from divisions*}
paulson@14293
   994
declare minus_divide_left  [symmetric, simp]
paulson@14293
   995
declare minus_divide_right [symmetric, simp]
paulson@14293
   996
paulson@14387
   997
text{*Also, extract signs from products*}
paulson@14387
   998
declare minus_mult_left [symmetric, simp]
paulson@14387
   999
declare minus_mult_right [symmetric, simp]
paulson@14387
  1000
paulson@14293
  1001
lemma minus_divide_divide [simp]:
paulson@14293
  1002
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
paulson@14293
  1003
apply (case_tac "b=0", simp) 
paulson@14293
  1004
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1005
done
paulson@14293
  1006
paulson@14387
  1007
lemma diff_divide_distrib:
paulson@14387
  1008
     "(a-b)/(c::'a::{field,division_by_zero}) = a/c - b/c"
paulson@14387
  1009
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
  1010
paulson@14293
  1011
paulson@14268
  1012
subsection {* Ordered Fields *}
paulson@14268
  1013
paulson@14277
  1014
lemma positive_imp_inverse_positive: 
paulson@14269
  1015
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
paulson@14268
  1016
  proof -
paulson@14268
  1017
  have "0 < a * inverse a" 
paulson@14268
  1018
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1019
  thus "0 < inverse a" 
paulson@14268
  1020
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
paulson@14268
  1021
  qed
paulson@14268
  1022
paulson@14277
  1023
lemma negative_imp_inverse_negative:
paulson@14268
  1024
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
paulson@14277
  1025
  by (insert positive_imp_inverse_positive [of "-a"], 
paulson@14268
  1026
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
paulson@14268
  1027
paulson@14268
  1028
lemma inverse_le_imp_le:
paulson@14269
  1029
      assumes invle: "inverse a \<le> inverse b"
paulson@14269
  1030
	  and apos:  "0 < a"
paulson@14269
  1031
	 shows "b \<le> (a::'a::ordered_field)"
paulson@14268
  1032
  proof (rule classical)
paulson@14268
  1033
  assume "~ b \<le> a"
paulson@14268
  1034
  hence "a < b"
paulson@14268
  1035
    by (simp add: linorder_not_le)
paulson@14268
  1036
  hence bpos: "0 < b"
paulson@14268
  1037
    by (blast intro: apos order_less_trans)
paulson@14268
  1038
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1039
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1040
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1041
    by (simp add: bpos order_less_imp_le mult_right_mono)
paulson@14268
  1042
  thus "b \<le> a"
paulson@14268
  1043
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
paulson@14268
  1044
  qed
paulson@14268
  1045
paulson@14277
  1046
lemma inverse_positive_imp_positive:
paulson@14277
  1047
      assumes inv_gt_0: "0 < inverse a"
paulson@14277
  1048
          and [simp]:   "a \<noteq> 0"
paulson@14277
  1049
        shows "0 < (a::'a::ordered_field)"
paulson@14277
  1050
  proof -
paulson@14277
  1051
  have "0 < inverse (inverse a)"
paulson@14277
  1052
    by (rule positive_imp_inverse_positive)
paulson@14277
  1053
  thus "0 < a"
paulson@14277
  1054
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
  1055
  qed
paulson@14277
  1056
paulson@14277
  1057
lemma inverse_positive_iff_positive [simp]:
paulson@14277
  1058
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1059
apply (case_tac "a = 0", simp)
paulson@14277
  1060
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1061
done
paulson@14277
  1062
paulson@14277
  1063
lemma inverse_negative_imp_negative:
paulson@14277
  1064
      assumes inv_less_0: "inverse a < 0"
paulson@14277
  1065
          and [simp]:   "a \<noteq> 0"
paulson@14277
  1066
        shows "a < (0::'a::ordered_field)"
paulson@14277
  1067
  proof -
paulson@14277
  1068
  have "inverse (inverse a) < 0"
paulson@14277
  1069
    by (rule negative_imp_inverse_negative)
paulson@14277
  1070
  thus "a < 0"
paulson@14277
  1071
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
  1072
  qed
paulson@14277
  1073
paulson@14277
  1074
lemma inverse_negative_iff_negative [simp]:
paulson@14277
  1075
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1076
apply (case_tac "a = 0", simp)
paulson@14277
  1077
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1078
done
paulson@14277
  1079
paulson@14277
  1080
lemma inverse_nonnegative_iff_nonnegative [simp]:
paulson@14277
  1081
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1082
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1083
paulson@14277
  1084
lemma inverse_nonpositive_iff_nonpositive [simp]:
paulson@14277
  1085
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1086
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1087
paulson@14277
  1088
paulson@14277
  1089
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1090
paulson@14268
  1091
lemma less_imp_inverse_less:
paulson@14269
  1092
      assumes less: "a < b"
paulson@14269
  1093
	  and apos:  "0 < a"
paulson@14269
  1094
	shows "inverse b < inverse (a::'a::ordered_field)"
paulson@14268
  1095
  proof (rule ccontr)
paulson@14268
  1096
  assume "~ inverse b < inverse a"
paulson@14268
  1097
  hence "inverse a \<le> inverse b"
paulson@14268
  1098
    by (simp add: linorder_not_less)
paulson@14268
  1099
  hence "~ (a < b)"
paulson@14268
  1100
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
  1101
  thus False
paulson@14268
  1102
    by (rule notE [OF _ less])
paulson@14268
  1103
  qed
paulson@14268
  1104
paulson@14268
  1105
lemma inverse_less_imp_less:
paulson@14268
  1106
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1107
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1108
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1109
done
paulson@14268
  1110
paulson@14268
  1111
text{*Both premises are essential. Consider -1 and 1.*}
paulson@14268
  1112
lemma inverse_less_iff_less [simp]:
paulson@14268
  1113
     "[|0 < a; 0 < b|] 
paulson@14268
  1114
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1115
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1116
paulson@14268
  1117
lemma le_imp_inverse_le:
paulson@14268
  1118
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
  1119
  by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1120
paulson@14268
  1121
lemma inverse_le_iff_le [simp]:
paulson@14268
  1122
     "[|0 < a; 0 < b|] 
paulson@14268
  1123
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1124
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1125
paulson@14268
  1126
paulson@14268
  1127
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1128
case is trivial, since inverse preserves signs.*}
paulson@14268
  1129
lemma inverse_le_imp_le_neg:
paulson@14268
  1130
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
paulson@14268
  1131
  apply (rule classical) 
paulson@14268
  1132
  apply (subgoal_tac "a < 0") 
paulson@14268
  1133
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
paulson@14268
  1134
  apply (insert inverse_le_imp_le [of "-b" "-a"])
paulson@14268
  1135
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1136
  done
paulson@14268
  1137
paulson@14268
  1138
lemma less_imp_inverse_less_neg:
paulson@14268
  1139
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
paulson@14268
  1140
  apply (subgoal_tac "a < 0") 
paulson@14268
  1141
   prefer 2 apply (blast intro: order_less_trans) 
paulson@14268
  1142
  apply (insert less_imp_inverse_less [of "-b" "-a"])
paulson@14268
  1143
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1144
  done
paulson@14268
  1145
paulson@14268
  1146
lemma inverse_less_imp_less_neg:
paulson@14268
  1147
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1148
  apply (rule classical) 
paulson@14268
  1149
  apply (subgoal_tac "a < 0") 
paulson@14268
  1150
   prefer 2
paulson@14268
  1151
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
paulson@14268
  1152
  apply (insert inverse_less_imp_less [of "-b" "-a"])
paulson@14268
  1153
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1154
  done
paulson@14268
  1155
paulson@14268
  1156
lemma inverse_less_iff_less_neg [simp]:
paulson@14268
  1157
     "[|a < 0; b < 0|] 
paulson@14268
  1158
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1159
  apply (insert inverse_less_iff_less [of "-b" "-a"])
paulson@14268
  1160
  apply (simp del: inverse_less_iff_less 
paulson@14268
  1161
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1162
  done
paulson@14268
  1163
paulson@14268
  1164
lemma le_imp_inverse_le_neg:
paulson@14268
  1165
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
  1166
  by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1167
paulson@14268
  1168
lemma inverse_le_iff_le_neg [simp]:
paulson@14268
  1169
     "[|a < 0; b < 0|] 
paulson@14268
  1170
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1171
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1172
paulson@14277
  1173
paulson@14365
  1174
subsection{*Inverses and the Number One*}
paulson@14365
  1175
paulson@14365
  1176
lemma one_less_inverse_iff:
paulson@14365
  1177
    "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
paulson@14365
  1178
  assume "0 < x"
paulson@14365
  1179
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1180
    show ?thesis by simp
paulson@14365
  1181
next
paulson@14365
  1182
  assume notless: "~ (0 < x)"
paulson@14365
  1183
  have "~ (1 < inverse x)"
paulson@14365
  1184
  proof
paulson@14365
  1185
    assume "1 < inverse x"
paulson@14365
  1186
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1187
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1188
    finally show False by auto
paulson@14365
  1189
  qed
paulson@14365
  1190
  with notless show ?thesis by simp
paulson@14365
  1191
qed
paulson@14365
  1192
paulson@14365
  1193
lemma inverse_eq_1_iff [simp]:
paulson@14365
  1194
    "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1195
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1196
paulson@14365
  1197
lemma one_le_inverse_iff:
paulson@14365
  1198
   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1199
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1200
                    eq_commute [of 1]) 
paulson@14365
  1201
paulson@14365
  1202
lemma inverse_less_1_iff:
paulson@14365
  1203
   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1204
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1205
paulson@14365
  1206
lemma inverse_le_1_iff:
paulson@14365
  1207
   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1208
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1209
paulson@14365
  1210
paulson@14277
  1211
subsection{*Division and Signs*}
paulson@14277
  1212
paulson@14277
  1213
lemma zero_less_divide_iff:
paulson@14277
  1214
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
paulson@14277
  1215
by (simp add: divide_inverse_zero zero_less_mult_iff)
paulson@14277
  1216
paulson@14277
  1217
lemma divide_less_0_iff:
paulson@14277
  1218
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
paulson@14277
  1219
      (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14277
  1220
by (simp add: divide_inverse_zero mult_less_0_iff)
paulson@14277
  1221
paulson@14277
  1222
lemma zero_le_divide_iff:
paulson@14277
  1223
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
paulson@14277
  1224
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14277
  1225
by (simp add: divide_inverse_zero zero_le_mult_iff)
paulson@14277
  1226
paulson@14277
  1227
lemma divide_le_0_iff:
paulson@14288
  1228
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
paulson@14288
  1229
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14277
  1230
by (simp add: divide_inverse_zero mult_le_0_iff)
paulson@14277
  1231
paulson@14277
  1232
lemma divide_eq_0_iff [simp]:
paulson@14277
  1233
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
paulson@14277
  1234
by (simp add: divide_inverse_zero field_mult_eq_0_iff)
paulson@14277
  1235
paulson@14288
  1236
paulson@14288
  1237
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1238
paulson@14288
  1239
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1240
proof -
paulson@14288
  1241
  assume less: "0<c"
paulson@14288
  1242
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1243
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1244
  also have "... = (a*c \<le> b)"
paulson@14288
  1245
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1246
  finally show ?thesis .
paulson@14288
  1247
qed
paulson@14288
  1248
paulson@14288
  1249
paulson@14288
  1250
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1251
proof -
paulson@14288
  1252
  assume less: "c<0"
paulson@14288
  1253
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1254
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1255
  also have "... = (b \<le> a*c)"
paulson@14288
  1256
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1257
  finally show ?thesis .
paulson@14288
  1258
qed
paulson@14288
  1259
paulson@14288
  1260
lemma le_divide_eq:
paulson@14288
  1261
  "(a \<le> b/c) = 
paulson@14288
  1262
   (if 0 < c then a*c \<le> b
paulson@14288
  1263
             else if c < 0 then b \<le> a*c
paulson@14288
  1264
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1265
apply (case_tac "c=0", simp) 
paulson@14288
  1266
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1267
done
paulson@14288
  1268
paulson@14288
  1269
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1270
proof -
paulson@14288
  1271
  assume less: "0<c"
paulson@14288
  1272
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1273
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1274
  also have "... = (b \<le> a*c)"
paulson@14288
  1275
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1276
  finally show ?thesis .
paulson@14288
  1277
qed
paulson@14288
  1278
paulson@14288
  1279
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1280
proof -
paulson@14288
  1281
  assume less: "c<0"
paulson@14288
  1282
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1283
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1284
  also have "... = (a*c \<le> b)"
paulson@14288
  1285
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1286
  finally show ?thesis .
paulson@14288
  1287
qed
paulson@14288
  1288
paulson@14288
  1289
lemma divide_le_eq:
paulson@14288
  1290
  "(b/c \<le> a) = 
paulson@14288
  1291
   (if 0 < c then b \<le> a*c
paulson@14288
  1292
             else if c < 0 then a*c \<le> b
paulson@14288
  1293
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1294
apply (case_tac "c=0", simp) 
paulson@14288
  1295
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1296
done
paulson@14288
  1297
paulson@14288
  1298
paulson@14288
  1299
lemma pos_less_divide_eq:
paulson@14288
  1300
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1301
proof -
paulson@14288
  1302
  assume less: "0<c"
paulson@14288
  1303
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@14288
  1304
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1305
  also have "... = (a*c < b)"
paulson@14288
  1306
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1307
  finally show ?thesis .
paulson@14288
  1308
qed
paulson@14288
  1309
paulson@14288
  1310
lemma neg_less_divide_eq:
paulson@14288
  1311
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1312
proof -
paulson@14288
  1313
  assume less: "c<0"
paulson@14288
  1314
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@14288
  1315
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1316
  also have "... = (b < a*c)"
paulson@14288
  1317
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1318
  finally show ?thesis .
paulson@14288
  1319
qed
paulson@14288
  1320
paulson@14288
  1321
lemma less_divide_eq:
paulson@14288
  1322
  "(a < b/c) = 
paulson@14288
  1323
   (if 0 < c then a*c < b
paulson@14288
  1324
             else if c < 0 then b < a*c
paulson@14288
  1325
             else  a < (0::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1326
apply (case_tac "c=0", simp) 
paulson@14288
  1327
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1328
done
paulson@14288
  1329
paulson@14288
  1330
lemma pos_divide_less_eq:
paulson@14288
  1331
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1332
proof -
paulson@14288
  1333
  assume less: "0<c"
paulson@14288
  1334
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@14288
  1335
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1336
  also have "... = (b < a*c)"
paulson@14288
  1337
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1338
  finally show ?thesis .
paulson@14288
  1339
qed
paulson@14288
  1340
paulson@14288
  1341
lemma neg_divide_less_eq:
paulson@14288
  1342
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1343
proof -
paulson@14288
  1344
  assume less: "c<0"
paulson@14288
  1345
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@14288
  1346
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1347
  also have "... = (a*c < b)"
paulson@14288
  1348
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1349
  finally show ?thesis .
paulson@14288
  1350
qed
paulson@14288
  1351
paulson@14288
  1352
lemma divide_less_eq:
paulson@14288
  1353
  "(b/c < a) = 
paulson@14288
  1354
   (if 0 < c then b < a*c
paulson@14288
  1355
             else if c < 0 then a*c < b
paulson@14288
  1356
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1357
apply (case_tac "c=0", simp) 
paulson@14288
  1358
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1359
done
paulson@14288
  1360
paulson@14288
  1361
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
paulson@14288
  1362
proof -
paulson@14288
  1363
  assume [simp]: "c\<noteq>0"
paulson@14288
  1364
  have "(a = b/c) = (a*c = (b/c)*c)"
paulson@14288
  1365
    by (simp add: field_mult_cancel_right)
paulson@14288
  1366
  also have "... = (a*c = b)"
paulson@14288
  1367
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1368
  finally show ?thesis .
paulson@14288
  1369
qed
paulson@14288
  1370
paulson@14288
  1371
lemma eq_divide_eq:
paulson@14288
  1372
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
paulson@14288
  1373
by (simp add: nonzero_eq_divide_eq) 
paulson@14288
  1374
paulson@14288
  1375
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
paulson@14288
  1376
proof -
paulson@14288
  1377
  assume [simp]: "c\<noteq>0"
paulson@14288
  1378
  have "(b/c = a) = ((b/c)*c = a*c)"
paulson@14288
  1379
    by (simp add: field_mult_cancel_right)
paulson@14288
  1380
  also have "... = (b = a*c)"
paulson@14288
  1381
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1382
  finally show ?thesis .
paulson@14288
  1383
qed
paulson@14288
  1384
paulson@14288
  1385
lemma divide_eq_eq:
paulson@14288
  1386
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
paulson@14288
  1387
by (force simp add: nonzero_divide_eq_eq) 
paulson@14288
  1388
paulson@14288
  1389
subsection{*Cancellation Laws for Division*}
paulson@14288
  1390
paulson@14288
  1391
lemma divide_cancel_right [simp]:
paulson@14288
  1392
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
paulson@14288
  1393
apply (case_tac "c=0", simp) 
paulson@14288
  1394
apply (simp add: divide_inverse_zero field_mult_cancel_right) 
paulson@14288
  1395
done
paulson@14288
  1396
paulson@14288
  1397
lemma divide_cancel_left [simp]:
paulson@14288
  1398
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
paulson@14288
  1399
apply (case_tac "c=0", simp) 
paulson@14288
  1400
apply (simp add: divide_inverse_zero field_mult_cancel_left) 
paulson@14288
  1401
done
paulson@14288
  1402
paulson@14353
  1403
subsection {* Division and the Number One *}
paulson@14353
  1404
paulson@14353
  1405
text{*Simplify expressions equated with 1*}
paulson@14353
  1406
lemma divide_eq_1_iff [simp]:
paulson@14353
  1407
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1408
apply (case_tac "b=0", simp) 
paulson@14353
  1409
apply (simp add: right_inverse_eq) 
paulson@14353
  1410
done
paulson@14353
  1411
paulson@14353
  1412
paulson@14353
  1413
lemma one_eq_divide_iff [simp]:
paulson@14353
  1414
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1415
by (simp add: eq_commute [of 1])  
paulson@14353
  1416
paulson@14353
  1417
lemma zero_eq_1_divide_iff [simp]:
paulson@14353
  1418
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
paulson@14353
  1419
apply (case_tac "a=0", simp) 
paulson@14353
  1420
apply (auto simp add: nonzero_eq_divide_eq) 
paulson@14353
  1421
done
paulson@14353
  1422
paulson@14353
  1423
lemma one_divide_eq_0_iff [simp]:
paulson@14353
  1424
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
paulson@14353
  1425
apply (case_tac "a=0", simp) 
paulson@14353
  1426
apply (insert zero_neq_one [THEN not_sym]) 
paulson@14353
  1427
apply (auto simp add: nonzero_divide_eq_eq) 
paulson@14353
  1428
done
paulson@14353
  1429
paulson@14353
  1430
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@14353
  1431
declare zero_less_divide_iff [of "1", simp]
paulson@14353
  1432
declare divide_less_0_iff [of "1", simp]
paulson@14353
  1433
declare zero_le_divide_iff [of "1", simp]
paulson@14353
  1434
declare divide_le_0_iff [of "1", simp]
paulson@14353
  1435
paulson@14288
  1436
paulson@14293
  1437
subsection {* Ordering Rules for Division *}
paulson@14293
  1438
paulson@14293
  1439
lemma divide_strict_right_mono:
paulson@14293
  1440
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1441
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
paulson@14293
  1442
              positive_imp_inverse_positive) 
paulson@14293
  1443
paulson@14293
  1444
lemma divide_right_mono:
paulson@14293
  1445
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
paulson@14293
  1446
  by (force simp add: divide_strict_right_mono order_le_less) 
paulson@14293
  1447
paulson@14293
  1448
paulson@14293
  1449
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1450
      have the same sign*}
paulson@14293
  1451
lemma divide_strict_left_mono:
paulson@14293
  1452
       "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1453
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono 
paulson@14293
  1454
      order_less_imp_not_eq order_less_imp_not_eq2  
paulson@14293
  1455
      less_imp_inverse_less less_imp_inverse_less_neg) 
paulson@14293
  1456
paulson@14293
  1457
lemma divide_left_mono:
paulson@14293
  1458
     "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
paulson@14293
  1459
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1460
   prefer 2 
paulson@14293
  1461
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
paulson@14293
  1462
  apply (case_tac "c=0", simp add: divide_inverse)
paulson@14293
  1463
  apply (force simp add: divide_strict_left_mono order_le_less) 
paulson@14293
  1464
  done
paulson@14293
  1465
paulson@14293
  1466
lemma divide_strict_left_mono_neg:
paulson@14293
  1467
     "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1468
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1469
   prefer 2 
paulson@14293
  1470
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
paulson@14293
  1471
  apply (drule divide_strict_left_mono [of _ _ "-c"]) 
paulson@14293
  1472
   apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) 
paulson@14293
  1473
  done
paulson@14293
  1474
paulson@14293
  1475
lemma divide_strict_right_mono_neg:
paulson@14293
  1476
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1477
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) 
paulson@14293
  1478
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
paulson@14293
  1479
done
paulson@14293
  1480
paulson@14293
  1481
paulson@14293
  1482
subsection {* Ordered Fields are Dense *}
paulson@14293
  1483
paulson@14365
  1484
lemma less_add_one: "a < (a+1::'a::ordered_semiring)"
paulson@14293
  1485
proof -
paulson@14365
  1486
  have "a+0 < (a+1::'a::ordered_semiring)"
paulson@14365
  1487
    by (blast intro: zero_less_one add_strict_left_mono) 
paulson@14293
  1488
  thus ?thesis by simp
paulson@14293
  1489
qed
paulson@14293
  1490
paulson@14365
  1491
lemma zero_less_two: "0 < (1+1::'a::ordered_semiring)"
paulson@14365
  1492
  by (blast intro: order_less_trans zero_less_one less_add_one) 
paulson@14365
  1493
paulson@14293
  1494
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
paulson@14293
  1495
by (simp add: zero_less_two pos_less_divide_eq right_distrib) 
paulson@14293
  1496
paulson@14293
  1497
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
paulson@14293
  1498
by (simp add: zero_less_two pos_divide_less_eq right_distrib) 
paulson@14293
  1499
paulson@14293
  1500
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
paulson@14293
  1501
by (blast intro!: less_half_sum gt_half_sum)
paulson@14293
  1502
paulson@14293
  1503
paulson@14293
  1504
subsection {* Absolute Value *}
paulson@14293
  1505
paulson@14293
  1506
lemma abs_zero [simp]: "abs 0 = (0::'a::ordered_ring)"
paulson@14293
  1507
by (simp add: abs_if)
paulson@14293
  1508
paulson@14294
  1509
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_ring)"
paulson@14294
  1510
  by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) 
paulson@14294
  1511
paulson@14294
  1512
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_ring)" 
paulson@14294
  1513
apply (case_tac "a=0 | b=0", force) 
paulson@14293
  1514
apply (auto elim: order_less_asym
paulson@14293
  1515
            simp add: abs_if mult_less_0_iff linorder_neq_iff
paulson@14293
  1516
                  minus_mult_left [symmetric] minus_mult_right [symmetric])  
paulson@14293
  1517
done
paulson@14293
  1518
paulson@14348
  1519
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_ring)"
paulson@14348
  1520
by (simp add: abs_if) 
paulson@14348
  1521
paulson@14294
  1522
lemma abs_eq_0 [simp]: "(abs a = 0) = (a = (0::'a::ordered_ring))"
paulson@14294
  1523
by (simp add: abs_if)
paulson@14294
  1524
paulson@14294
  1525
lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::ordered_ring))"
paulson@14294
  1526
by (simp add: abs_if linorder_neq_iff)
paulson@14294
  1527
paulson@14294
  1528
lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::ordered_ring)"
paulson@14294
  1529
by (simp add: abs_if  order_less_not_sym [of a 0])
paulson@14294
  1530
paulson@14294
  1531
lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::ordered_ring)) = (a = 0)" 
paulson@14294
  1532
by (simp add: order_le_less) 
paulson@14294
  1533
paulson@14294
  1534
lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::ordered_ring)"
paulson@14294
  1535
apply (auto simp add: abs_if linorder_not_less order_less_not_sym [of 0 a])  
paulson@14294
  1536
apply (drule order_antisym, assumption, simp) 
paulson@14294
  1537
done
paulson@14294
  1538
paulson@14294
  1539
lemma abs_ge_zero [simp]: "(0::'a::ordered_ring) \<le> abs a"
paulson@14294
  1540
apply (simp add: abs_if order_less_imp_le)
paulson@14294
  1541
apply (simp add: linorder_not_less) 
paulson@14294
  1542
done
paulson@14294
  1543
paulson@14294
  1544
lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::ordered_ring)"
paulson@14294
  1545
  by (force elim: order_less_asym simp add: abs_if)
paulson@14294
  1546
paulson@14305
  1547
lemma abs_zero_iff [simp]: "(abs a = 0) = (a = (0::'a::ordered_ring))"
paulson@14293
  1548
by (simp add: abs_if)
paulson@14293
  1549
paulson@14294
  1550
lemma abs_ge_self: "a \<le> abs (a::'a::ordered_ring)"
paulson@14294
  1551
apply (simp add: abs_if)
paulson@14294
  1552
apply (simp add: order_less_imp_le order_trans [of _ 0])
paulson@14294
  1553
done
paulson@14294
  1554
paulson@14294
  1555
lemma abs_ge_minus_self: "-a \<le> abs (a::'a::ordered_ring)"
paulson@14294
  1556
by (insert abs_ge_self [of "-a"], simp)
paulson@14294
  1557
paulson@14294
  1558
lemma nonzero_abs_inverse:
paulson@14294
  1559
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  1560
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  1561
                      negative_imp_inverse_negative)
paulson@14294
  1562
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  1563
done
paulson@14294
  1564
paulson@14294
  1565
lemma abs_inverse [simp]:
paulson@14294
  1566
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  1567
      inverse (abs a)"
paulson@14294
  1568
apply (case_tac "a=0", simp) 
paulson@14294
  1569
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1570
done
paulson@14294
  1571
paulson@14294
  1572
lemma nonzero_abs_divide:
paulson@14294
  1573
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  1574
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  1575
paulson@14294
  1576
lemma abs_divide:
paulson@14294
  1577
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
paulson@14294
  1578
apply (case_tac "b=0", simp) 
paulson@14294
  1579
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1580
done
paulson@14294
  1581
paulson@14295
  1582
lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::ordered_ring)"
paulson@14295
  1583
by (simp add: abs_if)
paulson@14295
  1584
paulson@14295
  1585
lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::ordered_ring))"
paulson@14295
  1586
proof 
paulson@14295
  1587
  assume ale: "a \<le> -a"
paulson@14295
  1588
  show "a\<le>0"
paulson@14295
  1589
    apply (rule classical) 
paulson@14295
  1590
    apply (simp add: linorder_not_le) 
paulson@14295
  1591
    apply (blast intro: ale order_trans order_less_imp_le
paulson@14295
  1592
                        neg_0_le_iff_le [THEN iffD1]) 
paulson@14295
  1593
    done
paulson@14295
  1594
next
paulson@14295
  1595
  assume "a\<le>0"
paulson@14295
  1596
  hence "0 \<le> -a" by (simp only: neg_0_le_iff_le)
paulson@14295
  1597
  thus "a \<le> -a"  by (blast intro: prems order_trans) 
paulson@14295
  1598
qed
paulson@14295
  1599
paulson@14295
  1600
lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::ordered_ring))"
paulson@14295
  1601
by (insert le_minus_self_iff [of "-a"], simp)
paulson@14295
  1602
paulson@14295
  1603
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_ring))"
paulson@14295
  1604
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
paulson@14295
  1605
paulson@14295
  1606
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_ring))"
paulson@14295
  1607
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
paulson@14295
  1608
paulson@14295
  1609
lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::ordered_ring)"
paulson@14295
  1610
apply (simp add: abs_if split: split_if_asm)
paulson@14295
  1611
apply (rule order_trans [of _ "-a"]) 
paulson@14295
  1612
 apply (simp add: less_minus_self_iff order_less_imp_le, assumption)
paulson@14295
  1613
done
paulson@14295
  1614
paulson@14295
  1615
lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::ordered_ring)"
paulson@14295
  1616
by (insert abs_le_D1 [of "-a"], simp)
paulson@14295
  1617
paulson@14295
  1618
lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::ordered_ring))"
paulson@14295
  1619
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
paulson@14295
  1620
paulson@14295
  1621
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_ring))" 
paulson@14295
  1622
apply (simp add: order_less_le abs_le_iff)  
paulson@14295
  1623
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff) 
paulson@14295
  1624
 apply (simp add:  linorder_not_less [symmetric]) 
paulson@14295
  1625
apply (simp add: le_minus_self_iff linorder_neq_iff) 
paulson@14295
  1626
apply (simp add:  linorder_not_less [symmetric]) 
paulson@14295
  1627
done
paulson@14295
  1628
paulson@14294
  1629
lemma abs_triangle_ineq: "abs (a+b) \<le> abs a + abs (b::'a::ordered_ring)"
paulson@14295
  1630
by (force simp add: abs_le_iff abs_ge_self abs_ge_minus_self add_mono)
paulson@14294
  1631
paulson@14294
  1632
lemma abs_mult_less:
paulson@14294
  1633
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_ring)"
paulson@14294
  1634
proof -
paulson@14294
  1635
  assume ac: "abs a < c"
paulson@14294
  1636
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  1637
  assume "abs b < d"
paulson@14294
  1638
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1639
qed
paulson@14293
  1640
paulson@14331
  1641
ML
paulson@14331
  1642
{*
paulson@14334
  1643
val add_assoc = thm"add_assoc";
paulson@14334
  1644
val add_commute = thm"add_commute";
paulson@14334
  1645
val mult_assoc = thm"mult_assoc";
paulson@14334
  1646
val mult_commute = thm"mult_commute";
paulson@14334
  1647
val zero_neq_one = thm"zero_neq_one";
paulson@14334
  1648
val diff_minus = thm"diff_minus";
paulson@14334
  1649
val abs_if = thm"abs_if";
paulson@14334
  1650
val divide_inverse = thm"divide_inverse";
paulson@14334
  1651
val inverse_zero = thm"inverse_zero";
paulson@14334
  1652
val divide_zero = thm"divide_zero";
paulson@14368
  1653
paulson@14334
  1654
val add_0 = thm"add_0";
paulson@14331
  1655
val add_0_right = thm"add_0_right";
paulson@14368
  1656
val add_zero_left = thm"add_0";
paulson@14368
  1657
val add_zero_right = thm"add_0_right";
paulson@14368
  1658
paulson@14331
  1659
val add_left_commute = thm"add_left_commute";
paulson@14334
  1660
val left_minus = thm"left_minus";
paulson@14331
  1661
val right_minus = thm"right_minus";
paulson@14331
  1662
val right_minus_eq = thm"right_minus_eq";
paulson@14331
  1663
val add_left_cancel = thm"add_left_cancel";
paulson@14331
  1664
val add_right_cancel = thm"add_right_cancel";
paulson@14331
  1665
val minus_minus = thm"minus_minus";
paulson@14331
  1666
val equals_zero_I = thm"equals_zero_I";
paulson@14331
  1667
val minus_zero = thm"minus_zero";
paulson@14331
  1668
val diff_self = thm"diff_self";
paulson@14331
  1669
val diff_0 = thm"diff_0";
paulson@14331
  1670
val diff_0_right = thm"diff_0_right";
paulson@14331
  1671
val diff_minus_eq_add = thm"diff_minus_eq_add";
paulson@14331
  1672
val neg_equal_iff_equal = thm"neg_equal_iff_equal";
paulson@14331
  1673
val neg_equal_0_iff_equal = thm"neg_equal_0_iff_equal";
paulson@14331
  1674
val neg_0_equal_iff_equal = thm"neg_0_equal_iff_equal";
paulson@14331
  1675
val equation_minus_iff = thm"equation_minus_iff";
paulson@14331
  1676
val minus_equation_iff = thm"minus_equation_iff";
paulson@14334
  1677
val mult_1 = thm"mult_1";
paulson@14331
  1678
val mult_1_right = thm"mult_1_right";
paulson@14331
  1679
val mult_left_commute = thm"mult_left_commute";
paulson@14353
  1680
val mult_zero_left = thm"mult_zero_left";
paulson@14353
  1681
val mult_zero_right = thm"mult_zero_right";
paulson@14334
  1682
val left_distrib = thm "left_distrib";
paulson@14331
  1683
val right_distrib = thm"right_distrib";
paulson@14331
  1684
val combine_common_factor = thm"combine_common_factor";
paulson@14331
  1685
val minus_add_distrib = thm"minus_add_distrib";
paulson@14331
  1686
val minus_mult_left = thm"minus_mult_left";
paulson@14331
  1687
val minus_mult_right = thm"minus_mult_right";
paulson@14331
  1688
val minus_mult_minus = thm"minus_mult_minus";
paulson@14365
  1689
val minus_mult_commute = thm"minus_mult_commute";
paulson@14331
  1690
val right_diff_distrib = thm"right_diff_distrib";
paulson@14331
  1691
val left_diff_distrib = thm"left_diff_distrib";
paulson@14331
  1692
val minus_diff_eq = thm"minus_diff_eq";
paulson@14334
  1693
val add_left_mono = thm"add_left_mono";
paulson@14331
  1694
val add_right_mono = thm"add_right_mono";
paulson@14331
  1695
val add_mono = thm"add_mono";
paulson@14331
  1696
val add_strict_left_mono = thm"add_strict_left_mono";
paulson@14331
  1697
val add_strict_right_mono = thm"add_strict_right_mono";
paulson@14331
  1698
val add_strict_mono = thm"add_strict_mono";
paulson@14341
  1699
val add_less_le_mono = thm"add_less_le_mono";
paulson@14341
  1700
val add_le_less_mono = thm"add_le_less_mono";
paulson@14331
  1701
val add_less_imp_less_left = thm"add_less_imp_less_left";
paulson@14331
  1702
val add_less_imp_less_right = thm"add_less_imp_less_right";
paulson@14331
  1703
val add_less_cancel_left = thm"add_less_cancel_left";
paulson@14331
  1704
val add_less_cancel_right = thm"add_less_cancel_right";
paulson@14331
  1705
val add_le_cancel_left = thm"add_le_cancel_left";
paulson@14331
  1706
val add_le_cancel_right = thm"add_le_cancel_right";
paulson@14331
  1707
val add_le_imp_le_left = thm"add_le_imp_le_left";
paulson@14331
  1708
val add_le_imp_le_right = thm"add_le_imp_le_right";
paulson@14331
  1709
val le_imp_neg_le = thm"le_imp_neg_le";
paulson@14331
  1710
val neg_le_iff_le = thm"neg_le_iff_le";
paulson@14331
  1711
val neg_le_0_iff_le = thm"neg_le_0_iff_le";
paulson@14331
  1712
val neg_0_le_iff_le = thm"neg_0_le_iff_le";
paulson@14331
  1713
val neg_less_iff_less = thm"neg_less_iff_less";
paulson@14331
  1714
val neg_less_0_iff_less = thm"neg_less_0_iff_less";
paulson@14331
  1715
val neg_0_less_iff_less = thm"neg_0_less_iff_less";
paulson@14331
  1716
val less_minus_iff = thm"less_minus_iff";
paulson@14331
  1717
val minus_less_iff = thm"minus_less_iff";
paulson@14331
  1718
val le_minus_iff = thm"le_minus_iff";
paulson@14331
  1719
val minus_le_iff = thm"minus_le_iff";
paulson@14331
  1720
val add_diff_eq = thm"add_diff_eq";
paulson@14331
  1721
val diff_add_eq = thm"diff_add_eq";
paulson@14331
  1722
val diff_eq_eq = thm"diff_eq_eq";
paulson@14331
  1723
val eq_diff_eq = thm"eq_diff_eq";
paulson@14331
  1724
val diff_diff_eq = thm"diff_diff_eq";
paulson@14331
  1725
val diff_diff_eq2 = thm"diff_diff_eq2";
paulson@14331
  1726
val less_iff_diff_less_0 = thm"less_iff_diff_less_0";
paulson@14331
  1727
val diff_less_eq = thm"diff_less_eq";
paulson@14331
  1728
val less_diff_eq = thm"less_diff_eq";
paulson@14331
  1729
val diff_le_eq = thm"diff_le_eq";
paulson@14331
  1730
val le_diff_eq = thm"le_diff_eq";
paulson@14331
  1731
val eq_iff_diff_eq_0 = thm"eq_iff_diff_eq_0";
paulson@14331
  1732
val le_iff_diff_le_0 = thm"le_iff_diff_le_0";
paulson@14331
  1733
val eq_add_iff1 = thm"eq_add_iff1";
paulson@14331
  1734
val eq_add_iff2 = thm"eq_add_iff2";
paulson@14331
  1735
val less_add_iff1 = thm"less_add_iff1";
paulson@14331
  1736
val less_add_iff2 = thm"less_add_iff2";
paulson@14331
  1737
val le_add_iff1 = thm"le_add_iff1";
paulson@14331
  1738
val le_add_iff2 = thm"le_add_iff2";
paulson@14334
  1739
val mult_strict_left_mono = thm"mult_strict_left_mono";
paulson@14331
  1740
val mult_strict_right_mono = thm"mult_strict_right_mono";
paulson@14331
  1741
val mult_left_mono = thm"mult_left_mono";
paulson@14331
  1742
val mult_right_mono = thm"mult_right_mono";
paulson@14348
  1743
val mult_left_le_imp_le = thm"mult_left_le_imp_le";
paulson@14348
  1744
val mult_right_le_imp_le = thm"mult_right_le_imp_le";
paulson@14348
  1745
val mult_left_less_imp_less = thm"mult_left_less_imp_less";
paulson@14348
  1746
val mult_right_less_imp_less = thm"mult_right_less_imp_less";
paulson@14331
  1747
val mult_strict_left_mono_neg = thm"mult_strict_left_mono_neg";
paulson@14331
  1748
val mult_strict_right_mono_neg = thm"mult_strict_right_mono_neg";
paulson@14331
  1749
val mult_pos = thm"mult_pos";
paulson@14331
  1750
val mult_pos_neg = thm"mult_pos_neg";
paulson@14331
  1751
val mult_neg = thm"mult_neg";
paulson@14331
  1752
val zero_less_mult_pos = thm"zero_less_mult_pos";
paulson@14331
  1753
val zero_less_mult_iff = thm"zero_less_mult_iff";
paulson@14331
  1754
val mult_eq_0_iff = thm"mult_eq_0_iff";
paulson@14331
  1755
val zero_le_mult_iff = thm"zero_le_mult_iff";
paulson@14331
  1756
val mult_less_0_iff = thm"mult_less_0_iff";
paulson@14331
  1757
val mult_le_0_iff = thm"mult_le_0_iff";
paulson@14331
  1758
val zero_le_square = thm"zero_le_square";
paulson@14331
  1759
val zero_less_one = thm"zero_less_one";
paulson@14331
  1760
val zero_le_one = thm"zero_le_one";
paulson@14387
  1761
val not_one_less_zero = thm"not_one_less_zero";
paulson@14387
  1762
val not_one_le_zero = thm"not_one_le_zero";
paulson@14331
  1763
val mult_left_mono_neg = thm"mult_left_mono_neg";
paulson@14331
  1764
val mult_right_mono_neg = thm"mult_right_mono_neg";
paulson@14331
  1765
val mult_strict_mono = thm"mult_strict_mono";
paulson@14331
  1766
val mult_strict_mono' = thm"mult_strict_mono'";
paulson@14331
  1767
val mult_mono = thm"mult_mono";
paulson@14331
  1768
val mult_less_cancel_right = thm"mult_less_cancel_right";
paulson@14331
  1769
val mult_less_cancel_left = thm"mult_less_cancel_left";
paulson@14331
  1770
val mult_le_cancel_right = thm"mult_le_cancel_right";
paulson@14331
  1771
val mult_le_cancel_left = thm"mult_le_cancel_left";
paulson@14331
  1772
val mult_less_imp_less_left = thm"mult_less_imp_less_left";
paulson@14331
  1773
val mult_less_imp_less_right = thm"mult_less_imp_less_right";
paulson@14331
  1774
val mult_cancel_right = thm"mult_cancel_right";
paulson@14331
  1775
val mult_cancel_left = thm"mult_cancel_left";
paulson@14331
  1776
val left_inverse = thm "left_inverse";
paulson@14331
  1777
val right_inverse = thm"right_inverse";
paulson@14331
  1778
val right_inverse_eq = thm"right_inverse_eq";
paulson@14331
  1779
val nonzero_inverse_eq_divide = thm"nonzero_inverse_eq_divide";
paulson@14331
  1780
val divide_self = thm"divide_self";
paulson@14331
  1781
val divide_inverse_zero = thm"divide_inverse_zero";
paulson@14365
  1782
val inverse_divide = thm"inverse_divide";
paulson@14331
  1783
val divide_zero_left = thm"divide_zero_left";
paulson@14331
  1784
val inverse_eq_divide = thm"inverse_eq_divide";
paulson@14331
  1785
val nonzero_add_divide_distrib = thm"nonzero_add_divide_distrib";
paulson@14331
  1786
val add_divide_distrib = thm"add_divide_distrib";
paulson@14331
  1787
val field_mult_eq_0_iff = thm"field_mult_eq_0_iff";
paulson@14331
  1788
val field_mult_cancel_right = thm"field_mult_cancel_right";
paulson@14331
  1789
val field_mult_cancel_left = thm"field_mult_cancel_left";
paulson@14331
  1790
val nonzero_imp_inverse_nonzero = thm"nonzero_imp_inverse_nonzero";
paulson@14331
  1791
val inverse_zero_imp_zero = thm"inverse_zero_imp_zero";
paulson@14331
  1792
val inverse_nonzero_imp_nonzero = thm"inverse_nonzero_imp_nonzero";
paulson@14331
  1793
val inverse_nonzero_iff_nonzero = thm"inverse_nonzero_iff_nonzero";
paulson@14331
  1794
val nonzero_inverse_minus_eq = thm"nonzero_inverse_minus_eq";
paulson@14331
  1795
val inverse_minus_eq = thm"inverse_minus_eq";
paulson@14331
  1796
val nonzero_inverse_eq_imp_eq = thm"nonzero_inverse_eq_imp_eq";
paulson@14331
  1797
val inverse_eq_imp_eq = thm"inverse_eq_imp_eq";
paulson@14331
  1798
val inverse_eq_iff_eq = thm"inverse_eq_iff_eq";
paulson@14331
  1799
val nonzero_inverse_inverse_eq = thm"nonzero_inverse_inverse_eq";
paulson@14331
  1800
val inverse_inverse_eq = thm"inverse_inverse_eq";
paulson@14331
  1801
val inverse_1 = thm"inverse_1";
paulson@14331
  1802
val nonzero_inverse_mult_distrib = thm"nonzero_inverse_mult_distrib";
paulson@14331
  1803
val inverse_mult_distrib = thm"inverse_mult_distrib";
paulson@14331
  1804
val inverse_add = thm"inverse_add";
paulson@14331
  1805
val nonzero_mult_divide_cancel_left = thm"nonzero_mult_divide_cancel_left";
paulson@14331
  1806
val mult_divide_cancel_left = thm"mult_divide_cancel_left";
paulson@14331
  1807
val nonzero_mult_divide_cancel_right = thm"nonzero_mult_divide_cancel_right";
paulson@14331
  1808
val mult_divide_cancel_right = thm"mult_divide_cancel_right";
paulson@14331
  1809
val mult_divide_cancel_eq_if = thm"mult_divide_cancel_eq_if";
paulson@14331
  1810
val divide_1 = thm"divide_1";
paulson@14331
  1811
val times_divide_eq_right = thm"times_divide_eq_right";
paulson@14331
  1812
val times_divide_eq_left = thm"times_divide_eq_left";
paulson@14331
  1813
val divide_divide_eq_right = thm"divide_divide_eq_right";
paulson@14331
  1814
val divide_divide_eq_left = thm"divide_divide_eq_left";
paulson@14331
  1815
val nonzero_minus_divide_left = thm"nonzero_minus_divide_left";
paulson@14331
  1816
val nonzero_minus_divide_right = thm"nonzero_minus_divide_right";
paulson@14331
  1817
val nonzero_minus_divide_divide = thm"nonzero_minus_divide_divide";
paulson@14331
  1818
val minus_divide_left = thm"minus_divide_left";
paulson@14331
  1819
val minus_divide_right = thm"minus_divide_right";
paulson@14331
  1820
val minus_divide_divide = thm"minus_divide_divide";
paulson@14331
  1821
val positive_imp_inverse_positive = thm"positive_imp_inverse_positive";
paulson@14331
  1822
val negative_imp_inverse_negative = thm"negative_imp_inverse_negative";
paulson@14331
  1823
val inverse_le_imp_le = thm"inverse_le_imp_le";
paulson@14331
  1824
val inverse_positive_imp_positive = thm"inverse_positive_imp_positive";
paulson@14331
  1825
val inverse_positive_iff_positive = thm"inverse_positive_iff_positive";
paulson@14331
  1826
val inverse_negative_imp_negative = thm"inverse_negative_imp_negative";
paulson@14331
  1827
val inverse_negative_iff_negative = thm"inverse_negative_iff_negative";
paulson@14331
  1828
val inverse_nonnegative_iff_nonnegative = thm"inverse_nonnegative_iff_nonnegative";
paulson@14331
  1829
val inverse_nonpositive_iff_nonpositive = thm"inverse_nonpositive_iff_nonpositive";
paulson@14331
  1830
val less_imp_inverse_less = thm"less_imp_inverse_less";
paulson@14331
  1831
val inverse_less_imp_less = thm"inverse_less_imp_less";
paulson@14331
  1832
val inverse_less_iff_less = thm"inverse_less_iff_less";
paulson@14331
  1833
val le_imp_inverse_le = thm"le_imp_inverse_le";
paulson@14331
  1834
val inverse_le_iff_le = thm"inverse_le_iff_le";
paulson@14331
  1835
val inverse_le_imp_le_neg = thm"inverse_le_imp_le_neg";
paulson@14331
  1836
val less_imp_inverse_less_neg = thm"less_imp_inverse_less_neg";
paulson@14331
  1837
val inverse_less_imp_less_neg = thm"inverse_less_imp_less_neg";
paulson@14331
  1838
val inverse_less_iff_less_neg = thm"inverse_less_iff_less_neg";
paulson@14331
  1839
val le_imp_inverse_le_neg = thm"le_imp_inverse_le_neg";
paulson@14331
  1840
val inverse_le_iff_le_neg = thm"inverse_le_iff_le_neg";
paulson@14331
  1841
val zero_less_divide_iff = thm"zero_less_divide_iff";
paulson@14331
  1842
val divide_less_0_iff = thm"divide_less_0_iff";
paulson@14331
  1843
val zero_le_divide_iff = thm"zero_le_divide_iff";
paulson@14331
  1844
val divide_le_0_iff = thm"divide_le_0_iff";
paulson@14331
  1845
val divide_eq_0_iff = thm"divide_eq_0_iff";
paulson@14331
  1846
val pos_le_divide_eq = thm"pos_le_divide_eq";
paulson@14331
  1847
val neg_le_divide_eq = thm"neg_le_divide_eq";
paulson@14331
  1848
val le_divide_eq = thm"le_divide_eq";
paulson@14331
  1849
val pos_divide_le_eq = thm"pos_divide_le_eq";
paulson@14331
  1850
val neg_divide_le_eq = thm"neg_divide_le_eq";
paulson@14331
  1851
val divide_le_eq = thm"divide_le_eq";
paulson@14331
  1852
val pos_less_divide_eq = thm"pos_less_divide_eq";
paulson@14331
  1853
val neg_less_divide_eq = thm"neg_less_divide_eq";
paulson@14331
  1854
val less_divide_eq = thm"less_divide_eq";
paulson@14331
  1855
val pos_divide_less_eq = thm"pos_divide_less_eq";
paulson@14331
  1856
val neg_divide_less_eq = thm"neg_divide_less_eq";
paulson@14331
  1857
val divide_less_eq = thm"divide_less_eq";
paulson@14331
  1858
val nonzero_eq_divide_eq = thm"nonzero_eq_divide_eq";
paulson@14331
  1859
val eq_divide_eq = thm"eq_divide_eq";
paulson@14331
  1860
val nonzero_divide_eq_eq = thm"nonzero_divide_eq_eq";
paulson@14331
  1861
val divide_eq_eq = thm"divide_eq_eq";
paulson@14331
  1862
val divide_cancel_right = thm"divide_cancel_right";
paulson@14331
  1863
val divide_cancel_left = thm"divide_cancel_left";
paulson@14331
  1864
val divide_strict_right_mono = thm"divide_strict_right_mono";
paulson@14331
  1865
val divide_right_mono = thm"divide_right_mono";
paulson@14331
  1866
val divide_strict_left_mono = thm"divide_strict_left_mono";
paulson@14331
  1867
val divide_left_mono = thm"divide_left_mono";
paulson@14331
  1868
val divide_strict_left_mono_neg = thm"divide_strict_left_mono_neg";
paulson@14331
  1869
val divide_strict_right_mono_neg = thm"divide_strict_right_mono_neg";
paulson@14331
  1870
val zero_less_two = thm"zero_less_two";
paulson@14331
  1871
val less_half_sum = thm"less_half_sum";
paulson@14331
  1872
val gt_half_sum = thm"gt_half_sum";
paulson@14331
  1873
val dense = thm"dense";
paulson@14331
  1874
val abs_zero = thm"abs_zero";
paulson@14331
  1875
val abs_one = thm"abs_one";
paulson@14331
  1876
val abs_mult = thm"abs_mult";
paulson@14348
  1877
val abs_mult_self = thm"abs_mult_self";
paulson@14331
  1878
val abs_eq_0 = thm"abs_eq_0";
paulson@14331
  1879
val zero_less_abs_iff = thm"zero_less_abs_iff";
paulson@14331
  1880
val abs_not_less_zero = thm"abs_not_less_zero";
paulson@14331
  1881
val abs_le_zero_iff = thm"abs_le_zero_iff";
paulson@14331
  1882
val abs_minus_cancel = thm"abs_minus_cancel";
paulson@14331
  1883
val abs_ge_zero = thm"abs_ge_zero";
paulson@14331
  1884
val abs_idempotent = thm"abs_idempotent";
paulson@14331
  1885
val abs_zero_iff = thm"abs_zero_iff";
paulson@14331
  1886
val abs_ge_self = thm"abs_ge_self";
paulson@14331
  1887
val abs_ge_minus_self = thm"abs_ge_minus_self";
paulson@14331
  1888
val nonzero_abs_inverse = thm"nonzero_abs_inverse";
paulson@14331
  1889
val abs_inverse = thm"abs_inverse";
paulson@14331
  1890
val nonzero_abs_divide = thm"nonzero_abs_divide";
paulson@14331
  1891
val abs_divide = thm"abs_divide";
paulson@14331
  1892
val abs_leI = thm"abs_leI";
paulson@14331
  1893
val le_minus_self_iff = thm"le_minus_self_iff";
paulson@14331
  1894
val minus_le_self_iff = thm"minus_le_self_iff";
paulson@14331
  1895
val eq_minus_self_iff = thm"eq_minus_self_iff";
paulson@14331
  1896
val less_minus_self_iff = thm"less_minus_self_iff";
paulson@14331
  1897
val abs_le_D1 = thm"abs_le_D1";
paulson@14331
  1898
val abs_le_D2 = thm"abs_le_D2";
paulson@14331
  1899
val abs_le_iff = thm"abs_le_iff";
paulson@14331
  1900
val abs_less_iff = thm"abs_less_iff";
paulson@14331
  1901
val abs_triangle_ineq = thm"abs_triangle_ineq";
paulson@14331
  1902
val abs_mult_less = thm"abs_mult_less";
paulson@14331
  1903
paulson@14331
  1904
val compare_rls = thms"compare_rls";
paulson@14331
  1905
*}
paulson@14331
  1906
paulson@14293
  1907
paulson@14265
  1908
end