src/HOL/Ring_and_Field.thy
author paulson
Mon Oct 04 15:28:03 2004 +0200 (2004-10-04)
changeset 15228 4d332d10fa3d
parent 15197 19e735596e51
child 15229 1eb23f805c06
permissions -rw-r--r--
revised simprules for division
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson and Markus Wenzel
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*)
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header {* (Ordered) Rings and Fields *}
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theory Ring_and_Field 
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imports OrderedGroup
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begin
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text {*
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  The theory of partially ordered rings is taken from the books:
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  \begin{itemize}
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  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
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  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
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  \end{itemize}
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  Most of the used notions can also be looked up in 
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  \begin{itemize}
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  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
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  \item \emph{Algebra I} by van der Waerden, Springer.
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  \end{itemize}
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*}
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axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult
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  left_distrib: "(a + b) * c = a * c + b * c"
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  right_distrib: "a * (b + c) = a * b + a * c"
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axclass semiring_0 \<subseteq> semiring, comm_monoid_add
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axclass semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add
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axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult  
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  mult_commute: "a * b = b * a"
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  distrib: "(a + b) * c = a * c + b * c"
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instance comm_semiring \<subseteq> semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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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: distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add
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instance comm_semiring_0 \<subseteq> semiring_0 ..
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axclass comm_semiring_0_cancel \<subseteq> comm_semiring_0, cancel_ab_semigroup_add
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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..
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axclass axclass_0_neq_1 \<subseteq> zero, one
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  zero_neq_one [simp]: "0 \<noteq> 1"
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axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult
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axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *)
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instance comm_semiring_1 \<subseteq> semiring_1 ..
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axclass axclass_no_zero_divisors \<subseteq> zero, times
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  no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
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axclass semiring_1_cancel \<subseteq> semiring_1, cancel_ab_semigroup_add
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instance semiring_1_cancel \<subseteq> semiring_0_cancel ..
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axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)
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instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel ..
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel ..
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axclass ring \<subseteq> semiring, ab_group_add
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instance ring \<subseteq> semiring_0_cancel ..
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axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add
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instance comm_ring \<subseteq> ring ..
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instance comm_ring \<subseteq> comm_semiring_0_cancel ..
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axclass ring_1 \<subseteq> ring, semiring_1
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instance ring_1 \<subseteq> semiring_1_cancel ..
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axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *)
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instance comm_ring_1 \<subseteq> ring_1 ..
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instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..
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axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors
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axclass field \<subseteq> comm_ring_1, inverse
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  left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"
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  divide_inverse:      "a / b = a * inverse b"
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lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring_0_cancel)"
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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 
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    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_0_cancel)"
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proof -
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  have "a*0 + a*0 = a*0 + 0"
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    by (simp add: right_distrib [symmetric])
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  thus ?thesis 
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    by (simp only: add_left_cancel)
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qed
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lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
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proof cases
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  assume "a=0" thus ?thesis by simp
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next
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  assume anz [simp]: "a\<noteq>0"
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  { assume "a * b = 0"
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    hence "inverse a * (a * b) = 0" by simp
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    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
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  thus ?thesis by force
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qed
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instance field \<subseteq> idom
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by (intro_classes, simp)
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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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subsection {* Distribution rules *}
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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:
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     "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_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: left_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add 
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  mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
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  mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c"
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axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add
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instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..
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axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add
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  mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
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  mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
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instance ordered_semiring_strict \<subseteq> semiring_0_cancel ..
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instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring
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apply intro_classes
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apply (case_tac "a < b & 0 < c")
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apply (auto simp add: mult_strict_left_mono order_less_le)
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apply (auto simp add: mult_strict_left_mono order_le_less)
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apply (simp add: mult_strict_right_mono)
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done
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axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add
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  mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
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axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add
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instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
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axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add
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  mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
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instance pordered_comm_semiring \<subseteq> pordered_semiring
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by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+)
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instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..
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instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict
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by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+)
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instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
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apply (intro_classes)
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apply (case_tac "a < b & 0 < c")
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apply (auto simp add: mult_strict_left_mono order_less_le)
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apply (auto simp add: mult_strict_left_mono order_le_less)
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done
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axclass pordered_ring \<subseteq> ring, pordered_semiring 
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instance pordered_ring \<subseteq> pordered_ab_group_add ..
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instance pordered_ring \<subseteq> pordered_cancel_semiring ..
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axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs
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instance lordered_ring \<subseteq> lordered_ab_group_meet ..
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instance lordered_ring \<subseteq> lordered_ab_group_join ..
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axclass axclass_abs_if \<subseteq> minus, ord, zero
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  abs_if: "abs a = (if (a < 0) then (-a) else a)"
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axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if
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instance ordered_ring_strict \<subseteq> lordered_ab_group ..
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instance ordered_ring_strict \<subseteq> lordered_ring
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by (intro_classes, simp add: abs_if join_eq_if)
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axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring
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axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *)
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  zero_less_one [simp]: "0 < 1"
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axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *)
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instance ordered_idom \<subseteq> ordered_ring_strict ..
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axclass ordered_field \<subseteq> field, ordered_idom
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lemma eq_add_iff1:
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     "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
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apply (simp add: diff_minus left_distrib)
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric])
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done
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lemma eq_add_iff2:
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     "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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lemma less_add_iff1:
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     "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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lemma less_add_iff2:
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     "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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lemma le_add_iff1:
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     "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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lemma le_add_iff2:
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     "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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subsection {* Ordering Rules for Multiplication *}
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lemma mult_left_le_imp_le:
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     "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
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  by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
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lemma mult_right_le_imp_le:
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     "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
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  by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
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lemma mult_left_less_imp_less:
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     "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
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  by (force simp add: mult_left_mono linorder_not_le [symmetric])
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lemma mult_right_less_imp_less:
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     "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
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  by (force simp add: mult_right_mono linorder_not_le [symmetric])
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lemma mult_strict_left_mono_neg:
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     "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
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apply (drule mult_strict_left_mono [of _ _ "-c"])
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apply (simp_all add: minus_mult_left [symmetric]) 
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done
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lemma mult_left_mono_neg:
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     "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
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apply (drule mult_left_mono [of _ _ "-c"])
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apply (simp_all add: minus_mult_left [symmetric]) 
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done
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lemma mult_strict_right_mono_neg:
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     "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
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apply (drule mult_strict_right_mono [of _ _ "-c"])
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apply (simp_all add: minus_mult_right [symmetric]) 
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done
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   321
obua@14738
   322
lemma mult_right_mono_neg:
obua@14738
   323
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
obua@14738
   324
apply (drule mult_right_mono [of _ _ "-c"])
obua@14738
   325
apply (simp)
obua@14738
   326
apply (simp_all add: minus_mult_right [symmetric]) 
obua@14738
   327
done
paulson@14265
   328
paulson@14265
   329
subsection{* Products of Signs *}
paulson@14265
   330
obua@14738
   331
lemma mult_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
paulson@14265
   332
by (drule mult_strict_left_mono [of 0 b], auto)
paulson@14265
   333
obua@14738
   334
lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
obua@14738
   335
by (drule mult_left_mono [of 0 b], auto)
obua@14738
   336
obua@14738
   337
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
paulson@14265
   338
by (drule mult_strict_left_mono [of b 0], auto)
paulson@14265
   339
obua@14738
   340
lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
obua@14738
   341
by (drule mult_left_mono [of b 0], auto)
obua@14738
   342
obua@14738
   343
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
obua@14738
   344
by (drule mult_strict_right_mono[of b 0], auto)
obua@14738
   345
obua@14738
   346
lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
obua@14738
   347
by (drule mult_right_mono[of b 0], auto)
obua@14738
   348
obua@14738
   349
lemma mult_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
paulson@14265
   350
by (drule mult_strict_right_mono_neg, auto)
paulson@14265
   351
obua@14738
   352
lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
obua@14738
   353
by (drule mult_right_mono_neg[of a 0 b ], auto)
obua@14738
   354
paulson@14341
   355
lemma zero_less_mult_pos:
obua@14738
   356
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
paulson@14265
   357
apply (case_tac "b\<le>0") 
paulson@14265
   358
 apply (auto simp add: order_le_less linorder_not_less)
paulson@14265
   359
apply (drule_tac mult_pos_neg [of a b]) 
paulson@14265
   360
 apply (auto dest: order_less_not_sym)
paulson@14265
   361
done
paulson@14265
   362
obua@14738
   363
lemma zero_less_mult_pos2:
obua@14738
   364
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
obua@14738
   365
apply (case_tac "b\<le>0") 
obua@14738
   366
 apply (auto simp add: order_le_less linorder_not_less)
obua@14738
   367
apply (drule_tac mult_pos_neg2 [of a b]) 
obua@14738
   368
 apply (auto dest: order_less_not_sym)
obua@14738
   369
done
obua@14738
   370
paulson@14265
   371
lemma zero_less_mult_iff:
obua@14738
   372
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
paulson@14265
   373
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
paulson@14265
   374
apply (blast dest: zero_less_mult_pos) 
obua@14738
   375
apply (blast dest: zero_less_mult_pos2)
paulson@14265
   376
done
paulson@14265
   377
paulson@14341
   378
text{*A field has no "zero divisors", and this theorem holds without the
paulson@14277
   379
      assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}
obua@14738
   380
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0 | b = 0)"
paulson@14265
   381
apply (case_tac "a < 0")
paulson@14265
   382
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
paulson@14265
   383
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
paulson@14265
   384
done
paulson@14265
   385
paulson@14265
   386
lemma zero_le_mult_iff:
obua@14738
   387
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14265
   388
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
paulson@14265
   389
                   zero_less_mult_iff)
paulson@14265
   390
paulson@14265
   391
lemma mult_less_0_iff:
obua@14738
   392
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14265
   393
apply (insert zero_less_mult_iff [of "-a" b]) 
paulson@14265
   394
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   395
done
paulson@14265
   396
paulson@14265
   397
lemma mult_le_0_iff:
obua@14738
   398
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14265
   399
apply (insert zero_le_mult_iff [of "-a" b]) 
paulson@14265
   400
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   401
done
paulson@14265
   402
obua@14738
   403
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"
obua@14738
   404
by (auto simp add: mult_pos_le mult_neg_le)
obua@14738
   405
obua@14738
   406
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> (0::_::pordered_cancel_semiring)" 
obua@14738
   407
by (auto simp add: mult_pos_neg_le mult_pos_neg2_le)
obua@14738
   408
obua@14738
   409
lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a"
paulson@14265
   410
by (simp add: zero_le_mult_iff linorder_linear) 
paulson@14265
   411
obua@14738
   412
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
obua@14738
   413
      theorems available to members of @{term ordered_idom} *}
obua@14738
   414
obua@14738
   415
instance ordered_idom \<subseteq> ordered_semidom
paulson@14421
   416
proof
paulson@14421
   417
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
paulson@14430
   418
  thus "(0::'a) < 1" by (simp add: order_le_less) 
paulson@14421
   419
qed
paulson@14421
   420
obua@14738
   421
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 
obua@14738
   422
by (intro_classes, simp)
obua@14738
   423
obua@14738
   424
instance ordered_idom \<subseteq> idom ..
obua@14738
   425
paulson@14387
   426
text{*All three types of comparision involving 0 and 1 are covered.*}
paulson@14387
   427
paulson@14387
   428
declare zero_neq_one [THEN not_sym, simp]
paulson@14387
   429
obua@14738
   430
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
paulson@14268
   431
  by (rule zero_less_one [THEN order_less_imp_le]) 
paulson@14268
   432
obua@14738
   433
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
obua@14738
   434
by (simp add: linorder_not_le) 
paulson@14387
   435
obua@14738
   436
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
obua@14738
   437
by (simp add: linorder_not_less) 
paulson@14268
   438
paulson@14268
   439
subsection{*More Monotonicity*}
paulson@14268
   440
paulson@14268
   441
text{*Strict monotonicity in both arguments*}
paulson@14268
   442
lemma mult_strict_mono:
obua@14738
   443
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   444
apply (case_tac "c=0")
paulson@14268
   445
 apply (simp add: mult_pos) 
paulson@14268
   446
apply (erule mult_strict_right_mono [THEN order_less_trans])
paulson@14268
   447
 apply (force simp add: order_le_less) 
paulson@14268
   448
apply (erule mult_strict_left_mono, assumption)
paulson@14268
   449
done
paulson@14268
   450
paulson@14268
   451
text{*This weaker variant has more natural premises*}
paulson@14268
   452
lemma mult_strict_mono':
obua@14738
   453
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   454
apply (rule mult_strict_mono)
paulson@14268
   455
apply (blast intro: order_le_less_trans)+
paulson@14268
   456
done
paulson@14268
   457
paulson@14268
   458
lemma mult_mono:
paulson@14268
   459
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
obua@14738
   460
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
paulson@14268
   461
apply (erule mult_right_mono [THEN order_trans], assumption)
paulson@14268
   462
apply (erule mult_left_mono, assumption)
paulson@14268
   463
done
paulson@14268
   464
obua@14738
   465
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
paulson@14387
   466
apply (insert mult_strict_mono [of 1 m 1 n]) 
paulson@14430
   467
apply (simp add:  order_less_trans [OF zero_less_one]) 
paulson@14387
   468
done
paulson@14387
   469
paulson@14268
   470
subsection{*Cancellation Laws for Relationships With a Common Factor*}
paulson@14268
   471
paulson@14268
   472
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
paulson@14268
   473
   also with the relations @{text "\<le>"} and equality.*}
paulson@14268
   474
paulson@14268
   475
lemma mult_less_cancel_right:
obua@14738
   476
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
paulson@14268
   477
apply (case_tac "c = 0")
paulson@14268
   478
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
paulson@14268
   479
                      mult_strict_right_mono_neg)
paulson@14268
   480
apply (auto simp add: linorder_not_less 
paulson@14268
   481
                      linorder_not_le [symmetric, of "a*c"]
paulson@14268
   482
                      linorder_not_le [symmetric, of a])
paulson@14268
   483
apply (erule_tac [!] notE)
paulson@14268
   484
apply (auto simp add: order_less_imp_le mult_right_mono 
paulson@14268
   485
                      mult_right_mono_neg)
paulson@14268
   486
done
paulson@14268
   487
paulson@14268
   488
lemma mult_less_cancel_left:
obua@14738
   489
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
obua@14738
   490
apply (case_tac "c = 0")
obua@14738
   491
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
obua@14738
   492
                      mult_strict_left_mono_neg)
obua@14738
   493
apply (auto simp add: linorder_not_less 
obua@14738
   494
                      linorder_not_le [symmetric, of "c*a"]
obua@14738
   495
                      linorder_not_le [symmetric, of a])
obua@14738
   496
apply (erule_tac [!] notE)
obua@14738
   497
apply (auto simp add: order_less_imp_le mult_left_mono 
obua@14738
   498
                      mult_left_mono_neg)
obua@14738
   499
done
paulson@14268
   500
paulson@14268
   501
lemma mult_le_cancel_right:
obua@14738
   502
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@14268
   503
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right)
paulson@14268
   504
paulson@14268
   505
lemma mult_le_cancel_left:
obua@14738
   506
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
obua@14738
   507
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left)
paulson@14268
   508
paulson@14268
   509
lemma mult_less_imp_less_left:
paulson@14341
   510
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
obua@14738
   511
      shows "a < (b::'a::ordered_semiring_strict)"
paulson@14377
   512
proof (rule ccontr)
paulson@14377
   513
  assume "~ a < b"
paulson@14377
   514
  hence "b \<le> a" by (simp add: linorder_not_less)
paulson@14377
   515
  hence "c*b \<le> c*a" by (rule mult_left_mono)
paulson@14377
   516
  with this and less show False 
paulson@14377
   517
    by (simp add: linorder_not_less [symmetric])
paulson@14377
   518
qed
paulson@14268
   519
paulson@14268
   520
lemma mult_less_imp_less_right:
obua@14738
   521
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
obua@14738
   522
  shows "a < (b::'a::ordered_semiring_strict)"
obua@14738
   523
proof (rule ccontr)
obua@14738
   524
  assume "~ a < b"
obua@14738
   525
  hence "b \<le> a" by (simp add: linorder_not_less)
obua@14738
   526
  hence "b*c \<le> a*c" by (rule mult_right_mono)
obua@14738
   527
  with this and less show False 
obua@14738
   528
    by (simp add: linorder_not_less [symmetric])
obua@14738
   529
qed  
paulson@14268
   530
paulson@14268
   531
text{*Cancellation of equalities with a common factor*}
paulson@14268
   532
lemma mult_cancel_right [simp]:
obua@14738
   533
     "(a*c = b*c) = (c = (0::'a::ordered_ring_strict) | a=b)"
paulson@14268
   534
apply (cut_tac linorder_less_linear [of 0 c])
paulson@14268
   535
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono
paulson@14268
   536
             simp add: linorder_neq_iff)
paulson@14268
   537
done
paulson@14268
   538
paulson@14268
   539
text{*These cancellation theorems require an ordering. Versions are proved
paulson@14268
   540
      below that work for fields without an ordering.*}
paulson@14268
   541
lemma mult_cancel_left [simp]:
obua@14738
   542
     "(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)"
obua@14738
   543
apply (cut_tac linorder_less_linear [of 0 c])
obua@14738
   544
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono
obua@14738
   545
             simp add: linorder_neq_iff)
obua@14738
   546
done
paulson@14268
   547
obua@14738
   548
text{*This list of rewrites decides ring equalities by ordered rewriting.*}
obua@15178
   549
lemmas ring_eq_simps =  
obua@15178
   550
(*  mult_ac*)
obua@14738
   551
  left_distrib right_distrib left_diff_distrib right_diff_distrib
obua@15178
   552
  group_eq_simps
obua@15178
   553
(*  add_ac
obua@14738
   554
  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
obua@15178
   555
  diff_eq_eq eq_diff_eq *)
obua@14738
   556
    
paulson@14265
   557
subsection {* Fields *}
paulson@14265
   558
paulson@14288
   559
lemma right_inverse [simp]:
paulson@14288
   560
      assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"
paulson@14288
   561
proof -
paulson@14288
   562
  have "a * inverse a = inverse a * a" by (simp add: mult_ac)
paulson@14288
   563
  also have "... = 1" using not0 by simp
paulson@14288
   564
  finally show ?thesis .
paulson@14288
   565
qed
paulson@14288
   566
paulson@14288
   567
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
paulson@14288
   568
proof
paulson@14288
   569
  assume neq: "b \<noteq> 0"
paulson@14288
   570
  {
paulson@14288
   571
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
paulson@14288
   572
    also assume "a / b = 1"
paulson@14288
   573
    finally show "a = b" by simp
paulson@14288
   574
  next
paulson@14288
   575
    assume "a = b"
paulson@14288
   576
    with neq show "a / b = 1" by (simp add: divide_inverse)
paulson@14288
   577
  }
paulson@14288
   578
qed
paulson@14288
   579
paulson@14288
   580
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
paulson@14288
   581
by (simp add: divide_inverse)
paulson@14288
   582
paulson@15228
   583
lemma divide_self: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
paulson@14288
   584
  by (simp add: divide_inverse)
paulson@14288
   585
paulson@14430
   586
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
paulson@14430
   587
by (simp add: divide_inverse)
paulson@14277
   588
paulson@15228
   589
lemma divide_self_if [simp]:
paulson@15228
   590
     "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
paulson@15228
   591
  by (simp add: divide_self)
paulson@15228
   592
paulson@14430
   593
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
paulson@14430
   594
by (simp add: divide_inverse)
paulson@14277
   595
paulson@14430
   596
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
paulson@14430
   597
by (simp add: divide_inverse)
paulson@14277
   598
paulson@14430
   599
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
paulson@14293
   600
by (simp add: divide_inverse left_distrib) 
paulson@14293
   601
paulson@14293
   602
paulson@14270
   603
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
paulson@14270
   604
      of an ordering.*}
paulson@14348
   605
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
paulson@14377
   606
proof cases
paulson@14377
   607
  assume "a=0" thus ?thesis by simp
paulson@14377
   608
next
paulson@14377
   609
  assume anz [simp]: "a\<noteq>0"
paulson@14377
   610
  { assume "a * b = 0"
paulson@14377
   611
    hence "inverse a * (a * b) = 0" by simp
paulson@14377
   612
    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
paulson@14377
   613
  thus ?thesis by force
paulson@14377
   614
qed
paulson@14270
   615
paulson@14268
   616
text{*Cancellation of equalities with a common factor*}
paulson@14268
   617
lemma field_mult_cancel_right_lemma:
paulson@14269
   618
      assumes cnz: "c \<noteq> (0::'a::field)"
paulson@14269
   619
	  and eq:  "a*c = b*c"
paulson@14269
   620
	 shows "a=b"
paulson@14377
   621
proof -
paulson@14268
   622
  have "(a * c) * inverse c = (b * c) * inverse c"
paulson@14268
   623
    by (simp add: eq)
paulson@14268
   624
  thus "a=b"
paulson@14268
   625
    by (simp add: mult_assoc cnz)
paulson@14377
   626
qed
paulson@14268
   627
paulson@14348
   628
lemma field_mult_cancel_right [simp]:
paulson@14268
   629
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
paulson@14377
   630
proof cases
paulson@14377
   631
  assume "c=0" thus ?thesis by simp
paulson@14377
   632
next
paulson@14377
   633
  assume "c\<noteq>0" 
paulson@14377
   634
  thus ?thesis by (force dest: field_mult_cancel_right_lemma)
paulson@14377
   635
qed
paulson@14268
   636
paulson@14348
   637
lemma field_mult_cancel_left [simp]:
paulson@14268
   638
     "(c*a = c*b) = (c = (0::'a::field) | a=b)"
paulson@14268
   639
  by (simp add: mult_commute [of c] field_mult_cancel_right) 
paulson@14268
   640
paulson@14268
   641
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
paulson@14377
   642
proof
paulson@14268
   643
  assume ianz: "inverse a = 0"
paulson@14268
   644
  assume "a \<noteq> 0"
paulson@14268
   645
  hence "1 = a * inverse a" by simp
paulson@14268
   646
  also have "... = 0" by (simp add: ianz)
paulson@14268
   647
  finally have "1 = (0::'a::field)" .
paulson@14268
   648
  thus False by (simp add: eq_commute)
paulson@14377
   649
qed
paulson@14268
   650
paulson@14277
   651
paulson@14277
   652
subsection{*Basic Properties of @{term inverse}*}
paulson@14277
   653
paulson@14268
   654
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"
paulson@14268
   655
apply (rule ccontr) 
paulson@14268
   656
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   657
done
paulson@14268
   658
paulson@14268
   659
lemma inverse_nonzero_imp_nonzero:
paulson@14268
   660
   "inverse a = 0 ==> a = (0::'a::field)"
paulson@14268
   661
apply (rule ccontr) 
paulson@14268
   662
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   663
done
paulson@14268
   664
paulson@14268
   665
lemma inverse_nonzero_iff_nonzero [simp]:
paulson@14268
   666
   "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"
paulson@14268
   667
by (force dest: inverse_nonzero_imp_nonzero) 
paulson@14268
   668
paulson@14268
   669
lemma nonzero_inverse_minus_eq:
paulson@14269
   670
      assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
paulson@14377
   671
proof -
paulson@14377
   672
  have "-a * inverse (- a) = -a * - inverse a"
paulson@14377
   673
    by simp
paulson@14377
   674
  thus ?thesis 
paulson@14377
   675
    by (simp only: field_mult_cancel_left, simp)
paulson@14377
   676
qed
paulson@14268
   677
paulson@14268
   678
lemma inverse_minus_eq [simp]:
paulson@14377
   679
   "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
paulson@14377
   680
proof cases
paulson@14377
   681
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
   682
next
paulson@14377
   683
  assume "a\<noteq>0" 
paulson@14377
   684
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
   685
qed
paulson@14268
   686
paulson@14268
   687
lemma nonzero_inverse_eq_imp_eq:
paulson@14269
   688
      assumes inveq: "inverse a = inverse b"
paulson@14269
   689
	  and anz:  "a \<noteq> 0"
paulson@14269
   690
	  and bnz:  "b \<noteq> 0"
paulson@14269
   691
	 shows "a = (b::'a::field)"
paulson@14377
   692
proof -
paulson@14268
   693
  have "a * inverse b = a * inverse a"
paulson@14268
   694
    by (simp add: inveq)
paulson@14268
   695
  hence "(a * inverse b) * b = (a * inverse a) * b"
paulson@14268
   696
    by simp
paulson@14268
   697
  thus "a = b"
paulson@14268
   698
    by (simp add: mult_assoc anz bnz)
paulson@14377
   699
qed
paulson@14268
   700
paulson@14268
   701
lemma inverse_eq_imp_eq:
paulson@14268
   702
     "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"
paulson@14268
   703
apply (case_tac "a=0 | b=0") 
paulson@14268
   704
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
   705
              simp add: eq_commute [of "0::'a"])
paulson@14268
   706
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
   707
done
paulson@14268
   708
paulson@14268
   709
lemma inverse_eq_iff_eq [simp]:
paulson@14268
   710
     "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"
paulson@14268
   711
by (force dest!: inverse_eq_imp_eq) 
paulson@14268
   712
paulson@14270
   713
lemma nonzero_inverse_inverse_eq:
paulson@14270
   714
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
paulson@14270
   715
  proof -
paulson@14270
   716
  have "(inverse (inverse a) * inverse a) * a = a" 
paulson@14270
   717
    by (simp add: nonzero_imp_inverse_nonzero)
paulson@14270
   718
  thus ?thesis
paulson@14270
   719
    by (simp add: mult_assoc)
paulson@14270
   720
  qed
paulson@14270
   721
paulson@14270
   722
lemma inverse_inverse_eq [simp]:
paulson@14270
   723
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
paulson@14270
   724
  proof cases
paulson@14270
   725
    assume "a=0" thus ?thesis by simp
paulson@14270
   726
  next
paulson@14270
   727
    assume "a\<noteq>0" 
paulson@14270
   728
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
   729
  qed
paulson@14270
   730
paulson@14270
   731
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
paulson@14270
   732
  proof -
paulson@14270
   733
  have "inverse 1 * 1 = (1::'a::field)" 
paulson@14270
   734
    by (rule left_inverse [OF zero_neq_one [symmetric]])
paulson@14270
   735
  thus ?thesis  by simp
paulson@14270
   736
  qed
paulson@14270
   737
paulson@15077
   738
lemma inverse_unique: 
paulson@15077
   739
  assumes ab: "a*b = 1"
paulson@15077
   740
  shows "inverse a = (b::'a::field)"
paulson@15077
   741
proof -
paulson@15077
   742
  have "a \<noteq> 0" using ab by auto
paulson@15077
   743
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
paulson@15077
   744
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
paulson@15077
   745
qed
paulson@15077
   746
paulson@14270
   747
lemma nonzero_inverse_mult_distrib: 
paulson@14270
   748
      assumes anz: "a \<noteq> 0"
paulson@14270
   749
          and bnz: "b \<noteq> 0"
paulson@14270
   750
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
paulson@14270
   751
  proof -
paulson@14270
   752
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
paulson@14270
   753
    by (simp add: field_mult_eq_0_iff anz bnz)
paulson@14270
   754
  hence "inverse(a*b) * a = inverse(b)" 
paulson@14270
   755
    by (simp add: mult_assoc bnz)
paulson@14270
   756
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
paulson@14270
   757
    by simp
paulson@14270
   758
  thus ?thesis
paulson@14270
   759
    by (simp add: mult_assoc anz)
paulson@14270
   760
  qed
paulson@14270
   761
paulson@14270
   762
text{*This version builds in division by zero while also re-orienting
paulson@14270
   763
      the right-hand side.*}
paulson@14270
   764
lemma inverse_mult_distrib [simp]:
paulson@14270
   765
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
   766
  proof cases
paulson@14270
   767
    assume "a \<noteq> 0 & b \<noteq> 0" 
paulson@14270
   768
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
   769
  next
paulson@14270
   770
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
paulson@14270
   771
    thus ?thesis  by force
paulson@14270
   772
  qed
paulson@14270
   773
paulson@14270
   774
text{*There is no slick version using division by zero.*}
paulson@14270
   775
lemma inverse_add:
paulson@14270
   776
     "[|a \<noteq> 0;  b \<noteq> 0|]
paulson@14270
   777
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
paulson@14270
   778
apply (simp add: left_distrib mult_assoc)
paulson@14270
   779
apply (simp add: mult_commute [of "inverse a"]) 
paulson@14270
   780
apply (simp add: mult_assoc [symmetric] add_commute)
paulson@14270
   781
done
paulson@14270
   782
paulson@14365
   783
lemma inverse_divide [simp]:
paulson@14365
   784
      "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
paulson@14430
   785
  by (simp add: divide_inverse mult_commute)
paulson@14365
   786
paulson@14277
   787
lemma nonzero_mult_divide_cancel_left:
paulson@14277
   788
  assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
paulson@14277
   789
    shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
   790
proof -
paulson@14277
   791
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
paulson@14277
   792
    by (simp add: field_mult_eq_0_iff divide_inverse 
paulson@14277
   793
                  nonzero_inverse_mult_distrib)
paulson@14277
   794
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
   795
    by (simp only: mult_ac)
paulson@14277
   796
  also have "... =  a * inverse b"
paulson@14277
   797
    by simp
paulson@14277
   798
    finally show ?thesis 
paulson@14277
   799
    by (simp add: divide_inverse)
paulson@14277
   800
qed
paulson@14277
   801
paulson@14277
   802
lemma mult_divide_cancel_left:
paulson@14277
   803
     "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
paulson@14277
   804
apply (case_tac "b = 0")
paulson@14277
   805
apply (simp_all add: nonzero_mult_divide_cancel_left)
paulson@14277
   806
done
paulson@14277
   807
paulson@14321
   808
lemma nonzero_mult_divide_cancel_right:
paulson@14321
   809
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
paulson@14321
   810
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
paulson@14321
   811
paulson@14321
   812
lemma mult_divide_cancel_right:
paulson@14321
   813
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
paulson@14321
   814
apply (case_tac "b = 0")
paulson@14321
   815
apply (simp_all add: nonzero_mult_divide_cancel_right)
paulson@14321
   816
done
paulson@14321
   817
paulson@14277
   818
(*For ExtractCommonTerm*)
paulson@14277
   819
lemma mult_divide_cancel_eq_if:
paulson@14277
   820
     "(c*a) / (c*b) = 
paulson@14277
   821
      (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
paulson@14277
   822
  by (simp add: mult_divide_cancel_left)
paulson@14277
   823
paulson@14284
   824
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
paulson@14430
   825
  by (simp add: divide_inverse)
paulson@14284
   826
paulson@14430
   827
lemma times_divide_eq_right [simp]: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
   828
by (simp add: divide_inverse mult_assoc)
paulson@14288
   829
paulson@14430
   830
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
   831
by (simp add: divide_inverse mult_ac)
paulson@14288
   832
paulson@14288
   833
lemma divide_divide_eq_right [simp]:
paulson@14288
   834
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
   835
by (simp add: divide_inverse mult_ac)
paulson@14288
   836
paulson@14288
   837
lemma divide_divide_eq_left [simp]:
paulson@14288
   838
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
   839
by (simp add: divide_inverse mult_assoc)
paulson@14288
   840
paulson@14268
   841
paulson@14293
   842
subsection {* Division and Unary Minus *}
paulson@14293
   843
paulson@14293
   844
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
   845
by (simp add: divide_inverse minus_mult_left)
paulson@14293
   846
paulson@14293
   847
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
   848
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
   849
paulson@14293
   850
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
   851
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
   852
paulson@14430
   853
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
paulson@14430
   854
by (simp add: divide_inverse minus_mult_left [symmetric])
paulson@14293
   855
paulson@14293
   856
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14430
   857
by (simp add: divide_inverse minus_mult_right [symmetric])
paulson@14430
   858
paulson@14293
   859
paulson@14293
   860
text{*The effect is to extract signs from divisions*}
paulson@14293
   861
declare minus_divide_left  [symmetric, simp]
paulson@14293
   862
declare minus_divide_right [symmetric, simp]
paulson@14293
   863
paulson@14387
   864
text{*Also, extract signs from products*}
paulson@14387
   865
declare minus_mult_left [symmetric, simp]
paulson@14387
   866
declare minus_mult_right [symmetric, simp]
paulson@14387
   867
paulson@14293
   868
lemma minus_divide_divide [simp]:
paulson@14293
   869
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
paulson@14293
   870
apply (case_tac "b=0", simp) 
paulson@14293
   871
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
   872
done
paulson@14293
   873
paulson@14430
   874
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
   875
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
   876
paulson@14293
   877
paulson@14268
   878
subsection {* Ordered Fields *}
paulson@14268
   879
paulson@14277
   880
lemma positive_imp_inverse_positive: 
paulson@14269
   881
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
paulson@14268
   882
  proof -
paulson@14268
   883
  have "0 < a * inverse a" 
paulson@14268
   884
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
   885
  thus "0 < inverse a" 
paulson@14268
   886
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
paulson@14268
   887
  qed
paulson@14268
   888
paulson@14277
   889
lemma negative_imp_inverse_negative:
paulson@14268
   890
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
paulson@14277
   891
  by (insert positive_imp_inverse_positive [of "-a"], 
paulson@14268
   892
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
paulson@14268
   893
paulson@14268
   894
lemma inverse_le_imp_le:
paulson@14269
   895
      assumes invle: "inverse a \<le> inverse b"
paulson@14269
   896
	  and apos:  "0 < a"
paulson@14269
   897
	 shows "b \<le> (a::'a::ordered_field)"
paulson@14268
   898
  proof (rule classical)
paulson@14268
   899
  assume "~ b \<le> a"
paulson@14268
   900
  hence "a < b"
paulson@14268
   901
    by (simp add: linorder_not_le)
paulson@14268
   902
  hence bpos: "0 < b"
paulson@14268
   903
    by (blast intro: apos order_less_trans)
paulson@14268
   904
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
   905
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
   906
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
   907
    by (simp add: bpos order_less_imp_le mult_right_mono)
paulson@14268
   908
  thus "b \<le> a"
paulson@14268
   909
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
paulson@14268
   910
  qed
paulson@14268
   911
paulson@14277
   912
lemma inverse_positive_imp_positive:
paulson@14277
   913
      assumes inv_gt_0: "0 < inverse a"
paulson@14277
   914
          and [simp]:   "a \<noteq> 0"
paulson@14277
   915
        shows "0 < (a::'a::ordered_field)"
paulson@14277
   916
  proof -
paulson@14277
   917
  have "0 < inverse (inverse a)"
paulson@14277
   918
    by (rule positive_imp_inverse_positive)
paulson@14277
   919
  thus "0 < a"
paulson@14277
   920
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
   921
  qed
paulson@14277
   922
paulson@14277
   923
lemma inverse_positive_iff_positive [simp]:
paulson@14277
   924
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
   925
apply (case_tac "a = 0", simp)
paulson@14277
   926
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
   927
done
paulson@14277
   928
paulson@14277
   929
lemma inverse_negative_imp_negative:
paulson@14277
   930
      assumes inv_less_0: "inverse a < 0"
paulson@14277
   931
          and [simp]:   "a \<noteq> 0"
paulson@14277
   932
        shows "a < (0::'a::ordered_field)"
paulson@14277
   933
  proof -
paulson@14277
   934
  have "inverse (inverse a) < 0"
paulson@14277
   935
    by (rule negative_imp_inverse_negative)
paulson@14277
   936
  thus "a < 0"
paulson@14277
   937
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
   938
  qed
paulson@14277
   939
paulson@14277
   940
lemma inverse_negative_iff_negative [simp]:
paulson@14277
   941
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
   942
apply (case_tac "a = 0", simp)
paulson@14277
   943
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
   944
done
paulson@14277
   945
paulson@14277
   946
lemma inverse_nonnegative_iff_nonnegative [simp]:
paulson@14277
   947
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
   948
by (simp add: linorder_not_less [symmetric])
paulson@14277
   949
paulson@14277
   950
lemma inverse_nonpositive_iff_nonpositive [simp]:
paulson@14277
   951
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
   952
by (simp add: linorder_not_less [symmetric])
paulson@14277
   953
paulson@14277
   954
paulson@14277
   955
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
   956
paulson@14268
   957
lemma less_imp_inverse_less:
paulson@14269
   958
      assumes less: "a < b"
paulson@14269
   959
	  and apos:  "0 < a"
paulson@14269
   960
	shows "inverse b < inverse (a::'a::ordered_field)"
paulson@14268
   961
  proof (rule ccontr)
paulson@14268
   962
  assume "~ inverse b < inverse a"
paulson@14268
   963
  hence "inverse a \<le> inverse b"
paulson@14268
   964
    by (simp add: linorder_not_less)
paulson@14268
   965
  hence "~ (a < b)"
paulson@14268
   966
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
   967
  thus False
paulson@14268
   968
    by (rule notE [OF _ less])
paulson@14268
   969
  qed
paulson@14268
   970
paulson@14268
   971
lemma inverse_less_imp_less:
paulson@14268
   972
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
   973
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
   974
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
   975
done
paulson@14268
   976
paulson@14268
   977
text{*Both premises are essential. Consider -1 and 1.*}
paulson@14268
   978
lemma inverse_less_iff_less [simp]:
paulson@14268
   979
     "[|0 < a; 0 < b|] 
paulson@14268
   980
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
   981
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
   982
paulson@14268
   983
lemma le_imp_inverse_le:
paulson@14268
   984
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
   985
  by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
   986
paulson@14268
   987
lemma inverse_le_iff_le [simp]:
paulson@14268
   988
     "[|0 < a; 0 < b|] 
paulson@14268
   989
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
   990
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
   991
paulson@14268
   992
paulson@14268
   993
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
   994
case is trivial, since inverse preserves signs.*}
paulson@14268
   995
lemma inverse_le_imp_le_neg:
paulson@14268
   996
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
paulson@14268
   997
  apply (rule classical) 
paulson@14268
   998
  apply (subgoal_tac "a < 0") 
paulson@14268
   999
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
paulson@14268
  1000
  apply (insert inverse_le_imp_le [of "-b" "-a"])
paulson@14268
  1001
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1002
  done
paulson@14268
  1003
paulson@14268
  1004
lemma less_imp_inverse_less_neg:
paulson@14268
  1005
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
paulson@14268
  1006
  apply (subgoal_tac "a < 0") 
paulson@14268
  1007
   prefer 2 apply (blast intro: order_less_trans) 
paulson@14268
  1008
  apply (insert less_imp_inverse_less [of "-b" "-a"])
paulson@14268
  1009
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1010
  done
paulson@14268
  1011
paulson@14268
  1012
lemma inverse_less_imp_less_neg:
paulson@14268
  1013
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1014
  apply (rule classical) 
paulson@14268
  1015
  apply (subgoal_tac "a < 0") 
paulson@14268
  1016
   prefer 2
paulson@14268
  1017
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
paulson@14268
  1018
  apply (insert inverse_less_imp_less [of "-b" "-a"])
paulson@14268
  1019
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1020
  done
paulson@14268
  1021
paulson@14268
  1022
lemma inverse_less_iff_less_neg [simp]:
paulson@14268
  1023
     "[|a < 0; b < 0|] 
paulson@14268
  1024
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1025
  apply (insert inverse_less_iff_less [of "-b" "-a"])
paulson@14268
  1026
  apply (simp del: inverse_less_iff_less 
paulson@14268
  1027
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1028
  done
paulson@14268
  1029
paulson@14268
  1030
lemma le_imp_inverse_le_neg:
paulson@14268
  1031
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
  1032
  by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1033
paulson@14268
  1034
lemma inverse_le_iff_le_neg [simp]:
paulson@14268
  1035
     "[|a < 0; b < 0|] 
paulson@14268
  1036
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1037
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1038
paulson@14277
  1039
paulson@14365
  1040
subsection{*Inverses and the Number One*}
paulson@14365
  1041
paulson@14365
  1042
lemma one_less_inverse_iff:
paulson@14365
  1043
    "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
paulson@14365
  1044
  assume "0 < x"
paulson@14365
  1045
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1046
    show ?thesis by simp
paulson@14365
  1047
next
paulson@14365
  1048
  assume notless: "~ (0 < x)"
paulson@14365
  1049
  have "~ (1 < inverse x)"
paulson@14365
  1050
  proof
paulson@14365
  1051
    assume "1 < inverse x"
paulson@14365
  1052
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1053
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1054
    finally show False by auto
paulson@14365
  1055
  qed
paulson@14365
  1056
  with notless show ?thesis by simp
paulson@14365
  1057
qed
paulson@14365
  1058
paulson@14365
  1059
lemma inverse_eq_1_iff [simp]:
paulson@14365
  1060
    "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1061
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1062
paulson@14365
  1063
lemma one_le_inverse_iff:
paulson@14365
  1064
   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1065
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1066
                    eq_commute [of 1]) 
paulson@14365
  1067
paulson@14365
  1068
lemma inverse_less_1_iff:
paulson@14365
  1069
   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1070
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1071
paulson@14365
  1072
lemma inverse_le_1_iff:
paulson@14365
  1073
   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1074
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1075
paulson@14365
  1076
paulson@14277
  1077
subsection{*Division and Signs*}
paulson@14277
  1078
paulson@14277
  1079
lemma zero_less_divide_iff:
paulson@14277
  1080
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
paulson@14430
  1081
by (simp add: divide_inverse zero_less_mult_iff)
paulson@14277
  1082
paulson@14277
  1083
lemma divide_less_0_iff:
paulson@14277
  1084
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
paulson@14277
  1085
      (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14430
  1086
by (simp add: divide_inverse mult_less_0_iff)
paulson@14277
  1087
paulson@14277
  1088
lemma zero_le_divide_iff:
paulson@14277
  1089
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
paulson@14277
  1090
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14430
  1091
by (simp add: divide_inverse zero_le_mult_iff)
paulson@14277
  1092
paulson@14277
  1093
lemma divide_le_0_iff:
paulson@14288
  1094
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
paulson@14288
  1095
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14430
  1096
by (simp add: divide_inverse mult_le_0_iff)
paulson@14277
  1097
paulson@14277
  1098
lemma divide_eq_0_iff [simp]:
paulson@14277
  1099
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
paulson@14430
  1100
by (simp add: divide_inverse field_mult_eq_0_iff)
paulson@14277
  1101
paulson@14288
  1102
paulson@14288
  1103
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1104
paulson@14288
  1105
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1106
proof -
paulson@14288
  1107
  assume less: "0<c"
paulson@14288
  1108
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1109
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1110
  also have "... = (a*c \<le> b)"
paulson@14288
  1111
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1112
  finally show ?thesis .
paulson@14288
  1113
qed
paulson@14288
  1114
paulson@14288
  1115
paulson@14288
  1116
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1117
proof -
paulson@14288
  1118
  assume less: "c<0"
paulson@14288
  1119
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1120
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1121
  also have "... = (b \<le> a*c)"
paulson@14288
  1122
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1123
  finally show ?thesis .
paulson@14288
  1124
qed
paulson@14288
  1125
paulson@14288
  1126
lemma le_divide_eq:
paulson@14288
  1127
  "(a \<le> b/c) = 
paulson@14288
  1128
   (if 0 < c then a*c \<le> b
paulson@14288
  1129
             else if c < 0 then b \<le> a*c
paulson@14288
  1130
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1131
apply (case_tac "c=0", simp) 
paulson@14288
  1132
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1133
done
paulson@14288
  1134
paulson@14288
  1135
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1136
proof -
paulson@14288
  1137
  assume less: "0<c"
paulson@14288
  1138
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1139
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1140
  also have "... = (b \<le> a*c)"
paulson@14288
  1141
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1142
  finally show ?thesis .
paulson@14288
  1143
qed
paulson@14288
  1144
paulson@14288
  1145
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1146
proof -
paulson@14288
  1147
  assume less: "c<0"
paulson@14288
  1148
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1149
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1150
  also have "... = (a*c \<le> b)"
paulson@14288
  1151
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1152
  finally show ?thesis .
paulson@14288
  1153
qed
paulson@14288
  1154
paulson@14288
  1155
lemma divide_le_eq:
paulson@14288
  1156
  "(b/c \<le> a) = 
paulson@14288
  1157
   (if 0 < c then b \<le> a*c
paulson@14288
  1158
             else if c < 0 then a*c \<le> b
paulson@14288
  1159
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1160
apply (case_tac "c=0", simp) 
paulson@14288
  1161
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1162
done
paulson@14288
  1163
paulson@14288
  1164
paulson@14288
  1165
lemma pos_less_divide_eq:
paulson@14288
  1166
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1167
proof -
paulson@14288
  1168
  assume less: "0<c"
paulson@14288
  1169
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@14288
  1170
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1171
  also have "... = (a*c < b)"
paulson@14288
  1172
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1173
  finally show ?thesis .
paulson@14288
  1174
qed
paulson@14288
  1175
paulson@14288
  1176
lemma neg_less_divide_eq:
paulson@14288
  1177
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1178
proof -
paulson@14288
  1179
  assume less: "c<0"
paulson@14288
  1180
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@14288
  1181
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1182
  also have "... = (b < a*c)"
paulson@14288
  1183
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1184
  finally show ?thesis .
paulson@14288
  1185
qed
paulson@14288
  1186
paulson@14288
  1187
lemma less_divide_eq:
paulson@14288
  1188
  "(a < b/c) = 
paulson@14288
  1189
   (if 0 < c then a*c < b
paulson@14288
  1190
             else if c < 0 then b < a*c
paulson@14288
  1191
             else  a < (0::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1192
apply (case_tac "c=0", simp) 
paulson@14288
  1193
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1194
done
paulson@14288
  1195
paulson@14288
  1196
lemma pos_divide_less_eq:
paulson@14288
  1197
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1198
proof -
paulson@14288
  1199
  assume less: "0<c"
paulson@14288
  1200
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@14288
  1201
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1202
  also have "... = (b < a*c)"
paulson@14288
  1203
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1204
  finally show ?thesis .
paulson@14288
  1205
qed
paulson@14288
  1206
paulson@14288
  1207
lemma neg_divide_less_eq:
paulson@14288
  1208
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1209
proof -
paulson@14288
  1210
  assume less: "c<0"
paulson@14288
  1211
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@14288
  1212
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1213
  also have "... = (a*c < b)"
paulson@14288
  1214
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1215
  finally show ?thesis .
paulson@14288
  1216
qed
paulson@14288
  1217
paulson@14288
  1218
lemma divide_less_eq:
paulson@14288
  1219
  "(b/c < a) = 
paulson@14288
  1220
   (if 0 < c then b < a*c
paulson@14288
  1221
             else if c < 0 then a*c < b
paulson@14288
  1222
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1223
apply (case_tac "c=0", simp) 
paulson@14288
  1224
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1225
done
paulson@14288
  1226
paulson@14288
  1227
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
paulson@14288
  1228
proof -
paulson@14288
  1229
  assume [simp]: "c\<noteq>0"
paulson@14288
  1230
  have "(a = b/c) = (a*c = (b/c)*c)"
paulson@14288
  1231
    by (simp add: field_mult_cancel_right)
paulson@14288
  1232
  also have "... = (a*c = b)"
paulson@14288
  1233
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1234
  finally show ?thesis .
paulson@14288
  1235
qed
paulson@14288
  1236
paulson@14288
  1237
lemma eq_divide_eq:
paulson@14288
  1238
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
paulson@14288
  1239
by (simp add: nonzero_eq_divide_eq) 
paulson@14288
  1240
paulson@14288
  1241
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
paulson@14288
  1242
proof -
paulson@14288
  1243
  assume [simp]: "c\<noteq>0"
paulson@14288
  1244
  have "(b/c = a) = ((b/c)*c = a*c)"
paulson@14288
  1245
    by (simp add: field_mult_cancel_right)
paulson@14288
  1246
  also have "... = (b = a*c)"
paulson@14288
  1247
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1248
  finally show ?thesis .
paulson@14288
  1249
qed
paulson@14288
  1250
paulson@14288
  1251
lemma divide_eq_eq:
paulson@14288
  1252
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
paulson@14288
  1253
by (force simp add: nonzero_divide_eq_eq) 
paulson@14288
  1254
paulson@14288
  1255
subsection{*Cancellation Laws for Division*}
paulson@14288
  1256
paulson@14288
  1257
lemma divide_cancel_right [simp]:
paulson@14288
  1258
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
paulson@14288
  1259
apply (case_tac "c=0", simp) 
paulson@14430
  1260
apply (simp add: divide_inverse field_mult_cancel_right) 
paulson@14288
  1261
done
paulson@14288
  1262
paulson@14288
  1263
lemma divide_cancel_left [simp]:
paulson@14288
  1264
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
paulson@14288
  1265
apply (case_tac "c=0", simp) 
paulson@14430
  1266
apply (simp add: divide_inverse field_mult_cancel_left) 
paulson@14288
  1267
done
paulson@14288
  1268
paulson@14353
  1269
subsection {* Division and the Number One *}
paulson@14353
  1270
paulson@14353
  1271
text{*Simplify expressions equated with 1*}
paulson@14353
  1272
lemma divide_eq_1_iff [simp]:
paulson@14353
  1273
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1274
apply (case_tac "b=0", simp) 
paulson@14353
  1275
apply (simp add: right_inverse_eq) 
paulson@14353
  1276
done
paulson@14353
  1277
paulson@14353
  1278
paulson@14353
  1279
lemma one_eq_divide_iff [simp]:
paulson@14353
  1280
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1281
by (simp add: eq_commute [of 1])  
paulson@14353
  1282
paulson@14353
  1283
lemma zero_eq_1_divide_iff [simp]:
paulson@14353
  1284
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
paulson@14353
  1285
apply (case_tac "a=0", simp) 
paulson@14353
  1286
apply (auto simp add: nonzero_eq_divide_eq) 
paulson@14353
  1287
done
paulson@14353
  1288
paulson@14353
  1289
lemma one_divide_eq_0_iff [simp]:
paulson@14353
  1290
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
paulson@14353
  1291
apply (case_tac "a=0", simp) 
paulson@14353
  1292
apply (insert zero_neq_one [THEN not_sym]) 
paulson@14353
  1293
apply (auto simp add: nonzero_divide_eq_eq) 
paulson@14353
  1294
done
paulson@14353
  1295
paulson@14353
  1296
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@14353
  1297
declare zero_less_divide_iff [of "1", simp]
paulson@14353
  1298
declare divide_less_0_iff [of "1", simp]
paulson@14353
  1299
declare zero_le_divide_iff [of "1", simp]
paulson@14353
  1300
declare divide_le_0_iff [of "1", simp]
paulson@14353
  1301
paulson@14288
  1302
paulson@14293
  1303
subsection {* Ordering Rules for Division *}
paulson@14293
  1304
paulson@14293
  1305
lemma divide_strict_right_mono:
paulson@14293
  1306
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1307
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
paulson@14293
  1308
              positive_imp_inverse_positive) 
paulson@14293
  1309
paulson@14293
  1310
lemma divide_right_mono:
paulson@14293
  1311
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
paulson@14293
  1312
  by (force simp add: divide_strict_right_mono order_le_less) 
paulson@14293
  1313
paulson@14293
  1314
paulson@14293
  1315
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1316
      have the same sign*}
paulson@14293
  1317
lemma divide_strict_left_mono:
paulson@14293
  1318
       "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1319
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono 
paulson@14293
  1320
      order_less_imp_not_eq order_less_imp_not_eq2  
paulson@14293
  1321
      less_imp_inverse_less less_imp_inverse_less_neg) 
paulson@14293
  1322
paulson@14293
  1323
lemma divide_left_mono:
paulson@14293
  1324
     "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
paulson@14293
  1325
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1326
   prefer 2 
paulson@14293
  1327
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
paulson@14293
  1328
  apply (case_tac "c=0", simp add: divide_inverse)
paulson@14293
  1329
  apply (force simp add: divide_strict_left_mono order_le_less) 
paulson@14293
  1330
  done
paulson@14293
  1331
paulson@14293
  1332
lemma divide_strict_left_mono_neg:
paulson@14293
  1333
     "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1334
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1335
   prefer 2 
paulson@14293
  1336
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
paulson@14293
  1337
  apply (drule divide_strict_left_mono [of _ _ "-c"]) 
paulson@14293
  1338
   apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) 
paulson@14293
  1339
  done
paulson@14293
  1340
paulson@14293
  1341
lemma divide_strict_right_mono_neg:
paulson@14293
  1342
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1343
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) 
paulson@14293
  1344
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
paulson@14293
  1345
done
paulson@14293
  1346
paulson@14293
  1347
paulson@14293
  1348
subsection {* Ordered Fields are Dense *}
paulson@14293
  1349
obua@14738
  1350
lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
paulson@14293
  1351
proof -
obua@14738
  1352
  have "a+0 < (a+1::'a::ordered_semidom)"
paulson@14365
  1353
    by (blast intro: zero_less_one add_strict_left_mono) 
paulson@14293
  1354
  thus ?thesis by simp
paulson@14293
  1355
qed
paulson@14293
  1356
obua@14738
  1357
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
paulson@14365
  1358
  by (blast intro: order_less_trans zero_less_one less_add_one) 
paulson@14365
  1359
paulson@14293
  1360
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
paulson@14293
  1361
by (simp add: zero_less_two pos_less_divide_eq right_distrib) 
paulson@14293
  1362
paulson@14293
  1363
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
paulson@14293
  1364
by (simp add: zero_less_two pos_divide_less_eq right_distrib) 
paulson@14293
  1365
paulson@14293
  1366
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
paulson@14293
  1367
by (blast intro!: less_half_sum gt_half_sum)
paulson@14293
  1368
paulson@14293
  1369
subsection {* Absolute Value *}
paulson@14293
  1370
obua@14738
  1371
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
paulson@14294
  1372
  by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) 
paulson@14294
  1373
obua@14738
  1374
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  1375
proof -
obua@14738
  1376
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  1377
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  1378
  have a: "(abs a) * (abs b) = ?x"
obua@14738
  1379
    by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps)
obua@14738
  1380
  {
obua@14738
  1381
    fix u v :: 'a
obua@14738
  1382
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> u * v = ?y"
obua@14738
  1383
      apply (subst prts[of u], subst prts[of v])
obua@14738
  1384
      apply (simp add: left_distrib right_distrib add_ac) 
obua@14738
  1385
      done
obua@14738
  1386
  }
obua@14738
  1387
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  1388
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  1389
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  1390
  have xy: "- ?x <= ?y"
obua@14754
  1391
    apply (simp)
obua@14754
  1392
    apply (rule_tac y="0::'a" in order_trans)
obua@14754
  1393
    apply (rule addm2)+
obua@14754
  1394
    apply (simp_all add: mult_pos_le mult_neg_le)
obua@14754
  1395
    apply (rule addm)+
obua@14754
  1396
    apply (simp_all add: mult_pos_le mult_neg_le)
obua@14754
  1397
    done
obua@14738
  1398
  have yx: "?y <= ?x"
obua@14738
  1399
    apply (simp add: add_ac)
obua@14754
  1400
    apply (rule_tac y=0 in order_trans)
obua@14754
  1401
    apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)
obua@14754
  1402
    apply (rule addm, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)
obua@14738
  1403
    done
obua@14738
  1404
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  1405
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  1406
  show ?thesis
obua@14738
  1407
    apply (rule abs_leI)
obua@14738
  1408
    apply (simp add: i1)
obua@14738
  1409
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  1410
    done
obua@14738
  1411
qed
paulson@14294
  1412
obua@14738
  1413
lemma abs_eq_mult: 
obua@14738
  1414
  assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
obua@14738
  1415
  shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
obua@14738
  1416
proof -
obua@14738
  1417
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  1418
    apply (auto)    
obua@14738
  1419
    apply (rule_tac split_mult_pos_le)
obua@14738
  1420
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  1421
    apply (simp)
obua@14738
  1422
    apply (rule_tac split_mult_neg_le)
obua@14738
  1423
    apply (insert prems)
obua@14738
  1424
    apply (blast)
obua@14738
  1425
    done
obua@14738
  1426
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  1427
    by (simp add: prts[symmetric])
obua@14738
  1428
  show ?thesis
obua@14738
  1429
  proof cases
obua@14738
  1430
    assume "0 <= a * b"
obua@14738
  1431
    then show ?thesis
obua@14738
  1432
      apply (simp_all add: mulprts abs_prts)
obua@14754
  1433
      apply (simp add: 
obua@14754
  1434
	iff2imp[OF zero_le_iff_zero_nprt]
obua@14754
  1435
	iff2imp[OF le_zero_iff_pprt_id]
obua@14754
  1436
      )
obua@14738
  1437
      apply (insert prems)
obua@14754
  1438
      apply (auto simp add: 
obua@14754
  1439
	ring_eq_simps 
obua@14754
  1440
	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
nipkow@15197
  1441
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])
nipkow@15197
  1442
	apply(drule (1) mult_pos_neg_le[of a b], simp)
nipkow@15197
  1443
	apply(drule (1) mult_pos_neg2_le[of b a], simp)
obua@14738
  1444
      done
obua@14738
  1445
  next
obua@14738
  1446
    assume "~(0 <= a*b)"
obua@14738
  1447
    with s have "a*b <= 0" by simp
obua@14738
  1448
    then show ?thesis
obua@14738
  1449
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1450
      apply (insert prems)
obua@14738
  1451
      apply (auto simp add: ring_eq_simps iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
nipkow@15197
  1452
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])
nipkow@15197
  1453
      apply(drule (1) mult_pos_le[of a b],simp)
nipkow@15197
  1454
      apply(drule (1) mult_neg_le[of a b],simp)
obua@14738
  1455
      done
obua@14738
  1456
  qed
obua@14738
  1457
qed
paulson@14294
  1458
obua@14738
  1459
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
obua@14738
  1460
by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  1461
obua@14738
  1462
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
obua@14738
  1463
by (simp add: abs_if) 
paulson@14294
  1464
paulson@14294
  1465
lemma nonzero_abs_inverse:
paulson@14294
  1466
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  1467
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  1468
                      negative_imp_inverse_negative)
paulson@14294
  1469
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  1470
done
paulson@14294
  1471
paulson@14294
  1472
lemma abs_inverse [simp]:
paulson@14294
  1473
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  1474
      inverse (abs a)"
paulson@14294
  1475
apply (case_tac "a=0", simp) 
paulson@14294
  1476
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1477
done
paulson@14294
  1478
paulson@14294
  1479
lemma nonzero_abs_divide:
paulson@14294
  1480
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  1481
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  1482
paulson@14294
  1483
lemma abs_divide:
paulson@14294
  1484
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
paulson@14294
  1485
apply (case_tac "b=0", simp) 
paulson@14294
  1486
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1487
done
paulson@14294
  1488
paulson@14294
  1489
lemma abs_mult_less:
obua@14738
  1490
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  1491
proof -
paulson@14294
  1492
  assume ac: "abs a < c"
paulson@14294
  1493
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  1494
  assume "abs b < d"
paulson@14294
  1495
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1496
qed
paulson@14293
  1497
obua@14738
  1498
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
obua@14738
  1499
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
obua@14738
  1500
obua@14738
  1501
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
obua@14738
  1502
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
obua@14738
  1503
obua@14738
  1504
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  1505
apply (simp add: order_less_le abs_le_iff)  
obua@14738
  1506
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
obua@14738
  1507
apply (simp add: le_minus_self_iff linorder_neq_iff) 
obua@14738
  1508
done
obua@14738
  1509
paulson@14430
  1510
text{*Moving this up spoils many proofs using @{text mult_le_cancel_right}*}
paulson@14430
  1511
declare times_divide_eq_left [simp]
paulson@14430
  1512
obua@15178
  1513
lemma linprog_dual_estimate:
obua@15178
  1514
  assumes
obua@15178
  1515
  "A * x \<le> (b::'a::lordered_ring)"
obua@15178
  1516
  "0 \<le> y"
obua@15178
  1517
  "abs (A - A') \<le> \<delta>A"
obua@15178
  1518
  "b \<le> b'"
obua@15178
  1519
  "abs (c - c') \<le> \<delta>c"
obua@15178
  1520
  "abs x \<le> r"
obua@15178
  1521
  shows
obua@15178
  1522
  "c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r"
obua@15178
  1523
proof -
obua@15178
  1524
  from prems have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
obua@15178
  1525
  from prems have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono) 
obua@15178
  1526
  have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: ring_eq_simps)  
obua@15178
  1527
  from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp
obua@15178
  1528
  have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"
obua@15178
  1529
    by (simp only: 4 estimate_by_abs)  
obua@15178
  1530
  have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x"
obua@15178
  1531
    by (simp add: abs_le_mult)
obua@15178
  1532
  have 7: "(abs (y * (A - A') + (y * A' - c') + (c'-c))) * abs x <= (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x"
obua@15178
  1533
    by (simp add: abs_triangle_ineq mult_right_mono)
obua@15178
  1534
  have 8: " (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x <=  (abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x"
obua@15178
  1535
    by (simp add: abs_triangle_ineq mult_right_mono)    
obua@15178
  1536
  have 9: "(abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x"
obua@15178
  1537
    by (simp add: abs_le_mult mult_right_mono)  
obua@15178
  1538
  have 10: "c'-c = -(c-c')" by (simp add: ring_eq_simps)
obua@15178
  1539
  have 11: "abs (c'-c) = abs (c-c')" 
obua@15178
  1540
    by (subst 10, subst abs_minus_cancel, simp)
obua@15178
  1541
  have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x"
obua@15178
  1542
    by (simp add: 11 prems mult_right_mono)
obua@15178
  1543
  have 13: "(abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x"
obua@15178
  1544
    by (simp add: prems mult_right_mono mult_left_mono)  
obua@15178
  1545
  have r: "(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x <=  (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
obua@15178
  1546
    apply (rule mult_left_mono)
obua@15178
  1547
    apply (simp add: prems)
obua@15178
  1548
    apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+
obua@15178
  1549
    apply (rule mult_left_mono[of "0" "\<delta>A", simplified])
obua@15178
  1550
    apply (simp_all)
obua@15178
  1551
    apply (rule order_trans[where y="abs (A-A')"], simp_all add: prems)
obua@15178
  1552
    apply (rule order_trans[where y="abs (c-c')"], simp_all add: prems)
obua@15178
  1553
    done    
obua@15178
  1554
  from 6 7 8 9 12 13 r have 14:" abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <=(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"     
obua@15178
  1555
    by (simp)
obua@15178
  1556
  show ?thesis 
obua@15178
  1557
    apply (rule_tac le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])
obua@15178
  1558
    apply (simp_all add: 5 14[simplified abs_of_ge_0[of y, simplified prems]])
obua@15178
  1559
    done
obua@15178
  1560
qed
obua@15178
  1561
obua@15178
  1562
lemma le_ge_imp_abs_diff_1:
obua@15178
  1563
  assumes
obua@15178
  1564
  "A1 <= (A::'a::lordered_ring)"
obua@15178
  1565
  "A <= A2" 
obua@15178
  1566
  shows "abs (A-A1) <= A2-A1"
obua@15178
  1567
proof -
obua@15178
  1568
  have "0 <= A - A1"    
obua@15178
  1569
  proof -
obua@15178
  1570
    have 1: "A - A1 = A + (- A1)" by simp
obua@15178
  1571
    show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified prems])
obua@15178
  1572
  qed
obua@15178
  1573
  then have "abs (A-A1) = A-A1" by (rule abs_of_ge_0)
obua@15178
  1574
  with prems show "abs (A-A1) <= (A2-A1)" by simp
obua@15178
  1575
qed
obua@15178
  1576
obua@15178
  1577
lemma linprog_dual_estimate_1:
obua@15178
  1578
  assumes
obua@15178
  1579
  "A * x \<le> (b::'a::lordered_ring)"
obua@15178
  1580
  "0 \<le> y"
obua@15178
  1581
  "A1 <= A"
obua@15178
  1582
  "A <= A2"
obua@15178
  1583
  "c1 <= c"
obua@15178
  1584
  "c <= c2"
obua@15178
  1585
  "abs x \<le> r"
obua@15178
  1586
  shows
obua@15178
  1587
  "c * x \<le> y * b + (y * (A2 - A1) + abs (y * A1 - c1) + (c2 - c1)) * r"
obua@15178
  1588
proof -
obua@15178
  1589
  from prems have delta_A: "abs (A-A1) <= (A2-A1)" by (simp add: le_ge_imp_abs_diff_1)
obua@15178
  1590
  from prems have delta_c: "abs (c-c1) <= (c2-c1)" by (simp add: le_ge_imp_abs_diff_1)
obua@15178
  1591
  show ?thesis
obua@15178
  1592
    apply (rule_tac linprog_dual_estimate)
obua@15178
  1593
    apply (auto intro: delta_A delta_c simp add: prems)
obua@15178
  1594
    done
obua@15178
  1595
qed
obua@15178
  1596
obua@14738
  1597
ML {*
paulson@14334
  1598
val left_distrib = thm "left_distrib";
obua@14738
  1599
val right_distrib = thm "right_distrib";
obua@14738
  1600
val mult_commute = thm "mult_commute";
obua@14738
  1601
val distrib = thm "distrib";
obua@14738
  1602
val zero_neq_one = thm "zero_neq_one";
obua@14738
  1603
val no_zero_divisors = thm "no_zero_divisors";
paulson@14331
  1604
val left_inverse = thm "left_inverse";
obua@14738
  1605
val divide_inverse = thm "divide_inverse";
obua@14738
  1606
val mult_zero_left = thm "mult_zero_left";
obua@14738
  1607
val mult_zero_right = thm "mult_zero_right";
obua@14738
  1608
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";
obua@14738
  1609
val inverse_zero = thm "inverse_zero";
obua@14738
  1610
val ring_distrib = thms "ring_distrib";
obua@14738
  1611
val combine_common_factor = thm "combine_common_factor";
obua@14738
  1612
val minus_mult_left = thm "minus_mult_left";
obua@14738
  1613
val minus_mult_right = thm "minus_mult_right";
obua@14738
  1614
val minus_mult_minus = thm "minus_mult_minus";
obua@14738
  1615
val minus_mult_commute = thm "minus_mult_commute";
obua@14738
  1616
val right_diff_distrib = thm "right_diff_distrib";
obua@14738
  1617
val left_diff_distrib = thm "left_diff_distrib";
obua@14738
  1618
val mult_left_mono = thm "mult_left_mono";
obua@14738
  1619
val mult_right_mono = thm "mult_right_mono";
obua@14738
  1620
val mult_strict_left_mono = thm "mult_strict_left_mono";
obua@14738
  1621
val mult_strict_right_mono = thm "mult_strict_right_mono";
obua@14738
  1622
val mult_mono = thm "mult_mono";
obua@14738
  1623
val mult_strict_mono = thm "mult_strict_mono";
obua@14738
  1624
val abs_if = thm "abs_if";
obua@14738
  1625
val zero_less_one = thm "zero_less_one";
obua@14738
  1626
val eq_add_iff1 = thm "eq_add_iff1";
obua@14738
  1627
val eq_add_iff2 = thm "eq_add_iff2";
obua@14738
  1628
val less_add_iff1 = thm "less_add_iff1";
obua@14738
  1629
val less_add_iff2 = thm "less_add_iff2";
obua@14738
  1630
val le_add_iff1 = thm "le_add_iff1";
obua@14738
  1631
val le_add_iff2 = thm "le_add_iff2";
obua@14738
  1632
val mult_left_le_imp_le = thm "mult_left_le_imp_le";
obua@14738
  1633
val mult_right_le_imp_le = thm "mult_right_le_imp_le";
obua@14738
  1634
val mult_left_less_imp_less = thm "mult_left_less_imp_less";
obua@14738
  1635
val mult_right_less_imp_less = thm "mult_right_less_imp_less";
obua@14738
  1636
val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg";
obua@14738
  1637
val mult_left_mono_neg = thm "mult_left_mono_neg";
obua@14738
  1638
val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg";
obua@14738
  1639
val mult_right_mono_neg = thm "mult_right_mono_neg";
obua@14738
  1640
val mult_pos = thm "mult_pos";
obua@14738
  1641
val mult_pos_le = thm "mult_pos_le";
obua@14738
  1642
val mult_pos_neg = thm "mult_pos_neg";
obua@14738
  1643
val mult_pos_neg_le = thm "mult_pos_neg_le";
obua@14738
  1644
val mult_pos_neg2 = thm "mult_pos_neg2";
obua@14738
  1645
val mult_pos_neg2_le = thm "mult_pos_neg2_le";
obua@14738
  1646
val mult_neg = thm "mult_neg";
obua@14738
  1647
val mult_neg_le = thm "mult_neg_le";
obua@14738
  1648
val zero_less_mult_pos = thm "zero_less_mult_pos";
obua@14738
  1649
val zero_less_mult_pos2 = thm "zero_less_mult_pos2";
obua@14738
  1650
val zero_less_mult_iff = thm "zero_less_mult_iff";
obua@14738
  1651
val mult_eq_0_iff = thm "mult_eq_0_iff";
obua@14738
  1652
val zero_le_mult_iff = thm "zero_le_mult_iff";
obua@14738
  1653
val mult_less_0_iff = thm "mult_less_0_iff";
obua@14738
  1654
val mult_le_0_iff = thm "mult_le_0_iff";
obua@14738
  1655
val split_mult_pos_le = thm "split_mult_pos_le";
obua@14738
  1656
val split_mult_neg_le = thm "split_mult_neg_le";
obua@14738
  1657
val zero_le_square = thm "zero_le_square";
obua@14738
  1658
val zero_le_one = thm "zero_le_one";
obua@14738
  1659
val not_one_le_zero = thm "not_one_le_zero";
obua@14738
  1660
val not_one_less_zero = thm "not_one_less_zero";
obua@14738
  1661
val mult_left_mono_neg = thm "mult_left_mono_neg";
obua@14738
  1662
val mult_right_mono_neg = thm "mult_right_mono_neg";
obua@14738
  1663
val mult_strict_mono = thm "mult_strict_mono";
obua@14738
  1664
val mult_strict_mono' = thm "mult_strict_mono'";
obua@14738
  1665
val mult_mono = thm "mult_mono";
obua@14738
  1666
val less_1_mult = thm "less_1_mult";
obua@14738
  1667
val mult_less_cancel_right = thm "mult_less_cancel_right";
obua@14738
  1668
val mult_less_cancel_left = thm "mult_less_cancel_left";
obua@14738
  1669
val mult_le_cancel_right = thm "mult_le_cancel_right";
obua@14738
  1670
val mult_le_cancel_left = thm "mult_le_cancel_left";
obua@14738
  1671
val mult_less_imp_less_left = thm "mult_less_imp_less_left";
obua@14738
  1672
val mult_less_imp_less_right = thm "mult_less_imp_less_right";
obua@14738
  1673
val mult_cancel_right = thm "mult_cancel_right";
obua@14738
  1674
val mult_cancel_left = thm "mult_cancel_left";
obua@14738
  1675
val ring_eq_simps = thms "ring_eq_simps";
obua@14738
  1676
val right_inverse = thm "right_inverse";
obua@14738
  1677
val right_inverse_eq = thm "right_inverse_eq";
obua@14738
  1678
val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide";
obua@14738
  1679
val divide_self = thm "divide_self";
obua@14738
  1680
val divide_zero = thm "divide_zero";
obua@14738
  1681
val divide_zero_left = thm "divide_zero_left";
obua@14738
  1682
val inverse_eq_divide = thm "inverse_eq_divide";
obua@14738
  1683
val add_divide_distrib = thm "add_divide_distrib";
obua@14738
  1684
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";
obua@14738
  1685
val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma";
obua@14738
  1686
val field_mult_cancel_right = thm "field_mult_cancel_right";
obua@14738
  1687
val field_mult_cancel_left = thm "field_mult_cancel_left";
obua@14738
  1688
val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero";
obua@14738
  1689
val inverse_zero_imp_zero = thm "inverse_zero_imp_zero";
obua@14738
  1690
val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero";
obua@14738
  1691
val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero";
obua@14738
  1692
val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq";
obua@14738
  1693
val inverse_minus_eq = thm "inverse_minus_eq";
obua@14738
  1694
val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq";
obua@14738
  1695
val inverse_eq_imp_eq = thm "inverse_eq_imp_eq";
obua@14738
  1696
val inverse_eq_iff_eq = thm "inverse_eq_iff_eq";
obua@14738
  1697
val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq";
obua@14738
  1698
val inverse_inverse_eq = thm "inverse_inverse_eq";
obua@14738
  1699
val inverse_1 = thm "inverse_1";
obua@14738
  1700
val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib";
obua@14738
  1701
val inverse_mult_distrib = thm "inverse_mult_distrib";
obua@14738
  1702
val inverse_add = thm "inverse_add";
obua@14738
  1703
val inverse_divide = thm "inverse_divide";
obua@14738
  1704
val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left";
obua@14738
  1705
val mult_divide_cancel_left = thm "mult_divide_cancel_left";
obua@14738
  1706
val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right";
obua@14738
  1707
val mult_divide_cancel_right = thm "mult_divide_cancel_right";
obua@14738
  1708
val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if";
obua@14738
  1709
val divide_1 = thm "divide_1";
obua@14738
  1710
val times_divide_eq_right = thm "times_divide_eq_right";
obua@14738
  1711
val times_divide_eq_left = thm "times_divide_eq_left";
obua@14738
  1712
val divide_divide_eq_right = thm "divide_divide_eq_right";
obua@14738
  1713
val divide_divide_eq_left = thm "divide_divide_eq_left";
obua@14738
  1714
val nonzero_minus_divide_left = thm "nonzero_minus_divide_left";
obua@14738
  1715
val nonzero_minus_divide_right = thm "nonzero_minus_divide_right";
obua@14738
  1716
val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide";
obua@14738
  1717
val minus_divide_left = thm "minus_divide_left";
obua@14738
  1718
val minus_divide_right = thm "minus_divide_right";
obua@14738
  1719
val minus_divide_divide = thm "minus_divide_divide";
obua@14738
  1720
val diff_divide_distrib = thm "diff_divide_distrib";
obua@14738
  1721
val positive_imp_inverse_positive = thm "positive_imp_inverse_positive";
obua@14738
  1722
val negative_imp_inverse_negative = thm "negative_imp_inverse_negative";
obua@14738
  1723
val inverse_le_imp_le = thm "inverse_le_imp_le";
obua@14738
  1724
val inverse_positive_imp_positive = thm "inverse_positive_imp_positive";
obua@14738
  1725
val inverse_positive_iff_positive = thm "inverse_positive_iff_positive";
obua@14738
  1726
val inverse_negative_imp_negative = thm "inverse_negative_imp_negative";
obua@14738
  1727
val inverse_negative_iff_negative = thm "inverse_negative_iff_negative";
obua@14738
  1728
val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative";
obua@14738
  1729
val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive";
obua@14738
  1730
val less_imp_inverse_less = thm "less_imp_inverse_less";
obua@14738
  1731
val inverse_less_imp_less = thm "inverse_less_imp_less";
obua@14738
  1732
val inverse_less_iff_less = thm "inverse_less_iff_less";
obua@14738
  1733
val le_imp_inverse_le = thm "le_imp_inverse_le";
obua@14738
  1734
val inverse_le_iff_le = thm "inverse_le_iff_le";
obua@14738
  1735
val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg";
obua@14738
  1736
val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg";
obua@14738
  1737
val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg";
obua@14738
  1738
val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg";
obua@14738
  1739
val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg";
obua@14738
  1740
val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg";
obua@14738
  1741
val one_less_inverse_iff = thm "one_less_inverse_iff";
obua@14738
  1742
val inverse_eq_1_iff = thm "inverse_eq_1_iff";
obua@14738
  1743
val one_le_inverse_iff = thm "one_le_inverse_iff";
obua@14738
  1744
val inverse_less_1_iff = thm "inverse_less_1_iff";
obua@14738
  1745
val inverse_le_1_iff = thm "inverse_le_1_iff";
obua@14738
  1746
val zero_less_divide_iff = thm "zero_less_divide_iff";
obua@14738
  1747
val divide_less_0_iff = thm "divide_less_0_iff";
obua@14738
  1748
val zero_le_divide_iff = thm "zero_le_divide_iff";
obua@14738
  1749
val divide_le_0_iff = thm "divide_le_0_iff";
obua@14738
  1750
val divide_eq_0_iff = thm "divide_eq_0_iff";
obua@14738
  1751
val pos_le_divide_eq = thm "pos_le_divide_eq";
obua@14738
  1752
val neg_le_divide_eq = thm "neg_le_divide_eq";
obua@14738
  1753
val le_divide_eq = thm "le_divide_eq";
obua@14738
  1754
val pos_divide_le_eq = thm "pos_divide_le_eq";
obua@14738
  1755
val neg_divide_le_eq = thm "neg_divide_le_eq";
obua@14738
  1756
val divide_le_eq = thm "divide_le_eq";
obua@14738
  1757
val pos_less_divide_eq = thm "pos_less_divide_eq";
obua@14738
  1758
val neg_less_divide_eq = thm "neg_less_divide_eq";
obua@14738
  1759
val less_divide_eq = thm "less_divide_eq";
obua@14738
  1760
val pos_divide_less_eq = thm "pos_divide_less_eq";
obua@14738
  1761
val neg_divide_less_eq = thm "neg_divide_less_eq";
obua@14738
  1762
val divide_less_eq = thm "divide_less_eq";
obua@14738
  1763
val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq";
obua@14738
  1764
val eq_divide_eq = thm "eq_divide_eq";
obua@14738
  1765
val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq";
obua@14738
  1766
val divide_eq_eq = thm "divide_eq_eq";
obua@14738
  1767
val divide_cancel_right = thm "divide_cancel_right";
obua@14738
  1768
val divide_cancel_left = thm "divide_cancel_left";
obua@14738
  1769
val divide_eq_1_iff = thm "divide_eq_1_iff";
obua@14738
  1770
val one_eq_divide_iff = thm "one_eq_divide_iff";
obua@14738
  1771
val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff";
obua@14738
  1772
val one_divide_eq_0_iff = thm "one_divide_eq_0_iff";
obua@14738
  1773
val divide_strict_right_mono = thm "divide_strict_right_mono";
obua@14738
  1774
val divide_right_mono = thm "divide_right_mono";
obua@14738
  1775
val divide_strict_left_mono = thm "divide_strict_left_mono";
obua@14738
  1776
val divide_left_mono = thm "divide_left_mono";
obua@14738
  1777
val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg";
obua@14738
  1778
val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg";
obua@14738
  1779
val less_add_one = thm "less_add_one";
obua@14738
  1780
val zero_less_two = thm "zero_less_two";
obua@14738
  1781
val less_half_sum = thm "less_half_sum";
obua@14738
  1782
val gt_half_sum = thm "gt_half_sum";
obua@14738
  1783
val dense = thm "dense";
obua@14738
  1784
val abs_one = thm "abs_one";
obua@14738
  1785
val abs_le_mult = thm "abs_le_mult";
obua@14738
  1786
val abs_eq_mult = thm "abs_eq_mult";
obua@14738
  1787
val abs_mult = thm "abs_mult";
obua@14738
  1788
val abs_mult_self = thm "abs_mult_self";
obua@14738
  1789
val nonzero_abs_inverse = thm "nonzero_abs_inverse";
obua@14738
  1790
val abs_inverse = thm "abs_inverse";
obua@14738
  1791
val nonzero_abs_divide = thm "nonzero_abs_divide";
obua@14738
  1792
val abs_divide = thm "abs_divide";
obua@14738
  1793
val abs_mult_less = thm "abs_mult_less";
obua@14738
  1794
val eq_minus_self_iff = thm "eq_minus_self_iff";
obua@14738
  1795
val less_minus_self_iff = thm "less_minus_self_iff";
obua@14738
  1796
val abs_less_iff = thm "abs_less_iff";
paulson@14331
  1797
*}
paulson@14331
  1798
paulson@14265
  1799
end