src/HOL/Ring_and_Field.thy
author obua
Tue May 11 20:11:08 2004 +0200 (2004-05-11)
changeset 14738 83f1a514dcb4
parent 14603 985eb6708207
child 14754 a080eeeaec14
permissions -rw-r--r--
changes made due to new Ring_and_Field theory
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer and Markus Wenzel, TU Muenchen
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             Lawrence C Paulson, University of Cambridge
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             Revised and splitted into Ring_and_Field.thy and Group.thy 
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             by Steven Obua, TU Muenchen, in May 2004
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    License: GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* (Ordered) Rings and Fields *}
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theory Ring_and_Field = OrderedGroup:
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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 \emph{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 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 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 comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)
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axclass ring \<subseteq> semiring, ab_group_add
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instance ring \<subseteq> semiring_0 ..
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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 ..
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axclass ring_1 \<subseteq> ring, semiring_1
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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_semigroup_add})"
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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_semigroup_add})"
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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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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> 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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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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lemma mult_right_mono_neg:
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     "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
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apply (drule mult_right_mono [of _ _ "-c"])
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apply (simp)
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apply (simp_all add: minus_mult_right [symmetric]) 
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done
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subsection{* Products of Signs *}
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lemma mult_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
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by (drule mult_strict_left_mono [of 0 b], auto)
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lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
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by (drule mult_left_mono [of 0 b], auto)
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   315
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
paulson@14265
   316
by (drule mult_strict_left_mono [of b 0], auto)
paulson@14265
   317
obua@14738
   318
lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
obua@14738
   319
by (drule mult_left_mono [of b 0], auto)
obua@14738
   320
obua@14738
   321
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
obua@14738
   322
by (drule mult_strict_right_mono[of b 0], auto)
obua@14738
   323
obua@14738
   324
lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
obua@14738
   325
by (drule mult_right_mono[of b 0], auto)
obua@14738
   326
obua@14738
   327
lemma mult_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
paulson@14265
   328
by (drule mult_strict_right_mono_neg, auto)
paulson@14265
   329
obua@14738
   330
lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
obua@14738
   331
by (drule mult_right_mono_neg[of a 0 b ], auto)
obua@14738
   332
paulson@14341
   333
lemma zero_less_mult_pos:
obua@14738
   334
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
paulson@14265
   335
apply (case_tac "b\<le>0") 
paulson@14265
   336
 apply (auto simp add: order_le_less linorder_not_less)
paulson@14265
   337
apply (drule_tac mult_pos_neg [of a b]) 
paulson@14265
   338
 apply (auto dest: order_less_not_sym)
paulson@14265
   339
done
paulson@14265
   340
obua@14738
   341
lemma zero_less_mult_pos2:
obua@14738
   342
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
obua@14738
   343
apply (case_tac "b\<le>0") 
obua@14738
   344
 apply (auto simp add: order_le_less linorder_not_less)
obua@14738
   345
apply (drule_tac mult_pos_neg2 [of a b]) 
obua@14738
   346
 apply (auto dest: order_less_not_sym)
obua@14738
   347
done
obua@14738
   348
paulson@14265
   349
lemma zero_less_mult_iff:
obua@14738
   350
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
paulson@14265
   351
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
paulson@14265
   352
apply (blast dest: zero_less_mult_pos) 
obua@14738
   353
apply (blast dest: zero_less_mult_pos2)
paulson@14265
   354
done
paulson@14265
   355
paulson@14341
   356
text{*A field has no "zero divisors", and this theorem holds without the
paulson@14277
   357
      assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}
obua@14738
   358
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0 | b = 0)"
paulson@14265
   359
apply (case_tac "a < 0")
paulson@14265
   360
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
paulson@14265
   361
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
paulson@14265
   362
done
paulson@14265
   363
paulson@14265
   364
lemma zero_le_mult_iff:
obua@14738
   365
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14265
   366
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
paulson@14265
   367
                   zero_less_mult_iff)
paulson@14265
   368
paulson@14265
   369
lemma mult_less_0_iff:
obua@14738
   370
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14265
   371
apply (insert zero_less_mult_iff [of "-a" b]) 
paulson@14265
   372
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   373
done
paulson@14265
   374
paulson@14265
   375
lemma mult_le_0_iff:
obua@14738
   376
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14265
   377
apply (insert zero_le_mult_iff [of "-a" b]) 
paulson@14265
   378
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   379
done
paulson@14265
   380
obua@14738
   381
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
   382
by (auto simp add: mult_pos_le mult_neg_le)
obua@14738
   383
obua@14738
   384
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
   385
by (auto simp add: mult_pos_neg_le mult_pos_neg2_le)
obua@14738
   386
obua@14738
   387
lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a"
paulson@14265
   388
by (simp add: zero_le_mult_iff linorder_linear) 
paulson@14265
   389
obua@14738
   390
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
obua@14738
   391
      theorems available to members of @{term ordered_idom} *}
obua@14738
   392
obua@14738
   393
instance ordered_idom \<subseteq> ordered_semidom
paulson@14421
   394
proof
paulson@14421
   395
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
paulson@14430
   396
  thus "(0::'a) < 1" by (simp add: order_le_less) 
paulson@14421
   397
qed
paulson@14421
   398
obua@14738
   399
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 
obua@14738
   400
by (intro_classes, simp)
obua@14738
   401
obua@14738
   402
instance ordered_idom \<subseteq> idom ..
obua@14738
   403
paulson@14387
   404
text{*All three types of comparision involving 0 and 1 are covered.*}
paulson@14387
   405
paulson@14387
   406
declare zero_neq_one [THEN not_sym, simp]
paulson@14387
   407
obua@14738
   408
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
paulson@14268
   409
  by (rule zero_less_one [THEN order_less_imp_le]) 
paulson@14268
   410
obua@14738
   411
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
obua@14738
   412
by (simp add: linorder_not_le) 
paulson@14387
   413
obua@14738
   414
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
obua@14738
   415
by (simp add: linorder_not_less) 
paulson@14268
   416
paulson@14268
   417
subsection{*More Monotonicity*}
paulson@14268
   418
paulson@14268
   419
lemma mult_left_mono_neg:
obua@14738
   420
     "[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)"
paulson@14268
   421
apply (drule mult_left_mono [of _ _ "-c"]) 
paulson@14268
   422
apply (simp_all add: minus_mult_left [symmetric]) 
paulson@14268
   423
done
paulson@14268
   424
paulson@14268
   425
lemma mult_right_mono_neg:
obua@14738
   426
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le> b * (c::'a::pordered_ring)"
obua@14738
   427
apply (drule mult_right_mono [of _ _ "-c"]) 
obua@14738
   428
apply (simp_all add: minus_mult_right [symmetric]) 
obua@14738
   429
done  
paulson@14268
   430
paulson@14268
   431
text{*Strict monotonicity in both arguments*}
paulson@14268
   432
lemma mult_strict_mono:
obua@14738
   433
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   434
apply (case_tac "c=0")
paulson@14268
   435
 apply (simp add: mult_pos) 
paulson@14268
   436
apply (erule mult_strict_right_mono [THEN order_less_trans])
paulson@14268
   437
 apply (force simp add: order_le_less) 
paulson@14268
   438
apply (erule mult_strict_left_mono, assumption)
paulson@14268
   439
done
paulson@14268
   440
paulson@14268
   441
text{*This weaker variant has more natural premises*}
paulson@14268
   442
lemma mult_strict_mono':
obua@14738
   443
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   444
apply (rule mult_strict_mono)
paulson@14268
   445
apply (blast intro: order_le_less_trans)+
paulson@14268
   446
done
paulson@14268
   447
paulson@14268
   448
lemma mult_mono:
paulson@14268
   449
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
obua@14738
   450
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
paulson@14268
   451
apply (erule mult_right_mono [THEN order_trans], assumption)
paulson@14268
   452
apply (erule mult_left_mono, assumption)
paulson@14268
   453
done
paulson@14268
   454
obua@14738
   455
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
paulson@14387
   456
apply (insert mult_strict_mono [of 1 m 1 n]) 
paulson@14430
   457
apply (simp add:  order_less_trans [OF zero_less_one]) 
paulson@14387
   458
done
paulson@14387
   459
paulson@14268
   460
subsection{*Cancellation Laws for Relationships With a Common Factor*}
paulson@14268
   461
paulson@14268
   462
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
paulson@14268
   463
   also with the relations @{text "\<le>"} and equality.*}
paulson@14268
   464
paulson@14268
   465
lemma mult_less_cancel_right:
obua@14738
   466
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
paulson@14268
   467
apply (case_tac "c = 0")
paulson@14268
   468
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
paulson@14268
   469
                      mult_strict_right_mono_neg)
paulson@14268
   470
apply (auto simp add: linorder_not_less 
paulson@14268
   471
                      linorder_not_le [symmetric, of "a*c"]
paulson@14268
   472
                      linorder_not_le [symmetric, of a])
paulson@14268
   473
apply (erule_tac [!] notE)
paulson@14268
   474
apply (auto simp add: order_less_imp_le mult_right_mono 
paulson@14268
   475
                      mult_right_mono_neg)
paulson@14268
   476
done
paulson@14268
   477
paulson@14268
   478
lemma mult_less_cancel_left:
obua@14738
   479
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
obua@14738
   480
apply (case_tac "c = 0")
obua@14738
   481
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
obua@14738
   482
                      mult_strict_left_mono_neg)
obua@14738
   483
apply (auto simp add: linorder_not_less 
obua@14738
   484
                      linorder_not_le [symmetric, of "c*a"]
obua@14738
   485
                      linorder_not_le [symmetric, of a])
obua@14738
   486
apply (erule_tac [!] notE)
obua@14738
   487
apply (auto simp add: order_less_imp_le mult_left_mono 
obua@14738
   488
                      mult_left_mono_neg)
obua@14738
   489
done
paulson@14268
   490
paulson@14268
   491
lemma mult_le_cancel_right:
obua@14738
   492
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@14268
   493
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right)
paulson@14268
   494
paulson@14268
   495
lemma mult_le_cancel_left:
obua@14738
   496
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
obua@14738
   497
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left)
paulson@14268
   498
paulson@14268
   499
lemma mult_less_imp_less_left:
paulson@14341
   500
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
obua@14738
   501
      shows "a < (b::'a::ordered_semiring_strict)"
paulson@14377
   502
proof (rule ccontr)
paulson@14377
   503
  assume "~ a < b"
paulson@14377
   504
  hence "b \<le> a" by (simp add: linorder_not_less)
paulson@14377
   505
  hence "c*b \<le> c*a" by (rule mult_left_mono)
paulson@14377
   506
  with this and less show False 
paulson@14377
   507
    by (simp add: linorder_not_less [symmetric])
paulson@14377
   508
qed
paulson@14268
   509
paulson@14268
   510
lemma mult_less_imp_less_right:
obua@14738
   511
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
obua@14738
   512
  shows "a < (b::'a::ordered_semiring_strict)"
obua@14738
   513
proof (rule ccontr)
obua@14738
   514
  assume "~ a < b"
obua@14738
   515
  hence "b \<le> a" by (simp add: linorder_not_less)
obua@14738
   516
  hence "b*c \<le> a*c" by (rule mult_right_mono)
obua@14738
   517
  with this and less show False 
obua@14738
   518
    by (simp add: linorder_not_less [symmetric])
obua@14738
   519
qed  
paulson@14268
   520
paulson@14268
   521
text{*Cancellation of equalities with a common factor*}
paulson@14268
   522
lemma mult_cancel_right [simp]:
obua@14738
   523
     "(a*c = b*c) = (c = (0::'a::ordered_ring_strict) | a=b)"
paulson@14268
   524
apply (cut_tac linorder_less_linear [of 0 c])
paulson@14268
   525
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono
paulson@14268
   526
             simp add: linorder_neq_iff)
paulson@14268
   527
done
paulson@14268
   528
paulson@14268
   529
text{*These cancellation theorems require an ordering. Versions are proved
paulson@14268
   530
      below that work for fields without an ordering.*}
paulson@14268
   531
lemma mult_cancel_left [simp]:
obua@14738
   532
     "(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)"
obua@14738
   533
apply (cut_tac linorder_less_linear [of 0 c])
obua@14738
   534
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono
obua@14738
   535
             simp add: linorder_neq_iff)
obua@14738
   536
done
paulson@14268
   537
obua@14738
   538
text{*This list of rewrites decides ring equalities by ordered rewriting.*}
obua@14738
   539
lemmas ring_eq_simps =
obua@14738
   540
  mult_ac
obua@14738
   541
  left_distrib right_distrib left_diff_distrib right_diff_distrib
obua@14738
   542
  add_ac
obua@14738
   543
  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
obua@14738
   544
  diff_eq_eq eq_diff_eq
obua@14738
   545
    
obua@14738
   546
thm ring_eq_simps
paulson@14265
   547
subsection {* Fields *}
paulson@14265
   548
paulson@14288
   549
lemma right_inverse [simp]:
paulson@14288
   550
      assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"
paulson@14288
   551
proof -
paulson@14288
   552
  have "a * inverse a = inverse a * a" by (simp add: mult_ac)
paulson@14288
   553
  also have "... = 1" using not0 by simp
paulson@14288
   554
  finally show ?thesis .
paulson@14288
   555
qed
paulson@14288
   556
paulson@14288
   557
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
paulson@14288
   558
proof
paulson@14288
   559
  assume neq: "b \<noteq> 0"
paulson@14288
   560
  {
paulson@14288
   561
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
paulson@14288
   562
    also assume "a / b = 1"
paulson@14288
   563
    finally show "a = b" by simp
paulson@14288
   564
  next
paulson@14288
   565
    assume "a = b"
paulson@14288
   566
    with neq show "a / b = 1" by (simp add: divide_inverse)
paulson@14288
   567
  }
paulson@14288
   568
qed
paulson@14288
   569
paulson@14288
   570
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
paulson@14288
   571
by (simp add: divide_inverse)
paulson@14288
   572
paulson@14288
   573
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
paulson@14288
   574
  by (simp add: divide_inverse)
paulson@14288
   575
paulson@14430
   576
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
paulson@14430
   577
by (simp add: divide_inverse)
paulson@14277
   578
paulson@14430
   579
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
paulson@14430
   580
by (simp add: divide_inverse)
paulson@14277
   581
paulson@14430
   582
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
paulson@14430
   583
by (simp add: divide_inverse)
paulson@14277
   584
paulson@14430
   585
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
paulson@14293
   586
by (simp add: divide_inverse left_distrib) 
paulson@14293
   587
paulson@14293
   588
paulson@14270
   589
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
paulson@14270
   590
      of an ordering.*}
paulson@14348
   591
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
paulson@14377
   592
proof cases
paulson@14377
   593
  assume "a=0" thus ?thesis by simp
paulson@14377
   594
next
paulson@14377
   595
  assume anz [simp]: "a\<noteq>0"
paulson@14377
   596
  { assume "a * b = 0"
paulson@14377
   597
    hence "inverse a * (a * b) = 0" by simp
paulson@14377
   598
    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
paulson@14377
   599
  thus ?thesis by force
paulson@14377
   600
qed
paulson@14270
   601
paulson@14268
   602
text{*Cancellation of equalities with a common factor*}
paulson@14268
   603
lemma field_mult_cancel_right_lemma:
paulson@14269
   604
      assumes cnz: "c \<noteq> (0::'a::field)"
paulson@14269
   605
	  and eq:  "a*c = b*c"
paulson@14269
   606
	 shows "a=b"
paulson@14377
   607
proof -
paulson@14268
   608
  have "(a * c) * inverse c = (b * c) * inverse c"
paulson@14268
   609
    by (simp add: eq)
paulson@14268
   610
  thus "a=b"
paulson@14268
   611
    by (simp add: mult_assoc cnz)
paulson@14377
   612
qed
paulson@14268
   613
paulson@14348
   614
lemma field_mult_cancel_right [simp]:
paulson@14268
   615
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
paulson@14377
   616
proof cases
paulson@14377
   617
  assume "c=0" thus ?thesis by simp
paulson@14377
   618
next
paulson@14377
   619
  assume "c\<noteq>0" 
paulson@14377
   620
  thus ?thesis by (force dest: field_mult_cancel_right_lemma)
paulson@14377
   621
qed
paulson@14268
   622
paulson@14348
   623
lemma field_mult_cancel_left [simp]:
paulson@14268
   624
     "(c*a = c*b) = (c = (0::'a::field) | a=b)"
paulson@14268
   625
  by (simp add: mult_commute [of c] field_mult_cancel_right) 
paulson@14268
   626
paulson@14268
   627
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
paulson@14377
   628
proof
paulson@14268
   629
  assume ianz: "inverse a = 0"
paulson@14268
   630
  assume "a \<noteq> 0"
paulson@14268
   631
  hence "1 = a * inverse a" by simp
paulson@14268
   632
  also have "... = 0" by (simp add: ianz)
paulson@14268
   633
  finally have "1 = (0::'a::field)" .
paulson@14268
   634
  thus False by (simp add: eq_commute)
paulson@14377
   635
qed
paulson@14268
   636
paulson@14277
   637
paulson@14277
   638
subsection{*Basic Properties of @{term inverse}*}
paulson@14277
   639
paulson@14268
   640
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"
paulson@14268
   641
apply (rule ccontr) 
paulson@14268
   642
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   643
done
paulson@14268
   644
paulson@14268
   645
lemma inverse_nonzero_imp_nonzero:
paulson@14268
   646
   "inverse a = 0 ==> a = (0::'a::field)"
paulson@14268
   647
apply (rule ccontr) 
paulson@14268
   648
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   649
done
paulson@14268
   650
paulson@14268
   651
lemma inverse_nonzero_iff_nonzero [simp]:
paulson@14268
   652
   "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"
paulson@14268
   653
by (force dest: inverse_nonzero_imp_nonzero) 
paulson@14268
   654
paulson@14268
   655
lemma nonzero_inverse_minus_eq:
paulson@14269
   656
      assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
paulson@14377
   657
proof -
paulson@14377
   658
  have "-a * inverse (- a) = -a * - inverse a"
paulson@14377
   659
    by simp
paulson@14377
   660
  thus ?thesis 
paulson@14377
   661
    by (simp only: field_mult_cancel_left, simp)
paulson@14377
   662
qed
paulson@14268
   663
paulson@14268
   664
lemma inverse_minus_eq [simp]:
paulson@14377
   665
   "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
paulson@14377
   666
proof cases
paulson@14377
   667
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
   668
next
paulson@14377
   669
  assume "a\<noteq>0" 
paulson@14377
   670
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
   671
qed
paulson@14268
   672
paulson@14268
   673
lemma nonzero_inverse_eq_imp_eq:
paulson@14269
   674
      assumes inveq: "inverse a = inverse b"
paulson@14269
   675
	  and anz:  "a \<noteq> 0"
paulson@14269
   676
	  and bnz:  "b \<noteq> 0"
paulson@14269
   677
	 shows "a = (b::'a::field)"
paulson@14377
   678
proof -
paulson@14268
   679
  have "a * inverse b = a * inverse a"
paulson@14268
   680
    by (simp add: inveq)
paulson@14268
   681
  hence "(a * inverse b) * b = (a * inverse a) * b"
paulson@14268
   682
    by simp
paulson@14268
   683
  thus "a = b"
paulson@14268
   684
    by (simp add: mult_assoc anz bnz)
paulson@14377
   685
qed
paulson@14268
   686
paulson@14268
   687
lemma inverse_eq_imp_eq:
paulson@14268
   688
     "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"
paulson@14268
   689
apply (case_tac "a=0 | b=0") 
paulson@14268
   690
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
   691
              simp add: eq_commute [of "0::'a"])
paulson@14268
   692
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
   693
done
paulson@14268
   694
paulson@14268
   695
lemma inverse_eq_iff_eq [simp]:
paulson@14268
   696
     "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"
paulson@14268
   697
by (force dest!: inverse_eq_imp_eq) 
paulson@14268
   698
paulson@14270
   699
lemma nonzero_inverse_inverse_eq:
paulson@14270
   700
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
paulson@14270
   701
  proof -
paulson@14270
   702
  have "(inverse (inverse a) * inverse a) * a = a" 
paulson@14270
   703
    by (simp add: nonzero_imp_inverse_nonzero)
paulson@14270
   704
  thus ?thesis
paulson@14270
   705
    by (simp add: mult_assoc)
paulson@14270
   706
  qed
paulson@14270
   707
paulson@14270
   708
lemma inverse_inverse_eq [simp]:
paulson@14270
   709
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
paulson@14270
   710
  proof cases
paulson@14270
   711
    assume "a=0" thus ?thesis by simp
paulson@14270
   712
  next
paulson@14270
   713
    assume "a\<noteq>0" 
paulson@14270
   714
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
   715
  qed
paulson@14270
   716
paulson@14270
   717
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
paulson@14270
   718
  proof -
paulson@14270
   719
  have "inverse 1 * 1 = (1::'a::field)" 
paulson@14270
   720
    by (rule left_inverse [OF zero_neq_one [symmetric]])
paulson@14270
   721
  thus ?thesis  by simp
paulson@14270
   722
  qed
paulson@14270
   723
paulson@14270
   724
lemma nonzero_inverse_mult_distrib: 
paulson@14270
   725
      assumes anz: "a \<noteq> 0"
paulson@14270
   726
          and bnz: "b \<noteq> 0"
paulson@14270
   727
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
paulson@14270
   728
  proof -
paulson@14270
   729
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
paulson@14270
   730
    by (simp add: field_mult_eq_0_iff anz bnz)
paulson@14270
   731
  hence "inverse(a*b) * a = inverse(b)" 
paulson@14270
   732
    by (simp add: mult_assoc bnz)
paulson@14270
   733
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
paulson@14270
   734
    by simp
paulson@14270
   735
  thus ?thesis
paulson@14270
   736
    by (simp add: mult_assoc anz)
paulson@14270
   737
  qed
paulson@14270
   738
paulson@14270
   739
text{*This version builds in division by zero while also re-orienting
paulson@14270
   740
      the right-hand side.*}
paulson@14270
   741
lemma inverse_mult_distrib [simp]:
paulson@14270
   742
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
   743
  proof cases
paulson@14270
   744
    assume "a \<noteq> 0 & b \<noteq> 0" 
paulson@14270
   745
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
   746
  next
paulson@14270
   747
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
paulson@14270
   748
    thus ?thesis  by force
paulson@14270
   749
  qed
paulson@14270
   750
paulson@14270
   751
text{*There is no slick version using division by zero.*}
paulson@14270
   752
lemma inverse_add:
paulson@14270
   753
     "[|a \<noteq> 0;  b \<noteq> 0|]
paulson@14270
   754
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
paulson@14270
   755
apply (simp add: left_distrib mult_assoc)
paulson@14270
   756
apply (simp add: mult_commute [of "inverse a"]) 
paulson@14270
   757
apply (simp add: mult_assoc [symmetric] add_commute)
paulson@14270
   758
done
paulson@14270
   759
paulson@14365
   760
lemma inverse_divide [simp]:
paulson@14365
   761
      "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
paulson@14430
   762
  by (simp add: divide_inverse mult_commute)
paulson@14365
   763
paulson@14277
   764
lemma nonzero_mult_divide_cancel_left:
paulson@14277
   765
  assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
paulson@14277
   766
    shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
   767
proof -
paulson@14277
   768
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
paulson@14277
   769
    by (simp add: field_mult_eq_0_iff divide_inverse 
paulson@14277
   770
                  nonzero_inverse_mult_distrib)
paulson@14277
   771
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
   772
    by (simp only: mult_ac)
paulson@14277
   773
  also have "... =  a * inverse b"
paulson@14277
   774
    by simp
paulson@14277
   775
    finally show ?thesis 
paulson@14277
   776
    by (simp add: divide_inverse)
paulson@14277
   777
qed
paulson@14277
   778
paulson@14277
   779
lemma mult_divide_cancel_left:
paulson@14277
   780
     "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
paulson@14277
   781
apply (case_tac "b = 0")
paulson@14277
   782
apply (simp_all add: nonzero_mult_divide_cancel_left)
paulson@14277
   783
done
paulson@14277
   784
paulson@14321
   785
lemma nonzero_mult_divide_cancel_right:
paulson@14321
   786
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
paulson@14321
   787
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
paulson@14321
   788
paulson@14321
   789
lemma mult_divide_cancel_right:
paulson@14321
   790
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
paulson@14321
   791
apply (case_tac "b = 0")
paulson@14321
   792
apply (simp_all add: nonzero_mult_divide_cancel_right)
paulson@14321
   793
done
paulson@14321
   794
paulson@14277
   795
(*For ExtractCommonTerm*)
paulson@14277
   796
lemma mult_divide_cancel_eq_if:
paulson@14277
   797
     "(c*a) / (c*b) = 
paulson@14277
   798
      (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
paulson@14277
   799
  by (simp add: mult_divide_cancel_left)
paulson@14277
   800
paulson@14284
   801
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
paulson@14430
   802
  by (simp add: divide_inverse)
paulson@14284
   803
paulson@14430
   804
lemma times_divide_eq_right [simp]: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
   805
by (simp add: divide_inverse mult_assoc)
paulson@14288
   806
paulson@14430
   807
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
   808
by (simp add: divide_inverse mult_ac)
paulson@14288
   809
paulson@14288
   810
lemma divide_divide_eq_right [simp]:
paulson@14288
   811
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
   812
by (simp add: divide_inverse mult_ac)
paulson@14288
   813
paulson@14288
   814
lemma divide_divide_eq_left [simp]:
paulson@14288
   815
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
   816
by (simp add: divide_inverse mult_assoc)
paulson@14288
   817
paulson@14268
   818
paulson@14293
   819
subsection {* Division and Unary Minus *}
paulson@14293
   820
paulson@14293
   821
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
   822
by (simp add: divide_inverse minus_mult_left)
paulson@14293
   823
paulson@14293
   824
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
   825
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
   826
paulson@14293
   827
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
   828
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
   829
paulson@14430
   830
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
paulson@14430
   831
by (simp add: divide_inverse minus_mult_left [symmetric])
paulson@14293
   832
paulson@14293
   833
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14430
   834
by (simp add: divide_inverse minus_mult_right [symmetric])
paulson@14430
   835
paulson@14293
   836
paulson@14293
   837
text{*The effect is to extract signs from divisions*}
paulson@14293
   838
declare minus_divide_left  [symmetric, simp]
paulson@14293
   839
declare minus_divide_right [symmetric, simp]
paulson@14293
   840
paulson@14387
   841
text{*Also, extract signs from products*}
paulson@14387
   842
declare minus_mult_left [symmetric, simp]
paulson@14387
   843
declare minus_mult_right [symmetric, simp]
paulson@14387
   844
paulson@14293
   845
lemma minus_divide_divide [simp]:
paulson@14293
   846
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
paulson@14293
   847
apply (case_tac "b=0", simp) 
paulson@14293
   848
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
   849
done
paulson@14293
   850
paulson@14430
   851
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
   852
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
   853
paulson@14293
   854
paulson@14268
   855
subsection {* Ordered Fields *}
paulson@14268
   856
paulson@14277
   857
lemma positive_imp_inverse_positive: 
paulson@14269
   858
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
paulson@14268
   859
  proof -
paulson@14268
   860
  have "0 < a * inverse a" 
paulson@14268
   861
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
   862
  thus "0 < inverse a" 
paulson@14268
   863
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
paulson@14268
   864
  qed
paulson@14268
   865
paulson@14277
   866
lemma negative_imp_inverse_negative:
paulson@14268
   867
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
paulson@14277
   868
  by (insert positive_imp_inverse_positive [of "-a"], 
paulson@14268
   869
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
paulson@14268
   870
paulson@14268
   871
lemma inverse_le_imp_le:
paulson@14269
   872
      assumes invle: "inverse a \<le> inverse b"
paulson@14269
   873
	  and apos:  "0 < a"
paulson@14269
   874
	 shows "b \<le> (a::'a::ordered_field)"
paulson@14268
   875
  proof (rule classical)
paulson@14268
   876
  assume "~ b \<le> a"
paulson@14268
   877
  hence "a < b"
paulson@14268
   878
    by (simp add: linorder_not_le)
paulson@14268
   879
  hence bpos: "0 < b"
paulson@14268
   880
    by (blast intro: apos order_less_trans)
paulson@14268
   881
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
   882
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
   883
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
   884
    by (simp add: bpos order_less_imp_le mult_right_mono)
paulson@14268
   885
  thus "b \<le> a"
paulson@14268
   886
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
paulson@14268
   887
  qed
paulson@14268
   888
paulson@14277
   889
lemma inverse_positive_imp_positive:
paulson@14277
   890
      assumes inv_gt_0: "0 < inverse a"
paulson@14277
   891
          and [simp]:   "a \<noteq> 0"
paulson@14277
   892
        shows "0 < (a::'a::ordered_field)"
paulson@14277
   893
  proof -
paulson@14277
   894
  have "0 < inverse (inverse a)"
paulson@14277
   895
    by (rule positive_imp_inverse_positive)
paulson@14277
   896
  thus "0 < a"
paulson@14277
   897
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
   898
  qed
paulson@14277
   899
paulson@14277
   900
lemma inverse_positive_iff_positive [simp]:
paulson@14277
   901
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
   902
apply (case_tac "a = 0", simp)
paulson@14277
   903
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
   904
done
paulson@14277
   905
paulson@14277
   906
lemma inverse_negative_imp_negative:
paulson@14277
   907
      assumes inv_less_0: "inverse a < 0"
paulson@14277
   908
          and [simp]:   "a \<noteq> 0"
paulson@14277
   909
        shows "a < (0::'a::ordered_field)"
paulson@14277
   910
  proof -
paulson@14277
   911
  have "inverse (inverse a) < 0"
paulson@14277
   912
    by (rule negative_imp_inverse_negative)
paulson@14277
   913
  thus "a < 0"
paulson@14277
   914
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
   915
  qed
paulson@14277
   916
paulson@14277
   917
lemma inverse_negative_iff_negative [simp]:
paulson@14277
   918
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
   919
apply (case_tac "a = 0", simp)
paulson@14277
   920
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
   921
done
paulson@14277
   922
paulson@14277
   923
lemma inverse_nonnegative_iff_nonnegative [simp]:
paulson@14277
   924
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
   925
by (simp add: linorder_not_less [symmetric])
paulson@14277
   926
paulson@14277
   927
lemma inverse_nonpositive_iff_nonpositive [simp]:
paulson@14277
   928
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
   929
by (simp add: linorder_not_less [symmetric])
paulson@14277
   930
paulson@14277
   931
paulson@14277
   932
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
   933
paulson@14268
   934
lemma less_imp_inverse_less:
paulson@14269
   935
      assumes less: "a < b"
paulson@14269
   936
	  and apos:  "0 < a"
paulson@14269
   937
	shows "inverse b < inverse (a::'a::ordered_field)"
paulson@14268
   938
  proof (rule ccontr)
paulson@14268
   939
  assume "~ inverse b < inverse a"
paulson@14268
   940
  hence "inverse a \<le> inverse b"
paulson@14268
   941
    by (simp add: linorder_not_less)
paulson@14268
   942
  hence "~ (a < b)"
paulson@14268
   943
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
   944
  thus False
paulson@14268
   945
    by (rule notE [OF _ less])
paulson@14268
   946
  qed
paulson@14268
   947
paulson@14268
   948
lemma inverse_less_imp_less:
paulson@14268
   949
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
   950
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
   951
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
   952
done
paulson@14268
   953
paulson@14268
   954
text{*Both premises are essential. Consider -1 and 1.*}
paulson@14268
   955
lemma inverse_less_iff_less [simp]:
paulson@14268
   956
     "[|0 < a; 0 < b|] 
paulson@14268
   957
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
   958
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
   959
paulson@14268
   960
lemma le_imp_inverse_le:
paulson@14268
   961
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
   962
  by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
   963
paulson@14268
   964
lemma inverse_le_iff_le [simp]:
paulson@14268
   965
     "[|0 < a; 0 < b|] 
paulson@14268
   966
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
   967
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
   968
paulson@14268
   969
paulson@14268
   970
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
   971
case is trivial, since inverse preserves signs.*}
paulson@14268
   972
lemma inverse_le_imp_le_neg:
paulson@14268
   973
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
paulson@14268
   974
  apply (rule classical) 
paulson@14268
   975
  apply (subgoal_tac "a < 0") 
paulson@14268
   976
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
paulson@14268
   977
  apply (insert inverse_le_imp_le [of "-b" "-a"])
paulson@14268
   978
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
   979
  done
paulson@14268
   980
paulson@14268
   981
lemma less_imp_inverse_less_neg:
paulson@14268
   982
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
paulson@14268
   983
  apply (subgoal_tac "a < 0") 
paulson@14268
   984
   prefer 2 apply (blast intro: order_less_trans) 
paulson@14268
   985
  apply (insert less_imp_inverse_less [of "-b" "-a"])
paulson@14268
   986
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
   987
  done
paulson@14268
   988
paulson@14268
   989
lemma inverse_less_imp_less_neg:
paulson@14268
   990
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
paulson@14268
   991
  apply (rule classical) 
paulson@14268
   992
  apply (subgoal_tac "a < 0") 
paulson@14268
   993
   prefer 2
paulson@14268
   994
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
paulson@14268
   995
  apply (insert inverse_less_imp_less [of "-b" "-a"])
paulson@14268
   996
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
   997
  done
paulson@14268
   998
paulson@14268
   999
lemma inverse_less_iff_less_neg [simp]:
paulson@14268
  1000
     "[|a < 0; b < 0|] 
paulson@14268
  1001
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1002
  apply (insert inverse_less_iff_less [of "-b" "-a"])
paulson@14268
  1003
  apply (simp del: inverse_less_iff_less 
paulson@14268
  1004
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1005
  done
paulson@14268
  1006
paulson@14268
  1007
lemma le_imp_inverse_le_neg:
paulson@14268
  1008
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
  1009
  by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1010
paulson@14268
  1011
lemma inverse_le_iff_le_neg [simp]:
paulson@14268
  1012
     "[|a < 0; b < 0|] 
paulson@14268
  1013
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1014
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1015
paulson@14277
  1016
paulson@14365
  1017
subsection{*Inverses and the Number One*}
paulson@14365
  1018
paulson@14365
  1019
lemma one_less_inverse_iff:
paulson@14365
  1020
    "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
paulson@14365
  1021
  assume "0 < x"
paulson@14365
  1022
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1023
    show ?thesis by simp
paulson@14365
  1024
next
paulson@14365
  1025
  assume notless: "~ (0 < x)"
paulson@14365
  1026
  have "~ (1 < inverse x)"
paulson@14365
  1027
  proof
paulson@14365
  1028
    assume "1 < inverse x"
paulson@14365
  1029
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1030
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1031
    finally show False by auto
paulson@14365
  1032
  qed
paulson@14365
  1033
  with notless show ?thesis by simp
paulson@14365
  1034
qed
paulson@14365
  1035
paulson@14365
  1036
lemma inverse_eq_1_iff [simp]:
paulson@14365
  1037
    "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1038
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1039
paulson@14365
  1040
lemma one_le_inverse_iff:
paulson@14365
  1041
   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1042
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1043
                    eq_commute [of 1]) 
paulson@14365
  1044
paulson@14365
  1045
lemma inverse_less_1_iff:
paulson@14365
  1046
   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1047
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1048
paulson@14365
  1049
lemma inverse_le_1_iff:
paulson@14365
  1050
   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1051
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1052
paulson@14365
  1053
paulson@14277
  1054
subsection{*Division and Signs*}
paulson@14277
  1055
paulson@14277
  1056
lemma zero_less_divide_iff:
paulson@14277
  1057
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
paulson@14430
  1058
by (simp add: divide_inverse zero_less_mult_iff)
paulson@14277
  1059
paulson@14277
  1060
lemma divide_less_0_iff:
paulson@14277
  1061
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
paulson@14277
  1062
      (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14430
  1063
by (simp add: divide_inverse mult_less_0_iff)
paulson@14277
  1064
paulson@14277
  1065
lemma zero_le_divide_iff:
paulson@14277
  1066
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
paulson@14277
  1067
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14430
  1068
by (simp add: divide_inverse zero_le_mult_iff)
paulson@14277
  1069
paulson@14277
  1070
lemma divide_le_0_iff:
paulson@14288
  1071
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
paulson@14288
  1072
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14430
  1073
by (simp add: divide_inverse mult_le_0_iff)
paulson@14277
  1074
paulson@14277
  1075
lemma divide_eq_0_iff [simp]:
paulson@14277
  1076
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
paulson@14430
  1077
by (simp add: divide_inverse field_mult_eq_0_iff)
paulson@14277
  1078
paulson@14288
  1079
paulson@14288
  1080
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1081
paulson@14288
  1082
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1083
proof -
paulson@14288
  1084
  assume less: "0<c"
paulson@14288
  1085
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1086
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1087
  also have "... = (a*c \<le> b)"
paulson@14288
  1088
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1089
  finally show ?thesis .
paulson@14288
  1090
qed
paulson@14288
  1091
paulson@14288
  1092
paulson@14288
  1093
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1094
proof -
paulson@14288
  1095
  assume less: "c<0"
paulson@14288
  1096
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1097
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1098
  also have "... = (b \<le> a*c)"
paulson@14288
  1099
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1100
  finally show ?thesis .
paulson@14288
  1101
qed
paulson@14288
  1102
paulson@14288
  1103
lemma le_divide_eq:
paulson@14288
  1104
  "(a \<le> b/c) = 
paulson@14288
  1105
   (if 0 < c then a*c \<le> b
paulson@14288
  1106
             else if c < 0 then b \<le> a*c
paulson@14288
  1107
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1108
apply (case_tac "c=0", simp) 
paulson@14288
  1109
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1110
done
paulson@14288
  1111
paulson@14288
  1112
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1113
proof -
paulson@14288
  1114
  assume less: "0<c"
paulson@14288
  1115
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1116
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1117
  also have "... = (b \<le> a*c)"
paulson@14288
  1118
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1119
  finally show ?thesis .
paulson@14288
  1120
qed
paulson@14288
  1121
paulson@14288
  1122
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1123
proof -
paulson@14288
  1124
  assume less: "c<0"
paulson@14288
  1125
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1126
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1127
  also have "... = (a*c \<le> b)"
paulson@14288
  1128
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1129
  finally show ?thesis .
paulson@14288
  1130
qed
paulson@14288
  1131
paulson@14288
  1132
lemma divide_le_eq:
paulson@14288
  1133
  "(b/c \<le> a) = 
paulson@14288
  1134
   (if 0 < c then b \<le> a*c
paulson@14288
  1135
             else if c < 0 then a*c \<le> b
paulson@14288
  1136
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1137
apply (case_tac "c=0", simp) 
paulson@14288
  1138
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1139
done
paulson@14288
  1140
paulson@14288
  1141
paulson@14288
  1142
lemma pos_less_divide_eq:
paulson@14288
  1143
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1144
proof -
paulson@14288
  1145
  assume less: "0<c"
paulson@14288
  1146
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@14288
  1147
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1148
  also have "... = (a*c < b)"
paulson@14288
  1149
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1150
  finally show ?thesis .
paulson@14288
  1151
qed
paulson@14288
  1152
paulson@14288
  1153
lemma neg_less_divide_eq:
paulson@14288
  1154
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1155
proof -
paulson@14288
  1156
  assume less: "c<0"
paulson@14288
  1157
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@14288
  1158
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1159
  also have "... = (b < a*c)"
paulson@14288
  1160
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1161
  finally show ?thesis .
paulson@14288
  1162
qed
paulson@14288
  1163
paulson@14288
  1164
lemma less_divide_eq:
paulson@14288
  1165
  "(a < b/c) = 
paulson@14288
  1166
   (if 0 < c then a*c < b
paulson@14288
  1167
             else if c < 0 then b < a*c
paulson@14288
  1168
             else  a < (0::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1169
apply (case_tac "c=0", simp) 
paulson@14288
  1170
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1171
done
paulson@14288
  1172
paulson@14288
  1173
lemma pos_divide_less_eq:
paulson@14288
  1174
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1175
proof -
paulson@14288
  1176
  assume less: "0<c"
paulson@14288
  1177
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@14288
  1178
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1179
  also have "... = (b < a*c)"
paulson@14288
  1180
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1181
  finally show ?thesis .
paulson@14288
  1182
qed
paulson@14288
  1183
paulson@14288
  1184
lemma neg_divide_less_eq:
paulson@14288
  1185
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1186
proof -
paulson@14288
  1187
  assume less: "c<0"
paulson@14288
  1188
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@14288
  1189
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1190
  also have "... = (a*c < b)"
paulson@14288
  1191
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1192
  finally show ?thesis .
paulson@14288
  1193
qed
paulson@14288
  1194
paulson@14288
  1195
lemma divide_less_eq:
paulson@14288
  1196
  "(b/c < a) = 
paulson@14288
  1197
   (if 0 < c then b < a*c
paulson@14288
  1198
             else if c < 0 then a*c < b
paulson@14288
  1199
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1200
apply (case_tac "c=0", simp) 
paulson@14288
  1201
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1202
done
paulson@14288
  1203
paulson@14288
  1204
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
paulson@14288
  1205
proof -
paulson@14288
  1206
  assume [simp]: "c\<noteq>0"
paulson@14288
  1207
  have "(a = b/c) = (a*c = (b/c)*c)"
paulson@14288
  1208
    by (simp add: field_mult_cancel_right)
paulson@14288
  1209
  also have "... = (a*c = b)"
paulson@14288
  1210
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1211
  finally show ?thesis .
paulson@14288
  1212
qed
paulson@14288
  1213
paulson@14288
  1214
lemma eq_divide_eq:
paulson@14288
  1215
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
paulson@14288
  1216
by (simp add: nonzero_eq_divide_eq) 
paulson@14288
  1217
paulson@14288
  1218
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
paulson@14288
  1219
proof -
paulson@14288
  1220
  assume [simp]: "c\<noteq>0"
paulson@14288
  1221
  have "(b/c = a) = ((b/c)*c = a*c)"
paulson@14288
  1222
    by (simp add: field_mult_cancel_right)
paulson@14288
  1223
  also have "... = (b = a*c)"
paulson@14288
  1224
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1225
  finally show ?thesis .
paulson@14288
  1226
qed
paulson@14288
  1227
paulson@14288
  1228
lemma divide_eq_eq:
paulson@14288
  1229
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
paulson@14288
  1230
by (force simp add: nonzero_divide_eq_eq) 
paulson@14288
  1231
paulson@14288
  1232
subsection{*Cancellation Laws for Division*}
paulson@14288
  1233
paulson@14288
  1234
lemma divide_cancel_right [simp]:
paulson@14288
  1235
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
paulson@14288
  1236
apply (case_tac "c=0", simp) 
paulson@14430
  1237
apply (simp add: divide_inverse field_mult_cancel_right) 
paulson@14288
  1238
done
paulson@14288
  1239
paulson@14288
  1240
lemma divide_cancel_left [simp]:
paulson@14288
  1241
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
paulson@14288
  1242
apply (case_tac "c=0", simp) 
paulson@14430
  1243
apply (simp add: divide_inverse field_mult_cancel_left) 
paulson@14288
  1244
done
paulson@14288
  1245
paulson@14353
  1246
subsection {* Division and the Number One *}
paulson@14353
  1247
paulson@14353
  1248
text{*Simplify expressions equated with 1*}
paulson@14353
  1249
lemma divide_eq_1_iff [simp]:
paulson@14353
  1250
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1251
apply (case_tac "b=0", simp) 
paulson@14353
  1252
apply (simp add: right_inverse_eq) 
paulson@14353
  1253
done
paulson@14353
  1254
paulson@14353
  1255
paulson@14353
  1256
lemma one_eq_divide_iff [simp]:
paulson@14353
  1257
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1258
by (simp add: eq_commute [of 1])  
paulson@14353
  1259
paulson@14353
  1260
lemma zero_eq_1_divide_iff [simp]:
paulson@14353
  1261
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
paulson@14353
  1262
apply (case_tac "a=0", simp) 
paulson@14353
  1263
apply (auto simp add: nonzero_eq_divide_eq) 
paulson@14353
  1264
done
paulson@14353
  1265
paulson@14353
  1266
lemma one_divide_eq_0_iff [simp]:
paulson@14353
  1267
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
paulson@14353
  1268
apply (case_tac "a=0", simp) 
paulson@14353
  1269
apply (insert zero_neq_one [THEN not_sym]) 
paulson@14353
  1270
apply (auto simp add: nonzero_divide_eq_eq) 
paulson@14353
  1271
done
paulson@14353
  1272
paulson@14353
  1273
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@14353
  1274
declare zero_less_divide_iff [of "1", simp]
paulson@14353
  1275
declare divide_less_0_iff [of "1", simp]
paulson@14353
  1276
declare zero_le_divide_iff [of "1", simp]
paulson@14353
  1277
declare divide_le_0_iff [of "1", simp]
paulson@14353
  1278
paulson@14288
  1279
paulson@14293
  1280
subsection {* Ordering Rules for Division *}
paulson@14293
  1281
paulson@14293
  1282
lemma divide_strict_right_mono:
paulson@14293
  1283
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1284
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
paulson@14293
  1285
              positive_imp_inverse_positive) 
paulson@14293
  1286
paulson@14293
  1287
lemma divide_right_mono:
paulson@14293
  1288
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
paulson@14293
  1289
  by (force simp add: divide_strict_right_mono order_le_less) 
paulson@14293
  1290
paulson@14293
  1291
paulson@14293
  1292
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1293
      have the same sign*}
paulson@14293
  1294
lemma divide_strict_left_mono:
paulson@14293
  1295
       "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1296
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono 
paulson@14293
  1297
      order_less_imp_not_eq order_less_imp_not_eq2  
paulson@14293
  1298
      less_imp_inverse_less less_imp_inverse_less_neg) 
paulson@14293
  1299
paulson@14293
  1300
lemma divide_left_mono:
paulson@14293
  1301
     "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
paulson@14293
  1302
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1303
   prefer 2 
paulson@14293
  1304
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
paulson@14293
  1305
  apply (case_tac "c=0", simp add: divide_inverse)
paulson@14293
  1306
  apply (force simp add: divide_strict_left_mono order_le_less) 
paulson@14293
  1307
  done
paulson@14293
  1308
paulson@14293
  1309
lemma divide_strict_left_mono_neg:
paulson@14293
  1310
     "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1311
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1312
   prefer 2 
paulson@14293
  1313
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
paulson@14293
  1314
  apply (drule divide_strict_left_mono [of _ _ "-c"]) 
paulson@14293
  1315
   apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) 
paulson@14293
  1316
  done
paulson@14293
  1317
paulson@14293
  1318
lemma divide_strict_right_mono_neg:
paulson@14293
  1319
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1320
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) 
paulson@14293
  1321
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
paulson@14293
  1322
done
paulson@14293
  1323
paulson@14293
  1324
paulson@14293
  1325
subsection {* Ordered Fields are Dense *}
paulson@14293
  1326
obua@14738
  1327
lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
paulson@14293
  1328
proof -
obua@14738
  1329
  have "a+0 < (a+1::'a::ordered_semidom)"
paulson@14365
  1330
    by (blast intro: zero_less_one add_strict_left_mono) 
paulson@14293
  1331
  thus ?thesis by simp
paulson@14293
  1332
qed
paulson@14293
  1333
obua@14738
  1334
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
paulson@14365
  1335
  by (blast intro: order_less_trans zero_less_one less_add_one) 
paulson@14365
  1336
paulson@14293
  1337
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
paulson@14293
  1338
by (simp add: zero_less_two pos_less_divide_eq right_distrib) 
paulson@14293
  1339
paulson@14293
  1340
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
paulson@14293
  1341
by (simp add: zero_less_two pos_divide_less_eq right_distrib) 
paulson@14293
  1342
paulson@14293
  1343
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
paulson@14293
  1344
by (blast intro!: less_half_sum gt_half_sum)
paulson@14293
  1345
paulson@14293
  1346
subsection {* Absolute Value *}
paulson@14293
  1347
obua@14738
  1348
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
paulson@14294
  1349
  by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) 
paulson@14294
  1350
obua@14738
  1351
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  1352
proof -
obua@14738
  1353
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  1354
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  1355
  have a: "(abs a) * (abs b) = ?x"
obua@14738
  1356
    by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps)
obua@14738
  1357
  {
obua@14738
  1358
    fix u v :: 'a
obua@14738
  1359
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> u * v = ?y"
obua@14738
  1360
      apply (subst prts[of u], subst prts[of v])
obua@14738
  1361
      apply (simp add: left_distrib right_distrib add_ac) 
obua@14738
  1362
      done
obua@14738
  1363
  }
obua@14738
  1364
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  1365
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  1366
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  1367
  have xy: "- ?x <= ?y"
obua@14738
  1368
    apply (simp add: compare_rls)
obua@14738
  1369
    apply (rule add_le_imp_le_left[of "-(pprt a * nprt b + nprt a * pprt b)"])
obua@14738
  1370
    apply (simp add: add_ac)
obua@14738
  1371
    proof -
obua@14738
  1372
      let ?r = "nprt a * nprt b +(nprt a * nprt b + (nprt a * pprt b + (pprt a * nprt b + (pprt a * pprt b + (pprt a * pprt b +
obua@14738
  1373
	(- (nprt a * pprt b) + - (pprt a * nprt b)))))))"
obua@14738
  1374
      let ?rr = "nprt a * nprt b + nprt a * nprt b + ((nprt a * pprt b) + (- (nprt a * pprt b))) + ((pprt a * nprt b) + - (pprt a * nprt b))
obua@14738
  1375
	+ pprt a * pprt b + pprt a * pprt b"
obua@14738
  1376
      have a:"?r = ?rr" by (simp only: add_ac)      
obua@14738
  1377
      have "0 <= ?rr"
obua@14738
  1378
	apply (simp)
obua@14738
  1379
	apply (rule addm)+
obua@14738
  1380
	apply (simp_all add: mult_neg_le mult_pos_le)
obua@14738
  1381
	done
obua@14738
  1382
      with a show "0 <= ?r" by simp
obua@14738
  1383
    qed
obua@14738
  1384
  have yx: "?y <= ?x"
obua@14738
  1385
    apply (simp add: add_ac)
obua@14738
  1386
    apply (simp add: compare_rls)
obua@14738
  1387
    apply (rule add_le_imp_le_right[of _ "-(pprt a * pprt b)"])
obua@14738
  1388
    apply (simp add: add_ac)
obua@14738
  1389
    apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)+
obua@14738
  1390
    done
obua@14738
  1391
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  1392
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  1393
  show ?thesis
obua@14738
  1394
    apply (rule abs_leI)
obua@14738
  1395
    apply (simp add: i1)
obua@14738
  1396
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  1397
    done
obua@14738
  1398
qed
paulson@14294
  1399
obua@14738
  1400
lemma abs_eq_mult: 
obua@14738
  1401
  assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
obua@14738
  1402
  shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
obua@14738
  1403
proof -
obua@14738
  1404
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  1405
    apply (auto)    
obua@14738
  1406
    apply (rule_tac split_mult_pos_le)
obua@14738
  1407
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  1408
    apply (simp)
obua@14738
  1409
    apply (rule_tac split_mult_neg_le)
obua@14738
  1410
    apply (insert prems)
obua@14738
  1411
    apply (blast)
obua@14738
  1412
    done
obua@14738
  1413
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  1414
    by (simp add: prts[symmetric])
obua@14738
  1415
  show ?thesis
obua@14738
  1416
  proof cases
obua@14738
  1417
    assume "0 <= a * b"
obua@14738
  1418
    then show ?thesis
obua@14738
  1419
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1420
      apply (insert prems)
obua@14738
  1421
      apply (auto simp add: ring_eq_simps iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
obua@14738
  1422
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id] order_antisym mult_pos_neg_le[of a b] mult_pos_neg2_le[of b a])
obua@14738
  1423
      done
obua@14738
  1424
  next
obua@14738
  1425
    assume "~(0 <= a*b)"
obua@14738
  1426
    with s have "a*b <= 0" by simp
obua@14738
  1427
    then show ?thesis
obua@14738
  1428
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1429
      apply (insert prems)
obua@14738
  1430
      apply (auto simp add: ring_eq_simps iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
obua@14738
  1431
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id] order_antisym mult_pos_le[of a b] mult_neg_le[of a b])
obua@14738
  1432
      done
obua@14738
  1433
  qed
obua@14738
  1434
qed
paulson@14294
  1435
obua@14738
  1436
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
obua@14738
  1437
by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  1438
obua@14738
  1439
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
obua@14738
  1440
by (simp add: abs_if) 
paulson@14294
  1441
paulson@14294
  1442
lemma nonzero_abs_inverse:
paulson@14294
  1443
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  1444
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  1445
                      negative_imp_inverse_negative)
paulson@14294
  1446
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  1447
done
paulson@14294
  1448
paulson@14294
  1449
lemma abs_inverse [simp]:
paulson@14294
  1450
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  1451
      inverse (abs a)"
paulson@14294
  1452
apply (case_tac "a=0", simp) 
paulson@14294
  1453
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1454
done
paulson@14294
  1455
paulson@14294
  1456
lemma nonzero_abs_divide:
paulson@14294
  1457
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  1458
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  1459
paulson@14294
  1460
lemma abs_divide:
paulson@14294
  1461
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
paulson@14294
  1462
apply (case_tac "b=0", simp) 
paulson@14294
  1463
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1464
done
paulson@14294
  1465
paulson@14294
  1466
lemma abs_mult_less:
obua@14738
  1467
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  1468
proof -
paulson@14294
  1469
  assume ac: "abs a < c"
paulson@14294
  1470
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  1471
  assume "abs b < d"
paulson@14294
  1472
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1473
qed
paulson@14293
  1474
obua@14738
  1475
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
obua@14738
  1476
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
obua@14738
  1477
obua@14738
  1478
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
obua@14738
  1479
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
obua@14738
  1480
obua@14738
  1481
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  1482
apply (simp add: order_less_le abs_le_iff)  
obua@14738
  1483
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
obua@14738
  1484
apply (simp add: le_minus_self_iff linorder_neq_iff) 
obua@14738
  1485
done
obua@14738
  1486
paulson@14430
  1487
text{*Moving this up spoils many proofs using @{text mult_le_cancel_right}*}
paulson@14430
  1488
declare times_divide_eq_left [simp]
paulson@14430
  1489
obua@14738
  1490
ML {*
paulson@14334
  1491
val left_distrib = thm "left_distrib";
obua@14738
  1492
val right_distrib = thm "right_distrib";
obua@14738
  1493
val mult_commute = thm "mult_commute";
obua@14738
  1494
val distrib = thm "distrib";
obua@14738
  1495
val zero_neq_one = thm "zero_neq_one";
obua@14738
  1496
val no_zero_divisors = thm "no_zero_divisors";
paulson@14331
  1497
val left_inverse = thm "left_inverse";
obua@14738
  1498
val divide_inverse = thm "divide_inverse";
obua@14738
  1499
val mult_zero_left = thm "mult_zero_left";
obua@14738
  1500
val mult_zero_right = thm "mult_zero_right";
obua@14738
  1501
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";
obua@14738
  1502
val inverse_zero = thm "inverse_zero";
obua@14738
  1503
val ring_distrib = thms "ring_distrib";
obua@14738
  1504
val combine_common_factor = thm "combine_common_factor";
obua@14738
  1505
val minus_mult_left = thm "minus_mult_left";
obua@14738
  1506
val minus_mult_right = thm "minus_mult_right";
obua@14738
  1507
val minus_mult_minus = thm "minus_mult_minus";
obua@14738
  1508
val minus_mult_commute = thm "minus_mult_commute";
obua@14738
  1509
val right_diff_distrib = thm "right_diff_distrib";
obua@14738
  1510
val left_diff_distrib = thm "left_diff_distrib";
obua@14738
  1511
val mult_left_mono = thm "mult_left_mono";
obua@14738
  1512
val mult_right_mono = thm "mult_right_mono";
obua@14738
  1513
val mult_strict_left_mono = thm "mult_strict_left_mono";
obua@14738
  1514
val mult_strict_right_mono = thm "mult_strict_right_mono";
obua@14738
  1515
val mult_mono = thm "mult_mono";
obua@14738
  1516
val mult_strict_mono = thm "mult_strict_mono";
obua@14738
  1517
val abs_if = thm "abs_if";
obua@14738
  1518
val zero_less_one = thm "zero_less_one";
obua@14738
  1519
val eq_add_iff1 = thm "eq_add_iff1";
obua@14738
  1520
val eq_add_iff2 = thm "eq_add_iff2";
obua@14738
  1521
val less_add_iff1 = thm "less_add_iff1";
obua@14738
  1522
val less_add_iff2 = thm "less_add_iff2";
obua@14738
  1523
val le_add_iff1 = thm "le_add_iff1";
obua@14738
  1524
val le_add_iff2 = thm "le_add_iff2";
obua@14738
  1525
val mult_left_le_imp_le = thm "mult_left_le_imp_le";
obua@14738
  1526
val mult_right_le_imp_le = thm "mult_right_le_imp_le";
obua@14738
  1527
val mult_left_less_imp_less = thm "mult_left_less_imp_less";
obua@14738
  1528
val mult_right_less_imp_less = thm "mult_right_less_imp_less";
obua@14738
  1529
val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg";
obua@14738
  1530
val mult_left_mono_neg = thm "mult_left_mono_neg";
obua@14738
  1531
val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg";
obua@14738
  1532
val mult_right_mono_neg = thm "mult_right_mono_neg";
obua@14738
  1533
val mult_pos = thm "mult_pos";
obua@14738
  1534
val mult_pos_le = thm "mult_pos_le";
obua@14738
  1535
val mult_pos_neg = thm "mult_pos_neg";
obua@14738
  1536
val mult_pos_neg_le = thm "mult_pos_neg_le";
obua@14738
  1537
val mult_pos_neg2 = thm "mult_pos_neg2";
obua@14738
  1538
val mult_pos_neg2_le = thm "mult_pos_neg2_le";
obua@14738
  1539
val mult_neg = thm "mult_neg";
obua@14738
  1540
val mult_neg_le = thm "mult_neg_le";
obua@14738
  1541
val zero_less_mult_pos = thm "zero_less_mult_pos";
obua@14738
  1542
val zero_less_mult_pos2 = thm "zero_less_mult_pos2";
obua@14738
  1543
val zero_less_mult_iff = thm "zero_less_mult_iff";
obua@14738
  1544
val mult_eq_0_iff = thm "mult_eq_0_iff";
obua@14738
  1545
val zero_le_mult_iff = thm "zero_le_mult_iff";
obua@14738
  1546
val mult_less_0_iff = thm "mult_less_0_iff";
obua@14738
  1547
val mult_le_0_iff = thm "mult_le_0_iff";
obua@14738
  1548
val split_mult_pos_le = thm "split_mult_pos_le";
obua@14738
  1549
val split_mult_neg_le = thm "split_mult_neg_le";
obua@14738
  1550
val zero_le_square = thm "zero_le_square";
obua@14738
  1551
val zero_le_one = thm "zero_le_one";
obua@14738
  1552
val not_one_le_zero = thm "not_one_le_zero";
obua@14738
  1553
val not_one_less_zero = thm "not_one_less_zero";
obua@14738
  1554
val mult_left_mono_neg = thm "mult_left_mono_neg";
obua@14738
  1555
val mult_right_mono_neg = thm "mult_right_mono_neg";
obua@14738
  1556
val mult_strict_mono = thm "mult_strict_mono";
obua@14738
  1557
val mult_strict_mono' = thm "mult_strict_mono'";
obua@14738
  1558
val mult_mono = thm "mult_mono";
obua@14738
  1559
val less_1_mult = thm "less_1_mult";
obua@14738
  1560
val mult_less_cancel_right = thm "mult_less_cancel_right";
obua@14738
  1561
val mult_less_cancel_left = thm "mult_less_cancel_left";
obua@14738
  1562
val mult_le_cancel_right = thm "mult_le_cancel_right";
obua@14738
  1563
val mult_le_cancel_left = thm "mult_le_cancel_left";
obua@14738
  1564
val mult_less_imp_less_left = thm "mult_less_imp_less_left";
obua@14738
  1565
val mult_less_imp_less_right = thm "mult_less_imp_less_right";
obua@14738
  1566
val mult_cancel_right = thm "mult_cancel_right";
obua@14738
  1567
val mult_cancel_left = thm "mult_cancel_left";
obua@14738
  1568
val ring_eq_simps = thms "ring_eq_simps";
obua@14738
  1569
val right_inverse = thm "right_inverse";
obua@14738
  1570
val right_inverse_eq = thm "right_inverse_eq";
obua@14738
  1571
val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide";
obua@14738
  1572
val divide_self = thm "divide_self";
obua@14738
  1573
val divide_zero = thm "divide_zero";
obua@14738
  1574
val divide_zero_left = thm "divide_zero_left";
obua@14738
  1575
val inverse_eq_divide = thm "inverse_eq_divide";
obua@14738
  1576
val add_divide_distrib = thm "add_divide_distrib";
obua@14738
  1577
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";
obua@14738
  1578
val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma";
obua@14738
  1579
val field_mult_cancel_right = thm "field_mult_cancel_right";
obua@14738
  1580
val field_mult_cancel_left = thm "field_mult_cancel_left";
obua@14738
  1581
val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero";
obua@14738
  1582
val inverse_zero_imp_zero = thm "inverse_zero_imp_zero";
obua@14738
  1583
val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero";
obua@14738
  1584
val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero";
obua@14738
  1585
val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq";
obua@14738
  1586
val inverse_minus_eq = thm "inverse_minus_eq";
obua@14738
  1587
val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq";
obua@14738
  1588
val inverse_eq_imp_eq = thm "inverse_eq_imp_eq";
obua@14738
  1589
val inverse_eq_iff_eq = thm "inverse_eq_iff_eq";
obua@14738
  1590
val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq";
obua@14738
  1591
val inverse_inverse_eq = thm "inverse_inverse_eq";
obua@14738
  1592
val inverse_1 = thm "inverse_1";
obua@14738
  1593
val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib";
obua@14738
  1594
val inverse_mult_distrib = thm "inverse_mult_distrib";
obua@14738
  1595
val inverse_add = thm "inverse_add";
obua@14738
  1596
val inverse_divide = thm "inverse_divide";
obua@14738
  1597
val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left";
obua@14738
  1598
val mult_divide_cancel_left = thm "mult_divide_cancel_left";
obua@14738
  1599
val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right";
obua@14738
  1600
val mult_divide_cancel_right = thm "mult_divide_cancel_right";
obua@14738
  1601
val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if";
obua@14738
  1602
val divide_1 = thm "divide_1";
obua@14738
  1603
val times_divide_eq_right = thm "times_divide_eq_right";
obua@14738
  1604
val times_divide_eq_left = thm "times_divide_eq_left";
obua@14738
  1605
val divide_divide_eq_right = thm "divide_divide_eq_right";
obua@14738
  1606
val divide_divide_eq_left = thm "divide_divide_eq_left";
obua@14738
  1607
val nonzero_minus_divide_left = thm "nonzero_minus_divide_left";
obua@14738
  1608
val nonzero_minus_divide_right = thm "nonzero_minus_divide_right";
obua@14738
  1609
val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide";
obua@14738
  1610
val minus_divide_left = thm "minus_divide_left";
obua@14738
  1611
val minus_divide_right = thm "minus_divide_right";
obua@14738
  1612
val minus_divide_divide = thm "minus_divide_divide";
obua@14738
  1613
val diff_divide_distrib = thm "diff_divide_distrib";
obua@14738
  1614
val positive_imp_inverse_positive = thm "positive_imp_inverse_positive";
obua@14738
  1615
val negative_imp_inverse_negative = thm "negative_imp_inverse_negative";
obua@14738
  1616
val inverse_le_imp_le = thm "inverse_le_imp_le";
obua@14738
  1617
val inverse_positive_imp_positive = thm "inverse_positive_imp_positive";
obua@14738
  1618
val inverse_positive_iff_positive = thm "inverse_positive_iff_positive";
obua@14738
  1619
val inverse_negative_imp_negative = thm "inverse_negative_imp_negative";
obua@14738
  1620
val inverse_negative_iff_negative = thm "inverse_negative_iff_negative";
obua@14738
  1621
val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative";
obua@14738
  1622
val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive";
obua@14738
  1623
val less_imp_inverse_less = thm "less_imp_inverse_less";
obua@14738
  1624
val inverse_less_imp_less = thm "inverse_less_imp_less";
obua@14738
  1625
val inverse_less_iff_less = thm "inverse_less_iff_less";
obua@14738
  1626
val le_imp_inverse_le = thm "le_imp_inverse_le";
obua@14738
  1627
val inverse_le_iff_le = thm "inverse_le_iff_le";
obua@14738
  1628
val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg";
obua@14738
  1629
val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg";
obua@14738
  1630
val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg";
obua@14738
  1631
val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg";
obua@14738
  1632
val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg";
obua@14738
  1633
val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg";
obua@14738
  1634
val one_less_inverse_iff = thm "one_less_inverse_iff";
obua@14738
  1635
val inverse_eq_1_iff = thm "inverse_eq_1_iff";
obua@14738
  1636
val one_le_inverse_iff = thm "one_le_inverse_iff";
obua@14738
  1637
val inverse_less_1_iff = thm "inverse_less_1_iff";
obua@14738
  1638
val inverse_le_1_iff = thm "inverse_le_1_iff";
obua@14738
  1639
val zero_less_divide_iff = thm "zero_less_divide_iff";
obua@14738
  1640
val divide_less_0_iff = thm "divide_less_0_iff";
obua@14738
  1641
val zero_le_divide_iff = thm "zero_le_divide_iff";
obua@14738
  1642
val divide_le_0_iff = thm "divide_le_0_iff";
obua@14738
  1643
val divide_eq_0_iff = thm "divide_eq_0_iff";
obua@14738
  1644
val pos_le_divide_eq = thm "pos_le_divide_eq";
obua@14738
  1645
val neg_le_divide_eq = thm "neg_le_divide_eq";
obua@14738
  1646
val le_divide_eq = thm "le_divide_eq";
obua@14738
  1647
val pos_divide_le_eq = thm "pos_divide_le_eq";
obua@14738
  1648
val neg_divide_le_eq = thm "neg_divide_le_eq";
obua@14738
  1649
val divide_le_eq = thm "divide_le_eq";
obua@14738
  1650
val pos_less_divide_eq = thm "pos_less_divide_eq";
obua@14738
  1651
val neg_less_divide_eq = thm "neg_less_divide_eq";
obua@14738
  1652
val less_divide_eq = thm "less_divide_eq";
obua@14738
  1653
val pos_divide_less_eq = thm "pos_divide_less_eq";
obua@14738
  1654
val neg_divide_less_eq = thm "neg_divide_less_eq";
obua@14738
  1655
val divide_less_eq = thm "divide_less_eq";
obua@14738
  1656
val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq";
obua@14738
  1657
val eq_divide_eq = thm "eq_divide_eq";
obua@14738
  1658
val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq";
obua@14738
  1659
val divide_eq_eq = thm "divide_eq_eq";
obua@14738
  1660
val divide_cancel_right = thm "divide_cancel_right";
obua@14738
  1661
val divide_cancel_left = thm "divide_cancel_left";
obua@14738
  1662
val divide_eq_1_iff = thm "divide_eq_1_iff";
obua@14738
  1663
val one_eq_divide_iff = thm "one_eq_divide_iff";
obua@14738
  1664
val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff";
obua@14738
  1665
val one_divide_eq_0_iff = thm "one_divide_eq_0_iff";
obua@14738
  1666
val divide_strict_right_mono = thm "divide_strict_right_mono";
obua@14738
  1667
val divide_right_mono = thm "divide_right_mono";
obua@14738
  1668
val divide_strict_left_mono = thm "divide_strict_left_mono";
obua@14738
  1669
val divide_left_mono = thm "divide_left_mono";
obua@14738
  1670
val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg";
obua@14738
  1671
val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg";
obua@14738
  1672
val less_add_one = thm "less_add_one";
obua@14738
  1673
val zero_less_two = thm "zero_less_two";
obua@14738
  1674
val less_half_sum = thm "less_half_sum";
obua@14738
  1675
val gt_half_sum = thm "gt_half_sum";
obua@14738
  1676
val dense = thm "dense";
obua@14738
  1677
val abs_one = thm "abs_one";
obua@14738
  1678
val abs_le_mult = thm "abs_le_mult";
obua@14738
  1679
val abs_eq_mult = thm "abs_eq_mult";
obua@14738
  1680
val abs_mult = thm "abs_mult";
obua@14738
  1681
val abs_mult_self = thm "abs_mult_self";
obua@14738
  1682
val nonzero_abs_inverse = thm "nonzero_abs_inverse";
obua@14738
  1683
val abs_inverse = thm "abs_inverse";
obua@14738
  1684
val nonzero_abs_divide = thm "nonzero_abs_divide";
obua@14738
  1685
val abs_divide = thm "abs_divide";
obua@14738
  1686
val abs_mult_less = thm "abs_mult_less";
obua@14738
  1687
val eq_minus_self_iff = thm "eq_minus_self_iff";
obua@14738
  1688
val less_minus_self_iff = thm "less_minus_self_iff";
obua@14738
  1689
val abs_less_iff = thm "abs_less_iff";
paulson@14331
  1690
*}
paulson@14331
  1691
paulson@14265
  1692
end