src/HOL/Ring_and_Field.thy
author obua
Mon Jun 14 14:20:55 2004 +0200 (2004-06-14)
changeset 14940 b9ab8babd8b3
parent 14770 fe9504ba63d5
child 15010 72fbe711e414
permissions -rw-r--r--
Further development of matrix theory
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson and Markus Wenzel
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    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 \url{http://www.mathworld.com} by Eric Weisstein et. al.
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  \item \emph{Algebra I} by van der Waerden, Springer.
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  \end{itemize}
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*}
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axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult
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  left_distrib: "(a + b) * c = a * c + b * c"
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  right_distrib: "a * (b + c) = a * b + a * c"
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axclass semiring_0 \<subseteq> semiring, comm_monoid_add
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axclass semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add
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axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult  
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  mult_commute: "a * b = b * a"
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  distrib: "(a + b) * c = a * c + b * c"
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instance comm_semiring \<subseteq> semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add
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instance comm_semiring_0 \<subseteq> semiring_0 ..
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axclass comm_semiring_0_cancel \<subseteq> comm_semiring_0, cancel_ab_semigroup_add
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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..
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axclass axclass_0_neq_1 \<subseteq> zero, one
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  zero_neq_one [simp]: "0 \<noteq> 1"
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axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult
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axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *)
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instance comm_semiring_1 \<subseteq> semiring_1 ..
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axclass axclass_no_zero_divisors \<subseteq> zero, times
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  no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
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axclass semiring_1_cancel \<subseteq> semiring_1, cancel_ab_semigroup_add
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instance semiring_1_cancel \<subseteq> semiring_0_cancel ..
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axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)
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instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel ..
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel ..
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axclass ring \<subseteq> semiring, ab_group_add
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instance ring \<subseteq> semiring_0_cancel ..
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axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add
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instance comm_ring \<subseteq> ring ..
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instance comm_ring \<subseteq> comm_semiring_0_cancel ..
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axclass ring_1 \<subseteq> ring, semiring_1
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instance ring_1 \<subseteq> semiring_1_cancel ..
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axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *)
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instance comm_ring_1 \<subseteq> ring_1 ..
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instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..
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axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors
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axclass field \<subseteq> comm_ring_1, inverse
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  left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"
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  divide_inverse:      "a / b = a * inverse b"
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lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring_0_cancel)"
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proof -
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  have "0*a + 0*a = 0*a + 0"
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    by (simp add: left_distrib [symmetric])
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  thus ?thesis 
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    by (simp only: add_left_cancel)
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qed
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lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring_0_cancel)"
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proof -
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  have "a*0 + a*0 = a*0 + 0"
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    by (simp add: right_distrib [symmetric])
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  thus ?thesis 
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    by (simp only: add_left_cancel)
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qed
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lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
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proof cases
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  assume "a=0" thus ?thesis by simp
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next
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  assume anz [simp]: "a\<noteq>0"
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  { assume "a * b = 0"
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    hence "inverse a * (a * b) = 0" by simp
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    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
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  thus ?thesis by force
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qed
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instance field \<subseteq> idom
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by (intro_classes, simp)
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axclass division_by_zero \<subseteq> zero, inverse
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  inverse_zero [simp]: "inverse 0 = 0"
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subsection {* Distribution rules *}
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theorems ring_distrib = right_distrib left_distrib
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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     "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
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by (simp add: left_distrib add_ac)
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lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: left_distrib [symmetric]) 
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done
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lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: right_distrib [symmetric]) 
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done
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lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
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by (simp add: right_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
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by (simp add: left_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add 
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  mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
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  mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c"
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axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add
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instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..
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axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add
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  mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
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  mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
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instance ordered_semiring_strict \<subseteq> semiring_0_cancel ..
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instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring
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apply intro_classes
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apply (case_tac "a < b & 0 < c")
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apply (auto simp add: mult_strict_left_mono order_less_le)
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apply (auto simp add: mult_strict_left_mono order_le_less)
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apply (simp add: mult_strict_right_mono)
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done
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axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add
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  mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
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axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add
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instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
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axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add
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  mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
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instance pordered_comm_semiring \<subseteq> pordered_semiring
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by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+)
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instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..
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instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict
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by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+)
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instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
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apply (intro_classes)
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apply (case_tac "a < b & 0 < c")
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apply (auto simp add: mult_strict_left_mono order_less_le)
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apply (auto simp add: mult_strict_left_mono order_le_less)
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done
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axclass pordered_ring \<subseteq> ring, pordered_semiring 
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instance pordered_ring \<subseteq> pordered_ab_group_add ..
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instance pordered_ring \<subseteq> pordered_cancel_semiring ..
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axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs
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instance lordered_ring \<subseteq> lordered_ab_group_meet ..
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instance lordered_ring \<subseteq> lordered_ab_group_join ..
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axclass axclass_abs_if \<subseteq> minus, ord, zero
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  abs_if: "abs a = (if (a < 0) then (-a) else a)"
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axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if
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instance ordered_ring_strict \<subseteq> lordered_ab_group ..
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instance ordered_ring_strict \<subseteq> lordered_ring
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by (intro_classes, simp add: abs_if join_eq_if)
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axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring
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axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *)
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  zero_less_one [simp]: "0 < 1"
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axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *)
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instance ordered_idom \<subseteq> ordered_ring_strict ..
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axclass ordered_field \<subseteq> field, ordered_idom
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lemma eq_add_iff1:
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     "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
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apply (simp add: diff_minus left_distrib)
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric])
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done
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lemma eq_add_iff2:
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     "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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lemma less_add_iff1:
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     "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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lemma less_add_iff2:
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     "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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lemma le_add_iff1:
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     "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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lemma le_add_iff2:
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     "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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subsection {* Ordering Rules for Multiplication *}
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lemma mult_left_le_imp_le:
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     "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
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  by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
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lemma mult_right_le_imp_le:
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     "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
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  by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
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lemma mult_left_less_imp_less:
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     "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
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  by (force simp add: mult_left_mono linorder_not_le [symmetric])
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lemma mult_right_less_imp_less:
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     "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
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  by (force simp add: mult_right_mono linorder_not_le [symmetric])
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lemma mult_strict_left_mono_neg:
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     "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
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apply (drule mult_strict_left_mono [of _ _ "-c"])
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apply (simp_all add: minus_mult_left [symmetric]) 
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done
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lemma mult_left_mono_neg:
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     "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
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apply (drule mult_left_mono [of _ _ "-c"])
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apply (simp_all add: minus_mult_left [symmetric]) 
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done
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lemma mult_strict_right_mono_neg:
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     "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
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apply (drule mult_strict_right_mono [of _ _ "-c"])
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apply (simp_all add: minus_mult_right [symmetric]) 
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done
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   320
obua@14738
   321
lemma mult_right_mono_neg:
obua@14738
   322
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
obua@14738
   323
apply (drule mult_right_mono [of _ _ "-c"])
obua@14738
   324
apply (simp)
obua@14738
   325
apply (simp_all add: minus_mult_right [symmetric]) 
obua@14738
   326
done
paulson@14265
   327
paulson@14265
   328
subsection{* Products of Signs *}
paulson@14265
   329
obua@14738
   330
lemma mult_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
paulson@14265
   331
by (drule mult_strict_left_mono [of 0 b], auto)
paulson@14265
   332
obua@14738
   333
lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
obua@14738
   334
by (drule mult_left_mono [of 0 b], auto)
obua@14738
   335
obua@14738
   336
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
paulson@14265
   337
by (drule mult_strict_left_mono [of b 0], auto)
paulson@14265
   338
obua@14738
   339
lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
obua@14738
   340
by (drule mult_left_mono [of b 0], auto)
obua@14738
   341
obua@14738
   342
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
obua@14738
   343
by (drule mult_strict_right_mono[of b 0], auto)
obua@14738
   344
obua@14738
   345
lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
obua@14738
   346
by (drule mult_right_mono[of b 0], auto)
obua@14738
   347
obua@14738
   348
lemma mult_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
paulson@14265
   349
by (drule mult_strict_right_mono_neg, auto)
paulson@14265
   350
obua@14738
   351
lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
obua@14738
   352
by (drule mult_right_mono_neg[of a 0 b ], auto)
obua@14738
   353
paulson@14341
   354
lemma zero_less_mult_pos:
obua@14738
   355
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
paulson@14265
   356
apply (case_tac "b\<le>0") 
paulson@14265
   357
 apply (auto simp add: order_le_less linorder_not_less)
paulson@14265
   358
apply (drule_tac mult_pos_neg [of a b]) 
paulson@14265
   359
 apply (auto dest: order_less_not_sym)
paulson@14265
   360
done
paulson@14265
   361
obua@14738
   362
lemma zero_less_mult_pos2:
obua@14738
   363
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
obua@14738
   364
apply (case_tac "b\<le>0") 
obua@14738
   365
 apply (auto simp add: order_le_less linorder_not_less)
obua@14738
   366
apply (drule_tac mult_pos_neg2 [of a b]) 
obua@14738
   367
 apply (auto dest: order_less_not_sym)
obua@14738
   368
done
obua@14738
   369
paulson@14265
   370
lemma zero_less_mult_iff:
obua@14738
   371
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
paulson@14265
   372
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
paulson@14265
   373
apply (blast dest: zero_less_mult_pos) 
obua@14738
   374
apply (blast dest: zero_less_mult_pos2)
paulson@14265
   375
done
paulson@14265
   376
paulson@14341
   377
text{*A field has no "zero divisors", and this theorem holds without the
paulson@14277
   378
      assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}
obua@14738
   379
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0 | b = 0)"
paulson@14265
   380
apply (case_tac "a < 0")
paulson@14265
   381
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
paulson@14265
   382
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
paulson@14265
   383
done
paulson@14265
   384
paulson@14265
   385
lemma zero_le_mult_iff:
obua@14738
   386
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14265
   387
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
paulson@14265
   388
                   zero_less_mult_iff)
paulson@14265
   389
paulson@14265
   390
lemma mult_less_0_iff:
obua@14738
   391
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14265
   392
apply (insert zero_less_mult_iff [of "-a" b]) 
paulson@14265
   393
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   394
done
paulson@14265
   395
paulson@14265
   396
lemma mult_le_0_iff:
obua@14738
   397
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14265
   398
apply (insert zero_le_mult_iff [of "-a" b]) 
paulson@14265
   399
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   400
done
paulson@14265
   401
obua@14738
   402
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
   403
by (auto simp add: mult_pos_le mult_neg_le)
obua@14738
   404
obua@14738
   405
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
   406
by (auto simp add: mult_pos_neg_le mult_pos_neg2_le)
obua@14738
   407
obua@14738
   408
lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a"
paulson@14265
   409
by (simp add: zero_le_mult_iff linorder_linear) 
paulson@14265
   410
obua@14738
   411
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
obua@14738
   412
      theorems available to members of @{term ordered_idom} *}
obua@14738
   413
obua@14738
   414
instance ordered_idom \<subseteq> ordered_semidom
paulson@14421
   415
proof
paulson@14421
   416
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
paulson@14430
   417
  thus "(0::'a) < 1" by (simp add: order_le_less) 
paulson@14421
   418
qed
paulson@14421
   419
obua@14738
   420
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 
obua@14738
   421
by (intro_classes, simp)
obua@14738
   422
obua@14738
   423
instance ordered_idom \<subseteq> idom ..
obua@14738
   424
paulson@14387
   425
text{*All three types of comparision involving 0 and 1 are covered.*}
paulson@14387
   426
paulson@14387
   427
declare zero_neq_one [THEN not_sym, simp]
paulson@14387
   428
obua@14738
   429
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
paulson@14268
   430
  by (rule zero_less_one [THEN order_less_imp_le]) 
paulson@14268
   431
obua@14738
   432
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
obua@14738
   433
by (simp add: linorder_not_le) 
paulson@14387
   434
obua@14738
   435
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
obua@14738
   436
by (simp add: linorder_not_less) 
paulson@14268
   437
paulson@14268
   438
subsection{*More Monotonicity*}
paulson@14268
   439
paulson@14268
   440
text{*Strict monotonicity in both arguments*}
paulson@14268
   441
lemma mult_strict_mono:
obua@14738
   442
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   443
apply (case_tac "c=0")
paulson@14268
   444
 apply (simp add: mult_pos) 
paulson@14268
   445
apply (erule mult_strict_right_mono [THEN order_less_trans])
paulson@14268
   446
 apply (force simp add: order_le_less) 
paulson@14268
   447
apply (erule mult_strict_left_mono, assumption)
paulson@14268
   448
done
paulson@14268
   449
paulson@14268
   450
text{*This weaker variant has more natural premises*}
paulson@14268
   451
lemma mult_strict_mono':
obua@14738
   452
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   453
apply (rule mult_strict_mono)
paulson@14268
   454
apply (blast intro: order_le_less_trans)+
paulson@14268
   455
done
paulson@14268
   456
paulson@14268
   457
lemma mult_mono:
paulson@14268
   458
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
obua@14738
   459
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
paulson@14268
   460
apply (erule mult_right_mono [THEN order_trans], assumption)
paulson@14268
   461
apply (erule mult_left_mono, assumption)
paulson@14268
   462
done
paulson@14268
   463
obua@14738
   464
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
paulson@14387
   465
apply (insert mult_strict_mono [of 1 m 1 n]) 
paulson@14430
   466
apply (simp add:  order_less_trans [OF zero_less_one]) 
paulson@14387
   467
done
paulson@14387
   468
paulson@14268
   469
subsection{*Cancellation Laws for Relationships With a Common Factor*}
paulson@14268
   470
paulson@14268
   471
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
paulson@14268
   472
   also with the relations @{text "\<le>"} and equality.*}
paulson@14268
   473
paulson@14268
   474
lemma mult_less_cancel_right:
obua@14738
   475
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
paulson@14268
   476
apply (case_tac "c = 0")
paulson@14268
   477
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
paulson@14268
   478
                      mult_strict_right_mono_neg)
paulson@14268
   479
apply (auto simp add: linorder_not_less 
paulson@14268
   480
                      linorder_not_le [symmetric, of "a*c"]
paulson@14268
   481
                      linorder_not_le [symmetric, of a])
paulson@14268
   482
apply (erule_tac [!] notE)
paulson@14268
   483
apply (auto simp add: order_less_imp_le mult_right_mono 
paulson@14268
   484
                      mult_right_mono_neg)
paulson@14268
   485
done
paulson@14268
   486
paulson@14268
   487
lemma mult_less_cancel_left:
obua@14738
   488
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
obua@14738
   489
apply (case_tac "c = 0")
obua@14738
   490
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
obua@14738
   491
                      mult_strict_left_mono_neg)
obua@14738
   492
apply (auto simp add: linorder_not_less 
obua@14738
   493
                      linorder_not_le [symmetric, of "c*a"]
obua@14738
   494
                      linorder_not_le [symmetric, of a])
obua@14738
   495
apply (erule_tac [!] notE)
obua@14738
   496
apply (auto simp add: order_less_imp_le mult_left_mono 
obua@14738
   497
                      mult_left_mono_neg)
obua@14738
   498
done
paulson@14268
   499
paulson@14268
   500
lemma mult_le_cancel_right:
obua@14738
   501
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@14268
   502
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right)
paulson@14268
   503
paulson@14268
   504
lemma mult_le_cancel_left:
obua@14738
   505
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
obua@14738
   506
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left)
paulson@14268
   507
paulson@14268
   508
lemma mult_less_imp_less_left:
paulson@14341
   509
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
obua@14738
   510
      shows "a < (b::'a::ordered_semiring_strict)"
paulson@14377
   511
proof (rule ccontr)
paulson@14377
   512
  assume "~ a < b"
paulson@14377
   513
  hence "b \<le> a" by (simp add: linorder_not_less)
paulson@14377
   514
  hence "c*b \<le> c*a" by (rule mult_left_mono)
paulson@14377
   515
  with this and less show False 
paulson@14377
   516
    by (simp add: linorder_not_less [symmetric])
paulson@14377
   517
qed
paulson@14268
   518
paulson@14268
   519
lemma mult_less_imp_less_right:
obua@14738
   520
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
obua@14738
   521
  shows "a < (b::'a::ordered_semiring_strict)"
obua@14738
   522
proof (rule ccontr)
obua@14738
   523
  assume "~ a < b"
obua@14738
   524
  hence "b \<le> a" by (simp add: linorder_not_less)
obua@14738
   525
  hence "b*c \<le> a*c" by (rule mult_right_mono)
obua@14738
   526
  with this and less show False 
obua@14738
   527
    by (simp add: linorder_not_less [symmetric])
obua@14738
   528
qed  
paulson@14268
   529
paulson@14268
   530
text{*Cancellation of equalities with a common factor*}
paulson@14268
   531
lemma mult_cancel_right [simp]:
obua@14738
   532
     "(a*c = b*c) = (c = (0::'a::ordered_ring_strict) | a=b)"
paulson@14268
   533
apply (cut_tac linorder_less_linear [of 0 c])
paulson@14268
   534
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono
paulson@14268
   535
             simp add: linorder_neq_iff)
paulson@14268
   536
done
paulson@14268
   537
paulson@14268
   538
text{*These cancellation theorems require an ordering. Versions are proved
paulson@14268
   539
      below that work for fields without an ordering.*}
paulson@14268
   540
lemma mult_cancel_left [simp]:
obua@14738
   541
     "(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)"
obua@14738
   542
apply (cut_tac linorder_less_linear [of 0 c])
obua@14738
   543
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono
obua@14738
   544
             simp add: linorder_neq_iff)
obua@14738
   545
done
paulson@14268
   546
obua@14738
   547
text{*This list of rewrites decides ring equalities by ordered rewriting.*}
obua@14738
   548
lemmas ring_eq_simps =
obua@14738
   549
  mult_ac
obua@14738
   550
  left_distrib right_distrib left_diff_distrib right_diff_distrib
obua@14738
   551
  add_ac
obua@14738
   552
  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
obua@14738
   553
  diff_eq_eq eq_diff_eq
obua@14738
   554
    
wenzelm@14770
   555
paulson@14265
   556
subsection {* Fields *}
paulson@14265
   557
paulson@14288
   558
lemma right_inverse [simp]:
paulson@14288
   559
      assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"
paulson@14288
   560
proof -
paulson@14288
   561
  have "a * inverse a = inverse a * a" by (simp add: mult_ac)
paulson@14288
   562
  also have "... = 1" using not0 by simp
paulson@14288
   563
  finally show ?thesis .
paulson@14288
   564
qed
paulson@14288
   565
paulson@14288
   566
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
paulson@14288
   567
proof
paulson@14288
   568
  assume neq: "b \<noteq> 0"
paulson@14288
   569
  {
paulson@14288
   570
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
paulson@14288
   571
    also assume "a / b = 1"
paulson@14288
   572
    finally show "a = b" by simp
paulson@14288
   573
  next
paulson@14288
   574
    assume "a = b"
paulson@14288
   575
    with neq show "a / b = 1" by (simp add: divide_inverse)
paulson@14288
   576
  }
paulson@14288
   577
qed
paulson@14288
   578
paulson@14288
   579
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
paulson@14288
   580
by (simp add: divide_inverse)
paulson@14288
   581
paulson@14288
   582
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
paulson@14288
   583
  by (simp add: divide_inverse)
paulson@14288
   584
paulson@14430
   585
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
paulson@14430
   586
by (simp add: divide_inverse)
paulson@14277
   587
paulson@14430
   588
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
paulson@14430
   589
by (simp add: divide_inverse)
paulson@14277
   590
paulson@14430
   591
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
paulson@14430
   592
by (simp add: divide_inverse)
paulson@14277
   593
paulson@14430
   594
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
paulson@14293
   595
by (simp add: divide_inverse left_distrib) 
paulson@14293
   596
paulson@14293
   597
paulson@14270
   598
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
paulson@14270
   599
      of an ordering.*}
paulson@14348
   600
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
paulson@14377
   601
proof cases
paulson@14377
   602
  assume "a=0" thus ?thesis by simp
paulson@14377
   603
next
paulson@14377
   604
  assume anz [simp]: "a\<noteq>0"
paulson@14377
   605
  { assume "a * b = 0"
paulson@14377
   606
    hence "inverse a * (a * b) = 0" by simp
paulson@14377
   607
    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
paulson@14377
   608
  thus ?thesis by force
paulson@14377
   609
qed
paulson@14270
   610
paulson@14268
   611
text{*Cancellation of equalities with a common factor*}
paulson@14268
   612
lemma field_mult_cancel_right_lemma:
paulson@14269
   613
      assumes cnz: "c \<noteq> (0::'a::field)"
paulson@14269
   614
	  and eq:  "a*c = b*c"
paulson@14269
   615
	 shows "a=b"
paulson@14377
   616
proof -
paulson@14268
   617
  have "(a * c) * inverse c = (b * c) * inverse c"
paulson@14268
   618
    by (simp add: eq)
paulson@14268
   619
  thus "a=b"
paulson@14268
   620
    by (simp add: mult_assoc cnz)
paulson@14377
   621
qed
paulson@14268
   622
paulson@14348
   623
lemma field_mult_cancel_right [simp]:
paulson@14268
   624
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
paulson@14377
   625
proof cases
paulson@14377
   626
  assume "c=0" thus ?thesis by simp
paulson@14377
   627
next
paulson@14377
   628
  assume "c\<noteq>0" 
paulson@14377
   629
  thus ?thesis by (force dest: field_mult_cancel_right_lemma)
paulson@14377
   630
qed
paulson@14268
   631
paulson@14348
   632
lemma field_mult_cancel_left [simp]:
paulson@14268
   633
     "(c*a = c*b) = (c = (0::'a::field) | a=b)"
paulson@14268
   634
  by (simp add: mult_commute [of c] field_mult_cancel_right) 
paulson@14268
   635
paulson@14268
   636
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
paulson@14377
   637
proof
paulson@14268
   638
  assume ianz: "inverse a = 0"
paulson@14268
   639
  assume "a \<noteq> 0"
paulson@14268
   640
  hence "1 = a * inverse a" by simp
paulson@14268
   641
  also have "... = 0" by (simp add: ianz)
paulson@14268
   642
  finally have "1 = (0::'a::field)" .
paulson@14268
   643
  thus False by (simp add: eq_commute)
paulson@14377
   644
qed
paulson@14268
   645
paulson@14277
   646
paulson@14277
   647
subsection{*Basic Properties of @{term inverse}*}
paulson@14277
   648
paulson@14268
   649
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"
paulson@14268
   650
apply (rule ccontr) 
paulson@14268
   651
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   652
done
paulson@14268
   653
paulson@14268
   654
lemma inverse_nonzero_imp_nonzero:
paulson@14268
   655
   "inverse a = 0 ==> a = (0::'a::field)"
paulson@14268
   656
apply (rule ccontr) 
paulson@14268
   657
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   658
done
paulson@14268
   659
paulson@14268
   660
lemma inverse_nonzero_iff_nonzero [simp]:
paulson@14268
   661
   "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"
paulson@14268
   662
by (force dest: inverse_nonzero_imp_nonzero) 
paulson@14268
   663
paulson@14268
   664
lemma nonzero_inverse_minus_eq:
paulson@14269
   665
      assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
paulson@14377
   666
proof -
paulson@14377
   667
  have "-a * inverse (- a) = -a * - inverse a"
paulson@14377
   668
    by simp
paulson@14377
   669
  thus ?thesis 
paulson@14377
   670
    by (simp only: field_mult_cancel_left, simp)
paulson@14377
   671
qed
paulson@14268
   672
paulson@14268
   673
lemma inverse_minus_eq [simp]:
paulson@14377
   674
   "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
paulson@14377
   675
proof cases
paulson@14377
   676
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
   677
next
paulson@14377
   678
  assume "a\<noteq>0" 
paulson@14377
   679
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
   680
qed
paulson@14268
   681
paulson@14268
   682
lemma nonzero_inverse_eq_imp_eq:
paulson@14269
   683
      assumes inveq: "inverse a = inverse b"
paulson@14269
   684
	  and anz:  "a \<noteq> 0"
paulson@14269
   685
	  and bnz:  "b \<noteq> 0"
paulson@14269
   686
	 shows "a = (b::'a::field)"
paulson@14377
   687
proof -
paulson@14268
   688
  have "a * inverse b = a * inverse a"
paulson@14268
   689
    by (simp add: inveq)
paulson@14268
   690
  hence "(a * inverse b) * b = (a * inverse a) * b"
paulson@14268
   691
    by simp
paulson@14268
   692
  thus "a = b"
paulson@14268
   693
    by (simp add: mult_assoc anz bnz)
paulson@14377
   694
qed
paulson@14268
   695
paulson@14268
   696
lemma inverse_eq_imp_eq:
paulson@14268
   697
     "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"
paulson@14268
   698
apply (case_tac "a=0 | b=0") 
paulson@14268
   699
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
   700
              simp add: eq_commute [of "0::'a"])
paulson@14268
   701
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
   702
done
paulson@14268
   703
paulson@14268
   704
lemma inverse_eq_iff_eq [simp]:
paulson@14268
   705
     "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"
paulson@14268
   706
by (force dest!: inverse_eq_imp_eq) 
paulson@14268
   707
paulson@14270
   708
lemma nonzero_inverse_inverse_eq:
paulson@14270
   709
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
paulson@14270
   710
  proof -
paulson@14270
   711
  have "(inverse (inverse a) * inverse a) * a = a" 
paulson@14270
   712
    by (simp add: nonzero_imp_inverse_nonzero)
paulson@14270
   713
  thus ?thesis
paulson@14270
   714
    by (simp add: mult_assoc)
paulson@14270
   715
  qed
paulson@14270
   716
paulson@14270
   717
lemma inverse_inverse_eq [simp]:
paulson@14270
   718
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
paulson@14270
   719
  proof cases
paulson@14270
   720
    assume "a=0" thus ?thesis by simp
paulson@14270
   721
  next
paulson@14270
   722
    assume "a\<noteq>0" 
paulson@14270
   723
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
   724
  qed
paulson@14270
   725
paulson@14270
   726
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
paulson@14270
   727
  proof -
paulson@14270
   728
  have "inverse 1 * 1 = (1::'a::field)" 
paulson@14270
   729
    by (rule left_inverse [OF zero_neq_one [symmetric]])
paulson@14270
   730
  thus ?thesis  by simp
paulson@14270
   731
  qed
paulson@14270
   732
paulson@14270
   733
lemma nonzero_inverse_mult_distrib: 
paulson@14270
   734
      assumes anz: "a \<noteq> 0"
paulson@14270
   735
          and bnz: "b \<noteq> 0"
paulson@14270
   736
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
paulson@14270
   737
  proof -
paulson@14270
   738
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
paulson@14270
   739
    by (simp add: field_mult_eq_0_iff anz bnz)
paulson@14270
   740
  hence "inverse(a*b) * a = inverse(b)" 
paulson@14270
   741
    by (simp add: mult_assoc bnz)
paulson@14270
   742
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
paulson@14270
   743
    by simp
paulson@14270
   744
  thus ?thesis
paulson@14270
   745
    by (simp add: mult_assoc anz)
paulson@14270
   746
  qed
paulson@14270
   747
paulson@14270
   748
text{*This version builds in division by zero while also re-orienting
paulson@14270
   749
      the right-hand side.*}
paulson@14270
   750
lemma inverse_mult_distrib [simp]:
paulson@14270
   751
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
   752
  proof cases
paulson@14270
   753
    assume "a \<noteq> 0 & b \<noteq> 0" 
paulson@14270
   754
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
   755
  next
paulson@14270
   756
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
paulson@14270
   757
    thus ?thesis  by force
paulson@14270
   758
  qed
paulson@14270
   759
paulson@14270
   760
text{*There is no slick version using division by zero.*}
paulson@14270
   761
lemma inverse_add:
paulson@14270
   762
     "[|a \<noteq> 0;  b \<noteq> 0|]
paulson@14270
   763
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
paulson@14270
   764
apply (simp add: left_distrib mult_assoc)
paulson@14270
   765
apply (simp add: mult_commute [of "inverse a"]) 
paulson@14270
   766
apply (simp add: mult_assoc [symmetric] add_commute)
paulson@14270
   767
done
paulson@14270
   768
paulson@14365
   769
lemma inverse_divide [simp]:
paulson@14365
   770
      "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
paulson@14430
   771
  by (simp add: divide_inverse mult_commute)
paulson@14365
   772
paulson@14277
   773
lemma nonzero_mult_divide_cancel_left:
paulson@14277
   774
  assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
paulson@14277
   775
    shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
   776
proof -
paulson@14277
   777
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
paulson@14277
   778
    by (simp add: field_mult_eq_0_iff divide_inverse 
paulson@14277
   779
                  nonzero_inverse_mult_distrib)
paulson@14277
   780
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
   781
    by (simp only: mult_ac)
paulson@14277
   782
  also have "... =  a * inverse b"
paulson@14277
   783
    by simp
paulson@14277
   784
    finally show ?thesis 
paulson@14277
   785
    by (simp add: divide_inverse)
paulson@14277
   786
qed
paulson@14277
   787
paulson@14277
   788
lemma mult_divide_cancel_left:
paulson@14277
   789
     "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
paulson@14277
   790
apply (case_tac "b = 0")
paulson@14277
   791
apply (simp_all add: nonzero_mult_divide_cancel_left)
paulson@14277
   792
done
paulson@14277
   793
paulson@14321
   794
lemma nonzero_mult_divide_cancel_right:
paulson@14321
   795
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
paulson@14321
   796
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
paulson@14321
   797
paulson@14321
   798
lemma mult_divide_cancel_right:
paulson@14321
   799
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
paulson@14321
   800
apply (case_tac "b = 0")
paulson@14321
   801
apply (simp_all add: nonzero_mult_divide_cancel_right)
paulson@14321
   802
done
paulson@14321
   803
paulson@14277
   804
(*For ExtractCommonTerm*)
paulson@14277
   805
lemma mult_divide_cancel_eq_if:
paulson@14277
   806
     "(c*a) / (c*b) = 
paulson@14277
   807
      (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
paulson@14277
   808
  by (simp add: mult_divide_cancel_left)
paulson@14277
   809
paulson@14284
   810
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
paulson@14430
   811
  by (simp add: divide_inverse)
paulson@14284
   812
paulson@14430
   813
lemma times_divide_eq_right [simp]: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
   814
by (simp add: divide_inverse mult_assoc)
paulson@14288
   815
paulson@14430
   816
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
   817
by (simp add: divide_inverse mult_ac)
paulson@14288
   818
paulson@14288
   819
lemma divide_divide_eq_right [simp]:
paulson@14288
   820
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
   821
by (simp add: divide_inverse mult_ac)
paulson@14288
   822
paulson@14288
   823
lemma divide_divide_eq_left [simp]:
paulson@14288
   824
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
   825
by (simp add: divide_inverse mult_assoc)
paulson@14288
   826
paulson@14268
   827
paulson@14293
   828
subsection {* Division and Unary Minus *}
paulson@14293
   829
paulson@14293
   830
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
   831
by (simp add: divide_inverse minus_mult_left)
paulson@14293
   832
paulson@14293
   833
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
   834
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
   835
paulson@14293
   836
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
   837
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
   838
paulson@14430
   839
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
paulson@14430
   840
by (simp add: divide_inverse minus_mult_left [symmetric])
paulson@14293
   841
paulson@14293
   842
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14430
   843
by (simp add: divide_inverse minus_mult_right [symmetric])
paulson@14430
   844
paulson@14293
   845
paulson@14293
   846
text{*The effect is to extract signs from divisions*}
paulson@14293
   847
declare minus_divide_left  [symmetric, simp]
paulson@14293
   848
declare minus_divide_right [symmetric, simp]
paulson@14293
   849
paulson@14387
   850
text{*Also, extract signs from products*}
paulson@14387
   851
declare minus_mult_left [symmetric, simp]
paulson@14387
   852
declare minus_mult_right [symmetric, simp]
paulson@14387
   853
paulson@14293
   854
lemma minus_divide_divide [simp]:
paulson@14293
   855
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
paulson@14293
   856
apply (case_tac "b=0", simp) 
paulson@14293
   857
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
   858
done
paulson@14293
   859
paulson@14430
   860
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
   861
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
   862
paulson@14293
   863
paulson@14268
   864
subsection {* Ordered Fields *}
paulson@14268
   865
paulson@14277
   866
lemma positive_imp_inverse_positive: 
paulson@14269
   867
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
paulson@14268
   868
  proof -
paulson@14268
   869
  have "0 < a * inverse a" 
paulson@14268
   870
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
   871
  thus "0 < inverse a" 
paulson@14268
   872
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
paulson@14268
   873
  qed
paulson@14268
   874
paulson@14277
   875
lemma negative_imp_inverse_negative:
paulson@14268
   876
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
paulson@14277
   877
  by (insert positive_imp_inverse_positive [of "-a"], 
paulson@14268
   878
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
paulson@14268
   879
paulson@14268
   880
lemma inverse_le_imp_le:
paulson@14269
   881
      assumes invle: "inverse a \<le> inverse b"
paulson@14269
   882
	  and apos:  "0 < a"
paulson@14269
   883
	 shows "b \<le> (a::'a::ordered_field)"
paulson@14268
   884
  proof (rule classical)
paulson@14268
   885
  assume "~ b \<le> a"
paulson@14268
   886
  hence "a < b"
paulson@14268
   887
    by (simp add: linorder_not_le)
paulson@14268
   888
  hence bpos: "0 < b"
paulson@14268
   889
    by (blast intro: apos order_less_trans)
paulson@14268
   890
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
   891
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
   892
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
   893
    by (simp add: bpos order_less_imp_le mult_right_mono)
paulson@14268
   894
  thus "b \<le> a"
paulson@14268
   895
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
paulson@14268
   896
  qed
paulson@14268
   897
paulson@14277
   898
lemma inverse_positive_imp_positive:
paulson@14277
   899
      assumes inv_gt_0: "0 < inverse a"
paulson@14277
   900
          and [simp]:   "a \<noteq> 0"
paulson@14277
   901
        shows "0 < (a::'a::ordered_field)"
paulson@14277
   902
  proof -
paulson@14277
   903
  have "0 < inverse (inverse a)"
paulson@14277
   904
    by (rule positive_imp_inverse_positive)
paulson@14277
   905
  thus "0 < a"
paulson@14277
   906
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
   907
  qed
paulson@14277
   908
paulson@14277
   909
lemma inverse_positive_iff_positive [simp]:
paulson@14277
   910
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
   911
apply (case_tac "a = 0", simp)
paulson@14277
   912
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
   913
done
paulson@14277
   914
paulson@14277
   915
lemma inverse_negative_imp_negative:
paulson@14277
   916
      assumes inv_less_0: "inverse a < 0"
paulson@14277
   917
          and [simp]:   "a \<noteq> 0"
paulson@14277
   918
        shows "a < (0::'a::ordered_field)"
paulson@14277
   919
  proof -
paulson@14277
   920
  have "inverse (inverse a) < 0"
paulson@14277
   921
    by (rule negative_imp_inverse_negative)
paulson@14277
   922
  thus "a < 0"
paulson@14277
   923
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
   924
  qed
paulson@14277
   925
paulson@14277
   926
lemma inverse_negative_iff_negative [simp]:
paulson@14277
   927
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
   928
apply (case_tac "a = 0", simp)
paulson@14277
   929
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
   930
done
paulson@14277
   931
paulson@14277
   932
lemma inverse_nonnegative_iff_nonnegative [simp]:
paulson@14277
   933
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
   934
by (simp add: linorder_not_less [symmetric])
paulson@14277
   935
paulson@14277
   936
lemma inverse_nonpositive_iff_nonpositive [simp]:
paulson@14277
   937
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
   938
by (simp add: linorder_not_less [symmetric])
paulson@14277
   939
paulson@14277
   940
paulson@14277
   941
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
   942
paulson@14268
   943
lemma less_imp_inverse_less:
paulson@14269
   944
      assumes less: "a < b"
paulson@14269
   945
	  and apos:  "0 < a"
paulson@14269
   946
	shows "inverse b < inverse (a::'a::ordered_field)"
paulson@14268
   947
  proof (rule ccontr)
paulson@14268
   948
  assume "~ inverse b < inverse a"
paulson@14268
   949
  hence "inverse a \<le> inverse b"
paulson@14268
   950
    by (simp add: linorder_not_less)
paulson@14268
   951
  hence "~ (a < b)"
paulson@14268
   952
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
   953
  thus False
paulson@14268
   954
    by (rule notE [OF _ less])
paulson@14268
   955
  qed
paulson@14268
   956
paulson@14268
   957
lemma inverse_less_imp_less:
paulson@14268
   958
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
   959
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
   960
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
   961
done
paulson@14268
   962
paulson@14268
   963
text{*Both premises are essential. Consider -1 and 1.*}
paulson@14268
   964
lemma inverse_less_iff_less [simp]:
paulson@14268
   965
     "[|0 < a; 0 < b|] 
paulson@14268
   966
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
   967
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
   968
paulson@14268
   969
lemma le_imp_inverse_le:
paulson@14268
   970
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
   971
  by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
   972
paulson@14268
   973
lemma inverse_le_iff_le [simp]:
paulson@14268
   974
     "[|0 < a; 0 < b|] 
paulson@14268
   975
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
   976
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
   977
paulson@14268
   978
paulson@14268
   979
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
   980
case is trivial, since inverse preserves signs.*}
paulson@14268
   981
lemma inverse_le_imp_le_neg:
paulson@14268
   982
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
paulson@14268
   983
  apply (rule classical) 
paulson@14268
   984
  apply (subgoal_tac "a < 0") 
paulson@14268
   985
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
paulson@14268
   986
  apply (insert inverse_le_imp_le [of "-b" "-a"])
paulson@14268
   987
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
   988
  done
paulson@14268
   989
paulson@14268
   990
lemma less_imp_inverse_less_neg:
paulson@14268
   991
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
paulson@14268
   992
  apply (subgoal_tac "a < 0") 
paulson@14268
   993
   prefer 2 apply (blast intro: order_less_trans) 
paulson@14268
   994
  apply (insert less_imp_inverse_less [of "-b" "-a"])
paulson@14268
   995
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
   996
  done
paulson@14268
   997
paulson@14268
   998
lemma inverse_less_imp_less_neg:
paulson@14268
   999
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1000
  apply (rule classical) 
paulson@14268
  1001
  apply (subgoal_tac "a < 0") 
paulson@14268
  1002
   prefer 2
paulson@14268
  1003
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
paulson@14268
  1004
  apply (insert inverse_less_imp_less [of "-b" "-a"])
paulson@14268
  1005
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1006
  done
paulson@14268
  1007
paulson@14268
  1008
lemma inverse_less_iff_less_neg [simp]:
paulson@14268
  1009
     "[|a < 0; b < 0|] 
paulson@14268
  1010
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1011
  apply (insert inverse_less_iff_less [of "-b" "-a"])
paulson@14268
  1012
  apply (simp del: inverse_less_iff_less 
paulson@14268
  1013
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1014
  done
paulson@14268
  1015
paulson@14268
  1016
lemma le_imp_inverse_le_neg:
paulson@14268
  1017
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
  1018
  by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1019
paulson@14268
  1020
lemma inverse_le_iff_le_neg [simp]:
paulson@14268
  1021
     "[|a < 0; b < 0|] 
paulson@14268
  1022
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1023
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1024
paulson@14277
  1025
paulson@14365
  1026
subsection{*Inverses and the Number One*}
paulson@14365
  1027
paulson@14365
  1028
lemma one_less_inverse_iff:
paulson@14365
  1029
    "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
paulson@14365
  1030
  assume "0 < x"
paulson@14365
  1031
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1032
    show ?thesis by simp
paulson@14365
  1033
next
paulson@14365
  1034
  assume notless: "~ (0 < x)"
paulson@14365
  1035
  have "~ (1 < inverse x)"
paulson@14365
  1036
  proof
paulson@14365
  1037
    assume "1 < inverse x"
paulson@14365
  1038
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1039
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1040
    finally show False by auto
paulson@14365
  1041
  qed
paulson@14365
  1042
  with notless show ?thesis by simp
paulson@14365
  1043
qed
paulson@14365
  1044
paulson@14365
  1045
lemma inverse_eq_1_iff [simp]:
paulson@14365
  1046
    "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1047
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1048
paulson@14365
  1049
lemma one_le_inverse_iff:
paulson@14365
  1050
   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1051
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1052
                    eq_commute [of 1]) 
paulson@14365
  1053
paulson@14365
  1054
lemma inverse_less_1_iff:
paulson@14365
  1055
   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1056
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1057
paulson@14365
  1058
lemma inverse_le_1_iff:
paulson@14365
  1059
   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1060
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1061
paulson@14365
  1062
paulson@14277
  1063
subsection{*Division and Signs*}
paulson@14277
  1064
paulson@14277
  1065
lemma zero_less_divide_iff:
paulson@14277
  1066
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
paulson@14430
  1067
by (simp add: divide_inverse zero_less_mult_iff)
paulson@14277
  1068
paulson@14277
  1069
lemma divide_less_0_iff:
paulson@14277
  1070
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
paulson@14277
  1071
      (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14430
  1072
by (simp add: divide_inverse mult_less_0_iff)
paulson@14277
  1073
paulson@14277
  1074
lemma zero_le_divide_iff:
paulson@14277
  1075
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
paulson@14277
  1076
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14430
  1077
by (simp add: divide_inverse zero_le_mult_iff)
paulson@14277
  1078
paulson@14277
  1079
lemma divide_le_0_iff:
paulson@14288
  1080
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
paulson@14288
  1081
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14430
  1082
by (simp add: divide_inverse mult_le_0_iff)
paulson@14277
  1083
paulson@14277
  1084
lemma divide_eq_0_iff [simp]:
paulson@14277
  1085
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
paulson@14430
  1086
by (simp add: divide_inverse field_mult_eq_0_iff)
paulson@14277
  1087
paulson@14288
  1088
paulson@14288
  1089
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1090
paulson@14288
  1091
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1092
proof -
paulson@14288
  1093
  assume less: "0<c"
paulson@14288
  1094
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1095
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1096
  also have "... = (a*c \<le> b)"
paulson@14288
  1097
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1098
  finally show ?thesis .
paulson@14288
  1099
qed
paulson@14288
  1100
paulson@14288
  1101
paulson@14288
  1102
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1103
proof -
paulson@14288
  1104
  assume less: "c<0"
paulson@14288
  1105
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1106
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1107
  also have "... = (b \<le> a*c)"
paulson@14288
  1108
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1109
  finally show ?thesis .
paulson@14288
  1110
qed
paulson@14288
  1111
paulson@14288
  1112
lemma le_divide_eq:
paulson@14288
  1113
  "(a \<le> b/c) = 
paulson@14288
  1114
   (if 0 < c then a*c \<le> b
paulson@14288
  1115
             else if c < 0 then b \<le> a*c
paulson@14288
  1116
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1117
apply (case_tac "c=0", simp) 
paulson@14288
  1118
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1119
done
paulson@14288
  1120
paulson@14288
  1121
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1122
proof -
paulson@14288
  1123
  assume less: "0<c"
paulson@14288
  1124
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1125
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1126
  also have "... = (b \<le> a*c)"
paulson@14288
  1127
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1128
  finally show ?thesis .
paulson@14288
  1129
qed
paulson@14288
  1130
paulson@14288
  1131
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1132
proof -
paulson@14288
  1133
  assume less: "c<0"
paulson@14288
  1134
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1135
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1136
  also have "... = (a*c \<le> b)"
paulson@14288
  1137
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1138
  finally show ?thesis .
paulson@14288
  1139
qed
paulson@14288
  1140
paulson@14288
  1141
lemma divide_le_eq:
paulson@14288
  1142
  "(b/c \<le> a) = 
paulson@14288
  1143
   (if 0 < c then b \<le> a*c
paulson@14288
  1144
             else if c < 0 then a*c \<le> b
paulson@14288
  1145
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1146
apply (case_tac "c=0", simp) 
paulson@14288
  1147
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1148
done
paulson@14288
  1149
paulson@14288
  1150
paulson@14288
  1151
lemma pos_less_divide_eq:
paulson@14288
  1152
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1153
proof -
paulson@14288
  1154
  assume less: "0<c"
paulson@14288
  1155
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@14288
  1156
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1157
  also have "... = (a*c < b)"
paulson@14288
  1158
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1159
  finally show ?thesis .
paulson@14288
  1160
qed
paulson@14288
  1161
paulson@14288
  1162
lemma neg_less_divide_eq:
paulson@14288
  1163
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1164
proof -
paulson@14288
  1165
  assume less: "c<0"
paulson@14288
  1166
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@14288
  1167
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1168
  also have "... = (b < a*c)"
paulson@14288
  1169
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1170
  finally show ?thesis .
paulson@14288
  1171
qed
paulson@14288
  1172
paulson@14288
  1173
lemma less_divide_eq:
paulson@14288
  1174
  "(a < b/c) = 
paulson@14288
  1175
   (if 0 < c then a*c < b
paulson@14288
  1176
             else if c < 0 then b < a*c
paulson@14288
  1177
             else  a < (0::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1178
apply (case_tac "c=0", simp) 
paulson@14288
  1179
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1180
done
paulson@14288
  1181
paulson@14288
  1182
lemma pos_divide_less_eq:
paulson@14288
  1183
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1184
proof -
paulson@14288
  1185
  assume less: "0<c"
paulson@14288
  1186
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@14288
  1187
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1188
  also have "... = (b < a*c)"
paulson@14288
  1189
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1190
  finally show ?thesis .
paulson@14288
  1191
qed
paulson@14288
  1192
paulson@14288
  1193
lemma neg_divide_less_eq:
paulson@14288
  1194
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1195
proof -
paulson@14288
  1196
  assume less: "c<0"
paulson@14288
  1197
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@14288
  1198
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
paulson@14288
  1199
  also have "... = (a*c < b)"
paulson@14288
  1200
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1201
  finally show ?thesis .
paulson@14288
  1202
qed
paulson@14288
  1203
paulson@14288
  1204
lemma divide_less_eq:
paulson@14288
  1205
  "(b/c < a) = 
paulson@14288
  1206
   (if 0 < c then b < a*c
paulson@14288
  1207
             else if c < 0 then a*c < b
paulson@14288
  1208
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
paulson@14288
  1209
apply (case_tac "c=0", simp) 
paulson@14288
  1210
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1211
done
paulson@14288
  1212
paulson@14288
  1213
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
paulson@14288
  1214
proof -
paulson@14288
  1215
  assume [simp]: "c\<noteq>0"
paulson@14288
  1216
  have "(a = b/c) = (a*c = (b/c)*c)"
paulson@14288
  1217
    by (simp add: field_mult_cancel_right)
paulson@14288
  1218
  also have "... = (a*c = b)"
paulson@14288
  1219
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1220
  finally show ?thesis .
paulson@14288
  1221
qed
paulson@14288
  1222
paulson@14288
  1223
lemma eq_divide_eq:
paulson@14288
  1224
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
paulson@14288
  1225
by (simp add: nonzero_eq_divide_eq) 
paulson@14288
  1226
paulson@14288
  1227
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
paulson@14288
  1228
proof -
paulson@14288
  1229
  assume [simp]: "c\<noteq>0"
paulson@14288
  1230
  have "(b/c = a) = ((b/c)*c = a*c)"
paulson@14288
  1231
    by (simp add: field_mult_cancel_right)
paulson@14288
  1232
  also have "... = (b = a*c)"
paulson@14288
  1233
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1234
  finally show ?thesis .
paulson@14288
  1235
qed
paulson@14288
  1236
paulson@14288
  1237
lemma divide_eq_eq:
paulson@14288
  1238
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
paulson@14288
  1239
by (force simp add: nonzero_divide_eq_eq) 
paulson@14288
  1240
paulson@14288
  1241
subsection{*Cancellation Laws for Division*}
paulson@14288
  1242
paulson@14288
  1243
lemma divide_cancel_right [simp]:
paulson@14288
  1244
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
paulson@14288
  1245
apply (case_tac "c=0", simp) 
paulson@14430
  1246
apply (simp add: divide_inverse field_mult_cancel_right) 
paulson@14288
  1247
done
paulson@14288
  1248
paulson@14288
  1249
lemma divide_cancel_left [simp]:
paulson@14288
  1250
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
paulson@14288
  1251
apply (case_tac "c=0", simp) 
paulson@14430
  1252
apply (simp add: divide_inverse field_mult_cancel_left) 
paulson@14288
  1253
done
paulson@14288
  1254
paulson@14353
  1255
subsection {* Division and the Number One *}
paulson@14353
  1256
paulson@14353
  1257
text{*Simplify expressions equated with 1*}
paulson@14353
  1258
lemma divide_eq_1_iff [simp]:
paulson@14353
  1259
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1260
apply (case_tac "b=0", simp) 
paulson@14353
  1261
apply (simp add: right_inverse_eq) 
paulson@14353
  1262
done
paulson@14353
  1263
paulson@14353
  1264
paulson@14353
  1265
lemma one_eq_divide_iff [simp]:
paulson@14353
  1266
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1267
by (simp add: eq_commute [of 1])  
paulson@14353
  1268
paulson@14353
  1269
lemma zero_eq_1_divide_iff [simp]:
paulson@14353
  1270
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
paulson@14353
  1271
apply (case_tac "a=0", simp) 
paulson@14353
  1272
apply (auto simp add: nonzero_eq_divide_eq) 
paulson@14353
  1273
done
paulson@14353
  1274
paulson@14353
  1275
lemma one_divide_eq_0_iff [simp]:
paulson@14353
  1276
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
paulson@14353
  1277
apply (case_tac "a=0", simp) 
paulson@14353
  1278
apply (insert zero_neq_one [THEN not_sym]) 
paulson@14353
  1279
apply (auto simp add: nonzero_divide_eq_eq) 
paulson@14353
  1280
done
paulson@14353
  1281
paulson@14353
  1282
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@14353
  1283
declare zero_less_divide_iff [of "1", simp]
paulson@14353
  1284
declare divide_less_0_iff [of "1", simp]
paulson@14353
  1285
declare zero_le_divide_iff [of "1", simp]
paulson@14353
  1286
declare divide_le_0_iff [of "1", simp]
paulson@14353
  1287
paulson@14288
  1288
paulson@14293
  1289
subsection {* Ordering Rules for Division *}
paulson@14293
  1290
paulson@14293
  1291
lemma divide_strict_right_mono:
paulson@14293
  1292
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1293
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
paulson@14293
  1294
              positive_imp_inverse_positive) 
paulson@14293
  1295
paulson@14293
  1296
lemma divide_right_mono:
paulson@14293
  1297
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
paulson@14293
  1298
  by (force simp add: divide_strict_right_mono order_le_less) 
paulson@14293
  1299
paulson@14293
  1300
paulson@14293
  1301
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1302
      have the same sign*}
paulson@14293
  1303
lemma divide_strict_left_mono:
paulson@14293
  1304
       "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1305
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono 
paulson@14293
  1306
      order_less_imp_not_eq order_less_imp_not_eq2  
paulson@14293
  1307
      less_imp_inverse_less less_imp_inverse_less_neg) 
paulson@14293
  1308
paulson@14293
  1309
lemma divide_left_mono:
paulson@14293
  1310
     "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> 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 (case_tac "c=0", simp add: divide_inverse)
paulson@14293
  1315
  apply (force simp add: divide_strict_left_mono order_le_less) 
paulson@14293
  1316
  done
paulson@14293
  1317
paulson@14293
  1318
lemma divide_strict_left_mono_neg:
paulson@14293
  1319
     "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1320
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1321
   prefer 2 
paulson@14293
  1322
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
paulson@14293
  1323
  apply (drule divide_strict_left_mono [of _ _ "-c"]) 
paulson@14293
  1324
   apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) 
paulson@14293
  1325
  done
paulson@14293
  1326
paulson@14293
  1327
lemma divide_strict_right_mono_neg:
paulson@14293
  1328
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1329
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) 
paulson@14293
  1330
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
paulson@14293
  1331
done
paulson@14293
  1332
paulson@14293
  1333
paulson@14293
  1334
subsection {* Ordered Fields are Dense *}
paulson@14293
  1335
obua@14738
  1336
lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
paulson@14293
  1337
proof -
obua@14738
  1338
  have "a+0 < (a+1::'a::ordered_semidom)"
paulson@14365
  1339
    by (blast intro: zero_less_one add_strict_left_mono) 
paulson@14293
  1340
  thus ?thesis by simp
paulson@14293
  1341
qed
paulson@14293
  1342
obua@14738
  1343
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
paulson@14365
  1344
  by (blast intro: order_less_trans zero_less_one less_add_one) 
paulson@14365
  1345
paulson@14293
  1346
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
paulson@14293
  1347
by (simp add: zero_less_two pos_less_divide_eq right_distrib) 
paulson@14293
  1348
paulson@14293
  1349
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
paulson@14293
  1350
by (simp add: zero_less_two pos_divide_less_eq right_distrib) 
paulson@14293
  1351
paulson@14293
  1352
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
paulson@14293
  1353
by (blast intro!: less_half_sum gt_half_sum)
paulson@14293
  1354
paulson@14293
  1355
subsection {* Absolute Value *}
paulson@14293
  1356
obua@14738
  1357
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
paulson@14294
  1358
  by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) 
paulson@14294
  1359
obua@14738
  1360
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  1361
proof -
obua@14738
  1362
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  1363
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  1364
  have a: "(abs a) * (abs b) = ?x"
obua@14738
  1365
    by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps)
obua@14738
  1366
  {
obua@14738
  1367
    fix u v :: 'a
obua@14738
  1368
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> u * v = ?y"
obua@14738
  1369
      apply (subst prts[of u], subst prts[of v])
obua@14738
  1370
      apply (simp add: left_distrib right_distrib add_ac) 
obua@14738
  1371
      done
obua@14738
  1372
  }
obua@14738
  1373
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  1374
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  1375
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  1376
  have xy: "- ?x <= ?y"
obua@14754
  1377
    apply (simp)
obua@14754
  1378
    apply (rule_tac y="0::'a" in order_trans)
obua@14754
  1379
    apply (rule addm2)+
obua@14754
  1380
    apply (simp_all add: mult_pos_le mult_neg_le)
obua@14754
  1381
    apply (rule addm)+
obua@14754
  1382
    apply (simp_all add: mult_pos_le mult_neg_le)
obua@14754
  1383
    done
obua@14738
  1384
  have yx: "?y <= ?x"
obua@14738
  1385
    apply (simp add: add_ac)
obua@14754
  1386
    apply (rule_tac y=0 in order_trans)
obua@14754
  1387
    apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)
obua@14754
  1388
    apply (rule addm, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)
obua@14738
  1389
    done
obua@14738
  1390
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  1391
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  1392
  show ?thesis
obua@14738
  1393
    apply (rule abs_leI)
obua@14738
  1394
    apply (simp add: i1)
obua@14738
  1395
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  1396
    done
obua@14738
  1397
qed
paulson@14294
  1398
obua@14738
  1399
lemma abs_eq_mult: 
obua@14738
  1400
  assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
obua@14738
  1401
  shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
obua@14738
  1402
proof -
obua@14738
  1403
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  1404
    apply (auto)    
obua@14738
  1405
    apply (rule_tac split_mult_pos_le)
obua@14738
  1406
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  1407
    apply (simp)
obua@14738
  1408
    apply (rule_tac split_mult_neg_le)
obua@14738
  1409
    apply (insert prems)
obua@14738
  1410
    apply (blast)
obua@14738
  1411
    done
obua@14738
  1412
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  1413
    by (simp add: prts[symmetric])
obua@14738
  1414
  show ?thesis
obua@14738
  1415
  proof cases
obua@14738
  1416
    assume "0 <= a * b"
obua@14738
  1417
    then show ?thesis
obua@14738
  1418
      apply (simp_all add: mulprts abs_prts)
obua@14754
  1419
      apply (simp add: 
obua@14754
  1420
	iff2imp[OF zero_le_iff_zero_nprt]
obua@14754
  1421
	iff2imp[OF le_zero_iff_pprt_id]
obua@14754
  1422
      )
obua@14738
  1423
      apply (insert prems)
obua@14754
  1424
      apply (auto simp add: 
obua@14754
  1425
	ring_eq_simps 
obua@14754
  1426
	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
obua@14754
  1427
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id] 
obua@14754
  1428
	order_antisym mult_pos_neg_le[of a b] mult_pos_neg2_le[of b a])
obua@14738
  1429
      done
obua@14738
  1430
  next
obua@14738
  1431
    assume "~(0 <= a*b)"
obua@14738
  1432
    with s have "a*b <= 0" by simp
obua@14738
  1433
    then show ?thesis
obua@14738
  1434
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1435
      apply (insert prems)
obua@14738
  1436
      apply (auto simp add: ring_eq_simps iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
obua@14738
  1437
	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
  1438
      done
obua@14738
  1439
  qed
obua@14738
  1440
qed
paulson@14294
  1441
obua@14738
  1442
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
obua@14738
  1443
by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  1444
obua@14738
  1445
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
obua@14738
  1446
by (simp add: abs_if) 
paulson@14294
  1447
paulson@14294
  1448
lemma nonzero_abs_inverse:
paulson@14294
  1449
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  1450
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  1451
                      negative_imp_inverse_negative)
paulson@14294
  1452
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  1453
done
paulson@14294
  1454
paulson@14294
  1455
lemma abs_inverse [simp]:
paulson@14294
  1456
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  1457
      inverse (abs a)"
paulson@14294
  1458
apply (case_tac "a=0", simp) 
paulson@14294
  1459
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1460
done
paulson@14294
  1461
paulson@14294
  1462
lemma nonzero_abs_divide:
paulson@14294
  1463
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  1464
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  1465
paulson@14294
  1466
lemma abs_divide:
paulson@14294
  1467
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
paulson@14294
  1468
apply (case_tac "b=0", simp) 
paulson@14294
  1469
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1470
done
paulson@14294
  1471
paulson@14294
  1472
lemma abs_mult_less:
obua@14738
  1473
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  1474
proof -
paulson@14294
  1475
  assume ac: "abs a < c"
paulson@14294
  1476
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  1477
  assume "abs b < d"
paulson@14294
  1478
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1479
qed
paulson@14293
  1480
obua@14738
  1481
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
obua@14738
  1482
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
obua@14738
  1483
obua@14738
  1484
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
obua@14738
  1485
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
obua@14738
  1486
obua@14738
  1487
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  1488
apply (simp add: order_less_le abs_le_iff)  
obua@14738
  1489
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
obua@14738
  1490
apply (simp add: le_minus_self_iff linorder_neq_iff) 
obua@14738
  1491
done
obua@14738
  1492
paulson@14430
  1493
text{*Moving this up spoils many proofs using @{text mult_le_cancel_right}*}
paulson@14430
  1494
declare times_divide_eq_left [simp]
paulson@14430
  1495
obua@14738
  1496
ML {*
paulson@14334
  1497
val left_distrib = thm "left_distrib";
obua@14738
  1498
val right_distrib = thm "right_distrib";
obua@14738
  1499
val mult_commute = thm "mult_commute";
obua@14738
  1500
val distrib = thm "distrib";
obua@14738
  1501
val zero_neq_one = thm "zero_neq_one";
obua@14738
  1502
val no_zero_divisors = thm "no_zero_divisors";
paulson@14331
  1503
val left_inverse = thm "left_inverse";
obua@14738
  1504
val divide_inverse = thm "divide_inverse";
obua@14738
  1505
val mult_zero_left = thm "mult_zero_left";
obua@14738
  1506
val mult_zero_right = thm "mult_zero_right";
obua@14738
  1507
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";
obua@14738
  1508
val inverse_zero = thm "inverse_zero";
obua@14738
  1509
val ring_distrib = thms "ring_distrib";
obua@14738
  1510
val combine_common_factor = thm "combine_common_factor";
obua@14738
  1511
val minus_mult_left = thm "minus_mult_left";
obua@14738
  1512
val minus_mult_right = thm "minus_mult_right";
obua@14738
  1513
val minus_mult_minus = thm "minus_mult_minus";
obua@14738
  1514
val minus_mult_commute = thm "minus_mult_commute";
obua@14738
  1515
val right_diff_distrib = thm "right_diff_distrib";
obua@14738
  1516
val left_diff_distrib = thm "left_diff_distrib";
obua@14738
  1517
val mult_left_mono = thm "mult_left_mono";
obua@14738
  1518
val mult_right_mono = thm "mult_right_mono";
obua@14738
  1519
val mult_strict_left_mono = thm "mult_strict_left_mono";
obua@14738
  1520
val mult_strict_right_mono = thm "mult_strict_right_mono";
obua@14738
  1521
val mult_mono = thm "mult_mono";
obua@14738
  1522
val mult_strict_mono = thm "mult_strict_mono";
obua@14738
  1523
val abs_if = thm "abs_if";
obua@14738
  1524
val zero_less_one = thm "zero_less_one";
obua@14738
  1525
val eq_add_iff1 = thm "eq_add_iff1";
obua@14738
  1526
val eq_add_iff2 = thm "eq_add_iff2";
obua@14738
  1527
val less_add_iff1 = thm "less_add_iff1";
obua@14738
  1528
val less_add_iff2 = thm "less_add_iff2";
obua@14738
  1529
val le_add_iff1 = thm "le_add_iff1";
obua@14738
  1530
val le_add_iff2 = thm "le_add_iff2";
obua@14738
  1531
val mult_left_le_imp_le = thm "mult_left_le_imp_le";
obua@14738
  1532
val mult_right_le_imp_le = thm "mult_right_le_imp_le";
obua@14738
  1533
val mult_left_less_imp_less = thm "mult_left_less_imp_less";
obua@14738
  1534
val mult_right_less_imp_less = thm "mult_right_less_imp_less";
obua@14738
  1535
val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg";
obua@14738
  1536
val mult_left_mono_neg = thm "mult_left_mono_neg";
obua@14738
  1537
val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg";
obua@14738
  1538
val mult_right_mono_neg = thm "mult_right_mono_neg";
obua@14738
  1539
val mult_pos = thm "mult_pos";
obua@14738
  1540
val mult_pos_le = thm "mult_pos_le";
obua@14738
  1541
val mult_pos_neg = thm "mult_pos_neg";
obua@14738
  1542
val mult_pos_neg_le = thm "mult_pos_neg_le";
obua@14738
  1543
val mult_pos_neg2 = thm "mult_pos_neg2";
obua@14738
  1544
val mult_pos_neg2_le = thm "mult_pos_neg2_le";
obua@14738
  1545
val mult_neg = thm "mult_neg";
obua@14738
  1546
val mult_neg_le = thm "mult_neg_le";
obua@14738
  1547
val zero_less_mult_pos = thm "zero_less_mult_pos";
obua@14738
  1548
val zero_less_mult_pos2 = thm "zero_less_mult_pos2";
obua@14738
  1549
val zero_less_mult_iff = thm "zero_less_mult_iff";
obua@14738
  1550
val mult_eq_0_iff = thm "mult_eq_0_iff";
obua@14738
  1551
val zero_le_mult_iff = thm "zero_le_mult_iff";
obua@14738
  1552
val mult_less_0_iff = thm "mult_less_0_iff";
obua@14738
  1553
val mult_le_0_iff = thm "mult_le_0_iff";
obua@14738
  1554
val split_mult_pos_le = thm "split_mult_pos_le";
obua@14738
  1555
val split_mult_neg_le = thm "split_mult_neg_le";
obua@14738
  1556
val zero_le_square = thm "zero_le_square";
obua@14738
  1557
val zero_le_one = thm "zero_le_one";
obua@14738
  1558
val not_one_le_zero = thm "not_one_le_zero";
obua@14738
  1559
val not_one_less_zero = thm "not_one_less_zero";
obua@14738
  1560
val mult_left_mono_neg = thm "mult_left_mono_neg";
obua@14738
  1561
val mult_right_mono_neg = thm "mult_right_mono_neg";
obua@14738
  1562
val mult_strict_mono = thm "mult_strict_mono";
obua@14738
  1563
val mult_strict_mono' = thm "mult_strict_mono'";
obua@14738
  1564
val mult_mono = thm "mult_mono";
obua@14738
  1565
val less_1_mult = thm "less_1_mult";
obua@14738
  1566
val mult_less_cancel_right = thm "mult_less_cancel_right";
obua@14738
  1567
val mult_less_cancel_left = thm "mult_less_cancel_left";
obua@14738
  1568
val mult_le_cancel_right = thm "mult_le_cancel_right";
obua@14738
  1569
val mult_le_cancel_left = thm "mult_le_cancel_left";
obua@14738
  1570
val mult_less_imp_less_left = thm "mult_less_imp_less_left";
obua@14738
  1571
val mult_less_imp_less_right = thm "mult_less_imp_less_right";
obua@14738
  1572
val mult_cancel_right = thm "mult_cancel_right";
obua@14738
  1573
val mult_cancel_left = thm "mult_cancel_left";
obua@14738
  1574
val ring_eq_simps = thms "ring_eq_simps";
obua@14738
  1575
val right_inverse = thm "right_inverse";
obua@14738
  1576
val right_inverse_eq = thm "right_inverse_eq";
obua@14738
  1577
val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide";
obua@14738
  1578
val divide_self = thm "divide_self";
obua@14738
  1579
val divide_zero = thm "divide_zero";
obua@14738
  1580
val divide_zero_left = thm "divide_zero_left";
obua@14738
  1581
val inverse_eq_divide = thm "inverse_eq_divide";
obua@14738
  1582
val add_divide_distrib = thm "add_divide_distrib";
obua@14738
  1583
val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";
obua@14738
  1584
val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma";
obua@14738
  1585
val field_mult_cancel_right = thm "field_mult_cancel_right";
obua@14738
  1586
val field_mult_cancel_left = thm "field_mult_cancel_left";
obua@14738
  1587
val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero";
obua@14738
  1588
val inverse_zero_imp_zero = thm "inverse_zero_imp_zero";
obua@14738
  1589
val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero";
obua@14738
  1590
val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero";
obua@14738
  1591
val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq";
obua@14738
  1592
val inverse_minus_eq = thm "inverse_minus_eq";
obua@14738
  1593
val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq";
obua@14738
  1594
val inverse_eq_imp_eq = thm "inverse_eq_imp_eq";
obua@14738
  1595
val inverse_eq_iff_eq = thm "inverse_eq_iff_eq";
obua@14738
  1596
val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq";
obua@14738
  1597
val inverse_inverse_eq = thm "inverse_inverse_eq";
obua@14738
  1598
val inverse_1 = thm "inverse_1";
obua@14738
  1599
val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib";
obua@14738
  1600
val inverse_mult_distrib = thm "inverse_mult_distrib";
obua@14738
  1601
val inverse_add = thm "inverse_add";
obua@14738
  1602
val inverse_divide = thm "inverse_divide";
obua@14738
  1603
val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left";
obua@14738
  1604
val mult_divide_cancel_left = thm "mult_divide_cancel_left";
obua@14738
  1605
val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right";
obua@14738
  1606
val mult_divide_cancel_right = thm "mult_divide_cancel_right";
obua@14738
  1607
val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if";
obua@14738
  1608
val divide_1 = thm "divide_1";
obua@14738
  1609
val times_divide_eq_right = thm "times_divide_eq_right";
obua@14738
  1610
val times_divide_eq_left = thm "times_divide_eq_left";
obua@14738
  1611
val divide_divide_eq_right = thm "divide_divide_eq_right";
obua@14738
  1612
val divide_divide_eq_left = thm "divide_divide_eq_left";
obua@14738
  1613
val nonzero_minus_divide_left = thm "nonzero_minus_divide_left";
obua@14738
  1614
val nonzero_minus_divide_right = thm "nonzero_minus_divide_right";
obua@14738
  1615
val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide";
obua@14738
  1616
val minus_divide_left = thm "minus_divide_left";
obua@14738
  1617
val minus_divide_right = thm "minus_divide_right";
obua@14738
  1618
val minus_divide_divide = thm "minus_divide_divide";
obua@14738
  1619
val diff_divide_distrib = thm "diff_divide_distrib";
obua@14738
  1620
val positive_imp_inverse_positive = thm "positive_imp_inverse_positive";
obua@14738
  1621
val negative_imp_inverse_negative = thm "negative_imp_inverse_negative";
obua@14738
  1622
val inverse_le_imp_le = thm "inverse_le_imp_le";
obua@14738
  1623
val inverse_positive_imp_positive = thm "inverse_positive_imp_positive";
obua@14738
  1624
val inverse_positive_iff_positive = thm "inverse_positive_iff_positive";
obua@14738
  1625
val inverse_negative_imp_negative = thm "inverse_negative_imp_negative";
obua@14738
  1626
val inverse_negative_iff_negative = thm "inverse_negative_iff_negative";
obua@14738
  1627
val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative";
obua@14738
  1628
val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive";
obua@14738
  1629
val less_imp_inverse_less = thm "less_imp_inverse_less";
obua@14738
  1630
val inverse_less_imp_less = thm "inverse_less_imp_less";
obua@14738
  1631
val inverse_less_iff_less = thm "inverse_less_iff_less";
obua@14738
  1632
val le_imp_inverse_le = thm "le_imp_inverse_le";
obua@14738
  1633
val inverse_le_iff_le = thm "inverse_le_iff_le";
obua@14738
  1634
val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg";
obua@14738
  1635
val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg";
obua@14738
  1636
val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg";
obua@14738
  1637
val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg";
obua@14738
  1638
val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg";
obua@14738
  1639
val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg";
obua@14738
  1640
val one_less_inverse_iff = thm "one_less_inverse_iff";
obua@14738
  1641
val inverse_eq_1_iff = thm "inverse_eq_1_iff";
obua@14738
  1642
val one_le_inverse_iff = thm "one_le_inverse_iff";
obua@14738
  1643
val inverse_less_1_iff = thm "inverse_less_1_iff";
obua@14738
  1644
val inverse_le_1_iff = thm "inverse_le_1_iff";
obua@14738
  1645
val zero_less_divide_iff = thm "zero_less_divide_iff";
obua@14738
  1646
val divide_less_0_iff = thm "divide_less_0_iff";
obua@14738
  1647
val zero_le_divide_iff = thm "zero_le_divide_iff";
obua@14738
  1648
val divide_le_0_iff = thm "divide_le_0_iff";
obua@14738
  1649
val divide_eq_0_iff = thm "divide_eq_0_iff";
obua@14738
  1650
val pos_le_divide_eq = thm "pos_le_divide_eq";
obua@14738
  1651
val neg_le_divide_eq = thm "neg_le_divide_eq";
obua@14738
  1652
val le_divide_eq = thm "le_divide_eq";
obua@14738
  1653
val pos_divide_le_eq = thm "pos_divide_le_eq";
obua@14738
  1654
val neg_divide_le_eq = thm "neg_divide_le_eq";
obua@14738
  1655
val divide_le_eq = thm "divide_le_eq";
obua@14738
  1656
val pos_less_divide_eq = thm "pos_less_divide_eq";
obua@14738
  1657
val neg_less_divide_eq = thm "neg_less_divide_eq";
obua@14738
  1658
val less_divide_eq = thm "less_divide_eq";
obua@14738
  1659
val pos_divide_less_eq = thm "pos_divide_less_eq";
obua@14738
  1660
val neg_divide_less_eq = thm "neg_divide_less_eq";
obua@14738
  1661
val divide_less_eq = thm "divide_less_eq";
obua@14738
  1662
val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq";
obua@14738
  1663
val eq_divide_eq = thm "eq_divide_eq";
obua@14738
  1664
val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq";
obua@14738
  1665
val divide_eq_eq = thm "divide_eq_eq";
obua@14738
  1666
val divide_cancel_right = thm "divide_cancel_right";
obua@14738
  1667
val divide_cancel_left = thm "divide_cancel_left";
obua@14738
  1668
val divide_eq_1_iff = thm "divide_eq_1_iff";
obua@14738
  1669
val one_eq_divide_iff = thm "one_eq_divide_iff";
obua@14738
  1670
val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff";
obua@14738
  1671
val one_divide_eq_0_iff = thm "one_divide_eq_0_iff";
obua@14738
  1672
val divide_strict_right_mono = thm "divide_strict_right_mono";
obua@14738
  1673
val divide_right_mono = thm "divide_right_mono";
obua@14738
  1674
val divide_strict_left_mono = thm "divide_strict_left_mono";
obua@14738
  1675
val divide_left_mono = thm "divide_left_mono";
obua@14738
  1676
val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg";
obua@14738
  1677
val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg";
obua@14738
  1678
val less_add_one = thm "less_add_one";
obua@14738
  1679
val zero_less_two = thm "zero_less_two";
obua@14738
  1680
val less_half_sum = thm "less_half_sum";
obua@14738
  1681
val gt_half_sum = thm "gt_half_sum";
obua@14738
  1682
val dense = thm "dense";
obua@14738
  1683
val abs_one = thm "abs_one";
obua@14738
  1684
val abs_le_mult = thm "abs_le_mult";
obua@14738
  1685
val abs_eq_mult = thm "abs_eq_mult";
obua@14738
  1686
val abs_mult = thm "abs_mult";
obua@14738
  1687
val abs_mult_self = thm "abs_mult_self";
obua@14738
  1688
val nonzero_abs_inverse = thm "nonzero_abs_inverse";
obua@14738
  1689
val abs_inverse = thm "abs_inverse";
obua@14738
  1690
val nonzero_abs_divide = thm "nonzero_abs_divide";
obua@14738
  1691
val abs_divide = thm "abs_divide";
obua@14738
  1692
val abs_mult_less = thm "abs_mult_less";
obua@14738
  1693
val eq_minus_self_iff = thm "eq_minus_self_iff";
obua@14738
  1694
val less_minus_self_iff = thm "less_minus_self_iff";
obua@14738
  1695
val abs_less_iff = thm "abs_less_iff";
paulson@14331
  1696
*}
paulson@14331
  1697
paulson@14265
  1698
end