src/HOL/Ring_and_Field.thy
author nipkow
Thu Aug 30 21:43:31 2007 +0200 (2007-08-30)
changeset 24491 8d194c9198ae
parent 24427 bc5cf3b09ff3
child 24506 020db6ec334a
permissions -rw-r--r--
added constant sgn
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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, Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel,
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             with contributions by Jeremy Avigad
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*)
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header {* (Ordered) Rings and Fields *}
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theory Ring_and_Field
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imports OrderedGroup
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begin
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text {*
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  The theory of partially ordered rings is taken from the books:
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  \begin{itemize}
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  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
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  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
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  \end{itemize}
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  Most of the used notions can also be looked up in 
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  \begin{itemize}
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  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
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  \item \emph{Algebra I} by van der Waerden, Springer.
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  \end{itemize}
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*}
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes left_distrib: "(a \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* c"
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  assumes right_distrib: "a \<^loc>* (b \<^loc>+ c) = a \<^loc>* b \<^loc>+ a \<^loc>* c"
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "\<^loc>0 \<^loc>* a = \<^loc>0"
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  assumes mult_zero_right [simp]: "a \<^loc>* \<^loc>0 = \<^loc>0"
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + comm_monoid_add + cancel_ab_semigroup_add
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instance semiring_0_cancel \<subseteq> semiring_0
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proof
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  fix a :: 'a
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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 "0 * a = 0"
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    by (simp only: add_left_cancel)
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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 "a * 0 = 0"
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    by (simp only: add_left_cancel)
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qed
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* 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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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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instance comm_semiring_0 \<subseteq> semiring_0 ..
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class comm_semiring_0_cancel = comm_semiring + comm_monoid_add + cancel_ab_semigroup_add
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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..
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instance comm_semiring_0_cancel \<subseteq> comm_semiring_0 ..
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "\<^loc>0 \<noteq> \<^loc>1"
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
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  (*previously almost_semiring*)
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instance comm_semiring_1 \<subseteq> semiring_1 ..
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class no_zero_divisors = zero + times +
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  assumes no_zero_divisors: "a \<noteq> \<^loc>0 \<Longrightarrow> b \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* b \<noteq> \<^loc>0"
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class semiring_1_cancel = semiring + comm_monoid_add + zero_neq_one
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  + cancel_ab_semigroup_add + monoid_mult
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instance semiring_1_cancel \<subseteq> semiring_0_cancel ..
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instance semiring_1_cancel \<subseteq> semiring_1 ..
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class comm_semiring_1_cancel = comm_semiring + comm_monoid_add + comm_monoid_mult
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  + zero_neq_one + cancel_ab_semigroup_add
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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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instance comm_semiring_1_cancel \<subseteq> comm_semiring_1 ..
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class ring = semiring + ab_group_add
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instance ring \<subseteq> semiring_0_cancel ..
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class comm_ring = comm_semiring + 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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class ring_1 = ring + zero_neq_one + monoid_mult
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instance ring_1 \<subseteq> semiring_1_cancel ..
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class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
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  (*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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class ring_no_zero_divisors = ring + no_zero_divisors
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class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
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class idom = comm_ring_1 + no_zero_divisors
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instance idom \<subseteq> ring_1_no_zero_divisors ..
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class division_ring = ring_1 + inverse +
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  assumes left_inverse [simp]:  "a \<noteq> \<^loc>0 \<Longrightarrow> inverse a \<^loc>* a = \<^loc>1"
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  assumes right_inverse [simp]: "a \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* inverse a = \<^loc>1"
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instance division_ring \<subseteq> ring_1_no_zero_divisors
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proof
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  fix a b :: 'a
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  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
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  show "a * b \<noteq> 0"
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  proof
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    assume ab: "a * b = 0"
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    hence "0 = inverse a * (a * b) * inverse b"
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      by simp
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    also have "\<dots> = (inverse a * a) * (b * inverse b)"
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      by (simp only: mult_assoc)
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    also have "\<dots> = 1"
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      using a b by simp
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    finally show False
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      by simp
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  qed
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qed
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class field = comm_ring_1 + inverse +
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  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a \<^loc>* a = \<^loc>1"
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  assumes divide_inverse: "a \<^loc>/ b = a \<^loc>* inverse b"
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instance field \<subseteq> division_ring
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proof
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  fix a :: 'a
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  assume "a \<noteq> 0"
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  thus "inverse a * a = 1" by (rule field_inverse)
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  thus "a * inverse a = 1" by (simp only: mult_commute)
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qed
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instance field \<subseteq> idom ..
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class division_by_zero = zero + inverse +
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  assumes inverse_zero [simp]: "inverse \<^loc>0 = \<^loc>0"
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subsection {* Distribution rules *}
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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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lemmas ring_distribs =
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  right_distrib left_distrib left_diff_distrib right_diff_distrib
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text{*This list of rewrites simplifies ring terms by multiplying
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everything out and bringing sums and products into a canonical form
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(by ordered rewriting). As a result it decides ring equalities but
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also helps with inequalities. *}
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lemmas ring_simps = group_simps ring_distribs
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class mult_mono = times + zero + ord +
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  assumes mult_left_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b"
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  assumes mult_right_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> a \<^loc>* c \<sqsubseteq> b \<^loc>* c"
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class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
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class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
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  + semiring + comm_monoid_add + cancel_ab_semigroup_add
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instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..
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instance pordered_cancel_semiring \<subseteq> pordered_semiring .. 
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class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
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instance ordered_semiring \<subseteq> pordered_cancel_semiring ..
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class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
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  assumes mult_strict_left_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b"
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  assumes mult_strict_right_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> a \<^loc>* c \<sqsubset> b \<^loc>* c"
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instance ordered_semiring_strict \<subseteq> semiring_0_cancel ..
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instance ordered_semiring_strict \<subseteq> ordered_semiring
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proof
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  fix a b c :: 'a
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  assume A: "a \<le> b" "0 \<le> c"
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  from A show "c * a \<le> c * b"
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    unfolding order_le_less
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    using mult_strict_left_mono by auto
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  from A show "a * c \<le> b * c"
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    unfolding order_le_less
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    using mult_strict_right_mono by auto
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qed
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class mult_mono1 = times + zero + ord +
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  assumes mult_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b"
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class pordered_comm_semiring = comm_semiring_0
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  + pordered_ab_semigroup_add + mult_mono1
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class pordered_cancel_comm_semiring = comm_semiring_0_cancel
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  + pordered_ab_semigroup_add + mult_mono1
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instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
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class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
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  assumes mult_strict_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b"
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instance pordered_comm_semiring \<subseteq> pordered_semiring
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proof
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  fix a b c :: 'a
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  assume "a \<le> b" "0 \<le> c"
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  thus "c * a \<le> c * b" by (rule mult_mono)
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  thus "a * c \<le> b * c" by (simp only: mult_commute)
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qed
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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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proof
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  fix a b c :: 'a
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  assume "a < b" "0 < c"
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  thus "c * a < c * b" by (rule mult_strict_mono)
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  thus "a * c < b * c" by (simp only: mult_commute)
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qed
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instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
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proof
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  fix a b c :: 'a
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  assume "a \<le> b" "0 \<le> c"
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  thus "c * a \<le> c * b"
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    unfolding order_le_less
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    using mult_strict_mono by auto
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qed
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class pordered_ring = ring + pordered_cancel_semiring 
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instance pordered_ring \<subseteq> pordered_ab_group_add ..
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class lordered_ring = 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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class abs_if = minus + ord + zero + abs +
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  assumes abs_if: "abs a = (if a \<sqsubset> 0 then (uminus a) else a)"
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(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
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   Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
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 *)
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class ordered_ring = ring + ordered_semiring + lordered_ab_group + abs_if
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instance ordered_ring \<subseteq> lordered_ring
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proof
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  fix x :: 'a
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  show "\<bar>x\<bar> = sup x (- x)"
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    by (simp only: abs_if sup_eq_if)
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qed
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class ordered_ring_strict = ring + ordered_semiring_strict + lordered_ab_group + abs_if
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instance ordered_ring_strict \<subseteq> ordered_ring ..
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class pordered_comm_ring = comm_ring + pordered_comm_semiring
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instance pordered_comm_ring \<subseteq> pordered_ring ..
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instance pordered_comm_ring \<subseteq> pordered_cancel_comm_semiring ..
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class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
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  (*previously ordered_semiring*)
haftmann@22390
   326
  assumes zero_less_one [simp]: "\<^loc>0 \<sqsubset> \<^loc>1"
paulson@14270
   327
haftmann@24422
   328
lemma pos_add_strict:
haftmann@24422
   329
  fixes a b c :: "'a\<Colon>ordered_semidom"
haftmann@24422
   330
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@24422
   331
  using add_strict_mono [of 0 a b c] by simp
haftmann@24422
   332
obua@23521
   333
class ordered_idom = comm_ring_1 + ordered_comm_semiring_strict + lordered_ab_group + abs_if
haftmann@22390
   334
  (*previously ordered_ring*)
paulson@14270
   335
nipkow@24491
   336
definition (in ordered_idom) sgn where
nipkow@24491
   337
"sgn x = (if x = \<^loc>0 then \<^loc>0 else if \<^loc>0 \<sqsubset> x then \<^loc>1 else uminus \<^loc>1)"
nipkow@24491
   338
obua@14738
   339
instance ordered_idom \<subseteq> ordered_ring_strict ..
paulson@14272
   340
huffman@23073
   341
instance ordered_idom \<subseteq> pordered_comm_ring ..
huffman@23073
   342
haftmann@22390
   343
class ordered_field = field + ordered_idom
paulson@14272
   344
nipkow@15923
   345
lemmas linorder_neqE_ordered_idom =
nipkow@15923
   346
 linorder_neqE[where 'a = "?'b::ordered_idom"]
nipkow@15923
   347
paulson@14272
   348
lemma eq_add_iff1:
nipkow@23477
   349
  "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
nipkow@23477
   350
by (simp add: ring_simps)
paulson@14272
   351
paulson@14272
   352
lemma eq_add_iff2:
nipkow@23477
   353
  "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
nipkow@23477
   354
by (simp add: ring_simps)
paulson@14272
   355
paulson@14272
   356
lemma less_add_iff1:
nipkow@23477
   357
  "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
nipkow@23477
   358
by (simp add: ring_simps)
paulson@14272
   359
paulson@14272
   360
lemma less_add_iff2:
nipkow@23477
   361
  "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
nipkow@23477
   362
by (simp add: ring_simps)
paulson@14272
   363
paulson@14272
   364
lemma le_add_iff1:
nipkow@23477
   365
  "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
nipkow@23477
   366
by (simp add: ring_simps)
paulson@14272
   367
paulson@14272
   368
lemma le_add_iff2:
nipkow@23477
   369
  "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
nipkow@23477
   370
by (simp add: ring_simps)
paulson@14272
   371
wenzelm@23389
   372
paulson@14270
   373
subsection {* Ordering Rules for Multiplication *}
paulson@14270
   374
paulson@14348
   375
lemma mult_left_le_imp_le:
nipkow@23477
   376
  "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
nipkow@23477
   377
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
paulson@14348
   378
 
paulson@14348
   379
lemma mult_right_le_imp_le:
nipkow@23477
   380
  "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
nipkow@23477
   381
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
paulson@14348
   382
paulson@14348
   383
lemma mult_left_less_imp_less:
obua@23521
   384
  "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
nipkow@23477
   385
by (force simp add: mult_left_mono linorder_not_le [symmetric])
paulson@14348
   386
 
paulson@14348
   387
lemma mult_right_less_imp_less:
obua@23521
   388
  "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
nipkow@23477
   389
by (force simp add: mult_right_mono linorder_not_le [symmetric])
paulson@14348
   390
paulson@14265
   391
lemma mult_strict_left_mono_neg:
nipkow@23477
   392
  "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
paulson@14265
   393
apply (drule mult_strict_left_mono [of _ _ "-c"])
paulson@14265
   394
apply (simp_all add: minus_mult_left [symmetric]) 
paulson@14265
   395
done
paulson@14265
   396
obua@14738
   397
lemma mult_left_mono_neg:
nipkow@23477
   398
  "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
obua@14738
   399
apply (drule mult_left_mono [of _ _ "-c"])
obua@14738
   400
apply (simp_all add: minus_mult_left [symmetric]) 
obua@14738
   401
done
obua@14738
   402
paulson@14265
   403
lemma mult_strict_right_mono_neg:
nipkow@23477
   404
  "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
paulson@14265
   405
apply (drule mult_strict_right_mono [of _ _ "-c"])
paulson@14265
   406
apply (simp_all add: minus_mult_right [symmetric]) 
paulson@14265
   407
done
paulson@14265
   408
obua@14738
   409
lemma mult_right_mono_neg:
nipkow@23477
   410
  "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
obua@14738
   411
apply (drule mult_right_mono [of _ _ "-c"])
obua@14738
   412
apply (simp)
obua@14738
   413
apply (simp_all add: minus_mult_right [symmetric]) 
obua@14738
   414
done
paulson@14265
   415
wenzelm@23389
   416
paulson@14265
   417
subsection{* Products of Signs *}
paulson@14265
   418
avigad@16775
   419
lemma mult_pos_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
paulson@14265
   420
by (drule mult_strict_left_mono [of 0 b], auto)
paulson@14265
   421
avigad@16775
   422
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
obua@14738
   423
by (drule mult_left_mono [of 0 b], auto)
obua@14738
   424
obua@14738
   425
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
paulson@14265
   426
by (drule mult_strict_left_mono [of b 0], auto)
paulson@14265
   427
avigad@16775
   428
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
obua@14738
   429
by (drule mult_left_mono [of b 0], auto)
obua@14738
   430
obua@14738
   431
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
obua@14738
   432
by (drule mult_strict_right_mono[of b 0], auto)
obua@14738
   433
avigad@16775
   434
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
obua@14738
   435
by (drule mult_right_mono[of b 0], auto)
obua@14738
   436
avigad@16775
   437
lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
paulson@14265
   438
by (drule mult_strict_right_mono_neg, auto)
paulson@14265
   439
avigad@16775
   440
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
obua@14738
   441
by (drule mult_right_mono_neg[of a 0 b ], auto)
obua@14738
   442
paulson@14341
   443
lemma zero_less_mult_pos:
obua@14738
   444
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   445
apply (cases "b\<le>0") 
paulson@14265
   446
 apply (auto simp add: order_le_less linorder_not_less)
paulson@14265
   447
apply (drule_tac mult_pos_neg [of a b]) 
paulson@14265
   448
 apply (auto dest: order_less_not_sym)
paulson@14265
   449
done
paulson@14265
   450
obua@14738
   451
lemma zero_less_mult_pos2:
obua@14738
   452
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   453
apply (cases "b\<le>0") 
obua@14738
   454
 apply (auto simp add: order_le_less linorder_not_less)
obua@14738
   455
apply (drule_tac mult_pos_neg2 [of a b]) 
obua@14738
   456
 apply (auto dest: order_less_not_sym)
obua@14738
   457
done
obua@14738
   458
paulson@14265
   459
lemma zero_less_mult_iff:
obua@14738
   460
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
   461
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos 
avigad@16775
   462
  mult_neg_neg)
paulson@14265
   463
apply (blast dest: zero_less_mult_pos) 
obua@14738
   464
apply (blast dest: zero_less_mult_pos2)
paulson@14265
   465
done
paulson@14265
   466
huffman@22990
   467
lemma mult_eq_0_iff [simp]:
huffman@22990
   468
  fixes a b :: "'a::ring_no_zero_divisors"
huffman@22990
   469
  shows "(a * b = 0) = (a = 0 \<or> b = 0)"
huffman@22990
   470
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors)
huffman@22990
   471
huffman@22990
   472
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
huffman@22990
   473
apply intro_classes
paulson@14265
   474
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
paulson@14265
   475
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
paulson@14265
   476
done
paulson@14265
   477
paulson@14265
   478
lemma zero_le_mult_iff:
obua@14738
   479
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14265
   480
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
paulson@14265
   481
                   zero_less_mult_iff)
paulson@14265
   482
paulson@14265
   483
lemma mult_less_0_iff:
obua@14738
   484
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14265
   485
apply (insert zero_less_mult_iff [of "-a" b]) 
paulson@14265
   486
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   487
done
paulson@14265
   488
paulson@14265
   489
lemma mult_le_0_iff:
obua@14738
   490
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14265
   491
apply (insert zero_le_mult_iff [of "-a" b]) 
paulson@14265
   492
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   493
done
paulson@14265
   494
obua@14738
   495
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"
avigad@16775
   496
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14738
   497
obua@14738
   498
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)" 
avigad@16775
   499
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
obua@14738
   500
obua@23095
   501
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a"
obua@23095
   502
by (simp add: zero_le_mult_iff linorder_linear)
obua@23095
   503
obua@23095
   504
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))"
obua@23095
   505
by (simp add: not_less)
paulson@14265
   506
obua@14738
   507
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
obua@14738
   508
      theorems available to members of @{term ordered_idom} *}
obua@14738
   509
obua@14738
   510
instance ordered_idom \<subseteq> ordered_semidom
paulson@14421
   511
proof
paulson@14421
   512
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
paulson@14430
   513
  thus "(0::'a) < 1" by (simp add: order_le_less) 
paulson@14421
   514
qed
paulson@14421
   515
obua@14738
   516
instance ordered_idom \<subseteq> idom ..
obua@14738
   517
paulson@14387
   518
text{*All three types of comparision involving 0 and 1 are covered.*}
paulson@14387
   519
paulson@17085
   520
lemmas one_neq_zero = zero_neq_one [THEN not_sym]
paulson@17085
   521
declare one_neq_zero [simp]
paulson@14387
   522
obua@14738
   523
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
paulson@14268
   524
  by (rule zero_less_one [THEN order_less_imp_le]) 
paulson@14268
   525
obua@14738
   526
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
obua@14738
   527
by (simp add: linorder_not_le) 
paulson@14387
   528
obua@14738
   529
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
obua@14738
   530
by (simp add: linorder_not_less) 
paulson@14268
   531
wenzelm@23389
   532
paulson@14268
   533
subsection{*More Monotonicity*}
paulson@14268
   534
paulson@14268
   535
text{*Strict monotonicity in both arguments*}
paulson@14268
   536
lemma mult_strict_mono:
obua@14738
   537
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
haftmann@21328
   538
apply (cases "c=0")
avigad@16775
   539
 apply (simp add: mult_pos_pos) 
paulson@14268
   540
apply (erule mult_strict_right_mono [THEN order_less_trans])
paulson@14268
   541
 apply (force simp add: order_le_less) 
paulson@14268
   542
apply (erule mult_strict_left_mono, assumption)
paulson@14268
   543
done
paulson@14268
   544
paulson@14268
   545
text{*This weaker variant has more natural premises*}
paulson@14268
   546
lemma mult_strict_mono':
obua@14738
   547
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   548
apply (rule mult_strict_mono)
paulson@14268
   549
apply (blast intro: order_le_less_trans)+
paulson@14268
   550
done
paulson@14268
   551
paulson@14268
   552
lemma mult_mono:
paulson@14268
   553
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
obua@14738
   554
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
paulson@14268
   555
apply (erule mult_right_mono [THEN order_trans], assumption)
paulson@14268
   556
apply (erule mult_left_mono, assumption)
paulson@14268
   557
done
paulson@14268
   558
huffman@21258
   559
lemma mult_mono':
huffman@21258
   560
     "[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|] 
huffman@21258
   561
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
huffman@21258
   562
apply (rule mult_mono)
huffman@21258
   563
apply (fast intro: order_trans)+
huffman@21258
   564
done
huffman@21258
   565
obua@14738
   566
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
paulson@14387
   567
apply (insert mult_strict_mono [of 1 m 1 n]) 
paulson@14430
   568
apply (simp add:  order_less_trans [OF zero_less_one]) 
paulson@14387
   569
done
paulson@14387
   570
avigad@16775
   571
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
avigad@16775
   572
    c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
avigad@16775
   573
  apply (subgoal_tac "a * c < b * c")
avigad@16775
   574
  apply (erule order_less_le_trans)
avigad@16775
   575
  apply (erule mult_left_mono)
avigad@16775
   576
  apply simp
avigad@16775
   577
  apply (erule mult_strict_right_mono)
avigad@16775
   578
  apply assumption
avigad@16775
   579
done
avigad@16775
   580
avigad@16775
   581
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
avigad@16775
   582
    c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
avigad@16775
   583
  apply (subgoal_tac "a * c <= b * c")
avigad@16775
   584
  apply (erule order_le_less_trans)
avigad@16775
   585
  apply (erule mult_strict_left_mono)
avigad@16775
   586
  apply simp
avigad@16775
   587
  apply (erule mult_right_mono)
avigad@16775
   588
  apply simp
avigad@16775
   589
done
avigad@16775
   590
wenzelm@23389
   591
paulson@14268
   592
subsection{*Cancellation Laws for Relationships With a Common Factor*}
paulson@14268
   593
paulson@14268
   594
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
paulson@14268
   595
   also with the relations @{text "\<le>"} and equality.*}
paulson@14268
   596
paulson@15234
   597
text{*These ``disjunction'' versions produce two cases when the comparison is
paulson@15234
   598
 an assumption, but effectively four when the comparison is a goal.*}
paulson@15234
   599
paulson@15234
   600
lemma mult_less_cancel_right_disj:
obua@14738
   601
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   602
apply (cases "c = 0")
paulson@14268
   603
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
paulson@14268
   604
                      mult_strict_right_mono_neg)
paulson@14268
   605
apply (auto simp add: linorder_not_less 
paulson@14268
   606
                      linorder_not_le [symmetric, of "a*c"]
paulson@14268
   607
                      linorder_not_le [symmetric, of a])
paulson@14268
   608
apply (erule_tac [!] notE)
paulson@14268
   609
apply (auto simp add: order_less_imp_le mult_right_mono 
paulson@14268
   610
                      mult_right_mono_neg)
paulson@14268
   611
done
paulson@14268
   612
paulson@15234
   613
lemma mult_less_cancel_left_disj:
obua@14738
   614
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   615
apply (cases "c = 0")
obua@14738
   616
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
obua@14738
   617
                      mult_strict_left_mono_neg)
obua@14738
   618
apply (auto simp add: linorder_not_less 
obua@14738
   619
                      linorder_not_le [symmetric, of "c*a"]
obua@14738
   620
                      linorder_not_le [symmetric, of a])
obua@14738
   621
apply (erule_tac [!] notE)
obua@14738
   622
apply (auto simp add: order_less_imp_le mult_left_mono 
obua@14738
   623
                      mult_left_mono_neg)
obua@14738
   624
done
paulson@14268
   625
paulson@15234
   626
paulson@15234
   627
text{*The ``conjunction of implication'' lemmas produce two cases when the
paulson@15234
   628
comparison is a goal, but give four when the comparison is an assumption.*}
paulson@15234
   629
paulson@15234
   630
lemma mult_less_cancel_right:
paulson@15234
   631
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   632
  shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   633
by (insert mult_less_cancel_right_disj [of a c b], auto)
paulson@15234
   634
paulson@15234
   635
lemma mult_less_cancel_left:
paulson@15234
   636
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   637
  shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   638
by (insert mult_less_cancel_left_disj [of c a b], auto)
paulson@15234
   639
paulson@14268
   640
lemma mult_le_cancel_right:
obua@14738
   641
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   642
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
paulson@14268
   643
paulson@14268
   644
lemma mult_le_cancel_left:
obua@14738
   645
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   646
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
paulson@14268
   647
paulson@14268
   648
lemma mult_less_imp_less_left:
paulson@14341
   649
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
obua@14738
   650
      shows "a < (b::'a::ordered_semiring_strict)"
paulson@14377
   651
proof (rule ccontr)
paulson@14377
   652
  assume "~ a < b"
paulson@14377
   653
  hence "b \<le> a" by (simp add: linorder_not_less)
wenzelm@23389
   654
  hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono)
paulson@14377
   655
  with this and less show False 
paulson@14377
   656
    by (simp add: linorder_not_less [symmetric])
paulson@14377
   657
qed
paulson@14268
   658
paulson@14268
   659
lemma mult_less_imp_less_right:
obua@14738
   660
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
obua@14738
   661
  shows "a < (b::'a::ordered_semiring_strict)"
obua@14738
   662
proof (rule ccontr)
obua@14738
   663
  assume "~ a < b"
obua@14738
   664
  hence "b \<le> a" by (simp add: linorder_not_less)
wenzelm@23389
   665
  hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono)
obua@14738
   666
  with this and less show False 
obua@14738
   667
    by (simp add: linorder_not_less [symmetric])
obua@14738
   668
qed  
paulson@14268
   669
paulson@14268
   670
text{*Cancellation of equalities with a common factor*}
paulson@24286
   671
lemma mult_cancel_right [simp,noatp]:
huffman@22990
   672
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   673
  shows "(a * c = b * c) = (c = 0 \<or> a = b)"
huffman@22990
   674
proof -
huffman@22990
   675
  have "(a * c = b * c) = ((a - b) * c = 0)"
nipkow@23477
   676
    by (simp add: ring_distribs)
huffman@22990
   677
  thus ?thesis
huffman@22990
   678
    by (simp add: disj_commute)
huffman@22990
   679
qed
paulson@14268
   680
paulson@24286
   681
lemma mult_cancel_left [simp,noatp]:
huffman@22990
   682
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   683
  shows "(c * a = c * b) = (c = 0 \<or> a = b)"
huffman@22990
   684
proof -
huffman@22990
   685
  have "(c * a = c * b) = (c * (a - b) = 0)"
nipkow@23477
   686
    by (simp add: ring_distribs)
huffman@22990
   687
  thus ?thesis
huffman@22990
   688
    by simp
huffman@22990
   689
qed
paulson@14268
   690
paulson@15234
   691
paulson@15234
   692
subsubsection{*Special Cancellation Simprules for Multiplication*}
paulson@15234
   693
paulson@15234
   694
text{*These also produce two cases when the comparison is a goal.*}
paulson@15234
   695
paulson@15234
   696
lemma mult_le_cancel_right1:
paulson@15234
   697
  fixes c :: "'a :: ordered_idom"
paulson@15234
   698
  shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   699
by (insert mult_le_cancel_right [of 1 c b], simp)
paulson@15234
   700
paulson@15234
   701
lemma mult_le_cancel_right2:
paulson@15234
   702
  fixes c :: "'a :: ordered_idom"
paulson@15234
   703
  shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   704
by (insert mult_le_cancel_right [of a c 1], simp)
paulson@15234
   705
paulson@15234
   706
lemma mult_le_cancel_left1:
paulson@15234
   707
  fixes c :: "'a :: ordered_idom"
paulson@15234
   708
  shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   709
by (insert mult_le_cancel_left [of c 1 b], simp)
paulson@15234
   710
paulson@15234
   711
lemma mult_le_cancel_left2:
paulson@15234
   712
  fixes c :: "'a :: ordered_idom"
paulson@15234
   713
  shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   714
by (insert mult_le_cancel_left [of c a 1], simp)
paulson@15234
   715
paulson@15234
   716
lemma mult_less_cancel_right1:
paulson@15234
   717
  fixes c :: "'a :: ordered_idom"
paulson@15234
   718
  shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   719
by (insert mult_less_cancel_right [of 1 c b], simp)
paulson@15234
   720
paulson@15234
   721
lemma mult_less_cancel_right2:
paulson@15234
   722
  fixes c :: "'a :: ordered_idom"
paulson@15234
   723
  shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   724
by (insert mult_less_cancel_right [of a c 1], simp)
paulson@15234
   725
paulson@15234
   726
lemma mult_less_cancel_left1:
paulson@15234
   727
  fixes c :: "'a :: ordered_idom"
paulson@15234
   728
  shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   729
by (insert mult_less_cancel_left [of c 1 b], simp)
paulson@15234
   730
paulson@15234
   731
lemma mult_less_cancel_left2:
paulson@15234
   732
  fixes c :: "'a :: ordered_idom"
paulson@15234
   733
  shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   734
by (insert mult_less_cancel_left [of c a 1], simp)
paulson@15234
   735
paulson@15234
   736
lemma mult_cancel_right1 [simp]:
huffman@23544
   737
  fixes c :: "'a :: ring_1_no_zero_divisors"
paulson@15234
   738
  shows "(c = b*c) = (c = 0 | b=1)"
paulson@15234
   739
by (insert mult_cancel_right [of 1 c b], force)
paulson@15234
   740
paulson@15234
   741
lemma mult_cancel_right2 [simp]:
huffman@23544
   742
  fixes c :: "'a :: ring_1_no_zero_divisors"
paulson@15234
   743
  shows "(a*c = c) = (c = 0 | a=1)"
paulson@15234
   744
by (insert mult_cancel_right [of a c 1], simp)
paulson@15234
   745
 
paulson@15234
   746
lemma mult_cancel_left1 [simp]:
huffman@23544
   747
  fixes c :: "'a :: ring_1_no_zero_divisors"
paulson@15234
   748
  shows "(c = c*b) = (c = 0 | b=1)"
paulson@15234
   749
by (insert mult_cancel_left [of c 1 b], force)
paulson@15234
   750
paulson@15234
   751
lemma mult_cancel_left2 [simp]:
huffman@23544
   752
  fixes c :: "'a :: ring_1_no_zero_divisors"
paulson@15234
   753
  shows "(c*a = c) = (c = 0 | a=1)"
paulson@15234
   754
by (insert mult_cancel_left [of c a 1], simp)
paulson@15234
   755
paulson@15234
   756
paulson@15234
   757
text{*Simprules for comparisons where common factors can be cancelled.*}
paulson@15234
   758
lemmas mult_compare_simps =
paulson@15234
   759
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
   760
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
   761
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
   762
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
   763
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
   764
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
   765
    mult_cancel_right mult_cancel_left
paulson@15234
   766
    mult_cancel_right1 mult_cancel_right2
paulson@15234
   767
    mult_cancel_left1 mult_cancel_left2
paulson@15234
   768
paulson@15234
   769
paulson@14265
   770
subsection {* Fields *}
paulson@14265
   771
paulson@14288
   772
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
paulson@14288
   773
proof
paulson@14288
   774
  assume neq: "b \<noteq> 0"
paulson@14288
   775
  {
paulson@14288
   776
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
paulson@14288
   777
    also assume "a / b = 1"
paulson@14288
   778
    finally show "a = b" by simp
paulson@14288
   779
  next
paulson@14288
   780
    assume "a = b"
paulson@14288
   781
    with neq show "a / b = 1" by (simp add: divide_inverse)
paulson@14288
   782
  }
paulson@14288
   783
qed
paulson@14288
   784
paulson@14288
   785
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
paulson@14288
   786
by (simp add: divide_inverse)
paulson@14288
   787
nipkow@23398
   788
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
paulson@14288
   789
  by (simp add: divide_inverse)
paulson@14288
   790
paulson@14430
   791
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
paulson@14430
   792
by (simp add: divide_inverse)
paulson@14277
   793
paulson@15228
   794
lemma divide_self_if [simp]:
paulson@15228
   795
     "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
paulson@15228
   796
  by (simp add: divide_self)
paulson@15228
   797
paulson@14430
   798
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
paulson@14430
   799
by (simp add: divide_inverse)
paulson@14277
   800
paulson@14430
   801
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
paulson@14430
   802
by (simp add: divide_inverse)
paulson@14277
   803
paulson@14430
   804
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
nipkow@23477
   805
by (simp add: divide_inverse ring_distribs) 
paulson@14293
   806
nipkow@23482
   807
(* what ordering?? this is a straight instance of mult_eq_0_iff
paulson@14270
   808
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
paulson@14270
   809
      of an ordering.*}
huffman@20496
   810
lemma field_mult_eq_0_iff [simp]:
huffman@20496
   811
  "(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
huffman@22990
   812
by simp
nipkow@23482
   813
*)
nipkow@23496
   814
(* subsumed by mult_cancel lemmas on ring_no_zero_divisors
paulson@14268
   815
text{*Cancellation of equalities with a common factor*}
paulson@14268
   816
lemma field_mult_cancel_right_lemma:
huffman@20496
   817
      assumes cnz: "c \<noteq> (0::'a::division_ring)"
huffman@20496
   818
         and eq:  "a*c = b*c"
huffman@20496
   819
        shows "a=b"
paulson@14377
   820
proof -
paulson@14268
   821
  have "(a * c) * inverse c = (b * c) * inverse c"
paulson@14268
   822
    by (simp add: eq)
paulson@14268
   823
  thus "a=b"
paulson@14268
   824
    by (simp add: mult_assoc cnz)
paulson@14377
   825
qed
paulson@14268
   826
paulson@14348
   827
lemma field_mult_cancel_right [simp]:
huffman@20496
   828
     "(a*c = b*c) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   829
by simp
paulson@14268
   830
paulson@14348
   831
lemma field_mult_cancel_left [simp]:
huffman@20496
   832
     "(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   833
by simp
nipkow@23496
   834
*)
huffman@20496
   835
lemma nonzero_imp_inverse_nonzero:
huffman@20496
   836
  "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
paulson@14377
   837
proof
paulson@14268
   838
  assume ianz: "inverse a = 0"
paulson@14268
   839
  assume "a \<noteq> 0"
paulson@14268
   840
  hence "1 = a * inverse a" by simp
paulson@14268
   841
  also have "... = 0" by (simp add: ianz)
huffman@20496
   842
  finally have "1 = (0::'a::division_ring)" .
paulson@14268
   843
  thus False by (simp add: eq_commute)
paulson@14377
   844
qed
paulson@14268
   845
paulson@14277
   846
paulson@14277
   847
subsection{*Basic Properties of @{term inverse}*}
paulson@14277
   848
huffman@20496
   849
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   850
apply (rule ccontr) 
paulson@14268
   851
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   852
done
paulson@14268
   853
paulson@14268
   854
lemma inverse_nonzero_imp_nonzero:
huffman@20496
   855
   "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   856
apply (rule ccontr) 
paulson@14268
   857
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   858
done
paulson@14268
   859
paulson@14268
   860
lemma inverse_nonzero_iff_nonzero [simp]:
huffman@20496
   861
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
paulson@14268
   862
by (force dest: inverse_nonzero_imp_nonzero) 
paulson@14268
   863
paulson@14268
   864
lemma nonzero_inverse_minus_eq:
huffman@20496
   865
      assumes [simp]: "a\<noteq>0"
huffman@20496
   866
      shows "inverse(-a) = -inverse(a::'a::division_ring)"
paulson@14377
   867
proof -
paulson@14377
   868
  have "-a * inverse (- a) = -a * - inverse a"
paulson@14377
   869
    by simp
paulson@14377
   870
  thus ?thesis 
nipkow@23496
   871
    by (simp only: mult_cancel_left, simp)
paulson@14377
   872
qed
paulson@14268
   873
paulson@14268
   874
lemma inverse_minus_eq [simp]:
huffman@20496
   875
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
paulson@14377
   876
proof cases
paulson@14377
   877
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
   878
next
paulson@14377
   879
  assume "a\<noteq>0" 
paulson@14377
   880
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
   881
qed
paulson@14268
   882
paulson@14268
   883
lemma nonzero_inverse_eq_imp_eq:
paulson@14269
   884
      assumes inveq: "inverse a = inverse b"
paulson@14269
   885
	  and anz:  "a \<noteq> 0"
paulson@14269
   886
	  and bnz:  "b \<noteq> 0"
huffman@20496
   887
	 shows "a = (b::'a::division_ring)"
paulson@14377
   888
proof -
paulson@14268
   889
  have "a * inverse b = a * inverse a"
paulson@14268
   890
    by (simp add: inveq)
paulson@14268
   891
  hence "(a * inverse b) * b = (a * inverse a) * b"
paulson@14268
   892
    by simp
paulson@14268
   893
  thus "a = b"
paulson@14268
   894
    by (simp add: mult_assoc anz bnz)
paulson@14377
   895
qed
paulson@14268
   896
paulson@14268
   897
lemma inverse_eq_imp_eq:
huffman@20496
   898
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
haftmann@21328
   899
apply (cases "a=0 | b=0") 
paulson@14268
   900
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
   901
              simp add: eq_commute [of "0::'a"])
paulson@14268
   902
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
   903
done
paulson@14268
   904
paulson@14268
   905
lemma inverse_eq_iff_eq [simp]:
huffman@20496
   906
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
huffman@20496
   907
by (force dest!: inverse_eq_imp_eq)
paulson@14268
   908
paulson@14270
   909
lemma nonzero_inverse_inverse_eq:
huffman@20496
   910
      assumes [simp]: "a \<noteq> 0"
huffman@20496
   911
      shows "inverse(inverse (a::'a::division_ring)) = a"
paulson@14270
   912
  proof -
paulson@14270
   913
  have "(inverse (inverse a) * inverse a) * a = a" 
paulson@14270
   914
    by (simp add: nonzero_imp_inverse_nonzero)
paulson@14270
   915
  thus ?thesis
paulson@14270
   916
    by (simp add: mult_assoc)
paulson@14270
   917
  qed
paulson@14270
   918
paulson@14270
   919
lemma inverse_inverse_eq [simp]:
huffman@20496
   920
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
paulson@14270
   921
  proof cases
paulson@14270
   922
    assume "a=0" thus ?thesis by simp
paulson@14270
   923
  next
paulson@14270
   924
    assume "a\<noteq>0" 
paulson@14270
   925
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
   926
  qed
paulson@14270
   927
huffman@20496
   928
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)"
paulson@14270
   929
  proof -
huffman@20496
   930
  have "inverse 1 * 1 = (1::'a::division_ring)" 
paulson@14270
   931
    by (rule left_inverse [OF zero_neq_one [symmetric]])
paulson@14270
   932
  thus ?thesis  by simp
paulson@14270
   933
  qed
paulson@14270
   934
paulson@15077
   935
lemma inverse_unique: 
paulson@15077
   936
  assumes ab: "a*b = 1"
huffman@20496
   937
  shows "inverse a = (b::'a::division_ring)"
paulson@15077
   938
proof -
paulson@15077
   939
  have "a \<noteq> 0" using ab by auto
paulson@15077
   940
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
paulson@15077
   941
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
paulson@15077
   942
qed
paulson@15077
   943
paulson@14270
   944
lemma nonzero_inverse_mult_distrib: 
paulson@14270
   945
      assumes anz: "a \<noteq> 0"
paulson@14270
   946
          and bnz: "b \<noteq> 0"
huffman@20496
   947
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
paulson@14270
   948
  proof -
paulson@14270
   949
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
nipkow@23482
   950
    by (simp add: anz bnz)
paulson@14270
   951
  hence "inverse(a*b) * a = inverse(b)" 
paulson@14270
   952
    by (simp add: mult_assoc bnz)
paulson@14270
   953
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
paulson@14270
   954
    by simp
paulson@14270
   955
  thus ?thesis
paulson@14270
   956
    by (simp add: mult_assoc anz)
paulson@14270
   957
  qed
paulson@14270
   958
paulson@14270
   959
text{*This version builds in division by zero while also re-orienting
paulson@14270
   960
      the right-hand side.*}
paulson@14270
   961
lemma inverse_mult_distrib [simp]:
paulson@14270
   962
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
   963
  proof cases
paulson@14270
   964
    assume "a \<noteq> 0 & b \<noteq> 0" 
haftmann@22993
   965
    thus ?thesis
haftmann@22993
   966
      by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
   967
  next
paulson@14270
   968
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
haftmann@22993
   969
    thus ?thesis
haftmann@22993
   970
      by force
paulson@14270
   971
  qed
paulson@14270
   972
huffman@20496
   973
lemma division_ring_inverse_add:
huffman@20496
   974
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
huffman@20496
   975
   ==> inverse a + inverse b = inverse a * (a+b) * inverse b"
nipkow@23477
   976
by (simp add: ring_simps)
huffman@20496
   977
huffman@20496
   978
lemma division_ring_inverse_diff:
huffman@20496
   979
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
huffman@20496
   980
   ==> inverse a - inverse b = inverse a * (b-a) * inverse b"
nipkow@23477
   981
by (simp add: ring_simps)
huffman@20496
   982
paulson@14270
   983
text{*There is no slick version using division by zero.*}
paulson@14270
   984
lemma inverse_add:
nipkow@23477
   985
  "[|a \<noteq> 0;  b \<noteq> 0|]
nipkow@23477
   986
   ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
huffman@20496
   987
by (simp add: division_ring_inverse_add mult_ac)
paulson@14270
   988
paulson@14365
   989
lemma inverse_divide [simp]:
nipkow@23477
   990
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
nipkow@23477
   991
by (simp add: divide_inverse mult_commute)
paulson@14365
   992
wenzelm@23389
   993
avigad@16775
   994
subsection {* Calculations with fractions *}
avigad@16775
   995
nipkow@23413
   996
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
   997
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
   998
because the latter are covered by a simproc. *}
nipkow@23413
   999
paulson@24427
  1000
lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]:
nipkow@23477
  1001
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
  1002
proof -
paulson@14277
  1003
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
nipkow@23482
  1004
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
paulson@14277
  1005
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
  1006
    by (simp only: mult_ac)
paulson@14277
  1007
  also have "... =  a * inverse b"
paulson@14277
  1008
    by simp
paulson@14277
  1009
    finally show ?thesis 
paulson@14277
  1010
    by (simp add: divide_inverse)
paulson@14277
  1011
qed
paulson@14277
  1012
nipkow@23413
  1013
lemma mult_divide_mult_cancel_left:
nipkow@23477
  1014
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1015
apply (cases "b = 0")
nipkow@23413
  1016
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
paulson@14277
  1017
done
paulson@14277
  1018
paulson@24427
  1019
lemma nonzero_mult_divide_mult_cancel_right [noatp]:
nipkow@23477
  1020
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1021
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
paulson@14321
  1022
nipkow@23413
  1023
lemma mult_divide_mult_cancel_right:
nipkow@23477
  1024
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1025
apply (cases "b = 0")
nipkow@23413
  1026
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
paulson@14321
  1027
done
nipkow@23413
  1028
paulson@14284
  1029
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
nipkow@23477
  1030
by (simp add: divide_inverse)
paulson@14284
  1031
paulson@15234
  1032
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
  1033
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1034
paulson@14430
  1035
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
  1036
by (simp add: divide_inverse mult_ac)
paulson@14288
  1037
nipkow@23482
  1038
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
nipkow@23482
  1039
paulson@24286
  1040
lemma divide_divide_eq_right [simp,noatp]:
nipkow@23477
  1041
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
  1042
by (simp add: divide_inverse mult_ac)
paulson@14288
  1043
paulson@24286
  1044
lemma divide_divide_eq_left [simp,noatp]:
nipkow@23477
  1045
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
  1046
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1047
avigad@16775
  1048
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1049
    x / y + w / z = (x * z + w * y) / (y * z)"
nipkow@23477
  1050
apply (subgoal_tac "x / y = (x * z) / (y * z)")
nipkow@23477
  1051
apply (erule ssubst)
nipkow@23477
  1052
apply (subgoal_tac "w / z = (w * y) / (y * z)")
nipkow@23477
  1053
apply (erule ssubst)
nipkow@23477
  1054
apply (rule add_divide_distrib [THEN sym])
nipkow@23477
  1055
apply (subst mult_commute)
nipkow@23477
  1056
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
nipkow@23477
  1057
apply assumption
nipkow@23477
  1058
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
nipkow@23477
  1059
apply assumption
avigad@16775
  1060
done
paulson@14268
  1061
wenzelm@23389
  1062
paulson@15234
  1063
subsubsection{*Special Cancellation Simprules for Division*}
paulson@15234
  1064
paulson@24427
  1065
lemma mult_divide_mult_cancel_left_if[simp,noatp]:
nipkow@23477
  1066
fixes c :: "'a :: {field,division_by_zero}"
nipkow@23477
  1067
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
nipkow@23413
  1068
by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
  1069
paulson@24427
  1070
lemma nonzero_mult_divide_cancel_right[simp,noatp]:
nipkow@23413
  1071
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
nipkow@23413
  1072
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
nipkow@23413
  1073
paulson@24427
  1074
lemma nonzero_mult_divide_cancel_left[simp,noatp]:
nipkow@23413
  1075
  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
nipkow@23413
  1076
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
nipkow@23413
  1077
nipkow@23413
  1078
paulson@24427
  1079
lemma nonzero_divide_mult_cancel_right[simp,noatp]:
nipkow@23413
  1080
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
nipkow@23413
  1081
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
nipkow@23413
  1082
paulson@24427
  1083
lemma nonzero_divide_mult_cancel_left[simp,noatp]:
nipkow@23413
  1084
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
nipkow@23413
  1085
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
nipkow@23413
  1086
nipkow@23413
  1087
paulson@24427
  1088
lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]:
nipkow@23477
  1089
  "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1090
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
nipkow@23413
  1091
paulson@24427
  1092
lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]:
nipkow@23477
  1093
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
nipkow@23413
  1094
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
nipkow@23413
  1095
paulson@15234
  1096
paulson@14293
  1097
subsection {* Division and Unary Minus *}
paulson@14293
  1098
paulson@14293
  1099
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
  1100
by (simp add: divide_inverse minus_mult_left)
paulson@14293
  1101
paulson@14293
  1102
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
  1103
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
  1104
paulson@14293
  1105
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
  1106
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1107
paulson@14430
  1108
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
paulson@14430
  1109
by (simp add: divide_inverse minus_mult_left [symmetric])
paulson@14293
  1110
paulson@14293
  1111
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14430
  1112
by (simp add: divide_inverse minus_mult_right [symmetric])
paulson@14430
  1113
paulson@14293
  1114
paulson@14293
  1115
text{*The effect is to extract signs from divisions*}
paulson@17085
  1116
lemmas divide_minus_left = minus_divide_left [symmetric]
paulson@17085
  1117
lemmas divide_minus_right = minus_divide_right [symmetric]
paulson@17085
  1118
declare divide_minus_left [simp]   divide_minus_right [simp]
paulson@14293
  1119
paulson@14387
  1120
text{*Also, extract signs from products*}
paulson@17085
  1121
lemmas mult_minus_left = minus_mult_left [symmetric]
paulson@17085
  1122
lemmas mult_minus_right = minus_mult_right [symmetric]
paulson@17085
  1123
declare mult_minus_left [simp]   mult_minus_right [simp]
paulson@14387
  1124
paulson@14293
  1125
lemma minus_divide_divide [simp]:
nipkow@23477
  1126
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1127
apply (cases "b=0", simp) 
paulson@14293
  1128
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1129
done
paulson@14293
  1130
paulson@14430
  1131
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
  1132
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
  1133
nipkow@23482
  1134
lemma add_divide_eq_iff:
nipkow@23482
  1135
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
nipkow@23482
  1136
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1137
nipkow@23482
  1138
lemma divide_add_eq_iff:
nipkow@23482
  1139
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
nipkow@23482
  1140
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1141
nipkow@23482
  1142
lemma diff_divide_eq_iff:
nipkow@23482
  1143
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
nipkow@23482
  1144
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1145
nipkow@23482
  1146
lemma divide_diff_eq_iff:
nipkow@23482
  1147
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
nipkow@23482
  1148
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1149
nipkow@23482
  1150
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
nipkow@23482
  1151
proof -
nipkow@23482
  1152
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1153
  have "(a = b/c) = (a*c = (b/c)*c)" by simp
nipkow@23496
  1154
  also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
nipkow@23482
  1155
  finally show ?thesis .
nipkow@23482
  1156
qed
nipkow@23482
  1157
nipkow@23482
  1158
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
nipkow@23482
  1159
proof -
nipkow@23482
  1160
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1161
  have "(b/c = a) = ((b/c)*c = a*c)"  by simp
nipkow@23496
  1162
  also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
nipkow@23482
  1163
  finally show ?thesis .
nipkow@23482
  1164
qed
nipkow@23482
  1165
nipkow@23482
  1166
lemma eq_divide_eq:
nipkow@23482
  1167
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
nipkow@23482
  1168
by (simp add: nonzero_eq_divide_eq) 
nipkow@23482
  1169
nipkow@23482
  1170
lemma divide_eq_eq:
nipkow@23482
  1171
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
nipkow@23482
  1172
by (force simp add: nonzero_divide_eq_eq) 
nipkow@23482
  1173
nipkow@23482
  1174
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1175
    b = a * c ==> b / c = a"
nipkow@23482
  1176
  by (subst divide_eq_eq, simp)
nipkow@23482
  1177
nipkow@23482
  1178
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1179
    a * c = b ==> a = b / c"
nipkow@23482
  1180
  by (subst eq_divide_eq, simp)
nipkow@23482
  1181
nipkow@23482
  1182
nipkow@23482
  1183
lemmas field_eq_simps = ring_simps
nipkow@23482
  1184
  (* pull / out*)
nipkow@23482
  1185
  add_divide_eq_iff divide_add_eq_iff
nipkow@23482
  1186
  diff_divide_eq_iff divide_diff_eq_iff
nipkow@23482
  1187
  (* multiply eqn *)
nipkow@23482
  1188
  nonzero_eq_divide_eq nonzero_divide_eq_eq
nipkow@23482
  1189
(* is added later:
nipkow@23482
  1190
  times_divide_eq_left times_divide_eq_right
nipkow@23482
  1191
*)
nipkow@23482
  1192
nipkow@23482
  1193
text{*An example:*}
nipkow@23482
  1194
lemma fixes a b c d e f :: "'a::field"
nipkow@23482
  1195
shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"
nipkow@23482
  1196
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
nipkow@23482
  1197
 apply(simp add:field_eq_simps)
nipkow@23482
  1198
apply(simp)
nipkow@23482
  1199
done
nipkow@23482
  1200
nipkow@23482
  1201
avigad@16775
  1202
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1203
    x / y - w / z = (x * z - w * y) / (y * z)"
nipkow@23482
  1204
by (simp add:field_eq_simps times_divide_eq)
nipkow@23482
  1205
nipkow@23482
  1206
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
nipkow@23482
  1207
    (x / y = w / z) = (x * z = w * y)"
nipkow@23482
  1208
by (simp add:field_eq_simps times_divide_eq)
paulson@14293
  1209
wenzelm@23389
  1210
paulson@14268
  1211
subsection {* Ordered Fields *}
paulson@14268
  1212
paulson@14277
  1213
lemma positive_imp_inverse_positive: 
nipkow@23482
  1214
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
nipkow@23482
  1215
proof -
paulson@14268
  1216
  have "0 < a * inverse a" 
paulson@14268
  1217
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1218
  thus "0 < inverse a" 
paulson@14268
  1219
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
nipkow@23482
  1220
qed
paulson@14268
  1221
paulson@14277
  1222
lemma negative_imp_inverse_negative:
nipkow@23482
  1223
  "a < 0 ==> inverse a < (0::'a::ordered_field)"
nipkow@23482
  1224
by (insert positive_imp_inverse_positive [of "-a"], 
nipkow@23482
  1225
    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
paulson@14268
  1226
paulson@14268
  1227
lemma inverse_le_imp_le:
nipkow@23482
  1228
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
nipkow@23482
  1229
shows "b \<le> (a::'a::ordered_field)"
nipkow@23482
  1230
proof (rule classical)
paulson@14268
  1231
  assume "~ b \<le> a"
nipkow@23482
  1232
  hence "a < b"  by (simp add: linorder_not_le)
nipkow@23482
  1233
  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
paulson@14268
  1234
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1235
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1236
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1237
    by (simp add: bpos order_less_imp_le mult_right_mono)
nipkow@23482
  1238
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
nipkow@23482
  1239
qed
paulson@14268
  1240
paulson@14277
  1241
lemma inverse_positive_imp_positive:
nipkow@23482
  1242
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
nipkow@23482
  1243
shows "0 < (a::'a::ordered_field)"
wenzelm@23389
  1244
proof -
paulson@14277
  1245
  have "0 < inverse (inverse a)"
wenzelm@23389
  1246
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
  1247
  thus "0 < a"
wenzelm@23389
  1248
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1249
qed
paulson@14277
  1250
paulson@14277
  1251
lemma inverse_positive_iff_positive [simp]:
nipkow@23482
  1252
  "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1253
apply (cases "a = 0", simp)
paulson@14277
  1254
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1255
done
paulson@14277
  1256
paulson@14277
  1257
lemma inverse_negative_imp_negative:
nipkow@23482
  1258
assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
nipkow@23482
  1259
shows "a < (0::'a::ordered_field)"
wenzelm@23389
  1260
proof -
paulson@14277
  1261
  have "inverse (inverse a) < 0"
wenzelm@23389
  1262
    using inv_less_0 by (rule negative_imp_inverse_negative)
nipkow@23482
  1263
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1264
qed
paulson@14277
  1265
paulson@14277
  1266
lemma inverse_negative_iff_negative [simp]:
nipkow@23482
  1267
  "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1268
apply (cases "a = 0", simp)
paulson@14277
  1269
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1270
done
paulson@14277
  1271
paulson@14277
  1272
lemma inverse_nonnegative_iff_nonnegative [simp]:
nipkow@23482
  1273
  "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1274
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1275
paulson@14277
  1276
lemma inverse_nonpositive_iff_nonpositive [simp]:
nipkow@23482
  1277
  "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1278
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1279
chaieb@23406
  1280
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
chaieb@23406
  1281
proof
chaieb@23406
  1282
  fix x::'a
chaieb@23406
  1283
  have m1: "- (1::'a) < 0" by simp
chaieb@23406
  1284
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1285
  have "(- 1) + x < x" by simp
chaieb@23406
  1286
  thus "\<exists>y. y < x" by blast
chaieb@23406
  1287
qed
chaieb@23406
  1288
chaieb@23406
  1289
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
chaieb@23406
  1290
proof
chaieb@23406
  1291
  fix x::'a
chaieb@23406
  1292
  have m1: " (1::'a) > 0" by simp
chaieb@23406
  1293
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1294
  have "1 + x > x" by simp
chaieb@23406
  1295
  thus "\<exists>y. y > x" by blast
chaieb@23406
  1296
qed
paulson@14277
  1297
paulson@14277
  1298
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1299
paulson@14268
  1300
lemma less_imp_inverse_less:
nipkow@23482
  1301
assumes less: "a < b" and apos:  "0 < a"
nipkow@23482
  1302
shows "inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1303
proof (rule ccontr)
paulson@14268
  1304
  assume "~ inverse b < inverse a"
paulson@14268
  1305
  hence "inverse a \<le> inverse b"
paulson@14268
  1306
    by (simp add: linorder_not_less)
paulson@14268
  1307
  hence "~ (a < b)"
paulson@14268
  1308
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
  1309
  thus False
paulson@14268
  1310
    by (rule notE [OF _ less])
nipkow@23482
  1311
qed
paulson@14268
  1312
paulson@14268
  1313
lemma inverse_less_imp_less:
nipkow@23482
  1314
  "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1315
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1316
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1317
done
paulson@14268
  1318
paulson@14268
  1319
text{*Both premises are essential. Consider -1 and 1.*}
paulson@24286
  1320
lemma inverse_less_iff_less [simp,noatp]:
nipkow@23482
  1321
  "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1322
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1323
paulson@14268
  1324
lemma le_imp_inverse_le:
nipkow@23482
  1325
  "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1326
by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1327
paulson@24286
  1328
lemma inverse_le_iff_le [simp,noatp]:
nipkow@23482
  1329
 "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1330
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1331
paulson@14268
  1332
paulson@14268
  1333
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1334
case is trivial, since inverse preserves signs.*}
paulson@14268
  1335
lemma inverse_le_imp_le_neg:
nipkow@23482
  1336
  "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
nipkow@23482
  1337
apply (rule classical) 
nipkow@23482
  1338
apply (subgoal_tac "a < 0") 
nipkow@23482
  1339
 prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
nipkow@23482
  1340
apply (insert inverse_le_imp_le [of "-b" "-a"])
nipkow@23482
  1341
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1342
done
paulson@14268
  1343
paulson@14268
  1344
lemma less_imp_inverse_less_neg:
paulson@14268
  1345
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1346
apply (subgoal_tac "a < 0") 
nipkow@23482
  1347
 prefer 2 apply (blast intro: order_less_trans) 
nipkow@23482
  1348
apply (insert less_imp_inverse_less [of "-b" "-a"])
nipkow@23482
  1349
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1350
done
paulson@14268
  1351
paulson@14268
  1352
lemma inverse_less_imp_less_neg:
paulson@14268
  1353
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
nipkow@23482
  1354
apply (rule classical) 
nipkow@23482
  1355
apply (subgoal_tac "a < 0") 
nipkow@23482
  1356
 prefer 2
nipkow@23482
  1357
 apply (force simp add: linorder_not_less intro: order_le_less_trans) 
nipkow@23482
  1358
apply (insert inverse_less_imp_less [of "-b" "-a"])
nipkow@23482
  1359
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1360
done
paulson@14268
  1361
paulson@24286
  1362
lemma inverse_less_iff_less_neg [simp,noatp]:
nipkow@23482
  1363
  "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
nipkow@23482
  1364
apply (insert inverse_less_iff_less [of "-b" "-a"])
nipkow@23482
  1365
apply (simp del: inverse_less_iff_less 
nipkow@23482
  1366
            add: order_less_imp_not_eq nonzero_inverse_minus_eq)
nipkow@23482
  1367
done
paulson@14268
  1368
paulson@14268
  1369
lemma le_imp_inverse_le_neg:
nipkow@23482
  1370
  "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1371
by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1372
paulson@24286
  1373
lemma inverse_le_iff_le_neg [simp,noatp]:
nipkow@23482
  1374
 "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1375
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1376
paulson@14277
  1377
paulson@14365
  1378
subsection{*Inverses and the Number One*}
paulson@14365
  1379
paulson@14365
  1380
lemma one_less_inverse_iff:
nipkow@23482
  1381
  "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
nipkow@23482
  1382
proof cases
paulson@14365
  1383
  assume "0 < x"
paulson@14365
  1384
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1385
    show ?thesis by simp
paulson@14365
  1386
next
paulson@14365
  1387
  assume notless: "~ (0 < x)"
paulson@14365
  1388
  have "~ (1 < inverse x)"
paulson@14365
  1389
  proof
paulson@14365
  1390
    assume "1 < inverse x"
paulson@14365
  1391
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1392
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1393
    finally show False by auto
paulson@14365
  1394
  qed
paulson@14365
  1395
  with notless show ?thesis by simp
paulson@14365
  1396
qed
paulson@14365
  1397
paulson@14365
  1398
lemma inverse_eq_1_iff [simp]:
nipkow@23482
  1399
  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1400
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1401
paulson@14365
  1402
lemma one_le_inverse_iff:
nipkow@23482
  1403
  "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1404
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1405
                    eq_commute [of 1]) 
paulson@14365
  1406
paulson@14365
  1407
lemma inverse_less_1_iff:
nipkow@23482
  1408
  "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1409
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1410
paulson@14365
  1411
lemma inverse_le_1_iff:
nipkow@23482
  1412
  "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1413
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1414
wenzelm@23389
  1415
paulson@14288
  1416
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1417
paulson@14288
  1418
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1419
proof -
paulson@14288
  1420
  assume less: "0<c"
paulson@14288
  1421
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1422
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1423
  also have "... = (a*c \<le> b)"
paulson@14288
  1424
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1425
  finally show ?thesis .
paulson@14288
  1426
qed
paulson@14288
  1427
paulson@14288
  1428
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1429
proof -
paulson@14288
  1430
  assume less: "c<0"
paulson@14288
  1431
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1432
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1433
  also have "... = (b \<le> a*c)"
paulson@14288
  1434
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1435
  finally show ?thesis .
paulson@14288
  1436
qed
paulson@14288
  1437
paulson@14288
  1438
lemma le_divide_eq:
paulson@14288
  1439
  "(a \<le> b/c) = 
paulson@14288
  1440
   (if 0 < c then a*c \<le> b
paulson@14288
  1441
             else if c < 0 then b \<le> a*c
paulson@14288
  1442
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1443
apply (cases "c=0", simp) 
paulson@14288
  1444
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1445
done
paulson@14288
  1446
paulson@14288
  1447
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1448
proof -
paulson@14288
  1449
  assume less: "0<c"
paulson@14288
  1450
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1451
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1452
  also have "... = (b \<le> a*c)"
paulson@14288
  1453
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1454
  finally show ?thesis .
paulson@14288
  1455
qed
paulson@14288
  1456
paulson@14288
  1457
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1458
proof -
paulson@14288
  1459
  assume less: "c<0"
paulson@14288
  1460
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1461
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1462
  also have "... = (a*c \<le> b)"
paulson@14288
  1463
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1464
  finally show ?thesis .
paulson@14288
  1465
qed
paulson@14288
  1466
paulson@14288
  1467
lemma divide_le_eq:
paulson@14288
  1468
  "(b/c \<le> a) = 
paulson@14288
  1469
   (if 0 < c then b \<le> a*c
paulson@14288
  1470
             else if c < 0 then a*c \<le> b
paulson@14288
  1471
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1472
apply (cases "c=0", simp) 
paulson@14288
  1473
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1474
done
paulson@14288
  1475
paulson@14288
  1476
lemma pos_less_divide_eq:
paulson@14288
  1477
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1478
proof -
paulson@14288
  1479
  assume less: "0<c"
paulson@14288
  1480
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@15234
  1481
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1482
  also have "... = (a*c < b)"
paulson@14288
  1483
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1484
  finally show ?thesis .
paulson@14288
  1485
qed
paulson@14288
  1486
paulson@14288
  1487
lemma neg_less_divide_eq:
paulson@14288
  1488
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1489
proof -
paulson@14288
  1490
  assume less: "c<0"
paulson@14288
  1491
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@15234
  1492
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1493
  also have "... = (b < a*c)"
paulson@14288
  1494
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1495
  finally show ?thesis .
paulson@14288
  1496
qed
paulson@14288
  1497
paulson@14288
  1498
lemma less_divide_eq:
paulson@14288
  1499
  "(a < b/c) = 
paulson@14288
  1500
   (if 0 < c then a*c < b
paulson@14288
  1501
             else if c < 0 then b < a*c
paulson@14288
  1502
             else  a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1503
apply (cases "c=0", simp) 
paulson@14288
  1504
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1505
done
paulson@14288
  1506
paulson@14288
  1507
lemma pos_divide_less_eq:
paulson@14288
  1508
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1509
proof -
paulson@14288
  1510
  assume less: "0<c"
paulson@14288
  1511
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@15234
  1512
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1513
  also have "... = (b < a*c)"
paulson@14288
  1514
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1515
  finally show ?thesis .
paulson@14288
  1516
qed
paulson@14288
  1517
paulson@14288
  1518
lemma neg_divide_less_eq:
paulson@14288
  1519
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1520
proof -
paulson@14288
  1521
  assume less: "c<0"
paulson@14288
  1522
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@15234
  1523
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1524
  also have "... = (a*c < b)"
paulson@14288
  1525
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1526
  finally show ?thesis .
paulson@14288
  1527
qed
paulson@14288
  1528
paulson@14288
  1529
lemma divide_less_eq:
paulson@14288
  1530
  "(b/c < a) = 
paulson@14288
  1531
   (if 0 < c then b < a*c
paulson@14288
  1532
             else if c < 0 then a*c < b
paulson@14288
  1533
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1534
apply (cases "c=0", simp) 
paulson@14288
  1535
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1536
done
paulson@14288
  1537
nipkow@23482
  1538
nipkow@23482
  1539
subsection{*Field simplification*}
nipkow@23482
  1540
nipkow@23482
  1541
text{* Lemmas @{text field_simps} multiply with denominators in
nipkow@23482
  1542
in(equations) if they can be proved to be non-zero (for equations) or
nipkow@23482
  1543
positive/negative (for inequations). *}
paulson@14288
  1544
nipkow@23482
  1545
lemmas field_simps = field_eq_simps
nipkow@23482
  1546
  (* multiply ineqn *)
nipkow@23482
  1547
  pos_divide_less_eq neg_divide_less_eq
nipkow@23482
  1548
  pos_less_divide_eq neg_less_divide_eq
nipkow@23482
  1549
  pos_divide_le_eq neg_divide_le_eq
nipkow@23482
  1550
  pos_le_divide_eq neg_le_divide_eq
paulson@14288
  1551
nipkow@23482
  1552
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
nipkow@23483
  1553
of positivity/negativity needed for @{text field_simps}. Have not added @{text
nipkow@23482
  1554
sign_simps} to @{text field_simps} because the former can lead to case
nipkow@23482
  1555
explosions. *}
paulson@14288
  1556
nipkow@23482
  1557
lemmas sign_simps = group_simps
nipkow@23482
  1558
  zero_less_mult_iff  mult_less_0_iff
paulson@14288
  1559
nipkow@23482
  1560
(* Only works once linear arithmetic is installed:
nipkow@23482
  1561
text{*An example:*}
nipkow@23482
  1562
lemma fixes a b c d e f :: "'a::ordered_field"
nipkow@23482
  1563
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
nipkow@23482
  1564
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
nipkow@23482
  1565
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
nipkow@23482
  1566
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
nipkow@23482
  1567
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1568
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
nipkow@23482
  1569
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1570
apply(simp add:field_simps)
avigad@16775
  1571
done
nipkow@23482
  1572
*)
avigad@16775
  1573
wenzelm@23389
  1574
avigad@16775
  1575
subsection{*Division and Signs*}
avigad@16775
  1576
avigad@16775
  1577
lemma zero_less_divide_iff:
avigad@16775
  1578
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
  1579
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
  1580
avigad@16775
  1581
lemma divide_less_0_iff:
avigad@16775
  1582
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
avigad@16775
  1583
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
  1584
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
  1585
avigad@16775
  1586
lemma zero_le_divide_iff:
avigad@16775
  1587
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
avigad@16775
  1588
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
  1589
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1590
avigad@16775
  1591
lemma divide_le_0_iff:
avigad@16775
  1592
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
avigad@16775
  1593
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1594
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1595
paulson@24286
  1596
lemma divide_eq_0_iff [simp,noatp]:
avigad@16775
  1597
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
nipkow@23482
  1598
by (simp add: divide_inverse)
avigad@16775
  1599
nipkow@23482
  1600
lemma divide_pos_pos:
nipkow@23482
  1601
  "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
nipkow@23482
  1602
by(simp add:field_simps)
nipkow@23482
  1603
avigad@16775
  1604
nipkow@23482
  1605
lemma divide_nonneg_pos:
nipkow@23482
  1606
  "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
nipkow@23482
  1607
by(simp add:field_simps)
avigad@16775
  1608
nipkow@23482
  1609
lemma divide_neg_pos:
nipkow@23482
  1610
  "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
nipkow@23482
  1611
by(simp add:field_simps)
avigad@16775
  1612
nipkow@23482
  1613
lemma divide_nonpos_pos:
nipkow@23482
  1614
  "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
nipkow@23482
  1615
by(simp add:field_simps)
avigad@16775
  1616
nipkow@23482
  1617
lemma divide_pos_neg:
nipkow@23482
  1618
  "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
nipkow@23482
  1619
by(simp add:field_simps)
avigad@16775
  1620
nipkow@23482
  1621
lemma divide_nonneg_neg:
nipkow@23482
  1622
  "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" 
nipkow@23482
  1623
by(simp add:field_simps)
avigad@16775
  1624
nipkow@23482
  1625
lemma divide_neg_neg:
nipkow@23482
  1626
  "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
nipkow@23482
  1627
by(simp add:field_simps)
avigad@16775
  1628
nipkow@23482
  1629
lemma divide_nonpos_neg:
nipkow@23482
  1630
  "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
nipkow@23482
  1631
by(simp add:field_simps)
paulson@15234
  1632
wenzelm@23389
  1633
paulson@14288
  1634
subsection{*Cancellation Laws for Division*}
paulson@14288
  1635
paulson@24286
  1636
lemma divide_cancel_right [simp,noatp]:
paulson@14288
  1637
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1638
apply (cases "c=0", simp)
nipkow@23496
  1639
apply (simp add: divide_inverse)
paulson@14288
  1640
done
paulson@14288
  1641
paulson@24286
  1642
lemma divide_cancel_left [simp,noatp]:
paulson@14288
  1643
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
nipkow@23482
  1644
apply (cases "c=0", simp)
nipkow@23496
  1645
apply (simp add: divide_inverse)
paulson@14288
  1646
done
paulson@14288
  1647
wenzelm@23389
  1648
paulson@14353
  1649
subsection {* Division and the Number One *}
paulson@14353
  1650
paulson@14353
  1651
text{*Simplify expressions equated with 1*}
paulson@24286
  1652
lemma divide_eq_1_iff [simp,noatp]:
paulson@14353
  1653
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1654
apply (cases "b=0", simp)
nipkow@23482
  1655
apply (simp add: right_inverse_eq)
paulson@14353
  1656
done
paulson@14353
  1657
paulson@24286
  1658
lemma one_eq_divide_iff [simp,noatp]:
paulson@14353
  1659
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1660
by (simp add: eq_commute [of 1])
paulson@14353
  1661
paulson@24286
  1662
lemma zero_eq_1_divide_iff [simp,noatp]:
paulson@14353
  1663
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
nipkow@23482
  1664
apply (cases "a=0", simp)
nipkow@23482
  1665
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1666
done
paulson@14353
  1667
paulson@24286
  1668
lemma one_divide_eq_0_iff [simp,noatp]:
paulson@14353
  1669
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
nipkow@23482
  1670
apply (cases "a=0", simp)
nipkow@23482
  1671
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1672
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1673
done
paulson@14353
  1674
paulson@14353
  1675
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@18623
  1676
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
paulson@18623
  1677
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
paulson@18623
  1678
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
paulson@18623
  1679
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
paulson@17085
  1680
paulson@17085
  1681
declare zero_less_divide_1_iff [simp]
paulson@24286
  1682
declare divide_less_0_1_iff [simp,noatp]
paulson@17085
  1683
declare zero_le_divide_1_iff [simp]
paulson@24286
  1684
declare divide_le_0_1_iff [simp,noatp]
paulson@14353
  1685
wenzelm@23389
  1686
paulson@14293
  1687
subsection {* Ordering Rules for Division *}
paulson@14293
  1688
paulson@14293
  1689
lemma divide_strict_right_mono:
paulson@14293
  1690
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1691
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
nipkow@23482
  1692
              positive_imp_inverse_positive)
paulson@14293
  1693
paulson@14293
  1694
lemma divide_right_mono:
paulson@14293
  1695
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
nipkow@23482
  1696
by (force simp add: divide_strict_right_mono order_le_less)
paulson@14293
  1697
avigad@16775
  1698
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1699
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1700
apply (drule divide_right_mono [of _ _ "- c"])
nipkow@23482
  1701
apply auto
avigad@16775
  1702
done
avigad@16775
  1703
avigad@16775
  1704
lemma divide_strict_right_mono_neg:
avigad@16775
  1705
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
nipkow@23482
  1706
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
nipkow@23482
  1707
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
avigad@16775
  1708
done
paulson@14293
  1709
paulson@14293
  1710
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1711
      have the same sign*}
paulson@14293
  1712
lemma divide_strict_left_mono:
nipkow@23482
  1713
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1714
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
paulson@14293
  1715
paulson@14293
  1716
lemma divide_left_mono:
nipkow@23482
  1717
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
nipkow@23482
  1718
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
paulson@14293
  1719
avigad@16775
  1720
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1721
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1722
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1723
  apply (auto simp add: mult_commute)
avigad@16775
  1724
done
avigad@16775
  1725
paulson@14293
  1726
lemma divide_strict_left_mono_neg:
nipkow@23482
  1727
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1728
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
nipkow@23482
  1729
paulson@14293
  1730
avigad@16775
  1731
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1732
paulson@24286
  1733
lemma le_divide_eq_1 [noatp]:
avigad@16775
  1734
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1735
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1736
by (auto simp add: le_divide_eq)
avigad@16775
  1737
paulson@24286
  1738
lemma divide_le_eq_1 [noatp]:
avigad@16775
  1739
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1740
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1741
by (auto simp add: divide_le_eq)
avigad@16775
  1742
paulson@24286
  1743
lemma less_divide_eq_1 [noatp]:
avigad@16775
  1744
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1745
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1746
by (auto simp add: less_divide_eq)
avigad@16775
  1747
paulson@24286
  1748
lemma divide_less_eq_1 [noatp]:
avigad@16775
  1749
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1750
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1751
by (auto simp add: divide_less_eq)
avigad@16775
  1752
wenzelm@23389
  1753
avigad@16775
  1754
subsection{*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1755
paulson@24286
  1756
lemma le_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1757
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1758
  shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1759
by (auto simp add: le_divide_eq)
avigad@16775
  1760
paulson@24286
  1761
lemma le_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1762
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1763
  shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1764
by (auto simp add: le_divide_eq)
avigad@16775
  1765
paulson@24286
  1766
lemma divide_le_eq_1_pos [simp,noatp]:
avigad@16775
  1767
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1768
  shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1769
by (auto simp add: divide_le_eq)
avigad@16775
  1770
paulson@24286
  1771
lemma divide_le_eq_1_neg [simp,noatp]:
avigad@16775
  1772
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1773
  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1774
by (auto simp add: divide_le_eq)
avigad@16775
  1775
paulson@24286
  1776
lemma less_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1777
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1778
  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1779
by (auto simp add: less_divide_eq)
avigad@16775
  1780
paulson@24286
  1781
lemma less_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1782
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1783
  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1784
by (auto simp add: less_divide_eq)
avigad@16775
  1785
paulson@24286
  1786
lemma divide_less_eq_1_pos [simp,noatp]:
avigad@16775
  1787
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1788
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1789
by (auto simp add: divide_less_eq)
paulson@18649
  1790
paulson@24286
  1791
lemma divide_less_eq_1_neg [simp,noatp]:
paulson@18649
  1792
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1793
  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1794
by (auto simp add: divide_less_eq)
avigad@16775
  1795
paulson@24286
  1796
lemma eq_divide_eq_1 [simp,noatp]:
avigad@16775
  1797
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1798
  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1799
by (auto simp add: eq_divide_eq)
avigad@16775
  1800
paulson@24286
  1801
lemma divide_eq_eq_1 [simp,noatp]:
avigad@16775
  1802
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1803
  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1804
by (auto simp add: divide_eq_eq)
avigad@16775
  1805
wenzelm@23389
  1806
avigad@16775
  1807
subsection {* Reasoning about inequalities with division *}
avigad@16775
  1808
avigad@16775
  1809
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1810
    ==> x * y <= x"
avigad@16775
  1811
  by (auto simp add: mult_compare_simps);
avigad@16775
  1812
avigad@16775
  1813
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1814
    ==> y * x <= x"
avigad@16775
  1815
  by (auto simp add: mult_compare_simps);
avigad@16775
  1816
avigad@16775
  1817
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
avigad@16775
  1818
    x / y <= z";
avigad@16775
  1819
  by (subst pos_divide_le_eq, assumption+);
avigad@16775
  1820
avigad@16775
  1821
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
nipkow@23482
  1822
    z <= x / y"
nipkow@23482
  1823
by(simp add:field_simps)
avigad@16775
  1824
avigad@16775
  1825
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
avigad@16775
  1826
    x / y < z"
nipkow@23482
  1827
by(simp add:field_simps)
avigad@16775
  1828
avigad@16775
  1829
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
avigad@16775
  1830
    z < x / y"
nipkow@23482
  1831
by(simp add:field_simps)
avigad@16775
  1832
avigad@16775
  1833
lemma frac_le: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1834
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
avigad@16775
  1835
  apply (rule mult_imp_div_pos_le)
avigad@16775
  1836
  apply simp;
avigad@16775
  1837
  apply (subst times_divide_eq_left);
avigad@16775
  1838
  apply (rule mult_imp_le_div_pos, assumption)
avigad@16775
  1839
  apply (rule mult_mono)
avigad@16775
  1840
  apply simp_all
paulson@14293
  1841
done
paulson@14293
  1842
avigad@16775
  1843
lemma frac_less: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1844
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
avigad@16775
  1845
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1846
  apply simp;
avigad@16775
  1847
  apply (subst times_divide_eq_left);
avigad@16775
  1848
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1849
  apply (erule mult_less_le_imp_less)
avigad@16775
  1850
  apply simp_all
avigad@16775
  1851
done
avigad@16775
  1852
avigad@16775
  1853
lemma frac_less2: "(0::'a::ordered_field) < x ==> 
avigad@16775
  1854
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
avigad@16775
  1855
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1856
  apply simp_all
avigad@16775
  1857
  apply (subst times_divide_eq_left);
avigad@16775
  1858
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1859
  apply (erule mult_le_less_imp_less)
avigad@16775
  1860
  apply simp_all
avigad@16775
  1861
done
avigad@16775
  1862
avigad@16775
  1863
text{*It's not obvious whether these should be simprules or not. 
avigad@16775
  1864
  Their effect is to gather terms into one big fraction, like
avigad@16775
  1865
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
avigad@16775
  1866
  seem to need them.*}
avigad@16775
  1867
avigad@16775
  1868
declare times_divide_eq [simp]
paulson@14293
  1869
wenzelm@23389
  1870
paulson@14293
  1871
subsection {* Ordered Fields are Dense *}
paulson@14293
  1872
obua@14738
  1873
lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
paulson@14293
  1874
proof -
obua@14738
  1875
  have "a+0 < (a+1::'a::ordered_semidom)"
nipkow@23482
  1876
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1877
  thus ?thesis by simp
paulson@14293
  1878
qed
paulson@14293
  1879
obua@14738
  1880
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
nipkow@23482
  1881
by (blast intro: order_less_trans zero_less_one less_add_one)
paulson@14365
  1882
paulson@14293
  1883
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
nipkow@23482
  1884
by (simp add: field_simps zero_less_two)
paulson@14293
  1885
paulson@14293
  1886
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
nipkow@23482
  1887
by (simp add: field_simps zero_less_two)
paulson@14293
  1888
haftmann@24422
  1889
instance ordered_field < dense_linear_order
haftmann@24422
  1890
proof
haftmann@24422
  1891
  fix x y :: 'a
haftmann@24422
  1892
  have "x < x + 1" by simp
haftmann@24422
  1893
  then show "\<exists>y. x < y" .. 
haftmann@24422
  1894
  have "x - 1 < x" by simp
haftmann@24422
  1895
  then show "\<exists>y. y < x" ..
haftmann@24422
  1896
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
haftmann@24422
  1897
qed
paulson@14293
  1898
paulson@15234
  1899
paulson@14293
  1900
subsection {* Absolute Value *}
paulson@14293
  1901
nipkow@24491
  1902
lemma mult_sgn_abs: "sgn x * abs x = (x::'a::{ordered_idom,linorder})"
nipkow@24491
  1903
using less_linear[of x 0]
nipkow@24491
  1904
by(auto simp: sgn_def abs_if)
nipkow@24491
  1905
obua@14738
  1906
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
nipkow@23477
  1907
by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
paulson@14294
  1908
obua@14738
  1909
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  1910
proof -
obua@14738
  1911
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  1912
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  1913
  have a: "(abs a) * (abs b) = ?x"
nipkow@23477
  1914
    by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
obua@14738
  1915
  {
obua@14738
  1916
    fix u v :: 'a
paulson@15481
  1917
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
paulson@15481
  1918
              u * v = pprt a * pprt b + pprt a * nprt b + 
paulson@15481
  1919
                      nprt a * pprt b + nprt a * nprt b"
obua@14738
  1920
      apply (subst prts[of u], subst prts[of v])
nipkow@23477
  1921
      apply (simp add: ring_simps) 
obua@14738
  1922
      done
obua@14738
  1923
  }
obua@14738
  1924
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  1925
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  1926
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  1927
  have xy: "- ?x <= ?y"
obua@14754
  1928
    apply (simp)
obua@14754
  1929
    apply (rule_tac y="0::'a" in order_trans)
nipkow@16568
  1930
    apply (rule addm2)
avigad@16775
  1931
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
nipkow@16568
  1932
    apply (rule addm)
avigad@16775
  1933
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14754
  1934
    done
obua@14738
  1935
  have yx: "?y <= ?x"
nipkow@16568
  1936
    apply (simp add:diff_def)
obua@14754
  1937
    apply (rule_tac y=0 in order_trans)
avigad@16775
  1938
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
avigad@16775
  1939
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
obua@14738
  1940
    done
obua@14738
  1941
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  1942
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  1943
  show ?thesis
obua@14738
  1944
    apply (rule abs_leI)
obua@14738
  1945
    apply (simp add: i1)
obua@14738
  1946
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  1947
    done
obua@14738
  1948
qed
paulson@14294
  1949
obua@14738
  1950
lemma abs_eq_mult: 
obua@14738
  1951
  assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
obua@14738
  1952
  shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
obua@14738
  1953
proof -
obua@14738
  1954
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  1955
    apply (auto)    
obua@14738
  1956
    apply (rule_tac split_mult_pos_le)
obua@14738
  1957
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  1958
    apply (simp)
obua@14738
  1959
    apply (rule_tac split_mult_neg_le)
obua@14738
  1960
    apply (insert prems)
obua@14738
  1961
    apply (blast)
obua@14738
  1962
    done
obua@14738
  1963
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  1964
    by (simp add: prts[symmetric])
obua@14738
  1965
  show ?thesis
obua@14738
  1966
  proof cases
obua@14738
  1967
    assume "0 <= a * b"
obua@14738
  1968
    then show ?thesis
obua@14738
  1969
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1970
      apply (insert prems)
obua@14754
  1971
      apply (auto simp add: 
nipkow@23477
  1972
	ring_simps 
obua@14754
  1973
	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
nipkow@15197
  1974
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])
avigad@16775
  1975
	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
avigad@16775
  1976
	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
obua@14738
  1977
      done
obua@14738
  1978
  next
obua@14738
  1979
    assume "~(0 <= a*b)"
obua@14738
  1980
    with s have "a*b <= 0" by simp
obua@14738
  1981
    then show ?thesis
obua@14738
  1982
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1983
      apply (insert prems)
nipkow@23477
  1984
      apply (auto simp add: ring_simps)
avigad@16775
  1985
      apply(drule (1) mult_nonneg_nonneg[of a b],simp)
avigad@16775
  1986
      apply(drule (1) mult_nonpos_nonpos[of a b],simp)
obua@14738
  1987
      done
obua@14738
  1988
  qed
obua@14738
  1989
qed
paulson@14294
  1990
obua@14738
  1991
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
obua@14738
  1992
by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  1993
obua@14738
  1994
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
obua@14738
  1995
by (simp add: abs_if) 
paulson@14294
  1996
paulson@14294
  1997
lemma nonzero_abs_inverse:
paulson@14294
  1998
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  1999
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  2000
                      negative_imp_inverse_negative)
paulson@14294
  2001
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  2002
done
paulson@14294
  2003
paulson@14294
  2004
lemma abs_inverse [simp]:
paulson@14294
  2005
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  2006
      inverse (abs a)"
haftmann@21328
  2007
apply (cases "a=0", simp) 
paulson@14294
  2008
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  2009
done
paulson@14294
  2010
paulson@14294
  2011
lemma nonzero_abs_divide:
paulson@14294
  2012
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  2013
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  2014
paulson@15234
  2015
lemma abs_divide [simp]:
paulson@14294
  2016
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
haftmann@21328
  2017
apply (cases "b=0", simp) 
paulson@14294
  2018
apply (simp add: nonzero_abs_divide) 
paulson@14294
  2019
done
paulson@14294
  2020
paulson@14294
  2021
lemma abs_mult_less:
obua@14738
  2022
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  2023
proof -
paulson@14294
  2024
  assume ac: "abs a < c"
paulson@14294
  2025
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  2026
  assume "abs b < d"
paulson@14294
  2027
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  2028
qed
paulson@14293
  2029
obua@14738
  2030
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
obua@14738
  2031
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
obua@14738
  2032
obua@14738
  2033
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
obua@14738
  2034
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
obua@14738
  2035
obua@14738
  2036
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  2037
apply (simp add: order_less_le abs_le_iff)  
obua@14738
  2038
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
obua@14738
  2039
apply (simp add: le_minus_self_iff linorder_neq_iff) 
obua@14738
  2040
done
obua@14738
  2041
avigad@16775
  2042
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
avigad@16775
  2043
    (abs y) * x = abs (y * x)";
avigad@16775
  2044
  apply (subst abs_mult);
avigad@16775
  2045
  apply simp;
avigad@16775
  2046
done;
avigad@16775
  2047
avigad@16775
  2048
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
avigad@16775
  2049
    abs x / y = abs (x / y)";
avigad@16775
  2050
  apply (subst abs_divide);
avigad@16775
  2051
  apply (simp add: order_less_imp_le);
avigad@16775
  2052
done;
avigad@16775
  2053
wenzelm@23389
  2054
obua@19404
  2055
subsection {* Bounds of products via negative and positive Part *}
obua@15178
  2056
obua@15580
  2057
lemma mult_le_prts:
obua@15580
  2058
  assumes
obua@15580
  2059
  "a1 <= (a::'a::lordered_ring)"
obua@15580
  2060
  "a <= a2"
obua@15580
  2061
  "b1 <= b"
obua@15580
  2062
  "b <= b2"
obua@15580
  2063
  shows
obua@15580
  2064
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
obua@15580
  2065
proof - 
obua@15580
  2066
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
obua@15580
  2067
    apply (subst prts[symmetric])+
obua@15580
  2068
    apply simp
obua@15580
  2069
    done
obua@15580
  2070
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
nipkow@23477
  2071
    by (simp add: ring_simps)
obua@15580
  2072
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
obua@15580
  2073
    by (simp_all add: prems mult_mono)
obua@15580
  2074
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
obua@15580
  2075
  proof -
obua@15580
  2076
    have "pprt a * nprt b <= pprt a * nprt b2"
obua@15580
  2077
      by (simp add: mult_left_mono prems)
obua@15580
  2078
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
obua@15580
  2079
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2080
    ultimately show ?thesis
obua@15580
  2081
      by simp
obua@15580
  2082
  qed
obua@15580
  2083
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
obua@15580
  2084
  proof - 
obua@15580
  2085
    have "nprt a * pprt b <= nprt a2 * pprt b"
obua@15580
  2086
      by (simp add: mult_right_mono prems)
obua@15580
  2087
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
obua@15580
  2088
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2089
    ultimately show ?thesis
obua@15580
  2090
      by simp
obua@15580
  2091
  qed
obua@15580
  2092
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
obua@15580
  2093
  proof -
obua@15580
  2094
    have "nprt a * nprt b <= nprt a * nprt b1"
obua@15580
  2095
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2096
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
obua@15580
  2097
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2098
    ultimately show ?thesis
obua@15580
  2099
      by simp
obua@15580
  2100
  qed
obua@15580
  2101
  ultimately show ?thesis
obua@15580
  2102
    by - (rule add_mono | simp)+
obua@15580
  2103
qed
obua@19404
  2104
obua@19404
  2105
lemma mult_ge_prts:
obua@15178
  2106
  assumes
obua@19404
  2107
  "a1 <= (a::'a::lordered_ring)"
obua@19404
  2108
  "a <= a2"
obua@19404
  2109
  "b1 <= b"
obua@19404
  2110
  "b <= b2"
obua@15178
  2111
  shows
obua@19404
  2112
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
obua@19404
  2113
proof - 
obua@19404
  2114
  from prems have a1:"- a2 <= -a" by auto
obua@19404
  2115
  from prems have a2: "-a <= -a1" by auto
obua@19404
  2116
  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg] 
obua@19404
  2117
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
obua@19404
  2118
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
obua@19404
  2119
    by (simp only: minus_le_iff)
obua@19404
  2120
  then show ?thesis by simp
obua@15178
  2121
qed
obua@15178
  2122
wenzelm@23389
  2123
haftmann@22842
  2124
subsection {* Theorems for proof tools *}
haftmann@22842
  2125
paulson@24427
  2126
lemma add_mono_thms_ordered_semiring [noatp]:
haftmann@22842
  2127
  fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add"
haftmann@22842
  2128
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2129
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2130
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2131
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@22842
  2132
by (rule add_mono, clarify+)+
haftmann@22842
  2133
paulson@24427
  2134
lemma add_mono_thms_ordered_field [noatp]:
haftmann@22842
  2135
  fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add"
haftmann@22842
  2136
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2137
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2138
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2139
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2140
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2141
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@22842
  2142
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@22842
  2143
paulson@14265
  2144
end