src/HOL/Ring_and_Field.thy
author obua
Sat Jun 30 17:30:10 2007 +0200 (2007-06-30)
changeset 23521 195fe3fe2831
parent 23496 84e9216a6d0e
child 23527 c1d2fa4b76df
permissions -rw-r--r--
added ordered_ring and ordered_semiring
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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 dom = ring_1 + ring_no_zero_divisors
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hide const dom
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class idom = comm_ring_1 + no_zero_divisors
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instance idom \<subseteq> dom ..
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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> dom
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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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  apply (intro_classes)
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  apply (cases "a < b & 0 < c")
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  apply (auto simp add: mult_strict_left_mono order_less_le)
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  apply (auto simp add: mult_strict_left_mono order_le_less)
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  apply (simp add: mult_strict_right_mono)
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  done
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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: "a <= b" "0 <= c"
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  with mult_mono show "c * a <= c * b" .
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  from mult_commute have "a * c = c * a" ..
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  also from mult_mono A have "\<dots> <= c * b" .
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  also from mult_commute have "c * b = b * c" ..
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  finally show "a * c <= b * c" .
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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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by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+)
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instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
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apply (intro_classes)
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apply (cases "a < b & 0 < c")
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apply (auto simp add: mult_strict_left_mono order_less_le)
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apply (auto simp add: mult_strict_left_mono order_le_less)
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done
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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 +
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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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  apply (intro_classes)
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  by (simp add: abs_if sup_eq_if)
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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_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*)
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  assumes zero_less_one [simp]: "\<^loc>0 \<sqsubset> \<^loc>1"
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class ordered_idom = comm_ring_1 + ordered_comm_semiring_strict + lordered_ab_group + abs_if
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  (*previously ordered_ring*)
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instance ordered_idom \<subseteq> ordered_ring_strict ..
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instance ordered_idom \<subseteq> pordered_comm_ring ..
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haftmann@22390
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class ordered_field = field + ordered_idom
paulson@14272
   325
nipkow@15923
   326
lemmas linorder_neqE_ordered_idom =
nipkow@15923
   327
 linorder_neqE[where 'a = "?'b::ordered_idom"]
nipkow@15923
   328
paulson@14272
   329
lemma eq_add_iff1:
nipkow@23477
   330
  "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
nipkow@23477
   331
by (simp add: ring_simps)
paulson@14272
   332
paulson@14272
   333
lemma eq_add_iff2:
nipkow@23477
   334
  "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
nipkow@23477
   335
by (simp add: ring_simps)
paulson@14272
   336
paulson@14272
   337
lemma less_add_iff1:
nipkow@23477
   338
  "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
nipkow@23477
   339
by (simp add: ring_simps)
paulson@14272
   340
paulson@14272
   341
lemma less_add_iff2:
nipkow@23477
   342
  "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
nipkow@23477
   343
by (simp add: ring_simps)
paulson@14272
   344
paulson@14272
   345
lemma le_add_iff1:
nipkow@23477
   346
  "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
nipkow@23477
   347
by (simp add: ring_simps)
paulson@14272
   348
paulson@14272
   349
lemma le_add_iff2:
nipkow@23477
   350
  "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
nipkow@23477
   351
by (simp add: ring_simps)
paulson@14272
   352
wenzelm@23389
   353
paulson@14270
   354
subsection {* Ordering Rules for Multiplication *}
paulson@14270
   355
paulson@14348
   356
lemma mult_left_le_imp_le:
nipkow@23477
   357
  "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
nipkow@23477
   358
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
paulson@14348
   359
 
paulson@14348
   360
lemma mult_right_le_imp_le:
nipkow@23477
   361
  "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
nipkow@23477
   362
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
paulson@14348
   363
paulson@14348
   364
lemma mult_left_less_imp_less:
obua@23521
   365
  "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
nipkow@23477
   366
by (force simp add: mult_left_mono linorder_not_le [symmetric])
paulson@14348
   367
 
paulson@14348
   368
lemma mult_right_less_imp_less:
obua@23521
   369
  "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
nipkow@23477
   370
by (force simp add: mult_right_mono linorder_not_le [symmetric])
paulson@14348
   371
paulson@14265
   372
lemma mult_strict_left_mono_neg:
nipkow@23477
   373
  "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
paulson@14265
   374
apply (drule mult_strict_left_mono [of _ _ "-c"])
paulson@14265
   375
apply (simp_all add: minus_mult_left [symmetric]) 
paulson@14265
   376
done
paulson@14265
   377
obua@14738
   378
lemma mult_left_mono_neg:
nipkow@23477
   379
  "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
obua@14738
   380
apply (drule mult_left_mono [of _ _ "-c"])
obua@14738
   381
apply (simp_all add: minus_mult_left [symmetric]) 
obua@14738
   382
done
obua@14738
   383
paulson@14265
   384
lemma mult_strict_right_mono_neg:
nipkow@23477
   385
  "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
paulson@14265
   386
apply (drule mult_strict_right_mono [of _ _ "-c"])
paulson@14265
   387
apply (simp_all add: minus_mult_right [symmetric]) 
paulson@14265
   388
done
paulson@14265
   389
obua@14738
   390
lemma mult_right_mono_neg:
nipkow@23477
   391
  "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
obua@14738
   392
apply (drule mult_right_mono [of _ _ "-c"])
obua@14738
   393
apply (simp)
obua@14738
   394
apply (simp_all add: minus_mult_right [symmetric]) 
obua@14738
   395
done
paulson@14265
   396
wenzelm@23389
   397
paulson@14265
   398
subsection{* Products of Signs *}
paulson@14265
   399
avigad@16775
   400
lemma mult_pos_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
paulson@14265
   401
by (drule mult_strict_left_mono [of 0 b], auto)
paulson@14265
   402
avigad@16775
   403
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
obua@14738
   404
by (drule mult_left_mono [of 0 b], auto)
obua@14738
   405
obua@14738
   406
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
paulson@14265
   407
by (drule mult_strict_left_mono [of b 0], auto)
paulson@14265
   408
avigad@16775
   409
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
obua@14738
   410
by (drule mult_left_mono [of b 0], auto)
obua@14738
   411
obua@14738
   412
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
obua@14738
   413
by (drule mult_strict_right_mono[of b 0], auto)
obua@14738
   414
avigad@16775
   415
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
obua@14738
   416
by (drule mult_right_mono[of b 0], auto)
obua@14738
   417
avigad@16775
   418
lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
paulson@14265
   419
by (drule mult_strict_right_mono_neg, auto)
paulson@14265
   420
avigad@16775
   421
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
obua@14738
   422
by (drule mult_right_mono_neg[of a 0 b ], auto)
obua@14738
   423
paulson@14341
   424
lemma zero_less_mult_pos:
obua@14738
   425
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   426
apply (cases "b\<le>0") 
paulson@14265
   427
 apply (auto simp add: order_le_less linorder_not_less)
paulson@14265
   428
apply (drule_tac mult_pos_neg [of a b]) 
paulson@14265
   429
 apply (auto dest: order_less_not_sym)
paulson@14265
   430
done
paulson@14265
   431
obua@14738
   432
lemma zero_less_mult_pos2:
obua@14738
   433
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   434
apply (cases "b\<le>0") 
obua@14738
   435
 apply (auto simp add: order_le_less linorder_not_less)
obua@14738
   436
apply (drule_tac mult_pos_neg2 [of a b]) 
obua@14738
   437
 apply (auto dest: order_less_not_sym)
obua@14738
   438
done
obua@14738
   439
paulson@14265
   440
lemma zero_less_mult_iff:
obua@14738
   441
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
   442
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos 
avigad@16775
   443
  mult_neg_neg)
paulson@14265
   444
apply (blast dest: zero_less_mult_pos) 
obua@14738
   445
apply (blast dest: zero_less_mult_pos2)
paulson@14265
   446
done
paulson@14265
   447
huffman@22990
   448
lemma mult_eq_0_iff [simp]:
huffman@22990
   449
  fixes a b :: "'a::ring_no_zero_divisors"
huffman@22990
   450
  shows "(a * b = 0) = (a = 0 \<or> b = 0)"
huffman@22990
   451
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors)
huffman@22990
   452
huffman@22990
   453
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
huffman@22990
   454
apply intro_classes
paulson@14265
   455
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
paulson@14265
   456
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
paulson@14265
   457
done
paulson@14265
   458
paulson@14265
   459
lemma zero_le_mult_iff:
obua@14738
   460
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14265
   461
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
paulson@14265
   462
                   zero_less_mult_iff)
paulson@14265
   463
paulson@14265
   464
lemma mult_less_0_iff:
obua@14738
   465
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14265
   466
apply (insert zero_less_mult_iff [of "-a" b]) 
paulson@14265
   467
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   468
done
paulson@14265
   469
paulson@14265
   470
lemma mult_le_0_iff:
obua@14738
   471
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14265
   472
apply (insert zero_le_mult_iff [of "-a" b]) 
paulson@14265
   473
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   474
done
paulson@14265
   475
obua@14738
   476
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
   477
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14738
   478
obua@14738
   479
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
   480
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
obua@14738
   481
obua@23095
   482
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a"
obua@23095
   483
by (simp add: zero_le_mult_iff linorder_linear)
obua@23095
   484
obua@23095
   485
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))"
obua@23095
   486
by (simp add: not_less)
paulson@14265
   487
obua@14738
   488
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
obua@14738
   489
      theorems available to members of @{term ordered_idom} *}
obua@14738
   490
obua@14738
   491
instance ordered_idom \<subseteq> ordered_semidom
paulson@14421
   492
proof
paulson@14421
   493
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
paulson@14430
   494
  thus "(0::'a) < 1" by (simp add: order_le_less) 
paulson@14421
   495
qed
paulson@14421
   496
obua@14738
   497
instance ordered_idom \<subseteq> idom ..
obua@14738
   498
paulson@14387
   499
text{*All three types of comparision involving 0 and 1 are covered.*}
paulson@14387
   500
paulson@17085
   501
lemmas one_neq_zero = zero_neq_one [THEN not_sym]
paulson@17085
   502
declare one_neq_zero [simp]
paulson@14387
   503
obua@14738
   504
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
paulson@14268
   505
  by (rule zero_less_one [THEN order_less_imp_le]) 
paulson@14268
   506
obua@14738
   507
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
obua@14738
   508
by (simp add: linorder_not_le) 
paulson@14387
   509
obua@14738
   510
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
obua@14738
   511
by (simp add: linorder_not_less) 
paulson@14268
   512
wenzelm@23389
   513
paulson@14268
   514
subsection{*More Monotonicity*}
paulson@14268
   515
paulson@14268
   516
text{*Strict monotonicity in both arguments*}
paulson@14268
   517
lemma mult_strict_mono:
obua@14738
   518
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
haftmann@21328
   519
apply (cases "c=0")
avigad@16775
   520
 apply (simp add: mult_pos_pos) 
paulson@14268
   521
apply (erule mult_strict_right_mono [THEN order_less_trans])
paulson@14268
   522
 apply (force simp add: order_le_less) 
paulson@14268
   523
apply (erule mult_strict_left_mono, assumption)
paulson@14268
   524
done
paulson@14268
   525
paulson@14268
   526
text{*This weaker variant has more natural premises*}
paulson@14268
   527
lemma mult_strict_mono':
obua@14738
   528
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   529
apply (rule mult_strict_mono)
paulson@14268
   530
apply (blast intro: order_le_less_trans)+
paulson@14268
   531
done
paulson@14268
   532
paulson@14268
   533
lemma mult_mono:
paulson@14268
   534
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
obua@14738
   535
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
paulson@14268
   536
apply (erule mult_right_mono [THEN order_trans], assumption)
paulson@14268
   537
apply (erule mult_left_mono, assumption)
paulson@14268
   538
done
paulson@14268
   539
huffman@21258
   540
lemma mult_mono':
huffman@21258
   541
     "[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|] 
huffman@21258
   542
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
huffman@21258
   543
apply (rule mult_mono)
huffman@21258
   544
apply (fast intro: order_trans)+
huffman@21258
   545
done
huffman@21258
   546
obua@14738
   547
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
paulson@14387
   548
apply (insert mult_strict_mono [of 1 m 1 n]) 
paulson@14430
   549
apply (simp add:  order_less_trans [OF zero_less_one]) 
paulson@14387
   550
done
paulson@14387
   551
avigad@16775
   552
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
avigad@16775
   553
    c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
avigad@16775
   554
  apply (subgoal_tac "a * c < b * c")
avigad@16775
   555
  apply (erule order_less_le_trans)
avigad@16775
   556
  apply (erule mult_left_mono)
avigad@16775
   557
  apply simp
avigad@16775
   558
  apply (erule mult_strict_right_mono)
avigad@16775
   559
  apply assumption
avigad@16775
   560
done
avigad@16775
   561
avigad@16775
   562
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
avigad@16775
   563
    c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
avigad@16775
   564
  apply (subgoal_tac "a * c <= b * c")
avigad@16775
   565
  apply (erule order_le_less_trans)
avigad@16775
   566
  apply (erule mult_strict_left_mono)
avigad@16775
   567
  apply simp
avigad@16775
   568
  apply (erule mult_right_mono)
avigad@16775
   569
  apply simp
avigad@16775
   570
done
avigad@16775
   571
wenzelm@23389
   572
paulson@14268
   573
subsection{*Cancellation Laws for Relationships With a Common Factor*}
paulson@14268
   574
paulson@14268
   575
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
paulson@14268
   576
   also with the relations @{text "\<le>"} and equality.*}
paulson@14268
   577
paulson@15234
   578
text{*These ``disjunction'' versions produce two cases when the comparison is
paulson@15234
   579
 an assumption, but effectively four when the comparison is a goal.*}
paulson@15234
   580
paulson@15234
   581
lemma mult_less_cancel_right_disj:
obua@14738
   582
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   583
apply (cases "c = 0")
paulson@14268
   584
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
paulson@14268
   585
                      mult_strict_right_mono_neg)
paulson@14268
   586
apply (auto simp add: linorder_not_less 
paulson@14268
   587
                      linorder_not_le [symmetric, of "a*c"]
paulson@14268
   588
                      linorder_not_le [symmetric, of a])
paulson@14268
   589
apply (erule_tac [!] notE)
paulson@14268
   590
apply (auto simp add: order_less_imp_le mult_right_mono 
paulson@14268
   591
                      mult_right_mono_neg)
paulson@14268
   592
done
paulson@14268
   593
paulson@15234
   594
lemma mult_less_cancel_left_disj:
obua@14738
   595
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   596
apply (cases "c = 0")
obua@14738
   597
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
obua@14738
   598
                      mult_strict_left_mono_neg)
obua@14738
   599
apply (auto simp add: linorder_not_less 
obua@14738
   600
                      linorder_not_le [symmetric, of "c*a"]
obua@14738
   601
                      linorder_not_le [symmetric, of a])
obua@14738
   602
apply (erule_tac [!] notE)
obua@14738
   603
apply (auto simp add: order_less_imp_le mult_left_mono 
obua@14738
   604
                      mult_left_mono_neg)
obua@14738
   605
done
paulson@14268
   606
paulson@15234
   607
paulson@15234
   608
text{*The ``conjunction of implication'' lemmas produce two cases when the
paulson@15234
   609
comparison is a goal, but give four when the comparison is an assumption.*}
paulson@15234
   610
paulson@15234
   611
lemma mult_less_cancel_right:
paulson@15234
   612
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   613
  shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   614
by (insert mult_less_cancel_right_disj [of a c b], auto)
paulson@15234
   615
paulson@15234
   616
lemma mult_less_cancel_left:
paulson@15234
   617
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   618
  shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   619
by (insert mult_less_cancel_left_disj [of c a b], auto)
paulson@15234
   620
paulson@14268
   621
lemma mult_le_cancel_right:
obua@14738
   622
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   623
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
paulson@14268
   624
paulson@14268
   625
lemma mult_le_cancel_left:
obua@14738
   626
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   627
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
paulson@14268
   628
paulson@14268
   629
lemma mult_less_imp_less_left:
paulson@14341
   630
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
obua@14738
   631
      shows "a < (b::'a::ordered_semiring_strict)"
paulson@14377
   632
proof (rule ccontr)
paulson@14377
   633
  assume "~ a < b"
paulson@14377
   634
  hence "b \<le> a" by (simp add: linorder_not_less)
wenzelm@23389
   635
  hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono)
paulson@14377
   636
  with this and less show False 
paulson@14377
   637
    by (simp add: linorder_not_less [symmetric])
paulson@14377
   638
qed
paulson@14268
   639
paulson@14268
   640
lemma mult_less_imp_less_right:
obua@14738
   641
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
obua@14738
   642
  shows "a < (b::'a::ordered_semiring_strict)"
obua@14738
   643
proof (rule ccontr)
obua@14738
   644
  assume "~ a < b"
obua@14738
   645
  hence "b \<le> a" by (simp add: linorder_not_less)
wenzelm@23389
   646
  hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono)
obua@14738
   647
  with this and less show False 
obua@14738
   648
    by (simp add: linorder_not_less [symmetric])
obua@14738
   649
qed  
paulson@14268
   650
paulson@14268
   651
text{*Cancellation of equalities with a common factor*}
paulson@14268
   652
lemma mult_cancel_right [simp]:
huffman@22990
   653
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   654
  shows "(a * c = b * c) = (c = 0 \<or> a = b)"
huffman@22990
   655
proof -
huffman@22990
   656
  have "(a * c = b * c) = ((a - b) * c = 0)"
nipkow@23477
   657
    by (simp add: ring_distribs)
huffman@22990
   658
  thus ?thesis
huffman@22990
   659
    by (simp add: disj_commute)
huffman@22990
   660
qed
paulson@14268
   661
paulson@14268
   662
lemma mult_cancel_left [simp]:
huffman@22990
   663
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   664
  shows "(c * a = c * b) = (c = 0 \<or> a = b)"
huffman@22990
   665
proof -
huffman@22990
   666
  have "(c * a = c * b) = (c * (a - b) = 0)"
nipkow@23477
   667
    by (simp add: ring_distribs)
huffman@22990
   668
  thus ?thesis
huffman@22990
   669
    by simp
huffman@22990
   670
qed
paulson@14268
   671
paulson@15234
   672
paulson@15234
   673
subsubsection{*Special Cancellation Simprules for Multiplication*}
paulson@15234
   674
paulson@15234
   675
text{*These also produce two cases when the comparison is a goal.*}
paulson@15234
   676
paulson@15234
   677
lemma mult_le_cancel_right1:
paulson@15234
   678
  fixes c :: "'a :: ordered_idom"
paulson@15234
   679
  shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   680
by (insert mult_le_cancel_right [of 1 c b], simp)
paulson@15234
   681
paulson@15234
   682
lemma mult_le_cancel_right2:
paulson@15234
   683
  fixes c :: "'a :: ordered_idom"
paulson@15234
   684
  shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   685
by (insert mult_le_cancel_right [of a c 1], simp)
paulson@15234
   686
paulson@15234
   687
lemma mult_le_cancel_left1:
paulson@15234
   688
  fixes c :: "'a :: ordered_idom"
paulson@15234
   689
  shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   690
by (insert mult_le_cancel_left [of c 1 b], simp)
paulson@15234
   691
paulson@15234
   692
lemma mult_le_cancel_left2:
paulson@15234
   693
  fixes c :: "'a :: ordered_idom"
paulson@15234
   694
  shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   695
by (insert mult_le_cancel_left [of c a 1], simp)
paulson@15234
   696
paulson@15234
   697
lemma mult_less_cancel_right1:
paulson@15234
   698
  fixes c :: "'a :: ordered_idom"
paulson@15234
   699
  shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   700
by (insert mult_less_cancel_right [of 1 c b], simp)
paulson@15234
   701
paulson@15234
   702
lemma mult_less_cancel_right2:
paulson@15234
   703
  fixes c :: "'a :: ordered_idom"
paulson@15234
   704
  shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   705
by (insert mult_less_cancel_right [of a c 1], simp)
paulson@15234
   706
paulson@15234
   707
lemma mult_less_cancel_left1:
paulson@15234
   708
  fixes c :: "'a :: ordered_idom"
paulson@15234
   709
  shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   710
by (insert mult_less_cancel_left [of c 1 b], simp)
paulson@15234
   711
paulson@15234
   712
lemma mult_less_cancel_left2:
paulson@15234
   713
  fixes c :: "'a :: ordered_idom"
paulson@15234
   714
  shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   715
by (insert mult_less_cancel_left [of c a 1], simp)
paulson@15234
   716
paulson@15234
   717
lemma mult_cancel_right1 [simp]:
huffman@22990
   718
  fixes c :: "'a :: dom"
paulson@15234
   719
  shows "(c = b*c) = (c = 0 | b=1)"
paulson@15234
   720
by (insert mult_cancel_right [of 1 c b], force)
paulson@15234
   721
paulson@15234
   722
lemma mult_cancel_right2 [simp]:
huffman@22990
   723
  fixes c :: "'a :: dom"
paulson@15234
   724
  shows "(a*c = c) = (c = 0 | a=1)"
paulson@15234
   725
by (insert mult_cancel_right [of a c 1], simp)
paulson@15234
   726
 
paulson@15234
   727
lemma mult_cancel_left1 [simp]:
huffman@22990
   728
  fixes c :: "'a :: dom"
paulson@15234
   729
  shows "(c = c*b) = (c = 0 | b=1)"
paulson@15234
   730
by (insert mult_cancel_left [of c 1 b], force)
paulson@15234
   731
paulson@15234
   732
lemma mult_cancel_left2 [simp]:
huffman@22990
   733
  fixes c :: "'a :: dom"
paulson@15234
   734
  shows "(c*a = c) = (c = 0 | a=1)"
paulson@15234
   735
by (insert mult_cancel_left [of c a 1], simp)
paulson@15234
   736
paulson@15234
   737
paulson@15234
   738
text{*Simprules for comparisons where common factors can be cancelled.*}
paulson@15234
   739
lemmas mult_compare_simps =
paulson@15234
   740
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
   741
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
   742
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
   743
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
   744
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
   745
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
   746
    mult_cancel_right mult_cancel_left
paulson@15234
   747
    mult_cancel_right1 mult_cancel_right2
paulson@15234
   748
    mult_cancel_left1 mult_cancel_left2
paulson@15234
   749
paulson@15234
   750
paulson@14265
   751
subsection {* Fields *}
paulson@14265
   752
paulson@14288
   753
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
paulson@14288
   754
proof
paulson@14288
   755
  assume neq: "b \<noteq> 0"
paulson@14288
   756
  {
paulson@14288
   757
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
paulson@14288
   758
    also assume "a / b = 1"
paulson@14288
   759
    finally show "a = b" by simp
paulson@14288
   760
  next
paulson@14288
   761
    assume "a = b"
paulson@14288
   762
    with neq show "a / b = 1" by (simp add: divide_inverse)
paulson@14288
   763
  }
paulson@14288
   764
qed
paulson@14288
   765
paulson@14288
   766
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
paulson@14288
   767
by (simp add: divide_inverse)
paulson@14288
   768
nipkow@23398
   769
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
paulson@14288
   770
  by (simp add: divide_inverse)
paulson@14288
   771
paulson@14430
   772
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
paulson@14430
   773
by (simp add: divide_inverse)
paulson@14277
   774
paulson@15228
   775
lemma divide_self_if [simp]:
paulson@15228
   776
     "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
paulson@15228
   777
  by (simp add: divide_self)
paulson@15228
   778
paulson@14430
   779
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
paulson@14430
   780
by (simp add: divide_inverse)
paulson@14277
   781
paulson@14430
   782
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
paulson@14430
   783
by (simp add: divide_inverse)
paulson@14277
   784
paulson@14430
   785
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
nipkow@23477
   786
by (simp add: divide_inverse ring_distribs) 
paulson@14293
   787
nipkow@23482
   788
(* what ordering?? this is a straight instance of mult_eq_0_iff
paulson@14270
   789
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
paulson@14270
   790
      of an ordering.*}
huffman@20496
   791
lemma field_mult_eq_0_iff [simp]:
huffman@20496
   792
  "(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
huffman@22990
   793
by simp
nipkow@23482
   794
*)
nipkow@23496
   795
(* subsumed by mult_cancel lemmas on ring_no_zero_divisors
paulson@14268
   796
text{*Cancellation of equalities with a common factor*}
paulson@14268
   797
lemma field_mult_cancel_right_lemma:
huffman@20496
   798
      assumes cnz: "c \<noteq> (0::'a::division_ring)"
huffman@20496
   799
         and eq:  "a*c = b*c"
huffman@20496
   800
        shows "a=b"
paulson@14377
   801
proof -
paulson@14268
   802
  have "(a * c) * inverse c = (b * c) * inverse c"
paulson@14268
   803
    by (simp add: eq)
paulson@14268
   804
  thus "a=b"
paulson@14268
   805
    by (simp add: mult_assoc cnz)
paulson@14377
   806
qed
paulson@14268
   807
paulson@14348
   808
lemma field_mult_cancel_right [simp]:
huffman@20496
   809
     "(a*c = b*c) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   810
by simp
paulson@14268
   811
paulson@14348
   812
lemma field_mult_cancel_left [simp]:
huffman@20496
   813
     "(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   814
by simp
nipkow@23496
   815
*)
huffman@20496
   816
lemma nonzero_imp_inverse_nonzero:
huffman@20496
   817
  "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
paulson@14377
   818
proof
paulson@14268
   819
  assume ianz: "inverse a = 0"
paulson@14268
   820
  assume "a \<noteq> 0"
paulson@14268
   821
  hence "1 = a * inverse a" by simp
paulson@14268
   822
  also have "... = 0" by (simp add: ianz)
huffman@20496
   823
  finally have "1 = (0::'a::division_ring)" .
paulson@14268
   824
  thus False by (simp add: eq_commute)
paulson@14377
   825
qed
paulson@14268
   826
paulson@14277
   827
paulson@14277
   828
subsection{*Basic Properties of @{term inverse}*}
paulson@14277
   829
huffman@20496
   830
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   831
apply (rule ccontr) 
paulson@14268
   832
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   833
done
paulson@14268
   834
paulson@14268
   835
lemma inverse_nonzero_imp_nonzero:
huffman@20496
   836
   "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   837
apply (rule ccontr) 
paulson@14268
   838
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   839
done
paulson@14268
   840
paulson@14268
   841
lemma inverse_nonzero_iff_nonzero [simp]:
huffman@20496
   842
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
paulson@14268
   843
by (force dest: inverse_nonzero_imp_nonzero) 
paulson@14268
   844
paulson@14268
   845
lemma nonzero_inverse_minus_eq:
huffman@20496
   846
      assumes [simp]: "a\<noteq>0"
huffman@20496
   847
      shows "inverse(-a) = -inverse(a::'a::division_ring)"
paulson@14377
   848
proof -
paulson@14377
   849
  have "-a * inverse (- a) = -a * - inverse a"
paulson@14377
   850
    by simp
paulson@14377
   851
  thus ?thesis 
nipkow@23496
   852
    by (simp only: mult_cancel_left, simp)
paulson@14377
   853
qed
paulson@14268
   854
paulson@14268
   855
lemma inverse_minus_eq [simp]:
huffman@20496
   856
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
paulson@14377
   857
proof cases
paulson@14377
   858
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
   859
next
paulson@14377
   860
  assume "a\<noteq>0" 
paulson@14377
   861
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
   862
qed
paulson@14268
   863
paulson@14268
   864
lemma nonzero_inverse_eq_imp_eq:
paulson@14269
   865
      assumes inveq: "inverse a = inverse b"
paulson@14269
   866
	  and anz:  "a \<noteq> 0"
paulson@14269
   867
	  and bnz:  "b \<noteq> 0"
huffman@20496
   868
	 shows "a = (b::'a::division_ring)"
paulson@14377
   869
proof -
paulson@14268
   870
  have "a * inverse b = a * inverse a"
paulson@14268
   871
    by (simp add: inveq)
paulson@14268
   872
  hence "(a * inverse b) * b = (a * inverse a) * b"
paulson@14268
   873
    by simp
paulson@14268
   874
  thus "a = b"
paulson@14268
   875
    by (simp add: mult_assoc anz bnz)
paulson@14377
   876
qed
paulson@14268
   877
paulson@14268
   878
lemma inverse_eq_imp_eq:
huffman@20496
   879
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
haftmann@21328
   880
apply (cases "a=0 | b=0") 
paulson@14268
   881
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
   882
              simp add: eq_commute [of "0::'a"])
paulson@14268
   883
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
   884
done
paulson@14268
   885
paulson@14268
   886
lemma inverse_eq_iff_eq [simp]:
huffman@20496
   887
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
huffman@20496
   888
by (force dest!: inverse_eq_imp_eq)
paulson@14268
   889
paulson@14270
   890
lemma nonzero_inverse_inverse_eq:
huffman@20496
   891
      assumes [simp]: "a \<noteq> 0"
huffman@20496
   892
      shows "inverse(inverse (a::'a::division_ring)) = a"
paulson@14270
   893
  proof -
paulson@14270
   894
  have "(inverse (inverse a) * inverse a) * a = a" 
paulson@14270
   895
    by (simp add: nonzero_imp_inverse_nonzero)
paulson@14270
   896
  thus ?thesis
paulson@14270
   897
    by (simp add: mult_assoc)
paulson@14270
   898
  qed
paulson@14270
   899
paulson@14270
   900
lemma inverse_inverse_eq [simp]:
huffman@20496
   901
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
paulson@14270
   902
  proof cases
paulson@14270
   903
    assume "a=0" thus ?thesis by simp
paulson@14270
   904
  next
paulson@14270
   905
    assume "a\<noteq>0" 
paulson@14270
   906
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
   907
  qed
paulson@14270
   908
huffman@20496
   909
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)"
paulson@14270
   910
  proof -
huffman@20496
   911
  have "inverse 1 * 1 = (1::'a::division_ring)" 
paulson@14270
   912
    by (rule left_inverse [OF zero_neq_one [symmetric]])
paulson@14270
   913
  thus ?thesis  by simp
paulson@14270
   914
  qed
paulson@14270
   915
paulson@15077
   916
lemma inverse_unique: 
paulson@15077
   917
  assumes ab: "a*b = 1"
huffman@20496
   918
  shows "inverse a = (b::'a::division_ring)"
paulson@15077
   919
proof -
paulson@15077
   920
  have "a \<noteq> 0" using ab by auto
paulson@15077
   921
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
paulson@15077
   922
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
paulson@15077
   923
qed
paulson@15077
   924
paulson@14270
   925
lemma nonzero_inverse_mult_distrib: 
paulson@14270
   926
      assumes anz: "a \<noteq> 0"
paulson@14270
   927
          and bnz: "b \<noteq> 0"
huffman@20496
   928
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
paulson@14270
   929
  proof -
paulson@14270
   930
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
nipkow@23482
   931
    by (simp add: anz bnz)
paulson@14270
   932
  hence "inverse(a*b) * a = inverse(b)" 
paulson@14270
   933
    by (simp add: mult_assoc bnz)
paulson@14270
   934
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
paulson@14270
   935
    by simp
paulson@14270
   936
  thus ?thesis
paulson@14270
   937
    by (simp add: mult_assoc anz)
paulson@14270
   938
  qed
paulson@14270
   939
paulson@14270
   940
text{*This version builds in division by zero while also re-orienting
paulson@14270
   941
      the right-hand side.*}
paulson@14270
   942
lemma inverse_mult_distrib [simp]:
paulson@14270
   943
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
   944
  proof cases
paulson@14270
   945
    assume "a \<noteq> 0 & b \<noteq> 0" 
haftmann@22993
   946
    thus ?thesis
haftmann@22993
   947
      by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
   948
  next
paulson@14270
   949
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
haftmann@22993
   950
    thus ?thesis
haftmann@22993
   951
      by force
paulson@14270
   952
  qed
paulson@14270
   953
huffman@20496
   954
lemma division_ring_inverse_add:
huffman@20496
   955
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
huffman@20496
   956
   ==> inverse a + inverse b = inverse a * (a+b) * inverse b"
nipkow@23477
   957
by (simp add: ring_simps)
huffman@20496
   958
huffman@20496
   959
lemma division_ring_inverse_diff:
huffman@20496
   960
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
huffman@20496
   961
   ==> inverse a - inverse b = inverse a * (b-a) * inverse b"
nipkow@23477
   962
by (simp add: ring_simps)
huffman@20496
   963
paulson@14270
   964
text{*There is no slick version using division by zero.*}
paulson@14270
   965
lemma inverse_add:
nipkow@23477
   966
  "[|a \<noteq> 0;  b \<noteq> 0|]
nipkow@23477
   967
   ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
huffman@20496
   968
by (simp add: division_ring_inverse_add mult_ac)
paulson@14270
   969
paulson@14365
   970
lemma inverse_divide [simp]:
nipkow@23477
   971
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
nipkow@23477
   972
by (simp add: divide_inverse mult_commute)
paulson@14365
   973
wenzelm@23389
   974
avigad@16775
   975
subsection {* Calculations with fractions *}
avigad@16775
   976
nipkow@23413
   977
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
   978
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
   979
because the latter are covered by a simproc. *}
nipkow@23413
   980
nipkow@23413
   981
lemma nonzero_mult_divide_mult_cancel_left[simp]:
nipkow@23477
   982
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
   983
proof -
paulson@14277
   984
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
nipkow@23482
   985
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
paulson@14277
   986
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
   987
    by (simp only: mult_ac)
paulson@14277
   988
  also have "... =  a * inverse b"
paulson@14277
   989
    by simp
paulson@14277
   990
    finally show ?thesis 
paulson@14277
   991
    by (simp add: divide_inverse)
paulson@14277
   992
qed
paulson@14277
   993
nipkow@23413
   994
lemma mult_divide_mult_cancel_left:
nipkow@23477
   995
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
   996
apply (cases "b = 0")
nipkow@23413
   997
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
paulson@14277
   998
done
paulson@14277
   999
nipkow@23413
  1000
lemma nonzero_mult_divide_mult_cancel_right:
nipkow@23477
  1001
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1002
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
paulson@14321
  1003
nipkow@23413
  1004
lemma mult_divide_mult_cancel_right:
nipkow@23477
  1005
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1006
apply (cases "b = 0")
nipkow@23413
  1007
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
paulson@14321
  1008
done
nipkow@23413
  1009
paulson@14284
  1010
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
nipkow@23477
  1011
by (simp add: divide_inverse)
paulson@14284
  1012
paulson@15234
  1013
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
  1014
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1015
paulson@14430
  1016
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
  1017
by (simp add: divide_inverse mult_ac)
paulson@14288
  1018
nipkow@23482
  1019
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
nipkow@23482
  1020
paulson@14288
  1021
lemma divide_divide_eq_right [simp]:
nipkow@23477
  1022
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
  1023
by (simp add: divide_inverse mult_ac)
paulson@14288
  1024
paulson@14288
  1025
lemma divide_divide_eq_left [simp]:
nipkow@23477
  1026
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
  1027
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1028
avigad@16775
  1029
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1030
    x / y + w / z = (x * z + w * y) / (y * z)"
nipkow@23477
  1031
apply (subgoal_tac "x / y = (x * z) / (y * z)")
nipkow@23477
  1032
apply (erule ssubst)
nipkow@23477
  1033
apply (subgoal_tac "w / z = (w * y) / (y * z)")
nipkow@23477
  1034
apply (erule ssubst)
nipkow@23477
  1035
apply (rule add_divide_distrib [THEN sym])
nipkow@23477
  1036
apply (subst mult_commute)
nipkow@23477
  1037
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
nipkow@23477
  1038
apply assumption
nipkow@23477
  1039
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
nipkow@23477
  1040
apply assumption
avigad@16775
  1041
done
paulson@14268
  1042
wenzelm@23389
  1043
paulson@15234
  1044
subsubsection{*Special Cancellation Simprules for Division*}
paulson@15234
  1045
nipkow@23413
  1046
lemma mult_divide_mult_cancel_left_if[simp]:
nipkow@23477
  1047
fixes c :: "'a :: {field,division_by_zero}"
nipkow@23477
  1048
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
nipkow@23413
  1049
by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
  1050
nipkow@23413
  1051
lemma nonzero_mult_divide_cancel_right[simp]:
nipkow@23413
  1052
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
nipkow@23413
  1053
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
nipkow@23413
  1054
nipkow@23413
  1055
lemma nonzero_mult_divide_cancel_left[simp]:
nipkow@23413
  1056
  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
nipkow@23413
  1057
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
nipkow@23413
  1058
nipkow@23413
  1059
nipkow@23413
  1060
lemma nonzero_divide_mult_cancel_right[simp]:
nipkow@23413
  1061
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
nipkow@23413
  1062
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
nipkow@23413
  1063
nipkow@23413
  1064
lemma nonzero_divide_mult_cancel_left[simp]:
nipkow@23413
  1065
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
nipkow@23413
  1066
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
nipkow@23413
  1067
nipkow@23413
  1068
nipkow@23413
  1069
lemma nonzero_mult_divide_mult_cancel_left2[simp]:
nipkow@23477
  1070
  "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1071
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
nipkow@23413
  1072
nipkow@23413
  1073
lemma nonzero_mult_divide_mult_cancel_right2[simp]:
nipkow@23477
  1074
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
nipkow@23413
  1075
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
nipkow@23413
  1076
paulson@15234
  1077
paulson@14293
  1078
subsection {* Division and Unary Minus *}
paulson@14293
  1079
paulson@14293
  1080
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
  1081
by (simp add: divide_inverse minus_mult_left)
paulson@14293
  1082
paulson@14293
  1083
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
  1084
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
  1085
paulson@14293
  1086
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
  1087
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1088
paulson@14430
  1089
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
paulson@14430
  1090
by (simp add: divide_inverse minus_mult_left [symmetric])
paulson@14293
  1091
paulson@14293
  1092
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14430
  1093
by (simp add: divide_inverse minus_mult_right [symmetric])
paulson@14430
  1094
paulson@14293
  1095
paulson@14293
  1096
text{*The effect is to extract signs from divisions*}
paulson@17085
  1097
lemmas divide_minus_left = minus_divide_left [symmetric]
paulson@17085
  1098
lemmas divide_minus_right = minus_divide_right [symmetric]
paulson@17085
  1099
declare divide_minus_left [simp]   divide_minus_right [simp]
paulson@14293
  1100
paulson@14387
  1101
text{*Also, extract signs from products*}
paulson@17085
  1102
lemmas mult_minus_left = minus_mult_left [symmetric]
paulson@17085
  1103
lemmas mult_minus_right = minus_mult_right [symmetric]
paulson@17085
  1104
declare mult_minus_left [simp]   mult_minus_right [simp]
paulson@14387
  1105
paulson@14293
  1106
lemma minus_divide_divide [simp]:
nipkow@23477
  1107
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1108
apply (cases "b=0", simp) 
paulson@14293
  1109
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1110
done
paulson@14293
  1111
paulson@14430
  1112
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
  1113
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
  1114
nipkow@23482
  1115
lemma add_divide_eq_iff:
nipkow@23482
  1116
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
nipkow@23482
  1117
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1118
nipkow@23482
  1119
lemma divide_add_eq_iff:
nipkow@23482
  1120
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
nipkow@23482
  1121
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1122
nipkow@23482
  1123
lemma diff_divide_eq_iff:
nipkow@23482
  1124
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
nipkow@23482
  1125
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1126
nipkow@23482
  1127
lemma divide_diff_eq_iff:
nipkow@23482
  1128
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
nipkow@23482
  1129
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1130
nipkow@23482
  1131
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
nipkow@23482
  1132
proof -
nipkow@23482
  1133
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1134
  have "(a = b/c) = (a*c = (b/c)*c)" by simp
nipkow@23496
  1135
  also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
nipkow@23482
  1136
  finally show ?thesis .
nipkow@23482
  1137
qed
nipkow@23482
  1138
nipkow@23482
  1139
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
nipkow@23482
  1140
proof -
nipkow@23482
  1141
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1142
  have "(b/c = a) = ((b/c)*c = a*c)"  by simp
nipkow@23496
  1143
  also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
nipkow@23482
  1144
  finally show ?thesis .
nipkow@23482
  1145
qed
nipkow@23482
  1146
nipkow@23482
  1147
lemma eq_divide_eq:
nipkow@23482
  1148
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
nipkow@23482
  1149
by (simp add: nonzero_eq_divide_eq) 
nipkow@23482
  1150
nipkow@23482
  1151
lemma divide_eq_eq:
nipkow@23482
  1152
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
nipkow@23482
  1153
by (force simp add: nonzero_divide_eq_eq) 
nipkow@23482
  1154
nipkow@23482
  1155
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1156
    b = a * c ==> b / c = a"
nipkow@23482
  1157
  by (subst divide_eq_eq, simp)
nipkow@23482
  1158
nipkow@23482
  1159
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1160
    a * c = b ==> a = b / c"
nipkow@23482
  1161
  by (subst eq_divide_eq, simp)
nipkow@23482
  1162
nipkow@23482
  1163
nipkow@23482
  1164
lemmas field_eq_simps = ring_simps
nipkow@23482
  1165
  (* pull / out*)
nipkow@23482
  1166
  add_divide_eq_iff divide_add_eq_iff
nipkow@23482
  1167
  diff_divide_eq_iff divide_diff_eq_iff
nipkow@23482
  1168
  (* multiply eqn *)
nipkow@23482
  1169
  nonzero_eq_divide_eq nonzero_divide_eq_eq
nipkow@23482
  1170
(* is added later:
nipkow@23482
  1171
  times_divide_eq_left times_divide_eq_right
nipkow@23482
  1172
*)
nipkow@23482
  1173
nipkow@23482
  1174
text{*An example:*}
nipkow@23482
  1175
lemma fixes a b c d e f :: "'a::field"
nipkow@23482
  1176
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
  1177
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
nipkow@23482
  1178
 apply(simp add:field_eq_simps)
nipkow@23482
  1179
apply(simp)
nipkow@23482
  1180
done
nipkow@23482
  1181
nipkow@23482
  1182
avigad@16775
  1183
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1184
    x / y - w / z = (x * z - w * y) / (y * z)"
nipkow@23482
  1185
by (simp add:field_eq_simps times_divide_eq)
nipkow@23482
  1186
nipkow@23482
  1187
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
nipkow@23482
  1188
    (x / y = w / z) = (x * z = w * y)"
nipkow@23482
  1189
by (simp add:field_eq_simps times_divide_eq)
paulson@14293
  1190
wenzelm@23389
  1191
paulson@14268
  1192
subsection {* Ordered Fields *}
paulson@14268
  1193
paulson@14277
  1194
lemma positive_imp_inverse_positive: 
nipkow@23482
  1195
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
nipkow@23482
  1196
proof -
paulson@14268
  1197
  have "0 < a * inverse a" 
paulson@14268
  1198
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1199
  thus "0 < inverse a" 
paulson@14268
  1200
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
nipkow@23482
  1201
qed
paulson@14268
  1202
paulson@14277
  1203
lemma negative_imp_inverse_negative:
nipkow@23482
  1204
  "a < 0 ==> inverse a < (0::'a::ordered_field)"
nipkow@23482
  1205
by (insert positive_imp_inverse_positive [of "-a"], 
nipkow@23482
  1206
    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
paulson@14268
  1207
paulson@14268
  1208
lemma inverse_le_imp_le:
nipkow@23482
  1209
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
nipkow@23482
  1210
shows "b \<le> (a::'a::ordered_field)"
nipkow@23482
  1211
proof (rule classical)
paulson@14268
  1212
  assume "~ b \<le> a"
nipkow@23482
  1213
  hence "a < b"  by (simp add: linorder_not_le)
nipkow@23482
  1214
  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
paulson@14268
  1215
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1216
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1217
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1218
    by (simp add: bpos order_less_imp_le mult_right_mono)
nipkow@23482
  1219
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
nipkow@23482
  1220
qed
paulson@14268
  1221
paulson@14277
  1222
lemma inverse_positive_imp_positive:
nipkow@23482
  1223
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
nipkow@23482
  1224
shows "0 < (a::'a::ordered_field)"
wenzelm@23389
  1225
proof -
paulson@14277
  1226
  have "0 < inverse (inverse a)"
wenzelm@23389
  1227
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
  1228
  thus "0 < a"
wenzelm@23389
  1229
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1230
qed
paulson@14277
  1231
paulson@14277
  1232
lemma inverse_positive_iff_positive [simp]:
nipkow@23482
  1233
  "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1234
apply (cases "a = 0", simp)
paulson@14277
  1235
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1236
done
paulson@14277
  1237
paulson@14277
  1238
lemma inverse_negative_imp_negative:
nipkow@23482
  1239
assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
nipkow@23482
  1240
shows "a < (0::'a::ordered_field)"
wenzelm@23389
  1241
proof -
paulson@14277
  1242
  have "inverse (inverse a) < 0"
wenzelm@23389
  1243
    using inv_less_0 by (rule negative_imp_inverse_negative)
nipkow@23482
  1244
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1245
qed
paulson@14277
  1246
paulson@14277
  1247
lemma inverse_negative_iff_negative [simp]:
nipkow@23482
  1248
  "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1249
apply (cases "a = 0", simp)
paulson@14277
  1250
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1251
done
paulson@14277
  1252
paulson@14277
  1253
lemma inverse_nonnegative_iff_nonnegative [simp]:
nipkow@23482
  1254
  "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1255
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1256
paulson@14277
  1257
lemma inverse_nonpositive_iff_nonpositive [simp]:
nipkow@23482
  1258
  "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1259
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1260
chaieb@23406
  1261
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
chaieb@23406
  1262
proof
chaieb@23406
  1263
  fix x::'a
chaieb@23406
  1264
  have m1: "- (1::'a) < 0" by simp
chaieb@23406
  1265
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1266
  have "(- 1) + x < x" by simp
chaieb@23406
  1267
  thus "\<exists>y. y < x" by blast
chaieb@23406
  1268
qed
chaieb@23406
  1269
chaieb@23406
  1270
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
chaieb@23406
  1271
proof
chaieb@23406
  1272
  fix x::'a
chaieb@23406
  1273
  have m1: " (1::'a) > 0" by simp
chaieb@23406
  1274
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1275
  have "1 + x > x" by simp
chaieb@23406
  1276
  thus "\<exists>y. y > x" by blast
chaieb@23406
  1277
qed
paulson@14277
  1278
paulson@14277
  1279
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1280
paulson@14268
  1281
lemma less_imp_inverse_less:
nipkow@23482
  1282
assumes less: "a < b" and apos:  "0 < a"
nipkow@23482
  1283
shows "inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1284
proof (rule ccontr)
paulson@14268
  1285
  assume "~ inverse b < inverse a"
paulson@14268
  1286
  hence "inverse a \<le> inverse b"
paulson@14268
  1287
    by (simp add: linorder_not_less)
paulson@14268
  1288
  hence "~ (a < b)"
paulson@14268
  1289
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
  1290
  thus False
paulson@14268
  1291
    by (rule notE [OF _ less])
nipkow@23482
  1292
qed
paulson@14268
  1293
paulson@14268
  1294
lemma inverse_less_imp_less:
nipkow@23482
  1295
  "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1296
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1297
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1298
done
paulson@14268
  1299
paulson@14268
  1300
text{*Both premises are essential. Consider -1 and 1.*}
paulson@14268
  1301
lemma inverse_less_iff_less [simp]:
nipkow@23482
  1302
  "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1303
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1304
paulson@14268
  1305
lemma le_imp_inverse_le:
nipkow@23482
  1306
  "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1307
by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1308
paulson@14268
  1309
lemma inverse_le_iff_le [simp]:
nipkow@23482
  1310
 "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1311
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1312
paulson@14268
  1313
paulson@14268
  1314
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1315
case is trivial, since inverse preserves signs.*}
paulson@14268
  1316
lemma inverse_le_imp_le_neg:
nipkow@23482
  1317
  "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
nipkow@23482
  1318
apply (rule classical) 
nipkow@23482
  1319
apply (subgoal_tac "a < 0") 
nipkow@23482
  1320
 prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
nipkow@23482
  1321
apply (insert inverse_le_imp_le [of "-b" "-a"])
nipkow@23482
  1322
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1323
done
paulson@14268
  1324
paulson@14268
  1325
lemma less_imp_inverse_less_neg:
paulson@14268
  1326
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1327
apply (subgoal_tac "a < 0") 
nipkow@23482
  1328
 prefer 2 apply (blast intro: order_less_trans) 
nipkow@23482
  1329
apply (insert less_imp_inverse_less [of "-b" "-a"])
nipkow@23482
  1330
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1331
done
paulson@14268
  1332
paulson@14268
  1333
lemma inverse_less_imp_less_neg:
paulson@14268
  1334
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
nipkow@23482
  1335
apply (rule classical) 
nipkow@23482
  1336
apply (subgoal_tac "a < 0") 
nipkow@23482
  1337
 prefer 2
nipkow@23482
  1338
 apply (force simp add: linorder_not_less intro: order_le_less_trans) 
nipkow@23482
  1339
apply (insert inverse_less_imp_less [of "-b" "-a"])
nipkow@23482
  1340
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1341
done
paulson@14268
  1342
paulson@14268
  1343
lemma inverse_less_iff_less_neg [simp]:
nipkow@23482
  1344
  "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
nipkow@23482
  1345
apply (insert inverse_less_iff_less [of "-b" "-a"])
nipkow@23482
  1346
apply (simp del: inverse_less_iff_less 
nipkow@23482
  1347
            add: order_less_imp_not_eq nonzero_inverse_minus_eq)
nipkow@23482
  1348
done
paulson@14268
  1349
paulson@14268
  1350
lemma le_imp_inverse_le_neg:
nipkow@23482
  1351
  "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1352
by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1353
paulson@14268
  1354
lemma inverse_le_iff_le_neg [simp]:
nipkow@23482
  1355
 "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1356
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1357
paulson@14277
  1358
paulson@14365
  1359
subsection{*Inverses and the Number One*}
paulson@14365
  1360
paulson@14365
  1361
lemma one_less_inverse_iff:
nipkow@23482
  1362
  "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
nipkow@23482
  1363
proof cases
paulson@14365
  1364
  assume "0 < x"
paulson@14365
  1365
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1366
    show ?thesis by simp
paulson@14365
  1367
next
paulson@14365
  1368
  assume notless: "~ (0 < x)"
paulson@14365
  1369
  have "~ (1 < inverse x)"
paulson@14365
  1370
  proof
paulson@14365
  1371
    assume "1 < inverse x"
paulson@14365
  1372
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1373
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1374
    finally show False by auto
paulson@14365
  1375
  qed
paulson@14365
  1376
  with notless show ?thesis by simp
paulson@14365
  1377
qed
paulson@14365
  1378
paulson@14365
  1379
lemma inverse_eq_1_iff [simp]:
nipkow@23482
  1380
  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1381
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1382
paulson@14365
  1383
lemma one_le_inverse_iff:
nipkow@23482
  1384
  "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1385
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1386
                    eq_commute [of 1]) 
paulson@14365
  1387
paulson@14365
  1388
lemma inverse_less_1_iff:
nipkow@23482
  1389
  "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1390
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1391
paulson@14365
  1392
lemma inverse_le_1_iff:
nipkow@23482
  1393
  "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1394
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1395
wenzelm@23389
  1396
paulson@14288
  1397
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1398
paulson@14288
  1399
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1400
proof -
paulson@14288
  1401
  assume less: "0<c"
paulson@14288
  1402
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1403
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1404
  also have "... = (a*c \<le> b)"
paulson@14288
  1405
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1406
  finally show ?thesis .
paulson@14288
  1407
qed
paulson@14288
  1408
paulson@14288
  1409
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1410
proof -
paulson@14288
  1411
  assume less: "c<0"
paulson@14288
  1412
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1413
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1414
  also have "... = (b \<le> a*c)"
paulson@14288
  1415
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1416
  finally show ?thesis .
paulson@14288
  1417
qed
paulson@14288
  1418
paulson@14288
  1419
lemma le_divide_eq:
paulson@14288
  1420
  "(a \<le> b/c) = 
paulson@14288
  1421
   (if 0 < c then a*c \<le> b
paulson@14288
  1422
             else if c < 0 then b \<le> a*c
paulson@14288
  1423
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1424
apply (cases "c=0", simp) 
paulson@14288
  1425
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1426
done
paulson@14288
  1427
paulson@14288
  1428
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1429
proof -
paulson@14288
  1430
  assume less: "0<c"
paulson@14288
  1431
  hence "(b/c \<le> a) = ((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_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1435
  finally show ?thesis .
paulson@14288
  1436
qed
paulson@14288
  1437
paulson@14288
  1438
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1439
proof -
paulson@14288
  1440
  assume less: "c<0"
paulson@14288
  1441
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1442
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1443
  also have "... = (a*c \<le> b)"
paulson@14288
  1444
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1445
  finally show ?thesis .
paulson@14288
  1446
qed
paulson@14288
  1447
paulson@14288
  1448
lemma divide_le_eq:
paulson@14288
  1449
  "(b/c \<le> a) = 
paulson@14288
  1450
   (if 0 < c then b \<le> a*c
paulson@14288
  1451
             else if c < 0 then a*c \<le> b
paulson@14288
  1452
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1453
apply (cases "c=0", simp) 
paulson@14288
  1454
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1455
done
paulson@14288
  1456
paulson@14288
  1457
lemma pos_less_divide_eq:
paulson@14288
  1458
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1459
proof -
paulson@14288
  1460
  assume less: "0<c"
paulson@14288
  1461
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@15234
  1462
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1463
  also have "... = (a*c < b)"
paulson@14288
  1464
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1465
  finally show ?thesis .
paulson@14288
  1466
qed
paulson@14288
  1467
paulson@14288
  1468
lemma neg_less_divide_eq:
paulson@14288
  1469
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1470
proof -
paulson@14288
  1471
  assume less: "c<0"
paulson@14288
  1472
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@15234
  1473
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1474
  also have "... = (b < a*c)"
paulson@14288
  1475
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1476
  finally show ?thesis .
paulson@14288
  1477
qed
paulson@14288
  1478
paulson@14288
  1479
lemma less_divide_eq:
paulson@14288
  1480
  "(a < b/c) = 
paulson@14288
  1481
   (if 0 < c then a*c < b
paulson@14288
  1482
             else if c < 0 then b < a*c
paulson@14288
  1483
             else  a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1484
apply (cases "c=0", simp) 
paulson@14288
  1485
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1486
done
paulson@14288
  1487
paulson@14288
  1488
lemma pos_divide_less_eq:
paulson@14288
  1489
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1490
proof -
paulson@14288
  1491
  assume less: "0<c"
paulson@14288
  1492
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@15234
  1493
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1494
  also have "... = (b < a*c)"
paulson@14288
  1495
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1496
  finally show ?thesis .
paulson@14288
  1497
qed
paulson@14288
  1498
paulson@14288
  1499
lemma neg_divide_less_eq:
paulson@14288
  1500
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1501
proof -
paulson@14288
  1502
  assume less: "c<0"
paulson@14288
  1503
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@15234
  1504
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1505
  also have "... = (a*c < b)"
paulson@14288
  1506
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1507
  finally show ?thesis .
paulson@14288
  1508
qed
paulson@14288
  1509
paulson@14288
  1510
lemma divide_less_eq:
paulson@14288
  1511
  "(b/c < a) = 
paulson@14288
  1512
   (if 0 < c then b < a*c
paulson@14288
  1513
             else if c < 0 then a*c < b
paulson@14288
  1514
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1515
apply (cases "c=0", simp) 
paulson@14288
  1516
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1517
done
paulson@14288
  1518
nipkow@23482
  1519
nipkow@23482
  1520
subsection{*Field simplification*}
nipkow@23482
  1521
nipkow@23482
  1522
text{* Lemmas @{text field_simps} multiply with denominators in
nipkow@23482
  1523
in(equations) if they can be proved to be non-zero (for equations) or
nipkow@23482
  1524
positive/negative (for inequations). *}
paulson@14288
  1525
nipkow@23482
  1526
lemmas field_simps = field_eq_simps
nipkow@23482
  1527
  (* multiply ineqn *)
nipkow@23482
  1528
  pos_divide_less_eq neg_divide_less_eq
nipkow@23482
  1529
  pos_less_divide_eq neg_less_divide_eq
nipkow@23482
  1530
  pos_divide_le_eq neg_divide_le_eq
nipkow@23482
  1531
  pos_le_divide_eq neg_le_divide_eq
paulson@14288
  1532
nipkow@23482
  1533
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
nipkow@23483
  1534
of positivity/negativity needed for @{text field_simps}. Have not added @{text
nipkow@23482
  1535
sign_simps} to @{text field_simps} because the former can lead to case
nipkow@23482
  1536
explosions. *}
paulson@14288
  1537
nipkow@23482
  1538
lemmas sign_simps = group_simps
nipkow@23482
  1539
  zero_less_mult_iff  mult_less_0_iff
paulson@14288
  1540
nipkow@23482
  1541
(* Only works once linear arithmetic is installed:
nipkow@23482
  1542
text{*An example:*}
nipkow@23482
  1543
lemma fixes a b c d e f :: "'a::ordered_field"
nipkow@23482
  1544
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
nipkow@23482
  1545
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
nipkow@23482
  1546
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
nipkow@23482
  1547
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
nipkow@23482
  1548
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1549
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
nipkow@23482
  1550
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1551
apply(simp add:field_simps)
avigad@16775
  1552
done
nipkow@23482
  1553
*)
avigad@16775
  1554
wenzelm@23389
  1555
avigad@16775
  1556
subsection{*Division and Signs*}
avigad@16775
  1557
avigad@16775
  1558
lemma zero_less_divide_iff:
avigad@16775
  1559
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
  1560
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
  1561
avigad@16775
  1562
lemma divide_less_0_iff:
avigad@16775
  1563
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
avigad@16775
  1564
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
  1565
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
  1566
avigad@16775
  1567
lemma zero_le_divide_iff:
avigad@16775
  1568
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
avigad@16775
  1569
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
  1570
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1571
avigad@16775
  1572
lemma divide_le_0_iff:
avigad@16775
  1573
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
avigad@16775
  1574
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1575
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1576
avigad@16775
  1577
lemma divide_eq_0_iff [simp]:
avigad@16775
  1578
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
nipkow@23482
  1579
by (simp add: divide_inverse)
avigad@16775
  1580
nipkow@23482
  1581
lemma divide_pos_pos:
nipkow@23482
  1582
  "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
nipkow@23482
  1583
by(simp add:field_simps)
nipkow@23482
  1584
avigad@16775
  1585
nipkow@23482
  1586
lemma divide_nonneg_pos:
nipkow@23482
  1587
  "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
nipkow@23482
  1588
by(simp add:field_simps)
avigad@16775
  1589
nipkow@23482
  1590
lemma divide_neg_pos:
nipkow@23482
  1591
  "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
nipkow@23482
  1592
by(simp add:field_simps)
avigad@16775
  1593
nipkow@23482
  1594
lemma divide_nonpos_pos:
nipkow@23482
  1595
  "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
nipkow@23482
  1596
by(simp add:field_simps)
avigad@16775
  1597
nipkow@23482
  1598
lemma divide_pos_neg:
nipkow@23482
  1599
  "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
nipkow@23482
  1600
by(simp add:field_simps)
avigad@16775
  1601
nipkow@23482
  1602
lemma divide_nonneg_neg:
nipkow@23482
  1603
  "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" 
nipkow@23482
  1604
by(simp add:field_simps)
avigad@16775
  1605
nipkow@23482
  1606
lemma divide_neg_neg:
nipkow@23482
  1607
  "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
nipkow@23482
  1608
by(simp add:field_simps)
avigad@16775
  1609
nipkow@23482
  1610
lemma divide_nonpos_neg:
nipkow@23482
  1611
  "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
nipkow@23482
  1612
by(simp add:field_simps)
paulson@15234
  1613
wenzelm@23389
  1614
paulson@14288
  1615
subsection{*Cancellation Laws for Division*}
paulson@14288
  1616
paulson@14288
  1617
lemma divide_cancel_right [simp]:
paulson@14288
  1618
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1619
apply (cases "c=0", simp)
nipkow@23496
  1620
apply (simp add: divide_inverse)
paulson@14288
  1621
done
paulson@14288
  1622
paulson@14288
  1623
lemma divide_cancel_left [simp]:
paulson@14288
  1624
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
nipkow@23482
  1625
apply (cases "c=0", simp)
nipkow@23496
  1626
apply (simp add: divide_inverse)
paulson@14288
  1627
done
paulson@14288
  1628
wenzelm@23389
  1629
paulson@14353
  1630
subsection {* Division and the Number One *}
paulson@14353
  1631
paulson@14353
  1632
text{*Simplify expressions equated with 1*}
paulson@14353
  1633
lemma divide_eq_1_iff [simp]:
paulson@14353
  1634
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1635
apply (cases "b=0", simp)
nipkow@23482
  1636
apply (simp add: right_inverse_eq)
paulson@14353
  1637
done
paulson@14353
  1638
paulson@14353
  1639
lemma one_eq_divide_iff [simp]:
paulson@14353
  1640
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1641
by (simp add: eq_commute [of 1])
paulson@14353
  1642
paulson@14353
  1643
lemma zero_eq_1_divide_iff [simp]:
paulson@14353
  1644
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
nipkow@23482
  1645
apply (cases "a=0", simp)
nipkow@23482
  1646
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1647
done
paulson@14353
  1648
paulson@14353
  1649
lemma one_divide_eq_0_iff [simp]:
paulson@14353
  1650
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
nipkow@23482
  1651
apply (cases "a=0", simp)
nipkow@23482
  1652
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1653
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1654
done
paulson@14353
  1655
paulson@14353
  1656
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@18623
  1657
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
paulson@18623
  1658
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
paulson@18623
  1659
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
paulson@18623
  1660
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
paulson@17085
  1661
paulson@17085
  1662
declare zero_less_divide_1_iff [simp]
paulson@17085
  1663
declare divide_less_0_1_iff [simp]
paulson@17085
  1664
declare zero_le_divide_1_iff [simp]
paulson@17085
  1665
declare divide_le_0_1_iff [simp]
paulson@14353
  1666
wenzelm@23389
  1667
paulson@14293
  1668
subsection {* Ordering Rules for Division *}
paulson@14293
  1669
paulson@14293
  1670
lemma divide_strict_right_mono:
paulson@14293
  1671
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1672
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
nipkow@23482
  1673
              positive_imp_inverse_positive)
paulson@14293
  1674
paulson@14293
  1675
lemma divide_right_mono:
paulson@14293
  1676
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
nipkow@23482
  1677
by (force simp add: divide_strict_right_mono order_le_less)
paulson@14293
  1678
avigad@16775
  1679
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1680
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1681
apply (drule divide_right_mono [of _ _ "- c"])
nipkow@23482
  1682
apply auto
avigad@16775
  1683
done
avigad@16775
  1684
avigad@16775
  1685
lemma divide_strict_right_mono_neg:
avigad@16775
  1686
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
nipkow@23482
  1687
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
nipkow@23482
  1688
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
avigad@16775
  1689
done
paulson@14293
  1690
paulson@14293
  1691
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1692
      have the same sign*}
paulson@14293
  1693
lemma divide_strict_left_mono:
nipkow@23482
  1694
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1695
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
paulson@14293
  1696
paulson@14293
  1697
lemma divide_left_mono:
nipkow@23482
  1698
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
nipkow@23482
  1699
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
paulson@14293
  1700
avigad@16775
  1701
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1702
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1703
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1704
  apply (auto simp add: mult_commute)
avigad@16775
  1705
done
avigad@16775
  1706
paulson@14293
  1707
lemma divide_strict_left_mono_neg:
nipkow@23482
  1708
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1709
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
nipkow@23482
  1710
paulson@14293
  1711
avigad@16775
  1712
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1713
avigad@16775
  1714
lemma le_divide_eq_1:
avigad@16775
  1715
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1716
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1717
by (auto simp add: le_divide_eq)
avigad@16775
  1718
avigad@16775
  1719
lemma divide_le_eq_1:
avigad@16775
  1720
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1721
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1722
by (auto simp add: divide_le_eq)
avigad@16775
  1723
avigad@16775
  1724
lemma less_divide_eq_1:
avigad@16775
  1725
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1726
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1727
by (auto simp add: less_divide_eq)
avigad@16775
  1728
avigad@16775
  1729
lemma divide_less_eq_1:
avigad@16775
  1730
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1731
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1732
by (auto simp add: divide_less_eq)
avigad@16775
  1733
wenzelm@23389
  1734
avigad@16775
  1735
subsection{*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1736
avigad@16775
  1737
lemma le_divide_eq_1_pos [simp]:
avigad@16775
  1738
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1739
  shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1740
by (auto simp add: le_divide_eq)
avigad@16775
  1741
avigad@16775
  1742
lemma le_divide_eq_1_neg [simp]:
avigad@16775
  1743
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1744
  shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1745
by (auto simp add: le_divide_eq)
avigad@16775
  1746
avigad@16775
  1747
lemma divide_le_eq_1_pos [simp]:
avigad@16775
  1748
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1749
  shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1750
by (auto simp add: divide_le_eq)
avigad@16775
  1751
avigad@16775
  1752
lemma divide_le_eq_1_neg [simp]:
avigad@16775
  1753
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1754
  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1755
by (auto simp add: divide_le_eq)
avigad@16775
  1756
avigad@16775
  1757
lemma less_divide_eq_1_pos [simp]:
avigad@16775
  1758
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1759
  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1760
by (auto simp add: less_divide_eq)
avigad@16775
  1761
avigad@16775
  1762
lemma less_divide_eq_1_neg [simp]:
avigad@16775
  1763
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1764
  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1765
by (auto simp add: less_divide_eq)
avigad@16775
  1766
avigad@16775
  1767
lemma divide_less_eq_1_pos [simp]:
avigad@16775
  1768
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1769
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1770
by (auto simp add: divide_less_eq)
paulson@18649
  1771
paulson@18649
  1772
lemma divide_less_eq_1_neg [simp]:
paulson@18649
  1773
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1774
  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1775
by (auto simp add: divide_less_eq)
avigad@16775
  1776
avigad@16775
  1777
lemma eq_divide_eq_1 [simp]:
avigad@16775
  1778
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1779
  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1780
by (auto simp add: eq_divide_eq)
avigad@16775
  1781
avigad@16775
  1782
lemma divide_eq_eq_1 [simp]:
avigad@16775
  1783
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1784
  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1785
by (auto simp add: divide_eq_eq)
avigad@16775
  1786
wenzelm@23389
  1787
avigad@16775
  1788
subsection {* Reasoning about inequalities with division *}
avigad@16775
  1789
avigad@16775
  1790
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1791
    ==> x * y <= x"
avigad@16775
  1792
  by (auto simp add: mult_compare_simps);
avigad@16775
  1793
avigad@16775
  1794
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1795
    ==> y * x <= x"
avigad@16775
  1796
  by (auto simp add: mult_compare_simps);
avigad@16775
  1797
avigad@16775
  1798
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
avigad@16775
  1799
    x / y <= z";
avigad@16775
  1800
  by (subst pos_divide_le_eq, assumption+);
avigad@16775
  1801
avigad@16775
  1802
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
nipkow@23482
  1803
    z <= x / y"
nipkow@23482
  1804
by(simp add:field_simps)
avigad@16775
  1805
avigad@16775
  1806
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
avigad@16775
  1807
    x / y < z"
nipkow@23482
  1808
by(simp add:field_simps)
avigad@16775
  1809
avigad@16775
  1810
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
avigad@16775
  1811
    z < x / y"
nipkow@23482
  1812
by(simp add:field_simps)
avigad@16775
  1813
avigad@16775
  1814
lemma frac_le: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1815
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
avigad@16775
  1816
  apply (rule mult_imp_div_pos_le)
avigad@16775
  1817
  apply simp;
avigad@16775
  1818
  apply (subst times_divide_eq_left);
avigad@16775
  1819
  apply (rule mult_imp_le_div_pos, assumption)
avigad@16775
  1820
  apply (rule mult_mono)
avigad@16775
  1821
  apply simp_all
paulson@14293
  1822
done
paulson@14293
  1823
avigad@16775
  1824
lemma frac_less: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1825
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
avigad@16775
  1826
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1827
  apply simp;
avigad@16775
  1828
  apply (subst times_divide_eq_left);
avigad@16775
  1829
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1830
  apply (erule mult_less_le_imp_less)
avigad@16775
  1831
  apply simp_all
avigad@16775
  1832
done
avigad@16775
  1833
avigad@16775
  1834
lemma frac_less2: "(0::'a::ordered_field) < x ==> 
avigad@16775
  1835
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
avigad@16775
  1836
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1837
  apply simp_all
avigad@16775
  1838
  apply (subst times_divide_eq_left);
avigad@16775
  1839
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1840
  apply (erule mult_le_less_imp_less)
avigad@16775
  1841
  apply simp_all
avigad@16775
  1842
done
avigad@16775
  1843
avigad@16775
  1844
text{*It's not obvious whether these should be simprules or not. 
avigad@16775
  1845
  Their effect is to gather terms into one big fraction, like
avigad@16775
  1846
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
avigad@16775
  1847
  seem to need them.*}
avigad@16775
  1848
avigad@16775
  1849
declare times_divide_eq [simp]
paulson@14293
  1850
wenzelm@23389
  1851
paulson@14293
  1852
subsection {* Ordered Fields are Dense *}
paulson@14293
  1853
obua@14738
  1854
lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
paulson@14293
  1855
proof -
obua@14738
  1856
  have "a+0 < (a+1::'a::ordered_semidom)"
nipkow@23482
  1857
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1858
  thus ?thesis by simp
paulson@14293
  1859
qed
paulson@14293
  1860
obua@14738
  1861
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
nipkow@23482
  1862
by (blast intro: order_less_trans zero_less_one less_add_one)
paulson@14365
  1863
paulson@14293
  1864
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
nipkow@23482
  1865
by (simp add: field_simps zero_less_two)
paulson@14293
  1866
paulson@14293
  1867
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
nipkow@23482
  1868
by (simp add: field_simps zero_less_two)
paulson@14293
  1869
paulson@14293
  1870
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
paulson@14293
  1871
by (blast intro!: less_half_sum gt_half_sum)
paulson@14293
  1872
paulson@15234
  1873
paulson@14293
  1874
subsection {* Absolute Value *}
paulson@14293
  1875
obua@14738
  1876
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
nipkow@23477
  1877
by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
paulson@14294
  1878
obua@14738
  1879
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  1880
proof -
obua@14738
  1881
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  1882
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  1883
  have a: "(abs a) * (abs b) = ?x"
nipkow@23477
  1884
    by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
obua@14738
  1885
  {
obua@14738
  1886
    fix u v :: 'a
paulson@15481
  1887
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
paulson@15481
  1888
              u * v = pprt a * pprt b + pprt a * nprt b + 
paulson@15481
  1889
                      nprt a * pprt b + nprt a * nprt b"
obua@14738
  1890
      apply (subst prts[of u], subst prts[of v])
nipkow@23477
  1891
      apply (simp add: ring_simps) 
obua@14738
  1892
      done
obua@14738
  1893
  }
obua@14738
  1894
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  1895
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  1896
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  1897
  have xy: "- ?x <= ?y"
obua@14754
  1898
    apply (simp)
obua@14754
  1899
    apply (rule_tac y="0::'a" in order_trans)
nipkow@16568
  1900
    apply (rule addm2)
avigad@16775
  1901
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
nipkow@16568
  1902
    apply (rule addm)
avigad@16775
  1903
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14754
  1904
    done
obua@14738
  1905
  have yx: "?y <= ?x"
nipkow@16568
  1906
    apply (simp add:diff_def)
obua@14754
  1907
    apply (rule_tac y=0 in order_trans)
avigad@16775
  1908
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
avigad@16775
  1909
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
obua@14738
  1910
    done
obua@14738
  1911
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  1912
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  1913
  show ?thesis
obua@14738
  1914
    apply (rule abs_leI)
obua@14738
  1915
    apply (simp add: i1)
obua@14738
  1916
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  1917
    done
obua@14738
  1918
qed
paulson@14294
  1919
obua@14738
  1920
lemma abs_eq_mult: 
obua@14738
  1921
  assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
obua@14738
  1922
  shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
obua@14738
  1923
proof -
obua@14738
  1924
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  1925
    apply (auto)    
obua@14738
  1926
    apply (rule_tac split_mult_pos_le)
obua@14738
  1927
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  1928
    apply (simp)
obua@14738
  1929
    apply (rule_tac split_mult_neg_le)
obua@14738
  1930
    apply (insert prems)
obua@14738
  1931
    apply (blast)
obua@14738
  1932
    done
obua@14738
  1933
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  1934
    by (simp add: prts[symmetric])
obua@14738
  1935
  show ?thesis
obua@14738
  1936
  proof cases
obua@14738
  1937
    assume "0 <= a * b"
obua@14738
  1938
    then show ?thesis
obua@14738
  1939
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1940
      apply (insert prems)
obua@14754
  1941
      apply (auto simp add: 
nipkow@23477
  1942
	ring_simps 
obua@14754
  1943
	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
nipkow@15197
  1944
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])
avigad@16775
  1945
	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
avigad@16775
  1946
	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
obua@14738
  1947
      done
obua@14738
  1948
  next
obua@14738
  1949
    assume "~(0 <= a*b)"
obua@14738
  1950
    with s have "a*b <= 0" by simp
obua@14738
  1951
    then show ?thesis
obua@14738
  1952
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1953
      apply (insert prems)
nipkow@23477
  1954
      apply (auto simp add: ring_simps)
avigad@16775
  1955
      apply(drule (1) mult_nonneg_nonneg[of a b],simp)
avigad@16775
  1956
      apply(drule (1) mult_nonpos_nonpos[of a b],simp)
obua@14738
  1957
      done
obua@14738
  1958
  qed
obua@14738
  1959
qed
paulson@14294
  1960
obua@14738
  1961
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
obua@14738
  1962
by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  1963
obua@14738
  1964
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
obua@14738
  1965
by (simp add: abs_if) 
paulson@14294
  1966
paulson@14294
  1967
lemma nonzero_abs_inverse:
paulson@14294
  1968
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  1969
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  1970
                      negative_imp_inverse_negative)
paulson@14294
  1971
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  1972
done
paulson@14294
  1973
paulson@14294
  1974
lemma abs_inverse [simp]:
paulson@14294
  1975
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  1976
      inverse (abs a)"
haftmann@21328
  1977
apply (cases "a=0", simp) 
paulson@14294
  1978
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1979
done
paulson@14294
  1980
paulson@14294
  1981
lemma nonzero_abs_divide:
paulson@14294
  1982
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  1983
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  1984
paulson@15234
  1985
lemma abs_divide [simp]:
paulson@14294
  1986
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
haftmann@21328
  1987
apply (cases "b=0", simp) 
paulson@14294
  1988
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1989
done
paulson@14294
  1990
paulson@14294
  1991
lemma abs_mult_less:
obua@14738
  1992
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  1993
proof -
paulson@14294
  1994
  assume ac: "abs a < c"
paulson@14294
  1995
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  1996
  assume "abs b < d"
paulson@14294
  1997
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1998
qed
paulson@14293
  1999
obua@14738
  2000
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
obua@14738
  2001
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
obua@14738
  2002
obua@14738
  2003
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
obua@14738
  2004
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
obua@14738
  2005
obua@14738
  2006
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  2007
apply (simp add: order_less_le abs_le_iff)  
obua@14738
  2008
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
obua@14738
  2009
apply (simp add: le_minus_self_iff linorder_neq_iff) 
obua@14738
  2010
done
obua@14738
  2011
avigad@16775
  2012
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
avigad@16775
  2013
    (abs y) * x = abs (y * x)";
avigad@16775
  2014
  apply (subst abs_mult);
avigad@16775
  2015
  apply simp;
avigad@16775
  2016
done;
avigad@16775
  2017
avigad@16775
  2018
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
avigad@16775
  2019
    abs x / y = abs (x / y)";
avigad@16775
  2020
  apply (subst abs_divide);
avigad@16775
  2021
  apply (simp add: order_less_imp_le);
avigad@16775
  2022
done;
avigad@16775
  2023
wenzelm@23389
  2024
obua@19404
  2025
subsection {* Bounds of products via negative and positive Part *}
obua@15178
  2026
obua@15580
  2027
lemma mult_le_prts:
obua@15580
  2028
  assumes
obua@15580
  2029
  "a1 <= (a::'a::lordered_ring)"
obua@15580
  2030
  "a <= a2"
obua@15580
  2031
  "b1 <= b"
obua@15580
  2032
  "b <= b2"
obua@15580
  2033
  shows
obua@15580
  2034
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
obua@15580
  2035
proof - 
obua@15580
  2036
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
obua@15580
  2037
    apply (subst prts[symmetric])+
obua@15580
  2038
    apply simp
obua@15580
  2039
    done
obua@15580
  2040
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
nipkow@23477
  2041
    by (simp add: ring_simps)
obua@15580
  2042
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
obua@15580
  2043
    by (simp_all add: prems mult_mono)
obua@15580
  2044
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
obua@15580
  2045
  proof -
obua@15580
  2046
    have "pprt a * nprt b <= pprt a * nprt b2"
obua@15580
  2047
      by (simp add: mult_left_mono prems)
obua@15580
  2048
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
obua@15580
  2049
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2050
    ultimately show ?thesis
obua@15580
  2051
      by simp
obua@15580
  2052
  qed
obua@15580
  2053
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
obua@15580
  2054
  proof - 
obua@15580
  2055
    have "nprt a * pprt b <= nprt a2 * pprt b"
obua@15580
  2056
      by (simp add: mult_right_mono prems)
obua@15580
  2057
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
obua@15580
  2058
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2059
    ultimately show ?thesis
obua@15580
  2060
      by simp
obua@15580
  2061
  qed
obua@15580
  2062
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
obua@15580
  2063
  proof -
obua@15580
  2064
    have "nprt a * nprt b <= nprt a * nprt b1"
obua@15580
  2065
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2066
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
obua@15580
  2067
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2068
    ultimately show ?thesis
obua@15580
  2069
      by simp
obua@15580
  2070
  qed
obua@15580
  2071
  ultimately show ?thesis
obua@15580
  2072
    by - (rule add_mono | simp)+
obua@15580
  2073
qed
obua@19404
  2074
obua@19404
  2075
lemma mult_ge_prts:
obua@15178
  2076
  assumes
obua@19404
  2077
  "a1 <= (a::'a::lordered_ring)"
obua@19404
  2078
  "a <= a2"
obua@19404
  2079
  "b1 <= b"
obua@19404
  2080
  "b <= b2"
obua@15178
  2081
  shows
obua@19404
  2082
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
obua@19404
  2083
proof - 
obua@19404
  2084
  from prems have a1:"- a2 <= -a" by auto
obua@19404
  2085
  from prems have a2: "-a <= -a1" by auto
obua@19404
  2086
  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
  2087
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
obua@19404
  2088
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
obua@19404
  2089
    by (simp only: minus_le_iff)
obua@19404
  2090
  then show ?thesis by simp
obua@15178
  2091
qed
obua@15178
  2092
wenzelm@23389
  2093
haftmann@22842
  2094
subsection {* Theorems for proof tools *}
haftmann@22842
  2095
haftmann@22842
  2096
lemma add_mono_thms_ordered_semiring:
haftmann@22842
  2097
  fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add"
haftmann@22842
  2098
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2099
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2100
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2101
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@22842
  2102
by (rule add_mono, clarify+)+
haftmann@22842
  2103
haftmann@22842
  2104
lemma add_mono_thms_ordered_field:
haftmann@22842
  2105
  fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add"
haftmann@22842
  2106
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2107
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2108
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2109
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2110
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2111
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@22842
  2112
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@22842
  2113
paulson@14265
  2114
end